Content-Type: multipart/mixed; boundary="-------------0507081518339" This is a multi-part message in MIME format. ---------------0507081518339 Content-Type: text/plain; name="05-239.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-239.keywords" open quantum systems, Liouville operator, spectral deformation, heat reservoir, non-equilibrium stationary states, KMS, Fermi Golden Rule, level shift operator ---------------0507081518339 Content-Type: application/postscript; name="nessI.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nessI.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: nessI.dvi %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips nessI -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.07.08:1513 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict 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b(>)g(\026)20 b Fq(\000)f Ft(1,)29 b(where)g Fs(\026)h Ft(is)f(the)h(parameter)e(in)i(the)g(preceding)e (paragraph\).)41 b(This)29 b(is)g(the)h(same)f(infra-red)260 1828 y(condition)f(as)g(in)h([14],)g(and)f(it)i(presen)n(ts)d(an)i (impro)n(v)n(emen)n(t)e(of)i(the)g(one)g(in)g([8)o(],)g(since)g([8])g (requires)e Fs(p)e(>)f Ft(2,)260 1953 y(though)j(with)h(less)f (restrictions)g(on)g(the)h(regularit)n(y)e(of)h Fs(k)f Fq(7!)d Fs(G)2285 1965 y Fn(j)2321 1953 y Ft(\()p Fs(k)s Ft(\).)384 2077 y(Our)h(approac)n(h)e(is)i(based)g(on)g(the)h(c)n (haracterization)c(of)j(equilibrium)h(states)f(in)g(terms)g(of)g(eigen) n(v)n(ectors)260 2202 y(corresp)r(onding)16 b(to)j(the)f(eigen)n(v)-5 b(alue)18 b(zero)f(of)h(certain)g(selfadjoin)n(t)g(op)r(erators)f Fs(L)p Ft(,)j(called)e Fo(Liouvil)t(le)23 b(op)l(er)l(ators)p Ft(,)260 2326 y(whic)n(h)31 b(act)g(on)g(the)g(GNS)h(represen)n(tation) d(Hilb)r(ert)j(space)e(\(p)r(ositiv)n(e)h(temp)r(erature)g(Hilb)r(ert)g (space\))g(\(see)260 2451 y([12)o(,)d(5)o(,)g(13)o(,)g(10)o(]\).)384 2575 y(P)n(arts)22 b(of)i(our)f(tec)n(hniques)h(can)f(b)r(e)h(view)n (ed)f(as)g(a)h(p)r(erturbation)f(theory)g(in)h(the)g(temp)r(eratures,)g (around)260 2700 y Fs(\016)s(\014)37 b Ft(:=)32 b Fq(j)p Fs(T)588 2664 y Fk(\000)p Fr(1)576 2722 y(1)699 2700 y Fq(\000)22 b Fs(T)847 2664 y Fk(\000)p Fr(2)835 2722 y(2)935 2700 y Fq(j)33 b Ft(=)f(0.)54 b(This)33 b(is)h(a)f Fo(singular)i(p)l(erturb)l(ation)g(the)l(ory)f Ft(in)g(the)g(sense)f (that)h(the)f(Hilb)r(ert)260 2824 y(spaces)23 b(represen)n(tations)f (of)h(the)i(system)e(for)h Fs(\016)s(\014)j Ft(=)c(0)g(and)h Fs(\016)s(\014)j(>)c Ft(0)g(are)g(not)h(normal)f(with)h(resp)r(ect)g (to)g(eac)n(h)260 2949 y(other)j(\([20)o(,)h(6)o(,)g(7]\).)384 3073 y(Our)g(tec)n(hniques)f(are)g(applicable)h(to)g(a)f(wide)i(class)e (of)h(non-selfadjoin)n(t)f(op)r(erators)f Fs(K)6 b Ft(,)28 b(con)n(taining)f(in)260 3198 y(particular)h(the)i(Liouville)f(op)r (erators)e(men)n(tioned)j(ab)r(o)n(v)n(e,)f(but)h(also)e(con)n(taining) g(non-selfadjoin)n(t)h(gener-)260 3323 y(ators)f(of)i(the)g(dynamics)f (used)h(in)f(the)h(examination)f(of)h(non-equilibrium)f(stationary)f (states)h(\([14,)g(16)o(]\).)260 3447 y(W)-7 b(e)28 b(th)n(us)f(carry)f (out)i(our)f(analysis)f(for)h(this)h(more)f(general)f(class)h(of)g(op)r (erators.)384 3572 y(In)39 b(order)e(to)i(study)g(the)g(sp)r(ectrum)f (of)h(the)g(op)r(erators)d Fs(K)6 b Ft(,)41 b(w)n(e)e(dev)n(elop)e(a)i (new)f(t)n(yp)r(e)h(of)f(sp)r(ectral)260 3696 y(deformation,)e Fs(K)42 b Fq(7!)36 b Fs(K)1059 3708 y Fn(\022)1096 3696 y Ft(,)i(with)e(a)f(sp)r(ectral)f(deformation)h(parameter)f Fs(\022)k Fq(2)f Fh(C)2850 3666 y Fr(2)2893 3696 y Ft(,)g(whic)n(h)f (com)n(bines)f(the)260 3821 y(deformations)17 b(in)n(tro)r(duced)i(in)g ([12)o(])f(and)h(in)g([5)o(],)i(hence)d Fs(\022)j Ft(is)e(in)g Fh(C)2259 3791 y Fr(2)2321 3821 y Ft(rather)e(than)i(in)g Fh(C)c Ft(.)40 b(\(Suc)n(h)19 b(a)f(com)n(bination)260 3945 y(w)n(as)28 b(already)h(men)n(tioned)g(in)h([5].\))43 b(In)30 b(order)e(to)h(establish)h(the)g(desired)f(sp)r(ectral)g(c)n (haracteristics)e(of)j(the)260 4070 y(op)r(erator)23 b(family)h Fs(K)913 4082 y Fn(\022)950 4070 y Ft(,)h(w)n(e)f(use)h(the) g(metho)r(d)g(of)f(the)h(F)-7 b(esh)n(bac)n(h)23 b(map,)j(and)e(p)r (erform)g(the)h(basic)f(step)g(of)h(the)260 4194 y(sp)r(ectral)i (renormalization)e(group)i(approac)n(h)e(as)i(dev)n(elop)r(ed)h(in)f ([2,)h(3)o(,)g(4)o(].)384 4319 y(Already)21 b(a)g(single)g(application) g(of)g(the)h(F)-7 b(esh)n(bac)n(h)21 b(map,)h(considered)f(in)g(this)h (pap)r(er,)g(yields)g(the)f(results)260 4443 y(men)n(tioned)30 b(ab)r(o)n(v)n(e.)45 b(Adapting)31 b(ideas)f(of)g([2,)h(4)o(,)g(5])f (on)h(the)g(full)g(renormalization)d(group)i(approac)n(h,)g(the)260 4568 y(restriction)f(on)g(the)i(temp)r(eratures)e(can)h(b)r(e)g(remo)n (v)n(ed.)42 b(W)-7 b(e)31 b(presen)n(t)e(in)h([17)o(])g(a)g(detailed)g (analysis)f(of)h(the)260 4692 y(R)n(G)c(to)g(the)g(sp)r(eci\014c)g(mo)r (del)g(at)g(hand.)36 b(It)27 b(relies)e(on)h([3)o(,)g(4,)g(5])g(and)f (features)h(some)f(simpli\014cations)h(due)g(to)260 4817 y(the)i(sp)r(eci\014cit)n(y)f(of)h(our)f(problem)g(and)g(some)g(recen)n (t)g(dev)n(elopmen)n(ts)g([2].)384 4941 y(In)33 b(con)n(trast)e(to)h (the)h(case)e(of)h(quan)n(tum)h(Hamiltonians)f(for)f(zero)h(temp)r (erature)f(systems,)i(the)g(sp)r(ec-)260 5066 y(tral)d(theory)f(of)h (time-translation)g(generators)e(of)i(op)r(en)g(quan)n(tum)h(systems)e (is)i(at)f(an)g(early)f(stage)g(of)i(its)260 5191 y(dev)n(elopmen)n(t.) 36 b(Our)27 b(pap)r(er)g(is)h(a)f(con)n(tribution)g(to)g(this)h(theory) -7 b(.)1949 5476 y(2)p eop %%Page: 3 3 3 2 bop 384 210 a Ft(This)39 b(pap)r(er)g(is)f(organized)f(as)h(follo)n (ws.)70 b(In)39 b(Section)g(2)f(w)n(e)g(describ)r(e)h(our)f(mo)r(del)h (and)f(de\014ne)i(the)260 335 y(dynamics)26 b(of)h(it.)37 b(\(The)27 b(de\014nition)g(of)f(the)h(dynamics)f(is)h(a)f(somewhat)g (subtle)h(matter.\))37 b(In)26 b(Section)h(3)f(w)n(e)260 459 y(giv)n(e)32 b(a)g(precise)g(form)n(ulation)f(of)i(our)f (assumptions,)h(state)f(the)h(results)g(and)f(discuss)g(assumptions)g (and)260 584 y(results.)47 b(In)32 b(Section)f(4)g(w)n(e)g(presen)n(t)g (the)g(Araki-W)-7 b(o)r(o)r(ds)30 b(construction)h(whic)n(h)g(w)n(e)g (use)g(throughout)g(this)260 708 y(pap)r(er.)36 b(In)25 b(Section)g(5)g(w)n(e)g(de\014ne)h(a)f(sp)r(ectral)f(deformation)h(of)g (a)g(family)g(of)h(op)r(erators)d Fs(K)31 b Ft(whic)n(h)25 b(con)n(tains)260 833 y(the)j(generator)d(of)i(the)h(ev)n(olution,)e (and)h(w)n(e)g(establish)g(some)g(basic)g(analyticit)n(y)f(and)h(sp)r (ectral)g(prop)r(erties)260 957 y(of)i(those)f(op)r(erators.)38 b(In)29 b(Section)g(6)f(w)n(e)h(carry)e(out)h(a)h(more)f(re\014ned)g (sp)r(ectral)g(analysis,)g(preparing)f(for)h(a)260 1082 y(pro)r(of)e(of)g(absence)f(of)i(normal)e(in)n(v)-5 b(arian)n(t)25 b(states,)h(whic)n(h)h(is)f(giv)n(en)f(in)i(Section)f(7.)36 b(Finally)-7 b(,)27 b(in)f(App)r(endices)260 1206 y(A{C)h(w)n(e)g (collect)h(some)f(tec)n(hnical)g(results.)260 1477 y Fu(2)135 b(Mo)t(del)44 b(and)h(Mathematical)i(F)-11 b(ramew)l(ork)260 1684 y Ft(W)k(e)37 b(consider)f(a)g(system)h(consisting)f(of)h(a)g (particle)f(system,)j(describ)r(ed)e(b)n(y)f(a)h(Hamiltonian)f Fs(H)3438 1696 y Fn(p)3514 1684 y Ft(on)g(a)260 1809 y(Hilb)r(ert)31 b(space)e Fq(H)843 1821 y Fn(p)882 1809 y Ft(,)i(and)f(t)n(w)n(o)f(\(thermal\))i(reserv)n(oirs,)d(at)i(in)n(v)n (erse)f(temp)r(eratures)g Fs(\014)2970 1821 y Fr(1)3038 1809 y Ft(and)h Fs(\014)3249 1821 y Fr(2)3286 1809 y Ft(,)h(describ)r(ed)260 1933 y(b)n(y)25 b(the)g(Hamiltonians)g Fs(H)1090 1945 y Fn(r)r Fr(1)1185 1933 y Ft(and)g Fs(H)1413 1945 y Fn(r)r Fr(2)1508 1933 y Ft(acting)g(on)g(Hilb)r(ert)h(spaces)e Fq(H)2474 1945 y Fn(r)r Fr(1)2569 1933 y Ft(and)h Fq(H)2798 1945 y Fn(r)r Fr(2)2868 1933 y Ft(,)h(resp)r(ectiv)n(ely)-7 b(.)35 b(The)25 b(full)260 2058 y(Hamiltonian)i(is)1693 2182 y Fs(H)j Ft(:=)22 b Fs(H)1971 2194 y Fr(0)2027 2182 y Ft(+)c 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Ft(\))3061 4716 y Fk(\003)3099 4750 y Fs(:)387 b Ft(\(2.5\))384 4925 y(If)28 b(the)g(coupling)f(op)r (erators)f Fs(G)1375 4937 y Fn(j)1438 4925 y Ft(are)h(suc)n(h)g(that) 1005 5132 y Fs(g)1048 5097 y Fr(2)1099 5019 y Fp(Z)1084 5260 y Ff(R)1131 5244 y Fm(3)1195 5064 y Fp(\000)1233 5132 y Ft(1)18 b(+)g Fq(j)p Fs(k)s Fq(j)1468 5097 y Fk(\000)p Fr(1)1558 5064 y Fp(\001)1609 5132 y Fq(k)p Fs(G)1716 5144 y Fn(j)1751 5132 y Ft(\()p Fs(k)s Ft(\))p Fq(k)1903 5090 y Fr(2)1968 5132 y Fs(dk)169 b Ft(is)28 b(su\016cien)n(tly)f (small)p Fs(;)574 b Ft(\(2.6\))1949 5476 y(3)p eop %%Page: 4 4 4 3 bop 260 210 a Ft(then)28 b(the)g(op)r(erator)e Fs(H)34 b Ft(is)28 b(self-adjoin)n(t)f(\(see)g(e.g.)37 b([5)o(]\).)384 335 y(No)n(w)32 b(w)n(e)g(set)h(up)f(a)g(mathematical)g(framew)n(ork)f (for)g(non-equilibrium)h(statistical)g(mec)n(hanics.)50 b(Op-)260 459 y(erators)36 b(on)j(the)f(Hilb)r(ert)i(space)d Fq(H)1428 471 y Fr(0)1504 459 y Ft(will)i(b)r(e)g(called)f(observ)-5 b(ables.)68 b(\(Strictly)39 b(sp)r(eaking)e(only)i(certain)260 584 y(self-adjoin)n(t)26 b(op)r(erators)f(on)h Fq(H)1237 596 y Fr(0)1302 584 y Ft(are)f(ph)n(ysical)h(observ)-5 b(ables.\))36 b(As)26 b(an)h(algebra)e(of)i(observ)-5 b(ables)25 b(describing)260 708 y(the)j(system)f(w)n(e)g(tak)n(e)g(the) h Fs(C)1188 678 y Fk(\003)1227 708 y Ft(-algebra)1408 916 y Fq(A)23 b Ft(=)g Fq(B)s Ft(\()p Fq(H)1745 928 y Fn(p)1783 916 y Ft(\))c Fq(\012)f Fe(W)p Ft(\()p Fs(L)2093 882 y Fr(2)2093 936 y(0)2130 916 y Ft(\))g Fq(\012)g Fe(W)p Ft(\()p Fs(L)2439 882 y Fr(2)2439 936 y(0)2476 916 y Ft(\))p Fs(;)978 b Ft(\(2.7\))260 1123 y(where)20 b Fe(W)p Ft(\()p Fs(L)669 1093 y Fr(2)669 1144 y(0)705 1123 y Ft(\))h(denotes)f(the)h(W)-7 b(eyl)20 b(CCR)h(algebra)d(o)n(v)n (er)h(the)h(space)g Fs(L)2452 1093 y Fr(2)2452 1144 y(0)2512 1123 y Ft(:=)i Fs(L)2679 1093 y Fr(2)2716 1123 y Ft(\()p Fh(R)2802 1093 y Fr(3)2846 1123 y Fs(;)14 b Ft(\(1)t(+)t Fq(j)p Fs(k)s Fq(j)3122 1093 y Fk(\000)p Fr(1)3210 1123 y Ft(\))p Fs(d)3285 1093 y Fr(3)3322 1123 y Fs(k)s Ft(\).)35 b(States)260 1248 y(of)27 b(the)h(system)g(are)e(p)r(ositiv)n(e)i (linear)e(functionals,)i Fs( )s Ft(,)g(on)f(the)h(algebra)e Fq(A)i Ft(normalized)e(as)h Fs( )s Ft(\()p Fg(1)p Ft(\))d(=)e(1.)384 1373 y(The)32 b(reason)d(w)n(e)i(c)n(hose)f Fq(A)h Ft(rather)f(than)i Fq(B)s Ft(\()p Fq(H)1878 1385 y Fr(0)1914 1373 y Ft(\))g(is)f(that)g (the)h(algebra)d Fq(A)i Ft(supp)r(orts)g(states)g(in)g(whic)n(h)260 1497 y(eac)n(h)26 b(reserv)n(oir)d(is)k(at)f(a)g(thermal)h(equilibrium) f(at)h(its)f(o)n(wn)g(temp)r(erature.)36 b(More)26 b(precisely)-7 b(,)26 b(consider)g(the)260 1622 y(ev)n(olution)h(for)g(the)h Fs(i)p Ft(-th)f(reserv)n(oir)e(giv)n(en)i(b)n(y)1513 1829 y Fs(\013)1566 1795 y Fn(t)1566 1850 y(r)r(i)1626 1829 y Ft(\()p Fs(A)p Ft(\))d(:=)f Fs(e)1926 1795 y Fn(iH)2003 1803 y Fl(r)q(i)2059 1795 y Fn(t)2088 1829 y Fs(Ae)2189 1795 y Fk(\000)p Fn(iH)2318 1803 y Fl(r)q(i)2374 1795 y Fn(t)2404 1829 y Fs(:)1082 b Ft(\(2.8\))260 2037 y(Then)36 b(there)g(are)e(stationary)h(states)g(on)h(the)g Fs(i)p Ft(-th)f(reserv)n(oir)f(algebra)g(of)h(observ)-5 b(ables,)37 b Fe(W)p Ft(\()p Fs(L)3340 2007 y Fr(2)3340 2057 y(0)3377 2037 y Ft(\),)h(whic)n(h)260 2161 y(describ)r(e)31 b(thermal)g (equilibria.)47 b(These)31 b(states)g(are)g(parametrized)f(b)n(y)h(the) g(in)n(v)n(erse)f(temp)r(erature)h Fs(\014)36 b Ft(and)260 2286 y(their)27 b(generating)g(functional)g(is)h(giv)n(en)f(b)n(y)1053 2523 y Fs(!)1108 2480 y Fr(\()p Fn(\014)s Fr(\))1105 2546 y Fn(r)r(i)1218 2523 y Ft(\()q Fs(W)1329 2535 y Fn(i)1357 2523 y Ft(\()p Fs(f)9 b Ft(\)\))23 b(=)g(exp)1755 2406 y Fp(\032)1817 2523 y Fq(\000)1892 2467 y Ft(1)p 1892 2504 42 4 v 1892 2580 a(4)1957 2410 y Fp(Z)2003 2599 y Ff(R)2050 2582 y Fm(3)2105 2467 y Fs(e)2144 2437 y Fn(\014)s Fk(j)p Fn(k)q Fk(j)2283 2467 y Ft(+)18 b(1)p 2105 2504 303 4 v 2105 2582 a Fs(e)2144 2558 y Fn(\014)s Fk(j)p Fn(k)q Fk(j)2283 2582 y Fq(\000)g Ft(1)2418 2523 y Fq(j)p Fs(f)9 b Ft(\()p Fs(k)s Ft(\))p Fq(j)2624 2489 y Fr(2)2661 2523 y Fs(d)2704 2489 y Fr(3)2741 2523 y Fs(k)2787 2406 y Fp(\033)2863 2523 y Fs(;)623 b Ft(\(2.9\))260 2767 y(where)24 b Fs(W)575 2779 y Fn(j)610 2767 y Ft(\()p Fs(f)9 b Ft(\))24 b(:=)e Fs(e)897 2737 y Fn(i\036)960 2745 y Fl(j)991 2737 y Fr(\()p Fn(f)7 b Fr(\))1086 2767 y Ft(,)26 b(with)f Fs(\036)1370 2779 y Fn(j)1405 2767 y Ft(\()p Fs(f)9 b Ft(\))23 b(:=)1690 2734 y Fr(1)p 1663 2748 88 4 v 1663 2757 a Fk(p)p 1718 2757 34 3 v 48 x Fr(2)1774 2700 y Fp(\000)1812 2767 y Fs(a)1856 2737 y Fk(\003)1856 2789 y Fn(j)1895 2767 y Ft(\()p Fs(f)9 b Ft(\))18 b(+)g Fs(a)2154 2779 y Fn(j)2189 2767 y Ft(\()p Fs(f)9 b Ft(\))2303 2700 y Fp(\001)2341 2767 y Ft(,)26 b(is)e(a)g(W)-7 b(eyl)25 b(op)r(erator,)f(see)g(e.g.)35 b([7].)h(The)260 2892 y(c)n(hoice)26 b(of)h(the)h(space)e Fs(L)1021 2861 y Fr(2)1021 2912 y(0)1085 2892 y Ft(ab)r(o)n(v)n(e)g(is) h(dictated)g(b)n(y)g(the)h(need)f(to)g(ha)n(v)n(e)f(the)h(r.h.s.)37 b(of)27 b(this)g(functional)g(\014nite.)260 3016 y(These)g(states)g 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Fn(n)2068 3390 y Ft(\))28 b(\(see)g([7)o(],)g(Section)g(5.2.3)e(and)h (\(2.15\)\).)384 3514 y(Consider)g(states)g(\(on)h Fq(A)p Ft(\))g(of)g(the)g(form)1520 3722 y Fs(!)1572 3734 y Fr(0)1632 3722 y Ft(:=)22 b Fs(!)1794 3734 y Fn(p)1851 3722 y Fq(\012)c Fs(!)1989 3679 y Fr(\()p Fn(\014)2053 3687 y Fm(1)2085 3679 y Fr(\))1986 3744 y Fn(r)r Fr(1)2133 3722 y Fq(\012)g Fs(!)2271 3679 y Fr(\()p Fn(\014)2335 3687 y Fm(2)2367 3679 y Fr(\))2268 3744 y Fn(r)r Fr(2)2397 3722 y Fs(;)1048 b Ft(\(2.10\))260 3929 y(where)20 b Fs(!)545 3941 y Fn(p)604 3929 y Ft(is)h(a)g(state)g(of)g(the)g (particle)g(system)f(and)h Fs(!)1937 3886 y Fr(\()p Fn(\014)s Fr(\))1934 3952 y Fn(r)r(i)2055 3929 y Ft(is)g(the)g(\()p Fs(\013)2353 3899 y Fn(t)2353 3951 y(r)r(i)2414 3929 y Fs(;)14 b(\014)t Ft(\)-KMS)21 b(state)f(of)h(the)h Fs(i)p Ft(-th)f(reserv)n(oir.)260 4054 y(The)27 b(set)g(of)f(states)h (whic)n(h)g(are)f(normal)f(w.r.t.)37 b Fs(!)1826 4066 y Fr(0)1890 4054 y Ft(is)27 b(the)g(same)f(for)g(an)n(y)h(c)n(hoice)f (of)g Fs(!)2996 4066 y Fn(p)3034 4054 y Ft(.)37 b(A)27 b(state)g Fs( )j Ft(whic)n(h)260 4178 y(is)d(normal)e(w.r.t.)37 b Fs(!)906 4190 y Fr(0)970 4178 y Ft(\(i.e.,)27 b(whic)n(h)g(is)g (represen)n(ted)e(b)n(y)i(a)f(densit)n(y)h(matrix)f Fs(\032)h Ft(in)g(the)g(GNS)h(represen)n(tation)260 4303 y(\()p Fq(H)q Fs(;)14 b(\031)s(;)g Ft(\012)547 4315 y Fr(0)584 4303 y Ft(\))28 b(of)g(\()p Fe(A)p Fs(;)14 b(!)920 4315 y Fr(0)957 4303 y Ft(\),)28 b(according)d(to)j Fs( )s Ft(\()p Fs(A)p Ft(\))c(=)e(T)-7 b(r\()p Fs(\032\031)s Ft(\()p Fs(A)p Ft(\)\)\))30 b(will)d(b)r(e)h(called)g(a)f Fs(\014)2861 4315 y Fr(1)2898 4303 y Fs(\014)2945 4315 y Fr(2)2982 4303 y Fo(-normal)j(state)p Ft(.)384 4427 y(In)e(the)g(particular)e(case)h Fs(!)1242 4439 y Fn(p)1280 4427 y Ft(\()p Fq(\001)p Ft(\))d(=)f(T)-7 b(r)o(\()p Fs(e)1635 4397 y Fk(\000)p Fn(\014)1725 4405 y Fl(p)1759 4397 y Fn(H)1813 4405 y Fl(p)1867 4427 y Fq(\001)p Ft(\))p Fs(=)p Ft(T)g(r)o(\()p Fs(e)2120 4397 y Fk(\000)p Fn(\014)2210 4405 y Fl(p)2244 4397 y Fn(H)2298 4405 y Fl(p)2338 4427 y Ft(\))28 b(w)n(e)f(call)g Fs(!)2724 4439 y Fr(0)2789 4427 y Ft(a)g Fo(r)l(efer)l(enc)l(e)j(state)p Ft(.)384 4552 y(The)25 b(Hamiltonian)f Fs(H)31 b Ft(generates)23 b(the)h(dynamics)g(of)g(observ)-5 b(ables)23 b Fs(A)g Fq(2)h(B)s Ft(\()p Fq(H)2840 4564 y Fr(0)2876 4552 y Ft(\))h(according)d(to)j(the)f(rule)1470 4760 y Fs(A)f Fq(7!)g Fs(\013)1714 4725 y Fn(t)1744 4760 y Ft(\()p Fs(A)p Ft(\))h(:=)f Fs(e)2044 4725 y Fn(iH)t(t)2155 4760 y Fs(Ae)2256 4725 y Fk(\000)p Fn(iH)t(t)2447 4760 y Fs(:)998 b Ft(\(2.11\))260 4967 y(Eqn)27 b(\(2.11\))f(de\014nes)i(a)e(group)h (of)g(*-automorphisms)e(of)i Fq(B)s Ft(\()p Fq(H)2249 4979 y Fr(0)2286 4967 y Ft(\).)37 b(Ho)n(w)n(ev)n(er,)26 b Fs(\013)2789 4937 y Fn(t)2846 4967 y Ft(is)h(not)g(exp)r(ected)h(to)f (map)260 5092 y(the)35 b(algebra)d Fq(A)j Ft(in)n(to)f(itself.)58 b(T)-7 b(o)34 b(circum)n(v)n(en)n(t)g(this)g(problem)g(w)n(e)g (de\014ne)h(the)g(in)n(teracting)e(ev)n(olution)h(of)1949 5476 y(4)p eop %%Page: 5 5 5 4 bop 260 210 a Ft(states)37 b(on)f Fq(A)i Ft(b)n(y)f(using)g(the)g (Araki-Dyson)f(expansion.)65 b(Namely)-7 b(,)39 b(for)e(a)f(state)h Fs( )k Ft(on)c(the)g(algebra)f Fq(A)260 335 y Ft(normal)26 b(w.r.t.)37 b(the)28 b(state)g Fs(!)1171 347 y Fr(0)1208 335 y Ft(,)f(w)n(e)h(de\014ne)f(the)h(ev)n(olution)f(b)n(y)889 560 y Fs( )946 526 y Fn(t)975 560 y Ft(\()p Fs(A)p Ft(\))d(:=)52 b(lim)1236 610 y Fn(n)p Fk(!1)1462 456 y(1)1435 481 y Fp(X)1423 657 y Fn(m)p Fr(=0)1566 560 y Ft(\()p Fs(ig)s Ft(\))1702 526 y Fn(m)1779 447 y Fp(Z)1862 468 y Fn(t)1825 636 y Fr(0)1905 560 y Fs(dt)1978 572 y Fr(1)2029 560 y Fq(\001)14 b(\001)g(\001)2140 447 y Fp(Z)2223 468 y Fn(t)2248 476 y Fl(m)p Fd(\000)p Fm(1)2186 636 y Fr(0)2394 560 y Fs(dt)2467 572 y Fn(m)2558 560 y Fs( )2615 526 y Fn(t;t)2685 534 y Fm(1)2718 526 y Fn(;:::)o(;t)2842 534 y Fl(m)2612 581 y Fn(n)2901 560 y Ft(\()p Fs(A)p Ft(\))p Fs(;)418 b Ft(\(2.12\))260 804 y(where)27 b(the)h(term)f(with)i Fs(m)23 b Ft(=)f(0)27 b(is)h Fs( )s Ft(\()p Fs(\013)1509 774 y Fn(t)1509 824 y Fr(0)1547 804 y Ft(\()p Fs(A)p Ft(\)\),)g(and,)g(for)f Fs(m)c Fq(\025)g Ft(1,)968 992 y Fs( )1025 958 y Fn(t;t)1095 966 y Fm(1)1127 958 y Fn(;:::)o(;t)1251 966 y Fl(m)1022 1013 y Fn(n)1310 992 y Ft(\()p Fs(A)p Ft(\))h(:=)f Fs( )1642 925 y Fp(\000)1680 992 y Ft([)p Fs(\013)1756 955 y Fn(t)1781 963 y Fl(m)1756 1014 y Fr(0)1840 992 y Ft(\()p Fs(v)1912 1004 y Fn(n)1958 992 y Ft(\))p Fs(;)14 b Ft([)p Fq(\001)g(\001)g(\001)g Ft([)p Fs(\013)2237 955 y Fn(t)2262 963 y Fm(1)2237 1014 y Fr(0)2299 992 y Ft(\()p Fs(v)2371 1004 y Fn(n)2417 992 y Ft(\))p Fs(;)g(\013)2539 958 y Fn(t)2539 1013 y Fr(0)2576 992 y Ft(\()p Fs(A)p Ft(\)])g Fq(\001)g(\001)g(\001)h Ft(]])2897 925 y Fp(\001)2949 992 y Fs(:)260 1181 y Ft(Here,)27 b Fs(v)519 1193 y Fn(n)588 1181 y Fq(2)c(A)28 b Ft(is)g(an)f(appro)n(ximating)f(sequence)h(for)g (the)h(op)r(erator)e Fs(v)s Ft(,)h(satisfying)g(the)h(relation)1346 1369 y(lim)1317 1419 y Fn(n)p Fk(!1)1504 1369 y Fs(!)1556 1381 y Fr(0)1593 1369 y Ft(\()p Fs(A)1687 1335 y Fk(\003)1726 1369 y Ft(\()p Fs(v)1801 1335 y Fk(\003)1798 1390 y Fn(n)1862 1369 y Fq(\000)18 b Fs(v)1988 1335 y Fk(\003)2026 1369 y Ft(\)\()p Fs(v)2130 1381 y Fn(n)2195 1369 y Fq(\000)g Fs(v)s Ft(\))p Fs(A)p Ft(\))24 b(=)e(0)p Fs(;)845 b Ft(\(2.13\))260 1567 y Fq(8)p Fs(A)27 b Fq(2)i(A)i Ft(of)f(the)h(form)g Fs(A)d Ft(=)g Fs(B)c Fq(\012)d Fs(W)1454 1579 y Fr(1)1491 1567 y Ft(\()p Fs(f)1564 1579 y Fr(1)1601 1567 y Ft(\))g Fq(\012)f Fs(W)1817 1579 y Fr(2)1855 1567 y Ft(\()p Fs(f)1928 1579 y Fr(2)1965 1567 y Ft(\))31 b(with)g Fs(B)h Fq(2)d(B)s Ft(\()p Fq(H)2559 1579 y Fr(0)2596 1567 y Ft(\),)i Fs(f)2723 1579 y Fr(1)p Fn(;)p Fr(2)2841 1567 y Fq(2)e Fs(L)2982 1537 y Fr(2)2982 1588 y(0)3018 1567 y Ft(.)47 b(Suc)n(h)30 b(a)g(sequence)260 1692 y(is)c(constructed)g(as)g(follo)n(ws.)36 b(Let)27 b Fq(f)p Fs(e)1424 1704 y Fn(m)1486 1692 y Fq(g)f Ft(b)r(e)h(an)f(orthonormal)f(basis)h(of)g Fs(L)2608 1662 y Fr(2)2608 1712 y(0)2645 1692 y Ft(.)37 b(W)-7 b(e)27 b(de\014ne)g(the)g(appro)n(ximate)260 1816 y(creation)f(op)r (erators)1383 1993 y Fs(a)1427 1958 y Fk(\003)1427 2013 y Fn(j;n)1519 1993 y Ft(\()p Fs(G)1616 2005 y Fn(j)1652 1993 y Ft(\))d(=)1831 1889 y Fn(M)1806 1914 y Fp(X)1795 2089 y Fn(m)p Fr(=1)1938 1993 y Fq(h)p Fs(e)2009 2005 y Fn(m)2072 1993 y Fs(;)14 b(G)2174 2005 y Fn(j)2209 1993 y Fq(i)p Fs(b)2277 1958 y Fk(\003)2277 2013 y Fn(j;\025)2367 1993 y Ft(\()p Fs(e)2438 2005 y Fn(m)2501 1993 y Ft(\))p Fs(;)912 b Ft(\(2.14\))260 2212 y(where)27 b Fs(n)c Ft(=)f(\()p Fs(\025;)14 b(M)9 b Ft(\),)29 b(and,)e(for)g(an)n(y)g Fs(f)32 b Fq(2)23 b Fs(L)1627 2182 y Fr(2)1664 2212 y Ft(\()p Fh(R)1750 2182 y Fr(3)1793 2212 y Ft(\))28 b(and)g Fs(\025)23 b(>)g Ft(0,)1076 2419 y Fs(b)1112 2385 y Fk(\003)1112 2439 y Fn(j;\025)1202 2419 y Ft(\()p Fs(f)9 b Ft(\))23 b(:=)1505 2363 y Fs(\025)p 1460 2400 140 4 v 1460 2416 a Fq(p)p 1529 2416 42 4 v 69 x Ft(2)p Fs(i)1623 2419 y Fq(f)p Fs(W)1743 2431 y Fn(j)1778 2419 y Ft(\()p Fs(f)9 b(=\025)p Ft(\))19 b Fq(\000)f Fg(1)g Fq(\000)g Fs(iW)2340 2431 y Fn(j)2375 2419 y Ft(\()p Fs(if)9 b(=\025)p Ft(\))18 b(+)g Fs(i)p Fg(1)p Fq(g)13 b Fs(:)604 b Ft(\(2.15\))260 2631 y(Similarly)18 b(w)n(e)h(de\014ne)g(the)h(appro)n(ximate)d (annihilation)i(op)r(erators)e Fs(a)2411 2643 y Fn(j;n)2503 2631 y Ft(\()p Fs(G)2600 2643 y Fn(j)2635 2631 y Ft(\).)35 b(Via)19 b(the)g(ab)r(o)n(v)n(e)f(construction)260 2755 y(w)n(e)h(obtain)h(the)h(family)f(of)g(in)n(teractions)e Fs(v)1576 2767 y Fn(n)1645 2755 y Fq(2)23 b(A)p Ft(.)35 b(Using)20 b(\(2.9\),)h(one)f(easily)f(sho)n(ws)g(that)h(\(2.13\))f(is) h(satis\014ed.)384 2880 y(In)31 b(App)r(endix)g(A)f(w)n(e)g(sho)n(w)f (that)i(under)f(condition)g(\(2.13\))f(the)i(in)n(tegrands)e(on)h(the)g (r.h.s.)45 b(of)30 b(\(2.12\))260 3004 y(are)21 b(con)n(tin)n(uous)g (functions)h(in)g Fs(t)1276 3016 y Fr(1)1313 3004 y Fs(;)14 b(:)g(:)g(:)g(;)g(t)1528 3016 y Fn(m)1591 3004 y Ft(,)23 b(that)f(the)g(series)f(is)h(absolutely)f(con)n(v)n(ergen)n(t)f(and)i (that)g(the)g(limit)260 3129 y(exists)27 b(and)h(is)f(indep)r(enden)n (t)h(of)g(the)g(appro)n(ximating)e(sequence)h Fs(v)2380 3141 y Fn(n)2425 3129 y Ft(.)384 3253 y(A)e Fs(\014)518 3265 y Fr(1)555 3253 y Fs(\014)602 3265 y Fr(2)640 3253 y Ft(-normal)e(state)h Fs( )k Ft(is)c(called)g(in)n(v)-5 b(arian)n(t)23 b(\(under)i(the)g(in)n(teracting)e(dynamics\),)i(or)f (stationary)-7 b(,)23 b(if)260 3378 y Fs( )317 3348 y Fn(t)346 3378 y Ft(\()p Fs(A)p Ft(\))h(=)f Fs( )s Ft(\()p Fs(A)p Ft(\))28 b(for)e(all)h Fs(A)d Fq(2)f(A)p Ft(,)28 b Fs(t)23 b Fq(2)g Fh(R)p Ft(,)33 b(see)27 b(\(2.12\).)36 b(Our)27 b(goal)e(is)i(to)g(sho)n(w)g(that,)g(if)h Fs(\014)3024 3390 y Fr(1)3084 3378 y Fq(6)p Ft(=)23 b Fs(\014)3219 3390 y Fr(2)3256 3378 y Ft(,)k(then)h(there)260 3502 y(are)j(no)h Fs(\014)570 3514 y Fr(1)607 3502 y Fs(\014)654 3514 y Fr(2)691 3502 y Ft(-normal)f(states)h(whic)n(h)g(are)f(in)n(v)-5 b(arian)n(t.)50 b(In)32 b(particular,)g(there)g(are)f(no)h(equilibrium) g(states)260 3627 y(\(see)27 b(Theorem)g(3.1\).)384 3751 y(T)-7 b(o)25 b(pass)f(to)g(a)h(Hilb)r(ert)g(space)f(framew)n(ork)f (one)i(uses)f(the)h(GNS)h(represen)n(tation)d(of)h(\()p Fq(A)p Fs(;)14 b(!)3235 3763 y Fr(0)3273 3751 y Ft(\),)26 b(where)e Fs(!)3643 3763 y Fr(0)260 3876 y Ft(is)j(giv)n(en)g(in)h (\(2.10\):)1587 4001 y(\()p Fq(A)p Fs(;)14 b(!)1774 4013 y Fr(0)1811 4001 y Ft(\))24 b Fq(!)f Ft(\()p Fq(H)q Fs(;)14 b(\031)s(;)g Ft(\012)2260 4013 y Fr(0)2297 4001 y Ft(\))p Fs(:)260 4163 y Ft(Here)25 b Fq(H)q Ft(,)i Fs(\031)i Ft(and)c(\012)870 4175 y Fr(0)933 4163 y Ft(are)g(a)h(Hilb)r(ert)g (space,)g(a)f(represen)n(tation)f(of)i(the)g(algebra)e Fq(A)i Ft(b)n(y)g(b)r(ounded)g(op)r(erators)260 4288 y(on)h Fq(H)q Ft(,)h(and)f(a)g(cyclic)h(elemen)n(t)f(in)h Fq(H)h Ft(\(i.e.)p 1633 4216 279 4 v 37 w Fs(\031)s Ft(\()p Fq(A)p Ft(\)\012)1873 4300 y Fr(0)1935 4288 y Ft(=)22 b Fq(H)q Ft(\))28 b(s.t.)1538 4477 y Fs(!)1590 4489 y Fr(0)1627 4477 y Ft(\()p Fs(A)p Ft(\))c(=)e Fq(h)q Ft(\012)1957 4489 y Fr(0)1994 4477 y Fs(;)14 b(\031)s Ft(\()p Fs(A)p Ft(\)\012)2267 4489 y Fr(0)2305 4477 y Fq(i)42 b Fs(:)260 4665 y Ft(\(In)89 b(this)h(pap)r(er)e(w)n(e)h(use)g(the)g(Araki-W)-7 b(o)r(o)r(ds)88 b(GNS)h(represen)n(tation)e(with)j Fs(!)3302 4677 y Fn(p)3340 4665 y Ft(\()p Fs(A)p Ft(\))126 b(:=)260 4790 y(T)-7 b(r)o(\()p Fs(e)416 4759 y Fk(\000)p Fn(\014)506 4767 y Fl(p)541 4759 y Fn(H)595 4767 y Fl(p)634 4790 y Fs(A)p Ft(\))p Fs(=)p Ft(T)g(r\()p Fs(e)927 4759 y Fk(\000)p Fn(\014)1017 4767 y Fl(p)1051 4759 y Fn(H)1105 4767 y Fl(p)1144 4790 y Ft(\))28 b(in)g(\(2.10\),)f(see)g(Section)h (4.\))384 4914 y(With)35 b(the)f(free)g(ev)n(olution)f Fs(\013)1339 4884 y Fn(t)1339 4935 y Fr(0)1377 4914 y Ft(\()p Fs(A)p Ft(\))h(:=)f Fs(e)1697 4884 y Fn(itH)1799 4892 y Fm(0)1836 4914 y Fs(Ae)1937 4884 y Fk(\000)p Fn(itH)2091 4892 y Fm(0)2163 4914 y Ft(one)g(asso)r(ciates)f(the)j(unitary)e (one-parameter)260 5039 y(group,)26 b Fs(U)575 5051 y Fr(0)612 5039 y Ft(\()p Fs(t)p Ft(\))e(=)f Fs(e)857 5009 y Fn(itL)951 5017 y Fm(0)987 5039 y Ft(,)28 b(on)f Fq(H)i Ft(s.t.)1427 5227 y Fs(\031)s Ft(\()p Fs(\013)1562 5193 y Fn(t)1562 5248 y Fr(0)1600 5227 y Ft(\()p Fs(A)p Ft(\)\))24 b(=)f Fs(U)1927 5239 y Fr(0)1964 5227 y Ft(\()p Fs(t)p Ft(\))p Fs(\031)s Ft(\()p Fs(A)p Ft(\))p Fs(U)2291 5239 y Fr(0)2329 5227 y Ft(\()p Fs(t)p Ft(\))2423 5193 y Fk(\000)p Fr(1)3468 5227 y Ft(\(2.16\))1949 5476 y(5)p eop %%Page: 6 6 6 5 bop 260 210 a Ft(and)27 b Fs(U)478 222 y Fr(0)515 210 y Ft(\()p Fs(t)p Ft(\)\012)669 222 y Fr(0)730 210 y Ft(=)c(\012)878 222 y Fr(0)915 210 y Ft(.)37 b(De\014ne)28 b(the)g Fo(standar)l(d)i(Liouvil)t(le)i(op)l(er)l(ator)1503 398 y Fs(L)22 b Ft(:=)h Fs(L)1750 410 y Fr(0)1805 398 y Ft(+)18 b Fs(g)s(\031)s Ft(\()p Fs(v)s Ft(\))h Fq(\000)f Fs(g)s(\031)2283 364 y Fk(0)2306 398 y Ft(\()p Fs(v)s Ft(\))p Fs(;)1032 b Ft(\(2.17\))260 586 y(de\014ned)27 b(on)g(the)g(dense)f(domain)h Fq(D)r Ft(\()p Fs(L)1477 598 y Fr(0)1514 586 y Ft(\))17 b Fq(\\)g(D)r Ft(\()p Fs(\031)s Ft(\()p Fs(v)s Ft(\)\))i Fq(\\)e(D)r Ft(\()p Fs(\031)2161 556 y Fk(0)2186 586 y Ft(\()p Fs(v)s Ft(\)\).)37 b(Here,)27 b Fs(\031)s Ft(\()p Fs(v)s Ft(\))h(and)f Fs(\031)3000 556 y Fk(0)3023 586 y Ft(\()p Fs(v)s Ft(\))h(can)e(b)r(e)i(de\014ned) 260 711 y(either)37 b(using)h(explicit)g(form)n(ulae)e(for)i Fs(\031)j Ft(and)c Fs(\031)1834 681 y Fk(0)1896 711 y Ft(in)h(the)g(Araki-W)-7 b(o)r(o)r(ds)36 b(represen)n(tation)g(giv)n (en)h(b)r(elo)n(w,)260 835 y(or)31 b(b)n(y)h(using)g(the)h(appro)n (ximation)d Fs(v)1454 847 y Fn(n)1562 835 y Fq(2)i(A)g Ft(of)g Fs(v)s Ft(,)i(constructed)e(ab)r(o)n(v)n(e.)49 b(By)32 b(the)h(Glimm-Ja\013e-Nelson)260 960 y(comm)n(utator)20 b(theorem,)i(the)g(op)r(erator)e Fs(L)h Ft(is)g(essen)n(tially)g (self-adjoin)n(t;)i(w)n(e)e(denote)g(its)h(self-adjoin)n(t)f(closure) 260 1085 y(again)28 b(b)n(y)h Fs(L)p Ft(.)42 b(The)29 b(op)r(erator)f Fs(L)h Ft(generates)f(the)h(one-parameter)e(group)i(of) g Fq(\003)p Ft(automorphisms)e Fs(\033)3388 1054 y Fn(t)3447 1085 y Ft(on)j(the)260 1209 y(v)n(on)d(Neumann)h(algebra)d Fs(\031)s Ft(\()p Fq(A)p Ft(\))1263 1179 y Fk(00)1335 1209 y Ft(\(the)j(w)n(eak)e(closure)h(of)g Fs(\031)s Ft(\()p Fq(A)p Ft(\)\),)1589 1397 y Fs(\033)1639 1363 y Fn(t)1669 1397 y Ft(\()p Fs(B)t Ft(\))d(:=)e Fs(e)1973 1363 y Fn(itL)2071 1397 y Fs(B)t(e)2177 1363 y Fk(\000)p Fn(itL)2327 1397 y Fs(;)1118 b Ft(\(2.18\))260 1585 y(where)27 b Fs(B)g Fq(2)d Fs(\031)s Ft(\()p Fq(A)p Ft(\))849 1555 y Fk(00)892 1585 y Ft(.)37 b(Let)28 b Fs( )j Ft(b)r(e)d(a)f(state)g(on) g(the)h(algebra)e Fq(A)i Ft(normal)f(w.r.t.)37 b(the)28 b(state)f Fs(!)3125 1597 y Fr(0)3162 1585 y Ft(,)h(i.e.)1637 1773 y Fs( )s Ft(\()p Fs(A)p Ft(\))c(=)f(T)-7 b(r\()p Fs(\032\031)s Ft(\()p Fs(A)p Ft(\)\))1167 b(\(2.19\))260 1961 y(for)27 b(some)g(p)r(ositiv)n(e)g(trace)f(class)h(op)r(erator)f Fs(\032)h Ft(on)g Fq(H)i Ft(of)e(trace)g(one.)36 b(It)28 b(is)f(sho)n(wn)g(in)h(App)r(endix)g(A)g(that)f(for)260 2086 y Fs( )k Ft(as)c(ab)r(o)n(v)n(e)f(the)i(limit)g(on)f(the)h(r.h.s.) 37 b(of)27 b(\(2.12\))g(exists)g(and)h(equals)1539 2274 y Fs( )1596 2240 y Fn(t)1625 2274 y Ft(\()p Fs(A)p Ft(\))c(=)f(T)-7 b(r\()p Fs(\032\033)2074 2240 y Fn(t)2103 2274 y Ft(\()p Fs(\031)s Ft(\()p Fs(A)p Ft(\)\)\))p Fs(:)1070 b Ft(\(2.20\))260 2462 y(In)28 b(particular,)e(the)i(limit)g(is)g(indep)r(enden)n(t)g(of) g(the)g(c)n(hoice)f(of)g(the)h(appro)n(ximating)e(family)h Fs(v)3234 2474 y Fn(n)3280 2462 y Ft(.)384 2587 y(The)f(follo)n(wing)e (result)h(connects)g(the)g(existence)g(of)h(normal)e(in)n(v)-5 b(arian)n(t)24 b(states)h(to)g(sp)r(ectral)g(prop)r(erties)260 2711 y(of)i(the)h(standard)f(Liouvillian)g Fs(L)p Ft(:)260 2887 y Fg(Theorem)j(2.1)h(\([13)o(,)h(10)o(]\))41 b Fo(A)35 b(normal)g Fs(\033)1674 2856 y Fn(t)1704 2887 y Fo(-invariant)g(state)g (on)f Fs(\031)s Ft(\()p Fq(A)p Ft(\))2606 2856 y Fk(00)2685 2887 y Fo(exists)g(if)i(and)f(only)h(if)f(zer)l(o)g(is)260 3011 y(an)30 b(eigenvalue)h(of)f Fs(L)p Fo(.)384 3186 y Ft(In)g(order)e(to)h(obtain)f(rather)g(subtle)i(sp)r(ectral)f (information)f(on)h(the)g(op)r(erator)f Fs(L)p 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b(order)f(to)i(form)n(ulate)260 4163 y(them,)26 b(w)n(e)e(\014rst)h(in) n(tro)r(duce)g(some)f(de\014nitions.)36 b(W)-7 b(e)25 b(refer)g(the)g(reader)e(to)i(the)g(remarks)f(at)g(the)i(end)f(of)g (this)260 4287 y(section)i(for)g(a)g(discussion)g(of)h(the)g (de\014nitions)f(and)h(conditions.)384 4412 y(W)-7 b(e)28 b(de\014ne)g(the)g(map)g Fs(\015)f Ft(:)c Fs(L)1268 4381 y Fr(2)1305 4412 y Ft(\()p Fh(R)1391 4381 y Fr(3)1435 4412 y Ft(\))g Fq(!)g Fs(L)1653 4381 y Fr(2)1690 4412 y Ft(\()p Fh(R)i Fq(\002)18 b Fs(S)1940 4381 y Fr(2)1977 4412 y Ft(\),)1212 4627 y(\()p Fs(\015)5 b(f)k Ft(\)\()p Fs(u;)14 b(\033)s Ft(\))23 b(=)1684 4552 y Fp(p)p 1767 4552 94 4 v 75 x Fq(j)p Fs(u)p Fq(j)1902 4510 y Fp(\032)2006 4574 y Fs(f)9 b Ft(\()p Fs(u\033)s Ft(\))p Fs(;)212 b(u)23 b Fq(\025)g Ft(0)p Fs(;)2006 4678 y Fq(\000)p 2071 4611 50 4 v Fs(f)8 b Ft(\()p Fq(\000)p Fs(u\033)s Ft(\))p Fs(;)83 b(u)23 b(<)g Ft(0)p Fs(:)3509 4627 y Ft(\(3.1\))260 4854 y(Let)29 b Fs(j)444 4866 y Fn(\022)482 4854 y Ft(\()p 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3775 y Fr(1)p Fn(=)p Fr(2)1245 3805 y Ft(\))e Fq(\032)f Fs(D)r Ft(\()p Fs(I)1530 3817 y Fn(\022)1569 3805 y Ft(\))k(and)h Fs(I)1828 3817 y Fn(\022)1866 3805 y Fs(f)37 b Ft(is)28 b(analytic)g Fq(8)p Fs(f)k Fq(2)25 b Fs(D)r Ft(\(\003)2706 3775 y Fr(1)p Fn(=)p Fr(2)2810 3805 y Ft(\).)40 b(Here)28 b(Condition)h(\(AA\))260 3930 y(is)d(used.)37 b(Hence)27 b(the)g(family)f Fs(K)1275 3942 y Fn(\022)1339 3930 y Ft(for)g Fs(\022)g Fq(2)d(f)p Fs(\022)i Fq(2)e Fh(C)1846 3899 y Fr(2)1890 3859 y Fp(\014)1890 3909 y(\014)1917 3930 y Fq(j)p Ft(Im)14 b Fs(\022)r Fq(j)24 b Fs(<)e(\022)2267 3942 y Fr(0)2305 3930 y Fq(g)k Ft(is)g(b)r(ounded)h (from)g Fs(D)r Ft(\(\003\))16 b Fq(\\)h Fs(D)r Ft(\()p Fs(N)9 b Ft(\))27 b(to)g Fq(H)260 4054 y Ft(\(and)d Fs(K)521 4066 y Fn(\022)558 4054 y Fs(f)33 b Ft(is)24 b(analytic)f(in)i Fs(\022)g Fq(2)e(f)p Fs(\022)i Fq(2)f Fh(C)1499 4024 y Fr(2)1542 3984 y Fp(\014)1542 4033 y(\014)1570 4054 y Fq(j)p Ft(Im)14 b Fs(\022)r Fq(j)23 b Fs(<)g(\022)1920 4066 y Fr(0)1957 4054 y Fq(g)p Ft(,)h Fq(8)p Fs(f)31 b Fq(2)24 b(D)r Ft(\(\003\))11 b Fq(\\)g(D)r Ft(\()p Fs(N)e Ft(\)\).)38 b(Moreo)n(v)n(er,)22 b(for)i Fq(j)p Ft(Im)g Fs(\022)r Fq(j)g Fs(>)e Ft(0)260 4179 y(the)28 b(op)r(erators)d Fs(K)841 4191 y Fn(\022)906 4179 y Ft(are)i(closed)g (on)g(the)h(domain)f Fs(D)r Ft(\(\003\))19 b Fq(\\)g Fs(D)r Ft(\()p Fs(N)9 b Ft(\).)384 4303 y(Ho)n(w)n(ev)n(er,)36 b Fq(f)p Fs(K)865 4315 y Fn(\022)901 4303 y Fq(j)14 b(j)p Ft(Im)p Fs(\022)r Fq(j)37 b Fs(<)e(\022)1300 4315 y Fr(0)1337 4303 y Fq(g)g Ft(is)g Fo(not)g Ft(an)g(analytic)g(family)g(in)h(the)f (sense)g(of)g(Kato.)59 b(The)35 b(problem)260 4428 y(here)25 b(is)h(the)g(lac)n(k)f(of)g(co)r(ercivit)n(y)g({)g(the)i(p)r (erturbation)e Fs(I)33 b Ft(is)25 b(not)h(b)r(ounded)g(relativ)n(ely)f (to)g(the)h(unp)r(erturb)r(ed)260 4552 y(op)r(erator)c Fs(L)648 4564 y Fr(0)685 4552 y Ft(.)36 b(T)-7 b(o)24 b(comp)r(ensate)g(for)f(this)i(w)n(e)f(ha)n(v)n(e)f(c)n(hosen)g(the)i (deformation)e Fs(U)2818 4564 y Fn(\022)2880 4552 y Ft(in)h(suc)n(h)g (a)g(w)n(a)n(y)f(that)i(the)260 4677 y(op)r(erator)f Fs(M)674 4689 y Fn(\022)734 4677 y Ft(:=)f(Im)p Fs(L)1001 4689 y Fr(0)p Fn(;\022)1116 4677 y Ft(is)j(co)r(erciv)n(e)e(for)h(Im)p Fs(\022)g(>)e Ft(0)i(,)h(i.e.,)h(the)f(p)r(erturbation)f Fs(I)2822 4689 y Fn(\022)2860 4677 y Ft(,)h(as)f(w)n(ell)g(as)g(Re)p Fs(L)3430 4689 y Fr(0)p Fn(;\022)3520 4677 y Ft(,)h(are)260 4801 y(b)r(ounded)g(relativ)n(e)f(to)g(this)h(op)r(erator.)35 b(The)25 b(problem)h(here)f(is)h(that)g Fs(M)2526 4813 y Fn(\022)2586 4801 y Fq(!)d Ft(0)i(as)g(Im)p Fs(\022)h Fq(!)d Ft(0)i(so)g(w)n(e)h(ha)n(v)n(e)e(to)260 4926 y(pro)r(ceed)j (carefully)-7 b(.)384 5050 y(The)25 b(next)g(result)f(is)h(similar)e (to)i(one)f(in)h([5],)g(but)g(the)g(pro)r(of)f(giv)n(en)g(b)r(elo)n(w)g (is)h(simpler)f(than)h(that)g(of)f([5].)1928 5476 y(12)p eop %%Page: 13 13 13 12 bop 260 210 a Fg(Theorem)30 b(5.1)41 b Fo(Assume)26 b(that)h(Condition)i(\(AA\))d(holds)j(and)e(let)g Fs(\022)2428 222 y Fr(0)2489 210 y Ft(=)22 b(\()p Fs(\016)2645 222 y Fr(0)2683 210 y Fs(;)14 b(\034)2756 222 y Fr(0)2793 210 y Ft(\))27 b Fo(b)l(e)h(as)f(in)g(that)g(c)l(ondition.)260 335 y(T)-6 b(ake)30 b(an)1301 530 y Fs(a)23 b(>)1578 474 y(g)1621 444 y Fr(2)p 1465 511 306 4 v 1465 587 a Ft(sin\(Im)q Fs(\016)s Ft(\))1781 530 y Fs(C)1846 496 y Fr(2)1840 551 y(0)1897 363 y Fp(0)1897 513 y(@)1994 451 y(X)1970 628 y Fn(j)s Fr(=1)p Fn(;)p Fr(2)2151 530 y Fq(k)p Fs(G)2258 542 y Fn(j)2293 530 y Fq(k)2335 545 y Fr(1)p Fn(=)p Fr(2)p Fn(;\022)2492 363 y Fp(1)2492 513 y(A)2565 380 y Fr(2)2616 530 y Fs(;)829 b Ft(\(5.24\))260 775 y Fo(wher)l(e)1448 899 y Fs(C)1507 911 y Fr(0)1568 899 y Ft(:=)23 b Fs(C)6 b Ft(\(1)18 b(+)g Fs(\014)1970 856 y Fk(\000)p Fr(1)p Fn(=)p Fr(2)1966 921 y(1)2145 899 y Ft(+)g Fs(\014)2279 856 y Fk(\000)p Fr(1)p Fn(=)p Fr(2)2275 921 y(2)2436 899 y Ft(\))p Fs(;)977 b Ft(\(5.25\))260 1073 y Fo(and)30 b(wher)l(e)h Fs(C)36 b Fo(is)30 b(a)g(c)l(onstant)f (dep)l(ending)i(only)f(on)g Ft(tan)14 b Fs(\016)2099 1085 y Fr(0)2136 1073 y Fo(.)39 b(Then)30 b(we)g(have:)338 1264 y Ft(\(i\))42 b Fq(f)p Fs(z)26 b Fq(2)e Fh(C)15 b Fq(j)p Ft(Im)20 b Fs(z)26 b Fq(\024)d(\000)p Fs(a)p Fq(g)j(\022)g Fs(\032)p Ft(\()p Fs(K)1416 1276 y Fn(\022)1453 1264 y Ft(\))33 b Fo(\(the)f(r)l(esolvent)g(set)f(of)i Fs(K)2343 1276 y Fn(\022)2380 1264 y Fo(\))f(if)h Fs(\022)c Fq(2)e Fs(S)2735 1229 y Fr(+)2730 1289 y Fn(\022)2762 1297 y Fm(0)2798 1264 y Fo(;)34 b(if)e(in)g(addition)i Fs(K)f Ft(=)26 b Fs(K)3642 1234 y Fk(\003)467 1389 y Fo(then)k(we)g(c)l(an)g(take)g Fs(\022)25 b Fq(2)p 1247 1308 120 4 v 24 w Fs(S)1303 1353 y Fr(+)1298 1414 y Fn(\022)1330 1422 y Fm(0)1366 1389 y Fo(;)315 1580 y Ft(\(ii\))42 b Fo(The)31 b(family)h Fs(K)961 1592 y Fn(\022)1027 1580 y Fo(is)e(analytic)i(of)e(typ)l(e)g(A)f(\(in)h(the)g(sense)g(of)g (Kato\))g(in)g Fs(\022)c Fq(2)d Fs(S)2926 1544 y Fr(+)2921 1605 y Fn(\022)2953 1613 y Fm(0)2989 1580 y Fo(;)292 1771 y Ft(\(iii\))42 b Fo(If)23 b Fs(K)28 b Ft(=)23 b Fs(K)811 1741 y Fk(\003)849 1771 y Fo(,)h(then,)g(for)f(any)f Fs(u)g Fo(and)g Fs(v)j Fo(which)f(ar)l(e)f Fs(U)2083 1783 y Fn(\022)2120 1771 y Fo(-analytic)g(in)f(a)h(strip)2801 1704 y Fp(\010)2850 1771 y Fs(\022)i Fq(2)e Fh(C)3046 1741 y Fr(2)3089 1771 y Fq(j)14 b Ft(0)23 b Fq(\024)g Ft(Im)p Fs(\022)i(<)e(\022)3569 1783 y Fr(1)3606 1704 y Fp(\011)3654 1771 y Fo(,)467 1895 y(for)31 b(some)f Fs(\022)851 1907 y Fr(1)911 1895 y Ft(=)23 b(\()p Fs(\016)1068 1907 y Fr(1)1106 1895 y Fs(;)14 b(\034)1179 1907 y Fr(0)1216 1895 y Ft(\))p Fs(;)g(\016)1322 1907 y Fr(1)1382 1895 y Fq(2)24 b Ft([0)p Fs(;)14 b Ft(min)p Fq(f)p Fs(\031)s(=)p Ft(3)p Fs(;)g(\022)1953 1907 y Fr(0)1989 1895 y Fq(g)p Ft(\))p Fo(,)30 b(the)f(fol)t(lowing)k(r)l(elation)d(holds:)1368 2036 y Fp(\012)1407 2103 y Fs(u;)14 b Ft(\()p Fs(K)24 b Fq(\000)18 b Fs(z)t Ft(\))1777 2069 y Fk(\000)p Fr(1)1866 2103 y Fs(v)1909 2036 y Fp(\013)1971 2103 y Ft(=)2059 2036 y Fp(\012)2098 2103 y Fs(u)p 2146 2080 38 3 v 26 x Fn(\022)2183 2103 y Fs(;)c Ft(\()p Fs(K)2323 2115 y Fn(\022)2379 2103 y Fq(\000)k Fs(z)t Ft(\))2537 2069 y Fk(\000)p Fr(1)2625 2103 y Fs(v)2665 2115 y Fn(\022)2703 2036 y Fp(\013)2756 2103 y Fs(;)689 b Ft(\(5.26\))467 2311 y Fo(wher)l(e)31 b Fs(u)750 2323 y Fn(\022)810 2311 y Ft(=)23 b Fs(U)955 2323 y Fn(\022)992 2311 y Fs(u)p Fo(,)30 b(etc.,)g(for)h Ft(Im)14 b Fs(z)26 b Fq(\024)d(\000)p Fs(a)29 b Fo(and)h Ft(0)23 b Fs(<)f Ft(Im)14 b Fs(\022)26 b(<)c(\022)2433 2323 y Fr(1)2471 2311 y Fs(=)p Ft(2)p Fo(.)384 2501 y(Similar)31 b(statements)e(hold)i(also)g(for)g Fq(\000)p Fs(\022)1677 2513 y Fr(0)1736 2501 y Fs(<)23 b Ft(Im)p Fs(\022)j Fq(\024)c Ft(0)p Fo(.)384 2709 y(Pr)l(o)l(of.)338 2900 y Ft(\(i\))42 b(This)31 b(statemen)n(t)g(is)g(a)f(sp)r(ecial)h (case)f(of)g(the)i(follo)n(wing)d(prop)r(osition)h(\(estimate)h (\(5.35\))f(b)r(elo)n(w)h(suf-)467 3025 y(\014ces\).)37 b(Let)28 b Fs(C)920 3037 y Fn(a;b)1037 3025 y Ft(b)r(e)g(the)g (truncated)g(w)n(edge)662 3232 y Fs(C)721 3244 y Fn(a;b)833 3232 y Ft(:=)2547 b(\(5.27\))828 3314 y Fp(\010)876 3382 y Fs(z)27 b Fq(2)c Fh(C)49 b Fq(j)27 b Ft(Im)14 b Fs(z)27 b(>)22 b Fq(\000)p Fs(a=)p Ft(2)p Fs(;)40 b Fq(j)p Ft(Re)14 b Fs(z)t Fq(j)23 b Fs(<)f Ft(2[\(sin)14 b Fs(b)p Ft(\))2272 3347 y Fk(\000)p Fr(1)2380 3382 y Ft(+)k Fs(a=)p Ft(4]\(Im)13 b Fs(z)21 b Ft(+)e Fs(a)p Ft(\))f(+)g Fq(k)p Fs(L)3178 3394 y Fn(p)3216 3382 y Fq(k)g Ft(+)g(1)3401 3314 y Fp(\011)3462 3382 y Fs(:)467 3622 y Fg(Prop)s(osition)31 b(5.2)40 b Fo(L)l(et)32 b Fs(\022)f Fq(2)d Fs(S)1502 3587 y Fr(+)1497 3647 y Fn(\022)1529 3655 y Fm(0)1565 3622 y Fo(,)34 b(and)f(take)g Fs(a)f Fo(as)h(in)g(\(5.24\).)49 b(Then)33 b Fs(\033)s Ft(\()p Fs(K)2922 3634 y Fn(\022)2960 3622 y Ft(\))28 b Fq(\032)g Fs(C)3172 3634 y Fn(a;)p Fr(Im)11 b Fn(\016)3354 3622 y Fo(,)34 b(and)f(for)467 3747 y Fs(z)27 b Fq(2)c Fh(C)15 b Fq(n)p Fs(C)766 3759 y Fn(a;)p Fr(Im)i Fn(\016)984 3747 y Fo(we)30 b(have)1428 3954 y Fq(k)p Ft(\()p Fs(K)1573 3966 y Fn(\022)1629 3954 y Fq(\000)18 b Fs(z)t Ft(\))1787 3920 y Fk(\000)p Fr(1)1875 3954 y Fq(k)23 b(\024)f Ft([dist)15 b(\()p Fs(z)t(;)f(C)2370 3966 y Fn(a;)p Fr(Im)c Fn(\016)2551 3954 y Ft(\)])2606 3920 y Fk(\000)p Fr(1)2696 3954 y Fs(:)749 b Ft(\(5.28\))467 4195 y Fo(Pr)l(o)l(of.)64 b Ft(T)-7 b(o)24 b(k)n(eep)g(notation)g(simple)g(w)n(e)g(pro)n(v)n(e)e (the)j(prop)r(osition)e(for)h Fs(\022)i Ft(purely)e(imaginary:)34 b Fs(\022)25 b Ft(=)e Fs(i\022)3634 4165 y Fk(0)3657 4195 y Ft(,)467 4320 y Fs(\022)508 4290 y Fk(0)555 4320 y Ft(=)g(\()p Fs(\016)715 4290 y Fk(0)738 4320 y Fs(;)14 b(\034)820 4290 y Fk(0)844 4320 y Ft(\))24 b Fq(2)f Fh(R)1032 4290 y Fr(2)1075 4320 y Ft(.)37 b(In)28 b(this)g(case)e(the)i(op)r (erator)e Fs(M)2135 4332 y Fn(\022)2200 4320 y Ft(is)i(of)f(the)h(form) 1709 4527 y Fs(M)1790 4539 y Fn(\022)1850 4527 y Ft(=)22 b(sin)14 b Fs(\016)2093 4493 y Fk(0)2116 4527 y Ft(\003)19 b(+)f Fs(\034)2321 4493 y Fk(0)2345 4527 y Fs(N)t(:)1029 b Ft(\(5.29\))467 4735 y(The)30 b(pro)r(of)f(b)r(elo)n(w)g(is)h(based)f (on)h(the)g(follo)n(wing)e(b)r(ounds)i(on)f(the)h(in)n(teraction,)g (whic)n(h,)g(to)f(simplify)467 4859 y(the)f(notation,)g(w)n(e)f(form)n (ulate)g(for)g(the)h(case)e Fs(\022)g Ft(=)c Fs(i\022)2129 4829 y Fk(0)2180 4859 y Ft(only)-7 b(.)1928 5476 y(13)p eop %%Page: 14 14 14 13 bop 467 210 a Fg(Lemma)30 b(5.3)40 b Fo(L)l(et)30 b Fs(\026)f Fo(b)l(e)h(the)g(same)g(as)g(in)g(Condition)i(\(AA\))d(ab)l (ove.)39 b(We)30 b(have)1048 358 y Fp(\015)1048 407 y(\015)1048 457 y(\015)1094 453 y Ft(\()p Fs(M)1207 465 y Fn(\022)1263 453 y Ft(+)18 b Fs(a)p Ft(\))1422 419 y Fk(\000)p Fr(1)p Fn(=)p Fr(2)1579 453 y Fs(I)1615 465 y Fn(\022)1653 453 y Ft(\()p Fs(M)1766 465 y Fn(\022)1822 453 y Ft(+)g Fs(a)p Ft(\))1981 419 y Fk(\000)p Fr(1)p Fn(=)p Fr(2)2137 358 y Fp(\015)2137 407 y(\015)2137 457 y(\015)2206 453 y Fq(\024)23 b Fs(C)2353 465 y Fr(0)2400 316 y Fp(P)2488 337 y Fr(2)2488 403 y Fn(j)s Fr(=1)2621 378 y Fq(k)o Fs(G)2727 390 y Fn(j)2763 378 y Fq(k)2804 406 y Fr(1)p Fn(=)p Fr(2)p Fn(;\022)p 2400 434 562 4 v 2528 451 a Fq(p)p 2597 451 237 4 v 70 x Fs(a)14 b Ft(sin)g Fs(\016)2811 497 y Fk(0)2972 453 y Fs(;)473 b Ft(\(5.30\))1048 713 y Fq(k)p Fs(\037)1142 725 y Fn(M)1205 734 y Fl(\022)1239 725 y Fk(\024)p Fn(\032)1329 713 y Fs(I)1365 725 y Fn(\022)1403 713 y Fs(\037)1455 725 y Fn(M)1518 734 y Fl(\022)1552 725 y Fk(\024)p Fn(\032)1643 713 y Fq(k)22 b(\024)h Fs(C)1854 725 y Fr(0)1905 596 y Fp(\022)2024 657 y Ft(2)p Fs(\032)p 1976 694 180 4 v 1976 770 a Ft(sin)14 b Fs(\016)2132 746 y Fk(0)2165 596 y Fp(\023)2215 613 y Fn(\026)2317 609 y Fr(2)2274 634 y Fp(X)2276 811 y Fn(j)s Fr(=1)2407 713 y Fq(k)p Fs(G)2514 725 y Fn(j)2549 713 y Fq(k)2591 741 y Fn(\026;\022)2702 713 y Fs(;)743 b Ft(\(5.31\))1048 1051 y Fq(jh)q Fs( )s(;)14 b(I)1234 1063 y Fn(\022)1272 1051 y Fs( )s Fq(ij)23 b(\024)1575 995 y Fs(")p 1505 1032 V 1505 1108 a Ft(sin)13 b Fs(\016)1660 1084 y Fk(0)1694 1051 y Fs(C)1759 1017 y Fr(2)1753 1071 y(0)1810 884 y Fp(0)1810 1034 y(@)1926 947 y Fr(2)1883 972 y Fp(X)1885 1149 y Fn(j)s Fr(=1)2016 1051 y Fq(k)p Fs(G)2123 1063 y Fn(j)2158 1051 y Fq(k)2200 1066 y Fr(1)p Fn(=)p Fr(2)p Fn(;\022)2357 884 y Fp(1)2357 1034 y(A)2430 901 y Fr(2)2481 1051 y Fq(h)p Fs( )s(;)h(M)2688 1063 y Fn(\022)2725 1051 y Fs( )s Fq(i)19 b Ft(+)2926 995 y(1)p 2926 1032 42 4 v 2927 1108 a Fs(")2991 1051 y Fq(k)p Fs( )s Fq(k)3131 1009 y Fr(2)3183 1051 y Fs(;)262 b Ft(\(5.32\))467 1310 y Fo(for)37 b(any)f Fs(a;)14 b(\032;)g(")32 b(>)h Ft(0)p Fo(,)k(and)f(wher)l(e)g Fs(C)1671 1322 y Fr(0)1744 1310 y Fo(is)g(given)g(in)g(\(5.25\).)57 b(Similar)37 b(estimates)f(hold)h (also)f(if)g(we)467 1435 y(r)l(eplac)l(e)31 b Fs(I)780 1447 y Fn(\022)848 1435 y Fo(by)f(either)h Ft(Re)p Fs(I)1327 1447 y Fn(\022)1395 1435 y Fo(or)f Ft(Im)p Fs(I)1637 1447 y Fn(\022)1675 1435 y Fo(.)467 1648 y Ft(This)23 b(lemma)g(follo)n(ws)f(from)h(Prop)r(osition)e(B.1)h(of)h(App)r(endix)h (B.3)f(and)f(equation)h(\(B.3.13\))f(\(cf.)36 b([4)o(]\).)467 1772 y(The)28 b(norms)f(on)g(the)h(r.h.s.)37 b(of)27 b(\(5.30\))g(-)g(\(5.32\))g(are)g(de\014ned)h(in)g(\(5.8\).)467 1925 y(No)n(w)d(w)n(e)g(use)f(the)i(lemma)f(ab)r(o)n(v)n(e)e(to)i(pro)n (v)n(e)e(Prop)r(osition)h(5.2.)35 b(First)25 b(w)n(e)f(determine)h(the) h(n)n(umerical)467 2050 y(range,)i(NR\()p Fs(K)944 2062 y Fn(\022)982 2050 y Ft(\),)h(of)g(the)g(op)r(erator)e Fs(K)1713 2062 y Fn(\022)1750 2050 y Ft(.)40 b(Let)28 b Fs(u)d Fq(2)g(D)r Ft(\()p Fs(M)2303 2007 y Fr(1)p Fn(=)p Fr(2)2294 2075 y Fn(\022)2407 2050 y Ft(\))k(and)g Fq(k)p Fs(u)p Fq(k)23 b Ft(=)i(1.)39 b(Recall)28 b(the)h(notation)467 2174 y Fq(j)p Fs(A)p Fq(j)h Ft(:=)f(\()p Fs(A)816 2144 y Fk(\003)855 2174 y Fs(A)p Ft(\))949 2144 y Fr(1)p Fn(=)p Fr(2)1085 2174 y Ft(and)i(remem)n(b)r(er)f(that)i(w)n(e)f(assumed)g (that)g Fs(\022)h Ft(=)c(\()p Fs(i\016)2736 2144 y Fk(0)2760 2174 y Fs(;)14 b(i\034)2871 2144 y Fk(0)2894 2174 y Ft(\).)48 b(By)31 b(estimate)h(\(5.32\))467 2299 y(and)c Fq(j)p Ft(Re)p Fs(L)807 2311 y Fr(0)p Fn(;\022)897 2299 y Fq(j)23 b(\024)g(k)p Fs(L)1130 2311 y Fn(p)1167 2299 y Fq(k)18 b Ft(+)g(cos)13 b Fs(\016)1475 2269 y Fk(0)1498 2299 y Ft(\003)28 b(w)n(e)f(ha)n(v)n(e)1271 2515 y Fq(j)p Ft(Re)14 b Fq(h)p Fs(K)1509 2527 y Fn(\022)1546 2515 y Fq(i)1579 2539 y Fn(u)1636 2515 y Fq(j)23 b(\024)1770 2397 y Fp(\034)1832 2515 y Ft(\003)18 b(+)2001 2458 y Fs(C)2066 2428 y Fr(2)2060 2479 y(1)2103 2458 y Fs(g)2146 2428 y Fr(2)p 2001 2495 183 4 v 2003 2571 a Ft(sin)13 b Fs(\016)2158 2547 y Fk(0)2193 2515 y Fs(M)2274 2527 y Fn(\022)2330 2515 y Ft(+)18 b Fq(k)p Fs(L)2512 2527 y Fn(p)2549 2515 y Fq(k)g Ft(+)g(1)2734 2397 y Fp(\035)2796 2598 y Fn(u)2853 2515 y Fs(;)592 b Ft(\(5.33\))467 2745 y(where)38 b Fq(h)p Fs(A)p Fq(i)844 2757 y Fn(u)930 2745 y Ft(:=)i Fq(h)q Fs(u;)14 b(Au)o Fq(i)p Ft(,)41 b(and)e(w)n(e)f(ha)n(v) n(e)f(set)h Fs(C)2088 2757 y Fr(1)2167 2745 y Ft(:=)j Fs(C)2355 2757 y Fr(0)2406 2683 y Fp(P)2494 2703 y Fr(2)2494 2770 y Fn(j)s Fr(=1)2627 2745 y Fq(k)p Fs(G)2734 2757 y Fn(j)2769 2745 y Fq(k)2811 2760 y Fr(1)p Fn(=)p Fr(2)p Fn(;\022)2968 2745 y Ft(.)69 b(Next,)42 b(using)c(that)467 2869 y(Im)q Fs(K)638 2881 y Fn(\022)698 2869 y Ft(=)22 b Fs(M)866 2881 y Fn(\022)922 2869 y Ft(+)c Fs(g)s Ft(Im)p Fs(I)1183 2881 y Fn(\022)1221 2869 y Ft(,)27 b(w)n(e)h(write)1318 3054 y(Im)14 b Fq(h)p Fs(K)1534 3066 y Fn(\022)1590 3054 y Ft(+)k Fs(ia)p Fq(i)1777 3079 y Fn(u)1844 3054 y Ft(=)1932 2962 y Fp(D)p 1982 2987 119 4 v 1982 3054 a Fs(M)2063 3066 y Fn(\022)2100 3001 y Fr(1)p Fn(=)p Fr(2)2205 3054 y Ft(\(1)g(+)g Fs(R)q Ft(\))p 2476 2987 V Fs(M)2557 3066 y Fn(\022)2594 3001 y Fr(1)p Fn(=)p Fr(2)2698 2962 y Fp(E)2749 3112 y Fn(u)2806 3054 y Fs(;)467 3277 y Ft(where)p 715 3210 V 35 w Fs(M)796 3289 y Fn(\022)868 3277 y Ft(=)35 b Fs(M)1049 3289 y Fn(\022)1110 3277 y Ft(+)23 b Fs(a)35 b Ft(and)g Fs(R)h Ft(=)f Fs(g)p 1688 3210 V 3 w(M)1769 3289 y Fn(\022)1806 3224 y Fk(\000)p Fr(1)p Fn(=)p Fr(2)1962 3277 y Ft(Im)p Fs(I)2097 3289 y Fn(\022)p 2135 3210 V 2135 3277 a Fs(M)2216 3289 y Fn(\022)2253 3224 y Fk(\000)p Fr(1)p Fn(=)p Fr(2)2409 3277 y Ft(.)60 b(Using)34 b(estimate)h (\(5.30\))g(w)n(e)f(obtain)467 3401 y Fq(k)p Fs(R)q Fq(k)22 b(\024)802 3364 y Fn(g)r(C)884 3372 y Fm(1)p 735 3382 250 4 v 735 3391 a Fk(p)p 789 3391 196 3 v 789 3441 a Fn(a)12 b Fr(sin)f Fn(\016)962 3424 y Fd(0)994 3401 y Ft(.)37 b(Hence)28 b(if)1753 3564 y Fs(g)s(C)1855 3576 y Fr(1)1916 3564 y Fs(<)2013 3508 y Ft(1)p 2013 3545 42 4 v 2013 3621 a(2)2065 3489 y Fq(p)p 2134 3489 237 4 v 75 x Fs(a)14 b Ft(sin)f Fs(\016)2347 3540 y Fk(0)2371 3564 y Fs(;)1074 b Ft(\(5.34\))467 3725 y(then)29 b(w)n(e)e(ha)n(v)n(e) 1493 3849 y(Im)14 b Fq(h)p Fs(K)1709 3861 y Fn(\022)1747 3849 y Fq(i)1779 3874 y Fn(u)1841 3849 y Ft(+)k Fs(a)23 b Fq(\025)2088 3793 y Ft(1)p 2088 3830 42 4 v 2088 3906 a(2)2154 3782 y Fp(\012)p 2193 3782 119 4 v 67 x Fs(M)2274 3861 y Fn(\022)2311 3782 y Fp(\013)2350 3882 y Fn(u)2416 3849 y Fq(\025)g Fs(a=)p Ft(2)p Fs(:)813 b Ft(\(5.35\))467 4010 y(This)31 b(sho)n(ws)f(that)h(Im)14 b Fq(h)p Fs(K)1301 4022 y Fn(\022)1338 4010 y Fq(i)1370 4035 y Fn(u)1442 4010 y Fq(\025)28 b(\000)p Fs(a=)p Ft(2.)44 b(It)31 b(follo)n(ws)f (from)g Fs(M)2444 4022 y Fn(\022)2510 4010 y Ft(=)d(sin)14 b Fs(\016)2758 3980 y Fk(0)2781 4010 y Ft(\003)21 b(+)f Fs(\034)2990 3980 y Fk(0)3014 4010 y Fs(N)39 b Ft(and)31 b(\(5.33\))f(that)467 4134 y Fq(j)p Ft(Re)14 b Fq(h)q Fs(K)706 4146 y Fn(\022)743 4134 y Fq(i)775 4159 y Fn(u)832 4134 y Fq(j)23 b(\024)966 4042 y Fp(D)1027 4092 y Fr(1+)p Fn(C)1163 4067 y Fm(2)1159 4108 y(1)1195 4092 y Fn(g)1229 4067 y Fm(2)p 1027 4115 236 4 v 1070 4163 a Fr(sin)11 b Fn(\016)1195 4146 y Fd(0)1272 4134 y Fs(M)1353 4146 y Fn(\022)1408 4134 y Ft(+)18 b Fq(k)p Fs(L)1590 4146 y Fn(p)1628 4134 y Fq(k)g Ft(+)g(1)1813 4042 y Fp(E)1863 4192 y Fn(u)1920 4134 y Fs(;)28 b Ft(and)f(hence,)h(b)n(y)f Fq(h)q Fs(M)2615 4146 y Fn(\022)2652 4134 y Fq(i)2684 4159 y Fn(u)2750 4134 y Fq(\024)2838 4067 y Fp(\012)p 2877 4067 119 4 v 67 x Fs(M)2958 4146 y Fn(\022)2995 4067 y Fp(\013)3035 4167 y Fn(u)3106 4134 y Ft(and)g(\(5.35\),)1146 4384 y Fq(j)p Ft(Re)14 b Fq(h)p Fs(K)1384 4396 y Fn(\022)1421 4384 y Fq(i)1453 4409 y Fn(u)1497 4384 y Fq(j)23 b(\024)f Ft(2)1682 4328 y(1)c(+)g Fs(C)1890 4298 y Fr(2)1884 4349 y(1)1927 4328 y Fs(g)1970 4298 y Fr(2)p 1682 4365 326 4 v 1755 4441 a Ft(sin)c Fs(\016)1911 4417 y Fk(0)2017 4384 y Ft(\(Im)g Fq(h)q Fs(K)2266 4396 y Fn(\022)2303 4384 y Fq(i)2335 4409 y Fn(u)2397 4384 y Ft(+)k Fs(a)p Ft(\))h(+)f Fq(k)p Fs(L)2757 4396 y Fn(p)2794 4384 y Fq(k)g Ft(+)g(1)p Fs(:)466 b Ft(\(5.36\))467 4569 y(Using)33 b(the)g(b)r(ound)g(\(5.34\))f(in)h(the)g(last)f(expression)g(sho)n(ws)f (that)i(NR\()p Fs(K)2832 4581 y Fn(\022)2870 4569 y Ft(\))f Fq(\032)f Fs(C)3089 4581 y Fn(a;\016)3177 4564 y Fd(0)3204 4569 y Ft(,)j(where)e Fs(C)3565 4581 y Fn(a;\016)3653 4564 y Fd(0)467 4693 y Ft(is)i(the)h(truncated)e(w)n(edge)g(\(5.27\),)i (pro)n(vided)e(condition)h(\(5.34\))f(is)h(satis\014ed.)55 b(In)34 b(particluar,)h(the)467 4818 y(sp)r(ectrum)c(of)g(the)f(op)r (erator)f Fs(K)1485 4830 y Fn(\022)1553 4818 y Ft(is)h(inside)h Fs(C)1937 4830 y Fn(a;\016)2025 4814 y Fd(0)2052 4818 y Ft(.)45 b(Moreo)n(v)n(er,)29 b(for)h Fs(z)40 b(=)-51 b Fq(2)28 b Fs(C)2854 4830 y Fn(a;\016)2942 4814 y Fd(0)3000 4818 y Ft(and)i Fs(u)g Ft(as)g(ab)r(o)n(v)n(e)f(w)n(e)467 4942 y(ha)n(v)n(e)e(the)h(estimate)1292 5067 y Fq(k)p Ft(\()p Fs(K)1437 5079 y Fn(\022)1492 5067 y Fq(\000)18 b Fs(z)t Ft(\))p Fs(u)p Fq(k)k(\025)h(j)14 b(h)p Fs(K)1990 5079 y Fn(\022)2027 5067 y Fq(i)2059 5092 y Fn(u)2121 5067 y Fq(\000)k Fs(z)t Fq(j)23 b(\025)f Ft(dist\()p Fs(z)t(;)14 b(C)2685 5079 y Fn(a;\016)2773 5063 y Fd(0)2800 5067 y Ft(\))p Fs(;)613 b Ft(\(5.37\))467 5227 y(whic)n(h,)28 b(b)n(y)f(taking)g Fs(u)c Ft(=)g(\()p Fs(K)1361 5239 y Fn(\022)1416 5227 y Fq(\000)18 b Fs(z)t Ft(\))1574 5197 y Fk(\000)p Fr(1)1663 5227 y Fs(v)s(=)p Fq(k)p Ft(\()p Fs(K)1893 5239 y Fn(\022)1948 5227 y Fq(\000)g Fs(z)t Ft(\))2106 5197 y Fk(\000)p Fr(1)2195 5227 y Fs(v)s Fq(k)p Ft(,)27 b(implies)h(\(5.28\).)1928 5476 y(14)p eop %%Page: 15 15 15 14 bop 315 210 a Ft(\(ii\))42 b(Estimates)27 b Fq(k)p Fs(u)p Fq(kk)p Ft(\()p Fs(K)1129 222 y Fn(\022)1183 210 y Fq(\000)18 b Fs(z)t Ft(\))p Fs(u)p Fq(k)k(\025)g Ft(Im)14 b Fq(h)p Fs(u;)g Ft(\()p Fs(K)1873 222 y Fn(\022)1929 210 y Fq(\000)k Fs(z)t Ft(\))p Fs(u)p Fq(i)27 b Ft(and)g(\(5.35\))g (imply)h(for)f(Im)14 b Fs(z)27 b Fq(\024)22 b(\000)p Fs(a)p Ft(:)1541 445 y Fq(k)p Ft(\()p Fs(K)1686 457 y Fn(\022)1741 445 y Fq(\000)c Fs(z)t Ft(\))p Fs(u)p Fq(k)k(\025)2108 329 y(p)p 2177 329 44 4 v 60 x Fs(a)p 2108 426 114 4 v 2144 502 a Ft(2)2231 445 y Fq(k)p Fs(M)2363 402 y Fr(1)p Fn(=)p Fr(2)2354 470 y Fn(\022)2467 445 y Fs(u)p Fq(k)k Fs(:)862 b Ft(\(5.38\))467 686 y(The)28 b(last)f(estimate)h(can)f(b)r (e)h(rewritten)f(as)1585 893 y Fq(k)p Fs(M)1717 850 y Fr(1)p Fn(=)p Fr(2)1708 918 y Fn(\022)1821 893 y Ft(\()p Fs(K)1924 905 y Fn(\022)1979 893 y Fq(\000)18 b Fs(z)t Ft(\))2137 859 y Fk(\000)p Fr(1)2226 893 y Fq(k)23 b(\024)2424 837 y Ft(2)p 2388 874 V 2388 891 a Fq(p)p 2457 891 44 4 v 60 x Fs(a)2539 893 y(:)906 b Ft(\(5.39\))467 1119 y(Similarly)27 b(w)n(e)h(ha)n(v)n(e)1594 1244 y Fq(k)p Fs(M)1726 1201 y Fk(\000)p Fr(1)p Fn(=)p Fr(2)1717 1269 y Fn(\022)1882 1244 y Fs(@)1926 1256 y Fn(\022)1963 1244 y Fs(K)2034 1256 y Fn(\022)2071 1244 y Fs(M)2161 1201 y Fk(\000)p Fr(1)p Fn(=)p Fr(2)2152 1269 y Fn(\022)2317 1244 y Fq(k)23 b(\024)f Fs(C)q(;)916 b Ft(\(5.40\))467 1418 y(where)27 b Fs(@)751 1430 y Fn(\022)817 1418 y Ft(stands)g(for)g Fs(@)1247 1430 y Fn(\016)1284 1418 y Ft(,)g Fs(@)1378 1430 y Fn(\034)1420 1418 y Ft(.)37 b(The)27 b(last)h(t)n(w)n(o)f(estimates)g(and)g(the)h(computation)1242 1626 y Fs(@)1286 1638 y Fn(\022)1324 1626 y Ft(\()p Fs(K)1427 1638 y Fn(\022)1482 1626 y Fq(\000)19 b Fs(z)t Ft(\))1641 1592 y Fk(\000)p Fr(1)1752 1626 y Ft(=)k Fq(\000)p Ft(\()p Fs(K)2008 1638 y Fn(\022)2063 1626 y Fq(\000)18 b Fs(z)t Ft(\))2221 1592 y Fk(\000)p Fr(1)2310 1626 y Fs(@)2354 1638 y Fn(\022)2392 1626 y Fs(K)2463 1638 y Fn(\022)2500 1626 y Ft(\()p Fs(K)2603 1638 y Fn(\022)2658 1626 y Fq(\000)h Fs(z)t Ft(\))2817 1592 y Fk(\000)p Fr(1)3468 1626 y Ft(\(5.41\))467 1833 y(imply)28 b(that)g(\()p Fs(K)983 1845 y Fn(\022)1039 1833 y Fq(\000)18 b Fs(z)t Ft(\))1197 1803 y Fk(\000)p Fr(1)1313 1833 y Ft(is)28 b(analytic)f(in)h Fs(\022)d Fq(2)e Fs(S)2008 1798 y Fk(\006)2003 1858 y Fn(\022)2035 1866 y Fm(0)2072 1833 y Ft(,)k(pro)n(vided)g(Im)14 b Fs(z)26 b Fq(\024)d(\000)p Fs(a)p Ft(.)292 2024 y(\(iii\))42 b(No)n(w)19 b(to)f(\014x)h(ideas)f(w)n(e)h(assume)f(that)h(Im)14 b Fs(\022)25 b Fq(\025)e Ft(0)18 b(and)g(Im)c Fs(z)27 b(<)22 b Fq(\000)p Fs(a)p Ft(.)34 b(F)-7 b(or)18 b Fs(\013)23 b(>)g Ft(0)18 b(w)n(e)h(de\014ne)g Fs(K)3306 1994 y Fr(\()p Fn(\013)p Fr(\))3427 2024 y Ft(:=)k Fs(K)7 b Ft(+)467 2149 y Fs(i\013N)i Ft(.)34 b(Then)20 b Fs(K)968 2106 y Fr(\()p Fn(\013)p Fr(\))962 2174 y Fn(\022)1089 2149 y Ft(:=)j Fs(U)1257 2161 y Fn(\022)1294 2149 y Fs(K)1371 2119 y Fr(\()p Fn(\013)p Fr(\))1470 2149 y Fs(U)1536 2113 y Fk(\000)p Fr(1)1527 2174 y Fn(\022)1648 2149 y Ft(=)g Fs(K)1807 2161 y Fn(\022)1845 2149 y Ft(+)r 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5011 y Ft(])24 b Fq(\032)e Fh(R)p Fo(.)1928 5476 y Ft(16)p eop %%Page: 17 17 17 16 bop 384 210 a Fo(R)l(emark.)75 b Ft(The)30 b(analysis)e(leading)h (to)h(Theorem)f(6.1)f(w)n(orks)g(also)h(for)g(in\014nite)h(dimensional) f(particle)260 335 y(systems.)77 b(W)-7 b(e)42 b(need)f(dim)15 b Fq(H)1228 347 y Fn(p)1312 335 y Fs(<)46 b Fq(1)41 b Ft(in)h(order)e(to)h(v)n(erify)f(the)i(assumptions)f Fs(\015)2934 347 y Fn(e)3015 335 y Fs(>)46 b Ft(0,)e Fs(\016)3272 347 y Fn(e)3353 335 y Fs(>)i Ft(0,)e(see)260 459 y(Prop)r(osition)26 b(7.2)h(and)g(Assumption)h(\(B\),)g(\(3.5\).) 384 584 y Fo(Pr)l(o)l(of)j(of)g(The)l(or)l(em)g(6.1.)68 b Ft(1.)27 b(W)-7 b(e)28 b(use)g(the)g(op)r(erator)e Fs(M)2190 596 y Fn(\022)2250 584 y Ft(:=)c(Im)q Fs(L)2517 596 y Fr(0)p Fn(;\022)2629 584 y Fs(>)h Ft(0)k(and)h(the)g(represen)n (tation)1197 776 y Fs(K)1268 788 y Fn(\022)1323 776 y Fq(\000)18 b Fs(z)27 b Ft(=)22 b(\()p Fs(M)1672 788 y Fn(\022)1728 776 y Ft(+)c Fs(a)p Ft(\))1887 742 y Fr(1)p Fn(=)p Fr(2)1992 776 y 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b(w)n(e)f(use)g(F)-7 b(esh)n(bac)n(h)27 b(maps)g Fs(F)1489 1547 y Fn(P)1531 1555 y Fl(e\032)1626 1535 y Ft(with)h(pro)5 b(jections)27 b Fs(P)2296 1547 y Fn(e\032)2394 1535 y Ft(de\014ned)h(as)1550 1740 y Fs(P)1603 1752 y Fn(e\032)1696 1740 y Ft(:=)23 b Fs(\037)1859 1752 y Fn(L)1905 1760 y Fl(p)1939 1752 y Fr(=)p Fn(e)2044 1740 y Fq(\012)18 b Fs(\037)2179 1752 y Fn(M)2242 1761 y Fl(\022)2276 1752 y Fk(\024)p Fn(\032)2367 1740 y Fs(:)1078 b Ft(\(6.14\))260 1944 y(Here,)32 b(recall,)g Fs(\037)786 1956 y Fn(L)832 1964 y Fl(p)867 1956 y Fr(=)p Fn(e)985 1944 y Ft(is)g(the)g(eigenpro)5 b(jection)30 b(for)i(the)g(op)r(erator)e Fs(L)2478 1956 y Fn(p)2548 1944 y Ft(corresp)r(onding)g(to)h(an)h(eigen)n(v)-5 b(alue)260 2068 y Fs(e)23 b Fq(2)g Fs(\033)s Ft(\()p Fs(L)539 2080 y Fn(p)578 2068 y Ft(\))k(and)g Fs(\037)850 2080 y Fn(M)913 2089 y Fl(\022)947 2080 y Fk(\024)p Fn(\032)1065 2068 y Ft(is)h(the)f(sp)r(ectral)g(pro)5 b(jection)26 b(for)h(the)h (self-adjoin)n(t)f(op)r(erator)e Fs(M)3109 2080 y Fn(\022)3174 2068 y Ft(corresp)r(onding)260 2193 y(to)i(the)h(sp)r(ectral)f(in)n (terv)-5 b(al)27 b([0)p Fs(;)14 b(\032)p Ft(])27 b(\(remem)n(b)r(er)h (that)g Fs(M)1993 2205 y Fn(\022)2057 2193 y Ft(is)g(a)f(p)r(ositiv)n (e)g(op)r(erator\).)260 2397 y Fg(Lemma)i(6.3)41 b Fo(Assume)26 b(Condition)j(\(AA\))d(holds.)39 b(L)l(et)27 b Fq(j)p Fs(g)s Fq(j)c Fs(<)2259 2345 y Fq(p)p 2328 2345 81 4 v 52 x Fs(\032)2371 2409 y Fr(0)2422 2397 y Fs(g)2462 2409 y Fr(0)2499 2397 y Fo(.)37 b(If)28 b Fs(z)e Fq(2)e Fs(S)2841 2409 y Fn(e)2903 2397 y Fo(then)j Fs(K)3156 2409 y Fn(\022)r(z)3250 2397 y Ft(:=)c Fs(K)3432 2409 y Fn(\022)3482 2397 y Fq(\000)13 b Fs(z)25 b Fq(2)260 2521 y(D)r Ft(\()p Fs(F)411 2533 y Fn(P)453 2541 y Fl(e\032)511 2553 y Fm(0)553 2521 y Ft(\))p Fo(,)31 b(and)f(the)g(op)l(er)l(ator)g Fs(K)1343 2478 y Fr(\(1\))1337 2547 y Fn(\022)r(z)1455 2521 y Ft(:=)23 b Fs(F)1619 2533 y Fn(P)1661 2541 y Fl(e\032)1719 2553 y Fm(0)1760 2521 y Ft(\()p Fs(K)1863 2533 y Fn(\022)r(z)1935 2521 y Ft(\))30 b Fo(acting)g(on)g Ft(Ran)p Fs(P)2566 2533 y Fn(e\032)2631 2541 y Fm(0)2698 2521 y Fo(is)g(of)h(the)f(form) 1177 2726 y Fs(K)1254 2683 y Fr(\(1\))1248 2751 y Fn(\022)r(z)1365 2726 y Ft(=)23 b(\()p Fs(e)18 b Fq(\000)g Fs(z)t Ft(\))p Fg(1)g Ft(+)g Fs(L)1906 2738 y Fn(r)r(\022)1994 2726 y Ft(+)g Fs(g)2120 2691 y Fr(2)2157 2726 y Ft(\003)2215 2738 y Fn(e)2269 2726 y Ft(+)g Fs(O)r Ft(\()p Fs(\017)p Ft(\()p Fs(g)s(;)c(\032)2638 2738 y Fr(0)2675 2726 y Ft(\)\))p Fs(;)706 b Ft(\(6.15\))260 2930 y Fo(wher)l(e)30 b Fs(\032)537 2942 y Fr(0)598 2930 y Fq(2)23 b Ft(\(0)p Fs(;)14 b(\033)s(=)p Ft(2\))p Fo(,)30 b(the)f(r)l(emainder)i(is)f (estimate)l(d)g(in)g(op)l(er)l(ator)h(norm,)f(and)g(for)h(any)f Fq(j)p Fs(g)s Fq(j)p Fs(;)14 b(\032)23 b(>)f Ft(0)p Fo(,)1263 3134 y Fs(\017)p Ft(\()p Fs(g)s(;)14 b(\032)p Ft(\))23 b(:=)g Fq(j)p Fs(g)s Fq(j)p Fs(\032)1750 3100 y Fn(\026)1812 3134 y Ft(+)18 b Fq(j)p Fs(g)s Fq(j)1984 3100 y Fr(3)2021 3134 y Fs(\032)2064 3100 y Fk(\000)p Fr(1)p Fn(=)p Fr(2)2239 3134 y Ft(+)g Fq(j)p Fs(g)s Fq(j)2411 3100 y Fr(2)2448 3134 y Fs(\032)2491 3100 y Fr(2)p Fn(\026)p Fk(\000)p Fr(1)2653 3134 y Fs(:)792 b Ft(\(6.16\))384 3463 y(W)-7 b(e)27 b(giv)n(e)d(here)i(a)f(short)g(pro)r(of)g(of)h(Lemma)g(6.3.)35 b(Another)26 b(pro)r(of)f(is)h(obtained)f(b)n(y)h(an)f(easy)g (translation)260 3587 y(of)i(Theorem)g(V.6)h(and)f(Lemma)g(V.9)h(of)g ([5)o(].)384 3712 y Fo(Pr)l(o)l(of)h(of)f(L)l(emma)g(6.3.)65 b Ft(In)26 b(this)f(pro)r(of)g(w)n(e)g(write)g Fs(\032)g Ft(for)g Fs(\032)2233 3724 y Fr(0)2270 3712 y Ft(.)36 b(In)25 b(order)f(to)h(pro)n(v)n(e)f(that)h Fs(K)3214 3724 y Fn(\022)r(z)3308 3712 y Fq(2)f(D)r Ft(\()p Fs(F)3538 3724 y Fn(P)3580 3732 y Fl(e\032)3648 3712 y Ft(\))260 3836 y(w)n(e)j(let)h Fs(W)35 b Ft(:=)23 b Fs(M)807 3848 y Fn(\022)862 3836 y Ft(+)18 b Fs(\032)p Ft(,)28 b(note)f(that)h Fs(W)40 b Ft(comm)n(utes)27 b(with)h Fs(L)2157 3848 y Fr(0)p Fn(\022)2227 3836 y Ft(,)g(and)g(write)p 1229 3974 65 4 v 1229 4040 a Fs(P)1294 4052 y Fn(e\032)1364 4040 y Fs(K)1435 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533 4427 V Fs(P)597 4506 y Fn(e\032)667 4494 y Ft(,)j(and)899 4423 y Fp(\015)899 4473 y(\015)945 4494 y Fs(A)1007 4464 y Fk(\000)p Fr(1)1097 4423 y Fp(\015)1097 4473 y(\015)1180 4494 y Fq(\024)e Fs(C)6 b Ft(,)39 b(uniformly)d(in)g Fs(\032)h Ft(and)f Fs(g)j Ft(\(cf.)63 b(the)37 b(pro)r(of)e(of)h (\(6.7\))g(giv)n(en)g(after)g(that)260 4618 y(equation\).)g(By)28 b(Lemma)f(5.3,)1453 4771 y Fq(k)p Fs(B)t Fq(k)22 b(\024)h Fs(C)1773 4783 y Fr(0)1811 4771 y Fq(j)p Fs(g)s Fq(j)1910 4710 y Ft(max)2064 4722 y Fn(j)2113 4710 y Fq(k)p Fs(G)2220 4722 y Fn(j)2254 4710 y Fq(k)2296 4725 y Fr(1)p Fn(=)p Fr(2)p Fn(;\022)p 1909 4752 545 4 v 2029 4769 a Fq(p)p 2098 4769 236 4 v 63 x Fs(\032)14 b Ft(sin)g Fs(\016)2311 4808 y Fk(0)2463 4771 y Fs(:)982 b Ft(\(6.18\))260 4978 y(Hence,)39 b(for)d Fq(j)p Fs(g)s Fq(j)h Fs(<)905 4926 y Fq(p)p 975 4926 43 4 v 975 4978 a Fs(\032)13 b(g)1071 4990 y Fr(0)1108 4978 y Ft(,)39 b(the)e(op)r(erator)e Fs(A)24 b Ft(+)g Fs(B)41 b Ft(is)36 b(in)n(v)n(ertible)g(and)g (therefore)g(so)f(is)p 3144 4911 65 4 v 37 w Fs(P)3208 4990 y Fn(e\032)3279 4978 y Fs(K)3350 4990 y Fn(\022)r(z)p 3421 4911 V 3421 4978 a Fs(P)3486 4990 y Fn(e\032)3592 4978 y Ft(on)260 5103 y(Ran)p 409 5036 V Fs(P)473 5115 y Fn(e\032)544 5103 y Ft(.)g(It)24 b(is)f(easy)g(to)g(see)h(that)f(the) h(other)f(conditions)g(in)h(the)g(de\014nition)g(of)f Fs(D)r Ft(\()p Fs(F)2937 5115 y Fn(P)2979 5123 y Fl(e\032)3047 5103 y Ft(\))h(\(see)f(Eqn)g(\(6.13\)\))260 5227 y(are)j(satis\014ed)i (and)f(therefore)g Fs(K)1299 5239 y Fn(\022)r(z)1393 5227 y Fq(2)c(D)r Ft(\()p Fs(F)1622 5239 y Fn(P)1664 5247 y Fl(e\032)1732 5227 y Ft(\).)1928 5476 y(18)p eop %%Page: 19 19 19 18 bop 384 210 a Ft(Next,)28 b(in)g(view)g(of)f(de\014nition)h (\(6.12\))f(w)n(e)g(compute)1226 405 y Fs(P)1279 417 y Fn(e\032)1350 405 y Fs(K)1421 417 y Fn(\022)r(z)1492 405 y Fs(P)1545 417 y Fn(e\032)1638 405 y Ft(=)c(\()p Fs(e)18 b Fq(\000)g Fs(z)t Ft(\))p Fg(1)g Ft(+)g Fs(L)2179 417 y Fn(r)r(\022)2267 405 y Ft(+)g Fs(g)s(P)2446 417 y Fn(e\032)2516 405 y Fs(I)2552 417 y Fn(\022)2590 405 y Fs(P)2643 417 y Fn(e\032)3468 405 y Ft(\(6.19\))260 600 y(acting)27 b(on)g(Ran)p Fs(P)826 612 y Fn(e\032)896 600 y Ft(.)37 b(By)28 b(\(5.31\))e(and)i(with)g Fs(\026)g Ft(as)f(in)g(Condition)h(\(AA\))1591 795 y Fs(g)s(P)1687 807 y Fn(e\032)1757 795 y Fs(I)1793 807 y Fn(\022)1831 795 y Fs(P)1884 807 y Fn(e\032)1978 795 y Ft(=)22 b Fs(O)r Ft(\()p Fs(g)s(\032)2248 761 y Fn(\026)2293 795 y Ft(\))p Fs(:)1120 b Ft(\(6.20\))384 990 y(Using)24 b(\(6.17\),)f(expanding)p 1268 923 65 4 v 23 w Fs(P)1333 1002 y Fn(e\032)1403 990 y Ft(\()p 1435 923 V Fs(P)1500 1002 y Fn(e\032)1570 990 y Fs(K)1641 1002 y Fn(\022)r(z)p 1713 923 V 1713 990 a Fs(P)1777 1002 y Fn(e\032)1848 990 y Ft(\))1880 960 y Fk(\000)p Fr(1)p 1969 923 V 1969 990 a Fs(P)2034 1002 y Fn(e\032)2128 990 y Ft(in)g(the)h(Neumann)g(series)f(in)g Fs(B)t Ft(,)i(and)e(using)h(that)260 1115 y Fq(k)o Fs(B)t Fq(k)f(\024)g Fs(C)6 b Fq(j)p Fs(g)s Fq(j)p Fs(\032)718 1084 y Fk(\000)p Fr(1)p 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1504 y Ft(.)384 1629 y(T)-7 b(o)24 b(estimate)h(the)f(op)r(erator)f (\003)1362 1641 y Fn(e\032\022)1490 1629 y Ft(w)n(e)h(use)g(the)h (expression)d(of)j Fs(I)2413 1641 y Fn(\022)2475 1629 y Ft(in)g(terms)f(of)g(creation)f(and)i(annihila-)260 1754 y(tion)e(op)r(erators,)g(pull)h(through)f(the)h(annihilation)f(op) r(erators)e(to)j(the)g(righ)n(t)e(un)n(til)i(they)g(either)f(con)n (tract)g(or)260 1878 y(hit)30 b(the)g(pro)5 b(jections)29 b Fs(P)1019 1890 y Fn(e\032)1089 1878 y Ft(,)i(and)f(use)f(estimates)h (\(B.3.4\))f(and)h(\(B.3.8\))f(for)h Fs(a)2722 1890 y Fn(j)s(`;r)2837 1878 y Ft(\()p Fs(k)s Ft(\))p Fs(P)3000 1890 y Fn(e\032)3100 1878 y Ft(and)g Fs(P)3317 1890 y Fn(e\032)3387 1878 y Fs(a)3431 1848 y Fk(\003)3431 1902 y Fn(j)s(`;r)3546 1878 y Ft(\()p Fs(k)s Ft(\).)260 2003 y(As)e(a)f(result)g(w)n(e)g(obtain)1495 2127 y(\003)1553 2139 y Fn(e\032\022)1680 2127 y Ft(=)22 b(\003)1825 2139 y Fn(e)1861 2127 y Fs(P)1914 2139 y Fn(e\032)2002 2127 y Ft(+)c Fs(O)r Ft(\()p Fs(\032)2225 2093 y 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Fo(L)l(et)27 b Fs(A)g Fo(b)l(e)g(a)g(normal)g(op)l(er)l(ator)h(on)f(a)g(Hilb)l(ert)g (sp)l(ac)l(e)h Fq(H)2712 3978 y Fr(1)2749 3966 y Fo(,)g(and)f(let)g Fs(B)k Fo(b)l(e)c(an)g(op)l(er)l(ator)467 4090 y(on)j(a)g(Hilb)l(ert)g (sp)l(ac)l(e)h Fq(H)1218 4102 y Fr(2)1255 4090 y Fo(,)f Ft(dim)14 b Fq(H)1532 4102 y Fr(2)1593 4090 y Ft(=)22 b Fs(d)i(<)e Fq(1)p Fo(.)39 b(Then)467 4215 y Ft(\(i\))31 b Fs(\033)s Ft(\()p Fs(A)19 b Fq(\012)f Ft(1)-18 b(l)18 b(+)g(1)-18 b(l)17 b Fq(\012)h Fs(B)t Ft(\))24 b(=)e Fs(\033)s Ft(\()p Fs(A)p Ft(\))e(+)e Fs(\033)s Ft(\()p Fs(B)t Ft(\))p Fo(,)467 4339 y Ft(\(ii\))31 b Fo(for)f Fs(z)36 b(=)-51 b Fq(2)23 b Fs(\033)s Ft(\()p Fs(A)p Ft(\))d(+)e Fs(\033)s Ft(\()p Fs(B)t Ft(\))31 b Fo(we)f(have)1061 4464 y Fp(\015)1061 4513 y(\015)1108 4534 y Ft(\()p Fs(A)19 b Fq(\012)f Ft(1)-18 b(l)17 b(+)h(1)-18 b(l)18 b Fq(\012)g Fs(B)k Fq(\000)c Fs(z)t Ft(\))1842 4500 y Fk(\000)p Fr(1)1931 4464 y Fp(\015)1931 4513 y(\015)2000 4534 y Fq(\024)23 b Fs(C)d Ft([dist\()p Fs(\033)s 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b(The)27 b(\(R)n(iesz\))g(pr)l(oje)l(ction)h(onto)f Fs(c)f Fo(is)2291 5040 y Fp(P)2379 5061 y Fn(p)2379 5128 y(j)s Fr(=1)2512 5103 y Fs(\037)2564 5115 y Fn(A)p Fr(=)p Fn(a)2701 5123 y Fl(j)2748 5103 y Fq(\012)12 b Fs(\037)2877 5115 y Fn(B)s Fr(=)p Fn(b)3010 5123 y Fl(j)3044 5103 y Fo(,)28 b(wher)l(e)g Fs(\037)3381 5115 y Fn(A)p Fr(=)p Fn(a)3548 5103 y Fo(and)467 5227 y Fs(\037)519 5239 y Fn(B)s Fr(=)p Fn(b)686 5227 y Fo(ar)l(e)i(the)g(\(R)n(iesz\))g(pr)l (oje)l(ctions)h(onto)f Fs(a)f Fo(and)i Fs(b)p Fo(,)e(r)l(esp)l(e)l (ctively.)1928 5476 y Ft(19)p eop %%Page: 20 20 20 19 bop 467 210 a Ft(W)-7 b(e)35 b(pro)n(v)n(e)d(part)i(3)f(of)h (Theorem)f(6.1)h(using)f(Lemma)h(6.4)f(and)h(refer)f(to)h(the)h(end)f (of)g(this)g(section)467 335 y(for)j(a)g(pro)r(of)g(of)h(the)f(lemma.) 67 b(W)-7 b(e)38 b(appro)n(ximate)d(the)j(op)r(erator)e(\003)2695 347 y Fn(e)2768 335 y Ft(b)n(y)h(a)g(family)h(of)f(op)r(erators)467 459 y(\003)525 416 y Fr(\()p Fn(\021)r Fr(\))525 469 y Fn(e)617 459 y 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(the)g(op)r(erator)467 1914 y Fs(L)524 1926 y Fn(r)r(\022)615 1914 y Ft(+)20 b Fs(g)743 1884 y Fr(2)780 1914 y Ft(\003)838 1871 y Fr(\()p Fn(\021)r Fr(\))838 1923 y Fn(e)961 1914 y Ft(has)30 b(a)g(simple)h(eigen)n(v)-5 b(alue)30 b(at)h Fs(g)1994 1884 y Fr(2)2031 1914 y Fs(\025)2079 1926 y Fn(e)2146 1914 y Ft(and)g(the)g(rest)f(of)h(the)g(sp)r(ectrum)g(is)g (lo)r(cated)g(in)467 2038 y Fq(f)p Fs(z)26 b Fq(2)e Fh(C)29 b Fq(j)14 b Ft(Im)19 b Fs(z)27 b Fq(\025)22 b Fs(g)1072 2008 y Fr(2)1109 2038 y Ft(Im)p Fs(\025)1256 2050 y Fn(e)1311 2038 y Ft(+)c(min\()p Fs(g)1607 2008 y Fr(2)1644 2038 y Fs(\016)1681 2050 y Fn(e)1717 2038 y Fs(;)c(\034)1799 2008 y Fk(0)1823 2038 y Ft(\))p Fq(g)p Ft(.)467 2196 y(W)-7 b(e)35 b(use)e(relation)g(\(6.23\))g(to)h(in)n(v)n(estigate)f (the)h(sp)r(ectrum)g(of)g Fs(K)2541 2208 y Fn(\022)2612 2196 y Ft(inside)g Fs(S)2905 2208 y Fn(e)2940 2196 y Ft(.)56 b(The)34 b(error)e(term)i(in)467 2321 y(\(6.23\))h(satis\014es) g Fs(O)r Ft(\()p Fs(\017)p Ft(\()p Fs(g)s(;)14 b(\032)1315 2333 y Fr(0)1353 2321 y Ft(\)\))37 b(=)f Fs(O)r Ft(\()p Fq(j)p Fs(g)s Fq(j)1741 2290 y Fr(2+)p Fn(\013)1873 2321 y Ft(\).)61 b(F)-7 b(rom)36 b(\(6.24\))f(\(with)h Fs(n)g Ft(=)h(1\))e(and)h(an)f(elemen)n(tary)467 2445 y(Neumann)i(series)e (estimate)i(it)g(follo)n(ws)e(that)i(the)f(sp)r(ectrum)h(of)f Fs(L)2670 2457 y Fn(r)r(\022)2764 2445 y Ft(+)24 b Fs(g)2896 2415 y Fr(2)2933 2445 y Ft(\003)2991 2457 y Fn(e)3051 2445 y Ft(+)f Fs(O)r Ft(\()p Fs(\017)p Ft(\()p Fs(g)s(;)14 b(\032)3425 2457 y Fr(0)3463 2445 y Ft(\)\))37 b(lies)467 2570 y(in)32 b(a)e(neigh)n(b)r(ourho)r(o)r(d)g(of)h(order)f Fs(O)r Ft(\()p Fq(j)p Fs(g)s Fq(j)1716 2540 y Fr(2+)p Fn(\013)1869 2570 y Ft(+)20 b Fs(g)1997 2540 y Fr(2)2034 2570 y Fq(k)p Ft(\003)2134 2527 y Fr(\()p Fn(\021)r Fr(\))2134 2579 y Fn(e)2246 2570 y Fq(\000)h Ft(\003)2390 2582 y Fn(e)2425 2570 y Fq(k)p Ft(\))28 b(=)h Fs(O)r Ft(\()p Fq(j)p Fs(g)s Fq(j)2807 2540 y Fr(2+)p Fn(\013)2939 2570 y Ft(\))i(of)g(the)h(sp)r(ectrum)f(of)467 2694 y Fs(L)524 2706 y Fn(r)r(\022)613 2694 y Ft(+)18 b Fs(g)739 2664 y Fr(2)775 2694 y Ft(\003)833 2651 y Fr(\()p Fn(\021)r Fr(\))833 2704 y Fn(e)953 2694 y Ft(\(for)27 b Fs(\021)k Ft(small)d(enough\).)36 b(Moreo)n(v)n(er,)25 b(since)i(b)n(y)h(our)e (assumptions)1659 2902 y Fq(j)p Fs(g)s Fq(j)1748 2867 y Fr(2+)p Fn(\013)1902 2902 y Fs(<)-28 b(<)23 b Ft(min\()p Fs(g)2240 2867 y Fr(2)2277 2902 y Fs(\016)2314 2914 y Fn(e)2350 2902 y Fs(;)14 b(\034)2432 2867 y Fk(0)2456 2902 y Ft(\))980 b(\(6.26\))467 3109 y(\(see)30 b(\(6.5\)\),)g(one)f (easily)g(pro)n(v)n(es,)f(using)h(Riesz)h(pro)5 b(jections,)29 b(that)h Fs(L)2691 3121 y Fn(r)r(\022)2780 3109 y Ft(+)19 b Fs(g)2907 3079 y Fr(2)2944 3109 y Ft(\003)3002 3121 y Fn(e)3057 3109 y Ft(+)h Fs(O)r Ft(\()p Fs(\017)p Ft(\()p Fs(g)s(;)14 b(\032)3428 3121 y Fr(0)3465 3109 y Ft(\)\))30 b(has)467 3234 y(a)e(simple)g(eigen)n(v)-5 b(alue)27 b Fs(z)1232 3246 y Fr(0)1296 3234 y Ft(in)h(an)g Fs(O)r Ft(\()p Fq(j)p Fs(g)s Fq(j)1695 3204 y Fr(2+)p Fn(\013)1827 3234 y Ft(\)-neigh)n(b)r(ourho)r(o)r(d)f(of)g Fs(g)2592 3204 y Fr(2)2629 3234 y Fs(\025)2677 3246 y Fn(e)2713 3234 y Ft(.)38 b(The)28 b(rest)f(of)h(the)g(sp)r(ectrum)467 3358 y(of)i Fs(L)621 3370 y Fn(r)r(\022)710 3358 y Ft(+)19 b Fs(g)837 3328 y Fr(2)874 3358 y Ft(\003)932 3370 y Fn(e)987 3358 y Ft(+)g Fs(O)r Ft(\()p Fs(\017)p Ft(\()p Fs(g)s(;)14 b(\032)1357 3370 y Fr(0)1394 3358 y Ft(\)\))30 b(is)g(lo)r(cated)f(in)g Fq(f)p Fs(z)g Fq(2)d Fh(C)j Fq(j)14 b Ft(Im)20 b Fs(z)29 b(>)c(g)2578 3328 y Fr(2)2615 3358 y Ft(Im)p Fs(\025)2762 3370 y Fn(e)2818 3358 y Ft(+)2912 3326 y Fr(1)p 2912 3340 34 4 v 2912 3387 a(2)2969 3358 y Ft(min\()p Fs(g)3182 3328 y Fr(2)3220 3358 y Fs(\016)3257 3370 y Fn(e)3292 3358 y Fs(;)14 b(\034)3374 3328 y Fk(0)3398 3358 y Ft(\))p Fq(g)p Ft(.)42 b(The)467 3483 y(result)28 b(\(6.6\))f(follo)n(ws)g(from)g(the)h(isosp)r(ectralit)n(y)-7 b(,)26 b(\(6.23\).)467 3641 y(Fix)i(an)g(arbitrary)e Fs(g)1129 3611 y Fk(0)1151 3641 y Ft(,)i(0)23 b Fs(<)g Fq(j)p Fs(g)1421 3611 y Fk(0)1444 3641 y Fq(j)h Fs(<)f Ft(min[\()p Fs(g)1812 3653 y Fr(0)1849 3641 y Ft(\))1881 3611 y Fr(1)p Fn(=\013)1996 3641 y Fs(;)14 b(g)2073 3653 y Fr(2)2110 3641 y Ft(].)37 b(By)28 b(the)g(Kato-Rellic)n(h)e(Theorem,) h Fs(g)f Fq(7!)e Fs(z)3536 3653 y Fr(0)3573 3641 y Ft(\()p Fs(g)s Ft(\))467 3765 y(is)30 b(analytic)e(in)i(a)f(complex)f(neigh)n (b)r(ourho)r(o)r(d)h(of)g Fs(g)2075 3735 y Fk(0)2098 3765 y Ft(.)42 b(This)29 b(completes)g(the)g(pro)r(of)g(of)g(Theorem)g (6.1,)467 3890 y(p)r(oin)n(t)f(3,)f(and)h(hence)f(the)h(en)n(tire)g (pro)r(of)f(of)g(Theorem)g(6.1.)1278 b Fc(\004)467 4047 y Fo(Pr)l(o)l(of)32 b(of)g(L)l(emma)e(6.4.)72 b Ft(By)29 b(using)f(the)h(sp)r(ectral)f(represen)n(tation)f(of)h Fs(A)h Ft(and)f(the)h(normal)f(form)g(of)467 4172 y(the)g(op)r(erator)e Fs(B)t Ft(,)i([15)o(])g(I.5.3,)f(one)g(obtains)782 4443 y(\()p Fs(A)19 b Fq(\012)f Ft(1)-18 b(l)18 b(+)g(1)-18 b(l)17 b Fq(\012)h Fs(B)23 b Fq(\000)18 b Fs(z)t Ft(\))1517 4408 y Fk(\000)p Fr(1)1629 4443 y Ft(=)1716 4364 y Fp(X)1761 4541 y Fn(j)1850 4330 y(m)1909 4338 y Fl(j)1940 4330 y Fk(\000)p Fr(1)1878 4364 y Fp(X)1875 4540 y Fn(n)p Fr(=0)2025 4443 y Ft(\()p Fq(\000)p Ft(1\))2196 4408 y Fn(n)2241 4443 y Ft(\()p Fs(A)h Ft(+)f Fs(b)2473 4455 y Fn(j)2526 4443 y Fq(\000)g Fs(z)t Ft(\))2684 4408 y Fk(\000)p Fn(n)p Fk(\000)p Fr(1)2884 4443 y Fq(\012)g Fs(Q)3033 4399 y Fr(\()p Fn(n)p Fr(\))3033 4466 y Fn(j)3130 4443 y Fs(;)315 b Ft(\(6.27\))467 4738 y(where)32 b Fs(b)748 4750 y Fn(j)814 4738 y Ft(are)f(the)i(eigen)n(v)-5 b(alues)30 b(of)i Fs(B)t Ft(,)i Fs(Q)1827 4695 y Fr(\(0\))1827 4761 y Fn(j)1946 4738 y Ft(=)29 b Fs(\037)2092 4750 y Fn(B)s Fr(=)p Fn(b)2225 4758 y Fl(j)2293 4738 y Ft(is)j(the)g(pro)5 b(jection)31 b(\(Riesz)h(in)n(tegral\))f(on)n(to)467 4863 y(the)h(eigen)n(v)-5 b(alue)31 b Fs(b)1051 4875 y Fn(j)1085 4863 y Ft(,)i(and,)f(for)f Fs(n)e Fq(\025)g Ft(1,)j Fs(Q)1797 4820 y Fr(\()p Fn(n)p Fr(\))1797 4886 y Fn(j)1923 4863 y Ft(=)d Fs(N)2093 4833 y Fn(n)2084 4884 y(j)2138 4863 y Ft(,)k(with)f Fs(N)2454 4875 y Fn(j)2518 4863 y Ft(=)d Fs(Q)2678 4820 y Fr(\(0\))2678 4886 y Fn(j)2766 4863 y Fs(N)2833 4875 y Fn(j)2897 4863 y Ft(=)g Fs(N)3058 4875 y Fn(j)3093 4863 y Fs(Q)3159 4820 y Fr(\(0\))3159 4886 y Fn(j)3279 4863 y Ft(a)i(nilp)r(oten)n(t)467 4987 y(matrix,)d Fs(N)837 4942 y Fn(m)896 4950 y Fl(j)828 5010 y Fn(j)953 4987 y Ft(=)23 b(0.)37 b(Assertions)26 b(\(i\),)i(\(ii\))h(follo)n(w.)1928 5476 y(20)p eop %%Page: 21 21 21 20 bop 467 210 a Ft(Let)28 b Fs(C)34 b Ft(b)r(e)28 b(a)f(circle)g(of)h(radius)f Fs(r)e(<)e Ft(dist)14 b([)p Fs(c;)g Ft(\()p Fs(\033)s Ft(\()p Fs(A)p Ft(\))20 b(+)e Fs(\033)s Ft(\()p Fs(B)t Ft(\)\))p Fq(nf)p Fs(c)p Fq(g)p Ft(])28 b(around)e Fs(c)p Ft(.)37 b(F)-7 b(rom)27 b(\(6.27\),)678 423 y(1)p 639 460 121 4 v 639 537 a(2)p Fs(\031)s(i)783 367 y Fp(I)829 555 y Fn(C)899 480 y Fs(dz)t Ft(\()p Fs(A)18 b Fq(\012)g Ft(1)-18 b(l)18 b(+)g(1)-18 b(l)17 b Fq(\012)h Fs(B)23 b Fq(\000)18 b Fs(z)t Ft(\))1719 445 y Fk(\000)p Fr(1)1831 480 y Ft(=)1968 423 y(1)p 1928 460 V 1928 537 a(2)p Fs(\031)s(i)2072 367 y Fp(I)2119 555 y Fn(C)2188 480 y Fs(dz)2288 401 y Fp(X)2332 578 y Fn(j)2421 367 y(m)2480 375 y Fl(j)2511 367 y Fk(\000)p Fr(1)2449 401 y Fp(X)2446 577 y Fn(n)p Fr(=0)2596 480 y Ft(\()p Fq(\000)p Ft(1\))2767 445 y Fn(n)795 726 y Fq(\002)874 634 y Fp(h)912 726 y Ft(\()p Fs(c)h Fq(\000)f Fs(z)t Ft(\))1157 692 y Fk(\000)p Fn(n)p Fk(\000)p Fr(1)1339 726 y Fs(\037)1391 738 y Fn(A)p Fr(=)p Fn(a)1528 746 y Fl(j)1581 726 y Fq(\012)g Fs(Q)1730 683 y Fr(\()p Fn(n)p Fr(\))1730 749 y Fn(j)1845 726 y Ft(+)g(\()p Fs(A)h Ft(+)f Fs(b)2160 738 y Fn(j)2213 726 y Fq(\000)g Fs(z)t Ft(\))2371 692 y Fk(\000)p Fn(n)p Fk(\000)p Fr(1)2553 726 y Ft(\(1)g Fq(\000)g Fs(\037)2780 738 y Fn(A)p Fr(=)p Fn(a)2917 746 y Fl(j)2952 726 y Ft(\))h Fq(\012)f Fs(Q)3152 683 y Fr(\()p Fn(n)p Fr(\))3152 749 y Fn(j)3249 634 y Fp(i)3302 726 y Fs(:)143 b Ft(\(6.28\))467 932 y(The)30 b(\014rst)e(term)i(on)e(the)i(r.h.s.)41 b(of)29 b(\(6.28\))f(con)n(tributes)h(only)g(for)f Fs(n)e Ft(=)f(0)j(\(for)h(eac)n(h)g Fs(j)34 b Ft(\014xed\),)c(while)467 1057 y(the)e(second)f(term)h(do)r(es)f(not)h(con)n(tribute)f(at)g(all.) 37 b(This)28 b(concludes)f(the)h(pro)r(of)f(of)g(Lemma)h(6.4.)110 b Fc(\004)260 1331 y Fu(7)135 b(Absence)44 b(of)h Fb(\014)1212 1349 y Ft(1)1258 1331 y Fb(\014)1324 1349 y Ft(2)1370 1331 y Fu(-normal)g(stationary)i(states)260 1538 y Ft(In)22 b(this)f(section)g(w)n(e)h(pro)n(v)n(e)e(Theorem)g(3.1.)34 b(Let)22 b Fs(L)g Ft(=)h Fs(L)1994 1550 y Fr(0)2037 1538 y Ft(+)6 b Fs(g)s(\031)s Ft(\()p Fs(v)s Ft(\))g Fq(\000)g Fs(g)s(\031)2478 1508 y Fk(0)2502 1538 y Ft(\()p Fs(v)s Ft(\))22 b(b)r(e)g(the)g(standard)f(\(self-adjoin)n(t\))260 1662 y(Liouville)28 b(op)r(erator,)g(\(2.17\),)h(and)f(let)i Fs(L)1567 1674 y Fn(\022)1633 1662 y Ft(b)r(e)f(its)g Fs(U)1921 1674 y Fn(\022)1958 1662 y Ft(-deformation.)40 b(If)30 b(Condition)e(\(C\))i(is)f(satis\014ed)f(then)260 1787 y(the)e(op)r(erator)f(\003)793 1799 y Fr(0)853 1787 y Ft(=)d Fs(i)p Ft(\000)1021 1799 y Fr(0)1084 1787 y Ft(is)27 b(an)n(ti-selfadjoin)n(t,)e(with)i(\000)1996 1799 y Fr(0)2056 1787 y Fq(\025)c Ft(0)j(\(see)g(also)f(Prop)r(osition) f(7.2)i(b)r(elo)n(w,)g(and)g([5)o(]\).)260 1911 y(Let)j Fs(\015)453 1923 y Fr(0)514 1911 y Fq(\025)c Ft(0)j(b)r(e)h(the)g(lo)n (w)n(est)e(eigen)n(v)-5 b(alue)28 b(of)g(\000)1727 1923 y Fr(0)1765 1911 y Ft(,)g(and)h(let)g Fs(\016)2137 1923 y Fr(0)2199 1911 y Fs(>)24 b Ft(0)k(denote)g(the)h(distance)g(of)f Fs(\015)3233 1923 y Fr(0)3299 1911 y Ft(to)g(the)h(rest)260 2036 y(of)e(the)h(sp)r(ectum)h(of)e(\000)974 2048 y Fr(0)1011 2036 y Ft(.)260 2242 y Fg(Theorem)j(7.1)41 b Fo(Assume)28 b(that)h(c)l(onditions)h(\(A\),)f(\(B\))g(and)h(\(C\))f(ar)l(e)h(ob)l (eye)l(d)g(for)g(some)f Ft(0)23 b Fs(<)f(\014)3302 2254 y Fr(1)3340 2242 y Fs(;)14 b(\014)3424 2254 y Fr(2)3484 2242 y Fs(<)22 b Fq(1)p Fo(,)260 2367 y Fs(\026)k(>)f Ft(1)p Fs(=)p Ft(2)p Fo(,)30 b(and)i(set)f Fs(\013)26 b Ft(=)g(\()p Fs(\026)19 b Fq(\000)h Ft(1)p Fs(=)p Ft(2\))p Fs(=)p Ft(\()p Fs(\026)e Ft(+)h(1)p Fs(=)p 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Fr(0)1564 5067 y Ft(\)\))23 b(=)g Fs(O)r Ft(\()p Fq(j)p Fs(g)s Fq(j)1925 5033 y Fr(2+)p Fn(\013)2057 5067 y Ft(\))g Fs(<)-28 b(<)23 b Ft(min\()p Fs(g)2450 5033 y Fr(2)2487 5067 y Fs(\016)2524 5079 y Fr(0)2561 5067 y Fs(;)14 b(\034)2643 5033 y Fk(0)2667 5067 y Ft(\))1928 5476 y(22)p eop %%Page: 23 23 23 22 bop 260 210 a Ft(\(see)26 b(also)g(\(7.1\)\))h(in)g(order)e(to)h (b)r(e)h(able)g(to)f(trace)g(the)h(simple)g(isolated)f(eigen)n(v)-5 b(alue)26 b Fs(z)2966 222 y Fr(0)3029 210 y Ft(\(c.f.)37 b(\(6.26\),)27 b(in)g(the)260 335 y(setting)33 b(of)g(Theorem)f(7.1,)h (where)g Fq(j)p Fs(g)s Fq(j)1492 305 y Fr(2+)p Fn(\013)1656 335 y Ft(represen)n(ts)e(the)j(error)d(term)h Fs(O)r Ft(\()p Fs(\017)p Ft(\()p Fs(g)s(;)14 b(\032)2899 347 y Fr(0)2937 335 y Ft(\)\))34 b(in)f(\(6.15\)\).)52 b(If)34 b(this)260 459 y(condition)d(fails)g(then)h(w)n(e)f(use)g(the)h(F)-7 b(esh)n(bac)n(h)30 b(map)i(iterativ)n(ely)e(un)n(til)i(the)f(error)f (term)h(in)h(the)g(equation)260 584 y(for)41 b(the)g(\014nal)h 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4263 y Fn(t)450 4314 y Fr(0)488 4294 y Ft(\()p Fs(\031)s Ft(\()p Fs(A)p Ft(\)\))c(:=)f Fs(\031)s Ft(\()p Fs(\013)998 4263 y Fn(t)998 4314 y Fr(0)1036 4294 y Ft(\()p Fs(A)p Ft(\)\))29 b(and)e(let)h Fs( )j Ft(b)r(e)d(an)f Fs(!)1869 4306 y Fr(0)1906 4294 y Ft(-normal)f(state)i(on)f Fq(A)p Ft(,)h(i.e.)1637 4501 y Fs( )s Ft(\()p Fs(A)p Ft(\))c(=)f(T)-7 b(r\()p Fs(\032\031)s Ft(\()p Fs(A)p Ft(\)\))1187 b(\(A.3\))260 4709 y(for)25 b(some)h(p)r(ositiv)n(e,)g(trace)g(class)f(op)r(erator)f Fs(\032)i Ft(on)g Fq(H)h Ft(of)g(trace)e(1.)36 b(Then)26 b(using)g(the)h(de\014nition)f Fs(V)3323 4721 y Fn(n)3392 4709 y Ft(=)d Fs(\031)s Ft(\()p Fs(v)3602 4721 y Fn(n)3648 4709 y Ft(\))260 4833 y(w)n(e)k(\014nd)476 5041 y Fs( )s Ft(\([)p Fs(\013)641 5004 y Fn(t)666 5012 y Fl(m)641 5063 y Fr(0)726 5041 y Ft(\()p Fs(v)798 5053 y Fn(n)844 5041 y Ft(\))p Fs(;)14 b Fq(\001)g(\001)g(\001)g Ft([)p Fs(\013)1100 5004 y Fn(t)1125 5012 y Fm(1)1100 5063 y Fr(0)1162 5041 y Ft(\()p Fs(v)1234 5053 y Fn(n)1280 5041 y Ft(\))p Fs(;)g(\013)1402 5006 y Fn(t)1402 5061 y Fr(0)1439 5041 y Ft(\()p Fs(A)p Ft(\)])g Fq(\001)g(\001)g(\001)h Ft(]\))23 b(=)g(T)-7 b(r)o(\()p Fs(\032)p Ft([)p Fs(\033)2113 5004 y Fn(t)2138 5012 y Fl(m)2110 5063 y Fr(0)2198 5041 y Ft(\()p Fs(V)2278 5053 y Fn(n)2324 5041 y Ft(\))p Fs(;)14 b Fq(\001)g(\001)g(\001)g Ft([)p Fs(\033)2577 5004 y Fn(t)2602 5012 y Fm(1)2574 5063 y Fr(0)2639 5041 y Ft(\()p Fs(V)2719 5053 y Fn(n)2765 5041 y Ft(\))p Fs(;)g(\033)2884 5006 y Fn(t)2881 5061 y Fr(0)2919 5041 y Ft(\()p Fs(A)p Ft(\)])g Fq(\001)g(\001)g(\001)h Ft(]\))p Fs(:)216 b Ft(\(A.4\))1928 5476 y(23)p eop %%Page: 24 24 24 23 bop 260 210 a Ft(Clearly)19 b(the)i(r.h.s.)34 b(is)21 b(con)n(tin)n(uous)e(in)i Fs(t)1497 222 y Fr(1)1534 210 y Fs(;)14 b Fq(\001)g(\001)g(\001)28 b Fs(;)14 b(t)1763 222 y Fn(m)1846 210 y Ft(and)21 b(therefore)e(the)i(in)n(tegrals)e(in)i (\(2.12\))f(are)f(w)n(ell)i(de\014ned)260 335 y(and,)35 b(b)n(y)e(a)h(standard)e(estimate,)j(the)f(series)f(on)g(the)h(r.h.s.) 55 b(of)34 b(\(2.12\))f(con)n(v)n(erges)e(absolutely)-7 b(.)54 b(In)34 b(fact,)260 459 y(using)27 b(the)h(Araki-Dyson)e(series) 809 687 y Fs(\033)859 653 y Fn(t)856 710 y Fr(\()p Fn(n)p Fr(\))953 687 y Ft(\()p Fs(\031)s Ft(\()p Fs(A)p Ft(\)\))85 b(=)1464 583 y Fk(1)1437 608 y Fp(X)1426 784 y Fn(m)p Fr(=0)1569 687 y Ft(\()p Fs(ig)s Ft(\))1705 653 y Fn(m)1781 574 y Fp(Z)1864 595 y Fn(t)1827 763 y Fr(0)1907 687 y Fs(dt)1980 699 y Fr(1)2032 687 y Fq(\001)14 b(\001)g(\001)2142 574 y Fp(Z)2225 595 y Fn(t)2250 603 y Fl(m)p Fd(\000)p Fm(1)2188 763 y Fr(0)2397 687 y Fs(dt)2470 699 y Fn(m)2561 687 y Ft([)p Fs(\033)2634 650 y Fn(t)2659 658 y Fl(m)2631 709 y Fr(0)2718 687 y Ft(\()p Fs(\031)s Ft(\()p Fs(v)2872 699 y Fn(n)2919 687 y Ft(\)\))p Fs(;)g Fq(\001)g(\001)g(\001)1426 898 y Ft([)p Fs(\033)1499 861 y Fn(t)1524 869 y Fm(1)1496 920 y Fr(0)1561 898 y Ft(\()p Fs(\031)s Ft(\()p Fs(v)1715 910 y Fn(n)1761 898 y Ft(\)\))p Fs(;)g(\033)1912 864 y Fn(t)1909 919 y Fr(0)1947 898 y Ft(\()p Fs(\031)s Ft(\()p Fs(A)p Ft(\)\)])g Fq(\001)g(\001)g(\001)h Ft(])p Fs(;)1106 b Ft(\(A.5\))260 1089 y(one)57 b(can)h(easily)f(see)g(that)h(this)h (series)d(is)i(nothing)g(but)g(the)g(Araki-Dyson)f(expansion)g(of)h (the)260 1214 y(function)53 b(T)-7 b(r\()p Fs(\032\033)821 1183 y Fn(t)818 1240 y Fr(\()p Fn(n)p Fr(\))916 1214 y Ft(\()p Fs(\031)s Ft(\()p Fs(A)p Ft(\)\)\).)114 b(Th)n(us)53 b(w)n(e)f(ha)n(v)n(e)g(sho)n(wn)g(that)h(the)g(r.h.s.)112 b(of)53 b(\(2.12\))f(is)h(equal)f(to)260 1338 y(lim)375 1350 y Fn(n)p Fk(!1)567 1338 y Ft(T)-7 b(r)o(\()p Fs(\032\033)777 1308 y Fn(t)774 1365 y Fr(\()p Fn(n)p Fr(\))872 1338 y Ft(\()p Fs(\031)s Ft(\()p Fs(A)p Ft(\)\)\).)384 1463 y(No)n(w,)34 b Fs(V)651 1475 y Fn(n)729 1463 y Ft(con)n(v)n(erges)29 b(to)k Fs(V)51 b Ft(strongly)31 b(on)h(the)h(dense)f(set)h(Span)p Fq(f)p Fs(\031)s Ft(\()p Fs(B)25 b Fq(\012)d Fs(W)2828 1475 y Fr(1)2865 1463 y Ft(\()p Fs(f)2938 1475 y Fr(1)2975 1463 y Ft(\))g Fq(\012)g Fs(W)3194 1475 y Fr(2)3231 1463 y Ft(\()p Fs(f)3304 1475 y Fr(2)3341 1463 y Ft(\)\)\012)3465 1475 y Fr(0)3503 1463 y Fq(j)p Fs(B)35 b Fq(2)260 1587 y(B)s Ft(\()p Fq(H)420 1599 y Fr(0)457 1587 y Ft(\))p Fs(;)14 b(f)567 1599 y Fr(1)p Fn(;)p Fr(2)679 1587 y Fq(2)24 b Fs(L)815 1557 y Fr(2)815 1608 y(0)852 1587 y Fq(g)j Ft(as)g(follo)n(ws)f(from)i(\(2.13\))e(and)i(the)g(relation) 1071 1778 y Fq(k)p Ft(\()p Fs(V)1193 1790 y Fn(n)1256 1778 y Fq(\000)18 b Fs(V)h Ft(\))p Fs(\031)s Ft(\()p Fs(A)p Ft(\)\012)1674 1790 y Fr(0)1713 1778 y Fq(k)1755 1744 y Fr(2)1815 1778 y Ft(=)j Fs(!)1954 1790 y Fr(0)1991 1778 y Ft(\()p Fs(A)2085 1744 y Fk(\003)2124 1778 y Ft(\()p Fs(v)2199 1744 y Fk(\003)2196 1798 y Fn(n)2260 1778 y Fq(\000)c Fs(v)2386 1744 y Fk(\003)2425 1778 y Ft(\)\()p Fs(v)2529 1790 y Fn(n)2593 1778 y Fq(\000)g Fs(v)s Ft(\))p Fs(A)p Ft(\))p Fs(:)620 b Ft(\(A.6\))260 1969 y(Hence)27 b Fs(L)563 1926 y Fr(\()p Fn(`)p Fr(\))563 1978 y Fn(n)673 1969 y Ft(con)n(v)n(erges)c(to)k Fs(L)1201 1939 y Fr(\()p Fn(`)p Fr(\))1311 1969 y Ft(strongly)e(on)h(the)h(same)f(set.)37 b(Since)27 b(this)f(set)h(is)f(a)h(core)e(for)h Fs(L)3269 1926 y Fr(\()p Fn(`)p Fr(\))3269 1978 y Fn(n)3379 1969 y Ft(and)h Fs(L)3597 1939 y Fr(\()p Fn(`)p Fr(\))260 2093 y Ft(w)n(e)h(conclude)g(that)h Fs(L)963 2050 y Fr(\()p Fn(`)p Fr(\))963 2103 y Fn(n)1075 2093 y Ft(con)n(v)n(erge)d(to)i Fs(L)1574 2063 y Fr(\()p Fn(`)p Fr(\))1686 2093 y Ft(in)h(the)g(strong) e(resolv)n(en)n(t)g(sense)h(as)g Fs(n)c Fq(!)h(1)j Ft(\([19],)h (Theorem)260 2218 y(VI)r(I)r(I.25\),)d(and)f(therefore,)g Fs(e)1172 2188 y Fn(itL)1266 2163 y Fm(\()p Fl(`)p Fm(\))1266 2204 y Fl(n)1367 2218 y Fq(!)e Fs(e)1512 2188 y Fn(itL)1606 2163 y Fm(\()p Fl(`)p Fm(\))1709 2218 y Ft(strongly)-7 b(.)35 b(Hence)26 b(the)f(functions)h(T)-7 b(r)o(\()p Fs(\032\033)3006 2188 y Fn(t)3003 2244 y Fr(\()p Fn(n)p Fr(\))3101 2218 y Ft(\()p Fs(\031)s Ft(\()p Fs(A)p Ft(\)\))27 b(con)n(v)n(erge)260 2342 y(to)g(T)-7 b(r\()p Fs(\032\033)572 2312 y Fn(t)602 2342 y Ft(\()p Fs(\031)s Ft(\()p Fs(A)p Ft(\)\)\))29 b(whic)n(h,)f(in)g(particular,)e(sho)n(ws)h(\(A.1\).)p 3613 2354 59 71 v 260 2614 a Fu(B)134 b(P)l(ositiv)l(e)97 b(T)-11 b(emp)t(erature)96 b(Represen)l(tation)i(and)d(Relativ)l(e)490 2763 y(Bounds)260 2962 y Fa(B.1)112 b(Jak)-6 b(\025)-50 b(si)m(\023)d(c-Pillet)34 b(Gluing)260 3140 y Ft(In)22 b(this)h(app)r(endix,)h(w)n(e)e(represen)n(t)f(the)i(Hilb)r(ert)g (space)e Fq(H)j Ft(in)e(a)g(form)g(whic)n(h)h(is)f(w)n(ell)g(suited)h (for)f(a)g(de\014nition)260 3265 y(of)27 b(the)h(translation)f (transformation.)35 b(This)28 b(represen)n(tation)d(is)j(due)g(to)f ([12)o(].)384 3389 y(Consider)g(the)h(F)-7 b(o)r(c)n(k)27 b(space)1254 3580 y Fq(F)k Ft(:=)22 b Fq(F)8 b Ft(\()p Fs(L)1612 3546 y Fr(2)1649 3580 y Ft(\()p Fs(X)25 b Fq(\002)18 b(f)p Ft(1)p Fs(;)c Ft(2)p Fq(g)p Ft(\)\))p Fs(;)96 b(X)29 b Ft(=)23 b Fh(R)h Fq(\002)18 b Fs(S)2649 3546 y Fr(2)3427 3580 y Ft(\(B.1.1\))260 3771 y(and)33 b(denote)f Fs(x)h Ft(=)e(\()p Fs(u;)14 b(\033)s Ft(\))32 b Fq(2)h Fs(X)7 b Ft(.)52 b(The)33 b(v)-5 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y Fd(0)2266 4530 y Fs(\016)s Ft(\()p Fs(x)d Fq(\000)e Fs(x)2535 4496 y Fk(0)2558 4530 y Ft(\))p Fs(:)260 4721 y Ft(F)-7 b(ollo)n(wing)26 b([12)o(],)i(w)n(e)f (in)n(tro)r(duce)h(the)g(unitary)f(map)379 4912 y Fs(U)32 b Ft(:)514 4845 y Fp(\002)548 4912 y Fq(F)8 b Ft(\()p Fs(L)705 4878 y Fr(2)742 4912 y Ft(\()p Fh(R)829 4878 y Fr(3)872 4912 y Ft(\)\))19 b Fq(\012)f(F)8 b Ft(\()p Fs(L)1195 4878 y Fr(2)1232 4912 y Ft(\()p Fh(R)1318 4878 y Fr(3)1361 4912 y Ft(\)\))1425 4845 y Fp(\003)1479 4912 y Fq(\012)1562 4845 y Fp(\002)1597 4912 y Fq(F)g Ft(\()p Fs(L)1754 4878 y Fr(2)1790 4912 y Ft(\()p Fh(R)1877 4878 y Fr(3)1920 4912 y Ft(\)\))19 b Fq(\012)f(F)8 b Ft(\()p Fs(L)2243 4878 y Fr(2)2280 4912 y Ft(\()p Fh(R)2366 4878 y Fr(3)2409 4912 y Ft(\)\))2473 4845 y Fp(\003)2532 4912 y Fq(!)23 b(F)8 b Ft(\()p Fs(L)2795 4878 y Fr(2)2832 4912 y Ft(\()p Fs(X)25 b Fq(\002)18 b(f)p Ft(1)p Fs(;)c Ft(2)p Fq(g)p Ft(\)\))117 b(\(B.1.2\))260 5103 y(de\014ned)28 b(b)n(y)1325 5227 y Fs(U)22 b Ft(\([\012)1519 5239 y Fn(r)r Fr(1)1608 5227 y Fq(\012)c Ft(\012)1751 5239 y Fn(r)r Fr(1)1821 5227 y Ft(])g Fq(\012)g Ft([\012)2028 5239 y Fn(r)r Fr(2)2116 5227 y Fq(\012)g Ft(\012)2259 5239 y Fn(r)r Fr(2)2329 5227 y Ft(]\))24 b(:=)2528 5206 y(~)2518 5227 y(\012)2578 5239 y Fn(r)3427 5227 y Ft(\(B.1.3\))1928 5476 y(24)p eop %%Page: 25 25 25 24 bop 260 210 a Ft(and)866 407 y Fs(U)932 315 y Fp(\020)995 407 y Ft([)p Fs(a)1062 373 y Fk(\003)1100 407 y Ft(\()p Fs(f)1173 419 y Fr(1)1210 407 y Ft(\))19 b Fq(\012)f Fg(1)g Ft(+)g Fg(1)h Fq(\012)f Fs(a)1687 373 y Fk(\003)1725 407 y Ft(\()p Fs(g)1797 419 y Fr(1)1834 407 y Ft(\)])h Fq(\012)f Fg(1)g Fq(\012)g Fg(1)981 590 y Ft(+)p Fg(1)g Fq(\012)g Fg(1)g Fq(\012)g Ft([)p Fs(a)1411 555 y Fk(\003)1449 590 y Ft(\()p Fs(f)1522 602 y Fr(2)1560 590 y Ft(\))g Fq(\012)g Fg(1)h Ft(+)f Fg(1)g Fq(\012)g Fs(a)2036 555 y Fk(\003)2074 590 y Ft(\()p Fs(g)2146 602 y Fr(2)2183 590 y Ft(\)])2252 497 y Fp(\021)2302 590 y Fs(U)2368 555 y Fk(\000)p Fr(1)2540 590 y Ft(:=)82 b Fs(a)2754 555 y Fk(\003)2792 590 y Ft(\()p Fs(f)28 b Fq(\010)18 b Fs(g)s Ft(\))p Fs(;)353 b Ft(\(B.1.4\))260 786 y(where,)27 b(for)g Fs(x)d Ft(=)e(\()p Fs(u;)14 b(\033)s Ft(\))24 b Fq(2)f Fs(X)7 b Ft(,)1243 1033 y([)p Fs(f)27 b Fq(\010)18 b Fs(g)s Ft(])c(\()p Fs(u;)g(\033)n(;)g(\013)p Ft(\))24 b(:=)1916 891 y Fp(\()1982 976 y Fs(u)14 b(f)2085 988 y Fn(\013)2132 976 y Ft(\()p Fs(u\033)s Ft(\))p Fs(;)147 b(u)22 b Fq(\025)h Ft(0)p Fs(;)1982 1096 y(u)14 b(g)2084 1108 y Fn(\013)2131 1096 y Ft(\()p Fq(\000)p Fs(u\033)s Ft(\))p Fs(;)83 b(u)22 b(<)h Ft(0)p Fs(:)3427 1033 y Ft(\(B.1.5\))260 1274 y(This)j(map)h(is)f(extended)h(to)g(the)f(Hilb)r (ert)i(space)d Fq(H)f Ft(=)f Fq(H)2067 1244 y Fn(p)2122 1274 y Fq(\012)16 b(F)34 b Ft(in)27 b(the)g(ob)n(vious)e(w)n(a)n(y)-7 b(.)36 b(W)-7 b(e)26 b(k)n(eep)h(the)f(same)260 1399 y(notation)h(for)g(its)h(extension.)384 1524 y(The)e(op)r(erators)e Fs(L)976 1536 y Fn(r)r Fr(1)1060 1524 y Fq(\012)14 b Fg(1)1187 1536 y Fn(r)r Fr(2)1271 1524 y Ft(+)g Fg(1)1398 1536 y Fn(r)r 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Ft(\))24 b(=)f Fs(e)1912 5171 y Fm(1)p 1911 5180 29 3 v 1911 5213 a(2)1950 5193 y Fn(\016)r Fr(sgn\()p Fn(u)p Fr(\))2173 5227 y Fs(F)12 b Ft(\()p Fs(j)2304 5239 y Fn(\022)2342 5227 y Ft(\()p Fs(u)p Ft(\))p Fs(;)i(\033)n(;)g(\013)p Ft(\))p Fs(;)746 b Ft(\(B.2.1\))1928 5476 y(25)p eop %%Page: 26 26 26 25 bop 260 210 a Ft(where)1585 335 y Fs(j)1619 347 y Fn(\022)1657 335 y Ft(\()p Fs(u)p Ft(\))23 b(=)f Fs(e)1918 300 y Fn(\016)r Fr(sgn\()p Fn(u)p Fr(\))2142 335 y Fs(u)c Ft(+)g Fs(\034)5 b(;)1072 b Ft(\(B.2.2\))260 485 y(and)38 b(sgn)f(is)h(the)g(sign)g(function,)j(sgn)o(\()p Fs(u)p Ft(\))f(=)g(1)e(if)g Fs(u)i Fq(\025)g Ft(0,)g(sgn)o(\()p Fq(\000)p Fs(u)p Ft(\))g(=)g Fq(\000)p Ft(sgn)o(\()p Fs(u)p Ft(\).)68 b(Next,)41 b(w)n(e)d(lift)h(the)260 610 y(op)r(erator)24 b(family)32 b(~)-48 b Fs(u)893 622 y Fn(\022)957 610 y Ft(from)25 b Fs(L)1208 580 y Fr(2)1245 610 y Ft(\()p Fs(X)d Fq(\002)15 b(f)p Ft(1)p Fs(;)f Ft(2)p Fq(g)p Ft(\))25 b(to)h(the)g(op)r(erator)e(family)-7 b(,)2567 589 y(~)2553 610 y Fs(U)2610 622 y Fn(\022)2647 610 y Ft(,)27 b(on)f Fq(H)2882 580 y Fn(p)2935 610 y Fq(\012)15 b(F)8 b Ft(\()p Fs(L)3172 580 y Fr(2)3209 610 y Ft(\()p Fs(X)22 b Fq(\002)15 b(f)p Ft(1)p Fs(;)f Ft(2)p Fq(g)p Ft(\)\))260 734 y(in)25 b(a)f(standard)g(w)n(a)n(y)f (\(cf.)37 b(\(5.9\)\).)e(The)25 b(family)1777 713 y(~)1763 734 y Fs(U)1820 746 y Fn(\022)1882 734 y Ft(is)f(related)g(to)h(the)g (family)g Fs(U)2783 746 y Fn(\022)2844 734 y Ft(in)n(tro)r(duced)g(in)g (Section)f(5)260 859 y(as)1698 983 y Fs(U)1755 995 y Fn(\022)1816 983 y Ft(=)e Fs(U)1984 962 y Ft(~)1969 983 y Fs(U)2026 995 y Fn(\022)2063 983 y Fs(U)2129 949 y Fk(\000)p Fr(1)2218 983 y Fs(:)384 1134 y Ft(The)28 b(op)r(erator)912 1113 y(~)890 1134 y Fs(K)33 b Ft(b)r(ecomes)27 b(after)h(sp)r(ectral)f (deformation)1470 1281 y(~)1448 1302 y Fs(K)1519 1314 y Fn(\022)1579 1302 y Ft(:=)1705 1281 y(~)1690 1302 y Fs(U)1747 1314 y Fn(\022)1784 1302 y Fs(K)1875 1281 y Ft(~)1861 1302 y Fs(U)1927 1266 y Fk(\000)p Fr(1)1918 1327 y Fn(\022)2039 1302 y Ft(=)2136 1281 y(~)2126 1302 y Fs(L)2183 1314 y Fr(0)p Fn(;\022)2292 1302 y Ft(+)18 b Fs(g)2428 1281 y Ft(~)2418 1302 y Fs(I)2454 1314 y Fn(\022)3427 1302 y Ft(\(B.2.3\))260 1470 y(where)1069 1616 y(~)1059 1637 y Fs(L)1116 1649 y Fr(0)p Fn(;\022)1289 1637 y Ft(=)82 b Fs(L)1493 1649 y Fn(p)1550 1637 y Ft(+)18 b(cosh)13 b Fs(\016)31 b(L)1929 1649 y Fn(f)1990 1637 y Ft(+)18 b(sinh)13 b Fs(\016)31 b Ft(\003)2360 1649 y Fn(f)2421 1637 y Ft(+)18 b Fs(\034)9 b(N)t(;)784 b Ft(\(B.2.4\))1148 1807 y(\003)83 b(=)f(d\000\()p Fq(j)p Fs(u)p Fq(j)p Ft(\))24 b(=)1803 1728 y Fp(X)1842 1903 y Fn(\013)1937 1694 y Fp(Z)1983 1883 y Fn(X)2060 1807 y Fs(a)2104 1773 y Fk(\003)2142 1807 y Ft(\()p Fs(x;)14 b(\013)p Ft(\))p Fq(j)p Fs(u)p Fq(j)p Fs(a)p Ft(\()p Fs(x;)g(\013)p Ft(\))p Fs(;)1142 2000 y Ft(~)1132 2021 y Fs(I)1168 2033 y Fn(\022)1289 2021 y Ft(=)82 b Fs(a)1480 1986 y Fk(\003)1518 2021 y Ft(\()p Fs(F)1603 2033 y Fr(1)p Fn(;\022)1695 2021 y Ft(\))18 b(+)g Fs(a)p Ft(\()p Fs(F)1957 2033 y Fr(2)p 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Fq(j)p Fs(u)p Fq(j)18 b Ft(+)g(1)28 b Fs(G)2445 1428 y Fn(\013k)2529 1324 y Fp(i)2582 1416 y Ft(\()p Fs(j)2648 1428 y Fn(\022)2686 1416 y Ft(\()p Fs(u)p Ft(\))p Fs(;)14 b(\033)s Ft(\))2917 1320 y Fp(\015)2917 1370 y(\015)2917 1420 y(\015)2964 1341 y Fr(2)3015 1416 y Fs(;)260 1642 y Ft(where)27 b(w)n(e)g(recall)g (that)h Fs(\015)k Ft(w)n(as)27 b(de\014ned)g(in)h(\(5.5\).)37 b(Estimate)27 b(\(B.3.12\))g(implies)581 1915 y Fq(jjj)p Fs(F)703 1927 y Fr(1)740 1915 y Fq(jjj)809 1881 y Fr(2)809 1936 y Fn(\027)934 1915 y Fq(\024)83 b Fs(C)1185 1836 y Fp(X)1161 2013 y Fn(j)s Fr(=1)p Fn(;)p Fr(2)1369 1836 y Fp(X)1342 2015 y Fn(k)q Fr(=1)p Fn(;)p Fr(3)1516 1915 y Ft(\(1)18 b(+)g(1)p Fs(=\014)1822 1927 y Fn(j)1856 1915 y Ft(\))1902 1802 y Fp(Z)1948 1991 y Ff(R)p Fk(\002)p Fn(S)2091 1974 y Fm(2)2136 1915 y Fs(dud\033)2335 1770 y Fp(\015)2335 1820 y(\015)2335 1869 y(\015)2335 1919 y(\015)2335 1969 y(\015)2381 1915 y Fs(\015)2424 1927 y Fn(\022)2475 1773 y Fp(")2534 1786 y(p)p 2617 1786 V 71 x Fq(j)p Fs(u)p Fq(j)g Ft(+)g(1)p 2534 1896 320 4 v 2626 1972 a Fq(j)p Fs(u)p Fq(j)2720 1948 y Fn(\027)2863 1915 y Fs(G)2928 1927 y Fn(j)s(k)3000 1773 y Fp(#)3062 1915 y Ft(\()p Fs(u;)c(\033)s Ft(\))3261 1770 y Fp(\015)3261 1820 y(\015)3261 1869 y(\015)3261 1919 y(\015)3261 1969 y(\015)3308 1790 y Fr(2)934 2155 y Fq(\024)83 b Fs(C)1185 2076 y Fp(X)1161 2253 y Fn(j)s Fr(=1)p Fn(;)p Fr(2)1328 2155 y Ft(\(1)19 b(+)f(1)p Fs(=\014)1635 2167 y Fn(j)1669 2155 y Ft(\))p Fq(k)p Fs(G)1808 2167 y Fn(j)1843 2155 y Fq(k)1885 2121 y Fr(2)1885 2176 y Fn(\027;\022)1975 2155 y Fs(;)1388 b Ft(\(B.3.13\))260 2430 y(where)27 b Fq(k)p Fs(G)607 2442 y Fn(j)642 2430 y Fq(k)684 2442 y Fn(\027;\022)802 2430 y Ft(is)g(giv)n(en)g(in)h(\(5.8\).)36 b(The)28 b(same)f(b)r(ound)h(is)g(obtained)f(for)g Fq(jjj)p Fs(F)2738 2442 y Fr(2)2776 2430 y Fq(jjj)2845 2400 y Fr(2)2845 2451 y Fn(\027)2886 2430 y Ft(.)260 2705 y Fu(C)134 b(Lev)l(el)46 b(Shift)f(Op)t(erator)260 2911 y Ft(W)-7 b(e)28 b(pro)n(v)n(e)d(estimate)j(\(6.22\).)36 b(W)-7 b(e)28 b(pass)e(to)i(the)g(Jak)l(\025)-37 b(si)n(\023)e (c-Pillet)23 b(glued)28 b(Hilb)r(ert)g(space)e(represen)n(tation)g (\(see)260 3036 y(App)r(endices)i(B.1)f(and)h(B.2\))f(and)g(omit)h(the) g(tilde)g(o)n(v)n(er)e(the)i(op)r(erators.)35 b(In)28 b(the)g(de\014nition)1417 3242 y(\003)1475 3254 y Fn(e\032\022)1601 3242 y Ft(:=)23 b Fs(P)1765 3254 y Fn(e\032)1835 3242 y Fs(I)1871 3254 y Fn(\022)p 1910 3175 65 4 v 1910 3242 a Fs(P)1974 3254 y Fn(e\032)2044 3242 y Fs(L)2101 3207 y Fk(\000)p Fr(1)2101 3267 y(0)p Fn(\022)p 2190 3175 V 2190 3242 a Fs(P)2255 3254 y Fn(e\032)2325 3242 y Fs(I)2361 3254 y Fn(\022)2399 3242 y Fs(P)2452 3254 y Fn(e\032)3491 3242 y Ft(\(C.1\))260 3448 y(w)n(e)k(substitute)h(expression)d (\(B.2.5\))i(for)g(the)g(op)r(erator)f Fs(I)2089 3460 y Fn(\022)2154 3448 y Ft(and,)h(using)g(the)h(pull-through)e(form)n (ulae,)g(pull)260 3573 y(the)31 b(annihilation)f(op)r(erators)e(to)i (the)h(righ)n(t)e(and)i(the)f(creation)f(op)r(erators)g(to)h(the)h (left)g(un)n(til)g(they)f(stand)260 3697 y(next)e(to)f(the)h(op)r (erators)e Fs(P)1112 3709 y Fn(e\032)1182 3697 y Ft(.)37 b(As)28 b(a)f(result)g(w)n(e)g(obtain)h(the)g(decomp)r(osition)1543 3904 y(\003)1601 3916 y Fn(e\032\022)1727 3904 y Ft(=)23 b(\003)1873 3870 y Fr(con)n(tracted)1873 3924 y Fn(e\032\022)2199 3904 y Ft(+)18 b Fs(R)29 b(;)1094 b Ft(\(C.2\))260 4110 y(where)32 b(\003)563 4080 y Fr(con)n(tracted)563 4134 y Fn(e\032\022)902 4110 y Ft(:=)f Fs(P)1074 4122 y Fn(e\032)1158 4043 y Fp(\012)1197 4110 y Fs(I)1233 4122 y Fn(\022)p 1271 4043 V 1271 4110 a Fs(P)1336 4122 y Fn(e\032)1406 4110 y Fs(L)1463 4075 y Fk(\000)p Fr(1)1463 4135 y(0)p Fn(\022)1552 4110 y Fs(I)1588 4122 y Fn(\022)1626 4043 y Fp(\013)1679 4110 y Fs(P)1732 4122 y Fn(e\032)1835 4110 y Ft(is)h(the)h(con)n(tracted)f(term)g(and)h(the)f(term)h Fs(R)g Ft(consists)f(of)260 4235 y(remaining)27 b(terms.)36 b(Here,)27 b(w)n(e)h(use)f(the)h(notation)1042 4441 y Fq(h)q Fs(I)1111 4453 y Fn(\022)1149 4441 y Fs(f)9 b Ft(\(\003)p Fs(;)14 b(L)1383 4453 y Fn(r)1418 4441 y Ft(\))p Fs(I)1486 4453 y Fn(\022)1525 4441 y Fq(i)23 b Ft(=)g Fq(h)p Fs(I)1736 4453 y Fn(\022)1774 4441 y Fs(f)9 b Ft(\(\003)18 b(+)g Fs(\025;)c(L)2157 4453 y Fn(r)2212 4441 y Ft(+)k Fs(`)p Ft(\))p Fs(I)2398 4453 y Fn(\022)2436 4441 y Fq(i)2468 4466 y Fr(\012)2534 4441 y Fq(j)2557 4453 y Fn(\025)p Fr(=\003)p Fn(;`)p Fr(=)p Fn(L)2837 4461 y Fl(r)2874 4441 y Fs(;)260 4647 y Ft(where)29 b Fq(h\001i)589 4672 y Fr(\012)667 4647 y Ft(=)d(T)-7 b(r)843 4659 y Fk(F)901 4647 y Ft(\()p Fq(\001)p Fs(P)1009 4659 y Fr(\012)1061 4647 y Ft(\),)30 b Fs(P)1199 4659 y Fr(\012)1281 4647 y Ft(is)f(the)h(pro)5 b(jection)28 b(on)n(to)h Fh(C)15 b Ft(\012)35 b(\(the)30 b(v)-5 b(acuum)30 b(sector)e(in)i Fq(F)8 b Ft(\),)30 b(and)f(where)g Fs(f)260 4772 y Ft(is)e(a)h(function)g(of)f(t)n(w)n(o)g(v)-5 b(ariables.)384 4896 y(The)26 b(remaining)e(terms,)h Fs(R)q Ft(,)g(are)g(estimated)g (using)f(\(B.3.4\))h(and)g(\(B.3.7\))g(and)g Fq(k)p Fs(P)2998 4908 y Fn(e\032)3068 4896 y Fs(L)3125 4861 y Fk(\000)p Fr(1)3125 4921 y(0)p Fn(\022)3213 4896 y Fs(P)3266 4908 y Fn(e\032)3337 4896 y Fq(k)d(\024)h Fs(c\032)3568 4866 y Fk(\000)p Fr(1)3657 4896 y Ft(.)260 5021 y(F)-7 b(or)27 b(instance)g(one)g(of)h(the)g(terms)f(app)r(earing)g(in)g Fs(R)i Ft(is)e(of)h(the)g(form)1393 5227 y Fs(P)1446 5239 y Fn(e\032)1516 5227 y Fs(a)1560 5193 y Fk(\003)1598 5227 y Ft(\()p Fs(F)1683 5239 y Fn(i\022)1745 5227 y Ft(\))p 1777 5160 V Fs(P)1842 5239 y Fn(e\032)1912 5227 y Fs(L)1969 5192 y Fk(\000)p Fr(1)1969 5252 y(0)p Fn(\022)p 2058 5160 V 2058 5227 a Fs(P)2123 5239 y Fn(e\032)2193 5227 y Fs(a)p Ft(\()p Fs(F)2322 5239 y Fn(j)s(\022)2391 5227 y Ft(\))p Fs(P)2476 5239 y Fn(e\032)3491 5227 y Ft(\(C.3\))1928 5476 y(28)p eop %%Page: 29 29 29 28 bop 260 210 a Ft(whic)n(h)27 b(is)h(b)r(ounded)g(b)n(y)f(\(see)h (\(B.3.4\))f(and)g(\(B.3.7\)\))535 417 y Fq(k)p Fs(P)630 429 y Fn(e\032)700 417 y Fs(a)744 383 y Fk(\003)782 417 y Ft(\()p Fs(F)867 429 y Fn(i\022)929 417 y Ft(\))p Fq(k)g(k)p 1072 350 65 4 v Fs(P)1137 429 y Fn(e\032)1207 417 y Fs(L)1264 381 y Fk(\000)p Fr(1)1264 442 y(0)p Fn(\022)p 1352 350 V 1352 417 a Fs(P)1417 429 y Fn(e\032)1487 417 y Fq(k)g(k)p Fs(a)p Ft(\()p Fs(F)1727 429 y Fn(j)s(\022)1796 417 y Ft(\))p Fs(P)1881 429 y Fn(e\032)1951 417 y Fq(k)618 619 y(\024)766 502 y Fp(\022)937 563 y Fs(c)p 837 600 236 4 v 837 676 a(\032)14 b Ft(sin)f Fs(\016)1049 652 y Fk(0)1083 502 y Fp(\023)1144 519 y Fr(1)p Fn(=)p Fr(2)1262 619 y Fq(k)p Fs(F)1357 631 y Fn(i)1384 619 y Fq(k)1426 631 y Fn(p)1492 619 y Fs(c\032)1571 584 y Fk(\000)p Fr(1)1674 502 y Fp(\022)1845 563 y Fs(c)p 1745 600 V 1745 676 a(\032)h Ft(sin)f Fs(\016)1957 652 y Fk(0)1991 502 y Fp(\023)2052 519 y Fr(1)p Fn(=)p Fr(2)2170 619 y Fq(k)p Fs(F)2265 631 y Fn(j)2300 619 y Fq(k)2342 631 y Fn(\032)618 874 y Fq(\024)766 757 y Fp(\022)937 818 y Fs(c)p 837 855 V 837 931 a(\032)h Ft(sin)f Fs(\016)1049 907 y Fk(0)1083 757 y Fp(\023)1144 774 y Fr(1)p Fn(=)p Fr(2)1262 782 y Fp(\020)1393 818 y Fs(c)p 1321 855 180 4 v 1321 931 a Ft(sin)h Fs(\016)1477 907 y Fk(0)1510 782 y Fp(\021)1560 799 y Fn(\026)p Fk(\000)p Fr(1)p Fn(=)p Fr(2)1770 874 y Fq(k)p Fs(F)1865 886 y Fn(i)1893 874 y Fq(k)1935 886 y Fn(\026)1979 874 y Fs(c\032)2058 840 y Fk(\000)p Fr(1)2161 757 y Fp(\022)2332 818 y Fs(c)p 2232 855 236 4 v 2232 931 a(\032)g Ft(sin)f Fs(\016)2444 907 y Fk(0)2478 757 y Fp(\023)2539 774 y Fr(1)p Fn(=)p Fr(2)2657 782 y Fp(\020)2788 818 y Fs(c)p 2717 855 180 4 v 2717 931 a Ft(sin)g Fs(\016)2872 907 y Fk(0)2906 782 y Fp(\021)2955 799 y Fn(\026)p Fk(\000)p Fr(1)p Fn(=)p Fr(2)3166 874 y Fq(k)p Fs(F)3261 886 y Fn(j)3296 874 y Fq(k)3338 886 y Fn(\026)3382 874 y Fs(:)260 1117 y Ft(Similarly)-7 b(,)25 b(w)n(e)g(estimate)g(other)f(terms)h(in)h Fs(R)f Ft(to)g(obtain)g Fs(R)f Ft(=)e Fs(O)r Ft(\()p Fs(g)2372 1087 y Fr(2)p Fn(\026)p Fk(\000)p Fr(1)2536 1117 y Ft(\).)36 b(No)n(w,)25 b(using)p 3052 1050 65 4 v 25 w Fs(P)3117 1129 y Fn(e\032)3210 1117 y Ft(=)e Fg(1)13 b Fq(\000)g Fs(P)3490 1129 y Fn(e\032)3585 1117 y Ft(w)n(e)260 1241 y(write)27 b(the)h(op)r(erator)e(\003)1008 1211 y Fr(con)n(tracted)1008 1265 y Fn(e\032\022)1344 1241 y Ft(as)1520 1448 y(\003)1578 1414 y Fr(con)n(tracted)1578 1469 y Fn(e\032\022)1909 1448 y Ft(=)c(\003)2054 1414 y Fk(0)2054 1469 y Fn(e\032\022)2176 1448 y Ft(+)c(\003)2317 1414 y Fk(00)2317 1469 y Fn(e\032\022)3491 1448 y Ft(\(C.4\))260 1655 y(where)27 b(\003)558 1625 y Fk(0)558 1679 y Fn(e\032\022)684 1655 y Ft(:=)c Fs(P)848 1667 y Fn(e\032)932 1588 y Fp(\012)971 1655 y Fs(I)1007 1667 y Fn(\022)1045 1655 y Fs(L)1102 1620 y Fk(\000)p Fr(1)1102 1680 y(0)p Fn(\022)1191 1655 y Fs(I)1227 1667 y Fn(\022)1265 1588 y Fp(\013)1318 1655 y Fs(P)1371 1667 y Fn(e\032)1469 1655 y Ft(and)1391 1862 y(\003)1449 1828 y Fk(00)1449 1882 y Fn(e\032\022)1575 1862 y Ft(=)g Fq(\000)p Fs(P)1781 1874 y Fn(e\032)1865 1795 y Fp(\012)1904 1862 y Fs(I)1940 1874 y Fn(\022)1978 1862 y Fs(P)2031 1874 y Fn(e\032)2102 1862 y Fs(L)2159 1826 y Fk(\000)p Fr(1)2159 1887 y(0)p Fn(\022)2247 1862 y Fs(I)2283 1874 y Fn(\022)2321 1795 y Fp(\013)2374 1862 y Fs(P)2427 1874 y Fn(e\032)2525 1862 y Fs(:)943 b Ft(\(C.5\))260 2069 y(Note)31 b(that)h(b)r(oth)g(terms)f(on)g(the)h(r.h.s.)48 b(of)32 b(\(C.4\))g(are)e(w)n(ell-de\014ned)h(since)h Fs(I)2742 2081 y Fn(\022)2780 2069 y Ft(\()p Fs( )24 b Fq(\012)d Ft(\012\))31 b(is)h(orthogonal)d(to)260 2193 y(Null\()p Fs(L)503 2205 y Fr(0)p Fn(\022)574 2193 y Ft(\),)i(for)f(all)g Fs( )g Fq(2)e(H)1145 2205 y Fn(p)1203 2193 y Fq(\012)20 b(H)1358 2205 y Fn(p)1397 2193 y Ft(.)44 b(A)31 b(simple)f(computation)g(sho)n(ws)f(that)i(\003)2787 2163 y Fk(00)2787 2217 y Fn(e\032\022)2920 2193 y Ft(is)f(equal)g(to)g Fs(P)3385 2205 y Fn(e\032)3486 2193 y Ft(times)260 2318 y(an)g(in)n(tegral)f(o)n(v)n(er)f Fs(!)i Fq(\024)d Fs(\032)k Ft(of)f(the)g(trace)g(of)g(the)h(pro)r(duct)f(of)g(t)n(w)n(o)f (coupling)h(functions)h Fs(F)3126 2330 y Fn(j)s(\022)3225 2318 y Ft(divided)f(b)n(y)g(a)260 2442 y(function)g(of)f(the)h(form)f Fq(\006)14 b Ft(cosh)f Fs(\016)s(!)22 b Ft(+)d(sinh)14 b Fs(\016)s(!)23 b Ft(+)c Fs(\034)39 b Ft(whic)n(h)30 b(is)f(b)r(ounded)h(b)r(elo)n(w)f(b)n(y)g Fs(c)14 b Ft(sin)g Fs(\016)3136 2412 y Fk(0)3159 2442 y Fs(!)s Ft(.)42 b(Hence)30 b(that)260 2579 y(in)n(tegral)c(is)i(b)r(ounded)g(b)n(y)f Fs(c\032)1179 2549 y Fr(2)p Fn(\026)p Fk(\000)p Fr(1)1355 2487 y Fp(\020)1405 2517 y(P)1492 2604 y Fn(j)1541 2579 y Fq(k)p Fs(G)1648 2591 y Fn(j)1683 2579 y Fq(k)1725 2591 y Fn(\026;\022)1822 2487 y Fp(\021)1872 2504 y Fr(2)1937 2579 y Ft(and,)g(consequen)n(tly)-7 b(,)27 b(\003)2687 2549 y Fk(00)2687 2603 y Fn(e\032\022)2813 2579 y Ft(=)c Fs(O)r Ft(\()p Fs(\032)3041 2549 y Fr(2)p Fn(\026)p Fk(\000)p Fr(1)3204 2579 y Ft(\))384 2714 y(A)33 b(simple)e(consideration)g(sho)n (ws)f(that)1680 2647 y Fp(\012)1720 2714 y Fs(I)1756 2726 y Fn(\022)1794 2714 y Fs(L)1851 2679 y Fk(\000)p Fr(1)1851 2739 y(0)p Fn(\022)1939 2714 y Fs(I)1975 2726 y Fn(\022)2013 2647 y Fp(\013)2084 2714 y Ft(is)i(indep)r(enden)n(t)h (of)f Fs(\022)r Ft(,)h(and)e(\003)3066 2684 y Fk(0)3066 2738 y Fn(e\032\022)3191 2714 y Fq(\000)21 b Ft(\003)3335 2726 y Fn(e)3370 2714 y Fs(P)3423 2726 y Fn(e\032)3525 2714 y Ft(is)32 b(of)260 2839 y(order)26 b Fs(O)r Ft(\()p Fs(\032)617 2809 y Fr(2)p Fn(\026)p Fk(\000)p Fr(1)781 2839 y Ft(\))h(as)g(w)n(ell.)37 b(Hence,)1482 3046 y(\003)1540 3058 y Fn(e\032\022)1666 3046 y Ft(=)22 b(\003)1811 3058 y Fn(e)1847 3046 y Fs(P)1900 3058 y Fn(e\032)1989 3046 y Ft(+)c Fs(O)r Ft(\()p Fs(\032)2212 3011 y Fr(2)p Fn(\026)p Fk(\000)p Fr(1)2375 3046 y Ft(\))28 b Fs(:)1033 b Ft(\(C.6\))260 3320 y Fu(References)301 3527 y Ft([1])42 b(H.)25 b(Araki)f(and)h(E.J.) f(W)-7 b(o)r(o)r(ds.)32 b(Represen)n(tation)24 b(of)h(the)g(canonical)f (comm)n(utation)g(relations)g(describ-)431 3651 y(ing)j(a)g (nonrelativistic)g(in\014nite)h(free)f(b)r(ose)h(gas.)35 b Fo(J.)30 b(Math.)h(Phys.)e Fg(4)f Ft(\(1963\),)e(637{662)301 3842 y([2])42 b(V.)24 b(Bac)n(h,)f(T.)h(Chen,)h(J.)e(F)-7 b(r\177)-42 b(ohlic)n(h,)24 b(and)g(I.M.)g(Sigal.)30 b(Smo)r(oth)24 b(F)-7 b(esh)n(bac)n(h)22 b(map)i(and)g(op)r(erator)e (theoric)431 3966 y(renormalization)j(group)h(metho)r(d.)38 b Fo(J.)29 b(F)-6 b(unc.)30 b(A)n(nalysis)e Fg(203)f Ft(\(2003\),)f(no.)i(1,)f(44{92)301 4157 y([3])42 b(V.)23 b(Bac)n(h,)f(J.)h(F)-7 b(r\177)-42 b(ohlic)n(h,)23 b(and)f(I.M.)h (Sigal.)28 b(Quan)n(tum)23 b(electro)r(dynamics)e(of)i(con\014ned)f (non-relativistic)431 4282 y(particles.)35 b Fo(A)l(dv.)c(in)f(Math.)f Fg(137)d Ft(\(1998\),)h(299{395)301 4472 y([4])42 b(V.)26 b(Bac)n(h,)g(J.)g(F)-7 b(r\177)-42 b(ohlic)n(h,)26 b(and)g(I.M.)g (Sigal.)34 b(Renormalization)25 b(group)g(analysis)g(of)h(sp)r(ectral)g (problems)431 4597 y(in)h(quan)n(tum)h(\014eld)g(theory)-7 b(.)36 b Fo(A)l(dv.)30 b(in)g(Math.)f Fg(137)e Ft(\(1998\),)f(205{298) 301 4787 y([5])42 b(V.)32 b(Bac)n(h,)h(J.)f(F)-7 b(r\177)-42 b(ohlic)n(h,)34 b(and)e(I.M.)h(Sigal.)50 b(Return)33 b(to)f(equilibrium.)51 b Fo(J.)35 b(Math.)g(Phys.)f Fg(41)e Ft(\(2000\),)431 4912 y(3985{4060)301 5103 y([6])42 b(O.)31 b(Bratteli)h(and)g(D.W.)h(Robinson.)50 b Fo(Op)l(er)l(ator)34 b(A)n(lgebr)l(as)g(and)h(Quantum)d(Statistic)l(al)i(Me)l(chanics)p Ft(,)431 5227 y(v)n(olume)26 b(1.)37 b(Springer)26 b(V)-7 b(erlag,)27 b(New)h(Y)-7 b(ork,)27 b(Heidelb)r(erg,)g(Berlin,)g(1979.) 1928 5476 y(29)p eop %%Page: 30 30 30 29 bop 301 210 a Ft([7])42 b(O.)31 b(Bratteli)h(and)g(D.W.)h (Robinson.)50 b Fo(Op)l(er)l(ator)34 b(A)n(lgebr)l(as)g(and)h(Quantum)d (Statistic)l(al)i(Me)l(chanics)p Ft(,)431 335 y(v)n(olume)26 b(2.)37 b(Springer)26 b(V)-7 b(erlag,)27 b(New)h(Y)-7 b(ork,)27 b(Heidelb)r(erg,)g(Berlin,)g(1981.)301 526 y([8])42 b(J.)31 b(Derezi)r(\023)-44 b(nski)31 b(and)g(V.)h(Jak)l(\024) -37 b(si)n(\023)e(c.)46 b(Return)32 b(to)f(equilibrium)h(for)f(P)n (auli-Fierz)f(systems.)48 b Fo(A)n(nn.)33 b(Henri)431 650 y(Poinc)l(ar)n(\023)-40 b(e)29 b Fg(4)e Ft(\(2003\),)f(no.)i(4,)f (739{793)301 841 y([9])42 b(J.)29 b(Derezi)r(\023)-44 b(nski,)30 b(V.)g(Jak)l(\024)-37 b(si)n(\023)e(c,)28 b(and)h(C.-A.)h(Pillet.)44 b(P)n(erturbation)28 b(theory)h(of)h Fs(W)2941 811 y Fk(\003)2979 841 y Ft(-dynamics,)g(Liouvil-)431 966 y(leans)d(and)g(KMS-states.)36 b Fo(R)l(ev.)30 b(Math.)h(Phys.)e Fg(15)e Ft(\(2003\),)f(no.)i(5,)f(447{489)260 1157 y([10])41 b(J.)g(F)-7 b(r\177)-42 b(ohlic)n(h)40 b(and)i(M.)f(Merkli.)78 b(Thermal)40 b(ionization.)77 b Fo(Mathematic)l(al)45 b(Physics,)i(A)n(nalysis)c(and)431 1281 y(Ge)l(ometry)28 b Fg(7)f Ft(\(2004\),)g(no.)g(3,)g(239-287)260 1472 y([11])41 b(R.)23 b(Haag,)f(N.)i(Hugenholz,)f(and)g(M.)g(Winnink.)29 b(On)23 b(the)g(equilibrium)g(states)g(in)g(quan)n(tum)f(statistical) 431 1597 y(mec)n(hanics.)36 b Fo(Comm.)30 b(Math.)h(Phys.)e Fg(5)f Ft(\(1967\))e(215{236)260 1788 y([12])41 b(V.)31 b(Jak)l(\024)-37 b(si)n(\023)e(c)28 b(and)j(C.-A.)g(Pillet.)47 b(On)31 b(a)g(mo)r(del)g(for)f(quan)n(tum)i(friction)f(I)r(I.)g(F)-7 b(ermi's)31 b(golden)f(rule)h(and)431 1912 y(dynamics)c(at)g(p)r (ositiv)n(e)g(temp)r(erature.)37 b Fo(Comm.)31 b(Math.)g(Phys.)e Fg(176)e Ft(\(1996\),)f(619{644)260 2103 y([13])41 b(V.)33 b(Jak)l(\024)-37 b(si)n(\023)e(c)29 b(and)j(C.-A.)h(Pillet.)52 b(Mathematical)32 b(theory)g(of)h(non-equilibrium)f(quan)n(tum)g (statistical)431 2228 y(mec)n(hanics.)k Fo(J.)29 b(Stat.)h(Phys.)f Fg(108)e Ft(\(2002\))f(787{829)260 2419 y([14])41 b(V.)19 b(Jak)l(\024)-37 b(si)n(\023)e(c)16 b(and)i(C.-A.)h(Pillet.)k (Non-equilibrium)c(steady)f(states)g(of)h(\014nite)h(quan)n(tum)f (systems)f(coupled)431 2543 y(to)27 b(thermal)g(reserv)n(oirs.)34 b Fo(Comm.)d(Math.)g(Phys.)e Fg(226)e Ft(\(2002\))f(131{162)260 2734 y([15])41 b(T.)24 b(Kato.)30 b(P)n(erturbation)23 b(Theory)g(for)h(Linear)f(Op)r(erators)30 b Fo(Die)d(Grund)t(lehr)l(en) g(der)h(mathematischen)431 2859 y(Wissenschaften,)j(Band)f(132,)39 b Ft(Springer-V)-7 b(erlag)25 b(New)j(Y)-7 b(ork)27 b(Inc.)g(1966)260 3050 y([16])41 b(M.)22 b(Merkli,)h(M.)g(M)r(\177)-44 b(uc)n(k,)23 b(I.M.)g(Sigal.)28 b(Theory)21 b(of)i(Non-Equilibrium)e (Stationary)h(States)g(as)g(a)g(Theory)431 3174 y(of)27 b(Resonances)f(I:)i(Existence)f(and)g(Prop)r(erties)f(of)i(NESS,)37 b(in)28 b(preparation)260 3365 y([17])41 b(M.)22 b(Merkli,)h(M.)g(M)r (\177)-44 b(uc)n(k,)23 b(I.M.)g(Sigal.)28 b(Theory)21 b(of)i(Non-Equilibrium)e(Stationary)h(States)g(as)g(a)g(Theory)431 3490 y(of)27 b(Resonances)f(I)r(I:)i(Renormalization)e(Group)h (Analysis,)37 b(in)28 b(preparation)260 3681 y([18])41 b(M.)33 b(M)r(\177)-44 b(uc)n(k.)53 b(Thermal)33 b(Relaxation)f(for)g (P)n(article)g(Systems)h(in)h(In)n(teraction)e(with)h(Sev)n(eral)f (Bosonic)431 3805 y(Heat)i(Reserv)n(oirs.)56 b(Ph.D.)35 b(Dissertation,)g(Departmen)n(t)g(of)g(Mathematics,)h(Johannes-Guten)n (b)r(erg)431 3930 y(Univ)n(ersit)n(y)-7 b(,)26 b(Mainz,)i(July)f(2004,) f(ISBN)i(3-8334-1866-4)260 4121 y([19])41 b(M.)f(Reed,)j(B.)d(Simon.)g Fo(Metho)l(ds)i(of)g(Mo)l(dern)h(Mathematic)l(al)g(Physics)f(I,)g(F)-6 b(unctional)41 b(A)n(nalysis)p Ft(,)431 4245 y(Academic)27 b(Press,)f(1980)260 4436 y([20])41 b(M.)27 b(T)-7 b(ak)n(esaki.)26 b(Disjoin)n(tness)i(of)f(the)h(KMS-States)f(of)h(Di\013eren)n(t)g(T)-7 b(emp)r(eratures,)35 b Fo(Commun.)30 b(Math.)431 4561 y(Phys.)f Fg(17)e Ft(\(1970\),)f(33-41)1928 5476 y(30)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0507081518339--