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Fv(related)i(to)e(asymptotic)i(completeness,)k(and)39 b(this)g(tends)257 2017 y(to)33 b(b)s(e)f(a)h(v)m(ery)h(hard)e (analytical)h(problem.\))404 2138 y(The)c(idea)g(that)g(initial)g (excited)h(b)s(ound)f(states)g(of)f(an)h(atom)f(approac)m(h)h(a)g (ground-)257 2258 y(state)45 b(through)g(emission)h(of)e(photons,)49 b(as)44 b(time)i(tends)f(to)g Fr(1)p Fv(,)i(\014rst)e(app)s(eared)g(in) 257 2379 y(Bohr's)30 b(theory)f(of)g(the)g(h)m(ydrogen)h(atom)f (coupled)h(to)f(the)g(quan)m(tized)i(radiation)d(\014eld,)257 2499 y(at)g(zero)g(temp)s(erature,)i(and)e(remained)h(a)e(guiding)h (idea)h(in)f(later)f Fs(p)-5 b(erturb)g(ative)28 b Fv(analy-)257 2619 y(ses)37 b(of)d(the)i(quan)m(tum)g(theory)g(of)f(atoms)g(coupled)h (to)f(the)h(electromagnetic)g(\014eld)g(b)m(y)257 2740 y(some)46 b(of)e(the)h(founding)g(fathers)g(of)f(quan)m(tum)i(mec)m (hanics.)82 b(See,)48 b(e.g.,)g([14])c(for)g(a)257 2860 y(review)39 b(of)d(results.)56 b(Mathematically)38 b(rigorous,)g(non-p) 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b(Decem)m(b)s(er)i(21,)e(1999)2054 b Fv(3)257 573 y(presen)m(t)31 b(the)f(general)f(formalism)h(in)f(a)g (fashion)h(that)f(will)g(mak)m(e)i(future)e(applications)257 693 y(to)41 b(more)g(complicated)h(problems,)i(e.g.,)f(to)d(the)h (problem)g(of)g Fs(stationary)h(states)f Fv(of)257 814 y(in\014nite)34 b(quan)m(tum)g(systems,)h(or)d(to)g(some)h(problems)h (of)e(transp)s(ort)g(theory)-8 b(,)33 b(feasible.)257 934 y(Unfortunately)-8 b(,)39 b(some)f(of)f(these)h(purp)s(oses)h(of)d (our)h(w)m(ork)h(mak)m(e)h(the)e(presen)m(t)i(pap)s(er)257 1054 y(quite)34 b(long.)404 1175 y(Next,)42 b(w)m(e)f(describ)s(e)h (the)e(class)h(of)e(op)s(en)i(quan)m(tum)g(systms)h(considered)f(in)g (this)257 1295 y(pap)s(er.)g(The)25 b Fs(smal)5 b(l,)27 b(b)-5 b(ounde)g(d)26 b(subsystem)e Fv(consists)h(of)e(a)h(con\014ned)h (atom)e(or)h(molecule.)257 1416 y(In)31 b(this)g(pap)s(er,)g(an)f (idealized)h(description)h(of)e(the)g(small)h(subsystem)i(as)d(a)g (quan)m(tum-)257 1536 y(mec)m(hanical)35 b(system)f(with)f(a)g Fs(\014nite-dimensional)f(state)j(sp)-5 b(ac)g(e)p Fv(,)1324 1739 y Fr(H)1406 1757 y Fq(el)1525 1712 y Fr(\030)1525 1743 y Fv(=)1660 1739 y Fp(C)1726 1696 y Fq(N)1821 1739 y Ft(;)134 b(N)70 b(<)60 b Fr(1)21 b Ft(;)896 b Fv(\(I.1\))257 1942 y(is)24 b(c)m(hosen,)i(i.e.,)g(w)m(e)e(consider)g(an)f(atom)g(or)f (molecule)j(with)e(only)h(\014nitely)g(man)m(y)g(energy)257 2063 y(lev)m(els.)45 b(When)29 b(the)g(coupling)g(b)s(et)m(w)m(een)i (the)e(small)g(subsystem)i(and)e(the)g(reserv)m(oir,)i(or)257 2183 y(heat)h(bath,)g(is)g(turned)g(o\013,)f(the)h(dynamics)h(is)f(giv) m(en)h(b)m(y)f(a)f(Hamiltonian,)h Ft(H)3131 2201 y Fq(el)3190 2183 y Fv(,)g(whic)m(h)257 2304 y(is)h(a)f(selfadjoin)m(t)h(op)s (erator)f(on)g Fr(H)1515 2321 y Fq(el)1574 2304 y Fv(.)44 b(The)33 b(sp)s(ectrum)h(of)d Ft(H)2462 2321 y Fq(el)2553 2304 y Fv(consists)j(of)e(eigen)m(v)-5 b(alues)257 2424 y Ft(E)329 2442 y Fo(0)369 2424 y Ft(;)16 b(E)484 2442 y Fo(1)524 2424 y Ft(;)g(:)g(:)g(:)34 b(;)16 b(E)829 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b(the)g(natural)g(scalar)g(pro)s(duct,)f Fr(h)16 b(\001)g(j)g(\001)g(i)p Fv(,)34 b(on)f Fr(H)2050 3495 y Fq(el)2109 3478 y Fv(,)1547 3681 y Fr(h)p Ft(')1649 3699 y Fq(i)1677 3681 y Fr(j)p Ft(')1768 3699 y Fq(j)1804 3681 y Fr(i)60 b Fv(=)f Ft(\016)2080 3699 y Fq(ij)2163 3681 y Ft(;)1118 b Fv(\(I.4\))257 3884 y(for)32 b(all)h Ft(i;)16 b(j)34 b Fv(=)27 b(0)p Ft(;)16 b Fv(1)p Ft(;)g(:)g(:)g(:)33 b(;)16 b(N)1246 3902 y Fo(1)1286 3884 y Fv(.)404 4005 y(The)25 b Fs(r)-5 b(eservoir)25 b Fv(is)g(c)m(hosen)h(to)f(consist)h (of)f(the)g(quan)m(tized)h(electromagnetic)h(\014eld)f(or)257 4125 y(of)33 b(the)g(quan)m(tized)i(vibrations,)f(or)f(phonons,)h(of)e (an)h(in\014nitely)h(extended,)i(harmonic)257 4246 y(material)i (medium.)61 b(The)39 b(mo)s(des)g(of)e(the)h(reserv)m(oir)i(are)d (indexed)j(b)m(y)f(w)m(a)m(v)m(e)h(v)m(ectors)252 4340 y Ft(~)257 4366 y(k)31 b Fr(2)c Fp(R)497 4327 y Fo(3)573 4366 y Fv(and,)32 b(for)e(photons,)i(a)f(helicit)m(y)i Ft(\025)27 b Fv(=)g Fr(\006)p Fv(1.)42 b(The)32 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5310 y Ft(~)2050 5336 y(k)s Fv(\))22 b Ft(d)2215 5297 y Fo(3)2254 5336 y Ft(k)i(;)84 b Fv(for)32 b(photons.)3308 5264 y(\(I.6\))p eop %%Page: 4 5 4 4 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2054 b Fv(4)404 573 y(A)m(t)31 b(zero)g(temp)s(erature,)i(the)e(Hilb)s(ert)g (space)i(of)d(pure)i(state)f(v)m(ectors)i(of)d(the)i(reser-)257 693 y(v)m(oir)h(is)h(c)m(hosen)g(to)e(b)s(e)h(the)g Fs(F)-7 b(o)i(ck)33 b(sp)-5 b(ac)g(e)1563 977 y Fr(F)37 b Fv(:=)1840 853 y Fn(1)1800 883 y Fm(M)1809 1092 y Fq(n)p Fo(=0)1967 977 y Ft(h)2023 934 y Fo(\()p Fq(n)p Fo(\))2147 977 y Ft(;)1134 b Fv(\(I.7\))257 1293 y(where)45 b Ft(h)606 1253 y Fo(\(0\))746 1293 y Fv(:=)h Fp(C)20 b Fv(,)52 b(and)44 b Ft(h)1297 1253 y Fo(\()p Fq(n)p Fo(\))1445 1293 y Fv(:=)h Ft(h)1649 1253 y Fn(\012)1704 1261 y Fj(s)1738 1253 y Fq(n)1785 1293 y Fv(,)h Ft(n)g Fr(\025)f Fv(1,)i(denotes)e(the)f (symmetric)i(tensor)257 1413 y(pro)s(duct)23 b(appropriate)g(for)f(the) 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Ft(f)11 b Fr(j)p Ft(g)t Fr(i)2675 5406 y Fq(h)2735 5388 y Fi(1)22 b Ft(;)420 b Fv(\(I.14\))p eop %%Page: 5 6 5 5 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2054 b Fv(5)257 573 y(where)44 b Fr(h)16 b(\001)g(j)g(\001)g(i)770 590 y Fq(h)859 573 y Fv(denotes)44 b(the)f(scalar)h(pro)s(duct)f(on)f Ft(h)p Fv(.)74 b(W)-8 b(e)44 b(note)f(that)f(the)h Fs(vacuum)257 693 y(ve)-5 b(ctor)p Fv(,)33 b(\012)27 b(=)g(\(1)p Ft(;)16 b Fv(0)p Ft(;)g Fv(0)p Ft(;)g Fr(\001)g(\001)g(\001)i Fv(\),)32 b(spans)i Ft(h)1635 654 y Fo(\(0\))1762 693 y Fv(and)f(has)g(the)g(prop)s(ert)m(y)g(that)1254 908 y Ft(a)p Fv(\()p Ft(f)11 b Fv(\))16 b(\012)60 b(=)f(0)22 b Ft(;)134 b Fv(for)32 b(all)h Ft(f)38 b Fr(2)27 b Ft(h)p Fv(.)761 b(\(I.15\))257 1123 y(The)24 b Fs(dynamics)d Fv(of)h(the)g(reserv)m(oir)i(at)e Fs(zer)-5 b(o)25 b(temp)-5 b(er)g(atur)g(e)23 b Fv(is)f(determined)i(b)m(y)g(the)e(Hamil-)257 1243 y(tonian)1256 1482 y Ft(H)1337 1499 y Fq(f)1442 1482 y Fv(=)1577 1346 y Fm(Z)1693 1482 y Ft(dk)i(a)1870 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5373 y Fq(i;j)1320 5352 y Fv(\()p Ft(e)1403 5309 y Fn(\000)p Fq(\022)1497 5352 y Ft(k)s Fv(\))62 b(:=)p 1816 5255 726 4 v 1816 5271 a Fm(\012)1863 5352 y Ft(')1927 5370 y Fq(j)1985 5267 y Fm(\014)1985 5327 y(\014)2040 5352 y Ft(G)p Fv(\()p Ft(e)2200 5323 y Fn(\000)2261 5305 y Fo(\026)2255 5323 y Fq(\022)2294 5352 y Ft(k)s Fv(\))16 b Ft(')2466 5370 y Fq(i)2494 5271 y Fm(\013)2563 5352 y Ft(:)670 b Fv(\(I.25\))p eop %%Page: 7 8 7 7 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2054 b Fv(7)404 573 y(W)-8 b(e)29 b(also)h(require)g(a)f(v)-5 b(ariet)m(y)30 b(of)f Fs(b)-5 b(ounde)g(dness)30 b(c)-5 b(onditions)28 b Fv(on)i(the)f(coupling)h(func-)257 693 y(tions)j Fr(f)p Ft(G)622 711 y Fq(i;j)702 693 y Fv(\()p Ft(e)785 654 y Fn(\000)p Fq(\022)879 693 y Ft(k)s Fv(\))p Fr(g)p Fv(.)257 897 y Fi(Hyp)s(othesis)k(H-)h(3.)49 b Fs(We)44 b(assume)g(that)h(ther)-5 b(e)44 b(exists)g Fv(p)s(ositiv)m(e)i Fs(c)-5 b(onstants)44 b Ft(\026)g(>)g Fv(0)257 1017 y Fs(and)35 b Ft(M)j(<)27 b Fr(1)p 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b(the)f(iden)m(tit)m(y)h(op)s(erator)d Fi(1)p Fv(.)44 b(The)33 b(dynamics)h(of)e(the)h(system)h(is)f (determined)257 3977 y(b)m(y)45 b(a)e(Hamiltonian,)k Ft(H)8 b Fv(,)46 b(whic)m(h)f(is)f(a)g(semib)s(ounded,)k(selfadjoin)m (t)c(op)s(erator)f(on)h Fr(H)257 4097 y Fv(with)34 b(discrete)g(sp)s (ectrum.)404 4218 y(Let)f(T)-8 b(r)q([)p Fr(\001)p Fv(])33 b(denote)i(the)e(usual)i(trace)e(on)h Ft(B)5 b Fv(\()p Fr(H)q Fv(\).)46 b(W)-8 b(e)34 b(assume)h(that)e(exp)q(\()p Fr(\000)p Ft(\014)6 b(H)i Fv(\))33 b(is)257 4338 y(trace-class,)h (i.e.,)1501 4558 y(T)-8 b(r)1601 4477 y Fm(\002)1642 4558 y Ft(e)1687 4515 y Fn(\000)p Fq(\014)s(H)1853 4477 y Fm(\003)1954 4558 y Ft(<)59 b Fr(1)22 b Ft(;)1034 b Fv(\(I)s(I.1\))257 4778 y(for)32 b(arbitrary)h Ft(\014)g(>)27 b Fv(0.)404 4898 y(W)-8 b(e)41 b(are)f(in)m(terested)j(in)e(describing) h(general)g(mixed)g(states)f(of)g(the)g(system,)j(in-)257 5019 y(cluding)54 b(its)f(thermal)g(equilibrium)i(states,)j(for)53 b(arbitrary)f(in)m(v)m(erse)j(temp)s(erature)257 5139 y Ft(\014)33 b Fv(=)27 b(\()p Ft(k)537 5157 y Fq(B)597 5139 y Ft(T)14 b Fv(\))706 5100 y Fn(\000)p Fo(1)800 5139 y Fv(,)32 b(where)h Ft(k)1191 5157 y Fq(B)1282 5139 y Fv(is)f(Boltzmann's)h(constan)m(t)f(and)g Ft(T)45 b Fv(denotes)33 b(the)e(absolute)257 5260 y(temp)s(erature.)62 b(F)-8 b(urthermore,)40 b(w)m(e)g(wish)f(to)f(study)i(the)e(time)h(ev)m (olution)h(of)e(general)257 5380 y(mixed)c(states,)g(as)f(determined)h (b)m(y)g(the)f(Hamiltonian)f Ft(H)8 b Fv(.)p eop %%Page: 11 12 11 11 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(11)404 573 y(According)45 b(to)f(Landau)g(and)h(v)m(on)g(Neumann,) j(a)c(mixed)i(state)f(of)e(the)i(system)257 693 y(corresp)s(onds)31 b(to)f(a)f(densit)m(y)j(matrix,)f(i.e.,)g(to)e(a)h(p)s(ositiv)m(e,)i (selfadjoin)m(t)e(op)s(erator)f(on)h Fr(H)257 814 y Fv(of)i(unit)h (trace.)404 934 y(The)27 b(t)m(w)m(o-sided)g(ideal)g(of)f(trace-class)h (op)s(erators)f(in)g Ft(B)5 b Fv(\()p Fr(H)q Fv(\))26 b(is)h(denoted)g(b)m(y)g Fr(L)3269 895 y Fo(1)3309 934 y Fv(\()p Fr(H)q Fv(\),)257 1054 y(the)c(t)m(w)m(o-sided)h(ideal)e(of)g (Hilb)s(ert-Sc)m(hmitt)h(op)s(erators)f(in)g Ft(B)5 b Fv(\()p Fr(H)q Fv(\))23 b(b)m(y)g Fr(L)2830 1015 y Fo(2)2870 1054 y Fv(\()p Fr(H)q Fv(\).)40 b(A)22 b(densit)m(y)257 1175 y(matrix)33 b Ft(\032)g Fv(is)g(a)g(p)s(ositiv)m(e)h(elemen)m(t)g (of)e Fr(L)1736 1135 y Fo(1)1776 1175 y Fv(\()p Fr(H)q Fv(\))h(of)f(unit)h(trace.)43 b(Then)1657 1391 y Ft(\024)60 b Fv(:=)f Ft(\032)1985 1348 y Fo(1)p Fq(=)p Fo(2)3270 1391 y Fv(\(I)s(I.2\))257 1608 y(is)33 b(Hilb)s(ert-Sc)m(hmitt,)h(with) 1126 1825 y(T)-8 b(r)1226 1744 y Fm(\002)1267 1825 y Ft(\024)1323 1781 y Fo(2)1363 1744 y Fm(\003)1464 1825 y Fv(=)60 b(T)-8 b(r)1700 1744 y Fm(\002)1741 1825 y Ft(\024)1797 1781 y Fn(\003)1837 1825 y Ft(\024)1893 1744 y Fm(\003)1994 1825 y Fv(=)60 b(T)-8 b(r[)p Ft(\032)p Fv(])60 b(=)g(1)21 b Ft(:)643 b Fv(\(I)s(I.3\))257 2042 y(As)33 b(a)g Fs(line)-5 b(ar)34 b(sp)-5 b(ac)g(e)p Fv(,)32 b Fr(L)1104 2002 y Fo(2)1144 2042 y Fv(\()p Fr(H)q Fv(\))g(is)h(a)g (Hilb)s(ert)g(space)g(with)h(scalar)f(pro)s(duct)f(giv)m(en)i(b)m(y)526 2258 y Fr(h\001j\001i)54 b Fv(:)60 b Fr(L)891 2215 y Fo(2)930 2258 y Fv(\()p Fr(H)q Fv(\))22 b Fr(\002)g(L)1276 2215 y Fo(2)1316 2258 y Fv(\()p Fr(H)q Fv(\))27 b Fr(!)g Fp(C)47 b Ft(;)82 b Fv(\()p Ft(\024;)16 b(\033)t Fv(\))27 b Fr(7!)g(h)p Ft(\024)p Fr(j)p Ft(\033)t Fr(i)54 b Fv(:=)g(T)-8 b(r)q([)p Ft(\024)2828 2215 y Fn(\003)2867 2258 y Ft(\033)t Fv(])22 b Ft(:)268 b Fv(\(I)s(I.4\))257 2475 y(F)-8 b(or)44 b(brevit)m(y)-8 b(,)50 b(w)m(e)c(denote)f Fr(L)1361 2436 y Fo(2)1401 2475 y Fv(\()p Fr(H)q Fv(\))g(b)m(y)h Fr(K)r Fv(.)80 b(This)46 b(Hilb)s(ert)f(space)h(is)g(isomorphic)g(to)257 2595 y Fr(H)23 b(\012)f(H)q Fv(.)44 b(It)32 b(carries)i(a)e(represen)m (tation)j Ft(`)d Fv(of)g(the)h(algebra)g Fr(A)f Fv(giv)m(en)i(b)m(y) 1431 2812 y Ft(`)p Fv([)p Ft(a)p Fv(])16 b Ft(\024)61 b Fv(:=)e Ft(a)16 b(\024)61 b Fr(2)e(K)24 b Ft(;)965 b Fv(\(I)s(I.5\))257 3029 y(for)28 b(arbitrary)g Ft(\024)f Fr(2)g(K)r Fv(,)j Ft(a)d Fr(2)g(A)p Fv(.)42 b(T)-8 b(o)28 b(ev)m(ery)i(elemen)m(t)g Ft(\024)d Fr(2)g(K)r Fv(,)j(w)m(e)f(can)f (asso)s(ciate)h(a)f(state)257 3149 y(of)k(the)h(system)i(giv)m(en)f(b)m (y)f(the)g(densit)m(y)i(matrix)1469 3366 y Ft(\032)60 b Fv(:=)f Fr(h)p Ft(\024)p Fr(j)p Ft(\024)p Fr(i)1956 3323 y Fn(\000)p Fo(1)2067 3366 y Ft(\024)16 b(\024)2195 3323 y Fn(\003)2257 3366 y Ft(:)986 b Fv(\(I)s(I.6\))257 3583 y(The)34 b(exp)s(ectation)g(v)-5 b(alue)33 b(of)f(an)g(observ)-5 b(able)34 b Ft(a)27 b Fr(2)g(A)33 b Fv(in)g(the)g(state)g Ft(\032)f Fv(is)h(giv)m(en)h(b)m(y)1060 3799 y Fr(h)p Ft(a)p Fr(i)1187 3817 y Fq(\032)1311 3799 y Fv(:=)83 b(T)-8 b(r[)p Ft(\032)16 b(a)p Fv(])60 b(=)g Fr(h)p Ft(\024)p Fr(j)p Ft(\024)p Fr(i)2179 3756 y Fn(\000)p Fo(1)2290 3799 y Fv(T)-8 b(r)2390 3719 y Fm(\002)2432 3799 y Ft(\024\024)2544 3756 y Fn(\003)2600 3799 y Ft(a)2651 3719 y Fm(\003)1324 3954 y Fv(=)97 b Fr(h)p Ft(\024)p Fr(j)p 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b(jections)44 b Ft(P)58 b Fr(2)44 b(K)h Fv(of)257 5135 y(rank)33 b(1,)g(i.e.,)1138 5352 y Ft(P)73 b Fv(=)60 b Ft( )20 b Fv(\()p Ft( )t Fr(j\001)p Fv(\))60 b(=)f Fr(j)p Ft( )t Fr(ih)p Ft( )t Fr(j)22 b Ft(;)82 b( )31 b Fr(2)c(H)c Ft(;)624 b Fv(\(I)s(I.10\))p eop %%Page: 12 13 12 12 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(12)257 573 y(in)33 b(Dirac's)g(bra-k)m(et)g(notation.)43 b(Then)1112 784 y Fr(h)p Ft(P)14 b Fr(j)p Ft(`)p Fv([)p Ft(a)p Fv(])p Ft(P)g Fr(i)82 b Fv(=)h(T)-8 b(r[)p Ft(P)30 b(a)16 b(P)e Fv(])60 b(=)f(T)-8 b(r[)p Ft(P)30 b(a)p Fv(])1597 930 y(=)83 b(\()p Ft( )t Fr(j)16 b Ft(a)g( )t Fv(\))22 b Ft(:)1097 b Fv(\(I)s(I.11\))404 1141 y(As)35 b(an)f(algebra,)i Fr(K)c Fv(=)f Fr(L)1339 1102 y Fo(2)1378 1141 y Fv(\()p Fr(H)q Fv(\))k(is)g(what)g(is)h(called)f(a)f Fs(Hilb)-5 b(ert)37 b(algebr)-5 b(a)p Fv(,)35 b(i.e.,)h(it)e(is)h(a)257 1262 y Fr(\003)p Fv(-algebra,)29 b(and,)g(as)g(a)f(linear)g(space,)j (it)d(is)h(a)f(Hilb)s(ert)g(space,)j(with)e(the)f(prop)s(ert)m(y)h (that)630 1473 y Fr(h)p Ft(\024)724 1491 y Fo(1)780 1473 y Ft(\024)836 1491 y Fo(2)875 1473 y Fr(j)16 b Ft(\024)974 1491 y Fo(3)1014 1473 y Fr(i)60 b Fv(=)f Fr(h)p Ft(\024)1341 1491 y Fo(2)1381 1473 y Fr(j)16 b Ft(\024)1480 1430 y Fn(\003)1480 1494 y Fo(1)1536 1473 y Ft(\024)1592 1491 y Fo(3)1631 1473 y Fr(i)71 b Fv(and)g Fr(h)p Ft(\024)2062 1491 y Fo(1)2102 1473 y Fr(j)16 b Ft(\024)2201 1491 y Fo(2)2241 1473 y Fr(i)59 b Fv(=)h Fr(h)p Ft(\024)2568 1430 y Fn(\003)2568 1494 y Fo(2)2607 1473 y Fr(j)16 b Ft(\024)2706 1430 y Fn(\003)2706 1494 y Fo(1)2746 1473 y Fr(i)22 b Ft(;)389 b Fv(\(I)s(I.12\))257 1685 y(whic)m(h)34 b(follo)m(ws)g(from)e(\(I)s(I.4\),)h(see)g([22,)g(18)o(].)404 1805 y(The)39 b(time)h(ev)m(olution)f(of)g(an)f(observ)-5 b(able)40 b Ft(a)d Fr(2)h(A)g Fv(in)h(the)g Fs(Heisenb)-5 b(er)g(g)40 b(pictur)-5 b(e)39 b Fv(is)257 1926 y(de\014ned,)34 b(as)f(usual,)h(b)m(y)1395 2137 y Ft(\013)1457 2155 y Fq(t)1486 2137 y Fv(\()p Ft(a)p Fv(\))60 b(:=)g Ft(e)1881 2094 y Fq(itH)2019 2137 y Ft(a)22 b(e)2137 2094 y Fn(\000)p Fq(itH)2331 2137 y Ft(:)864 b Fv(\(I)s(I.13\))257 2349 y(Then,)34 b(for)e Ft(\024;)16 b(\033)32 b Fr(2)27 b(K)r Fv(,)775 2560 y Fr(h)p Ft(\024)p Fr(j)16 b Ft(`)p Fv([)p Ft(\013)1042 2578 y Fq(t)1072 2560 y Fv(\()p Ft(a)p Fv(\)])g Ft(\033)t Fr(i)84 b Fv(=)f(T)-8 b(r)1682 2480 y Fm(\002)1723 2560 y Ft(\024)1779 2517 y Fn(\003)1835 2560 y Ft(\013)1897 2578 y Fq(t)1927 2560 y Fv(\()p Ft(a)p Fv(\))16 b Ft(\033)2129 2480 y Fm(\003)1423 2717 y Fv(=)83 b(T)-8 b(r)1682 2636 y Fm(\002)1723 2717 y Ft(\024)1779 2674 y Fn(\003)1835 2717 y Ft(e)1880 2674 y Fq(itH)2013 2717 y Ft(a)16 b(e)2125 2674 y Fn(\000)p Fq(itH)2314 2717 y Ft(\033)2373 2636 y Fm(\003)3222 2717 y Fv(\(I)s(I.14\))1423 3008 y(=)83 b(T)-8 b(r)1682 2927 y Fm(\002)1723 3008 y Fv(\()p Ft(e)1806 2964 y Fn(\000)p Fq(itH)1994 3008 y Ft(\024)16 b(e)2111 2964 y Fq(itH)2229 3008 y Fv(\))2267 2964 y Fn(\003)2322 3008 y Ft(a)g Fv(\()p 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Fv(\(I)s(I.17\))257 4053 y(for)30 b Ft(t)d Fr(2)g Fp(R)5 b Fv(.)49 b(W)-8 b(e)30 b(de\014ne)h(a)f(selfadjoin)m(t)h(linear)f(op)s(erator)f Fr(L)p Fv(,)i(the)g Fs(Liouvil)5 b(lian)p Fv(,)29 b(on)h Fr(K)i Fv(b)m(y)257 4174 y(setting)1509 4385 y Fr(L)16 b Ft(\024)61 b Fv(:=)e([)16 b Ft(H)24 b(;)33 b(\024)16 b Fv(])22 b Ft(;)995 b Fv(\(I)s(I.18\))257 4597 y(where)36 b([)p Fr(\001)p Ft(;)16 b Fr(\001)p Fv(])35 b(denotes)h(the)f(comm)m (utator.)50 b(Under)35 b(our)g(h)m(yp)s(otheses)i(on)d Ft(H)8 b Fv(,)35 b(the)g(op)s(er-)257 4717 y(ator)d Fr(L)h Fv(is)g(essen)m(tially)i(selfadjoin)m(t)f(on)e(the)h(follo)m(wing)g (core)g Fr(D)i Fv(dense)g(in)d Fr(K)r Fv(,)974 4929 y Fr(D)62 b Fv(:=)e(span)1470 4848 y Fm(\010)1544 4929 y Fr(j)p Ft( )1634 4946 y Fq(i)1663 4929 y Fr(ih)p Ft( )1802 4946 y Fq(j)1839 4929 y Fr(j)1898 4844 y Fm(\014)1898 4904 y(\014)1964 4929 y Ft(i;)16 b(j)50 b Fv(=)43 b(0)p Ft(;)16 b Fv(1)p Ft(;)g Fv(2)p Ft(;)g(:)g(:)g(:)2656 4848 y Fm(\011)2735 4929 y Ft(;)460 b Fv(\(I)s(I.19\))257 5140 y(where)34 b Fr(f)p Ft( )651 5158 y Fq(i)680 5140 y Fr(g)729 5101 y Fn(1)729 5161 y Fq(i)p Fo(=0)879 5140 y Fv(is)f(a)g(complete)g(orthonormal)g(system)h(of)e(eigen)m(v)m (ectors)j(of)e Ft(H)8 b Fv(,)32 b(i.e.,)1194 5352 y Ft(H)8 b( )1346 5370 y Fq(i)1434 5352 y Fv(=)59 b Ft(E)1641 5370 y Fq(i)1670 5352 y Ft( )1733 5370 y Fq(i)1783 5352 y Ft(;)81 b(i)44 b Fv(=)f(0)p Ft(;)16 b Fv(1)p Ft(;)g Fv(2)p Ft(;)g(:)g(:)g(:)39 b(;)680 b Fv(\(I)s(I.20\))p eop %%Page: 13 14 13 13 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(13)257 573 y(with)1359 793 y Ft(E)1431 810 y Fo(0)1520 793 y Fr(\024)48 b Ft(E)1716 810 y Fo(1)1805 793 y Fr(\024)g Ft(E)2001 810 y Fo(2)2090 793 y Fr(\024)g(\001)16 b(\001)g(\001)39 b Ft(:)829 b Fv(\(I)s(I.21\))257 1013 y(It)44 b(is)g(instructiv)m(e)h (to)e(v)m(erify)i(that)e Fr(L)h Fv(is)g(a)f(symmetric)i(op)s(erator)e (on)g Fr(D)48 b(3)d Ft(\024;)16 b(\033)48 b Fv(b)m(y)257 1133 y(computation,)851 1273 y Fm(\012)898 1353 y Fr(L)p Ft(\024)1021 1269 y Fm(\014)1021 1328 y(\014)1071 1353 y Ft(\033)1130 1273 y Fm(\013)1260 1353 y Fv(=)83 b(T)-8 b(r)1520 1273 y Fm(\002)1561 1353 y Fv(\()p Fr(L)p Ft(\024)p Fv(\))1760 1310 y Fn(\003)1816 1353 y Ft(\033)1875 1273 y Fm(\003)1976 1353 y Fv(=)60 b(T)-8 b(r)2212 1273 y Fm(\000)2258 1353 y Fv([)16 b Ft(H)24 b(;)33 b(\024)16 b Fv(])2565 1310 y Fn(\003)2621 1353 y Ft(\033)2680 1273 y Fm(\001)1260 1499 y Fv(=)83 b Fr(\000)p Fv(T)-8 b(r)1596 1418 y Fm(\000)1641 1499 y Fv([)16 b Ft(H)24 b(;)33 b(\024)1905 1455 y Fn(\003)1961 1499 y Fv(])16 b Ft(\033)2063 1418 y Fm(\001)2168 1499 y Fv(=)60 b(T)-8 b(r)2404 1418 y Fm(\000)2450 1499 y Ft(\024)2506 1455 y Fn(\003)2545 1499 y Fv([)16 b Ft(H)25 b(;)32 b(\033)20 b Fv(])2855 1418 y Fm(\001)1260 1644 y Fv(=)1419 1563 y Fm(\012)1466 1644 y Ft(\024)1522 1559 y Fm(\014)1522 1619 y(\014)1572 1644 y Fr(L)p Ft(\033)1698 1563 y Fm(\013)1767 1644 y Ft(;)1428 b Fv(\(I)s(I.22\))257 1864 y(using)34 b(the)f(cyclicit)m(y)i (of)d(the)h(trace.)44 b(Eq.)33 b(\(I)s(I.17\))f(can)h(no)m(w)g(b)s(e)g (rewritten)h(as)1563 2084 y Ft(\024)1619 2101 y Fq(t)1709 2084 y Fv(=)59 b Ft(e)1889 2040 y Fn(\000)p Fq(it)p Fn(L)2068 2084 y Ft(\024)22 b(;)1049 b Fv(\(I)s(I.23\))257 2304 y(and)33 b(one)g(easily)h(v)m(eri\014es)g(that)1322 2524 y Ft(e)1367 2480 y Fq(it)p Fn(L)1486 2524 y Ft(`)p Fv([)p Ft(a)p Fv(])16 b Ft(e)1693 2480 y Fn(\000)p Fq(it)p Fn(L)1910 2524 y Fv(=)60 b Ft(`)p Fv(\()p Ft(\013)2187 2541 y Fq(t)2217 2524 y Fv(\()p Ft(a)p Fv(\)\))21 b Ft(:)792 b Fv(\(I)s(I.24\))257 2813 y Fh(I)t(I.2)132 b(Equilibrium)47 b(States)e(of)f(Bounded)h (Systems)257 2997 y Fv(Since)32 b(w)m(e)g(are)f(in)m(terested)i(in)e (studying)h(systems)h(in)e(thermal)g(equilibrium,)i(w)m(e)f(m)m(ust)257 3118 y(iden)m(tify)h(those)f(v)m(ectors)h(in)e Fr(K)j Fv(whic)m(h)f(describ)s(e)g(equilibrium)g(states)f(at)f(an)g(arbitrary) 257 3238 y(in)m(v)m(erse)f(temp)s(erature)e Ft(\014)6 b Fv(;)28 b(see)g([22,)f(18)o(,)g(12,)g(13])g(for)f(more)h(details.)43 b(Let)27 b 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Fv(is)33 b(giv)m(en)h(b)m(y)f(a)f(densit)m (y)j(matrix)1183 4301 y Ft(\032)1233 4318 y Fq(\014)s(;Q)1415 4301 y Fv(:=)59 b Ft(Z)1651 4257 y Fn(\000)p Fo(1)1644 4326 y Fq(\014)s(;Q)1804 4301 y Fv(exp)q([)p Fr(\000)p Ft(\014)6 b Fv(\()p Ft(H)29 b Fr(\000)22 b Ft(Q)p Fv(\)])g Ft(;)668 b Fv(\(I)s(I.25\))257 4521 y(where)42 b Ft(Q)e Fv(is)h(an)f(arbitrary)h(selfadjoin)m(t)g(op)s(erator)e(on)i Fr(H)g Fv(a\016liated)g(with)g Fr(A)3197 4482 y Fn(0)3220 4521 y Fv(,)h(com-)257 4641 y(m)m(uting)34 b(with)f Ft(H)8 b Fv(,)32 b(and)h(suc)m(h)h(that)1054 4861 y Ft(Z)1128 4818 y Fn(\000)p Fo(1)1121 4887 y Fq(\014)s(;Q)1303 4861 y Fv(:=)59 b(T)-8 b(r)1565 4781 y Fm(\000)1611 4861 y Fv(exp)q([)p Fr(\000)p Ft(\014)6 b Fv(\()p Ft(H)29 b Fr(\000)21 b Ft(Q)p Fv(\)])2311 4781 y Fm(\001)2417 4861 y Ft(<)59 b Fr(1)22 b Ft(:)523 b Fv(\(I)s(I.26\))257 5081 y(The)32 b(ph)m(ysical)h(in)m(terpretation)f(of)f Ft(Q)g Fv(is)h(that)e(of)h(a)g Fs(c)-5 b(onserve)g(d)32 b(char)-5 b(ge)30 b Fv(of)h(the)g(system.)257 5202 y(Since)d Ft(Q)e Fv(is)g(a\016liated)h(with)g Fr(A)1389 5162 y Fn(0)1412 5202 y Fv(,)g(observ)-5 b(ables)28 b(\(i.e.,)g(elemen)m(ts)g (of)e Fr(A)p Fv(\))f(are)h Fs(neutr)-5 b(al)27 b Fv(with)257 5322 y(resp)s(ect)34 b(to)e Ft(Q)p Fv(.)p eop %%Page: 14 15 14 14 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(14)404 573 y(An)29 b(elemen)m(t)j Ft(\024)d Fv(of)g Fr(K)j Fv(describ)s(es)f(a)f(thermal)g(equilibrium)h(state)f(of)f(the)h (system)h(i\013)950 770 y Ft(\024)60 b Fr(\021)f Ft(\024)1257 787 y Fq(\014)s(;Q)1439 770 y Fv(=)h Ft(Z)1649 719 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1642 798 y Fq(\014)s(;Q)1846 770 y Fv(exp)q([)p Fr(\000)p Ft(\014)6 b Fv(\()p Ft(H)29 b Fr(\000)22 b Ft(Q)p Fv(\))p Ft(=)p Fv(2])16 b Ft(U)32 b(;)436 b Fv(\(I)s(I.27\))257 967 y(for)32 b(an)h(arbitrary)f(unitary)i (op)s(erator)e Ft(U)43 b Fv(on)32 b Fr(H)q Fv(.)44 b(F)-8 b(or,)607 1164 y Fr(h)p Ft(\024)701 1181 y Fq(\014)s(;Q)823 1164 y Fr(j)16 b Ft(`)p Fv([)p Ft(a)p Fv(])g Ft(\024)1084 1181 y Fq(\014)s(;Q)1208 1164 y Fr(i)83 b Fv(=)g(T)-8 b(r)1588 1083 y Fm(\002)1629 1164 y Ft(\024)1685 1120 y Fn(\003)1685 1184 y Fq(\014)s(;Q)1824 1164 y Ft(a)16 b(\024)1947 1181 y Fq(\014)s(;Q)2070 1083 y Fm(\003)3222 1164 y Fv(\(I)s(I.28\))1329 1318 y(=)83 b Ft(Z)1562 1275 y Fn(\000)p Fo(1)1555 1344 y Fq(\014)s(;Q)1693 1318 y Fv(T)-8 b(r)1793 1237 y Fm(\002)1835 1318 y Ft(\024)1891 1336 y Fq(\014)s(;Q)2029 1318 y Ft(\024)2085 1275 y Fn(\003)2085 1339 y Fq(\014)s(;Q)2224 1318 y Ft(a)2275 1237 y Fm(\003)1329 1476 y Fv(=)83 b Ft(Z)1562 1433 y Fn(\000)p Fo(1)1555 1501 y Fq(\014)s(;Q)1693 1476 y Fv(T)-8 b(r)1793 1395 y Fm(\000)1839 1476 y Fv(exp)q([)p Fr(\000)p Ft(\014)6 b Fv(\()p Ft(H)28 b Fr(\000)22 b Ft(Q)p Fv(\)])16 b Ft(a)2606 1395 y Fm(\001)2712 1476 y Fv(=)60 b Fr(h)p Ft(a)p Fr(i)2975 1493 y Fq(\014)s(;Q)3119 1476 y Ft(:)257 1673 y Fv(Next,)31 b(w)m(e)g(recall)f(some)g(general)g(prop)s(erties)g(of)f(equilibrium)i (states)f(\(see)h([22)o(,)f(21)o(,)g(12)o(,)257 1793 y(13]\).)404 1914 y(First,)d(since)h Ft(\032)946 1931 y Fq(\014)s(;Q)1094 1914 y Fv(is)f(strictly)h(p)s(ositiv)m(e,)g(for)e Ft(\014)33 b(<)27 b Fr(1)p Fv(,)g(w)m(e)g(ha)m(v)m(e)g(that,)h(for)d (arbitrary)257 2034 y Ft(a)j Fr(2)f(A)p Fv(,)1222 2231 y Fr(h)16 b Ft(a)1327 2188 y Fn(\003)1383 2231 y Ft(a)g Fr(i)1488 2249 y Fq(\014)s(;Q)1665 2231 y Fv(=)54 b(0)98 b(=)-16 b Fr(\))97 b Ft(a)55 b Fv(=)f(0)21 b Ft(:)692 b Fv(\(I)s(I.29\))257 2428 y(Equiv)-5 b(alen)m(tly)d(,)1246 2625 y Ft(`)p Fv([)p Ft(a)p Fv(])16 b Ft(\024)1464 2642 y Fq(\014)s(;Q)1641 2625 y Fv(=)54 b(0)98 b(=)-16 b Fr(\))97 b Ft(a)55 b Fv(=)f(0)21 b Ft(:)716 b Fv(\(I)s(I.30\))257 2822 y(Th)m(us,)35 b(the)e(v)m(ector)g Ft(\024)1048 2839 y Fq(\014)s(;Q)1198 2822 y Fr(2)27 b(K)35 b Fv(is)e Fs(sep)-5 b(ar)g(ating)31 b Fv(for)h(the)h(algebra)g Ft(`)p Fv([)p Fr(A)p Fv(].)404 2942 y(Second,)39 b(the)f(state)f Ft(\032)1233 2960 y Fq(\014)s(;Q)1393 2942 y Fv(satis\014es)i(the)e Fs(Kub)-5 b(o-Martin-Schwinger)38 b(\(KMS\))h(c)-5 b(on-)257 3063 y(dition)p Fv(:)689 3260 y Fr(h)16 b Ft(a)g(\013)872 3277 y Fq(t)902 3260 y Fv(\()p Ft(b)p Fv(\))g Fr(i)1073 3277 y Fq(\014)s(;Q)1279 3260 y Fv(=)83 b Ft(Z)1512 3216 y Fn(\000)p Fo(1)1505 3285 y Fq(\014)s(;Q)1643 3260 y Fv(T)-8 b(r)1744 3179 y Fm(\002)1785 3260 y Fv(exp)q([)p Fr(\000)p Ft(\014)6 b Fv(\()p Ft(H)29 b Fr(\000)21 b Ft(Q)p Fv(\)])16 b Ft(a)g(e)2613 3216 y Fq(itH)2748 3260 y Ft(b)g(e)2850 3216 y Fn(\000)p Fq(itH)3022 3179 y Fm(\003)1279 3425 y Fv(=)83 b Ft(Z)1512 3381 y Fn(\000)p Fo(1)1505 3450 y Fq(\014)s(;Q)1643 3425 y Fv(T)-8 b(r)1744 3344 y Fm(\002)1785 3425 y Ft(e)1830 3381 y Fq(\014)s(Q)1949 3425 y Ft(e)1994 3381 y Fq(itH)2127 3425 y Ft(b)16 b(e)2229 3381 y Fn(\000)p Fo(\()p Fq(\014)s Fo(+)p Fq(it)p Fo(\))p Fq(H)2571 3425 y Ft(a)2622 3344 y Fm(\003)1279 3589 y Fv(=)83 b Ft(Z)1512 3546 y Fn(\000)p Fo(1)1505 3615 y Fq(\014)s(;Q)1643 3589 y Fv(T)-8 b(r)1744 3509 y Fm(\002)1785 3589 y Ft(e)1830 3546 y Fn(\000)p Fq(\014)s Fo(\()p Fq(H)5 b Fn(\000)p Fq(Q)p Fo(\))2177 3589 y 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b(ha)m(v)m(e)h(used)f(\(I)s(I.38\))f(in)h(the)f(\014rst)h(and)f (the)h(KMS)g(condition)g(\(I)s(I.31\))f(in)g(the)h(third)257 3167 y(equation.)404 3287 y(W)-8 b(e)44 b(ha)m(v)m(e)h(no)m(w)f (summarized)h(all)f(imp)s(ortan)m(t)f(elemen)m(ts)j(of)d(the)h(quan)m (tum)h(me-)257 3407 y(c)m(hanics)g(of)e(\014nite)h(systems)i(in)d(or)g (close)h(to)f(thermal)h(equilibrium.)77 b(Ho)m(w)m(ev)m(er,)48 b(w)m(e)257 3528 y(shall)40 b(shortly)f(pass)h(to)e(the)h(study)h(of)e (in\014nite)i(systems)h(whic)m(h)f(ma)m(y)f(b)s(e)g(view)m(ed)i(as)257 3648 y(thermo)s(dynamic)33 b(limits)g(of)d(\014nite)i(systems.)46 b(W)-8 b(e)32 b(shall)g(analyze)g(their)g(prop)s(erties)g(in)257 3768 y(or)40 b(close)h(to)f(thermal)g(equilibrium)i(and)e(their)h(b)s (eha)m(viour)g(under)g(small)f(p)s(erturba-)257 3889 y(tions)e(of)f(their)h(dynamics)i(b)m(y)e(coupling)g(them)h(to)e (\014nite)h(subsystems.)62 b(In)38 b(order)g(to)257 4009 y(prepare)32 b(the)g(ground)g(for)e(our)i(analysis,)h(w)m(e)f(m)m(ust)h 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y(\(I)s(I.67\))32 b(is)i(closed\).)404 1558 y(If)22 b(w)m(e)h(prefer)g(to)f(w)m(ork)h(on) f(the)h(Hilb)s(ert)f(space)h Fr(H)r(\012)q(H)q Fv(,)i(instead)f(of)d Fr(K)r Fv(,)k(the)d(op)s(erators)257 1678 y Fr(S)40 b Fv(and)33 b Fr(L)g Fv(are)g(replaced)g(b)m(y)1156 1870 y Ft(S)65 b Fv(=)60 b Ft(I)1460 1887 y Fq(C)1535 1870 y Fr(S)24 b Ft(I)1669 1826 y Fn(\000)p Fo(1)1661 1895 y Fq(C)1785 1870 y Ft(;)111 b(L)60 b Fv(=)f Ft(I)2227 1887 y Fq(C)2303 1870 y Fr(L)16 b Ft(I)2437 1826 y Fn(\000)p Fo(1)2429 1895 y Fq(C)2553 1870 y Ft(;)642 b Fv(\(I)s(I.69\))257 2061 y(and)33 b Ft(J)58 b Fv(=)49 b Ft(I)727 2079 y Fq(C)802 2061 y Fr(J)35 b Ft(I)954 2020 y Fn(\000)p Fo(1)946 2089 y Fq(C)1080 2061 y Fv(is)f(giv)m(en)f(b)m(y)h(\(I)s(I.56\).)404 2182 y(It)k(is)h(instructiv)m(e)i(to)d(determine)i(the)f(explicit)g (forms)g(of)f Ft(S)44 b Fv(and)39 b Ft(L)p Fv(.)61 b(By)39 b(\(I)s(I.51\))257 2302 y(and)33 b(\(I)s(I.69\),)801 2494 y Ft(S)22 b(`)p Fv([)p Ft(a)p Fv(])16 b Ft(I)1088 2511 y Fq(C)1164 2494 y Ft(\024)1220 2511 y Fq(\032)1344 2494 y Fv(=)83 b Ft(S)22 b(I)1628 2511 y Fq(C)1703 2494 y Ft(`)p Fv([)p Ft(a)p Fv(])16 b Ft(\024)1921 2511 y Fq(\032)2049 2494 y Fv(=)86 b Ft(I)2254 2511 y Fq(C)2330 2494 y Fr(S)24 b Ft(I)2464 2450 y Fn(\000)p Fo(1)2456 2519 y Fq(C)2574 2494 y Ft(I)2617 2511 y Fq(C)2692 2494 y Ft(`)p Fv([)p Ft(a)p Fv(])16 b Ft(\024)2910 2511 y Fq(\032)1344 2639 y Fv(=)83 b Ft(I)1546 2657 y Fq(C)1621 2639 y Ft(`)p Fv([)p Ft(a)1740 2596 y Fn(\003)1780 2639 y Fv(])16 b Ft(\024)1879 2657 y Fq(\032)2006 2639 y Fv(=)87 b Ft(`)p Fv([)p Ft(a)2288 2596 y Fn(\003)2327 2639 y Fv(])16 b Ft(I)2413 2657 y Fq(C)2489 2639 y Ft(\024)2545 2657 y Fq(\032)2607 2639 y Ft(:)588 b Fv(\(I)s(I.70\))257 2831 y(Next,)34 b(for)e Ft(\024)27 b Fv(=)g Fr(j)p Ft( )t Fr(ih)p Ft(')p Fr(j)p Fv(,)33 b(with)g Ft( )t(;)16 b(')27 b Fr(2)g(H)q Fv(,)1246 3022 y Fr(L)16 b Ft(\024)60 b Fv(=)f Fr(j)p Ft(H)8 b( )t Fr(ih)p Ft(')p Fr(j)43 b(\000)h(j)p Ft( )t Fr(ih)p Ft(H)8 b(')p Fr(j)21 b Ft(;)731 b Fv(\(I)s(I.71\))257 3214 y(b)m(y)34 b(\(I)s(I.18\).)43 b(Th)m(us)893 3405 y Ft(I)936 3423 y Fq(C)1011 3405 y Fr(L)16 b Ft(\024)84 b Fv(=)f(\()p Ft(H)8 b( )t Fv(\))21 b Fr(\012)h Fv(\()p Ft(C)7 b(')p Fv(\))43 b Fr(\000)g Ft( )26 b Fr(\012)21 b Fv(\()p Ft(C)7 b(H)h(')p Fv(\))1234 3551 y(=)83 b(\()p Ft(H)8 b( )t Fv(\))21 b Fr(\012)h Fv(\()p Ft(C)7 b(')p Fv(\))43 b Fr(\000)g Ft( )26 b Fr(\012)2309 3470 y Fm(\000)2355 3551 y Fv(\()p Ft(C)7 b(H)h(C)f Fv(\))p Ft(C)g(')2815 3470 y Fm(\001)3222 3551 y Fv(\(I)s(I.72\))1234 3696 y(=)83 b(\()p Ft(H)29 b Fr(\012)22 b Fi(1)43 b Fr(\000)g Fi(1)22 b Fr(\012)g Ft(C)7 b(H)h(C)f Fv(\))16 b(\()p Ft( )25 b Fr(\012)c Ft(C)7 b(')p Fv(\))1234 3841 y(=)83 b(\()p Ft(H)29 b Fr(\012)22 b Fi(1)43 b Fr(\000)g Fi(1)22 b Fr(\012)g Ft(C)7 b(H)h(C)f Fv(\))16 b Ft(I)2373 3859 y Fq(C)2448 3841 y Ft(\024)21 b(:)257 4033 y Fv(W)-8 b(e)33 b(conclude)h(that)1294 4225 y Ft(L)60 b Fv(=)f Ft(H)29 b Fr(\012)22 b Fi(1)49 b Fr(\000)f Fi(1)22 b Fr(\012)g Ft(C)7 b(H)h(C)28 b(:)763 b Fv(\(I)s(I.73\))257 4416 y(In)49 b(their)f(imp)s(ortan)m(t)h(pap)s(er)f([3])g(on)g(the)h (equilibrium)h(states)f(of)f(non-in)m(teracting)257 4536 y(b)s(osons,)c(Araki)d(and)f(W)-8 b(o)s(o)s(ds)41 b(mak)m(e)g(a)g(sp)s (ecial)g(c)m(hoice)h(for)e Ft(C)7 b Fv(:)60 b(They)42 b(c)m(ho)s(ose)f Ft(C)48 b Fv(to)257 4657 y(b)s(e)43 b(giv)m(en)h(b)m(y)g(the)f Fs(time-r)-5 b(eversal)43 b(op)-5 b(er)g(ator)p Fv(,)46 b Ft(T)14 b Fv(,)45 b(whic)m(h,)i (according)c(to)f(a)h(result)h(of)257 4777 y(Wigner,)34 b(is)f(indeed)h(an)e(an)m(ti-unitary)h(in)m(v)m(olution)h(on)e Fr(H)q Fv(.)44 b(F)-8 b(or)32 b(this)h(c)m(hoice,)1571 4969 y Ft(T)d(H)23 b(T)74 b Fv(=)59 b Ft(H)30 b(;)1056 b Fv(\(I)s(I.74\))257 5160 y(and)33 b(hence,)h(for)e(the)h(Araki-W)-8 b(o)s(o)s(ds)32 b(isomorphism)i Ft(I)2266 5178 y Fq(T)2349 5160 y Fv(:)27 b Fr(K)i(!)e(H)c(\012)e(H)q Fv(,)1371 5352 y Ft(L)60 b Fv(=)f Ft(H)29 b Fr(\012)22 b Fi(1)49 b Fr(\000)f Fi(1)22 b Fr(\012)g Ft(H)29 b(:)840 b Fv(\(I)s(I.75\))p eop %%Page: 20 21 20 20 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(20)257 573 y Fh(I)t(I.5)132 b(P)l(erturbation)46 b(Theory)e(for)g(Equilibrium)i(States)257 758 y Fv(Next,)25 b(w)m(e)f(consider)f(a)f(\014nite)h(system)h(with)e(dynamics)i (determined)g(b)m(y)f(a)f(Hamiltonian)257 878 y Ft(H)40 b Fv(of)33 b(the)g(form)1535 1076 y Ft(H)67 b Fv(=)60 b Ft(H)1900 1094 y Fo(0)1982 1076 y Fv(+)44 b Ft(I)29 b(;)1021 b Fv(\(I)s(I.76\))257 1275 y(where)42 b Ft(H)628 1292 y Fo(0)708 1275 y Fv(is)f(the)g(Hamiltonian)g(of)f(an)h(\(unp)s (erturb)s(ed\))g Fs(r)-5 b(efer)g(enc)g(e)40 b Fv(system,)45 b(and)40 b Ft(I)257 1395 y Fv(is)35 b(a)f(p)s(erturbation.)48 b(In)35 b(this)f(section,)i(w)m(e)f(assume)h(that)e Ft(I)42 b Fv(is)34 b(a)g Fs(b)-5 b(ounde)g(d)33 b Fv(selfadjoin)m(t)257 1515 y(op)s(erator,)d(\(b)s(ecause)h(w)m(e)g(w)m(an)m(t)f(to)g(a)m(v)m (oid)g(obscuring)h(the)f(general)g(theory)g(with)h(op)s(er-)257 1636 y(ator)i(domain)g(problems\).)47 b(The)34 b(equilibrium)h(state)e (of)g(the)h(system)h(is)e(giv)m(en)i(b)m(y)f(the)257 1756 y(densit)m(y)h(matrix)1231 1955 y Ft(\032)1281 1972 y Fq(\014)s(;Q)1464 1955 y Fv(=)59 b Ft(Z)1673 1911 y Fn(\000)p Fo(1)1666 1980 y Fq(\014)s(;Q)1821 1955 y Fv(exp)q([)p Fr(\000)p Ft(\014)6 b Fv(\()p Ft(H)28 b Fr(\000)22 b Ft(Q)p Fv(\)])701 b(\(I)s(I.77\))257 2153 y(whic)m(h)34 b(corresp)s(onds)g(to)f(the)g(v)m(ector)1449 2351 y Ft(\024)1505 2369 y Fq(\014)s(;Q)1687 2351 y Fv(=)59 b Ft(\032)1872 2300 y Fo(1)p Fq(=)p Fo(2)1872 2379 y Fq(\014)s(;Q)2054 2351 y Fr(2)h(K)24 b Ft(:)918 b Fv(\(I)s(I.78\))257 2550 y(W)-8 b(e)38 b(assume)h(that)e Ft(Q)h Fv(is)g(a\016liated)g(with)g Fr(A)1917 2510 y Fn(0)1977 2550 y Fv(and)g(that)f(it)g(comm)m(utes)i (with)f Ft(H)3261 2567 y Fo(0)3338 2550 y Fv(and)257 2670 y(with)c Ft(I)40 b Fv(and)33 b(th)m(us)g(with)g Ft(H)8 b Fv(.)43 b(Then)1282 2869 y Ft(\024)1338 2886 y Fq(\014)s(;Q)1520 2869 y Fv(=)59 b Ft(e)1700 2825 y Fq(\014)s(Q=)p Fo(2)1890 2869 y Ft(Z)1964 2818 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1957 2896 y Fq(\014)s(;Q)2145 2869 y Ft(e)2190 2825 y Fn(\000)p Fq(\014)s(H)q(=)p Fo(2)2444 2869 y Ft(:)751 b Fv(\(I)s(I.79\))257 3067 y(The)34 b(equilibrium)g (state)f(of)g(the)g(reference)h(system)g(is)f(giv)m(en)h(b)m(y)1186 3281 y Ft(\024)1242 3237 y Fo(0)1242 3301 y Fq(\014)s(;Q)1424 3281 y Fv(=)1560 3200 y Fm(\000)1605 3281 y Ft(Z)1679 3237 y Fo(0)1672 3301 y Fq(\014)s(;Q)1794 3200 y Fm(\001)1840 3222 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2021 3281 y Ft(e)2066 3237 y Fq(\014)s(Q=)p Fo(2)2256 3281 y Ft(e)2301 3237 y Fn(\000)p Fq(\014)s(H)2457 3246 y Fk(0)2492 3237 y Fq(=)p Fo(2)3222 3281 y Fv(\(I)s(I.80\))257 3479 y(corresp)s(onding)g (to)e(the)h(densit)m(y)i(matrix)1513 3686 y Ft(\032)1563 3643 y Fo(0)1563 3707 y Fq(\014)s(;Q)1746 3686 y Fv(=)1881 3605 y Fm(\000)1927 3686 y Ft(\024)1983 3643 y Fo(0)1983 3707 y Fq(\014)s(;Q)2105 3605 y Fm(\001)2151 3628 y Fo(2)2212 3686 y Ft(:)983 b Fv(\(I)s(I.81\))257 3884 y(Note)33 b(that)453 4083 y Ft(Z)520 4100 y Fq(\014)s(;Q)725 4083 y Fv(=)83 b(T)-8 b(r)984 4002 y Fm(\002)1026 4083 y Ft(e)1071 4039 y Fn(\000)p Fq(\014)s Fo(\()p Fq(H)5 b Fn(\000)p Fq(Q)p Fo(\))1401 4002 y Fm(\003)1502 4083 y Fv(=)60 b(T)-8 b(r)1738 4002 y Fm(\002)1780 4083 y Ft(e)1825 4039 y Fn(\000)p Fq(\014)s Fo(\()p Fq(H)2008 4048 y Fk(0)2043 4039 y Fn(\000)p Fq(Q)p Fo(+)p Fq(I)5 b Fo(\))2275 4002 y Fm(\003)2377 4083 y Fr(\024)59 b Fv(T)-8 b(r)2612 4002 y Fm(\002)2654 4083 y Ft(e)2699 4039 y Fn(\000)p Fq(\014)s Fo(\()p Fq(H)2882 4048 y Fk(0)2917 4039 y Fn(\000)p Fq(Q)p Fo(\))3075 4083 y Ft(e)3120 4039 y Fn(\000)p Fq(\014)s(I)3258 4002 y Fm(\003)725 4244 y Fr(\024)83 b Ft(e)929 4201 y Fq(\014)s Fn(k)p Fq(I)5 b Fn(k)1104 4244 y Fv(T)-8 b(r)1205 4163 y Fm(\002)1246 4244 y Ft(e)1291 4201 y Fn(\000)p Fq(\014)s Fo(\()p Fq(H)1474 4210 y Fk(0)1509 4201 y Fn(\000)p Fq(Q)p Fo(\))1651 4163 y Fm(\003)1752 4244 y Fv(=)60 b Ft(e)1933 4201 y Fq(\014)s Fn(k)p Fq(I)5 b Fn(k)2108 4244 y Ft(Z)2182 4201 y Fo(0)2175 4265 y Fq(\014)s(;Q)2319 4244 y Ft(;)876 b Fv(\(I)s(I.82\))257 4443 y(where)25 b(the)e(\014rst)h(inequalit)m(y)h(is)f(the)g(so-called) f(Golden-Thompson)i(inequalit)m(y)g(\(whic)m(h)257 4563 y(follo)m(ws)k(from)f(the)h(T)-8 b(rotter)28 b(pro)s(duct)g(form)m(ula) g(and)h(the)f(H\177)-49 b(older)29 b(inequalit)m(y)-8 b(,)31 b(see,)f(e.g.,)257 4683 y([13,)37 b(37)o(]\),)h(and)f(the)g (second)h(inequalit)m(y)h(follo)m(ws)e(from)g Fr(k)p Ft(e)2468 4644 y Fn(\000)p Fq(\014)s(I)2606 4683 y Fr(k)c(\024)h Ft(e)2843 4644 y Fq(\014)s Fn(k)p Fq(I)5 b Fn(k)2997 4683 y Fv(,)38 b(for)e Ft(\014)j(>)34 b Fv(0.)257 4804 y(Th)m(us,)h(when)e Ft(I)41 b Fv(is)33 b(b)s(ounded,)g Ft(Z)1460 4821 y Fq(\014)s(;Q)1615 4804 y Fv(is)g(\014nite)g(i\013)g Ft(Z)2153 4764 y Fo(0)2146 4826 y Fq(\014)s(;Q)2300 4804 y Fv(is.)404 4924 y(The)k(Liouvillian)g(of)e(the)i(reference)h(system)g (and)e(the)h(Liouvillian)g(of)e(the)i(in)m(ter-)257 5044 y(acting)c(system)i(are)d(giv)m(en)i(b)m(y)945 5243 y Fr(L)1012 5260 y Fo(0)1134 5243 y Fv(:=)83 b Ft(`)p Fv([)p Ft(H)1469 5260 y Fo(0)1509 5243 y Fv(])43 b Fr(\000)g Ft(r)s Fv([)p Ft(H)1853 5260 y Fo(0)1892 5243 y Fv(])22 b Ft(;)1254 b Fv(\(I)s(I.83\))984 5388 y Fr(L)83 b Fv(:=)g Ft(`)p Fv([)p Ft(H)8 b Fv(])43 b Fr(\000)h Ft(r)s Fv([)p Ft(H)8 b Fv(])59 b(=)g Fr(L)2118 5406 y Fo(0)2196 5388 y Fv(+)38 b Ft(`)p Fv([)p Ft(I)8 b Fv(])38 b Fr(\000)f Ft(r)s Fv([)p Ft(I)8 b Fv(])22 b Ft(;)414 b Fv(\(I)s(I.84\))p eop %%Page: 21 22 21 21 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(21)257 573 y(resp)s(ectiv)m(ely)-8 b(.)71 b(W)-8 b(e)41 b(also)f(de\014ne)i(t)m(w)m(o)f Fs(R)-5 b(adon-Niko)g(dym)42 b(op)-5 b(er)g(ators)p Fv(,)42 b Fr(L)2946 590 y Fq(`)3020 573 y Fv(and)e Fr(L)3284 590 y Fq(r)3323 573 y Fv(,)i(b)m(y)257 693 y(setting)1040 904 y Fr(L)1107 921 y Fq(`)1200 904 y Fv(:=)59 b Fr(L)1429 921 y Fo(0)1507 904 y Fv(+)38 b Ft(`)p Fv([)p Ft(I)8 b Fv(])21 b Ft(;)111 b Fr(L)1993 921 y Fq(r)2091 904 y Fv(:=)60 b Fr(L)2321 921 y Fo(0)2399 904 y Fr(\000)37 b Ft(r)s Fv([)p Ft(I)8 b Fv(])22 b Ft(:)509 b Fv(\(I)s(I.85\))257 1115 y(Note)33 b(that)1115 1325 y Ft(e)1160 1282 y Fq(z)s(`)p Fo([)p Fq(a)p Fo(])1365 1325 y Fv(=)59 b Ft(`)1541 1245 y Fm(\002)1583 1325 y Ft(e)1628 1282 y Fq(z)15 b(a)1717 1245 y Fm(\003)1780 1325 y Ft(;)111 b(e)1963 1282 y Fq(z)s(r)r Fo([)p Fq(a)p Fo(])2173 1325 y Fv(=)60 b Ft(r)2356 1245 y Fm(\002)2397 1325 y Ft(e)2446 1282 y Fo(\026)-39 b Fq(z)15 b(a)2531 1245 y Fm(\003)2594 1325 y Ft(;)601 b Fv(\(I)s(I.86\))257 1536 y(for)23 b Ft(a)k Fr(2)g(A)c Fv(and)g Ft(z)32 b Fr(2)27 b Fp(C)20 b Fv(,)31 b(as)23 b(follo)m(ws)h(from)f(the)g(fact)g (that)g Ft(`)g Fv(is)h(a)e(linear)i(homomorphism,)257 1656 y(while)34 b Ft(r)h Fv(is)e(an)g(an)m(ti-linear)g(homomorphism.)44 b(By)33 b(\(I)s(I.85\))g(and)f(\(I)s(I.86\),)508 1910 y Ft(e)553 1867 y Fn(\000)p Fq(\014)s Fn(L)700 1879 y Fj(`)730 1867 y Fq(=)p Fo(2)820 1910 y Ft(\024)876 1867 y Fo(0)876 1931 y Fq(\014)s(;Q)1082 1910 y Fv(=)83 b Ft(e)1286 1865 y Fn(\000)1351 1834 y Fj(\014)p 1351 1849 38 3 v 1354 1891 a Fk(2)1398 1784 y Fm(\000)1444 1865 y Fq(`)p Fo([)p Fq(H)1551 1874 y Fk(0)1585 1865 y Fo(]+)p Fq(`)p Fo([)p Fq(I)5 b Fo(])p Fn(\000)p Fq(r)r Fo([)p Fq(H)1932 1874 y Fk(0)1964 1865 y Fo(])1984 1784 y Fm(\001)2050 1910 y Ft(\024)2106 1867 y Fo(0)2106 1931 y Fq(\014)s(;Q)2288 1910 y Fv(=)59 b Ft(e)2468 1867 y Fn(\000)2533 1836 y Fj(\014)p 2533 1852 V 2536 1893 a Fk(2)2581 1867 y Fo(\()p Fq(H)2666 1876 y Fk(0)2701 1867 y Fo(+)p Fq(I)5 b Fo(\))2839 1910 y Ft(\024)2895 1867 y Fo(0)2895 1931 y Fq(\014)s(;Q)3034 1910 y Ft(e)3089 1836 y Fj(\014)p 3089 1852 V 3092 1893 a Fk(2)3148 1867 y Fq(H)3206 1876 y Fk(0)1082 2086 y Fv(=)1241 2005 y Fm(\000)1286 2086 y Ft(Z)1360 2042 y Fo(0)1353 2106 y Fq(\014)s(;Q)1476 2005 y Fm(\001)1521 2027 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1702 2086 y Ft(e)1747 2042 y Fn(\000)1812 2011 y Fj(\014)p 1812 2027 V 1815 2068 a Fk(2)1871 2042 y Fq(H)1955 2086 y Ft(e)2010 2011 y Fj(\014)p 2010 2027 V 2013 2068 a Fk(2)2069 2042 y Fq(Q)2145 2086 y Ft(e)2190 2042 y Fn(\000)2255 2011 y Fj(\014)p 2255 2027 V 2258 2068 a Fk(2)2314 2042 y Fq(H)2372 2051 y Fk(0)2427 2086 y Ft(e)2482 2011 y Fj(\014)p 2482 2027 V 2485 2068 a Fk(2)2541 2042 y Fq(H)2599 2051 y Fk(0)1082 2262 y Fv(=)1241 2182 y Fm(\000)1286 2262 y Ft(Z)1353 2280 y Fq(\014)s(;Q)1476 2262 y Ft(=)-5 b(Z)1594 2219 y Fo(0)1587 2283 y Fq(\014)s(;Q)1708 2182 y Fm(\001)1754 2204 y Fo(1)p Fq(=)p Fo(2)1880 2182 y Fm(\000)1926 2262 y Ft(Z)1993 2280 y Fq(\014)s(;Q)2115 2182 y Fm(\001)2160 2204 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2341 2262 y Ft(e)2386 2219 y Fn(\000)2451 2188 y Fj(\014)p 2451 2204 V 2454 2245 a Fk(2)2499 2219 y Fo(\()p Fq(H)5 b Fn(\000)p Fq(Q)p Fo(\))1082 2439 y Fv(=)1241 2358 y Fm(\000)1286 2439 y Ft(Z)1353 2456 y Fq(\014)s(;Q)1476 2439 y Ft(=)-5 b(Z)1594 2395 y Fo(0)1587 2459 y Fq(\014)s(;Q)1708 2358 y Fm(\001)1754 2380 y Fo(1)p Fq(=)p Fo(2)1880 2439 y Ft(\024)1936 2456 y Fq(\014)s(;Q)2080 2439 y Ft(;)1115 b Fv(\(I)s(I.87\))257 2649 y(and)33 b(a)f(similar)i(calculation)f(yields)1098 2876 y Ft(e)1143 2832 y Fq(\014)s Fn(L)1235 2840 y Fj(r)1269 2832 y Fq(=)p Fo(2)1360 2876 y Ft(\024)1416 2832 y Fo(0)1416 2896 y Fq(\014)s(;Q)1598 2876 y Fv(=)1734 2795 y Fm(\000)1779 2876 y Ft(Z)1846 2893 y Fq(\014)s(;Q)1968 2876 y Ft(=)-5 b(Z)2086 2832 y Fo(0)2079 2896 y Fq(\014)s(;Q)2201 2795 y Fm(\001)2246 2817 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2428 2876 y Ft(\024)2484 2893 y Fq(\014)s(;Q)2628 2876 y Ft(:)567 b Fv(\(I)s(I.88\))257 3086 y(It)35 b(follo)m(ws)g(from)f(\(I)s(I.82\))g (and)h(\(I)s(I.87\),)f(\(I)s(I.88\))g(that)g Ft(\024)2331 3047 y Fo(0)2331 3109 y Fq(\014)s(;Q)2488 3086 y Fv(is)h(in)g(the)g (domain)f(of)g(def-)257 3207 y(inition)f(of)f(the)h(\(generally)g Fs(unb)-5 b(ounde)g(d)p Fv(\))32 b(op)s(erators)g(exp)q([)p Fr(\000)p Ft(\014)6 b Fr(L)2622 3224 y Fq(`)2655 3207 y Ft(=)p Fv(2])32 b(and)g(exp)q([)p Ft(\014)6 b Fr(L)3305 3224 y Fq(r)3343 3207 y Ft(=)p Fv(2],)257 3327 y(and)370 3538 y Ft(\024)426 3555 y Fq(\014)s(;Q)608 3538 y Fv(=)744 3457 y Fm(\000)790 3538 y Ft(Z)864 3494 y Fo(0)857 3558 y 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4007 y Fq(`)1826 3989 y Fv(,)g(and)g Fr(L)2143 4007 y Fq(r)2213 3989 y Fv(are)g(mapp)s(ed)g(to)543 4200 y Ft(L)83 b Fv(:=)g Ft(I)921 4217 y Fq(C)997 4200 y Fr(L)16 b Ft(I)1131 4156 y Fn(\000)p Fo(1)1123 4225 y Fq(C)1312 4200 y Fv(=)87 b Ft(L)1541 4217 y Fo(0)1618 4200 y Fv(+)38 b Ft(I)30 b Fr(\012)21 b Fi(1)38 b Fr(\000)g Fi(1)22 b Fr(\012)f Ft(C)7 b(I)h(C)94 b Fv(=:)86 b Ft(L)2832 4217 y Fo(0)2910 4200 y Fv(+)38 b Ft(W)106 b Fv(\(I)s(I.90\))510 4345 y Ft(L)576 4363 y Fq(`)692 4345 y Fv(:=)83 b Ft(I)921 4363 y Fq(C)997 4345 y Fr(L)1064 4363 y Fq(`)1113 4345 y Ft(I)1164 4302 y Fn(\000)p Fo(1)1156 4370 y Fq(C)1345 4345 y Fv(=)k Ft(L)1574 4363 y Fo(0)1651 4345 y Fv(+)38 b Ft(I)30 b Fr(\012)21 b Fi(1)h Ft(;)1182 b Fv(\(I)s(I.91\))505 4490 y Ft(L)571 4508 y Fq(r)692 4490 y Fv(:=)83 b Ft(I)921 4508 y Fq(C)997 4490 y Fr(L)1064 4508 y Fq(r)1118 4490 y Ft(I)1169 4447 y Fn(\000)p Fo(1)1161 4515 y Fq(C)1350 4490 y Fv(=)k Ft(L)1579 4508 y Fo(0)1656 4490 y Fr(\000)38 b Fi(1)22 b Fr(\012)f Ft(C)7 b(I)h(C)29 b(;)1023 b Fv(\(I)s(I.92\))257 4701 y(with)1208 4912 y Ft(L)1274 4929 y Fo(0)1397 4912 y Fv(:=)83 b Ft(H)1664 4929 y Fo(0)1725 4912 y Fr(\012)21 b Fi(1)38 b Fr(\000)g Fi(1)22 b Fr(\012)f Ft(C)7 b(H)2363 4929 y Fo(0)2403 4912 y Ft(C)28 b(;)694 b Fv(\(I)s(I.93\))1208 5057 y Ft(W)97 b Fv(:=)83 b Ft(I)29 b Fr(\012)22 b Fi(1)38 b Fr(\000)g Fi(1)21 b Fr(\012)h Ft(C)7 b(I)h(C)28 b(:)833 b Fv(\(I)s(I.94\))257 5268 y(These)44 b(form)m(ulae)e(will)g(turn)g (out)f(to)g(b)s(e)h(v)m(ery)h(useful)g(in)f(our)f(analysis)i(of)e (concrete)257 5388 y(systems.)p eop %%Page: 22 23 22 22 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(22)257 573 y Fu(I)t(I)t(I)157 b(Equilibrium)48 b(States)593 756 y(in)i(the)h(Thermo)t(dynamic)g(Limit)257 1004 y Fh(I)t(I)t(I.1)133 b(Thermo)t(dynamic)44 b(Limit)257 1188 y Fv(In)e(this)g(section,)j(w)m(e)d(recapitulate)g(the)g(general)g (theory)f(of)g(in\014nite)h(systems,)k(i.e.,)257 1309 y(systems)34 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934 y Fv(\()p Ft(a)p Fv(\))e(is)g(th)m(us)h(w)m (ell-de\014ned,)j(for)c Ft({)f Fv(su\016cien)m(tly)257 1054 y(large.)42 b(In)28 b(a)f(general)g(theory)h(of)f(thermal)h (equilibrium)h(states,)g(one)f(will)g(assume)g(that)1445 1260 y(lim)1501 1320 y Fq({)1597 1260 y Ft(\013)1660 1217 y Fo(\003)1709 1225 y Fj({)1659 1282 y Fq(t)1739 1260 y Fv(\()p Ft(a)p Fv(\))60 b(=:)f Ft(\013)2150 1278 y Fq(t)2180 1260 y Fv(\()p Ft(a)p Fv(\))925 b(\(I)s(I)s(I.5\))257 1499 y(exists,)45 b(for)40 b(all)g Ft(a)g Fr(2)h(A)1139 1516 y Fn(1)1254 1499 y Fv(and)f(all)h Ft(t)f Fr(2)g Fp(R)5 b Fv(,)49 b(and)40 b(that)g Fr(f)p Ft(\013)2444 1516 y Fq(t)2474 1499 y Fr(g)2523 1516 y Fq(t)p Fn(2)p Fl(R)2688 1499 y Fv(is)h(a)f(one-parameter)257 1619 y Fr(\003)p Fv(-automorphism)d(group)g(of)f(the)h(algebra)f Fr(A)p Fv(.)56 b(It)37 b(describ)s(es)h(the)g(time)f(ev)m(olution)h(of) 257 1740 y(the)33 b Fs(observables)e Fv(of)h(the)h(in\014nite)h(system) h(in)d(the)h(Heisen)m(b)s(erg)i(picture.)257 1997 y Fi(I)s(I)s(I.1.3) 112 b(Equilibrium)34 b(States)j([40)q(])257 2182 y Fv(As)42 b(discussed)h(in)e(Subsect.)h(I)s(I)s(I.1.2,)h(Assumptions)f(\(I)s(I)s (I.3\))f(and)f(\(I)s(I)s(I.4\))h(guaran)m(tee)257 2303 y(that)31 b(a)g(\014nite)h(system)h(con\014ned)g(to)e(a)g(region)g (\003)g(has)g(equilibrium)j(states)e Fr(h)16 b(\001)g(i)3189 2263 y Fo(\003)3189 2325 y Fq(\014)s(;Q)3312 2303 y Fv(,)31 b(see)257 2423 y(Eq.)j(\(I)s(I.28\),)e(whic)m(h)i(satisfy)g(the)f(KMS)g (condition,)g(Eqs.)h(\(I)s(I.31\),)e(\(I)s(I.35\),)g(\(I)s(I.36\).)404 2543 y(F)-8 b(or)33 b Ft(a)d Fr(2)g(A)834 2561 y Fn(1)909 2543 y Fv(,)35 b(w)m(e)g(ma)m(y)g(consider)h(the)e(sequence)j(of)d(exp) s(ectation)i(v)-5 b(alues)35 b Fr(h)p Ft(a)p Fr(i)3346 2501 y Fo(\003)3395 2509 y Fj({)3346 2571 y Fq(\014)s(;Q)3468 2543 y Fv(,)257 2664 y(whic)m(h)c(are)e(w)m(ell)h(de\014ned)g(if)f Ft({)g Fv(is)g(large)g(enough,)h(dep)s(ending)g(on)f Ft(a)p Fv(.)42 b(Let)29 b Ft(!)3002 2681 y Fq(\014)3049 2664 y Fv(\()16 b Fr(\001)g Fv(\))29 b(denote)257 2784 y(a)39 b(limit)g(of)g(a)g(\(suitable)h(subsequence)i(of)7 b(\))39 b(the)g(sequence)j Fr(h)16 b(\001)g(i)2638 2741 y Fo(\003)2687 2749 y Fj({)2638 2812 y Fq(\014)s(;Q)2799 2784 y Fv(of)39 b(states)h(on)f Fr(A)3421 2802 y Fn(1)257 2904 y Fv(Then)34 b Ft(!)573 2922 y Fq(\014)653 2904 y Fv(is)f(a)f Fs(time-tr)-5 b(anslation)34 b(invariant)e Fv(KMS)h Fs(state)g Fv(on)f Fr(A)2660 2922 y Fn(1)2735 2904 y Fv(,)g(i.e.,)1420 3111 y Ft(!)1481 3128 y Fq(\014)1528 3030 y Fm(\000)1573 3111 y Ft(\013)1635 3128 y Fq(t)1665 3111 y Fv(\()p Ft(a)p Fv(\))1792 3030 y Fm(\001)1897 3111 y Fv(=)60 b Ft(!)2094 3128 y Fq(\014)2141 3111 y Fv(\()p Ft(a)p Fv(\))21 b Ft(;)916 b Fv(\(I)s(I)s(I.6\))257 3317 y(and,)33 b(de\014ning)1355 3523 y Ft(F)1418 3540 y Fq(ab)1490 3523 y Fv(\()p Ft(t)p Fv(\))83 b(:=)g Ft(!)1931 3540 y Fq(\014)1978 3442 y Fm(\000)2024 3523 y Ft(a)16 b(\013)2153 3540 y Fq(t)2183 3523 y Fv(\()p Ft(b)p Fv(\))2300 3442 y Fm(\001)2368 3523 y Ft(;)837 b Fv(\(I)s(I)s(I.7\))1341 3668 y Ft(G)1418 3686 y 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b(e)33 b(de\014ne)g(a)g Fr(\003)p Fv(-algebra)1417 4620 y Fn(\016)1379 4708 y Fr(A)g Fv(b)m(y)605 4869 y Fn(\016)567 4957 y Fr(A)f Fv(:=)840 4846 y Fm(n)923 4957 y Ft(a)974 4974 y Fq(f)1068 4957 y Fv(=)1193 4821 y Fm(Z)1309 4957 y Ft(dt)16 b(f)11 b Fv(\()p Ft(t)p Fv(\))16 b Ft(\013)1659 4974 y Fq(t)1689 4957 y Fv(\()p Ft(a)p Fv(\))1848 4842 y Fm(\014)1848 4902 y(\014)1848 4962 y(\014)1914 4957 y Ft(a)27 b Fr(2)g(A)33 b Ft(;)2292 4931 y Fv(^)2271 4957 y Ft(f)k Fr(2)27 b Ft(C)2525 4913 y Fn(1)2518 4977 y Fo(0)2600 4957 y Fv(\()p Fp(R)5 b Fv(\))2758 4846 y Fm(o)2852 4957 y Ft(;)305 b Fv(\(I)s(I)s(I.11\))257 5232 y(where)564 5206 y(^)543 5232 y Ft(f)47 b Fv(denotes)38 b(the)f(F)-8 b(ourier)36 b(transform)g(of)g Ft(f)11 b Fv(.)55 b(Since)37 b Ft(\013)2535 5250 y Fq(t)2601 5232 y Fv(has)g(b)s(een)g(assumed)i(to)257 5388 y(b)s(e)g(a)f Fr(\003)p Fv(-automorphism)g(group)g(of)g Fr(A)p Fv(,)1818 5300 y Fn(\016)1780 5388 y Fr(A)g Fv(is)h(a)f Fr(\003)p Fv(-subalgebra)g(of)f Fr(A)p Fv(,)j(and)e Ft(\013)3180 5406 y Fq(t)3248 5388 y Fv(lea)m(v)m(es)p eop %%Page: 24 25 24 24 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(24)317 514 y Fn(\016)279 602 y Fr(A)41 b Fv(in)m(v)-5 b(arian)m(t.)70 b(F)-8 b(or)40 b(an)m(y)i Ft(a)g Fr(2)1464 514 y Fn(\016)1426 602 y Fr(A)p Fv(,)i Ft(\013)1637 619 y Fq(t)1666 602 y Fv(\()p Ft(a)p Fv(\))d(is)h(the)g(b)s(oundary)f(v)-5 b(alue)42 b(of)e(an)3149 514 y Fn(\016)3111 602 y Fr(A)p Fv(-v)-5 b(alued)257 763 y(en)m(tire)34 b(function)f Ft(\013)975 781 y Fq(z)1015 763 y Fv(\()p Ft(a)p Fv(\),)g Ft(z)f Fr(2)c Fp(C)20 b Fv(.)50 b(F)-8 b(or)32 b Ft(a)27 b Fr(2)h(A)k Fv(and)h Ft(b)28 b Fr(2)2354 675 y Fn(\016)2316 763 y Fr(A)p Fv(,)33 b(w)m(e)h(can)f(rewrite)h(the)f(KMS)257 883 y(condition)h(as)e(the)h(equation)1190 1103 y Ft(!)1251 1121 y Fq(\014)1298 1023 y Fm(\000)1344 1103 y Ft(a)16 b(\013)1473 1121 y Fq(t)1503 1103 y Fv(\()p Ft(b)p Fv(\))1620 1023 y Fm(\001)1726 1103 y Fv(=)59 b Ft(!)1922 1121 y Fq(\014)1969 1023 y Fm(\000)2015 1103 y Ft(\013)2077 1121 y Fn(\000)p Fq(i\014)s Fo(+)p Fq(t)2283 1103 y Fv(\()p Ft(b)p Fv(\))16 b Ft(a)2467 1023 y Fm(\001)2535 1103 y Ft(:)622 b Fv(\(I)s(I)s(I.12\))257 1323 y(Note)33 b(that)1478 1543 y Ft(\013)1540 1561 y Fq(z)1580 1543 y Fv(\()p Ft(a)p Fv(\))1707 1500 y Fn(\003)1806 1543 y Fv(=)59 b Ft(\013)2007 1561 y Fo(\026)-39 b Fq(z)2043 1543 y Fv(\()p Ft(a)2132 1500 y Fn(\003)2172 1543 y Fv(\))21 b Ft(;)926 b Fv(\(I)s(I)s(I.13\)) 257 1816 y(for)39 b Ft(z)k Fr(2)37 b Fp(C)20 b Fv(,)47 b Ft(a)38 b Fr(2)957 1728 y Fn(\016)919 1816 y Fr(A)p Fv(.)63 b(The)40 b(in)m(v)-5 b(ariance)40 b(\(I)s(I)s(I.6\),)g(the)g (KMS)f(condition)h(\(I)s(I)s(I.12\),)g(and)257 1977 y(\(I)s(I)s(I.13\)) 32 b(then)i(imply)f(that,)g(for)f Ft(a;)16 b(b)28 b Fr(2)1773 1889 y Fn(\016)1735 1977 y Fr(A)o Fv(,)478 2197 y Ft(!)539 2215 y Fq(\014)586 2117 y Fm(\000)632 2197 y Ft(a)683 2154 y Fn(\003)739 2197 y Ft(b)780 2117 y Fm(\001)885 2197 y Fv(=)60 b Ft(!)1082 2215 y Fq(\014)1129 2117 y Fm(\000)1174 2197 y 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Fv(suc)m(h)i(that,)f(for)f(ev)m (ery)i Ft(a)e Fr(2)g(A)p Fv(,)h(there)g(exists)257 998 y(a)33 b(sequence)i Fr(f)p Ft(a)843 1016 y Fq(\025)888 998 y Fr(g)937 1016 y Fq(\025)p Fn(2)p Fl(N)1104 998 y Fr(\022)1238 973 y Fm(e)1207 998 y Fr(A)e Fv(with)g(the)g(prop)s(ert) m(y)g(that)1176 1195 y(lim)1152 1257 y Fq(\025)p Fn(!1)1351 1195 y Ft(!)1412 1212 y Fq(\014)1459 1114 y Fm(\000)1504 1195 y Fv(\()p Ft(a)22 b Fr(\000)g Ft(a)1764 1212 y Fq(\025)1809 1195 y Fv(\))1847 1151 y Fn(\003)1903 1195 y Fv(\()p Ft(a)g Fr(\000)f Ft(a)2162 1212 y Fq(\025)2208 1195 y Fv(\))2246 1114 y Fm(\001)2351 1195 y Fv(=)60 b(0)21 b Ft(;)600 b Fv(\(I)s(I)s(I.19\))257 1420 y(and)33 b(that)1276 1617 y Ft(!)1337 1634 y Fq(\014)1383 1536 y Fm(\000)1429 1617 y Ft(a)16 b(\013)1558 1634 y Fq(t)1588 1617 y Fv(\()p Ft(b)p Fv(\))1705 1536 y Fm(\001)1811 1617 y Fv(=)59 b Ft(!)2007 1634 y Fq(\014)2054 1536 y Fm(\000)2100 1617 y Ft(\013)2162 1634 y Fn(\000)p Fq(t)2247 1617 y Fv(\()p Ft(a)p Fv(\))16 b Ft(b)2431 1536 y Fm(\001)3184 1617 y Fv(\(I)s(I)s(I.20\))257 1834 y(is)33 b Fs(c)-5 b(ontinuous)33 b Fv(in)g Ft(t)p Fv(,)f(for)g(arbitrary)h Ft(a;)16 b(b)28 b Fr(2)1902 1809 y Fm(e)1871 1834 y Fr(A)p Fv(.)404 1954 y(Next,)35 b(w)m(e)g(recapitulate)g(the)f(GNS)g Fs(c)-5 b(onstruction)33 b Fv(in)h(a)g(situation)g(where)h(\(I)s(I)s(I.6\),)257 2075 y(\(I)s(I)s(I.12\),)53 b(\(I)s(I)s(I.19\),)g(and)c(\(I)s(I)s (I.20\))f(hold:)77 b(T)-8 b(o)49 b(the)g(data)g(\()p Fr(A)p Ft(;)16 b(\013)2699 2092 y Fq(t)2729 2075 y Ft(;)g(!)2833 2092 y Fq(\014)2880 2075 y Fv(\),)53 b(as)c(sp)s(eci\014ed)257 2195 y(ab)s(o)m(v)m(e,)d(one)c(can)h(asso)s(ciate)g(a)f(separable)h (Hilb)s(ert)g(space)g Fr(H)2597 2213 y Fq(\014)2645 2195 y Fv(,)h(a)e(represen)m(tation)i Ft(`)257 2316 y Fv(of)e Fr(A)f Fv(on)h Fr(H)724 2333 y Fq(\014)771 2316 y Fv(,)j(a)c(v)m(ector) i(\012)1305 2333 y Fq(\014)1395 2316 y Fr(2)g(H)1585 2333 y Fq(\014)1674 2316 y Fv(whic)m(h)g(is)g(cyclic)g(for)f Ft(`)p Fv([)p Fr(A)p Fv(],)i(and)e(a)f(con)m(tin)m(uous)257 2436 y(one-parameter)28 b(group)f(of)g(unitary)h(op)s(erators)f Fr(f)p Ft(e)2133 2397 y Fn(\000)p Fq(it)p Fn(L)2290 2436 y Fr(g)2339 2453 y Fq(t)p Fn(2)p Fl(R)2463 2436 y Fv(,)i(where)f Fr(L)g Fv(is)f(a)g(selfadjoin)m(t)257 2556 y(op)s(erator)32 b(on)h Fr(H)868 2574 y Fq(\014)915 2556 y Fv(,)g(suc)m(h)h(that,)e(for) g(all)h Ft(a)27 b Fr(2)g(A)p Fv(,)1384 2753 y Ft(!)1445 2770 y Fq(\014)1492 2753 y Fv(\()p Ft(a)p Fv(\))83 b(=)g Fr(h)p Fv(\012)1969 2770 y Fq(\014)2033 2753 y Fr(j)16 b Ft(`)p Fv([)p Ft(a)p Fv(])g(\012)2308 2770 y Fq(\014)2356 2753 y Fr(i)22 b Ft(;)741 b Fv(\(I)s(I)s(I.21\))1276 2898 y Ft(`)1317 2817 y Fm(\002)1359 2898 y Ft(\013)1421 2915 y Fq(t)1450 2898 y Fv(\()p Ft(a)p Fv(\))1577 2817 y Fm(\003)1702 2898 y Fv(=)83 b Ft(e)1906 2854 y Fq(it)p Fn(L)2035 2898 y Ft(`)p Fv([)p Ft(a)p Fv(])28 b Ft(e)2254 2854 y Fn(\000)p Fq(it)p Fn(L)2433 2898 y Ft(;)724 b Fv(\(I)s(I)s(I.22\))1418 3043 y Fr(L)16 b Fv(\012)1571 3061 y Fq(\014)1702 3043 y Fv(=)83 b(0)21 b Ft(:)1226 b Fv(\(I)s(I)s(I.23\))257 3239 y(T)-8 b(o)35 b(construct)i Fr(H)917 3257 y Fq(\014)964 3239 y Fv(,)f(\012)1097 3257 y Fq(\014)1144 3239 y Fv(,)g Ft(`)p Fv(,)g(and)f Fr(L)p Fv(,)h(let)f Fr(N)46 b Fv(:=)32 b Fr(f)p Ft(a)f Fr(2)g(A)k(j)g Ft(!)2500 3257 y Fq(\014)2547 3239 y Fv(\()p Ft(a)2636 3200 y Fn(\003)2676 3239 y Ft(a)p Fv(\))c(=)g(0)g Fr(g)p Fv(.)51 b(As)36 b(noted)257 3360 y(ab)s(o)m(v)m(e,)42 b Fr(N)54 b Fv(is)40 b(trivial)g(if)f Fr(A)g Fv(is)h(simple,)j(b)m(y)d (the)g(KMS)g(condition.)64 b(W)-8 b(e)40 b(set)g([)p Ft(a)p Fv(])f(:=)f Ft(a)257 3480 y Fv(mo)s(d)33 b Fr(N)15 b Fv(,)40 b(for)e(all)h Ft(a)f Fr(2)f(A)p Fv(.)62 b(Clearly)-8 b(,)42 b Fr(D)e Fv(:=)e Fr(f)p Fv([)p Ft(a)p Fv(])h Fr(j)g Ft(a)e Fr(2)h(Ag)g Fv(is)i(a)e(linear)h(space.)64 b(It)39 b(is)257 3600 y(equipp)s(ed)c(with)e(a)f(scalar)h(pro)s(duct)1382 3797 y Fr(h)16 b Fv([)p Ft(a)p Fv(])g Fr(j)g Fv([)p Ft(b)p Fv(])g Fr(i)62 b Fv(:=)e Ft(!)2035 3814 y Fq(\014)2082 3716 y Fm(\000)2127 3797 y Ft(a)2178 3753 y Fn(\003)2234 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y Fq(t)p Fn(2)p Fl(R)1335 4746 y Fv(on)g Fr(H)1552 4764 y Fq(\014)1632 4746 y Fv(b)m(y)1428 4942 y Ft(e)1473 4899 y Fn(\000)p Fq(it)p Fn(L)1646 4942 y Fv([)p Ft(a)p Fv(])61 b(:=)1974 4862 y Fm(\002)2016 4942 y Ft(\013)2078 4960 y Fq(t)2107 4942 y Fv(\()p Ft(a)p Fv(\))2234 4862 y Fm(\003)2298 4942 y Ft(:)859 b Fv(\(I)s(I)s(I.26\))257 5147 y(Unitarit)m(y)42 b(follo)m(ws)g(from)e(the)i(in)m(v)-5 b(ariance)42 b(of)e Ft(!)2084 5165 y Fq(\014)2172 5147 y Fv(under)h Ft(\013)2518 5165 y Fq(t)2548 5147 y Fv(.)69 b(By)41 b(\(I)s(I)s(I.20\),)i Ft(e)3232 5108 y Fn(\000)p Fq(it)p Fn(L)3430 5147 y Fv(is)257 5268 y(strongly)35 b(con)m(tin)m(uous)h(on)e(the)h(separable)g(Hilb)s (ert)g(space)g Fr(H)2545 5285 y Fq(\014)2592 5268 y Fv(,)g(and)f(hence) i(it)e(is)g(gen-)257 5388 y(erated)f(b)m(y)h(a)e(selfadjoin)m(t)i(op)s (erator)d Fr(L)p Fv(,)i(the)g Fs(Liouvil)5 b(lian)p Fv(;)32 b(\(Stone's)i(theorem\).)p eop %%Page: 26 27 26 26 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(26)257 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Fv([)p Ft(a)p Fv(])16 b(\012)1873 3524 y Fq(\014)1943 3506 y Ft(;)1214 b Fv(\(I)s(I)s(I.37\))257 3706 y(and)37 b(w)m(e)i(ha)m(v)m(e)f(used)g(\(I)s(I)s(I.13\).)56 b(As)38 b(a)f(corollary)g(of)f(\(I)s(I)s(I.36\),)i(\(I)s(I)s(I.31\),)g (and)f(\(I)s(I)s(I.26\),)257 3827 y(w)m(e)d(ha)m(v)m(e)g(that)1290 4026 y Ft(e)1335 3983 y Fq(it)p Fn(L)1454 4026 y Ft(`)p Fv([)p Ft(a)p Fv(])16 b Ft(e)1661 3983 y Fn(\000)p Fq(it)p Fn(L)1902 4026 y Fv(=)83 b Ft(`)2102 3946 y Fm(\002)2143 4026 y Ft(\013)2205 4044 y Fq(t)2235 4026 y Fv(\()p Ft(a)p Fv(\))2362 3946 y Fm(\003)2425 4026 y Ft(;)732 b Fv(\(I)s(I)s(I.38\)) 1284 4183 y Ft(e)1329 4140 y Fq(it)p Fn(L)1448 4183 y Ft(r)s Fv([)p Ft(a)p Fv(])16 b Ft(e)1661 4140 y Fn(\000)p Fq(it)p Fn(L)1902 4183 y Fv(=)83 b Ft(r)2108 4102 y Fm(\002)2149 4183 y Ft(\013)2211 4201 y Fq(t)2240 4183 y Fv(\()p Ft(a)p Fv(\))2367 4102 y Fm(\003)2431 4183 y Ft(:)726 b Fv(\(I)s(I)s(I.39\)) 257 4392 y(Eq.)42 b(\(I)s(I)s(I.36\))e(implies)i(that)e Fr(J)35 b Ft(i)16 b Fr(L)41 b Fv(=)g 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Fq(T)10 b Fn(!1)1439 4792 y Fv(1)p 1428 4841 71 4 v 1428 4929 a Ft(T)1525 4728 y Fm(Z)1624 4754 y Fq(T)1580 4953 y Fo(0)1696 4864 y Ft(\032)1746 4881 y Fn(\006)p Fq(t)1830 4864 y Fv(\()p Ft(a)p Fv(\))16 b Ft(dt)63 b Fv(=)e Ft(!)2320 4881 y Fq(\014)2367 4864 y Fv(\()p Ft(a)p Fv(\))22 b Ft(:)641 b Fv(\(I)s(I)s(I.47\))257 5114 y Fs(If)35 b Ft(\033)t Fv(\()p Fr(L)p Fv(\))22 b Fr(n)f(f)p Fv(0)p Fr(g)34 b Fs(is)h(absolutely)g(c)-5 b(ontinuous)34 b(then)1455 5311 y Fv(lim)1412 5371 y Fq(t)p Fn(!\0061)1650 5311 y Ft(\032)1700 5329 y Fq(t)1730 5311 y Fv(\()p Ft(a)p Fv(\))62 b(=)g Ft(!)2118 5329 y Fq(\014)2165 5311 y Fv(\()p Ft(a)p Fv(\))21 b Ft(:)844 b Fv(\(I)s(I)s(I.48\))p eop %%Page: 29 30 29 29 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(29)257 573 y Fs(Pr)-5 b(o)g(of.)49 b Fv(First,)32 b(the)i(con)m(tin)m(uit)m(y)g(of)e(the)h(sp)s(ectrum)h(a)m(w)m(a)m(y)g (from)e(0)h(implies)h(that)1081 817 y Ft(w)24 b Fr(\000)50 b Fv(lim)1272 879 y Fq(T)10 b Fn(!1)1501 746 y Fv(1)p 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Ft(I)8 b Fv(\))22 b Ft(:)982 b Fv(\(I)s(I)s(I.55\))p eop %%Page: 30 31 30 30 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(30)257 573 y(Then)34 b(the)f(equation)738 863 y Ft(\013)801 812 y Fo(\()p Fq(I)5 b Fo(\))800 886 y Fq(t)895 863 y Fv(\()p Ft(a)p Fv(\))87 b(:=)1335 739 y Fn(1)1299 769 y Fm(X)1304 978 y Fq(n)p Fo(=0)1459 863 y Ft(i)1492 820 y Fq(n)1555 728 y Fm(Z)1655 754 y Fq(t)1611 953 y Fo(0)1701 863 y Ft(dt)1787 881 y Fo(1)1842 728 y Fm(Z)1942 754 y Fq(t)1967 763 y Fk(1)1898 953 y Fo(0)2022 863 y Ft(dt)2108 881 y Fo(2)3184 863 y Fv(\(I)s(I)s(I.56\))904 1171 y Fr(\001)16 b(\001)g(\001)1034 1036 y Fm(Z)1133 1062 y Fq(t)1158 1071 y Fj(n)p Fe(\000)p Fk(1)1089 1261 y Fo(0)1300 1171 y Ft(dt)1386 1189 y Fq(n)1449 1061 y Fm(h)1496 1171 y Ft(I)8 b Fv(\()p Ft(t)1620 1189 y Fq(n)1667 1171 y Fv(\))16 b Ft(;)33 b Fr(\001)16 b(\001)g(\001)1911 1090 y Fm(\002)1953 1171 y Ft(I)8 b Fv(\()p Ft(t)2077 1189 y Fo(2)2116 1171 y Fv(\))16 b Ft(;)33 b Fv([)p Ft(I)8 b Fv(\()p 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Fr(L)1873 1893 y Fo(\()p Fq(`)p Fo(\))1873 1971 y Fq(I)1962 1863 y Fm(\001)2034 1944 y Ft(`)p Fv([)p Ft(a)p Fv(])44 b(exp)2373 1863 y Fm(\000)2419 1944 y Fr(\000)p Ft(it)p Fr(L)2630 1893 y Fo(\()p Fq(`)p Fo(\))2630 1971 y Fq(I)2718 1863 y Fm(\001)2786 1944 y Ft(:)371 b Fv(\(I)s(I)s(I.57\))257 2164 y(Using)37 b(\(I)s(I)s(I.38\))e(and)g(applying)i(the)f(represen)m (tation)h Ft(`)e Fv(to)h(\(I)s(I)s(I.56\),)g(w)m(e)g(readily)h(\014nd) 257 2284 y(that)1358 2504 y Fr(L)1425 2454 y Fo(\()p Fq(`)p Fo(\))1425 2531 y Fq(I)1573 2504 y Fv(=)59 b Fr(L)44 b Fv(+)f Ft(`)p Fv([)p Ft(I)8 b Fv(])44 b(+)f Fr(R)22 b Ft(;)806 b Fv(\(I)s(I)s(I.58\))257 2749 y(where)38 b Fr(R)f Fv(is)h(an)e(arbitrary)h(op)s(erator)f(in)h Ft(`)p Fv([)p Fr(A)p Fv(])2011 2710 y Fn(0)2068 2749 y Fv(=)d Ft(r)s Fv([)p Fr(A)p Fv(])2357 2710 y Fn(00)2399 2749 y Fv(.)56 b(Of)36 b(course,)j Ft(\013)3018 2698 y Fo(\()p Fq(I)5 b Fo(\))3017 2772 y Fq(t)3149 2749 y Fv(can)37 b(also)257 2870 y(b)s(e)c(implemen)m(ted)i(unitarily)f(on)e 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b(It)36 b(w)m(ould)i(b)s(e)f(pleasan)m(t)h (to)257 4512 y(ha)m(v)m(e)c Fr(L)549 4461 y Fo(\()p Fq(`)p Fo(\))549 4539 y Fq(I)665 4512 y Fv(=)27 b Fr(L)835 4461 y Fo(\()p Fq(r)r Fo(\))835 4539 y Fq(I)928 4512 y Fv(.)43 b(This)34 b(equation)f(has)g(the)g(unique)h(solution)721 4747 y Fr(L)788 4696 y Fo(\()p Fq(`)p Fo(\))788 4774 y Fq(I)936 4747 y Fv(=)59 b Fr(L)1138 4696 y Fo(\()p Fq(r)r Fo(\))1138 4774 y Fq(I)1291 4747 y Fv(=:)h Fr(L)1521 4765 y Fq(I)1621 4747 y Fv(=)f Fr(L)38 b Fv(+)g Ft(`)p Fv([)p Ft(I)8 b Fv(])38 b Fr(\000)g Ft(r)s Fv([)p Ft(I)8 b Fv(])38 b(+)f Fr(Z)30 b Ft(;)480 b Fv(\(I)s(I)s(I.62\))257 4967 y(where)38 b Fr(Z)44 b Fv(is)37 b(in)g(the)f(cen)m(ter)i(of)e(the) h(v)m(on)g(Neumann)g(algebra)f Ft(`)p Fv([)p Fr(A)p Fv(])2781 4928 y Fn(0)q(0)2824 4967 y Fv(,)h(whic)m(h)h(is)f(giv)m(en)257 5088 y(b)m(y)d Ft(`)p Fv([)p Fr(A)p Fv(])566 5048 y Fn(00)630 5088 y Fr(\\)22 b Ft(r)s Fv([)p Fr(A)p Fv(])896 5048 y Fn(00)938 5088 y Fv(.)43 b(Without)33 b(loss)g(of)g(generalit)m(y)-8 b(,)33 b(w)m(e)h(set)f Fr(Z)i Fv(=)27 b(0.)43 b(Then)1508 5308 y Fr(L)1575 5325 y Fq(I)1675 5308 y Fv(=)60 b Fr(\000J)34 b(L)2054 5325 y Fq(I)2111 5308 y Fr(J)40 b Ft(:)940 b Fv(\(I)s(I)s(I.63\))p eop %%Page: 31 32 31 31 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(31)257 573 y(In)29 b(our)g(applications)h(of)e(the)h(general)g (theory)-8 b(,)30 b(w)m(e)g(shall)g(use)f(the)g(follo)m(wing)g (notation:)1071 793 y Fr(L)1138 810 y Fq(I)1238 793 y Fv(:=)60 b Fr(L)38 b Fv(+)g Ft(W)73 b Fv(:=)60 b Fr(L)38 b Fv(+)g Ft(`)p Fv([)p Ft(I)8 b Fv(])38 b Fr(\000)g Ft(r)s Fv([)p Ft(I)8 b Fv(])21 b Ft(;)519 b Fv(\(I)s(I)s(I.64\))257 1013 y(with)48 b Ft(W)65 b Fv(:=)51 b Ft(`)p Fv([)p Ft(I)8 b Fv(])31 b Fr(\000)h Ft(r)s Fv([)p Ft(I)8 b Fv(].)86 b(It)47 b(is)h(natural)f(to)g(ask)g(whether)i(w)m(e)f(can)f(construct)h (a)257 1147 y(p)s(erturb)s(ed)34 b(KMS)f(state)g(for)g(the)g(dynamics)i (describ)s(ed)f(b)m(y)g Fr(f)p Ft(\013)2633 1096 y Fo(\()p Fq(I)5 b Fo(\))2632 1169 y Fq(t)2727 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Fu(IV)156 b(KMS)60 b(States)f(and)h(Liouvillians)c(for)61 b(Idealized)588 4503 y(A)l(toms)69 b(coupled)f(to)j(the)e(Quan)l(tized) f(Electro-)588 4685 y(magnetic)50 b(Field)257 4934 y Fh(IV.1)132 b(KMS)54 b(States)h(for)f(the)g(Quan)l(tized)i(F)-11 b(ree)53 b(Electromag-)640 5083 y(netic)45 b(Field)257 5268 y Fv(In)29 b(this)h(subsection,)h(w)m(e)f(illustrate)f(the)g (general)h(theory)f(dev)m(elop)s(ed)h(in)f(Sects.)h(I)s(I)f(and)257 5388 y(I)s(I)s(I)22 b(on)g(the)h(example)h(of)d(the)i(equilibrium)h (\(KMS\))e(states)h(of)f(the)g(free)h(electromagnetic)p eop %%Page: 34 35 34 34 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(34)257 573 y(\014eld)40 b(in)f(the)g(thermo)s(dynamic)h(limit,)h (as)d(describ)s(ed)j(b)m(y)e(Araki)g(and)g(W)-8 b(o)s(o)s(ds)38 b(in)h([3].)257 693 y(Similar)g(results)g(can)f(b)s(e)g(deriv)m(ed)i (for)d(gases)i(of)e(free)i(fermions)f(at)g(p)s(ositiv)m(e)h(densit)m(y) 257 814 y(and)33 b(temp)s(erature;)h(see)f([4].)404 934 y(It)f(is)g(con)m(v)m(enien)m(t)i(to)e(describ)s(e)h(the)f(free)g (electromagnetic)i(\014eld)e(in)g(terms)h(of)e(cre-)257 1054 y(ation)j(-and)f(annihilation)h(op)s(erators)g Ft(a)1758 1015 y Fn(\003)1758 1077 y Fq(\025)1804 1054 y Fv(\()1837 1028 y Ft(~)1842 1054 y(k)s Fv(\),)g Ft(a)2046 1072 y Fq(\025)2091 1054 y Fv(\()2124 1028 y Ft(~)2129 1054 y(k)s Fv(\))g(satisfying)h(the)f(canonical)g(com-)257 1175 y(m)m(utation)f(relations,)945 1368 y([)p Ft(a)1023 1325 y Fn(\003)1023 1389 y Fq(\025)1069 1368 y Fv(\()1102 1342 y Ft(~)1107 1368 y(k)s Fv(\))16 b Ft(;)33 b(a)1326 1325 y Fn(\003)1326 1389 y Fq(\025)1367 1370 y Fe(0)1393 1368 y Fv(\()1426 1342 y Ft(~)1431 1368 y(k)1485 1325 y Fn(0)1509 1368 y Fv(\)])83 b(=)g([)p Ft(a)1894 1386 y Fq(\025)1939 1368 y Fv(\()1972 1342 y Ft(~)1977 1368 y(k)s Fv(\))16 b Ft(;)33 b(a)2196 1386 y Fq(\025)2237 1367 y Fe(0)2264 1368 y Fv(\()2297 1342 y Ft(~)2302 1368 y(k)2356 1325 y Fn(0)2379 1368 y Fv(\)])87 b(=)g(0)21 b Ft(;)444 b Fv(\(IV.1\))945 1524 y([)p 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(er,)h(translate)d(our)g(results)h(in)m(to)g Ft(C)2480 2039 y Fn(\003)2519 2078 y Fv(-algebra)e(language.)43 b(But)257 2199 y(in)33 b(the)g(analysis)h(of)e(concrete)i(mo)s(dels,)f Fr(\003)p Fv(-algebras)g(are)f(more)h(con)m(v)m(enien)m(t.)404 2319 y(F)-8 b(or)35 b(notational)h(con)m(v)m(enience,)41 b(w)m(e)c(set)g Ft(k)f Fv(:=)e(\()2178 2293 y Ft(~)2183 2319 y(k)s(;)16 b(\025)p Fv(\))33 b Fr(2)h Fp(R)2573 2280 y Fo(3)2642 2319 y Fr(\002)24 b Fp(Z)2811 2336 y Fo(2)2848 2319 y Fv(,)37 b(where)3193 2293 y Ft(~)3198 2319 y(k)f Fr(2)e Fp(R)3450 2280 y Fo(3)257 2439 y Fv(is)46 b(the)g(w)m(a)m(v)m(e)h(v)m(ector)f(and)f Ft(\025)j Fv(=)g Fr(\006)p Fv(1)d(is)h(the)f(p)s(olarization)g(index,)50 b(and)45 b(w)m(e)h(denote)257 2479 y Fm(R)340 2560 y Ft(dk)30 b Fv(:=)602 2485 y Fm(P)707 2589 y Fq(\025)p Fo(=)p Fn(\006)p Fo(1)914 2479 y Fm(R)996 2560 y Ft(d)1047 2520 y Fo(3)1087 2560 y Ft(k)s Fv(.)43 b(W)-8 b(e)33 b(de\014ne)1260 2774 y Ft(\016)t Fv(\()p Ft(k)24 b Fr(\000)e Ft(k)1572 2731 y Fn(0)1595 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Ft(:)657 b Fv(\(IV.7\))257 5049 y(Then)34 b(\(IV.1\))f(and)f(\(IV.2\))h(can)g(b)s(e)g(written)g (as)1048 5243 y([)p Ft(a)1126 5199 y Fn(\003)1166 5243 y Fv(\()p Ft(f)11 b Fv(\))16 b Ft(;)32 b(a)1427 5199 y Fn(\003)1467 5243 y Fv(\()p Ft(g)t Fv(\)])82 b(=)h([)p Ft(a)p Fv(\()p Ft(f)11 b Fv(\))16 b Ft(;)33 b(a)p Fv(\()p Ft(g)t Fv(\)])86 b(=)h(0)21 b Ft(;)533 b Fv(\(IV.8\))1034 5388 y([)p Ft(a)p Fv(\()p Ft(\034)11 b(f)g Fv(\))16 b Ft(;)33 b(a)1427 5345 y Fn(\003)1467 5388 y Fv(\()p Ft(g)t Fv(\)])82 b(=)h(\()p Ft(f)5 b(;)16 b(g)t Fv(\))g Fi(1)22 b Ft(:)1029 b Fv(\(IV.9\))p eop %%Page: 35 36 35 35 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(35)257 573 y(W)-8 b(e)48 b(shall,)j(ho)m(w)m(ev)m(er,)h(con)m(tin) m(ue)d(to)d(also)h(use)h(the)f(op)s(erator-v)-5 b(alued)47 b(distributions)257 693 y Ft(a)p Fv(\()p Ft(k)s Fv(\),)33 b Ft(a)549 654 y Fn(\003)589 693 y Fv(\()p Ft(k)s Fv(\).)404 814 y(Let)f Fr(P)41 b Fv(denote)33 b(the)g(p)s(olynomial)g(algebra)g 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b(\012)g(arise)257 4409 y(b)m(y)e(GNS)e(construction)i(from)e(the)h Fs(quasi-fr)-5 b(e)g(e)36 b Fv(state)i Ft(!)2355 4426 y Fn(1)2467 4409 y Fv(on)g Fr(P)45 b Fv(giv)m(en)38 b(b)m(y)h(letting)f Ft(\014)257 4529 y Fv(tend)i(to)e Fr(1)g Fv(in)h(\(IV.15\))f(and)g (\(IV.16\).)62 b(In)39 b(our)f(notation,)i(w)m(e)f(iden)m(tify)h Fr(P)47 b Fv(with)39 b(its)257 4650 y(represen)m(tation)c(on)d Fr(F)10 b Fv(.)404 4770 y(W)-8 b(e)30 b(de\014ne)i(an)e(an)m (ti-unitary)g(op)s(erator)g Ft(T)43 b Fv(on)30 b Fr(F)10 b Fv(,)31 b(the)g Fs(se)-5 b(c)g(ond)31 b(quantization)f Fv(of)g Ft(\034)11 b Fv(,)257 4890 y(b)m(y)34 b(setting)1528 5098 y Ft(T)c Fv(\012)84 b(=)e(\012)23 b Ft(;)1139 b Fv(\(IV.22\))1252 5243 y Ft(T)35 b(a)1395 5199 y Fo(#)1458 5243 y Fv(\()p Ft(f)11 b Fv(\))22 b Ft(T)97 b Fv(=)82 b Ft(a)1978 5199 y Fo(#)2042 5243 y Fv(\()p Ft(\034)11 b(f)g Fv(\))21 b Ft(;)908 b Fv(\(IV.23\))1615 5388 y Ft(T)97 b Fv(=)82 b Ft(T)1998 5345 y Fn(\003)2124 5388 y Fv(=)87 b Ft(T)2358 5345 y Fn(\000)p Fo(1)2474 5388 y Ft(:)685 b Fv(\(IV.24\))p eop %%Page: 37 38 37 37 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(37)257 573 y(Ph)m(ysically)-8 b(,)36 b Ft(T)46 b Fv(describ)s(es)34 b Fs(time)h(r)-5 b(eversal)p Fv(;)32 b(\(compare)h(\(IV.12\))g(and)f(\(IV.23\)\).)404 693 y(Next,)38 b(w)m(e)g(describ)s(e)g(an)e(isomorphism,)k Ft(I)1988 711 y Fq(T)2043 693 y Fv(,)d(b)s(et)m(w)m(een)i Fr(H)2571 646 y Fq(f)2570 721 y(\014)2654 693 y Fv(and)d Fr(F)f(\012)24 b(F)47 b Fv(and)36 b(b)s(e-)257 814 y(t)m(w)m(een)43 b(appropriate)f(represen)m(tations.)72 b(On)41 b Fr(F)d(\012)27 b(F)10 b Fv(,)44 b(w)m(e)e(in)m(tro)s(duce)h(the)e(creation-)257 934 y(and)33 b(annihilation)g(op)s(erators)947 1126 y Ft(a)998 1079 y Fo(#)998 1154 y Fq(`)1061 1126 y Fv(\()p Ft(f)11 b Fv(\))83 b(:=)g Ft(a)1516 1082 y Fo(#)1579 1126 y Fv(\()p Ft(f)11 b Fv(\))21 b Fr(\012)h Fi(1)f Ft(;)1249 b Fv(\(IV.25\))947 1271 y Ft(a)998 1228 y Fo(#)998 1292 y Fq(r)1061 1271 y Fv(\()p Ft(f)11 b Fv(\))83 b(:=)g 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3325 y Fv(])p Ft(I)1939 3283 y Fn(\000)p Fo(1)1931 3352 y Fq(T)2059 3325 y Fv(and)h Ft(I)2285 3342 y Fq(T)2340 3325 y Ft(r)s Fv([)p Ft(a)2465 3285 y Fo(#)2528 3325 y Fv(])p Ft(I)2606 3283 y Fn(\000)p Fo(1)2598 3352 y Fq(T)2727 3325 y Fv(satisfy)h(the)f(canoni-)257 3445 y(cal)j(comm)m(utation)g(relations)f(\(IV.8\){\(IV.9\),)h(b)s (ecause)h Ft(a)2418 3398 y Fo(#)2418 3473 y Fq(`)2509 3445 y Fv(and)e Ft(a)2745 3406 y Fo(#)2745 3466 y Fq(r)2836 3445 y Fv(satisfy)h(them.)43 b(It)257 3565 y(follo)m(ws)33 b(that)f Ft(`)p Fv([)p Fr(P)8 b Fv(])32 b(is)g Fr(\003)p Fs(-homomorphic)e Fv(to)h Ft(I)1939 3583 y Fq(T)1994 3565 y Ft(`)p Fv([)p Fr(P)8 b Fv(])p Ft(I)2216 3524 y Fn(\000)p Fo(1)2208 3592 y Fq(T)2342 3565 y Fv(and)32 b Ft(r)s Fv([)p Fr(P)8 b Fv(])32 b(is)g Fr(\003)p Fs(-homomorphic)257 3686 y Fv(to)h Ft(I)420 3703 y Fq(T)475 3686 y Ft(r)s Fv([)p Fr(P)8 b Fv(])p Ft(I)703 3645 y Fn(\000)p Fo(1)695 3713 y Fq(T)797 3686 y Fv(.)44 b(F)-8 b(urthermore,)33 b(since)i Ft(a)p Fv(\()p Ft(f)11 b 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Ft(a)2257 4553 y Fn(\003)2297 4597 y Fv(\()p Ft(k)s Fv(\)])g Ft(`)p Fv([)p Ft(a)p Fv(\()p Ft(k)2681 4553 y Fn(0)2705 4597 y Fv(\)])g(\012)2856 4549 y Fq(f)2856 4624 y(\014)2904 4516 y Fm(\013)2973 4597 y Ft(;)186 b Fv(\(IV.31\))257 4788 y(where)39 b(the)f(\014rst)h(equation)f(follo)m(ws)h(from)e (\(IV.29\))h(and)f(\(IV.21\),)i(the)f(second)h(from)257 4909 y(\(IV.9\))34 b(and)g(\(IV.21\),)g(and)g(the)g(remaining)h (equations)g(from)e(\(IV.27\),)h(\(IV.15\),)g(and)257 5029 y(\(IV.17\).)44 b(Lik)m(ewise,)375 5140 y Fm(\012)422 5221 y Fv(\012)22 b Fr(\012)f Fv(\012)709 5136 y Fm(\014)709 5196 y(\014)769 5221 y Ft(I)812 5238 y Fq(T)883 5221 y Ft(r)s Fv([)p Ft(a)1008 5177 y Fn(\003)1048 5221 y Fv(\()p Ft(k)s Fv(\)])16 b Ft(r)s Fv([)p Ft(a)p Fv(\()p Ft(k)1438 5177 y Fn(0)1461 5221 y Fv(\)])g Ft(I)1593 5177 y Fn(\000)p Fo(1)1585 5246 y Fq(T)1704 5221 y Fv(\012)22 b Fr(\012)g Fv(\012)1964 5140 y Fm(\013)2094 5221 y Fv(=)83 b Ft(!)2318 5174 y Fq(f)2314 5249 y(\014)2363 5140 y Fm(\000)2409 5221 y Ft(a)2460 5177 y Fn(\003)2499 5221 y Fv(\()p Ft(k)s Fv(\))16 b Ft(a)p Fv(\()p Ft(k)2788 5177 y Fn(0)2812 5221 y Fv(\))2850 5140 y Fm(\001)3186 5221 y Fv(\(IV.32\))2094 5388 y(=)2253 5307 y Fm(\012)2300 5388 y Fv(\012)2370 5341 y Fq(f)2370 5416 y(\014)2445 5303 y Fm(\014)2445 5363 y(\014)2505 5388 y Ft(r)s Fv([)p Ft(a)2630 5345 y Fn(\003)2670 5388 y Fv(\()p Ft(k)s Fv(\)])g Ft(r)s Fv([)p Ft(a)p Fv(\()p Ft(k)3060 5345 y Fn(0)3083 5388 y Fv(\)])g(\012)3234 5341 y Fq(f)3234 5416 y(\014)3282 5307 y Fm(\013)3351 5388 y Ft(:)p eop %%Page: 38 39 38 38 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(38)257 573 y(It)32 b(follo)m(ws)g(that)f Ft(I)934 590 y Fq(T)1016 573 y Fv(:)c Fr(H)1153 526 y Fq(f)1152 601 y(\014)1227 573 y Fr(!)f(F)k(\012)19 b(F)10 b Fv(,)32 b(de\014ned)g(b)m(y)h(\(IV.28\){\(IV.30\),)e(is)h(an)f(isometry)-8 b(.)404 706 y(Next,)33 b(w)m(e)h(calculate)f Ft(L)1285 723 y Fq(f)1358 706 y Fv(:=)27 b Ft(I)1531 723 y Fq(T)1586 706 y Fr(L)1653 723 y Fq(f)1699 706 y Ft(I)1750 665 y Fn(\000)p Fo(1)1742 733 y Fq(T)1844 706 y Fv(.)44 b(W)-8 b(e)33 b(claim)g(that)909 980 y Ft(L)975 997 y Fq(f)1081 980 y Fv(=)1216 844 y Fm(Z)1332 980 y Ft(dk)d(!)t Fv(\()p Ft(k)s Fv(\))1675 899 y Fm(\002)1716 980 y Ft(a)1767 936 y Fn(\003)1767 1000 y Fq(`)1807 980 y Fv(\()p Ft(k)s Fv(\))p Ft(a)1988 997 y Fq(`)2021 980 y Fv(\()p Ft(k)s Fv(\))43 b Fr(\000)g Ft(a)2364 936 y Fn(\003)2364 1000 y Fq(r)2404 980 y Fv(\()p Ft(k)s Fv(\))p Ft(a)2585 997 y Fq(r)2623 980 y Fv(\()p Ft(k)s Fv(\))2753 899 y Fm(\003)2816 980 y Ft(:)343 b Fv(\(IV.33\))257 1243 y(Then)1126 1463 y Ft(e)1171 1419 y Fq(itL)1268 1431 y Fj(f)1329 1463 y Ft(a)1380 1480 y Fq(`)1413 1463 y Fv(\()p Ft(k)s Fv(\))16 b Ft(e)1604 1419 y Fn(\000)p Fq(itL)1756 1431 y Fj(f)1884 1463 y Fv(=)83 b Ft(e)2088 1419 y Fn(\000)p Fq(it!)r Fo(\()p Fq(k)r Fo(\))2352 1463 y Ft(a)2403 1480 y Fq(`)2437 1463 y Fv(\()p Ft(k)s Fv(\))21 b Ft(;)571 b Fv(\(IV.34\))1121 1608 y Ft(e)1166 1565 y Fq(itL)1263 1577 y Fj(f)1324 1608 y Ft(a)1375 1626 y Fq(r)1413 1608 y Fv(\()p Ft(k)s Fv(\))16 b Ft(e)1604 1565 y Fn(\000)p Fq(itL)1756 1577 y Fj(f)1884 1608 y Fv(=)83 b Ft(e)2088 1565 y Fq(it!)r Fo(\()p Fq(k)r Fo(\))2298 1608 y Ft(a)2349 1626 y Fq(r)2387 1608 y Fv(\()p Ft(k)s Fv(\))21 b Ft(:)621 b Fv(\(IV.35\))257 1828 y(Th)m(us,)35 b(using)e(\(IV.29\),)f(w)m(e)i(\014nd)f(that)321 2048 y Ft(e)366 2005 y Fq(itL)463 2017 y Fj(f)524 2048 y Ft(I)567 2066 y Fq(T)639 2048 y Ft(`)p Fv([)p Ft(a)p Fv(\()p Ft(k)s Fv(\)])16 b Ft(I)982 2005 y Fn(\000)p Fo(1)974 2073 y Fq(T)1093 2048 y Ft(e)1138 2005 y Fn(\000)p Fq(itL)1290 2017 y Fj(f)1394 2048 y Fv(=)59 b Ft(e)1574 2005 y Fn(\000)p Fq(it!)r Fo(\()p Fq(k)r Fo(\))1839 2048 y Ft(I)1882 2066 y Fq(T)1953 2048 y Ft(`)p Fv([)p Ft(a)p Fv(\()p Ft(k)s Fv(\)])16 b Ft(I)2296 2005 y Fn(\000)p Fo(1)2288 2073 y Fq(T)2451 2048 y Fv(=)59 b Ft(I)2629 2066 y Fq(T)2700 2048 y Ft(`)2741 1967 y Fm(\002)2783 2048 y Ft(\013)2846 2001 y Fq(f)2845 2071 y(t)2891 1967 y Fm(\000)2936 2048 y Ft(a)p Fv(\()p Ft(k)s 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Fq(T)1650 2769 y Ft(r)1697 2688 y Fm(\002)1738 2769 y Ft(\013)1801 2721 y Fq(f)1800 2791 y(t)1846 2688 y Fm(\000)1892 2769 y Ft(a)p Fv(\()p Ft(k)s Fv(\))2073 2688 y Fm(\001\003)2176 2769 y Ft(I)2227 2725 y Fn(\000)p Fo(1)2219 2793 y Fq(T)2343 2769 y Ft(;)816 b Fv(\(IV.37\))257 2989 y(b)s(ecause)37 b Ft(r)h Fv(is)e(an)m(ti-linear.)52 b(This,)37 b(\(IV.18\),)f(and)f (the)h(corresp)s(onding)g(relation)g(for)f Ft(r)257 3109 y Fv(pro)m(v)m(e)f(\(IV.33\).)404 3229 y(F)-8 b(or)31 b Ft( )26 b Fr(\012)c Ft(')27 b Fr(2)g(F)32 b(\012)21 b(F)10 b Fv(,)33 b(w)m(e)h(de\014ne)1446 3449 y Ft(E)28 b( )e Fr(\012)21 b Ft(')60 b Fv(=)f Ft(')22 b Fr(\012)f Ft( )26 b(;)896 b Fv(\(IV.38\))257 3669 y(and)33 b(w)m(e)h(set)1529 3889 y Ft(J)69 b Fv(:=)59 b Ft(E)28 b(T)36 b Fr(\012)21 b Ft(T)35 b(:)963 b Fv(\(IV.39\))257 4109 y(Then)1391 4329 y Ft(J)26 b(a)1522 4282 y Fo(#)1522 4357 y Fq(`)1585 4329 y Fv(\()p Ft(f)11 b Fv(\))16 b Ft(J)68 b Fv(=)60 b Ft(a)2045 4286 y Fo(#)2045 4350 y Fq(r)2108 4329 y Fv(\()p Ft(\034)11 b(f)g Fv(\))22 b Ft(;)841 b Fv(\(IV.40\))257 4559 y(and,)29 b(using)f(that)g Fr(J)18 b Ft(`)p Fv([)p Ft(a)1130 4520 y Fo(#)1193 4559 y Fv(\()p Ft(f)11 b Fv(\)])p Fr(J)46 b Fv(=)27 b Ft(r)s Fv([)p Ft(a)1695 4520 y Fo(#)1758 4559 y Fv(\()p Ft(\034)11 b(f)g Fv(\)])27 b(and)h(Eqs.)h (\(IV.29\){\(IV.30\),)f(w)m(e)h(v)m(erify)257 4679 y(that)385 4899 y Ft(J)c(I)507 4917 y Fq(T)578 4899 y Ft(`)p Fv([)p Ft(a)697 4856 y Fo(#)760 4899 y Fv(\()p Ft(f)11 b Fv(\)])16 b Ft(I)989 4856 y Fn(\000)p Fo(1)981 4924 y Fq(T)1100 4899 y Ft(J)69 b Fv(=)59 b Ft(I)1401 4917 y Fq(T)1472 4899 y Ft(r)s Fv([)p Ft(a)1597 4856 y Fo(#)1660 4899 y Fv(\()p Ft(\034)11 b(f)g Fv(\)])16 b Ft(I)1942 4856 y Fn(\000)p Fo(1)1934 4924 y Fq(T)2097 4899 y Fv(=)59 b Ft(I)2275 4917 y Fq(T)2346 4899 y Ft(J)26 b(`)p Fv([)p Ft(a)2545 4856 y Fo(#)2608 4899 y Fv(\()p Ft(f)11 b Fv(\)])16 b Ft(J)25 b(I)2916 4856 y Fn(\000)p Fo(1)2908 4924 y Fq(T)3032 4899 y Ft(:)127 b Fv(\(IV.41\))257 5119 y(By)33 b(\(IV.39\),)1446 5339 y Ft(J)26 b Fv(\012)c Fr(\012)f Fv(\012)61 b(=)e(\012)22 b Fr(\012)g Fv(\012)g Ft(;)896 b Fv(\(IV.42\))p eop %%Page: 39 40 39 39 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(39)257 573 y(and)33 b Ft(J)j Fv(=)27 b Ft(J)703 534 y Fn(\003)770 573 y Fv(=)g Ft(J)936 534 y Fn(\000)p Fo(1)1030 573 y Fv(.)44 b(It)32 b(follo)m(ws)i(from)e (\(IV.41\){\(IV.42\))g(that)1535 768 y Ft(J)68 b Fv(=)60 b Ft(I)1836 785 y Fq(T)1907 768 y Fr(J)35 b Ft(I)2059 724 y Fn(\000)p Fo(1)2051 792 y Fq(T)2175 768 y Ft(;)984 b Fv(\(IV.43\))257 962 y(i.e.,)36 b Ft(J)44 b Fv(is)35 b(the)g(mo)s(dular)f(conjugation)h(in)g(the)g(Araki-W)-8 b(o)s(o)s(ds)34 b(represen)m(tation.)51 b(Note)257 1083 y(that)1315 1278 y Ft(J)26 b Fv(\()p Fr(P)j(\012)22 b Fi(1)p Fv(\))16 b Ft(J)69 b Fv(=)59 b Fi(1)22 b Fr(\012)f Ft(T)14 b Fr(P)8 b Ft(T)35 b(:)749 b Fv(\(IV.44\))404 1472 y(Our)36 b(accoun)m(t)h(summarizes)i(all)d(essen)m(tial)i (features)f(of)f(the)h(quan)m(tum)h(theory)f(of)257 1593 y(the)c(free)g(electromagnetic)h(\014eld)g(in)f(thermal)g(equilibrium.) 257 1878 y Fh(IV.2)132 b(An)47 b(Idealized)i(A)l(tom)e(and)h(the)g (Quan)l(tized)h(F)-11 b(ree)46 b(Elec-)640 2027 y(tromagnetic)f(Field) 257 2212 y Fv(As)c(a)e(next)h(step,)j(w)m(e)e(consider)f(a)g(system)h (consisting)g(of)f(an)f(idealized)i(atom)f(with)257 2333 y(\014nitely)32 b(man)m(y)g(lev)m(els)h(and)e(the)g(electromagnetic)h (\014eld,)g(coupled)g(to)f(eac)m(h)g(other)g(and)257 2453 y(in)i(thermal)g(equilibrium.)404 2573 y(W)-8 b(e)30 b(b)s(egin)g(b)m(y)h(describing)h(an)e(idealized)h(atom)f(with)h (\014nitely)g(man)m(y)g(lev)m(els.)45 b(This)257 2694 y(system)33 b(is)e(a)g(sp)s(ecial)h(example)g(of)e(those)i(describ)s (ed)g(in)f(Subsection)i(I)s(I.1.)43 b(W)-8 b(e)31 b(brie\015y)257 2814 y(recall)48 b(the)f(main)g(ob)5 b(jects)49 b(and)e(notions.)86 b(The)48 b(state)g(space)g Fr(H)2773 2832 y Fq(el)2878 2814 y Fv(has)g(dimension)257 2934 y Ft(N)38 b(<)27 b Fr(1)p Fv(,)1606 3129 y Fr(H)1688 3147 y Fq(el)1807 3129 y Fv(=)59 b Fp(C)2008 3086 y Fq(N)2103 3129 y Ft(;)1056 b Fv(\(IV.45\))257 3324 y(and)34 b(the)h(Hamiltonian,)g Ft(H)1289 3342 y Fq(el)1347 3324 y Fv(,)g(is)f(a)g(selfadjoin)m(t)g Ft(N)g Fr(\002)22 b Ft(N)45 b Fv(matrix)34 b(on)g Fr(H)2931 3342 y Fq(el)2990 3324 y Fv(.)47 b(According)257 3444 y(to)39 b(Hyp)s(othesis)h(H-1,)f(the)g(eigen)m(v)-5 b(alues)41 b(of)d Ft(H)1995 3462 y Fq(el)2092 3444 y Fv(are)g(simple.)63 b(W)-8 b(e)39 b(c)m(ho)s(ose)g(the)g(stan-)257 3565 y(dard)33 b(basis)h(in)f Fr(H)921 3582 y Fq(el)1013 3565 y Fv(to)f(consist)i(of)f (eigen)m(v)m(ectors)i Fr(f)p Ft(')2223 3582 y Fq(j)2259 3565 y Fr(g)2308 3524 y Fq(N)7 b Fn(\000)p Fo(1)2308 3590 y Fq(j)t Fo(=0)2498 3565 y Fv(of)32 b Ft(H)2690 3582 y Fq(el)2782 3565 y Fv(corresp)s(onding)i(to)257 3685 y(the)44 b(eigen)m(v)-5 b(alues)45 b Ft(E)1023 3703 y Fo(0)1107 3685 y Ft(<)f(E)1299 3703 y Fo(1)1383 3685 y Ft(<)g Fr(\001)16 b(\001)g(\001)46 b Ft(<)e(E)1854 3703 y Fq(N)7 b Fn(\000)p Fo(1)2012 3685 y Fv(,)45 b(i.e.,)i Ft(H)2363 3703 y Fq(el)2421 3685 y Ft(')2485 3703 y Fq(j)2566 3685 y Fv(=)d Ft(E)2758 3703 y Fq(j)2795 3685 y Ft(')2859 3703 y Fq(j)2895 3685 y Fv(.)75 b(The)44 b(atomic)257 3806 y(Liouvillian)34 b(acts)f(on)f Fr(K)1154 3823 y Fq(el)1241 3806 y Fv(=)26 b Fr(M)1460 3823 y Fq(N)1555 3806 y Fr(3)h Ft(\024)33 b Fv(b)m(y)1470 4000 y Ft(L)1536 4018 y Fq(el)1612 4000 y Ft(\024)59 b Fv(=)1863 3920 y Fm(\002)1904 4000 y Ft(H)1985 4018 y Fq(el)2060 4000 y Ft(;)33 b(\024)2176 3920 y Fm(\003)2239 4000 y Ft(;)920 b Fv(\(IV.46\))257 4195 y(where)34 b Fr(M)656 4213 y Fq(N)755 4195 y Fv(denotes)f(the)g(algebra)f(of)f(complex)j Ft(N)d Fr(\002)20 b Ft(N)43 b Fv(matrices,)33 b(and)g(the)f(atomic)257 4316 y(KMS)h(state)g(is)g(giv)m(en)h(b)m(y)1217 4598 y Ft(\032)1267 4554 y Fq(el)1267 4618 y(\014)1386 4598 y Fv(=)59 b Ft(Z)1595 4554 y Fn(\000)p Fo(1)1588 4623 y Fq(\014)s(;)p Fo(0)1733 4473 y Fq(N)7 b Fn(\000)p Fo(1)1738 4503 y Fm(X)1749 4713 y Fq(j)t Fo(=0)1903 4598 y Ft(e)1948 4554 y Fn(\000)p Fq(\014)s(E)2098 4564 y Fj(j)2156 4598 y Fr(j)p Ft(')2247 4615 y Fq(j)2283 4598 y Fr(ih)p Ft(')2423 4615 y Fq(j)2460 4598 y Fr(j)22 b Ft(:)650 b Fv(\(IV.47\))257 4912 y(Recall)41 b(from)g(\(IV.17\))f(that)h Ft(!)1432 4865 y Fq(f)1428 4940 y(\014)1517 4912 y Fv(denotes)h(the)f(unique)h (KMS)f(state)g(of)f(the)h(electro-)257 5032 y(magnetic)35 b(\014eld)h(at)e(in)m(v)m(erse)i(temp)s(erature)g Ft(\014)6 b Fv(.)48 b(The)35 b(reference)h(state)f(of)f(the)h(systems)257 5153 y(consisting)f(of)e(an)h(atom)f(and)h(the)g(quan)m(tized)h (radiation)f(\014eld)g(is)1523 5347 y Ft(\032)1573 5304 y Fo(0)1573 5368 y Fq(\014)1680 5347 y Fv(:=)60 b Ft(\032)1893 5304 y Fq(el)1893 5368 y(\014)1973 5347 y Fr(\012)22 b Ft(!)2136 5300 y Fq(f)2132 5375 y(\014)2202 5347 y Ft(:)957 b Fv(\(IV.48\))p eop %%Page: 40 41 40 40 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(40)257 573 y(Let)36 b(\()p Fr(H)555 590 y Fq(\014)602 573 y Ft(;)16 b Fv(\012)715 590 y Fq(\014)s(;)p Fo(0)818 573 y Ft(;)g(`;)g(r)s Fv(\))35 b(denote)h(the)g(GNS)f(Hilb)s(ert)h (space,)h(the)e(cyclic)i(v)m(ector,)g(the)f(left)257 693 y(rep)s(esen)m(tation)50 b(of)e Fr(A)54 b Fv(:=)g Fr(M)1408 711 y Fq(N)1507 693 y Fr(\012)32 b(P)8 b Fv(,)53 b(and)c(the)f(righ)m(t)h(an)m(ti-represen)m(tation)h(of)e Fr(A)p Fv(,)257 814 y(resp)s(ectiv)m(ely)-8 b(,)38 b(asso)s(ciated)d (to)e(\()p Ft(\032)1484 774 y Fo(0)1484 836 y Fq(\014)1532 814 y Ft(;)16 b Fr(A)p Fv(\).)47 b(The)35 b(Liouvillian)g(of)e(the)i (uncoupled)g(system)257 934 y(is)1498 1154 y Fr(L)1565 1172 y Fo(0)1664 1154 y Fv(=)59 b Fr(L)1866 1172 y Fq(el)1963 1154 y Fv(+)38 b Fr(L)2144 1172 y Fq(f)2212 1154 y Ft(;)947 b Fv(\(IV.49\))257 1374 y(where)42 b Fr(L)614 1392 y Fq(f)699 1374 y Fv(=)e Ft(I)866 1333 y Fn(\000)p Fo(1)858 1401 y Fq(T)960 1374 y Ft(L)1026 1392 y Fq(f)1072 1374 y Ft(I)1115 1392 y Fq(T)1170 1374 y Fv(,)i(with)e Ft(L)1534 1392 y Fq(f)1620 1374 y Fv(as)g(in)h(\(IV.33\))f(and)g Ft(I)2458 1392 y Fq(T)2553 1374 y Fv(as)g(in)h(\(IV.28\){\(IV.30\).)257 1494 y(Note)33 b(that)f Ft(\032)754 1455 y Fo(0)754 1517 y Fq(\014)834 1494 y Fv(is)i(the)f(unique)h(KMS)e(state)h(of)g(the)g (uncoupled)h(system.)404 1615 y(W)-8 b(e)33 b(de\014ne)g Ft(I)i Fr(2)28 b(A)k Fv(b)m(y)900 1878 y Ft(I)67 b Fv(:=)1173 1742 y Fm(Z)1289 1878 y Ft(dk)1415 1797 y Fm(\010)1473 1878 y Ft(G)p Fv(\()p Ft(k)s Fv(\))22 b Fr(\012)g Ft(a)1851 1834 y Fn(\003)1890 1878 y Fv(\()p Ft(k)s Fv(\))44 b(+)f Ft(G)p Fv(\()p Ft(k)s Fv(\))2390 1834 y Fn(\003)2451 1878 y Fr(\012)21 b Ft(a)p Fv(\()p Ft(k)s Fv(\))2729 1797 y Fm(\011)2809 1878 y Ft(;)350 b Fv(\(IV.50\))257 2154 y(where,)42 b(as)d(in)g(Hyp)s(othesis)i(H-3,)f(the)f(coupling)g (function)h Ft(G)d Fv(:)h Fp(R)2755 2114 y Fo(3)2826 2154 y Fr(\002)26 b Fp(Z)2997 2171 y Fo(2)3071 2154 y Fr(!)37 b(M)3323 2171 y Fq(N)3430 2154 y Fv(is)257 2274 y(assumed)e(to)d(ob)s(ey)1474 2494 y Fr(k)p Ft(G)p Fv(\()p Ft(k)s Fv(\))p Fr(k)59 b(\024)g Ft(!)t Fv(\()p Ft(k)s Fv(\))2168 2450 y Fq(\026)2236 2494 y Ft(;)923 b Fv(\(IV.51\))257 2714 y(for)32 b(some)i Ft(\026)27 b(>)g Fv(0.)43 b(The)33 b(Liouvillian)h(of)e(the)h(in)m(teracting)g(system)i(is)e(de\014ned)h (b)m(y)1246 2934 y Fr(L)1313 2951 y Fq(g)1413 2934 y Fv(:=)60 b Fr(L)1643 2951 y Fo(0)1726 2934 y Fv(+)43 b Ft(g)1896 2853 y Fm(\010)1954 2934 y Ft(`)p Fv([)p Ft(I)8 b Fv(])22 b Fr(\000)f Ft(\031)t Fv([)p Ft(I)8 b Fv(])2383 2853 y Fm(\011)2463 2934 y Ft(;)696 b Fv(\(IV.52\))257 3154 y(and)33 b(the)g(\\Radon-Nik)m(o)s(dym")g(op)s(erators)f(b)m(y)876 3374 y Fr(L)943 3391 y Fq(g)r(;`)1091 3374 y Fv(:=)60 b Fr(L)1321 3391 y Fo(0)1404 3374 y Fv(+)43 b Ft(g)t(`)p Fv([)p Ft(I)8 b Fv(])21 b Ft(;)206 b Fr(L)2041 3391 y Fq(g)r(;r)2194 3374 y Fv(:=)60 b Fr(L)2424 3391 y Fo(0)2507 3374 y Fr(\000)43 b Ft(g)t(r)s Fv([)p Ft(I)8 b Fv(])21 b Ft(:)309 b Fv(\(IV.53\))404 3594 y(F)-8 b(or)42 b(the)i(purp)s(ose)h (of)e(our)g(analysis)i(it)f(is)g(con)m(v)m(enien)m(t)i(to)d(w)m(ork)i (in)e(the)h(Araki-)257 3714 y(W)-8 b(o)s(o)s(ds)34 b(represen)m (tation,)i(i.e.,)f(to)e(conjugate)h(the)h(ab)s(o)m(v)m(e)g(op)s (erators)e(b)m(y)i(the)f(isomor-)257 3835 y(phism)595 4055 y Ft(I)638 4072 y Fo(0)737 4055 y Fv(:=)59 b Ft(I)942 4072 y Fq(C)1023 4055 y Fr(\012)22 b Ft(I)1164 4072 y Fq(T)1278 4055 y Fv(:)60 b Fr(H)1447 4072 y Fq(\014)1554 4055 y Fr(!)1736 4030 y Fm(b)1711 4055 y Fr(H)45 b Fv(:=)e Fr(H)2066 4072 y Fq(el)2147 4055 y Fr(\012)21 b(H)2326 4072 y Fq(el)2407 4055 y Fr(\012)h(F)31 b(\012)22 b(F)32 b Ft(;)353 b Fv(\(IV.54\))257 4275 y(where)26 b(conjugation)e(b)m(y)h Ft(I)1221 4292 y Fq(C)1304 4275 y Fv(denotes)g(complex)h(conjugation)e (in)g Fr(M)2764 4292 y Fq(N)2856 4275 y Fv(in)g(the)h(standard)257 4395 y(basis)34 b Fr(f)p Ft(')610 4413 y Fq(i)659 4395 y Fr(\012)22 b Ft(')821 4413 y Fq(j)857 4395 y Fr(g)906 4354 y Fq(N)7 b Fn(\000)p Fo(1)906 4421 y Fq(i;j)t Fo(=0)1077 4395 y Fv(.)43 b(W)-8 b(e)33 b(set)1551 4613 y Fm(b)1543 4639 y Fv(\012)1613 4656 y Fq(\014)s(;)p Fo(0)1775 4639 y Fv(:=)60 b Ft(I)1981 4656 y Fo(0)2037 4639 y Fv(\012)2107 4656 y Fq(\014)s(;)p Fo(0)3186 4639 y Fv(\(IV.55\))257 4859 y(and)33 b(note)g(that)1014 5121 y Fm(b)1006 5146 y Fv(\012)1076 5164 y Fq(\014)s(;)p Fo(0)1238 5146 y Fv(=)60 b Ft(Z)1448 5103 y Fn(\000)p Fo(1)1441 5172 y Fq(\014)s(;)p Fo(0)1575 5022 y Fq(N)7 b Fn(\000)p Fo(1)1580 5052 y Fm(X)1591 5262 y Fq(j)t Fo(=0)1745 5146 y Ft(e)1790 5103 y Fn(\000)p Fq(\014)s(E)1940 5113 y Fj(j)1998 5146 y Ft(')2062 5164 y Fq(j)2120 5146 y Fr(\012)22 b Ft(')2282 5164 y Fq(j)2340 5146 y Fr(\012)f Fv(\012)h Fr(\012)g Fv(\012)g Ft(:)440 b Fv(\(IV.56\))p eop %%Page: 41 42 41 41 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(41)257 573 y(De\014ning)1032 780 y Ft(L)1098 797 y Fo(0)1197 780 y Fv(:=)60 b Ft(I)1403 797 y Fo(0)1459 780 y Fr(L)1526 797 y Fo(0)1582 780 y Ft(I)1633 736 y Fn(\000)p Fo(1)1625 802 y(0)1774 780 y Ft(;)158 b(L)2025 797 y Fq(g)2125 780 y Fv(:=)60 b Ft(I)2331 797 y Fo(0)2387 780 y Fr(L)2454 797 y Fq(g)2510 780 y Ft(I)2561 736 y Fn(\000)p Fo(1)2553 802 y(0)2677 780 y Ft(;)482 b Fv(\(IV.57\))257 986 y(w)m(e)34 b(\014nd)f(that)1486 1193 y Ft(L)1552 1211 y Fq(g)1652 1193 y Fv(=)60 b Ft(L)1854 1211 y Fo(0)1932 1193 y Fv(+)37 b Ft(g)t(W)e(;)936 b Fv(\(IV.58\))257 1400 y(where)1255 1606 y Ft(L)1321 1624 y Fo(0)1443 1606 y Fv(=)83 b Ft(L)1668 1624 y Fq(el)1749 1606 y Fr(\012)22 b Fi(1)1903 1563 y Fq(f)1986 1606 y Fv(+)38 b Fi(1)2156 1563 y Fq(el)2236 1606 y Fr(\012)22 b Ft(L)2400 1624 y Fq(f)2467 1606 y Ft(;)692 b Fv(\(IV.59\))1235 1752 y Ft(L)1301 1769 y Fq(el)1443 1752 y Fv(=)83 b Ft(H)1683 1769 y Fq(el)1763 1752 y Fr(\012)22 b Fi(1)1917 1769 y Fq(el)1997 1752 y Fr(\000)g Fi(1)2151 1769 y Fq(el)2231 1752 y Fr(\012)g Ft(H)2410 1769 y Fq(el)2490 1752 y Ft(;)669 b Fv(\(IV.60\))257 1958 y Ft(L)323 1976 y Fq(f)402 1958 y Fv(is)33 b(de\014ned)h(in)f(\(IV.33\),)f(and)h(the)g(in)m(teraction)g (is)g(giv)m(en)h(b)m(y)1308 2165 y Ft(W)73 b Fv(:=)59 b Ft(I)1678 2183 y Fo(0)1718 2084 y Fm(\010)1776 2165 y Ft(`)p Fv([)p Ft(I)8 b Fv(])22 b Fr(\000)f Ft(r)s Fv([)p Ft(I)8 b Fv(])2193 2084 y Fm(\011)2251 2165 y Ft(I)2302 2122 y Fn(\000)p Fo(1)2294 2187 y(0)2418 2165 y Ft(:)741 b Fv(\(IV.61\))257 2372 y(By)33 b(means)h(of)e(\(IV.50\),)h(and)f(Eqs.) i(\(IV.29\),)f(\(IV.30\),)f(w)m(e)i(readily)f(\014nd)g(that)618 2627 y Ft(W)97 b Fv(=)966 2491 y Fm(Z)1082 2627 y Ft(dk)1208 2516 y Fm(n)1274 2546 y(\000)1320 2537 y(p)p 1420 2537 349 4 v 90 x Fv(1)21 b(+)h Ft(\032)p Fv(\()p Ft(k)s Fv(\))16 b Ft(G)1861 2645 y Fq(`)1894 2627 y Fv(\()p Ft(k)s Fv(\))43 b Fr(\000)2187 2537 y Fm(p)p 2286 2537 181 4 v 2286 2627 a Ft(\032)p Fv(\()p Ft(k)s Fv(\))p 2494 2547 117 4 v 28 w Ft(G)2571 2598 y Fn(\003)2571 2648 y Fq(r)2610 2627 y Fv(\()p Ft(k)s Fv(\))2740 2546 y Fm(\001)2802 2627 y Ft(a)2853 2584 y Fn(\003)2853 2648 y Fq(`)2893 2627 y Fv(\()p Ft(k)s Fv(\))163 b(\(IV.62\))1273 2763 y Fm(\000)1318 2754 y(p)p 1418 2754 349 4 v 90 x Fv(1)21 b(+)h Ft(\032)p Fv(\()p Ft(k)s Fv(\))17 b Ft(G)1860 2801 y Fn(\003)1860 2865 y Fq(`)1899 2844 y Fv(\()p Ft(k)s Fv(\))43 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Fn(\003)2569 3213 y Fq(r)2608 3193 y Fv(\()p Ft(k)s Fv(\))2738 3112 y Fm(\001)2800 3193 y Ft(a)2851 3210 y Fq(r)2889 3193 y Fv(\()p Ft(k)s Fv(\))3019 3082 y Fm(o)3107 3193 y Ft(;)257 3421 y Fv(whic)m(h)34 b(w)m(e)g(ma)m(y)f(rewrite)h(as)480 3628 y Ft(W)97 b Fv(=)82 b Ft(a)878 3585 y Fn(\003)878 3649 y Fq(`)918 3547 y Fm(\000)964 3540 y(p)p 1063 3540 219 4 v 1063 3628 a Fv(1)22 b(+)f Ft(\032)c(G)1375 3646 y Fq(`)1446 3628 y Fr(\000)1560 3561 y(p)p 1641 3561 51 4 v 67 x Ft(\032)p 1707 3548 117 4 v 16 w(G)1784 3599 y Fn(\003)1784 3649 y Fq(r)1840 3547 y Fm(\001)1935 3628 y Fv(+)48 b Ft(a)2110 3646 y Fq(`)2143 3547 y Fm(\000)2189 3540 y(p)p 2289 3540 219 4 v 88 x Fv(1)21 b(+)h Ft(\032)16 b(G)2600 3585 y Fn(\003)2600 3649 y Fq(`)2678 3628 y Fr(\000)2791 3561 y(p)p 2873 3561 51 4 v 2873 3628 a Ft(\032)p 2939 3548 115 4 v 16 w(G)3016 3646 y Fq(r)3070 3547 y Fm(\001)3186 3628 y Fv(\(IV.63\))827 3792 y(+)27 b Ft(a)981 3748 y Fn(\003)981 3812 y Fq(r)1021 3711 y Fm(\000)1067 3725 y Fr(p)p 1148 3725 51 4 v 67 x Ft(\032)17 b(G)1292 3748 y Fn(\003)1292 3812 y Fq(`)1369 3792 y Fr(\000)1483 3704 y Fm(p)p 1582 3704 219 4 v 1582 3792 a Fv(1)22 b(+)f Ft(\032)p 1817 3712 115 4 v 17 w(G)1894 3809 y Fq(r)1948 3711 y Fm(\001)2042 3792 y Fv(+)49 b Ft(a)2218 3809 y Fq(r)2256 3711 y Fm(\000)2302 3725 y Fr(p)p 2383 3725 51 4 v 67 x Ft(\032)17 b(G)2527 3809 y Fq(`)2598 3792 y Fr(\000)2711 3704 y Fm(p)p 2811 3704 219 4 v 88 x Fv(1)22 b(+)f Ft(\032)p 3046 3712 117 4 v 17 w(G)3123 3763 y Fn(\003)3123 3812 y Fq(r)3178 3711 y Fm(\001)3246 3792 y Ft(;)257 3998 y Fv(using)36 b(the)g(shorthand)f (notation)p 1542 3943 86 4 v 35 w Ft(m)c Fv(:=)h Ft(I)1836 4016 y Fq(C)1895 3998 y Ft(mI)2031 3957 y Fn(\000)p Fo(1)2023 4025 y Fq(C)2125 3998 y Fv(,)k Ft(m)2273 4016 y Fq(`)2337 3998 y Fv(:=)c Ft(m)23 b Fr(\012)g Fi(1)2735 4016 y Fq(el)2794 3998 y Fv(,)36 b Ft(m)2942 4016 y Fq(r)3011 3998 y Fv(:=)c Fi(1)3202 4016 y Fq(el)3284 3998 y Fr(\012)23 b Ft(m)p Fv(,)257 4119 y(and)314 4349 y Ft(a)365 4306 y Fn(\003)365 4369 y Fq(\033)412 4349 y Fv(\()p Ft(m)535 4366 y Fq(\026)582 4349 y Fv(\))59 b(:=)842 4213 y Fm(Z)958 4349 y Ft(dk)19 b(m)1164 4366 y Fq(\026)1211 4349 y Fv(\()p Ft(k)s Fv(\))d Ft(a)1408 4306 y Fn(\003)1408 4369 y Fq(\033)1455 4349 y Fv(\()p Ft(k)s Fv(\))71 b Ft(;)111 b(a)1845 4366 y Fq(\033)1892 4349 y Fv(\()p Ft(m)2015 4366 y Fq(\026)2062 4349 y Fv(\))59 b(:=)2322 4213 y Fm(Z)2438 4349 y Ft(dk)19 b(m)2644 4366 y Fq(\026)2691 4349 y Fv(\()p Ft(k)s Fv(\))d Ft(a)2888 4366 y Fq(\033)2935 4349 y Fv(\()p Ft(k)s Fv(\))22 b Ft(;)72 b Fv(\(IV.64\))257 4611 y(for)32 b Ft(\033)n(;)16 b(\026)28 b Fr(2)f(f)p Ft(`;)16 b(r)s Fr(g)32 b Fv(and)h(matrix-v)-5 b(alued)33 b(functions)h Ft(m)27 b Fv(:)g Fp(R)2408 4572 y Fo(3)2475 4611 y Fr(\002)22 b Fp(Z)2642 4629 y Fo(2)2706 4611 y Fr(!)27 b(B)s Fv(\()p Fr(H)3018 4629 y Fq(el)3077 4611 y Fv(\).)404 4732 y(Similarly)-8 b(,)33 b(setting)929 4939 y Ft(L)995 4956 y Fq(g)r(;`)1144 4939 y Fv(:=)60 b Ft(I)1350 4956 y Fo(0)1405 4939 y Fr(L)1472 4956 y Fq(g)r(;`)1577 4939 y Ft(I)1628 4895 y Fn(\000)p Fo(1)1620 4961 y(0)1770 4939 y Ft(;)158 b(L)2021 4956 y Fq(g)r(;r)2174 4939 y Fv(:=)60 b Ft(I)2380 4956 y Fo(0)2436 4939 y Fr(L)2503 4956 y Fq(g)r(;r)2613 4939 y Ft(I)2664 4895 y Fn(\000)p Fo(1)2656 4961 y(0)2780 4939 y Ft(;)379 b Fv(\(IV.65\))257 5145 y(w)m(e)34 b(ha)m(v)m(e)g(that)940 5352 y Ft(L)1006 5370 y Fq(g)r(;`)1154 5352 y Fv(=)60 b Ft(L)1356 5370 y Fo(0)1433 5352 y Fv(+)38 b Ft(g)t(W)1690 5370 y Fq(`)1770 5352 y Ft(;)158 b(L)2021 5370 y Fq(g)r(;r)2174 5352 y Fv(=)60 b Ft(L)2376 5370 y Fo(0)2454 5352 y Fv(+)38 b Ft(g)t(W)2711 5370 y Fq(r)2770 5352 y Ft(;)389 b Fv(\(IV.66\))p eop %%Page: 42 43 42 42 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(42)257 573 y(where)300 793 y Ft(W)392 810 y Fq(`)508 793 y Fv(=)83 b Ft(I)710 810 y Fo(0)765 793 y Ft(`)p Fv([)p Ft(I)8 b Fv(])16 b Ft(I)978 749 y Fn(\000)p Fo(1)970 815 y(0)3186 793 y Fv(\(IV.67\))508 946 y(=)83 b Ft(a)718 903 y Fn(\003)718 967 y Fq(`)757 866 y Fm(\000)803 859 y(p)p 902 859 219 4 v 902 946 a Fv(1)22 b(+)f Ft(\032)c(G)1214 964 y Fq(`)1247 866 y Fm(\001)1341 946 y Fv(+)49 b Ft(a)1517 964 y Fq(`)1550 866 y Fm(\000)1596 859 y(p)p 1695 859 V 1695 946 a Fv(1)22 b(+)g Ft(\032)16 b(G)2007 903 y Fn(\003)2007 967 y Fq(`)2046 866 y Fm(\001)2141 946 y Fv(+)49 b Ft(a)2317 903 y Fn(\003)2317 967 y Fq(r)2356 866 y Fm(\000)2402 879 y Fr(p)p 2483 879 51 4 v 67 x Ft(\032)17 b(G)2627 903 y Fn(\003)2627 967 y Fq(`)2666 866 y Fm(\001)2761 946 y Fv(+)48 b Ft(a)2936 964 y Fq(r)2974 866 y Fm(\000)3020 879 y Fr(p)p 3101 879 V 67 x Ft(\032)17 b(G)3245 964 y Fq(`)3278 866 y Fm(\001)3345 946 y Ft(;)295 1101 y(W)387 1118 y Fq(r)508 1101 y Fv(=)83 b Fr(\000)p Ft(I)786 1118 y Fo(0)841 1101 y Ft(r)s Fv([)p Ft(I)8 b Fv(])16 b Ft(I)1060 1057 y Fn(\000)p Fo(1)1052 1123 y(0)3186 1101 y Fv(\(IV.68\))508 1254 y(=)83 b Fr(\000)p Ft(a)794 1211 y Fn(\003)794 1275 y Fq(`)833 1174 y Fm(\000)879 1187 y Fr(p)p 960 1187 V 67 x Ft(\032)p 1027 1174 117 4 v 17 w(G)1104 1225 y Fn(\003)1104 1275 y Fq(r)1143 1174 y Fm(\001)1237 1254 y Fr(\000)49 b Ft(a)1413 1272 y Fq(`)1446 1174 y Fm(\000)1492 1187 y Fr(p)p 1573 1187 51 4 v 67 x Ft(\032)p 1640 1174 115 4 v 17 w(G)1717 1272 y Fq(r)1755 1174 y Fm(\001)1849 1254 y Fr(\000)g Ft(a)2025 1211 y Fn(\003)2025 1275 y Fq(r)2064 1174 y Fm(\000)2110 1167 y(p)p 2210 1167 219 4 v 87 x Fv(1)21 b(+)h Ft(\032)p 2444 1174 115 4 v 16 w(G)2521 1272 y Fq(r)2559 1174 y Fm(\001)2654 1254 y Fr(\000)48 b Ft(a)2829 1272 y Fq(r)2867 1174 y Fm(\000)2913 1167 y(p)p 3013 1167 219 4 v 87 x Fv(1)21 b(+)h Ft(\032)p 3247 1174 117 4 v 16 w(G)3324 1225 y Fn(\003)3324 1275 y Fq(r)3364 1174 y Fm(\001)3431 1254 y Ft(:)404 1474 y Fv(Besides)34 b Ft(L)816 1492 y Fq(f)862 1474 y Fv(,)f(the)g(follo)m(wing)f(p)s(ositiv)m(e)i(op)s (erator,)1224 1694 y Ft(L)1290 1712 y Fq(aux)1472 1694 y Fv(=)59 b Ft(H)1688 1712 y Fq(f)1755 1694 y Fr(\012)22 b Fi(1)1909 1712 y Fq(f)1998 1694 y Fv(+)43 b Fi(1)2173 1712 y Fq(f)2240 1694 y Fr(\012)22 b Ft(H)2419 1712 y Fq(f)2485 1694 y Ft(;)674 b Fv(\(IV.69\))257 1915 y(pla)m(ys)29 b(an)f(imp)s(ortan)m(t)g(role)g(in)g(our)f(analysis.)44 b(It)27 b(is)i(straigthforw)m(ard)f(to)f(see)i(that)f Ft(L)3373 1932 y Fq(aux)257 2035 y Fv(is)33 b(selfadjoin)m(t)h(on)e (its)h(natural)g(domain)925 2255 y(Dom)o([)p Ft(L)1222 2272 y Fq(aux)1345 2255 y Fv(])60 b(=)1567 2174 y Fm(\010)1625 2255 y Ft( )31 b Fr(2)c(F)32 b(\012)22 b(F)2123 2170 y Fm(\014)2123 2230 y(\014)2189 2255 y Fr(k)p Ft(L)2304 2272 y Fq(aux)2427 2255 y Ft( )t Fr(k)k Ft(<)h Fr(1)2770 2174 y Fm(\011)3186 2255 y Fv(\(IV.70\))257 2475 y(and)33 b(that)f Ft(L)724 2492 y Fq(aux)880 2475 y Fv(dominates)h Ft(L)1412 2492 y Fq(f)1458 2475 y Fv(,)f(in)h(the)g(sense)i(that)d (Dom[)p Ft(L)2559 2492 y Fq(f)2604 2475 y Fv(])c Fr(\023)f Fv(Dom)o([)p Ft(L)3059 2492 y Fq(aux)3182 2475 y Fv(])32 b(and)1577 2695 y Fr(j)p Ft(L)1670 2712 y Fq(f)1716 2695 y Fr(j)60 b(\024)f Ft(L)2004 2712 y Fq(aux)2148 2695 y Ft(:)1011 b Fv(\(IV.71\))257 2915 y(Moreo)m(v)m(er,)35 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y(>)1444 3790 y(;)1595 3576 y Fr(\024)1733 3435 y Fm(\022)1806 3440 y(Z)1922 3576 y Ft(dk)2043 3465 y Fm(\020)2102 3576 y Fv(1)h(+)2353 3504 y(1)p 2280 3553 195 4 v 2280 3641 a Ft(!)t Fv(\()p Ft(k)s Fv(\))2484 3465 y Fm(\021)2560 3576 y Fr(k)p Ft(f)11 b Fv(\()p Ft(k)s Fv(\))p Fr(k)2847 3532 y Fo(2)2886 3435 y Fm(\023)2959 3458 y Fo(1)p Fq(=)p Fo(2)3091 3576 y Ft(;)68 b Fv(\(IV.72\))257 4066 y Fs(for)35 b Ft(\033)n(;)16 b(\026)28 b Fr(2)f(f)p Ft(`;)16 b(r)s Fr(g)p Fs(.)404 4237 y Fv(F)-8 b(rom)32 b(these)i(relativ)m(e)f(b)s (ounds)h(and)e(Hyp)s(othesis)j(H-4,)d(i.e.,)638 4391 y Fm(\022)711 4396 y(Z)827 4532 y Ft(dk)948 4451 y Fm(\000)994 4532 y Ft(!)t Fv(\()p Ft(k)s Fv(\))21 b(+)g Ft(!)t Fv(\()p Ft(k)s Fv(\))1502 4489 y Fn(\000)p Fo(3)1596 4451 y Fm(\001)1658 4532 y Fr(k)p Ft(G)p Fv(\()p Ft(k)s Fv(\))p Fr(k)1963 4489 y Fo(2)2002 4391 y Fm(\023)2075 4414 y Fo(1)p Fq(=)p Fo(2)2245 4532 y Fr(\024)59 b Fv(\003)h Ft(<)f Fr(1)21 b Ft(;)397 b Fv(\(IV.73\))257 4804 y(w)m(e)34 b(conclude)g(selfadjoin)m (tness)h(of)d Ft(L)1623 4821 y Fq(g)1696 4804 y Fv(b)m(y)h(Nelson's)i (comm)m(utator)e(theorem.)257 4974 y Fi(Theorem)k(IV.2.)49 b Fs(The)34 b(op)-5 b(er)g(ators)34 b Ft(L)1714 4991 y Fq(g)1754 4974 y Fs(,)h Ft(L)1885 4991 y Fq(g)r(;`)1974 4974 y Fs(,)f(and)g Ft(L)2293 4991 y Fq(g)r(;r)2421 4974 y Fs(ar)-5 b(e)35 b(essential)5 b(ly)33 b(selfadjoint)257 5094 y(on)i Fv(Dom)o([)p Ft(L)694 5112 y Fq(aux)817 5094 y Fv(])p Fs(.)p eop %%Page: 43 44 43 43 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(43)257 573 y Fs(Pr)-5 b(o)g(of.)49 b Fv(By)33 b(\(IV.71\))f(and)h (\(IV.72\),)f(w)m(e)i(ha)m(v)m(e,)g(for)e Ft(';)16 b( )31 b Fr(2)c Fv(Dom[)p Ft(L)2751 590 y Fq(aux)2874 573 y Fv(],)441 771 y Fr(jh)p Ft( )t Fr(j)22 b Ft(L)688 789 y Fq(g)r(;)p Fo(#)823 771 y Ft(')p Fr(ij)60 b(\024)2098 b Fv(\(IV.74\))607 847 y Fm(\000)653 928 y Fv(1)43 b(+)g(16)p Ft(g)t(\014)1074 885 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1238 928 y Fv(\(1)21 b(+)h Ft(\014)6 b Fv(\))1543 885 y Fo(1)p Fq(=)p Fo(2)1668 928 y Fv(\003)1736 847 y Fm(\001)1804 843 y(\015)1804 903 y(\015)1859 928 y Fv(\()p Ft(L)1963 946 y Fq(aux)2107 928 y Fv(+)22 b(1\))2292 885 y Fo(1)p Fq(=)p Fo(2)2401 928 y Ft( )2468 843 y Fm(\015)2468 903 y(\015)2545 843 y(\015)2545 903 y(\015)2601 928 y Fv(\()p Ft(L)2705 946 y Fq(aux)2849 928 y Fv(+)f(1\))3033 885 y Fo(1)p Fq(=)p Fo(2)3143 928 y Ft(')3207 843 y Fm(\015)3207 903 y(\015)3284 928 y Ft(;)257 1126 y Fv(where)34 b Ft(L)605 1144 y Fq(g)r(;)p Fo(#)751 1126 y Fv(=)27 b Ft(L)920 1144 y Fq(g)961 1126 y Fv(,)32 b Ft(L)1086 1144 y Fq(g)r(;`)1175 1126 y Fv(,)h(or)f Ft(L)1420 1144 y Fq(g)r(;r)1514 1126 y Fv(.)44 b(Next,)33 b(w)m(e)h(observ)m(e)g(that)f Ft(L)2619 1144 y Fq(f)2697 1126 y Fv(and)g Ft(L)2953 1144 y Fq(aux)3108 1126 y Fv(comm)m(ute)257 1266 y(and)g(that)f(on)h(Dom)o([)p Ft(L)1091 1215 y Fo(3)p Fq(=)p Fo(2)1091 1276 y Fq(aux)1214 1266 y Fv(],)g(w)m(e)g(ha)m(v)m(e)1272 1383 y Fm(\002)1313 1464 y Ft(a)1364 1482 y Fq(\033)1411 1464 y Fv(\()p 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y Fo(1)p Fq(=)p Fo(2)2274 2047 y Ft( )2341 1962 y Fm(\015)2341 2022 y(\015)2418 1962 y(\015)2418 2022 y(\015)2474 2047 y Fv(\()p Ft(L)2578 2064 y Fq(aux)2722 2047 y Fv(+)g(1\))2907 2003 y Fo(1)p Fq(=)p Fo(2)3016 2047 y Ft(')3080 1962 y Fm(\015)3080 2022 y(\015)3157 2047 y Ft(:)257 2245 y Fv(Th)m(us,)39 b Ft(L)602 2262 y Fq(g)679 2245 y Fv(is)e(essen)m (tially)i(selfadjoin)m(t)e(b)m(y)g(a)f(v)-5 b(arian)m(t)36 b([33,)h(Thm.)h(X.36'])e(of)g(Nelson's)257 2365 y(comm)m(utator)d (theorem.)p 3425 2365 4 66 v 3429 2303 59 4 v 3429 2365 V 3487 2365 4 66 v 404 2606 a(Next,)g(w)m(e)h(recall)f(the)g (expression)h(for)e(the)h(mo)s(dular)f(conjugation)h(in)f(the)h(Araki-) 257 2726 y(W)-8 b(o)s(o)s(ds)33 b(represen)m(tation,)1017 2925 y Ft(J)69 b Fv(=)59 b Ft(I)1318 2942 y Fo(0)1374 2925 y Fr(J)35 b Ft(I)1526 2881 y Fn(\000)p Fo(1)1518 2947 y(0)1679 2925 y Fv(=)60 b Ft(E)22 b Fv(\()16 b Ft(C)29 b Fr(\012)22 b Ft(C)28 b Fr(\012)22 b Ft(T)35 b Fr(\012)22 b Ft(T)30 b Fv(\))21 b Ft(;)467 b 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Fo(0)1854 3875 y Fv(=)1997 3850 y Fm(b)1989 3875 y Fv(\012)2059 3892 y Fq(\014)s(;)p Fo(0)2183 3875 y Ft(:)976 b Fv(\(IV.79\))257 4160 y Fh(IV.3)132 b(KMS)41 b(States)g(for)f(an)g(Idealized)i(A)l(tom)f (coupled)g(to)g(the)640 4310 y(Quan)l(tized)46 b(Electromagnetic)f (Field)257 4495 y Fv(The)28 b(selfadjoin)m(tness)h(of)d(the)h(in)m (teracting)g(Liouvillean)h Ft(L)2404 4512 y Fq(g)2471 4495 y Fv(guaran)m(tees)g(the)f(existence)257 4615 y(of)35 b(the)h(dynamics)h(as)f(a)f(strongly)h(con)m(tin)m(uous)h(unitary)f (group)g Fr(f)p Fv(exp[)p Fr(\000)p Ft(itL)3115 4632 y Fq(g)3156 4615 y Fv(])p Fr(g)3232 4632 y Fq(t)p Fn(2)p Fl(R)3392 4615 y Fv(on)282 4726 y Fm(b)257 4751 y Fr(H)r Fv(.)54 b(W)-8 b(e)37 b(de\014ne)g(the)g(Heisen)m(b)s(erg-picture)h (time)f(ev)m(olution)g(of)f(a)g(b)s(ounded)h(op)s(erator)257 4871 y Ft(b)c Fv(on)492 4846 y Fm(b)467 4871 y Fr(H)h Fv(b)m(y)1370 5069 y Ft(\013)1433 5022 y Fq(g)1432 5092 y(t)1473 4989 y Fm(\000)1519 5069 y Ft(b)1560 4989 y Fm(\001)1665 5069 y Fv(:=)60 b Ft(e)1873 5026 y Fq(itL)1970 5034 y Fj(g)2033 5069 y Ft(b)22 b(e)2141 5026 y Fn(\000)p Fq(itL)2293 5034 y Fj(g)2355 5069 y Ft(:)804 b Fv(\(IV.80\))404 5268 y(In)29 b(the)g(follo)m(wing)g(theorem)h(w)m(e)g(construct)g(a)e (p)s(erturb)s(ed)i(KMS)f(state)g(for)g(the)g(dy-)257 5388 y(namics)d(describ)s(ed)g(b)m(y)f Fr(f)p Ft(\013)1237 5341 y Fq(g)1236 5411 y(t)1276 5388 y Fr(g)1325 5406 y Fq(t)p Fn(2)p Fl(R)1450 5388 y Fv(,)h(starting)e(from)g(the)g(data)g (\()2516 5363 y Fm(b)2491 5388 y Fr(H)r Ft(;)16 b(`)p Fv([)p Fr(M)2803 5406 y Fq(N)2875 5388 y Fr(\012)5 b(P)j Fv(])p Ft(;)16 b(J)n(;)g(L)3263 5406 y Fq(g)3304 5388 y Ft(;)g(I)3390 5406 y Fo(0)3430 5388 y Fv(\).)p eop %%Page: 44 45 44 44 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(44)257 573 y Fi(Theorem)37 b(IV.3.)49 b Fs(Assume)d(that)h Ft(G)g Fs(ful\014l)5 b(ls)46 b(Hyp)-5 b(othesis)46 b(H-4.)80 b(Then)46 b(the)g(ve)-5 b(ctor)265 668 y Fm(b)257 693 y Fv(\012)327 711 y Fq(\014)s(;)p Fo(0)479 693 y Fs(is)48 b(in)h(the)g(domain)f(of)g(the)h(two)g(unb)-5 b(ounde)g(d)48 b(op)-5 b(er)g(ators)49 b Fv(exp)q([)p Fr(\000)p Ft(\014)6 b(L)3079 711 y Fq(g)r(;`)3167 693 y Ft(=)p Fv(2])49 b Fs(and)257 814 y Fv(exp)q([)p Ft(\014)6 b(L)560 831 y Fq(g)r(;r)654 814 y Ft(=)p Fv(2])p Fs(,)34 b(and)h(the)g(ve)-5 b(ctor)723 1024 y Fm(b)716 1049 y Fv(\012)786 1067 y Fq(\014)s(;g)951 1049 y Fv(:=)62 b Ft(Z)1190 1006 y Fn(\000)p Fo(1)1183 1075 y Fq(\014)s(;)p Fo(0)1306 1049 y Ft(e)1351 1006 y Fn(\000)p Fq(\014)s(L)1497 1018 y Fj(g)r(;`)1578 1006 y Fq(=)p Fo(2)1682 1024 y Fm(b)1675 1049 y Fv(\012)1745 1067 y Fq(\014)s(;)p Fo(0)1909 1049 y Fv(=)g Ft(Z)2121 1006 y Fn(\000)p Fo(1)2114 1075 y Fq(\014)s(;)p Fo(0)2237 1049 y Ft(e)2282 1006 y Fq(\014)s(L)2373 1014 y Fj(g)r(;r)2459 1006 y Fq(=)p Fo(2)2563 1024 y Fm(b)2556 1049 y Fv(\012)2626 1067 y Fq(\014)s(;)p Fo(0)3186 1049 y Fv(\(IV.81\))257 1293 y Fs(wher)-5 b(e)35 b Ft(Z)600 1310 y Fq(\014)s(;g)731 1293 y Fv(=)836 1208 y Fm(\015)836 1268 y(\015)891 1293 y Ft(e)936 1253 y Fn(\000)p Fq(\014)s(L)1082 1265 y Fj(g)r(;`)1163 1253 y Fq(=)p Fo(2)1268 1267 y Fm(b)1260 1293 y Fv(\012)1330 1310 y Fq(\014)s(;)p Fo(0)1433 1208 y Fm(\015)1433 1268 y(\015)1517 1293 y Fv(=)1621 1208 y Fm(\015)1621 1268 y(\015)1677 1293 y Ft(e)1722 1253 y Fq(\014)s(L)1813 1261 y Fj(g)r(;r)1898 1253 y Fq(=)p Fo(2)2003 1267 y Fm(b)1995 1293 y Fv(\012)2065 1310 y Fq(\014)s(;)p Fo(0)2168 1208 y Fm(\015)2168 1268 y(\015)2223 1293 y Fs(,)h(de\014nes)f(a)g(KMS) h(state,)g Ft(!)3282 1245 y Fq(g)3278 1320 y(\014)3325 1293 y Fs(,)g(on)257 1413 y Fr(M)374 1431 y Fq(N)460 1413 y Fr(\012)18 b(P)8 b Fs(,)34 b(for)f(the)h(time)f(evolution)g (given)f(by)i Fr(f)p Ft(\013)2136 1366 y Fq(g)2135 1435 y(t)2175 1413 y Fr(g)2224 1431 y Fq(t)p Fn(2)p Fl(R)2349 1413 y Fs(.)44 b(Mor)-5 b(e)g(over,)33 b(the)h(normaliza-)257 1533 y(tion)h(factor)g Ft(Z)809 1551 y Fq(\014)s(;g)946 1533 y Fs(ob)-5 b(eys)35 b(the)g(estimate)368 1753 y Fv(0)83 b Ft(<)g(Z)726 1771 y Fq(\014)s(;g)3186 1753 y Fv(\(IV.82\))500 1953 y Fr(\024)659 1833 y Fm(q)p 759 1833 497 4 v 120 x Fv(T)-8 b(r)859 1872 y Fm(\010)917 1953 y Ft(e)962 1924 y Fn(\000)p Fq(\014)s(H)1118 1936 y Fj(el)1176 1872 y Fm(\011)1288 1953 y Fv(exp)1437 1842 y Fm(\020)1497 1953 y Ft(g)1548 1909 y Fo(2)1603 1953 y Ft(\014)1664 1909 y Fo(2)1735 1817 y Fm(Z)1835 1872 y(\000)1880 1953 y Fv(1)22 b(+)f([)p Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\)])2358 1909 y Fn(\000)p Fo(1)2452 1872 y Fm(\001)2514 1953 y Fr(k)p Ft(G)p Fv(\()p Ft(k)s Fv(\))p Fr(k)2819 1909 y Fo(2)2874 1953 y Ft(dk)2979 1842 y Fm(\021)3100 1953 y Ft(<)62 b Fr(1)21 b Ft(:)257 2244 y Fs(In)35 b(p)-5 b(articular,)34 b Ft(L)926 2261 y Fq(g)974 2218 y Fm(b)967 2244 y Fv(\012)1037 2261 y Fq(\014)s(;)p Fo(0)1166 2244 y Fv(=)27 b(0)p Fs(.)257 2439 y(Pr)-5 b(o)g(of.)49 b Fv(W)-8 b(e)33 b(\014rst)g(note)g(that)f (due)h(to)f(the)h(de\014nition)h(of)e Ft(Z)2443 2456 y Fq(\014)s(;g)2546 2439 y Fv(,)631 2674 y Ft(Z)705 2631 y Fo(2)698 2695 y Fq(\014)s(;g)860 2674 y Fv(=)996 2593 y Fm(\012)1050 2649 y(b)1043 2674 y Fv(\012)1113 2692 y Fq(\014)s(;)p Fo(0)1215 2589 y Fm(\014)1215 2649 y(\014)1270 2674 y Ft(e)1315 2631 y Fn(\000)p Fq(\014)s(L)1461 2643 y Fj(g)r(;`)1570 2649 y Fm(b)1562 2674 y Fv(\012)1632 2692 y Fq(\014)s(;)p Fo(0)1735 2593 y Fm(\013)1842 2674 y Fv(=)1977 2593 y Fm(\012)2024 2674 y Fv(\012)2094 2692 y Fq(\014)s(;)p Fo(0)2197 2589 y Fm(\014)2197 2649 y(\014)2252 2674 y Ft(e)2297 2631 y Fn(\000)p Fq(\014)s Fn(L)2444 2643 y Fj(g)r(;`)2545 2674 y Fv(\012)2615 2692 y Fq(\014)s(;)p Fo(0)2717 2593 y Fm(\013)2786 2674 y Ft(:)373 b Fv(\(IV.83\))257 2923 y(Th)m(us,)50 b(if)45 b(w)m(e)g(can)h(pro)m(v)m(e)g(\(IV.82\))e (then)i(\012)1929 2940 y Fq(\014)s(;)p Fo(0)2076 2923 y Fv(is)f(in)g(the)h(domain)f(of)f Ft(e)3020 2884 y Fn(\000)3085 2853 y Fj(\014)p 3085 2868 38 3 v 3088 2910 a Fk(2)3133 2884 y Fn(L)3182 2896 y Fj(g)r(;`)3311 2923 y Fv(and,)257 3066 y(similarly)-8 b(,)50 b(in)45 b(the)g(domain)g(of)f Ft(e)1534 2996 y Fj(\014)p 1535 3012 V 1538 3053 a Fk(2)1582 3027 y Fn(L)1631 3035 y Fj(g)r(;r)1720 3066 y Fv(.)81 b(T)-8 b(o)44 b(demonstrate)i(\(IV.82\),)i(w)m(e)e(in)m(tro)s(duce)257 3187 y Fr(I)7 b Fv(\()p Ft(\034)k Fv(\))28 b(:=)f Ft(e)649 3147 y Fn(\000)p Fq(\034)8 b(L)791 3156 y Fk(0)830 3187 y Ft(`)p Fv([)p Ft(I)g Fv(])p Ft(e)1021 3147 y Fq(\034)g(L)1108 3156 y Fk(0)1179 3187 y Fv(and)33 b(observ)m(e)h(that)374 3326 y Fm(\012)421 3407 y Fv(\012)491 3424 y Fq(\014)s(;)p Fo(0)593 3322 y Fm(\014)593 3382 y(\014)648 3407 y Ft(e)693 3363 y Fn(\000)p Fq(\014)s Fn(L)840 3375 y Fj(g)r(;`)941 3407 y Fv(\012)1011 3424 y Fq(\014)s(;)p Fo(0)1114 3326 y Fm(\013)3186 3407 y Fv(\(IV.84\))457 3638 y(=)652 3513 y Fn(1)616 3543 y Fm(X)621 3752 y Fq(n)p Fo(=0)776 3638 y Ft(g)827 3594 y Fq(n)889 3502 y Fm(Z)989 3528 y Fq(\014)945 3728 y Fo(0)1052 3638 y Ft(d\034)1145 3655 y Fo(1)1201 3502 y Fm(Z)1301 3528 y Fq(\034)1332 3537 y Fk(1)1256 3728 y Fo(0)1387 3638 y Ft(d\034)1480 3655 y Fo(2)1536 3638 y Fr(\001)16 b(\001)g(\001)1666 3502 y Fm(Z)1765 3528 y Fq(\034)1796 3537 y Fj(n)p Fe(\000)p Fk(1)1721 3728 y Fo(0)1938 3638 y Ft(d\034)2031 3655 y Fq(n)2094 3557 y Fm(\012)2141 3638 y Fv(\012)2211 3655 y Fq(\014)s(;)p Fo(0)2314 3553 y Fm(\014)2314 3613 y(\014)2369 3638 y Fr(I)7 b Fv(\()p Ft(\034)2509 3655 y Fo(1)2549 3638 y Fv(\))16 b Fr(\001)g(\001)g(\001)h(I)7 b Fv(\()p Ft(\034)2873 3655 y Fq(n)2920 3638 y Fv(\))16 b(\012)3044 3655 y Fq(\014)s(;)p Fo(0)3147 3557 y Fm(\013)457 3959 y Fv(=)652 3834 y Fn(1)616 3864 y Fm(X)621 4073 y Fq(n)p Fo(=0)776 3959 y Ft(g)827 3915 y Fo(2)p Fq(n)925 3823 y Fm(Z)1024 3849 y Fq(\014)980 4049 y Fo(0)1088 3959 y Ft(d\034)1181 3976 y Fo(1)1236 3823 y Fm(Z)1336 3849 y Fq(\034)1367 3858 y Fk(1)1292 4049 y Fo(0)1422 3959 y Ft(d\034)1515 3976 y Fo(2)1571 3959 y Fr(\001)g(\001)g(\001)1701 3823 y Fm(Z)1801 3849 y Fq(\034)1832 3858 y Fk(2)p Fj(n)p Fe(\000)p Fk(1)1756 4049 y Fo(0)2004 3959 y Ft(d\034)2097 3976 y Fo(2)p Fq(n)2195 3878 y Fm(\012)2242 3959 y Fv(\012)2312 3976 y Fq(\014)s(;)p Fo(0)2415 3874 y Fm(\014)2415 3934 y(\014)2470 3959 y Fr(I)7 b Fv(\()p Ft(\034)2610 3976 y Fo(1)2650 3959 y Fv(\))16 b Fr(\001)g(\001)g(\001)h(I)7 b Fv(\()p Ft(\034)2974 3976 y Fo(2)p Fq(n)3056 3959 y Fv(\))16 b(\012)3180 3976 y Fq(\014)s(;)p Fo(0)3283 3878 y Fm(\013)3352 3959 y Ft(;)257 4306 y Fv(using)39 b(that)735 4231 y Fm(Q)829 4257 y Fq(n)829 4335 y(j)t Fo(=1)956 4195 y Fm(\020)1015 4306 y Ft(e)1060 4266 y Fq(\027)1095 4274 y Fj(n)1138 4266 y Fn(L)1187 4275 y Fk(0)1225 4306 y Ft(`)p Fv([)p Ft(I)8 b Fv(])1371 4195 y Fm(\021)1430 4306 y Fv(\012)1500 4323 y Fq(\014)s(;)p Fo(0)1639 4306 y Fr(2)36 b(H)1822 4323 y Fq(\014)1869 4306 y Fv(,)k(for)d Ft(\027)2138 4323 y Fo(1)2178 4306 y Ft(;)16 b(:)g(:)g(:)34 b(;)16 b(\027)2459 4323 y Fq(n)2542 4306 y Fr(2)36 b Fp(R)2709 4323 y Fo(+)2812 4306 y Fv(with)3040 4231 y Fm(P)3145 4257 y Fq(n)3145 4335 y(j)t Fo(=1)3288 4306 y Ft(\027)3336 4323 y Fq(n)3419 4306 y Fr(\024)257 4468 y Ft(\014)6 b Fv(.)76 b(Abbreviating)44 b Ft(e)1067 4429 y Fn(\000)p Fq(\034)8 b(!)r Fo(\()p Fq(k)r Fo(\))1305 4468 y Ft(a)1356 4429 y Fn(\003)1395 4468 y Fv(\()p Ft(k)s Fv(\))45 b(=:)g Ft(a)1769 4429 y Fo(+)1829 4468 y Fv(\()p Ft(k)s(;)16 b(\034)11 b Fv(\),)46 b Ft(e)2173 4429 y Fq(\034)8 b(!)r Fo(\()p Fq(k)r Fo(\))2356 4468 y Ft(a)p Fv(\()p Ft(k)s Fv(\))45 b(=:)g Ft(a)2781 4429 y Fn(\000)2841 4468 y Fv(\()p Ft(k)s(;)16 b(\034)11 b Fv(\),)46 b Ft(G)p Fv(\()p Ft(k)s Fv(\))f(=:)257 4588 y Ft(G)334 4549 y Fo(+)393 4588 y Fv(\()p Ft(k)s Fv(\),)31 b(and)f Ft(G)845 4549 y Fn(\003)884 4588 y Fv(\()p Ft(k)s Fv(\))d(=:)h Ft(G)1249 4549 y Fn(\000)1307 4588 y Fv(\()p Ft(k)s Fv(\),)j(and)f(using)h (\(IV.36\),)f(with)h Ft(t)c Fv(=)g Ft(i\034)11 b Fv(,)32 b(and)e(\(IV.50\),)g(w)m(e)257 4709 y(ma)m(y)k(write)662 4972 y Fr(I)7 b Fv(\()p Ft(\034)k Fv(\))60 b(=)1050 4877 y Fm(X)1046 5086 y Fq(\033)r Fo(=)p Fn(\006)1215 4836 y Fm(Z)1331 4972 y Ft(dk)24 b(`)1498 4891 y Fm(\002)1540 4972 y Ft(e)1585 4928 y Fn(\000)p Fq(\034)8 b(H)1737 4940 y Fj(el)1794 4972 y Ft(G)1871 4928 y Fq(\033)1918 4972 y Fv(\()p Ft(k)s Fv(\))p Ft(e)2093 4928 y Fq(\034)g(H)2190 4940 y Fj(el)2270 4972 y Fr(\012)21 b Ft(a)2418 4928 y Fq(\033)2465 4972 y Fv(\()p Ft(k)s(;)16 b(\034)11 b Fv(\))2691 4891 y Fm(\003)2755 4972 y Ft(:)404 b Fv(\(IV.85\))p eop %%Page: 45 46 45 45 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(45)257 573 y(Inserting)34 b(this)f(expression)i(in)e(\(IV.84\),)f (w)m(e)i(then)f(obtain)319 694 y Fm(\012)367 774 y Fv(\012)437 731 y Fo(0)437 795 y Fq(\014)484 690 y Fm(\014)484 749 y(\014)539 774 y Ft(e)584 731 y Fn(\000)p Fq(\014)s Fn(L)731 743 y Fj(g)r(;`)832 774 y Fv(\012)902 731 y Fo(0)902 795 y Fq(\014)950 694 y Fm(\013)3186 774 y Fv(\(IV.86\))403 1006 y(=)598 881 y Fn(1)561 911 y Fm(X)567 1120 y Fq(n)p Fo(=0)831 911 y Fm(X)722 1128 y Fq(\033)p 722 1140 43 3 v 2 w Fn(2f)p Fo(+)p Fq(;)p Fn(\000g)1011 1109 y Fk(2)p Fj(n)1100 1006 y Ft(g)1151 962 y Fo(2)p Fq(n)1249 870 y Fm(Z)1349 896 y Fq(\014)1305 1096 y Fo(0)1412 1006 y Ft(d\034)1505 1023 y Fo(1)1561 870 y Fm(Z)1661 896 y Fq(\034)1692 905 y Fk(1)1616 1096 y Fo(0)1747 1006 y Ft(d\034)1840 1023 y Fo(2)1896 1006 y Fr(\001)16 b(\001)g(\001)2026 870 y Fm(Z)2125 896 y Fq(\034)2156 905 y Fk(2)p Fj(n)p Fe(\000)p Fk(1)2081 1096 y Fo(0)2329 1006 y Ft(d\034)2422 1023 y Fo(2)p Fq(n)2520 870 y Fm(Z)2636 1006 y Ft(dk)2738 1023 y Fo(1)2793 1006 y Fr(\001)g(\001)g(\001)h Ft(dk)3025 1023 y Fo(2)p Fq(n)798 1281 y Fv(T)-8 b(r)898 1200 y Fm(\002)939 1281 y Ft(e)984 1238 y Fn(\000)p Fo(\()p Fq(\014)s Fo(+)p Fq(\034)1195 1247 y Fk(1)1231 1238 y Fn(\000)p Fq(\034)1317 1247 y Fk(2)p Fj(n)1390 1238 y Fo(\))p Fq(H)1475 1250 y Fj(el)1533 1281 y Ft(G)1610 1238 y Fq(\033)1650 1247 y Fk(1)1689 1281 y Fv(\()p Ft(k)1778 1299 y Fo(1)1817 1281 y Fv(\))p Ft(e)1900 1238 y Fo(\()p Fq(\034)1958 1247 y Fk(1)1994 1238 y Fn(\000)p Fq(\034)2080 1247 y Fk(2)2114 1238 y Fo(\))p Fq(H)2199 1250 y Fj(el)2274 1281 y Fr(\001)16 b(\001)g(\001)h Ft(e)2449 1238 y Fo(\()p Fq(\034)2507 1247 y Fk(2)p Fj(n)p Fe(\000)p Fk(1)2659 1238 y Fn(\000)p Fq(\034)2745 1247 y Fk(2)p Fj(n)2818 1238 y Fo(\))p Fq(H)2903 1250 y Fj(el)2961 1281 y Ft(G)3038 1238 y Fq(\033)3078 1247 y Fk(2)p Fj(n)3156 1281 y Fv(\()p Ft(k)3245 1299 y Fo(2)p Fq(n)3327 1281 y Fv(\))3365 1200 y Fm(\003)1034 1435 y Ft(!)1099 1392 y Fo(0)1095 1456 y Fq(\014)1142 1355 y Fm(\000)1187 1435 y Ft(a)1238 1392 y Fq(\033)1278 1401 y Fk(1)1318 1435 y Fv(\()p Ft(k)1407 1453 y Fo(1)1446 1435 y Ft(;)f(\034)1531 1453 y Fo(1)1571 1435 y Fv(\))g Fr(\001)g(\001)g(\001)h Ft(a)1806 1392 y Fq(\033)1846 1401 y Fk(2)p Fj(n)1924 1435 y Fv(\()p Ft(k)2013 1453 y Fo(2)p Fq(n)2095 1435 y Ft(;)f(\034)2180 1453 y Fo(2)p Fq(n)2263 1435 y Fv(\))2301 1355 y Fm(\001)2368 1435 y Ft(:)257 1637 y Fv(Applying)34 b(H\177)-49 b(older's)34 b(inequalit)m(y)g(for)e(the)h(trace,)583 1909 y(T)-8 b(r[)p Ft(A)783 1926 y Fo(1)822 1909 y Ft(B)896 1926 y Fo(1)952 1909 y Fr(\001)16 b(\001)g(\001)h Ft(A)1155 1926 y Fq(n)1202 1909 y Ft(B)1276 1926 y Fq(n)1323 1909 y Fv(])60 b Fr(\024)1588 1784 y Fq(n)1546 1814 y Fm(Y)1548 2024 y Fq(j)t Fo(=1)1689 1909 y Fr(k)p Ft(B)1812 1926 y Fq(j)1849 1909 y Fr(k)1983 1784 y Fq(n)1941 1814 y Fm(Y)1943 2024 y Fq(j)t Fo(=1)2068 1798 y Fm(\020)2128 1909 y Fv(T)-8 b(r)p Fr(f)16 b(j)p Ft(A)2393 1926 y Fq(j)2430 1909 y Fr(j)2457 1865 y Fq(p)2493 1875 y Fj(j)2545 1909 y Fr(g)2594 1798 y Fm(\021)2653 1821 y Fo(1)p Fq(=p)2759 1831 y Fj(j)2818 1909 y Ft(;)341 b Fv(\(IV.87\))257 2211 y(where)34 b Ft(p)588 2229 y Fq(j)652 2211 y Fr(\025)27 b Fv(0)32 b(and)h(1)p Ft(=p)1173 2229 y Fo(1)1233 2211 y Fv(+)22 b Fr(\001)16 b(\001)g(\001)22 b Fv(+)g(1)p Ft(=p)1711 2229 y Fq(n)1784 2211 y Fv(=)27 b(1,)33 b(w)m(e)g(observ)m (e)i(that)601 2424 y(T)-8 b(r)701 2344 y Fm(\002)743 2424 y Ft(e)788 2381 y Fn(\000)p Fo(\()p Fq(\014)s Fo(+)p Fq(\034)999 2390 y Fk(1)1034 2381 y Fn(\000)p Fq(\034)1120 2390 y Fk(2)p Fj(n)1193 2381 y Fo(\))p Fq(H)1278 2393 y Fj(el)1336 2424 y Ft(G)1413 2381 y Fq(\033)1453 2390 y Fk(1)1492 2424 y Fv(\()p Ft(k)1581 2442 y Fo(1)1620 2424 y Fv(\))p Ft(e)1703 2381 y Fo(\()p Fq(\034)1761 2390 y Fk(1)1797 2381 y Fn(\000)p Fq(\034)1883 2390 y Fk(2)1918 2381 y Fo(\))p Fq(H)2003 2393 y Fj(el)2077 2424 y Fr(\001)16 b(\001)g(\001)h Ft(e)2252 2381 y Fo(\()p Fq(\034)2310 2390 y Fk(2)p Fj(n)p Fe(\000)p Fk(1)2462 2381 y Fn(\000)p Fq(\034)2548 2390 y Fk(2)p Fj(n)2621 2381 y Fo(\))p Fq(H)2706 2393 y Fj(el)2764 2424 y Ft(G)2841 2381 y Fq(\033)2881 2390 y Fk(2)p Fj(n)2959 2424 y Fv(\()p Ft(k)3048 2442 y Fo(2)p Fq(n)3130 2424 y Fv(\))3168 2344 y Fm(\003)1511 2670 y Fr(\024)82 b Fv(T)-8 b(r)1770 2589 y Fm(\002)1811 2670 y Ft(e)1856 2626 y Fn(\000)p Fq(\014)s(H)2012 2638 y Fj(el)2070 2589 y Fm(\003)2174 2545 y Fo(2)p Fq(n)2149 2575 y Fm(Y)2152 2785 y Fq(j)t Fo(=1)2293 2670 y Fr(k)p Ft(G)p Fv(\()p Ft(k)2508 2687 y Fq(j)2544 2670 y Fv(\))p Fr(k)21 b Ft(:)507 b Fv(\(IV.88\))257 2980 y(Next,)34 b(since)g Ft(!)827 2940 y Fo(0)823 3002 y Fq(\014)902 2980 y Fv(is)f(quasi-free,)h(Wic)m(k's)h(theorem)e(implies)h(that)550 3191 y Ft(!)615 3148 y Fo(0)611 3212 y Fq(\014)658 3110 y Fm(\000)703 3191 y Ft(a)754 3148 y Fq(\033)794 3157 y Fk(1)833 3191 y Fv(\()p Ft(k)922 3209 y Fo(1)962 3191 y Ft(;)16 b(\034)1047 3209 y Fo(1)1087 3191 y Fv(\))g Fr(\001)g(\001)g(\001)h Ft(a)1322 3148 y Fq(\033)1362 3157 y Fk(2)p Fj(n)1440 3191 y Fv(\()p Ft(k)1529 3209 y Fo(2)p Fq(n)1611 3191 y Ft(;)f(\034)1696 3209 y Fo(2)p Fq(n)1778 3191 y Fv(\))1816 3110 y Fm(\001)1949 3191 y Fv(=)1161 b(\(IV.89\))750 3328 y Fm(X)716 3539 y Fq(\031)r Fn(2P)855 3548 y Fk(2)p Fj(n)986 3298 y Fq(n)944 3328 y Fm(Y)947 3537 y Fq(j)t Fo(=1)1088 3422 y Ft(!)1153 3379 y Fo(0)1149 3443 y Fq(\014)1196 3341 y Fm(\000)1241 3422 y Ft(a)1292 3379 y Fq(\033)1332 3393 y Fj(\031)r Fk(\(2)p Fj(j)s Fe(\000)p Fk(1\))1564 3422 y Fv(\()p Ft(k)1653 3440 y Fq(\031)r Fo(\(2)p Fq(j)t Fn(\000)p Fo(1\))1913 3422 y Ft(;)16 b(\034)1998 3440 y Fq(\031)r Fo(\(2)p Fq(j)t Fn(\000)p Fo(1\))2258 3422 y Fv(\))g Ft(a)2363 3379 y Fq(\033)2403 3393 y Fj(\031)r Fk(\(2)p Fj(j)s Fk(\))2557 3422 y Fv(\()p Ft(k)2646 3440 y Fq(\031)r Fo(\(2)p Fq(j)t Fo(\))2815 3422 y Ft(;)g(\034)2900 3440 y Fq(\031)r Fo(\(2)p Fq(j)t Fo(\))3071 3422 y Fv(\))3109 3341 y Fm(\001)3176 3422 y Ft(;)257 3719 y Fv(where)46 b Fr(P)619 3737 y Fo(2)p Fq(n)744 3719 y Fv(is)f(the)f(set)h(of)f (pairings,)j(i.e.,)h(all)43 b(p)s(erm)m(utations)j Ft(\031)k Fr(2)c(S)2959 3737 y Fo(2)p Fq(n)3085 3719 y Fv(suc)m(h)f(that)257 3839 y Ft(\031)t Fv(\(1\))37 b Ft(<)g(\031)t Fv(\(3\))g Ft(<)h Fr(\001)16 b(\001)g(\001)38 b Ft(<)f(\031)t Fv(\(2)p Ft(n)25 b Fr(\000)h Fv(1\))38 b(and)h Ft(\031)t Fv(\(2)p Ft(j)32 b Fr(\000)25 b Fv(1\))37 b Ft(<)h(\031)t Fv(\(2)p Ft(j)6 b Fv(\).)61 b(Since)40 b Ft(!)3043 3800 y Fo(0)3039 3862 y Fq(\014)3086 3839 y Fv(\()p Ft(a)3175 3800 y Fo(+)3234 3839 y Ft(a)3285 3800 y Fo(+)3344 3839 y Fv(\))d(=)257 3960 y Ft(!)322 3920 y Fo(0)318 3983 y Fq(\014)365 3960 y Fv(\()p Ft(a)454 3920 y Fn(\000)513 3960 y Ft(a)564 3920 y Fn(\000)624 3960 y Fv(\))27 b(=)g(0,)32 b(the)h(only)g(non)m(v) -5 b(anishing)34 b(con)m(tributions)g(in)f(\(IV.89\))f(come)h(from)818 4247 y Ft(!)883 4203 y Fo(0)879 4267 y Fq(\014)926 4166 y Fm(\000)971 4247 y Ft(a)1022 4203 y Fn(\000)1081 4247 y Fv(\()p Ft(k)s(;)16 b(\034)11 b Fv(\))16 b Ft(a)1374 4203 y Fo(+)1435 4247 y Fv(\()p Ft(k)1527 4203 y Fn(0)1550 4247 y Ft(;)g(\034)1646 4203 y Fn(0)1670 4247 y Fv(\))1708 4166 y Fm(\001)1837 4247 y Fv(=)83 b Ft(\016)t Fv(\()p Ft(k)24 b Fr(\000)e Ft(k)2308 4203 y Fn(0)2331 4247 y Fv(\))2396 4175 y Ft(e)2441 4136 y Fo(\()p Fq(\034)8 b Fn(\000)p Fq(\034)2601 4113 y Fe(0)2624 4136 y Fo(\))p Fq(!)r Fo(\()p Fq(k)r Fo(\))p 2379 4224 433 4 v 2379 4315 a Ft(e)2424 4286 y Fq(\014)s(!)r Fo(\()p Fq(k)r Fo(\))2649 4315 y Fr(\000)38 b Fv(1)2843 4247 y Ft(;)316 b Fv(\(IV.90\))818 4517 y Ft(!)883 4474 y Fo(0)879 4538 y Fq(\014)926 4437 y Fm(\000)971 4517 y Ft(a)1022 4474 y Fo(+)1081 4517 y Fv(\()p Ft(k)s(;)16 b(\034)11 b Fv(\))16 b Ft(a)1374 4474 y Fn(\000)1435 4517 y Fv(\()p Ft(k)1527 4474 y Fn(0)1550 4517 y Ft(;)g(\034)1646 4474 y Fn(0)1670 4517 y Fv(\))1708 4437 y Fm(\001)1837 4517 y Fv(=)83 b Ft(\016)t Fv(\()p Ft(k)24 b Fr(\000)e Ft(k)2308 4474 y Fn(0)2331 4517 y Fv(\))2379 4446 y Ft(e)2424 4407 y Fo(\()p Fq(\014)s Fn(\000)p Fq(\034)8 b Fo(+)p Fq(\034)2682 4383 y Fe(0)2705 4407 y Fo(\))p Fq(!)r Fo(\()p Fq(k)r Fo(\))p 2379 4494 498 4 v 2411 4585 a Ft(e)2456 4556 y Fq(\014)s(!)r Fo(\()p Fq(k)r Fo(\))2681 4585 y Fr(\000)38 b Fv(1)2908 4517 y Ft(:)251 b Fv(\(IV.91\))257 4758 y(Th)m(us,)35 b(for)d(0)26 b Fr(\024)h Ft(\034)912 4719 y Fn(0)963 4758 y Fr(\024)g Ft(\034)39 b Fr(\024)27 b Ft(\014)6 b Fv(,)32 b(w)m(e)i(ha)m(v)m(e)g(the)f(estimate)751 5028 y Ft(\016)t Fv(\()p Ft(k)24 b Fr(\000)e Ft(k)1063 4984 y Fn(0)1086 5028 y Fv(\))1166 4956 y(2)16 b Ft(e)1276 4917 y Fq(\014)s(!)r Fo(\()p Fq(k)r Fo(\))p Fq(=)p Fo(2)p 1134 5005 433 4 v 1134 5096 a Ft(e)1179 5067 y Fq(\014)s(!)r Fo(\()p Fq(k)r Fo(\))1404 5096 y Fr(\000)38 b Fv(1)1660 5028 y Fr(\024)1863 4933 y Fm(X)1818 5146 y Fq(\033)n(;\033)1918 5127 y Fe(0)1942 5146 y Fo(=)p Fn(\006)2068 5028 y Ft(!)2133 4984 y Fo(0)2129 5048 y Fq(\014)2176 4947 y Fm(\000)2221 5028 y Ft(a)2272 4984 y Fq(\033)2320 5028 y Fv(\()p Ft(k)s(;)16 b(\034)11 b Fv(\))16 b Ft(a)2613 4984 y Fq(\033)2655 4961 y Fe(0)2683 5028 y Fv(\()p Ft(k)2775 4984 y Fn(0)2798 5028 y Ft(;)g(\034)2894 4984 y Fn(0)2918 5028 y Fv(\))2956 4947 y Fm(\001)1660 5362 y Fr(\024)82 b Ft(\016)t Fv(\()p Ft(k)25 b Fr(\000)d Ft(k)2131 5319 y Fn(0)2154 5362 y Fv(\))2202 5291 y Ft(e)2247 5252 y Fq(\014)s(!)r Fo(\()p Fq(k)r Fo(\))2472 5291 y Fv(+)38 b(1)p 2202 5340 V 2202 5430 a Ft(e)2247 5401 y Fq(\014)s(!)r Fo(\()p Fq(k)r Fo(\))2472 5430 y Fr(\000)g Fv(1)2666 5362 y Ft(:)493 b Fv(\(IV.92\))p eop %%Page: 46 47 46 46 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(46)257 573 y(The)34 b(n)m(um)m(b)s(er)g(of)e(pairings)h(is)991 846 y Fr(jP)1086 864 y Fo(2)p Fq(n)1168 846 y Fr(j)60 b Fv(=)g(\(2)p Ft(n)21 b Fr(\000)h Fv(1\)\(2)p Ft(n)f Fr(\000)h Fv(3\))16 b Fr(\001)g(\001)g(\001)h Fv(1)59 b(=)2493 775 y(\(2)p Ft(n)p Fv(\)!)p 2493 823 210 4 v 2497 912 a(2)2546 883 y Fq(n)2614 912 y Ft(n)p Fv(!)2734 846 y Ft(:)425 b Fv(\(IV.93\))257 1102 y(Hence,)43 b(\014rst)d(summing) h(o)m(v)m(er)g(all)f Ft(\033)p 1575 1118 60 4 v 43 w Fr(2)f(f)p Fv(+)p Ft(;)16 b Fr(\000g)2070 1062 y Fo(2)p Fq(n)2152 1102 y Fv(,)41 b(taking)f(\(IV.89\))g(and)g(the)g(upp)s(er) 257 1222 y(b)s(ound)33 b(in)g(\(IV.92\))f(in)m(to)h(accoun)m(t,)h(and)e (using)i(\(IV.93\),)e(w)m(e)i(obtain)257 1375 y Fm(\012)304 1456 y Fv(\012)374 1474 y Fq(\014)s(;)p Fo(0)477 1371 y Fm(\014)477 1431 y(\014)532 1456 y Ft(e)577 1413 y Fn(\000)p Fq(\014)s Fn(L)724 1425 y Fj(g)r(;`)825 1456 y Fv(\012)895 1474 y Fq(\014)s(;)p Fo(0)997 1375 y Fm(\013)27 b(\000)1117 1456 y Fv(T)-8 b(r)p Fr(f)p Ft(e)1311 1413 y Fn(\000)p Fq(\014)s(H)1467 1425 y Fj(el)1525 1456 y Fr(g)1574 1375 y Fm(\001)1619 1398 y Fn(\000)p Fo(1)270 1687 y Fr(\024)465 1563 y Fn(1)428 1593 y Fm(X)434 1802 y Fq(n)p Fo(=0)599 1616 y Fv(\(2)p Ft(n)p Fv(\)!)p 599 1664 210 4 v 603 1753 a(2)652 1723 y Fq(n)720 1753 y Ft(n)p Fv(!)835 1577 y Fm(\020)894 1687 y Ft(g)945 1644 y Fo(2)1000 1552 y Fm(Z)1116 1687 y Fv(coth[)p Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\))p Ft(=)p Fv(2])31 b Fr(k)p Ft(G)p Fv(\()p Ft(k)s Fv(\))p Fr(k)2044 1644 y Fo(2)2100 1687 y Ft(dk)2205 1577 y Fm(\021)2264 1599 y Fq(n)2327 1552 y Fm(Z)2427 1578 y Fq(\014)2382 1777 y Fo(0)2467 1687 y Ft(d\034)2560 1705 y Fo(1)2615 1552 y Fm(Z)2715 1578 y Fq(\034)2746 1587 y Fk(1)2671 1777 y Fo(0)2778 1687 y Ft(d\034)2871 1705 y Fo(2)2926 1687 y Fr(\001)16 b(\001)g(\001)3056 1552 y Fm(Z)3156 1578 y Fq(\034)3187 1587 y Fk(2)p Fj(n)p Fe(\000)p Fk(1)3112 1777 y Fo(0)3336 1687 y Ft(d\034)3429 1705 y Fo(2)p Fq(n)270 2008 y Fv(=)465 1884 y Fn(1)428 1913 y Fm(X)434 2123 y Fq(n)p Fo(=0)617 1937 y Fv(1)p 599 1985 86 4 v 599 2073 a Ft(n)p Fv(!)710 1897 y Fm(\020)780 1937 y Ft(g)831 1897 y Fo(2)886 1937 y Ft(\014)947 1897 y Fo(2)p 780 1985 207 4 v 858 2073 a Fv(2)1028 1872 y Fm(Z)1144 2008 y Fv(coth[)p Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\))p Ft(=)p Fv(2])16 b Fr(k)p Ft(G)p Fv(\()p Ft(k)s Fv(\))p Fr(k)2057 1965 y Fo(2)2111 2008 y Ft(dk)2216 1897 y Fm(\021)2275 1920 y Fq(n)270 2319 y Fv(=)82 b(exp)577 2209 y Fm(\020)647 2248 y Ft(g)698 2209 y Fo(2)753 2248 y Ft(\014)814 2209 y Fo(2)p 647 2296 V 726 2385 a Fv(2)895 2184 y Fm(Z)1011 2319 y Fv(coth)q([)p Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\))p Ft(=)p Fv(2])16 b Fr(k)p Ft(G)p Fv(\()p Ft(k)s Fv(\))p Fr(k)1925 2276 y Fo(2)1979 2319 y Ft(dk)2084 2209 y Fm(\021)270 2553 y Fr(\024)82 b Fv(exp)577 2442 y Fm(\020)637 2553 y Ft(g)688 2509 y Fo(2)743 2553 y Fv(\(1)21 b(+)h Ft(\014)6 b Fv(\))p Ft(\014)21 b Fv(\003)1192 2442 y Fm(\021)1338 2553 y Ft(<)87 b Fr(1)21 b Ft(;)1555 b Fv(\(IV.94\))257 2800 y(due)34 b(to)e(\(IV.73\),)g(and)h(using)g(coth)17 b Ft(x)27 b Fr(\024)g Fv(2)22 b(+)f(2)p Ft(=x)p Fv(.)p 3425 2800 4 66 v 3429 2738 59 4 v 3429 2800 V 3487 2800 4 66 v 257 3253 a Fu(V)156 b(Sp)t(ectral)43 b(Analysis)g(of)i(the)f(in) l(teracting)e(Liouvil-)531 3436 y(lian)257 3684 y Fh(V.1)132 b(Main)45 b(Results)g(and)g(Outline)h(of)f(Pro)t(ofs)257 3869 y Fv(In)27 b(this)h(section)g(w)m(e)f(presen)m(t)i(our)d(main)h (results)h(on)f(the)g(sp)s(ectrum)h(of)e(the)h(in)m(teracting)257 3989 y(Liouvillian)j Ft(L)806 4007 y Fq(g)875 3989 y Fv(in)m(tro)s(duced)h(in)e(Eqs.)h(\(IV.52\),)g(\(IV.58\).)42 b(Throughout)29 b(our)g(analysis,)257 4110 y(w)m(e)35 b(assume)f(that)f(Hyp)s(otheses)i(\(H-1\))d(through)i(\(H-5\))e(stated) i(in)f(the)g(in)m(tro)s(duction,)257 4230 y(Sect.)48 b(I,)j(are)c(satis\014ed.)88 b(Our)46 b(goal)h(is)g(to)g(pro)m(v)m(e)h (that)e Ft(L)2505 4248 y Fq(g)2593 4230 y Fv(has)h Fs(pur)-5 b(ely)48 b(absolutely)257 4350 y(c)-5 b(ontinuous)34 b(sp)-5 b(e)g(ctrum)30 b Fv(co)m(v)m(ering)j(the)f(real)f(axis,)h (except)h(for)d(a)h Fs(simple)f Fv(eigen)m(v)-5 b(alue)33 b(at)257 4471 y(0.)50 b(The)36 b(eigen)m(v)m(ector)h(corresp)s(onding)f (to)e(this)i(eigen)m(v)-5 b(alue)36 b(is)g(the)f(p)s(erturb)s(ed)h Ft(K)7 b(M)j(S)257 4591 y Fv(state)504 4566 y Fm(b)496 4591 y Fv(\012)566 4609 y Fq(\014)s(;g)702 4591 y Fv(constructed)34 b(in)f(Theorem)h(IV.3.)404 4712 y(Our)25 b(metho)s(d)g(to)g(pro)m(v)m (e)i(this)f(result)g(in)m(v)m(olv)m(es)i Fs(two)g(key)g(ingr)-5 b(e)g(dients:)38 b Fv(a)25 b(no)m(v)m(el)i(v)-5 b(ari-)257 4832 y(an)m(t)28 b(of)f(the)i(tec)m(hnique)h(of)d Fs(c)-5 b(omplex)29 b(sp)-5 b(e)g(ctr)g(al)30 b(deformations)c Fv(\(see,)k(e.g.,)f([15,)f(33)o(]\),)h(and)257 4952 y(the)h Fs(isosp)-5 b(e)g(ctr)g(al)32 b(F)-7 b(eshb)i(ach)30 b(map)f Fv(in)m(tro)s(duced)h(in)g([7,)g(8)o(].)43 b(W)-8 b(e)30 b(shall)g(\014rst)g(qualitativ)m(ely)257 5073 y(describ)s(e)h(these)f(ingredien)m(ts)g(and)f(then)h(outline)f(the)g (basic)h(strategy)f(underlying)h(our)257 5193 y(metho)s(d.)p eop %%Page: 47 48 47 47 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(47)404 573 y(Recall)33 b(that)f(the)h(Hilb)s(ert)f(space)i(of)e (temp)s(erature)h(states)h(of)e(the)h(system,)h(in)f(the)257 693 y(Araki-W)-8 b(o)s(o)s(ds)33 b(represen)m(tation,)h(is)f(giv)m(en)h (b)m(y)1330 888 y Fm(b)1305 913 y Fr(H)61 b Fv(=)e Fr(H)1665 931 y Fq(el)1746 913 y Fr(\012)21 b(H)1925 931 y Fq(el)2006 913 y Fr(\012)g(F)32 b(\012)22 b(F)32 b Ft(;)838 b Fv(\(V.1\))257 1133 y(see)32 b(Eq.)f(\(IV.54\).)43 b(The)31 b(w)m(a)m(v)m(e)h (function)f(of)f(a)g(photon)g(in)h(momen)m(tum)h(space)f(is)g(giv)m(en) 257 1264 y(b)m(y)40 b(a)e(pair)h(of)f(functions)i(\()p Ft(f)1323 1281 y Fo(+)1382 1264 y Fv(\()1415 1237 y Ft(~)1420 1264 y(k)s Fv(\))p Ft(;)16 b(f)1603 1281 y Fn(\000)1662 1264 y Fv(\()1695 1237 y Ft(~)1700 1264 y(k)s Fv(\)\),)1893 1237 y Ft(~)1898 1264 y(k)40 b Fr(2)d Fp(R)2157 1224 y Fo(3)2203 1264 y Fv(.)61 b(On)39 b(the)g(space)h(of)e(one-photon)257 1384 y(w)m(a)m(v)m(e)d(functions)e(w)m(e)h(de\014ne)g Fs(dilatations)p Fv(,)e Fr(f)p Ft(u)p Fv(\()p Ft(\022)s Fv(\))p Fr(g)2123 1402 y Fq(\022)r Fn(2)p Fl(R)2256 1384 y Fv(,)h(b)m(y)g(setting)764 1614 y Ft(u)p Fv(\()p Ft(\022)s Fv(\))944 1534 y Fm(\000)990 1614 y Ft(f)1038 1632 y 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8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2497 m -1000 -1000 l 6624 -1000 l 6624 2497 l cp clip 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Ft(#)g(>)g Fv(0.)257 2515 y(where)32 b Ft(L)603 2532 y Fq(aux)756 2515 y Fv(is)f(the)g Fs(p)-5 b(ositive)30 b Fv(op)s(erator)g(de\014ned)i(in)e(\(IV.69\).)43 b(The)32 b(op)s(erator)d Ft(L)3235 2532 y Fo(0)3275 2515 y Fv(\()p Ft(\022)s Fv(\))i(is)257 2635 y(clearly)36 b(analytic)g(in)f Ft(\022)j Fv(on)d(the)g(strip)g(\006)1752 2653 y Fq(\031)r(=)p Fo(2)1870 2635 y Fv(.)51 b(If)34 b Ft(\022)g Fr(62)d Fp(R)46 b Fv(then)36 b(the)f(sp)s(ectrum)h(of)e Ft(L)3331 2653 y Fo(0)3371 2635 y Fv(\()p Ft(\022)s Fv(\))257 2756 y(in)m(tersects)f(the)d(real)g(axis)g Fs(only)g Fv(in)g(the)g(eigen)m (v)-5 b(alues)32 b Fr(f)p Ft(E)2340 2773 y Fq(i)2384 2756 y Fr(\000)16 b Ft(E)2548 2773 y Fq(j)2612 2756 y Fr(j)27 b Ft(i;)16 b(j)34 b Fv(=)27 b(0)p Ft(;)16 b(:)g(:)g(:)34 b(;)16 b(N)27 b Fr(\000)16 b Fv(1)p Fr(g)257 2876 y Fv(of)35 b Ft(L)437 2894 y Fq(el)496 2876 y Fv(.)53 b(If)35 b(Im)17 b Ft(\022)35 b Fv(=:)d Ft(#)g(>)g Fv(0)j(it)h(is)f(con)m(tained)i(in)f (the)g(closed)g(lo)m(w)m(er)h(half)e(plane,)i(while)257 2996 y(if)42 b Ft(#)i(<)f Fv(0)e(it)h(lies)h(in)g(the)f(closed)i(upp)s (er)e(half)g(plane.)73 b(In)42 b(deriving)h(Eq.)g(\(V.6\))f(and)257 3117 y(establishing)47 b(these)f(prop)s(erties)h(of)d Ft(\033)t Fv(\()p Ft(L)1822 3134 y Fo(0)1862 3117 y Fv(\()p Ft(\022)s Fv(\)\),)k(the)e Fs(r)-5 b(elative)46 b(minus)g(sign)e Fv(b)s(et)m(w)m(een)257 3237 y(the)31 b(third)f(and)g(the)g(fourth)g (argumen)m(t)g(on)g(the)g(R.S.)g(of)g(Eq.)g(\(V.4\))g(is)g Fs(crucial)p Fv(!)43 b(In)30 b(the)257 3358 y(example)41 b(considered)f(ab)s(o)m(v)m(e,)h(the)f(sp)s(ectrum)g(of)e Ft(L)2230 3375 y Fo(0)2270 3358 y Fv(\()p Ft(\022)s Fv(\))g(for)h(Re)16 b Ft(\022)41 b Fv(=)c(0)h(and)h(Im)17 b Ft(\022)40 b Fv(=)257 3478 y Ft(#)28 b(>)f Fv(0)32 b(is)h(depicted)i(in)d(Fig.)g(2.) 404 3598 y(The)45 b(absolutely)g(con)m(tin)m(uous)h(nature)e(of)g(the)h (sp)s(ectrum)g(of)f Ft(L)2840 3616 y Fo(0)2926 3598 y Fv(=)i Ft(L)3114 3616 y Fo(0)3154 3598 y Fv(\()p Ft(\022)j Fv(=)e(0\))257 3719 y(a)m(w)m(a)m(y)h(from)e(its)g(eigen)m(v)-5 b(alues)48 b(can)f(b)s(e)f(inferred)h(from)f(the)h(sp)s(ectral)g(prop)s (erties)g(of)257 3839 y Ft(L)323 3857 y Fo(0)363 3839 y Fv(\()p Ft(\022)s Fv(\),)33 b Ft(\022)e Fr(62)d Fp(R)5 b Fv(,)39 b(b)m(y)34 b(considering)g(matrix)g(elemen)m(ts)h(of)d(the)i (resolv)m(en)m(t)h(of)d Ft(L)3078 3857 y Fo(0)3151 3839 y Fv(b)s(et)m(w)m(een)257 3959 y(v)m(ectors)i(in)f(the)g(dense)h (domain)f Fr(D)j Fv(of)c(dilatation-analytic)g(v)m(ectors)i(and)f (using)g(that)716 4085 y Fm(\012)763 4166 y Ft(')27 b Fr(j)g Fv(\()p Ft(L)1012 4184 y Fo(0)1074 4166 y Fr(\000)21 b Ft(z)t Fv(\))1258 4123 y Fn(\000)p Fo(1)1369 4166 y Ft( )1436 4085 y Fm(\013)1543 4166 y Fv(=)1679 4085 y Fm(\012)1739 4141 y(b)1726 4166 y Ft(U)10 b Fv(\()1848 4140 y(\026)1840 4166 y Ft(\022)s Fv(\))p Ft(')27 b Fr(j)h Fv(\()p Ft(L)2176 4184 y Fo(0)2215 4166 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)g Ft(z)t Fv(\))2546 4123 y Fn(\000)p Fo(1)2670 4141 y Fm(b)2657 4166 y Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))p Ft( )2924 4085 y Fm(\013)2993 4166 y Ft(;)250 b Fv(\(V.7\))257 4373 y(for)26 b Ft(')g Fv(and)g Ft( )k Fv(in)c Fr(D)s Fv(.)41 b(Clearly)28 b(the)e(R.S.)g(of)g(\(V.7\))g(is)g (analytic)h(in)g Ft(z)j Fv(in)d(the)f(complemen)m(t)257 4493 y(of)41 b Ft(\033)t Fv(\()p Ft(L)540 4511 y Fo(0)580 4493 y Fv(\()p Ft(\022)s Fv(\)\),)i(and)e(this)g(pro)m(vides)i(an)e (analytic)h(con)m(tin)m(uation)g(in)f Ft(z)46 b Fv(of)40 b(the)h(L.S.)h(of)257 4614 y(\(V.7\))33 b(to)f(the)h(complemen)m(t)i (of)d Ft(\033)t Fv(\()p Ft(L)1621 4631 y Fo(0)1661 4614 y Fv(\()p Ft(\022)s Fv(\)\).)404 4734 y(The)38 b(idea)f(is)h(no)m(w)g (to)f(study)h(what)g(happ)s(ens)g(to)f(the)h(sp)s(ectrum)g(of)f Ft(L)3072 4752 y Fo(0)3112 4734 y Fv(\()p Ft(\022)s Fv(\))g(when)257 4854 y(the)c(p)s(erturbation)1275 5061 y Ft(g)t(W)14 b Fv(\()p Ft(\022)s Fv(\))59 b(:=)g Ft(g)1857 5036 y Fm(b)1844 5061 y Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))16 b Ft(W)2196 5036 y Fm(b)2183 5061 y Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))2383 5018 y Fn(\000)p Fo(1)3270 5061 y Fv(\(V.8\))257 5268 y(is)36 b(added)g(to)f Ft(L)836 5285 y Fo(0)876 5268 y Fv(\()p Ft(\022)s Fv(\).)52 b(In)36 b(de\014ning)g(the)g(op)s(erator)e Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))35 b(for)g(complex)i(v)-5 b(alues)36 b(of)f Ft(\022)g Fr(2)257 5388 y Fv(\006)327 5406 y Fq(#)369 5415 y Fk(0)409 5388 y Fv(,)29 b(w)m(e)h(shall)f(mak)m(e)h(use)f(of)f (Hyp)s(othesis)j(\(H-2\))d(stated)h(in)g(Sect.)g(I.)43 b(There)30 b(are)e(some)p eop %%Page: 49 50 49 49 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(49)257 573 y(imp)s(ortan)m(t)30 b(tec)m(hnical)i(p)s(oin)m(ts)f (in)f(the)g(construction)h(of)f(an)g(analytic)h(con)m(tin)m(uation)g (of)257 693 y(the)i(op)s(erator)f Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))32 b(and)h(of)f(the)h(in)m(teracting)h(Liouvillian)922 873 y Ft(L)988 890 y Fq(g)1028 873 y Fv(\()p Ft(\022)s Fv(\))60 b(=)1361 848 y Fm(b)1347 873 y Ft(U)11 b Fv(\()p Ft(\022)s Fv(\))16 b Ft(L)1630 890 y Fq(g)1700 848 y Fm(b)1687 873 y Ft(U)11 b Fv(\()p Fr(\000)p Ft(\022)s Fv(\))59 b(=)h Ft(L)2225 890 y Fo(0)2264 873 y Fv(\()p Ft(\022)s Fv(\))44 b(+)f Ft(g)t(W)14 b Fv(\()p Ft(\022)s Fv(\))438 b(\(V.9\))257 1043 y(that)40 b(require)h(careful)g (examination.)66 b(They)41 b(are)f(dealt)h(with)f(in)g(Sect.)h(V.3)f (and)g(in)257 1163 y(App)s(endix)29 b(A.)42 b(The)28 b(upshot)g(is)g(that)f(there)h(exists)h(a)e(natural)g(domain)h Fr(D)2987 1180 y Fo(0)3054 1163 y Fr(\022)f(D)j Fv(dense)257 1283 y(in)396 1258 y Fm(b)371 1283 y Fr(H)k Fv(suc)m(h)g(that,)f(for)f (arbitrary)g(v)m(ectors)i Ft(')f Fv(and)f Ft( )37 b Fv(in)c Fr(D)2415 1301 y Fo(0)2454 1283 y Fv(,)g(the)g(matrix)g(elemen)m(ts)574 1372 y Fm(\012)634 1428 y(b)621 1453 y Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))p Ft(')912 1368 y Fm(\014)912 1428 y(\014)973 1453 y Fv(\()p Ft(L)1077 1471 y Fq(g)1117 1453 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)t Fv(\))1447 1410 y Fn(\000)p Fo(1)1572 1428 y Fm(b)1559 1453 y Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))p Ft( )1826 1372 y Fm(\013)1895 1453 y Ft(;)134 b Fv(Im)17 b Ft(z)32 b Fr(6)p Fv(=)27 b(0)22 b Ft(;)134 b(\022)30 b Fr(2)d Fp(R)33 b Ft(;)333 b Fv(\(V.10\))257 1623 y(are)33 b(the)g(b)s(oundary)g(v)-5 b(alues)33 b(on)g(the)g(real)g(axis)g(of)f(the)h(function)1194 1712 y Fm(\012)1254 1767 y(b)1241 1792 y Ft(U)11 b Fv(\()1364 1766 y(\026)1356 1792 y Ft(\022)s Fv(\))p Ft(')1533 1708 y Fm(\014)1533 1768 y(\014)1593 1792 y Fv(\()p Ft(L)1697 1810 y Fq(g)1737 1792 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)g Ft(z)t Fv(\))2068 1749 y Fn(\000)p Fo(1)2192 1767 y Fm(b)2179 1792 y Ft(U)11 b Fv(\()p Ft(\022)s Fv(\))p Ft( )2447 1712 y Fm(\013)2515 1792 y Ft(;)680 b Fv(\(V.11\))257 1962 y(whic)m(h)50 b(is)f(analytic)f(in)h Ft(\022)i Fv(on)d(the)h (strip)f(\006)1926 1980 y Fq(#)1968 1989 y Fk(0)2008 1962 y Fv(,)k(de\014ned)d(in)g(Eq.)g(\(I.24\),)i(thanks)f(to)257 2082 y(Hyp)s(othesis)c(H-2.)78 b(Since)45 b Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))1497 2043 y Fn(\000)p Fo(1)1638 2082 y Fv(=)46 b Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))1960 2043 y Fn(\003)2046 2082 y Fv(=)47 b Ft(U)10 b Fv(\()p Fr(\000)p Ft(\022)s Fv(\),)47 b(for)d Ft(\022)49 b Fr(2)d Fp(R)5 b Fv(,)53 b(the)45 b(matrix)257 2203 y(elemen)m(ts)35 b(\(V.10\))d(are)h(indep)s(enden)m(t)i(of)d Ft(\022)s Fv(,)h(and)f(hence)572 2302 y Fm(\012)619 2383 y Ft(')710 2298 y Fm(\014)710 2358 y(\014)771 2383 y Fv(\()p Ft(L)875 2400 y Fq(g)937 2383 y Fr(\000)21 b Ft(z)t Fv(\))1121 2339 y Fn(\000)p Fo(1)1233 2383 y Ft( )1300 2302 y Fm(\013)1406 2383 y Fv(=)1542 2302 y Fm(\012)1602 2357 y(b)1589 2383 y Ft(U)10 b Fv(\()1711 2356 y(\026)1703 2383 y Ft(\022)t Fv(\))p Ft(')1881 2298 y Fm(\014)1881 2358 y(\014)1941 2383 y Fv(\()p Ft(L)2045 2400 y Fq(g)2085 2383 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)g Ft(z)t Fv(\))2416 2339 y Fn(\000)p Fo(1)2540 2357 y Fm(b)2527 2383 y Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))p Ft( )2794 2302 y Fm(\013)2863 2383 y Ft(;)332 b Fv(\(V.12\))257 2552 y(for)32 b Ft(';)16 b( )32 b Fr(2)27 b(D)775 2570 y Fo(0)814 2552 y Fv(,)33 b(and)g(Im)17 b Ft(z)31 b(>)c Fv(0,)33 b(0)27 b Ft(<)g Fv(Im)16 b Ft(\022)31 b(<)c(#)2032 2570 y Fo(0)2071 2552 y Fv(.)404 2673 y(If)38 b(w)m(e)i(are)e(able)h(to)f(\014nd)h(out)f (where)i(the)f(sp)s(ectrum,)i Ft(\033)t Fv(\()p Ft(L)2627 2690 y Fq(g)2668 2673 y Fv(\()p Ft(\022)s Fv(\)\),)f(of)e Ft(L)3080 2690 y Fq(g)3120 2673 y Fv(\()p Ft(\022)s Fv(\))h(is)g(lo-) 257 2793 y(cated)e(for,)g(e.g.,)h(purely)f(imaginary)g Ft(\022)f Fv(=)e Ft(i#)p Fv(,)k(with)f(0)c Ft(<)g(#)h(<)f(#)2699 2811 y Fo(0)2739 2793 y Fv(,)k(then)g(w)m(e)h(can)e(use)257 2913 y(Eq.)h(\(V.12\))e(to)g(construct)h(an)g(analytic)g(con)m(tin)m (uation)g(in)g Ft(z)k Fv(of)35 b(matrix)h(elemen)m(ts)i(of)257 3034 y(\()p Ft(L)361 3051 y Fq(g)406 3034 y Fr(\000)t Ft(z)t Fv(\))573 2994 y Fn(\000)p Fo(1)692 3034 y Fv(b)s(et)m(w)m(een) 26 b(v)m(ectors)g(in)e Fr(D)1563 3051 y Fo(0)1626 3034 y Fv(from)g(the)g(upp)s(er)h(half)f(plane)g(to)g(the)g(complemen)m(t) 257 3154 y(of)32 b Ft(\033)t Fv(\()p Ft(L)531 3172 y Fq(g)572 3154 y Fv(\()p Ft(\022)s Fv(\)\))g(in)h(the)g(lo)m(w)m(er)h (half)e(plane.)404 3275 y(W)-8 b(e)35 b(shall)g(attempt)g(to)g(lo)s (cate)g(the)g(sp)s(ectrum)h(of)e Ft(L)2366 3292 y Fq(g)2407 3275 y Fv(\()p Ft(i#)p Fv(\))h(with)h(the)f(help)g(of)g Fs(p)-5 b(er-)257 3395 y(turb)g(ative)48 b(metho)-5 b(ds)p Fv(,)49 b(using)e(that)g(w)m(e)g(kno)m(w)h Ft(\033)t Fv(\()p Ft(L)2174 3412 y Fo(0)2213 3395 y Fv(\()p Ft(i#)p Fv(\)\))f(explicitly)-8 b(.)87 b(The)48 b(form)e(of)257 3515 y Ft(\033)t Fv(\()p Ft(L)420 3533 y Fo(0)460 3515 y Fv(\()p Ft(i#)p Fv(\)\),)32 b(for)e Ft(#)d(>)g Fv(0,)k(see)h(Fig.)e (2,)h(F)-8 b(orm)m(ula)30 b(\(V.6\),)h(and)g(the)g(b)s(ounds)g(presen)m (ted)i(in)257 3636 y(Lemma)27 b(IV.1)f(suggest)h(to)f(apply)g(the)h (metho)s(d)f(of)g(the)g Fs(isosp)-5 b(e)g(ctr)g(al)28 b(F)-7 b(eshb)i(ach)27 b(map)e Fv(de-)257 3756 y(v)m(elop)s(ed)32 b(in)f([7,)f(8],)h(in)g(order)g(to)f(explore)h(the)g(prop)s(erties)h (of)e Ft(\033)t Fv(\()p Ft(L)2702 3774 y Fq(g)2742 3756 y Fv(\()p Ft(i#)p Fv(\)\),)i(0)27 b 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b(to)g(corrections)g(of)f Ft(o)p Fv(\()p Ft(g)1258 1180 y Fo(2)1297 1219 y Fv(\);)g(\(see)i(\(V.19\)\).)404 1340 y(Since,)g(for)e Ft(z)g Fr(2)27 b(S)1063 1357 y Fq(i;j)1144 1340 y Fv(,)32 b(the)i(resolv)m(en)m(t)h(\(V.31\))d(has)h (a)f(norm-con)m(v)m(ergen)m(t)j(Neumann)257 1460 y(series)f(in)e Ft(g)p 682 1380 106 4 v 4 w(W)13 b Fv(\()p Ft(i#)p Fv(\),)33 b(the)f(leading)g(con)m(tribution)h(to)f(the)g(second)h(term)g(on)e (the)i(R.S.)f(of)257 1581 y(\(V.34\))h(is)g(seen)h(to)e(b)s(e)h(giv)m (en)g(b)m(y)808 1815 y Fr(\000)p Ft(g)935 1771 y Fo(2)990 1815 y Ft(P)1053 1832 y Fq(i;j)1149 1815 y Ft(W)14 b Fv(\()p Ft(i#)p Fv(\))p 1437 1735 77 4 v 16 w Ft(P)1514 1832 y Fq(i;j)1594 1734 y Fm(\000)p 1640 1735 67 4 v 81 x Ft(L)1706 1832 y Fo(0)1746 1815 y Fv(\()p Ft(i#)p Fv(\))22 b Fr(\000)g Ft(E)2104 1832 y Fq(i;j)2184 1734 y Fm(\001)2230 1756 y Fn(\000)p Fo(1)p 2340 1735 77 4 v 2340 1815 a Ft(P)2417 1832 y Fq(i;j)2513 1815 y Ft(W)14 b Fv(\()p Ft(i#)p Fv(\))i Ft(P)2864 1832 y Fq(i;j)3222 1815 y 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b(real)g(\(selfadjoin)m(t\))g(part)f (is)h(denoted)g(b)m(y)g(\001)p Ft(E)2648 2654 y Fq(i;j)2729 2636 y Fv(\()p Ft(g)t Fv(\))25 b Fr(\012)g Ft(P)3045 2654 y Fo(\012)p Fn(\012)p Fo(\012)3206 2636 y Fv(,)40 b(while)257 2757 y(the)35 b(imaginary)g(part)f(is)g(written)h(as)g Ft(ig)1748 2718 y Fo(2)1787 2757 y Fv(\000)1848 2718 y Fo(\()p Fq(i;j)t Fo(\))2006 2757 y Fr(\012)22 b Ft(P)2167 2774 y Fo(\012)p Fn(\012)p Fo(\012)2362 2757 y Fv(where)36 b(\000)2707 2718 y Fo(\()p Fq(i;j)t Fo(\))2876 2757 y Fv(is)e(a)g(selfadjoin)m(t)257 2877 y(op)s(erator.)41 b(Since)28 b(w)m(e)g(temp)s(orarily)f(assumed)i Ft(E)2056 2895 y Fq(i;j)2163 2877 y Fv(to)e(b)s(e)g(a)f(simple)i(eigen)m(v)-5 b(alue)29 b(of)d Ft(L)3409 2895 y Fq(el)3468 2877 y Fv(,)257 2998 y(the)34 b(rank)f(of)g Ft(P)840 2958 y Fq(el)826 3020 y(E)878 3030 y Fj(i;j)988 2998 y Fv(is)h(one,)g(and)f(\001)p Ft(E)1637 3015 y Fq(i;j)1717 2998 y Fv(\()p Ft(g)t Fv(\))28 b(=)f Ft(g)2026 2958 y Fo(2)2081 2998 y Fv(\001)p Ft(e)2207 3015 y Fq(i;j)2304 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/acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 3172 m -1000 -1000 l 6669 -1000 l 6669 3172 l cp clip 0.06299 0.06299 sc % Polyline 7.500 slw n 2025 1890 m 2925 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4254 y Fr(\032)1384 4229 y Fm(b)1359 4254 y Fr(H)32 b Fs(r)-5 b(eturns)35 b(to)f(e)-5 b(quilibrium)33 b(exp)-5 b(onential)5 b(ly)33 b(fast)p Fv(,)e(with)h(a)257 4374 y Fs(r)-5 b(ate)33 b Fv(of)f Fr(O)s Fv(\()p Ft(\014)744 4335 y Fn(\000)p Fo(1)837 4374 y Fv(\),)h(for)f Ft(g)e(>)d Fv(0)33 b(su\016cien)m(tly)i(small.)404 4494 y(In)45 b(the)h(remaining)g(subsections)h(and)f(in)f(t)m(w)m(o)h(app)s (endices,)k(w)m(e)c(presen)m(t)h(some)257 4615 y(analytical)40 b(details)h(required)g(to)e(render)h(the)g(analysis)h(presen)m(ted)g (in)f(this)g(and)g(the)257 4735 y(last)33 b(subsection)i (mathematically)f(rigorous.)257 5024 y Fh(V.3)132 b(Complex)45 b(Dilatation)i(of)d(the)h(Liouvillian)257 5209 y Fv(In)e(this)f (section,)k(w)m(e)d(discuss)h(the)f(dilatation)e(analyticit)m(y)j(of)d (the)i(Liouvillian)g Ft(L)3428 5226 y Fq(g)3468 5209 y Fv(.)257 5329 y(The)34 b(tec)m(hnical)g(details)g(of)e(this)h (discussion)i(are)d(giv)m(en)i(in)f(App)s(endix)h(A.)p eop %%Page: 65 66 65 65 bop 257 266 a 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Ft(\032)p Fv(\()p Ft(e)2409 2854 y Fq(\022)2449 2883 y Ft(k)s Fv(\))p 2568 2803 117 4 v 27 w Ft(G)2645 2854 y Fn(\003)2645 2904 y Fq(r)2684 2883 y Fv(\()p Ft(e)2767 2840 y Fq(\022)2806 2883 y Ft(k)s Fv(\))2898 2803 y Fm(\001)2960 2883 y Ft(a)3011 2901 y Fq(r)3050 2883 y Fv(\()p Ft(k)s Fv(\))3180 2773 y Fm(o)257 3116 y Fv(and)33 b Ft(e)492 3076 y Fn(\000)p Fq(\022)586 3116 y Ft(k)d Fv(:=)d(\()p Ft(e)880 3076 y Fn(\000)p Fq(\022)969 3089 y Ft(~)974 3116 y(k)t(;)16 b(\025)p Fv(\).)404 3236 y(In)49 b(order)f(to)g(obtain) h(an)f(analytic)i(con)m(tin)m(uation)f(of)f(the)h(dilated)g(in)m (teraction)257 3356 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))28 b(in)g(Eq.)h(\(V.104\))e(from)h(real)g(to)f(complex)j Ft(\022)g Fr(2)d Fv(\006)2287 3374 y Fq(#)2329 3383 y Fk(0)2368 3356 y Fv(,)i(w)m(e)g(recall)g(that)e(Hyp)s(othesis)257 3477 y(H-2)48 b(insures)h(the)f(dilatation)g(analyticit)m(y)h(of)f Ft(G)p Fv(\()p Ft(e)2231 3438 y Fn(\000)p Fq(\022)2325 3477 y Ft(k)s Fv(\),)k(for)47 b Ft(\022)56 b Fr(2)c Fv(\006)2948 3494 y Fq(#)2990 3503 y Fk(0)3029 3477 y Fv(,)g(and)c(th)m(us)257 3597 y(also)39 b(insures)h(the)e(dilatation)g(analyticit)m(y)i(of)e Ft(e)2067 3558 y Fq(\022)2106 3597 y Ft(!)t Fv(\()p Ft(k)s Fv(\))2301 3558 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2465 3597 y Ft(G)p Fv(\()p Ft(e)2625 3558 y Fn(\000)p Fq(\022)2719 3597 y Ft(k)s Fv(\).)61 b(W)-8 b(e)38 b(follo)m(w)h(the)257 3718 y(con)m(v)m(en)m(tion)c(that,)e(for)f(a)g(matrix-v)-5 b(alued)33 b(function)g Ft(M)10 b Fv(\()p Ft(z)t Fv(\),)34 b Ft(z)e Fr(2)27 b Fp(C)2744 3678 y Fo(3)2790 3718 y Fv(,)1133 3837 y Fm(\012)1180 3917 y Ft(')1244 3935 y Fq(i)1272 3833 y Fm(\014)1272 3892 y(\014)p 1321 3837 105 4 v 1321 3917 a Ft(M)11 b Fv(\()p Ft(z)t Fv(\))16 b Ft(')1631 3935 y Fq(j)1668 3837 y Fm(\013)1798 3917 y Fv(:=)p 1984 3821 627 4 v 1984 3837 a Fm(\012)2032 3917 y Ft(')2096 3935 y Fq(i)2124 3833 y Fm(\014)2124 3892 y(\014)2178 3917 y Ft(M)10 b Fv(\()p 2336 3862 50 4 v 16 w Ft(z)22 b Fv(\))g Ft(')2527 3935 y Fq(j)2563 3837 y Fm(\013)3173 3917 y Fv(\(V.105\))1093 4009 y Fm(\012)1140 4090 y Ft(')1204 4108 y Fq(i)1232 4005 y Fm(\014)1232 4065 y(\014)1282 4090 y Ft(M)1386 4047 y Fn(\003)1426 4090 y Fv(\()p Ft(z)t Fv(\))16 b Ft(')1631 4108 y Fq(j)1668 4009 y Fm(\013)1798 4090 y Fv(:=)p 1984 3993 627 4 v 1984 4009 a Fm(\012)2032 4090 y Ft(')2096 4108 y Fq(j)2132 4005 y Fm(\014)2132 4065 y(\014)2187 4090 y Ft(M)10 b Fv(\()p 2345 4035 50 4 v 16 w Ft(z)22 b Fv(\))f Ft(')2535 4108 y Fq(i)2563 4009 y Fm(\013)2632 4090 y Ft(:)514 b Fv(\(V.106\))257 4272 y(F)-8 b(urthermore,)33 b(for)f Ft(\022)f Fr(2)c Fp(R)5 b Fv(,)38 b(w)m(e)c(set)388 4498 y Ft(\026)447 4516 y Fq(\022)485 4498 y Fv(\()p Ft(k)s Fv(\))83 b(:=)884 4408 y Fm(p)p 984 4408 552 4 v 90 x Ft(!)t Fv(\()p Ft(k)s Fv(\))16 b Ft(\032)p Fv(\()p Ft(e)1328 4469 y Fn(\000)p Fq(\022)1422 4498 y Ft(k)s Fv(\))108 b(=)1785 4370 y Fm(q)p 1884 4370 1227 4 v 1884 4498 a Ft(!)t Fv(\()p Ft(k)s Fv(\))2095 4417 y Fm(\000)2141 4498 y Fv(exp)q([)p Ft(e)2362 4469 y Fn(\000)p 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Fo(2)1567 5027 y Fv(=)g Fr(fj)p Fv(Im)17 b Ft(z)t Fr(j)45 b Ft(<)f(\031)t(=)p Fv(2)p Fr(g)e Fv(ab)s(out)g Fp(R)54 b Fv(b)m(y)44 b(analytic)f(con-)257 5147 y(tin)m(uation.)404 5268 y(With)35 b(Eqs.)h(\(V.105\){\(V.108\),)e (w)m(e)h(see)h(that)e(the)i(in)m(teraction)f(is)g(dilatation)g(an-)257 5388 y(alytic.)p eop %%Page: 66 67 66 66 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(66)257 573 y Fi(Lemma)37 b(V.2.)49 b Fs(Assume)35 b(Hyp)-5 b(otheses)35 b(H-1,)g(H-2,)g(and)f(H-4.)45 b(Then)34 b(the)h(map)928 793 y Ft(W)76 b Fv(:)62 b(\006)1255 810 y Fq(#)1297 819 y Fk(0)1363 793 y Fr(!)27 b(B)1555 712 y Fm(\002)1596 793 y Fv(Dom[)p Ft(L)1894 810 y Fq(aux)2017 793 y Fv(])p Ft(;)16 b Fr(H)2170 712 y Fm(\003)2247 793 y Ft(;)121 b(\022)30 b Fr(7!)d Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))348 b(\(V.109\))257 1013 y Fs(is)35 b(analytic.)257 1208 y(Pr)-5 b(o)g(of.)49 b Fv(W)-8 b(e)33 b(\014rst)g(write)g Ft(\013)28 b Fv(:=)f(Re)16 b Ft(\022)s Fv(,)33 b Ft(#)28 b Fv(:=)f(Im)17 b Ft(\022)35 b Fv(and)e(observ)m(e)h(that)684 1343 y Fm(\014)684 1403 y(\014)717 1428 y Fv(exp)q([)p Ft(e)938 1385 y Fn(\000)p Fq(\022)1032 1428 y Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\)])43 b Fr(\000)g Fv(1)1526 1343 y Fm(\014)1526 1403 y(\014)3173 1428 y Fv(\(V.110\))767 1573 y(=)926 1489 y Fm(\014)926 1548 y(\014)959 1573 y Fv(exp)q([)p Ft(e)1180 1530 y Fn(\000)p Fq(\013)1301 1573 y Fv(cos)q(\()p Ft(#)p Fv(\))p Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\)])16 b(exp[)p Ft(ie)2117 1530 y Fn(\000)p Fq(\013)2238 1573 y Fv(sin)q(\()p Ft(#)p Fv(\))p Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\)])43 b Fr(\000)g Fv(1)2985 1489 y Fm(\014)2985 1548 y(\014)767 1719 y Fr(\025)926 1634 y Fm(\014)926 1694 y(\014)959 1719 y Fv(exp)q([)p Ft(e)1180 1675 y Fn(\000)p Fq(\013)1301 1719 y Fv(cos)q(\()p Ft(#)p Fv(\))p Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\)])42 b Fr(\000)h Fv(1)2058 1634 y Fm(\014)2058 1694 y(\014)2151 1719 y Fr(\025)60 b Ft(e)2332 1675 y Fn(\000)p Fq(\013)2469 1719 y Fv(cos\()p Ft(#)p Fv(\))16 b Ft(\014)23 b(!)t Fv(\()p Ft(k)s Fv(\))e Ft(;)257 1939 y Fv(and)33 b(hence)555 2207 y Fr(j)p Ft(\026)641 2225 y Fq(\022)680 2207 y Fv(\()p Ft(k)s Fv(\))p Fr(j)f Ft(;)49 b Fr(j)p Ft(\027)1020 2225 y Fq(\022)1059 2207 y Fv(\()p Ft(k)s Fv(\))p Fr(j)87 b(\024)1466 2117 y Fm(p)p 1565 2117 216 4 v 1565 2207 a Ft(!)t Fv(\()p Ft(k)s Fv(\))1798 2067 y Fm(\022)2124 2136 y Ft(e)2169 2096 y Fq(\013=)p Fo(2)p 1881 2184 651 4 v 1881 2204 a Fm(p)p 1981 2204 552 4 v 85 x Fv(cos\()p Ft(#)p Fv(\))16 b Ft(\014)23 b(!)t Fv(\()p Ft(k)s Fv(\))2585 2207 y(+)43 b(1)2753 2067 y Fm(\023)2848 2207 y Ft(:)298 b Fv(\(V.111\))257 2497 y(So,)33 b(de\014ning)542 2717 y Ft(w)615 2666 y Fo(\()p Fq(\022)r Fo(\))612 2744 y(+)p Fq(;`)720 2717 y Fv(\()p Ft(k)s Fv(\))83 b(:=)g Ft(e)1164 2673 y Fn(\000)p Fq(\022)1274 2717 y Ft(!)t Fv(\()p Ft(k)s Fv(\))1469 2673 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1649 2636 y Fm(\000)1695 2717 y Ft(\027)1743 2734 y Fq(\022)1782 2717 y Fv(\()p Ft(k)s Fv(\))16 b Ft(G)2005 2734 y Fq(`)2038 2717 y Fv(\()p Ft(e)2121 2673 y Fn(\000)p Fq(\022)2215 2717 y Ft(k)s Fv(\))44 b Fr(\000)f Ft(\026)2529 2734 y Fq(\022)2568 2717 y Fv(\()p Ft(k)s Fv(\))p 2714 2637 117 4 v 16 w Ft(G)2791 2688 y Fn(\003)2791 2737 y Fq(r)2830 2717 y Fv(\()p Ft(e)2913 2673 y Fn(\000)p Fq(\022)3007 2717 y Ft(k)s Fv(\))3099 2636 y Fm(\001)3167 2717 y Ft(;)-21 b Fv(\(V.112\))542 2891 y Ft(w)615 2840 y Fo(\()p Fq(\022)r Fo(\))612 2919 y Fn(\000)p Fq(;`)720 2891 y Fv(\()p Ft(k)s Fv(\))83 b(:=)g Ft(e)1164 2848 y Fn(\000)p Fq(\022)1274 2891 y Ft(!)t Fv(\()p Ft(k)s Fv(\))1469 2848 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1649 2811 y Fm(\000)1695 2891 y Ft(\027)1743 2909 y Fq(\022)1782 2891 y Fv(\()p Ft(k)s Fv(\))16 b Ft(G)2005 2848 y Fn(\003)2005 2912 y Fq(`)2045 2891 y Fv(\()p Ft(e)2128 2848 y Fn(\000)p Fq(\022)2222 2891 y Ft(k)s Fv(\))43 b Fr(\000)h Ft(\026)2536 2909 y Fq(\022)2574 2891 y Fv(\()p Ft(k)s Fv(\))p 2720 2811 115 4 v 16 w Ft(G)2797 2909 y Fq(r)2835 2891 y Fv(\()p Ft(e)2918 2848 y Fn(\000)p Fq(\022)3012 2891 y Ft(k)s Fv(\))3104 2811 y Fm(\001)3172 2891 y Ft(;)-26 b Fv(\(V.113\))537 3066 y Ft(w)610 3015 y Fo(\()p Fq(\022)r Fo(\))607 3088 y(+)p Fq(;r)720 3066 y Fv(\()p Ft(k)s Fv(\))83 b(:=)g Ft(e)1164 3022 y Fq(\022)1219 3066 y Ft(!)t Fv(\()p Ft(k)s Fv(\))1414 3022 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1595 2985 y Fm(\000)1640 3066 y Ft(\026)1699 3083 y Fn(\000)p Fq(\022)1793 3066 y Fv(\()p Ft(k)s Fv(\))16 b Ft(G)2016 3022 y Fn(\003)2016 3086 y Fq(`)2055 3066 y Fv(\()p Ft(e)2138 3022 y Fq(\022)2178 3066 y Ft(k)s Fv(\))43 b Fr(\000)g Ft(\027)2480 3083 y Fn(\000)p Fq(\022)2574 3066 y Fv(\()p Ft(k)s Fv(\))p 2720 2986 V 16 w Ft(G)2797 3083 y Fq(r)2835 3066 y Fv(\()p Ft(e)2918 3022 y Fq(\022)2958 3066 y Ft(k)s Fv(\))3050 2985 y Fm(\001)3117 3066 y Ft(;)29 b Fv(\(V.114\))537 3235 y Ft(w)610 3184 y Fo(\()p Fq(\022)r Fo(\))607 3256 y Fn(\000)p Fq(;r)720 3235 y Fv(\()p Ft(k)s Fv(\))83 b(:=)g Ft(e)1164 3191 y Fq(\022)1219 3235 y Ft(!)t Fv(\()p Ft(k)s Fv(\))1414 3191 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1595 3154 y Fm(\000)1640 3235 y Ft(\026)1699 3252 y Fn(\000)p Fq(\022)1793 3235 y Fv(\()p Ft(k)s Fv(\))16 b Ft(G)2016 3252 y Fq(`)2049 3235 y Fv(\()p Ft(e)2132 3191 y Fq(\022)2171 3235 y Ft(k)s Fv(\))43 b Fr(\000)h Ft(\027)2474 3252 y Fn(\000)p Fq(\022)2568 3235 y Fv(\()p Ft(k)s Fv(\))p 2714 3155 117 4 v 16 w Ft(G)2791 3206 y Fn(\003)2791 3255 y Fq(r)2830 3235 y Fv(\()p Ft(e)2913 3191 y Fq(\022)2953 3235 y Ft(k)s Fv(\))3045 3154 y Fm(\001)3112 3235 y Ft(;)34 b Fv(\(V.115\))257 3455 y(Hyp)s(otheses)h(H-3)d(and)h(H-4)f(insure)h (the)g(follo)m(wing)g(estimate,)551 3665 y Fm(\015)551 3725 y(\015)606 3750 y Ft(w)679 3707 y Fo(\()p Fq(\022)r Fo(\))676 3771 y Fq(\033)n(;\034)778 3750 y Fv(\()p Ft(k)s Fv(\))908 3665 y Fm(\015)908 3725 y(\015)1023 3750 y Fr(\024)1168 3679 y Fv(2)p Ft(e)1262 3639 y Fo(\()p Fq(M)7 b Fo(+1\))p Fn(j)p Fo(Re)13 b Fq(\022)r Fn(j)p 1168 3727 488 4 v 1230 3747 a Fm(p)p 1329 3747 264 4 v 1329 3832 a Fv(cos)q(\()p Ft(#)p Fv(\))1687 3610 y Fm(\022)1760 3750 y Fv(1)43 b(+)2085 3679 y(1)p 1982 3727 255 4 v 1982 3815 a Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\))2246 3610 y Fm(\023)2320 3632 y Fo(1)p Fq(=)p Fo(2)2451 3750 y Fr(j)2473 3724 y Ft(~)2478 3750 y(k)t Fr(j)2560 3707 y Fq(\026)2628 3750 y Ft(\024)p Fv(\()p Ft(k)s Fv(\))22 b Ft(;)310 b Fv(\(V.116\))257 4040 y(for)40 b(an)m(y)h Ft(\022)s(in)p Fv(\006)815 4057 y Fq(#)857 4066 y Fk(0)897 4040 y Fv(,)h(with)e Ft(#)h Fv(:=)e(Im)17 b Ft(\022)43 b(M)51 b(<)39 b Fr(1)h Fv(as)g(in)g(Hyp)s(otheses)j(H-3)c(and)h(4,)i (and)257 4160 y Ft(\024)28 b Fr(\024)f Fv(1)32 b(is)h(a)f(function)h (suc)m(h)h(that)979 4280 y Fm(Z)1078 4305 y(\020)1138 4416 y Ft(!)t Fv(\()p Ft(k)s Fv(\))21 b(+)g Ft(!)t Fv(\()p Ft(k)s Fv(\))1646 4372 y Fn(\000)p Fo(3)1740 4305 y Fm(\021)1816 4416 y Fr(j)1838 4389 y Ft(~)1843 4416 y(k)s Fr(j)1924 4372 y Fo(2)p Fq(\026)2022 4416 y Fr(j)p Ft(\024)p Fv(\()p Ft(k)s Fv(\))p Fr(j)2262 4372 y Fo(2)2318 4416 y Ft(dk)62 b Fr(\024)e Fv(\003)2686 4372 y Fo(2)2747 4416 y Ft(:)399 b Fv(\(V.117\))257 4710 y(F)-8 b(urthermore,)41 b(the)f(matrix-v)-5 b(alued)40 b(functions)g Ft(\022)h Fr(7!)c Ft(w)2383 4659 y Fo(\()p Fq(\022)r Fo(\))2380 4720 y Fq(\033)n(;\034)2481 4710 y Fv(\()p Ft(k)s Fv(\))i(are)g(analytic)h(in)f(\006)3387 4727 y Fq(#)3429 4736 y Fk(0)3468 4710 y Fv(.)257 4830 y(Hence)34 b(the)f(standard)g(b)s(ounds)h(giv)m(en)f(in)g(Lemma)g(IV.1) g(insure)h(that)265 5062 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))59 b(=)g Ft(a)740 5019 y Fn(\003)740 5083 y Fq(`)780 5062 y Fv(\()p Ft(w)891 5011 y Fo(\()p Fq(\022)r Fo(\))888 5090 y(+)p Fq(;`)995 5062 y Fv(\))38 b(+)g Ft(a)1236 5080 y Fq(`)1269 5062 y Fv(\()p Ft(w)1380 5011 y Fo(\()p Fq(\022)r Fo(\))1377 5090 y Fn(\000)p Fq(;`)1484 5062 y Fv(\))g(+)g Ft(a)1725 5019 y Fn(\003)1725 5083 y Fq(r)1765 5062 y Fv(\()p Ft(w)1876 5011 y Fo(\()p Fq(\022)r Fo(\))1873 5084 y(+)p Fq(;r)1985 5062 y Fv(\))g(+)f Ft(a)2225 5080 y Fq(r)2263 5062 y Fv(\()p Ft(w)2374 5011 y Fo(\()p Fq(\022)r Fo(\))2371 5084 y Fn(\000)p Fq(;r)2484 5062 y Fv(\))59 b(=)2721 4968 y Fm(X)2717 5177 y Fq(\033)r Fo(=)p Fn(\006)2901 4968 y Fm(X)2885 5180 y Fq(\034)8 b Fo(=)p Fq(`;r)3078 5062 y Ft(a)3129 5019 y Fq(\033)3129 5083 y(\034)3176 5062 y Fv(\()p Ft(w)3287 5019 y Fo(\()p Fq(\022)r Fo(\))3284 5083 y Fq(\033)n(;\034)3385 5062 y Fv(\))22 b Ft(;)3173 5295 y Fv(\(V.118\))p eop %%Page: 67 68 67 67 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(67)257 573 y(with)35 b Ft(a)532 532 y Fo(+)532 601 y(#)626 573 y Fv(:=)30 b Ft(a)810 534 y Fn(\003)810 596 y Fo(#)907 573 y Fv(and)k Ft(a)1149 532 y Fn(\000)1149 601 y Fo(#)1243 573 y Fv(:=)c Ft(a)1427 590 y Fo(#)1490 573 y Fv(,)35 b(is)f(dilatation)h(analytic,)g(and)g(the)f(follo)m(wing) h(norm)257 693 y(b)s(ound)e(holds,)g(for)g(some)g(constan)m(t)g Ft(C)h(<)27 b Fr(1)p Fv(,)745 896 y Fm(\015)745 956 y(\015)800 981 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\)\()p Ft(L)1134 998 y Fq(aux)1278 981 y Fv(+)21 b(1\))1462 937 y Fn(\000)p Fo(1)1556 896 y Fm(\015)1556 956 y(\015)1671 981 y Fr(\024)1817 909 y Ft(C)i(e)1955 870 y Fo(\()p Fq(M)7 b Fo(+1\))p Fn(j)p Fo(Re)13 b Fq(\022)r Fn(j)2365 909 y Fv(\()p Ft(\014)27 b Fv(+)21 b(1\))p 1817 958 853 4 v 2072 1046 a Ft(\014)h Fv(cos)q(\()p Ft(#)p Fv(\))2701 981 y(\003)f Ft(:)p 3292 981 4 66 v 3296 918 59 4 v 3296 981 V 3354 981 4 66 v 3173 1163 a Fv(\(V.119\))404 1376 y(Our)32 b(next)i(goal)e(is)h(to)f (establish)i(the)f(analyticit)m(y)h(of)e(the)h(resolv)m(en)m(t)1245 1604 y Ft(R)1319 1622 y Fq(g)1359 1604 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))61 b(:=)1798 1524 y Fm(\000)1860 1604 y Ft(L)1926 1622 y Fq(g)1967 1604 y Fv(\()p Ft(\022)s Fv(\))43 b Fr(\000)g Ft(z)2302 1524 y Fm(\001)2349 1546 y Fn(\000)p Fo(1)2465 1604 y Ft(;)681 b Fv(\(V.120\))257 1824 y(as)43 b(an)g(op)s(erator-v)-5 b(alued)42 b(function)h(of)f Ft(\022)j Fv(and)e Ft(z)t Fv(.)74 b(This)44 b(is)f(not)f(immediate)i (from)e(a)257 1944 y(direct)j(application)f(of)g(standard)g(tec)m (hniques)j(in)d(dilatation)f(analyticit)m(y)j(b)s(ecause)257 2065 y Ft(\022)31 b Fr(7!)d Ft(L)525 2082 y Fq(g)565 2065 y Fv(\()p Ft(\022)s Fv(\))33 b(is)h Fs(not)e Fv(a)h(family)h(of)e (t)m(yp)s(e)i(A,)f(for)g Ft(\022)e Fr(2)c Fv(\006)2223 2082 y Fq(#)2265 2091 y Fk(0)2305 2065 y Fv(.)44 b(Indeed,)35 b(as)e(w)m(e)h(p)s(oin)m(t)f(out)g(in)257 2185 y(App)s(endix)j(A,)g Ft(\022)d Fr(7!)d Ft(L)1114 2203 y Fq(g)1155 2185 y Fv(\()p Ft(\022)s Fv(\))k(is)h(not)f(ev)m(en)j(an)d(analytic)h(family)g(on)g Fr(H)h Fv(in)e(the)h(sense)i(of)257 2306 y(Kato)32 b(\(see,)i(e.g.,)f ([33]\).)404 2426 y(T)-8 b(o)32 b(mak)m(e)i(a)e(precise)j(statemen)m (t,)f(w)m(e)f(in)m(tro)s(duce)h(a)e(dense)i(set)g(of)e(v)m(ectors,)1386 2646 y Fr(D)62 b Fv(:=)e Fr(D)1761 2663 y Fo(1)1822 2646 y Fr(\\)22 b(D)1984 2663 y Fo(2)2083 2646 y Fr(\022)59 b(H)24 b Ft(;)822 b Fv(\(V.121\))257 2865 y(where)1100 3085 y Fr(D)1175 3102 y Fo(1)1275 3085 y Fv(:=)1480 2990 y Fm(\\)1437 3206 y Fn(j)p Fq(\022)r Fn(j\024)p Fq(\022)1600 3215 y Fk(0)1666 3085 y Fv(Dom)1871 3004 y Fm(\002)1912 3085 y Fv(\()p Ft(L)2016 3102 y Fq(aux)2161 3085 y Fv(+)21 b(1\))2358 3060 y Fm(b)2345 3085 y Ft(U)11 b Fv(\()p Ft(\022)s Fv(\))2546 3004 y Fm(\003)2609 3085 y Ft(;)537 b Fv(\(V.122\))257 3409 y(and)33 b Fr(D)522 3427 y Fo(2)594 3409 y Fv(consists)h(of)e(all)g(v)m(ectors)i(whic)m(h)g(are)e(analytic) h(w.r.t.)g Ft(\022)d Fr(2)e Fv(\006)2840 3427 y Fq(#)2882 3436 y Fk(0)2921 3409 y Fv(.)43 b(W)-8 b(e)33 b(see)g(that)257 3530 y Fr(D)d(\022)d(H)34 b Fv(is)f(dense)h(b)m(y)g(observing)g(that) 396 3771 y Fr(D)63 b(\023)c Fv(span)865 3661 y Fm(n)931 3771 y Ft(')995 3789 y Fq(`)1050 3771 y Fr(\012)21 b Ft(')1211 3789 y Fq(r)1271 3771 y Fr(\012)g Ft( )1431 3789 y Fq(`)1487 3771 y Fr(\012)g Ft( )1647 3789 y Fq(r)1718 3657 y Fm(\014)1718 3716 y(\014)1718 3776 y(\014)1784 3771 y Ft(')1848 3789 y Fq(`=r)1977 3771 y Fr(2)27 b(H)2151 3789 y Fq(el)2231 3771 y Ft(;)82 b( )2403 3789 y Fq(`=r)2532 3771 y Fr(2)28 b(D)2700 3789 y Fq(Gauss)2903 3661 y Fm(o)2991 3771 y Ft(;)155 b Fv(\(V.123\))257 4013 y(where)34 b Fr(D)614 4030 y Fq(Gauss)850 4013 y Fv(consists)g(of)e(all)h (translates)g(and)g(dilates)g(of)f(Gaussians,)400 4254 y Fr(D)475 4272 y Fq(Gauss)761 4254 y Fv(:=)947 4144 y Fm(n)1013 4254 y Ft(a)1064 4211 y Fn(\003)1104 4254 y Fv(\()p Ft(f)1190 4272 y Fo(1)1229 4254 y Fv(\))16 b Fr(\001)g(\001)g(\001)h Ft(a)1464 4211 y Fn(\003)1504 4254 y Fv(\()p Ft(f)1590 4272 y Fq(n)1637 4254 y Fv(\)\012)1778 4140 y Fm(\014)1778 4199 y(\014)1778 4259 y(\014)1844 4254 y Ft(f)1892 4272 y Fq(j)1928 4254 y Fv(\()1961 4228 y Ft(~)1966 4254 y(k)s(;)f(\025)p Fv(\))28 b(=)f(exp)q([)p Fr(\000)p Fv(\()2574 4228 y Ft(~)2579 4254 y(k)d Fr(\000)2747 4228 y Ft(~)2752 4254 y(k)2803 4272 y Fq(j)2839 4254 y Fv(\))2877 4211 y Fo(2)2916 4254 y Ft(=\033)3020 4272 y Fq(j)3057 4254 y Fv(])e Ft(\016)3149 4272 y Fq(\025)3190 4282 y Fj(j)3223 4272 y Fq(;\025)3309 4254 y Ft(;)1419 4467 y(n)27 b Fr(2)h Fp(N)1663 4484 y Fo(0)1730 4467 y Ft(;)1801 4440 y(~)1806 4467 y(k)1857 4484 y Fq(j)1920 4467 y Fr(2)f Fp(R)2078 4423 y Fo(3)2145 4467 y Ft(;)49 b(\033)2276 4484 y Fq(j)2340 4467 y Ft(>)27 b Fv(0)21 b Ft(;)49 b(\025)2646 4484 y Fq(j)2710 4467 y Fr(2)27 b(f\006)p Fv(1)p Fr(g)3025 4356 y Fm(o)3112 4467 y Ft(:)34 b Fv(\(V.124\))404 4708 y(F)-8 b(or)34 b Ft(';)16 b( )34 b Fr(2)d(D)s Fv(,)k(w)m(e)h(establish)g(the)f(existence)i(of)e(the)g (desired)h(analytic)f(con)m(tin)m(u-)257 4829 y(ation)e(of)514 5048 y(\()p Ft(\022)s(;)16 b(z)t Fv(\))61 b Fr(7!)e Ft(F)1011 5066 y Fq('; )1129 5048 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))61 b(:=)1569 4967 y Fm(\012)1629 5023 y(b)1616 5048 y Ft(U)10 b Fv(\()1738 5022 y(\026)1730 5048 y Ft(\022)s Fv(\))p Ft(')1880 4963 y Fm(\014)1880 5023 y(\014)1935 4967 y(\000)1997 5048 y Ft(L)2063 5066 y Fq(g)2104 5048 y Fv(\()p Ft(\022)s Fv(\))43 b Fr(\000)g Ft(z)2439 4967 y Fm(\001)2486 4990 y Fn(\000)p Fo(1)2615 5023 y Fm(b)2602 5048 y Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))p Ft( )2869 4967 y Fm(\013)3173 5048 y Fv(\(V.125\))257 5268 y(in)26 b(the)g(follo)m(wing)g(theorem,)i(whic)m(h)g(is)e(an)f(immediate)i (consequence)i(of)c(Theorem)i(A.4)257 5388 y(pro)m(v)m(en)34 b(in)f(App)s(endix)h(A.)p eop %%Page: 68 69 68 68 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(68)257 573 y Fi(Theorem)37 b(V.3.)49 b Fs(L)-5 b(et)32 b Fv(0)27 b Ft(<)f(#)1384 534 y Fn(0)1384 593 y Fo(0)1452 573 y Ft(<)g(#)1611 590 y Fo(0)1679 573 y Ft(<)g(\031)t(=)p Fv(2)p Fs(,)32 b(and)e(assume)h(that)g Fv(Im)17 b Ft(z)32 b(>)27 b Fv(0)p Fs(,)k Fv(0)c Ft(<)g(g)k Fr(\034)257 693 y Ft(#)314 654 y Fn(0)314 714 y Fo(0)354 693 y Fv(\()p Ft(\031)t(=)p Fv(2)21 b Fr(\000)h Ft(#)725 711 y Fo(0)765 693 y Fv(\))p Fs(,)35 b(and)f Ft(\022)c Fr(2)d Fv(\006)1294 711 y Fq(#)1336 720 y Fk(0)1375 693 y Fs(,)35 b Fv(Im)17 b Ft(\022)30 b Fr(\025)d Ft(#)1808 654 y Fn(0)1808 714 y Fo(0)1848 693 y Fs(.)45 b(Then)257 884 y Fi(\(i\))i Fs(F)-7 b(or)46 b(al)5 b(l)47 b Ft(';)16 b( )53 b Fr(2)c(D)s Fs(,)h(the)d(function)f Ft(z)54 b Fr(7!)49 b(h)p Ft(')p Fr(j)p Fv(\()p Ft(L)2313 901 y Fq(g)2383 884 y Fr(\000)31 b Ft(z)t Fv(\))2577 844 y Fn(\000)p Fo(1)2672 884 y Ft( )t Fr(i)47 b Fs(has)f(an)g(analytic)501 1004 y(c)-5 b(ontinuation)35 b(fr)-5 b(om)34 b(the)h(upp)-5 b(er)35 b(half-plane)e(into)i Fr(G)2441 965 y Fn(0)2499 1004 y Fs(given)f(by)628 1137 y Fm(\012)675 1218 y Ft(')739 1133 y Fm(\014)739 1193 y(\014)794 1137 y(\000)855 1218 y Ft(L)921 1235 y Fq(g)1005 1218 y Fr(\000)44 b Ft(z)1174 1137 y Fm(\001)1220 1159 y Fn(\000)p Fo(1)1336 1218 y Ft( )1403 1137 y Fm(\013)1512 1218 y Fv(=)1650 1137 y Fm(\012)1710 1192 y(b)1697 1218 y Ft(U)11 b Fv(\()1820 1191 y(\026)1812 1218 y Ft(\022)s Fv(\))p Ft(')1962 1133 y Fm(\014)1962 1193 y(\014)2016 1137 y(\000)2078 1218 y Ft(L)2144 1235 y Fq(g)2185 1218 y Fv(\()p Ft(\022)s Fv(\))43 b Fr(\000)g Ft(z)2520 1137 y Fm(\001)2567 1159 y Fn(\000)p Fo(1)2696 1192 y Fm(b)2683 1218 y Ft(U)10 b Fv(\()p Ft(\022)s Fv(\))p Ft( )2950 1137 y Fm(\013)3019 1218 y Ft(:)127 b Fv(\(V.126\))257 1471 y Fi(\(ii\))47 b Fs(On)34 b(the)g(r)-5 b(e)g(ctangular)35 b(domain)e Ft(R)c Fv(:=)1868 1390 y Fm(\010)1926 1471 y Ft(\022)h Fr(2)d 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2171 y Fn(\000)1234 2200 y Fr(n)31 b(G)1378 2161 y Fn(0)1401 2200 y Fs(,)51 b(wher)-5 b(e)48 b Fr(G)1835 2161 y Fn(0)1906 2200 y Fs(is)g(the)g(c)-5 b(onne)g(cte)g(d)48 b(c)-5 b(omp)g(onent)47 b(of)g Fr(f)p Ft(z)56 b Fr(2)501 2320 y Fp(C)20 b Fr(j)28 b(k)p Ft(R)745 2338 y Fq(g)785 2320 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))p Fr(k)28 b Ft(<)f Fr(1g)34 b Fs(c)-5 b(ontaining)34 b Fp(C)1904 2281 y Fo(+)1969 2320 y Fs(.)257 2607 y Fh(V.4)132 b(In)l(v)l(ertibilit)l(y)48 b(of)c Fg(L)1550 2630 y Fc(g)1597 2607 y FF(\()p Fg(\022)s FF(\))26 b Ff(\000)g Fg(z)49 b Fh(on)c Ff(S)2237 2630 y Fc(>)257 2792 y Fv(In)25 b(this)g (subsection)i(w)m(e)e(study)h(the)f(sp)s(ectral)g(prop)s(erties)g(of)f Ft(L)2551 2809 y Fq(g)2592 2792 y Fv(\()p Ft(\022)s Fv(\))g(on)h Fr(S)2927 2809 y Fq(>)2986 2792 y Fv(.)41 b(In)24 b(the)h(pre-)257 2912 y(vious)j(subsection,)h(w)m(e)f(established)g(the)f(analytic)g (con)m(tin)m(uation)g(of)f(matrix)h(elemen)m(ts)257 3032 y(of)34 b(\()p Ft(L)474 3050 y Fq(g)514 3032 y Fv(\()p Ft(\022)s Fv(\))23 b Fr(\000)g Ft(z)t Fv(\))847 2993 y Fn(\000)p Fo(1)976 3032 y Fv(in)34 b Ft(\022)j Fv(and)d Ft(z)t Fv(.)49 b(This)35 b(allo)m(ws)g(us)f(no)m(w)h(to)f(c)m(ho)s(ose) h(purely)g(imaginary)257 3153 y(v)-5 b(alues)34 b(of)e Ft(\022)s Fv(,)h(and)f(w)m(e)i(shall)f(henceforth)h(assume)g(that)e(0) 27 b Ft(<)g(#)2592 3113 y Fn(0)2592 3173 y Fo(0)2659 3153 y Ft(<)g(#)2819 3170 y Fo(0)2886 3153 y Ft(<)g(\031)t(=)p Fv(2,)1167 3358 y Ft(\022)63 b Fv(=)d Ft(i#)119 b Fv(and)f Ft(#)1952 3314 y Fn(0)1952 3378 y Fo(0)2051 3358 y Fr(\024)60 b Ft(#)g Fr(\024)f Ft(#)2496 3375 y Fo(0)2558 3358 y Ft(:)588 b Fv(\(V.128\))404 3563 y(W)-8 b(e)22 b(\014rst)h(demonstrate) g(that)f Fr(S)1560 3580 y Fq(>)1646 3563 y Fr(\022)27 b Ft(\032)p Fv([)p Ft(L)1892 3580 y Fq(g)1933 3563 y Fv(\()p Ft(\022)s Fv(\)],)d(the)e(resolv)m(en)m(t)j(set)d(of)g Ft(L)3000 3580 y Fq(g)3040 3563 y Fv(\()p Ft(\022)s Fv(\),)j(simply)257 3683 y(b)m(y)34 b(expanding)g(\()p Ft(L)966 3700 y Fq(g)1006 3683 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)t Fv(\))1336 3644 y Fn(\000)p Fo(1)1464 3683 y Fv(in)33 b(a)f(norm-con)m(v)m(ergen)m(t)i(Neumann)g(series.)257 3853 y Fi(Theorem)j(V.4.)49 b Fs(F)-7 b(or)34 b Ft(g)c(>)d Fv(0)35 b Fs(su\016ciently)g(smal)5 b(l,)34 b Fr(S)2270 3871 y Fq(>)2356 3853 y Fr(\022)27 b Ft(\032)p Fv([)p Ft(L)2602 3871 y Fq(g)2643 3853 y Fv(\()p Ft(\022)s Fv(\)])p Fs(,)35 b(and)1245 3973 y Fm(\015)1245 4033 y(\015)1301 4058 y Fv(\()p Ft(L)1405 4076 y Fq(g)1445 4058 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)t Fv(\))1775 4015 y Fn(\000)p Fo(1)1870 3973 y Fm(\015)1870 4033 y(\015)1988 4058 y Fr(\024)62 b(O)2207 3977 y Fm(\000)2252 4058 y Ft(\032)2302 4015 y Fn(\000)p Fo(1)2302 4080 y(0)2396 3977 y Fm(\001)2464 4058 y Ft(;)682 b Fv(\(V.129\))257 4263 y Fs(for)35 b(any)g Ft(z)d Fr(2)27 b(S)828 4280 y Fq(>)887 4263 y Fs(.)257 4454 y(Pr)-5 b(o)g(of.)49 b Fv(W)-8 b(e)33 b(expand)g(the)g(in)m(v)m(erse)i(of)e Ft(L)1738 4471 y Fq(g)1778 4454 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)38 b Fv(in)32 b(a)h(Neumann)g(series,)360 4659 y Fm(\000)406 4740 y Ft(L)472 4758 y Fq(g)512 4740 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)804 4659 y Fm(\001)851 4682 y Fn(\000)p Fo(1)1005 4740 y Fv(=)1177 4616 y Fn(1)1140 4645 y Fm(X)1146 4855 y Fq(n)p Fo(=0)1284 4659 y Fm(\000)1330 4740 y Ft(L)1396 4758 y Fo(0)1436 4740 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)h Ft(z)1728 4659 y Fm(\001)1774 4682 y Fn(\000)p Fo(1)1869 4630 y Fm(n)1935 4740 y Fr(\000)p Ft(g)t(W)14 b Fv(\()p Ft(\022)s Fv(\))2308 4659 y Fm(\000)2353 4740 y Ft(L)2419 4758 y Fo(0)2459 4740 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)2751 4659 y Fm(\001)2797 4682 y Fn(\000)p Fo(1)2892 4630 y Fm(o)2958 4652 y Fq(n)3027 4740 y Ft(;)119 b Fv(\(V.130\))257 5027 y(whic)m(h)34 b(is)g(easily)f(seen)h(to)e(b)s(e)h(norm-con)m(v)m (ergen)m(t)i(since,)f(b)m(y)f(Lemma)g(IV.1,)g(w)m(e)h(ha)m(v)m(e)426 5147 y Fm(\015)426 5207 y(\015)498 5232 y Fv(\()p Ft(L)602 5249 y Fq(aux)746 5232 y Fv(+)22 b Ft(\032)894 5249 y Fo(0)933 5232 y Fv(\))971 5188 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1152 5232 y Ft(g)t(W)14 b Fv(\()p Ft(\022)s Fv(\))i(\()p Ft(L)1553 5249 y Fq(aux)1697 5232 y Fv(+)21 b Ft(\032)1844 5249 y Fo(0)1884 5232 y Fv(\))1922 5188 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2103 5147 y Fm(\015)2103 5207 y(\015)2218 5232 y Fv(=)60 b Fr(O)2435 5151 y Fm(\000)2480 5232 y Ft(g)19 b(\032)2596 5181 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2596 5256 y(0)2762 5151 y Fm(\001)2867 5232 y Fv(=)59 b Fr(O)3083 5151 y Fm(\000)3129 5232 y Ft(g)3180 5188 y Fq(")3216 5151 y Fm(\001)3283 5232 y Ft(;)3173 5352 y Fv(\(V.131\))p eop %%Page: 69 70 69 69 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(69)257 573 y(and)381 681 y Fm(\015)381 741 y(\015)381 800 y(\015)381 860 y(\015)463 754 y Ft(L)529 771 y Fq(aux)673 754 y Fv(+)22 b Ft(\032)821 771 y Fo(0)p 462 802 399 4 v 462 891 a Ft(L)528 908 y Fo(0)568 891 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)f Ft(z)871 681 y Fm(\015)871 741 y(\015)871 800 y(\015)871 860 y(\015)986 825 y Fv(=)181 b(sup)1131 899 y Fk(0)p Fe(\024j)p Fj(r)1257 916 y(f)1297 899 y Fe(j\024)p Fj(r)1393 907 y(aux)1194 972 y(\025)p Fe(2)p Fj(\033)r Fk([)p Fj(L)1371 989 y(el)1423 972 y Fk(])1512 681 y Fm(\014)1512 741 y(\014)1512 800 y(\014)1512 860 y(\014)2027 754 y Ft(r)2071 771 y Fq(aux)2215 754 y Fv(+)22 b Ft(\032)2363 771 y Fo(0)p 1572 802 1287 4 v 1572 891 a Ft(\025)f Fv(+)h(cos)q(\()p Ft(#)p Fv(\))p Ft(r)2056 908 y Fq(f)2123 891 y Fr(\000)f Ft(i)16 b Fv(sin)r(\()p Ft(#)p Fv(\))p Ft(r)2567 908 y Fq(aux)2711 891 y Fr(\000)22 b Ft(z)2868 681 y Fm(\014)2868 741 y(\014)2868 800 y(\014)2868 860 y(\014)2961 825 y Fr(\024)60 b Ft(C)7 b Fv(\()p Ft(#)p Fv(\))22 b Ft(;)3173 1092 y Fv(\(V.132\))257 1311 y(where)39 b Ft(C)7 b Fv(\()p Ft(#)p Fv(\))36 b Ft(<)g Fr(1)h Fv(is)h(a)g Ft(#)p Fv(-dep)s(enden)m(t)i(constan)m(t,)g(and)d Ft(r)2459 1328 y Fq(aux)2582 1311 y Fv(,)i Ft(r)2692 1328 y Fq(f)2775 1311 y Fv(are)e(p)s(oin)m(ts)i(in)f(the)257 1431 y(join)m(t)33 b(sp)s(ectrum)h(of)e Ft(L)1088 1449 y Fq(aux)1211 1431 y Fv(,)g Ft(L)1336 1449 y Fq(f)1382 1431 y Fv(,)h(resp)s(ectiv)m(ely)-8 b(.)404 1552 y(T)g(o)40 b(c)m(hec)m(k)j(\(V.132\),)e(w)m(e)h (distinguish)g(the)f(cases)g Ft(r)2343 1569 y Fq(aux)2506 1552 y Ft(<)f(\034)11 b(\032)2725 1569 y Fo(0)2805 1552 y Fv(and)41 b Ft(r)3047 1569 y Fq(aux)3209 1552 y Fr(\025)f Ft(\034)11 b(\032)3428 1569 y Fo(0)3468 1552 y Fv(,)257 1672 y(with)34 b Ft(\034)k Fr(\025)27 b Fv(0)32 b(to)h(b)s(e)f(pic)m(k) m(ed)j(later.)44 b(F)-8 b(or)32 b Ft(r)1782 1689 y Fq(aux)1931 1672 y Ft(<)27 b(\034)11 b(\032)2137 1689 y Fo(0)2177 1672 y Fv(,)33 b(w)m(e)g(ha)m(v)m(e)466 1892 y Fr(j)p Ft(\025)21 b Fv(+)h(cos\()p Ft(#)p Fv(\))p Ft(r)976 1909 y Fq(f)1044 1892 y Fr(\000)f Ft(i)16 b Fv(sin)r(\()p Ft(#)p Fv(\))p Ft(r)1488 1909 y Fq(aux)1632 1892 y Fr(\000)21 b Ft(z)t Fr(j)88 b(\025)f(j)p Ft(\025)21 b Fv(+)h(cos\()p Ft(#)p Fv(\))p Ft(r)2566 1909 y Fq(f)2634 1892 y Fr(\000)f Fv(Re\()p Ft(z)t Fv(\))p Fr(j)175 b Fv(\(V.133\))903 2102 y Fr(\025)1062 1991 y Fm(\020)1131 2030 y Ft(\032)1181 2048 y Fo(0)p 1131 2079 90 4 v 1152 2167 a Fv(2)1253 2102 y Fr(\000)21 b Fv(cos)q(\()p Ft(#)p Fv(\))p Ft(\034)11 b(\032)1717 2119 y Fo(0)1757 1991 y Fm(\021)1817 2171 y Fo(+)1963 2102 y Fr(\025)2135 2030 y Fv(\(1)22 b Fr(\000)f Fv(2)p Ft(\034)28 b Fv(cos)17 b Ft(#)p Fv(\))2702 2048 y Fo(+)p 2135 2079 626 4 v 2275 2167 a Fv(2\(1)k(+)h Ft(\034)11 b Fv(\))2787 2102 y(\()p Ft(r)2869 2119 y Fq(aux)3013 2102 y Fv(+)22 b Ft(\032)3161 2119 y Fo(0)3200 2102 y Fv(\))g Ft(;)257 2369 y Fv(and,)33 b(for)f Ft(r)667 2387 y Fq(aux)816 2369 y Fr(\025)27 b Ft(\034)11 b(\032)1022 2387 y Fo(0)1063 2369 y Fv(,)395 2589 y Fr(j)p Ft(\025)22 b Fv(+)f(cos)q(\()p Ft(#)p Fv(\))p Ft(r)906 2607 y Fq(f)973 2589 y Fr(\000)h Ft(i)16 b Fv(sin)q(\()p Ft(#)p Fv(\))p Ft(r)1417 2607 y Fq(aux)1561 2589 y Fr(\000)22 b Ft(z)t Fr(j)87 b(\025)g(j)16 b Fv(sin)q(\()p Ft(#)p Fv(\))p Ft(r)2325 2607 y Fq(aux)2469 2589 y Fv(+)22 b(Im\()p Ft(z)t Fv(\))p Fr(j)338 b Fv(\(V.134\))714 2799 y Fr(\025)873 2688 y Fm(\020)932 2799 y Fv(sin)q(\()p Ft(#)p Fv(\))22 b Fr(\000)1315 2727 y Fv(tan)16 b Ft(#)p 1315 2776 215 4 v 1371 2864 a Fv(4)p Ft(\034)1539 2688 y Fm(\021)1599 2868 y Fo(+)1674 2799 y Ft(r)1718 2816 y Fq(aux)1927 2799 y Fr(\025)2100 2727 y Fv(\()p Ft(\034)27 b Fv(sin)17 b Ft(#)23 b Fr(\000)e Fv(tan)16 b Ft(#)p Fv(\))2772 2745 y Fo(+)p 2100 2776 733 4 v 2293 2864 a Fv(4\(1)21 b(+)g Ft(\034)11 b Fv(\))2858 2799 y(\()p Ft(r)2940 2816 y Fq(aux)3084 2799 y Fv(+)21 b Ft(\032)3231 2816 y Fo(0)3271 2799 y Fv(\))h Ft(:)257 3079 y Fv(Cho)s(osing)33 b Ft(\034)39 b Fv(:=)27 b(\(sin)17 b Ft(#)k Fv(+)g(2)16 b(cos)h Ft(#)p Fv(\)\(4)f(cos)h Ft(#)p Fv(\))1895 3040 y Fn(\000)p Fo(1)1990 3079 y Fv(\(sin)g Ft(#)22 b Fv(+)e(cos)d Ft(#)p Fv(\))2581 3040 y Fn(\000)p Fo(1)2676 3079 y Fv(,)32 b(w)m(e)i(obtain)e Ft(C)7 b Fv(\()p Ft(#)p Fv(\))27 b(=)257 3199 y(\(sin)17 b Ft(#=)p Fv(2\)\(sin)g Ft(#)22 b Fv(+)g(cos)17 b Ft(#)p Fv(\))1217 3160 y Fn(\000)p Fo(1)1312 3199 y Fv(.)p 3425 3199 4 66 v 3429 3137 59 4 v 3429 3199 V 3487 3199 4 66 v 257 3609 a Fh(V.5)132 b(In)l(v)l(ertibilit)l(y)55 b(of)c Fg(L)1564 3632 y Fc(g)1611 3609 y FF(\()p Fg(\022)s FF(\))30 b Ff(\000)g Fg(z)56 b Fh(in)c(the)g(Vicinit)l(y)g(of)g(A)l (tomic)592 3758 y(Eigen)l(v)-7 b(alues)45 b(a)l(w)l(a)l(y)g(from)f (Zero)257 3943 y Fv(In)31 b(this)h(subsection)h(w)m(e)e(in)m(v)m (estigate)i(the)e(in)m(v)m(ertibilit)m(y)j(of)c Ft(L)2530 3960 y Fq(g)2571 3943 y Fv(\()p Ft(\022)s Fv(\))18 b Fr(\000)g Ft(z)35 b Fv(in)c Fr(S)3058 3960 y Fq(i;j)3138 3943 y Fv(,)g(for)f(an)m(y)257 4063 y Ft(i)j Fr(6)p Fv(=)f Ft(j)6 b Fv(.)54 b(In)36 b(Theorem)h(V.7)e(b)s(elo)m(w,)j(w)m(e)f(sho)m (w)g(the)f(existence)i(of)e(a)f(p)s(ositiv)m(e)i(constan)m(t)257 4184 y Ft(\015)308 4201 y Fn(6)p Fo(=0)429 4184 y Ft(>)27 b Fv(0)33 b(suc)m(h)h(that)726 4404 y Fr(S)785 4421 y Fq(i;j)909 4404 y Fr(\\)43 b(f)p Ft(z)32 b Fr(2)27 b Fp(C)53 b Fv(:)60 b(Im)16 b Ft(z)32 b(>)27 b Fr(\000)p Ft(g)1860 4360 y Fo(2)1915 4404 y Ft(\015)1966 4421 y Fn(6)p Fo(=0)2060 4404 y Fr(g)59 b(\022)h Ft(\032)p Fv([)p Ft(L)2447 4421 y Fq(g)2488 4404 y Fv(\()p Ft(\022)s Fv(\)])22 b Ft(;)485 b Fv(\(V.135\))257 4624 y(pro)m(vided)42 b Ft(g)h(>)c Fv(0)h(is)h(su\016cien)m(tly)i(small,)f(and)e Ft(")g(>)f Fv(0)h(is)h(as)f(in)g(Eq.)h(\(V.21\).)66 b(Theo-)257 4744 y(rem)37 b(V.7)g(and)f(Theorem)i(V.4)e(imply)-8 b(,)39 b(in)d(particular,)i(that)e(the)h(sp)s(ectrum)h(of)e Ft(L)3331 4762 y Fq(g)3371 4744 y Fv(\()p Ft(\022)s Fv(\))257 4864 y(in)d(\()p Fr(\0001)16 b Ft(;)32 b Fr(\000)p Ft(\032)784 4882 y Fo(0)824 4864 y Fv(])43 b Fr([)h Fv([)p Ft(\032)1080 4882 y Fo(0)1136 4864 y Ft(;)33 b Fr(1)p Fv(\))f(is)h(absolutely)h(con) m(tin)m(uous.)404 4985 y(Let)e Ft(z)g Fr(2)27 b(S)806 5002 y Fq(i;j)887 4985 y Fv(.)43 b(W)-8 b(e)33 b(recall)g(from)g(the)g (de\014nition)g(\(V.24\))f(of)g Fr(S)2693 5002 y Fq(i;j)2806 4985 y Fv(that)645 5269 y Fr(j)p Fv(Re)q(\()p Ft(z)t Fv(\))22 b Fr(\000)g Ft(\021)t Fr(j)59 b(\024)g Ft(\032)1356 5287 y Fo(0)1396 5269 y Ft(=)p Fv(2)118 b(and)g(Im)17 b Ft(z)64 b Fr(\025)c(\000)2351 5198 y Fv(sin\()p Ft(#)p Fv(\))p 2351 5246 254 4 v 2453 5335 a(4)2630 5269 y Ft(\032)2680 5287 y Fo(0)2741 5269 y Ft(;)405 b Fv(\(V.136\))p eop %%Page: 70 71 70 70 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(70)257 573 y(where)37 b Ft(\021)e Fv(:=)d Ft(E)832 590 y Fq(i;j)944 573 y Fr(6)p Fv(=)f(0)k(is)h(a)f(nonzero)h(eigen)m(v) -5 b(alue)37 b(of)e Ft(L)2335 590 y Fq(el)2394 573 y Fv(.)51 b(The)37 b(pro)m(ximit)m(y)g(of)e Ft(z)40 b Fv(to)35 b Ft(\021)257 693 y Fv(implies)f(that)f Ft(z)k Fv(is)c(uniformly)h(in)e Ft(g)k Fv(a)m(w)m(a)m(y)e(from)e(an)m(y)i(other)e(eigen)m(v)-5 b(alue)34 b(of)f Ft(L)3178 711 y Fq(el)3237 693 y Fv(,)656 923 y(dist)814 843 y Fm(\000)881 923 y Ft(z)21 b(;)43 b(\033)t Fv([)p Ft(L)1169 941 y Fq(el)1229 923 y Fv(])21 b Fr(n)h(f)p Ft(\021)t Fr(g)1519 813 y Fm(\021)1638 923 y Fr(\025)1783 852 y Fv(1)p 1783 901 49 4 v 1783 989 a(2)1937 923 y(min)1868 977 y Fj(\026;\027)s Fe(2)p Fj(\033)r Fk([)p Fj(L)2098 995 y(el)2151 977 y Fk(])1959 1038 y Fj(\026)p Fe(6)p Fk(=)p Fj(\027)2195 923 y Fr(j)p Ft(\026)f Fr(\000)h Ft(\027)6 b Fr(j)60 b Ft(>)g Fv(0)21 b Ft(:)399 b Fv(\(V.137\))404 1208 y(As)39 b(in)h([7,)h(Sect.)f(IV])f(and)h(in)f ([9,)i(Sect.)f(I)s(I)s(I],)f(w)m(e)h(establish)h(the)f(in)m(v)m (ertibilit)m(y)i(of)257 1328 y Ft(L)323 1346 y Fq(g)364 1328 y Fv(\()p Ft(\022)s Fv(\))25 b Fr(\000)g Ft(z)43 b Fv(b)m(y)c(means)g(of)f(the)h Fs(F)-7 b(eshb)i(ach)37 b(map)p Fv(,)i Fr(F)2168 1346 y Fq(P)2227 1328 y Fv(\()p Ft(L)2331 1346 y Fq(g)2372 1328 y Fv(\()p Ft(\022)s Fv(\))25 b Fr(\000)g Ft(z)t Fv(\),)41 b(corresp)s(onding)e(to)257 1449 y(the)33 b(pro)5 b(jection)1027 1636 y Ft(P)1090 1654 y Fq(\021)1192 1636 y Fv(:=)59 b Ft(P)1431 1593 y Fq(el)1417 1657 y(\021)1511 1636 y Fr(\012)22 b Ft(P)1686 1593 y Fq(aux)1672 1657 y(\032)1708 1666 y Fk(0)1878 1636 y Ft(;)p 2040 1556 77 4 v 135 w(P)2116 1654 y Fq(\021)2218 1636 y Fv(:=)59 b Fi(1)22 b Fr(\000)f Ft(P)2618 1654 y Fq(\021)2682 1636 y Ft(;)464 b Fv(\(V.138\))257 1833 y(where)35 b Ft(P)617 1794 y Fq(el)603 1853 y(\021)705 1833 y Fv(:=)29 b Ft(\037)898 1850 y Fn(f)p Fq(\021)r Fn(g)1011 1833 y Fv([)p Ft(L)1104 1850 y Fq(el)1163 1833 y Fv(])34 b(is)g(the)h(pro)5 b(jection)34 b(of)g Ft(L)2136 1850 y Fq(el)2228 1833 y Fv(on)m(to)g Ft(\021)t Fv(,)g(and)g Ft(P)2830 1794 y Fq(aux)2816 1853 y(\032)2852 1862 y Fk(0)2981 1833 y Fv(:=)29 b Ft(\037)p Fv([)p Ft(L)3267 1850 y Fq(aux)3419 1833 y Ft(<)257 1953 y(\032)307 1971 y Fo(0)347 1953 y Fv(])23 b(is)h(the)g(pro)5 b(jection)24 b(on)m(to)f(sp)s(ectral)h(v)-5 b(alues)24 b(of)f Ft(L)2115 1971 y Fq(aux)2260 1953 y Fv(strictly)i(less)f(than)g Ft(\032)3024 1971 y Fo(0)3063 1953 y Fv(.)41 b(T)-8 b(o)23 b(pro)m(v)m(e)257 2074 y(existence)35 b(of)e Fr(F)854 2091 y Fq(P)899 2099 y Fj(\021)941 2074 y Fv(\()p Ft(L)1045 2091 y Fq(g)1085 2074 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)t Fv(\),)34 b(w)m(e)f(require)h(the)f(follo)m(wing)g (preparatory)g(lemma.)257 2244 y Fi(Lemma)k(V.5.)49 b Fs(Assume)43 b(\(V.128\))f(and)h(\(V.136\).)68 b(Then,)44 b(for)e Ft(g)j(>)d Fv(0)g Fs(su\016ciently)257 2364 y(smal)5 b(l,)p 546 2284 V 34 w Ft(P)623 2382 y Fq(\021)665 2284 y Fm(\000)710 2364 y Ft(L)776 2382 y Fq(g)817 2364 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)1109 2284 y Fm(\001)p 1155 2284 V 80 x Ft(P)1232 2382 y Fq(\021)1309 2364 y Fs(is)34 b(invertible)g(on)h Fv(Ran)p 2171 2284 V 16 w Ft(P)2247 2382 y Fq(\021)2289 2364 y Fs(.)257 2561 y(Pr)-5 b(o)g(of.)49 b Fv(W)-8 b(e)25 b(construct)i(the)e(in)m(v)m (erse)j(of)p 1724 2481 V 24 w Ft(P)1801 2578 y Fq(\021)1843 2561 y Ft(L)1909 2578 y Fq(g)1949 2561 y Fv(\()p Ft(\022)s Fv(\))p 2073 2481 V Ft(P)2150 2578 y Fq(\021)2198 2561 y Fr(\000)7 b Ft(z)30 b Fv(on)25 b(Ran)p 2675 2481 V 16 w Ft(P)2752 2578 y Fq(\021)2819 2561 y Fv(b)m(y)h(an)f(expansion)257 2681 y(in)33 b(a)f(Neumann)i(series,)391 2858 y Fm(\000)p 436 2859 V 436 2939 a Ft(P)513 2957 y Fq(\021)555 2939 y Ft(L)621 2957 y Fq(g)661 2939 y Fv(\()p Ft(\022)s Fv(\))p 785 2859 V Ft(P)862 2957 y Fq(\021)925 2939 y Fr(\000)22 b Ft(z)1072 2858 y Fm(\001)1118 2881 y Fn(\000)p Fo(1)p 1229 2859 V 1229 2939 a Ft(P)1305 2957 y Fq(\021)1434 2939 y Fv(=)1633 2814 y Fn(1)1597 2844 y Fm(X)1602 3054 y Fq(n)p Fo(=0)p 1907 2788 V 1907 2868 a Ft(P)1983 2885 y Fq(\021)p 1767 2916 399 4 v 1767 3004 a Ft(L)1833 3022 y Fo(0)1872 3004 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)g Ft(z)2175 2799 y Fm(\022)2249 2939 y Fr(\000)p Ft(g)t(W)14 b Fv(\()p Ft(\022)s Fv(\))p 2771 2788 77 4 v 2771 2868 a Ft(P)2848 2885 y Fq(\021)p 2631 2916 399 4 v 2631 3004 a Ft(L)2697 3022 y Fo(0)2737 3004 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)3040 2799 y Fm(\023)3113 2821 y Fq(n)474 3260 y Fv(=)643 3188 y(\()p Ft(L)747 3206 y Fq(aux)891 3188 y Fv(+)f Ft(\032)1038 3206 y Fo(0)1078 3188 y Fv(\))1116 3149 y Fo(1)p Fq(=)p Fo(2)p 1226 3108 77 4 v 1226 3188 a Ft(P)1302 3206 y Fq(\021)p 643 3237 702 4 v 794 3325 a Ft(L)860 3343 y Fo(0)900 3325 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)1407 3135 y Fn(1)1371 3165 y Fm(X)1376 3374 y Fq(n)p Fo(=0)1515 3119 y Fm(\032)1589 3260 y Fv(\()p Ft(L)1693 3277 y Fq(aux)1837 3260 y Fv(+)h Ft(\032)1985 3277 y Fo(0)2025 3260 y Fv(\))2063 3216 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2244 3179 y Fm(\000)2290 3260 y Fr(\000)p Ft(g)t(W)14 b Fv(\()p Ft(\022)s Fv(\))2647 3179 y Fm(\001)2707 3260 y Fv(\()p Ft(L)2811 3277 y Fq(aux)2955 3260 y Fv(+)22 b Ft(\032)3103 3277 y Fo(0)3143 3260 y Fv(\))3181 3216 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1105 3436 y Fm(\022)1188 3506 y Fv(\()p Ft(L)1292 3523 y Fq(aux)1437 3506 y Fv(+)f Ft(\032)1584 3523 y Fo(0)1624 3506 y Fv(\))p 1662 3426 77 4 v Ft(P)1738 3523 y Fq(\021)p 1188 3554 592 4 v 1285 3642 a Ft(L)1351 3660 y Fo(0)1391 3642 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)h Ft(z)1790 3436 y Fm(\023)17 b(\033)1954 3459 y Fq(n)2018 3577 y Fv(\()p Ft(L)2122 3594 y Fq(aux)2266 3577 y Fv(+)k Ft(\032)2413 3594 y Fo(0)2453 3577 y Fv(\))2491 3534 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2678 3577 y Ft(:)468 b Fv(\(V.139\))257 3816 y(By)33 b(Lemma)h(IV.1,)e(w)m(e)i(ha)m(v)m(e)508 3919 y Fm(\015)508 3978 y(\015)580 4003 y Fv(\()p Ft(L)684 4021 y Fq(aux)828 4003 y Fv(+)22 b Ft(\032)976 4021 y Fo(0)1015 4003 y Fv(\))1053 3960 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1234 4003 y Ft(g)t(W)14 b Fv(\()p Ft(\022)s Fv(\))i(\()p Ft(L)1635 4021 y Fq(aux)1779 4003 y Fv(+)21 b Ft(\032)1926 4021 y Fo(0)1966 4003 y Fv(\))2004 3960 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2169 3919 y Fm(\015)2169 3978 y(\015)2284 4003 y Fv(=)60 b Fr(O)2501 3923 y Fm(\000)2546 4003 y Ft(g)19 b(\032)2662 3952 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2662 4028 y(0)2827 3923 y Fm(\001)2895 4003 y Ft(:)251 b Fv(\(V.140\))257 4191 y(W)-8 b(e)33 b(mak)m(e)h(use)g(of)p 1465 4298 77 4 v 1465 4378 a Ft(P)1541 4396 y Fq(\021)1643 4378 y Fv(=)p 1778 4298 V 59 w Ft(P)1855 4316 y Fo(\(1\))1855 4399 y Fq(\021)1971 4378 y Fv(+)p 2068 4298 V 21 w Ft(P)2145 4316 y Fo(\(2\))2145 4399 y Fq(\021)2261 4378 y Ft(;)885 b Fv(\(V.141\))p 753 4476 V 753 4556 a Ft(P)830 4495 y Fo(\(1\))830 4577 y Fq(\021)984 4556 y Fv(:=)p 1146 4469 136 4 v 59 w Ft(P)1223 4527 y Fq(el)1209 4577 y(\021)1303 4556 y Fr(\012)22 b Fi(1)1457 4574 y Fq(f)1524 4556 y Ft(;)p 1685 4476 77 4 v 134 w(P)1762 4495 y Fo(\(2\))1762 4577 y Fq(\021)1915 4556 y Fv(:=)60 b Ft(P)2155 4517 y Fq(el)2141 4577 y(\021)2235 4556 y Fr(\012)21 b Ft(\037)p Fv([)p Ft(L)2486 4574 y Fq(aux)2636 4556 y Fr(\025)27 b Ft(\032)2789 4574 y Fo(0)2829 4556 y Fv(])22 b Ft(;)268 b Fv(\(V.142\))257 4760 y(where)p 534 4673 136 4 v 29 w Ft(P)611 4731 y Fq(el)597 4781 y(\021)697 4760 y Fv(:=)27 b Ft(\037)888 4778 y Fl(R)p Fn(nf)p Fq(\021)r Fn(g)1084 4760 y Fv([)p Ft(L)1177 4778 y Fq(el)1236 4760 y Fv(])h(is)g(the)g(pro) 5 b(jection)29 b(on)m(to)f(the)g(eigen)m(v)-5 b(alues)30 b(of)d Ft(L)3056 4778 y Fq(el)3143 4760 y Fv(di\013eren)m(t)257 4914 y(from)41 b Ft(\021)t Fv(.)69 b(Since)42 b(b)s(oth)p 1145 4834 77 4 v 40 w Ft(P)1222 4852 y Fo(\(1\))1222 4935 y Fq(\021)1357 4914 y Fv(and)p 1555 4834 V 41 w Ft(P)1632 4852 y Fo(\(2\))1632 4935 y Fq(\021)1767 4914 y Fv(comm)m(ute)g(with)g Ft(L)2492 4932 y Fq(aux)2655 4914 y Fv(and)g(with)f Ft(L)3150 4932 y Fo(0)3190 4914 y Fv(\()p Ft(\022)s Fv(\),)i(w)m(e)257 5034 y(ha)m(v)m(e)34 b(that)916 5149 y Fm(\015)916 5209 y(\015)916 5269 y(\015)916 5328 y(\015)982 5222 y Fv(\()p Ft(L)1086 5240 y Fq(aux)1230 5222 y Fv(+)21 b Ft(\032)1377 5240 y Fo(0)1417 5222 y Fv(\))p 1455 5142 V Ft(P)1532 5240 y Fq(\021)p 982 5271 592 4 v 1078 5359 a Ft(L)1144 5376 y Fo(0)1184 5359 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)1584 5149 y Fm(\015)1584 5209 y(\015)1584 5269 y(\015)1584 5328 y(\015)1698 5293 y Fv(=)60 b(max)1836 5354 y Fq(j)t Fo(=1)p Fq(;)p Fo(2)2016 5149 y Fm(\015)2016 5209 y(\015)2016 5269 y(\015)2016 5328 y(\015)2081 5221 y Fv(\()p Ft(L)2185 5239 y Fq(aux)2329 5221 y Fv(+)22 b Ft(\032)2477 5239 y Fo(0)2516 5221 y Fv(\))p 2554 5141 77 4 v Ft(P)2631 5159 y Fo(\()p Fq(j)t Fo(\))2631 5242 y Fq(\021)p 2081 5271 642 4 v 2202 5359 a Ft(L)2268 5376 y Fo(0)2308 5359 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)f Ft(z)2732 5149 y Fm(\015)2732 5209 y(\015)2732 5269 y(\015)2732 5328 y(\015)2809 5293 y Ft(:)337 b Fv(\(V.143\))p eop %%Page: 71 72 71 71 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(71)257 585 y(On)33 b(Ran)p 611 505 77 4 v 16 w Ft(P)687 523 y Fo(\(1\))687 605 y Fq(\021)782 585 y Fv(,)f(w)m(e)i(observ)m(e)g (that)335 765 y Fm(\015)335 825 y(\015)335 885 y(\015)335 945 y(\015)400 838 y Fv(\()p Ft(L)504 855 y Fq(aux)649 838 y Fv(+)21 b Ft(\032)796 855 y Fo(0)836 838 y Fv(\))p 874 758 V Ft(P)950 776 y Fo(\(1\))950 858 y Fq(\021)p 400 887 645 4 v 523 975 a Ft(L)589 993 y Fo(0)629 975 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)1055 765 y Fm(\015)1055 825 y(\015)1055 885 y(\015)1055 945 y(\015)1169 910 y Fv(=)60 b(sup)1452 769 y Fm(\032)1548 795 y(\014)1548 855 y(\014)1548 915 y(\014)2108 839 y Ft(r)24 b Fv(+)e Ft(\032)2324 856 y Fo(0)p 1591 887 1289 4 v 1591 975 a Ft(\026)g Fv(+)f(cos)q(\()p Ft(#)p Fv(\))p Ft(r)2077 993 y Fq(f)2144 975 y Fr(\000)h Ft(i)16 b Fv(sin)q(\()p Ft(#)p Fv(\))p Ft(r)2588 993 y Fq(aux)2732 975 y Fr(\000)22 b Ft(z)2890 795 y Fm(\014)2890 855 y(\014)2890 915 y(\014)2955 769 y(\033)3052 910 y Ft(;)94 b Fv(\(V.144\))257 1188 y(where)34 b(the)g(suprem)m(um)g(is)g(tak)m(en)f(o)m(v)m(er)h Ft(\026)27 b Fr(2)h Ft(\033)t Fv([)p Ft(L)2086 1206 y Fq(el)2145 1188 y Fv(])22 b Fr(n)g(f)p Ft(\021)t Fr(g)32 b Fv(and)g(0)c Fr(\024)f Ft(r)2860 1206 y Fq(f)2933 1188 y Fr(\024)g Ft(r)3080 1206 y Fq(aux)3202 1188 y Fv(.)44 b(Since)257 1309 y(sin)17 b Ft(#)28 b Fr(\025)f Fv(sin)17 b Ft(#)774 1269 y Fn(0)774 1329 y Fo(0)841 1309 y Ft(>)27 b Fv(0)32 b(and)h Fr(j)p Ft(\026)21 b Fr(\000)h Fv(Re)16 b Ft(z)t Fr(j)28 b(\025)f Ft(c)p Fv(,)33 b(for)f(some)h(constan)m(t)h Ft(c)27 b(>)g Fv(0,)32 b(w)m(e)i(ha)m(v)m(e)699 1529 y Fr(j)p Ft(\026)21 b Fv(+)h(cos\()p Ft(#)p Fv(\))p Ft(r)1211 1546 y Fq(f)1279 1529 y Fr(\000)f Ft(i)16 b Fv(sin)q(\()p Ft(#)p Fv(\))p Ft(r)1722 1546 y Fq(aux)1867 1529 y Fr(\000)21 b Ft(z)t Fr(j)1133 b Fv(\(V.145\))782 1728 y Fr(\025)950 1657 y Fv(1)p 950 1705 49 4 v 950 1793 a(2)1009 1647 y Fm(\000)1055 1728 y Ft(c)22 b Fr(\000)f Fv(cos)q(\()p Ft(#)p Fv(\))p Ft(r)1524 1746 y Fq(f)1569 1647 y Fm(\001)1615 1767 y Fo(+)1717 1728 y Fv(+)1847 1657 y(1)p 1847 1705 V 1847 1793 a(4)1905 1647 y Fm(\000)1951 1728 y Ft(r)1995 1746 y Fq(aux)2139 1728 y Fr(\000)g Fv(sin)q(\()p Ft(#)p Fv(\))p Ft(\032)2539 1746 y Fo(0)2579 1647 y Fm(\001)2625 1767 y Fo(+)782 1962 y Fr(\025)950 1891 y Fv(1)p 950 1939 V 950 2027 a(2)1009 1881 y Fm(\000)1055 1962 y Fv(2)p Ft(\032)1154 1980 y Fo(0)1215 1962 y Fr(\000)h Ft(r)1357 1980 y Fq(aux)1479 1881 y Fm(\001)1524 2001 y Fo(+)1627 1962 y Fv(+)1756 1891 y(1)p 1756 1939 V 1756 2027 a(4)1815 1881 y Fm(\000)1860 1962 y Ft(r)1904 1980 y Fq(aux)2048 1962 y Fr(\000)g Ft(\032)2196 1980 y Fo(0)2236 1881 y Fm(\001)2281 2001 y Fo(+)2427 1962 y Fr(\025)2600 1891 y Fv(1)p 2600 1939 V 2600 2027 a(8)2658 1881 y Fm(\000)2704 1962 y Ft(\032)2754 1980 y Fo(0)2815 1962 y Fv(+)g Ft(r)2960 1881 y Fm(\001)3027 1962 y Ft(;)257 2205 y Fv(and)33 b(hence)1343 2352 y Fm(\015)1343 2412 y(\015)1343 2471 y(\015)1343 2531 y(\015)1408 2424 y Fv(\()p Ft(L)1512 2442 y Fq(aux)1656 2424 y Fv(+)22 b Ft(\032)1804 2442 y Fo(0)1843 2424 y Fv(\))p 1881 2344 77 4 v Ft(P)1958 2362 y Fo(\(1\))1958 2445 y Fq(\021)p 1408 2473 645 4 v 1531 2562 a Ft(L)1597 2579 y Fo(0)1636 2562 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)g Ft(z)2062 2352 y Fm(\015)2062 2412 y(\015)2062 2471 y(\015)2062 2531 y(\015)2177 2496 y Fr(\024)59 b Fv(8)22 b Ft(:)763 b Fv(\(V.146\))257 2810 y(On)33 b(Ran)p 611 2730 77 4 v 16 w Ft(P)687 2749 y Fo(\(2\))687 2831 y Fq(\021)782 2810 y Fv(,)f(w)m(e)i(estimate)279 2991 y Fm(\015)279 3051 y(\015)279 3111 y(\015)279 3170 y(\015)344 3063 y Fv(\()p Ft(L)448 3081 y Fq(aux)593 3063 y Fv(+)21 b Ft(\032)740 3081 y Fo(0)780 3063 y Fv(\))p 818 2983 V Ft(P)894 3002 y Fo(\(2\))894 3084 y Fq(\021)p 344 3113 645 4 v 467 3201 a Ft(L)533 3218 y Fo(0)573 3201 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)998 2991 y Fm(\015)998 3051 y(\015)998 3111 y(\015)998 3170 y(\015)1113 3136 y Fr(\024)66 b Fv(sup)1249 3216 y Fq(r)r Fn(\025)p Fq(\032)1374 3225 y Fk(0)1408 2995 y Fm(\032)1685 3064 y Ft(r)24 b Fv(+)e Ft(\032)1901 3082 y Fo(0)p 1515 3113 597 4 v 1515 3201 a Fr(j)p Ft(r)d Fv(sin)e Ft(#)22 b Fv(+)f(Im)q Ft(z)t Fr(j)2142 2995 y Fm(\033)2250 3136 y Fr(\024)65 b Fv(sup)2385 3216 y Fq(r)r Fn(\025)p Fq(\032)2510 3225 y Fk(0)2545 2995 y Fm(\032)2651 3064 y Fv(2\()p Ft(r)24 b Fv(+)e Ft(\032)2954 3082 y Fo(0)2993 3064 y Fv(\))p 2651 3113 381 4 v 2665 3201 a Ft(r)i Fr(\000)d Ft(\032)2880 3218 y Fo(0)2920 3201 y Ft(=)p Fv(2)3063 2995 y Fm(\033)3197 3136 y Fv(=)87 b(8)21 b Ft(;)3173 3331 y Fv(\(V.147\))257 3550 y(and)33 b(putting)g(together)g(\(V.146\)) f(and)g(\(V.147\),)g(w)m(e)i(obtain)1369 3698 y Fm(\015)1369 3758 y(\015)1369 3817 y(\015)1369 3877 y(\015)1434 3771 y Fv(\()p Ft(L)1538 3789 y Fq(aux)1682 3771 y Fv(+)22 b Ft(\032)1830 3789 y Fo(0)1870 3771 y Fv(\))p 1908 3691 77 4 v Ft(P)1984 3789 y Fq(\021)p 1434 3819 592 4 v 1531 3908 a Ft(L)1597 3925 y Fo(0)1636 3908 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)g Ft(z)2036 3698 y Fm(\015)2036 3758 y(\015)2036 3817 y(\015)2036 3877 y(\015)2151 3842 y Fr(\024)59 b Fv(8)22 b Ft(:)789 b Fv(\(V.148\))257 4120 y(Inserting)40 b(\(V.148\))e(and)h(\(V.140\),)g(the)g(Neumann)h(series) g(\(V.139\))e(is)h(seen)h(to)e(con-)257 4254 y(v)m(erge)43 b(in)e(norm)g(since)i Fr(O)s Fv(\()p Ft(g)19 b(\032)1389 4203 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1389 4278 y(0)1554 4254 y Fv(\))41 b(=)g Fr(O)s Fv(\()p Ft(g)1920 4214 y Fq(")1956 4254 y Fv(\))g Fr(\034)g Fv(1,)i(for)d Ft(g)45 b(>)c Fv(0)g(su\016cien)m(tly)i(small.)p 3425 4374 4 66 v 3429 4311 59 4 v 3429 4374 V 3487 4374 4 66 v 404 4615 a(Lemma)33 b(V.5)f(establishes)i(the)f(existence)i(of)d(the)g (follo)m(wing)h(F)-8 b(esh)m(bac)m(h)34 b(op)s(erator,)681 4835 y Fr(F)751 4852 y Fq(P)796 4860 y Fj(\021)922 4835 y Fv(:=)83 b Fr(F)1178 4852 y Fq(P)1223 4860 y Fj(\021)1265 4754 y Fm(\000)1310 4835 y Ft(L)1376 4852 y Fq(g)1417 4835 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)1709 4754 y Fm(\001)922 4980 y Fv(:=)1108 4899 y Fm(\000)1153 4980 y Ft(L)1219 4998 y Fo(0)1259 4980 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)1551 4899 y Fm(\001)1598 4980 y Ft(P)1661 4998 y Fq(\021)1746 4980 y Fv(+)43 b Ft(g)t(P)1979 4998 y Fq(\021)2020 4980 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Ft(P)2313 4998 y Fq(\021)3173 4980 y Fv(\(V.149\))1129 5149 y Fr(\000)22 b Ft(g)1278 5106 y Fo(2)1316 5149 y Ft(P)1379 5167 y Fq(\021)1421 5149 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p 1651 5069 77 4 v Ft(P)1727 5167 y Fq(\021)1769 5069 y Fm(\000)p 1815 5069 V 80 x Ft(P)1891 5167 y Fq(\021)1933 5149 y Ft(L)1999 5167 y Fq(g)2040 5149 y Fv(\()p Ft(\022)s Fv(\))p 2164 5069 V Ft(P)2240 5167 y Fq(\021)2304 5149 y Fr(\000)22 b Ft(z)2451 5069 y Fm(\001)2497 5091 y Fn(\000)p Fo(1)p 2591 5069 V 2591 5149 a Ft(P)2668 5167 y Fq(\021)2710 5149 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Ft(P)3003 5167 y Fq(\021)3044 5149 y Ft(:)p eop %%Page: 72 73 72 72 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(72)257 573 y(The)34 b(imp)s(ortance)f(of)f Fr(F)1151 590 y Fq(P)1196 598 y Fj(\021)1271 573 y Fv(lies)h(in)g(the)g(follo)m (wing)g(iden)m(tit)m(y)h(\(see)g([7,)f(8)o(,)g(9]\),)511 704 y Fm(\000)556 785 y Ft(L)622 802 y Fq(g)663 785 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)955 704 y Fm(\001)1001 727 y Fn(\000)p Fo(1)1179 785 y Fv(=)1338 674 y Fm(h)1385 785 y Ft(P)1448 802 y Fq(\021)1533 785 y Fr(\000)43 b Ft(g)1703 704 y Fm(\000)p 1748 705 77 4 v 1748 785 a Ft(P)1824 802 y Fq(\021)1866 785 y Ft(L)1932 802 y Fq(g)1973 785 y Fv(\()p Ft(\022)s Fv(\))p 2097 705 V Ft(P)2173 802 y Fq(\021)2237 785 y Fr(\000)22 b Ft(z)2384 704 y Fm(\001)2430 727 y Fn(\000)p Fo(1)p 2524 705 V 2524 785 a Ft(P)2601 802 y Fq(\021)2643 785 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Ft(P)2936 802 y Fq(\021)2977 674 y Fm(i)1359 998 y Fr(F)1439 954 y Fn(\000)p Fo(1)1429 1022 y Fq(P)1474 1030 y Fj(\021)1555 887 y Fm(h)1603 998 y Ft(P)1666 1015 y Fq(\021)1751 998 y Fr(\000)43 b Ft(g)t(P)1984 1015 y Fq(\021)2025 998 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p 2255 917 V Ft(P)2331 1015 y Fq(\021)2373 917 y Fm(\000)p 2419 917 V 81 x Ft(P)2495 1015 y Fq(\021)2537 998 y Ft(L)2603 1015 y Fq(g)2644 998 y Fv(\()p Ft(\022)s Fv(\))p 2768 917 V Ft(P)2844 1015 y Fq(\021)2908 998 y Fr(\000)21 b Ft(z)3054 917 y Fm(\001)3101 939 y Fn(\000)p Fo(1)3195 887 y Fm(i)1365 1197 y Fv(+)p 1468 1117 V 27 w Ft(P)1544 1214 y Fq(\021)1586 1116 y Fm(\000)p 1632 1117 V 81 x Ft(P)1708 1214 y Fq(\021)1750 1197 y Ft(L)1816 1214 y Fq(g)1857 1197 y Fv(\()p Ft(\022)s Fv(\))p 1981 1117 V Ft(P)2057 1214 y Fq(\021)2121 1197 y Fr(\000)g Ft(z)2267 1116 y Fm(\001)2314 1138 y Fn(\000)p Fo(1)p 2408 1117 V 2408 1197 a Ft(P)2484 1214 y Fq(\021)2548 1197 y Ft(;)598 b Fv(\(V.150\))257 1371 y(whic)m(h)50 b(manifestly)h(sho)m(ws)f(that)e Ft(L)1630 1388 y Fq(g)1671 1371 y Fv(\()p Ft(\022)s Fv(\))32 b Fr(\000)g Ft(z)54 b Fv(is)49 b(in)m(v)m(ertible)i(i\013)d Fr(F)2797 1388 y Fq(P)2842 1396 y Fj(\021)2884 1371 y Fv(\()p Ft(L)2988 1388 y Fq(g)3029 1371 y Fv(\()p Ft(\022)s Fv(\))32 b Fr(\000)g Ft(z)t Fv(\))50 b(is)257 1491 y(in)m(v)m(ertible)35 b(on)e(Ran)16 b Ft(P)1078 1509 y Fq(\021)1120 1491 y Fv(.)43 b(Indeed,)303 1614 y Fm(\015)303 1673 y(\015)380 1618 y(\000)426 1698 y Ft(L)492 1716 y Fq(g)533 1698 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)825 1618 y Fm(\001)871 1640 y Fn(\000)p Fo(1)987 1614 y Fm(\015)987 1673 y(\015)1102 1698 y Fr(\024)1238 1588 y Fm(\020)1297 1698 y Fv(1)38 b(+)g Fr(O)1579 1618 y Fm(\000)1624 1698 y Ft(g)19 b(\032)1740 1647 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1740 1723 y(0)1906 1618 y Fm(\001)1951 1588 y(\021)2032 1614 y(\015)2032 1673 y(\015)2104 1698 y Fr(F)2174 1716 y Fq(P)2219 1724 y Fj(\021)2261 1618 y Fm(\000)2307 1698 y Ft(L)2373 1716 y Fq(g)2413 1698 y Fv(\()p Ft(\022)s Fv(\))j Fr(\000)g Ft(z)2706 1618 y Fm(\001)2768 1614 y(\015)2768 1673 y(\015)2872 1698 y Fv(+)49 b Fr(O)3078 1618 y Fm(\000)3123 1698 y Ft(\032)3173 1647 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)3173 1723 y(0)3339 1618 y Fm(\001)3406 1698 y Ft(;)3173 1856 y Fv(\(V.151\))257 2023 y(b)m(y)34 b(Lemma)f(V.5)g(and)f(the)h(estimates)h(in)f(its)g(pro)s(of.)404 2143 y(T)-8 b(o)32 b(apply)i(the)f(F)-8 b(esh)m(bac)m(h)34 b(map,)f(w)m(e)g(in)m(tro)s(duce)h(the)f(lev)m(el-shift)h(op)s(erator,) 509 2317 y Ft(Q)586 2335 y Fq(E)668 2317 y Fr(\012)21 b Ft(P)828 2335 y Fo(\012)943 2317 y Fv(:=)61 b(lim)1106 2380 y Fq(")p Fn(&)p Fo(0)1244 2237 y Fm(\010)1318 2317 y Ft(P)1381 2335 y Fo(\012)1453 2317 y Ft(W)p 1575 2237 V 30 w(P)1651 2335 y Fo(\012)1722 2317 y Fv(\()p Ft(L)1826 2335 y Fo(0)1888 2317 y Fr(\000)21 b Ft(E)28 b Fr(\000)22 b Ft(i")p Fv(\))2300 2274 y Fn(\000)p Fo(1)p 2410 2237 V 2410 2317 a Ft(P)2486 2335 y Fo(\012)2558 2317 y Ft(W)14 b(P)2727 2335 y Fo(\012)2798 2237 y Fm(\011)2878 2317 y Ft(;)268 b Fv(\(V.152\))257 2540 y(where)38 b Ft(E)g Fr(2)33 b Fp(R)47 b Fv(and)36 b Ft(P)1115 2557 y Fo(\012)1203 2540 y Fv(:=)c Fr(j)p Fv(\012)25 b Fr(\012)f Fv(\012)p Fr(ih)p Fv(\012)g Fr(\012)g Fv(\012)p Fr(j)37 b Fv(is)f(the)g(pro)5 b(jection)37 b(on)m(to)f(the)g(v)-5 b(acuum)257 2660 y(\012)26 b Fr(\012)g Fv(\012)39 b(in)g Fr(F)d(\012)25 b(F)10 b Fv(.)62 b(An)39 b(explicit)h(computation)f(sho)m(ws)h(that)e Ft(Q)2731 2678 y Fq(E)2830 2660 y Fv(is)h(b)s(ounded)g(and)257 2780 y(has)33 b(a)g(nonnegativ)m(e)h(imaginary)f(part)319 2955 y(\000)380 2972 y Fq(E)461 2955 y Fr(\012)22 b Ft(P)622 2972 y Fo(\012)736 2955 y Fv(:=)60 b(Im)17 b Ft(Q)1109 2972 y Fq(E)1190 2955 y Fr(\012)22 b Ft(P)1351 2972 y Fo(\012)1465 2955 y Fv(=)60 b Ft(P)1664 2972 y Fo(\012)1735 2955 y Ft(W)p 1857 2875 V 30 w(P)1934 2972 y Fo(\012)2005 2955 y Ft(\016)t Fv(\()p Ft(L)2156 2972 y Fo(0)2217 2955 y Fr(\000)22 b Ft(E)6 b Fv(\))p 2447 2875 V 16 w Ft(P)2523 2972 y Fo(\012)2595 2955 y Ft(W)14 b(P)2764 2972 y Fo(\012)2878 2955 y Fr(\025)60 b Fv(0)21 b Ft(:)62 b Fv(\(V.153\))257 3129 y(It)31 b(is)g(con)m(v)m(enien)m(t)i(to)e(omit)f(the)h(trivial)g (tensor)g(factor)g Fr(\012)p Ft(P)2442 3146 y Fo(\012)2527 3129 y Fv(in)g(our)f(notation,)h(i.e.,)h(to)257 3249 y(iden)m(tify)i Ft(Q)689 3267 y Fq(E)782 3249 y Fv(with)f Ft(Q)1081 3267 y Fq(E)1162 3249 y Fr(\012)22 b Ft(P)1323 3267 y Fo(\012)1410 3249 y Fv(and)33 b(\000)1661 3267 y Fq(E)1753 3249 y Fv(with)g(\000)2036 3267 y Fq(E)2118 3249 y Fr(\012)21 b Ft(P)2278 3267 y Fo(\012)2333 3249 y Fv(.)404 3369 y(In)33 b(Theorem)g(B.1)g(of)f(App)s(endix)i(B,)f(w)m (e)g(sho)m(w)h(that)e(if)h Ft(\021)e Fv(=)c Ft(E)2742 3387 y Fq(i)2792 3369 y Fr(\000)21 b Ft(E)2961 3387 y Fq(j)3025 3369 y Fr(6)p Fv(=)27 b(0)32 b(then)1418 3555 y Ft(P)1481 3573 y Fq(\021)1539 3555 y Fv(\000)1600 3573 y Fq(\021)1658 3555 y Ft(P)1721 3573 y Fq(\021)1822 3555 y Fr(\025)60 b Ft(\015)2014 3512 y Fo(\()p Fq(i;j)t Fo(\))2165 3555 y Ft(P)2228 3573 y Fq(\021)2292 3555 y Ft(;)854 b Fv(\(V.154\))257 3742 y(for)32 b(some)i(strictly)g(p)s(ositiv)m(e)g (constan)m(t)f Ft(\015)1795 3703 y Fo(\()p Fq(i;j)t Fo(\))1957 3742 y Ft(>)27 b Fv(0.)257 3913 y Fi(Theorem)37 b(V.6.)49 b Fs(Assume)35 b(\(V.128\).)44 b(F)-7 b(or)34 b(any)h Fv(0)27 b Ft(<)g(")f(<)h Fv(2)p Ft(=)p Fv(3)p Fs(,)687 4002 y Fm(\015)687 4062 y(\015)742 4087 y Fr(F)812 4104 y Fq(P)857 4112 y Fj(\021)943 4087 y Fr(\000)43 b Ft(P)1125 4104 y Fq(\021)1167 4006 y Fm(\000)1213 4087 y Ft(L)1279 4104 y Fo(0)1318 4087 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)g Ft(z)k Fr(\000)c Ft(g)1782 4043 y Fo(2)1820 4087 y Ft(Q)1897 4104 y Fq(\021)1940 4006 y Fm(\001)1985 4087 y Ft(P)2048 4104 y Fq(\021)2090 4002 y Fm(\015)2090 4062 y(\015)2207 4087 y Fv(=)62 b Fr(O)2426 4006 y Fm(\000)2471 4087 y Ft(g)2522 4043 y Fo(2+)p Fq(")2649 4006 y Fm(\001)2716 4087 y Ft(:)430 b Fv(\(V.155\))257 4261 y Fs(Pr)-5 b(o)g(of.)49 b Fv(Denoting)32 b Ft(\037)1045 4278 y Fq(r)1083 4261 y Fv(\()p Ft(!)t Fv(\))26 b(:=)i Ft(\037)p Fv([)p Ft(!)i(<)d(r)s Fv(],)33 b(w)m(e)g(\014rst)g(observ)m(e)i(that,)d(b)m(y)i(Lemma)f (IV.1,)286 4350 y Fm(\015)286 4410 y(\015)342 4435 y Ft(a)393 4453 y Fq(`)426 4435 y Fv(\()p Ft(w)537 4392 y Fo(\()p Fq(\022)r Fo(\))534 4456 y Fq(\033)n(;\027)635 4435 y Fv(\))16 b Ft(P)752 4453 y Fq(\021)794 4350 y Fm(\015)794 4410 y(\015)936 4435 y Fv(=)1099 4350 y Fm(\015)1099 4410 y(\015)1154 4435 y Ft(a)1205 4453 y Fq(`)1238 4435 y Fv(\()p Ft(\037)1337 4453 y Fq(\032)1373 4462 y Fk(0)1429 4435 y Ft(w)1502 4392 y Fo(\()p Fq(\022)r Fo(\))1499 4456 y Fq(\033)n(;\027)1600 4435 y Fv(\))g Ft(P)1717 4453 y Fq(\021)1759 4350 y Fm(\015)1759 4410 y(\015)369 4604 y Fr(\024)83 b(O)609 4523 y Fm(\000)655 4604 y Ft(\032)705 4553 y Fo(1)p Fq(=)p Fo(2)705 4628 y(0)815 4523 y Fm(\001)877 4519 y(\015)877 4579 y(\015)932 4604 y Ft(a)983 4621 y Fq(`)1016 4604 y Fv(\()p Ft(\037)1115 4621 y Fq(\032)1151 4630 y Fk(0)1207 4604 y Ft(w)1280 4560 y Fo(\()p Fq(\022)r Fo(\))1277 4624 y Fq(\033)n(;\027)1378 4604 y Fv(\))16 b(\()p Ft(L)1536 4621 y Fq(aux)1680 4604 y Fv(+)22 b Ft(\032)1828 4621 y Fo(0)1868 4604 y Fv(\))1906 4560 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2087 4604 y Ft(P)2150 4621 y Fq(\021)2191 4519 y Fm(\015)2191 4579 y(\015)369 4840 y Fr(\024)83 b(O)609 4759 y Fm(\000)655 4840 y Ft(\032)705 4789 y Fo(1)p Fq(=)p Fo(2)705 4864 y(0)815 4759 y Fm(\001)877 4699 y(\022)950 4704 y(Z)1005 4930 y Fq(!)r Fo(\()p Fq(k)r Fo(\))p Fq(<\032)1235 4939 y Fk(0)1291 4840 y Ft(dk)1412 4729 y Fm(\020)1472 4840 y Fv(1)21 b(+)1722 4769 y(1)p 1650 4817 195 4 v 1650 4905 a Ft(!)t Fv(\()p Ft(k)s Fv(\))1854 4729 y Fm(\021)1930 4755 y(\015)1930 4815 y(\015)1985 4840 y Ft(w)2058 4796 y Fo(\()p Fq(\022)r Fo(\))2055 4860 y Fq(\033)n(;\027)2156 4840 y Fv(\()p Ft(k)s Fv(\))2286 4755 y Fm(\015)2286 4815 y(\015)2342 4782 y Fo(2)2381 4699 y Fm(\023)2454 4722 y Fo(1)p Fq(=)p Fo(2)369 5148 y Fr(\024)83 b(O)609 5067 y Fm(\000)655 5148 y Ft(\032)705 5097 y Fo(\(1+)p Fq(\034)8 b Fo(\))p Fq(=)p Fo(2)705 5172 y(0)980 5148 y Ft(\014)1041 5104 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1205 5148 y Fv(\()p Ft(\014)28 b Fv(+)21 b(1\))1510 5104 y Fo(1)p Fq(=)p Fo(2)1620 5067 y Fm(\001)1682 5007 y(\022)1755 5012 y(Z)1810 5238 y Fq(!)r Fo(\()p Fq(k)r Fo(\))p Fq(<\032)2040 5247 y Fk(0)2096 5148 y Ft(dk)2217 5037 y Fm(\020)2277 5148 y Fv(1)g(+)2594 5076 y(1)p 2455 5125 328 4 v 2455 5213 a Ft(!)t Fv(\()p Ft(k)s Fv(\))2650 5184 y Fo(2+)p Fq(\034)2792 5037 y Fm(\021)2878 5076 y Fv(4)16 b Ft(e)2988 5037 y Fo(2)p Fq(\031)3087 5076 y Ft(\024)p Fv(\()p Ft(k)s Fv(\))p 2878 5125 395 4 v 2953 5213 a(cos)3084 5184 y Fo(2)3140 5213 y Ft(#)3283 5007 y Fm(\023)3356 5029 y Fo(1)p Fq(=)p Fo(2)369 5388 y Fv(=)83 b Fr(O)609 5307 y Fm(\000)655 5388 y Ft(#)712 5345 y Fn(\000)p Fo(4)823 5388 y Ft(\032)873 5337 y Fo(\(1+)p Fq(\034)8 b Fo(\))p Fq(=)p Fo(2)873 5413 y(0)1148 5388 y Ft(\014)1209 5345 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1373 5388 y Fv(\()p Ft(\014)28 b Fv(+)21 b(1\))1678 5345 y Fo(1)p Fq(=)p Fo(2)1788 5307 y Fm(\001)1855 5388 y Ft(;)1291 b Fv(\(V.156\))p eop %%Page: 73 74 73 73 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(73)257 573 y(for)32 b(an)m(y)i(0)27 b Ft(<)g(\034)38 b(<)27 b Fv(1.)43 b(Using)33 b(similar)h(estimates)g(for)e(the)h(other) g(terms,)g(w)m(e)h(obtain)534 708 y Fm(\015)534 768 y(\015)605 793 y Ft(g)20 b(P)735 810 y Fq(\021)793 793 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))i Ft(P)1102 810 y Fq(\021)1160 708 y Fm(\015)1160 768 y(\015)1275 793 y Fv(=)60 b Fr(O)1492 712 y Fm(\000)1537 793 y Ft(g)19 b(\032)1653 742 y Fo(\(1+)p Fq(\034)8 b Fo(\))p Fq(=)p Fo(2)1653 817 y(0)1912 712 y Fm(\001)2018 793 y Fv(=)59 b Fr(O)2234 712 y Fm(\000)2279 793 y Ft(g)2330 749 y Fo(1+\(1)p Fn(\000)p Fq(")p Fo(\)\(1+)p Fq(\034)8 b Fo(\))2786 712 y Fm(\001)2853 793 y Ft(;)293 b Fv(\(V.157\))257 1013 y(Compare)23 b(to)e([7,)j(\(IV.101\)].)40 b(Next,)24 b(the)f(second)g(resolv)m(en)m(t)g(equation)g(and)f(Lemma)g (V.5)257 1133 y(yield)650 1261 y Fm(\015)650 1320 y(\015)650 1380 y(\015)705 1375 y Ft(g)756 1332 y Fo(2)795 1375 y Ft(P)858 1393 y Fq(\021)900 1375 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p 1130 1295 77 4 v Ft(P)1206 1393 y Fq(\021)1248 1294 y Fm(\000)p 1293 1295 V 1293 1375 a Ft(P)1370 1393 y Fq(\021)1412 1375 y Ft(L)1478 1393 y Fq(g)1518 1375 y Fv(\()p Ft(\022)s Fv(\))p 1642 1295 V Ft(P)1719 1393 y Fq(\021)1782 1375 y Fr(\000)22 b Ft(z)1929 1294 y Fm(\001)1975 1317 y Fn(\000)p Fo(1)p 2070 1295 V 2070 1375 a Ft(P)2146 1393 y Fq(\021)2188 1375 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Ft(P)2481 1393 y Fq(\021)1128 1588 y Fr(\000)27 b Ft(g)1282 1544 y Fo(2)1321 1588 y Ft(P)1384 1605 y Fq(\021)1425 1588 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p 1655 1508 V Ft(P)1731 1605 y Fq(\021)1773 1507 y Fm(\000)p 1819 1508 V 81 x Ft(P)1896 1605 y Fq(\021)1937 1588 y Ft(L)2003 1605 y Fo(0)2043 1588 y Fv(\()p Ft(\022)s Fv(\))p 2167 1508 V Ft(P)2244 1605 y Fq(\021)2307 1588 y Fr(\000)22 b Ft(z)2454 1507 y Fm(\001)2500 1529 y Fn(\000)p Fo(1)p 2595 1508 V 2595 1588 a Ft(P)2671 1605 y Fq(\021)2713 1588 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Ft(P)3006 1605 y Fq(\021)3048 1473 y Fm(\015)3048 1533 y(\015)3048 1593 y(\015)733 1786 y Fv(=)83 b Fr(O)973 1706 y Fm(\000)1018 1786 y Ft(g)1069 1743 y Fo(3)1124 1786 y Ft(\032)1174 1735 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1174 1811 y(0)1339 1706 y Fm(\001)1471 1786 y Fv(=)k Fr(O)1715 1706 y Fm(\000)1760 1786 y Ft(g)1811 1743 y Fo(2+)p Fq(")1937 1706 y Fm(\001)2005 1786 y Ft(;)1141 b Fv(\(V.158\))257 2017 y(Compare)24 b(to)f([7,)i(\(IV.101\)].)40 b(Third,)26 b(w)m(e)e(de\014ne)p 2058 1937 V 25 w Ft(P)2134 2035 y Fq(\021)2176 2017 y Fv(\()p Ft(!)t Fv(\))j(:=)g Fi(1)s Fr(\000)s Ft(\037)2673 2035 y Fn(f)p Fq(\021)r Fn(g)2785 2017 y Fv([)p Ft(L)2878 2035 y Fq(el)2937 2017 y Fv(])s Fr(\012)s Ft(\037)p Fv([)p Ft(L)3200 2035 y Fq(f)3249 2017 y Fv(+)s Ft(!)j(<)257 2138 y(\032)307 2155 y Fo(0)347 2138 y Fv(])j(and)424 2430 y Ft(M)10 b Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))85 b(:=)1015 2294 y Fm(Z)1131 2430 y Ft(dk)1236 2289 y Fm(\032)1311 2430 y Ft(w)1384 2379 y Fo(\()p Fq(\022)r Fo(\))1381 2458 y Fn(\000)p Fq(;`)1488 2430 y Fv(\()p Ft(k)s Fv(\))1618 2289 y Fm(\022)p 1948 2278 V 1948 2358 a Ft(P)2024 2376 y Fq(\021)2066 2358 y Fv(\()p Ft(!)t Fv(\()p Ft(k)s Fv(\)\))p 1701 2407 883 4 v 1701 2495 a Ft(L)1767 2513 y Fo(0)1807 2495 y Fv(\()p Ft(\022)s Fv(\))22 b(+)f Ft(e)2095 2466 y Fn(\000)p Fq(i#)2220 2495 y Ft(!)t Fv(\()p Ft(k)s Fv(\))g Fr(\000)h Ft(z)2594 2289 y Fm(\023)2667 2430 y Ft(w)2740 2379 y Fo(\()p Fq(\022)r Fo(\))2737 2458 y(+)p Fq(;`)2844 2430 y Fv(\()p Ft(k)s Fv(\))199 b(\(V.159\))1273 2717 y Fr(\000)22 b Ft(w)1444 2666 y Fo(\()p Fq(\022)r Fo(\))1441 2739 y Fn(\000)p Fq(;r)1553 2717 y Fv(\()p Ft(k)s Fv(\))1683 2577 y Fm(\022)p 1986 2566 77 4 v 1986 2646 a Ft(P)2062 2663 y Fq(\021)2104 2646 y Fv(\()p Ft(!)t Fv(\()p Ft(k)s Fv(\)\))p 1766 2694 828 4 v 1766 2782 a Ft(L)1832 2800 y Fo(0)1872 2782 y Fv(\()p Ft(\022)s Fv(\))g(+)f Ft(e)2160 2753 y Fq(i#)2230 2782 y Ft(!)t Fv(\()p Ft(k)s Fv(\))g Fr(\000)h Ft(z)2604 2577 y Fm(\023)2677 2717 y Ft(w)2750 2666 y Fo(\()p Fq(\022)r Fo(\))2747 2739 y(+)p Fq(;r)2859 2717 y Fv(\()p Ft(k)s Fv(\))2989 2577 y Fm(\033)3086 2717 y Ft(;)257 2994 y Fv(similar)36 b(to)e Ft(Q)h Fv(in)g([7,)h (\(IV.67\)].)50 b(A)35 b(normal-ordering)f(pro)s(cedure)i(as)f(in)g ([7,)h(\(IV.66\){)257 3115 y(\(IV.76\)])d(then)g(giv)m(es,)h(for)e(an)m (y)h(0)27 b Ft(<)g(\034)1679 3075 y Fn(0)1730 3115 y Ft(<)g Fv(1,)524 3248 y Fm(\015)524 3307 y(\015)524 3367 y(\015)579 3362 y Ft(g)630 3319 y Fo(2)669 3362 y Ft(P)732 3380 y Fq(\021)773 3362 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p 1003 3282 77 4 v Ft(P)1080 3380 y Fq(\021)1122 3281 y Fm(\000)p 1167 3282 V 1167 3362 a Ft(P)1244 3380 y Fq(\021)1286 3362 y Ft(L)1352 3380 y Fo(0)1391 3362 y Fv(\()p Ft(\022)s Fv(\))p 1515 3282 V Ft(P)1592 3380 y Fq(\021)1655 3362 y Fr(\000)22 b Ft(z)1802 3281 y Fm(\001)1848 3304 y Fn(\000)p Fo(1)p 1943 3282 V 1943 3362 a Ft(P)2019 3380 y Fq(\021)2061 3362 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Ft(P)2354 3380 y Fq(\021)2439 3362 y Fr(\000)43 b Ft(g)2609 3319 y Fo(2)2648 3362 y Ft(P)2711 3380 y Fq(\021)2753 3362 y Ft(M)10 b Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))p Ft(P)3136 3380 y Fq(\021)3180 3248 y Fm(\015)3180 3307 y(\015)3180 3367 y(\015)1551 3555 y Fv(=)83 b Fr(O)1791 3474 y Fm(\000)1837 3555 y Ft(g)1888 3511 y Fo(2)1943 3555 y Ft(\032)1993 3511 y Fq(\034)2032 3488 y Fe(0)1993 3575 y Fo(0)2058 3474 y Fm(\001)2191 3555 y Fv(=)j Fr(O)2434 3474 y Fm(\000)2479 3555 y Ft(g)2530 3511 y Fo(2+2)p Fq(\034)2694 3488 y Fe(0)2717 3511 y Fo(\(1)p Fn(\000)p Fq(")p Fo(\))2898 3474 y Fm(\001)2966 3555 y Ft(:)180 b Fv(\(V.160\))257 3775 y(F)-8 b(ourth,)33 b(using)g(the)g(\014rst)g (resolv)m(en)m(t)i(equation,)e(w)m(e)h(obtain)651 3902 y Fm(\015)651 3962 y(\015)651 4022 y(\015)706 4017 y Ft(g)757 3973 y Fo(2)796 4017 y Ft(P)859 4034 y Fq(\021)901 3936 y Fm(\000)946 4017 y Ft(M)10 b Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))40 b Fr(\000)e Ft(M)10 b Fv(\()p Ft(\022)s(;)16 b(\021)t Fv(\))1743 3936 y Fm(\001)1790 4017 y Ft(P)1853 4034 y Fq(\021)1894 3902 y Fm(\015)1894 3962 y(\015)1894 4022 y(\015)852 4275 y Fr(\024)83 b Ft(g)1062 4232 y Fo(2)1117 4275 y Fr(j)p Fv(Im)p Ft(z)t Fr(j)1370 4140 y Fm(Z)1486 4275 y Ft(dk)1642 4204 y Fv(8)16 b Ft(e)1752 4165 y Fo(2)p Fq(\031)p 1617 4253 244 4 v 1617 4341 a Fv(cos)1747 4312 y Fo(2)1803 4341 y Ft(#)1887 4135 y Fm(\022)1960 4275 y Fv(1)22 b(+)2241 4204 y(1)p 2138 4253 255 4 v 2138 4341 a Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\))2403 4135 y Fm(\023)16 b(\022)2684 4204 y Ft(\024)p Fv(\()p Ft(k)s Fv(\))p 2576 4253 404 4 v 2576 4341 a Ft(\032)2626 4358 y Fo(0)2687 4341 y Fv(+)22 b Ft(!)t Fv(\()p Ft(k)s Fv(\))2989 4135 y Fm(\023)3062 4157 y Fo(2)852 4498 y Fv(=)83 b Fr(O)1092 4417 y Fm(\000)1137 4498 y Ft(g)1188 4454 y Fo(2)1243 4498 y Ft(\032)1293 4454 y Fo(1)p Fn(\000)p Fq(\034)1422 4431 y Fe(0)q(0)1293 4520 y Fo(0)1467 4417 y Fm(\001)1600 4498 y Fv(=)j Fr(O)1843 4417 y Fm(\000)1888 4498 y Ft(g)1939 4454 y Fo(2+\(1)p Fn(\000)p Fq(")p Fo(\)\(1)p Fn(\000)p Fq(\034)2362 4431 y Fe(0)q(0)2404 4454 y Fo(\))2435 4417 y Fm(\001)2503 4498 y Ft(;)643 b Fv(\(V.161\))257 4718 y(for)41 b(an)m(y)h(0)g Ft(<)g(\034)870 4679 y Fn(0)q(0)955 4718 y Ft(<)g Fv(1.)70 b(Fifth,)43 b(a)e(similar)h(estimate)h(as)f(\(V.161\))f(and)g(an)h (analytic)257 4838 y(con)m(tin)m(uation)34 b Ft(#)28 b Fr(!)e Fv(0)33 b(giv)m(e)287 4974 y Fm(\015)287 5033 y(\015)342 5058 y Ft(g)393 5015 y Fo(2)448 5058 y Ft(P)511 5076 y Fq(\021)569 5058 y Ft(M)10 b Fv(\()p Ft(\022)s(;)16 b(\021)t Fv(\))g Ft(P)971 5076 y Fq(\021)1057 5058 y Fr(\000)44 b Ft(g)1228 5015 y Fo(2)1283 5058 y Ft(P)1346 5076 y Fq(\021)1404 5058 y Ft(Q)1481 5076 y Fq(\021)1539 5058 y Ft(P)1602 5076 y Fq(\021)1644 4974 y Fm(\015)1644 5033 y(\015)1759 5058 y Fv(=)59 b Fr(O)1975 4978 y Fm(\000)2021 5058 y Ft(g)2072 5015 y Fo(2)2127 5058 y Ft(\032)2177 5015 y Fo(1)p Fn(\000)p Fq(\034)2306 4991 y Fe(000)2177 5081 y Fo(0)2369 4978 y Fm(\001)2501 5058 y Fv(=)87 b Fr(O)2745 4978 y Fm(\000)2790 5058 y Ft(g)2841 5015 y Fo(2+\(1)p Fn(\000)p Fq(")p Fo(\)\(1)p Fn(\000)p Fq(\034)3264 4991 y Fe(0)q(00)3324 5015 y Fo(\))3355 4978 y Fm(\001)3423 5058 y Ft(;)3173 5179 y Fv(\(V.162\))p eop %%Page: 74 75 74 74 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(74)257 573 y(for)49 b(an)m(y)g(0)54 b Ft(<)h(\034)910 534 y Fn(000)1026 573 y Ft(<)f Fv(1.)92 b(Estimates)51 b(\(V.161\))d(and)h(\(V.162\))f(are)h(similar)h(to)e([7,)257 693 y(Lemmata)32 b(IV.11,)f(IV.12].)43 b(Cho)s(osing)31 b Ft(\034)1789 654 y Fn(0)q(00)1878 693 y Fv(:=)c Ft(\034)2061 654 y Fn(0)q(0)2131 693 y Fv(:=)h(1)18 b Fr(\000)g Ft(\034)2476 654 y Fn(0)2527 693 y Fv(:=)27 b(1)18 b Fr(\000)g Ft(\034)43 b Fv(and)31 b(observing)257 814 y(that)i(1)21 b(+)h(2)p Ft(\034)11 b Fv(\(1)21 b Fr(\000)h Ft(")p Fv(\))27 b Fr(\024)g Fv(min)1321 733 y Fm(\010)1379 814 y Fv(1)22 b(+)f(\(1)h Fr(\000)f Ft(")p Fv(\)\(1)g(+)h Ft(\034)11 b Fv(\))27 b Ft(;)44 b Fv(2)21 b(+)h(2)p Ft(\034)11 b Fv(\(1)21 b Fr(\000)h Ft(")p Fv(\))2792 733 y Fm(\011)2850 814 y Fv(,)32 b(w)m(e)i(arriv)m(e)f(at)407 965 y Fm(\015)407 1025 y(\015)462 1050 y Fr(F)532 1067 y Fq(P)577 1075 y Fj(\021)663 1050 y Fr(\000)43 b Ft(P)845 1067 y Fq(\021)887 969 y Fm(\000)933 1050 y Ft(L)999 1067 y Fo(0)1039 1050 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)k Fr(\000)c Ft(g)1502 1006 y Fo(2)1541 1050 y Ft(Q)1618 1067 y Fq(\021)1660 969 y Fm(\001)1705 1050 y Ft(P)1768 1067 y Fq(\021)1810 965 y Fm(\015)1810 1025 y(\015)1925 1050 y Fv(=)60 b Fr(O)2142 969 y Fm(\000)2187 1050 y Ft(g)2238 1006 y Fo(2+)p Fq(")2402 1050 y Fv(+)38 b Ft(g)2567 1006 y Fo(1+2)p Fq(\034)8 b Fo(\(1)p Fn(\000)p Fq(")p Fo(\))2912 969 y Fm(\001)2980 1050 y Ft(;)166 b Fv(\(V.163\))257 1270 y(from)33 b(whic)m(h)h(\(V.155\))e(follo)m(ws)h(up)s(on)g(c)m(ho)s (osing)g Ft(\034)38 b Fv(:=)27 b(2)p Ft(")p Fv(\(1)21 b Fr(\000)h Ft(")p Fv(\))2680 1230 y Fn(\000)p Fo(1)2774 1270 y Fv(.)p 3425 1270 4 66 v 3429 1207 59 4 v 3429 1270 V 3487 1270 4 66 v 404 1510 a(Next,)33 b(thanks)h(to)e(\(V.154\),) g(w)m(e)i(ha)m(v)m(e)648 1660 y Fm(\015)648 1720 y(\015)703 1664 y(\000)749 1744 y Ft(L)815 1762 y Fo(0)855 1744 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)k Fr(\000)c Ft(g)1318 1701 y Fo(2)1357 1744 y Ft(Q)1434 1762 y Fq(\021)1476 1664 y Fm(\001)1521 1686 y Fn(\000)p Fo(1)1632 1744 y Ft(P)1695 1762 y Fq(\021)1737 1660 y Fm(\015)1737 1720 y(\015)1875 1744 y Fr(\024)83 b Fv(dist)2192 1664 y Fm(\010)2250 1744 y Ft(\033)t Fv([)p Ft(L)2402 1762 y Fo(0)2442 1744 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(g)2736 1701 y Fo(2)2775 1744 y Ft(Q)2852 1762 y Fq(\021)2894 1744 y Fv(])16 b Ft(;)33 b(z)3046 1664 y Fm(\011)1875 1914 y Fr(\024)83 b Fv(2)2099 1833 y Fm(\000)2145 1914 y Ft(\015)2201 1870 y Fo(\()p Fq(i;j)t Fo(\))2336 1833 y Fm(\001)2381 1855 y Fn(\000)p Fo(1)2492 1914 y Ft(g)2543 1870 y Fn(\000)p Fo(2)2658 1914 y Ft(;)488 b Fv(\(V.164\))257 2146 y(whenev)m(er)36 b Fr(j)p Fv(Im)q Ft(z)t Fr(j)29 b(\024)g Ft(\015)1096 2106 y Fo(\()p Fq(i;j)t Fo(\))1231 2146 y Ft(g)1282 2106 y Fo(2)1321 2146 y Ft(=)p Fv(2.)45 b(Com)m(bining)35 b(this)f(with)g(\(V.155\),)f(w)m(e)i(obtain)e(a)g(con-)257 2266 y(v)m(ergen)m(t)i(Neumann)e(series)h(expansion)g(for)e Fr(F)1986 2225 y Fn(\000)p Fo(1)1976 2293 y Fq(P)2021 2301 y Fj(\021)2081 2266 y Fv(,)907 2497 y Fm(\015)907 2557 y(\015)962 2582 y Fr(F)1042 2539 y Fn(\000)p Fo(1)1032 2607 y Fq(P)1077 2615 y Fj(\021)1137 2497 y Fm(\015)1137 2557 y(\015)1252 2582 y Fr(\024)59 b Ft(C)23 b(g)1531 2539 y Fn(\000)p Fo(2)1694 2458 y Fn(1)1658 2487 y Fm(X)1663 2697 y Fq(n)p Fo(=0)1801 2442 y Fm(\022)1885 2511 y Ft(C)g(g)2029 2471 y Fo(2+)p Fq(")p 1885 2559 271 4 v 1975 2647 a Ft(g)2026 2618 y Fo(2)2165 2442 y Fm(\023)2238 2464 y Fq(n)2345 2582 y Fv(=)59 b Fr(O)2561 2501 y Fm(\000)2607 2582 y Ft(g)2658 2539 y Fn(\000)p Fo(2)2751 2501 y Fm(\001)2819 2582 y Ft(:)327 b Fv(\(V.165\))257 2884 y(De\014ning)1136 3104 y Ft(\015)1187 3121 y Fn(6)p Fo(=0)1341 3104 y Fv(:=)59 b(min)1666 3023 y Fm(\010)1724 3104 y Ft(\015)1780 3060 y Fo(\()p Fq(i;j)t Fo(\))1947 3019 y Fm(\014)1947 3079 y(\014)2013 3104 y Ft(i)28 b Fr(6)p Fv(=)e Ft(j)2250 3023 y Fm(\011)2367 3104 y Ft(>)60 b Fv(0)21 b Ft(;)573 b Fv(\(V.166\))257 3324 y(w)m(e)34 b(th)m(us)g(arriv)m(e)f(at)f(the)h (follo)m(wing)g(theorem.)257 3494 y Fi(Theorem)k(V.7.)49 b Fs(Assume)40 b(\(V.128\),)h Ft(z)h Fr(2)36 b(S)2003 3511 y Fq(i;j)2084 3494 y Fs(,)41 b Ft(i)c Fr(6)p Fv(=)g Ft(j)6 b Fs(,)41 b(and)f Fv(Im\()p Ft(z)t Fv(\))d Fr(\025)g(\000)p Ft(\015)3168 3511 y Fn(6)p Fo(=0)3278 3494 y Ft(g)3329 3455 y Fo(2)3368 3494 y Ft(=)p Fv(2)p Fs(.)257 3614 y(F)-7 b(or)46 b(su\016ciently)h(smal)5 b(l)46 b Ft(g)52 b(>)d Fv(0)p Fs(,)g(the)e(dilate)-5 b(d)46 b(Liouvil)5 b(lian)46 b Ft(L)2645 3632 y Fq(g)2685 3614 y Fv(\()p Ft(\022)s Fv(\))31 b Fr(\000)f Ft(z)51 b Fs(is)c(b)-5 b(ounde)g(d)257 3735 y(invertible,)394 3955 y Ft(\033)t Fv([)p Ft(L)546 3972 y Fq(g)586 3955 y Fv(\()p Ft(\022)s Fv(\)])44 b Fr(\\)889 3874 y Fm(\010)947 3955 y Ft(z)32 b Fr(2)27 b Fp(C)1223 3870 y Fm(\014)1223 3930 y(\014)1291 3955 y Fv(Re)17 b Ft(z)32 b Fr(\025)27 b Ft(\032)1653 3972 y Fo(0)1692 3955 y Ft(=)p Fv(2)22 b Ft(;)51 b Fv(Im)17 b Ft(z)31 b Fr(\025)c(\000)p Ft(\015)2329 3972 y Fn(6)p Fo(=0)2439 3955 y Ft(g)2490 3911 y Fo(2)2529 3955 y Ft(=)p Fv(2)2627 3874 y Fm(\011)2774 3955 y Fv(=)89 b Fr(;)21 b Ft(:)137 b Fv(\(V.167\))257 4244 y Fh(V.6)132 b(In)l(v)l(ertibilit)l (y)68 b(of)d Fg(L)1591 4267 y Fc(g)1638 4244 y FF(\()p Fg(\022)s FF(\))38 b Ff(\000)f Fg(z)70 b Fh(in)c Ff(S)2313 4267 y Fx(0)p Fc(;>)2511 4244 y Fh(outside)g(the)f(cone)592 4393 y Ff(C)7 b FF(\()p Fg(#)770 4359 y FE(0)798 4393 y FF(\))32 b(=)994 4296 y Fb(\010)1064 4393 y Ff(j)p FF(Re)19 b Fg(z)5 b Ff(j)32 b(\024)g(\000)19 b FF(cot\()p Fg(#)1864 4359 y FE(0)1891 4393 y FF(\)Im)g Fg(z)2149 4296 y Fb(\011)2219 4393 y Fh(,)44 b(for)g Fg(#)2571 4359 y FE(0)2630 4393 y Fg(<)32 b(#)257 4578 y Fv(The)41 b(purp)s(ose)f(of)f(this)h(subsection)h(is)f(to)f(study)h(the)g(in)m(v) m(ertibilit)m(y)i(of)d Ft(L)3032 4595 y Fq(g)3073 4578 y Fv(\()p Ft(\022)s Fv(\))26 b Fr(\000)g Ft(z)44 b Fv(in)257 4698 y Fr(S)316 4716 y Fo(0)p Fq(;>)430 4698 y Fv(.)i(Th)m(us)35 b(w)m(e)f(henceforth)h(assume)g(\(V.128\))d(and)i Fr(j)p Ft(z)t Fr(j)29 b(\024)g Ft(\032)2547 4716 y Fo(0)2586 4698 y Ft(=)p Fv(2.)46 b(W)-8 b(e)33 b(in)m(tro)s(duce)i(the)257 4818 y(pro)5 b(jections)1263 5038 y Ft(P)1326 5056 y Fo(0)1448 5038 y Fv(:=)83 b Ft(P)1711 4995 y Fq(el)1697 5059 y Fo(0)1791 5038 y Fr(\012)22 b Ft(\037)p Fv([)p Ft(L)2043 5056 y Fq(aux)2193 5038 y Ft(<)27 b(\032)2346 5056 y Fo(0)2385 5038 y Fv(])22 b Ft(;)712 b Fv(\(V.168\))p 1250 5128 77 4 v 1250 5208 a Ft(P)1326 5226 y Fo(0)1448 5208 y Fv(:=)83 b Fi(1)22 b Fr(\000)g Ft(P)1873 5226 y Fo(0)1999 5208 y Fv(=)p 2161 5128 V 86 w Ft(P)2238 5146 y Fo(\(1\))2238 5229 y(0)2354 5208 y Fv(+)p 2451 5128 V 21 w Ft(P)2528 5146 y Fo(\(2\))2528 5229 y(0)2644 5208 y Ft(;)502 b Fv(\(V.169\))p 662 5298 V 662 5378 a Ft(P)739 5316 y Fo(\(1\))739 5398 y(0)892 5378 y Fv(:=)p 1055 5285 136 4 v 60 w Ft(P)1132 5343 y Fq(el)1118 5402 y Fo(0)1212 5378 y Fr(\012)21 b Fi(1)121 b Ft(;)p 1634 5298 77 4 v 121 w(P)1711 5316 y Fo(\(2\))1711 5398 y(0)1865 5378 y Fv(:=)59 b Ft(P)2104 5334 y Fq(el)2090 5398 y Fo(0)2184 5378 y Fr(\012)22 b Ft(\037)p Fv([)p Ft(L)2436 5395 y Fq(aux)2586 5378 y Fr(\025)27 b Ft(\032)2739 5395 y Fo(0)2778 5378 y Fv(])22 b Ft(;)319 b Fv(\(V.170\))p eop %%Page: 75 76 75 75 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(75)257 573 y(where)34 b Ft(P)616 534 y Fq(el)602 593 y Fo(0)707 573 y Fv(is)f(the)g(pro)5 b(jection)33 b(of)f(rank)h Ft(N)43 b Fv(on)m(to)33 b(Ker)16 b Ft(L)2352 590 y Fq(el)2411 573 y Fv(,)907 874 y Ft(P)984 831 y Fq(el)970 895 y Fo(0)1102 874 y Fv(=)59 b Ft(\037)1298 892 y Fn(f)p Fo(0)p Fn(g)1408 874 y Fv([)p Ft(L)1501 892 y Fq(el)1561 874 y Fv(])g(=)1783 750 y Fq(N)7 b Fn(\000)p Fo(1)1787 780 y Fm(X)1793 989 y Fq(n)p Fo(=0)1952 874 y Fr(j)p Ft(')2043 892 y Fq(n)2112 874 y Fr(\012)21 b Ft(')2273 892 y Fq(n)2320 874 y Fr(i)16 b(h)p Ft(')2476 892 y Fq(n)2545 874 y Fr(\012)22 b Ft(')2707 892 y Fq(n)2753 874 y Fr(j)g Ft(;)344 b Fv(\(V.171\))257 1174 y(and)38 b Ft(\037)p Fv([)p Ft(L)606 1192 y Fq(aux)764 1174 y Ft(<)e(\032)926 1192 y Fo(0)966 1174 y Fv(])h(is)i(the)f(pro)5 b(jection)38 b(on)m(to)g(sp)s(ectral)g(v)-5 b(alues)39 b(of)e Ft(L)2849 1192 y Fq(aux)3009 1174 y Fv(strictly)i(less)257 1295 y(than)33 b Ft(\032)535 1312 y Fo(0)575 1295 y Fv(.)43 b(Again,)33 b(w)m(e)g(\014rst)g(establish)i(the)e(applicabilit)m(y)h (of)e(the)h(F)-8 b(esh)m(bac)m(h)34 b(map.)257 1465 y Fi(Lemma)j(V.8.)49 b Fs(Assume)36 b(\(V.128\))f(and)g Fr(j)p Ft(z)t Fr(j)29 b(\024)g Ft(\032)2115 1483 y Fo(0)2155 1465 y Ft(=)p Fv(2)p Fs(.)46 b(Then,)35 b(for)h Ft(g)31 b(>)e Fv(0)35 b Fs(su\016ciently)257 1585 y(smal)5 b(l,)p 546 1505 77 4 v 34 w Ft(P)623 1603 y Fo(0)662 1505 y Fm(\000)708 1585 y Ft(L)774 1603 y Fq(g)814 1585 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)g Ft(z)1107 1505 y Fm(\001)p 1153 1505 V 80 x Ft(P)1229 1603 y Fo(0)1304 1585 y Fs(is)34 b(invertible)g(on)h Fv(Ran)p 2166 1505 V 16 w Ft(P)2242 1603 y Fo(0)2282 1585 y Fs(.)257 1778 y(Pr)-5 b(o)g(of.)49 b Fv(Analogous)32 b(to)h(Lemma)g(V.5.)p 3425 1778 4 66 v 3429 1716 59 4 v 3429 1778 V 3487 1778 4 66 v 404 2019 a(By)c(Lemma)h(V.8,)g Ft(L)1170 2037 y Fq(g)1210 2019 y Fv(\()p Ft(\022)s Fv(\))15 b Fr(\000)g Ft(z)34 b Fv(is)29 b(in)m(v)m(ertible)j(\(on)d Fr(H)q Fv(\))g(i\013)g Fr(F)2544 2037 y Fq(P)2589 2046 y Fk(0)2655 2019 y Fv(:=)e Fr(F)2855 2037 y Fq(P)2900 2046 y Fk(0)2939 2019 y Fv(\()p Ft(L)3043 2037 y Fq(g)3083 2019 y Fv(\()p Ft(\022)s Fv(\))15 b Fr(\000)g Ft(z)t Fv(\))30 b(is)257 2140 y(in)m(v)m(ertible)37 b(on)d(Ran)16 b Ft(P)1081 2157 y Fo(0)1121 2140 y Fv(.)48 b(As)35 b(in)g(the)g(previous)g(subsection,)i(the)e(lev)m(el)h(shift)f (op)s(erator)257 2260 y Ft(Q)334 2278 y Fq(E)394 2260 y Fv(,)e(in)m(tro)s(duced)g(in)g(\(V.152\),)f(pla)m(ys)i(an)e(imp)s (ortan)m(t)g(role.)44 b(Note)32 b(that)h Ft(J)9 b Fv(\()p Ft(')3122 2278 y Fq(n)3190 2260 y Fr(\012)21 b Ft(')3351 2278 y Fq(n)3419 2260 y Fr(\012)257 2380 y Fv(\012)h Fr(\012)g Fv(\012\))28 b(=)f Ft(')750 2398 y Fq(n)818 2380 y Fr(\012)22 b Ft(')980 2398 y Fq(n)1048 2380 y Fr(\012)g Fv(\012)g Fr(\012)f Fv(\012,)33 b(and)g(hence)1005 2593 y Ft(P)1082 2550 y Fq(el)1068 2614 y Fo(0)1162 2593 y Fr(\012)22 b Ft(P)1323 2611 y Fo(\012)1437 2593 y Fv(=)60 b Ft(J)31 b(P)1735 2550 y Fq(el)1721 2614 y Fo(0)1814 2593 y Fr(\012)22 b Ft(P)1975 2611 y Fo(\012)2090 2593 y Fv(=)59 b Ft(P)2302 2550 y Fq(el)2288 2614 y Fo(0)2382 2593 y Fr(\012)21 b Ft(P)2542 2611 y Fo(\012)2619 2593 y Ft(J)31 b(;)442 b Fv(\(V.172\))257 2806 y(where)53 b Ft(J)61 b Fv(is)53 b(de\014ned)g(in)f(\(IV.77\).)102 b(Since)53 b(furthermore,)k Ft(J)25 b(L)2726 2824 y Fo(0)2783 2806 y Ft(J)84 b Fv(=)76 b Fr(\000)p Ft(L)3215 2824 y Fo(0)3254 2806 y Fv(,)57 b(and)257 2927 y Ft(J)26 b(W)k(J)52 b Fv(=)43 b Fr(\000)p Ft(W)14 b Fv(,)33 b(w)m(e)g(ha)m(v)m(e)h(that)662 3140 y Ft(P)739 3096 y Fq(el)725 3160 y Fo(0)819 3140 y Fr(\012)22 b Ft(P)980 3157 y Fo(\012)1051 3140 y Ft(W)p 1173 3060 77 4 v 30 w(P)1249 3157 y Fo(\012)1321 3140 y Fv(\()p Ft(L)1425 3157 y Fo(0)1486 3140 y Fr(\000)g Ft(i")p Fv(\))1701 3096 y Fn(\000)p Fo(1)p 1811 3060 V 1811 3140 a Ft(P)1888 3157 y Fo(\012)1959 3140 y Ft(W)30 b(P)2158 3096 y Fq(el)2144 3160 y Fo(0)2238 3140 y Fr(\012)21 b Ft(P)2398 3157 y Fo(\012)3173 3140 y Fv(\(V.173\))982 3285 y(=)83 b Ft(P)1218 3242 y Fq(el)1204 3306 y Fo(0)1297 3285 y Fr(\012)22 b Ft(P)1458 3303 y Fo(\012)1529 3285 y Ft(J)k(W)p 1730 3205 V 29 w(P)1807 3303 y Fo(\012)1878 3285 y Fv(\()p Ft(L)1982 3303 y Fo(0)2044 3285 y Fr(\000)21 b Ft(i")p Fv(\))2258 3242 y Fn(\000)p Fo(1)p 2369 3205 V 2369 3285 a Ft(P)2445 3303 y Fo(\012)2517 3285 y Ft(W)29 b(J)d(P)2795 3242 y Fq(el)2781 3306 y Fo(0)2875 3285 y Fr(\012)21 b Ft(P)3035 3303 y Fo(\012)982 3430 y Fv(=)83 b Fr(\000)27 b Ft(P)1321 3387 y Fq(el)1307 3451 y Fo(0)1400 3430 y Fr(\012)22 b Ft(P)1561 3448 y Fo(\012)1632 3430 y Ft(W)p 1754 3350 V 30 w(P)1830 3448 y Fo(\012)1902 3430 y Fv(\()p Ft(L)2006 3448 y Fo(0)2067 3430 y Fv(+)g Ft(i")p Fv(\))2282 3387 y Fn(\000)p Fo(1)p 2392 3350 V 2392 3430 a Ft(P)2469 3448 y Fo(\012)2540 3430 y Ft(W)30 b(P)2739 3387 y Fq(el)2725 3451 y Fo(0)2819 3430 y Fr(\012)21 b Ft(P)2979 3448 y Fo(\012)3056 3430 y Ft(;)257 3643 y Fv(and)33 b(therefore)1527 3857 y Ft(Q)1604 3874 y Fo(0)1660 3857 y Ft(P)1723 3874 y Fo(0)1822 3857 y Fv(=)59 b Ft(i)16 b Fv(\000)2067 3874 y Fo(0)2123 3857 y Ft(P)2186 3874 y Fo(0)3173 3857 y Fv(\(V.174\))257 4070 y(is)33 b(purely)h(imaginary)-8 b(.)257 4240 y Fi(Lemma)37 b(V.9.)49 b Fs(Assume)36 b(\(V.128\))f(and)g Fr(j)p Ft(z)t Fr(j)29 b(\024)g Ft(\032)2115 4257 y Fo(0)2155 4240 y Ft(=)p Fv(2)p Fs(.)46 b(Then,)35 b(for)h Ft(g)31 b(>)e Fv(0)35 b Fs(su\016ciently)257 4360 y(smal)5 b(l)34 b(and)h(any)f Fv(0)27 b Ft(<)g(")g(<)g Fv(2)p Ft(=)p Fv(3)p Fs(,)684 4488 y Fm(\015)684 4548 y(\015)739 4573 y Fr(F)809 4591 y Fq(P)854 4600 y Fk(0)937 4573 y Fr(\000)43 b Ft(P)1119 4591 y Fo(0)1158 4492 y Fm(\000)1204 4573 y Ft(L)1270 4591 y Fo(0)1310 4573 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)k Fr(\000)c Ft(ig)1806 4530 y Fo(2)1845 4573 y Fv(\000)1906 4591 y Fo(0)1945 4492 y Fm(\001)1991 4573 y Ft(P)2054 4591 y Fo(0)2093 4488 y Fm(\015)2093 4548 y(\015)2210 4573 y Fv(=)62 b Fr(O)2429 4492 y Fm(\000)2474 4573 y Ft(g)2525 4530 y Fo(2+)p Fq(")2652 4492 y Fm(\001)2719 4573 y Ft(:)427 b Fv(\(V.175\))257 4786 y Fs(Pr)-5 b(o)g(of.)49 b Fv(Analogous)25 b(to)f(Theorem)i(V.6,)g(taking)f(in)m(to)g(accoun)m (t)g(\(V.174\).)p 3425 4786 4 66 v 3429 4724 59 4 v 3429 4786 V 3487 4786 4 66 v 404 5027 a(W)-8 b(e)42 b(no)m(w)h(distinguish)h (b)s(et)m(w)m(een)h(sp)s(ectral)e(parameters,)i Ft(z)t Fv(,)h(v)m(ery)e(close)f(to)f(zero)257 5147 y(and)34 b(those)h(whic)m(h)g(are)f(at)f(least)h(of)g(magnitude)g Fr(O)s Fv(\()p Ft(g)2264 5108 y Fo(2+)p Fq(")2390 5147 y Fv(\).)47 b(The)34 b(latter)g(can)g(b)s(e)g(dealt)257 5268 y(with)29 b(b)m(y)g(a)f(standard)h(Neumann)h(series)f(expansion,)i (pro)m(vided)f(they)f(are)f(outside)i(the)257 5388 y(cone)36 b Fr(C)6 b Fv(\()p Ft(#)634 5349 y Fn(0)658 5388 y Fv(\),)35 b Ft(#)815 5349 y Fn(0)869 5388 y Ft(<)c(#)p Fv(,)k(while)h(for)e(the)i (sp)s(ectral)f(parameters)h(close)f(to)g(zero)g(w)m(e)g(apply)p eop %%Page: 76 77 76 76 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(76)257 573 y(the)44 b(renormalization)f(group)g(argumen)m(ts)i (dev)m(elop)s(ed)g(in)e([7,)g(8].)75 b(This)44 b(is)f(done)h(in)257 693 y(Subsection)35 b(V.7.)404 814 y(W)-8 b(e)41 b(turn)g(to)f(pro)m (ving)h(the)g(in)m(v)m(ertibilit)m(y)j(of)c(the)h(resolv)m(en)m(t)i(of) d Ft(L)2903 831 y Fq(g)2944 814 y Fv(\()p Ft(\022)s Fv(\))27 b Fr(\000)g Ft(z)45 b Fv(for)40 b Ft(z)257 934 y Fv(outside)34 b(of)e Fr(C)6 b Fv(\()p Ft(#)857 895 y Fn(0)881 934 y Fv(\))32 b(and)h(of)f(magnitude)i(b)s(et)m(w)m(een)g Fr(O)s Fv(\()p Ft(g)2281 895 y Fo(2+)p Fq(")2407 934 y Fv(\))e(and)h Ft(\032)2717 952 y Fo(0)2757 934 y Ft(=)p Fv(2,)f(see)i(Fig.)e(4.)257 1104 y Fi(Theorem)37 b(V.10.)49 b Fs(Assume)39 b(\(V.128\),)g Fv(0)33 b Ft(<)h(#)2058 1065 y Fn(0)2116 1104 y Ft(<)f(#)p Fs(,)40 b(and)e Fv(0)c Ft(<)g(")f(<)h Fv(1)p Ft(=)p Fv(3)p Fs(.)55 b(Supp)-5 b(ose)257 1225 y(that)37 b Ft(C)528 1242 y Fo(0)584 1225 y Ft(g)635 1185 y Fo(2+)p Fq(")791 1225 y Fr(\024)30 b(j)p Ft(z)t Fr(j)h(\024)f Ft(\032)1187 1242 y Fo(0)1227 1225 y Ft(=)p Fv(2)36 b Fs(and)g Ft(z)46 b(=)-60 b Fr(2)31 b(C)6 b Fv(\()p Ft(#)1880 1185 y Fn(0)1904 1225 y Fv(\))p Fs(,)36 b(wher)-5 b(e)36 b Ft(C)2355 1242 y Fo(0)2425 1225 y Ft(<)30 b Fr(1)36 b Fs(is)g(su\016ciently)h(lar)-5 b(ge.)257 1345 y(Then,)34 b(for)h Ft(g)30 b(>)d Fv(0)35 b Fs(su\016ciently)g(smal)5 b(l,)34 b Ft(L)1808 1362 y Fq(g)1848 1345 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)g Ft(z)39 b Fs(is)c(b)-5 b(ounde)g(d)34 b(invertible.)257 1540 y(Pr)-5 b(o)g(of.)49 b Fv(W)-8 b(e)36 b(\014rst)h(observ)m(e)h (that)1505 1459 y Fm(\000)1550 1540 y Ft(L)1616 1558 y Fo(0)1656 1540 y Fv(\()p Ft(\022)s Fv(\))24 b Fr(\000)g Ft(ig)1988 1501 y Fo(2)2027 1540 y Fv(\000)2088 1558 y Fo(0)2127 1459 y Fm(\001)2173 1540 y Ft(P)2236 1558 y Fo(0)2311 1540 y Fv(is)37 b(a)f(normal)g(op)s(erator.)53 b(Since)257 1660 y(\000)318 1678 y Fo(0)385 1660 y Fr(\025)27 b Fv(0,)32 b(w)m(e)i(ha)m(v)m(e)g(that)e Ft(\033)t Fv([)p Ft(L)1328 1678 y Fq(f)1374 1660 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(ig)1701 1621 y Fo(2)1740 1660 y Fv(\000)1801 1678 y Fo(0)1841 1660 y Fv(])27 b Fr(\022)g Ft(\033)t Fv([)p Ft(L)2150 1678 y Fq(f)2196 1660 y Fv(\()p Ft(\022)s Fv(\)])g(=)g Fr(C)6 b Fv(\()p Ft(#)p Fv(\).)44 b(Hence)34 b(w)m(e)f(obtain)284 1799 y Fm(\015)284 1859 y(\015)284 1918 y(\015)340 1833 y(\000)385 1913 y Ft(L)451 1931 y Fo(0)491 1913 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)27 b Fr(\000)21 b Ft(ig)987 1870 y Fo(2)1026 1913 y Fv(\000)1087 1931 y Fo(0)1127 1833 y Fm(\001)1172 1855 y Fn(\000)p Fo(1)1283 1913 y Ft(P)1346 1931 y Fo(0)1385 1799 y Fm(\015)1385 1859 y(\015)1385 1918 y(\015)1500 1913 y Fr(\024)60 b Fv(dist)1794 1833 y Fm(\010)1852 1913 y Ft(z)37 b(;)49 b Fr(C)6 b Fv(\()p Ft(#)p Fv(\))2200 1833 y Fm(\011)2259 1855 y Fn(\000)p Fo(1)2412 1913 y Fr(\024)2548 1833 y Fm(\000)2594 1913 y Ft(C)2664 1931 y Fo(0)2735 1913 y Fv(sin)q(\()p Ft(#=)p Fv(2\))16 b Ft(g)3153 1870 y Fo(2+)p Fq(")3279 1833 y Fm(\001)3325 1855 y Fn(\000)p Fo(1)3441 1913 y Ft(:)3173 2071 y Fv(\(V.176\))257 2289 y(Inserting)32 b(this)f(estimate)h(in)m(to)e(a)h(Neumann)g(series) h(expansion)g(and)f(using)g(\(the)g(ana-)257 2410 y(logue)i(of)7 b(\))32 b(\(V.155\),)g(w)m(e)i(arriv)m(e)f(at)f(the)h(assertion,)507 2621 y Fm(\015)507 2681 y(\015)563 2706 y Fr(F)633 2723 y Fq(P)678 2732 y Fk(0)717 2621 y Fm(\015)717 2681 y(\015)831 2706 y Fr(\024)60 b(O)1048 2625 y Fm(\000)1093 2706 y Ft(g)1144 2662 y Fn(\000)p Fo(2)p Fn(\000)p Fq(")1325 2625 y Fm(\001)1445 2581 y Fn(1)1409 2611 y Fm(X)1414 2820 y Fq(n)p Fo(=0)1553 2565 y Fm(\022)1771 2634 y Fv(const)p 1636 2683 493 4 v 1636 2771 a Ft(C)1706 2789 y Fo(0)1778 2771 y Fv(sin)q(\()p Ft(#=)p Fv(2\))2138 2565 y Fm(\023)2212 2588 y Fq(n)2318 2706 y Fv(=)g Fr(O)2535 2625 y Fm(\000)2580 2706 y Ft(g)2631 2662 y Fn(\000)p Fo(2)p Fn(\000)p Fq(")2812 2625 y Fm(\001)2879 2706 y Ft(;)267 b Fv(\(V.177\))257 3007 y(pro)m(vided)34 b Ft(C)728 3025 y Fo(0)800 3007 y Fv(is)f(c)m(hosen)h(su\016cien)m(tly)i(large.)p 3425 3007 4 66 v 3429 2945 59 4 v 3429 3007 V 3487 3007 4 66 v 257 3417 a Fh(V.7)132 b(Renormalization)49 b(Group)f(Study)h(of)f (the)g(Sp)t(ectrum)g(of)592 3566 y Fg(L)670 3590 y Fc(g)716 3566 y FF(\()p Fg(\022)s FF(\))d Fh(in)g Ff(S)1128 3590 y Fx(0)p Fc(;<)257 3761 y Fv(Ha)m(ving)31 b(dealt)f(with)g(the)h(sp)s (ectral)f(parameters)h(of)e(magnitude)i(larger)f(than)f Ft(C)3235 3778 y Fo(0)3291 3761 y Ft(g)3342 3721 y Fo(2+)p Fq(")3468 3761 y Fv(,)257 3881 y(w)m(e)45 b(shall)f(henceforth)g (assume)h(that)f Ft(z)50 b Fr(2)45 b(S)1976 3899 y Fo(0)p Fq(;<)2090 3881 y Fv(,)h(i.e.,)h(that)c Fr(j)p Ft(z)t Fr(j)j(\024)f Fv(sin)q(\()p Ft(#)p Fv(\))16 b Ft(g)3173 3842 y Fo(2+)p Fq("=)p Fo(2)3371 3881 y Ft(=)p Fv(2.)257 4001 y(The)29 b(analysis)f(of)f(the)h(sp)s(ectrum)h(of)e Ft(L)1676 4019 y Fq(g)1716 4001 y Fv(\()p Ft(\022)s Fv(\))g(in)h Fr(S)2035 4019 y Fo(0)p Fq(;<)2176 4001 y Fv(is)g(the)g(most)g(in)m(v)m (olv)m(ed)h(part)e(of)g(our)257 4122 y(analysis,)36 b(as)e(it)f (requires)j(an)e(application)g(of)f(the)h(renormalization)g (transformation)257 4242 y(dev)m(elop)s(ed)h(in)e([7,)f(8].)404 4363 y(T)-8 b(o)27 b(apply)h(the)g(renormalization)g(group)f(map,)i(it) f(is)g(necessary)h(to)e(con)m(v)m(ert)i Fr(F)3308 4380 y Fq(P)3353 4389 y Fk(0)3419 4363 y Fr(\021)257 4483 y(F)327 4500 y Fq(P)372 4509 y Fk(0)411 4483 y Fv(\()p Ft(L)515 4500 y Fq(g)556 4483 y Fv(\()p Ft(\022)s Fv(\))23 b Fr(\000)h Ft(z)t Fv(\))36 b(in)m(to)g(a)g(normal-ordered)g(form.)53 b(More)36 b(precisely)-8 b(,)39 b(w)m(e)e(expand)f Fr(F)3411 4500 y Fq(P)3456 4509 y Fk(0)257 4603 y Fv(in)d(a)f(Neumann)i(series,) 284 4894 y Fr(F)354 4911 y Fq(P)399 4920 y Fk(0)497 4894 y Fv(=)60 b(\()p Ft(L)737 4911 y Fo(0)776 4894 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)g Ft(z)t Fv(\))p Ft(P)1170 4911 y Fo(0)1253 4894 y Fv(+)1409 4769 y Fn(1)1372 4799 y Fm(X)1380 5008 y Fq(\027)t Fo(=1)1516 4894 y Fv(\()p Fr(\000)p Fv(1\))1717 4850 y Fq(\027)t Fn(\000)p Fo(1)1866 4894 y Ft(g)1917 4850 y Fq(\027)1976 4894 y Ft(P)2039 4911 y Fo(0)2095 4894 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))2341 4753 y Fm(\022)p 2565 4742 77 4 v 2565 4822 a Ft(P)2642 4840 y Fo(0)p 2424 4871 399 4 v 2424 4959 a Ft(L)2490 4977 y Fo(0)2530 4959 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)2849 4894 y(W)14 b Fv(\()p Ft(\022)s Fv(\))i Ft(P)3158 4911 y Fo(0)3197 4753 y Fm(\023)3270 4776 y Fq(\027)t Fn(\000)p Fo(1)3425 4894 y Ft(;)3173 5111 y Fv(\(V.178\))257 5324 y(whic)m(h)34 b(is)g(norm-con)m(v)m(ergen)m(t,)g (as)f(w)m(e)h(ha)m(v)m(e)f(seen)h(in)f(the)g(previous)h(subsection.)p eop %%Page: 77 78 77 77 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(77)404 573 y(T)-8 b(o)39 b(con)m(v)m(ert)j Fr(F)976 590 y Fq(P)1021 599 y Fk(0)1099 573 y Fv(in)m(to)e(its)g (normal-ordered)g(form.)64 b(it)40 b(is)g(con)m(v)m(enien)m(t)i(to)d (adopt)257 693 y(the)33 b(follo)m(wing)g(notation.)43 b(W)-8 b(e)33 b(henceforth)h(denote)1180 913 y Ft(k)86 b Fv(:=)d(\()1536 887 y Ft(~)1541 913 y(k)s(;)16 b(\025;)g(\034)11 b Fv(\))87 b Fr(2)g Fp(R)2134 870 y Fo(3)2201 913 y Fr(\002)22 b(f)p Fv(1)p Ft(;)16 b Fv(2)p Fr(g)21 b(\002)g(f)p Ft(`;)16 b(r)s Fr(g)22 b Ft(;)239 b Fv(\(V.179\))803 971 y Fm(Z)918 1107 y Ft(dk)25 b(f)11 b Fv(\()p Ft(k)s Fv(\))83 b(:=)1524 1012 y Fm(X)1503 1224 y Fq(\025)p Fo(=1)p Fq(;)p Fo(2)1721 1012 y Fm(X)1705 1224 y Fq(\034)8 b Fo(=)p Fq(`;r)1897 971 y Fm(Z)1953 1197 y Fl(R)2001 1178 y Fk(3)2056 1107 y Ft(d)2107 1064 y Fo(3)2141 1081 y Ft(~)2146 1107 y(k)25 b(f)11 b Fv(\()2314 1081 y Ft(~)2319 1107 y(k)r(;)16 b(\025;)g(\034)11 b Fv(\))23 b Ft(;)517 b Fv(\(V.180\))257 1424 y(and)767 1644 y Ft(a)818 1605 y Fo(+)878 1644 y Fv(\()911 1618 y Ft(~)916 1644 y(k)s(;)16 b(\025;)g(\034)11 b Fv(\))60 b(:=)g Ft(a)1478 1605 y Fn(\003)1478 1665 y Fq(\034)1521 1644 y Fv(\()1554 1618 y Ft(~)1559 1644 y(k)s(;)16 b(\025)p Fv(\))55 b Ft(;)87 b(a)1971 1605 y Fn(\000)2030 1644 y Fv(\()2063 1618 y Ft(~)2068 1644 y(k)s(;)16 b(\025;)g(\034)11 b Fv(\))61 b(:=)e Ft(a)2630 1662 y Fq(\034)2674 1644 y Fv(\()2707 1618 y Ft(~)2712 1644 y(k)s(;)16 b(\025)p Fv(\))54 b Ft(;)188 b Fv(\(V.181\))1285 1803 y Ft(w)1358 1752 y Fo(\()p Fq(\022)r Fo(\))1355 1825 y Fn(\006)1452 1803 y Fv(\()1485 1777 y Ft(~)1490 1803 y(k)s(;)16 b(\025;)g(\034)11 b Fv(\))60 b(:=)g Ft(w)2074 1752 y Fo(\()p Fq(\022)r Fo(\))2071 1825 y Fn(\006)p Fq(;\034)2188 1803 y Fv(\()2221 1777 y Ft(~)2226 1803 y(k)s(;)16 b(\025)p Fv(\))22 b Ft(:)706 b Fv(\(V.182\))257 2043 y(F)-8 b(urthermore,)39 b(w)m(e)f(write)f Ft(!)t Fv(\()p Ft(k)s Fv(\))d(:=)g Fr(j)1644 2017 y Ft(~)1649 2043 y(k)t Fr(j)p Fv(,)k(etc.)57 b(In)37 b(this)h(new)f(notation)g(the) g(op)s(erators)257 2164 y(to)c(deal)g(with)g(app)s(ear)f(in)h(a)f(more) h(compact)g(form)g(as)1227 2384 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))60 b(=)f Ft(a)1703 2344 y Fo(+)1762 2384 y Fv(\()p Ft(w)1873 2333 y Fo(\()p Fq(\022)r Fo(\))1870 2405 y(+)1967 2384 y Fv(\))21 b(+)h Ft(a)2175 2344 y Fn(\000)2234 2384 y Fv(\()p Ft(w)2345 2333 y Fo(\()p Fq(\022)r Fo(\))2342 2405 y Fn(\000)2438 2384 y Fv(\))g Ft(;)648 b Fv(\(V.183\))372 2529 y Ft(L)438 2546 y Fq(aux)621 2529 y Fv(=)756 2449 y Fm(R)839 2529 y Ft(dk)24 b(!)t Fv(\()p Ft(k)s Fv(\))16 b Ft(a)1227 2490 y Fo(+)1286 2529 y Fv(\()p Ft(k)s Fv(\))p Ft(a)1467 2490 y Fn(\000)1526 2529 y Fv(\()p Ft(k)s Fv(\))54 b Ft(;)82 b(L)1885 2546 y Fq(f)1990 2529 y Fv(=)2125 2449 y Fm(R)2208 2529 y Ft(dk)25 b(\034)i(!)t Fv(\()p Ft(k)s Fv(\))16 b Ft(a)2666 2490 y Fo(+)2725 2529 y Fv(\()p Ft(k)s Fv(\))p Ft(a)2906 2490 y Fn(\000)2965 2529 y Fv(\()p Ft(k)s Fv(\))22 b Ft(:)29 b Fv(\(V.184\))257 2749 y(Th)m(us)34 b(the)f(term)h(in)e(\(V.178\))g(of)g(order)h Ft(g)1791 2710 y Fq(\027)1866 2749 y Fv(can)g(b)s(e)g(written)g(as)436 2947 y Fm(X)321 3156 y Fq(\033)361 3165 y Fk(1)396 3156 y Fq(;:::)10 b(;\033)546 3164 y Fj(\027)584 3156 y Fo(=)p Fn(\006)711 3041 y Ft(P)774 3059 y Fo(0)829 3041 y Ft(a)880 2998 y Fq(\033)920 3007 y Fk(1)959 3041 y Fv(\()p Ft(w)1070 2998 y Fo(\()p Fq(\022)r Fo(\))1067 3062 y Fq(\033)1107 3071 y Fk(1)1164 3041 y Fv(\))1218 2901 y Fm(\022)p 1443 2890 77 4 v 1443 2970 a Ft(P)1519 2987 y Fo(0)p 1301 3018 399 4 v 1301 3106 a Ft(L)1367 3124 y Fo(0)1407 3106 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)1710 2901 y Fm(\023)1827 3041 y Fr(\001)16 b(\001)g(\001)1984 2901 y Fm(\022)p 2209 2890 77 4 v 2209 2970 a Ft(P)2285 2987 y Fo(0)p 2067 3018 399 4 v 2067 3106 a Ft(L)2133 3124 y Fo(0)2173 3106 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)2476 2901 y Fm(\023)2565 3041 y Ft(a)2616 2998 y Fq(\033)2656 3006 y Fj(\027)2700 3041 y Fv(\()p Ft(w)2811 2998 y Fo(\()p Fq(\022)r Fo(\))2808 3062 y Fq(\033)2848 3070 y Fj(\027)2904 3041 y Fv(\))16 b Ft(P)3021 3059 y Fo(0)3082 3041 y Ft(:)64 b Fv(\(V.185\))404 3356 y(F)-8 b(or)42 b(future)i(purp)s(ose,)j(w)m(e)e(in)m(tro)s(duce)f(some)g(more) g(notation.)76 b(W)-8 b(e)43 b(collect)i(the)257 3476 y(eigen)m(v)-5 b(alues)37 b(of)d Ft(L)943 3494 y Fq(el)1037 3476 y Fv(in)g(a)h(set)g Fr(f)p Ft(\021)1487 3494 y Fo(0)1526 3476 y Ft(;)16 b(\021)1617 3494 y Fo(1)1657 3476 y Ft(;)g Fr(\001)g(\001)g(\001)34 b Ft(;)16 b(\021)1938 3494 y Fq(M)2017 3476 y Fr(g)31 b Fv(=)f Fr(f)p Ft(E)2324 3494 y Fq(i;j)2404 3476 y Fr(j)p Fv(0)g Fr(\024)h Ft(i;)16 b(j)37 b Fr(\024)30 b Ft(N)j Fr(\000)23 b Fv(1)p Fr(g)p Fv(,)35 b(where)257 3596 y Ft(\021)305 3614 y Fo(0)372 3596 y Fv(:=)27 b(0)32 b(and)h Ft(M)38 b Fr(\024)27 b Ft(N)10 b Fv(\()p Ft(N)33 b Fr(\000)21 b Fv(1\).)43 b(W)-8 b(e)33 b(then)g(in)m(tro)s(duce)826 3830 y Ft(\037)887 3779 y Fo(\()p Fq(\013)p Fo(\))887 3858 y Fq(el)1075 3830 y Fv(:=)83 b Ft(P)1338 3787 y Fq(el)1324 3851 y(\021)1359 3859 y Fj(\013)1494 3830 y Fv(=)k Ft(\037)1718 3848 y Fn(f)p Fq(\021)1788 3856 y Fj(\013)1833 3848 y Fn(g)1872 3830 y Fv([)p Ft(L)1965 3848 y Fq(el)2024 3830 y Fv(])22 b Ft(;)1073 b Fv(\(V.186\))698 4043 y(\026)-61 b Ft(\037)747 3992 y Fo(\()p Fq(\013)p Fo(\))747 4071 y Fq(f)852 4043 y Fv(\()p Ft(!)t Fv(\))82 b(:=)1261 3903 y Fm(\032)1377 3983 y Fi(1)1433 3944 y Fq(f)2167 3983 y Fv(for)32 b Ft(\013)c Fv(=)f(1)p Ft(;)16 b Fv(2)p Ft(;)g(:)g(:)g(:)34 b(;)16 b(M)10 b Fv(,)1377 4103 y Ft(\037)p Fv([)p Ft(L)1531 4121 y Fq(aux)1676 4103 y Fv(+)21 b Ft(!)31 b(<)26 b(\032)2017 4121 y Fo(0)2057 4103 y Fv(])83 b(for)32 b Ft(\013)c Fv(=)f(0,)3173 4043 y(\(V.187\))257 4316 y(and)33 b(w)m(e)h(observ)m(e) g(that)p 1318 4524 77 4 v 1318 4604 a Ft(P)1395 4622 y Fo(0)1494 4604 y Fv(=)1664 4480 y Fq(M)1629 4510 y Fm(X)1633 4719 y Fq(\013)p Fo(=0)1811 4604 y Ft(\037)1872 4553 y Fo(\()p Fq(\013)p Fo(\))1872 4632 y Fq(el)1998 4604 y Fr(\012)f Fv(\026)-60 b Ft(\037)2157 4553 y Fo(\()p Fq(\013)p Fo(\))2157 4632 y Fq(f)2261 4604 y Fv(\(0\))21 b Ft(:)739 b Fv(\(V.188\))p eop %%Page: 78 79 78 78 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(78)257 573 y(F)-8 b(or)32 b Ft(k)483 590 y Fo(1)522 573 y Ft(;)16 b(:)g(:)g(:)34 b(;)16 b(k)806 590 y Fq(m)873 573 y Ft(;)919 547 y Fv(~)916 573 y Ft(k)967 590 y Fo(1)1006 573 y Ft(;)g(:)g(:)g(:)34 b(;)1242 547 y Fv(~)1239 573 y Ft(k)1290 590 y Fq(n)1364 573 y Fr(2)27 b Fp(R)1522 534 y Fo(3)1589 573 y Fr(\002)22 b(f)p Fv(1)p Ft(;)16 b Fv(2)p Fr(g)21 b(\002)g(f)p Ft(`;)16 b(r)s Fr(g)p Fv(,)33 b(w)m(e)g(further)g(denote)846 741 y Ft(k)900 697 y Fo(\()p Fq(m)p Fo(\))1071 741 y Fv(:=)48 b(\()p Ft(k)1311 758 y Fo(1)1350 741 y Ft(;)16 b(:)g(:)g(:)34 b(;)16 b(k)1634 758 y Fq(m)1701 741 y Fv(\))21 b Ft(;)1956 714 y Fv(~)1954 741 y Ft(k)2008 697 y Fo(\()p Fq(n)p Fo(\))2158 741 y Fv(:=)49 b(\()2351 714 y(~)2348 741 y Ft(k)2399 758 y Fo(1)2438 741 y Ft(;)16 b(:)g(:)g(:)34 b(;)2674 714 y Fv(~)2671 741 y Ft(k)2722 758 y Fq(n)2769 741 y Fv(\))21 b Ft(;)318 b Fv(\(V.189\))1031 886 y Ft(L)p 1031 902 67 4 v 60 w Fv(:=)59 b(\()p Ft(L)1423 904 y Fq(f)1469 886 y Ft(;)16 b(L)1578 904 y Fq(aux)1701 886 y Fv(\))21 b Ft(;)167 b(r)p 1954 902 47 4 v 62 w Fv(:=)59 b(\()p Ft(r)2304 904 y Fq(f)2350 886 y Ft(;)16 b(r)2437 904 y Fq(aux)2559 886 y Fv(\))22 b Ft(;)527 b Fv(\(V.190\))738 1113 y Ft(a)789 1069 y Fo(+)848 1113 y Fv(\()p Ft(k)940 1069 y Fo(\()p Fq(m)p Fo(\))1062 1113 y Fv(\))59 b(:=)1354 988 y Fq(m)1322 1018 y Fm(Y)1324 1228 y Fq(j)t Fo(=1)1465 1113 y Ft(a)1516 1069 y Fo(+)1576 1113 y Fv(\()p Ft(k)1665 1130 y Fq(j)1701 1113 y Fv(\))21 b Ft(;)167 b(a)2005 1069 y Fn(\000)2064 1113 y Fv(\()2105 1086 y(~)2102 1113 y Ft(k)2156 1069 y Fo(\()p Fq(n)p Fo(\))2258 1113 y Fv(\))59 b(:=)2560 988 y Fq(n)2518 1018 y Fm(Y)2520 1228 y Fq(j)t Fo(=1)2661 1113 y Ft(a)2712 1069 y Fn(\000)2772 1113 y Fv(\()2813 1086 y(~)2810 1113 y Ft(k)2861 1130 y Fq(j)2897 1113 y Fv(\))21 b Ft(;)190 b Fv(\(V.191\))1009 1447 y Ft(dk)1114 1404 y Fo(\()p Fq(m)p Fo(\))1295 1447 y Fv(:=)1490 1323 y Fq(m)1457 1353 y Fm(Y)1460 1563 y Fq(j)t Fo(=1)1601 1447 y Ft(dk)1703 1465 y Fq(j)1760 1447 y Ft(;)167 b(d)2008 1421 y Fv(~)2005 1447 y Ft(k)2059 1404 y Fo(\()p Fq(n)p Fo(\))2220 1447 y Fv(:=)2425 1323 y Fq(n)2382 1353 y Fm(Y)2385 1563 y Fq(j)t Fo(=1)2526 1447 y Ft(d)2580 1421 y Fv(~)2577 1447 y Ft(k)2628 1465 y Fq(j)2686 1447 y Ft(;)460 b Fv(\(V.192\))776 1712 y Ft(K)866 1668 y Fo(\()p Fq(m;n)p Fo(\))1109 1712 y Fv(:=)60 b(\()p Ft(k)1364 1668 y Fo(\()p Fq(m)p Fo(\))1485 1712 y Ft(;)1548 1686 y Fv(~)1545 1712 y Ft(k)1599 1668 y Fo(\()p Fq(n)p Fo(\))1701 1712 y Fv(\))21 b Ft(;)167 b(dK)2095 1668 y Fo(\()p Fq(m;n)p Fo(\))2338 1712 y Fv(:=)59 b Ft(dk)2605 1668 y Fo(\()p Fq(m)p Fo(\))2743 1712 y Ft(d)2797 1686 y Fv(~)2794 1712 y Ft(k)2848 1668 y Fo(\()p Fq(n)p Fo(\))2971 1712 y Ft(;)175 b Fv(\(V.193\))813 1933 y Ft(!)t Fv(\()p Ft(k)970 1890 y Fo(\()p Fq(m)p Fo(\))1091 1933 y Fv(\))60 b(:=)1392 1809 y Fq(m)1351 1838 y Fm(X)1362 2048 y Fq(j)t Fo(=1)1511 1933 y Ft(!)t Fv(\()p Ft(k)1665 1951 y Fq(j)1701 1933 y Fv(\))21 b Ft(;)167 b(\037)2015 1951 y Fq(\032)2055 1933 y Fv([)p Ft(!)t Fv(])27 b(:=)g Ft(\037)2392 1951 y Fo([0)p Fq(;\032)p Fo(\))2534 1933 y Fv([)p Ft(!)t Fv(])21 b Ft(:)472 b Fv(\(V.194\))257 2202 y(Equipp)s(ed)35 b(with)e(this)g(notation,)f(w)m(e)i(rewrite)g(\(V.185\))e(as)487 2323 y Fm(X)386 2521 y Fj(\033)422 2536 y Fk(1)457 2521 y Fj(;:::)10 b(;\033)598 2529 y(\027)636 2521 y Fk(=)p Fe(\006)267 2579 y Fj(\013)307 2594 y Fk(1)342 2579 y Fj(;:::)g(;\013)487 2594 y(\027)s Fe(\000)p Fk(1)604 2579 y(=0)p Fj(;:::)g(;M)876 2282 y Fm(Z)992 2418 y Ft(dk)1097 2375 y Fo(\()p Fq(\027)t Fo(\))1195 2418 y Ft(\037)1256 2367 y Fo(\(0\))1256 2446 y Fq(el)1350 2418 y Ft(w)1423 2375 y Fo(\()p Fq(\022)r Fo(\))1420 2439 y Fq(\033)1460 2448 y Fk(1)1516 2418 y Fv(\()p Ft(k)1605 2436 y Fo(1)1644 2418 y Fv(\))p Ft(\037)1743 2367 y Fo(\()p Fq(\013)1815 2376 y Fk(1)1851 2367 y Fo(\))1743 2446 y Fq(el)1926 2418 y Fr(\001)16 b(\001)g(\001)44 b Ft(\037)2144 2367 y Fo(\()p Fq(\013)2216 2376 y Fj(\027)s Fe(\000)p Fk(1)2334 2367 y Fo(\))2144 2446 y Fq(el)2365 2418 y Ft(w)2438 2375 y Fo(\()p Fq(\022)r Fo(\))2435 2439 y Fq(\033)2475 2447 y Fj(\027)2532 2418 y Fv(\()p Ft(k)2621 2436 y Fq(\027)2664 2418 y Fv(\))p Ft(\037)2763 2367 y Fo(\(0\))2763 2446 y Fq(el)2857 2418 y Fr(\012)258 b Fv(\(V.195\))423 2825 y Ft(\037)484 2842 y Fq(\032)520 2851 y Fk(0)559 2825 y Fv([)p Ft(L)652 2842 y Fq(aux)775 2825 y Fv(])16 b Ft(a)869 2781 y Fq(\033)909 2790 y Fk(1)949 2825 y Fv(\()p Ft(k)1038 2842 y Fo(1)1077 2825 y Fv(\))1318 2745 y(\026)-60 b Ft(\037)1368 2694 y Fo(\()p Fq(\013)1440 2703 y Fk(1)1475 2694 y Fo(\))1368 2773 y Fq(f)1507 2745 y Fv(\(0\))p 1141 2802 657 4 v 1141 2890 a Ft(\021)1189 2907 y Fq(\013)1234 2916 y Fk(1)1295 2890 y Fv(+)21 b Ft(L)1458 2907 y Fq(f)1504 2890 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)1851 2825 y Fr(\001)16 b(\001)g(\001)2195 2745 y Fv(\026)-61 b Ft(\037)2244 2694 y Fo(\()p Fq(\013)2316 2703 y Fj(\027)s Fe(\000)p Fk(1)2434 2694 y Fo(\))2244 2773 y Fq(f)2466 2745 y Fv(\(0\))p 2018 2802 739 4 v 2018 2890 a Ft(\021)2066 2907 y Fq(\013)2111 2916 y Fj(\027)s Fe(\000)p Fk(1)2254 2890 y Fv(+)21 b Ft(L)2417 2907 y Fq(f)2463 2890 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)2766 2825 y(a)2817 2781 y Fq(\033)2857 2789 y Fj(\027)2900 2825 y Fv(\()p Ft(k)2989 2842 y Fq(\027)3032 2825 y Fv(\))16 b Ft(\037)3147 2842 y Fq(\032)3183 2851 y Fk(0)3222 2825 y Fv([)p Ft(L)3315 2842 y Fq(aux)3438 2825 y Fv(])22 b Ft(:)257 3048 y Fv(No)m(w)33 b(w)m(e)f(normal-order)g(the)g(pro)s(duct)g(of)f(creation-)h(and)f (annihilation)h(op)s(erators)g(in)257 3169 y(the)c(second)g(line)g(of)e (\(V.195\).)41 b(By)28 b(\(a)f(t)m(w)m(o-comp)s(onen)m(t)h(v)-5 b(arian)m(t)27 b(of)7 b(\))26 b([8,)i(Lemma)g(A.3],)257 3289 y(w)m(e)34 b(ha)m(v)m(e,)g(for)e(arbitrary)h(functions)g Ft(f)1685 3307 y Fo(1)1725 3289 y Ft(;)16 b(f)1816 3307 y Fo(2)1856 3289 y Ft(;)g(:)g(:)g(:)33 b(;)16 b(f)2136 3307 y Fq(\027)t Fn(\000)p Fo(1)2270 3289 y Fv(,)383 3457 y Ft(a)434 3414 y Fq(\033)474 3423 y Fk(1)513 3457 y Fv(\()p Ft(k)602 3475 y Fo(1)641 3457 y Fv(\))g Ft(f)743 3475 y Fq(\013)788 3484 y Fk(1)827 3457 y Fv([)p Ft(L)920 3475 y Fq(f)966 3457 y Ft(;)g(L)1075 3475 y Fq(aux)1198 3457 y Fv(])38 b Fr(\001)16 b(\001)g(\001)39 b Ft(f)1463 3475 y Fq(\013)1508 3484 y Fj(\027)s Fe(\000)p Fk(1)1629 3457 y Fv([)p Ft(L)1722 3475 y Fq(f)1768 3457 y Ft(;)16 b(L)1877 3475 y Fq(aux)2000 3457 y Fv(])g Ft(a)2094 3414 y Fq(\033)2134 3422 y Fj(\027)2178 3457 y Fv(\()p Ft(k)2267 3475 y Fq(\027)2310 3457 y Fv(\))825 b(\(V.196\))466 3687 y(=)643 3592 y Fm(X)625 3804 y Fn(Q\022N)851 3592 y Fm(Y)821 3804 y Fq(j)t Fn(2Q)959 3813 y Fk(+)1025 3687 y Ft(a)1076 3644 y Fo(+)1135 3687 y Fv(\()p Ft(k)1224 3705 y Fq(j)1260 3687 y Fv(\))1298 3547 y Fm(\034)1373 3687 y Fv(\012)1443 3543 y Fm(\014)1443 3602 y(\014)1443 3662 y(\014)1443 3722 y(\014)1521 3563 y Fq(\027)1477 3592 y Fm(Y)1479 3802 y Fq(j)t Fo(=1)1604 3547 y Fm(\032)1679 3687 y Fv([)p Ft(a)1757 3644 y Fq(\033)1797 3654 y Fj(j)1834 3687 y Fv(\()p Ft(k)1923 3705 y Fq(j)1959 3687 y Fv(\)])2024 3644 y Fq(\037)p Fo([)p Fq(j)11 b(=)-42 b Fn(2)o(Q)p Fo(])625 4045 y Ft(f)673 4063 y Fq(\013)718 4073 y Fj(j)755 3905 y Fm(\024)807 4045 y Ft(L)873 4063 y Fq(f)941 4045 y Fv(+)21 b Ft(r)1082 4063 y Fq(f)1149 4045 y Fv(+)1324 3917 y Fq(j)1268 3951 y Fm(X)1290 4150 y Fj(i)p Fk(=1)1257 4206 y Fj(i)p Fe(2Q)1371 4221 y(\000)1433 4045 y Fv(\()p Fr(\000)p Fv(1\))1634 4002 y Fq(\034)1665 4012 y Fj(j)1701 4045 y Ft(!)t Fv(\()p Ft(k)1855 4063 y Fq(i)1882 4045 y Fv(\))h(+)2154 3921 y Fq(\027)2102 3951 y Fm(X)2124 4150 y Fj(i)p Fk(=1)2050 4206 y Fj(j)s Fk(+1)p Fe(2Q)2247 4221 y Fk(+)2307 4045 y Fv(\()p Fr(\000)p Fv(1\))2508 4002 y Fq(\034)2539 4012 y Fj(j)2576 4045 y Ft(!)t Fv(\()p Ft(k)2730 4063 y Fq(i)2757 4045 y Fv(\))49 b Ft(;)743 4453 y(L)809 4471 y Fq(aux)953 4453 y Fv(+)21 b Ft(r)1094 4471 y Fq(aux)1238 4453 y Fv(+)1413 4325 y Fq(j)1357 4359 y Fm(X)1379 4558 y Fj(i)p Fk(=1)1346 4614 y Fj(i)p Fe(2Q)1460 4629 y(\000)1538 4453 y Ft(!)t Fv(\()p Ft(k)1692 4471 y Fq(i)1719 4453 y Fv(\))h(+)1991 4329 y Fq(\027)1938 4359 y Fm(X)1961 4558 y Fj(i)p Fk(=1)1886 4614 y Fj(j)s Fk(+1)p Fe(2Q)2083 4629 y Fk(+)2161 4453 y Ft(!)t Fv(\()p Ft(k)2315 4471 y Fq(i)2342 4453 y Fv(\))2380 4313 y Fm(\025)o(\033)2507 4453 y Fv(\012)2577 4313 y Fm(\035)2652 4309 y(\014)2652 4369 y(\014)2652 4428 y(\014)2652 4488 y(\014)2685 4552 y Fq(r)p 2685 4564 34 3 v 2 w Fo(=)p Fq(L)p 2774 4564 48 3 v 2873 4359 a Fm(Y)2842 4570 y Fq(j)t Fn(2Q)2980 4579 y Fe(\000)3048 4453 y Ft(a)3099 4410 y Fo(+)3158 4453 y Fv(\()p Ft(k)3247 4471 y Fq(j)3283 4453 y Fv(\))g Ft(;)257 4791 y Fv(where)46 b Fr(N)62 b Fv(:=)47 b Fr(f)p Fv(1)p Ft(;)16 b Fv(2)p Ft(;)g(:)g(:)g(:)33 b(;)16 b(\027)6 b Fr(g)p Fv(,)48 b Fr(Q)1523 4808 y Fn(\006)1629 4791 y Fv(:=)f Fr(f)p Ft(j)53 b Fr(2)47 b(Qj)p Ft(\033)2195 4808 y Fq(j)2279 4791 y Fv(=)g Fr(\006g)p Fv(,)g(and)e([)p Ft(a)2881 4751 y Fq(\033)2921 4761 y Fj(j)2958 4791 y Fv(\()p Ft(k)3047 4808 y Fq(j)3083 4791 y Fv(\)])3148 4751 y Fq(\037)p Fo([)p Fq(j)11 b(=)-42 b Fn(2)o(Q)p Fo(])3419 4791 y Fv(=)257 4911 y Ft(a)308 4872 y Fq(\033)348 4882 y Fj(j)386 4911 y Fv(\()p Ft(k)475 4929 y Fq(j)511 4911 y Fv(\),)35 b(for)f Ft(j)47 b(=)-60 b Fr(2)30 b(Q)p Fv(,)35 b(and)g([)p Ft(a)1345 4872 y Fq(\033)1385 4882 y Fj(j)1422 4911 y Fv(\()p Ft(k)1511 4929 y Fq(j)1547 4911 y Fv(\)])1612 4872 y Fq(\037)p Fo([)p Fq(j)11 b(=)-42 b Fn(2)o(Q)p Fo(])1867 4911 y Fv(=)30 b(1,)35 b(for)f Ft(j)i Fr(2)30 b(Q)p Fv(.)49 b(T)-8 b(o)35 b(apply)g(this)g(form)m(ula) 257 5031 y(to)e(the)g(second)g(line)h(of)e(\(V.195\),)g(w)m(e)h(c)m(ho) s(ose)681 5292 y Ft(f)729 5310 y Fq(\013)778 5292 y Fv([)p Ft(r)849 5310 y Fq(f)895 5292 y Ft(;)16 b(r)982 5310 y Fq(aux)1104 5292 y Fv(])60 b(:=)1835 5213 y(\026)-61 b Ft(\037)1884 5162 y Fo(\()p Fq(\013)p Fo(\))1884 5241 y Fq(f)1989 5213 y Fv(\()p Ft(r)2071 5230 y Fq(aux)2193 5213 y Fv(\))p 1364 5269 1328 4 v 1364 5358 a Ft(\021)1412 5375 y Fq(\013)1483 5358 y Fv(+)21 b(cos)q(\()p Ft(#)p Fv(\))p Ft(r)1888 5375 y Fq(f)1955 5358 y Fr(\000)h Ft(i)16 b Fv(sin)q(\()p Ft(#)p Fv(\))p Ft(r)2399 5375 y Fq(aux)2543 5358 y Fr(\000)22 b Ft(z)2722 5292 y(:)424 b Fv(\(V.197\))p eop %%Page: 79 80 79 79 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(79)257 573 y(Note)33 b(that,)g(similar)g(to)f (\(V.143\){\(V.148\),)f(w)m(e)j(ha)m(v)m(e)1060 778 y(sup)948 851 y Fk(0)p Fe(\024j)p Fj(r)1074 869 y(f)1114 851 y Fe(j\024)p Fj(r)1210 859 y(aux)991 922 y(\013)p Fk(=0)p Fj(;:::)10 b(;M)1329 667 y Fm(n)1411 697 y(\000)1457 778 y Ft(r)1501 796 y Fq(aux)1645 778 y Fv(+)22 b Ft(\032)1793 796 y Fo(0)1832 697 y Fm(\001)1894 693 y(\014)1894 753 y(\014)1927 778 y Ft(f)1975 796 y Fq(\013)2025 778 y Fv([)p Ft(r)2096 796 y Fq(f)2141 778 y Ft(;)16 b(r)2228 796 y Fq(aux)2351 778 y Fv(])2378 693 y Fm(\014)2378 753 y(\014)2411 667 y(o)2537 778 y Fr(\024)60 b Ft(C)28 b(;)375 b Fv(\(V.198\))257 1082 y(for)38 b(some)g Ft(C)43 b(<)35 b Fr(1)p Fv(.)58 b(Inserting)39 b(form)m(ula)f(\(V.196\))f(in)m (to)h(\(V.195\))f(and)h(summing)h(up)257 1202 y(the)46 b(con)m(tributions)h(to)e(all)g(orders)h(in)f Ft(g)t Fv(,)j(w)m(e)e(obtain)g(an)f(e\013ectiv)m(e)i(Liouvillian)f(on)257 1322 y(Ran)16 b Ft(P)511 1340 y Fo(0)551 1322 y Fv(,)591 1500 y Ft(L)657 1518 y Fo(\(0\))751 1500 y Fv([)p Ft(z)t Fv(])23 b Fr(\000)f Ft(z)87 b Fv(:=)c Fr(F)1363 1518 y Fq(P)1408 1527 y Fk(0)1447 1500 y Fv(\()p Ft(L)1551 1518 y Fq(g)1592 1500 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)t Fv(\))1251 b(\(V.199\))1121 1682 y(=)96 b Ft(P)1356 1700 y Fo(0)1412 1572 y Fm(\020)1488 1682 y Ft(E)1560 1700 y Fo(\(0\))1654 1682 y Fv([)p Ft(z)t Fv(])39 b Fr(\000)f Ft(z)43 b Fv(+)37 b Ft(T)2168 1700 y Fo(\(0\))2263 1682 y Fv([)p Ft(z)t(;)16 b(L)p 2382 1698 67 4 v 2 w Fv(])38 b(+)f Ft(W)2720 1700 y Fo(\(0\))2815 1682 y Fv([)p Ft(z)t Fv(])2934 1572 y Fm(\021)3011 1682 y Ft(P)3074 1700 y Fo(0)3135 1682 y Ft(;)257 1882 y Fv(where)605 2060 y Ft(E)677 2077 y Fo(\(0\))772 2060 y Fv([)p Ft(z)t Fv(])84 b(:=)f Ft(\037)1206 2009 y Fo(\(0\))1206 2087 y Fq(el)1322 2060 y Fr(\012)21 b Ft(P)1482 2077 y Fo(\012)1554 1949 y Fm(\020)1629 2060 y Fr(F)1699 2077 y Fq(P)1744 2086 y Fk(0)1783 2060 y Fv(\()p Ft(L)1887 2077 y Fq(g)1928 2060 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)h Ft(z)t Fv(\))38 b(+)g Ft(z)2476 1949 y Fm(\021)2552 2060 y Ft(\037)2613 2009 y Fo(\(0\))2613 2087 y Fq(el)2729 2060 y Fr(\012)22 b Ft(P)2890 2077 y Fo(\012)2966 2060 y Ft(;)180 b Fv(\(V.200\))972 2321 y(=)97 b Fr(\000)1273 2196 y Fn(1)1237 2226 y Fm(X)1244 2435 y Fq(\027)t Fo(=2)1381 2321 y Fv(\()p Fr(\000)p Ft(g)t Fv(\))1584 2277 y Fq(\027)1871 2226 y Fm(X)1771 2424 y Fj(\033)1807 2439 y Fk(1)1841 2424 y Fj(;:::)11 b(;\033)1983 2432 y(\027)2021 2424 y Fk(=)p Fe(\006)1652 2482 y Fj(\013)1692 2497 y Fk(1)1727 2482 y Fj(;:::)f(;\013)1872 2497 y(\027)s Fe(\000)p Fk(1)1989 2482 y(=0)p Fj(;:::)g(;M)1145 2547 y Fm(Z)1261 2682 y Ft(dk)1366 2639 y Fo(\()p Fq(\027)t Fo(\))1485 2682 y Ft(\037)1546 2632 y Fo(\(0\))1546 2710 y Fq(el)1641 2682 y Ft(w)1714 2639 y Fo(\()p Fq(\022)r Fo(\))1711 2703 y Fq(\033)1751 2712 y Fk(1)1807 2682 y Fv(\()p Ft(k)1896 2700 y Fo(1)1935 2682 y Fv(\))p Ft(\037)2034 2632 y Fo(\()p Fq(\013)2106 2641 y Fk(1)2141 2632 y Fo(\))2034 2710 y Fq(el)2216 2682 y Fr(\001)16 b(\001)g(\001)44 b Ft(\037)2434 2632 y Fo(\()p Fq(\013)2506 2641 y Fj(\027)s Fe(\000)p Fk(1)2624 2632 y Fo(\))2434 2710 y Fq(el)2656 2682 y Ft(w)2729 2639 y Fo(\()p Fq(\022)r Fo(\))2726 2703 y Fq(\033)2766 2711 y Fj(\027)2822 2682 y Fv(\()p Ft(k)2911 2700 y Fq(\027)2954 2682 y Fv(\))p Ft(\037)3053 2632 y Fo(\(0\))3053 2710 y Fq(el)1145 2916 y Fr(\002)1221 2805 y Fm(D)1281 2916 y Fv(\012)1351 2801 y Fm(\014)1351 2861 y(\014)1351 2921 y(\014)1412 2916 y Ft(a)1463 2873 y Fq(\033)1503 2882 y Fk(1)1542 2916 y Fv(\()p Ft(k)1631 2934 y Fo(1)1671 2916 y Fv(\))16 b Ft(f)1773 2934 y Fq(\013)1818 2943 y Fk(1)1857 2916 y Fv([)p Ft(L)p 1884 2932 V Fv(])44 b Fr(\001)16 b(\001)g(\001)44 b Ft(f)2226 2934 y Fq(\013)2271 2943 y Fj(\027)s Fe(\000)p Fk(1)2392 2916 y Fv([)p Ft(L)p 2419 2932 V 1 w Fv(])16 b Ft(a)2580 2873 y Fq(\033)2620 2881 y Fj(\027)2663 2916 y Fv(\()p Ft(k)2752 2934 y Fq(\027)2795 2916 y Fv(\))27 b(\012)2930 2805 y Fm(E)3013 2916 y Ft(;)383 3179 y(T)440 3196 y Fo(\(0\))535 3179 y Fv([)p Ft(z)t(;)16 b(r)p 654 3195 47 4 v 3 w Fv(])466 3361 y(=)83 b Ft(\037)686 3310 y Fo(\(0\))686 3388 y Fq(el)802 3361 y Fr(\012)22 b Ft(P)963 3378 y Fo(\012)1034 3250 y Fm(\020)1093 3361 y Fr(F)1163 3378 y Fq(P)1208 3387 y Fk(0)1247 3361 y Fv(\()p Ft(r)1329 3378 y Fq(f)1374 3361 y Fv(\()p Ft(\022)s Fv(\))g(+)g Ft(L)1684 3378 y Fq(g)1724 3361 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)f Ft(z)t Fv(\))39 b Fr(\000)f(F)2277 3378 y Fq(P)2322 3387 y Fk(0)2361 3361 y Fv(\()p Ft(L)2465 3378 y Fq(g)2505 3361 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)t Fv(\))2835 3250 y Fm(\021)2912 3361 y Ft(\037)2973 3310 y Fo(\(0\))2973 3388 y Fq(el)3089 3361 y Fr(\012)g Ft(P)3249 3378 y Fo(\012)3326 3361 y Ft(;)466 3622 y Fv(=)83 b Ft(r)669 3639 y Fq(f)714 3622 y Fv(\()p Ft(\022)s Fv(\))38 b Fr(\000)1027 3497 y Fn(1)990 3527 y Fm(X)997 3736 y Fq(\027)t Fo(=2)1134 3622 y Fv(\()p Fr(\000)p Ft(g)t Fv(\))1337 3578 y Fq(\027)1625 3527 y Fm(X)1524 3725 y Fj(\033)1560 3740 y Fk(1)1595 3725 y Fj(;:::)10 b(;\033)1736 3733 y(\027)1774 3725 y Fk(=)p Fe(\006)1405 3783 y Fj(\013)1445 3798 y Fk(1)1480 3783 y Fj(;:::)g(;\013)1625 3798 y(\027)s Fe(\000)p Fk(1)1742 3783 y(=0)p Fj(;:::)h(;M)3173 3622 y Fv(\(V.201\))625 3848 y Fm(Z)741 3983 y Ft(dk)846 3940 y Fo(\()p Fq(\027)t Fo(\))944 3983 y Ft(\037)1005 3933 y Fo(\(0\))1005 4011 y Fq(el)1099 3983 y Ft(w)1172 3940 y Fo(\()p Fq(\022)r Fo(\))1169 4004 y Fq(\033)1209 4013 y Fk(1)1265 3983 y Fv(\()p Ft(k)1354 4001 y Fo(1)1394 3983 y Fv(\))p Ft(\037)1493 3933 y Fo(\()p Fq(\013)1565 3942 y Fk(1)1600 3933 y Fo(\))1493 4011 y Fq(el)1675 3983 y Fr(\001)16 b(\001)g(\001)44 b Ft(\037)1893 3933 y Fo(\()p Fq(\013)1965 3942 y Fj(\027)s Fe(\000)p Fk(1)2083 3933 y Fo(\))1893 4011 y Fq(el)2115 3983 y Ft(w)2188 3940 y Fo(\()p Fq(\022)r Fo(\))2185 4004 y Fq(\033)2225 4012 y Fj(\027)2281 3983 y Fv(\()p Ft(k)2370 4001 y Fq(\027)2413 3983 y Fv(\))p Ft(\037)2512 3933 y Fo(\(0\))2512 4011 y Fq(el)625 4217 y Fr(\002)701 4106 y Fm(D)762 4217 y Fv(\012)832 4102 y Fm(\014)832 4162 y(\014)832 4222 y(\014)893 4106 y(n)975 4217 y Ft(a)1026 4174 y Fq(\033)1066 4183 y Fk(1)1105 4217 y Fv(\()p Ft(k)1194 4235 y Fo(1)1234 4217 y Fv(\))16 b Ft(f)1336 4235 y Fq(\013)1381 4244 y Fk(1)1420 4217 y Fv([)p Ft(L)p 1447 4233 67 4 v 22 w Fv(+)22 b Ft(r)p 1633 4233 47 4 v 2 w Fv(])44 b Fr(\001)16 b(\001)g(\001)44 b Ft(f)1955 4235 y Fq(\013)2000 4244 y Fj(\027)s Fe(\000)p Fk(1)2121 4217 y Fv([)p Ft(L)p 2148 4233 67 4 v 23 w Fv(+)21 b Ft(r)p 2334 4233 47 4 v 3 w Fv(])p Ft(a)2459 4174 y Fq(\033)2499 4182 y Fj(\027)2542 4217 y Fv(\()p Ft(k)2631 4235 y Fq(\027)2674 4217 y Fv(\))979 4430 y Fr(\000)27 b Ft(a)1133 4386 y Fq(\033)1173 4395 y Fk(1)1213 4430 y Fv(\()p Ft(k)1302 4447 y Fo(1)1341 4430 y Fv(\))16 b Ft(f)1443 4447 y Fq(\013)1488 4456 y Fk(1)1527 4430 y Fv([)p Ft(L)p 1554 4446 67 4 v Fv(])44 b Fr(\001)16 b(\001)g(\001)44 b Ft(f)1896 4447 y Fq(\013)1941 4456 y Fj(\027)s Fe(\000)p Fk(1)2063 4430 y Fv([)p Ft(L)p 2090 4446 V Fv(])p Ft(a)2234 4386 y Fq(\033)2274 4394 y Fj(\027)2317 4430 y Fv(\()p Ft(k)2406 4447 y Fq(\027)2449 4430 y Fv(\))27 b(\012)2584 4319 y Fm(E)2667 4430 y Ft(;)257 4635 y Fv(using)41 b Ft(L)p 520 4651 V 39 w Fv(:=)e(\()p Ft(L)871 4652 y Fq(f)917 4635 y Ft(;)16 b(L)1026 4652 y Fq(aux)1149 4635 y Fv(\),)41 b Ft(r)p 1255 4651 47 4 v 42 w Fv(:=)e(\()p Ft(r)1565 4652 y Fq(f)1610 4635 y Ft(;)16 b(r)1697 4652 y Fq(aux)1820 4635 y Fv(\),)41 b(and)f Ft(r)2167 4652 y Fq(f)2212 4635 y Fv(\()p Ft(\022)s Fv(\))f(:=)g(cos)q(\()p Ft(#)p Fv(\))p Ft(r)2825 4652 y Fq(f)2897 4635 y Fr(\000)26 b Ft(i)16 b Fv(sin)r(\()p Ft(#)p Fv(\))p Ft(r)3346 4652 y Fq(aux)3468 4635 y Fv(.)257 4755 y(F)-8 b(urthermore,)547 5003 y Ft(W)639 5021 y Fo(\(0\))733 5003 y Fv([)p Ft(z)t Fv(])84 b(:=)1189 4879 y Fn(1)1153 4909 y Fm(X)1130 5107 y Fj(m;n)p Fk(=0)1116 5163 y Fj(m)p Fk(+)p Fj(n)p Fe(\025)p Fk(1)1360 5003 y Ft(W)1466 4960 y Fo(\(0\))1452 5024 y Fq(m;n)1581 5003 y Fv([)p Ft(z)t Fv(])23 b Ft(;)1439 b Fv(\(V.202\))512 5352 y Ft(W)618 5309 y Fo(\(0\))604 5373 y Fq(m;n)733 5352 y Fv([)p Ft(z)t Fv(])84 b(:=)1106 5217 y Fm(Z)1222 5352 y Ft(dK)1363 5309 y Fo(\()p Fq(m;n)p Fo(\))1568 5352 y Ft(a)1619 5309 y Fo(+)1678 5352 y Fv(\()p Ft(k)1770 5309 y Fo(\()p Fq(m)p Fo(\))1892 5352 y Fv(\))22 b Ft(w)2025 5309 y Fo(\(0\))2022 5373 y Fq(m;n)2150 5271 y Fm(\002)2192 5352 y Ft(z)t(;)16 b(L)p 2284 5368 67 4 v 1 w(;)g(K)2484 5309 y Fo(\()p Fq(m;n)p Fo(\))2669 5271 y Fm(\003)2732 5352 y Ft(a)2783 5309 y Fn(\000)2842 5352 y Fv(\()2883 5326 y(~)2880 5352 y Ft(k)2934 5309 y Fo(\()p Fq(n)p Fo(\))3036 5352 y Fv(\))22 b Ft(;)50 b Fv(\(V.203\))p eop %%Page: 80 81 80 80 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(80)257 573 y(and)33 b(w)m(e)h(shall)g(displa)m(y)g(the)g(dep)s (endence)h(of)d Ft(L)2020 590 y Fo(\(0\))2115 573 y Fv(,)h Ft(E)2247 590 y Fo(\(0\))2342 573 y Fv(,)g Ft(T)2459 590 y Fo(\(0\))2553 573 y Fv(,)g(and)g Ft(W)2895 590 y Fo(\(0\))3023 573 y Fv(on)f Ft(\022)k Fv(unless)257 716 y(necessary)-8 b(.)59 b(Note)36 b(that)h Ft(E)1259 734 y Fo(\(0\))1390 716 y Fv(is)g(an)g(op)s(erator)f(on)h(Ran)16 b Ft(\037)2421 665 y Fo(\(0\))2421 744 y Fq(el)2515 716 y Fv(,)38 b(i.e.,)g Ft(E)2841 734 y Fo(\(0\))2970 716 y Fr(2)c(M)3186 734 y Fq(N)3290 716 y Fv(is)j(an)257 836 y Ft(N)30 b Fr(\002)19 b Ft(N)42 b Fv(matrix.)i(Similarly)-8 b(,)32 b Ft(r)p 1367 852 47 4 v 30 w Fr(7!)27 b Ft(T)1623 854 y Fo(\(0\))1717 836 y Fv([)p Ft(z)t(;)16 b(r)p 1836 852 V 4 w Fv(])32 b(is)f(an)h Ft(N)e Fr(\002)19 b Ft(N)42 b Fv(matrix-v)-5 b(alued)32 b(function,)257 980 y(and)c Ft(r)p 442 996 V 30 w Fr(7!)f Ft(w)714 929 y Fo(\(0\))711 990 y Fq(m;n)839 980 y Fv([)p Ft(z)t(;)16 b(r)p 958 996 V 4 w(;)g(K)1139 940 y Fo(\()p Fq(m;n)p Fo(\))1324 980 y Fv(])27 b(are)h Ft(N)22 b Fr(\002)12 b Ft(N)38 b Fv(matrix-v)-5 b(alued)28 b(functions,)i(for)d Ft(m)12 b Fv(+)g Ft(n)27 b Fr(\025)g Fv(1,)257 1100 y(p)s(oin)m(t)m(wise)34 b(in)e Ft(K)891 1061 y Fo(\()p Fq(m;n)p Fo(\))1075 1100 y Fv(.)43 b(Eq.)33 b(\(V.196\))e(yields)i(the)g(follo)m(wing)f(explicit)h (expressions)i(for)257 1234 y Ft(w)330 1183 y Fo(\(0\))327 1244 y Fq(m;n)489 1234 y Fv(\(compare)e(to)f([8,)g(Lemma)i(I)s(I)s (I.6]\),)358 1406 y Ft(w)431 1362 y Fo(\(0\))428 1426 y Fq(m;n)557 1325 y Fm(\002)598 1406 y Ft(z)t(;)16 b(L)p 690 1422 67 4 v 2 w(;)g(K)891 1362 y Fo(\()p Fq(m;n)p Fo(\))1075 1325 y Fm(\003)1203 1406 y Fv(=)1894 b(\(V.204\))524 1654 y Fr(\000)653 1529 y Fn(1)616 1559 y Fm(X)625 1768 y Fq(p)p Fo(=0)882 1559 y Fm(X)809 1771 y Fq(b)p Fn(2B)932 1779 y Fj(m;n;p)1331 1529 y Fq(M)1297 1559 y Fm(X)1148 1768 y Fq(\013)1193 1777 y Fk(1)1228 1768 y Fq(;:::)10 b(;\013)1383 1777 y Fj(\027)s Fe(\000)p Fk(1)1500 1768 y Fo(=0)1590 1654 y Fv(\()p Fr(\000)p Ft(g)t Fv(\))1793 1610 y Fq(\027)1851 1518 y Fm(Z)1967 1654 y Ft(dX)2107 1610 y Fo(\()p Fq(p;p)p Fo(\))2256 1654 y Fr(S)2315 1671 y Fq(m;n)2444 1543 y Fm(n)2527 1654 y Ft(F)2590 1671 y Fq(b)2624 1573 y Fm(\002)2666 1654 y Ft(X)2755 1610 y Fo(\()p Fq(p;p)p Fo(\))2904 1654 y Ft(;)16 b(K)3037 1610 y Fo(\()p Fq(m;n)p Fo(\))3222 1573 y Fm(\003)3280 1543 y(o)3368 1654 y Ft(;)257 1934 y Fv(where)32 b Fr(B)601 1951 y Fq(m;n;p)816 1934 y Fv(denotes)g(the)g(set)f(of)g(partitions)g Ft(b)c Fv(=)g(\()p Ft(b)2287 1951 y Fq(k)2330 1934 y Ft(;)16 b(b)2416 1944 y Fo(~)2414 1961 y Fq(k)2458 1934 y Ft(;)g(b)2542 1951 y Fq(x)2586 1934 y Ft(;)g(b)2674 1951 y Fo(~)-39 b Fq(x)2715 1934 y Fv(\))31 b(of)f Fr(f)p Fv(1)p Ft(;)16 b Fv(2)p Ft(;)g(:)g(:)g(:)34 b(;)16 b(m)i Fv(+)257 2054 y Ft(n)h Fv(+)f(2)p Ft(p)p Fr(g)30 b Fv(suc)m(h)j(that)d Fr(j)p Ft(b)1101 2072 y Fq(k)1145 2054 y Fr(j)d Fv(=)g Ft(m)p Fv(,)k Fr(j)p Ft(b)1515 2064 y Fo(~)1513 2082 y Fq(k)1557 2054 y Fr(j)c Fv(=)g Ft(n)p Fv(,)k(and)g Fr(j)p Ft(b)2086 2072 y Fq(x)2131 2054 y Fr(j)c Fv(=)g Fr(j)p Ft(b)2360 2072 y Fo(~)-39 b Fq(x)2427 2054 y Fv(=)27 b Ft(p)p Fv(,)32 b(i.e.,)g Ft(b)2862 2072 y Fq(k)2905 2054 y Fv(,)f Ft(b)3006 2064 y Fo(~)3004 2082 y Fq(k)3047 2054 y Fv(,)g Ft(b)3146 2072 y Fq(x)3191 2054 y Fv(,)g Ft(b)3294 2072 y Fo(~)-39 b Fq(x)3365 2054 y Fv(are)257 2174 y(ordered,)32 b(pairwise)g(disjoin)m(t)g(subsets)h(of)d Fr(f)p Fv(1)p Ft(;)16 b Fv(2)p Ft(;)g(:)g(:)g(:)34 b(;)16 b(m)i Fv(+)g Ft(n)h Fv(+)f(2)p Ft(p)p Fr(g)30 b Fv(whose)i(union)f(giv) m(e)257 2295 y Fr(f)p Fv(1)p Ft(;)16 b Fv(2)p Ft(;)g(:)g(:)g(:)34 b(;)16 b(m)22 b Fv(+)g Ft(n)g Fv(+)f(2)p Ft(p)p Fr(g)p Fv(.)404 2415 y(Giv)m(en)47 b Ft(b)k Fr(2)f(B)971 2433 y Fq(m;n;p)1202 2415 y Fv(and)c(denoting)h Ft(M)61 b Fv(:=)51 b Ft(m)31 b Fv(+)g Ft(n)g Fv(+)g(2)p Ft(p)p Fv(,)50 b(the)d(matrix-v)-5 b(alued)257 2536 y(function)33 b Ft(F)702 2553 y Fq(b)769 2536 y Fv(is)h(de\014ned)g(b)m(y)381 2707 y Ft(F)444 2725 y Fq(b)478 2627 y Fm(\002)520 2707 y Ft(X)609 2664 y Fo(\()p Fq(p;p)p Fo(\))758 2707 y Ft(;)16 b(K)891 2664 y Fo(\()p Fq(m;n)p Fo(\))1076 2627 y Fm(\003)1204 2707 y Fv(:=)1866 b(\(V.205\))547 2876 y Ft(\037)608 2825 y Fo(\(0\))608 2904 y Fq(el)702 2876 y Ft(w)775 2825 y Fo(\()p Fq(\022)r Fo(\))772 2904 y Fq(b)869 2876 y Fv(\(1)p Ft(;)16 b(X)1088 2833 y Fo(\()p Fq(p;p)p Fo(\))1237 2876 y Ft(;)g(K)1370 2833 y Fo(\()p Fq(m;n)p Fo(\))1555 2876 y Fv(\))p Ft(\037)1654 2825 y Fo(\()p Fq(\013)1726 2834 y Fk(1)1761 2825 y Fo(\))1654 2904 y Fq(el)1836 2876 y Fr(\001)g(\001)g(\001)44 b Ft(\037)2054 2825 y Fo(\()p Fq(\013)2126 2834 y Fj(\027)s Fe(\000)p Fk(1)2244 2825 y Fo(\))2054 2904 y Fq(el)2276 2876 y Ft(w)2349 2825 y Fo(\()p Fq(\022)r Fo(\))2346 2904 y Fq(b)2442 2876 y Fv(\()p Ft(M)5 b(;)16 b(X)2711 2833 y Fo(\()p Fq(p;p)p Fo(\))2861 2876 y Ft(;)g(K)2994 2833 y Fo(\()p Fq(m;n)p Fo(\))3178 2876 y Fv(\))p Ft(\037)3277 2825 y Fo(\(0\))3277 2904 y Fq(el)547 3052 y Fr(\001)574 2941 y Fm(D)635 3052 y Fv(\012)705 2937 y Fm(\014)705 2997 y(\014)705 3057 y(\014)766 3052 y Ft(a)817 3008 y Fq(b)851 3052 y Fv(\(1)p Ft(;)g(X)1070 3008 y Fo(\()p Fq(p;p)p Fo(\))1220 3052 y Ft(;)g(K)1353 3008 y Fo(\()p Fq(m;n)p Fo(\))1537 3052 y Fv(\))g Ft(f)1639 3069 y Fq(\013)1684 3078 y Fk(1)1723 3052 y Fv([)p Ft(L)p 1750 3068 V 23 w Fv(+)21 b Ft(\026)p 1936 3087 59 4 v 43 x Fo(1)2034 3052 y Fv(])1046 3264 y Fr(\001)16 b(\001)g(\001)45 b Ft(f)1252 3282 y Fq(\013)1297 3291 y Fj(\027)s Fe(\000)p Fk(1)1418 3264 y Fv([)p Ft(L)p 1445 3280 67 4 v 22 w Fv(+)22 b Ft(\026)p 1631 3300 59 4 v 1689 3308 a Fq(\027)t Fn(\000)p Fo(1)1823 3264 y Fv(])p Ft(a)1901 3221 y Fq(b)1935 3264 y Fv(\()p Ft(M)5 b(;)16 b(X)2204 3221 y Fo(\()p Fq(p;p)p Fo(\))2355 3264 y Ft(;)g(K)2488 3221 y Fo(\()p Fq(m;n)p Fo(\))2672 3264 y Fv(\))27 b(\012)2807 3154 y Fm(E)2890 3264 y Ft(;)257 3458 y Fv(where)415 3816 y Ft(w)488 3766 y Fo(\()p Fq(\022)r Fo(\))485 3844 y Fq(b)581 3816 y Fv(\()p Ft(j;)16 b(X)791 3773 y Fo(\()p Fq(p;p)p Fo(\))941 3816 y Ft(;)g(K)1074 3773 y Fo(\()p Fq(m;n)p Fo(\))1259 3816 y Fv(\))59 b(:=)1519 3523 y Fm(8)1519 3612 y(>)1519 3642 y(>)1519 3672 y(>)1519 3702 y(<)1519 3881 y(>)1519 3911 y(>)1519 3941 y(>)1519 3971 y(:)1649 3619 y Ft(w)1722 3568 y Fo(\()p Fq(\022)r Fo(\))1719 3640 y(+)1815 3619 y Fv(\()p Ft(k)1904 3636 y Fq(l)1930 3619 y Fv(\))22 b Ft(;)87 b Fv(if)33 b Ft(j)38 b Fv(is)33 b(the)g Ft(l)2569 3579 y Fq(th)2673 3619 y Fv(mem)m(b)s(er)g(of)g Ft(b)3199 3636 y Fq(k)3242 3619 y Fv(,)1649 3755 y Ft(w)1722 3704 y Fo(\()p Fq(\022)r Fo(\))1719 3777 y Fn(\000)1815 3755 y Fv(\()1856 3729 y(~)1853 3755 y Ft(k)1904 3773 y Fq(l)1930 3755 y Fv(\))22 b Ft(;)87 b Fv(if)33 b Ft(j)38 b Fv(is)33 b(the)g Ft(l)2569 3716 y Fq(th)2673 3755 y Fv(mem)m(b)s(er)g(of)g Ft(b)3201 3765 y Fo(~)3199 3783 y Fq(k)3242 3755 y Fv(,)1649 3892 y Ft(w)1722 3841 y Fo(\()p Fq(\022)r Fo(\))1719 3914 y(+)1815 3892 y Fv(\()p Ft(x)1908 3910 y Fq(l)1935 3892 y Fv(\))21 b Ft(;)83 b Fv(if)33 b Ft(j)38 b Fv(is)33 b(the)g Ft(l)2569 3853 y Fq(th)2673 3892 y Fv(mem)m(b)s(er)g(of)g Ft(b)3199 3910 y Fq(x)3243 3892 y Fv(,)1649 4029 y Ft(w)1722 3978 y Fo(\()p Fq(\022)r Fo(\))1719 4051 y Fn(\000)1815 4029 y Fv(\()6 b(~)-55 b Ft(x)1908 4047 y Fq(l)1935 4029 y Fv(\))21 b Ft(;)83 b Fv(if)33 b Ft(j)38 b Fv(is)33 b(the)g Ft(l)2569 3990 y Fq(th)2673 4029 y Fv(mem)m(b)s(er)g(of)g Ft(b)3203 4047 y Fo(~)-39 b Fq(x)3243 4029 y Fv(,)3173 4158 y(\(V.206\))257 4323 y(and)322 4589 y Ft(a)373 4546 y Fq(b)408 4589 y Fv(\()p Ft(j;)16 b(X)618 4546 y Fo(\()p Fq(p;p)p Fo(\))768 4589 y Ft(;)g(K)901 4546 y Fo(\()p Fq(m;n)p Fo(\))1085 4589 y Fv(\))60 b(:=)1345 4385 y Fm(8)1345 4475 y(<)1345 4654 y(:)1475 4467 y Fv(1)22 b Ft(;)302 b Fv(if)32 b Ft(j)h Fr(2)27 b Ft(b)2170 4484 y Fq(k)2235 4467 y Fr([)22 b Ft(b)2365 4477 y Fo(~)2363 4495 y Fq(k)2406 4467 y Fv(,)1475 4588 y Ft(a)1526 4549 y Fo(+)1586 4588 y Fv(\()p Ft(x)1679 4606 y Fq(l)1705 4588 y Fv(\))g Ft(;)83 b Fv(if)32 b Ft(j)39 b Fv(is)33 b(the)g Ft(l)2340 4549 y Fq(th)2443 4588 y Fv(mem)m(b)s(er)h(of)e Ft(b)2969 4606 y Fq(x)3013 4588 y Fv(,)1475 4710 y Ft(a)1526 4671 y Fn(\000)1586 4710 y Fv(\()6 b(~)-55 b Ft(x)1679 4728 y Fq(l)1705 4710 y Fv(\))22 b Ft(;)83 b Fv(if)32 b Ft(j)39 b Fv(is)33 b(the)g Ft(l)2340 4671 y Fq(th)2443 4710 y Fv(mem)m(b)s(er)h(of)e Ft(b)2973 4728 y Fo(~)-39 b Fq(x)3013 4710 y Fv(.)3173 4589 y(\(V.207\))257 4875 y(Moreo)m(v)m(er,)35 b Fr(S)772 4893 y Fq(m;n)933 4875 y Fv(denotes)f(the)f(symmetrization)h(op)s(erator,)653 5059 y Fr(S)712 5076 y Fq(m;n)841 5059 y Fr(f)p Ft(F)14 b Fr(g)p Fv([)p Ft(X)1132 5015 y Fo(\()p Fq(p;p)p Fo(\))1281 5059 y Ft(;)i(K)1414 5015 y Fo(\()p Fq(m;n)p Fo(\))1598 5059 y Fv(])60 b(:=)1385 b(\(V.208\))911 5186 y(1)p 829 5235 215 4 v 829 5323 a Ft(m)p Fv(!)16 b Ft(n)p Fv(!)1095 5163 y Fm(X)1079 5363 y Fj(\031)r Fe(2)p Fj(S)1197 5371 y(m)1091 5420 y Fk(~)-34 b Fj(\031)r Fe(2)p Fj(S)1205 5428 y(n)1281 5258 y Ft(F)1358 5177 y Fm(\002)1399 5258 y Ft(X)1488 5214 y Fo(\()p Fq(p;p)p Fo(\))1638 5258 y Ft(;)16 b(k)1732 5275 y Fq(\031)r Fo(\(1\))1869 5258 y Ft(;)g(:)g(:)g(:)34 b(;)16 b(k)2153 5275 y Fq(\031)r Fo(\()p Fq(m)p Fo(\))2333 5258 y Fv(;)2396 5232 y(~)2393 5258 y Ft(k)2448 5275 y Fo(~)-39 b Fq(\031)r Fo(\(1\))2581 5258 y Ft(;)16 b(:)g(:)g(:)34 b(;)2817 5232 y Fv(~)2814 5258 y Ft(k)2869 5275 y Fo(~)-39 b Fq(\031)r Fo(\()p Fq(n)p Fo(\))3010 5177 y Fm(\003)3073 5258 y Ft(:)p eop %%Page: 81 82 81 81 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(81)257 573 y(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m(wing)g (estimates)h(on)f(these)h(co)s(e\016cien)m(ts.)257 743 y Fi(Lemma)j(V.11.)49 b Fs(Assume)28 b(that)h Ft(z)j Fr(2)27 b(S)1740 761 y Fo(0)p Fq(;<)1854 743 y Fs(.)42 b(Then)28 b(ther)-5 b(e)28 b(exists)g(a)g(c)-5 b(onstant,)29 b Ft(C)34 b(<)27 b Fr(1)p Fs(,)257 863 y(such)35 b(that,)g(for)g Ft(g)30 b(>)d Fv(0)35 b Fs(su\016ciently)g(smal)5 b(l,)1300 997 y Fm(\015)1300 1056 y(\015)1355 1081 y Ft(E)1427 1099 y Fo(\(0\))1521 1081 y Fv([)p Ft(z)t Fv(])45 b(+)e Ft(i)16 b(g)1888 1038 y Fo(2)1944 1081 y Fv(\000)2005 1099 y Fo(0)2044 997 y Fm(\015)2044 1056 y(\015)2182 1081 y Fr(\024)83 b Ft(C)23 b(g)2485 1038 y Fo(2+2)p Fq(")2669 1081 y Ft(;)477 b Fv(\(V.209\))1041 1151 y Fm(\015)1041 1211 y(\015)1096 1236 y Ft(@)1147 1253 y Fq(r)1179 1261 y Fj(aux)1293 1236 y Ft(T)1350 1253 y Fo(\(0\))1444 1236 y Fv([)p Ft(z)t(;)16 b(r)p 1563 1252 47 4 v 4 w Fv(])44 b(+)f Ft(i)16 b Fv(sin)h Ft(#)2043 1151 y Fm(\015)2043 1211 y(\015)2182 1236 y Fr(\024)83 b Ft(C)23 b(g)2485 1192 y Fo(2)p Fq(")2578 1236 y Ft(;)568 b Fv(\(V.210\))366 1553 y Fm(\015)366 1613 y(\015)421 1638 y Ft(w)494 1594 y Fo(\(0\))491 1658 y Fq(m;n)620 1638 y Fv([)p Ft(z)t(;)16 b(r)p 739 1654 V 4 w(;)g(K)920 1594 y Fo(\()p Fq(m;n)p Fo(\))1104 1638 y Fv(])1131 1553 y Fm(\015)1131 1613 y(\015)1249 1638 y Fr(\024)61 b Ft(C)24 b(\032)1530 1655 y Fo(0)1586 1497 y Fm(\022)1669 1566 y Ft(C)f(g)p 1669 1615 144 4 v 1696 1703 a(\032)1746 1721 y Fo(0)1822 1497 y Fm(\023)1896 1520 y Fq(m)p Fo(+)p Fq(n)2125 1513 y(m)2093 1543 y Fm(Y)2095 1753 y Fq(j)t Fo(=1)2354 1566 y Ft(\024)p Fv(\()p Ft(k)2499 1584 y Fq(j)2535 1566 y Fv(\))p 2246 1615 435 4 v 2246 1706 a Ft(!)t Fv(\()p Ft(k)2400 1723 y Fq(j)2436 1706 y Fv(\))2474 1677 y Fo(1)p Fq(=)p Fo(2)p Fn(\000)p Fq(\026)2766 1513 y(n)2723 1543 y Fm(Y)2726 1753 y Fq(j)t Fo(=1)2985 1566 y Ft(\024)p Fv(\()3082 1540 y(~)3079 1566 y Ft(k)3130 1584 y Fq(j)3166 1566 y Fv(\))p 2877 1615 V 2877 1719 a Ft(!)t Fv(\()2983 1693 y(~)2980 1719 y Ft(k)3031 1737 y Fq(j)3066 1719 y Fv(\))3104 1690 y Fo(1)p Fq(=)p Fo(2)p Fn(\000)p Fq(\026)3343 1638 y Ft(;)3173 1869 y Fv(\(V.211\))540 2136 y Fm(Z)595 2362 y Fq(B)651 2332 y Fj(m)p Fk(+)p Fj(n)648 2372 y(\032)680 2387 y Fk(0)799 2187 y Fm(\015)799 2247 y(\015)855 2272 y Ft(@)906 2289 y Fq(r)938 2297 y Fj(aux)1052 2272 y Ft(w)1125 2228 y Fo(\(0\))1122 2292 y Fq(m;n)1250 2272 y Fv([)p Ft(z)t(;)16 b(r)p 1369 2288 47 4 v 4 w(;)g(K)1550 2228 y Fo(\()p Fq(m;n)p Fo(\))1735 2272 y Fv(])1762 2187 y Fm(\015)1762 2247 y(\015)1866 2147 y Fq(m)1833 2177 y Fm(Y)1836 2387 y Fq(j)t Fo(=1)1961 2131 y Fm(\022)2173 2200 y Ft(d)2224 2161 y Fo(3)2263 2200 y Ft(k)2314 2218 y Fq(j)p 2044 2249 435 4 v 2044 2340 a Ft(!)t Fv(\()p Ft(k)2198 2357 y Fq(j)2233 2340 y Fv(\))2271 2311 y Fo(3)p Fq(=)p Fo(2+)p Fq(\026)2489 2131 y Fm(\023)2642 2147 y Fq(n)2600 2177 y Fm(Y)2602 2387 y Fq(j)t Fo(=1)2727 2131 y Fm(\022)2939 2200 y Ft(d)2990 2161 y Fo(3)3032 2174 y Fv(~)3029 2200 y Ft(k)3080 2218 y Fq(j)p 2810 2249 V 2810 2353 a Ft(!)t Fv(\()2916 2327 y(~)2913 2353 y Ft(k)2964 2371 y Fq(j)3000 2353 y Fv(\))3038 2324 y Fo(3)p Fq(=)p Fo(2+)p Fq(\026)3255 2131 y Fm(\023)2040 2542 y Fr(\024)83 b Ft(C)2292 2461 y Fm(\000)2338 2542 y Ft(C)23 b(g)2482 2461 y Fm(\001)2527 2483 y Fq(m)p Fo(+)p Fq(n)2713 2542 y Ft(;)433 b Fv(\(V.212\))257 2760 y Fs(wher)-5 b(e)35 b Ft(B)607 2777 y Fq(r)672 2760 y Fv(:=)27 b Fr(f)p Ft(k)d Fr(j)e Ft(!)t Fv(\()p Ft(k)s Fv(\))k Ft(<)h(r)s Fr(g)p Fs(.)257 2954 y(Pr)-5 b(o)g(of.)49 b Fv(The)41 b(asserted)h(estimates)g(follo)m(w)f(from)f(adaption)h(of)f ([8,)i(Section)f(I)s(I)s(I].)g(F)-8 b(or)257 3075 y(illustration,)34 b(w)m(e)f(giv)m(e)h(a)e(pro)s(of)g(of)g(\(V.209\).)43 b(W)-8 b(e)33 b(\014rst)g(rewrite)g Ft(E)2739 3092 y Fo(\(0\))2834 3075 y Fv([)p Ft(z)t Fv(])g(as)610 3378 y Ft(E)682 3395 y Fo(\(0\))777 3378 y Fv([)p Ft(z)t Fv(])84 b(=)f Fr(\000)p Ft(g)1250 3334 y Fo(2)1304 3378 y Ft(M)10 b Fv([)p Ft(\022)s(;)16 b(z)t Fv(])52 b Fr(\000)1815 3253 y Fn(1)1778 3283 y Fm(X)1786 3492 y Fq(\027)t Fo(=4)1922 3378 y Fv(\()p Fr(\000)p Ft(g)t Fv(\))2125 3334 y Fq(\027)2413 3283 y Fm(X)2313 3481 y Fj(\033)2349 3496 y Fk(1)2383 3481 y Fj(;:::)10 b(;\033)2524 3489 y(\027)2562 3481 y Fk(=)p Fe(\006)2193 3539 y Fj(\013)2233 3554 y Fk(1)2268 3539 y Fj(;:::)g(;\013)2413 3554 y(\027)s Fe(\000)p Fk(1)2530 3539 y(=0)p Fj(;:::)h(;M)2802 3242 y Fm(Z)2918 3378 y Ft(dk)3023 3334 y Fo(\()p Fq(\027)t Fo(\))3173 3378 y Fv(\(V.213\))1123 3705 y Ft(\037)1184 3654 y Fo(\(0\))1184 3732 y Fq(el)1278 3705 y Ft(w)1351 3661 y Fo(\()p Fq(\022)r Fo(\))1348 3725 y Fq(\033)1388 3734 y Fk(1)1444 3705 y Fv(\()p Ft(k)1533 3722 y Fo(1)1573 3705 y Fv(\))p Ft(\037)1672 3654 y Fo(\()p Fq(\013)1744 3663 y Fk(1)1779 3654 y Fo(\))1672 3732 y Fq(el)1854 3705 y Fr(\001)16 b(\001)g(\001)44 b Ft(\037)2072 3654 y Fo(\()p Fq(\013)2144 3663 y Fj(\027)s Fe(\000)p Fk(1)2262 3654 y Fo(\))2072 3732 y Fq(el)2293 3705 y Ft(w)2366 3661 y Fo(\()p Fq(\022)r Fo(\))2363 3725 y Fq(\033)2403 3733 y Fj(\027)2460 3705 y Fv(\()p Ft(k)2549 3722 y Fq(\027)2592 3705 y Fv(\))p Ft(\037)2691 3654 y Fo(\(0\))2691 3732 y Fq(el)1123 3882 y Fr(\002)1199 3771 y Fm(D)1259 3882 y Fv(\012)1329 3767 y Fm(\014)1329 3827 y(\014)1329 3887 y(\014)1390 3882 y Ft(a)1441 3839 y Fq(\033)1481 3848 y Fk(1)1520 3882 y Fv(\()p Ft(k)1609 3900 y Fo(1)1649 3882 y Fv(\))16 b Ft(f)1751 3900 y Fq(\013)1796 3909 y Fk(1)1835 3882 y Fv([)p Ft(L)p 1862 3898 67 4 v Fv(])44 b Fr(\001)16 b(\001)g(\001)44 b Ft(f)2204 3900 y Fq(\013)2249 3909 y Fj(\027)s Fe(\000)p Fk(1)2370 3882 y Fv([)p Ft(L)p 2397 3898 V 1 w Fv(])16 b Ft(a)2558 3839 y Fq(\033)2598 3847 y Fj(\027)2641 3882 y Fv(\()p Ft(k)2730 3900 y Fq(\027)2773 3882 y Fv(\))27 b(\012)2908 3771 y Fm(E)2991 3882 y Ft(;)257 4122 y Fv(where)436 4408 y Ft(M)10 b Fv([)p Ft(\022)s(;)16 b(z)t Fv(])62 b(:=)993 4283 y Fq(M)958 4313 y Fm(X)963 4522 y Fq(\013)p Fo(=0)1119 4272 y Fm(Z)1234 4408 y Ft(dk)19 b(f)1403 4425 y Fq(\013)1453 4408 y Fv([\()p Fr(\000)p Fv(1\))1681 4364 y Fq(\034)1724 4408 y Ft(!)t Fv(\()p Ft(k)s Fv(\))p Ft(;)d(!)t Fv(\()p Ft(k)s Fv(\)])g Ft(\037)2261 4357 y Fo(\(0\))2261 4435 y Fq(el)2354 4408 y Ft(w)2427 4357 y Fo(\()p Fq(\022)r Fo(\))2424 4429 y Fn(\000)2521 4408 y Fv(\()p Ft(k)s Fv(\))p Ft(\037)2712 4357 y Fo(\()p Fq(\013)p Fo(\))2712 4435 y Fq(el)2816 4408 y Ft(w)2889 4357 y Fo(\()p Fq(\022)r Fo(\))2886 4429 y(+)2983 4408 y Fv(\()p Ft(k)s Fv(\))p Ft(\037)3174 4357 y Fo(\(0\))3174 4435 y Fq(el)3290 4408 y Ft(:)3173 4625 y Fv(\(V.214\))257 4864 y(Here,)44 b Ft(k)f Fv(=)e(\()769 4838 y Ft(~)774 4864 y(k)s(;)16 b(\025;)g(\034)11 b Fv(\))41 b Fr(2)g Fp(R)1275 4825 y Fo(3)1347 4864 y Fr(\002)27 b(f)p Fv(1)p Ft(;)16 b Fv(2)p Fr(g)26 b(\002)h(f)p Ft(`;)16 b(r)s Fr(g)p Fv(,)42 b(and)f(\()p Fr(\000)p Fv(1\))2515 4825 y Fq(\034)2598 4864 y Fv(:=)g(1,)h(for)e Ft(\034)52 b Fv(=)40 b Ft(`)p Fv(,)j(and)257 4984 y(\()p Fr(\000)p Fv(1\))458 4945 y Fq(\034)528 4984 y Fv(:=)27 b Fr(\000)p Fv(1,)32 b(for)g Ft(\034)39 b Fv(=)27 b Ft(r)s Fv(.)43 b(W)-8 b(e)33 b(recall)g(that) 1018 5295 y Ft(f)1066 5313 y Fq(\013)1115 5295 y Fv([\()p Fr(\000)p Fv(1\))1343 5252 y Fq(\034)1386 5295 y Ft(!)t(;)16 b(!)t Fv(])59 b(=)2127 5216 y(\026)-60 b Ft(\037)2177 5165 y Fo(\()p Fq(\013)p Fo(\))2177 5244 y Fq(f)2281 5216 y Fv([)p Ft(!)t Fv(])p 1790 5272 935 4 v 1790 5363 a Ft(\021)1838 5381 y Fq(\013)1910 5363 y Fv(+)21 b Ft(e)2052 5334 y Fn(\000)p Fq(i)p Fo(\()p Fn(\000)p Fo(1\))2275 5315 y Fj(\034)2315 5334 y Fq(#)2361 5363 y Ft(!)t Fv(\()p Ft(k)s Fv(\))g Fr(\000)h Ft(z)3173 5295 y Fv(\(V.215\))p eop %%Page: 82 83 82 82 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(82)257 573 y(and)33 b(that)f(an)h(analytic)g(con)m(tin)m(uation)h (from)e Ft(\022)e Fv(=)d(0)33 b(to)f Ft(\022)e Fv(=)d Ft(i#)33 b Fv(yields)539 866 y(\000)600 883 y Fo(0)722 866 y Fv(=)916 741 y Fq(M)881 771 y Fm(X)885 980 y Fq(\013)p Fo(=0)1041 730 y Fm(Z)1157 866 y Ft(dk)1646 794 y Fv(1)p 1288 843 766 4 v 1288 934 a Ft(\021)1336 951 y Fq(\013)1407 934 y Fv(+)22 b Ft(e)1550 905 y Fn(\000)p Fq(i)p Fo(\()p Fn(\000)p Fo(1\))1773 886 y Fj(\034)1813 905 y Fq(#)1859 934 y Ft(!)t Fv(\()p Ft(k)s Fv(\))2080 866 y Ft(\037)2141 815 y Fo(\(0\))2141 894 y Fq(el)2235 866 y Ft(w)2308 815 y Fo(\()p Fq(\022)r Fo(\))2305 888 y Fn(\000)2402 866 y Fv(\()p Ft(k)s Fv(\))p Ft(\037)2593 815 y Fo(\()p Fq(\013)p Fo(\))2593 894 y Fq(el)2697 866 y Ft(w)2770 815 y Fo(\()p Fq(\022)r Fo(\))2767 888 y(+)2863 866 y Fv(\()p Ft(k)s Fv(\))p Ft(\037)3054 815 y Fo(\(0\))3054 894 y Fq(el)3170 866 y Ft(;)722 1203 y Fv(=)916 1079 y Fq(M)881 1109 y Fm(X)885 1318 y Fq(\013)p Fo(=0)1041 1068 y Fm(Z)1157 1203 y Ft(dk)1570 1132 y Fv(1)p 1288 1181 613 4 v 1288 1269 a Ft(\021)1336 1286 y Fq(\013)1407 1269 y Fv(+)g Ft(!)t Fv(\()p Ft(k)s Fv(\))f Fr(\000)h Ft(i)p Fv(0)1927 1203 y Ft(\037)1988 1153 y Fo(\(0\))1988 1231 y Fq(el)2082 1203 y Ft(w)2155 1153 y Fo(\(0\))2152 1225 y Fn(\000)2249 1203 y Fv(\()p Ft(k)s Fv(\))p Ft(\037)2440 1153 y Fo(\()p Fq(\013)p Fo(\))2440 1231 y Fq(el)2544 1203 y Ft(w)2617 1153 y Fo(\(0\))2614 1225 y(+)2711 1203 y Fv(\()p Ft(k)s Fv(\))p Ft(\037)2902 1153 y Fo(\(0\))2902 1231 y Fq(el)3018 1203 y Ft(;)128 b Fv(\(V.216\))257 1491 y(since)34 b(\000)557 1508 y Fo(0)629 1491 y Fv(is)f(indep)s (enden)m(t)i(of)d Ft(\022)s Fv(.)44 b(In)33 b(view)h(of)e(the)h(fact)f (that)471 1640 y Fm(\014)471 1699 y(\014)471 1759 y(\014)471 1819 y(\014)786 1705 y Fv(\026)-61 b Ft(\037)835 1654 y Fo(\()p Fq(\013)p Fo(\))835 1732 y Fq(f)939 1705 y Fv([)p Ft(!)t Fv(\()p Ft(k)s Fv(\)])p 514 1761 935 4 v 514 1852 a Ft(\021)562 1870 y Fq(\013)633 1852 y Fv(+)22 b Ft(e)776 1823 y Fn(\000)p Fq(i)p Fo(\()p Fn(\000)p Fo(1\))999 1804 y Fj(\034)1039 1823 y Fq(#)1085 1852 y Ft(!)t Fv(\()p Ft(k)s Fv(\))f Fr(\000)g Ft(z)1501 1784 y Fr(\000)1989 1713 y Fv(1)p 1631 1761 766 4 v 1631 1852 a Ft(\021)1679 1870 y Fq(\013)1750 1852 y Fv(+)g Ft(e)1892 1823 y Fn(\000)p Fq(i)p Fo(\()p Fn(\000)p Fo(1\))2115 1804 y Fj(\034)2156 1823 y Fq(#)2202 1852 y Ft(!)t Fv(\()p Ft(k)s Fv(\))2406 1640 y Fm(\014)2406 1699 y(\014)2406 1759 y(\014)2406 1819 y(\014)2499 1784 y Fv(=)59 b Fr(O)s Fv(\()p Fr(j)p Ft(z)t Fr(j)p Fv(\))22 b Ft(;)230 b Fv(\(V.217\))257 2042 y(for)32 b Ft(\013)c Fr(\025)f Fv(1,)33 b(and)389 2191 y Fm(\014)389 2250 y(\014)389 2310 y(\014)389 2370 y(\014)601 2255 y Fv(\026)-61 b Ft(\037)650 2205 y Fo(\(0\))650 2283 y Fq(f)745 2255 y Fv([)p Ft(!)t Fv(\()p Ft(k)s Fv(\)])p 433 2312 718 4 v 433 2403 a Ft(e)478 2374 y Fn(\000)p Fq(i)p Fo(\()p Fn(\000)p Fo(1\))701 2355 y Fj(\034)741 2374 y Fq(#)787 2403 y Ft(!)t Fv(\()p Ft(k)s Fv(\))21 b Fr(\000)g Ft(z)1203 2335 y Fr(\000)1333 2264 y Ft(e)1378 2224 y Fq(i)p Fo(\()p Fn(\000)p Fo(1\))1546 2201 y Fj(\034)1586 2224 y Fq(#)p 1333 2312 300 4 v 1385 2400 a Ft(!)t Fv(\()p Ft(k)s Fv(\))1642 2191 y Fm(\014)1642 2250 y(\014)1642 2310 y(\014)1642 2370 y(\014)1735 2335 y Fr(\024)2055 2264 y Fv(2)16 b Fr(j)p Ft(z)t Fr(j)p 1880 2312 519 4 v 1880 2400 a Fv(\()p Ft(!)t Fv(\()p Ft(k)s Fv(\))21 b(+)h Ft(\032)2282 2418 y Fo(0)2321 2400 y Fv(\))2359 2371 y Fo(2)2452 2335 y Fv(+)2581 2264 y Ft(\037)2642 2281 y Fq(\032)2678 2290 y Fk(0)2717 2264 y Fv([)p Ft(!)t Fv(\()p Ft(k)s Fv(\)])p 2581 2312 385 4 v 2676 2400 a Ft(!)t Fv(\()p Ft(k)s Fv(\))2997 2335 y Ft(;)149 b Fv(\(V.218\))257 2593 y(w)m(e)34 b(obtain)e(that)1283 2714 y Fm(\015)1283 2773 y(\015)1338 2798 y Ft(K)7 b Fv([)p Ft(\022)s(;)16 b(z)t Fv(])40 b Fr(\000)d Ft(i)p Fv(\000)1869 2816 y Fo(0)1909 2714 y Fm(\015)1909 2773 y(\015)2024 2798 y Fv(=)60 b Fr(O)2241 2718 y Fm(\000)2286 2798 y Ft(g)2337 2755 y Fo(2)2375 2718 y Fm(\001)2443 2798 y Ft(:)703 b Fv(\(V.219\))404 3004 y(Secondly)-8 b(,)34 b(w)m(e)f(observ)m(e)i (that)337 3117 y Fm(\014)337 3177 y(\014)337 3237 y(\014)370 3121 y(D)431 3232 y Fv(\012)501 3117 y Fm(\014)501 3177 y(\014)501 3237 y(\014)562 3232 y Ft(a)613 3188 y Fq(\033)653 3197 y Fk(1)692 3232 y Fv(\()p Ft(k)781 3249 y Fo(1)820 3232 y Fv(\))16 b Ft(f)922 3249 y Fq(\013)967 3258 y Fk(1)1006 3232 y Fv([)p Ft(L)p 1033 3248 67 4 v 1 w Fv(])43 b Fr(\001)16 b(\001)g(\001)44 b Ft(f)1375 3249 y Fq(\013)1420 3258 y Fj(\027)s Fe(\000)p Fk(1)1542 3232 y Fv([)p Ft(L)p 1569 3248 V Fv(])16 b Ft(a)1729 3188 y Fq(\033)1769 3196 y Fj(\027)1813 3232 y Fv(\()p Ft(k)1902 3249 y Fq(\027)1945 3232 y Fv(\))27 b(\012)2080 3121 y Fm(E)2141 3117 y(\014)2141 3177 y(\014)2141 3237 y(\014)3173 3232 y Fv(\(V.220\))420 3444 y(=)83 b Ft(\032)629 3462 y Fo(0)685 3330 y Fm(\014)685 3389 y(\014)685 3449 y(\014)718 3334 y(D)779 3444 y Fv(\012)849 3330 y Fm(\014)849 3389 y(\014)849 3449 y(\014)910 3444 y Fv(\()p Ft(L)1014 3462 y Fq(aux)1158 3444 y Fv(+)21 b Ft(\032)1305 3462 y Fo(0)1345 3444 y Fv(\))1383 3401 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1564 3444 y Ft(a)1615 3401 y Fq(\033)1655 3410 y Fk(1)1694 3444 y Fv(\()p Ft(k)1783 3462 y Fo(1)1823 3444 y Fv(\))16 b(\()p Ft(L)1981 3462 y Fq(aux)2125 3444 y Fv(+)21 b Ft(\032)2272 3462 y Fo(0)2312 3444 y Fv(\))2350 3401 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2542 3444 y Fv(\()p Ft(L)2646 3462 y Fq(aux)2790 3444 y Fv(+)h Ft(\032)2938 3462 y Fo(0)2978 3444 y Fv(\))16 b Ft(f)3080 3462 y Fq(\013)3125 3471 y Fk(1)3164 3444 y Fv([)p Ft(L)p 3191 3460 V Fv(])606 3657 y Fr(\001)g(\001)g(\001)44 b Fv(\()p Ft(L)867 3674 y Fq(aux)1011 3657 y Fv(+)22 b Ft(\032)1159 3674 y Fo(0)1198 3657 y Fv(\))16 b Ft(f)1300 3674 y Fq(\013)1345 3683 y Fj(\027)s Fe(\000)p Fk(1)1467 3657 y Fv([)p Ft(L)p 1494 3673 V 1 w Fv(])27 b(\()p Ft(L)1719 3674 y Fq(aux)1863 3657 y Fv(+)22 b Ft(\032)2011 3674 y Fo(0)2050 3657 y Fv(\))2088 3613 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2269 3657 y Ft(a)2320 3613 y Fq(\033)2360 3621 y Fj(\027)2404 3657 y Fv(\()p Ft(k)2493 3674 y Fo(1)2532 3657 y Fv(\))16 b(\()p Ft(L)2690 3674 y Fq(aux)2834 3657 y Fv(+)22 b Ft(\032)2982 3674 y Fo(0)3021 3657 y Fv(\))3059 3613 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)3251 3657 y Fv(\012)3321 3546 y Fm(E)3383 3542 y(\014)3383 3602 y(\014)3383 3662 y(\014)420 3918 y Fr(\024)83 b Ft(\032)629 3935 y Fo(0)685 3918 y Fv(8)734 3874 y Fq(\027)t Fn(\000)p Fo(1)943 3793 y Fq(\027)899 3823 y Fm(Y)902 4033 y Fq(j)t Fo(=1)1027 3803 y Fm(\015)1027 3863 y(\015)1027 3923 y(\015)1082 3918 y Fv(\()p Ft(L)1186 3935 y Fq(aux)1330 3918 y Fv(+)22 b Ft(\032)1478 3935 y Fo(0)1517 3918 y Fv(\))1555 3874 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1737 3918 y Ft(a)1788 3874 y Fq(\033)1828 3884 y Fj(j)1865 3918 y Fv(\()p Ft(k)1954 3935 y Fq(j)1990 3918 y Fv(\))16 b(\()p Ft(L)2148 3935 y Fq(aux)2292 3918 y Fv(+)22 b Ft(\032)2440 3935 y Fo(0)2480 3918 y Fv(\))2518 3874 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2682 3803 y Fm(\015)2682 3863 y(\015)2682 3923 y(\015)2760 3918 y Ft(;)257 4224 y Fv(using)34 b(\(V.198\).)42 b(The)34 b(standard)f(estimate)h(from)e(Lemma)h(IV.1)g(implies)h(that)344 4343 y Fm(Z)460 4479 y Ft(dk)562 4496 y Fq(j)614 4394 y Fm(\015)614 4454 y(\015)670 4479 y Ft(w)743 4435 y Fo(\()p Fq(\022)r Fo(\))740 4499 y Fq(\033)780 4509 y Fj(j)836 4479 y Fv(\()p Ft(k)925 4496 y Fq(j)961 4479 y Fv(\))999 4394 y Fm(\015)999 4454 y(\015)1082 4364 y(\015)1082 4424 y(\015)1082 4484 y(\015)1137 4479 y Fv(\()p Ft(L)1241 4496 y Fq(aux)1385 4479 y Fv(+)22 b Ft(\032)1533 4496 y Fo(0)1572 4479 y Fv(\))1610 4435 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1792 4479 y Ft(a)1843 4435 y Fq(\033)1883 4445 y Fj(j)1920 4479 y Fv(\()p Ft(k)2009 4496 y Fq(j)2045 4479 y Fv(\))16 b(\()p Ft(L)2203 4496 y Fq(aux)2347 4479 y Fv(+)22 b Ft(\032)2495 4496 y Fo(0)2535 4479 y Fv(\))2573 4435 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2737 4364 y Fm(\015)2737 4424 y(\015)2737 4484 y(\015)2852 4479 y Fv(=)60 b Fr(O)s Fv(\()p Ft(\032)3157 4428 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)3157 4503 y(0)3322 4479 y Fv(\))21 b Ft(:)3173 4657 y Fv(\(V.221\))257 4862 y(Inserting)k (\(V.219\){\(V.221\))e(in)m(to)h(the)h(sum)g(in)f(\(V.213\))f(and)h (summing)h(up)g(the)f(terms)257 4982 y(of)32 b(order)h Ft(\027)h Fr(\025)27 b Fv(4)32 b(as)h(in)g([8)o(,)g(Lemma)g(I)s(I)s (I.7],)g(w)m(e)g(obtain)g(that)559 5264 y Fr(k)p Ft(E)680 5281 y Fo(\(0\))774 5264 y Fv([)p Ft(z)t Fv(])39 b(+)f Ft(ig)1114 5220 y Fo(2)1153 5264 y Fv(\000)1214 5281 y Fo(0)1253 5264 y Fr(k)60 b(\024)1534 5139 y Fn(1)1497 5169 y Fm(X)1505 5378 y Fq(\027)t Fo(=4)1657 5264 y Ft(C)24 b(\032)1801 5281 y Fo(0)1857 5123 y Fm(\022)1948 5192 y Ft(C)g(g)p 1940 5241 161 4 v 1940 5353 a(\032)1990 5302 y Fo(1)p Fq(=)p Fo(2)1990 5378 y(0)2110 5123 y Fm(\023)2184 5145 y Fq(\027)2286 5264 y Fv(=)60 b Fr(O)2503 5183 y Fm(\000)2548 5264 y Ft(g)2599 5220 y Fo(2+2)p Fq(")2760 5183 y Fm(\001)2828 5264 y Ft(;)318 b Fv(\(V.222\))p eop %%Page: 83 84 83 83 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(83)257 573 y(th)m(us)34 b(establishing)g(\(V.209\).)p 3425 573 4 66 v 3429 510 59 4 v 3429 573 V 3487 573 4 66 v 404 814 a(In)f(Theorem)g(B.2)g(in)g(App)s(endix)h(B)e(w)m(e)i(pro) m(v)m(e)g(that)e(Ker)17 b(\000)2630 831 y Fo(0)2696 814 y Fv(=)27 b Fp(C)36 b Ft(\024)2937 831 y Fq(\014)2991 814 y Fv(,)c(where)1164 1132 y Ft(\024)1220 1149 y Fq(\014)1327 1132 y Fv(=)59 b Ft(Z)1536 1088 y Fn(\000)p Fo(1)1663 1007 y Fq(N)7 b Fn(\000)p Fo(1)1668 1037 y Fm(X)1673 1246 y Fq(n)p Fo(=0)1849 1132 y Ft(e)1894 1088 y Fn(\000)p Fq(\014)s(E)2044 1096 y Fj(n)2086 1088 y Fq(=)p Fo(2)2183 1132 y Ft(')2247 1149 y Fq(n)2315 1132 y Fr(\012)22 b Ft(')2477 1149 y Fq(n)2545 1132 y Ft(;)601 b Fv(\(V.223\))257 1458 y Ft(Z)34 b Fv(=)461 1383 y Fm(P)567 1410 y Fq(N)7 b Fn(\000)p Fo(1)567 1487 y Fq(n)p Fo(=0)740 1458 y Ft(e)785 1419 y Fn(\000)p Fq(\014)s(E)935 1427 y Fj(n)1014 1458 y Fv(is)33 b(a)g(normalization)f(factor.)43 b(Hence,)34 b(denoting)f(b)m(y)1514 1678 y Ft(P)1591 1635 y Fq(el)1577 1699 y(\024)1618 1711 y Fj(\014)1723 1678 y Fv(:=)60 b Fr(j)p Ft(\024)1969 1696 y Fq(\014)2016 1678 y Fr(i)16 b(h)p Ft(\024)2164 1696 y Fq(\014)2211 1678 y Fr(j)935 b Fv(\(V.224\))257 1898 y(the)33 b(orthogonal)f(pro)5 b(jection)33 b(on)m(to)g Ft(\024)1652 1916 y Fq(\014)1699 1898 y Fv(,)g(w)m(e)h(ha)m(v)m(e)f(that)1338 2118 y(\000)1399 2136 y Fo(0)1498 2118 y Fv(=)59 b(\000)1694 2136 y Fo(0)1750 2118 y Ft(P)1827 2075 y Fq(el)1813 2139 y(\024)1854 2151 y Fj(\014)1959 2118 y Fr(\025)h Fm(b)-56 b Ft(\015)2145 2136 y Fo(0)2201 2118 y Ft(P)2278 2075 y Fq(el)2264 2139 y(\024)2305 2151 y Fj(\014)2371 2118 y Ft(;)775 b Fv(\(V.225\))257 2338 y(for)39 b(some)g(p)s(ositiv)m(e)h Fm(b)-55 b Ft(\015)1081 2356 y Fo(0)1157 2338 y Ft(>)38 b Fv(0.)61 b(Our)39 b(strategy)g(is)g (no)m(w)h(to)e(apply)i(the)f(F)-8 b(esh)m(bac)m(h)40 b(map)257 2459 y(again,)33 b(using)g(the)g(pro)5 b(jection)1285 2679 y Ft(P)1348 2696 y Fo(1)1447 2679 y Fv(:=)60 b Ft(P)1687 2635 y Fq(el)1673 2699 y(\024)1714 2711 y Fj(\014)1781 2679 y Fr(\012)21 b Ft(\037)p Fv([)p Ft(L)2032 2696 y Fq(aux)2182 2679 y Ft(<)27 b(\032)2335 2696 y Fo(1)2375 2679 y Fv(])22 b Ft(;)722 b Fv(\(V.226\))257 2899 y(where)1572 3119 y Ft(\032)1622 3136 y Fo(1)1721 3119 y Fv(:=)60 b Ft(g)1935 3075 y Fo(2+)p Fq("=)p Fo(2)2153 3119 y Ft(:)993 b Fv(\(V.227\))257 3339 y(W)-8 b(e)33 b(ha)m(v)m(e)h Ft(P)713 3356 y Fo(1)753 3339 y Ft(P)816 3356 y Fo(0)882 3339 y Fv(=)27 b Ft(P)1048 3356 y Fo(0)1087 3339 y Ft(P)1150 3356 y Fo(1)1217 3339 y Fv(=)g Ft(P)1383 3356 y Fo(1)1454 3339 y Fv(and)p 1208 3496 77 4 v 1208 3576 a Ft(P)1284 3593 y Fo(1)1384 3576 y Fv(:=)59 b Fi(1)22 b Fr(\000)f Ft(P)1784 3593 y Fo(1)1883 3576 y Fv(=)p 2019 3496 V 60 w Ft(P)2095 3514 y Fo(\(1\))2095 3596 y(1)2211 3576 y Fv(+)p 2309 3496 V 22 w Ft(P)2385 3514 y Fo(\(2\))2385 3596 y(1)2501 3576 y Ft(;)645 b Fv(\(V.228\))257 3796 y(where)p 574 3936 V 574 4016 a Ft(P)650 3954 y Fo(\(1\))650 4036 y(1)804 4016 y Fv(:=)p 967 3936 V 60 w Ft(P)1043 3954 y Fq(el)1043 4036 y(\024)1084 4048 y Fj(\014)1151 4016 y Fr(\012)22 b Fi(1)g Ft(;)p 1606 3936 V 252 w(P)1683 3954 y Fo(\(2\))1683 4036 y(1)1836 4016 y Fv(:=)60 b Ft(P)2076 3972 y Fq(el)2062 4036 y(\024)2103 4048 y Fj(\014)2170 4016 y Fr(\012)21 b Ft(\037)p Fv([)p Ft(L)2421 4033 y Fq(aux)2571 4016 y Fr(\025)27 b Ft(\032)2724 4033 y Fo(1)2764 4016 y Fv(])22 b Ft(;)333 b Fv(\(V.229\))257 4236 y(Again,)51 b(for)46 b(the)h(F)-8 b(esh)m(bac)m(h)48 b(map)f(to)g(b)s(e)f (de\014ned,)52 b(w)m(e)c(pro)m(v)m(e)g(the)f(in)m(v)m(ertibilit)m(y)j (of)257 4356 y(the)43 b(op)s(erator)e(restricted)i(to)e(Ran)p 1599 4276 V 16 w Ft(P)1676 4374 y Fo(1)1731 4356 y Ft(P)1794 4374 y Fo(0)1834 4356 y Fv(.)71 b(W)-8 b(e)42 b(divide)h(the)f(pro)s (of)f(in)m(to)h(a)f(series)j(of)257 4476 y(lemmata.)257 4647 y Fi(Lemma)37 b(V.12.)49 b Fs(Assume)35 b Ft(z)d Fr(2)27 b(S)1554 4664 y Fo(0)p Fq(;<)1668 4647 y Fs(.)45 b(Then)970 4801 y Fm(\015)970 4861 y(\015)970 4921 y(\015)970 4981 y(\015)1035 4873 y Fv(\()p Ft(T)1130 4890 y Fo(\(0\))1225 4873 y Fv([)p Ft(z)t(;)16 b(L)p 1344 4889 67 4 v 1 w Fv(])22 b(+)g Ft(E)1630 4890 y Fo(\(0\))1724 4873 y Fv([)p Ft(z)t Fv(])h Fr(\000)e Ft(z)t Fv(\))p 2050 4793 77 4 v 16 w Ft(P)2128 4890 y Fo(1)2183 4873 y Ft(P)2246 4890 y Fo(0)p 1035 4923 1251 4 v 1462 5011 a Ft(L)1528 5029 y Fq(aux)1672 5011 y Fv(+)h Ft(\032)1820 5029 y Fo(1)2296 4801 y Fm(\014)2296 4861 y(\014)2296 4921 y(\014)2296 4981 y(\014)2391 4946 y Fr(\024)62 b(O)s Fv(\(1\))21 b Ft(:)390 b Fv(\(V.230\))p eop %%Page: 84 85 84 84 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(84)257 573 y Fs(Pr)-5 b(o)g(of.)49 b Fv(According)33 b(to)f(\(V.209\),)g(\(V.210\))g(and)h(\(V.225\),)f(w)m(e)i(ha)m(v)m(e) 303 772 y Fr(\000)p Fv(Im)p Fr(f)p Ft(T)601 789 y Fo(\(0\))695 772 y Fv([)p Ft(z)t(;)16 b(L)p 814 788 67 4 v 2 w Fv(])22 b(+)f Ft(E)1100 789 y Fo(\(0\))1195 772 y Fv([)p Ft(z)t Fv(])h Fr(\000)g Ft(z)t Fr(g)60 b(\025)1712 691 y Fm(\000)1757 772 y Fv(sin)17 b Ft(#)22 b Fr(\000)g(O)s Fv(\()p Ft(g)2240 728 y Fo(2)p Fq(")2311 772 y Fv(\))2349 691 y Fm(\001)2394 772 y Ft(L)2460 789 y Fq(aux)2626 772 y Fv(+)2746 691 y Fm(\000)2792 772 y(b)-56 b Ft(\015)2842 789 y Fo(0)2903 772 y Fr(\000)21 b(O)s Fv(\()p Ft(g)3170 728 y Fq("=)p Fo(2)3277 772 y Fv(\))3315 691 y Fm(\001)3360 772 y Ft(g)3411 728 y Fo(2)3173 892 y Fv(\(V.231\))257 1126 y(on)33 b(Ran)p 568 1046 77 4 v Ft(P)644 1065 y Fo(\(1\))644 1147 y(1)738 1126 y Fv(.)44 b(Con)m(v)m(ersely)-8 b(,)35 b(on)e(Ran)p 1632 1046 V Ft(P)1708 1065 y Fo(\(2\))1802 1126 y Fv(,)g(w)m(e)h (estimate)832 1325 y Fr(\000)p Fv(Im)p Fr(f)p Ft(T)1130 1343 y Fo(\(0\))1225 1325 y Fv([)p Ft(z)t(;)16 b(L)p 1344 1341 67 4 v 1 w Fv(])22 b(+)f Ft(E)1629 1343 y Fo(\(0\))1724 1325 y Fv([)p Ft(z)t Fv(])i Fr(\000)e Ft(z)t Fr(g)1128 b Fv(\(V.232\))915 1533 y Fr(\025)1074 1452 y Fm(\000)1119 1533 y Fv(sin)17 b Ft(#)23 b Fr(\000)e(O)s Fv(\()p Ft(g)1602 1490 y Fo(2)p Fq(")1673 1533 y Fv(\))1711 1452 y Fm(\001)1757 1533 y Ft(L)1823 1551 y Fq(aux)1983 1533 y Fr(\000)2107 1462 y Fv(sin)c Ft(#)p 2107 1510 194 4 v 2179 1598 a Fv(2)2310 1533 y Ft(\032)2360 1551 y Fo(1)2438 1533 y Fr(\000)38 b(O)s Fv(\()p Ft(g)2722 1490 y Fo(2+2)p Fq(")2883 1533 y Fv(\))915 1725 y Fr(\025)1074 1644 y Fm(\000)1119 1725 y Fv(sin)q(\()p Ft(#)p Fv(\))p Ft(=)p Fv(2)g Fr(\000)g(O)s Fv(\()p Ft(g)1792 1681 y Fo(3)p Fq("=)p Fo(2)1933 1725 y Fv(\))1971 1644 y Fm(\001)2017 1725 y Ft(L)2083 1742 y Fq(aux)2227 1725 y Ft(;)257 1936 y Fv(since)c Fr(O)s Fv(\()p Ft(g)666 1896 y Fo(2+)p Fq(")792 1936 y Fv(\))27 b(=)g Ft(\032)1010 1953 y Fo(1)1050 1936 y Fr(O)s Fv(\()p Ft(g)1220 1896 y Fo(3)p Fq("=)p Fo(2)1361 1936 y Fv(\))g(=)g Ft(L)1595 1953 y Fq(aux)1718 1936 y Fr(O)s Fv(\()p Ft(g)1888 1896 y Fo(3)p Fq("=)p Fo(2)2029 1936 y Fv(\).)p 3425 1936 4 66 v 3429 1873 59 4 v 3429 1936 V 3487 1936 4 66 v 257 2226 a Fi(Lemma)37 b(V.13.)49 b Fs(Assume)35 b Ft(z)d Fr(2)27 b(S)1554 2244 y Fo(0)p Fq(;<)1668 2226 y Fs(.)45 b(Then)455 2352 y Fm(\015)455 2412 y(\015)511 2437 y Ft(P)574 2454 y Fo(0)629 2437 y Fv(\()p Ft(L)733 2454 y Fq(aux)878 2437 y Fv(+)21 b Ft(\032)1025 2454 y Fo(1)1065 2437 y Fv(\))1103 2393 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1284 2437 y Ft(W)1376 2454 y Fo(\(0\))1470 2437 y Fv([)p Ft(z)t Fv(])16 b(\()p Ft(L)1693 2454 y Fq(aux)1839 2437 y Fv(+)21 b Ft(\032)1986 2454 y Fo(1)2026 2437 y Fv(\))2064 2393 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2245 2437 y Ft(P)2308 2454 y Fo(0)2347 2352 y Fm(\015)2347 2412 y(\015)2486 2437 y Fv(=)83 b Fr(O)2726 2356 y Fm(\000)2771 2437 y Ft(g)19 b Fv(\()p Ft(\032)2925 2454 y Fo(0)2965 2437 y Ft(=\032)3064 2454 y Fo(1)3104 2437 y Fv(\))3142 2393 y Fo(1)p Fq(=)p Fo(2)3252 2356 y Fm(\001)2486 2598 y Fv(=)83 b Fr(O)2726 2517 y Fm(\000)2771 2598 y Ft(g)2822 2554 y Fo(1)p Fn(\000)p Fo(3)p Fq("=)p Fo(4)3054 2517 y Fm(\001)3121 2598 y Ft(:)25 b Fv(\(V.233\))257 2797 y Fs(Pr)-5 b(o)g(of.)49 b Fv(The)j(pro)s(of)f(is)h(an)f(adaption)g(of)g ([8,)56 b(Theorem)d(B.2],)j(using)d(the)e(b)s(ounds)257 2917 y(\(V.211\),)32 b(for)f(all)h Ft(m)20 b Fv(+)g Ft(n)27 b Fr(\025)g Fv(1,)32 b(and)g(summing)h(up)f(all)g(con)m(tributions.)p 3425 2917 V 3429 2855 59 4 v 3429 2917 V 3487 2917 4 66 v 404 3158 a(Putting)40 b(together)h(Lemma)g(V.12)f(and)h(Lemma)g (V.13,)h(w)m(e)f(obtain)f(the)h(in)m(v)m(ert-)257 3278 y(ibilit)m(y)c(of)d Ft(L)721 3296 y Fo(\(0\))816 3278 y Fv([)p Ft(z)t Fv(])24 b Fr(\000)f Ft(z)40 b Fv(restricted)c(to)f(Ran) p 1876 3198 77 4 v 16 w Ft(P)1952 3296 y Fo(1)2008 3278 y Ft(P)2071 3296 y Fo(0)2145 3278 y Fv(b)m(y)h(a)f(Neumann)h(series)h (expansion)257 3399 y(of)32 b(\()p Ft(L)472 3416 y Fo(\(0\))567 3399 y Fv([)p Ft(z)t Fv(])23 b Fr(\000)e Ft(z)t Fv(\))877 3359 y Fn(\000)p Fo(1)1005 3399 y Fv(around)32 b(\()p Ft(T)1430 3416 y Fo(\(0\))1525 3399 y Fv([)p Ft(z)t(;)16 b(L)p 1644 3415 67 4 v 1 w Fv(])22 b(+)g Ft(E)1930 3416 y Fo(\(0\))2024 3399 y Fv([)p Ft(z)t Fv(])h Fr(\000)e Ft(z)t Fv(\))2334 3359 y Fn(\000)p Fo(1)2429 3399 y Fv(.)257 3569 y Fi(Lemma)37 b(V.14.)49 b Fs(Assume)41 b(\(V.128\))g(and)f Fr(j)p Ft(z)t Fr(j)f(\024)g Ft(C)2227 3586 y Fo(0)2266 3569 y Ft(g)2317 3530 y Fo(2+)p Fq(")2443 3569 y Fs(.)63 b(Then,)42 b(for)f Ft(g)g(>)d Fv(0)j Fs(su\016-)257 3689 y(ciently)35 b(smal)5 b(l,)p 860 3609 77 4 v 34 w Ft(P)937 3707 y Fo(1)992 3689 y Fv(\()p Ft(L)1096 3707 y Fo(\(0\))1213 3689 y Fr(\000)21 b Ft(z)t Fv(\))p 1413 3609 V 16 w Ft(P)1490 3707 y Fo(1)1565 3689 y Fs(is)34 b(invertible)h(on)f Fv(Ran)p 2427 3609 V 16 w Ft(P)2503 3707 y Fo(1)2559 3689 y Ft(P)2622 3707 y Fo(0)2661 3689 y Fs(.)404 3859 y Fv(Lemma)48 b(V.14)g(justi\014es)h(a)e(second)i(application)g(of)e (the)h(F)-8 b(esh)m(bac)m(h)50 b(map)e(with)257 3980 y(pro)5 b(jection)47 b Ft(P)797 3997 y Fo(1)836 3980 y Fv(.)84 b(That)46 b(is,)k(the)d(op)s(erator)e Fr(F)2004 3997 y Fq(P)2049 4006 y Fk(1)2088 3980 y Fv(\()p Ft(L)2192 3997 y Fo(\(0\))2286 3980 y Fv([)p Ft(z)t Fv(])32 b Fr(\000)e Ft(z)t Fv(\))47 b(is)f(w)m(ell-de\014ned.)86 b(T)-8 b(o)257 4100 y(form)m(ulate)33 b(this)h(result,)f(w)m(e)h(de\014ne)g(a)e (bijection,)735 4347 y Ft(Z)802 4364 y Fo(\(0\))956 4347 y Fv(:)60 b Fr(S)1102 4364 y Fo(0)p Fq(;<)1243 4347 y Fr(!)26 b Ft(D)1448 4364 y Fo(1)p Fq(=)p Fo(2)1580 4347 y Ft(;)135 b(Z)1809 4364 y Fo(\(0\))1903 4347 y Fv(\()p Ft(\020)8 b Fv(\))58 b(:=)2391 4275 y Ft(i)16 b(\020)p 2261 4324 360 4 v 2261 4412 a Fv(sin)q(\()p Ft(#)p Fv(\))g Ft(\032)2580 4430 y Fo(1)2652 4347 y Ft(;)494 b Fv(\(V.234\))257 4593 y(and)33 b(w)m(e)h(in)m(tro)s(duce)f(the)g(unitary)g(dilatation) 423 4792 y Ft(U)489 4809 y Fo(\(0\))600 4792 y Fv(\012)60 b(:=)f(\012)23 b Ft(;)134 b(U)1212 4809 y Fo(\(0\))1323 4792 y Ft(a)1374 4748 y Fq(\033)1421 4792 y Fv(\()1454 4765 y Ft(~)1459 4792 y(k)s(;)16 b(\025;)g(\034)11 b Fv(\))16 b Ft(U)1839 4748 y Fn(\003)1829 4814 y Fo(\(0\))1984 4792 y Fv(:=)60 b Ft(\032)2197 4741 y Fn(\000)p Fo(3)p Fq(=)p Fo(2)2197 4816 y(1)2379 4792 y Ft(a)2430 4748 y Fq(\033)2477 4792 y Fv(\()2510 4765 y Ft(~)2515 4792 y(k)s(=\032)2668 4809 y Fo(1)2707 4792 y Ft(;)16 b(\025;)g(\034)11 b Fv(\))23 b Ft(;)182 b Fv(\(V.235\))257 4991 y(noting)33 b(that)416 5189 y Ft(U)482 5207 y Fo(\(0\))593 5189 y Ft(L)p 593 5205 67 4 v 17 w(U)752 5146 y Fn(\003)742 5212 y Fo(\(0\))896 5189 y Fv(=)59 b Ft(\032)1081 5207 y Fo(1)1137 5189 y Ft(L)p 1137 5205 V 1 w Fv(\()p Ft(\022)24 b(;)135 b(U)1539 5207 y Fo(\(0\))1650 5189 y Fv(Ran)16 b Ft(\037)1902 5207 y Fq(\032)1938 5216 y Fk(1)1977 5189 y Fv([)p Ft(L)2070 5207 y Fq(aux)2192 5189 y Fv(])60 b(=)g(Ran)16 b Ft(\037)2667 5207 y Fo(1)2706 5189 y Fv([)p Ft(L)2799 5207 y Fq(aux)2922 5189 y Fv(])22 b Ft(;)175 b Fv(\(V.236\))257 5388 y(where)34 b Ft(L)p 539 5404 V 28 w Fv(=)27 b(\()p Ft(L)840 5406 y Fq(f)885 5388 y Ft(;)16 b(L)994 5406 y Fq(aux)1117 5388 y Fv(\).)p eop %%Page: 85 86 85 85 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(85)257 573 y Fi(Theorem)37 b(V.15.)49 b Fs(L)-5 b(et)32 b Ft(z)g Fr(2)27 b Ft(D)1455 590 y Fo(1)p Fq(=)p Fo(2)1565 573 y Fs(.)44 b(Then,)31 b(for)h Ft(g)e(>)d Fv(0)32 b Fs(su\016ciently)g(smal)5 b(l,)31 b Ft(L)3175 590 y Fq(g)3216 573 y Fv(\()p Ft(\022)s Fv(\))15 b Fr(\000)g Ft(z)257 693 y Fs(is)35 b(isosp)-5 b(e)g(ctr)g(al)34 b(\(in)g(the)h(sense)f(of)h([7,)f(8]\))h(to)411 930 y Ft(L)477 947 y Fo(\(1\))571 930 y Fv([)p Ft(z)t Fv(])23 b Fr(\000)e Ft(z)67 b Fv(:=)1244 858 y Ft(i)p 1081 907 360 4 v 1081 995 a Fv(sin)q(\()p Ft(#)p Fv(\))16 b Ft(\032)1400 1013 y Fo(1)1466 930 y Ft(U)1532 947 y Fo(\(0\))1643 930 y Fr(F)1713 947 y Fq(P)1758 956 y Fk(1)1797 849 y Fm(\000)1843 930 y Ft(L)1909 947 y Fo(\(0\))2003 930 y Fv([)p Ft(Z)2104 886 y Fn(\000)p Fo(1)2097 959 y(\(0\))2199 930 y Fv(\()p Ft(z)t Fv(\)])38 b Fr(\000)g Ft(Z)2577 886 y Fn(\000)p Fo(1)2570 959 y(\(0\))2671 930 y Fv(\()p Ft(z)t Fv(\))2796 849 y Fm(\001)2859 930 y Ft(U)2935 886 y Fn(\003)2925 952 y Fo(\(0\))3173 930 y Fv(\(V.237\))257 1169 y Fs(de\014ne)-5 b(d)34 b(on)h Fr(H)808 1187 y Fq(r)r(ed)942 1169 y Fv(:=)27 b(Ran)16 b Ft(\037)1324 1187 y Fo(1)1364 1169 y Fv([)p Ft(L)1457 1187 y Fq(aux)1579 1169 y Fv(])p Fs(.)404 1340 y Fv(Similar)33 b(to)f Ft(L)925 1357 y Fo(\(0\))1020 1340 y Fv([)p Ft(z)t Fv(],)h(w)m(e)h(write)f Ft(L)1642 1357 y Fo(\(1\))1737 1340 y Fv([)p Ft(z)t Fv(])g(as)379 1563 y Ft(L)445 1581 y Fo(\(1\))539 1563 y Fv([)p Ft(z)t Fv(])23 b Fr(\000)e Ft(z)65 b Fv(=)59 b Ft(\037)1068 1581 y Fo(1)1108 1563 y Fv([)p Ft(L)1201 1581 y Fq(aux)1324 1563 y Fv(])1367 1453 y Fm(\020)1443 1563 y Ft(E)1515 1581 y Fo(\(1\))1609 1563 y Fv([)p Ft(z)t Fv(])39 b Fr(\000)f Ft(z)43 b Fv(+)37 b Ft(T)2123 1581 y Fo(\(1\))2218 1563 y Fv([)p Ft(z)t(;)16 b(L)p 2337 1579 67 4 v 1 w Fv(])38 b(+)g Ft(W)2675 1581 y Fo(\(1\))2770 1563 y Fv([)p Ft(z)t Fv(])2889 1453 y Fm(\021)2966 1563 y Ft(\037)3027 1581 y Fo(1)3066 1563 y Fv([)p Ft(L)3159 1581 y Fq(aux)3282 1563 y Fv(])22 b Ft(;)3173 1720 y Fv(\(V.238\))257 1901 y(where)416 2112 y Ft(E)488 2130 y Fo(\(1\))582 2112 y Fv([)p Ft(z)t Fv(])84 b(:=)1128 2041 y Ft(i)p 965 2089 360 4 v 965 2177 a Fv(sin)q(\()p Ft(#)p Fv(\))16 b Ft(\032)1284 2195 y Fo(1)1351 2031 y Fm(\012)1398 2112 y Fv(\012)1468 2027 y Fm(\014)1468 2087 y(\014)1528 2112 y Fr(F)1598 2130 y Fq(P)1643 2139 y Fk(1)1682 2031 y Fm(\000)1728 2112 y Ft(L)1794 2130 y Fo(\(0\))1889 2112 y Fv([)p Ft(Z)1990 2069 y Fn(\000)p Fo(1)2084 2112 y Fv(\()p Ft(z)t Fv(\)])39 b Fr(\000)e Ft(Z)2462 2069 y Fn(\000)p Fo(1)2557 2112 y Fv(\()p Ft(z)t Fv(\))2682 2031 y Fm(\001)2744 2112 y Fv(\012)2814 2031 y Fm(\013)2900 2112 y Fv(+)g Ft(z)27 b(;)61 b Fv(\(V.239\))341 2372 y Ft(T)398 2390 y Fo(\(1\))492 2372 y Fv([)p Ft(z)t(;)16 b(r)p 611 2388 47 4 v 4 w Fv(])83 b(:=)1128 2301 y Ft(i)p 965 2349 360 4 v 965 2438 a Fv(sin)q(\()p Ft(#)p Fv(\))16 b Ft(\032)1284 2455 y Fo(1)1351 2292 y Fm(\012)1398 2372 y Fv(\012)1468 2288 y Fm(\014)1468 2347 y(\014)1528 2372 y Fr(F)1598 2390 y Fq(P)1643 2399 y Fk(1)1704 2372 y Fr(\016)21 b(F)1844 2390 y Fq(P)1889 2399 y Fk(0)1928 2292 y Fm(\000)1974 2372 y Ft(r)p 1974 2388 47 4 v 24 w Fv(+)h Ft(L)2206 2390 y Fq(g)2246 2372 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)f Ft(Z)2563 2329 y Fn(\000)p Fo(1)2658 2372 y Fv(\()p Ft(z)t Fv(\))2783 2292 y Fm(\001)3173 2372 y Fv(\(V.240\))1664 2582 y Fr(\000)h(F)1832 2599 y Fq(P)1877 2608 y Fk(1)1937 2582 y Fr(\016)f(F)2077 2599 y Fq(P)2122 2608 y Fk(0)2161 2501 y Fm(\000)2207 2582 y Ft(r)p 2207 2598 V 24 w Fv(+)h Ft(L)2439 2599 y Fq(g)2479 2582 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)g Ft(Z)2797 2539 y Fn(\000)p Fo(1)2891 2582 y Fv(\()p Ft(z)t Fv(\))3016 2501 y Fm(\001)3078 2582 y Fv(\012)3148 2501 y Fm(\013)3218 2582 y Ft(;)396 2813 y(W)488 2831 y Fo(\(1\))582 2813 y Fv([)p Ft(z)t Fv(])84 b(:=)1039 2689 y Fn(1)1002 2718 y Fm(X)980 2917 y Fj(m;n)p Fk(=0)965 2973 y Fj(m)p Fk(+)p Fj(n)p Fe(\025)p Fk(1)1209 2813 y Ft(W)1315 2770 y Fo(\(1\))1301 2834 y Fq(m;n)1430 2813 y Fv([)p Ft(z)t Fv(])23 b Ft(;)1590 b Fv(\(V.241\))361 3162 y Ft(W)467 3118 y Fo(\(1\))453 3182 y Fq(m;n)582 3162 y Fv([)p Ft(z)t Fv(])84 b(:=)955 3026 y Fm(Z)1071 3162 y Ft(dK)1212 3118 y Fo(\()p Fq(m;n)p Fo(\))1418 3162 y Ft(a)1469 3118 y Fo(+)1528 3162 y Fv(\()p Ft(k)1620 3118 y Fo(\()p Fq(m)p Fo(\))1741 3162 y Fv(\))22 b Ft(w)1874 3118 y Fo(\(1\))1871 3182 y Fq(m;n)2000 3081 y Fm(\002)2041 3162 y Ft(z)t(;)16 b(L)p 2133 3178 67 4 v 2 w(;)g(K)2334 3118 y Fo(\()p Fq(m;n)p Fo(\))2518 3081 y Fm(\003)2581 3162 y Ft(a)2632 3118 y Fn(\000)2692 3162 y Fv(\()2733 3136 y(~)2730 3162 y Ft(k)2784 3118 y Fo(\()p Fq(n)p Fo(\))2885 3162 y Fv(\))22 b Ft(:)201 b Fv(\(V.242\))257 3397 y(F)-8 b(rom)33 b(Lemma)i(V.11)e (and)h(using)g(the)g(tec)m(hniques)i(from)e([8,)f(Section)i(IV],)f(w)m (e)h(deriv)m(e)257 3518 y(the)e(follo)m(wing)g(estimates)h(on)f(these)h (op)s(erators.)257 3688 y Fi(Lemma)j(V.16.)49 b Fs(Assume)43 b Ft(z)k Fr(2)41 b Ft(D)1613 3705 y Fo(1)p Fq(=)p Fo(2)1723 3688 y Fs(.)69 b(Then)43 b(ther)-5 b(e)43 b(exists)f(a)h(c)-5 b(onstant,)44 b Ft(C)49 b(<)42 b Fr(1)p Fs(,)257 3808 y(such)35 b(that,)g(for)g Ft(g)30 b(>)d Fv(0)35 b Fs(su\016ciently)g (smal)5 b(l,)1654 3995 y Fr(j)p Ft(E)1753 4013 y Fo(\(1\))1847 3995 y Fv([)p Ft(z)t Fv(])p Fr(j)84 b(\024)f Ft(C)23 b(g)2364 3952 y Fo(3)p Fq("=)p Fo(2)2528 3995 y Ft(;)618 b Fv(\(V.243\))1181 4149 y Fr(j)p Ft(@)1259 4166 y Fq(r)1291 4174 y Fj(aux)1405 4149 y Ft(T)1462 4166 y Fo(\(1\))1557 4149 y Fv([)p Ft(z)t(;)16 b(r)p 1676 4165 47 4 v 4 w Fv(])38 b Fr(\000)f Fv(1)p Fr(j)83 b(\024)g Ft(C)23 b(g)2364 4105 y Fo(2)p Fq(")2457 4149 y Ft(;)689 b Fv(\(V.244\))364 4403 y Fm(\014)364 4463 y(\014)397 4488 y Ft(w)470 4445 y Fo(\(1\))467 4509 y Fq(m;n)596 4488 y Fv([)p Ft(z)t(;)16 b(r)p 715 4504 V 4 w(;)g(K)896 4445 y Fo(\()p Fq(m;n)p Fo(\))1080 4488 y Fv(])1107 4403 y Fm(\014)1107 4463 y(\014)1203 4488 y Fr(\024)1340 4407 y Fm(\000)1386 4488 y Ft(C)23 b(g)1530 4407 y Fm(\001)1575 4430 y Fq(m)p Fo(+)p Fq(n)1804 4363 y(m)1772 4393 y Fm(Y)1774 4603 y Fq(j)t Fo(=1)2033 4417 y Ft(\024)p Fv(\()p Ft(k)2178 4434 y Fq(j)2215 4417 y Fv(\))p 1926 4465 435 4 v 1926 4556 a Ft(!)t Fv(\()p Ft(k)2080 4573 y Fq(j)2115 4556 y Fv(\))2153 4527 y Fo(1)p Fq(=)p Fo(2)p Fn(\000)p Fq(\026)2445 4363 y(n)2403 4393 y Fm(Y)2405 4603 y Fq(j)t Fo(=1)2664 4417 y Ft(\024)p Fv(\()2761 4390 y(~)2758 4417 y Ft(k)2809 4434 y Fq(j)2845 4417 y Fv(\))p 2556 4465 V 2556 4569 a Ft(!)t Fv(\()2662 4543 y(~)2659 4569 y Ft(k)2710 4587 y Fq(j)2746 4569 y Fv(\))2784 4540 y Fo(1)p Fq(=)p Fo(2)p Fn(\000)p Fq(\026)3023 4488 y Ft(;)123 b Fv(\(V.245\))540 4796 y Fm(Z)595 5021 y Fq(B)651 4991 y Fj(m)p Fk(+)p Fj(n)648 5042 y Fk(1)799 4847 y Fm(\014)799 4906 y(\014)833 4931 y Ft(@)884 4949 y Fq(r)916 4957 y Fj(aux)1029 4931 y Ft(w)1102 4888 y Fo(\(1\))1099 4952 y Fq(m;n)1228 4931 y Fv([)p Ft(z)t(;)16 b(r)p 1347 4947 47 4 v 4 w(;)g(K)1528 4888 y Fo(\()p Fq(m;n)p Fo(\))1712 4931 y Fv(])1739 4847 y Fm(\014)1739 4906 y(\014)1821 4807 y Fq(m)1789 4837 y Fm(Y)1791 5047 y Fq(j)t Fo(=1)1916 4791 y Fm(\022)2128 4860 y Ft(d)2179 4821 y Fo(3)2218 4860 y Ft(k)2269 4877 y Fq(j)p 2000 4908 435 4 v 2000 4999 a Ft(!)t Fv(\()p Ft(k)2154 5017 y Fq(j)2189 4999 y Fv(\))2227 4970 y Fo(3)p Fq(=)p Fo(2+)p Fq(\026)2444 4791 y Fm(\023)2598 4807 y Fq(n)2556 4837 y Fm(Y)2558 5047 y Fq(j)t Fo(=1)2683 4791 y Fm(\022)2895 4860 y Ft(d)2946 4821 y Fo(3)2988 4834 y Fv(~)2985 4860 y Ft(k)3036 4877 y Fq(j)p 2766 4908 V 2766 5013 a Ft(!)t Fv(\()2872 4986 y(~)2869 5013 y Ft(k)2920 5030 y Fq(j)2956 5013 y Fv(\))2994 4984 y Fo(3)p Fq(=)p Fo(2+)p Fq(\026)3211 4791 y Fm(\023)2040 5201 y Fr(\024)83 b Ft(C)2292 5121 y Fm(\000)2338 5201 y Ft(C)23 b(g)2482 5121 y Fm(\001)2527 5143 y Fq(m)p Fo(+)p Fq(n)2713 5201 y Ft(;)433 b Fv(\(V.246\))257 5388 y Fs(wher)-5 b(e)35 b Ft(B)607 5406 y Fq(r)672 5388 y Fv(:=)27 b Fr(f)p Ft(k)d Fr(j)e Ft(!)t Fv(\()p Ft(k)s Fv(\))k Ft(<)h(r)s Fr(g)p Fs(.)p eop %%Page: 86 87 86 86 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(86)257 573 y Fs(Pr)-5 b(o)g(of.)49 b Fv(T)-8 b(o)32 b(pro)m(v)m(e)i(\(V.243\),)e(w)m(e)i(observ)m(e)g(that)f(due)g(to)f (Lemma)h(V.13)g(w)m(e)g(ha)m(v)m(e)648 807 y Ft(\032)698 763 y Fn(\000)p Fo(1)698 829 y(1)793 722 y Fm(\014)793 782 y(\014)826 726 y(\012)873 807 y Fv(\012)943 722 y Fm(\014)943 782 y(\014)1004 807 y Ft(W)1096 824 y Fo(\(0\))1190 807 y Fv([)p Ft(\020)8 b Fv(])p 1311 727 77 4 v 16 w Ft(P)1387 824 y Fo(1)1443 726 y Fm(\000)p 1489 727 V 81 x Ft(P)1565 824 y Fo(1)1605 807 y Ft(L)1671 824 y Fo(\(0\))1765 807 y Fv([)p Ft(\020)g Fv(\))p 1881 727 V Ft(P)1957 824 y Fo(1)2018 807 y Fr(\000)22 b Ft(\020)2167 726 y Fm(\001)2211 748 y Fn(\000)p Fo(1)p 2322 727 V 2322 807 a Ft(P)2398 824 y Fo(1)2454 807 y Ft(W)2546 824 y Fo(\(0\))2640 807 y Fv([)p Ft(z)t Fv(])16 b(\012)2829 726 y Fm(\013)2878 722 y(\014)2878 782 y(\014)731 976 y Fr(\024)83 b(O)s Fv(\(1\))1117 891 y Fm(\015)1117 951 y(\015)1172 976 y Ft(P)1235 994 y Fo(0)1291 976 y Fv(\()p Ft(L)1395 994 y Fq(aux)1539 976 y Fv(+)22 b Ft(\032)1687 994 y Fo(1)1726 976 y Fv(\))1764 933 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)1946 976 y Ft(W)2038 994 y Fo(\(0\))2132 976 y Fv([)p Ft(\020)8 b Fv(])16 b(\()p Ft(L)2357 994 y Fq(aux)2501 976 y Fv(+)21 b Ft(\032)2648 994 y Fo(1)2688 976 y Fv(\))2726 933 y Fn(\000)p Fo(1)p Fq(=)p Fo(2)2907 976 y Ft(P)2970 994 y Fo(0)3009 891 y Fm(\015)3009 951 y(\015)3065 918 y Fo(2)731 1138 y Fv(=)83 b Fr(O)971 1057 y Fm(\000)1016 1138 y Ft(g)1067 1094 y Fo(2)p Fn(\000)p Fo(3)p Fq("=)p Fo(2)1299 1057 y Fm(\001)1367 1138 y Ft(:)1779 b Fv(\(V.247\))257 1358 y(Th)m(us,)35 b(using)e(\(V.222\),)f(w)m(e)h (obtain)496 1627 y Fr(j)p Ft(E)595 1645 y Fo(\(1\))690 1627 y Fv([)p Ft(z)t Fv(])p Fr(j)84 b Fv(=)1236 1556 y Ft(i)p 1073 1604 360 4 v 1073 1692 a Fv(sin)q(\()p Ft(#)p Fv(\))16 b Ft(\032)1392 1710 y Fo(1)1442 1627 y Ft(P)1505 1645 y Fo(1)1561 1627 y Ft(E)1633 1645 y Fo(\(0\))1727 1627 y Fv([)p Ft(Z)1828 1584 y Fn(\000)p Fo(1)1923 1627 y Fv(\()p Ft(z)t Fv(\)])g Ft(P)2154 1645 y Fo(1)2243 1627 y Fv(+)48 b Fr(O)2448 1546 y Fm(\000)2493 1627 y Ft(g)2544 1584 y Fo(2)p Fn(\000)p Fo(3)p Fq("=)p Fo(2)2777 1546 y Fm(\001)3173 1627 y Fv(\(V.248\))904 1887 y(=)1236 1816 y Ft(i)p 1073 1864 V 1073 1953 a Fv(sin)q(\()p Ft(#)p Fv(\))16 b Ft(\032)1392 1970 y Fo(1)1442 1887 y Ft(P)1505 1905 y Fo(1)1561 1807 y Fm(\000)1606 1887 y Ft(E)1678 1905 y Fo(\(0\))1773 1887 y Fv([)p Ft(Z)1874 1844 y Fn(\000)p Fo(1)1968 1887 y Fv(\()p Ft(z)t Fv(\)])23 b(+)e Ft(ig)2324 1844 y Fo(2)2363 1887 y Fv(\000)2424 1905 y Fo(0)2464 1807 y Fm(\001)2526 1887 y Ft(P)2589 1905 y Fo(1)2677 1887 y Fv(+)48 b Fr(O)2882 1807 y Fm(\000)2927 1887 y Ft(g)2978 1844 y Fo(2)p Fn(\000)p Fo(3)p Fq("=)p Fo(2)3210 1807 y Fm(\001)904 2104 y Fv(=)83 b Fr(O)1144 2023 y Fm(\000)1189 2104 y Ft(g)1240 2060 y Fo(3)p Fq("=)p Fo(2)1382 2023 y Fm(\001)1449 2104 y Ft(:)257 2324 y Fv(A)47 b(similar)g(argumen)m(t)f(yields)i(\(V.244\).)84 b(The)47 b(pro)s(of)e(of)h(\(V.245\))g(and)g(\(V.246\))g(is)257 2444 y(rather)h(length)m(y)-8 b(,)51 b(and)c(w)m(e)g(shall)g(only)g (examine)h(the)f(tree)g(lev)m(el)h(con)m(tributions)g(to)257 2565 y Ft(w)330 2514 y Fo(\(1\))327 2575 y Fq(m;n)456 2565 y Fv(,)f(i.e.,)g(those)e(resulting)g(from)e(rescaling)i Ft(w)2135 2514 y Fo(\(0\))2132 2575 y Fq(m;n)2261 2565 y Fv(.)78 b(It)43 b(actually)i(turns)g(out)e(that)257 2685 y(these)34 b(con)m(tributions)g(are)f(the)g(dominan)m(t)g(ones.)44 b(W)-8 b(e)33 b(set)436 2930 y Ft(w)509 2886 y Fo(\(1\))p Fq(;T)506 2950 y(m;n)674 2930 y Fv([)p Ft(z)t(;)16 b(r)p 793 2946 47 4 v 4 w(;)g(K)974 2886 y Fo(\()p Fq(m;n)p Fo(\))1158 2930 y Fv(])60 b(:=)1527 2858 y Ft(i)p 1417 2907 254 4 v 1417 2995 a Fv(sin)q(\()p Ft(#)p Fv(\))1702 2930 y Ft(\032)1762 2842 y Fk(3)p 1762 2854 31 3 v 1762 2895 a(2)1803 2869 y Fo(\()p Fq(m)p Fo(+)p Fq(n)p Fo(\))p Fn(\000)p Fo(1)1752 2954 y(1)2134 2930 y Ft(w)2207 2886 y Fo(\(0\))2204 2950 y Fq(m;n)2332 2930 y Fv([)p Ft(Z)2433 2886 y Fn(\000)p Fo(1)2528 2930 y Fv(\()p Ft(z)t Fv(\))p Ft(;)16 b(\032)2746 2947 y Fo(1)2786 2930 y Ft(r)p 2786 2946 47 4 v 3 w(;)g(\032)2926 2947 y Fo(1)2966 2930 y Ft(K)3056 2886 y Fo(\()p Fq(m;n)p Fo(\))3240 2930 y Fv(])22 b Ft(:)3173 3112 y Fv(\(V.249\))257 3331 y(One)36 b(then)f(easily)h(c)m (hec)m(ks)i(that)c(\(V.211\))h(and)g(\(V.212\))f(imply)i(the)f(b)s (ounds)h(\(V.245\))257 3465 y(and)d(\(V.246\),)f(with)h Ft(w)1124 3414 y Fo(\(1\))1121 3475 y Fq(m;n)1283 3465 y Fv(replaced)g(b)m(y)h Ft(w)1876 3414 y Fo(\(1\))p Fq(;T)1873 3475 y(m;n)2040 3465 y Fv(.)p 3425 3465 4 66 v 3429 3403 59 4 v 3429 3465 V 3487 3465 4 66 v 404 3706 a(Using)e(these)i(b)s (ounds,)f(it)f(is)h(not)f(di\016cult)h(to)f(v)m(erify)h(that,)f(for)g (a)g(suitable)h(c)m(hoice)257 3826 y(of)f Ft(\032)h Fv(and)g Ft(\030)5 b Fv(,)1317 4046 y Ft(D)1398 4064 y Fo(1)p Fq(=)p Fo(2)1535 4046 y Fr(3)27 b Ft(z)64 b Fr(7!)59 b Ft(L)1959 4064 y Fo(\(1\))2054 4046 y Fv([)p Ft(z)t Fv(])28 b Fr(2)f(W)2381 4003 y Fn(0)2373 4067 y Fo(\001)3173 4046 y Fv(\(V.250\))257 4266 y(de\014nes)f(an)e(analytic)h(family)-8 b(,)26 b(where)f Fr(W)1746 4227 y Fn(0)1738 4288 y Fo(\001)1825 4266 y Fv(is)g(the)f(Banac)m(h)h(space)g(of)f(op)s(erators)g(de\014ned) 257 4387 y(in)47 b([8,)j(\(I.46\)].)85 b(Hence)48 b Ft(L)1283 4404 y Fo(\(1\))1428 4387 y Fr(2)i(W)1639 4404 y Fo(\001)1702 4387 y Fv(,)g(and)d(Lemma)g(V.16)f(implies)i(the)f(follo)m(wing)257 4507 y(theorem.)257 4677 y Fi(Theorem)37 b(V.17.)49 b Fs(F)-7 b(or)41 b(some)g(c)-5 b(onstant)41 b Ft(C)46 b(<)40 b Fr(1)h Fs(and)g(su\016ciently)h(smal)5 b(l)41 b Ft(g)h(>)e Fv(0)p Fs(,)257 4798 y Ft(L)323 4815 y Fo(\(1\))453 4798 y Fs(b)-5 b(elongs)34 b(to)h(the)g(p)-5 b(olydisk)1290 5033 y Ft(L)1356 5051 y Fo(\(1\))1513 5033 y Fr(2)62 b(B)1707 4952 y Fm(\000)1753 5033 y Ft(C)23 b(g)1897 4990 y Fo(2)p Fq(")1995 5033 y Ft(;)44 b(C)23 b(g)2210 4990 y Fo(3)p Fq("=)p Fo(2)2352 4952 y Fm(\001)2419 5033 y Ft(;)727 b Fv(\(V.251\))257 5253 y Fs(de\014ne)-5 b(d)34 b(in)h([8)o(,)g(\(I.64\)].)p eop %%Page: 87 88 87 87 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(87)404 573 y(In)38 b(other)h(w)m(ords,)i Ft(L)1172 590 y Fo(\(1\))1305 573 y Fv(is)e(a)f(prop)s(er)h(initial)f(op)s (erator)g(for)g(the)h(renormalization)257 693 y(group)j(map)h Fr(R)852 711 y Fq(\032)935 693 y Fv(de\014ned)g(in)g([8].)72 b(W)-8 b(e)43 b(ma)m(y)g(th)m(us)g(in)m(v)m(ok)m(e)h([8,)h(Theorems)f (V.7)e(and)257 814 y(V.10])33 b(to)f(obtain)g(the)h(follo)m(wing)g (result.)257 984 y Fi(Theorem)k(V.18.)49 b Fs(L)-5 b(et)39 b Fv(0)32 b Ft(<)h(#)1459 945 y Fn(0)1516 984 y Ft(<)f(#)p Fs(.)55 b(F)-7 b(or)37 b(su\016ciently)h(smal)5 b(l)37 b Ft(g)f(>)d Fv(0)p Fs(,)39 b(ther)-5 b(e)38 b(exists)257 1104 y(a)d(numb)-5 b(er,)34 b Ft(E)791 1122 y Fo(\()p Fn(1)p Fo(\))948 1104 y Fr(2)27 b(S)1099 1122 y Fo(0)p Fq(;<)1213 1104 y Fs(,)35 b(such)g(that)257 1304 y Fi(\(i\))47 b Ft(E)496 1322 y Fo(\()p Fn(1)p Fo(\))661 1304 y Fs(is)35 b(a)f(simple)g(eigenvalue)g(of)h Ft(L)1806 1322 y Fq(g)1846 1304 y Fv(\()p Ft(\022)s Fv(\))p Fs(,)g(and)257 1506 y Fi(\(ii\))47 b Fs(the)34 b(sp)-5 b(e)g(ctrum)35 b(of)g Ft(L)1208 1524 y Fq(g)1248 1506 y Fv(\()p Ft(\022)s Fv(\))g Fs(ob)-5 b(eys)876 1722 y Ft(\033)t Fv([)p Ft(L)1028 1740 y Fq(g)1068 1722 y Fv(\()p Ft(\022)s Fv(\)])49 b Fr(\\)g(S)1441 1740 y Fo(0)p Fq(;<)1617 1722 y Fr(\022)1755 1642 y Fm(\000)1801 1722 y Ft(E)1873 1740 y Fo(\()p Fn(1)p Fo(\))2040 1722 y Fv(+)38 b Fr(C)6 b Fv(\()p Ft(#)2306 1679 y Fn(0)2330 1722 y Fv(\))2368 1642 y Fm(\001)2463 1722 y Fr(\\)48 b(S)2635 1740 y Fo(0)p Fq(;<)2771 1722 y Ft(:)375 b Fv(\(V.252\))404 1979 y(A)32 b(simple)i(corollary)f(\(see) h(Fig.)e(6\))g(is)257 2149 y Fi(Corollary)k(V.19.)49 b Fs(L)-5 b(et)35 b Fv(0)27 b Ft(<)g(#)1467 2110 y Fn(0)1518 2149 y Ft(<)g(#)p Fs(.)45 b(F)-7 b(or)34 b(su\016ciently)h(smal)5 b(l)34 b Ft(g)c(>)d Fv(0)p Fs(,)257 2349 y Fi(\(i\))47 b Fv(0)35 b Fs(is)g(a)f(simple)g(eigenvalue)g(of)h Ft(L)1653 2367 y Fq(g)1693 2349 y Fv(\()p Ft(\022)s Fv(\))p Fs(,)g(and)257 2552 y Fi(\(ii\))47 b Fs(the)34 b(sp)-5 b(e)g(ctrum)35 b(of)g Ft(L)1208 2569 y Fq(g)1248 2552 y Fv(\()p Ft(\022)s Fv(\))g Fs(ob)-5 b(eys)1260 2768 y Ft(\033)t Fv([)p Ft(L)1412 2785 y Fq(g)1452 2768 y Fv(\()p Ft(\022)s Fv(\)])49 b Fr(\\)g(S)1825 2785 y Fo(0)p Fq(;<)2001 2768 y Fr(\022)62 b(C)6 b Fv(\()p Ft(#)2291 2724 y Fn(0)2315 2768 y Fv(\))48 b Fr(\\)h(S)2574 2785 y Fo(0)p Fq(;<)2710 2768 y Ft(:)436 b Fv(\(V.253\))257 3037 y Fs(Pr)-5 b(o)g(of.)49 b Fv(W)-8 b(e)39 b(\014rst)g(note)g(that)f Ft(\033)t Fv([)p Ft(L)1536 3055 y Fq(g)1577 3037 y Fv(\()p Ft(\022)s Fv(\)])f Fr(\022)p 1878 2956 131 4 v 37 w Fp(C)1944 3055 y Fn(\000)2009 3037 y Fv(,)j(b)m(y)f(analytic)g(con)m(tin)m(uation)h(and)f(the)257 3157 y(fact)33 b(that)f(the)h(sp)s(ectrum)h(of)e Ft(L)1432 3175 y Fq(g)1505 3157 y Fv(is)h(real.)44 b(Th)m(us)34 b(Im)17 b Ft(E)2283 3175 y Fo(\()p Fn(1)p Fo(\))2440 3157 y Fr(\024)27 b Fv(0.)404 3278 y(Secondly)-8 b(,)36 b(0)d(is)i(an)f(eigen)m(v)-5 b(alue)35 b(of)e Ft(L)1798 3295 y Fq(g)1839 3278 y Fv(\()p Ft(\022)s Fv(\),)h(so)g(0)c Fr(2)f Ft(E)2390 3295 y Fo(\()p Fn(1)p Fo(\))2542 3278 y Fv(+)23 b Fr(C)6 b Fv(\()p Ft(#)2793 3238 y Fn(0)2817 3278 y Fv(\),)34 b(whic)m(h)i(implies)257 3398 y(that)d Ft(E)541 3416 y Fo(\()p Fn(1)p Fo(\))698 3398 y Fv(=)27 b(0.)p 3425 3398 4 66 v 3429 3336 59 4 v 3429 3398 V 3487 3398 4 66 v 257 3851 a Fu(A)156 b(Analytic)70 b(Con)l(tin)l (uation)e(of)k(the)f(Resolv)l(en)l(t)f(of)531 4033 y(the)51 b(Liouvillian)257 4281 y Fh(A.1)132 b(Outline)46 b(of)e(the)h(strategy) 257 4466 y Fv(Our)33 b(goal)f(in)h(this)g(app)s(endix)h(is)f(to)f (establish)i(the)f(analyticit)m(y)h(of)e(the)h(resolv)m(en)m(t)1245 4691 y Ft(R)1319 4708 y Fq(g)1359 4691 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))61 b(:=)1798 4610 y Fm(\000)1860 4691 y Ft(L)1926 4708 y Fq(g)1967 4691 y Fv(\()p Ft(\022)s Fv(\))43 b Fr(\000)g Ft(z)2302 4610 y Fm(\001)2349 4632 y Fn(\000)p Fo(1)2465 4691 y Ft(;)778 b Fv(\(A.1\))257 4907 y(as)29 b(an)g(op)s(erator-v)-5 b(alued)28 b(function)h(of)f Ft(\022)j Fv(and)e Ft(z)t Fv(.)43 b(This)30 b(do)s(es)f(not)f(follo)m (w)h(from)f(a)g(direct)257 5027 y(application)44 b(of)f(standard)g(tec) m(hniques)j(in)e(dilatation)f(analyticit)m(y)-8 b(,)47 b(in)d(con)m(trast)g(to)257 5147 y([7,)d(8,)g(9],)i(b)s(ecause)g Ft(\022)h Fr(7!)d Ft(L)1328 5165 y Fq(g)1368 5147 y Fv(\()p Ft(\022)s Fv(\))g(is)h(not)f(a)g(family)g(of)g(t)m(yp)s(e)h(A,)f(for)f Ft(\022)45 b Fr(2)c Ft(D)s Fv(\(0)p Ft(;)16 b(\031)t(=)p Fv(2\).)257 5268 y(Indeed,)37 b(as)e(w)m(e)g(p)s(oin)m(t)g(out)g(b)s (elo)m(w,)h Ft(\022)d Fr(7!)d Ft(L)1882 5285 y Fq(g)1923 5268 y Fv(\()p Ft(\022)s Fv(\))35 b(is)g(not)f(ev)m(en)i(an)f(analytic) g(family)g(on)257 5388 y Fr(H)f Fv(in)f(the)g(sense)h(of)f(Kato)e (\(see,)j(e.g.,)f([33]\).)p eop %%Page: 88 89 88 88 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(88)404 573 y(Note,)52 b(ho)m(w)m(ev)m(er,)j(that)49 b(w)m(e)g(are)g(not)f(really)h(in)m(terested)i(in)e(global)f (analyticit)m(y)257 693 y(prop)s(erties)26 b(of)e Ft(R)885 711 y Fq(g)925 693 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\).)42 b(F)-8 b(or)24 b(our)g(sp)s(ectral)h(analysis,)j(it)c(su\016ces)j(to)d (ha)m(v)m(e)i(an)f(analytic)257 814 y(con)m(tin)m(uation)42 b(of)e Ft(R)1022 831 y Fq(g)1062 814 y Fv(\(0)p Ft(;)16 b(\025)27 b Fv(+)f Ft(i")p Fv(\),)43 b(with)e Ft(\025)26 b Fv(+)h Ft(i")40 b Fr(2)g Fp(C)2271 774 y Fo(+)2336 814 y Fv(,)i(0)e Ft(<)g(")g Fr(\034)f Fv(1,)j(in)f(the)g(upp)s(er)257 934 y(half-plane,)k(to)c Ft(R)951 952 y Fq(g)992 934 y Fv(\()p Ft(i#;)16 b(z)1212 895 y Fn(0)1237 934 y Fv(\),)44 b(with)f Ft(z)1627 895 y Fn(0)1694 934 y Fr(2)g Fp(C)1868 895 y Fn(\000)1975 934 y Fv(in)f(the)g(lo)m(w)m(er)h(half-plane)f(and)g Ft(#)i(>)e Fv(0.)257 1054 y(Hence,)35 b(it)d(su\016ces)j(to)e(ha)m(v)m (e)h(a)f(connected)h(domain)f Fr(A)27 b(\022)h Fp(C)2508 1015 y Fo(2)2553 1054 y Fv(,)33 b(con)m(taining)h(\(0)p Ft(;)16 b(z)t Fv(\))33 b(and)257 1175 y(\()p Ft(i#;)16 b(z)477 1135 y Fn(0)502 1175 y Fv(\),)33 b(suc)m(h)h(that)e Fr(A)27 b(3)g Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))29 b Fr(7!)d Ft(R)1671 1192 y Fq(g)1712 1175 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))33 b(is)h(analytic.)404 1295 y(The)c(construction)g(of)f Fr(A)g Fv(or,)h(rather,)g(of)f(the)h(curv)m(e)h(in)e Fr(A)h Fv(linking)g(\(0)p Ft(;)16 b(\025)f Fv(+)g Ft(i")p Fv(\))29 b(and)257 1416 y(\()p Ft(i#;)16 b(z)477 1376 y Fn(0)502 1416 y Fv(\))33 b(is)g(as)g(follo)m(ws.)404 1616 y Fr(\017)48 b Fv(First,)g(using)d(the)g(selfadjoin)m(tness)i(of)d Ft(L)2073 1633 y Fq(g)2114 1616 y Fv(,)j(w)m(e)f(pass)f(from)g Ft(R)2885 1633 y Fq(g)2925 1616 y Fv(\(0)p Ft(;)16 b(\025)30 b Fv(+)f Ft(i")p Fv(\))45 b(to)501 1736 y Ft(R)575 1754 y Fq(g)616 1736 y Fv(\(0)p Ft(;)16 b(\025)21 b Fv(+)h(2)p Ft(i)p Fv(\),)32 b(b)m(y)i(usual)f(analytic)g(con)m(tin)m(uation)h(in)f Ft(z)t Fv(.)404 1939 y Fr(\017)48 b Fv(Second,)32 b(for)d(\()p Ft(\022)s(;)16 b(z)t Fv(\))28 b(=)f(\(0)p Ft(;)16 b(\025)g Fv(+)g(2)p Ft(i)p Fv(\),)30 b(w)m(e)h(pass)g(from)e Ft(R)2474 1956 y Fq(g)2514 1939 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))31 b(to)f(a)f(more)h(regular)501 2059 y(resolv)m(en)m(t,)1001 2250 y Fm(e)983 2275 y Ft(R)1057 2293 y Fq(g)1097 2275 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))61 b(:=)e(\()p Ft(L)1640 2293 y Fq(aux)1785 2275 y Fv(+)21 b(1\))1969 2232 y Fn(\000)p Fo(1)2085 2275 y Ft(R)2159 2293 y Fq(g)2199 2275 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))23 b(\()p Ft(L)2542 2293 y Fq(aux)2686 2275 y Fv(+)f(1\))2871 2232 y Fn(\000)p Fo(1)2987 2275 y Ft(:)256 b Fv(\(A.2\))501 2492 y(Note)26 b(that)f(the)h (restriction)h(of)e Ft(R)1733 2509 y Fq(g)1773 2492 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))27 b(to)e(Dom\()p Ft(L)2437 2509 y Fq(aux)2559 2492 y Fv(\))h(can)f(b)s(e)h (reconstructed)501 2627 y(from)750 2602 y Fm(e)732 2627 y Ft(R)806 2645 y Fq(g)846 2627 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\).)404 2830 y Fr(\017)48 b Fv(The)34 b(k)m(ey)g(step)f(of)f (our)h(construction)h(is)f(as)f(follo)m(ws.)45 b(In)m(tro)s(ducing)1295 3046 y(\006)1365 3003 y Fo(+)1365 3067 y Fq(r)1484 3046 y Fv(:=)1646 2965 y Fm(\010)1705 3046 y Ft(\022)30 b Fr(2)d Fp(C)1976 2961 y Fm(\014)1976 3021 y(\014)2042 3046 y Fv(0)g Fr(\024)g Fv(Im)17 b Ft(\022)30 b(<)d(r)2579 2965 y Fm(\011)2658 3046 y Ft(;)585 b Fv(\(A.3\))501 3262 y(w)m(e)34 b(pro)m(v)m(e)g(that)1461 3479 y(\006)1531 3435 y Fo(+)1531 3504 y Fq(#)1573 3513 y Fk(0)1639 3479 y Fr(3)27 b Ft(\022)63 b Fr(7!)2014 3454 y Fm(e)1996 3479 y Ft(R)2070 3496 y Fq(g)2110 3479 y Fv(\()p Ft(\022)s(;)16 b(\025)22 b Fv(+)g(2)p Ft(i)p Fv(\))734 b(\(A.4\))501 3695 y(de\014nes)35 b(an)d(analytic)h(family)g(with)746 3827 y Fm(\015)746 3887 y(\015)836 3886 y(e)818 3912 y Ft(R)892 3929 y Fq(g)932 3912 y Fv(\()p Ft(\022)s(;)16 b(\025)22 b Fv(+)g(2)p Ft(i)p Fv(\))1374 3827 y Fm(\015)1374 3887 y(\015)1489 3912 y Fr(\024)60 b Fv(dist)p Fr(f)p Fv(NumRan)q([)p Ft(L)2308 3929 y Fq(g)2348 3912 y Fv(\()p Ft(\022)s Fv(\)])16 b Ft(;)33 b(\025)22 b Fv(+)f(2)p Ft(i)p Fr(g)2882 3868 y Fn(\000)p Fo(1)2998 3912 y Ft(:)245 b Fv(\(A.5\))501 4128 y(The)41 b(main)f(di\016cult)m(y)i(w)m(e)f(are)f (facing)g(here)h(is)f(that)g(the)g(co)s(e\016cien)m(t)i(in)e(fron)m(t) 501 4248 y(of)h(the)h(dominan)m(t)g(op)s(erator)f Ft(L)1711 4266 y Fq(aux)1875 4248 y Fv(is)h(linearly)h(v)-5 b(anishing,)45 b(as)c Ft(\022)46 b Fr(!)41 b Fv(0,)j(since)501 4369 y Ft(L)567 4386 y Fq(g)608 4369 y Fv(\()p Ft(\022)s Fv(\))38 b(=)h(cosh\()p Ft(\022)s Fv(\))p Ft(L)1259 4386 y Fq(f)1332 4369 y Fr(\000)26 b Fv(sinh)q(\()p Ft(\022)s Fv(\))p Ft(L)1798 4386 y Fq(aux)1946 4369 y Fv(+)h Ft(g)t(W)14 b Fv(\()p Ft(\022)s Fv(\).)63 b(So,)41 b(while)f(all)g(other)f(terms) 501 4489 y(in)h Ft(L)688 4507 y Fq(g)729 4489 y Fv(\()p Ft(\022)s Fv(\))f(are)h(relativ)m(ely)i(b)s(ounded)e(w.r.t.)g Ft(L)2233 4507 y Fq(aux)2356 4489 y Fv(,)i(their)e(relativ)m(e)h(b)s (ounds)f(are)501 4609 y(div)m(ergen)m(t,)h(as)c Ft(\022)i Fr(!)34 b Fv(0.)58 b(Our)37 b(main)h(observ)-5 b(ation,)39 b(ho)m(w)m(ev)m(er,)i(is)d(that)f(w)m(e)i(only)501 4730 y(need)26 b(to)f(con)m(trol)g(the)g Fs(imaginary)i(p)-5 b(art)25 b Fv(of)g Ft(L)2131 4747 y Fq(g)2171 4730 y Fv(\()p Ft(\022)s Fv(\))g(and)g(that)g(the)g(imaginary)g(part)501 4868 y(is)j(asymptotically)g(of)f(the)g(form)g(Im)17 b Ft(L)1936 4886 y Fq(g)1976 4868 y Fv(\()p Ft(\022)s Fv(\))27 b(=)g Fr(\000)p Ft(i)p Fv(Im)q(\()p Ft(\022)s Fv(\))2580 4788 y Fm(\000)2626 4868 y Ft(L)2692 4886 y Fq(aux)2825 4868 y Fv(+)10 b Ft(g)2965 4843 y Fm(f)2962 4868 y Ft(W)j Fv(\()p Ft(\022)s Fv(\),)28 b(where)504 4986 y Fm(f)501 5011 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))40 b(is)g(relativ)m(ely)i Ft(L)1375 5029 y Fq(aux)1497 5011 y Fv(-b)s(ounded)e(with)h(zero)f(relativ)m(e)h(b)s(ound,)h(as)d Ft(\022)k Fr(!)38 b Fv(0.)501 5132 y(Hence,)43 b(for)38 b(su\016cien)m(tly)k(small)e Fr(j)p Ft(\022)s Fr(j)g Fv(with)g(Im)16 b Ft(\022)42 b Fr(\025)c Fv(0,)i(the)g(imaginary)g (part)f(of)501 5252 y Ft(L)567 5270 y Fq(g)608 5252 y Fv(\()p Ft(\022)s Fv(\))f(is)i(negativ)m(e)g(de\014nite.)63 b(The)39 b(assumption)i(Im)16 b Ft(z)42 b Fr(\025)c Fv(2)g(no)m(w)h (insures)i(the)501 5388 y(di\013eren)m(tiabilt)m(y)35 b(of)1286 5363 y Fm(e)1268 5388 y Ft(R)1342 5406 y Fq(g)1382 5388 y Fv(\()p Ft(\022)s(;)16 b(\025)23 b Fv(+)e(2)p Ft(i)p Fv(\))33 b(at)f Ft(\022)e Fv(=)d(0.)p eop %%Page: 89 90 89 89 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(89)404 573 y Fr(\017)48 b Fv(F)-8 b(ourth,)35 b(con)m(tin)m(uing)h (from)f Ft(\022)e Fv(=)e(0)j(to)h Ft(\022)f Fv(=)c Ft(i#)p Fv(,)36 b(0)31 b Ft(<)f(#)2535 534 y Fn(0)2535 593 y Fo(0)2606 573 y Fr(\024)g Ft(#)h Fr(\024)g Ft(#)2964 590 y Fo(0)3034 573 y Ft(<)g(\031)t(=)p Fv(2,)k(the)501 703 y(norm)g(estimate)g(\(A.5\))g(enables)h(us)f(to)f(analytically)h (con)m(tin)m(ue)2939 678 y Fm(e)2921 703 y Ft(R)2995 721 y Fq(g)3036 703 y Fv(\()p Ft(i#;)16 b(z)t Fv(\))36 b(in)e Ft(z)501 824 y Fv(from)f Ft(z)e Fv(=)c Ft(\025)22 b Fv(+)g(2)p Ft(i)32 b Fv(to)g(the)h(connected)i(comp)s(onen)m(t)e Fr(G)39 b Fv(of)1400 963 y Fm(\010)1458 1044 y Ft(z)32 b Fr(2)27 b Fp(C)1732 959 y Fm(\014)1732 1019 y(\014)1797 1044 y Fr(k)p Ft(L)1912 1061 y Fq(g)1953 1044 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)t Fr(k)28 b Ft(<)e Fr(1)2538 963 y Fm(\011)3270 1044 y Fv(\(A.6\))501 1264 y(con)m(taining)35 b Ft(\025)22 b Fv(+)g(2)p Ft(i)p Fv(.)47 b(Since,)35 b(on)f(Dom\()p Ft(L)2039 1281 y Fq(aux)2161 1264 y Fv(\),)g(w)m(e)h(can)f(obtain)g Ft(R)2964 1281 y Fq(g)3004 1264 y Fv(\()p Ft(i#;)16 b(z)t Fv(\))35 b(from)519 1374 y Fm(e)501 1399 y Ft(R)575 1417 y Fq(g)616 1399 y Fv(\()p Ft(i#;)16 b(z)t Fv(\),)36 b(w)m(e)e(arriv)m(e)h(at)e(the)i (desired)g(analytic)f(con)m(tin)m(uation)h(of)f(matrix)g(ele-)501 1520 y(men)m(ts)g Fr(h)p Ft(')p Fr(j)p Ft(R)989 1537 y Fq(g)1030 1520 y Fv(\(0)p Ft(;)16 b(z)t Fv(\))p Ft( )t Fr(i)p Fv(,)33 b(for)f Ft(';)16 b( )37 b Fv(in)c(the)g(dense)h(set)f Fr(D)i Fv(de\014ned)f(in)f(\(V.123\).)257 1809 y Fh(A.2)132 b(Key)44 b(Step)257 1993 y Fv(T)-8 b(o)31 b(carry)f(out)g(the)h(third)f (step)h(indicated)g(ab)s(o)m(v)m(e,)h(w)m(e)f(\014rst)g(pro)m(v)m(e)g (some)g(preparatory)257 2114 y(lemmata.)257 2284 y Fi(Lemma)37 b(A.1.)49 b Fs(L)-5 b(et)41 b Fv(0)36 b Ft(<)g(#)1334 2302 y Fo(0)1411 2284 y Ft(<)g(\031)t(=)p Fv(2)p Fs(,)41 b(and)f(assume)f(that)i Ft(\022)e Fr(2)e Fv(\006)2753 2243 y Fo(+)2753 2312 y Fq(#)2795 2321 y Fk(0)2834 2284 y Fs(,)k Fv(Im)q Ft(z)g Fr(\025)c Fv(2)p Fs(,)k(and)257 2404 y Fv(0)27 b Ft(<)g(g)k Fr(\034)26 b Ft(M)732 2422 y Fq(#)774 2431 y Fk(0)840 2404 y Fv(:=)h Ft(#)1027 2422 y Fo(0)1067 2404 y Fv(\()p Ft(\031)t(=)p Fv(2)22 b Fr(\000)f Ft(#)1438 2422 y Fo(0)1478 2404 y Fv(\))p Fs(.)45 b(Then)34 b Ft(L)1911 2422 y Fq(g)1951 2404 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)40 b Fs(is)35 b(invertible)f(and)816 2559 y Fm(\015)816 2619 y(\015)888 2563 y(\000)933 2644 y Ft(L)999 2661 y Fq(g)1040 2644 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)1332 2563 y Fm(\001)1379 2586 y Fn(\000)p Fo(1)1489 2559 y Fm(\015)1489 2619 y(\015)1607 2644 y Fr(\024)61 b Fv(dist)q Fr(f)p Fv(NumRan[)p Ft(L)2427 2661 y Fq(g)2468 2644 y Fv(\()p Ft(\022)s Fv(\)])16 b Ft(;)33 b(z)t Fr(g)2793 2600 y Fn(\000)p Fo(1)2909 2644 y Ft(:)334 b Fv(\(A.7\))404 2864 y Fs(Pr)-5 b(o)g(of:)56 b Fv(It)39 b(su\016ces)i(to)e(pro)m(v)m(e)h(\(A.7\))f(only)h(for)e (purely)j(imaginary)e Ft(\022)i Fv(=)d Ft(i#)p Fv(,)k(0)37 b Fr(\024)257 2984 y Ft(#)28 b(<)f(#)502 3002 y Fo(0)542 2984 y Fv(,)j(since)h(the)f(real)g(part)g(of)f Ft(\022)k Fv(giv)m(es)e(rise)f(to)g(a)f(unitary)h(dilatation)g(whic)m(h)h(lea)m (v)m(es)257 3105 y(norms)38 b(and)e(n)m(umerical)i(ranges)f(and)g (hence)h(b)s(oth)e(sides)i(of)e(\(A.7\))h(unc)m(hanged.)57 b(W)-8 b(e)257 3225 y(observ)m(e)35 b(that,)d(b)m(y)i(Cauc)m(h)m(y's)h (estimate)f(and)e(\(V.116\),)g(w)m(e)i(ha)m(v)m(e)331 3441 y Fm(\015)331 3501 y(\015)386 3526 y Ft(@)437 3543 y Fq(\022)476 3526 y Ft(w)549 3482 y Fo(\()p Fq(\022)r Fo(\))546 3546 y Fq(\033)n(;\034)648 3526 y Fv(\()p Ft(k)s Fv(\))778 3441 y Fm(\015)778 3501 y(\015)893 3526 y Fr(\024)1028 3385 y Fm(\022)1290 3455 y Fv(2)p 1111 3503 406 4 v 1111 3591 a Ft(\031)t(=)p Fv(2)k Fr(\000)g Ft(#)1477 3609 y Fo(0)1526 3385 y Fm(\023)1783 3455 y Fv(2)p 1626 3503 364 4 v 1626 3523 a Fm(p)p 1726 3523 264 4 v 85 x Fv(cos\()p Ft(#)p Fv(\))2021 3385 y Fm(\022)2095 3526 y Fv(1)43 b(+)2419 3455 y(1)p 2316 3503 255 4 v 2316 3591 a Ft(\014)6 b(!)t Fv(\()p Ft(k)s Fv(\))2581 3385 y Fm(\023)2654 3408 y Fo(1)p Fq(=)p Fo(2)2786 3526 y Fr(j)2808 3500 y Ft(~)2813 3526 y(k)s Fr(j)2894 3482 y Fq(\026)2962 3526 y Ft(\024)p Fv(\()p Ft(k)s Fv(\))22 b Ft(:)73 b Fv(\(A.8\))257 3816 y(This)34 b(and)f(Lemma)g(IV.1)g(imply)g(that)483 3951 y Fm(\015)483 4011 y(\015)538 4036 y Fv(Im)q Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Fr(g)i Fv(\()p Ft(L)1103 4053 y Fq(aux)1247 4036 y Fv(+)21 b(1\))1431 3992 y Fn(\000)p Fo(1)1525 3951 y Fm(\015)1525 4011 y(\015)1664 4036 y Fv(=)1822 3951 y Fm(\015)1822 4011 y(\015)1878 4036 y Fv(Im)p Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(W)14 b Fv(\(0\))p Fr(g)i Fv(\()p Ft(L)2792 4053 y Fq(aux)2936 4036 y Fv(+)21 b(1\))3120 3992 y Fn(\000)p Fo(1)3214 3951 y Fm(\015)3214 4011 y(\015)1664 4181 y Fv(=)82 b Fr(O)1903 4100 y Fm(\000)1965 4181 y Ft(#=)-5 b(M)2160 4198 y Fq(#)2202 4207 y Fk(0)2257 4100 y Fm(\001)2324 4181 y Ft(;)919 b Fv(\(A.9\))257 4401 y(and)33 b(similarly)971 4536 y Fm(\015)971 4596 y(\015)1026 4621 y Fv(Re)p Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Fr(g)i Fv(\()p Ft(L)1589 4638 y Fq(aux)1733 4621 y Fv(+)21 b(1\))1917 4577 y Fn(\000)p Fo(1)2011 4536 y Fm(\015)2011 4596 y(\015)2126 4621 y Fv(=)60 b Fr(O)2343 4540 y Fm(\000)2404 4621 y Fv(1)p Ft(=)-5 b(M)2591 4638 y Fq(#)2633 4647 y Fk(0)2688 4540 y Fm(\001)2755 4621 y Ft(:)440 b Fv(\(A.10\))257 4841 y(Next,)34 b(w)m(e)g(note)e(that)411 5061 y Ft(L)477 5078 y Fq(g)518 5061 y Fv(\()p Ft(\022)s Fv(\))642 5017 y Fn(\003)764 5061 y Fv(=)83 b(cos)17 b Ft(#)f(L)1209 5078 y Fq(f)1299 5061 y Fv(+)43 b Ft(i)33 b Fv(sin)17 b Ft(#)f(L)1759 5078 y Fq(aux)1926 5061 y Fv(+)43 b Ft(g)t(W)14 b Fv(\()p Ft(\022)s Fv(\))2326 5017 y Fn(\003)3222 5061 y Fv(\(A.11\))764 5206 y(=)83 b(cos)17 b Ft(#)f(L)1209 5224 y Fq(f)1299 5206 y Fv(+)43 b Ft(g)t Fv(Re)p Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Fr(g)42 b Fv(+)h Ft(i)2106 5126 y Fm(\000)2152 5206 y Fv(sin)17 b Ft(#)f(L)2427 5224 y Fq(aux)2594 5206 y Fv(+)43 b Ft(g)t Fv(Im)p Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))3159 5163 y Fn(\003)3198 5206 y Fr(g)3247 5126 y Fm(\001)3314 5206 y Ft(:)p eop %%Page: 90 91 90 90 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(90)257 573 y(Th)m(us,)35 b(for)d(an)m(y)h Ft( )e Fr(2)c Fv(Dom[)p Ft(L)1349 590 y Fq(aux)1471 573 y Fv(])h Fr(\022)f Fv(Dom)o([)p Ft(L)1926 590 y Fq(g)1967 573 y Fv(\()p Ft(\022)s Fv(\))2091 534 y Fn(\003)2130 573 y Fv(],)33 b(w)m(e)g(ha)m(v)m(e)344 818 y(Im)460 708 y Fm(D)521 818 y Fv(\()p Ft(L)625 836 y Fq(aux)769 818 y Fv(+)22 b(1\))954 775 y Fn(\000)p Fo(1)1064 818 y Ft( )1158 704 y Fm(\014)1158 763 y(\014)1158 823 y(\014)1219 818 y Ft(L)1285 836 y Fq(g)1325 818 y Fv(\()p Ft(\022)s Fv(\))1449 775 y Fn(\003)1505 818 y Ft( )1572 708 y Fm(E)3222 818 y Fv(\(A.12\))427 1056 y(=)596 984 y(1)p 596 1033 49 4 v 596 1121 a(2)671 945 y Fm(D)732 1056 y Ft( )826 941 y Fm(\014)826 1001 y(\014)826 1061 y(\014)886 975 y(\002)928 1056 y Fv(\()p Ft(L)1032 1073 y Fq(aux)1176 1056 y Fv(+)f(1\))1360 1012 y Fn(\000)p Fo(1)1471 1056 y Ft(;)32 b(g)t Fv(Re)p Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Fr(g)2024 975 y Fm(\003)2081 1056 y Ft( )2148 945 y Fm(E)2252 1056 y Fv(+)43 b(sin)17 b Ft(#)2580 945 y Fm(D)2642 1056 y Ft( )2736 941 y Fm(\014)2736 1001 y(\014)2736 1061 y(\014)2769 945 y(\020)2923 984 y Ft(L)2989 1002 y Fq(aux)p 2839 1033 357 4 v 2839 1121 a Ft(L)2905 1138 y Fq(aux)3049 1121 y Fv(+)k(1)3205 945 y Fm(\021)3281 1056 y Ft( )3348 945 y Fm(E)586 1286 y Fv(+)g(Re)798 1176 y Fm(D)859 1286 y Fv(\()p Ft(L)963 1304 y Fq(aux)1108 1286 y Fv(+)g(1\))1292 1243 y Fn(\000)p Fo(1)1402 1286 y Ft( )1497 1172 y Fm(\014)1497 1231 y(\014)1497 1291 y(\014)1557 1286 y Ft(g)t Fv(Im)p Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))2003 1243 y Fn(\003)2063 1286 y Fr(\000)22 b Ft(W)14 b Fv(\(0\))p Fr(g)i Ft( )2524 1176 y Fm(E)2606 1286 y Ft(:)257 1526 y Fv(As)33 b(in)g(\(IV.74\){\(IV.75\),)f(w)m(e)i(obtain)e(from)h(Lemma) g(IV.1)g(that)563 1651 y Fm(\014)563 1711 y(\014)563 1771 y(\014)613 1655 y(D)673 1766 y Ft( )767 1651 y Fm(\014)767 1711 y(\014)767 1771 y(\014)828 1685 y(\002)869 1766 y Fv(\()p Ft(L)973 1783 y Fq(aux)1118 1766 y Fv(+)21 b(1\))1302 1722 y Fn(\000)p Fo(1)1412 1766 y Ft(;)33 b(g)t Fv(Re)o Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Fr(g)1965 1685 y Fm(\003)2022 1766 y Ft( )2089 1655 y Fm(E)2167 1651 y(\014)2167 1711 y(\014)2167 1771 y(\014)3222 1766 y Fv(\(A.13\))646 1978 y(=)83 b Ft(g)856 1864 y Fm(\014)856 1924 y(\014)856 1983 y(\014)905 1868 y(D)966 1978 y Fv(\()p Ft(L)1070 1996 y Fq(aux)1214 1978 y Fv(+)21 b(1\))1398 1935 y Fn(\000)p Fo(1)1509 1978 y Ft( )1603 1864 y Fm(\014)1603 1924 y(\014)1603 1983 y(\014)1663 1898 y(\002)1705 1978 y Ft(L)1771 1996 y Fq(aux)1909 1978 y Ft(;)33 b Fv(Re)p Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))p Fr(g)2412 1898 y Fm(\003)2469 1978 y Fv(\()p Ft(L)2573 1996 y Fq(aux)2717 1978 y Fv(+)22 b(1\))2902 1935 y Fn(\000)p Fo(1)3012 1978 y Ft( )3079 1868 y Fm(E)3156 1864 y(\014)3156 1924 y(\014)3156 1983 y(\014)646 2163 y Fr(\024)83 b Ft(C)23 b(g)965 2082 y Fm(\012)1028 2163 y Ft( )1122 2078 y Fm(\014)1122 2138 y(\014)1182 2163 y Fv(\()p Ft(L)1286 2180 y Fq(aux)1431 2163 y Fv(+)e(1\))1615 2119 y Fn(\000)p Fo(1)1725 2163 y Ft( )1792 2082 y Fm(\013)1861 2163 y Ft(;)257 2375 y Fv(whereas)34 b(\(A.9\))f(implies)h(that)422 2500 y Fm(\014)422 2560 y(\014)422 2620 y(\014)471 2504 y(D)532 2615 y Fv(\()p Ft(L)636 2632 y Fq(aux)780 2615 y Fv(+)22 b(1\))965 2571 y Fn(\000)p Fo(1)1075 2615 y Ft( )1169 2500 y Fm(\014)1169 2560 y(\014)1169 2620 y(\014)1230 2615 y Ft(g)t Fv(Im)o Fr(f)p Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))1675 2571 y Fn(\003)1736 2615 y Fr(\000)22 b Ft(W)14 b Fv(\(0\))p Fr(g)i Ft( )2197 2504 y Fm(E)2273 2500 y(\014)2273 2560 y(\014)2273 2620 y(\014)2366 2615 y Fv(=)59 b Fr(O)2582 2534 y Fm(\000)2644 2615 y Ft(g)19 b(#)d(M)2887 2571 y Fn(\000)p Fo(1)2877 2640 y Fq(#)2919 2649 y Fk(0)2983 2534 y Fm(\001)3045 2615 y Fr(h)p Ft( )t Fr(j)p Ft( )t Fr(i)22 b Ft(:)3222 2772 y Fv(\(A.14\))257 2978 y(These)35 b(estimates)f(yield)436 3212 y(Im)553 3102 y Fm(D)614 3212 y Fv(\()p Ft(L)718 3230 y Fq(aux)862 3212 y Fv(+)21 b(1\))1046 3169 y Fn(\000)p Fo(1)1157 3212 y Ft( )1251 3098 y Fm(\014)1251 3157 y(\014)1251 3217 y(\014)1311 3212 y Ft(L)1377 3230 y Fq(g)1418 3212 y Fv(\()p Ft(\022)s Fv(\))1542 3169 y Fn(\003)1597 3212 y Ft( )1664 3102 y Fm(E)3222 3212 y Fv(\(A.15\))519 3450 y Fr(\025)83 b Fv(sin)17 b Ft(#)887 3339 y Fm(D)949 3450 y Ft( )1043 3335 y Fm(\014)1043 3395 y(\014)1043 3455 y(\014)1076 3339 y(\020)1230 3378 y Ft(L)1296 3396 y Fq(aux)p 1146 3427 V 1146 3515 a Ft(L)1212 3532 y Fq(aux)1356 3515 y Fv(+)k(1)1512 3339 y Fm(\021)1588 3450 y Ft( )1655 3339 y Fm(E)1759 3450 y Fr(\000)43 b(O)1959 3369 y Fm(\000)2005 3450 y Ft(g)19 b(#)d(M)2248 3406 y Fn(\000)p Fo(1)2238 3475 y Fq(#)2280 3484 y Fk(0)2344 3369 y Fm(\001)2411 3450 y Fr(h)p Ft( )t Fr(j)p Ft( )t Fr(i)1645 3652 y(\000)21 b Ft(C)i(g)1902 3571 y Fm(\012)1965 3652 y Ft( )2059 3567 y Fm(\014)2059 3627 y(\014)2120 3652 y Fv(\()p Ft(L)2224 3669 y Fq(aux)2368 3652 y Fv(+)e(1\))2552 3609 y Fn(\000)p Fo(1)2663 3652 y Ft( )2730 3571 y Fm(\013)519 3806 y Fr(\025)83 b(\000)p Fv(\(1)21 b(+)h Ft(C)7 b(g)t Fv(\))1142 3726 y Fm(\012)1205 3806 y Ft( )1299 3722 y Fm(\014)1299 3781 y(\014)1359 3806 y Fv(\()p Ft(L)1463 3824 y Fq(aux)1607 3806 y Fv(+)22 b(1\))1792 3763 y Fn(\000)p Fo(1)1902 3806 y Ft( )1969 3726 y Fm(\013)2060 3806 y Fv(+)2179 3726 y Fm(\000)2224 3806 y Fv(sin)17 b Ft(#)44 b Fr(\000)g Fv(\()p Ft(g)19 b(#)d(M)2862 3763 y Fn(\000)p Fo(1)2852 3832 y Fq(#)2894 3841 y Fk(0)2958 3806 y Fv(\))3012 3726 y Fm(\001)3079 3806 y Fr(h)p Ft( )t Fr(j)p Ft( )t Fr(i)519 3961 y(\025)83 b(\000)p Fv(\(1)21 b(+)h Ft(C)7 b(g)t Fv(\))1142 3880 y Fm(\012)1205 3961 y Ft( )1299 3876 y Fm(\014)1299 3936 y(\014)1359 3961 y Fv(\()p Ft(L)1463 3978 y Fq(aux)1607 3961 y Fv(+)22 b(1\))1792 3917 y Fn(\000)p Fo(1)1902 3961 y Ft( )1969 3880 y Fm(\013)2038 3961 y Ft(;)257 4173 y Fv(pro)m(vided)36 b(that)d Ft(g)g Fr(\034)c Ft(M)1173 4191 y Fq(#)1215 4200 y Fk(0)1254 4173 y Fv(.)47 b(Note)35 b(that)e(\(A.15\))h(extends)i(to)d(an)m(y)i Ft( )e Fr(2)c Fv(Dom[)p Ft(L)3237 4191 y Fq(g)3278 4173 y Fv(\()p Ft(\022)s Fv(\))3402 4134 y Fn(\003)3441 4173 y Fv(],)257 4293 y(b)m(y)34 b(con)m(tin)m(uit)m(y)-8 b(.)404 4414 y(Assuming)29 b(no)m(w)f(that)f Ft( )k Fr(2)d Fv(Ker)p Fr(f)p Ft(L)1710 4431 y Fq(g)1750 4414 y Fv(\()p Ft(\022)s Fv(\))1874 4375 y Fn(\003)1925 4414 y Fr(\000)18 b Fv(\026)-55 b Ft(z)t Fr(g)p Fv(,)29 b(w)m(e)f(deriv)m(e)i(from)d (Estimate)i(\(A.15\))257 4534 y(the)k(follo)m(wing)g(inequalit)m(y)-8 b(,)466 4747 y Fr(\000)p Fv(Im\()p Ft(z)t Fv(\))799 4666 y Fm(\012)847 4747 y Fv(\()p Ft(L)951 4764 y Fq(aux)1095 4747 y Fv(+)22 b(1\))1280 4703 y Fn(\000)p Fo(1)1390 4747 y Ft( )1484 4662 y Fm(\014)1484 4722 y(\014)1544 4747 y Ft( )1611 4666 y Fm(\013)1741 4747 y Fv(=)83 b(Im)2017 4666 y Fm(\012)2064 4747 y Fv(\()p Ft(L)2168 4764 y Fq(aux)2312 4747 y Fv(+)22 b(1\))2497 4703 y Fn(\000)p Fo(1)2607 4747 y Ft( )2701 4662 y Fm(\014)2701 4722 y(\014)2767 4747 y Fv(\026)-55 b Ft(z)21 b( )2894 4666 y Fm(\013)3222 4747 y Fv(\(A.16\))1741 4901 y(=)83 b(Im)2017 4820 y Fm(\012)2064 4901 y Fv(\()p Ft(L)2168 4919 y Fq(aux)2312 4901 y Fv(+)22 b(1\))2497 4858 y Fn(\000)p Fo(1)2607 4901 y Ft( )2701 4816 y Fm(\014)2701 4876 y(\014)2761 4901 y Ft(L)2827 4919 y Fq(g)2868 4901 y Fv(\()p Ft(\022)s Fv(\))2992 4858 y Fn(\003)3047 4901 y Ft( )3114 4820 y Fm(\013)1741 5055 y Fr(\025)83 b(\000)p Fv(\(1)22 b(+)f Ft(C)7 b(g)t Fv(\))2364 4975 y Fm(\012)2427 5055 y Ft( )2521 4971 y Fm(\014)2521 5030 y(\014)2581 5055 y Fv(\()p Ft(L)2685 5073 y Fq(aux)2829 5055 y Fv(+)22 b(1\))3014 5012 y Fn(\000)p Fo(1)3124 5055 y Ft( )3191 4975 y Fm(\013)3260 5055 y Ft(:)257 5268 y Fv(Since)27 b(Im)q Ft(z)k Fr(\025)c Fv(2,)g(this)f (estimate)h(implies)g(that)f Ft( )31 b Fv(=)c(0,)g(pro)m(vided)g Ft(g)j(>)d Fv(0)e(is)h(su\016cien)m(tly)257 5388 y(small.)41 b(Hence,)26 b(Ran)p Fr(f)p Ft(L)1146 5406 y Fq(g)1187 5388 y Fv(\()p Ft(\022)s Fv(\))r Fr(\000)r Ft(z)t Fr(g)d Fv(is)g(dense,)k(and)c(w)m(e)g(ma)m(y)h(de\014ne)g(an)f(in)m(v)m(erse,) 3141 5307 y Fm(\000)3187 5388 y Ft(L)3253 5406 y Fq(g)3293 5388 y Fv(\()p Ft(\022)s Fv(\))r Fr(\000)p eop %%Page: 91 92 91 91 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(91)257 574 y Ft(z)306 494 y Fm(\001)353 516 y Fn(\000)p Fo(1)447 574 y Fv(.)43 b(F)-8 b(urthermore,)31 b(the)h(densit)m(y)g(of) e(Ran)p Fr(f)p Ft(L)2007 592 y Fq(g)2047 574 y Fv(\()p Ft(\022)s Fv(\))18 b Fr(\000)g Ft(z)t Fr(g)31 b Fv(insures)i(the)e(v)-5 b(alidit)m(y)32 b(of)e(the)257 695 y(second)k(equation)g(in)e(the)h (follo)m(wing)g(n)m(umerical)h(range)f(estimate,)520 839 y Fm(\015)520 898 y(\015)591 843 y(\000)637 923 y Ft(L)703 941 y Fq(g)743 923 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)g Ft(z)1036 843 y Fm(\001)1082 865 y Fn(\000)p Fo(1)1193 839 y Fm(\015)1193 898 y(\015)3222 923 y Fv(\(A.17\))603 1106 y(=)82 b(sup)908 995 y Fm(n)975 1021 y(\015)975 1081 y(\015)1046 1025 y(\000)1092 1106 y Ft(L)1158 1123 y Fq(g)1199 1106 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)1491 1025 y Fm(\001)1537 1048 y Fn(\000)p Fo(1)1648 1106 y Ft( )1731 1021 y Fm(\015)1731 1081 y(\015)1819 991 y(\014)1819 1051 y(\014)1819 1111 y(\014)1885 1106 y Ft( )31 b Fr(2)c Fv(Dom)o([)p Ft(L)2368 1123 y Fq(g)2409 1106 y Fv(\()p Ft(\022)s Fv(\)])16 b Ft(;)44 b Fr(k)p Ft( )t Fr(k)27 b Fv(=)f(1)2990 995 y Fm(o)603 1318 y Fv(=)82 b(sup)908 1208 y Fm(n)975 1234 y(\015)975 1294 y(\015)1046 1238 y(\000)1092 1318 y Ft(L)1158 1336 y Fq(g)1199 1318 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)1491 1238 y Fm(\001)1553 1318 y Ft( )1637 1234 y Fm(\015)1637 1294 y(\015)1692 1260 y Fn(\000)p Fo(1)1819 1204 y Fm(\014)1819 1264 y(\014)1819 1323 y(\014)1885 1318 y Ft(')27 b Fr(2)g Fv(Ran)p Fr(f)p Ft(L)2358 1336 y Fq(g)2398 1318 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)t Fr(g)16 b Ft(;)44 b Fr(k)p Ft(')p Fr(k)27 b Fv(=)g(1)3167 1208 y Fm(o)603 1531 y Fr(\024)82 b Fv(sup)908 1420 y Fm(n)975 1446 y(\014)975 1506 y(\014)1008 1531 y Fr(h)p Ft(')p Fr(j)p Ft(L)1203 1549 y Fq(g)1243 1531 y Fv(\()p Ft(\022)s Fv(\))p Ft(')p Fr(i)22 b(\000)g Ft(z)1638 1446 y Fm(\014)1638 1506 y(\014)1671 1473 y Fn(\000)p Fo(1)1798 1416 y Fm(\014)1798 1476 y(\014)1798 1536 y(\014)1864 1531 y Ft(')27 b Fr(2)g Fv(Ran)p Fr(f)p Ft(L)2337 1549 y Fq(g)2377 1531 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)t Fr(g)16 b Ft(;)45 b Fr(k)p Ft(')p Fr(k)26 b Fv(=)h(1)3146 1420 y Fm(o)603 1730 y Fv(=)82 b(dist)919 1649 y Fm(\010)977 1730 y Fv(NumRan)q([)p Ft(L)1454 1748 y Fq(g)1494 1730 y Fv(\()p Ft(\022)s Fv(\)])16 b Ft(;)33 b(z)1770 1649 y Fm(\011)1829 1672 y Fn(\000)p Fo(1)1945 1730 y Ft(:)p 2447 1730 4 66 v 2451 1668 59 4 v 2451 1730 V 2509 1730 4 66 v 257 1950 a Fi(Lemma)k(A.2.)49 b Fs(L)-5 b(et)41 b Fv(0)36 b Ft(<)g(#)1334 1968 y Fo(0)1411 1950 y Ft(<)g(\031)t(=)p Fv(2)p Fs(,)41 b(and)f(assume)f(that)i Ft(\022)e Fr(2)e Fv(\006)2753 1909 y Fo(+)2753 1978 y Fq(#)2795 1987 y Fk(0)2834 1950 y Fs(,)k Fv(Im)q Ft(z)g Fr(\025)c Fv(2)p Fs(,)k(and)257 2071 y Fv(0)27 b Ft(<)g(g)k Fr(\034)26 b Ft(M)732 2088 y Fq(#)774 2097 y Fk(0)840 2071 y Fv(:=)h Ft(#)1027 2088 y Fo(0)1067 2071 y Fv(\()p Ft(\031)t(=)p Fv(2)22 b Fr(\000)f Ft(#)1438 2088 y Fo(0)1478 2071 y Fv(\))p Fs(.)45 b(Then)656 2306 y Ft(B)730 2324 y Fn(\006)790 2306 y Fv(\()p Ft(\022)s Fv(\))62 b(:=)f(\()p Ft(L)1244 2324 y Fq(aux)1389 2306 y Fv(+)21 b(1\))1573 2263 y Fn(\007)p Fo(1)1689 2226 y Fm(\000)1735 2306 y Ft(L)1801 2324 y Fq(g)1841 2306 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)2133 2226 y Fm(\001)2180 2248 y Fn(\000)p Fo(1)2296 2306 y Fv(\()p Ft(L)2400 2324 y Fq(aux)2544 2306 y Fv(+)h(1\))2729 2263 y Fn(\006)p Fo(1)3222 2306 y Fv(\(A.18\))257 2526 y Fs(is)35 b(de\014ne)-5 b(d)34 b(on)g Fv(Dom[)p Ft(L)1128 2544 y Fq(aux)1250 2526 y Fv(])h Fs(and)g(extends)f(to)h(a)g(b)-5 b(ounde)g(d)34 b(op)-5 b(er)g(ator)34 b(on)h Fr(H)h Fs(of)f(norm)371 2760 y Fr(k)p Ft(B)494 2778 y Fn(\006)553 2760 y Fv(\()p Ft(\022)s Fv(\))p Fr(k)62 b(\024)g Ft(e)971 2717 y Fo(2)p Fn(j)p Fo(Re)12 b Fq(\022)r Fn(j)1196 2680 y Fm(\000)1242 2760 y Fv(1)21 b(+)h Fr(O)s Fv(\()p Ft(g)t(=)-5 b(M)1718 2778 y Fq(#)1760 2787 y Fk(0)1797 2760 y Fv(\))1835 2680 y Fm(\001)1897 2760 y Fv(dist)2054 2680 y Fm(\010)2113 2760 y Fv(NumRan[)p Ft(L)2589 2778 y Fq(g)2630 2760 y Fv(\()p Ft(\022)s Fv(\)])16 b Ft(;)33 b(z)2906 2680 y Fm(\011)2964 2702 y Fn(\000)p Fo(1)3080 2760 y Ft(:)115 b Fv(\(A.19\))404 2980 y Fs(Pr)-5 b(o)g(of:)43 b Fv(W)-8 b(e)33 b(\014rst)g(notice)g(that)f(Dom[)p Ft(B)1877 2998 y Fn(\006)1936 2980 y Fv(\()p Ft(\022)s Fv(\)])27 b(=)g(Dom[)p Ft(B)2523 2998 y Fn(\006)2582 2980 y Fv(\()p Ft(i)p Fv(Im)17 b Ft(\022)s Fv(\)])33 b(and)1182 3201 y Fr(k)p Ft(B)1305 3218 y Fn(\006)1364 3201 y Fv(\()p Ft(\022)s Fv(\))p Fr(k)59 b(\024)g Ft(e)1776 3157 y Fo(2)p Fn(j)p Fo(Re)13 b Fq(\022)r Fn(j)2001 3201 y Fr(k)p Ft(B)2124 3218 y Fn(\006)2183 3201 y Fv(\()p Ft(i)p Fv(Im)k Ft(\022)s Fv(\))p Fr(k)22 b Ft(:)651 b Fv(\(A.20\))257 3421 y(Th)m(us,)45 b(it)c(su\016ces)j(to)c(pro)m(v)m(e)j(the)e(assertion)h(for)f Ft(\022)k Fv(=)c Ft(i#)p Fv(,)j(0)d Fr(\024)g Ft(#)h(<)f(#)2986 3438 y Fo(0)3026 3421 y Fv(,)i(whic)m(h)g(w)m(e)257 3541 y(henceforth)34 b(assume.)45 b(Next)33 b(w)m(e)h(observ)m(e)g(that)292 3801 y Fm(\000)338 3881 y Ft(L)404 3899 y Fq(g)444 3881 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)736 3801 y Fm(\001)783 3823 y Fn(\000)p Fo(1)893 3881 y Fv(\()p Ft(L)997 3899 y Fq(aux)1142 3881 y Fv(+)g(1\))1326 3838 y Fn(\000)p Fo(1)1469 3881 y Fr(\000)49 b Fv(\()p Ft(L)1698 3899 y Fq(aux)1842 3881 y Fv(+)21 b(1\))2026 3838 y Fn(\000)p Fo(1)2137 3801 y Fm(\000)2182 3881 y Ft(L)2248 3899 y Fq(g)2289 3881 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)2581 3801 y Fm(\001)2628 3823 y Fn(\000)p Fo(1)3222 3881 y Fv(\(A.21\))375 4051 y(=)534 3970 y Fm(\000)580 4051 y Ft(L)646 4068 y Fq(g)686 4051 y Fv(\()p Ft(\022)s Fv(\))h Fr(\000)f Ft(z)978 3970 y Fm(\001)1025 3992 y Fn(\000)p Fo(1)1135 4051 y Fv(\()p Ft(L)1239 4068 y Fq(aux)1384 4051 y Fv(+)g(1\))1568 4007 y Fn(\000)p Fo(1)1684 3970 y Fm(\002)1725 4051 y Ft(L)1791 4068 y Fq(aux)1930 4051 y Ft(;)33 b(L)2056 4068 y Fq(g)2096 4051 y Fv(\()p Ft(\022)s Fv(\))2220 3970 y Fm(\003)2284 4051 y Fv(\()p Ft(L)2388 4068 y Fq(aux)2532 4051 y Fv(+)21 b(1\))2716 4007 y Fn(\000)p Fo(1)2827 3970 y Fm(\000)2872 4051 y Ft(L)2938 4068 y Fq(g)2979 4051 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)h Ft(z)3271 3970 y Fm(\001)3317 3992 y Fn(\000)p Fo(1)3433 4051 y Ft(:)257 4271 y Fv(Indeed,)41 b(thanks)e(to)e(Lemma)h(A.1)g(and) g(Dom)o([)p Ft(L)2087 4288 y Fq(g)2128 4271 y Fv(\()p Ft(\022)s Fv(\)])d Fr(\023)h Fv(Dom[)p Ft(L)2724 4288 y Fq(aux)2846 4271 y Fv(],)j(b)s(oth)f(sides)h(in)257 4391 y(\(A.21\))33 b(de\014ne)g(a)g(b)s(ounded)g(op)s(erator.)43 b(Note)33 b(that)f(on)g(Dom[)p Ft(L)2636 4408 y Fq(aux)2759 4391 y Fv(],)1135 4530 y Fm(\002)1177 4611 y Ft(L)1243 4628 y Fq(aux)1382 4611 y Ft(;)g(L)1507 4628 y Fq(g)1548 4611 y Fv(\()p Ft(\022)s Fv(\))1672 4530 y Fm(\003)1773 4611 y Fv(=)59 b Ft(g)1975 4530 y Fm(\002)2017 4611 y Ft(L)2083 4628 y Fq(aux)2221 4611 y Ft(;)33 b(W)14 b Fv(\()p Ft(\022)s Fv(\))2511 4530 y Fm(\003)2574 4611 y Ft(;)621 b Fv(\(A.22\))257 4831 y(and)42 b(hence,)j(b)m(y)e(the)e (same)i(argumen)m(t)f(as)g(in)f(\(IV.73\),)j(\(IV.74\){\(IV.75\),)f(w)m (e)g(ev)m(en)257 4951 y(ha)m(v)m(e)34 b(that)1013 5057 y Fm(\015)1013 5117 y(\015)1013 5176 y(\015)1090 5171 y Fv(\()p Ft(L)1194 5189 y Fq(aux)1339 5171 y Fv(+)21 b(1\))1523 5128 y Fn(\000)p Fo(1)1634 5091 y Fm(\002)1675 5171 y Ft(L)1741 5189 y Fq(aux)1880 5171 y Ft(;)33 b(L)2006 5189 y Fq(g)2046 5171 y Fv(\()p Ft(\022)s Fv(\))2170 5091 y Fm(\003)2233 5057 y(\015)2233 5117 y(\015)2233 5176 y(\015)2348 5171 y Fv(=)60 b Fr(O)s Fv(\()p Ft(g)t Fv(\))20 b Ft(:)483 b Fv(\(A.23\))p eop %%Page: 92 93 92 92 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(92)257 573 y(On)51 b(the)g(other)f(hand,)56 b(Im)p Ft(z)62 b Fr(\025)57 b Fv(2)50 b(and)h Ft(g)60 b Fr(\034)d Ft(M)2209 590 y Fq(#)2251 599 y Fk(0)2340 573 y Fv(insures)52 b(the)f(condition)g(that)257 693 y(dist)q Fr(f)p Fv(NumRan[)p Ft(L)940 711 y Fq(g)981 693 y Fv(\()p Ft(\022)s Fv(\)])16 b Ft(;)33 b(z)t Fr(g)27 b(\025)g Fv(1.)44 b(Hence)34 b Fr(k)p Ft(X)8 b Fr(k)26 b Fv(=)h Fr(O)s Fv(\()p Ft(g)t Fv(\),)k(where)827 926 y Ft(X)67 b Fv(:=)1138 846 y Fm(\000)1184 926 y Ft(L)1250 944 y Fq(g)1290 926 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)1582 846 y Fm(\001)1629 868 y Fn(\000)p Fo(1)1739 926 y Fv(\()p Ft(L)1843 944 y Fq(aux)1988 926 y Fv(+)g(1\))2172 883 y Fn(\000)p Fo(1)2283 846 y Fm(\002)2324 926 y Ft(L)2390 944 y Fq(aux)2529 926 y Ft(;)32 b(L)2654 944 y Fq(g)2695 926 y Fv(\()p Ft(\022)s Fv(\))2819 846 y Fm(\003)2882 926 y Ft(;)313 b Fv(\(A.24\))257 1140 y(and)311 1353 y(\()p Fi(1)22 b Fr(\000)f Ft(X)8 b Fv(\))16 b(\()p Ft(L)771 1371 y Fq(aux)916 1353 y Fv(+)21 b(1\))1100 1310 y Fn(\000)p Fo(1)1210 1273 y Fm(\000)1256 1353 y Ft(L)1322 1371 y Fq(g)1363 1353 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)h Ft(z)1655 1273 y Fm(\001)1701 1295 y Fn(\000)p Fo(1)1812 1353 y Fv(\()p Ft(L)1916 1371 y Fq(aux)2060 1353 y Fv(+)g(1\))2245 1310 y Fn(\000)p Fo(1)2398 1353 y Fv(=)2534 1273 y Fm(\000)2579 1353 y Ft(L)2645 1371 y Fq(g)2686 1353 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)f Ft(z)2978 1273 y Fm(\001)3025 1295 y Fn(\000)p Fo(1)3141 1353 y Ft(:)54 b Fv(\(A.25\))257 1567 y(This)34 b(implies)g(that)f Fi(1)21 b Fr(\000)h Ft(X)40 b Fv(is)33 b(in)m(v)m(ertible)i(and)e(that) 1278 1696 y Fm(\015)1278 1755 y(\015)1334 1780 y Fv(\()p Fi(1)21 b Fr(\000)h Ft(X)8 b Fv(\))1674 1737 y Fn(\000)p Fo(1)1768 1696 y Fm(\015)1768 1755 y(\015)1883 1780 y Fv(=)59 b(1)38 b(+)g Fr(O)s Fv(\()p Ft(g)t Fv(\))20 b Ft(:)748 b Fv(\(A.26\))257 1994 y(Multiplying)30 b(\(A.25\))e(b)m(y)i (\()p Fi(1)13 b Fr(\000)g Ft(X)8 b Fv(\))1541 1954 y Fn(\000)p Fo(1)1663 1994 y Fv(and)29 b(using)g(\(A.26\),)g(w)m(e)g (arriv)m(e)g(at)f(the)h(assertion)257 2114 y(for)j Ft(B)480 2132 y Fo(+)540 2114 y Fv(\()p Ft(i#)p Fv(\).)44 b(The)33 b(pro)s(of)f(for)g Ft(B)1455 2132 y Fn(\000)1514 2114 y Fv(\()p Ft(i#)p Fv(\))h(is)g(similar.)p 3425 2114 4 66 v 3429 2052 59 4 v 3429 2114 V 3487 2114 4 66 v 404 2355 a(Putting)g(together)f(Lemma)i(A.1)e(and)h(Lemma)g(A.2,)g(w)m(e)g (arriv)m(e)h(at)257 2525 y Fi(Theorem)j(A.3.)49 b Fs(L)-5 b(et)32 b Fv(0)27 b Ft(<)g(#)1385 2543 y Fo(0)1452 2525 y Ft(<)g(\031)t(=)p Fv(2)k Fs(and)g Fv(0)c Ft(<)g(g)j Fr(\034)d Ft(M)2404 2543 y Fq(#)2446 2552 y Fk(0)2512 2525 y Fv(:=)g Ft(#)2699 2543 y Fo(0)2739 2525 y Fv(\()p Ft(\031)t(=)p Fv(2)14 b Fr(\000)g Ft(#)3095 2543 y Fo(0)3135 2525 y Fv(\))p Fs(.)43 b(Then,)257 2646 y(for)35 b(al)5 b(l)35 b Ft(z)c Fr(2)c(f)p Ft(z)32 b Fr(2)27 b Fp(C)48 b Fr(j)22 b Fv(Im)p Ft(z)32 b Fr(\025)27 b Fv(2)p Fr(g)p Fs(,)405 2844 y Fm(e)387 2869 y Ft(R)q Fv(\()16 b Fr(\001)g Ft(;)g(z)t Fv(\))84 b(:)f(\006)953 2826 y Fo(+)953 2894 y Fq(#)995 2903 y Fk(0)1124 2869 y Fr(!)88 b(B)s Fv([)p Fr(H)24 b(\012)d(H)q Fv(])i Ft(;)1455 b Fv(\(A.27\))883 3038 y Ft(\022)93 b Fr(7!)1225 3013 y Fm(e)1207 3038 y Ft(R)q Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))56 b(=)e(\()p Ft(L)1788 3056 y Fq(aux)1932 3038 y Fv(+)22 b(1\))2117 2995 y Fn(\000)p Fo(1)2232 2957 y Fm(\000)2278 3038 y Ft(L)2344 3056 y Fq(g)2384 3038 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)g Ft(z)2677 2957 y Fm(\001)2723 2980 y Fn(\000)p Fo(1)2839 3038 y Fv(\()p Ft(L)2943 3056 y Fq(aux)3087 3038 y Fv(+)g(1\))3272 2995 y Fn(\000)p Fo(1)257 3252 y Fs(is)35 b(analytic,)f(i.e.,)g Ft(\022)c Fr(7!)d Ft(L)1225 3269 y Fq(g)1265 3252 y Fv(\()p Ft(\022)s Fv(\))35 b Fs(de\014nes)e(an)i(analytic)f(family)g(on)g Fv(\006)2760 3210 y Fo(+)2760 3279 y Fq(#)2802 3288 y Fk(0)2876 3252 y Fs(in)g(the)h(sense)f(of)257 3372 y(Kato.)404 3542 y(Pr)-5 b(o)g(of:)43 b Fv(W)-8 b(e)33 b(start)f(with)h(the)g(observ)-5 b(ation)34 b(that)442 3778 y Ft(@)493 3795 y Fq(\022)532 3778 y Ft(L)598 3795 y Fq(g)639 3778 y Fv(\()p Ft(\022)s Fv(\))59 b(=)h Fr(\000)p Fi(1)1090 3734 y Fq(el)1170 3778 y Fr(\012)1268 3667 y Fm(\020)1327 3778 y Ft(e)1372 3734 y Fn(\000)p Fq(\022)1466 3778 y Ft(H)1547 3795 y Fq(f)1614 3778 y Fr(\012)22 b Fi(1)1768 3795 y Fq(f)1835 3778 y Fv(+)f Ft(e)1977 3734 y Fq(\022)2017 3778 y Fi(1)2073 3795 y Fq(f)2140 3778 y Fr(\012)g Ft(H)2318 3795 y Fq(f)2363 3667 y Fm(\021)2466 3778 y Fv(+)44 b Ft(g)19 b(@)2703 3795 y Fq(\022)2743 3778 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))21 b Ft(;)201 b Fv(\(A.28\))257 4018 y(where)34 b(\(see)g(\(V.118\))1010 4246 y Ft(@)1061 4263 y Fq(\022)1101 4246 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))83 b(=)g Ft(a)1624 4202 y Fn(\003)1624 4266 y Fq(`)1663 4246 y Fv(\()p Ft(@)1752 4263 y Fq(\022)1792 4246 y Ft(w)1865 4195 y Fo(\()p Fq(\022)r Fo(\))1862 4273 y(+)p Fq(;`)1969 4246 y Fv(\))38 b(+)g Ft(a)2210 4263 y Fq(`)2243 4246 y Fv(\()p Ft(@)2332 4263 y Fq(\022)2371 4246 y Ft(w)2444 4195 y Fo(\()p Fq(\022)r Fo(\))2441 4273 y Fn(\000)p Fq(;`)2548 4246 y Fv(\))1573 4420 y(+)16 b Ft(a)1716 4377 y Fn(\003)1716 4441 y Fq(r)1755 4420 y Fv(\()p Ft(@)1844 4438 y Fq(\022)1884 4420 y Ft(w)1957 4369 y Fo(\()p Fq(\022)r Fo(\))1954 4442 y(+)p Fq(;r)2066 4420 y Fv(\))38 b(+)g Ft(a)2307 4438 y Fq(r)2345 4420 y Fv(\()p Ft(@)2434 4438 y Fq(\022)2473 4420 y Ft(w)2546 4369 y Fo(\()p Fq(\022)r Fo(\))2543 4442 y Fn(\000)p Fq(;r)2655 4420 y Fv(\))22 b Ft(:)480 b Fv(\(A.29\))257 4634 y(By)33 b(Lemma)h(IV.1)e(and)h(\(A.8\),)g(w)m(e)g(clearly)h(ha)m (v)m(e)937 4760 y Fm(\015)937 4820 y(\015)937 4879 y(\015)1009 4875 y Ft(@)1060 4892 y Fq(\022)1100 4875 y Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))21 b(\()p Ft(L)1455 4892 y Fq(aux)1599 4875 y Fv(+)h(1\))1784 4831 y Fn(\000)p Fo(1)1894 4760 y Fm(\015)1894 4820 y(\015)1894 4879 y(\015)2009 4875 y Fv(=)59 b Fr(O)2225 4794 y Fm(\000)2271 4875 y Ft(e)2316 4831 y Fn(j)p Fo(Re)11 b Fq(\022)r Fn(j)2505 4875 y Ft(M)2609 4831 y Fn(\000)p Fo(1)2599 4900 y Fq(#)2641 4909 y Fk(0)2704 4794 y Fm(\001)2772 4875 y Ft(;)423 b Fv(\(A.30\))257 5110 y(and)33 b(hence)614 5209 y Fm(\015)614 5269 y(\015)614 5328 y(\015)686 5323 y Ft(@)737 5341 y Fq(\022)776 5323 y Ft(L)842 5341 y Fq(g)883 5323 y Fv(\()p Ft(\022)s Fv(\))21 b(\()p Ft(L)1132 5341 y Fq(aux)1276 5323 y Fv(+)h(1\))1461 5280 y Fn(\000)p Fo(1)1571 5209 y Fm(\015)1571 5269 y(\015)1571 5328 y(\015)1686 5323 y Fv(=)60 b Fr(O)1903 5213 y Fm(\020)1962 5323 y Ft(e)2007 5280 y Fo(2)p Fn(j)p Fo(Re)12 b Fq(\022)r Fn(j)2232 5243 y Fm(\000)2277 5323 y Fv(1)22 b(+)f Ft(g)f(M)2616 5280 y Fn(\000)p Fo(1)2606 5349 y Fq(#)2648 5358 y Fk(0)2711 5243 y Fm(\001)2757 5213 y(\021)2838 5323 y Ft(:)357 b Fv(\(A.31\))p eop %%Page: 93 94 93 93 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(93)257 573 y(Using)34 b(Lemmata)e(A.1,)h(A.2,)g(and)f(\(A.31\),)h (w)m(e)g(th)m(us)h(obtain)390 690 y Fm(\015)390 750 y(\015)390 809 y(\015)445 805 y Fv(\()p Ft(L)549 822 y Fq(aux)693 805 y Fv(+)22 b(1\))878 761 y Fn(\000)p Fo(1)988 805 y Ft(@)1039 822 y Fq(\022)1079 724 y Fm(\010\000)1182 805 y Ft(L)1248 822 y Fq(g)1289 805 y Fv(\()p Ft(\022)s Fv(\))f Fr(\000)h Ft(z)1581 724 y Fm(\001)1627 746 y Fn(\000)p Fo(1)1722 724 y Fm(\011)1796 805 y Fv(\()p Ft(L)1900 822 y Fq(aux)2044 805 y Fv(+)g(1\))2229 761 y Fn(\000)p Fo(1)2323 690 y Fm(\015)2323 750 y(\015)2323 809 y(\015)3222 805 y Fv(\(A.32\))473 1017 y(=)631 902 y Fm(\015)631 962 y(\015)631 1022 y(\015)687 1017 y Fv(\()p Ft(L)791 1035 y Fq(aux)935 1017 y Fv(+)g(1\))1120 974 y Fn(\000)p Fo(1)1230 936 y Fm(\000)1276 1017 y Ft(L)1342 1035 y Fq(g)1382 1017 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)f Ft(z)1674 936 y Fm(\001)1721 959 y Fn(\000)p Fo(1)1837 1017 y Ft(@)1888 1035 y Fq(\022)1927 1017 y Ft(L)1993 1035 y Fq(g)2034 1017 y Fv(\()p Ft(\022)s Fv(\))2179 936 y Fm(\000)2225 1017 y Ft(L)2291 1035 y Fq(g)2332 1017 y Fv(\()p Ft(\022)s Fv(\))g Fr(\000)h Ft(z)2624 936 y Fm(\001)2670 959 y Fn(\000)p Fo(1)2781 1017 y Fv(\()p Ft(L)2885 1035 y Fq(aux)3029 1017 y Fv(+)g(1\))3214 974 y Fn(\000)p Fo(1)3308 902 y Fm(\015)3308 962 y(\015)3308 1022 y(\015)473 1230 y Fv(=)631 1115 y Fm(\015)631 1175 y(\015)631 1235 y(\015)687 1230 y Ft(B)761 1247 y Fo(+)820 1230 y Fv(\()p Ft(\022)s Fv(\))16 b(\()p Ft(L)1064 1247 y Fq(aux)1208 1230 y Fv(+)22 b(1\))1393 1186 y Fn(\000)p Fo(1)1509 1230 y Ft(@)1560 1247 y Fq(\022)1599 1230 y Ft(L)1665 1247 y Fq(g)1706 1230 y Fv(\()p Ft(\022)s Fv(\))f(\()p Ft(L)1955 1247 y Fq(aux)2100 1230 y Fv(+)g(1\))2284 1186 y Fn(\000)p Fo(1)2395 1230 y Ft(B)2469 1247 y Fn(\000)2528 1230 y Fv(\()p Ft(\022)s Fv(\))2652 1115 y Fm(\015)2652 1175 y(\015)2652 1235 y(\015)473 1429 y Fr(\024)82 b Ft(C)35 b Fv(dist)893 1348 y Fm(\010)951 1429 y Fv(NumRan)q([)p Ft(L)1428 1446 y Fq(g)1468 1429 y Fv(\()p Ft(\022)s Fv(\)])16 b Ft(;)33 b(z)1744 1348 y Fm(\011)1803 1370 y Fn(\000)p Fo(2)1984 1429 y Fr(\024)87 b Ft(C)2224 1385 y Fn(0)2334 1429 y Ft(<)f Fr(1)22 b Ft(;)257 1645 y Fv(for)32 b(some)i(constan)m (ts)g Ft(C)r(;)16 b(C)1275 1606 y Fn(0)1325 1645 y Fv(=)27 b Fr(O)1509 1564 y Fm(\000)1554 1645 y Ft(e)1599 1606 y Fo(4)p Fn(j)p Fo(Re)12 b Fq(\022)r Fn(j)1808 1564 y Fm(\001)1881 1645 y Ft(<)27 b Fr(1)p Fv(.)p 3425 1645 4 66 v 3429 1582 59 4 v 3429 1645 V 3487 1645 4 66 v 257 1936 a Fi(Theorem)37 b(A.4.)49 b Fs(L)-5 b(et)44 b Fv(0)f Ft(<)f(#)1428 1896 y Fn(0)1428 1956 y Fo(0)1511 1936 y Ft(<)h(#)1687 1953 y Fo(0)1769 1936 y Ft(<)g(\031)t(=)p Fv(2)p Fs(,)i(and)e(assume)g(that)h Fv(Im)p Ft(z)k Fr(\025)42 b Fv(2)p Fs(,)k Fv(0)c Ft(<)257 2056 y(g)f Fr(\034)c Ft(#)537 2017 y Fn(0)537 2076 y Fo(0)577 2056 y Fv(\()p Ft(\031)t(=)p Fv(2)26 b Fr(\000)f Ft(#)956 2073 y Fo(0)996 2056 y Fv(\))p Fs(,)42 b(and)e Ft(\022)h Fr(2)d Fv(\006)1560 2015 y Fo(+)1560 2084 y Fq(#)1602 2093 y Fk(0)1667 2056 y Fr(\\)26 b(f)p Fv(Im)p Ft(\022)41 b Fr(\025)c Ft(#)2179 2017 y Fn(0)2179 2076 y Fo(0)2219 2056 y Fr(g)p Fs(.)62 b(De\014ne)39 b Ft(R)2748 2017 y Fn(0)2772 2056 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))38 b(:=)g(\()p Ft(L)3271 2073 y Fq(aux)3419 2056 y Fv(+)257 2198 y(1\))362 2173 y Fm(e)344 2198 y Ft(R)q Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\)\()p Ft(L)739 2216 y Fq(aux)885 2198 y Fv(+)21 b(1\))35 b Fs(on)f Fv(Dom[)p Ft(L)1541 2216 y Fq(aux)1663 2198 y Fv(])p Fs(.)45 b(Then)257 2388 y Fi(\(i\))i Ft(R)499 2349 y Fn(0)523 2388 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))36 b Fs(extends)e(to)h(a)g(b)-5 b(ounde)g(d)34 b(op)-5 b(er)g(ator)34 b(on)h Fr(H)23 b(\012)f(H)q Fs(.)257 2587 y Fi(\(ii\))47 b Ft(L)522 2605 y Fq(g)562 2587 y Fv(\()p Ft(\022)s Fv(\))5 b Fr(\000)g Ft(z)32 b Fs(is)27 b(invertible)g(on)g Fr(H)6 b(\012)f(H)q Fs(,)29 b(and)e(its)g(inverse,)h Ft(R)q Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))29 b(:=)e(\()p Ft(L)3033 2605 y Fq(g)3073 2587 y Fv(\()p Ft(\022)s Fv(\))5 b Fr(\000)g Ft(z)t Fv(\))3370 2548 y Fn(\000)p Fo(1)3465 2587 y Fs(,)501 2708 y(is)35 b(given)f(the)h(extension)f(of)g Ft(R)1642 2668 y Fn(0)1666 2708 y Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))p Fs(.)257 2906 y Fi(\(iii\))46 b Fs(On)31 b(the)g(r)-5 b(e)g(ctangular)31 b(domain)f Ft(R)e Fv(:=)1885 2826 y Fm(\010)1944 2906 y Ft(\022)i Fr(2)d Fp(C)2191 2822 y Fm(\014)2191 2882 y(\014)2257 2906 y Fr(j)p Fv(Re)17 b Ft(\022)s Fr(j)27 b(\024)g Ft(\013)15 b(;)47 b(#)2829 2867 y Fn(0)2829 2927 y Fo(0)2896 2906 y Fr(\024)27 b Fv(Im)17 b Ft(\022)30 b Fr(\024)d Ft(#)3367 2924 y Fo(0)3407 2826 y Fm(\011)3465 2906 y Fs(,)501 3027 y(wher)-5 b(e)34 b Ft(\013)28 b Fv(:=)f Fr(\000)1082 2989 y Fo(1)p 1082 3003 36 4 v 1082 3057 a(2)1144 3027 y Fv(ln)16 b(cos)q(\(2)p Ft(#)1516 2987 y Fn(0)1516 3047 y Fo(0)1555 3027 y Fv(\))27 b Ft(>)g Fv(0)p Fs(,)35 b(the)g(map)1079 3231 y Ft(R)44 b Fr(3)g Ft(\022)65 b Fr(7!)d Ft(L)1642 3249 y Fq(g)1682 3231 y Fv(\()p Ft(\022)s Fv(\))43 b Fr(2)h(B)2025 3150 y Fm(\002)2067 3231 y Fv(Dom)o([)p Ft(L)2364 3249 y Fq(aux)2487 3231 y Fv(])16 b Ft(;)33 b Fr(H)23 b(\012)e(H)2875 3150 y Fm(\003)3222 3231 y Fv(\(A.33\))501 3435 y Fs(de\014nes)34 b(an)h(analytic)f(family)h(of)f(typ)-5 b(e)35 b(A.)257 3634 y Fi(\(iv\))48 b Ft(\033)t Fv([)p Ft(L)636 3652 y Fq(g)676 3634 y Fv(\()p Ft(\022)s Fv(\)])81 b Fr(\022)p 1065 3553 131 4 v 81 w Fp(C)1131 3605 y Fn(\000)1227 3634 y Fr(\\)31 b Fv(\()p Fp(C)57 b Fr(n)30 b(G)1607 3595 y Fn(0)1631 3634 y Fv(\))p Fs(,)50 b(wher)-5 b(e)47 b Fr(G)2101 3595 y Fn(0)2172 3634 y Fs(is)g(the)h(c)-5 b(onne)g(cte)g(d)46 b(c)-5 b(omp)g(onent)47 b(of)501 3755 y Fr(f)p Ft(z)32 b Fr(2)27 b Fp(C)20 b Fr(j)28 b(k)p Ft(R)q Fv(\()p Ft(\022)s(;)16 b(z)t Fv(\))p Fr(k)28 b Ft(<)f Fr(1g)34 b Fs(c)-5 b(ontaining)33 b Fp(C)2082 3715 y Fo(+)2147 3755 y Fs(.)404 3945 y(Pr)-5 b(o)g(of:)44 b Fv(Statemen)m(ts)34 b(\(i\))f(and)h(\(ii\))f(easily)h(follo)m(w)f (from)g(Lemmata)h(A.1,)f(A.2,)g(and)257 4065 y(Theorem)h(A.3.)257 4185 y(F)-8 b(or)32 b(\(iii\),)h(w)m(e)h(set)f Ft(\013)28 b Fv(:=)f(Re)16 b Ft(\022)s Fv(,)33 b Ft(#)27 b Fv(:=)g(Im)17 b Ft(\022)s Fv(,)33 b(and)g(w)m(e)g(observ)m(e)i(that)357 4390 y Ft(L)423 4407 y Fq(f)469 4390 y Fv(\()p Ft(\022)s Fv(\))60 b(=)f(cosh)q(\()p Ft(\022)s Fv(\))16 b Ft(L)1179 4407 y Fq(f)1269 4390 y Fr(\000)43 b Fv(sinh)q(\()p Ft(\022)s Fv(\))16 b Ft(L)1768 4407 y Fq(aux)1950 4390 y Fv(=)60 b Fr(\000)16 b Fv(sinh)q(\()p Ft(\022)s Fv(\))2476 4309 y Fm(\002)2517 4390 y Ft(L)2583 4407 y Fq(aux)2749 4390 y Fr(\000)44 b Fv(coth\()p Ft(\022)s Fv(\))16 b Ft(L)3259 4407 y Fq(f)3305 4309 y Fm(\003)3368 4390 y Ft(:)3222 4510 y Fv(\(A.34\))257 4714 y(Since)34 b(sinh)17 b Ft(\022)31 b Fv(=)c(sinh)17 b Ft(\013)33 b Fv(cos)17 b Ft(#)44 b Fv(+)f Ft(i)33 b Fv(cosh)17 b Ft(\013)33 b Fv(sin)17 b Ft(#)33 b Fv(and)g Ft(#)27 b Fr(\025)g Ft(#)2555 4675 y Fn(0)2555 4735 y Fo(0)2622 4714 y Ft(>)g Fv(0,)33 b(w)m(e)g(ha)m(v)m (e)h(that)857 4919 y Fr(j)16 b Fv(sinh)i Ft(\022)s Fr(j)59 b(\025)h(j)p Fv(Im)17 b(sinh)g Ft(\022)s Fr(j)60 b(\025)f Fv(sin)17 b Ft(#)61 b Fr(\025)e Fv(sin)17 b Ft(#)2563 4875 y Fn(0)2563 4939 y Fo(0)2663 4919 y Ft(>)59 b Fv(0)21 b Ft(:)327 b Fv(\(A.35\))257 5148 y(Next,)34 b(w)m(e)g(observ)m(e)g (that,)e(for)g Ft( )f Fr(2)d Fv(Dom)o([)p Ft(L)1885 5097 y Fo(1)p Fq(=)p Fo(2)1885 5158 y Fq(aux)2008 5148 y Fv(],)567 5267 y Fm(\014)567 5327 y(\014)616 5271 y(\012)663 5352 y Ft( )747 5267 y Fm(\014)747 5327 y(\014)802 5352 y Ft(L)868 5370 y Fq(f)913 5352 y Fv(\()p Ft(\022)s Fv(\))16 b Ft( )1120 5271 y Fm(\013)1184 5267 y(\014)1184 5327 y(\014)1277 5352 y Fr(\024)59 b(j)16 b Fv(sinh)i Ft(\022)s Fr(j)1743 5271 y Fm(\000)1788 5352 y Fv(1)k(+)f Fr(j)16 b Fv(coth)h Ft(\022)s Fr(j)2275 5271 y Fm(\001)22 b(\012)2390 5352 y Ft( )2473 5267 y Fm(\014)2473 5327 y(\014)2528 5352 y Ft(L)2594 5370 y Fq(aux)2733 5352 y Ft( )2800 5271 y Fm(\013)2869 5352 y Ft(;)326 b Fv(\(A.36\))p eop %%Page: 94 95 94 94 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(94)257 573 y(b)s(ecause)34 b Fr(\006)p Ft(L)760 590 y Fq(f)833 573 y Fr(\024)27 b Ft(L)1002 590 y Fq(aux)1125 573 y Fv(.)43 b(T)-8 b(o)33 b(construct)g(a)g(lo)m(w)m(er)g(b)s(ound,)g (w)m(e)h(use)588 708 y Fm(\014)588 768 y(\014)637 712 y(\012)684 793 y Ft( )767 708 y Fm(\014)767 768 y(\014)822 793 y Ft(L)888 810 y Fq(f)934 793 y Fv(\()p Ft(\022)s Fv(\))16 b Ft( )1141 712 y Fm(\013)1205 708 y(\014)1205 768 y(\014)1321 793 y Fr(\024)83 b(j)16 b Fv(sinh)h Ft(\022)s Fr(j)1810 708 y Fm(\014)1810 768 y(\014)1843 793 y Fv(Re)1959 712 y Fm(\012)2006 793 y Ft( )2089 708 y Fm(\014)2089 768 y(\014)2144 712 y(\000)2189 793 y Ft(L)2255 810 y Fq(aux)2416 793 y Fr(\000)38 b Fv(coth\()p Ft(\022)s Fv(\))p Ft(L)2904 810 y Fq(f)2950 712 y Fm(\001)3012 793 y Ft( )3079 712 y Fm(\013)1321 938 y Fr(\024)83 b(j)16 b Fv(sinh)h Ft(\022)s Fr(j)1810 857 y Fm(\000)1856 938 y Fv(1)k Fr(\000)h Fv(Re)p Fr(f)p Fv(coth\()p Ft(\022)s Fv(\))p Fr(g)2545 857 y Fm(\001)f(\012)2659 938 y Ft( )2742 853 y Fm(\014)2742 913 y(\014)2797 938 y Ft(L)2863 956 y Fq(aux)3002 938 y Ft( )3069 857 y Fm(\013)3138 938 y Ft(:)57 b Fv(\(A.37\))257 1158 y(T)-8 b(o)33 b(ha)m(v)m(e)h(a)e(non)m (trivial)i(lo)m(w)m(er)f(b)s(ound,)g(w)m(e)h(therefore)f(require)h (that)951 1432 y(Re)1066 1351 y Fm(\010)1124 1432 y Fv(coth)16 b Ft(\022)1372 1351 y Fm(\011)1490 1432 y Fv(=)1879 1360 y(sinh)q(\(2)p Ft(\013)q Fv(\))p 1636 1409 848 4 v 1636 1497 a(cosh\(2)p Ft(\013)q Fv(\))43 b Fr(\000)g Fv(cos)q(\(2)p Ft(#)p Fv(\))2553 1432 y Ft(<)59 b Fv(1)22 b Ft(;)436 b Fv(\(A.38\))257 1704 y(or,)47 b(equiv)-5 b(alen)m(tly)d(,)49 b(cos\(2)p Ft(#)p Fv(\))e Ft(<)e Fv(cosh)q(\(2)p Ft(\013)q Fv(\))29 b Fr(\000)g(j)16 b Fv(sinh)q(\(2)p Ft(\013)q Fv(\))p Fr(j)p Fv(.)77 b(By)44 b(our)g(assumption)h(on)257 1825 y Ft(#)314 1786 y Fn(0)314 1845 y Fo(0)354 1825 y Fv(,)37 b(w)m(e)f(ha)m(v)m(e)h(that)e(cos)q(\(2)p Ft(#)p Fv(\))d Ft(<)g Fv(cosh)q(\(2)p Ft(\013)q Fv(\))23 b Fr(\000)h(j)16 b Fv(sinh)q(\(2)p Ft(\013)q Fv(\))p Fr(j)p Fv(,)36 b(and)f(\(A.38\))g (holds.)54 b(Th)m(us,)257 1945 y(for)45 b(all)f Ft(\022)51 b Fr(2)c Ft(D)s Fv(\(0)p Ft(;)16 b(#)1045 1963 y Fo(0)1085 1945 y Fv(\))30 b Fr(\\)f(f)p Fv(Im)q Ft(z)52 b Fr(\025)47 b Ft(#)1690 1906 y Fn(0)1690 1966 y Fo(0)1730 1945 y Fr(g)p Fv(,)h(the)d(quadratic)g(form)g Ft(L)2791 1963 y Fq(f)2836 1945 y Fv(\()p Ft(\022)s Fv(\))g(is)g(sectorial,)257 2066 y(and)38 b(Dom)656 2083 y Fn(Q)718 2066 y Fv([)p Ft(L)811 2083 y Fq(f)857 2066 y Fv(\()p Ft(\022)s Fv(\)])d(=)g(Dom)1359 2083 y Fn(Q)1421 2066 y Fv([)p Ft(L)1514 2083 y Fq(aux)1636 2066 y Fv(].)58 b(Since)39 b Ft(W)14 b Fv(\()p Ft(\022)s Fv(\))37 b(is)h(a)f(relativ)m(ely)i(b)s(ounded)f(form)257 2186 y(p)s(erturbation)29 b(w.r.t.)f Ft(L)1146 2203 y Fq(aux)1269 2186 y Fv(,)h(with)g(zero)f(relativ)m(e)h(b)s(ound,)h(so)e (it)g(is)g(with)h(resp)s(ect)g Ft(L)3325 2203 y Fq(f)3371 2186 y Fv(\()p Ft(\022)s Fv(\))257 2306 y(and)k(hence)h Ft(L)784 2324 y Fo(0)824 2306 y Fv(\()p Ft(\022)s Fv(\).)43 b(This)34 b(pro)m(v)m(es)g(\(iii\).)404 2427 y(T)-8 b(o)32 b(pro)m(v)m(e)i(\(iv\),)f(w)m(e)h(observ)m(e)g(that,)f(b)m(y)g (analytic)h(con)m(tin)m(uation,)f Ft(L)2917 2444 y Fq(g)2958 2427 y Fv(\()p Ft(\022)s Fv(\))21 b Fr(\000)h Ft(z)37 b Fv(is)c(in-)257 2547 y(v)m(ertible)i(in)d Fp(C)789 2508 y Fo(+)854 2547 y Fv(,)h(since)h Ft(L)1219 2565 y Fo(0)1259 2547 y Fv(\()p Ft(\022)s Fv(\))22 b Fr(\000)f Ft(z)37 b Fv(is.)p 3425 2547 4 66 v 3429 2485 59 4 v 3429 2547 V 3487 2547 4 66 v 257 3000 a Fu(B)156 b(P)l(ositivit)l(y)49 b(of)j(the)e(Lev)l(el-Shift)e(Op)t(erator)j Fa(\000)3217 3031 y Ft(E)257 3248 y Fh(B.1)132 b(De\014nitions)257 3433 y Fv(Recall)34 b(from)f(Hyp)s(othesis)i(H-1)d(that)h(the)h(sp)s (ectrum)h(of)d Ft(H)2473 3451 y Fq(el)2565 3433 y Fv(is)i(assumed)h(to) d(en)m(tirely)257 3553 y(consist)43 b(of)e(simple)i(eigen)m(v)-5 b(alues)44 b Ft(E)1606 3571 y Fq(m)1715 3553 y Ft(<)d(E)1904 3571 y Fq(m)p Fo(+1)2103 3553 y Fv(with)h(corresp)s(onding)h (orthonormal)257 3674 y(eigen)m(v)m(ectors)37 b Ft(')868 3691 y Fq(m)934 3674 y Fv(,)e Ft(m)30 b Fr(2)f(N)44 b Fv(:=)30 b Fr(f)p Fv(0)p Ft(;)16 b Fv(1)p Ft(;)g(:)g(:)g(:)33 b(;)16 b(N)33 b Fr(\000)23 b Fv(1)p Fr(g)p Fv(.)47 b(Th)m(us)36 b Fr(f)p Ft(')2628 3691 y Fq(i)2655 3674 y Fr(j)p Ft(i)30 b Fr(2)g(N)15 b(g)29 b(\022)g(H)3200 3691 y Fq(el)3293 3674 y Fv(is)34 b(an)257 3794 y(orthonormal)f(basis.)44 b(W)-8 b(e)33 b(also)g(ha)m(v)m(e)1220 4014 y Ft(L)1286 4032 y Fq(el)1428 4014 y Fv(:=)83 b Ft(H)1695 4032 y Fq(el)1776 4014 y Fr(\012)21 b Fi(1)1929 4032 y Fq(el)2010 4014 y Fr(\000)g Fi(1)2163 4032 y Fq(el)2244 4014 y Fr(\012)g Ft(H)2422 4032 y Fq(f)2489 4014 y Ft(;)758 b Fv(\(B.1\))1234 4160 y Ft(L)1300 4177 y Fq(f)1428 4160 y Fv(:=)83 b Ft(H)1695 4177 y Fq(f)1762 4160 y Fr(\012)22 b Fi(1)1916 4177 y Fq(f)1983 4160 y Fr(\000)f Fi(1)2136 4177 y Fq(f)2204 4160 y Fr(\012)g Ft(H)2382 4177 y Fq(f)2449 4160 y Ft(;)798 b Fv(\(B.2\))1240 4305 y Ft(L)1306 4322 y Fo(0)1428 4305 y Fv(:=)83 b Ft(L)1680 4322 y Fq(el)1761 4305 y Fr(\012)22 b Fi(1)1915 4261 y Fq(f)1982 4305 y Fv(+)f Fi(1)2135 4261 y Fq(el)2216 4305 y Fr(\012)g Ft(L)2379 4322 y Fq(f)2447 4305 y Ft(;)800 b Fv(\(B.3\))257 4525 y(where)42 b Fi(1)603 4486 y Fq(el)701 4525 y Fv(:=)d Fi(1)899 4542 y Fq(el)984 4525 y Fr(\012)27 b Fi(1)1143 4542 y Fq(el)1242 4525 y Fv(and)40 b Fi(1)1495 4486 y Fq(f)1579 4525 y Fv(:=)g Fi(1)1778 4542 y Fq(f)1850 4525 y Fr(\012)26 b Fi(1)2008 4542 y Fq(f)2054 4525 y Fv(.)65 b(Henceforth,)43 b(w)m(e)e(frequen)m (tly)h(omit)257 4645 y(trivial)33 b(tensor)g(factors,)g(lik)m(e)h Fr(\012)p Fi(1)p Fv(,)f(unless)h(they)f(clarify)g(the)g(exp)s(osition.) 404 4766 y(Recall)28 b(from)g(\(IV.62\))g(that)g(the)g(in)m(teraction)h (in)g(the)f(Liouvillian)h(at)f(in)m(v)m(erse)i(tem-)257 4886 y(p)s(erature)j Ft(\014)38 b Fv(in)33 b(the)g(Araki-W)-8 b(o)s(o)s(ds)32 b(represen)m(tation)j(is)e(giv)m(en)h(b)m(y)504 5106 y Ft(W)97 b Fv(=)83 b Ft(a)903 5063 y Fn(\003)903 5127 y Fq(`)942 5025 y Fm(\000)988 5018 y(p)p 1088 5018 219 4 v 88 x Fv(1)21 b(+)h Ft(\032)16 b(G)1399 5124 y Fq(`)1470 5106 y Fr(\000)1584 5039 y(p)p 1665 5039 51 4 v 67 x Ft(\032)p 1732 5026 117 4 v 17 w(G)1809 5077 y Fn(\003)1809 5127 y Fq(r)1848 5025 y Fm(\001)1943 5106 y Fv(+)48 b Ft(a)2118 5124 y Fq(`)2151 5025 y Fm(\000)2197 5018 y(p)p 2297 5018 219 4 v 88 x Fv(1)21 b(+)h Ft(\032)16 b(G)2608 5063 y Fn(\003)2608 5127 y Fq(`)2686 5106 y Fr(\000)2799 5039 y(p)p 2881 5039 51 4 v 2881 5106 a Ft(\032)p 2947 5026 115 4 v 16 w(G)3024 5124 y Fq(r)3062 5025 y Fm(\001)3274 5106 y Fv(\(B.4\))852 5270 y(+)27 b Ft(a)1006 5226 y Fn(\003)1006 5290 y Fq(r)1045 5189 y Fm(\000)1091 5202 y Fr(p)p 1172 5202 51 4 v 68 x Ft(\032)17 b(G)1316 5226 y Fn(\003)1316 5290 y Fq(`)1393 5270 y Fr(\000)1507 5182 y Fm(p)p 1607 5182 219 4 v 88 x Fv(1)k(+)h Ft(\032)p 1841 5190 115 4 v 16 w(G)1918 5287 y Fq(r)1956 5189 y Fm(\001)2051 5270 y Fv(+)48 b Ft(a)2226 5287 y Fq(r)2264 5189 y Fm(\000)2310 5202 y Fr(p)p 2391 5202 51 4 v 68 x Ft(\032)q(G)2519 5287 y Fq(`)2590 5270 y Fr(\000)2703 5182 y Fm(p)p 2803 5182 219 4 v 88 x Fv(1)21 b(+)h Ft(\032)p 3038 5190 117 4 v 17 w(G)3115 5241 y Fn(\003)3115 5290 y Fq(r)3154 5189 y Fm(\001)3221 5270 y Ft(;)p eop %%Page: 95 96 95 95 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(95)257 573 y(where)26 b(the)f(coupling)g(function)f Ft(G)j Fv(:)h Fp(R)1670 534 y Fo(3)1721 573 y Fr(\002)5 b Fp(Z)1871 590 y Fo(2)1935 573 y Fr(!)27 b(B)s Fv(\()p Fr(H)2247 590 y Fq(el)2306 573 y Fv(\))d(is)h(a)f(matrix-v)-5 b(alued)25 b(function)257 693 y(and)1238 911 y Ft(\032)60 b Fr(\021)g Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\))59 b(=)1852 830 y Fm(\000)1898 911 y Ft(e)1943 867 y Fq(\014)s(!)r Fo(\()p Fq(\030)s Fo(\))2163 911 y Fr(\000)38 b Fv(1)2326 830 y Fm(\001)2371 852 y Fn(\000)p Fo(1)2487 911 y Ft(:)760 b Fv(\(B.5\))404 1128 y(W)-8 b(e)38 b(further)g(recall)h(from)f (\(V.152\){\(V.153\))e(that,)j(for)f Ft(E)k Fr(2)36 b Fp(R)5 b Fv(,)45 b(the)39 b(imaginary)257 1249 y(part)33 b(\000)530 1266 y Fq(E)622 1249 y Fv(of)f(the)h(lev)m(el)h(shift)f(op)s (erator)f Ft(Q)1813 1266 y Fq(E)1905 1249 y Fv(is)h(giv)m(en)h(b)m(y) 840 1467 y(\000)901 1484 y Fq(E)982 1467 y Fr(\012)22 b Ft(P)1143 1484 y Fo(\012)1257 1467 y Fv(:=)60 b Ft(P)1483 1484 y Fo(\012)1554 1467 y Ft(W)30 b(P)1753 1423 y Fn(?)1739 1487 y Fo(\012)1828 1467 y Ft(\016)1875 1386 y Fm(\002)1916 1467 y Ft(L)1982 1484 y Fo(0)2022 1467 y Ft(P)2099 1423 y Fn(?)2085 1487 y Fo(\012)2201 1467 y Fr(\000)43 b Ft(E)2398 1386 y Fm(\003)2456 1467 y Ft(P)2533 1423 y Fn(?)2519 1487 y Fo(\012)2608 1467 y Ft(W)30 b(P)2793 1484 y Fo(\012)2869 1467 y Ft(;)378 b Fv(\(B.6\))257 1694 y(where,)28 b Ft(P)623 1711 y Fo(\012)703 1694 y Fv(is)e(the)f(orthogonal)g(pro)5 b(jection)26 b(on)m(to)f(the)g(v)-5 b(acuum)26 b(v)m(ector)g(\012)2971 1654 y Fq(f)3044 1694 y Fv(:=)h(\012)3244 1711 y Fq(f)3297 1694 y Fr(\012)7 b Fv(\012)3450 1711 y Fq(f)257 1814 y Fv(in)33 b Fr(F)441 1832 y Fq(f)509 1814 y Fr(\012)21 b(F)676 1832 y Fq(f)722 1814 y Fv(.)404 1934 y(Applying)i(the)g(mo)s (dular)f(conjugation)g Ft(J)37 b Fv(=)26 b Ft(E)6 b Fv(\()p Ft(C)i Fr(\012)q Ft(C)g Fr(\012)q Ft(T)15 b Fr(\012)q Ft(T)f Fv(\))23 b(\(see)g(Eq.)g(\(IV.77\)\),)257 2055 y(w)m(e)29 b(see)g(that)f Ft(J)9 b Fv(\(\000)918 2072 y Fq(E)989 2055 y Fr(\012)j Ft(P)1140 2072 y Fo(\012)1195 2055 y Fv(\))p Ft(J)36 b Fv(=)27 b(\000)1487 2072 y Fn(\000)p Fq(E)1614 2055 y Fr(\012)12 b Ft(P)1765 2072 y Fo(\012)1819 2055 y Fv(.)42 b(In)m(tro)s(ducing)29 b(the)f(corresp)s(onding)h (restric-)257 2175 y(tion,)35 b Ft(J)541 2193 y Fq(el)629 2175 y Fv(=)29 b Ft(E)806 2193 y Fq(el)865 2175 y Fv(\()p Ft(C)g Fr(\012)23 b Ft(C)7 b Fv(\),)34 b(of)f Ft(J)43 b Fv(to)34 b Fr(H)1689 2193 y Fq(el)1770 2175 y Fr(\012)23 b(H)1951 2193 y Fq(el)2010 2175 y Fv(,)34 b(where)h Ft(E)2426 2193 y Fq(el)2485 2175 y Fv(\()p Ft(')23 b Fr(\002)f Ft( )t Fv(\))29 b(:=)h Ft( )c Fr(\012)d Ft(')33 b Fv(is)i(the)257 2296 y(corresp)s(onding)28 b(exc)m(hange)g(op)s(erator,)f(w)m(e)g (\014nd)g(that)g(\000)2297 2313 y Fq(E)2382 2296 y Fv(is)g(an)m (ti-unitarily)g(equiv)-5 b(alen)m(t)257 2416 y(to)33 b(\000)438 2433 y Fn(\000)p Fq(E)552 2416 y Fv(,)1477 2634 y Ft(J)1531 2651 y Fq(el)1606 2634 y Fv(\000)1667 2651 y Fq(E)1743 2634 y Ft(J)1797 2651 y Fq(el)1915 2634 y Fv(=)60 b(\000)2112 2651 y Fn(\000)p Fq(E)2248 2634 y Ft(:)999 b Fv(\(B.7\))257 2851 y(In)33 b(particular,)g Ft(J)910 2869 y Fq(el)969 2851 y Fv(\000)1030 2869 y Fo(0)1069 2851 y Ft(J)1123 2869 y Fq(el)1209 2851 y Fv(=)27 b(\000)1373 2869 y Fo(0)1412 2851 y Fv(.)404 2972 y(F)-8 b(or)31 b(an)m(y)j(Borel)e(set)i(\001)27 b Fr(\022)g Fp(R)5 b Fv(,)39 b(w)m(e)33 b(denote)h Ft(\037)2036 2989 y Fo(\001)2126 2972 y Fv(:=)27 b Ft(\037)2317 2989 y Fo(\001)2380 2972 y Fv(\()p Ft(L)2484 2989 y Fq(el)2543 2972 y Fv(\))32 b(and)h(further)1071 3189 y Fr(A)1149 3207 y Fo(\001)1295 3189 y Fv(:=)1481 3108 y Fm(\010)1539 3189 y Fv(\()p Ft(i;)16 b(j)6 b Fv(\))28 b Fr(2)f(N)1952 3146 y Fo(2)2013 3105 y Fm(\014)2013 3164 y(\014)2095 3189 y Ft(E)2167 3207 y Fq(i;j)2274 3189 y Fr(2)h Fv(\001)2448 3108 y Fm(\011)2528 3189 y Ft(;)719 b Fv(\(B.8\))1040 3358 y Fr(A)1118 3307 y Fo(\(1\))1118 3385 y(\001)1295 3358 y Fv(:=)1481 3277 y Fm(\010)1539 3358 y Ft(i)28 b Fr(2)f(N)1808 3273 y Fm(\014)1808 3333 y(\014)1890 3358 y Fr(9)p Ft(j)34 b Fr(2)27 b(N)42 b Fv(:)70 b Ft(E)2401 3376 y Fq(i;j)2509 3358 y Fr(2)27 b Fv(\001)2682 3277 y Fm(\011)2762 3358 y Ft(;)485 b Fv(\(B.9\))1040 3527 y Fr(A)1118 3476 y Fo(\(2\))1118 3554 y(\001)1295 3527 y Fv(:=)1481 3446 y Fm(\010)1539 3527 y Ft(j)33 b Fr(2)28 b(N)1821 3442 y Fm(\014)1821 3502 y(\014)1903 3527 y Fr(9)p Ft(i)g Fr(2)f(N)42 b Fv(:)70 b Ft(E)2401 3544 y Fq(i;j)2509 3527 y Fr(2)27 b Fv(\001)2682 3446 y Fm(\011)2762 3527 y Ft(;)437 b Fv(\(B.10\))947 3696 y([)p Fr(A)1052 3645 y Fo(\()p Fq(\027)t Fo(\))1052 3723 y(\001)1150 3696 y Fv(])1177 3652 y Fq(c)1295 3696 y Fv(:=)83 b Fr(N)53 b(n)38 b(A)1779 3645 y Fo(\()p Fq(\027)t Fo(\))1779 3723 y(\001)1898 3696 y Ft(;)111 b(\027)34 b Fv(=)27 b(1)p Ft(;)16 b Fv(2)21 b Ft(:)816 b Fv(\(B.11\))257 3929 y(Note)33 b(that)f Fr(A)782 3878 y Fo(\(2\))782 3956 y(\001)904 3929 y Fv(=)27 b Fr(A)1085 3878 y Fo(\(1\))1085 3956 y Fn(\000)p Fo(\001)1235 3929 y Fv(and)33 b(that)f([)p Fr(A)1741 3878 y Fo(\(1\))1741 3956 y(\001)1835 3929 y Fv(])1862 3890 y Fq(c)1924 3929 y Fv(=)27 b([)p Fr(A)2132 3878 y Fo(\(2\))2132 3956 y(\001)2226 3929 y Fv(])2253 3890 y Fq(c)2315 3929 y Fv(=)g Fr(;)33 b Fv(if)f(\001)27 b Fr(3)g Fv(0.)44 b(Moreo)m(v)m(er,)716 4169 y(\001)27 b(=)g Fr(f)p Ft(E)1048 4187 y Fq(i;j)1129 4169 y Fr(g)f(\022)h Fv(\(0)p Ft(;)16 b Fr(1)p Fv(\))115 b(=)-16 b Fr(\))115 b(A)2039 4118 y Fo(\(1\))2039 4196 y(\001)2160 4169 y Fr(3)28 b Ft(i)k(;)82 b Fv([)p Fr(A)2532 4118 y Fo(\(1\))2532 4196 y(\001)2626 4169 y Fv(])2653 4126 y Fq(c)2715 4169 y Fr(3)27 b Fv(0)22 b Ft(;)321 b Fv(\(B.12\))640 4334 y(\001)28 b(=)f Fr(f)p Ft(E)973 4351 y Fq(i;j)1053 4334 y Fr(g)g(\022)g Fv(\()p Fr(\0001)p Ft(;)16 b Fv(0\))114 b(=)-16 b Fr(\))115 b(A)2039 4283 y Fo(\(1\))2039 4361 y(\001)2160 4334 y Fr(3)28 b Ft(i)k(;)82 b Fv([)p Fr(A)2532 4283 y Fo(\(1\))2532 4361 y(\001)2626 4334 y Fv(])2653 4290 y Fq(c)2715 4334 y Fr(3)27 b Ft(N)32 b Fr(\000)22 b Fv(1)f Ft(:)114 b Fv(\(B.13\))404 4551 y(F)-8 b(urthermore,)33 b(w)m(e)g(denote)h(the)f(corresp)s(onding)g(canonical)g(pro)5 b(jections)34 b(b)m(y)1169 4769 y Ft(p)1218 4725 y Fo(\(1\))1372 4769 y Fv(:)60 b Fr(A)1537 4786 y Fo(\001)1627 4769 y Fr(!)26 b(A)1829 4718 y Fo(\(1\))1829 4796 y(\001)1945 4769 y Ft(;)90 b Fv(\()p Ft(i;)16 b(j)6 b Fv(\))27 b Fr(7!)g Ft(i)22 b(;)732 b Fv(\(B.14\))1169 4930 y Ft(p)1218 4886 y Fo(\(2\))1372 4930 y Fv(:)60 b Fr(A)1537 4947 y Fo(\001)1627 4930 y Fr(!)26 b(A)1829 4879 y Fo(\(2\))1829 4957 y(\001)1945 4930 y Ft(;)83 b Fv(\()p Ft(i;)16 b(j)6 b Fv(\))28 b Fr(7!)f Ft(j)g(;)726 b Fv(\(B.15\))257 5147 y(and)28 b(w)m(e)g(call)g(a)f(Borel)h(set)g(\001)f Fr(\022)g Fp(R)39 b Fs(nonde)-5 b(gener)g(ate)26 b Fv(i\013)h Ft(p)2331 5108 y Fo(\(1\))2453 5147 y Fv(or)g Ft(p)2616 5108 y Fo(\(2\))2737 5147 y Fv(is)h(bijectiv)m(e.)44 b(\(Note)257 5268 y(that)23 b Ft(p)508 5228 y Fo(\(1\))624 5268 y Fv(is)g(bijectiv)m(e)i(i\013)d Ft(p)1245 5228 y Fo(\(2\))1362 5268 y Fv(is.\))40 b(The)24 b(assumption)g(of)e(simplicit)m(y)i(of)e (the)h(eigen)m(v)-5 b(alues)257 5388 y Ft(E)329 5406 y Fq(m)440 5388 y Ft(<)43 b(E)631 5406 y Fq(m)p Fo(+1)831 5388 y Fv(implies)h(that)e Fr(f)p Ft(E)6 b Fr(g)42 b Fv(is)h(nondegenerate,)j(for)c(an)m(y)h Ft(E)50 b Fr(2)44 b Fp(R)5 b Fv(.)79 b(F)-8 b(or)41 b(an)m(y)p eop %%Page: 96 97 96 96 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(96)257 574 y(nondegenerate)37 b(\001)31 b Fr(\022)g Fp(R)5 b Fv(,)41 b(there)36 b(exists)g(a)f(bijection)h Ft(N)41 b Fr(\021)31 b Ft(N)2570 591 y Fo(\001)2680 574 y Fv(:)62 b Fr(A)2847 523 y Fo(\(1\))2847 601 y(\001)2972 574 y Fr(!)31 b(A)3179 523 y Fo(\(2\))3179 601 y(\001)3308 574 y Fv(suc)m(h)257 694 y(that)1210 914 y Fr(A)1288 932 y Fo(\001)1411 914 y Fv(=)1546 834 y Fm(\010)1604 914 y Fv(\()p Ft(i;)16 b(N)1796 932 y Fo(\001)1860 914 y Fv(\()p Ft(i)p Fv(\)\))2029 830 y Fm(\014)2029 889 y(\014)2111 914 y Ft(i)27 b Fr(2)g(A)2341 863 y Fo(\(1\))2341 941 y(\001)2435 834 y Fm(\011)2515 914 y Ft(:)684 b Fv(\(B.16\))257 1134 y(The)34 b(case)f(\001)28 b(=)f Fr(f)p Ft(E)997 1152 y Fq(i;j)1077 1134 y Fr(g)32 b Fv(and)h Ft(E)g Fv(=)27 b Ft(E)1628 1152 y Fq(i;j)1741 1134 y Fv(is)33 b(of)f(sp)s(ecial)i(in)m (terest,)g(and)e(w)m(e)i(denote)1368 1387 y(\000)1429 1344 y Fo(\()p Fq(E)1508 1354 y Fj(i;j)1582 1344 y Fo(\))1673 1387 y Fv(:=)60 b(\000)1897 1405 y Fq(i;j)1977 1273 y Fm(\014)1977 1333 y(\014)1977 1392 y(\014)2010 1456 y Fo(Ran)12 b Fq(P)2203 1433 y Fj(el)2193 1479 y(E)2238 1495 y(i;j)2341 1387 y Ft(;)858 b Fv(\(B.17\))257 1698 y(recalling)34 b(that)e Ft(P)936 1659 y Fq(el)922 1719 y(i;j)1029 1698 y Fv(=)27 b Ft(\037)1193 1716 y Fn(f)p Fq(E)1280 1726 y Fj(i;j)1354 1716 y Fn(g)1393 1698 y Fv([)p Ft(L)1486 1716 y Fq(el)1545 1698 y Fv(].)404 1819 y(W)-8 b(e)23 b(conclude)h(this)f(section)h(with)f(a)f(computation)h (of)f(the)i(matrix)f(elemen)m(ts)h(of)f(\000)3409 1836 y Fq(E)3468 1819 y Fv(.)257 1939 y(T)-8 b(o)29 b(this)h(end)f(w)m(e)h (abbreviate)g Ft(')1444 1957 y Fq(i;j)1551 1939 y Fv(:=)d Ft(')1745 1957 y Fq(i)1788 1939 y Fr(\012)14 b Ft(')1942 1957 y Fq(j)1978 1939 y Fv(,)30 b Ft(E)2107 1957 y Fq(i;j)2214 1939 y Fv(:=)d Ft(E)2416 1957 y Fq(i)2459 1939 y Fr(\000)14 b Ft(E)2621 1957 y Fq(j)2658 1939 y Fv(,)30 b(and)f Ft(G)2978 1957 y Fq(i;j)3085 1939 y Fr(\021)e Ft(G)3265 1957 y Fq(i;j)3345 1939 y Fv(\()p Ft(\030)5 b Fv(\),)257 2075 y Ft(\030)32 b Fv(:=)27 b(\()502 2049 y Ft(~)500 2075 y(\030)t(;)16 b(\025)p Fv(\).)44 b(W)-8 b(e)33 b(obtain)271 2295 y Fr(h)p Ft(')373 2313 y Fq(i;j)453 2295 y Fr(j)16 b Fv(\000)557 2313 y Fq(E)633 2295 y Ft(')697 2313 y Fq(k)r(;`)788 2295 y Fr(i)2400 b Fv(\(B.18\))354 2454 y(=)512 2374 y Fm(\012)560 2454 y Ft(')624 2472 y Fq(i;j)725 2454 y Fr(\012)22 b Fv(\012)893 2411 y Fq(f)939 2370 y Fm(\014)939 2429 y(\014)994 2454 y Ft(a)1045 2472 y Fq(`)1078 2374 y Fm(\000)1123 2367 y(p)p 1223 2367 219 4 v 87 x Fv(1)f(+)h Ft(\032)17 b(G)1535 2411 y Fn(\003)1596 2454 y Fr(\012)k Fi(1)1749 2472 y Fq(el)1851 2454 y Fr(\000)1971 2387 y(p)p 2052 2387 51 4 v 67 x Ft(\032)16 b Fi(1)2174 2472 y Fq(el)2255 2454 y Fr(\012)p 2352 2374 77 4 v 21 w Ft(G)2429 2374 y Fm(\001)2497 2454 y Ft(P)2574 2411 y Fn(?)2560 2475 y Fo(\012)2632 2454 y Ft(\016)2679 2374 y Fm(\002)2721 2454 y Ft(L)2787 2472 y Fo(0)2826 2454 y Ft(P)2903 2411 y Fn(?)2889 2475 y Fo(\012)2984 2454 y Fr(\000)21 b Ft(E)3159 2374 y Fm(\003)3217 2454 y Ft(P)3294 2411 y Fn(?)3280 2475 y Fo(\012)985 2618 y Ft(a)1036 2574 y Fn(\003)1036 2638 y Fq(`)1076 2537 y Fm(\000)1121 2530 y(p)p 1221 2530 219 4 v 88 x Fv(1)g(+)h Ft(\032)16 b(G)22 b Fr(\012)g Fi(1)1708 2635 y Fq(el)1810 2618 y Fr(\000)1929 2551 y(p)p 2010 2551 51 4 v 67 x Ft(\032)17 b Fi(1)2133 2635 y Fq(el)2213 2618 y Fr(\012)p 2311 2538 77 4 v 22 w Ft(G)2388 2556 y Fn(\003)2427 2537 y Fm(\001)22 b(\000)2540 2618 y Ft(')2604 2635 y Fq(k)r(;`)2717 2618 y Fr(\012)f Fv(\012)2884 2574 y Fq(f)2930 2537 y Fm(\001\013)512 2781 y Fv(+)615 2701 y Fm(\012)662 2781 y Ft(')726 2799 y Fq(i;j)828 2781 y Fr(\012)h Fv(\012)996 2738 y Fq(f)1042 2697 y Fm(\014)1042 2757 y(\014)1096 2781 y Ft(a)1147 2799 y Fq(r)1186 2701 y Fm(\000)1231 2714 y Fr(p)p 1313 2714 51 4 v 1313 2781 a Ft(\032)16 b(G)22 b Fr(\012)f Fi(1)1631 2799 y Fq(el)1733 2781 y Fr(\000)1853 2694 y Fm(p)p 1952 2694 219 4 v 1952 2781 a Fv(1)h(+)f Ft(\032)c Fi(1)2243 2799 y Fq(el)2323 2781 y Fr(\012)p 2421 2701 77 4 v 22 w Ft(G)2498 2720 y Fn(\003)2537 2701 y Fm(\001)2605 2781 y Ft(P)2682 2738 y Fn(?)2668 2802 y Fo(\012)2740 2781 y Ft(\016)2787 2701 y Fm(\002)2829 2781 y Ft(L)2895 2799 y Fo(0)2934 2781 y Ft(P)3011 2738 y Fn(?)2997 2802 y Fo(\012)3092 2781 y Fr(\000)k Ft(E)3267 2701 y Fm(\003)3325 2781 y Ft(P)3402 2738 y Fn(?)3388 2802 y Fo(\012)985 2945 y Ft(a)1036 2902 y Fn(\003)1036 2966 y Fq(r)1076 2864 y Fm(\000)1121 2878 y Fr(p)p 1203 2878 51 4 v 1203 2945 a Ft(\032)16 b(G)1346 2902 y Fn(\003)1407 2945 y Fr(\012)22 b Fi(1)1561 2963 y Fq(el)1663 2945 y Fr(\000)1782 2857 y Fm(p)p 1882 2857 219 4 v 88 x Fv(1)f(+)h Ft(\032)16 b Fi(1)2172 2963 y Fq(el)2253 2945 y Fr(\012)p 2350 2865 77 4 v 21 w Ft(G)2427 2864 y Fm(\001)22 b(\000)2540 2945 y Ft(')2604 2963 y Fq(k)r(;`)2717 2945 y Fr(\012)f Fv(\012)2884 2902 y Fq(f)2930 2864 y Fm(\001\013)354 3149 y Fv(=)512 3013 y Fm(Z)628 3149 y Ft(d\030)727 3038 y Fm(n)793 3068 y(\012)840 3149 y Ft(')904 3166 y Fq(i;j)984 3064 y Fm(\014)984 3124 y(\014)1039 3068 y(\000)1085 3061 y(p)p 1184 3061 219 4 v 1184 3149 a Fv(1)h(+)f Ft(\032)c(G)1496 3105 y Fn(\003)1557 3149 y Fr(\012)k Fi(1)1710 3166 y Fq(el)1812 3149 y Fr(\000)1932 3082 y(p)p 2013 3082 51 4 v 67 x Ft(\032)c Fi(1)2136 3166 y Fq(el)2216 3149 y Fr(\012)p 2314 3069 77 4 v 22 w Ft(G)2391 3068 y Fm(\001)2458 3149 y Ft(\016)2505 3068 y Fm(\002)2546 3149 y Ft(L)2612 3166 y Fq(el)2693 3149 y Fr(\000)22 b Ft(E)27 b Fv(+)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))3177 3068 y Fm(\003)985 3283 y(\000)1031 3276 y(p)p 1130 3276 219 4 v 1130 3363 a Fv(1)22 b(+)f Ft(\032)c(G)k Fr(\012)h Fi(1)1617 3381 y Fq(el)1719 3363 y Fr(\000)1838 3296 y(p)p 1920 3296 51 4 v 1920 3363 a Ft(\032)16 b Fi(1)2042 3381 y Fq(el)2123 3363 y Fr(\012)p 2220 3283 77 4 v 21 w Ft(G)2297 3302 y Fn(\003)2337 3283 y Fm(\001)2404 3363 y Ft(')2468 3381 y Fq(k)r(;`)2559 3283 y Fm(\013)512 3527 y Fv(+)615 3446 y Fm(\012)662 3527 y Ft(')726 3544 y Fq(i;j)828 3527 y Fr(\012)22 b Fv(\012)996 3483 y Fq(f)1042 3442 y Fm(\014)1042 3502 y(\014)1096 3446 y(\000)1142 3460 y Fr(p)p 1223 3460 51 4 v 67 x Ft(\032)17 b(G)22 b Fr(\012)f Fi(1)1542 3544 y Fq(el)1644 3527 y Fr(\000)1764 3439 y Fm(p)p 1863 3439 219 4 v 1863 3527 a Fv(1)h(+)f Ft(\032)c Fi(1)2154 3544 y Fq(el)2234 3527 y Fr(\012)p 2332 3447 77 4 v 22 w Ft(G)2409 3465 y Fn(\003)2448 3446 y Fm(\001)2515 3527 y Ft(\016)2562 3446 y Fm(\002)2604 3527 y Ft(L)2670 3544 y Fq(el)2751 3527 y Fr(\000)k Ft(E)28 b Fr(\000)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))3235 3446 y Fm(\003)985 3629 y(\000)1031 3642 y Fr(p)p 1112 3642 51 4 v 67 x Ft(\032)16 b(G)1255 3666 y Fn(\003)1317 3709 y Fr(\012)21 b Fi(1)1470 3727 y Fq(el)1572 3709 y Fr(\000)1691 3622 y Fm(p)p 1791 3622 219 4 v 87 x Fv(1)h(+)f Ft(\032)c Fi(1)2082 3727 y Fq(el)2162 3709 y Fr(\012)p 2260 3629 77 4 v 22 w Ft(G)2337 3629 y Fm(\001)2404 3709 y Ft(')2468 3727 y Fq(k)r(;`)2559 3629 y Fm(\013)2628 3599 y(o)2716 3709 y Ft(;)257 3951 y Fv(and)33 b(hence)294 4239 y Fr(h)p Ft(')396 4257 y Fq(i;j)476 4239 y Fr(j)16 b Fv(\000)580 4257 y Fq(E)657 4239 y Ft(')721 4257 y Fq(k)r(;`)812 4239 y Fr(i)86 b Fv(=)1099 4104 y Fm(Z)1215 4239 y Ft(d\030)1360 4115 y Fq(N)7 b Fn(\000)p Fo(1)1365 4145 y Fm(X)1329 4354 y Fq(m;n)p Fo(=0)3226 4239 y Fv(\(B.19\))461 4414 y Fm(n)527 4524 y Ft(\016)574 4444 y Fm(\002)615 4524 y Ft(E)687 4542 y Fq(m;n)838 4524 y Fr(\000)22 b Ft(E)28 b Fv(+)21 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1322 4444 y Fm(\003)20 b(\012)1431 4524 y Ft(')1495 4542 y Fq(i;j)1575 4440 y Fm(\014)1575 4499 y(\014)1630 4444 y(\000)1676 4437 y(p)p 1775 4437 219 4 v 1775 4524 a Fv(1)i(+)f Ft(\032)c(G)2087 4481 y Fn(\003)2148 4524 y Fr(\012)k Fi(1)2301 4542 y Fq(el)2403 4524 y Fr(\000)2523 4457 y(p)p 2604 4457 51 4 v 67 x Ft(\032)c Fi(1)2727 4542 y Fq(el)2807 4524 y Fr(\012)p 2905 4444 77 4 v 22 w Ft(G)2982 4444 y Fm(\001)3049 4524 y Ft(')3113 4542 y Fq(m;n)3242 4444 y Fm(\013)1405 4637 y(\012)1452 4718 y Ft(')1516 4735 y Fq(m;n)1645 4633 y Fm(\014)1645 4693 y(\014)1700 4637 y(\000)1746 4630 y(p)p 1846 4630 219 4 v 88 x Fv(1)k(+)h Ft(\032)16 b(G)22 b Fr(\012)f Fi(1)2332 4735 y Fq(el)2434 4718 y Fr(\000)2554 4651 y(p)p 2635 4651 51 4 v 67 x Ft(\032)16 b Fi(1)2757 4735 y Fq(el)2838 4718 y Fr(\012)p 2935 4638 77 4 v 21 w Ft(G)3012 4656 y Fn(\003)3052 4637 y Fm(\001)3119 4718 y Ft(')3183 4735 y Fq(m;n)3312 4637 y Fm(\013)461 4881 y Fv(+)26 b Ft(\016)610 4801 y Fm(\002)652 4881 y Ft(E)724 4899 y Fq(m;n)875 4881 y Fr(\000)21 b Ft(E)28 b Fr(\000)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1359 4801 y Fm(\003)19 b(\012)1467 4881 y Ft(')1531 4899 y Fq(i;j)1612 4797 y Fm(\014)1612 4857 y(\014)1666 4801 y(\000)1712 4814 y Fr(p)p 1793 4814 51 4 v 67 x Ft(\032)e(G)22 b Fr(\012)f Fi(1)2112 4899 y Fq(el)2214 4881 y Fr(\000)2333 4794 y Fm(p)p 2433 4794 219 4 v 87 x Fv(1)h(+)f Ft(\032)c Fi(1)2724 4899 y Fq(el)2804 4881 y Fr(\012)p 2902 4801 77 4 v 22 w Ft(G)2979 4820 y Fn(\003)3018 4801 y Fm(\001)3085 4881 y Ft(')3149 4899 y Fq(k)r(;`)3240 4801 y Fm(\013)1405 4983 y(\012)1452 5064 y Ft(')1516 5082 y Fq(m;n)1645 4979 y Fm(\014)1645 5039 y(\014)1700 4983 y(\000)1746 4997 y Fr(p)p 1827 4997 51 4 v 67 x Ft(\032)g(G)1971 5021 y Fn(\003)2032 5064 y Fr(\012)k Fi(1)2185 5082 y Fq(el)2287 5064 y Fr(\000)2407 4977 y Fm(p)p 2506 4977 219 4 v 2506 5064 a Fv(1)h(+)f Ft(\032)c Fi(1)2797 5082 y Fq(el)2877 5064 y Fr(\012)p 2975 4984 77 4 v 22 w Ft(G)3052 4983 y Fm(\001)3119 5064 y Ft(')3183 5082 y Fq(k)r(;`)3274 4983 y Fm(\013)3343 4954 y(o)3431 5064 y Ft(:)p eop %%Page: 97 98 97 97 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(97)257 573 y(Hence,)34 b(the)f(\014nal)g(result)h(is)351 791 y Fr(h)p Ft(')453 809 y Fq(i;j)533 791 y Fr(j)16 b Fv(\000)637 809 y Fq(E)713 791 y Ft(')777 809 y Fq(k)r(;`)868 791 y Fr(i)2320 b Fv(\(B.20\))434 1035 y(=)592 899 y Fm(Z)708 1035 y Ft(d\030)854 910 y Fq(N)7 b Fn(\000)p Fo(1)858 940 y Fm(X)823 1149 y Fq(m;n)p Fo(=0)1038 924 y Fm(n)1104 1035 y Ft(\016)1151 954 y Fm(\002)1193 1035 y Ft(E)1265 1052 y Fq(m;n)1415 1035 y Fr(\000)22 b Ft(E)28 b Fv(+)21 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1899 954 y Fm(\003)20 b(\000)2007 947 y(p)p 2106 947 219 4 v 2106 1035 a Fv(1)i(+)f Ft(\032)p 2341 955 77 4 v 17 w(G)2418 1052 y Fq(m;i)2544 1035 y Ft(\016)2587 1052 y Fq(j;n)2726 1035 y Fr(\000)2845 968 y(p)p 2926 968 51 4 v 67 x Ft(\032)c(\016)3036 1052 y Fq(i;m)p 3163 955 77 4 v 3163 1035 a Ft(G)3240 1052 y Fq(j;n)3335 954 y Fm(\001)1537 1220 y(\000)1583 1213 y(p)p 1683 1213 219 4 v 88 x Fv(1)k(+)h Ft(\032)16 b(G)1994 1318 y Fq(m;k)2135 1301 y Ft(\016)2178 1318 y Fq(n;`)2317 1301 y Fr(\000)2436 1234 y(p)p 2518 1234 51 4 v 2518 1301 a Ft(\032)g(\016)2627 1318 y Fq(m;k)2768 1301 y Ft(G)2845 1318 y Fq(`;n)2941 1220 y Fm(\001)592 1465 y Fv(+)27 b Ft(\016)742 1384 y Fm(\002)784 1465 y Ft(E)856 1482 y Fq(m;n)1007 1465 y Fr(\000)21 b Ft(E)28 b Fr(\000)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1491 1384 y Fm(\003)19 b(\000)1598 1397 y Fr(p)p 1679 1397 V 68 x Ft(\032)e(G)1823 1482 y Fq(i;m)1949 1465 y Ft(\016)1992 1482 y Fq(j;n)2131 1465 y Fr(\000)2250 1377 y Fm(p)p 2350 1377 219 4 v 88 x Fv(1)k(+)h Ft(\032)16 b(\016)2627 1482 y Fq(i;m)2754 1465 y Ft(G)2831 1482 y Fq(n;j)2930 1384 y Fm(\001)1537 1567 y(\000)1583 1580 y Fr(p)p 1664 1580 51 4 v 67 x Ft(\032)p 1731 1567 77 4 v 17 w(G)1808 1665 y Fq(k)r(;m)1949 1647 y Ft(\016)1992 1665 y Fq(n;`)2131 1647 y Fr(\000)2250 1560 y Fm(p)p 2350 1560 219 4 v 87 x Fv(1)21 b(+)h Ft(\032)16 b(\016)2627 1665 y Fq(m;k)p 2768 1567 77 4 v 2768 1647 a Ft(G)2845 1665 y Fq(n;`)2941 1567 y Fm(\001)3008 1537 y(o)3096 1647 y Ft(:)257 1887 y Fv(In)33 b(particular,)g(for)f Ft(E)h Fv(=)27 b(0,)33 b Ft(i)27 b Fv(=)g Ft(j)6 b Fv(,)33 b(and)f Ft(k)f Fv(=)26 b Ft(`)p Fv(,)33 b(w)m(e)h(ha)m(v)m(e)525 2128 y Fr(h)p Ft(')627 2145 y Fq(i;i)699 2128 y Fr(j)16 b Fv(\000)803 2145 y Fo(0)859 2128 y Ft(')923 2145 y Fq(k)r(;k)1023 2128 y Fr(i)60 b Fv(=)f Ft(\016)1299 2145 y Fq(i;k)1402 2017 y Fm(\020)1462 2033 y(X)1471 2245 y Fq(j)t Fn(6)p Fo(=)p Fq(k)1622 2128 y Ft(\021)1670 2145 y Fq(k)r(;j)1781 2128 y Ft(e)1826 2084 y Fn(\000)p Fq(\014)s(E)1976 2096 y Fj(k)q(;j)2061 2084 y Fq(=)p Fo(2)2136 2017 y Fm(\021)2244 2128 y Fr(\000)49 b Fv(\(1)21 b Fr(\000)h Ft(\016)2618 2145 y Fq(i;k)2704 2128 y Fv(\))16 b Ft(\021)2806 2145 y Fq(i;k)2915 2128 y Ft(;)284 b Fv(\(B.21\))257 2444 y(where)257 2686 y Ft(\021)305 2703 y Fq(i;k)451 2686 y Fv(:=)60 b(2)679 2550 y Fm(Z)779 2575 y(\020)838 2686 y Ft(\016)885 2605 y Fm(\002)927 2686 y Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))20 b Fr(\000)h Ft(E)1305 2703 y Fq(i;k)1392 2605 y Fm(\003)1455 2686 y Fv(+)h Ft(\016)1600 2605 y Fm(\002)1641 2686 y Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))20 b(+)i Ft(E)2020 2703 y Fq(i;k)2106 2605 y Fm(\003)2148 2575 y(\021)2224 2595 y(p)p 2323 2595 608 4 v 2323 2686 a Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\)\(1)21 b(+)h Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\)\))2947 2601 y Fm(\014)2947 2661 y(\014)2980 2686 y Ft(G)3057 2703 y Fq(i;k)3143 2686 y Fv(\()p Ft(\030)g Fv(\))3267 2601 y Fm(\014)3267 2661 y(\014)3300 2627 y Fo(2)3356 2686 y Ft(d\030)25 b(:)3226 2864 y Fv(\(B.22\))257 3081 y(F)-8 b(orm)m(ula)45 b(\(B.21\))e(is)i(equiv)-5 b(alen)m(t)46 b(to)e(the)h(detailed-balance) h(equation)f(\(V.50\).)78 b(T)-8 b(o)257 3201 y(deriv)m(e)35 b(\(B.21\),)d(w)m(e)h(use)1142 3407 y(1)22 b(+)f Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\))p 1142 3456 342 4 v 1226 3544 a Ft(\032)p Fv(\()p Ft(\030)g Fv(\))1554 3479 y(=)59 b(1)21 b(+)1929 3407 y(1)p 1867 3456 174 4 v 1867 3544 a Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\))2110 3479 y(=)60 b Ft(e)2291 3435 y Fq(\014)15 b(!)r Fo(\()p Fq(\030)s Fo(\))p Fq(=)p Fo(2)2577 3479 y Ft(;)622 b Fv(\(B.23\))257 3744 y(whic)m(h)34 b(implies)h(the)e(follo)m(wing)f(t)m(w)m(o)i(iden)m (tities,)287 3963 y Ft(\016)334 3882 y Fm(\002)376 3963 y Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))20 b Fr(\000)h Ft(E)754 3980 y Fq(i;k)841 3882 y Fm(\003)16 b(\000)944 3963 y Fv(1)22 b(+)f Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\))1286 3882 y Fm(\001)1415 3963 y Fv(=)83 b Ft(\016)1621 3882 y Fm(\002)1662 3963 y Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))20 b Fr(\000)i Ft(E)2041 3980 y Fq(i;k)2127 3882 y Fm(\003)2185 3873 y(p)p 2285 3873 570 4 v 90 x Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\)\(1)21 b(+)g Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\))32 b Ft(e)2915 3919 y Fq(\014)s(E)3010 3931 y Fj(i;k)3090 3919 y Fq(=)p Fo(2)3186 3963 y Ft(;)13 b Fv(\(B.24\))546 4129 y Ft(\016)593 4048 y Fm(\002)635 4129 y Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))20 b(+)i Ft(E)1014 4146 y Fq(i;k)1100 4048 y Fm(\003)1158 4129 y Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\))83 b(=)g Ft(\016)1621 4048 y Fm(\002)1662 4129 y Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))20 b(+)i Ft(E)2041 4146 y Fq(i;k)2127 4048 y Fm(\003)2185 4039 y(p)p 2285 4039 V 90 x Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\)\(1)21 b(+)g Ft(\032)p Fv(\()p Ft(\030)5 b Fv(\))32 b Ft(e)2915 4085 y Fq(\014)s(E)3010 4097 y Fj(i;k)3090 4085 y Fq(=)p Fo(2)3186 4129 y Ft(:)13 b Fv(\(B.25\))257 4418 y Fh(B.2)132 b(Strict)45 b(P)l(ositivit)l(y)i(of)d FF(\000)1773 4441 y Fc(E)1887 4418 y Fh(for)g Fg(E)39 b Ff(6)p FF(=)32 b(0)257 4602 y Fv(Our)38 b(next)h(task)f(is)h(to)e(sho)m(w)i(that)f (under)g(certain)h(assumptions)g Ft(\037)2826 4620 y Fo(\001)2889 4602 y Fv(\000)2950 4620 y Fq(E)3010 4602 y Ft(\037)3071 4620 y Fo(\001)3170 4602 y Fr(\025)c Ft(\013)q(\037)3405 4620 y Fo(\001)3468 4602 y Fv(,)257 4723 y(for)f(some)h Ft(\013)30 b(>)f Fv(0)34 b(and)g(all)g Ft(E)i Fr(6)p Fv(=)29 b(0.)48 b(Since)35 b(\000)1916 4740 y Fq(E)2010 4723 y Fv(is)g(an)m(ti-unitarily)f(equiv)-5 b(alen)m(t)36 b(to)e(\000)3354 4740 y Fn(\000)p Fq(E)3468 4723 y Fv(,)257 4843 y(w)m(e)g(ma)m(y)f(restrict)h(ourselv)m(es)h(to)d(considering)i Ft(E)f(>)27 b Fv(0.)257 5013 y Fi(Theorem)37 b(B.1.)49 b Fs(Assume)33 b(Hyp)-5 b(otheses)34 b(H-1)f(and)f(H-5.)45 b(L)-5 b(et)33 b Ft(E)g(>)27 b Fv(0)33 b Fs(and)g Fv(\001)27 b Fr(\022)g Fp(R)45 b Fs(b)-5 b(e)257 5134 y(a)35 b(nonde)-5 b(gener)g(ate)33 b(Bor)-5 b(el)35 b(set.)44 b(Then)1339 5352 y Ft(\037)1400 5370 y Fo(\001)1480 5352 y Fv(\000)1541 5370 y Fq(E)1617 5352 y Ft(\037)1678 5370 y Fo(\001)1803 5352 y Fr(\025)61 b Ft(\015)1991 5370 y Fq(E)2051 5352 y Fv(\(\001\))16 b Ft(\037)2285 5370 y Fo(\001)2370 5352 y Ft(;)829 b Fv(\(B.26\))p eop %%Page: 98 99 98 98 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(98)257 573 y Fs(wher)-5 b(e)35 b(the)f(numb)-5 b(er)35 b Ft(\015)1093 590 y Fq(E)1152 573 y Fv(\(\001\))27 b Fr(\025)g Fv(0)35 b Fs(is)g(de\014ne)-5 b(d)33 b(by)446 810 y Ft(\015)497 828 y Fq(E)556 810 y Fv(\(\001\))83 b(:=)126 b(min)996 899 y Fo(~)-49 b Fq(m)p Fn(2A)1147 864 y Fk(\(1\))1147 923 y(\001)1247 675 y Fm(Z)1363 810 y Ft(d\030)1565 716 y Fm(X)1477 954 y Fq(m)p Fn(2)p Fo([)p Fn(A)1662 919 y Fk(\(1\))1662 977 y(\001)1745 954 y Fo(])1765 935 y Fj(c)1797 700 y Fm(n)1863 810 y Ft(\016)1910 729 y Fm(\002)1952 810 y Ft(E)2024 828 y Fq(m;N)7 b Fo(\()14 b(~)-49 b Fq(m)q Fo(\))2312 810 y Fr(\000)22 b Ft(E)27 b Fv(+)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))2796 729 y Fm(\003)2852 726 y(\014)2852 785 y(\014)2885 810 y Ft(G)2962 828 y Fq(m;)14 b Fo(~)-49 b Fq(m)3111 810 y Fv(\()p Ft(\030)5 b Fv(\))3235 726 y Fm(\014)3235 785 y(\014)3267 752 y Fo(2)982 1135 y Fv(+)27 b Ft(\016)1132 1054 y Fm(\002)1173 1135 y Ft(E)1259 1153 y Fo(~)-49 b Fq(m)q(;N)7 b Fo(\()p Fq(m)p Fo(\))1534 1135 y Fr(\000)21 b Ft(E)28 b Fr(\000)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))2018 1054 y Fm(\003)2074 1050 y(\014)2074 1110 y(\014)2107 1135 y Ft(G)2184 1153 y Fq(N)i Fo(\()14 b(~)-49 b Fq(m)q Fo(\))p Fq(;N)7 b Fo(\()p Fq(m)p Fo(\))2569 1135 y Fv(\()p Ft(\030)e Fv(\))2693 1050 y Fm(\014)2693 1110 y(\014)2725 1077 y Fo(2)2765 1025 y Fm(o)2853 1135 y Ft(:)346 b Fv(\(B.27\))257 1346 y Fs(F)-7 b(urthermor)i(e,)34 b(for)h Ft(E)1078 1363 y Fq(i;j)1185 1346 y Fr(6)p Fv(=)27 b(0)p Fs(,)1035 1546 y Fv(\000)1096 1503 y Fo(\()p Fq(E)1175 1513 y Fj(i;j)1249 1503 y Fo(\))1342 1546 y Fr(\025)62 b Ft(\015)1536 1503 y Fo(\()p Fq(i;j)t Fo(\))1733 1546 y Fv(:=)g Ft(\015)1949 1564 y Fq(E)2001 1574 y Fj(i;j)2078 1546 y Fv(\()p Fr(f)p Ft(E)2237 1564 y Fq(i;j)2317 1546 y Fr(g)p Fv(\))g Ft(>)g Fv(0)21 b Ft(;)525 b Fv(\(B.28\))257 1735 y Fs(uniformly)35 b(in)g Ft(\014)d Fr(!)27 b(1)p Fs(.)257 1922 y(Pr)-5 b(o)g(of.)49 b Fv(Since)26 b(\001)f(is)h(nondegenerate,)i Fr(f)p Ft(')1786 1939 y Fq(i;N)7 b Fo(\()p Fq(i)p Fo(\))1975 1922 y Fr(j)p Ft(i)28 b Fr(2)f(A)2233 1871 y Fo(\(1\))2233 1949 y(\001)2327 1922 y Fr(g)e Fv(is)g(an)g(ONB)g(in)g(Ran)p Ft(\037)3204 1939 y Fo(\001)3267 1922 y Fv(\()p Ft(L)3371 1939 y Fq(el)3430 1922 y Fv(\).)257 2042 y(W)-8 b(e)40 b(ma)m(y)f(therefore)h(write)g(an)m(y)f(normalized)h(v)m(ector)g(in)f (\010)g Fr(2)e Fv(Ran)p Ft(\037)2882 2059 y Fo(\001)2945 2042 y Fv(\()p Ft(L)3049 2059 y Fq(el)3108 2042 y Fv(\))i(as)g(\010)f (=)257 2101 y Fm(P)363 2205 y Fq(i)p Fn(2N)522 2176 y Ft(c)564 2193 y Fq(i)608 2176 y Ft(')672 2193 y Fq(i;N)7 b Fo(\()p Fq(i)p Fo(\))862 2176 y Fv(,)33 b(where)1204 2101 y Fm(P)1309 2205 y Fq(i)p Fn(2N)1468 2176 y Fr(j)p Ft(c)1537 2193 y Fq(i)1566 2176 y Fr(j)1593 2137 y Fo(2)1660 2176 y Fv(=)28 b(1)k(and)h Ft(c)2077 2193 y Fq(i)2133 2176 y Fv(=)28 b(0,)33 b(for)f(all)h Ft(i)28 b Fr(2)g Fv([)p Fr(A)2890 2125 y Fo(\(2\))2890 2203 y(\001)2985 2176 y Fv(])3012 2137 y Fq(c)3046 2176 y Fv(.)45 b(Inserting)257 2296 y(this)34 b(in)m(to)e(\(B.20\))h(yields)289 2578 y Fr(h)p Fv(\010)p Fr(j)16 b Fv(\000)501 2595 y Fq(E)578 2578 y Fv(\010)p Fr(i)87 b Fv(=)935 2442 y Fm(Z)1051 2578 y Ft(d\030)1197 2453 y Fq(N)7 b Fn(\000)p Fo(1)1201 2483 y Fm(X)1166 2692 y Fq(m;n)p Fo(=0)1381 2467 y Fm(n)3226 2578 y Fv(\(B.29\))455 2850 y Ft(\016)502 2769 y Fm(\002)544 2850 y Ft(E)616 2867 y Fq(m;n)767 2850 y Fr(\000)21 b Ft(E)28 b Fv(+)21 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1250 2769 y Fm(\003)1312 2765 y(\014)1312 2825 y(\014)1345 2762 y(p)p 1445 2762 219 4 v 88 x Fv(1)21 b(+)h Ft(\032)17 b(G)1757 2867 y Fq(m;N)1902 2848 y Fe(\000)p Fk(1)1984 2867 y Fo(\()p Fq(n)p Fo(\))2102 2850 y Ft(c)2144 2867 y Fq(N)2207 2848 y Fe(\000)p Fk(1)2290 2867 y Fo(\()p Fq(n)p Fo(\))2435 2850 y Fr(\000)2554 2783 y(p)p 2636 2783 51 4 v 2636 2850 a Ft(\032)f(G)2779 2867 y Fq(N)7 b Fo(\()p Fq(m)p Fo(\))p Fq(;n)3042 2850 y Ft(c)3084 2867 y Fq(m)3151 2765 y Fm(\014)3151 2825 y(\014)3184 2791 y Fo(2)455 3032 y Fv(+)27 b Ft(\016)605 2952 y Fm(\002)647 3032 y Ft(E)719 3050 y Fq(m;n)869 3032 y Fr(\000)22 b Ft(E)28 b Fr(\000)21 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1353 2952 y Fm(\003)1415 2948 y(\014)1415 3008 y(\014)1448 2965 y Fr(p)p 1530 2965 V 1530 3032 a Ft(\032)16 b(G)1673 3050 y Fq(N)1736 3031 y Fe(\000)p Fk(1)1819 3050 y Fo(\()p Fq(n)p Fo(\))p Fq(;m)2021 3032 y Fv(\026)-51 b Ft(c)2061 3050 y Fq(N)2124 3031 y Fe(\000)p Fk(1)2207 3050 y Fo(\()p Fq(n)p Fo(\))2352 3032 y Fr(\000)2471 2945 y Fm(p)p 2571 2945 219 4 v 87 x Fv(1)21 b(+)h Ft(\032)16 b(G)2882 3050 y Fq(n;N)7 b Fo(\()p Fq(m)p Fo(\))3147 3032 y Fv(\026)-51 b Ft(c)3187 3050 y Fq(m)3254 2948 y Fm(\014)3254 3008 y(\014)3287 2974 y Fo(2)3348 2922 y Fm(o)3436 3032 y Ft(:)257 3256 y Fv(No)m(w)32 b(w)m(e)g(observ)m(e)h(that)e(the)h(range) f Fr(N)1702 3217 y Fo(2)1772 3256 y Fv(of)g(summation)h(con)m(tains)g (the)g(follo)m(wing)f(t)m(w)m(o)257 3376 y(disjoin)m(t)j(subsets:)1172 3565 y Fr(A)1250 3514 y Fo(\(1\))1250 3592 y(\001)1366 3565 y Fr(\002)22 b Fv([)p Fr(A)1569 3514 y Fo(\(2\))1569 3592 y(\001)1663 3565 y Fv(])1690 3522 y Fq(c)1747 3565 y Ft(;)205 b Fv([)p Fr(A)2084 3514 y Fo(\(1\))2084 3592 y(\001)2178 3565 y Fv(])2205 3522 y Fq(c)2262 3565 y Fr(\002)21 b(A)2437 3514 y Fo(\(2\))2437 3592 y(\001)2553 3565 y Ft(:)646 b Fv(\(B.30\))257 3779 y(Using)26 b(this)h(and)e(the)h (fact)f(that)h Ft(c)1483 3797 y Fq(i)1536 3779 y Fv(v)-5 b(anishes)27 b(for)e Ft(i)j Fr(2)f Fv([)p Fr(A)2312 3728 y Fo(\(1\))2312 3806 y(\001)2406 3779 y Fv(])2433 3740 y Fq(c)2468 3779 y Fv(,)g(w)m(e)g(obtain)e(the)h(follo)m(wing)257 3899 y(lo)m(w)m(er)34 b(b)s(ound,)1383 4088 y Fr(h)p Fv(\010)p Fr(j)16 b Fv(\000)1595 4106 y Fq(E)1672 4088 y Fv(\010)p Fr(i)60 b(\025)f Ft(\015)2031 4045 y Fn(0)2092 4088 y Fv(+)38 b Ft(\015)2262 4045 y Fn(00)2326 4088 y Ft(;)873 b Fv(\(B.31\))257 4277 y(where)434 4490 y Ft(\015)490 4446 y Fn(0)596 4490 y Fv(:=)782 4354 y Fm(Z)898 4490 y Ft(d\030)1065 4395 y Fm(X)1013 4634 y Fq(m)p Fn(2A)1178 4598 y Fk(\(1\))1178 4657 y(\001)1277 4490 y Fr(j)p Ft(c)1346 4507 y Fq(m)1413 4490 y Fr(j)1440 4446 y Fo(2)1574 4395 y Fm(X)1496 4634 y Fq(n)p Fn(2)p Fo([)p Fn(A)1662 4598 y Fk(\(2\))1662 4657 y(\001)1744 4634 y Fo(])1764 4615 y Fj(c)1796 4379 y Fm(n)1862 4490 y Ft(\016)1909 4409 y Fm(\002)1951 4490 y Ft(E)2023 4507 y Fq(m;n)2174 4490 y Fr(\000)21 b Ft(E)28 b Fv(+)21 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))2657 4409 y Fm(\003)2719 4405 y(\014)2719 4465 y(\014)2752 4423 y Fr(p)p 2834 4423 51 4 v 2834 4490 a Ft(\032)16 b(G)2977 4507 y Fq(N)7 b Fo(\()p Fq(m)p Fo(\))p Fq(;n)3224 4405 y Fm(\014)3224 4465 y(\014)3257 4431 y Fo(2)782 4815 y Fv(+)27 b Ft(\016)932 4734 y Fm(\002)974 4815 y Ft(E)1046 4832 y Fq(m;n)1197 4815 y Fr(\000)21 b Ft(E)28 b Fr(\000)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1681 4734 y Fm(\003)1742 4730 y(\014)1742 4790 y(\014)1776 4727 y(p)p 1875 4727 219 4 v 1875 4815 a Fv(1)22 b(+)f Ft(\032)c(G)2187 4832 y Fq(N)7 b Fo(\()p Fq(m)p Fo(\))p Fq(;n)2434 4730 y Fm(\014)2434 4790 y(\014)2467 4756 y Fo(2)2528 4704 y Fm(o)3226 4815 y Fv(\(B.32\))596 5048 y(:=)782 4913 y Fm(Z)898 5048 y Ft(d\030)1065 4954 y Fm(X)1026 5192 y Fo(~)-48 b Fq(m)p Fn(2A)1178 5157 y Fk(\(1\))1178 5215 y(\001)1277 5048 y Fr(j)p Ft(c)1360 5066 y Fo(~)f Fq(m)1413 5048 y Fr(j)1440 5005 y Fo(2)1584 4954 y Fm(X)1496 5192 y Fq(m)p Fn(2)p Fo([)p Fn(A)1681 5157 y Fk(\(1\))1681 5215 y(\001)1764 5192 y Fo(])1784 5173 y Fj(c)1815 4938 y Fm(n)1882 5048 y Ft(\016)1929 4968 y Fm(\002)1970 5048 y Ft(E)2056 5066 y Fo(~)g Fq(m)q(;N)7 b Fo(\()p Fq(m)p Fo(\))2331 5048 y Fr(\000)21 b Ft(E)28 b Fv(+)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))2815 4968 y Fm(\003)2871 5048 y Ft(\032)782 5373 y Fv(+)27 b Ft(\016)932 5292 y Fm(\002)974 5373 y Ft(E)1060 5391 y Fo(~)-49 b Fq(m;N)7 b Fo(\()p Fq(m)p Fo(\))1334 5373 y Fr(\000)22 b Ft(E)28 b Fr(\000)21 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1818 5292 y Fm(\003)1874 5373 y Fv(\(1)22 b(+)f Ft(\032)p Fv(\))2168 5263 y Fm(o)2257 5289 y(\014)2257 5348 y(\014)2290 5373 y Ft(G)2367 5391 y Fq(N)7 b Fo(\()14 b(~)-49 b Fq(m)q Fo(\))p Fq(;N)7 b Fo(\()p Fq(m)p Fo(\))2751 5289 y Fm(\014)2751 5348 y(\014)2785 5315 y Fo(2)2846 5373 y Ft(;)353 b Fv(\(B.33\))p eop %%Page: 99 100 99 99 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)2006 b Fv(99)257 573 y(and)33 b(similarly)490 815 y Ft(\015)546 771 y Fn(00)672 815 y Fv(:=)858 679 y Fm(Z)974 815 y Ft(d\030)1140 720 y Fm(X)1102 959 y Fo(~)-49 b Fq(m)p Fn(2A)1253 924 y Fk(\(1\))1253 982 y(\001)1353 815 y Fr(j)p Ft(c)1436 832 y Fo(~)g Fq(m)1488 815 y Fr(j)1515 771 y Fo(2)1659 720 y Fm(X)1571 959 y Fq(m)p Fn(2)p Fo([)p Fn(A)1756 924 y Fk(\(1\))1756 982 y(\001)1839 959 y Fo(])1859 940 y Fj(c)1891 704 y Fm(n)1957 815 y Ft(\016)2004 734 y Fm(\002)2046 815 y Ft(E)2118 832 y Fq(m;N)7 b Fo(\()14 b(~)-49 b Fq(m)q Fo(\))2406 815 y Fr(\000)22 b Ft(E)27 b Fv(+)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))2890 734 y Fm(\003)2946 815 y Fv(\(1)22 b(+)f Ft(\032)p Fv(\))858 1140 y(+)27 b Ft(\016)1008 1059 y Fm(\002)1049 1140 y Ft(E)1121 1157 y Fq(m;N)7 b Fo(\()15 b(~)-50 b Fq(m)q Fo(\))1410 1140 y Fr(\000)21 b Ft(E)28 b Fr(\000)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1894 1059 y Fm(\003)1950 1140 y Ft(\032)2000 1029 y Fm(o)2088 1055 y(\014)2088 1115 y(\014)2122 1140 y Ft(G)2199 1157 y Fq(m;)13 b Fo(~)-48 b Fq(m)2347 1055 y Fm(\014)2347 1115 y(\014)2380 1081 y Fo(2)2441 1140 y Ft(:)758 b Fv(\(B.34\))257 1368 y(Adding)33 b Ft(\015)657 1329 y Fn(0)713 1368 y Fv(and)g Ft(\015)959 1329 y Fn(00)1001 1368 y Fv(,)g(w)m(e)h(arriv)m(e)f(at)f(the)h(\014rst) g(assertion,)h(Eq.)f(\(B.27\),)397 1616 y Ft(\015)453 1572 y Fn(0)498 1616 y Fv(+)21 b Ft(\015)651 1572 y Fn(0)q(0)777 1616 y Fr(\025)988 1521 y Fm(X)949 1760 y Fo(~)-48 b Fq(m)p Fn(2A)1101 1724 y Fk(\(1\))1101 1783 y(\001)1200 1616 y Fr(j)p Ft(c)1283 1633 y Fo(~)f Fq(m)1336 1616 y Fr(j)1363 1572 y Fo(2)1419 1480 y Fm(Z)1535 1616 y Ft(d\030)1737 1521 y Fm(X)1649 1760 y Fq(m)p Fn(2)p Fo([)p Fn(A)1834 1724 y Fk(\(1\))1834 1783 y(\001)1917 1760 y Fo(])1937 1741 y Fj(c)1969 1505 y Fm(n)2035 1616 y Ft(\016)2082 1535 y Fm(\002)2124 1616 y Ft(E)2196 1633 y Fq(m;N)7 b Fo(\()14 b(~)-49 b Fq(m)q Fo(\))2484 1616 y Fr(\000)22 b Ft(E)27 b Fv(+)22 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))2968 1535 y Fm(\003)3024 1531 y(\014)3024 1591 y(\014)3058 1616 y Ft(G)3135 1633 y Fq(m;)13 b Fo(~)-48 b Fq(m)3283 1531 y Fm(\014)3283 1591 y(\014)3316 1557 y Fo(2)936 1940 y Fv(+)27 b Ft(\016)1086 1860 y Fm(\002)1127 1940 y Ft(E)1213 1958 y Fo(~)-49 b Fq(m)q(;N)7 b Fo(\()p Fq(m)p Fo(\))1488 1940 y Fr(\000)21 b Ft(E)28 b Fr(\000)21 b Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))1971 1860 y Fm(\003)2028 1856 y(\014)2028 1916 y(\014)2061 1940 y Ft(G)2138 1958 y Fq(N)i Fo(\()14 b(~)-49 b Fq(m)q Fo(\))p Fq(;N)7 b Fo(\()p Fq(m)p Fo(\))2523 1856 y Fm(\014)2523 1916 y(\014)2556 1882 y Fo(2)2595 1830 y Fm(o)2683 1940 y Ft(:)516 b Fv(\(B.35\))257 2193 y(T)-8 b(o)26 b(pro)m(v)m(e)h(\(B.28\),)g(w)m(e)g(assume)g(that)e (\001)j(=)f Fr(f)p Ft(E)1978 2211 y Fq(i;j)2058 2193 y Fr(g)e Fv(and)h Ft(E)33 b Fv(=)27 b Ft(E)2595 2211 y Fq(i;j)2703 2193 y Ft(>)g Fv(0.)41 b(Then)27 b Ft(i)g Fr(2)g(A)3401 2142 y Fo(\(1\))3401 2220 y(\001)257 2333 y Fv(and)33 b(0)27 b Fr(2)g Fv([)p Fr(A)720 2282 y Fo(\(1\))720 2360 y(\001)814 2333 y Fv(])841 2293 y Fq(c)876 2333 y Fv(,)33 b(so)g(retaining)f(only)h(the)g(term)h Ft(m)27 b Fv(=)g(0)32 b(in)h(\(B.27\),)f(w)m(e)i(\014nd)f(that)526 2582 y Ft(\015)582 2539 y Fo(\()p Fq(i;j)t Fo(\))776 2582 y Fr(\025)103 b Fv(min)925 2672 y Fo(~)-48 b Fq(m)p Fn(2A)1077 2636 y Fk(\(1\))1077 2695 y(\001)1177 2447 y Fm(Z)1292 2582 y Ft(d\030)32 b(\016)1465 2502 y Fm(\002)1506 2582 y Ft(E)1578 2600 y Fo(0)p Fq(;N)7 b Fo(\()14 b(~)-49 b Fq(m)q Fo(\))1839 2582 y Fr(\000)22 b Ft(E)2009 2600 y Fq(i;j)2111 2582 y Fv(+)g Ft(!)t Fv(\()p Ft(\030)5 b Fv(\))2398 2502 y Fm(\003)2454 2498 y(\014)2454 2557 y(\014)2487 2582 y Ft(G)2564 2600 y Fq(m;)14 b Fo(~)-49 b Fq(m)2713 2582 y Fv(\()p Ft(\030)5 b Fv(\))2837 2498 y Fm(\014)2837 2557 y(\014)2869 2524 y Fo(2)2930 2582 y Ft(:)269 b Fv(\(B.36\))257 2861 y(Since)34 b Ft(E)584 2878 y Fo(0)p Fq(;N)7 b Fo(\()14 b(~)-49 b Fq(m)q Fo(\))850 2861 y Fr(\024)27 b Fv(0,)32 b(w)m(e)h(ha)m(v)m(e)h(that)e Ft(\034)1682 2878 y Fq(min)1843 2861 y Fr(\024)27 b Ft(E)2018 2878 y Fq(i;j)2119 2861 y Fr(\000)20 b Ft(E)2287 2878 y Fo(0)p Fq(;N)7 b Fo(\()14 b(~)-49 b Fq(m)q Fo(\))2554 2861 y Fr(\024)27 b Fv(2)p Ft(\034)2748 2878 y Fq(max)2892 2861 y Fv(,)32 b(and)g(Hyp)s(oth-)257 2992 y(esis)i(H-5)e(directly)i (yields)g Ft(\015)1309 2953 y Fo(\()p Fq(i;j)t Fo(\))1472 2992 y Ft(>)27 b Fv(0.)p 3425 2992 4 66 v 3429 2930 59 4 v 3429 2992 V 3487 2992 4 66 v 257 3399 a Fh(B.3)132 b(Sp)t(ectral)44 b(Gap)g(ab)t(o)l(v)l(e)h FF(0)f Fh(for)g FF(\000)2145 3364 y Fx(\(0\))257 3583 y Fv(In)38 b(this)f(section)h(w)m (e)g(assume)h(that)d(\001)f(=)f Fr(f)p Fv(0)p Fr(g)i Fv(and)h(that)g Ft(E)j Fv(=)34 b(0.)57 b(F)-8 b(urthermore,)38 b(w)m(e)257 3704 y(shall)h(mak)m(e)g(use)g(of)e(Hyp)s(othesis)j(H-3,)e (in)g(addition)g(to)g(Hyp)s(otheses)i(H-1)d(and)h(H-5.)257 3824 y(Note)30 b(that)g(since)h(the)f(eigen)m(v)-5 b(alues)32 b(of)d Ft(H)1791 3842 y Fq(el)1880 3824 y Fv(are)g(nondegenerate,)j Fr(f)p Fv(0)p Fr(g)d Fv(is)h(nondegener-)257 3945 y(ate)j(and)g (symmetric.)45 b(Moreo)m(v)m(er)34 b(\000)1611 3905 y Fo(\(0\))1733 3945 y Fv(=)27 b(\000)1897 3962 y Fo(0)1936 3860 y Fm(\014)1936 3920 y(\014)1969 3984 y Fo(Ran)13 b Fq(P)2163 3961 y Fj(el)2153 4005 y Fk(0)2252 3945 y Fv(comm)m(utes)35 b(with)e Ft(J)2987 3962 y Fq(el)3046 3945 y Fv(.)257 4152 y Fi(Theorem)k(B.2.)49 b Fs(The)36 b(in\014mum)f(of)g(the)h(sp)-5 b(e)g(ctrum)36 b(of)g Fv(\000)2433 4113 y Fo(\(0\))2563 4152 y Fs(is)g(a)f(simple)g (eigenvalue)257 4272 y(and)d(e)-5 b(quals)31 b Fv(0)p Fs(,)i(with)e(eigenve)-5 b(ctor)31 b Ft(\024)1607 4290 y Fq(\014)1682 4272 y Fv(=)1785 4198 y Fm(P)1890 4224 y Fq(N)7 b Fn(\000)p Fo(1)1890 4301 y Fq(m)p Fo(=0)2064 4272 y Ft(e)2109 4233 y Fn(\000)p Fq(\014)s(E)2259 4241 y Fj(m)2317 4233 y Fq(=)p Fo(2)2392 4272 y Ft(')2456 4290 y Fq(i;i)2527 4272 y Fs(.)44 b(F)-7 b(urthermor)i(e,)32 b(the)g(gap)257 4393 y(ab)-5 b(ove)34 b Fv(0)h Fs(c)-5 b(an)34 b(b)-5 b(e)35 b(estimate)-5 b(d)34 b(by)1194 4614 y Fv(0)27 b Ft(<)g Fm(b)-55 b Ft(\015)1424 4631 y Fo(0)1525 4614 y Fr(\024)62 b Fv(min)1825 4503 y Fm(n)1908 4614 y Ft(\033)1967 4533 y Fm(\002)2008 4614 y Fv(\000)2069 4570 y Fo(\(0\))2164 4533 y Fm(\003)2243 4614 y Fr(n)38 b(f)p Fv(0)p Fr(g)2493 4503 y Fm(o)3226 4614 y Fv(\(B.37\))257 4835 y Fs(wher)-5 b(e)590 5102 y Fm(b)-56 b Ft(\015)640 5119 y Fo(0)741 5102 y Fv(:=)62 b(2)971 4961 y Fm(\022)1045 4977 y Fq(N)7 b Fn(\000)p Fo(1)1049 5007 y Fm(X)1045 5216 y Fq(m)p Fo(=0)1214 5102 y Ft(e)1259 5058 y Fn(\000)p Fq(\014)s(E)1409 5066 y Fj(m)1472 4961 y Fm(\023)1731 5102 y Fv(min)1583 5164 y Fo(0)p Fn(\024)p Fq(m)27 b Fv(0)34 b Fs(is)h(de\014ne)-5 b(d)34 b(in)g(\(B.21\).)p eop %%Page: 100 101 100 100 bop 257 266 a Fw(BFS-4,)32 b(Decem)m(b)s(er)i(21,)e(1999)1957 b Fv(100)257 573 y Fi(Remark)37 b(B.3.)49 b Fs(Note)35 b(that)438 825 y Fv(lim)16 b(inf)481 888 y Fq(\014)s Fn(!1)725 825 y Fm(b)-56 b Ft(\015)775 843 y Fo(0)876 825 y Fr(\025)210 b Fv(min)1014 887 y Fo(0)p Fn(\024)p Fq(m)27 b Fv(0)p Fs(,)35 b(uniformly)f(in)h Ft(\014)d Fr(!)27 b(1)p Fs(,)34 b(thanks)h(to)g(Hyp)-5 b(othesis)35 b(H-5.)257 1345 y(Pr)-5 b(o)g(of.)49 b Fv(Since)h Fr(f)p Fv(0)p Fr(g)e Fv(is)i(nondegenerate,)k Fr(f)p Ft(')1949 1362 y Fq(i;i)2021 1345 y Fr(j)p Ft(i)g Fr(2)h(N)15 b(g)48 b Fv(is)i(an)f(orthonormal)f(basis)257 1465 y(in)43 b(Ran)16 b Ft(P)649 1426 y Fq(el)635 1486 y Fo(0)707 1465 y Fv(,)45 b(and)d(w)m(e)h(ma)m(y)g(write)g(an)m(y)g(normalized)g(v)m(ector)g Ft(\024)g Fr(2)g Fv(Ran)16 b Ft(P)3090 1426 y Fq(el)3076 1486 y Fo(0)3191 1465 y Fv(as)42 b Ft(\024)h Fv(=)257 1511 y Fm(P)363 1615 y Fq(i)p Fn(2N)522 1585 y Ft(\024)578 1603 y Fq(i)622 1585 y Ft(')686 1603 y Fq(i;i)758 1585 y Fv(,)37 b(where)1107 1511 y Fm(P)1212 1615 y Fq(i)p Fn(2N)1371 1585 y Fr(j)p Ft(\024)1454 1603 y Fq(i)1483 1585 y Fr(j)1510 1546 y Fo(2)1582 1585 y Fv(=)c(1.)53 b(Inserting)37 b(this)g(in)m(to)f(\(B.21\))g(and)g(denoting)257 1716 y Ft(\026)316 1734 y Fq(m;n)472 1716 y Fv(:=)27 b Ft(\021)650 1734 y Fq(m;n)796 1716 y Ft(e)841 1677 y Fq(\014)s Fo(\()p Fq(E)963 1685 y Fj(m)1022 1677 y Fo(+)p Fq(E)1129 1685 y Fj(n)1171 1677 y Fo(\))p Fq(=)p Fo(2)1273 1716 y Fv(,)32 b(w)m(e)i(obtain)530 2034 y Fr(h)p Ft(\024)p Fr(j)16 b Fv(\000)728 2052 y Fo(0)784 2034 y Ft(\024)p Fr(i)83 b Fv(=)1151 1910 y Fq(N)7 b Fn(\000)p Fo(1)1156 1940 y Fm(X)1120 2149 y Fq(m;n)p Fo(=0)1351 2034 y Ft(\021)1399 2052 y Fq(m;n)1545 1924 y Fm(\020)1604 2034 y Ft(e)1649 1991 y Fn(\000)p Fq(\014)s(E)1799 1999 y Fj(m;n)1915 1991 y Fq(=)p Fo(2)2006 2034 y Fr(j)p Ft(\024)2089 2052 y Fq(n)2136 2034 y Fr(j)2163 1991 y Fo(2)2246 2034 y Fr(\000)p 2365 1980 123 4 v 43 w Ft(\024)2421 2052 y Fq(m)2504 2034 y Ft(\024)2560 2052 y Fq(n)2608 1924 y Fm(\021)3226 2034 y Fv(\(B.40\))961 2385 y(=)1151 2260 y Fq(N)g Fn(\000)p Fo(1)1156 2290 y Fm(X)1120 2499 y Fq(m;n)p Fo(=0)1351 2385 y Ft(\026)1410 2402 y Fq(m;n)1555 2274 y Fm(\020)1615 2385 y Ft(e)1660 2341 y Fn(\000)p Fq(\014)s(E)1810 2349 y Fj(m)1889 2385 y Fr(j)p Ft(\024)1972 2402 y Fq(n)2019 2385 y Fr(j)2046 2341 y Fo(2)2129 2385 y Fr(\000)p 2248 2294 452 4 v 43 w Ft(e)2293 2356 y Fn(\000)p Fq(\014)s(E)2443 2364 y Fj(n)2485 2356 y Fq(=)p Fo(2)2576 2385 y Ft(\024)2632 2402 y Fq(m)2715 2385 y Ft(e)2760 2341 y Fn(\000)p Fq(\014)s(E)2910 2349 y Fj(m)2969 2341 y Fq(=)p Fo(2)3060 2385 y Ft(\024)3116 2402 y Fq(n)3163 2274 y Fm(\021)961 2735 y Fv(=)1151 2611 y Fq(N)g Fn(\000)p Fo(1)1156 2641 y Fm(X)1120 2850 y Fq(m;n)p Fo(=0)1351 2735 y Ft(\026)1410 2753 y Fq(m;n)1555 2621 y Fm(\014)1555 2681 y(\014)1555 2740 y(\014)1589 2735 y Ft(e)1634 2692 y Fn(\000)p Fq(\014)s(E)1784 2700 y Fj(m)1842 2692 y Fq(=)p Fo(2)1933 2735 y Ft(\024)1989 2753 y Fq(n)2080 2735 y Fr(\000)43 b Ft(e)2244 2692 y Fn(\000)p Fq(\014)s(E)2394 2700 y Fj(n)2436 2692 y Fq(=)p Fo(2)2527 2735 y Ft(\024)2583 2753 y Fq(m)2650 2621 y Fm(\014)2650 2681 y(\014)2650 2740 y(\014)2683 2647 y Fo(2)961 3086 y Fv(=)83 b(2)1250 2961 y Fq(N)7 b Fn(\000)p Fo(1)1255 2991 y Fm(X)1202 3202 y Fo(0)p Fn(\024)p Fq(m