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m(tin)m(uous)h(function)e(\012)g Fp(3)f Fq(z)k Fp(7!)27 b Fq(A)p Ft(\()p Fq(z)t Ft(\))k(extends)i(b)m(y)0 2276 y(con)m(tin)m(uit)m(y)g(to)f(\012)22 b Fp([)h Ft(\000)32 b(if)g(for)g(ev)m(ery)i Fq(z)1395 2291 y Fo(0)1463 2276 y Fp(2)28 b Ft(\000)k(the)h(limit)1423 2479 y Fq(A)p Ft(\()p Fq(z)1579 2494 y Fo(0)1618 2479 y Ft(\))28 b(:=)123 b(lim)1815 2538 y Fm(z)s Fl(!)p Fm(z)1955 2547 y Fk(0)1988 2538 y Fm(;z)s Fl(2)p Fo(\012)2158 2479 y Fq(A)p Ft(\()p Fq(z)t Ft(\))0 2719 y(exists.)81 b(W)-8 b(e)45 b(will)e(denote)j(the)f (functions)g(extended)h(b)m(y)g(con)m(tin)m(uit)m(y)f(with)g(the)g (same)g(letter.)80 b(If)0 2839 y(\012)28 b Fp(\032)g Fr(C)284 2854 y Fl(\006)372 2839 y Ft(and)h Fq(A)p Ft(\()p Fq(z)t Ft(\))h(extends)h(b)m(y)f(con)m(tin)m(uit)m(y)f(to)g(a)g(part)f (of)h(the)g(real)g(axis,)g(w)m(e)h(denote)g(b)m(y)g Fq(A)p Ft(\()p Fq(x)15 b Fp(\006)g Ft(i0\))0 2960 y(its)32 b(v)-5 b(alues)33 b(along)e Fr(R)p Ft(.)146 3080 y(In)40 b(the)g(dev)m (elopmen)m(t)g(of)f(Mourre)h(theory)-8 b(,)42 b(w)m(e)f(will)c(mak)m(e) j(use)g(of)f(the)h(follo)m(wing)d(t)m(w)m(o)j(simple)0 3200 y(facts.)0 3389 y Fr(Prop)s(osition)c(2.3)49 b Fi(L)-5 b(et)35 b Ft(\012)g Fi(b)-5 b(e)35 b(an)f(op)-5 b(en)34 b(c)-5 b(onvex)34 b(set,)h(and)f(let)p 1163 3513 V 1163 3591 a Fr(R)1247 3606 y Fo(+)1328 3591 y Fp(\002)22 b Ft(\012)29 b Fp(3)f Ft(\()p Fq(\017;)17 b(z)t Ft(\))28 b Fp(7!)f Fq(A)p Ft(\()p Fq(\017;)17 b(z)t Ft(\))29 b Fp(2)f(B)s Ft(\()p Fp(H)q Ft(\))0 3794 y Fi(b)-5 b(e)48 b(a)g(b)-5 b(ounde)g(d)48 b(function)g(which)g(is)g(c)-5 b(ontinuously)48 b(di\013er)-5 b(entiable)48 b(in)g(e)-5 b(ach)47 b(variable)h(sep)-5 b(ar)g(ately.)0 3914 y(Assume)35 b(further)g(that)g(for)g(some)f(c)-5 b(onstants)34 b Fq(C)42 b Fi(and)35 b Ft(0)27 b Fq(<)h(\022)i Fp(\024)f Ft(1)p Fi(,)926 4117 y Ft(sup)932 4195 y Fm(z)s Fl(2)p Fo(\012)1089 4117 y Fp(k)p Fq(@)1195 4076 y Fm(k)1190 4141 y(\017)1238 4117 y Fq(@)1294 4076 y Fm(l)1289 4141 y(z)1330 4117 y Fq(A)p Ft(\()p Fq(\017;)17 b(z)t Ft(\))p Fp(k)28 b Fq(<)g(C)7 b(\017)1909 4076 y Fl(\000)p Fo(1)2003 4117 y Fq(`)2044 4132 y Fm(\022)2083 4117 y Ft(\()p Fq(\017)p Ft(\))p Fq(;)216 b(k)26 b Ft(+)c Fq(l)30 b Ft(=)d(1)p Fq(:)0 4380 y Fi(Then,)34 b(the)h(function)f Ft(\012)29 b Fp(3)f Fq(z)k Fp(7!)27 b Fq(A)p Ft(\(0)p Fq(;)17 b(z)t Ft(\))35 b Fi(is)g(in)g(the)g(class)f Fq(C)2250 4344 y Fo(0)p Fm(;\022)2243 4405 y Fo(u)2343 4380 y Ft(\(\012\))p Fi(.)0 4569 y Fr(Pro)s(of.)65 b Ft(F)-8 b(or)32 b Fq(z)588 4584 y Fo(1)627 4569 y Fq(;)17 b(z)716 4584 y Fo(2)784 4569 y Fp(2)28 b Ft(\012)33 b(and)f Fq(\017)c(>)g Ft(0)k(w)m(e)i(ha)m (v)m(e)135 4766 y Fp(k)p Fq(A)p Ft(\(0)p Fq(;)17 b(z)434 4781 y Fo(1)474 4766 y Ft(\))22 b Fp(\000)g Fq(A)p Ft(\(0)p Fq(;)17 b(z)882 4781 y Fo(2)922 4766 y Ft(\))p Fp(k)82 b(\024)1198 4695 y Fj(R)1253 4721 y Fm(\017)1237 4792 y Fo(0)1302 4766 y Fp(k)p Fq(@)1403 4781 y Fm(\034)1447 4766 y Fq(A)p Ft(\()p Fq(\034)6 b(;)17 b(z)1695 4781 y Fo(1)1734 4766 y Ft(\))p Fp(k)p Ft(d)p Fq(\034)34 b Ft(+)22 b Fp(j)p Fq(z)2123 4781 y Fo(1)2184 4766 y Fp(\000)h Fq(z)2329 4781 y Fo(2)2369 4766 y Fp(j)2414 4695 y Fj(R)2468 4721 y Fo(1)2452 4792 y(0)2524 4766 y Fp(k)p Fq(@)2625 4781 y Fm(z)2665 4766 y Fq(A)p Ft(\()p Fq(\017;)17 b(z)2904 4781 y Fo(1)2967 4766 y Ft(+)22 b Fq(t)p Ft(\()p Fq(z)3183 4781 y Fo(2)3245 4766 y Fp(\000)g Fq(z)3389 4781 y Fo(1)3429 4766 y Ft(\)\))p Fp(k)p Ft(d)p Fq(t)1092 4957 y Ft(+)1185 4886 y Fj(R)1240 4913 y Fm(\017)1224 4983 y Fo(0)1289 4957 y Fp(k)p Fq(@)1390 4972 y Fm(\034)1434 4957 y Fq(A)p Ft(\()p Fq(\034)6 b(;)17 b(z)1682 4972 y Fo(2)1721 4957 y Ft(\))p Fp(k)p Ft(d)p Fq(\034)1092 5149 y Fp(\024)29 b Ft(2)p Fq(C)1340 5078 y Fj(R)1395 5104 y Fm(\017)1379 5174 y Fo(0)1444 5149 y Fq(\034)1497 5113 y Fl(\000)p Fo(1)1592 5149 y Fq(`)1633 5164 y Fm(\022)1672 5149 y Ft(\()p Fq(\034)11 b Ft(\)d)p Fq(\034)34 b Ft(+)22 b Fq(C)7 b Fp(j)p Fq(z)2179 5164 y Fo(1)2241 5149 y Fp(\000)23 b Fq(z)2386 5164 y Fo(2)2425 5149 y Fp(j)p Fq(\017)2492 5113 y Fl(\000)p Fo(1)2587 5149 y Fq(`)2628 5164 y Fm(\022)2667 5149 y Ft(\()p Fq(\017)p Ft(\))p Fq(:)3530 5268 y Ft(\(2.35\))1841 5753 y(20)p eop %%Page: 21 21 21 20 bop 0 407 a Ft(Using)31 b(the)h(form)e(of)i(functions)f Fq(`)1240 422 y Fm(\022)1311 407 y Ft(one)g(easily)g(sho)m(ws)i(that)f (for)f(some)g(constan)m(ts)i Fq(C)3138 422 y Fm(\022)3208 407 y Ft(and)f(all)d Fq(\017)f Fp(\025)h Ft(0,)1359 533 y Fj(Z)1442 559 y Fm(\017)1405 722 y Fo(0)1492 650 y Fq(\034)1545 609 y Fl(\000)p Fo(1)1639 650 y Fq(`)1680 665 y Fm(\022)1719 650 y Ft(\()p Fq(\034)11 b Ft(\)d)p Fq(\034)40 b Fp(\024)28 b Fq(C)2159 665 y Fm(\022)2198 650 y Fq(`)2239 665 y Fm(\022)2278 650 y Ft(\()p Fq(\017)p Ft(\))p Fq(:)0 901 y Ft(Com)m(bining)37 b(this)i(estimate)f(with)h (\(2.35\))f(and)h(setting)f Fq(\017)h Ft(=)g Fp(j)p Fq(z)2401 916 y Fo(1)2467 901 y Fp(\000)27 b Fq(z)2616 916 y Fo(2)2655 901 y Fp(j)39 b Ft(w)m(e)h(deriv)m(e)g(that)e Fq(A)p Ft(\(0)p Fq(;)17 b(z)t Ft(\))39 b Fp(2)0 1022 y Fq(C)77 985 y Fo(0)p Fm(;\022)70 1046 y Fo(u)171 1022 y Ft(\(\012\).)44 b Fe(2)0 1271 y Fr(Prop)s(osition)36 b(2.4)49 b Fi(L)-5 b(et)42 b Fq(A)1046 1286 y Fm(\025)1092 1271 y Fi(,)h Fq(\025)f Fp(2)f(I)7 b Fi(,)45 b(b)-5 b(e)42 b(a)g(family)f(of)h (functions)g(de\014ne)-5 b(d)41 b(on)h(op)-5 b(en)41 b(c)-5 b(onvex)42 b(sets)0 1391 y Ft(\012)70 1406 y Fm(\025)147 1391 y Fp(\032)32 b Fr(C)p Fi(.)50 b(If)36 b(the)h(family)f Fq(A)1056 1406 y Fm(\025)1138 1391 y Fi(is)h(of)f(the)h(class)f Fq(C)1838 1355 y Fm(n;`)1831 1416 y Fo(u)1933 1391 y Ft(\(\012)2041 1406 y Fm(\025)2087 1391 y Ft(\))h Fi(uniformly)f(in)g Fq(\025)p Fi(,)h(then)g(the)g(functions)f Fq(A)3734 1406 y Fm(\025)0 1512 y Fi(extend)e(by)h(c)-5 b(ontinuity)36 b(to)p 1009 1434 71 4 v 35 w Ft(\012)1079 1527 y Fm(\025)1159 1512 y Fi(and)f(the)g(family)f Fq(A)1880 1527 y Fm(\025)1960 1512 y Fi(is)h(of)f(the)h(class)f Fq(C)2652 1476 y Fm(n;`)2645 1536 y Fo(u)2748 1512 y Ft(\()p 2786 1434 V(\012)2856 1527 y Fm(\025)2902 1512 y Ft(\))g Fi(uniformly)h(in)f Fq(\025)p Fi(.)0 1712 y Ft(The)g(pro)s(of)d(of)h(this)h(prop)s(osition) d(is)i(elemen)m(tary)h(and)g(w)m(e)g(will)d(skip)j(it.)146 1833 y(Let)43 b Fq(H)50 b Ft(b)s(e)42 b(a)g(closed)h(op)s(erator)e(on)i Fp(H)q Ft(.)72 b(W)-8 b(e)43 b(denote)g(the)g(domain)d(of)i Fq(H)50 b Ft(b)m(y)43 b Fp(D)s Ft(\()p Fq(H)8 b Ft(\))42 b(and)g(the)0 1953 y(sp)s(ectrum)33 b(of)f Fq(H)40 b Ft(b)m(y)34 b Fq(\033)t Ft(\()p Fq(H)8 b Ft(\).)42 b(The)34 b(n)m(umerical)d(range)i(of)f Fq(H)40 b Ft(is)32 b(de\014ned)i(b)m(y) 966 2170 y Fh(N)p Ft(\()p Fq(H)8 b Ft(\))27 b(:=)h Fp(f)p Ft(\()p Fq( )t Fp(j)p Fq(H)8 b( )t Ft(\))59 b(:)h Fq( )32 b Fp(2)c(D)s Ft(\()p Fq(H)8 b Ft(\))p Fq(;)48 b Fp(k)p Fq( )t Fp(k)27 b Ft(=)h(1)p Fp(g)p Fq(:)0 2387 y Ft(The)34 b(v)m(ector)f(space)h Fp(D)s Ft(\()p Fq(H)8 b Ft(\))31 b(equipp)s(ed)j(with)e(the)h(graph)f(norm)1423 2603 y Fp(k)p Fq( )t Fp(k)1590 2618 y Fm(H)1684 2603 y Ft(=)c Fp(k)p Fq( )t Fp(k)22 b Ft(+)g Fp(k)p Fq(H)8 b( )t Fp(k)p Fq(;)0 2820 y Ft(is)33 b(a)h(Banac)m(h)g(space.)49 b(A)34 b(v)m(ector)h(space)f Fp(C)j(\032)30 b(D)s Ft(\()p Fq(H)8 b Ft(\))33 b(is)g(called)g(a)g(core)h(of)g Fq(H)41 b Ft(if)33 b Fp(C)40 b Ft(is)33 b(dense)i(in)f Fp(D)s Ft(\()p Fq(H)8 b Ft(\))0 2941 y(in)32 b(the)h(graph)f(norm.)43 b(The)33 b(follo)m(wing)d(useful)j(fact)f(is)g(w)m(ell)g(kno)m(wn:)0 3165 y Fr(Lemma)37 b(2.5)49 b Fi(L)-5 b(et)41 b Fq(A)h Fi(and)e Fq(B)46 b Fi(b)-5 b(e)41 b(close)-5 b(d)40 b(op)-5 b(er)g(ators)40 b(and)h(let)g Fp(D)s Ft(\()p Fq(B)5 b Ft(\))40 b Fi(b)-5 b(e)41 b(a)g(c)-5 b(or)g(e)41 b(of)f Fq(A)p Fi(.)63 b(Then)41 b(any)0 3285 y(c)-5 b(or)g(e)34 b(of)h Fq(B)40 b Fi(is)35 b(a)f(c)-5 b(or)g(e)35 b(of)f Fq(A)p Fi(.)146 3510 y Ft(F)-8 b(or)26 b(an)m(y)g(closed)g(op)s(erator) g Fq(H)8 b Ft(,)27 b(w)m(e)g(sa)m(y)g(that)e(\012)i(is)e(an)h(isolated) f(subset)i(of)f Fq(\033)t Ft(\()p Fq(H)8 b Ft(\))25 b(if)g(it)g(is)h (relativ)m(ely)0 3630 y(closed)35 b(and)g(op)s(en)h(subset)g(of)f Fq(\033)t Ft(\()p Fq(H)8 b Ft(\).)50 b(If)36 b(in)e(addition)f(\012)j (is)e(b)s(ounded,)j(then)f(there)f(exists)h(a)f(simple)0 3751 y(closed)c(path)g Fq(\015)36 b Ft(whic)m(h)c(separates)g(\012)g (and)f Fq(\033)t Ft(\()p Fq(H)8 b Ft(\))19 b Fp(n)f Ft(\012,)32 b(and)g(w)m(e)g(can)f(de\014ne)h(the)g(sp)s(ectral)f(pro)5 b(jection)0 3871 y(of)32 b Fq(H)40 b Ft(on)m(to)33 b(\012)g(b)m(y)g (the)g(form)m(ula)1275 4127 y Fr(1)1331 4142 y Fo(\012)1386 4127 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)1735 4059 y(1)p 1692 4104 135 4 v 1692 4195 a(2)p Fq(\031)t Ft(i)1853 4010 y Fj(I)1899 4198 y Fm(\015)1943 4127 y Ft(\()p Fq(z)g Fp(\000)c Fq(H)8 b Ft(\))2280 4086 y Fl(\000)p Fo(1)2374 4127 y Ft(d)p Fq(z)t(:)0 4385 y Ft(It)33 b(is)f(easy)h(to)g(sho)m(w)g (the)g(follo)m(wing)d(fact:)0 4586 y Fr(Lemma)37 b(2.6)49 b Fi(L)-5 b(et)37 b Ft(\012)g Fi(b)-5 b(e)37 b(a)f(b)-5 b(ounde)g(d)36 b(isolate)-5 b(d)36 b(subset)h(of)g Fq(\033)t Ft(\()p Fq(H)8 b Ft(\))36 b Fi(and)g Fq(p)h Fi(is)f(a)h(pr)-5 b(oje)g(ction)36 b(such)g(that)0 4706 y Ft(Ran)p Fr(1)231 4721 y Fo(\012)286 4706 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)h(Ran)o Fq(p)35 b Fi(and)f Ft([)p Fq(p;)17 b Ft(\()p Fq(z)27 b Fp(\000)c Fq(H)8 b Ft(\))1486 4670 y Fl(\000)p Fo(1)1579 4706 y Ft(])28 b(=)g(0)34 b Fi(for)h Fq(z)d Fp(62)d Fq(\033)t Ft(\()p Fq(H)8 b Ft(\))p Fi(.)44 b(Then)34 b Fq(p)27 b Ft(=)h Fr(1)2937 4721 y Fo(\012)2992 4706 y Ft(\()p Fq(H)8 b Ft(\))p Fi(.)146 4907 y Ft(An)45 b(isolated)d(p)s(oin)m(t)i (of)f Fq(\033)t Ft(\()p Fq(H)8 b Ft(\))44 b(will)e(b)s(e)i(called)f(an) h(isolated)f(eigen)m(v)-5 b(alue)43 b(of)h Fq(H)8 b Ft(.)77 b(An)45 b(isolated)0 5027 y(eigen)m(v)-5 b(alue)25 b Fq(z)504 5042 y Fo(0)570 5027 y Ft(of)g Fq(H)33 b Ft(is)25 b(called)g(semisimple)e(if)i(\()p Fq(z)13 b Fp(\000)8 b Fq(H)g Ft(\))2027 4991 y Fl(\000)p Fo(1)2147 5027 y Ft(has)26 b(a)f(simple)g(p)s(ole)f(at)i Fq(z)3045 5042 y Fo(0)3085 5027 y Ft(,)h(or)e(equiv)-5 b(alen)m(tly)d(,)0 5147 y(if)1278 5268 y(Ran)p Fr(1)1509 5283 y Fl(f)p Fm(z)1577 5292 y Fk(0)1611 5283 y Fl(g)1651 5268 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)g(Ker)q(\()p Fq(H)i Fp(\000)23 b Fq(z)2397 5283 y Fo(0)2437 5268 y Ft(\))p Fq(:)1841 5753 y Ft(21)p eop %%Page: 22 22 22 21 bop 146 407 a Ft(W)-8 b(e)46 b(denote)g(b)m(y)h Fq(\033)858 422 y Fo(disc)980 407 y Ft(\()p Fq(H)8 b Ft(\))45 b(the)h(discrete)g(sp)s(ectrum)g(of)f Fq(H)8 b Ft(,)49 b(that)c(is,)j(the)e(set)g(of)f(all)f(isolated)0 527 y(eigen)m(v)-5 b(alues)43 b Fq(e)f Ft(suc)m(h)i(that)e(dim)16 b Fr(1)1289 543 y Fl(f)p Fm(e)p Fl(g)1396 527 y Ft(\()p Fq(H)8 b Ft(\))44 b Fq(<)h Fp(1)p Ft(.)73 b(The)43 b(essen)m(tial)g(sp) s(ectrum)g(of)f Fq(H)50 b Ft(is)42 b(de\014ned)i(b)m(y)0 648 y Fq(\033)55 663 y Fo(ess)147 648 y Ft(\()p Fq(H)8 b Ft(\))27 b(:=)g Fq(\033)t Ft(\()p Fq(H)8 b Ft(\))22 b Fp(n)g Fq(\033)842 663 y Fo(disc)964 648 y Ft(\()p Fq(H)8 b Ft(\).)146 768 y(If)25 b Fq(H)33 b Ft(is)24 b(a)h(self-adjoin)m(t)e(op)s(erator,)j(w)m(e)g(denote)g(b)m(y)g Fq(\033)2048 783 y Fo(pp)2131 768 y Ft(\()p Fq(H)8 b Ft(\),)26 b Fq(\033)2404 783 y Fo(sc)2468 768 y Ft(\()p Fq(H)8 b Ft(\))24 b(and)h Fq(\033)2894 783 y Fo(ac)2966 768 y Ft(\()p Fq(H)8 b Ft(\))24 b(the)i(pure)f(p)s(oin)m(t,)0 888 y(singular)33 b(con)m(tin)m(uous)i(and)f(absolutely)g(con)m(tin)m (uous)h(sp)s(ectrum)g(of)f Fq(H)8 b Ft(.)48 b(The)35 b(singular)e(sp)s(ectrum)i(is)0 1009 y(de\014ned)d(b)m(y)f Fq(\033)522 1024 y Fo(sing)649 1009 y Ft(\()p Fq(H)8 b Ft(\))28 b(=)p 945 924 303 4 v 27 w Fq(\033)1000 1024 y Fo(pp)1083 1009 y Ft(\()p Fq(H)8 b Ft(\))17 b Fp([)h Fq(\033)1404 1024 y Fo(sc)1468 1009 y Ft(\()p Fq(H)8 b Ft(\).)42 b(If)30 b(\002)g(a)g(Borel)g(subset)i(of)e Fr(R)p Ft(,)g(then)h Fr(1)3062 1024 y Fo(\002)3121 1009 y Ft(\()p Fq(H)8 b Ft(\))30 b(will)e(denote)0 1129 y(the)35 b(sp)s(ectral)g(pro)5 b(jection)34 b(of)g Fq(H)42 b Ft(on)m(to)35 b(\002.)49 b(W)-8 b(e)36 b(denote)f(b)m(y)h Fr(1)2295 1082 y Fo(pp)2295 1153 y(\002)2377 1129 y Ft(\()p Fq(H)8 b Ft(\),)35 b Fr(1)2660 1093 y Fo(sc)2660 1154 y(\002)2723 1129 y Ft(\()p Fq(H)8 b Ft(\),)35 b Fr(1)3006 1093 y Fo(ac)3006 1154 y(\002)3077 1129 y Ft(\()p Fq(H)8 b Ft(\))34 b(the)h(sp)s(ectral)0 1249 y(pro)5 b(jections)32 b(of)e Fq(H)39 b Ft(on)m(to)31 b(\002)g(asso)s(ciated)h(to)f(the)g(pure)h(p)s (oin)m(t,)f(singular)f(con)m(tin)m(uous)i(and)f(absolutely)0 1370 y(con)m(tin)m(uous)i(sp)s(ectrum.)146 1490 y(W)-8 b(e)29 b(no)m(w)f(recall)f(some)g(standard)i(results)f(ab)s(out)g (linear)e(op)s(erators)i(that)f(will)f(b)s(e)i(used)h(through-)0 1611 y(out)j(the)h(pap)s(er.)0 1777 y Fr(Prop)s(osition)j(2.7)49 b Fi(L)-5 b(et)35 b Fq(H)42 b Fi(b)-5 b(e)35 b(a)f(self-adjoint)g(op)-5 b(er)g(ator.)44 b(If)35 b Fq(z)d Fp(2)c Fr(C)22 b Fp(n)g Fq(\033)t Ft(\()p Fq(H)8 b Ft(\))35 b Fi(then)1260 1998 y Fp(k)p Ft(\()p Fq(z)27 b Fp(\000)c Fq(H)8 b Ft(\))1647 1957 y Fl(\000)p Fo(1)1741 1998 y Fp(k)27 b Ft(=)2182 1930 y(1)p 1932 1975 551 4 v 1932 2066 a(dist)o(\()p Fq(z)t(;)17 b(\033)t Ft(\()p Fq(H)8 b Ft(\)\))2492 1998 y Fq(:)0 2225 y Ft(An)33 b(immediate)d(consequence)35 b(of)d(this)h(prop)s(osition)d(is)0 2391 y Fr(Prop)s(osition)36 b(2.8)49 b Fi(L)-5 b(et)34 b Fq(H)41 b Fi(b)-5 b(e)34 b(a)g(self-adjoint)f(and)g Fq(V)56 b Fi(a)33 b(b)-5 b(ounde)g(d)34 b(op)-5 b(er)g(ator.)44 b(Then,)33 b Fq(\033)t Ft(\()p Fq(H)28 b Ft(+)20 b Fq(V)h Ft(\))28 b Fp(\032)p 0 2433 80 4 v 0 2511 a Fq(B)5 b Ft(\()p Fq(\033)t Ft(\()p Fq(H)j Ft(\))p Fq(;)17 b Fp(k)p Fq(V)k Fp(k)p Ft(\))34 b Fi(and)g(for)h Fq(z)e Fp(2)28 b Fr(C)22 b Fp(n)p 1327 2433 V 22 w Fq(B)5 b Ft(\()p Fq(\033)t Ft(\()p Fq(H)j Ft(\))p Fq(;)17 b Fp(k)p Fq(V)j Fp(k)p Ft(\))35 b Fi(one)f(has)h(the)f(b)-5 b(ound)972 2732 y Fp(k)p Ft(\()p Fq(z)27 b Fp(\000)c Ft(\()p Fq(H)29 b Ft(+)22 b Fq(V)g Ft(\)\))1633 2691 y Fl(\000)p Fo(1)1727 2732 y Fp(k)28 b(\024)2320 2665 y Ft(1)p 1920 2709 851 4 v 1920 2801 a(dist)o(\()p Fq(z)t(;)17 b(\033)t Ft(\()p Fq(H)8 b Ft(\)\))22 b Fp(\000)h(k)p Fq(V)e Fp(k)2780 2732 y Fq(:)0 3075 y Ft(The)k(concept)f(of)f(the)h(n)m (umerical)e(range)i(allo)m(ws)e(to)i(form)m(ulate)e(related)h(results)h (for)f(closed)h(op)s(erators.)0 3241 y Fr(Prop)s(osition)36 b(2.9)49 b Fi(L)-5 b(et)38 b Fq(H)45 b Fi(b)-5 b(e)37 b(a)h(close)-5 b(d)37 b(op)-5 b(er)g(ator)37 b(such)h(that)g Fp(D)s Ft(\()p Fq(H)8 b Ft(\))32 b(=)h Fp(D)s Ft(\()p Fq(H)2998 3205 y Fl(\003)3037 3241 y Ft(\))p Fi(.)53 b(Then,)38 b Fq(\033)t Ft(\()p Fq(H)8 b Ft(\))32 b Fp(\032)p 0 3277 234 4 v 0 3362 a Fh(N)p Ft(\()p Fq(H)8 b Ft(\))34 b Fi(and,)h(for)f Fq(z)f Fp(2)28 b Fr(C)22 b Fp(n)p 990 3277 V 22 w Fh(N)p Ft(\()p Fq(H)8 b Ft(\))o Fi(,)35 b(one)f(has)h(the)g (b)-5 b(ound)1255 3583 y Fp(k)p Ft(\()p Fq(z)26 b Fp(\000)d Fq(H)8 b Ft(\))1641 3542 y Fl(\000)p Fo(1)1735 3583 y Fp(k)27 b(\024)2183 3515 y Ft(1)p 1927 3560 561 4 v 1927 3651 a(dist\()p Fq(z)t(;)17 b Fh(N)p Ft(\()p Fq(H)8 b Ft(\)\))2498 3583 y Fq(:)0 3938 y Fr(Prop)s(osition)36 b(2.10)49 b Fi(L)-5 b(et)35 b Fq(H)43 b Fi(b)-5 b(e)35 b(a)h(close)-5 b(d)34 b(op)-5 b(er)g(ator)35 b(such)g(that)h Fp(D)s Ft(\()p Fq(H)8 b Ft(\))28 b(=)g Fp(D)s Ft(\()p Fq(H)3026 3902 y Fl(\003)3065 3938 y Ft(\))p Fi(,)35 b(and)g(let)h Fq(V)57 b Fi(b)-5 b(e)35 b(a)0 4059 y(b)-5 b(ounde)g(d)33 b(op)-5 b(er)g(ator.)44 b(Then,)33 b Fq(\033)t Ft(\()p Fq(H)28 b Ft(+)19 b Fq(V)j Ft(\))28 b Fp(\032)p 1627 3981 80 4 v 28 w Fq(B)5 b Ft(\()p Fh(N)p Ft(\()p Fq(H)j Ft(\))p Fq(;)17 b Fp(k)p Fq(V)j Fp(k)p Ft(\))p Fi(,)34 b(and)f(for)h Fq(z)e Fp(2)c Fr(C)20 b Fp(n)p 2986 3981 V 20 w Fq(B)5 b Ft(\()p Fh(N)p Ft(\()p Fq(H)j Ft(\))p Fq(;)17 b Fp(k)p Fq(V)j Fp(k)p Ft(\))34 b Fi(one)0 4179 y(has)g(the)h(b)-5 b(ound)967 4319 y Fp(k)p Ft(\()p Fq(z)27 b Fp(\000)c Ft(\()p Fq(H)29 b Ft(+)22 b Fq(V)g Ft(\)\))1628 4278 y Fl(\000)p Fo(1)1722 4319 y Fp(k)28 b(\024)2320 4251 y Ft(1)p 1915 4296 861 4 v 1915 4387 a(dist)o(\()p Fq(z)t(;)17 b Fh(N)p Ft(\()p Fq(H)8 b Ft(\)\))22 b Fp(\000)h(k)p Fq(V)e Fp(k)2785 4319 y Fq(:)146 4553 y Ft(W)-8 b(e)33 b(sa)m(y)h(that)e(the)h(op)s(erator)f Fq(B)38 b Ft(is)32 b Fq(A)p Ft(-b)s(ounded)h(if)e Fp(D)s Ft(\()p Fq(B)5 b Ft(\))28 b Fp(\033)g(D)s Ft(\()p Fq(A)p Ft(\))k(and)1117 4730 y Fp(k)p Fq(B)5 b(\036)p Fp(k)27 b(\024)h Fq(a)p Fp(k)p Fq(A\036)p Fp(k)22 b Ft(+)g Fq(b)p Fp(k)p Fq(\036)p Fp(k)p Fq(;)114 b(\036)27 b Fp(2)h(D)s Ft(\()p Fq(A)p Ft(\))p Fq(:)867 b Ft(\(2.36\))0 4907 y(The)37 b(in\014m)m(um)e(of)g(p)s(ossible)h(v)-5 b(alues)35 b(of)h Fq(a)g Ft(in)g(\(2.36\))f(is)g(called)g(the)i Fq(A)p Ft(-b)s(ound)f(of)f Fq(B)5 b Ft(.)54 b(Recall)35 b(that)h(if)0 5027 y Fq(A)d Ft(is)g(closed)h(and)f(the)h Fq(A)p Ft(-b)s(ound)f(of)f Fq(B)39 b Ft(is)32 b(less)i(than)f(1,)h (then)f Fq(A)23 b Ft(+)g Fq(B)38 b Ft(is)33 b(closed)g(on)g Fp(D)s Ft(\()p Fq(A)p Ft(\).)45 b(Clearly)-8 b(,)0 5147 y(if)31 b(for)h(some)g Fq(z)527 5162 y Fo(0)595 5147 y Fp(62)c Fq(\033)t Ft(\()p Fq(A)p Ft(\),)k Fp(k)p Fq(B)1080 5162 y Fo(1)1120 5147 y Ft(\()p Fq(z)1203 5162 y Fo(0)1264 5147 y Fp(\000)22 b Fq(A)p Ft(\))1474 5111 y Fl(\000)p Fo(1)1569 5147 y Fp(k)27 b Ft(=)h Fq(a)p Ft(,)k(where)i Fq(B)f Ft(=)27 b Fq(B)2426 5162 y Fo(1)2488 5147 y Ft(+)21 b Fq(B)2659 5162 y Fo(2)2731 5147 y Ft(and)33 b Fq(B)2995 5162 y Fo(2)3067 5147 y Ft(is)f(b)s(ounded,)h(then)0 5268 y(the)g Fq(A)p Ft(-b)s(ound)f(of)h Fq(B)k Ft(is)32 b(less)h(than)g(or)f(equal)g(to)h Fq(a)p Ft(.)1841 5753 y(22)p eop %%Page: 23 23 23 22 bop 0 407 a Fr(Prop)s(osition)36 b(2.11)49 b Fi(Supp)-5 b(ose)31 b(that)h Fq(A)p Fi(,)h Fq(B)38 b Fi(ar)-5 b(e)31 b(op)-5 b(er)g(ators)32 b(such)g(that)h Fq(A)f Fi(is)g(close)-5 b(d,)31 b Fp(D)s Ft(\()p Fq(B)5 b Ft(\))28 b Fp(\033)g(D)s Ft(\()p Fq(A)p Ft(\))0 527 y Fi(and,)34 b(for)h(some)f Fq(z)669 542 y Fo(0)736 527 y Fp(2)28 b Fr(C)p Fi(,)35 b(we)g(have)942 747 y Fp(k)p Fq(B)1066 762 y Fo(1)1105 747 y Ft(\()p Fq(z)1188 762 y Fo(0)1250 747 y Fp(\000)22 b Fq(A)p Ft(\))1460 706 y Fl(\000)p Fo(1)1555 747 y Fp(k)27 b Fq(<)h Ft(1)p Fq(;)156 b Fp(k)p Ft(\()p Fq(z)2101 762 y Fo(0)2162 747 y Fp(\000)23 b Fq(A)p Ft(\))2373 706 y Fl(\000)p Fo(1)2467 747 y Fq(B)2541 762 y Fo(1)2581 747 y Fp(k)k Fq(<)h Ft(1)p Fq(;)0 967 y Fi(wher)-5 b(e)34 b Fq(B)f Ft(=)28 b Fq(B)560 982 y Fo(1)621 967 y Ft(+)22 b Fq(B)793 982 y Fo(2)868 967 y Fi(and)34 b Fq(B)1131 982 y Fo(2)1205 967 y Fi(is)h(b)-5 b(ounde)g(d.)44 b(Then)1441 1187 y Ft(\()p Fq(A)22 b Ft(+)g Fq(B)5 b Ft(\))1789 1146 y Fl(\003)1856 1187 y Ft(=)28 b Fq(A)2033 1146 y Fl(\003)2095 1187 y Ft(+)22 b Fq(B)2272 1146 y Fl(\003)2311 1187 y Fq(:)1192 b Ft(\(2.37\))0 1636 y Fr(Pro)s(of.)66 b Ft(Replacing)32 b Fq(A)h Ft(with)g Fq(A)23 b Fp(\000)g Fq(z)1391 1651 y Fo(0)1431 1636 y Ft(,)33 b(where)i Fq(z)1819 1651 y Fo(0)1888 1636 y Fp(62)29 b Fq(\033)t Ft(\()p Fq(A)p Ft(\),)k(w)m(e)i(can)e(assume)h(that)f Fq(z)3170 1651 y Fo(0)3239 1636 y Ft(=)28 b(0.)46 b(W)-8 b(e)33 b(can)0 1756 y(also)f(subtract)h(the)g(b)s(ounded)g(op)s(erator)f Fq(B)1614 1771 y Fo(2)1686 1756 y Ft(from)g Fq(B)37 b Ft(without)32 b(a\013ecting)g(\(2.37\).)146 1876 y(Clearly)-8 b(,)32 b Fp(D)s Ft(\()p Fq(A)695 1840 y Fl(\003)756 1876 y Ft(+)22 b Fq(B)933 1840 y Fl(\003)973 1876 y Ft(\))27 b Fp(\033)i(D)s Ft(\()p Fq(A)1335 1840 y Fl(\003)1374 1876 y Ft(\))j(and)1325 2109 y(\()p Fq(A)22 b Ft(+)g Fq(B)5 b Ft(\))1673 2068 y Fl(\003)1713 2009 y Fj(\014)1713 2059 y(\014)1713 2109 y(\014)1740 2163 y Fl(D)r Fo(\()p Fm(A)1877 2144 y Ff(\003)1913 2163 y Fo(\))1973 2109 y Ft(=)27 b Fq(A)2149 2068 y Fl(\003)2211 2109 y Ft(+)22 b Fq(B)2388 2068 y Fl(\003)2427 2109 y Fq(:)0 2356 y Ft(W)-8 b(e)33 b(w)m(an)m(t)g(to)g(sho)m(w)g(that)1404 2477 y Fp(D)s Ft(\()p Fq(A)1595 2436 y Fl(\003)1634 2477 y Ft(\))28 b Fp(\033)g(D)s Ft(\(\()p Fq(A)22 b Ft(+)g Fq(B)5 b Ft(\))2271 2436 y Fl(\003)2310 2477 y Ft(\))p Fq(:)1155 b Ft(\(2.38\))146 2651 y(Let)37 b Fq( )j Fp(2)35 b(D)s Ft(\()p Fq(A)p Ft(\).)56 b(Since)37 b Fp(k)p Fq(A)1223 2615 y Fl(\000)p Fo(1)1318 2651 y Fq(B)5 b Fp(k)35 b Fq(<)g Ft(1,)j(w)m(e)g(kno)m(w)g(that)f Fr(1)25 b Ft(+)g Fq(A)2585 2615 y Fl(\000)p Fo(1)2679 2651 y Fq(B)42 b Ft(has)c(a)e(b)s(ounded)i(in)m(v)m(erse.)0 2772 y(Hence)c Fq( )353 2787 y Fo(1)420 2772 y Ft(:=)28 b(\(1)22 b(+)g Fq(A)831 2735 y Fl(\000)p Fo(1)925 2772 y Fq(B)5 b Ft(\))1042 2735 y Fl(\000)p Fo(1)1137 2772 y Fq( )36 b Ft(satis\014es)1571 2992 y Fp(k)p Fq( )1684 3007 y Fo(1)1723 2992 y Fp(k)28 b(\024)g Fq(C)1976 3007 y Fo(1)2015 2992 y Fp(k)p Fq( )t Fp(k)p Fq(:)1321 b Ft(\(2.39\))146 3212 y(Since)30 b Fp(k)p Fq(B)5 b(A)600 3175 y Fl(\000)p Fo(1)694 3212 y Fp(k)28 b Fq(<)f Ft(1,)j(the)g(op)s(erator)f Fr(1)16 b Ft(+)g Fq(B)5 b(A)1852 3175 y Fl(\000)p Fo(1)1976 3212 y Ft(has)30 b(also)f(a)g(b)s(ounded)h(in)m(v)m(erse.)44 b(But,)30 b(on)g Fp(D)s Ft(\()p Fq(A)p Ft(\),)0 3332 y(\()p Fr(1)20 b Ft(+)g Fq(A)283 3296 y Fl(\000)p Fo(1)377 3332 y Fq(B)5 b Ft(\))494 3296 y Fl(\000)p Fo(1)616 3332 y Ft(=)28 b Fq(A)793 3296 y Fl(\000)p Fo(1)887 3332 y Ft(\()p Fr(1)20 b Ft(+)g Fq(B)5 b(A)1249 3296 y Fl(\000)p Fo(1)1344 3332 y Ft(\))1382 3296 y Fl(\000)p Fo(1)1476 3332 y Fq(A)p Ft(.)43 b(Therefore,)33 b Fr(1)20 b Ft(+)g Fq(B)5 b(A)2411 3296 y Fl(\000)p Fo(1)2537 3332 y Ft(is)31 b(b)s(ounded)i(as)e(an)h(op)s(erator)f(on)0 3452 y Fp(D)s Ft(\()p Fq(A)p Ft(\).)43 b(Th)m(us)34 b Fq( )609 3467 y Fo(1)677 3452 y Fp(2)28 b(D)s Ft(\()p Fq(A)p Ft(\).)146 3573 y(No)m(w)33 b(let)f Fq(\036)c Fp(2)g(D)s Ft(\(\()p Fq(A)22 b Ft(+)g Fq(B)5 b Ft(\))1155 3537 y Fl(\003)1194 3573 y Ft(\).)44 b(Using)32 b(that)g Fq( )1851 3588 y Fo(1)1919 3573 y Fp(2)c(D)s Ft(\()p Fq(A)p Ft(\))f(=)h Fp(D)s Ft(\()p Fq(A)21 b Ft(+)h Fq(B)5 b Ft(\),)33 b(w)m(e)h(ha)m(v)m (e)1336 3793 y Fp(j)p Ft(\()p Fq(\036)p Fp(j)p Ft(\()p Fq(A)21 b Ft(+)h Fq(B)5 b Ft(\))p Fq( )1898 3808 y Fo(1)1938 3793 y Ft(\))p Fp(j)28 b(\024)g Fq(C)7 b Fp(k)p Fq( )2327 3808 y Fo(1)2366 3793 y Fp(k)p Fq(:)1087 b Ft(\(2.40\))146 4013 y(Clearly)-8 b(,)1190 4125 y Fp(j)p Ft(\()p Fq(\036)p Fp(j)p Fq(A )t Ft(\))p Fp(j)82 b Ft(=)27 b Fp(j)p Ft(\()p Fq(\036)p Fp(j)p Fq(A)p Ft(\(1)21 b(+)h Fq(A)2237 4089 y Fl(\000)p Fo(1)2332 4125 y Fq(B)5 b Ft(\))p Fq( )2512 4140 y Fo(1)2552 4125 y Ft(\))1630 4316 y(=)27 b(\()p Fq(\036)p Fp(j)p Ft(\()p Fq(A)22 b Ft(+)g Fq(B)5 b Ft(\))p Fq( )2268 4331 y Fo(1)2308 4316 y Ft(\))1630 4507 y Fp(\024)28 b Fq(C)7 b Fp(k)p Fq( )1925 4522 y Fo(1)1964 4507 y Fp(k)28 b(\024)g Fq(C)2217 4522 y Fo(1)2256 4507 y Fp(k)p Fq( )t Fp(k)p Fq(;)0 4680 y Ft(where)35 b(in)e(the)i(last)e(steps)i(w)m(e)g (used)g(\(2.40\))e(and)h(\(2.39\).)47 b(This)34 b(sho)m(ws)h(that)f Fq(\036)c Fp(2)g(D)s Ft(\()p Fq(A)3260 4644 y Fl(\003)3299 4680 y Ft(\),)k(and)g(ends)0 4801 y(the)f(pro)s(of)f(of)g(the)h (inclusion)e(\(2.38\).)43 b Fe(2)1841 5753 y Ft(23)p eop %%Page: 24 24 24 23 bop 0 407 a Fs(3)161 b(General)54 b(theory)0 655 y Fn(3.1)135 b(Dissipativ)l(e)47 b(op)t(erators)0 840 y Ft(W)-8 b(e)33 b(b)s(egin)f(with)0 1021 y Fr(De\014nition)k(3.1)49 b Fi(A)35 b(close)-5 b(d)34 b(op)-5 b(er)g(ator)35 b Fq(B)40 b Fi(is)34 b(c)-5 b(al)5 b(le)-5 b(d)34 b(dissip)-5 b(ative)34 b(if)h Fh(N)p Ft(\()p Fq(B)5 b Ft(\))28 b Fp(\032)p 2900 943 81 4 v 28 w Fr(C)2981 1036 y Fl(\000)3040 1021 y Fi(.)0 1203 y Ft(On)33 b Fp(D)s Ft(\()p Fq(B)5 b Ft(\))21 b Fp(\\)i(D)s Ft(\()p Fq(B)705 1167 y Fl(\003)744 1203 y Ft(\))32 b(w)m(e)i(can)f(de\014ne)927 1442 y(Re)p Fq(B)g Ft(:=)1289 1374 y(1)p 1289 1419 49 4 v 1289 1510 a(2)1348 1442 y(\()p Fq(B)27 b Ft(+)22 b Fq(B)1664 1401 y Fl(\003)1704 1442 y Ft(\))p Fq(;)212 b Ft(Im)o Fq(B)33 b Ft(:=)2358 1374 y(1)p 2344 1419 76 4 v 2344 1510 a(2i)2430 1442 y(\()p Fq(B)28 b Fp(\000)22 b Fq(B)2748 1401 y Fl(\003)2788 1442 y Ft(\))p Fq(:)0 1662 y Ft(Clearly)-8 b(,)32 b(Re)p Fq(B)37 b Ft(and)c(Im)p Fq(B)k Ft(are)c(symmetric)f(op)s(erators.)146 1782 y(In)h(all)e(our)h(applications)f(the)i(follo)m(wing)d(condition)h (will)f(b)s(e)j(satis\014ed:)322 1977 y Fp(D)s Ft(\()p Fq(B)5 b Ft(\))27 b(=)h Fp(D)s Ft(\()p Fq(B)885 1936 y Fl(\003)924 1977 y Ft(\))p Fq(;)81 b Ft(Im)p Fq(B)37 b Ft(is)c(b)s(ounded)g(and)g(Re)p Fq(B)k Ft(is)32 b(self-adjoin)m(t)f (on)i Fp(D)s Ft(\()p Fq(B)5 b Ft(\))p Fq(:)321 b Ft(\(3.41\))0 2172 y(Clearly)-8 b(,)32 b(under)h(this)f(condition)f Fq(B)38 b Ft(is)32 b(dissipativ)m(e)g(i\013)g(Im)o Fq(B)h Fp(\024)28 b Ft(0.)0 2371 y Fr(Prop)s(osition)36 b(3.2)49 b Fi(L)-5 b(et)35 b Fq(B)40 b Fi(b)-5 b(e)34 b(a)h(dissip)-5 b(ative)34 b(op)-5 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Fp(2)e(D)s Ft(\()p Fq(B)5 b Ft(\))27 b(=)g Fp(D)s Ft(\()p Fq(B)2287 2895 y Fl(\003)2326 2931 y Ft(\))33 b(and)1141 3126 y(0)28 b(=)f(\()p Fq( )t Fp(j)p Ft(\()p Fq(B)g Fp(\000)c Fq(e)p Ft(\))p Fq( )t Ft(\))28 b(=)f(\()p Fq( )t Fp(j)p Ft(\()p Fq(B)2262 3085 y Fl(\003)2323 3126 y Fp(\000)c Fq(e)p Ft(\))p Fq( )t Ft(\))p Fq(:)0 3320 y Ft(Hence)32 b(0)c(=)f(\()p Fq( )t Fp(j)p Ft(Im)o Fq(B)5 b( )t Ft(\).)43 b(Since)31 b(Im)p Fq(B)h Fp(\024)d Ft(0,)i(w)m(e)g(deriv)m(e)h(that)e(0)e(=)f(Im)p Fq(B)5 b( )t Ft(.)43 b(This)31 b(and)g(0)c(=)h(\()p Fq(B)23 b Fp(\000)c Fq(e)p Ft(\))p Fq( )0 3441 y Ft(yield)32 b(that)g Fq(e )g Ft(=)c(Re)p Fq(B)5 b( )t Ft(.)43 b(Hence)897 3635 y(Ran)p Fr(1)1128 3651 y Fl(f)p Fm(e)p Fl(g)1236 3635 y Ft(\(Re)p Fq(B)5 b Ft(\))22 b Fp(\\)h Ft(Ran)o Fr(1)1847 3651 y Fl(f)p Fo(0)p Fl(g)1957 3635 y Ft(\(Im)p Fq(B)5 b Ft(\))28 b Fp(\033)g Ft(Ran)o Fr(1)2592 3651 y Fl(f)p Fm(e)p Fl(g)2700 3635 y Ft(\()p Fq(B)5 b Ft(\))p Fq(:)0 3830 y Ft(The)34 b(inclusion)d Fp(\032)i Ft(is)f(ob)m(vious,)h (and)f(P)m(art)h(\(i\))f(follo)m(ws.)146 3950 y(By)h(P)m(art)g(\(i\))f (w)m(e)h(ha)m(v)m(e)1112 4145 y Fq(p)28 b Fp(\024)g Fr(1)1350 4160 y Fl(f)p Fm(e)p Fl(g)1458 4145 y Ft(\(Re\()p Fq(B)5 b Ft(\)\))p Fq(;)114 b(p)28 b Fp(\024)g Fr(1)2183 4160 y Fl(f)p Fo(0)p Fl(g)2293 4145 y Ft(\(Im)o(\()p Fq(B)5 b Ft(\)\))p Fq(:)0 4340 y Ft(Hence)1379 4460 y(0)27 b(=)h([)p Fq(p;)17 b Ft(Re)p Fq(B)5 b Ft(])28 b(=)f([)p Fq(p;)17 b Ft(Im)o Fq(B)5 b Ft(])p Fq(:)0 4623 y Ft(This)33 b(implies)d(\(ii\).) 146 4744 y(T)-8 b(o)43 b(establish)g(P)m(art)g(\(iii\),)g(w)m(e)h(note) f(that)g(if)f Fq(e)k Fp(2)g Fq(\033)2135 4759 y Fo(disc)2257 4744 y Ft(\()p Fq(B)5 b Ft(\))43 b(then)h Fq(e)f Ft(is)g(a)g(p)s(ole)f (of)g(\()p Fq(z)34 b Fp(\000)c Fq(B)5 b Ft(\))3658 4708 y Fl(\000)p Fo(1)3752 4744 y Ft(.)0 4864 y(By)44 b(Prop)s(osition)e (2.9)h(this)h(p)s(ole)e(is)h(simple.)76 b(Hence)45 b Fq(e)e Ft(is)g(a)h(semisimple)d(eigen)m(v)-5 b(alue)43 b(of)g Fq(B)5 b Ft(.)77 b(An)0 4984 y(application)30 b(of)i(Lemma)f(2.6)h(completes)h(the)g(pro)s(of)f(of)g(\(iii\).)40 b Fe(2)146 5147 y Ft(If)c Fq(e)h Ft(is)e(an)h(isolated)f(real)h(eigen)m (v)-5 b(alue)35 b(of)h Fq(B)41 b Ft(with)36 b(dim)15 b Fr(1)2285 5163 y Fl(f)p Fm(e)p Fl(g)2393 5147 y Ft(\()p Fq(B)5 b Ft(\))33 b(=)h Fp(1)p Ft(,)j(then)g(Ker\()p Fq(B)29 b Fp(\000)c Fq(e)p Ft(\))37 b(ma)m(y)0 5268 y(b)s(e)c(strictly) f(smaller)e(than)j(Ran)p Fr(1)1262 5283 y Fl(f)p Fm(e)p Fl(g)1369 5268 y Ft(\()p Fq(B)5 b Ft(\).)43 b(\(W)-8 b(e)33 b(thank)g(E.)g(Skibsted)h(for)e(p)s(oin)m(ting)f(this)h(out)g (to)h(us\).)1841 5753 y(24)p eop %%Page: 25 25 25 24 bop 0 407 a Fr(Prop)s(osition)36 b(3.3)49 b Fi(L)-5 b(et)32 b Fq(B)37 b Fi(b)-5 b(e)32 b(a)g(b)-5 b(ounde)g(d)32 b(dissip)-5 b(ative)31 b(op)-5 b(er)g(ator)32 b(on)g(a)g(Hilb)-5 b(ert)32 b(sp)-5 b(ac)g(e)32 b Fp(H)h Fi(such)f(that)0 527 y Fq(\033)t Ft(\()p Fq(B)5 b Ft(\))22 b Fp(\\)h Fr(R)k Fp(\032)h Fq(\033)596 542 y Fo(disc)719 527 y Ft(\()p Fq(B)5 b Ft(\))p Fi(.)44 b(Then)1067 760 y Fq(c)28 b Ft(:=)50 b(sup)1267 838 y Fm(z)s Fl(2)p Fd(C)1409 847 y Fk(+)1476 660 y Fj(\015)1476 710 y(\015)1476 760 y(\015)p Ft(\()p Fq(z)27 b Fp(\000)c Fq(B)5 b Ft(\))1849 719 y Fl(\000)p Fo(1)1943 760 y Fr(1)1999 775 y Fm(\033)r Fo(\()p Fm(B)s Fo(\))p Fl(n)p Fd(R)2253 760 y Ft(\()p Fq(B)g Ft(\))2408 660 y Fj(\015)2408 710 y(\015)2408 760 y(\015)28 b Fq(<)g Fp(1)p Fq(;)817 b Ft(\(3.42\))0 1055 y Fi(and,)34 b(for)h Fq(z)d Fp(2)p 546 977 81 4 v 28 w Fr(C)627 1070 y Fo(+)708 1055 y Fp(n)22 b Fq(\033)t Ft(\()p Fq(B)5 b Ft(\))p Fi(,)980 1320 y Fp(k)p Ft(\()p Fq(z)27 b Fp(\000)c Fq(B)5 b Ft(\))1357 1279 y Fl(\000)p Fo(1)1451 1320 y Fp(k)28 b(\024)g Ft(max)1832 1224 y Fj(\020)2235 1253 y Ft(1)p 1891 1297 736 4 v 1891 1388 a(dist\()p Fq(z)t(;)17 b(\033)t Ft(\()p Fq(B)5 b Ft(\))23 b Fp(\\)f Fr(R)p Ft(\))2637 1320 y Fq(;)17 b(c)2723 1224 y Fj(\021)2772 1320 y Fq(:)731 b Ft(\(3.43\))0 1768 y Fr(Pro)s(of.)65 b Ft(By)33 b(Prop)s(osition)e (3.2)1311 1988 y Fr(1)1367 2004 y Fm(\033)r Fo(\()p Fm(B)s Fo(\))p Fl(\\)p Fd(R)1661 1988 y Ft(=)1875 1905 y Fj(X)1764 2093 y Fm(e)p Fl(2)p Fm(\033)r Fo(\()p Fm(B)s Fo(\))p Fl(\\)p Fd(R)2123 1988 y Fr(1)2179 2004 y Fl(f)p Fm(e)p Fl(g)2286 1988 y Ft(\()p Fq(B)5 b Ft(\))p Fq(;)0 2295 y Ft(is)32 b(an)h(orthogonal)d(pro)5 b(jection.)44 b(Therefore)263 2515 y Fp(k)p Ft(\()p Fq(z)27 b Fp(\000)c Fq(B)5 b Ft(\))640 2474 y Fl(\000)p Fo(1)734 2515 y Fp(k)28 b Ft(=)f(max)1113 2419 y Fj(\020)1163 2515 y Fp(k)p Ft(\()p Fq(z)g Fp(\000)22 b Fq(B)5 b Ft(\))1539 2474 y Fl(\000)p Fo(1)1634 2515 y Fr(1)1690 2531 y Fm(\033)r Fo(\()p Fm(B)s Fo(\))p Fl(\\)p Fd(R)1956 2515 y Ft(\()p Fq(B)g Ft(\))p Fp(k)p Fq(;)17 b Fp(k)p Ft(\()p Fq(z)26 b Fp(\000)c Fq(B)5 b Ft(\))2580 2474 y Fl(\000)p Fo(1)2675 2515 y Fr(1)2731 2531 y Fm(\033)r Fo(\()p Fm(B)s Fo(\))p Fl(n)p Fd(R)2985 2515 y Ft(\()p Fq(B)g Ft(\))p Fp(k)3190 2419 y Fj(\021)3240 2515 y Fq(:)263 b Ft(\(3.44\))146 2748 y(Also)32 b(b)m(y)i(Prop)s(osition)d(3.2,)h Fq(B)5 b Fr(1)1346 2763 y Fm(\033)r Fo(\()p Fm(B)s Fo(\))p Fl(\\)p Fd(R)1612 2748 y Ft(\()p Fq(B)g Ft(\))33 b(is)f(self-adjoin)m (t.)42 b(Hence)934 3020 y Fp(k)p Ft(\()p Fq(z)27 b Fp(\000)22 b Fq(B)5 b Ft(\))1310 2978 y Fl(\000)p Fo(1)1405 3020 y Fr(1)1461 3035 y Fm(\033)r Fo(\()p Fm(B)s Fo(\))p Fl(\\)p Fd(R)1727 3020 y Ft(\()p Fq(B)g Ft(\))p Fp(k)27 b Ft(=)2416 2952 y(1)p 2073 2996 V 2073 3088 a(dist\()p Fq(z)t(;)17 b(\033)t Ft(\()p Fq(B)5 b Ft(\))22 b Fp(\\)h Fr(R)p Ft(\))2818 3020 y Fq(:)685 b Ft(\(3.45\))146 3295 y(Since)33 b Fq(B)5 b Fr(1)536 3311 y Fm(\033)r Fo(\()p Fm(B)s Fo(\))p Fl(n)p Fd(R)791 3295 y Ft(\()p Fq(B)g Ft(\))33 b(is)f(a)h(b)s(ounded)h(op)s (erator)e(with)h(sp)s(ectrum)g(con)m(tained)g(in)f Fr(C)3237 3310 y Fl(\000)3296 3295 y Ft(,)h(w)m(e)h(clearly)0 3416 y(ha)m(v)m(e)g(\(3.42\).)43 b(No)m(w)33 b(\(3.44\),)f(\(3.42\))g(and)g (\(3.45\))g(imply)f(\(3.43\).)42 b Fe(2)146 3586 y Ft(The)34 b(follo)m(wing)c(result)i(is)g(an)h(immediate)d(consequence)35 b(of)d(the)h(previous)g(prop)s(osition.)0 3789 y Fr(Prop)s(osition)j (3.4)49 b Fi(L)-5 b(et)31 b Fq(B)36 b Fi(b)-5 b(e)30 b(a)h(b)-5 b(ounde)g(d)30 b(dissip)-5 b(ative)30 b(op)-5 b(er)g(ator)30 b(such)h(that)g Fq(\033)t Ft(\()p Fq(B)5 b Ft(\))13 b Fp(\\)g Fr(R)28 b Fp(\032)g Fq(\033)3472 3804 y Fo(disc)3595 3789 y Ft(\()p Fq(B)5 b Ft(\))p Fi(,)0 3910 y(and)29 b(let)h Fq(c)f Fi(b)-5 b(e)30 b(the)f(c)-5 b(onstant)29 b(de\014ne)-5 b(d)29 b(in)g(\(3.42\).)42 b(If)29 b Fq(V)51 b Fi(is)30 b(a)f(b)-5 b(ounde)g(d)29 b(op)-5 b(er)g(ator)29 b(such)h(that)f Fq(c)p Fp(k)p Fq(V)22 b Fp(k)27 b Fq(<)h Ft(1)p Fi(,)0 4030 y(then)1085 4151 y Fq(\033)t Ft(\()p Fq(B)g Ft(+)22 b Fq(V)f Ft(\))h Fp(\\)p 1609 4072 81 4 v 23 w Fr(C)1690 4166 y Fo(+)1777 4151 y Fp(\032)p 1882 4073 80 4 v 28 w Fq(B)5 b Ft(\()p Fq(\033)t Ft(\()p Fq(B)g Ft(\))22 b Fp(\\)h Fr(R)p Fq(;)17 b Fp(k)p Fq(V)j Fp(k)p Ft(\))p Fq(:)0 4325 y Fi(F)-7 b(urthermor)i(e,)34 b(for)h Fq(z)d Fp(2)p 920 4247 81 4 v 28 w Fr(C)1001 4340 y Fo(+)1082 4325 y Fp(n)p 1154 4247 80 4 v 22 w Fq(B)5 b Ft(\()p Fq(\033)t Ft(\()p Fq(B)g Ft(\))22 b Fp(\\)h Fr(R)p Fq(;)17 b Fp(k)p Fq(V)k Fp(k)p Ft(\))34 b Fi(one)h(has)f(the)h(b)-5 b(ound)922 4589 y Fp(k)p Ft(\()p Fq(z)26 b Fp(\000)d Fq(B)k Fp(\000)c Fq(V)f Ft(\))1499 4548 y Fl(\000)p Fo(1)1593 4589 y Fp(k)27 b(\024)2279 4522 y Ft(1)p 1785 4566 1036 4 v 1785 4658 a(dist\()p Fq(z)t(;)17 b(\033)t Ft(\()p Fq(B)5 b Ft(\))23 b Fp(\\)f Fr(R)p Ft(\))g Fp(\000)h(k)p Fq(V)e Fp(k)2831 4589 y Fq(:)672 b Ft(\(3.46\))1841 5753 y(25)p eop %%Page: 26 26 26 25 bop 0 407 a Fn(3.2)135 b(The)45 b(F)-11 b(esh)l(bac)l(h)44 b(form)l(ula)0 592 y Ft(Let)35 b Fp(H)g Ft(b)s(e)g(a)f(Hilb)s(ert)f (space)i(decomp)s(osed)g(in)m(to)f(a)g(direct)g(sum)g Fp(H)e Ft(=)f Fp(H)2738 555 y Fo(v)2804 592 y Fp(\010)23 b(H)p 2989 517 43 4 v 2989 555 a Fo(v)3032 592 y Ft(.)49 b(The)35 b(pro)5 b(jections)0 712 y(on)m(to)29 b Fp(H)301 676 y Fo(v)372 712 y Ft(and)g Fp(H)p 643 637 V 643 676 a Fo(v)714 712 y Ft(w)m(e)h(denote)f(b)m(y)h Fr(1)1352 676 y Fo(vv)1460 712 y Ft(and)f Fr(1)p 1702 637 39 4 v -36 x Fo(v)p 1740 637 V 1 w(v)1783 712 y Ft(.)42 b(In)29 b(this)f(section)h(w)m(e)h(study)g(op)s(erators)e(of)h(the)g(form)1457 985 y Fq(H)36 b Ft(=)1677 839 y Fj(")1768 912 y Fq(H)1857 876 y Fo(vv)2019 912 y Fq(H)2108 876 y Fo(v)p 2146 838 V 1 w(v)1767 1057 y Fq(H)p 1856 982 V 1856 1020 a Fo(v)q(v)2019 1057 y Fq(H)p 2108 982 V 2108 1020 a Fo(v)p 2146 982 V 1 w(v)2230 839 y Fj(#)2295 985 y Fq(;)1208 b Ft(\(3.47\))0 1271 y(W)-8 b(e)34 b(assume)h(that)e Fq(H)809 1235 y Fo(vv)922 1271 y Ft(and)h Fq(H)p 1202 1196 V 1202 1235 a Fo(v)p 1240 1196 V 1 w(v)1316 1271 y Ft(are)g(closed)g(op)s(erators)f (on)h Fp(H)2423 1235 y Fo(v)2499 1271 y Ft(and)g Fp(H)p 2775 1196 43 4 v 2775 1235 a Fo(v)2818 1271 y Ft(;)g(moreo)m(v)m(er)g (w)m(e)h(supp)s(ose)0 1391 y(that)j Fq(H)p 306 1317 39 4 v 306 1355 a Fo(v)q(v)424 1391 y Ft(:)f Fp(H)573 1355 y Fo(v)653 1391 y Fp(!)g(H)p 875 1317 43 4 v 875 1355 a Fo(v)956 1391 y Ft(and)h Fq(H)1240 1355 y Fo(v)p 1278 1317 39 4 v 1 w(v)1358 1391 y Ft(:)g Fp(H)p 1508 1317 43 4 v 1508 1355 a Fo(v)1587 1391 y Fp(!)g(H)1810 1355 y Fo(v)1890 1391 y Ft(are)h(b)s(ounded)g(op)s(erators.)60 b(Clearly)38 b Fq(H)45 b Ft(is)38 b(closed)0 1512 y(and)33 b Fp(D)s Ft(\()p Fq(H)8 b Ft(\))26 b(=)i Fp(D)s Ft(\()p Fq(H)772 1476 y Fo(vv)851 1512 y Ft(\))22 b Fp(\010)g(D)s Ft(\()p Fq(H)p 1217 1437 39 4 v 1217 1476 a Fo(v)p 1255 1437 V 1 w(v)1297 1512 y Ft(\).)146 1632 y(F)-8 b(or)32 b(an)m(y)h Fq(z)g Fp(62)28 b Fq(\033)t Ft(\()p Fq(H)p 863 1557 V 863 1596 a Fo(v)p 901 1557 V 1 w(v)943 1632 y Ft(\))k(w)m(e)i(de\014ne)1146 1838 y Fq(W)1238 1853 y Fo(v)1281 1838 y Ft(\()p Fq(z)t Ft(\))84 b(:=)27 b Fq(H)1709 1802 y Fo(v)p 1747 1763 V 1 w(v)1789 1838 y Ft(\()p Fq(z)t Fr(1)p 1932 1763 V -36 x Fo(v)p 1971 1763 V 2 w(v)2036 1838 y Fp(\000)22 b Fq(H)p 2224 1763 V 2224 1802 a Fo(v)p 2262 1763 V 1 w(v)2305 1838 y Ft(\))2343 1802 y Fl(\000)p Fo(1)2437 1838 y Fq(H)p 2526 1763 V 2526 1802 a Fo(v)q(v)2606 1838 y Fq(;)1162 2029 y(G)1239 2044 y Fo(v)1281 2029 y Ft(\()p Fq(z)t Ft(\))84 b(:=)27 b Fq(z)t 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3941 a Fo(v)p 1612 3902 V 1 w(v)1654 3982 y Ft(\))1692 3941 y Fl(\000)p Fo(1)1786 3982 y Fq(;)147 b(B)5 b Ft(\()p Fq(z)t Ft(\))29 b(:=)e Fr(1)22 b Ft(+)g(\()p Fq(z)t Fr(1)p 2642 3902 V -41 x Fo(v)p 2681 3902 V 2 w(v)2746 3982 y Fp(\000)g Fq(H)p 2934 3902 V 2934 3941 a Fo(v)p 2972 3902 V 1 w(v)3015 3982 y Ft(\))3053 3941 y Fl(\000)p Fo(1)3147 3982 y Fq(H)p 3236 3902 V 3236 3941 a Fo(v)q(v)3316 3982 y Fq(:)0 4191 y Ft(Both)33 b Fq(A)p Ft(\()p Fq(z)t Ft(\))g(and)f Fq(B)5 b Ft(\()p Fq(z)t Ft(\))34 b(ha)m(v)m(e)g(b)s(ounded)f(in)m(v)m(erses:)368 4400 y Fq(A)441 4359 y Fl(\000)p Fo(1)535 4400 y Ft(\()p Fq(z)t Ft(\))28 b(=)g Fr(1)22 b Fp(\000)h Fq(H)1059 4359 y Fo(v)p 1097 4320 V 1 w(v)1139 4400 y Ft(\()p Fq(z)t Fr(1)p 1282 4320 V -41 x Fo(v)p 1321 4320 V 2 w(v)1385 4400 y Fp(\000)g Fq(H)p 1574 4320 V 1574 4359 a Fo(v)p 1612 4320 V 1 w(v)1654 4400 y Ft(\))1692 4359 y Fl(\000)p Fo(1)1786 4400 y Fq(;)147 b(B)2039 4359 y Fl(\000)p Fo(1)2134 4400 y Ft(\()p Fq(z)t Ft(\))28 b(=)f Fr(1)c Fp(\000)f 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b Fp(62)39 b Fq(\033)t Ft(\()p Fq(H)8 b Ft(\),)41 b(then)f(the)f(left)g(hand)g(side)h(of)e(\(3.50\))h (is)g(in)m(v)m(ertible.)62 b(Hence)41 b(so)e(is)g(the)h(righ)m(t)e (hand)0 5268 y(side.)44 b(This)32 b(implies)e(that)j Fq(G)1076 5283 y Fo(v)1118 5268 y Ft(\()p Fq(z)t Ft(\))g(is)f(in)m(v)m (ertible.)1841 5753 y(26)p eop %%Page: 27 27 27 26 bop 146 407 a Ft(Next)31 b(w)m(e)g(will)c(use)k(the)f(iden)m(tit) m(y)g(\(also)f(understo)s(o)s(d)h(in)f(the)h(sense)i(of)d(op)s(erators) h(from)e Fp(D)s Ft(\()p Fq(H)8 b Ft(\))29 b(to)0 527 y Fp(H)q Ft(\):)887 648 y(\()p Fq(z)e Fp(\000)c Fq(H)8 b Ft(\))27 b(=)h Fq(A)1428 606 y Fl(\000)p Fo(1)1522 648 y Ft(\()p Fq(z)t Ft(\)\()p Fq(G)1762 663 y Fo(v)1805 648 y Ft(\()p Fq(z)t Ft(\))23 b(+)f Fq(z)t Fr(1)p 2156 568 39 4 v -42 x Fo(v)p 2195 568 V 2 w(v)2259 648 y Fp(\000)h Fq(H)p 2448 568 V 2448 606 a Fo(v)p 2486 568 V 1 w(v)2528 648 y Ft(\))p Fq(B)2645 606 y Fl(\000)p Fo(1)2739 648 y Ft(\()p Fq(z)t Ft(\))p Fq(:)639 b Ft(\(3.51\))0 812 y(If)33 b(0)c Fp(62)g Fq(\033)t Ft(\()p Fq(G)445 827 y Fo(v)488 812 y Ft(\()p Fq(z)t Ft(\)\),)34 b(then)g(the)g(righ)m(t)e (hand)i(side)f(of)g(\(3.51\))g(is)f(in)m(v)m(ertible.)45 b(Hence)35 b(so)f(is)e Fq(z)c Fp(\000)23 b Fq(H)8 b Ft(.)45 b(This)0 933 y(completes)32 b(the)h(pro)s(of)f(of)g(\(i\).)146 1053 y(T)-8 b(o)33 b(sho)m(w)h(\(ii\))c(w)m(e)k(note)f(that)f(if)f Fq(z)i Fp(62)28 b Fq(\033)t Ft(\()p Fq(H)8 b Ft(\),)32 b(\(3.50\))g(or)g(\(3.51\))g(implies)824 1251 y(\()p Fq(z)26 b Fp(\000)d Fq(H)8 b Ft(\))1160 1210 y Fl(\000)p Fo(1)1281 1251 y Ft(=)28 b Fq(B)5 b Ft(\()p Fq(z)t Ft(\)\()p Fq(G)1704 1210 y Fl(\000)p Fo(1)1704 1275 y(v)1799 1251 y Ft(\()p Fq(z)t Ft(\))23 b(+)f(\()p Fq(z)t Fr(1)p 2188 1171 V -41 x Fo(v)p 2227 1171 V 2 w(v)2291 1251 y Fp(\000)h Fq(H)p 2480 1171 V 2480 1210 a Fo(v)p 2518 1171 V 1 w(v)2560 1251 y Ft(\))2598 1210 y Fl(\000)p Fo(1)2692 1251 y Ft(\))p Fq(A)p Ft(\()p Fq(z)t Ft(\))p Fq(;)0 1449 y Ft(whic)m(h,)33 b(after)g(substituting)f(the)h(expressions)h(de\014ning)e Fq(A)p 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Fo(v)p 2531 1778 V 1 w(v)2573 1853 y Ft(\),)39 b(then)f Fq(W)2996 1868 y Fo(v)3038 1853 y Ft(\()p Fq(x)p Ft(\))g(is)f(self-adjoin)m(t)0 1973 y(and)1025 2051 y(d)p 998 2095 110 4 v 998 2187 a(d)p Fq(x)1117 2118 y(W)1209 2133 y Fo(v)1252 2118 y Ft(\()p Fq(x)p Ft(\))28 b(=)f Fp(\000)p Fq(H)1680 2077 y Fo(v)p 1718 2039 39 4 v 1 w(v)1761 2118 y Ft(\()p Fq(x)p Fr(1)p 1910 2039 V -41 x Fo(v)p 1948 2039 V 1 w(v)2013 2118 y Fp(\000)c Fq(H)p 2202 2039 V 2202 2077 a Fo(v)p 2239 2039 V(v)2282 2118 y Ft(\))2320 2077 y Fl(\000)p Fo(2)2414 2118 y Fq(H)p 2503 2039 V 2503 2077 a Fo(v)q(v)2611 2118 y Fp(\024)28 b Ft(0)p Fq(:)738 b Ft(\(3.52\))146 2309 y(The)33 b(rest)f(of)e(this)h (section)h(is)f(dev)m(oted)h(to)f(v)-5 b(arious)31 b(re\014nemen)m(ts)i (of)d(Prop)s(osition)g(3.5.)43 b(The)32 b(\014rst)0 2429 y(result)h(in)e(this)i(direction)e(is)0 2613 y Fr(Theorem)74 b(3.6)49 b Fi(Assume)35 b(that)g Fq(e)28 b Fp(2)g Fr(R)22 b Fp(n)g Fq(\033)t Ft(\()p Fq(H)p 1798 2539 V 1798 2577 a Fo(v)p 1836 2539 V 1 w(v)1878 2613 y Ft(\))p Fi(.)45 b(Then,)0 2734 y Ft(\(i\))34 b Fq(e)h Fi(is)g(an)f(eigenvalue)g(of)g(H) i(i\013)e Ft(0)h Fi(is)f(an)h(eigenvalue)f(of)g Fq(G)2269 2749 y Fo(v)2312 2734 y Ft(\()p Fq(e)p Ft(\))p Fi(.)0 2854 y Ft(\(ii\))f(dim)15 b Fr(1)400 2870 y Fl(f)p Fm(e)p Fl(g)508 2854 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)g(dim)15 b Fr(1)1038 2870 y Fl(f)p Fo(0)p Fl(g)1148 2854 y Ft(\()p Fq(G)1263 2869 y Fo(v)1306 2854 y Ft(\()p Fq(e)p Ft(\)\))p Fi(.)0 2974 y Ft(\(iii\))32 b Fi(Set)j Fq(p)28 b Ft(=)f Fr(1)595 2990 y Fl(f)p Fo(0)p Fl(g)705 2974 y Ft(\()p Fq(G)820 2989 y Fo(v)862 2974 y Ft(\()p Fq(e)p Ft(\)\))p Fi(.)45 b(Then)34 b Fq(pW)1505 2938 y Fl(0)1491 2999 y Fo(v)1534 2974 y Ft(\()p Fq(e)p Ft(\))p Fq(p)h Fi(is)f(a)h(ne)-5 b(gative)34 b(op)-5 b(er)g(ator)34 b(and)695 3192 y Fr(1)751 3207 y Fl(f)p Fm(e)p Fl(g)859 3192 y Ft(\()p Fq(H)8 b Ft(\))82 b(=)1210 3095 y Fj(\020)1259 3192 y Fq(p)22 b Ft(+)g(\()p Fq(e)p Fr(1)p 1567 3117 V -36 x Fo(v)p 1606 3117 V 2 w(v)1670 3192 y Fp(\000)h Fq(H)p 1859 3117 V 1859 3156 a Fo(v)p 1897 3117 V 1 w(v)1939 3192 y Ft(\))1977 3156 y Fl(\000)p Fo(1)2071 3192 y Fq(H)p 2160 3117 V 2160 3156 a Fo(v)q(v)2240 3192 y Fq(p)2289 3095 y Fj(\021)1106 3398 y Fp(\002)17 b Ft(\()p Fq(p)22 b Fp(\000)h Fq(pW)1564 3362 y Fl(0)1550 3423 y Fo(v)1592 3398 y Ft(\()p Fq(e)p Ft(\))p Fq(p)p Ft(\))1800 3350 y Fl(\000)p Fo(1)1911 3302 y Fj(\020)1961 3398 y Fq(p)f Ft(+)g Fq(pH)2268 3362 y Fo(v)p 2306 3324 V 1 w(v)2348 3398 y Ft(\()p Fq(e)p Fr(1)p 2487 3324 V -36 x Fo(v)p 2525 3324 V 1 w(v)2590 3398 y Fp(\000)h Fq(H)p 2779 3324 V 2779 3362 a Fo(v)p 2816 3324 V(v)2859 3398 y Ft(\))2897 3362 y Fl(\000)p Fo(1)2991 3302 y Fj(\021)3057 3398 y Fq(:)3530 3295 y Ft(\(3.53\))0 3608 y(\(iv\))34 b Fi(If)833 3742 y Fq(u)28 b Ft(:=)f(\()p Fq(p)22 b Fp(\000)h Fq(pW)1411 3701 y Fl(0)1397 3767 y Fo(v)1439 3742 y Ft(\()p Fq(e)p Ft(\))p Fq(p)p Ft(\))1647 3701 y Fl(\000)1712 3674 y Fk(1)p 1712 3686 31 4 v 1712 3727 a(2)1773 3646 y Fj(\020)1823 3742 y Fq(p)f Ft(+)g Fq(pH)2130 3701 y Fo(v)p 2168 3662 39 4 v 1 w(v)2210 3742 y Ft(\()p Fq(e)p Fr(1)p 2349 3662 V -41 x Fo(v)p 2387 3662 V 1 w(v)2452 3742 y Fp(\000)h Fq(H)p 2641 3662 V 2641 3701 a Fo(v)p 2678 3662 V(v)2721 3742 y Ft(\))2759 3701 y Fl(\000)p Fo(1)2853 3646 y Fj(\021)2919 3742 y Fq(;)0 3907 y Fi(then)35 b Fq(u)f Fi(is)h(a)f(p)-5 b(artial)35 b(isometry)g(and)1280 4105 y Fq(uu)1392 4063 y Fl(\003)1458 4105 y Ft(=)27 b Fq(p;)226 b(u)1919 4063 y Fl(\003)1958 4105 y Fq(u)27 b Ft(=)h Fr(1)2201 4120 y Fl(f)p Fm(e)p Fl(g)2308 4105 y Ft(\()p Fq(H)8 b Ft(\))p Fq(:)0 4487 y Fr(Pro)s(of.)65 b Ft(Let)33 b Fq(H)8 b( )31 b Ft(=)c Fq(e )t Ft(.)44 b(Then)1395 4676 y Fq(H)1484 4640 y Fo(vv)1563 4676 y Fq( )1630 4640 y Fo(v)1694 4676 y Ft(+)22 b Fq(H)1881 4640 y Fo(v)p 1919 4601 V 1 w(v)1962 4676 y Fq( )p 2029 4601 43 4 v 2029 4640 a Fo(v)2099 4676 y Ft(=)27 b Fq(e )2314 4640 y Fo(v)2357 4676 y Fq(;)1395 4867 y(H)p 1484 4792 39 4 v 1484 4831 a Fo(vv)1564 4867 y Fq( )1631 4831 y Fo(v)1695 4867 y Ft(+)22 b Fq(H)p 1882 4792 V 1882 4831 a Fo(v)p 1920 4792 V 1 w(v)1963 4867 y Fq( )p 2030 4792 43 4 v 2030 4831 a Fo(v)2100 4867 y Ft(=)27 b Fq(e )p 2315 4792 V 2315 4831 a Fo(v)2358 4867 y Fq(:)3530 4772 y Ft(\(3.54\))0 5070 y(The)34 b(second)f (equation)g(giv)m(es)g(\(recall)e(that)h Fq(e)c Fp(62)g Fq(\033)t Ft(\()p Fq(H)p 2015 4995 39 4 v 2015 5034 a Fo(v)p 2053 4995 V 1 w(v)2095 5070 y Ft(\)\))1295 5268 y Fq( )p 1362 5188 43 4 v 1362 5227 a Fo(v)1432 5268 y Ft(=)g(\()p Fq(e)p Fr(1)p 1675 5188 39 4 v -41 x Fo(v)p 1713 5188 V 1 w(v)1778 5268 y Fp(\000)22 b Fq(H)p 1966 5188 V 1966 5227 a Fo(v)p 2004 5188 V 1 w(v)2046 5268 y Ft(\))2084 5227 y Fl(\000)p Fo(1)2179 5268 y Fq(H)p 2268 5188 V 2268 5227 a Fo(vv)2348 5268 y Fq( )2415 5227 y Fo(v)2457 5268 y Fq(:)1046 b Ft(\(3.55\))1841 5753 y(27)p eop %%Page: 28 28 28 27 bop 0 407 a Ft(Inserting)33 b(this)f(iden)m(tit)m(y)g(in)m(to)g (the)h(\014rst)g(equation)f(of)h(\(3.54\))e(w)m(e)j(get)1611 627 y Fq(G)1688 642 y Fo(v)1731 627 y Ft(\()p Fq(e)p Ft(\))p Fq( )1919 586 y Fo(v)1989 627 y Ft(=)27 b(0)p Fq(:)1362 b Ft(\(3.56\))0 847 y(No)m(w,)50 b(if)45 b Fq( )436 811 y Fo(v)529 847 y Fp(2)51 b(D)s Ft(\()p Fq(H)853 811 y Fo(vv)932 847 y Ft(\))46 b(satis\014es)h(\(3.56\))e(and)h Fq( )p 1954 772 43 4 v 1954 811 a Fo(v)2043 847 y Ft(is)f(giv)m(en)i (in)e(terms)h(of)g Fq( )3027 811 y Fo(v)3115 847 y Ft(b)m(y)h (\(3.55\),)i(then)0 967 y Fq( )k Ft(=)48 b Fq( )307 931 y Fo(v)380 967 y Fp(\010)31 b Fq( )p 555 892 V 555 931 a Fo(v)642 967 y Ft(satis\014es)46 b Fq(H)8 b( )52 b Ft(=)c Fq(e )t Ft(,)g(whic)m(h)e(follo)m(ws)d(b)m(y)j(a)e(simple)g (computation.)78 b(Th)m(us,)50 b(the)0 1088 y(pro)5 b(jection)1724 1208 y Fq( )32 b Fp(7!)27 b Fq( )2013 1167 y Fo(v)3530 1208 y Ft(\(3.57\))0 1382 y(restricted)33 b(to)f(Ran)p Fr(1)784 1398 y Fl(f)p Fm(e)p Fl(g)892 1382 y Ft(\()p Fq(H)8 b Ft(\))31 b(is)i(a)f(bijection)f(on)m(to)i(Ran)o Fq(p)p Ft(.)44 b(This)33 b(yields)f(\(i\))f(and)i(\(ii\).)146 1503 y(De\014ne)g Fq(w)e Ft(:)c Fp(H)688 1467 y Fo(v)758 1503 y Fp(7!)h(H)971 1467 y Fo(v)1035 1503 y Fp(\010)23 b(H)p 1220 1428 V 1220 1467 a Fo(v)1295 1503 y Ft(b)m(y)33 b(setting)1246 1723 y Fq(w)d Ft(:=)e Fq(p)22 b Ft(+)g(\()p Fq(e)p Fr(1)p 1785 1643 39 4 v -41 x Fo(v)p 1823 1643 V 1 w(v)1888 1723 y Fp(\000)g Fq(H)p 2076 1643 V 2076 1682 a Fo(v)p 2114 1643 V 1 w(v)2156 1723 y Ft(\))2194 1682 y Fl(\000)p Fo(1)2289 1723 y Fq(H)p 2378 1643 V 2378 1682 a Fo(vv)2458 1723 y Fq(p:)0 1943 y Ft(Then)34 b Fq(w)h Ft(is)d(the)h(in)m(v)m(erse)h(of)e(the)h(map)f(\(3.57\).)42 b(Besides,)1108 2163 y Fq(w)1181 2122 y Fl(\003)1220 2163 y Fq(w)30 b Ft(=)d Fq(p)22 b Ft(+)h Fq(pH)1731 2122 y Fo(v)p 1769 2083 V 1 w(v)1811 2163 y Ft(\()p Fq(e)p Fr(1)p 1950 2083 V -41 x Fo(v)p 1988 2083 V 1 w(v)2053 2163 y Fp(\000)f Fq(H)p 2241 2083 V 2241 2122 a Fo(v)p 2279 2083 V 1 w(v)2321 2163 y Ft(\))2359 2122 y Fl(\000)p Fo(2)2454 2163 y Fq(H)p 2543 2083 V 2543 2122 a Fo(vv)2623 2163 y Fq(p)0 2383 y Ft(restricted)35 b(to)g(Ran)p Fq(p)f Ft(is)g(a)h(p)s(ositiv)m(e,)g(in)m(v)m(ertible)f(op)s(erator.)49 b(With)35 b(a)f(sligh)m(t)g(abuse)h(of)g(the)g(notation)0 2503 y(w)m(e)i(denote)g(its)f(in)m(v)m(erse)i(b)m(y)f(\()p Fq(w)1182 2467 y Fl(\003)1220 2503 y Fq(w)s Ft(\))1331 2467 y Fl(\000)p Fo(1)1425 2503 y Ft(.)55 b(One)36 b(easily)g(c)m(hec)m (ks)j(that)d Fq(w)s Ft(\()p Fq(w)2693 2467 y Fl(\003)2731 2503 y Fq(w)s Ft(\))2842 2467 y Fl(\000)p Fo(1)2935 2503 y Fq(w)3008 2467 y Fl(\003)3083 2503 y Ft(is)g(an)g(orthogonal)0 2623 y(pro)5 b(jection)32 b(on)h(Ran)p Fq(w)d Ft(=)d(Ran)p Fr(1)1208 2639 y Fl(f)p Fm(e)p Fl(g)1315 2623 y Ft(\()p Fq(H)8 b Ft(\).)43 b(Hence)1377 2844 y Fr(1)1433 2859 y Fl(f)p Fm(e)p Fl(g)1541 2844 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)g Fq(w)s Ft(\()p Fq(w)2020 2802 y Fl(\003)2059 2844 y Fq(w)s Ft(\))2170 2802 y Fl(\000)p Fo(1)2263 2844 y Fq(w)2336 2802 y Fl(\003)2375 2844 y Fq(:)0 3080 y Ft(This)33 b(sho)m(ws)h(\(iii\).)41 b(P)m(art)32 b(\(iv\))g(follo)m(ws)g(from)f (the)i(iden)m(tit)m(y)f Fq(u)c Ft(=)f(\()p Fq(w)2515 3043 y Fl(\003)2554 3080 y Fq(w)s Ft(\))2665 3043 y Fl(\000)2730 3016 y Fk(1)p 2729 3028 31 4 v 2729 3069 a(2)2774 3080 y Fq(w)2847 3043 y Fl(\003)2885 3080 y Ft(.)44 b Fe(2)146 3250 y Ft(Due)38 b(to)f(the)i(assumption)e Fq(e)f Fp(62)h Fq(\033)t Ft(\()p Fq(H)p 1545 3175 39 4 v 1545 3214 a Fo(v)p 1583 3175 V 1 w(v)1625 3250 y Ft(\),)i(the)f(pro)s(of)f(of)g (Theorem)h(3.6)g(w)m(as)g(relativ)m(ely)f(simple.)0 3370 y(Related)32 b(results)h(are)g(subtler)f(if)g Fq(e)c Fp(2)g Fq(\033)t Ft(\()p Fq(H)p 1600 3295 V 1600 3334 a Fo(v)p 1637 3295 V(v)1680 3370 y Ft(\).)43 b(As)33 b(a)g(w)m(arm)f(up,)h(w)m(e)g(pro)m(v)m(e)0 3574 y Fr(Prop)s(osition)j (3.7)49 b Fi(L)-5 b(et)35 b Fq(e)28 b Fp(2)g Fr(R)p Fi(.)44 b(Supp)-5 b(ose)34 b(that)h(the)g(limit)1282 3794 y Ft(lim)1296 3854 y Fm(y)r Fl(#)p Fo(0)1434 3794 y Fq(W)1526 3809 y Fo(v)1568 3794 y Ft(\()p Fq(e)23 b Ft(+)f(i)p Fq(y)t Ft(\))k(=:)h Fq(W)2138 3809 y Fo(v)2181 3794 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))1031 b(\(3.58\))0 4057 y Fi(exists.)44 b(Then)34 b Fq(W)652 4072 y Fo(v)695 4057 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))34 b Fi(is)g(dissip)-5 b(ative.)146 4178 y(Assume)35 b(mor)-5 b(e)g(over)34 b(that)h Fq(e)g Fi(is)g(not)g(an)f (eigenvalue)g(of)h Fq(H)p 2301 4103 V 2301 4142 a Fo(v)p 2338 4103 V(v)2381 4178 y Fi(.)45 b(Then)1090 4398 y Ft(dim)15 b(Ker\()p Fq(H)29 b Fp(\000)23 b Fq(e)p Ft(\))28 b Fp(\024)g Ft(dim)15 b(Ker)p Fq(G)2303 4413 y Fo(v)2346 4398 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))p Fq(:)839 b Ft(\(3.59\))0 4618 y Fi(In)34 b(p)-5 b(articular,)35 b(if)g Fq(e)g Fi(is)f(an)h(eigenvalue)e(of)i Fq(H)8 b Fi(,)34 b(then)h Ft(0)g Fi(is)f(an)h(eigenvalue)e(of)i Fq(G)2967 4633 y Fo(v)3009 4618 y Ft(\()p Fq(e)23 b Ft(+)f(i0\))p Fi(.)0 4821 y Fr(Pro)s(of.)83 b Ft(Assume)42 b(that)f Fq( )47 b Fp(2)c(D)s Ft(\()p Fq(H)8 b Ft(\))40 b(and)i Fq(H)8 b( )46 b Ft(=)d Fq(e )t Ft(.)70 b(Since)41 b Fq(e)h Ft(is)f(not)g(an)g (eigen)m(v)-5 b(alue)41 b(of)g Fq(H)p 3672 4746 V 3672 4785 a Fo(v)p 3710 4746 V 1 w(v)3752 4821 y Ft(,)0 4942 y Fq(e)p Fr(1)p 101 4867 V -37 x Fo(v)p 139 4867 V 1 w(v)204 4942 y Fp(\000)23 b Fq(H)p 393 4867 V 393 4905 a Fo(v)p 430 4867 V(v)505 4942 y Ft(is)32 b(injectiv)m(e)h(and)1295 5162 y Fq( )p 1362 5082 43 4 v 1362 5120 a Fo(v)1432 5162 y Ft(=)28 b(\()p Fq(e)p Fr(1)p 1675 5082 39 4 v -42 x Fo(v)p 1713 5082 V 1 w(v)1778 5162 y Fp(\000)22 b Fq(H)p 1966 5082 V 1966 5120 a Fo(v)p 2004 5082 V 1 w(v)2046 5162 y Ft(\))2084 5120 y Fl(\000)p Fo(1)2179 5162 y Fq(H)p 2268 5082 V 2268 5120 a Fo(vv)2348 5162 y Fq( )2415 5120 y Fo(v)2457 5162 y Fq(:)1046 b Ft(\(3.60\))1841 5753 y(28)p eop %%Page: 29 29 29 28 bop 0 407 a Ft(Moreo)m(v)m(er,)824 527 y(0)27 b(=)h Fr(1)1060 543 y Fl(f)p Fm(e)p Fl(g)1167 527 y Ft(\()p Fq(H)p 1294 447 39 4 v 1294 486 a Fo(v)p 1332 447 V 1 w(v)1374 527 y Ft(\))g(=)f Fp(\000)p Ft(s)d Fp(\000)e Ft(lim)1795 587 y Fm(y)r Fl(#)p Fo(0)1933 527 y Ft(i)p Fq(y)t Ft(\(\()p Fq(e)f Ft(+)h(i)p Fq(y)t Ft(\))p Fr(1)p 2427 447 V -41 x Fo(v)p 2463 447 V -1 w(v)2528 527 y Fp(\000)g Fq(H)p 2716 447 V 2716 486 a Fo(v)p 2754 447 V 1 w(v)2796 527 y Ft(\))2834 486 y Fl(\000)p Fo(1)2929 527 y Fq(:)574 b Ft(\(3.61\))0 730 y(Therefore,)373 935 y(s)22 b Fp(\000)h Ft(lim)669 950 y Fm(y)r Fl(#)p Fo(0)797 839 y Fj(\020)847 935 y Ft(\(\()p Fq(e)f Ft(+)g(i)p Fq(y)t Ft(\))p Fr(1)p 1262 861 V -36 x Fo(v)p 1298 861 V -1 w(v)1363 935 y Fp(\000)h Fq(H)p 1552 861 V 1552 899 a Fo(v)p 1589 861 V(v)1632 935 y Ft(\))1670 899 y Fl(\000)p Fo(1)1786 935 y Fp(\000)g Ft(\()p Fq(e)p Fr(1)p 2025 861 V -36 x Fo(v)p 2063 861 V 1 w(v)2128 935 y Fp(\000)f Fq(H)p 2316 861 V 2316 899 a Fo(v)p 2354 861 V 1 w(v)2396 935 y Ft(\))2434 899 y Fl(\000)p Fo(1)2529 839 y Fj(\021)2595 935 y Fq(H)p 2684 861 V 2684 899 a Fo(v)q(v)2764 935 y Fq( )2831 899 y Fo(v)373 1127 y Ft(=)27 b Fp(\000)p Ft(s)c Fp(\000)g Ft(lim)850 1142 y Fm(y)r Fl(#)p Fo(0)978 1127 y Ft(i)p Fq(y)t Ft(\(\()p Fq(e)e Ft(+)h(i)p Fq(y)t Ft(\))p Fr(1)p 1472 1052 V -36 x Fo(v)p 1508 1052 V -1 w(v)1573 1127 y Fp(\000)g Fq(H)p 1761 1052 V 1761 1091 a Fo(v)p 1799 1052 V 1 w(v)1841 1127 y Ft(\))1879 1091 y Fl(\000)p Fo(1)1974 1127 y Ft(\()p Fq(e)p Fr(1)p 2113 1052 V -36 x Fo(v)p 2151 1052 V 1 w(v)2216 1127 y Fp(\000)g Fq(H)p 2404 1052 V 2404 1091 a Fo(v)p 2442 1052 V 1 w(v)2484 1127 y Ft(\))2522 1091 y Fl(\000)p Fo(1)2617 1127 y Fq(H)p 2706 1052 V 2706 1091 a Fo(vv)2786 1127 y Fq( )2853 1091 y Fo(v)2978 1127 y Ft(=)28 b(0)p Fq(:)3530 1024 y Ft(\(3.62\))0 1324 y(Com)m(bining)j(\(3.60\))h(and)g(\(3.62\))g(w)m(e)i(get)1010 1522 y Fq( )p 1077 1442 43 4 v 1077 1481 a Fo(v)1147 1522 y Ft(=)28 b(s)22 b Fp(\000)h Ft(lim)1425 1582 y Fm(y)r Fl(#)p Fo(0)1547 1522 y Ft(\(\()p Fq(e)f Ft(+)g(i)p Fq(y)t Ft(\))p Fr(1)p 1962 1442 39 4 v -41 x Fo(v)p 1998 1442 V -1 w(v)2063 1522 y Fp(\000)g Fq(H)p 2251 1442 V 2251 1481 a Fo(v)p 2289 1442 V 1 w(v)2331 1522 y Ft(\))2369 1481 y Fl(\000)p Fo(1)2464 1522 y Fq(H)p 2553 1442 V 2553 1481 a Fo(vv)2633 1522 y Fq( )2700 1481 y Fo(v)2742 1522 y Fq(:)0 1764 y Ft(Substituting)32 b(this)g(iden)m(tit)m(y)g(in)m (to)g(the)h(\014rst)g(equation)g(in)e(\(3.54\))h(w)m(e)i(deriv)m(e)f (that)1513 1962 y Fq(G)1590 1977 y Fo(v)1633 1962 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))p Fq( )2018 1921 y Fo(v)2087 1962 y Ft(=)27 b(0)p Fq(:)146 2160 y Ft(No)m(w)33 b(assume)h(that)e Fq( )984 2124 y Fo(v)1054 2160 y Ft(=)c(0.)43 b(Then)33 b Fq( )f Fp(2)c(H)p 1805 2085 43 4 v 1805 2124 a Fo(v)1880 2160 y Ft(and)33 b(therefore)1493 2358 y Fq(e )f Ft(=)c Fq(H)8 b( )31 b Ft(=)c Fq(H)p 2112 2278 39 4 v 2112 2317 a Fo(v)p 2150 2278 V 1 w(v)2192 2358 y Fq( )t(:)0 2556 y Ft(Since)34 b Fq(e)d Fp(62)g Fq(\033)484 2571 y Fo(pp)567 2556 y Ft(\()p Fq(H)p 694 2481 V 694 2520 a Fo(v)p 732 2481 V 1 w(v)774 2556 y Ft(\),)k Fq( )f Ft(=)c(0.)48 b(Th)m(us,)37 b(the)d(pro)5 b(jection)34 b Fq( )h Fp(7!)30 b Fq( )2407 2520 y Fo(v)2483 2556 y Ft(restricted)35 b(to)f(Ran)p Fr(1)3271 2572 y Fl(f)p Fm(e)p Fl(g)3378 2556 y Ft(\()p Fq(H)8 b Ft(\))34 b(is)g(an)0 2676 y(injectiv)m(e)e(map) g(in)m(to)g(Ker)p Fq(G)1036 2691 y Fo(v)1079 2676 y Ft(\()p Fq(x)22 b Ft(+)g(i0\).)42 b Fe(2)146 2840 y Ft(The)34 b(follo)m(wing)c(theorem)i(is)g(the)h(principal)e(result)h(of)g(this)g (section.)0 3044 y Fr(Theorem)74 b(3.8)49 b Fi(Assume)39 b(that)g Fq(e)g Fi(is)f(not)g(an)h(eigenvalue)e(of)h Fq(H)p 2467 2969 V 2467 3008 a Fo(v)p 2505 2969 V 1 w(v)2547 3044 y Fi(,)i(that)f(the)f(limit)g(\(3.58\))g(exists,)0 3164 y(and)c(that)i(the)e(function)1399 3284 y Fr(C)1480 3299 y Fo(+)1561 3284 y Fp([)22 b(f)p Fq(e)p Fp(g)28 b(3)g Fq(z)k Fp(7!)c Fq(W)2213 3299 y Fo(v)2255 3284 y Ft(\()p Fq(z)t Ft(\))0 3449 y Fi(is)36 b(in)h(the)f(class)g Fq(C)704 3413 y Fo(1)744 3449 y Ft(\()p Fr(C)863 3464 y Fo(+)945 3449 y Fp([)24 b(f)p Fq(e)p Fp(g)p Ft(\))36 b Fi(\(its)h(derivative)f(at)h Fq(z)e Ft(=)c Fq(e)37 b Fi(we)f(denote)g(by)h Fq(W)2964 3413 y Fl(0)2950 3474 y Fo(v)2992 3449 y Ft(\()p Fq(e)24 b Ft(+)f(i0\))p Fi(\).)49 b(F)-7 b(urther,)0 3570 y(assume)34 b(that)h Ft(0)28 b Fp(2)g Fq(\033)766 3585 y Fo(disc)889 3570 y Ft(\()p Fq(G)1004 3585 y Fo(v)1046 3570 y Ft(\()p Fq(e)22 b Ft(+)g(i0\)\))p Fi(.)44 b(Then)34 b(the)h(fol)5 b(lowing)33 b(holds:)0 3690 y Ft(\(i\))h Fq(e)h Fi(is)g(an)f(eigenvalue)g(of)g Fq(H)8 b Fi(.)0 3811 y Ft(\(ii\))33 b(dim)15 b Fr(1)400 3826 y Fl(f)p Fm(e)p Fl(g)508 3811 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)g(dim)15 b(Ker)q Fq(G)1217 3826 y Fo(v)1259 3811 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))p Fi(.)0 3931 y Ft(\(iii\))34 b Fi(Set)j Fq(p)32 b Ft(=)f Fr(1)607 3946 y Fl(f)p Fo(0)p Fl(g)718 3931 y Ft(\()p Fq(G)833 3946 y Fo(v)875 3931 y Ft(\()p Fq(e)24 b Ft(+)f(i0\)\))p Fi(.)50 b(Then,)37 b Fq(p)g Fi(is)g(the)g(ortho)-5 b(gonal)36 b(pr)-5 b(oje)g(ction)36 b(onto)h Ft(Ker\()p Fq(G)3387 3946 y Fo(v)3429 3931 y Ft(\()p Fq(e)24 b Ft(+)g(i0\))p Fi(,)0 4051 y(the)35 b(op)-5 b(er)g(ator)1359 4172 y Fq(pW)1500 4187 y Fo(v)1543 4172 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))p Fq(p)k Ft(=:)i Fq(pW)2208 4187 y Fo(v)2250 4172 y Ft(\()p Fq(e)p Ft(\))p Fq(p)0 4337 y Fi(is)35 b(self-adjoint,)e(and)h(the)h(op)-5 b(er)g(ator)1359 4535 y Fq(pW)1514 4494 y Fl(0)1500 4559 y Fo(v)1543 4535 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))p Fq(p)k Ft(=:)i Fq(pW)2222 4494 y Fl(0)2208 4559 y Fo(v)2250 4535 y Ft(\()p Fq(e)p Ft(\))p Fq(p)0 4733 y Fi(is)35 b(ne)-5 b(gative.)0 4853 y Ft(\(iv\))34 b Fi(The)h(limits)712 5049 y Ft(lim)708 5105 y Fm(y)r Fl(!)p Fo(0)851 5049 y Ft(\(\()p Fq(e)22 b Ft(+)g(i)p Fq(y)t Ft(\))p Fr(1)p 1266 4974 V -37 x Fo(v)p 1303 4974 V(v)1367 5049 y Fp(\000)h Fq(H)p 1556 4974 V 1556 5012 a Fo(v)p 1594 4974 V 1 w(v)1636 5049 y Ft(\))1674 5012 y Fl(\000)p Fo(1)1768 5049 y Fq(H)p 1857 4974 V 1857 5012 a Fo(v)q(v)1937 5049 y Fq(p)83 b Ft(=:)28 b(\()p Fq(e)p Fr(1)p 2339 4974 V -37 x Fo(v)p 2377 4974 V 1 w(v)2442 5049 y Fp(\000)23 b Fq(H)p 2631 4974 V 2631 5012 a Fo(v)p 2668 4974 V(v)2711 5049 y Ft(\))2749 5012 y Fl(\000)p Fo(1)2843 5049 y Fq(H)p 2932 4974 V 2932 5012 a Fo(v)q(v)3012 5049 y Fq(p)695 5240 y Ft(lim)691 5296 y Fm(y)r Fl(!)p Fo(0)851 5240 y Fq(pH)989 5204 y Fo(v)p 1027 5165 V 1 w(v)1069 5240 y Ft(\(\()p Fq(e)f Ft(+)g(i)p Fq(y)t Ft(\))p Fr(1)p 1484 5165 V -36 x Fo(v)p 1521 5165 V(v)1585 5240 y Fp(\000)h Fq(H)p 1774 5165 V 1774 5204 a Fo(v)p 1812 5165 V 1 w(v)1854 5240 y Ft(\))1892 5204 y Fl(\000)p Fo(1)2069 5240 y Ft(=:)28 b Fq(pH)2338 5204 y Fo(v)p 2376 5165 V 1 w(v)2418 5240 y Ft(\()p Fq(e)p Fr(1)p 2557 5165 V -36 x Fo(v)p 2596 5165 V 2 w(v)2660 5240 y Fp(\000)23 b Fq(H)p 2849 5165 V 2849 5204 a Fo(v)p 2886 5165 V(v)2929 5240 y Ft(\))2967 5204 y Fl(\000)p Fo(1)3061 5240 y Fq(;)3530 5166 y Ft(\(3.63\))1841 5753 y(29)p eop %%Page: 30 30 30 29 bop 0 407 a Fi(exist)35 b(and)704 515 y Fr(1)760 530 y Fl(f)p Fm(e)p Fl(g)867 515 y Ft(\()p Fq(H)8 b Ft(\))82 b(=)1218 418 y Fj(\020)1268 515 y Fq(p)22 b Ft(+)g(\()p Fq(e)p Fr(1)p 1576 440 39 4 v -36 x Fo(v)p 1614 440 V 1 w(v)1679 515 y Fp(\000)g Fq(H)p 1867 440 V 1867 479 a Fo(v)p 1905 440 V 1 w(v)1947 515 y Ft(\))1985 479 y Fl(\000)p Fo(1)2080 515 y Fq(H)p 2169 440 V 2169 479 a Fo(vv)2249 515 y Fq(p)2298 418 y Fj(\021)1114 721 y Fp(\002)p Ft(\()p Fq(p)h Fp(\000)g Fq(pW)1556 685 y Fl(0)1542 746 y Fo(v)1584 721 y Ft(\()p Fq(e)p Ft(\))p Fq(p)p Ft(\))1792 685 y Fl(\000)p Fo(1)1903 625 y Fj(\020)1953 721 y Fq(p)f Ft(+)g Fq(pH)2260 685 y Fo(v)p 2298 647 V 1 w(v)2340 721 y Ft(\()p Fq(e)p Fr(1)p 2479 647 V -36 x Fo(v)p 2517 647 V 1 w(v)2582 721 y Fp(\000)g Fq(H)p 2770 647 V 2770 685 a Fo(v)p 2808 647 V 1 w(v)2850 721 y Ft(\))2888 685 y Fl(\000)p Fo(1)2983 625 y Fj(\021)3049 721 y Fq(:)3530 618 y Ft(\(3.64\))0 908 y(\(v\))35 b Fi(If)833 1042 y Fq(u)28 b Ft(:=)f(\()p Fq(p)22 b Fp(\000)h Fq(pW)1411 1000 y Fl(0)1397 1066 y Fo(v)1439 1042 y Ft(\()p Fq(e)p Ft(\))p Fq(p)p Ft(\))1647 1000 y Fl(\000)1712 973 y Fk(1)p 1712 985 31 4 v 1712 1027 a(2)1773 945 y Fj(\020)1823 1042 y Fq(p)f Ft(+)g Fq(pH)2130 1000 y Fo(v)p 2168 962 39 4 v 1 w(v)2210 1042 y Ft(\()p Fq(e)p Fr(1)p 2349 962 V -42 x Fo(v)p 2387 962 V 1 w(v)2452 1042 y Fp(\000)h Fq(H)p 2641 962 V 2641 1000 a Fo(v)p 2678 962 V(v)2721 1042 y Ft(\))2759 1000 y Fl(\000)p Fo(1)2853 945 y Fj(\021)2919 1042 y Fq(;)0 1216 y Fi(then)35 b Fq(u)f Fi(is)h(a)f(p)-5 b(artial)35 b(isometry)g(and)1280 1436 y Fq(uu)1392 1394 y Fl(\003)1458 1436 y Ft(=)27 b Fq(p;)226 b(u)1919 1394 y Fl(\003)1958 1436 y Fq(u)27 b Ft(=)h Fr(1)2201 1451 y Fl(f)p Fm(e)p Fl(g)2308 1436 y Ft(\()p Fq(H)8 b Ft(\))p Fq(:)0 1883 y Fr(Pro)s(of.)65 b Ft(W)-8 b(e)33 b(b)s(egin)f(with)g(pro) s(ofs)g(of)g(P)m(arts)i(\(iii\))29 b(and)k(\(iv\).)146 2004 y(W)-8 b(e)30 b(\014rst)g(observ)m(e)h(that)e(the)h(relation)d Fq(W)1686 1968 y Fl(\003)1672 2028 y Fo(v)1726 2004 y Ft(\()p Fq(z)t Ft(\))h(=)f Fq(W)2074 2019 y Fo(v)2117 2004 y Ft(\()p 2155 1951 50 4 v Fq(z)5 b Ft(\))29 b(yields)g(that)g (the)h(functions)f Fq(W)3425 2019 y Fo(v)3467 2004 y Ft(\()p Fq(z)t Ft(\))h(and)0 2124 y Fq(G)77 2139 y Fo(v)119 2124 y Ft(\()p Fq(z)t Ft(\))j(are)g(also)f(of)g(the)h(class)f Fq(C)1220 2088 y Fo(1)1260 2124 y Ft(\()p Fr(C)1379 2139 y Fl(\000)1460 2124 y Fp([)22 b(f)p Fq(e)p Fp(g)p Ft(\).)146 2244 y(Since)29 b(the)g(op)s(erator)g Fp(\000)p Fq(G)1105 2259 y Fo(v)1147 2244 y Ft(\()p Fq(e)14 b Ft(+)g(i0\))29 b(is)f(dissipativ)m(e)g(and)h(0)f Fp(2)g Fq(\033)2458 2259 y Fo(disc)2580 2244 y Ft(\()p Fq(G)2695 2259 y Fo(v)2738 2244 y Ft(\()p Fq(e)14 b Ft(+)g(i0\)\),)29 b(Prop)s(osition)e(3.2)0 2365 y(yields)32 b(that)h Fq(p)f Ft(is)g(an)h(orthogonal)d(pro)5 b(jection.)44 b(It)32 b(follo)m(ws)g(that)1447 2585 y Fr(1)1503 2600 y Fl(f)p Fo(0)p Fl(g)1613 2585 y Ft(\()p Fq(G)1728 2543 y Fl(\003)1728 2609 y Fo(v)1770 2585 y Ft(\()p Fq(e)23 b Ft(+)f(i0\)\))k(=)i Fq(p:)0 2804 y Ft(Since)33 b Fq(G)332 2768 y Fl(\003)332 2829 y Fo(v)374 2804 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))27 b(=)g Fq(G)899 2819 y Fo(v)942 2804 y Ft(\()p Fq(e)22 b Fp(\000)h Ft(i0\),)31 b(the)i(iden)m(tities)f Fq(pG)2035 2819 y Fo(v)2077 2804 y Ft(\()p Fq(e)22 b Fp(\006)h Ft(i0\))p Fq(p)j Ft(=)i(0)k(can)h(b)s(e)g (rewritten)f(as)1213 3024 y Fq(p)p Ft(\()p Fq(e)p Fr(1)1401 2983 y Fo(vv)1503 3024 y Fp(\000)22 b Fq(H)1691 2983 y Fo(vv)1770 3024 y Ft(\))p Fq(p)28 b Ft(=)g Fq(pW)2130 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Fq(pW)2222 5227 y Fl(0)2208 5292 y Fo(v)2250 5268 y Ft(\()p Fq(e)p Ft(\))p Fq(p)1110 b Ft(\(3.66\))1841 5753 y(30)p eop %%Page: 31 31 31 30 bop 0 407 a Ft(is)30 b(a)g(self-adjoin)m(t)e(op)s(erator.)42 b(This)31 b(pro)m(v)m(es)h(\(iii\))27 b(except)32 b(for)e(the)h(part)f (asserting)g(that)g Fq(pW)3406 371 y Fl(0)3392 431 y Fo(v)3435 407 y Ft(\()p Fq(e)p Ft(\))p Fq(p)g Ft(is)g(a)0 527 y(negativ)m(e)j(op)s(erator.)43 b(Using)32 b Fq(p)27 b Ft(=)h Fq(p)1319 491 y Fl(\003)1358 527 y Ft(,)33 b Fq(W)1524 491 y Fl(0)1510 552 y Fo(v)1552 527 y Ft(\()p Fq(e)23 b Fp(\000)f Ft(i0\))27 b(=)g Fq(W)2108 491 y Fl(0)2094 552 y Fo(v)2137 527 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))2455 491 y Fl(\003)2526 527 y Ft(and)33 b(\(3.66\),)f(w)m(e)h(also)f(ha)m(v) m(e)i(that)1260 724 y Fq(pW)1415 683 y Fl(0)1401 749 y Fo(v)1444 724 y Ft(\()p Fq(e)22 b Fp(\000)h Ft(i0\))p Fq(p)j Ft(=)i Fq(pW)2098 683 y Fl(0)2084 749 y Fo(v)2126 724 y Ft(\()p Fq(e)22 b Ft(+)g(i0\))p Fq(p:)1010 b Ft(\(3.67\))146 921 y(W)-8 b(e)33 b(no)m(w)h(sho)m(w)f(that)g(the)g(limit)654 1117 y(\()p Fq(e)p Fr(1)p 793 1038 39 4 v -41 x Fo(v)p 831 1038 V 1 w(v)896 1117 y Fp(\000)22 b Fq(H)p 1084 1038 V 1084 1076 a Fo(v)p 1122 1038 V 1 w(v)1165 1117 y Ft(\))1203 1076 y Fl(\000)p Fo(1)1297 1117 y Fq(H)p 1386 1038 V 1386 1076 a Fo(v)q(v)1466 1117 y Fq(p)28 b Ft(:=)105 b(lim)1673 1177 y Fm(y)r Fl(!)p Fo(0)p Fm(;y)r Fl(6)p Fo(=0)1963 1117 y Ft(\(\()p Fq(e)23 b Ft(+)f(i)p Fq(y)t Ft(\))p Fr(1)p 2379 1038 V -41 x Fo(v)p 2415 1038 V -1 w(v)2480 1117 y Fp(\000)g Fq(H)p 2668 1038 V 2668 1076 a Fo(v)p 2706 1038 V 1 w(v)2748 1117 y Ft(\))2786 1076 y Fl(\000)p Fo(1)2881 1117 y Fq(H)p 2970 1038 V 2970 1076 a Fo(vv)3050 1117 y Fq(p;)404 b Ft(\(3.68\))0 1358 y(exists.)43 b(Denote)29 b(the)h(expression)g(inside)f(the)h (limit)25 b(b)m(y)30 b Fq(L)p Ft(\()p Fq(y)t Ft(\).)42 b(Then,)31 b(the)f(resolv)m(en)m(t)g(iden)m(tit)m(y)f(yields)128 1552 y(\()p Fq(L)232 1516 y Fl(\003)271 1552 y Ft(\()p Fq(y)357 1567 y Fo(1)396 1552 y Ft(\))22 b Fp(\000)h 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Fk(2)1806 1969 y Fo(+)p Fm(y)1896 1978 y Fk(1)1930 1969 y Fo(\))1984 1934 y Ft(\()p Fq(pW)2163 1949 y Fo(v)2206 1934 y Ft(\()p Fq(e)g Fp(\000)g Ft(i)p Fq(y)2486 1949 y Fo(1)2525 1934 y Ft(\))p Fq(p)g Fp(\000)g Fq(pW)2874 1949 y Fo(v)2917 1934 y Ft(\()p Fq(e)g Ft(+)g(i)p Fq(y)3196 1949 y Fo(2)3234 1934 y Ft(\))p Fq(p)p Ft(\))1602 2126 y Fp(\000)1806 2086 y Fo(1)p 1689 2102 V 1689 2160 a(i\()p Fm(y)1771 2169 y Fk(1)1806 2160 y Fo(+)p Fm(y)1896 2169 y Fk(2)1930 2160 y Fo(\))1984 2126 y Ft(\()p Fq(pW)2163 2141 y Fo(v)2206 2126 y Ft(\()p Fq(e)g Fp(\000)g Ft(i)p Fq(y)2486 2141 y Fo(2)2525 2126 y Ft(\))p Fq(p)g Fp(\000)g Fq(pW)2874 2141 y Fo(v)2917 2126 y Ft(\()p Fq(e)g Ft(+)g(i)p Fq(y)3196 2141 y Fo(1)3234 2126 y Ft(\))p Fq(p)p Ft(\))16 b Fq(:)3530 1847 y Ft(\(3.69\))0 2336 y(Note)33 b(that)f(it)g(follo)m (ws)f(from)g(\(3.65\))h(and)h(\(3.67\))f(that)g(the)h(function)1211 2533 y Fr(C)1292 2548 y Fo(+)1373 2533 y Fp([)23 b Fr(C)1543 2548 y Fl(\000)1624 2533 y Fp([)f(f)p Fq(e)p Fp(g)28 b(3)g Fq(z)k Fp(7!)c Fq(pW)2325 2548 y Fo(v)2367 2533 y Ft(\()p Fq(z)t Ft(\))p Fq(p;)962 b Ft(\(3.70\))0 2729 y(is)25 b(con)m(tin)m(uously)i(di\013eren)m(tiable.)40 b(This)26 b(observ)-5 b(ation)25 b(and)h(iden)m(tit)m(y)g(\(3.69\))f (yield)h(that)f(the)i(sequence)0 2850 y Fq(L)p Ft(\()p Fq(y)152 2865 y Fm(n)199 2850 y Ft(\))43 b(is)f(Cauc)m(h)m(y)j(whenev)m (er)g Fq(y)1234 2865 y Fm(n)1326 2850 y Fp(!)g Ft(0.)74 b(Th)m(us,)47 b(the)d(limit)39 b(\(3.68\))j(exists)i(and)f(this)g (implies)d(the)0 2970 y(existence)34 b(of)e(the)h(limits)d(\(3.63\))i (\(for)g(the)h(second)g(limit)c(tak)m(e)34 b(the)f(adjoin)m(t)e(of)h (\(3.68\)\).)43 b(Since)676 3167 y Fq(pW)831 3126 y Fl(0)817 3192 y Fo(v)860 3167 y Ft(\()p Fq(e)p Ft(\))p Fq(p)28 b Ft(=)f Fp(\000)p Ft(\(\()p Fq(e)p Fr(1)p 1415 3087 39 4 v -41 x Fo(v)p 1454 3087 V 2 w(v)1518 3167 y Fp(\000)c Fq(H)p 1707 3087 V 1707 3126 a Fo(v)p 1745 3087 V 1 w(v)1787 3167 y Ft(\))1825 3126 y Fl(\000)p Fo(1)1920 3167 y Fq(H)p 2009 3087 V 2009 3126 a Fo(vv)2089 3167 y Fq(p)p Ft(\))2176 3126 y Fl(\003)2215 3167 y Ft(\()p Fq(e)p Fr(1)p 2354 3087 V -41 x Fo(v)p 2392 3087 V 1 w(v)2457 3167 y Fp(\000)g Fq(H)p 2646 3087 V 2646 3126 a Fo(v)p 2683 3087 V(v)2726 3167 y Ft(\))2764 3126 y Fl(\000)p Fo(1)2858 3167 y Fq(H)p 2947 3087 V 2947 3126 a Fo(v)q(v)3027 3167 y Fq(p;)0 3364 y Ft(it)32 b(follo)m(ws)f(that)h(the)h(op)s(erator)f Fq(pW)1345 3328 y Fl(0)1331 3388 y Fo(v)1374 3364 y Ft(\()p Fq(e)p Ft(\))p Fq(p)g Ft(is)g(negativ)m(e.)44 b(This)33 b(completes)f(the)h(pro)s(of)f(of)g(P)m(art)h(\(iii\).)146 3484 y(Set)f Fq(w)e Ft(:=)e Fq(p)20 b Ft(+)f(\()p Fq(e)p Fr(1)p 847 3409 V -36 x Fo(v)p 886 3409 V 2 w(v)948 3484 y Fp(\000)h Fq(H)p 1134 3409 V 1134 3448 a Fo(v)p 1172 3409 V 1 w(v)1214 3484 y Ft(\))1252 3448 y Fl(\000)p Fo(1)1346 3484 y Fq(H)p 1435 3409 V 1435 3448 a Fo(v)q(v)1516 3484 y Fq(p)31 b Ft(and)g Fq(w)s Ft(\()p Fq(z)t Ft(\))d(:=)g Fq(p)19 b Ft(+)h(\()p Fq(z)t Fr(1)p 2448 3409 V -36 x Fo(v)p 2487 3409 V 2 w(v)2549 3484 y Fp(\000)g Fq(H)p 2735 3409 V 2735 3448 a Fo(v)p 2773 3409 V 1 w(v)2815 3484 y Ft(\))2853 3448 y Fl(\000)p Fo(1)2948 3484 y Fq(H)p 3037 3409 V 3037 3448 a Fo(vv)3117 3484 y Fq(p)p Ft(.)43 b(By)32 b(\(3.63\))e(w)m(e)0 3604 y(ha)m(v)m(e)1502 3725 y(lim)1516 3785 y Fm(y)r Fl(#)p Fo(0)1655 3725 y Fq(w)s Ft(\()p Fq(e)21 b Ft(+)h(i)p Fq(y)t Ft(\))k(=)i Fq(w)s(:)1252 b Ft(\(3.71\))0 3927 y(W)-8 b(e)33 b(easily)f(compute)g(that)1336 4048 y Fq(H)8 b(w)s Ft(\()p Fq(z)t Ft(\))27 b(=)g Fq(z)t(w)s Ft(\()p Fq(z)t Ft(\))c Fp(\000)g Fq(G)2200 4063 y Fo(v)2242 4048 y Ft(\()p Fq(z)t Ft(\))p Fq(p:)0 4212 y Ft(Hence)1436 4333 y(lim)1449 4393 y Fm(y)r Fl(#)p Fo(0)1588 4333 y Fq(H)8 b(w)s Ft(\()p Fq(e)21 b Ft(+)h(i)p Fq(y)t Ft(\))k(=)h Fq(ew)s(:)1186 b Ft(\(3.72\))0 4541 y(\(3.71\))43 b(and)h(\(3.72\))f (imply)f(that)h(Ran)p Fq(w)49 b Fp(\032)f Ft(Ker\()p Fq(H)37 b Fp(\000)30 b Fq(e)p Ft(\).)78 b(Th)m(us)45 b Fq(w)h Ft(maps)e(Ker)p Fq(G)3196 4556 y Fo(v)3238 4541 y Ft(\()p Fq(e)30 b Ft(+)g(i0\))42 b(in)m(to)0 4661 y(Ker\()p Fq(H)32 b Fp(\000)24 b Fq(e)p Ft(\).)52 b(Clearly)-8 b(,)35 b(the)g(in)m(v)m(erse)i(of)e Fq(w)i Ft(is)e Fr(1)1806 4625 y Fo(vv)1921 4661 y Ft(restricted)h(to)f(Ker\()p Fq(H)c Fp(\000)25 b Fq(e)p Ft(\).)51 b(Therefore,)37 b Fq(w)h Ft(is)d(a)0 4782 y(bijectiv)m(e)f(map)f(from)f(Ker)p Fq(G)1072 4797 y Fo(v)1115 4782 y Ft(\()p Fq(e)23 b Ft(+)g(i0\))32 b(to)i(Ker\()p Fq(H)c Fp(\000)24 b Fq(e)p Ft(\).)47 b(Similarly)29 b(as)34 b(at)g(the)g(end)h(of)e(the)h(pro)s(of)f(of)0 4902 y(Theorem)g(3.6,)f(w)m(e)i(note)f(that)701 5090 y Fq(w)774 5054 y Fl(\003)813 5090 y Fq(w)85 b Ft(=)28 b Fq(p)22 b Ft(+)g(\(\()p Fq(e)p Fr(1)p 1418 5016 V -36 x Fo(v)p 1456 5016 V 1 w(v)1521 5090 y Fp(\000)h Fq(H)p 1710 5016 V 1710 5054 a Fo(v)p 1747 5016 V(v)1790 5090 y Ft(\))1828 5054 y Fl(\000)p Fo(1)1922 5090 y Fq(H)p 2011 5016 V 2011 5054 a Fo(v)q(v)2091 5090 y Fq(p)p Ft(\))2178 5054 y Fl(\003)2217 5090 y Ft(\()p Fq(e)p Fr(1)p 2356 5016 V -36 x Fo(v)p 2395 5016 V 2 w(v)2459 5090 y Fp(\000)g Fq(H)p 2648 5016 V 2648 5054 a Fo(v)p 2686 5016 V 1 w(v)2728 5090 y Ft(\))2766 5054 y Fl(\000)p Fo(1)2860 5090 y Fq(H)p 2949 5016 V 2949 5054 a Fo(v)q(v)3029 5090 y Fq(p)968 5282 y Ft(=)28 b Fq(p)22 b Fp(\000)h Fq(pW)1398 5245 y Fl(0)1384 5306 y Fo(v)1426 5282 y Ft(\()p Fq(e)p Ft(\))p Fq(p;)1841 5753 y Ft(31)p eop %%Page: 32 32 32 31 bop 0 407 a Ft(and)1377 527 y Fr(1)1433 543 y Fl(f)p Fm(e)p Fl(g)1541 527 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)g Fq(w)s Ft(\()p Fq(w)2020 486 y Fl(\003)2059 527 y Fq(w)s Ft(\))2170 486 y Fl(\000)p Fo(1)2263 527 y Fq(w)2336 486 y Fl(\003)2375 527 y Fq(:)0 696 y Ft(This)37 b(pro)m(v)m(es)h (Relation)d(\(3.64\))g(and)i(completes)f(the)h(pro)s(of)f(of)g(P)m(art) h(\(iv\).)55 b(P)m(art)37 b(\(v\))f(follo)m(ws)g(from)0 816 y(the)d(iden)m(tit)m(y)1542 937 y Fq(u)27 b Ft(=)h(\()p Fq(w)1840 896 y Fl(\003)1879 937 y Fq(w)s Ft(\))1990 896 y Fl(\000)2055 868 y Fk(1)p 2054 880 31 4 v 2054 922 a(2)2098 937 y Fq(w)2171 896 y Fl(\003)2210 937 y Fq(:)0 1105 y Fe(2)0 1438 y Fn(3.3)135 b(Coun)l(ting)46 b(the)f(eigen)l(v)-7 b(alues)0 1623 y Ft(Let)32 b([)p Fq(e)246 1638 y Fl(\000)305 1623 y Fq(;)17 b(e)394 1638 y Fo(+)453 1623 y Ft(])28 b Fp(3)g Fq(x)g Fp(7!)f Fq(G)p Ft(\()p Fq(x)p Ft(\))32 b(b)s(e)g(a)f(function)g(with)g(v)-5 b(alues)31 b(in)g(b)s(ounded)h(self-adjoin)m(t)e(op)s(erators)h(on)h(a) 0 1744 y(Hilb)s(ert)27 b(space)j Fp(H)q Ft(.)42 b(W)-8 b(e)29 b(sa)m(y)h(that)e(the)h(function)f Fq(G)h Ft(is)56 b(strictly)28 b(increasing)g(if)f(the)i(follo)m(wing)d(holds:)0 1864 y(If)33 b Fq(x)28 b(>)f(y)t Ft(,)32 b(then)h(there)g(is)g Fq(\017)28 b(>)f Ft(0)32 b(suc)m(h)i(that,)f(for)f(all)e Fq( )i Fp(2)c(H)q Ft(,)1184 2071 y(\()p Fq( )t Fp(j)p Fq(G)p Ft(\()p Fq(x)p Ft(\))p Fq( )t Ft(\))g Fq(>)f Ft(\()p Fq( )t Fp(j)p Fq(G)p Ft(\()p Fq(y)t Ft(\))p Fq( )t Ft(\))21 b(+)h Fq(\017)p Fp(k)p Fq( )t Fp(k)2529 2030 y Fo(2)2568 2071 y Fq(:)0 2278 y Fr(Prop)s(osition)36 b(3.9)49 b Fi(L)-5 b(et)35 b Ft([)p Fq(e)1038 2293 y Fl(\000)1097 2278 y Fq(;)17 b(e)1186 2293 y Fo(+)1245 2278 y Ft(])28 b Fp(3)g Fq(x)g Fp(7!)f Fq(G)p Ft(\()p Fq(x)p Ft(\))h Fp(2)g(B)s Ft(\()p Fp(H)q Ft(\))36 b Fi(b)-5 b(e)34 b(a)h(function)f (such)h(that:)0 2398 y Ft(\(a\))g Fi(F)-7 b(or)33 b(al)5 b(l)35 b Fq(x)28 b Fp(2)g Ft([)p Fq(e)730 2413 y Fl(\000)789 2398 y Fq(;)17 b(e)878 2413 y Fo(+)937 2398 y Ft(])p Fi(,)35 b Fq(G)p Ft(\()p Fq(x)p Ft(\))g Fi(is)g(self-adjoint.)0 2519 y Ft(\(b\))g([)p Fq(e)237 2534 y Fl(\000)296 2519 y Fq(;)17 b(e)385 2534 y Fo(+)444 2519 y Ft(])28 b Fp(3)g Fq(x)g Fp(7!)g Fq(G)p Ft(\()p Fq(x)p Ft(\))35 b Fi(is)f(c)-5 b(ontinuous)35 b(and)f(strictly)i(incr)-5 b(e)g(asing.)0 2639 y Ft(\(c\))35 b(dim)15 b Fr(1)389 2654 y Fo([0)p Fm(;)p Fl(1)p Fo([)558 2639 y Ft(\()p Fq(G)p Ft(\()p Fq(e)756 2654 y Fo(+)815 2639 y Ft(\)\))28 b Fq(<)f Fp(1)p Fq(:)0 2759 y Fi(Then:)0 2880 y(\(i\))i(The)g(set)h Fp(f)p Fq(x)d Fp(2)i Ft([)p Fq(e)779 2895 y Fl(\000)838 2880 y Fq(;)17 b(e)927 2895 y Fo(+)986 2880 y Ft(])57 b(:)g(0)28 b Fp(2)g Fq(\033)t Ft(\()p Fq(G)p Ft(\()p Fq(x)p Ft(\)\))p Fp(g)h Fi(is)h(a)f(\014nite)g(subset)h(of)f Ft([)p Fq(e)2637 2895 y Fl(\000)2696 2880 y Fq(;)17 b(e)2785 2895 y Fo(+)2844 2880 y Ft(])30 b Fi(and,)g(for)f Fq(x)f Fp(2)g Ft([)p Fq(e)3515 2895 y Fl(\000)3575 2880 y Fq(;)17 b(e)3664 2895 y Fo(+)3723 2880 y Ft(])p Fi(,)0 3000 y(we)34 b(have)h Ft(0)27 b Fp(2)h Fq(\033)t Ft(\()p Fq(G)p Ft(\()p Fq(x)p Ft(\)\))35 b Fi(i\013)g Ft(0)27 b Fp(2)h Fq(\033)1267 3015 y Fo(disc)1390 3000 y Ft(\()p Fq(G)p Ft(\()p Fq(x)p Ft(\)\))p Fi(.)0 3120 y(\(ii\))664 3158 y Fj(X)567 3346 y Fm(x)p Fl(2)p Fo([)p Fm(e)707 3355 y Ff(\000)758 3346 y Fm(;e)811 3355 y Fk(+)861 3346 y Fo(])897 3241 y Ft(dim)16 b(Ker)g Fq(G)p Ft(\()p Fq(x)p Ft(\))28 b(=)g(dim)15 b Fr(1)1825 3256 y Fo([0)p Fm(;)p Fl(1)p Fo([)1994 3241 y Ft(\()p Fq(G)p Ft(\()p Fq(e)2192 3256 y Fo(+)2251 3241 y Ft(\)\))22 b Fp(\000)h Ft(dim)15 b Fr(1)2684 3256 y Fo(]0)p Fm(;)p Fl(1)p Fo([)2853 3241 y Ft(\()p Fq(G)p Ft(\()p Fq(e)3051 3256 y Fl(\000)3110 3241 y Ft(\)\))p Fq(:)0 3620 y Fr(Pro)s(of.)95 b Ft(W)-8 b(e)47 b(will)e(use)k(the)e (Couran)m(t{W)-8 b(eyl)48 b(\(min-max\))d(principle)h(\([RS4],)51 b(Theorem)c(XI)s(I)s(I.1,)0 3740 y([W)-8 b(e)q(]\).)43 b(Let)1004 3861 y Fq(f)1052 3876 y Fm(n)1099 3861 y Ft(\()p Fq(x)p Ft(\))28 b(:=)142 b(inf)1389 3920 y Fm( )1435 3929 y Fk(1)1469 3920 y Fm(;:::)o(; )1614 3929 y Fg(n)p Ff(\000)p Fk(1)1953 3861 y Ft(sup)1935 3931 y Ff(k)p Fg( )r Ff(k)p Fk(=1)1761 4002 y Fg( )r Ff(2f)p Fg( )1919 4017 y Fk(1)1952 4002 y Fg(;:::; )2088 4017 y(n)p Ff(\000)p Fk(1)2209 4002 y Ff(g)2241 3988 y(?)2303 3861 y Ft(\()p Fq( )t Fp(j)p Fq(G)p Ft(\()p Fq(x)p Ft(\))p Fq( )t Ft(\))p Fq(:)0 4164 y Ft(W)-8 b(e)38 b(set)h(\006\()p Fq(x)p Ft(\))e(:=)g Fp(\0001)g Ft(if)g Fp(H)i Ft(is)e(\014nite)h(dimensional,) f(\006\()p Fq(x)p Ft(\))g(:=)f(inf)2565 4179 y Fm(n)2629 4164 y Fq(f)2677 4179 y Fm(n)2724 4164 y Ft(\()p Fq(x)p Ft(\))i(otherwise.)60 b(It)38 b(follo)m(ws)0 4284 y(from)g(the)i (min-max)d(principle)h(that)i(\006\()p Fq(x)p Ft(\))g(=)f(sup)18 b Fq(\033)2030 4299 y Fo(ess)2121 4284 y Ft(\()p Fq(G)p Ft(\()p Fq(x)p Ft(\)\))40 b(\(note)g(that)f(sup)17 b Fp(;)40 b Ft(=)f Fp(\0001)p Ft(\).)64 b(F)-8 b(ur-)0 4405 y(thermore,)34 b Fq(f)494 4420 y Fm(n)541 4405 y Ft(\()p Fq(x)p Ft(\))g(is)f(a)h(non-increasing)e(sequence)37 b(suc)m(h)e(that)f(if)e(\006\()p Fq(x)p Ft(\))f Fq(<)e(f)2849 4420 y Fm(n)2896 4405 y Ft(\()p Fq(x)p Ft(\))34 b(then)h Fq(f)3333 4420 y Fm(n)3380 4405 y Ft(\()p Fq(x)p Ft(\))f(is)f(the)0 4525 y Fq(n)p Ft(-th)f(eigen)m(v)-5 b(alue)32 b(of)h Fq(G)p Ft(\()p Fq(x)p Ft(\))f(\(in)g(the)h(non-increasing)f(order\))g (coun)m(ted)i(with)e(m)m(ultiplicites.)146 4645 y(Clearly)-8 b(,)44 b(since)f Fq(G)p Ft(\()p Fq(x)p Ft(\))g(is)f(a)g(strictly)g (increasing)g(con)m(tin)m(uous)h(function,)i(the)e(functions)f Fq(f)3601 4660 y Fm(n)3648 4645 y Ft(\()p Fq(x)p Ft(\))0 4766 y(are)36 b(also)e(strictly)h(increasing)g(and)g(con)m(tin)m(uous)h (in)f Fq(x)p Ft(.)53 b(In)36 b(particular,)e(it)h(follo)m(ws)f(that)h (\006\()p Fq(x)p Ft(\))i(is)e(an)0 4886 y(increasing)d(function.)43 b(By)33 b(\(c\),)g(\006\()p Fq(e)1361 4901 y Fo(+)1420 4886 y Ft(\))28 b Fq(<)f Ft(0,)33 b(hence)h(\006\()p Fq(x)p Ft(\))28 b Fq(<)g Ft(0)k(for)g(all)f Fq(x)d Fp(2)g Ft([)p Fq(e)2917 4901 y Fl(\000)2976 4886 y Fq(;)17 b(e)3065 4901 y Fo(+)3124 4886 y Ft(].)44 b(Therefore,)1069 5090 y(dim)15 b Fr(1)1304 5106 y Fo([0)p Fm(;)p Fl(1)p Fo([)1473 5090 y Ft(\()p Fq(G)p Ft(\()p Fq(x)p Ft(\)\))28 b(=)g(#)p Fp(f)p Fq(n)60 b Ft(:)g Fq(f)2273 5105 y Fm(n)2320 5090 y Ft(\()p Fq(x)p Ft(\))28 b Fp(\025)h Ft(0)p Fp(g)p Fq(;)1069 5282 y Ft(dim)15 b(Ker)i Fq(G)p Ft(\()p Fq(x)p Ft(\))28 b(=)g(#)p Fp(f)p Fq(n)60 b Ft(:)g Fq(f)2146 5297 y Fm(n)2193 5282 y Ft(\()p Fq(x)p Ft(\))28 b(=)g(0)p Fp(g)p Fq(;)1841 5753 y Ft(32)p eop %%Page: 33 33 33 32 bop 0 407 a Ft(and)36 b(the)h(result)f(follo)m(ws)f(from)g (elemen)m(tary)h(prop)s(erties)g(of)g(strictly)f(increasing)h(con)m (tin)m(uous)g(func-)0 527 y(tions.)43 b Fe(2)146 685 y Ft(The)36 b(follo)m(wing)d(t)m(w)m(o)i(theorems)h(follo)m(w)d(b)m(y)j (com)m(bining)d(the)j(last)e(prop)s(osition)f(with)i(Theorems)0 806 y(3.6)29 b(and)h(3.8.)42 b(W)-8 b(e)30 b(consider)g(a)f (self-adjoin)m(t)f(op)s(erator)h Fq(H)37 b Ft(whic)m(h)30 b(has)g(the)g(same)g(form)e(as)i(in)f(Section)0 926 y(3.2.)81 b(W)-8 b(e)46 b(will)d(assume)i(in)g(addition)e(that)i Fq(H)1797 890 y Fo(vv)1921 926 y Ft(is)g(a)g(b)s(ounded)h(op)s(erator.) 81 b(The)46 b(\014rst)g(theorem)0 1046 y(describ)s(es)34 b(ho)m(w)f(to)f(coun)m(t)h(eigen)m(v)-5 b(alues)33 b(outside)f Fq(\033)t Ft(\()p Fq(H)p 2033 972 39 4 v 2033 1010 a Fo(v)p 2071 972 V 1 w(v)2114 1046 y Ft(\).)0 1214 y Fr(Theorem)74 b(3.10)49 b Fi(Assume)42 b(that)g Ft([)p Fq(e)1409 1229 y Fl(\000)1468 1214 y Fq(;)17 b(e)1557 1229 y Fo(+)1616 1214 y Ft(])27 b Fp(\\)g Fq(\033)t Ft(\()p Fq(H)p 1949 1139 V 1949 1178 a Fo(v)p 1987 1139 V 1 w(v)2029 1214 y Ft(\))40 b(=)g Fp(;)h Fi(and)g(that)g Ft(dim)16 b Fr(1)2951 1229 y Fo([0)p Fm(;)p Fl(1)p Fo([)3119 1214 y Ft(\()p Fq(G)3234 1229 y Fo(v)3277 1214 y Ft(\()p Fq(e)3360 1229 y Fo(+)3419 1214 y Ft(\)\))39 b Fq(<)h Fp(1)p Fi(.)0 1334 y(Then,)274 1512 y Ft(dim)15 b Fr(1)509 1465 y Fo(pp)509 1541 y([)p Fm(e)562 1550 y Ff(\000)614 1541 y Fm(;e)667 1550 y Fk(+)717 1541 y Fo(])740 1512 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)h(dim)15 b Fr(1)1271 1527 y Fo([)p Fm(e)1324 1536 y Ff(\000)1376 1527 y Fm(;e)1429 1536 y Fk(+)1479 1527 y Fo(])1502 1512 y Ft(\()p Fq(H)8 b Ft(\))28 b(=)f(dim)15 b Fr(1)2033 1527 y Fo([0)p Fm(;)p Fl(1)p Fo([)2202 1512 y Ft(\()p Fq(G)2317 1527 y Fo(v)2359 1512 y Ft(\()p Fq(e)2442 1527 y Fo(+)2502 1512 y Ft(\)\))22 b Fp(\000)g Ft(dim)15 b Fr(1)2934 1527 y Fo(]0)p Fm(;)p Fl(1)p Fo([)3103 1512 y Ft(\()p Fq(G)3218 1527 y Fo(v)3261 1512 y Ft(\()p Fq(e)3344 1527 y Fl(\000)3403 1512 y Ft(\)\))p Fq(:)0 1690 y Fr(Pro)s(of.)65 b Ft(W)-8 b(e)33 b(kno)m(w)h(b)m(y)f(Prop)s(osition)e(3.5)h(\(i\))g (that)846 1868 y([)p Fq(e)918 1883 y Fl(\000)978 1868 y Fq(;)17 b(e)1067 1883 y Fo(+)1125 1868 y Ft(])23 b Fp(\\)f Fq(\033)t Ft(\()p Fq(H)8 b Ft(\))27 b Fp(\032)i(f)p Fq(x)e Fp(2)i Ft([)p Fq(e)1919 1883 y Fl(\000)1978 1868 y Fq(;)17 b(e)2067 1883 y Fo(+)2126 1868 y Ft(])60 b(:)g(0)28 b Fp(2)g Fq(\033)t Ft(\()p Fq(G)2645 1883 y Fo(v)2687 1868 y Ft(\()p Fq(x)p Ft(\)\))p Fp(g)p Fq(:)597 b Ft(\(3.73\))0 2046 y(Since)37 b Fq(G)336 2009 y Fl(0)336 2070 y Fo(v)379 2046 y Ft(\()p Fq(x)p Ft(\))e Fp(\025)h Fr(1)714 2009 y Fo(vv)831 2046 y Ft(w)m(e)i(see)g(that)f Fq(G)1434 2061 y Fo(v)1476 2046 y Ft(\()p Fq(x)p Ft(\))h(is)e(strictly)h (increasing.)56 b(Th)m(us)38 b(w)m(e)h(easily)d(see)i(that)f(the)0 2166 y(function)44 b Fq(G)p Ft(\()p Fq(x)p Ft(\))h(satis\014es)g(the)g (assumptions)f(of)g(Prop)s(osition)f(3.9,)k(whic)m(h)e(implies)d(that)i (the)h(set)0 2286 y(\(3.73\))32 b(is)g(\014nite.)43 b(Hence)961 2464 y Fr(1)1017 2480 y Fo([)p Fm(e)1070 2489 y Ff(\000)1121 2480 y Fm(;e)1174 2489 y Fk(+)1224 2480 y Fo(])1248 2464 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)h Fr(1)1600 2417 y Fo(pp)1600 2493 y([)p Fm(e)1653 2502 y Ff(\000)1704 2493 y Fm(;e)1757 2502 y Fk(+)1807 2493 y Fo(])1831 2464 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)2223 2381 y Fj(X)2126 2569 y Fm(x)p Fl(2)p Fo([)p Fm(e)2266 2578 y Ff(\000)2318 2569 y Fm(;e)2371 2578 y Fk(+)2421 2569 y Fo(])2457 2464 y Fr(1)2513 2480 y Fl(f)p Fm(x)p Fl(g)2627 2464 y Ft(\()p Fq(H)8 b Ft(\))p Fq(:)0 2735 y Ft(It)33 b(follo)m(ws)e(from)g(Theorem)i(3.6)f(that)h (for)f(all)e Fq(x)e Fp(2)g Ft([)p Fq(e)1970 2750 y Fl(\000)2030 2735 y Fq(;)17 b(e)2119 2750 y Fo(+)2178 2735 y Ft(],)1252 2913 y(dim)e Fr(1)1487 2928 y Fl(f)p Fm(x)p Fl(g)1601 2913 y Ft(\()p Fq(H)8 b Ft(\))28 b(=)f(dim)15 b(Ker)i Fq(G)2327 2928 y Fo(v)2369 2913 y Ft(\()p Fq(x)p Ft(\))p Fq(;)0 3090 y Ft(and)33 b(the)g(result)f(follo)m(ws)f(from)h(Prop)s (osition)f(3.9.)43 b Fe(2)0 3308 y Fr(Corollary)36 b(3.11)49 b Fi(Assume)35 b(that)g Ft([)p Fq(e)1381 3323 y Fl(\000)1440 3308 y Fq(;)17 b(e)1529 3323 y Fo(+)1588 3308 y Ft(])22 b Fp(\\)h Fq(\033)t Ft(\()p Fq(H)p 1912 3234 V 1912 3272 a Fo(v)p 1950 3234 V 1 w(v)1992 3308 y Ft(\))28 b(=)f Fp(;)p Fi(.)45 b(Then)1341 3486 y Ft(dim)15 b Fr(1)1576 3502 y Fo([)p Fm(e)1629 3511 y Ff(\000)1681 3502 y Fm(;e)1734 3511 y Fk(+)1784 3502 y Fo(])1808 3486 y Ft(\()p Fq(H)8 b Ft(\))27 b Fp(\024)h Ft(dim)15 b Fp(H)2369 3445 y Fo(v)2411 3486 y Fq(:)1092 b Ft(\(3.74\))146 3666 y(If)33 b(the)g(self-energy)g (is)f(di\013eren)m(tiable,)f(one)i(can)g(also)f(coun)m(t)h(eigen)m(v)-5 b(alues)32 b(of)g Fq(H)40 b Ft(inside)32 b Fq(\033)t Ft(\()p Fq(H)p 3626 3591 V 3626 3630 a Fo(v)p 3664 3591 V 1 w(v)3706 3666 y Ft(\).)0 3833 y Fr(Theorem)74 b(3.12)49 b Fi(Assume)35 b(that)h Fq(\033)1379 3848 y Fo(pp)1462 3833 y Ft(\()p Fq(H)p 1589 3759 V 1589 3797 a Fo(v)p 1626 3759 V(v)1669 3833 y Ft(\))22 b Fp(\\)g Ft([)p Fq(e)1889 3848 y Fl(\000)1949 3833 y Fq(;)17 b(e)2038 3848 y Fo(+)2097 3833 y Ft(])27 b(=)h Fp(;)p Fi(,)34 b(and)h(that)g(the)g(fol)5 b(lowing)33 b(holds:)0 3954 y Ft(\(a\))i Fi(The)f(limit)1272 4074 y Fq(W)1364 4089 y Fo(v)1406 4074 y Ft(\()p Fq(x)23 b Ft(+)f(i0\))k(:=)i(lim)1906 4134 y Fm(y)r Fl(#)p Fo(0)2044 4074 y Fq(W)2136 4089 y Fo(v)2178 4074 y Ft(\()p Fq(x)23 b Ft(+)f(i)p Fq(y)t Ft(\))1020 b(\(3.75\))0 4274 y Fi(exists)35 b(for)f(al)5 b(l)35 b Fq(x)28 b Fp(2)g Ft([)p Fq(e)811 4289 y Fl(\000)870 4274 y Fq(;)17 b(e)959 4289 y Fo(+)1018 4274 y Ft(])p Fi(.)0 4395 y Ft(\(b\))34 b Fi(The)e(function)i Ft(Re)p Fq(W)951 4410 y Fo(v)993 4395 y Ft(\()p Fq(x)20 b Ft(+)f(i0\))32 b Fi(is)h(di\013er)-5 b(entiable)32 b(and)h(for)h(some)e Fq(\017)c(>)g Ft(0)33 b Fi(and)g(al)5 b(l)33 b Fq(x)28 b Fp(2)g Ft([)p Fq(e)3449 4410 y Fl(\000)3509 4395 y Fq(;)17 b(e)3598 4410 y Fo(+)3657 4395 y Ft(])33 b Fi(it)0 4515 y(satis\014es)1057 4635 y Ft(Re)p Fq(G)1249 4594 y Fl(0)1249 4660 y Fo(v)1291 4635 y Ft(\()p Fq(x)23 b Ft(+)f(i0\))k(=)i Fr(1)1806 4594 y Fo(vv)1907 4635 y Fp(\000)23 b Ft(Re)p Fq(W)2228 4594 y Fl(0)2214 4660 y Fo(v)2257 4635 y Ft(\()p Fq(x)p Ft(\))28 b Fp(\025)g Fq(\017)p Fr(1)2616 4594 y Fo(vv)2696 4635 y Fq(:)0 4792 y Ft(\(c\))35 b(dim)15 b Fr(1)389 4807 y Fo([0)p Fm(;)p Fl(1)p Fo([)558 4792 y Ft(\(Re)i Fq(G)805 4807 y Fo(v)847 4792 y Ft(\()p Fq(e)930 4807 y Fo(+)1011 4792 y Ft(+)22 b(i0\)\))27 b Fq(<)g Fp(1)p Fi(.)0 4912 y(Then,)343 5090 y Ft(dim)15 b Fr(1)578 5043 y Fo(pp)578 5119 y([)p Fm(e)631 5128 y Ff(\000)683 5119 y Fm(;e)736 5128 y Fk(+)786 5119 y Fo(])809 5090 y Ft(\()p Fq(H)8 b Ft(\))28 b Fp(\024)g Ft(dim)15 b Fr(1)1342 5105 y Fo([0)p Fm(;)p Fl(1)p Fo([)1511 5090 y Ft(\(Re)p Fq(G)1741 5105 y Fo(v)1783 5090 y Ft(\()p Fq(e)1866 5105 y Fo(+)1947 5090 y Ft(+)22 b(i0\)\))f Fp(\000)i Ft(dim)15 b Fr(1)2554 5105 y Fo(]0)p Fm(;)p Fl(1)p Fo([)2723 5090 y Ft(\(Re)p Fq(G)2953 5105 y Fo(v)2995 5090 y Ft(\()p Fq(e)3078 5105 y Fl(\000)3160 5090 y Ft(+)22 b(i0\)\))p Fq(:)1841 5753 y Ft(33)p eop %%Page: 34 34 34 33 bop 0 407 a Fr(Pro)s(of.)95 b Ft(Assumptions)48 b(\(a\))g(and)g(Theorem)g(3.8,)j(and)d(then)g(Prop)s(osition)e(3.2,)52 b(yield)46 b(that)i(for)0 527 y Fq(x)28 b Fp(2)g Ft([)p Fq(e)249 542 y Fl(\000)309 527 y Fq(;)17 b(e)398 542 y Fo(+)456 527 y Ft(],)638 747 y(dim)e Fr(1)873 763 y Fl(f)p Fm(x)p Fl(g)987 747 y Ft(\()p Fq(H)8 b Ft(\))27 b Fp(\024)h Ft(dim)16 b(Ker)g Fq(G)1714 762 y Fo(v)1757 747 y Ft(\()p Fq(x)22 b Ft(+)g(i0\))27 b Fp(\024)h Ft(dim)15 b(KerRe)p Fq(G)2745 762 y Fo(v)2788 747 y Ft(\()p Fq(x)22 b Ft(+)g(i0\))p Fq(:)387 b Ft(\(3.76\))0 967 y(Assumption)36 b(\(b\))g(yields)g(that)g(the)h(function)f([)p Fq(e)1833 982 y Fl(\000)1892 967 y Fq(;)17 b(e)1981 982 y Fo(+)2040 967 y Ft(])35 b Fp(3)f Fq(x)h Fp(7!)e Ft(Re)p Fq(G)2617 982 y Fo(v)2660 967 y Ft(\()p Fq(x)25 b Ft(+)f(i0\))35 b(is)h(con)m(tin)m(uous)h(and)0 1088 y(strictly)43 b(increasing.)78 b(Using)44 b(also)f(\(c\))h(w)m(e)i(see)f(that)f(the)h(function)e(Re)p Fq(G)2843 1103 y Fo(v)2886 1088 y Ft(\()p Fq(x)30 b Ft(+)g(i0\))43 b(satis\014es)i(the)0 1208 y(conditions)36 b(of)h(Prop)s(osition)e (3.9.)57 b(This)37 b(prop)s(osition)e(and)j(Relation)d(\(3.76\))h (yield)g(the)i(statemen)m(t)0 1328 y(of)32 b(the)h(theorem.)43 b Fe(2)0 1606 y Fr(Corollary)36 b(3.13)49 b Fi(Assume)35 b(that)g Fq(\033)1364 1621 y Fo(pp)1447 1606 y Ft(\()p Fq(H)p 1574 1532 39 4 v 1574 1570 a Fo(v)p 1612 1532 V 1 w(v)1654 1606 y Ft(\))22 b Fp(\\)h Ft([)p Fq(e)1875 1621 y Fl(\000)1934 1606 y Fq(;)17 b(e)2023 1621 y Fo(+)2082 1606 y Ft(])28 b(=)f Fp(;)p Fi(,)35 b(and)f(that)h(the)g(fol)5 b(lowing)34 b(holds:)0 1727 y Ft(\(a\))h Fi(The)f(limit)1258 1847 y Fq(W)1350 1862 y Fo(v)1393 1847 y Ft(\()p Fq(x)22 b Ft(+)g(i0\))27 b(:=)g(lim)1892 1907 y Fm(y)r Fl(#)p Fo(0)2030 1847 y Fq(W)2122 1862 y Fo(v)2165 1847 y Ft(\()p Fq(x)22 b Ft(+)g(i)p Fq(y)t Ft(\))p Fq(;)1007 b Ft(\(3.77\))0 2065 y Fi(exists)35 b(for)f(al)5 b(l)35 b Fq(x)28 b Fp(2)g Ft([)p Fq(e)811 2080 y Fl(\000)870 2065 y Fq(;)17 b(e)959 2080 y Fo(+)1018 2065 y Ft(])p Fi(.)0 2186 y Ft(\(b\))38 b Fi(The)f(function)g Ft([)p Fq(e)828 2201 y Fl(\000)887 2186 y Fq(;)17 b(e)976 2201 y Fo(+)1035 2186 y Ft(])33 b Fp(3)g Fq(x)g Fp(7!)g Ft(Re)p Fq(W)1622 2201 y Fo(v)1664 2186 y Ft(\()p Fq(x)25 b Ft(+)f(i0\))36 b Fi(is)i(di\013er)-5 b(entiable,)37 b(and)g(for)g(some)g Fq(\017)c(>)g Ft(0)k Fi(and)0 2306 y(al)5 b(l)34 b Fq(x)29 b Fp(2)f Ft([)p Fq(e)389 2321 y Fl(\000)448 2306 y Fq(;)17 b(e)537 2321 y Fo(+)596 2306 y Ft(])p Fi(,)959 2427 y Ft(Re)p Fq(G)1151 2385 y Fl(0)1151 2451 y Fo(v)1193 2427 y Ft(\()p Fq(x)23 b Ft(+)f(i0\))k(=)i Fr(1)1708 2385 y Fo(vv)1809 2427 y Fp(\000)23 b Ft(Re)p Fq(W)2130 2385 y Fl(0)2116 2451 y Fo(v)2159 2427 y Ft(\()p Fq(x)f Ft(+)g(i0\))27 b Fp(\025)h Fq(\017)p Fr(1)2714 2385 y Fo(vv)2794 2427 y Fq(:)0 2721 y Fi(Then)1341 2842 y Ft(dim)15 b Fr(1)1576 2794 y Fo(pp)1576 2870 y([)p Fm(e)1629 2879 y Ff(\000)1681 2870 y Fm(;e)1734 2879 y Fk(+)1784 2870 y Fo(])1808 2842 y Ft(\()p Fq(H)8 b Ft(\))27 b Fp(\024)h Ft(dim)15 b Fp(H)2369 2801 y Fo(v)2411 2842 y Fq(:)1092 b Ft(\(3.78\))0 3349 y Fs(4)161 b(F)-13 b(o)t(c)l(k)54 b(spaces)e(and)i(all)i(that)0 3568 y Ft(In)49 b(this)e(section)i(w)m(e)g(describ)s(e)g(in)e(an)h(abstract)h(setting)f (some)g(Hilb)s(ert)f(spaces)i(and)g(op)s(erators)0 3688 y(of)c(the)g(quan)m(tum)g(\014eld)g(theory)-8 b(.)81 b(W)-8 b(e)46 b(ha)m(v)m(e)g(attempted)f(to)g(giv)m(e)g(an)g(essen)m (tially)f(self-con)m(tained)0 3809 y(presen)m(tation)24 b(of)g(the)g(topics)g(w)m(e)h(will)d(need.)42 b(F)-8 b(or)23 b(additional)e(information,)i(the)h(reader)h(ma)m(y)e(consult)0 3929 y([BSZ,)33 b(BR)o(,)g(De,)g(DG)o(,)f(GJ,)g(RS2].)146 4049 y(Let)39 b Fh(h)h Ft(b)s(e)f(a)f(Hilb)s(ert)g(space.)64 b(W)-8 b(e)39 b(set)h Fh(h)1673 4013 y Fo(0)p Fl(\012)1806 4049 y Ft(:=)e Fr(C)h Ft(and)g Fh(h)2306 4013 y Fm(n)p Fl(\012)2447 4049 y Ft(:=)f Fh(h)27 b Fp(\012)g Fq(:)17 b(:)g(:)26 b Fp(\012)h Fh(h)p Fq(:)39 b Ft(If)g Fq(A)g Ft(is)g(a)f(closed)0 4170 y(op)s(erator)e(on)i Fh(h)p Ft(,)g(w)m(e)g(denote)g(b)m(y)g Fq(A)1326 4134 y Fl(\012)p Fm(n)1465 4170 y Ft(the)g(closed)f(op)s(erator)g(on)g Fh(h)2511 4134 y Fm(n)p Fl(\012)2650 4170 y Ft(de\014ned)i(b)m(y)f Fq(A)25 b Fp(\012)h Fq(:)17 b(:)g(:)25 b Fp(\012)g Fq(A)38 b Ft(\(if)0 4290 y Fq(n)c Ft(=)g(0,)j Fq(A)388 4254 y Fl(\012)p Fo(0)517 4290 y Ft(=)d(1\).)54 b(Let)36 b Fq(S)1033 4305 y Fm(n)1117 4290 y Ft(b)s(e)g(the)h(group)f(of)g(p)s(erm)m (utations)f(of)h Fq(n)g Ft(elemen)m(ts.)55 b(F)-8 b(or)36 b(eac)m(h)h Fq(\033)h Fp(2)d Fq(S)3733 4305 y Fm(n)0 4410 y Ft(w)m(e)f(de\014ne)f(an)g(op)s(erator)f(\(whic)m(h)h(w)m(e)g (also)f(denote)h(b)m(y)h Fq(\033)t Ft(\))e(on)h(the)g(basis)f(elemen)m (ts)h(of)f Fh(h)3285 4374 y Fm(n)p Fl(\012)3420 4410 y Ft(b)m(y)1058 4631 y Fq(\033)t Ft(\()p Fq(f)1203 4646 y Fm(i)1227 4655 y Fk(1)1288 4631 y Fp(\012)23 b Fq(:)17 b(:)g(:)k Fp(\012)i Fq(f)1672 4646 y Fm(i)1696 4654 y Fg(n)1743 4631 y Ft(\))k(=)h Fq(f)1960 4646 y Fm(\033)r Fo(\()p Fm(i)2053 4655 y Fk(1)2089 4646 y Fo(\))2142 4631 y Fp(\012)23 b Fq(:)17 b(:)g(:)22 b Fp(\012)g Fq(f)2526 4646 y Fm(\033)r Fo(\()p Fm(i)2619 4654 y Fg(n)2663 4646 y Fo(\))2695 4631 y Fq(;)0 4851 y Ft(where)29 b Fp(f)p Fq(f)375 4866 y Fm(k)418 4851 y Fp(g)e Ft(is)h(a)f(basis)h(of)f Fh(h)p Ft(.)43 b Fq(\033)31 b Ft(extends)f(b)m(y)f(linearit)m(y)d(to)i (a)f(unitary)h(op)s(erator)f(on)h Fh(h)3160 4814 y Fl(\012)p Fm(n)3262 4851 y Ft(,)h(whic)m(h)f(do)s(es)0 4971 y(not)k(dep)s(end)i (on)f(a)f(basis.)43 b(W)-8 b(e)33 b(set)1559 5130 y Fq(P)1622 5145 y Fm(n)1696 5130 y Ft(:=)1855 5063 y(1)p 1837 5107 86 4 v 1837 5199 a Fq(n)p Ft(!)1976 5047 y Fj(X)1949 5231 y Fm(\033)r Fl(2)p Fm(S)2081 5239 y Fg(n)2140 5130 y Fq(\033)n(:)1841 5753 y Ft(34)p eop %%Page: 35 35 35 34 bop 0 407 a Ft(The)34 b(op)s(erator)d Fq(P)656 422 y Fm(n)736 407 y Ft(is)h(an)g(orthogonal)f(pro)5 b(jection.)43 b(Let)1541 627 y(\000)1602 642 y Fm(n)1649 627 y Ft(\()p Fh(h)p Ft(\))28 b(:=)g(Ran)o Fq(P)2164 642 y Fm(n)2211 627 y Fq(:)0 847 y Ft(This)33 b(Hilb)s(ert)e(space)i (is)f(commonly)f(called)h(the)h Fq(n)p Ft(-particle)e(b)s(osonic)h (space.)146 967 y(The)i(symmetric)e(\(or)g(b)s(oson\))g(F)-8 b(o)s(c)m(k)33 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b(ev)m(ery)f(form-factor)c(can)j(b)s(e)f (decomp)s(osed)h(in)m(to)f(a)g(sum)g(of)g(simple)f(form-factors.)50 b(One)36 b(can)0 5268 y(alternativ)m(ely)31 b(use)j(this)e(fact)g(to)h (de\014ne)h(the)f(op)s(erators)f Fq(a)2169 5232 y Fo(#)2232 5268 y Ft(\()p Fq(v)t Ft(\),)g(etc.)1841 5753 y(36)p eop %%Page: 37 37 37 36 bop 146 407 a Ft(If)33 b Fq(U)43 b Ft(is)32 b(a)g(unitary)h(op)s (erator)e(on)i Fh(h)p Ft(,)g(then)1321 598 y(\000\()p Fq(U)10 b Ft(\))p Fq(')p Ft(\()p Fq(v)t Ft(\)\000\()p Fq(U)1900 557 y Fl(\000)p Fo(1)1995 598 y Ft(\))28 b(=)f Fq(')p Ft(\()p Fq(U)10 b(v)t Ft(\))p Fq(:)1072 b Ft(\(4.81\))0 789 y(F)-8 b(rom)44 b(this)i(relation)e(it)g(follo)m(ws)h(that)h(if)e Fq(!)49 b Ft(is)d(a)f(self-adjoin)m(t)f(op)s(erator)h(on)h Fh(h)g Ft(suc)m(h)h(that)f Fq(!)t(v)53 b Fp(2)0 910 y(B)s Ft(\()p Fp(K)q Fq(;)17 b Fp(K)24 b(\012)f Fh(h)p Ft(\),)32 b(then)1384 1030 y(i[d\000\()p Fq(!)t Ft(\))p Fq(;)17 b(')p Ft(\()p Fq(v)t Ft(\)])26 b(=)h Fq(')p Ft(\(i)p Fq(!)t(v)t Ft(\))p Fq(:)1133 b Ft(\(4.82\))146 1192 y(W)-8 b(e)33 b(no)m(w)h(describ)s(e)f(sev)m(eral)g(tec)m(hnical)f(results)h (whic)m(h)g(will)e(b)s(e)h(used)i(in)e(the)h(sequel.)0 1371 y Fr(Prop)s(osition)j(4.1)49 b Fi(L)-5 b(et)36 b Fq(v)d Fp(2)d Fq(B)5 b Ft(\()p Fp(K)q Fq(;)17 b Fp(K)24 b(\012)f Fh(h)p Ft(\))36 b Fi(and)g Fq(!)j Fi(an)c(op)-5 b(er)g(ator)36 b(on)f Fh(h)p Fi(.)48 b(Assume)36 b(that)g Fq(!)d Fp(\025)d Ft(0)35 b Fi(and)0 1492 y(that)i Fq(!)k Fi(is)36 b(invertible)h(on)f(the)h(r)-5 b(ange)36 b(of)h Fq(v)t Fi(.)51 b(Set)37 b Fq(K)1912 1507 y Fo(0)1983 1492 y Ft(:=)31 b Fq(v)2168 1455 y Fl(\003)2207 1492 y Fq(v)t Fi(.)51 b(Then)36 b(the)h(fol)5 b(lowing)36 b(estimates)g(hold)0 1612 y(for)f(any)f Ft(\011)h Fi(in)g(the)g(quadr) -5 b(atic)34 b(form)h(domain)e(of)i Ft(d\000\()p Fq(!)t Ft(\))p Fi(:)877 1800 y Fp(k)p Fq(a)p Ft(\()p Fq(v)t Ft(\)\011)p Fp(k)1231 1764 y Fo(2)1353 1800 y Fp(\024)28 b(k)p Fq(v)1559 1764 y Fl(\003)1598 1800 y Fq(!)1663 1764 y Fl(\000)p Fo(1)1757 1800 y Fq(v)t Fp(k)p Ft(\(\011)p Fp(j)p Ft(d\000\()p Fq(!)t Ft(\)\011\))p Fq(;)838 1992 y Fp(k)p Fq(a)939 1955 y Fl(\003)978 1992 y Ft(\()p Fq(v)t Ft(\)\011)p Fp(k)1231 1955 y Fo(2)1353 1992 y Fp(\024)g Ft(\(\011)p Fp(j)p Fq(K)1683 2007 y Fo(0)1722 1992 y Ft(\011\))22 b(+)g Fp(k)p Fq(v)2057 1955 y Fl(\003)2096 1992 y Fq(!)2161 1955 y Fl(\000)p Fo(1)2255 1992 y Fq(v)t Fp(k)p Ft(\(\011)p Fp(j)p Ft(d\000\()p Fq(!)t Ft(\)\011\))p Fq(;)865 2183 y Fp(k)p Fq(')p Ft(\()p Fq(v)t Ft(\)\011)p Fp(k)1232 2147 y Fo(2)1353 2183 y Fp(\024)28 b Ft(\(\011)p Fp(j)p Fq(K)1683 2198 y Fo(0)1722 2183 y Ft(\011\))22 b(+)g(2)p Fp(k)p Fq(v)2106 2147 y Fl(\003)2145 2183 y Fq(!)2210 2147 y Fl(\000)p Fo(1)2303 2183 y Fq(v)t Fp(k)p Ft(\(\011)p Fp(j)p Ft(d\000\()p Fq(!)t Ft(\)\011\))p Fq(:)3530 1992 y Ft(\(4.83\))0 2373 y Fr(Remark.)42 b Ft(Results)30 b(of)f(this)h(genre)g(go)f(bac)m(k)i(to)f(the)g Fq(N)2067 2388 y Fm(\034)2110 2373 y Ft(-estimates)f(of)g(Glimm)d(and)k (Ja\013e)g([GJ],)g(see)0 2493 y(also)i([Ar,)g(BFS1].)0 2623 y Fr(Pro)s(of.)65 b Ft(Before)33 b(w)m(e)g(start,)g(w)m(e)h (remark)e(that)g Fq(v)1816 2586 y Fl(\003)1856 2623 y Fq(!)1921 2586 y Fl(\000)p Fo(1)2014 2623 y Fq(v)g Fp(2)c(K)34 b Ft(i\013)d Fq(v)2464 2586 y Fl(\003)2503 2623 y Fq(!)2568 2586 y Fl(\000)2633 2559 y Fk(1)p 2632 2571 31 4 v 2632 2612 a(2)2705 2623 y Fp(2)d(B)s Ft(\()p Fp(K)c(\012)e Fh(h)p Fq(;)17 b Fh(h)p Ft(\),)33 b(and)g(that)1162 2827 y Fp(k)p Fq(v)1263 2786 y Fl(\003)1302 2827 y Fq(!)1367 2786 y Fl(\000)p Fo(1)1461 2827 y Fq(v)t Fp(k)27 b Ft(=)h Fp(k)p Fq(v)1794 2786 y Fl(\003)1833 2827 y Fq(!)1898 2786 y Fl(\000)1963 2759 y Fk(1)p 1962 2771 V 1962 2812 a(2)2006 2827 y Fp(k)2056 2786 y Fo(2)2123 2827 y Ft(=)g Fp(k)p Fq(!)2342 2786 y Fl(\000)2407 2759 y Fk(1)p 2406 2771 V 2406 2812 a(2)2450 2827 y Fq(v)t Fp(k)2551 2786 y Fo(2)2590 2827 y Fq(:)146 3018 y Ft(Let)33 b Fp(Q)g Ft(denote)g(the)g(quadratic)g(form)e(domain)g(of)h(d\000\()p Fq(!)t Ft(\).)43 b(Since)1145 3210 y([)p Fq(N)5 b(;)17 b(a)1350 3169 y Fl(\003)1389 3210 y Ft(\()p Fq(v)t 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Fp(k)p Fq(v)1436 5245 y Fl(\003)1475 5282 y Fq(!)1540 5245 y Fl(\000)p Fo(1)1633 5282 y Fq(v)t Fp(k)p Ft(\(\011)p Fp(j)p Ft(d\000\()p Fq(!)t Ft(\)\011\))p Fq(:)1841 5753 y Ft(37)p eop %%Page: 38 38 38 37 bop 0 407 a Ft(This)39 b(pro)m(v)m(es)i(the)e(\014rst)g(relation) e(in)i(\(4.83\).)61 b(Before)40 b(w)m(e)f(pro)m(v)m(e)i(the)e(second,)j (it)c(is)g(con)m(v)m(enien)m(t)j(to)0 527 y(in)m(tro)s(duce)33 b(some)f(additional)e(notation.)146 648 y(F)-8 b(or)29 b Fq(i)f Ft(=)g(1)p Fq(;)17 b(:)g(:)g(:)e(;)i(n)p Ft(,)31 b(let)e Fq(\034)1057 597 y Fo(\()p Fm(n)p Fo(\))1046 670 y Fm(i)1189 648 y Ft(denote)h(the)h(transp)s(osition)d(of)h(1)h (and)g Fq(i)p Ft(.)42 b(Recall)29 b(that)g Fq(\034)3281 597 y Fo(\()p Fm(n)p Fo(\))3270 670 y Fm(i)3413 648 y Ft(de\014nes)j(a)0 779 y(unitary)i(op)s(erator)g(\(whic)m(h)h(w)m(e)h (denote)f(b)m(y)h(the)f(same)f(letter\))g(on)h Fh(h)2557 743 y Fl(\012)p Fm(n)2659 779 y Ft(.)50 b(Clearly)-8 b(,)34 b(\()p Fq(\034)3187 729 y Fo(\()p Fm(n)p Fo(\))3176 802 y Fm(i)3289 779 y Ft(\))3327 743 y Fo(2)3398 779 y Ft(=)d Fr(1)p Ft(.)49 b(F)-8 b(or)0 900 y(an)m(y)33 b Fq(h)28 b Fp(2)g(B)s Ft(\()p Fh(h)p Ft(\))33 b(w)m(e)h(de\014ne)696 1119 y Fq(h)752 1068 y Fo(\()p Fm(n)p Fo(\))752 1141 y Fm(i)882 1119 y Ft(:=)27 b Fr(1)1068 1077 y Fl(\012)p Fo(\()p Fm(i)p Fl(\000)p Fo(1\))1318 1119 y Fp(\012)c Fq(h)f Fp(\012)h Fr(1)1652 1077 y Fl(\012)p Fo(\()p Fm(n)p Fl(\000)p Fm(i)p Fo(\))1915 1119 y Ft(=)28 b Fq(\034)2072 1068 y Fo(\()p Fm(n)p Fo(\))2061 1141 y Fm(i)2174 1119 y Ft(\()p Fq(h)23 b Fp(\012)f Fr(1)2446 1077 y Fl(\012)p Fo(\()p Fm(n)p Fl(\000)p Fo(1\))2693 1119 y Ft(\)\()p Fq(\034)2822 1068 y Fo(\()p Fm(n)p Fo(\))2811 1141 y Fm(i)2924 1119 y Ft(\))2962 1077 y Fl(\000)p Fo(1)3057 1119 y Fq(:)0 1324 y Ft(Assume)33 b(that)g(\011)f(satis\014es)h (\(4.84\).)43 b(Then,)34 b(using)e(\(4.79\))g(and)h(\(4.85\),)f(w)m(e)h (deriv)m(e)h(that)129 1526 y(\()p Fq(a)218 1490 y Fl(\003)258 1526 y Ft(\()p Fq(v)t Ft(\)\011)p Fp(j)p Fq(a)540 1490 y Fl(\003)578 1526 y Ft(\()p Fq(v)t Ft(\)\011\))83 b(=)27 b(\()p Fq(n)c Ft(+)f(1\)\(\011)p Fp(j)p Ft(\()p Fq(v)1540 1490 y Fl(\003)1600 1526 y Fp(\012)h Fr(1)1756 1490 y Fl(\012)p Fm(n)1858 1526 y Ft(\))p Fq(P)1959 1541 y Fm(n)p Fo(+1)2096 1526 y Ft(\()p Fq(v)j Fp(\012)c Fr(1)2362 1490 y Fl(\012)p Fm(n)2464 1526 y Ft(\)\011\))902 1767 y(=)1005 1684 y Fm(n)p Fo(+1)1028 1701 y Fj(P)1015 1841 y Fm(i)p Fo(=1)1138 1767 y Ft(\(\011)p Fp(j)p Ft(\()p Fq(v)1369 1731 y Fl(\003)1430 1767 y Fp(\012)h Fr(1)1586 1731 y Fl(\012)p Fm(n)1688 1767 y Ft(\)\()p Fr(1)1820 1782 y Fl(K)1900 1767 y Fp(\012)g Fq(\034)2053 1716 y Fo(\()p Fm(n)p Fo(+1\))2042 1790 y Fm(i)2245 1767 y Ft(\)\()p Fq(v)j Fp(\012)d Fr(1)2550 1731 y Fl(\012)p Fm(n)2651 1767 y Ft(\)\011\))902 1975 y(=)k(\(\011)p Fp(j)p Fq(K)1230 1990 y Fo(0)1269 1975 y Ft(\011\))22 b(+)1503 1908 y Fj(P)1591 1935 y Fm(n)1591 1999 y(i)p Fo(=2)1709 1975 y Ft(\(\011)p Fp(j)p Ft(\()p Fq(v)1940 1939 y Fl(\003)2001 1975 y Fp(\012)h Fr(1)2157 1939 y Fl(\012)p Fm(n)2259 1975 y Ft(\)\()p Fr(1)2391 1990 y Fl(K)2471 1975 y Fp(\012)f Fq(\034)2623 1924 y Fo(\()p Fm(n)p Fo(+1\))2612 1998 y Fm(i)2816 1975 y Ft(\)\()p Fq(v)k Fp(\012)c Fr(1)3120 1939 y Fl(\012)p Fm(n)3222 1975 y Ft(\)\011\))p Fq(:)3530 1751 y Ft(\(4.86\))0 2173 y(Note)33 b(that)f(for)g Fq(i)c(>)g Ft(1,)1006 2378 y(\(\011)p Fp(j)p Ft(\()p Fq(v)1237 2337 y Fl(\003)1297 2378 y Fp(\012)23 b Fr(1)1453 2337 y Fl(\012)p Fm(n)1555 2378 y Ft(\)\()p Fr(1)1687 2393 y Fl(K)1767 2378 y Fp(\012)g Fq(\034)1920 2327 y Fo(\()p Fm(n)p Fo(+1\))1909 2401 y Fm(i)2112 2378 y Ft(\)\()p Fq(v)j Fp(\012)d Fr(1)2417 2337 y Fl(\012)p Fm(n)2519 2378 y Ft(\)\011\))k(=)267 2597 y(=)h(\(\()p Fq(!)522 2533 y Fk(1)p 521 2545 31 4 v 521 2586 a(2)566 2597 y Ft(\))604 2546 y Fo(\()p Fm(n)p Fo(\))604 2620 y Fm(i)p Fl(\000)p Fo(1)722 2597 y Ft(\011)p Fp(j)p Ft(\()p Fq(v)915 2561 y Fl(\003)976 2597 y Fp(\012)22 b Fr(1)1131 2561 y Fl(\012)p Fm(n)1233 2597 y Ft(\)\()p Fq(!)1374 2560 y Fl(\000)1439 2533 y Fk(1)p 1438 2545 V 1438 2586 a(2)1483 2597 y Ft(\))1521 2546 y Fo(\()p Fm(n)p Fo(+1\))1521 2620 y Fm(i)1713 2597 y Ft(\()p Fr(1)1807 2612 y Fl(K)1887 2597 y Fp(\012)h Fq(\034)2040 2546 y Fo(\()p Fm(n)p Fo(+1\))2029 2620 y Fm(i)2232 2597 y Ft(\)\()p Fq(!)2373 2560 y Fl(\000)2438 2533 y Fk(1)p 2437 2545 V 2437 2586 a(2)2482 2597 y Ft(\))2520 2546 y Fo(\()p Fm(n)p Fo(+1\))2520 2620 y Fm(i)2712 2597 y Ft(\)\()p Fq(v)j Fp(\012)c Fr(1)3016 2561 y Fl(\012)p Fm(n)3118 2597 y Ft(\)\()p Fq(!)3269 2533 y Fk(1)p 3268 2545 V 3268 2586 a(2)3313 2597 y Ft(\))3351 2546 y Fo(\()p Fm(n)p Fo(\))3351 2620 y Fm(i)p Fl(\000)p Fo(1)3469 2597 y Ft(\011\))267 2805 y(=)28 b(\(\()p Fq(!)522 2741 y Fk(1)p 521 2753 V 521 2794 a(2)566 2805 y Ft(\))604 2754 y Fo(\()p Fm(n)p Fo(\))604 2828 y Fm(i)p Fl(\000)p Fo(1)722 2805 y Ft(\011)p Fp(j)p Ft(\()p Fq(v)915 2769 y Fl(\003)954 2805 y Ft(\()p Fr(1)1048 2820 y Fl(K)1128 2805 y Fp(\012)23 b Fq(!)1293 2768 y Fl(\000)1358 2741 y Fk(1)p 1357 2753 V 1357 2794 a(2)1401 2805 y Ft(\))f Fp(\012)h Fr(1)1617 2769 y Fl(\012)p Fm(n)1719 2805 y Ft(\)\()p Fr(1)1851 2820 y Fl(K)1931 2805 y Fp(\012)g Fq(\034)2084 2754 y Fo(\()p Fm(n)p Fo(+1\))2073 2828 y Fm(i)2276 2805 y Ft(\)\()p Fr(1)2408 2820 y Fl(K)2489 2805 y Fp(\012)f Fq(!)2653 2768 y Fl(\000)2718 2741 y Fk(1)p 2717 2753 V 2717 2794 a(2)2762 2805 y Ft(\))p Fq(v)k Fp(\012)c Fr(1)3028 2769 y Fl(\012)p Fm(n)3130 2805 y Ft(\)\()p Fq(!)3281 2741 y Fk(1)p 3280 2753 V 3280 2794 a(2)3325 2805 y Ft(\))3363 2754 y Fo(\()p Fm(n)p Fo(\))3363 2828 y Fm(i)p Fl(\000)p Fo(1)3481 2805 y Ft(\011\))942 2997 y Fp(\024)29 b(k)p Fq(v)1149 2961 y Fl(\003)1188 2997 y Ft(\()p Fr(1)1282 3012 y Fl(K)1362 2997 y Fp(\012)23 b Fq(!)1527 2961 y Fl(\000)1592 2934 y Fk(1)p 1591 2946 V 1591 2987 a(2)1635 2997 y Ft(\))p Fp(k)17 b(k)p Ft(\()p Fr(1)1884 3012 y Fl(K)1964 2997 y Fp(\012)22 b Fq(!)2128 2961 y Fl(\000)2193 2934 y Fk(1)p 2193 2946 V 2193 2987 a(2)2237 2997 y Ft(\))p Fq(v)t Fp(k)17 b(k)p Ft(\()p Fq(!)2556 2934 y Fk(1)p 2554 2946 V 2554 2987 a(2)2599 2997 y Ft(\))2637 2947 y Fo(\()p Fm(n)p Fo(\))2637 3020 y Fm(i)p Fl(\000)p Fo(1)2755 2997 y Ft(\011)p Fp(k)2881 2961 y Fo(2)942 3190 y Ft(=)28 b Fp(k)p Fq(v)1147 3154 y Fl(\003)1186 3190 y Fq(!)1251 3154 y Fl(\000)p Fo(1)1345 3190 y Fq(v)t Fp(k)p Ft(\(\011)p Fp(j)p Fq(!)c Fp(\012)f Fr(1)1829 3154 y Fl(\012)p Fo(\()p Fm(n)p Fl(\000)p Fo(1\))2076 3190 y Ft(\011\))p Fq(:)0 3352 y Ft(Th)m(us,)250 3498 y Fj(P)338 3525 y Fm(n)p Fo(+1)338 3589 y Fm(i)p Fo(=2)475 3565 y Ft(\(\011)p Fp(j)p Ft(\()p Fq(v)706 3529 y Fl(\003)767 3565 y Fp(\012)f Fr(1)922 3529 y Fl(\012)p Fm(n)1024 3565 y Ft(\)\()p Fr(1)1156 3580 y Fl(K)1236 3565 y Fp(\012)h Fq(\034)1389 3514 y Fo(\()p Fm(n)p Fo(+1\))1378 3588 y Fm(i)1582 3565 y Ft(\)\()p Fq(v)i Fp(\012)e Fr(1)1886 3529 y Fl(\012)p Fm(n)1988 3565 y Ft(\)\011\))83 b Fp(\024)28 b Fq(n)p Fp(k)p Fq(v)2487 3529 y Fl(\003)2526 3565 y Fq(!)2591 3529 y Fl(\000)p Fo(1)2685 3565 y Fq(v)t Fp(k)p Ft(\(\011)p Fp(j)p Fq(!)c Fp(\012)f Fr(1)3169 3529 y Fl(\012)p Fo(\()p Fm(n)p Fl(\000)p Fo(1\))3416 3565 y Ft(\011\))2223 3756 y(=)k Fp(k)p Fq(v)2427 3720 y Fl(\003)2466 3756 y Fq(!)2531 3720 y Fl(\000)p Fo(1)2625 3756 y Fq(v)t Fp(k)p Ft(\(\011)p Fp(j)p Ft(d\000\()p Fq(!)t Ft(\)\011\))p Fq(:)0 3960 y Ft(Com)m(bining)e(this)h(inequalit)m(y)f(with)h(the)g(iden)m(tit)m(y) h(\(4.86\),)g(w)m(e)g(deriv)m(e)g(the)g(second)g(relation)e(in)g (\(4.83\).)146 4080 y(Finally)-8 b(,)30 b(the)j(third)f(relation)f (follo)m(ws)g(from)g(the)i(\014rst)g(t)m(w)m(o)h(and)e(the)h(simple)e (estimate)1135 4286 y Fp(k)p Fq(')p Ft(\()p Fq(v)t Ft(\)\011)p Fp(k)1502 4244 y Fo(2)1568 4286 y Fp(\024)d(k)p Fq(a)p Ft(\()p Fq(v)t Ft(\)\011)p Fp(k)2027 4244 y Fo(2)2088 4286 y Ft(+)22 b Fp(k)p Fq(a)2287 4244 y Fl(\003)2326 4286 y Ft(\()p Fq(v)t Ft(\)\011)p Fp(k)2579 4244 y Fo(2)2618 4286 y Fq(:)0 4491 y Fe(2)146 4657 y Ft(W)-8 b(e)33 b(will)e(also)g (mak)m(e)i(use)g(of)g(the)g(follo)m(wing)c(estimate.)0 4847 y Fr(Lemma)37 b(4.2)49 b Fi(L)-5 b(et)35 b Fq(v)d Fp(2)c Fq(B)5 b Ft(\()p Fp(K)q Fq(;)17 b Fp(K)23 b(\012)g Fh(h)p Ft(\))p Fi(,)35 b Ft(1)27 b Fp(\025)i Fq(\016)i Fp(\025)d Ft(0)35 b Fi(and)f Fq(N)2225 4862 y Fm(\016)2291 4847 y Ft(:=)28 b(1)22 b(+)g Fq(\016)t(N)10 b Fi(.)45 b(Then,)34 b(for)g(any)h Fq(\014)6 b Fi(,)1109 5063 y Fp(k)p Ft(\()p Fq(')p Ft(\()p Fq(v)t Ft(\))21 b Fp(\000)i Fq(N)1597 5015 y Fl(\000)p Fm(\014)1587 5088 y(\016)1699 5063 y Fq(')p Ft(\()p Fq(v)t Ft(\))p Fq(N)1978 5015 y Fm(\014)1968 5088 y(\016)2026 5063 y Ft(\))p Fp(k)k(\024)h Fq(C)2316 5078 y Fm(\014)2363 4973 y Fp(p)p 2446 4973 47 4 v 90 x Fq(\016)t Fp(k)p Fq(v)t Fp(k)p Fq(:)1841 5753 y Ft(38)p eop %%Page: 39 39 39 38 bop 0 407 a Fr(Pro)s(of.)65 b Ft(It)33 b(su\016ces)h(to)e (consider)h(the)g(case)h Fq(\014)f Fp(\025)28 b Ft(0.)44 b(F)-8 b(or)31 b(\011)d Fp(2)g(K)23 b(\012)g Ft(\000)2624 422 y Fm(n)2671 407 y Ft(\()p Fh(h)p Ft(\))33 b(w)m(e)g(ha)m(v)m(e)453 540 y Fj(\020)503 636 y Fq(a)554 600 y Fl(\003)593 636 y Ft(\()p Fq(v)t Ft(\))22 b Fp(\000)h Fq(N)930 589 y Fl(\000)p Fm(\014)920 662 y(\016)1032 636 y Fq(a)1083 600 y Fl(\003)1123 636 y Ft(\()p Fq(v)t Ft(\))p Fq(N)1338 589 y Fm(\014)1328 662 y(\016)1385 540 y Fj(\021)1451 636 y Ft(\011)28 b(=)1658 558 y Fp(p)p 1741 558 228 4 v 78 x Fq(n)23 b Ft(+)f(1)1969 540 y Fj(\020)2018 636 y Ft(1)g Fp(\000)h Ft(\()2309 597 y Fo(1+)p Fm(\016)r(n)p 2237 613 312 4 v 2237 671 a Fo(1+)p Fm(\016)r Fo(\()p Fm(n)p Fo(+1\))2558 636 y Ft(\))2596 588 y Fm(\014)2643 540 y Fj(\021)2693 636 y Fq(P)2756 651 y Fm(n)p Fo(+1)2893 636 y Fq(v)j Fp(\012)d Fr(1)3122 600 y Fl(\012)p Fm(n)3223 636 y Ft(\011)p Fq(:)0 875 y Ft(F)-8 b(or)32 b(0)27 b Fq(<)h(x)g Fp(\024)g Ft(1)k(w)m(e)i(ha)m(v)m(e)g Fp(j)p Ft(1)21 b Fp(\000)i Ft(\(1)f Fp(\000)h Fq(x)p Ft(\))1493 839 y Fm(\014)1540 875 y Fp(j)k(\024)i Fq(C)1771 890 y Fm(\014)1818 875 y Fq(x)p Ft(.)43 b(Hence)413 1008 y Fj(\015)413 1058 y(\015)413 1108 y(\015)p Fq(a)510 1071 y Fl(\003)550 1108 y Ft(\()p Fq(v)t Ft(\))22 b Fp(\000)g Fq(N)886 1060 y Fl(\000)p Fm(\014)876 1133 y(\016)989 1108 y Fq(a)1040 1071 y Fl(\003)1080 1108 y Ft(\()p Fq(v)t Ft(\))p Fq(N)1295 1060 y Fm(\014)1285 1133 y(\016)1342 1008 y Fj(\015)1342 1058 y(\015)1342 1108 y(\015)83 b Fp(\024)28 b Ft(sup)1723 1131 y Fm(n)p Fl(\025)p Fo(0)1877 1029 y Fp(p)p 1960 1029 228 4 v 79 x Fq(n)22 b Ft(+)g(1)2187 1008 y Fj(\014)2187 1058 y(\014)2187 1108 y(\014)2215 1011 y(\020)2264 1108 y Ft(1)g Fp(\000)h Ft(\()2555 1068 y Fo(1+)p Fm(\016)r(n)p 2483 1084 312 4 v 2483 1142 a Fo(1+)p Fm(\016)r Fo(\()p Fm(n)p Fo(+1\))2804 1108 y Ft(\))2842 1059 y Fm(\014)2889 1011 y Fj(\021)2939 1008 y(\014)2939 1058 y(\014)2939 1108 y(\014)p Fp(k)p Fq(v)t Fp(k)1471 1301 y(\024)28 b Ft(sup)1723 1324 y Fm(n)p Fl(\025)p Fo(0)1877 1301 y Fq(C)1947 1316 y Fm(\014)1994 1222 y Fp(p)p 2077 1222 228 4 v 79 x Fq(n)22 b Ft(+)g(1)2453 1261 y Fm(\016)p 2314 1277 312 4 v 2314 1335 a Fo(1+)p Fm(\016)r Fo(\()p Fm(n)p Fo(+1\))2636 1301 y Fp(k)p Fq(v)t Fp(k)1471 1500 y(\024)28 b Fq(C)1646 1515 y Fm(\014)1693 1416 y Fp(p)p 1776 1416 47 4 v 84 x Fq(\016)t Fp(k)p Fq(v)t Fp(k)p Fq(:)3530 1297 y Ft(\(4.87\))0 1716 y(After)33 b(taking)e(adjoin)m(ts,)i(\(4.87\))e(yields)1163 1846 y Fj(\015)1163 1896 y(\015)1163 1946 y(\015)p Fq(a)p Ft(\()p Fq(v)t Ft(\))22 b Fp(\000)h Fq(N)1597 1899 y Fl(\000)p Fm(\014)1587 1971 y(\016)1699 1946 y Fq(a)p Ft(\()p Fq(v)t Ft(\))p Fq(N)1965 1899 y Fm(\014)1955 1971 y(\016)2013 1846 y Fj(\015)2013 1896 y(\015)2013 1946 y(\015)28 b Fp(\024)g Fq(C)2262 1961 y Fm(\014)2309 1856 y Fp(p)p 2392 1856 V 90 x Fq(\016)t Fp(k)p Fq(v)t Fp(k)p Fq(:)0 2163 y Ft(Clearly)-8 b(,)32 b(the)h(ab)s(o)m(v)m(e)g(t)m (w)m(o)g(estimates)g(yield)e(the)i(statemen)m(t.)44 b Fe(2)146 2332 y Ft(The)d(\014nal)d(results)i(whic)m(h)g(w)m(e)h(need)f (is)f(the)h(exp)s(onen)m(tial)f(la)m(w)g(for)g(b)s(osonic)g(systems,)k (see)d(e.g.)0 2453 y([BSZ],)33 b(Section)f(3.2.)0 2654 y Fr(Theorem)74 b(4.3)49 b Fi(L)-5 b(et)36 b Fh(h)916 2669 y Fo(1)990 2654 y Fi(and)f Fh(h)1223 2669 y Fo(2)1297 2654 y Fi(b)-5 b(e)35 b(Hilb)-5 b(ert)35 b(sp)-5 b(ac)g(es.)44 b(Ther)-5 b(e)34 b(exist)g(a)h(unitary)h(mapping)1227 2871 y Fq(U)i Ft(:)28 b(\000\()p Fh(h)1528 2886 y Fo(1)1567 2871 y Ft(\))23 b Fp(\012)f Ft(\000\()p Fh(h)1869 2886 y Fo(2)1909 2871 y Ft(\))27 b Fp(7!)h Ft(\000\()p Fh(h)2244 2886 y Fo(1)2305 2871 y Fp(\010)23 b Fh(h)2448 2886 y Fo(2)2488 2871 y Ft(\))p Fq(;)0 3088 y Fi(with)35 b(the)g(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)0 3209 y Ft(\(i\))34 b Fi(If)g Fq(A)313 3224 y Fo(1)388 3209 y Fi(and)g Fq(A)650 3224 y Fo(2)724 3209 y Fi(ar)-5 b(e)35 b(op)-5 b(er)g(ators)34 b(on)h Fh(h)1496 3224 y Fo(1)1570 3209 y Fi(and)g Fh(h)1803 3224 y Fo(2)1877 3209 y Fi(then)1097 3426 y Fq(U)10 b Ft(\(\000\()p Fq(A)1383 3441 y Fo(1)1423 3426 y Ft(\))22 b Fp(\012)g Ft(\000\()p Fq(A)1754 3441 y Fo(2)1794 3426 y Ft(\)\))p Fq(U)1946 3385 y Fl(\000)p Fo(1)2068 3426 y Ft(=)28 b(\000\()p Fq(A)2344 3441 y Fo(1)2406 3426 y Fp(\010)22 b Fq(A)2578 3441 y Fo(2)2618 3426 y Ft(\))p Fq(:)0 3643 y Ft(\(ii\))33 b Fi(If)h Ft(\012)i Fi(denotes)e(the)h (vacuum)f(on)h Ft(\000\()p Fh(h)1526 3658 y Fo(1)1588 3643 y Fp(\010)22 b Fh(h)1730 3658 y Fo(2)1770 3643 y Ft(\))35 b Fi(and)f Ft(\012)2102 3658 y Fo(1)2142 3643 y Fi(,)h Ft(\012)2277 3658 y Fo(2)2351 3643 y Fi(the)g(vacua)g(on)f Ft(\000\()p Fh(h)3071 3658 y Fo(1)3111 3643 y Ft(\))p Fi(,)g Ft(\000\()p Fh(h)3355 3658 y Fo(2)3395 3643 y Ft(\))p Fi(,)h(then)1528 3860 y Fq(U)10 b Ft(\(\012)1712 3875 y Fo(1)1775 3860 y Fp(\012)23 b Ft(\012)1945 3875 y Fo(2)1984 3860 y Ft(\))28 b(=)g(\012)p Fq(:)0 4078 y Ft(\(iii\))k Fi(If)i Fq(f)342 4093 y Fo(1)410 4078 y Fp(2)28 b Fh(h)547 4093 y Fo(1)586 4078 y Fi(,)35 b Fq(f)699 4093 y Fo(2)766 4078 y Fp(2)28 b Fh(h)903 4093 y Fo(2)943 4078 y Fi(,)35 b(then)813 4295 y Fq(U)27 b Ft(exp)q(\(i)p Fq(')p Ft(\()p Fq(f)1271 4310 y Fo(1)1310 4295 y Ft(\))22 b Fp(\012)g Ft(exp)q(\(i)p Fq(')p Ft(\()p Fq(f)1834 4310 y Fo(2)1873 4295 y Ft(\)\))p Fq(U)2025 4254 y Fl(\000)p Fo(1)2147 4295 y Ft(=)28 b(exp)q(\(i)p Fq(')p Ft(\()p Fq(f)2616 4310 y Fo(1)2676 4295 y Fp(\010)23 b Fq(f)2824 4310 y Fo(2)2864 4295 y Ft(\)\))p Fq(:)0 4512 y Ft(\(iv\))35 b Fi(L)-5 b(et)35 b Fp(K)i Fi(b)-5 b(e)35 b(a)g(Hilb)-5 b(ert)35 b(sp)-5 b(ac)g(e.)45 b(Assume)35 b(that)h Fq(v)1902 4527 y Fo(1)1970 4512 y Fp(2)29 b Fq(B)5 b Ft(\()p Fp(K)q Fq(;)17 b Fp(K)24 b(\012)e Fh(h)2545 4527 y Fo(1)2585 4512 y Ft(\))p Fi(,)35 b Fq(f)2736 4527 y Fo(2)2804 4512 y Fp(2)29 b Fh(h)2942 4527 y Fo(2)3017 4512 y Fi(and)34 b Fq(v)3253 4527 y Fo(2)3321 4512 y Ft(=)29 b Fr(1)3482 4527 y Fl(K)3562 4512 y Fp(\012)23 b Fq(f)3710 4527 y Fo(2)3750 4512 y Fi(.)0 4632 y(Then)34 b Fq(v)301 4647 y Fo(1)363 4632 y Fp(\010)22 b Fq(v)509 4647 y Fo(2)584 4632 y Fi(c)-5 b(an)34 b(b)-5 b(e)35 b(viewe)-5 b(d)34 b(as)g(an)h(element)f(of)g Fq(B)5 b Ft(\()p Fp(K)q Fq(;)17 b Fp(K)24 b(\012)f Ft(\()p Fh(h)2450 4647 y Fo(1)2511 4632 y Fp(\010)g Fh(h)2654 4647 y Fo(2)2694 4632 y Ft(\)\))34 b Fi(and)484 4850 y Ft(\()p Fr(1)578 4865 y Fl(K)658 4850 y Fp(\012)23 b Fq(U)10 b Ft(\))17 b(exp)q(\(i)p Fq(')p Ft(\()p Fq(v)1253 4865 y Fo(1)1292 4850 y Ft(\)\))22 b Fp(\012)g Ft(exp)q(\(i)p 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5010 y(~)1588 5036 y Fq(H)7 b Ft(.)1841 5753 y(41)p eop %%Page: 42 42 42 41 bop 146 407 a Ft(Consider)34 b(an)f(op)s(erator)g(of)g(the)g (form)f(\(5.91\))h(and)g(assume)h(that)f(\(5.92\))f(holds.)46 b(This)33 b(op)s(erator)0 527 y(can)g(b)s(e)g(extended)h(to)e(act)h(on) g(the)g(Hilb)s(ert)d(space)1937 502 y(~)1908 527 y Fp(H)23 b(\012)g Ft(\000\()2211 512 y(~)2214 527 y Fh(h)p Ft(\))33 b(as)1295 726 y(~)1269 751 y Fq(H)1358 715 y Fo(ext)1350 776 y(fr)1486 751 y Ft(:=)1642 726 y(~)1616 751 y Fq(H)1697 766 y Fo(fr)1773 751 y Fp(\012)22 b Fr(1)g Fp(\000)h Fr(1)f Fp(\012)h Ft(d\000\()8 b(~)-57 b Fq(!)s Ft(\))p Fq(;)1295 924 y Ft(~)1269 950 y Fq(H)1358 913 y Fo(ext)1486 950 y Ft(:=)1642 924 y(~)1616 950 y Fq(H)30 b Fp(\012)23 b Fr(1)f Fp(\000)g Fr(1)h Fp(\012)f Ft(d\000\()8 b(~)-57 b Fq(!)t Ft(\))p Fq(:)0 1185 y Ft(Since)284 1160 y(~)255 1185 y Fp(H)23 b(\012)g Ft(\000)523 1200 y Fo(0)562 1185 y Ft(\()597 1170 y(~)600 1185 y Fh(h)p Ft(\))33 b(is)f(an)g(in)m(v)-5 b(arian)m(t)31 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y(~)2356 1808 y Fq(H)2445 1771 y Fo(ext)2545 1808 y Ft(\)\).)61 b(Let)39 b(us)g(sho)m(w)h(that)3512 1782 y(~)3487 1808 y Fq(H)3576 1771 y Fo(ext)3714 1808 y Ft(is)0 1928 y(unitarily)30 b(equiv)-5 b(alen)m(t)33 b(to)f(an)g(op)s(erator)g(of)g(the)h(form)f (\(5.88\))g(satisfying)f(Hyp)s(othesis)j(A.)146 2048 y(Let)f Fq(U)43 b Ft(b)s(e)33 b(the)g(map)f(from)f(\000\()1274 2033 y(~)1277 2048 y Fh(h)p Ft(\))22 b Fp(\012)h Ft(\000\()1576 2033 y(~)1579 2048 y Fh(h)p Ft(\))33 b(to)f(\000\()1908 2033 y(~)1911 2048 y Fh(h)22 b Fp(\010)2073 2033 y Ft(~)2076 2048 y Fh(h)p Ft(\))33 b(de\014ned)h(in)d(Theorem)i(4.3.)43 b(Clearly)-8 b(,)1149 2268 y Fr(1)1205 2283 y Fl(K)1286 2268 y Fp(\012)22 b Fq(U)38 b Ft(:)1573 2243 y(~)1544 2268 y Fp(H)23 b(\012)g Ft(\000\()p Fh(h)p Ft(\))28 b Fp(!)f(K)d(\012)e Ft(\000\()2381 2253 y(~)2384 2268 y Fh(h)h Fp(\010)2546 2253 y Ft(~)2549 2268 y Fh(h)p Ft(\))0 2488 y(is)32 b(a)g(unitary)h(map.)42 b(Next,)34 b(w)m(e)f(ha)m(v)m(e)h (the)f(unitary)f(map)982 2708 y Fq(L)1048 2667 y Fo(2)1088 2708 y Ft(\()p Fr(R)1210 2723 y Fo(+)1269 2708 y Ft(\))22 b Fp(\010)h Fq(L)1495 2667 y Fo(2)1534 2708 y Ft(\()p Fr(R)1656 2723 y Fo(+)1715 2708 y Ft(\))28 b Fp(3)g Ft(\()p Fq(f)1961 2723 y Fo(1)2000 2708 y Fq(;)17 b(f)2092 2723 y Fo(2)2131 2708 y Ft(\))28 b Fp(7!)f Fq(f)39 b Fp(2)28 b Fq(L)2571 2667 y Fo(2)2610 2708 y Ft(\()p Fr(R)p Ft(\))p Fq(;)733 b Ft(\(5.93\))0 2928 y(where)1293 3086 y Fq(f)11 b Ft(\()p Fq(!)t Ft(\))26 b(=)1623 2940 y Fj(\()1731 3025 y Fq(f)1779 3040 y Fo(1)1819 3025 y Ft(\()p Fq(!)t Ft(\))114 b(if)32 b Fq(!)f Fp(\025)d Ft(0)1731 3146 y Fq(f)1779 3161 y Fo(2)1819 3146 y Ft(\()p Fq(!)t Ft(\))114 b(if)32 b Fq(!)f(<)c Ft(0)p Fq(;)0 3317 y Ft(whic)m(h)33 b(induces)g(the)g(unitary)g(map)602 3537 y Fq(w)d Ft(:)754 3522 y(~)757 3537 y Fh(h)22 b Fp(\010)919 3522 y Ft(~)922 3537 y Fh(h)28 b Ft(=)1096 3441 y Fj(\020)1146 3537 y Fq(L)1212 3496 y Fo(2)1252 3537 y Ft(\()p Fr(R)1374 3552 y Fo(+)1432 3537 y Ft(\))22 b Fp(\012)h Fh(g)1634 3441 y Fj(\021)1706 3537 y Fp(\010)1805 3441 y Fj(\020)1855 3537 y Fq(L)1921 3496 y Fo(2)1961 3537 y Ft(\()p Fr(R)2083 3552 y Fo(+)2141 3537 y Ft(\))f Fp(\012)h Fh(g)2343 3441 y Fj(\021)2420 3537 y Fp(7!)k Fh(h)h Ft(=)g Fq(L)2788 3496 y Fo(2)2827 3537 y Ft(\()p Fr(R)p Ft(\))22 b Fp(\012)h Fh(g)p Fq(:)0 3757 y Ft(Set)1445 3878 y Fq(W)42 b Ft(:=)27 b Fr(1)1765 3893 y Fl(K)1846 3878 y Fp(\012)22 b Ft(\(\000\()p Fq(w)s Ft(\))p Fq(U)10 b Ft(\))p Fq(:)0 4052 y Ft(Clearly)-8 b(,)1469 4172 y Fq(W)41 b Ft(:)1686 4147 y(~)1657 4172 y Fp(H)23 b(\012)g Ft(\000\()1960 4157 y(~)1963 4172 y Fh(h)p Ft(\))k Fp(7!)h(H)q Fq(;)0 4347 y Ft(is)k(a)g(unitary)h(map.) 42 b(Let)33 b Fq(\013)28 b Fp(2)g Fq(B)5 b Ft(\()p Fp(K)q Fq(;)17 b Fp(K)24 b(\012)f Fh(h)p Ft(\))32 b(b)s(e)h(giv)m(en)g(b)m(y) 1649 4567 y Fq(\013)28 b Ft(:=)38 b(~)-59 b Fq(\013)23 b Fp(\010)g Ft(0)p Fq(:)0 4787 y Ft(W)-8 b(e)33 b(ha)m(v)m(e)1822 4879 y Fq(w)s Ft(\()8 b(~)-57 b Fq(!)r(;)17 b Fp(\000)8 b Ft(~)-57 b Fq(!)t Ft(\))p Fq(w)2293 4843 y Fl(\003)2415 4879 y Ft(=)27 b Fq(!)t(;)979 5071 y(W)14 b Ft(\(d\000\()8 b(~)-57 b Fq(!)s Ft(\)\))22 b Fp(\012)h Fr(1)f Fp(\000)g Fr(1)h Fp(\012)f Ft(d\000\()8 b(~)-57 b Fq(!)t Ft(\)\))p Fq(W)2293 5034 y Fl(\003)2415 5071 y Ft(=)27 b(d\000\()p Fq(!)t Ft(\))p Fq(;)1701 5262 y(W)14 b(')p Ft(\()c(~)-59 b Fq(\013)p Ft(\))22 b Fp(\012)g Fr(1)p Fq(W)2292 5226 y Fl(\003)2415 5262 y Ft(=)27 b Fq(')p Ft(\()p Fq(\013)q Ft(\))p Fq(:)1841 5753 y Ft(42)p eop %%Page: 43 43 43 42 bop 0 407 a Ft(Th)m(us)991 506 y Fq(W)1121 481 y Ft(~)1097 506 y Fq(H)1186 470 y Fo(ext)1178 531 y(fr)1285 506 y Fq(W)1391 470 y Fl(\003)1458 506 y Ft(=)27 b Fq(K)i Fp(\012)23 b Fr(1)f Ft(+)g Fr(1)g Fp(\012)h Ft(d\000\()p Fq(!)t Ft(\))p Fq(;)991 705 y(W)1121 680 y Ft(~)1097 705 y Fq(H)1186 669 y Fo(ext)1285 705 y Fq(W)1391 669 y Fl(\003)1458 705 y Ft(=)k Fq(K)i Fp(\012)23 b Fr(1)f Ft(+)g Fr(1)g Fp(\012)h Ft(d\000\()p Fq(!)t Ft(\))f(+)g Fq(\025')p Ft(\()p Fq(\013)q Ft(\))p Fq(:)0 884 y Ft(Hence)35 b Fq(W)422 859 y Ft(~)397 884 y Fq(H)486 848 y Fo(ext)478 908 y(fr)585 884 y Fq(W)691 848 y Fl(\003)763 884 y Ft(and)f Fq(W)1085 859 y Ft(~)1060 884 y Fq(H)1149 848 y Fo(ext)1248 884 y Fq(W)1354 848 y Fl(\003)1427 884 y Ft(ha)m(v)m(e)g(the)g(form)e (of)h(the)h(op)s(erators)f Fq(H)2846 899 y Fo(fr)2899 884 y Ft(,)h Fq(H)8 b Ft(.)45 b(F)-8 b(urthermore,)33 b(it)0 1004 y(follo)m(ws)e(from)h(Prop)s(osition)e(5.2)j(that)f(the)h (op)s(erator)f Fq(W)2136 979 y Ft(~)2111 1004 y Fq(H)2200 968 y Fo(ext)2299 1004 y Fq(W)2405 968 y Fl(\003)2477 1004 y Ft(is)g(self-adjoin)m(t)f(on)981 1199 y Fp(D)s Ft(\()p Fq(W)1230 1174 y Ft(~)1205 1199 y Fq(H)1294 1158 y Fo(ext)1286 1224 y(fr)1393 1199 y Fq(W)1499 1158 y Fl(\003)1538 1199 y Ft(\))d(=)g Fp(D)s Ft(\()p Fq(W)1956 1174 y Ft(~)1932 1199 y Fq(H)2021 1158 y Fo(ext)2013 1224 y(fr)2119 1199 y Fq(W)2225 1158 y Fl(\003)2265 1199 y Ft(\))22 b Fp(\\)g(D)s Ft(\()p Fq(')p Ft(\()p Fq(\013)q Ft(\)\))p Fq(:)0 1395 y Ft(Th)m(us)34 b(the)f(op)s(erator)f Fq(W)939 1369 y Ft(~)914 1395 y Fq(H)1003 1358 y Fo(ext)1102 1395 y Fq(W)1208 1358 y Fl(\003)1280 1395 y Ft(satis\014es)h(Hyp)s (othesis)h(A.)e Fe(2)0 1766 y Fs(6)161 b(Main)55 b(results)0 1986 y Ft(The)34 b(Hilb)s(ert)c(space)k Fp(H)29 b Ft(=)e Fp(K)d(\012)f Ft(\000\()p Fh(h)p Ft(\))32 b(has)h(a)f(natural)g(decomp) s(osition)f Fp(H)d Ft(=)g Fp(H)2966 1949 y Fo(v)3031 1986 y Fp(\010)22 b(H)p 3215 1911 43 4 v 3215 1949 a Fo(v)3258 1986 y Ft(,)32 b(where)1370 2178 y Fp(H)1455 2142 y Fo(v)1580 2178 y Ft(:=)c Fp(K)23 b(\012)g Ft(\000)1971 2193 y Fo(0)2010 2178 y Ft(\()p Fh(h)p Ft(\))p Fq(;)1370 2298 y Fp(H)p 1455 2224 V 1455 2262 a Fo(v)1580 2298 y Ft(:=)1711 2232 y Fj(L)1803 2258 y Fl(1)1803 2323 y Fm(n)p Fo(=1)1957 2298 y Fp(K)g(\012)g Ft(\000)2217 2313 y Fm(n)2264 2298 y Ft(\()p Fh(h)p Ft(\))p Fq(:)0 2499 y Ft(\(v)k(stands)g(for)d(the)i(v)-5 b(acuum\).)41 b(Note)26 b(that)f Fq(H)1672 2463 y Fo(vv)1664 2524 y(fr)1779 2499 y Ft(=)i Fq(K)7 b Ft(,)27 b Fp(D)s Ft(\()p Fq(H)2225 2514 y Fo(fr)2278 2499 y Ft(\))h(=)f Fp(D)s Ft(\()p Fq(K)7 b Ft(\))h Fp(\010)g(D)s Ft(\()p Fq(H)p 2993 2424 39 4 v 2993 2463 a Fo(v)p 3030 2424 V(v)2985 2524 y(fr)3072 2499 y Ft(\))25 b(and)h Fp(D)s Ft(\()p Fq(')p Ft(\()p Fq(\013)q Ft(\)\))h(=)0 2619 y Fp(K)13 b(\010)e(D)s Ft(\()p Fq(')p Ft(\()p Fq(\013)q Ft(\))p 498 2545 V -36 x Fo(v)p 535 2545 V(v)578 2619 y Ft(\).)41 b(Hence,)30 b(if)c Fp(D)k Ft(is)c(as)i(in)e(Hyp)s(othesis)i(A,)g(then)f Fp(D)k Ft(=)c Fp(D)s Ft(\()p Fq(K)7 b Ft(\))k Fp(\010)g(D)s Ft(\()p Fq(H)p 3107 2545 V 3107 2583 a Fo(v)p 3145 2545 V 1 w(v)3099 2644 y(fr)3187 2619 y Ft(\))g Fp(\\)g(D)s Ft(\()p Fq(')p Ft(\()p Fq(\013)q Ft(\))p 3634 2545 V -36 x Fo(v)p 3672 2545 V 1 w(v)3714 2619 y Ft(\).)0 2740 y(Using)34 b(Hyp)s(othesis)i(A,)f(the)g(fact)g(that)g Fq(H)p 1584 2665 V 1584 2703 a Fo(vv)1695 2740 y Ft(=)d Fq(\025')p Ft(\()p Fq(\013)q Ft(\))p 2063 2665 V -37 x Fo(vv)2177 2740 y Ft(is)j(a)f(b)s(ounded)i(op)s(erator)e(and)h(the)g (Kato{)0 2860 y(Rellic)m(h)30 b(theorem)h(w)m(e)i(see)g(that)e Fq(H)1300 2824 y Fo(vv)1400 2860 y Ft(+)20 b Fq(H)p 1585 2785 V 1585 2824 a Fo(v)p 1623 2785 V 1 w(v)1697 2860 y Ft(is)31 b(essen)m(tially)g(self-adjoin)m(t)f(on)h Fp(D)s Ft(.)43 b(Therefore,)33 b Fq(H)p 3602 2785 V 3602 2824 a Fo(v)p 3640 2785 V 1 w(v)3714 2860 y Ft(is)0 2980 y(essen)m(tially)25 b(self-adjoin)m(t)f(on)h Fp(D)s Ft(\()p Fq(H)p 1290 2906 V 1290 2944 a Fo(v)p 1328 2906 V 1 w(v)1282 3005 y(fr)1370 2980 y Ft(\))8 b Fp(\\)g(D)s Ft(\()p Fq(')p Ft(\()p Fq(\013)q Ft(\))p 1811 2906 V -36 x Fo(v)p 1848 2906 V(v)1890 2980 y Ft(\).)41 b(Th)m(us)27 b(the)f(formalism)c(and)k (results)g(of)f(Chapter)0 3101 y(3)34 b(can)h(b)s(e)f(applied)g(to)g (the)g(op)s(erator)g Fq(H)8 b Ft(.)48 b(W)-8 b(e)35 b(will)d(use)k(the) e(notation)f(in)m(tro)s(duced)i(in)f(Section)g(3.2.)0 3221 y(In)f(particular,)e(w)m(e)i(recall)f(that)g(the)h(self-energy)g (is)f(de\014ned)i(b)m(y)963 3408 y Fq(W)1055 3423 y Fo(v)1098 3408 y Ft(\()p Fq(z)t Ft(\))83 b(=)28 b Fq(H)1499 3372 y Fo(v)p 1537 3333 V 1 w(v)1579 3408 y Ft(\()p Fq(z)t Fr(1)p 1722 3333 V -36 x Fo(v)p 1761 3333 V 2 w(v)1826 3408 y Fp(\000)22 b Fq(H)p 2014 3333 V 2014 3372 a Fo(v)p 2052 3333 V 1 w(v)2094 3408 y Ft(\))2132 3372 y Fl(\000)p Fo(1)2227 3408 y Fq(H)p 2316 3333 V 2316 3372 a Fo(vv)1306 3599 y Ft(=)1420 3560 y Fo(1)p 1420 3576 36 4 v 1420 3634 a(2)1465 3599 y Fq(\025)1522 3563 y Fo(2)1562 3599 y Fq(a)p Ft(\()p Fq(\013)q Ft(\))1752 3563 y Fo(v)p 1790 3525 39 4 v 1 w(v)1832 3599 y Ft(\()p Fq(z)t Fr(1)p 1975 3525 V -36 x Fo(v)p 2014 3525 V 2 w(v)2078 3599 y Fp(\000)h Fq(H)p 2267 3525 V 2267 3563 a Fo(v)p 2305 3525 V 1 w(v)2347 3599 y Ft(\))2385 3563 y Fl(\000)p Fo(1)2479 3599 y Fq(a)2530 3563 y Fl(\003)2570 3599 y Ft(\()p Fq(\013)q Ft(\))p 2709 3525 V -36 x Fo(v)q(v)2789 3599 y Fq(:)146 3788 y Ft(W)-8 b(e)35 b(de\014ne)g(a)f(self-adjoin)m(t)f(op)s(erator)g Fq(s)i Ft(on)f(the)g(Hilb)s(ert)f(space)i Fh(h)g Ft(b)m(y)g Fq(s)30 b Ft(:=)h Fp(\000)p Ft(i)p Fq(@)3148 3803 y Fm(!)3222 3788 y Fp(\012)23 b Fr(1)3378 3803 y Fc(g)3418 3788 y Ft(,)34 b(so)h(that)0 3908 y([)p Fq(s;)17 b(!)t Ft(])27 b(=)g Fp(\000)p Ft(i.)43 b(The)34 b(conjugate)f(op)s(erator)f(is)g (de\014ned)i(b)m(y)1528 4103 y Fq(S)f Ft(:=)28 b Fr(1)1808 4118 y Fl(K)1888 4103 y Fp(\012)23 b Ft(d\000\()p Fq(s)p Ft(\))p Fq(:)1278 b Ft(\(6.94\))146 4299 y(F)-8 b(or)32 b(an)m(y)h Fq(\027)i Fp(\025)28 b Ft(0)k(w)m(e)i(in)m(tro)s(duce)e(the) h(follo)m(wing)d(h)m(yp)s(othesis:)840 4494 y(Hyp)s(othesis)66 b Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))p Fq(:)455 b Fp(h)p Fq(s)p Fp(i)2177 4453 y Fm(\027)2220 4494 y Fq(\013)28 b Fp(2)h(B)s Ft(\()p Fp(K)q Fq(;)17 b Fp(K)23 b(\012)g Fh(h)p Ft(\))p Fq(:)146 4689 y Ft(W)-8 b(e)33 b(will)e(compare)h Fq(W)981 4704 y Fo(v)1023 4689 y Ft(\()p Fq(z)t Ft(\))h(with)g(its)f (second-order)h(appro)m(ximation)e Fq(\025)2817 4653 y Fo(2)2856 4689 y Fq(w)s Ft(\()p Fq(z)t Ft(\),)i(where)1077 4881 y Fq(w)s Ft(\()p Fq(z)t Ft(\))83 b(:=)28 b Fq(')p Ft(\()p Fq(\013)q Ft(\))1692 4845 y Fo(v)p 1730 4807 V 1 w(v)1772 4881 y Ft(\()p Fq(z)t Fr(1)p 1915 4807 V -36 x Fo(v)p 1954 4807 V 2 w(v)2018 4881 y Fp(\000)23 b Fq(H)p 2207 4807 V 2207 4845 a Fo(v)p 2245 4807 V 1 w(v)2199 4906 y(fr)2287 4881 y Ft(\))2325 4845 y Fl(\000)p Fo(1)2419 4881 y Fq(')p Ft(\()p Fq(\013)q Ft(\))p 2622 4807 V -36 x Fo(v)q(v)1358 5090 y Ft(=)1472 5051 y Fo(1)p 1472 5067 36 4 v 1472 5125 a(2)1517 4994 y Fj(\020)1567 5090 y Fq(a)p Ft(\()p Fq(\013)q Ft(\)\()p Fq(z)j Fp(\000)d Fq(H)2047 5105 y Fo(fr)2100 5090 y Ft(\))2138 5054 y Fl(\000)p Fo(1)2232 5090 y Fq(a)2283 5054 y Fl(\003)2323 5090 y Ft(\()p Fq(\013)q Ft(\))2462 4994 y Fj(\021)2511 5017 y Fo(vv)1358 5282 y Ft(=)1472 5242 y Fo(1)p 1472 5258 V 1472 5316 a(2)1517 5282 y Fq(\013)1580 5245 y Fl(\003)1619 5282 y Ft(\()p Fq(z)k Fp(\000)c Fq(K)29 b Fp(\012)22 b Fr(1)h Fp(\000)f Fr(1)g Fp(\012)h Fq(!)t Ft(\))2499 5245 y Fl(\000)p Fo(1)2593 5282 y Fq(\013)q(:)1841 5753 y Ft(43)p eop %%Page: 44 44 44 43 bop 0 407 a Ft(W)-8 b(e)32 b(no)m(w)h(state)f(an)g(auxiliary)e (result)i(on)g(the)g(regularit)m(y)f(prop)s(erties)h(of)g(the)g (function)g Fq(w)s Ft(\()p Fq(z)t Ft(\).)43 b(Some)0 527 y(of)38 b(these)i(prop)s(erties)e(will)e(b)s(e)j(used)g(in)f(the)h (statemen)m(t)g(of)e(our)i(main)d(theorems,)41 b(but)d(w)m(e)i(remark)0 648 y(that)32 b(they)i(are)e(of)h(indep)s(enden)m(t)g(in)m(terest.)0 838 y Fr(Theorem)74 b(6.1)49 b Fi(Assume)31 b(that)h(Hyp)-5 b(othesis)31 b Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))31 b Fi(holds)f(with)h Fq(\027)j(>)2618 799 y Fo(1)p 2618 815 36 4 v 2618 873 a(2)2663 838 y Fq(:)d Fi(L)-5 b(et)32 b Fq(n)c Fp(2)g Fr(N)i Fi(and)h Ft(0)c Fq(<)h(\022)j Fp(\024)d Ft(1)0 959 y Fi(b)-5 b(e)36 b(such)g(that)g Fq(\027)h Ft(=)30 b Fq(n)23 b Ft(+)925 919 y Fo(1)p 925 935 V 925 993 a(2)993 959 y Ft(+)g Fq(\022)s Fi(.)49 b(Then,)35 b(the)h(function)g Fr(C)2133 974 y Fo(+)2222 959 y Fp(3)31 b Fq(z)j Fp(7!)c Fq(w)s Ft(\()p Fq(z)t Ft(\))36 b Fi(extends)g(by)g(c)-5 b(ontinuity)36 b(to)p 0 1001 81 4 v 0 1079 a Fr(C)81 1094 y Fo(+)175 1079 y Fi(and)e(is)h(in)f(the)h(class)f Fq(C)1061 1043 y Fm(n;\022)1054 1104 y Fo(u)1163 1079 y Ft(\()p 1201 1001 V Fr(C)1282 1094 y Fo(+)1341 1079 y Ft(\))p Fi(.)146 1269 y Ft(The)42 b(next)f(three)g(theorems)g(describ)s(e)g(our)f(main)e(results.)68 b(In)40 b(our)h(mo)s(del,)f(the)h(sp)s(ectrum)g(of)0 1390 y Fq(K)h Ft(pla)m(ys)35 b(a)g(role)f(of)h(the)g(threshold)g(set.) 51 b(The)36 b(\014rst)g(theorem)e(asserts)j(that,)e(a)m(w)m(a)m(y)h (from)e(an)h Fq(O)s Ft(\()p Fq(\025)3703 1354 y Fo(2)3742 1390 y Ft(\))0 1510 y(neigh)m(b)s(orho)s(o)s(d)c(of)h Fq(\033)t Ft(\()p Fq(K)7 b Ft(\),)33 b(the)g(Limiting)c(Absorption)j (Principle)g(holds.)0 1721 y Fr(Theorem)74 b(6.2)49 b Fi(Assume)43 b(that)f(Hyp)-5 b(otheses)42 b Ft(A)h Fi(and)e Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))43 b Fi(hold)e(with)h Fq(\027)48 b(>)41 b Ft(1)p Fi(.)67 b(L)-5 b(et)43 b Fq(\026)d(>)3538 1682 y Fo(1)p 3538 1698 36 4 v 3538 1756 a(2)3625 1721 y Fi(and)0 1854 y Ft(0)35 b Fq(<)f Ft(\003)262 1869 y Fo(1)337 1854 y Fq(<)g Ft(\()485 1772 y Fp(p)p 568 1772 49 4 v 82 x Ft(2)p Fp(k)p Fq(s\013)q Fp(k)p Ft(\))864 1818 y Fl(\000)p Fo(1)958 1854 y Fi(.)56 b(Then,)39 b(ther)-5 b(e)39 b(exists)f(a)h(c)-5 b(onstant)38 b Fq(\014)2386 1869 y Fo(1)2460 1854 y Fq(>)d Ft(0)k Fi(such)f(that)h(for)g Fp(j)p Fq(\025)p Fp(j)34 b(\024)i Ft(\003)3575 1869 y Fo(1)3653 1854 y Fi(the)0 1974 y(fol)5 b(lowing)33 b(holds:)0 2095 y Ft(\(i\))h Fi(Set)h(for)f(shortness)1377 2215 y Ft(\002)28 b(:=)f Fr(R)22 b Fp(n)g Fq(I)8 b Ft(\()p Fq(\033)t Ft(\()p Fq(K)f Ft(\))p Fq(;)17 b(\025)2204 2174 y Fo(2)2243 2215 y Fq(\014)2298 2230 y Fo(1)2337 2215 y Ft(\))p Fq(:)0 2383 y Fi(Then,)34 b(the)h(function)1328 2504 y Fq(z)d Fp(7!)27 b(h)p Fq(S)6 b Fp(i)1676 2462 y Fl(\000)p Fm(\026)1777 2504 y Ft(\()p Fq(z)27 b Fp(\000)22 b Fq(H)8 b Ft(\))2113 2462 y Fl(\000)p Fo(1)2207 2504 y Fp(h)p Fq(S)e Fp(i)2351 2462 y Fl(\000)p Fm(\026)3530 2504 y Ft(\(6.95\))0 2671 y Fi(extends)36 b(by)i(c)-5 b(ontinuity)37 b(to)h(a)f(function)g(on)f Fr(C)1753 2686 y Fo(+)1836 2671 y Fp([)24 b Ft(\002)p Fi(.)52 b(In)36 b(p)-5 b(articular,)38 b(the)f(sp)-5 b(e)g(ctrum)37 b(of)g Fq(H)45 b Fi(on)37 b(the)0 2792 y(set)e Ft(\002)g Fi(is)f(absolutely)h (c)-5 b(ontinuous.)0 2912 y Ft(\(ii\))33 b Fi(L)-5 b(et)35 b Fq(n)28 b Fp(2)g Fr(N)p Fi(,)34 b Ft(0)28 b Fq(<)f(\022)k Fp(\024)d Ft(1)p Fi(,)35 b(and)f Fq(\026)h Fi(b)-5 b(e)34 b(such)h(that)g Fq(\027)f Fp(\025)28 b Fq(\026)22 b Ft(+)2339 2873 y Fo(1)p 2339 2889 36 4 v 2339 2947 a(2)2412 2912 y Ft(=)27 b(1)22 b(+)g Fq(n)g Ft(+)g Fq(\022)s Fi(.)45 b(Then,)34 b(the)h(function)0 3033 y(\(6.95\))f(is)h(of)f(the)h(class)f Fq(C)986 2996 y Fm(n;\022)979 3057 y Fo(u)1122 3033 y Fi(of)h(the)g(set)p 1452 3159 81 4 v 1452 3238 a Fr(C)1533 3253 y Fo(+)1614 3238 y Fp(n)22 b Fq(B)5 b Ft(\()p Fq(\033)t Ft(\()p Fq(K)i Ft(\))p Fq(;)17 b(\025)2129 3197 y Fo(2)2168 3238 y Fq(\014)2223 3253 y Fo(1)2263 3238 y Ft(\))p Fq(:)146 3654 y Ft(Our)34 b(last)f(t)m(w)m(o)i(theorems)f(describ)s(e)h(the)f (structure)h(of)f(the)g(sp)s(ectrum)h(near)f(an)g(isolated)e(eigen-)0 3774 y(v)-5 b(alue)36 b Fq(k)k Ft(of)c Fq(K)7 b Ft(.)55 b(They)38 b(incorp)s(orate)e(the)h(notion)f(of)g(F)-8 b(ermi's)35 b(Golden)h(Rule.)55 b(W)-8 b(e)37 b(remark)f(that)g(in)0 3895 y(our)k(approac)m(h)g(the)g(eigen)m(v)-5 b(alue)40 b Fq(k)j Ft(ma)m(y)c(ha)m(v)m(e)j(an)d(in\014nite)g(m)m(ultiplicit)m(y) -8 b(.)62 b(Let)40 b Fq(p)3088 3910 y Fm(k)3171 3895 y Ft(=)g Fr(1)3343 3910 y Fl(f)p Fm(k)r Fl(g)3456 3895 y Ft(\()p Fq(K)7 b Ft(\).)66 b(If)0 4015 y Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))33 b(holds)f(for)g(some)g Fq(\027)j(>)1073 3976 y Fo(1)p 1073 3992 36 4 v 1073 4049 a(2)1151 4015 y Ft(then)e(it)f(follo)m(ws)f(from)g(Theorem)i(6.1)f(that)1464 4220 y Fq(w)1534 4235 y Fm(k)1604 4220 y Ft(:=)27 b Fq(p)1783 4235 y Fm(k)1826 4220 y Fq(w)s Ft(\()p Fq(k)d Ft(+)f(i0\))p Fq(p)2275 4235 y Fm(k)3530 4220 y Ft(\(6.96\))0 4425 y(is)44 b(a)f(b)s(ounded)i(dissipativ)m(e)f(op)s(erator.)77 b(W)-8 b(e)45 b(will)d(alw)m(a)m(ys)i(consider)h Fq(w)2728 4440 y Fm(k)2814 4425 y Ft(as)g(an)f(op)s(erator)f(on)h(the)0 4546 y(Hilb)s(ert)38 b(space)j(Ran)p Fq(p)835 4561 y Fm(k)877 4546 y Ft(.)65 b(In)40 b(the)h(standard)f(description)f(of)g (atomic)f(radiation,)i(the)g(sp)s(ectrum)g(of)0 4666 y(Im)o Fq(w)186 4681 y Fm(k)271 4666 y Ft(captures)j(the)g(emission)e (and)h(absorption)g(pro)s(cesses)i(and)e(radiativ)m(e)f(life-time)e(of) i(energy)0 4786 y(lev)m(el)d Fq(k)j Ft(\(of)c(order)i Fq(\025)792 4750 y Fo(2)831 4786 y Ft(\).)60 b(The)39 b(sp)s(ectrum)f(of)g(Re)p Fq(w)1895 4801 y Fm(k)1975 4786 y Ft(captures)i(the)e(line)f(shift)g(of)h(this)g(energy)h(lev)m (el)0 4907 y(\(of)32 b(order)h Fq(\025)461 4871 y Fo(2)500 4907 y Ft(\).)44 b(If)33 b Fq(\033)t Ft(\()p Fq(w)874 4922 y Fm(k)916 4907 y Ft(\))22 b Fp(\\)h Fr(R)k Ft(=)h Fp(;)p Ft(,)33 b(that)f(is,)h(if)e(Im)p Fq(w)2003 4922 y Fm(k)2073 4907 y Fq(<)d Ft(0,)k(one)h(exp)s(ects)i(that)d(the)h (energy)h(lev)m(el)e Fq(k)0 5027 y Ft(has)i(dissolv)m(ed)g(in)m(to)f (the)i(con)m(tin)m(uum,)e(and)h(that)g(the)g(sp)s(ectrum)g(of)g Fq(H)41 b Ft(in)33 b(a)g(neigh)m(b)s(orho)s(o)s(d)g(of)g Fq(k)k Ft(is)0 5147 y(purely)e(absolutely)f(con)m(tin)m(uous.)51 b(Among)33 b(other)i(things,)g(our)g(next)g(theorem)g(justi\014es)g (rigorously)0 5268 y(this)d(heuristic)g(exp)s(ectation.)1841 5753 y(44)p eop %%Page: 45 45 45 44 bop 0 407 a Fr(Theorem)74 b(6.3)49 b Fi(Assume)36 b(that)g(Hyp)-5 b(otheses)35 b Ft(A)h Fi(and)f Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))35 b Fi(hold)g(with)g Fq(\027)h(>)28 b Ft(1)36 b Fi(and)f(let)g Fq(\026)29 b(>)3524 368 y Fo(1)p 3524 384 36 4 v 3524 441 a(2)3570 407 y Fi(.)46 b(L)-5 b(et)0 527 y Fq(k)38 b Fi(b)-5 b(e)35 b(an)f(isolate)-5 b(d)34 b(eigenvalue)g(of)g Fq(K)7 b Fi(.)45 b(Assume)35 b(that)1278 747 y Fp(T)1332 762 y Fm(k)1403 747 y Ft(:=)28 b Fq(\033)t Ft(\()p Fq(w)1701 762 y Fm(k)1743 747 y Ft(\))22 b Fp(\\)h Fr(R)k Fp(\032)h Fq(\033)2163 762 y Fo(disc)2286 747 y Ft(\()p Fq(w)2394 762 y Fm(k)2436 747 y Ft(\))p Fq(:)0 967 y Fi(L)-5 b(et)44 b Fq(\014)232 982 y Fo(1)316 967 y Fi(b)-5 b(e)43 b(the)h(c)-5 b(onstant)44 b(fr)-5 b(om)44 b(the)f(pr)-5 b(evious)44 b(the)-5 b(or)g(em)43 b(and)h Fq(\024)g Ft(:=)h(1)29 b Fp(\000)g Fq(\027)2876 931 y Fl(\000)p Fo(1)2971 967 y Fi(.)72 b(Then)43 b(ther)-5 b(e)44 b(exist)0 1088 y(c)-5 b(onstants)34 b Ft(\003)496 1103 y Fo(2)563 1088 y Fq(>)28 b Ft(0)34 b Fi(and)h Fq(\014)995 1103 y Fo(2)1062 1088 y Fq(>)27 b Ft(0)35 b Fi(such)g(that)g(for)g Fp(j)p Fq(\025)p Fp(j)27 b(\024)h Ft(\003)2139 1103 y Fo(2)2213 1088 y Fi(the)35 b(fol)5 b(lowing)33 b(holds:)0 1208 y Ft(\(i\))h Fi(Set)h(for)f(shortness)950 1428 y Ft(\002\()p Fq(k)s Ft(\))27 b(:=)p 1314 1350 51 4 v 28 w Fq(I)8 b Ft(\()p Fq(k)s(;)17 b(\025)1558 1387 y Fo(2)1597 1428 y Fq(\014)1652 1443 y Fo(1)1691 1428 y Ft(\))22 b Fp(n)g Fq(I)8 b Ft(\()p Fp(f)p Fq(k)s Fp(g)22 b Ft(+)g Fq(\025)2243 1387 y Fo(2)2282 1428 y Fp(T)2336 1443 y Fm(k)2379 1428 y Fq(;)17 b Fp(j)p Fq(\025)p Fp(j)2536 1387 y Fo(2+)p Fm(\024)2670 1428 y Fq(\014)2725 1443 y Fo(2)2765 1428 y Ft(\))p Fq(:)0 1648 y Fi(Then,)34 b(the)h(function)1328 1768 y Fq(z)d Fp(7!)27 b(h)p Fq(S)6 b Fp(i)1676 1727 y Fl(\000)p Fm(\026)1777 1768 y Ft(\()p Fq(z)27 b Fp(\000)22 b Fq(H)8 b Ft(\))2113 1727 y Fl(\000)p Fo(1)2207 1768 y Fp(h)p Fq(S)e Fp(i)2351 1727 y Fl(\000)p Fm(\026)3530 1768 y Ft(\(6.97\))0 1943 y Fi(extends)38 b(by)i(c)-5 b(ontinuity)39 b(to)g(a)g(function)g(on)g Fr(C)1767 1958 y Fo(+)1851 1943 y Fp([)26 b Ft(\002\()p Fq(k)s Ft(\))p Fi(.)57 b(In)38 b(p)-5 b(articular,)40 b(the)g(sp)-5 b(e)g(ctrum)39 b(of)f Fq(H)47 b Fi(on)0 2063 y(the)35 b(set)g Ft(\002\()p Fq(k)s Ft(\))g Fi(is)f(absolutely)h (c)-5 b(ontinuous.)0 2183 y Ft(\(ii\))34 b Fi(L)-5 b(et)36 b Fq(n)29 b Fp(2)h Fr(N)p Fi(,)35 b Ft(0)29 b Fq(<)g(\022)j Fp(\024)e Ft(1)35 b Fi(and)g Fq(\026)g Fi(b)-5 b(e)36 b(such)f(that)h Fq(\027)g Fp(\025)29 b Fq(\026)23 b Ft(+)2329 2144 y Fo(1)p 2329 2160 36 4 v 2329 2218 a(2)2404 2183 y Ft(=)29 b Fq(n)22 b Ft(+)h(1)g(+)f Fq(\022)s Fi(.)47 b(Then,)35 b(the)h(function)0 2304 y(\(6.97\))e(is)h(in)f(the)h(class)f Fq(C)991 2268 y Fm(n;\022)984 2328 y Fo(u)1127 2304 y Fi(of)h(the)g(set)p 939 2458 81 4 v 939 2536 a Fr(C)1020 2551 y Fo(+)1101 2536 y Fp(\\)1190 2440 y Fj(\020)p 1240 2458 80 4 v 96 x Fq(B)5 b Ft(\()p Fq(k)s(;)17 b(\025)1512 2495 y Fo(2)1551 2536 y Fq(\014)1606 2551 y Fo(1)1645 2536 y Ft(\))22 b Fp(n)g Fq(B)5 b Ft(\()p Fp(f)p Fq(k)s Fp(g)22 b Ft(+)g Fq(\025)2225 2495 y Fo(2)2265 2536 y Fp(T)2319 2551 y Fm(k)2361 2536 y Fq(;)17 b Fp(j)p Fq(\025)p Fp(j)2518 2495 y Fo(2+)p Fm(\024)2652 2536 y Fq(\014)2707 2551 y Fo(2)2747 2440 y Fj(\021)2813 2536 y Fq(:)146 2985 y Ft(The)24 b(next)f(theorem)f(is)g(p)s(erhaps)h(our)f(deep)s(est) h(result.)40 b(It)23 b(concerns)h(the)e(situation)f(where)i Fp(T)3528 3000 y Fm(k)3599 2985 y Fp(6)p Ft(=)28 b Fp(;)p Ft(,)0 3105 y(and)d(describ)s(es)g(the)g(structure)h(of)e(the)h(sp)s (ectrum)g(of)f Fq(H)32 b Ft(around)24 b(a)g(p)s(oin)m(t)g Fq(m)k Fp(2)g Fq(\033)2960 3120 y Fo(disc)3083 3105 y Ft(\()p Fq(w)3191 3120 y Fm(k)3233 3105 y Ft(\))6 b Fp(\\)g Fr(R)p Ft(.)40 b(W)-8 b(e)25 b(set)0 3225 y Fq(p)49 3240 y Fm(k)r(;m)201 3225 y Ft(=)j Fr(1)361 3241 y Fl(f)p Fm(m)p Fl(g)498 3225 y Ft(\()p Fq(w)606 3240 y Fm(k)648 3225 y Ft(\).)41 b(It)24 b(follo)m(ws)f(from)g(Prop)s(osition)f(3.2)h (that)h Fq(p)2301 3240 y Fm(k)r(;m)2450 3225 y Ft(is)f(an)h(orthogonal) f(pro)5 b(jection.)40 b(W)-8 b(e)0 3346 y(emphasize)24 b(that)g(in)g(the)g(follo)m(wing)e(theorem)i(w)m(e)h(need)h(a)e (stronger)g(assumption)g(on)g(the)g(in)m(teraction,)0 3466 y(namely)32 b(w)m(e)h(need)h Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))33 b(with)f Fq(\027)i(>)28 b Ft(2.)0 3694 y Fr(Theorem)74 b(6.4)49 b Fi(Assume)29 b(that)g(Hyp)-5 b(otheses)28 b Ft(A)g Fi(and)g Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))29 b Fi(hold)f(with)g Fq(\027)34 b(>)28 b Ft(2)p Fi(.)42 b(L)-5 b(et)29 b Fq(k)i Fi(b)-5 b(e)28 b(an)g(isolate)-5 b(d)0 3815 y(eigenvalue)34 b(of)g Fq(K)7 b Fi(,)1278 3935 y Fp(T)1332 3950 y Fm(k)1403 3935 y Ft(:=)28 b Fq(\033)t Ft(\()p Fq(w)1701 3950 y Fm(k)1743 3935 y Ft(\))22 b Fp(\\)h Fr(R)k Fp(\032)h Fq(\033)2163 3950 y Fo(disc)2286 3935 y Ft(\()p Fq(w)2394 3950 y Fm(k)2436 3935 y Ft(\))p Fq(:)0 4110 y Fi(and)41 b(let)h Fq(m)e Fp(2)h(T)626 4125 y Fm(k)669 4110 y Fi(.)65 b(L)-5 b(et)42 b Fq(\014)994 4125 y Fo(1)1034 4110 y Fi(,)h Fq(\014)1162 4125 y Fo(2)1243 4110 y Fi(and)e Fq(\024)h Fi(b)-5 b(e)41 b(as)g(in)h(the)f(pr)-5 b(evious)42 b(the)-5 b(or)g(ems.)64 b(Then)41 b(ther)-5 b(e)41 b(exists)h(a)0 4230 y(c)-5 b(onstant)35 b Ft(\003)457 4245 y Fo(3)523 4230 y Fq(>)28 b Ft(0)35 b Fi(such)f(that)h(for)g Ft(0)28 b Fq(<)f Fp(j)p Fq(\025)p Fp(j)g(\024)h Ft(\003)1780 4245 y Fo(3)1854 4230 y Fi(the)35 b(fol)5 b(lowing)34 b(holds:)0 4350 y Ft(\(i\))g Fi(Set)h(for)f(shortness)933 4570 y Ft(\002\()p Fq(k)s(;)17 b(m)p Ft(\))27 b(:=)p 1426 4492 51 4 v 28 w Fq(I)8 b Ft(\()p Fq(k)25 b Ft(+)d Fq(\025)1746 4529 y Fo(2)1785 4570 y Fq(m;)17 b Fp(j)p Fq(\025)p Fp(j)2027 4529 y Fo(2+)p Fm(\024)2161 4570 y Fq(\014)2216 4585 y Fo(2)2256 4570 y Ft(\))22 b Fp(\\)p 2405 4492 V 23 w Fq(I)7 b Ft(\()p Fq(k)s(;)17 b(\025)2648 4529 y Fo(2)2687 4570 y Fq(\014)2742 4585 y Fo(1)2782 4570 y Ft(\))p Fq(:)0 4790 y Fi(Then)33 b Ft(dim)15 b Fr(1)488 4743 y Fo(pp)488 4819 y(\002\()p Fm(k)r(;m)p Fo(\))750 4790 y Fp(\024)29 b Ft(dim)15 b Fq(p)1084 4805 y Fm(k)r(;m)1208 4790 y Fi(.)45 b(In)33 b(p)-5 b(articular,)34 b Fq(\033)1939 4805 y Fo(pp)2022 4790 y Ft(\()p Fq(H)8 b Ft(\))19 b Fp(\\)i Ft(\002\()p Fq(k)s(;)c(m)p Ft(\))33 b Fi(is)h(a)g(\014nite)f(set)h(c)-5 b(onsisting)33 b(of)0 4911 y(eigenvalues)h(of)g(\014nite)h(multiplicity.)0 5031 y Ft(\(ii\))e Fi(If)h Fq(\026)28 b(>)467 4992 y Fo(1)p 467 5008 36 4 v 467 5065 a(2)512 5031 y Fi(,)35 b(then)g(the)g(function)1328 5251 y Fq(z)d Fp(7!)27 b(h)p Fq(S)6 b Fp(i)1676 5210 y Fl(\000)p Fm(\026)1777 5251 y Ft(\()p Fq(z)27 b Fp(\000)22 b Fq(H)8 b Ft(\))2113 5210 y Fl(\000)p Fo(1)2207 5251 y Fp(h)p Fq(S)e Fp(i)2351 5210 y Fl(\000)p Fm(\026)3530 5251 y Ft(\(6.98\))1841 5753 y(45)p eop %%Page: 46 46 46 45 bop 0 407 a Fi(extends)26 b(by)i(c)-5 b(ontinuity)27 b(to)g(a)g(function)g(on)f Fr(C)1682 422 y Fo(+)1746 407 y Fp([)5 b Ft(\(\002\()p Fq(k)s(;)17 b(m)p Ft(\))5 b Fp(n)g Fq(\033)2305 422 y Fo(pp)2387 407 y Ft(\()p Fq(H)j Ft(\)\))p Fi(.)42 b(In)26 b(p)-5 b(articular,)29 b(the)e(sp)-5 b(e)g(ctrum)0 527 y(of)35 b Fq(H)42 b Fi(on)34 b Ft(\002\()p Fq(k)s(;)17 b(m)p Ft(\))22 b Fp(n)g Fq(\033)861 542 y Fo(pp)944 527 y Ft(\()p Fq(H)8 b Ft(\))35 b Fi(is)f(absolutely)h (c)-5 b(ontinuous.)0 648 y Ft(\(iii\))32 b Fi(L)-5 b(et)36 b Fq(n)28 b Fp(2)h Fr(N)p Fi(,)34 b Ft(0)28 b Fq(<)g(\022)j Fp(\024)e Ft(1)35 b Fi(and)f Fq(\027)h Fp(\025)28 b Fq(\026)22 b Ft(+)1708 608 y Fo(1)p 1708 624 36 4 v 1708 682 a(2)1781 648 y Ft(=)28 b Fq(n)22 b Ft(+)h(1)f(+)g Fq(\022)s Fi(.)45 b(Then,)35 b(for)f(any)h Fq(\017)29 b(>)f Ft(0)p Fi(,)35 b(the)g(function)0 768 y(\(6.98\))f(is)h(in)f(the)h(class)f Fq(C)991 732 y Fm(n;\022)984 793 y Fo(u)1127 768 y Fi(of)h(the)g(set)p 650 909 81 4 v 650 988 a Fr(C)731 1003 y Fo(+)812 988 y Fp(\\)901 891 y Fj(\020)p 951 910 80 4 v 97 x Fq(B)5 b Ft(\()p Fq(k)25 b Ft(+)d Fq(\025)1299 947 y Fo(2)1338 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Fq(H)p 1747 2031 39 4 v 1747 2070 a Fo(v)p 1785 2031 V 1 w(v)1865 2106 y Ft(will)f(hold)i(uniformly)e(on)i Fr(R)p Ft(.)58 b(In)37 b(Section)g(7.3)g(w)m(e)0 2226 y(pro)m(v)m(e)27 b(Theorem)f(6.1.)40 b(In)26 b(Section)g(7.4)f(w)m(e)h (pro)m(v)m(e)h(an)e(estimate)g(on)h(the)f(di\013erence)i(\()p Fq(z)t Fr(1)p 3219 2152 V -36 x Fo(v)p 3258 2152 V 2 w(v)3308 2226 y Fp(\000)8 b Fq(H)p 3482 2152 V 3482 2190 a Fo(v)p 3520 2152 V 1 w(v)3562 2226 y Ft(\))3600 2190 y Fl(\000)p Fo(1)3702 2226 y Fp(\000)0 2347 y Ft(\()p Fq(z)t Fr(1)p 143 2272 V -36 x Fo(v)p 182 2272 V 2 w(v)246 2347 y Fp(\000)23 b Fq(H)p 435 2272 V 435 2311 a Fo(v)p 473 2272 V 1 w(v)427 2371 y(fr)515 2347 y Ft(\))553 2311 y Fl(\000)p Fo(1)647 2347 y Ft(.)0 2634 y Fn(7.1)135 b(Limiting)46 b(Absorption)f(Principle)0 2818 y Ft(This)33 b(section)f(is)h(dev)m(oted)h(to)e(the)h(pro)s(of)f(of)g(the)h(follo)m (wing)d(theorem:)0 3011 y Fr(Theorem)74 b(7.1)49 b Fi(Assume)43 b(that)f(Hyp)-5 b(otheses)42 b Ft(A)h Fi(and)e Fq(S)6 b Ft(\()p 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b(A)p Ft(])p Fq(;)1468 4659 y Ft(e)1511 4618 y Fo(i)p Fm(\034)8 b Fd(S)1619 4659 y Fq(A)28 b Ft(:=)g(e)1894 4618 y Fo(i)p Fm(\034)8 b(S)2004 4659 y Fq(A)p Ft(e)2120 4618 y Fl(\000)p Fo(i)p Fm(\034)g(S)2284 4659 y Fq(:)0 4828 y Ft(Note)33 b(that)1348 4920 y(e)1391 4884 y Fo(i)p Fm(\034)8 b Fd(S)1500 4920 y Ft(d\000\()p Fq(!)t Ft(\))82 b(=)28 b(d\000\()p Fq(!)t Ft(\))21 b(+)h Fq(\034)11 b(N)5 b(;)1515 5041 y Ft(e)1558 5005 y Fo(i)p Fm(\034)j Fd(S)1667 5041 y Fq(N)93 b Ft(=)28 b Fq(N)5 b(;)1338 5161 y Ft(e)1381 5125 y Fo(i)p Fm(\034)j Fd(S)1489 5161 y Ft(\()p Fq(a)p Ft(\()p Fq(\013)q Ft(\)\))83 b(=)28 b Fq(a)p Ft(\(e)2074 5125 y Fo(i)p Fm(\034)8 b(s)2170 5161 y Fq(\013)q Ft(\))p Fq(;)1299 5282 y Ft(e)1342 5245 y Fo(i)p Fm(\034)g Fd(S)1450 5282 y Ft(\()p Fq(a)1539 5245 y Fl(\003)1579 5282 y Ft(\()p Fq(\013)q Ft(\)\))82 b(=)28 b Fq(a)1993 5245 y Fl(\003)2032 5282 y Ft(\(e)2113 5245 y Fo(i)p Fm(\034)8 b(s)2209 5282 y Fq(\013)q Ft(\))p Fq(:)1841 5753 y Ft(46)p eop %%Page: 47 47 47 46 bop 0 407 a Ft(W)-8 b(e)33 b(c)m(ho)s(ose)g(a)g(real)e(function)h Fq(\020)j Fp(2)28 b Fq(C)1379 371 y Fl(1)1372 431 y Fo(0)1454 407 y Ft(\()p Fr(R)p Ft(\))k(suc)m(h)i(that)e Fq(\020)8 b Ft(\()p Fq(t)p Ft(\))27 b(=)h(1)k(in)g(a)g(neigh)m(b)s(orho)s(o)s(d)g (of)g(0)g(and)h(set)1601 627 y Fq(\030)5 b Ft(\()p Fq(t)p Ft(\))27 b(:=)h(e)1961 586 y Fm(t)1991 627 y Fq(\020)8 b Ft(\()p Fq(t)p Ft(\))p Fq(:)1350 b Ft(\(7.99\))0 847 y(Let)1440 967 y Fh(D)28 b Ft(:=)g Fp(D)s Ft(\()p Fq(H)1867 982 y Fo(fr)1919 967 y Ft(\))22 b Fp(\\)h(D)s Ft(\()p Fq(N)10 b Ft(\))p Fq(:)1142 b Ft(\(7.100\))0 1142 y(F)-8 b(or)32 b Fq(\017)c Fp(2)g Fr(R)k Ft(w)m(e)i(de\014ne)1066 1338 y Fq(H)1147 1353 y Fm(\017;)p Fo(fr)1332 1338 y Ft(:=)27 b Fq(H)1543 1353 y Fo(fr)1618 1338 y Fp(\000)c Ft(i)p Fq(\017N)5 b(;)1159 1529 y(V)1216 1544 y Fm(\017)1332 1529 y Ft(:=)1502 1490 y Fo(1)p 1472 1506 95 4 v 1472 1515 a Fl(p)p 1531 1515 36 3 v 55 x Fo(2)1576 1529 y Ft(\()p Fq(a)p Ft(\()p Fq(\030)g Ft(\()p Fp(\000)p Fq(\017s)p Ft(\))p 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Fp(f\000)p Ft(i)p Fq(n\017)22 b Ft(+)g Fr(R)44 b Ft(:)55 b Fq(n)28 b Ft(=)g(1)p Fq(;)17 b Ft(2)p Fq(;)g(:)g(:)g(:)n Fp(g)p Fq(:)146 4317 y Ft(The)34 b(next)f(lemma)e (giv)m(es)i(the)g(basic)f(prop)s(erties)h(of)f(the)h(op)s(erator)f Fq(H)2744 4332 y Fm(\017)2776 4317 y Ft(.)0 4545 y Fr(Lemma)37 b(7.3)49 b Fi(Assume)35 b(that)g(Hyp)-5 b(othesis)35 b Fq(S)6 b Ft(\(0\))34 b Fi(holds.)44 b(Then)34 b(the)h(fol)5 b(lowing)34 b(is)g(true:)0 4665 y Ft(\(i\))d Fi(F)-7 b(or)30 b(any)i Fq(\017)c Fp(6)p Ft(=)f(0)p Fi(,)32 b Fq(V)835 4680 y Fm(\017)899 4665 y Fi(is)g(an)f(in\014nitesimal)f(p)-5 b(erturb)g(ation)32 b(of)f Fq(H)2431 4680 y Fm(\017;)p Fo(fr)2532 4665 y Fi(.)44 b(In)31 b(p)-5 b(articular,)32 b Fq(H)3284 4680 y Fm(\017)3348 4665 y Fi(is)f(a)g(close)-5 b(d)0 4786 y(op)g(er)g(ator)35 b(with)f(domain)g Fh(D)p Fi(.)0 4906 y Ft(\(ii\))f Fi(F)-7 b(or)34 b(any)h Fq(\017)g Fi(we)f(have)h Fq(H)1066 4870 y Fl(\003)1058 4931 y Fm(\017)1132 4906 y Ft(=)28 b Fq(H)1317 4921 y Fl(\000)p Fm(\017)1404 4906 y Fi(.)1841 5753 y Ft(47)p eop %%Page: 48 48 48 47 bop 0 407 a Fr(Pro)s(of.)65 b Ft(It)33 b(follo)m(ws)e(from)g (estimate)h(\(4.80\))g(that)1400 628 y Fp(k)p Fq(N)1538 587 y Fl(\000)1603 560 y Fk(1)p 1603 572 31 4 v 1603 613 a(2)1648 628 y Fq(V)1705 643 y Fm(\017)1737 628 y Fp(k)c(\024)g Ft(2)p Fp(k)p Fq(\030)5 b Fp(k)2117 643 y Fl(1)2190 628 y Fp(k)p Fq(\013)q Fp(k)p Fq(;)1101 b Ft(\(7.104\))0 836 y(and)29 b(th)m(us)h Fq(V)454 851 y Fm(\017)515 836 y Ft(is)e(an)h(in\014nitesimal)c(p)s(erturbation)j (of)g Fq(N)10 b Ft(.)43 b(This)29 b(observ)-5 b(ation)28 b(and)h(\(7.103\))f(yield)g(that)0 956 y Fq(V)57 971 y Fm(\017)122 956 y Ft(is)k(also)g(an)g(in\014nitesimal)e(p)s (erturbation)h(of)h Fq(H)1868 971 y Fm(\017;)p Fo(fr)1970 956 y Ft(.)43 b(This)33 b(pro)m(v)m(es)h(\(i\).)146 1076 y(Let)d(us)f(sho)m(w)h(\(ii\).)41 b(Clearly)-8 b(,)30 b(w)m(e)h(can)f(assume)h(that)f Fq(\017)e Fp(6)p Ft(=)f(0.)43 b(It)30 b(is)f(easy)j(to)d(see)i(that)f(w)m(e)h(can)g(split)0 1197 y Fq(V)57 1212 y Fm(\017)118 1197 y Ft(as)e Fq(V)291 1212 y Fm(\017)352 1197 y Ft(=)e Fq(V)512 1212 y Fm(\017;)p Fo(1)614 1197 y Ft(+)14 b Fq(V)761 1212 y Fm(\017;)p Fo(2)878 1197 y Ft(where)30 b Fq(V)1213 1212 y Fm(\017;)p Fo(2)1329 1197 y Ft(is)e(b)s(ounded)i(and)f Fp(k)p Fq(N)2142 1161 y Fl(\000)p Fo(1)2236 1197 y Fq(V)2293 1212 y Fm(\017;)p Fo(1)2381 1197 y Fp(k)e Fq(<)h Ft(1,)h Fp(k)p Fq(V)2774 1212 y Fm(\017;)p Fo(1)2861 1197 y Fq(N)2949 1161 y Fl(\000)p Fo(1)3044 1197 y Fp(k)f Fq(<)f Ft(1.)42 b(Therefore,)0 1317 y(w)m(e)34 b(can)e(apply)h(Prop)s(osition)e(2.11.)42 b Fe(2)146 1484 y Ft(F)-8 b(or)32 b Fq(z)h Fp(62)28 b Fq(\033)t Ft(\()p Fq(H)671 1499 y Fm(\017)703 1484 y Ft(\))33 b(w)m(e)g(set)1452 1604 y Fq(G)1529 1619 y Fm(\017)1561 1604 y Ft(\()p Fq(z)t Ft(\))c(:=)e(\()p Fq(z)g Fp(\000)c Fq(H)2136 1619 y Fm(\017)2168 1604 y Ft(\))2206 1563 y Fl(\000)p Fo(1)2301 1604 y Fq(:)0 1773 y Ft(In)i(the)g(next)g(3)f (lemmas)f(w)m(e)i(describ)s(e)g(some)f(prop)s(erties)h(of)f Fq(G)2274 1788 y Fm(\017)2306 1773 y Ft(\()p Fq(z)t Ft(\))h(that)f (hold)g(under)h(the)g(assumption)0 1894 y(S\(0\))42 b(and)g Fq(\017)j Fp(6)p Ft(=)f(0.)72 b(Although)41 b(they)i(are)f(formally)e (ob)m(vious,)45 b(they)e(require)f(a)g(pro)s(of)f(due)i(to)f(the)0 2014 y(un)m(b)s(oundedness)36 b(of)c(some)g(of)g(the)h(op)s(erators.) 146 2135 y(F)-8 b(or)32 b Fq(n)c Ft(=)g(1)p Fq(;)17 b Ft(2)p Fq(;)g(:)g(:)g(:)31 b Ft(w)m(e)i(de\014ne)h(closed)f(op)s (erators)f Fq(H)2076 2098 y Fo(\()p Fm(n)p Fo(\))2068 2159 y Fm(\017)2210 2135 y Ft(b)m(y)h(the)g(form)m(ula)375 2392 y Fq(H)464 2350 y Fo(\()p Fm(n)p Fo(\))456 2416 y Fm(\017)593 2392 y Ft(=)28 b Fp(\000)p Ft(i)p Fq(\016)845 2407 y Fo(1)p Fm(n)927 2392 y Fq(N)k Ft(+)1183 2324 y Fq(\025)p 1145 2368 132 4 v 1145 2386 a Fp(p)p 1228 2386 49 4 v 82 x Ft(2)1304 2295 y Fj(\020)1353 2392 y Ft(\()p Fp(\000)p Ft(1\))1555 2350 y Fm(k)1598 2392 y Fq(a)p Ft(\()p Fq(s)1733 2350 y Fm(k)1776 2392 y Fq(\030)1824 2350 y Fo(\()p Fm(n)p Fo(\))1925 2392 y Ft(\()p Fp(\000)p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\))23 b(+)f Fq(a)2436 2350 y Fl(\003)2475 2392 y Ft(\()p Fq(s)2559 2350 y Fm(k)2602 2392 y Fq(\030)2650 2350 y Fo(\()p Fm(n)p Fo(\))2751 2392 y Ft(\()p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\))3013 2295 y Fj(\021)3079 2392 y Fq(;)375 b Ft(\(7.105\))0 2664 y(where)34 b Fq(\016)325 2679 y Fo(1)p Fm(n)436 2664 y Ft(=)27 b(1)33 b(if)f Fq(n)c Ft(=)g(1)k(and)h(0)g(otherwise.)44 b(Here,)34 b(of)e(course,)i Fq(\030)2469 2627 y Fo(\()p Fm(n)p Fo(\))2603 2664 y Ft(is)f(the)g Fq(n)p Ft(-th)f(deriv)-5 b(ativ)m(e)33 b(of)f(the)0 2784 y(function)g Fq(\030)5 b Ft(.)43 b(Note)33 b(that)f(for)g(an)m(y)h(\011)28 b Fp(2)g Fh(D)1532 2968 y Ft(d)1586 2932 y Fm(n)p 1513 3012 141 4 v 1513 3104 a Ft(d)p Fq(\017)1606 3075 y Fm(n)1663 3036 y Fq(H)1744 3051 y Fm(\017)1777 3036 y Ft(\011)f(=)h Fq(H)2073 2995 y Fo(\()p Fm(n)p Fo(\))2065 3060 y Fm(\017)2174 3036 y Ft(\011)p Fq(:)0 3281 y Fr(Lemma)37 b(7.4)49 b Fi(Assume)35 b(that)g(Hyp)-5 b(othesis)35 b(S\(0\))f(holds)g(and)h(let) g Fq(z)d Fp(62)c Fq(\033)t Ft(\()p Fq(H)2782 3296 y Fm(\017)2815 3281 y Ft(\))p Fi(.)44 b(Then)34 b(the)h(function)1529 3489 y Fr(R)1613 3504 y Fo(+)1700 3489 y Fp(3)28 b Fq(\017)g Fp(7!)f Fq(G)2065 3504 y Fm(\017)2098 3489 y Ft(\()p Fq(z)t Ft(\))p Fq(;)1231 b Ft(\(7.106\))0 3697 y Fi(is)35 b(in\014nitely)f(di\013er)-5 b(entiable)34 b(and)664 3881 y Ft(d)718 3845 y Fm(n)p 644 3925 V 644 4017 a Ft(d)p Fq(\017)737 3988 y Fm(n)795 3948 y Fq(G)872 3963 y Fm(\017)904 3948 y Ft(\()p Fq(z)t Ft(\))29 b(=)1306 3865 y Fj(X)1161 4044 y Fm(n)1204 4053 y Fk(1)1238 4044 y Fo(+)p Fl(\001\001\001)o Fo(+)p Fm(n)1450 4056 y Fg(l)1474 4044 y Fo(=)p Fm(n)1588 3948 y Fq(G)1665 3963 y Fm(\017)1698 3948 y Ft(\()p Fq(z)t Ft(\))p Fq(H)1912 3907 y Fo(\()p Fm(n)1982 3916 y Fk(1)2017 3907 y Fo(\))1904 3973 y Fm(\017)2049 3948 y Fq(G)2126 3963 y Fm(\017)2158 3948 y Ft(\()p Fq(z)t Ft(\))17 b Fp(\001)g(\001)g(\001)e Fq(G)2510 3963 y Fm(\017)2543 3948 y Ft(\()p Fq(z)t Ft(\))p Fq(H)2757 3907 y Fo(\()p Fm(n)2827 3919 y Fg(l)2851 3907 y Fo(\))2749 3973 y Fm(\017)2883 3948 y Fq(G)2960 3963 y Fm(\017)2993 3948 y Ft(\()p Fq(z)t Ft(\))p Fq(:)336 b Ft(\(7.107\))0 4371 y Fr(Remark.)41 b Ft(In)26 b(this)g(section)g(w)m(e)h(will)d(deal)i(only)f(with)h(the)g (\014rst)h(deriv)-5 b(ativ)m(e)26 b(of)f(the)i(function)e(\(7.106\).)0 4491 y(The)34 b(higher)e(deriv)-5 b(ativ)m(es)32 b(will)e(b)s(e)j(used) h(in)e(Section)g(7.2.)0 4611 y Fr(Pro)s(of.)65 b Ft(Let)33 b Fq(\017)28 b(>)f Ft(0)33 b(b)s(e)f(\014xed)i(and)f Fq(z)f Fp(62)c Fq(\033)t Ft(\()p Fq(H)1702 4626 y Fm(\017)1735 4611 y Ft(\).)43 b(First)32 b(note)h(that)1563 4819 y Fp(k)p Fq(N)10 b(G)1778 4834 y Fm(\017)1811 4819 y Ft(\()p Fq(z)t Ft(\))p Fp(k)28 b Fq(<)g(C)r(:)1264 b Ft(\(7.108\))0 5027 y(In)36 b(fact,)h(b)m(y)g(Lemma)e(7.3)g Fp(k)p Ft(\()p Fq(H)1169 5042 y Fm(\017;)p Fo(fr)1295 5027 y Ft(+)24 b(i\))p Fq(G)1538 5042 y Fm(\017)1570 5027 y Ft(\()p Fq(z)t Ft(\))p Fp(k)36 b Ft(is)g(b)s(ounded)h(and)f(hence)h(\(7.108\))e (follo)m(ws)g(from)g(the)0 5147 y(b)s(ound)884 5268 y Fp(k)p Fq(N)10 b(G)1099 5283 y Fm(\017)1132 5268 y Ft(\()p Fq(z)t Ft(\))p Fp(k)28 b(\024)g(k)p Fq(N)10 b Ft(\()p Fq(H)1697 5283 y Fm(\017;)p Fo(fr)1821 5268 y Ft(+)22 b(i\))1985 5227 y Fl(\000)p Fo(1)2078 5268 y Fp(kk)p Ft(\()p Fq(H)2297 5283 y Fm(\017;)p Fo(fr)2420 5268 y Ft(+)g(i\))p Fq(G)2661 5283 y Fm(\017)2693 5268 y Ft(\()p Fq(z)t Ft(\))p Fp(k)p Fq(:)1841 5753 y Ft(48)p eop %%Page: 49 49 49 48 bop 0 407 a Ft(Next)33 b(w)m(e)h(note)f(that)f(for)g Fp(j)p Fq(h)p Fp(j)27 b(\024)1217 368 y Fm(\017)p 1214 384 36 4 v 1214 441 a Fo(2)1291 407 y Ft(w)m(e)34 b(ha)m(v)m(e)1038 691 y Fp(k)p Ft(\()p Fq(V)1183 706 y Fm(\017)p Fo(+)p Fm(h)1333 691 y Fp(\000)23 b Fq(V)1490 706 y Fm(\017)1523 691 y Ft(\))p Fq(N)1649 650 y Fl(\000)p Fo(1)1743 691 y Fp(k)28 b(\024)g Fq(C)2013 624 y Fp(j)p Fq(h)p Fp(j)p 2013 668 112 4 v 2021 760 a(j)p Fq(\017)p Fp(j)2134 691 y(k)p Fq(\013)q Fp(k)17 b Ft(sup)2374 765 y Fm(t)2477 691 y Fp(j)p Fq(\030)2553 650 y Fl(0)2575 691 y Ft(\()p Fq(t)p Ft(\))p Fp(j)p Fq(:)740 b Ft(\(7.109\))0 967 y(Hence,)34 b(for)e Fp(j)p Fq(h)p Fp(j)27 b(\024)724 928 y Fm(\017)p 720 944 36 4 v 720 1001 a Fo(2)646 1187 y Fp(k)p Ft(\()p Fq(H)815 1202 y Fm(\017)p Fo(+)p Fm(h)965 1187 y Fp(\000)22 b Fq(H)1145 1202 y Fm(\017)1178 1187 y Ft(\))p Fq(G)1293 1202 y Fm(\017)1325 1187 y Ft(\()p Fq(z)t Ft(\))p Fp(k)28 b Ft(=)g Fp(k)p Ft(\()p Fp(\000)p Ft(i)p Fq(hN)k Ft(+)22 b Fq(V)2146 1202 y Fm(\017)p Fo(+)p Fm(h)2296 1187 y Fp(\000)h Fq(V)2453 1202 y Fm(\017)2486 1187 y Ft(\))p Fq(G)2601 1202 y Fm(\017)2633 1187 y Ft(\()p Fq(z)t Ft(\))p Fp(k)28 b(\024)g Fq(C)3011 1202 y Fo(1)3051 1187 y Fq(h:)0 1407 y Ft(Therefore,)g(for)d(small)f(enough)i Fq(h)p Ft(,)h Fq(z)32 b Fp(62)c Fq(\033)t Ft(\()p Fq(H)1642 1422 y Fm(\017)p Fo(+)p Fm(h)1770 1407 y Ft(\))e(and)f Fq(h)j Fp(7!)g Fq(G)2305 1422 y Fm(\017)p Fo(+)p Fm(h)2433 1407 y Ft(\()p Fq(z)t Ft(\))e(is)f(norm)g(con)m(tin)m(uous)h(at)f Fq(h)j Ft(=)g(0.)146 1528 y(W)-8 b(e)31 b(will)d(no)m(w)j(sho)m(w)g (that)f(the)h(function)f(\(7.106\))f(is)h(di\013eren)m(tiable)f(and)h (that)g(Relation)e(\(7.107\))0 1648 y(holds)37 b(for)f Fq(n)f Ft(=)g(1.)56 b(An)37 b(easy)h(inductiv)m(e)f(argumen)m(t)f(then) i(yields)e(that)h(this)f(function)h(is)f(in\014nitely)0 1768 y(di\013eren)m(tiable)31 b(and)i(that)f(Relation)f(\(7.107\))g (holds)i(for)f(all)e Fq(n)p Ft(.)146 1889 y(W)-8 b(e)33 b(ha)m(v)m(e)937 2009 y Fq(G)1014 2024 y Fm(\017)p Fo(+)p Fm(h)1142 2009 y Ft(\()p Fq(z)t Ft(\))23 b Fp(\000)f Fq(G)1466 2024 y Fm(\017)1499 2009 y Ft(\()p Fq(z)t Ft(\))h Fp(\000)f Fq(hG)1879 2024 y Fm(\017)1912 2009 y Ft(\()p Fq(z)t Ft(\))p Fq(H)2126 1968 y Fo(\(1\))2118 2034 y Fm(\017)2221 2009 y Fq(G)2298 2024 y Fm(\017)2330 2009 y Ft(\()p Fq(z)t Ft(\))29 b(=)e(I)22 b(+)g(I)s(I)q Fq(;)638 b Ft(\(7.110\))0 2183 y(where)428 2376 y(I)83 b(:=)28 b(\()p Fq(G)792 2391 y Fm(\017)p Fo(+)p Fm(h)920 2376 y Ft(\()p Fq(z)t Ft(\))23 b Fp(\000)f Fq(G)1244 2391 y Fm(\017)1277 2376 y Ft(\()p Fq(z)t Ft(\)\))17 b(\()p Fq(H)1576 2391 y Fm(\017)p Fo(+)p Fm(h)1726 2376 y Fp(\000)23 b Fq(H)1907 2391 y Fm(\017)1940 2376 y Ft(\))16 b Fq(G)2071 2391 y Fm(\017)2104 2376 y Ft(\()p Fq(z)t Ft(\))546 2567 y(=)28 b Fq(G)727 2582 y Fm(\017)p Fo(+)p Fm(h)855 2567 y Ft(\()p Fq(z)t Ft(\)\()p Fp(\000)p Ft(i)p Fq(hN)33 b Ft(+)22 b Fq(\025V)1502 2582 y Fm(\017)p Fo(+)p Fm(h)1652 2567 y Fp(\000)h Fq(\025V)1866 2582 y Fm(\017)1898 2567 y Ft(\))p Fq(G)2013 2582 y Fm(\017)2046 2567 y Ft(\()p Fq(z)t Ft(\)\()p Fp(\000)p Ft(i)p Fq(hN)33 b Ft(+)22 b Fq(\025V)2693 2582 y Fm(\017)p Fo(+)p Fm(h)2843 2567 y Fp(\000)h Fq(\025V)3057 2582 y Fm(\017)3089 2567 y Ft(\))p Fq(G)3204 2582 y Fm(\017)3237 2567 y Ft(\()p Fq(z)t Ft(\))p Fq(;)390 2760 y Ft(I)s(I)83 b(:=)28 b Fq(G)754 2775 y Fm(\017)787 2760 y Ft(\()p Fq(z)t Ft(\)\()p Fq(H)1031 2775 y Fm(\017)p Fo(+)p Fm(h)1181 2760 y Fp(\000)23 b Fq(H)1362 2775 y Fm(\017)1417 2760 y Fp(\000)f Fq(hH)1661 2724 y Fo(\(1\))1653 2784 y Fm(\017)1755 2760 y Ft(\))p Fq(G)1870 2775 y Fm(\017)1903 2760 y Ft(\()p Fq(z)t Ft(\))546 2953 y(=)28 b Fq(G)727 2968 y Fm(\017)760 2953 y Ft(\()p Fq(z)t Ft(\)\()p Fq(\025V)1037 2968 y Fm(\017)p Fo(+)p Fm(h)1187 2953 y Fp(\000)23 b Fq(\025V)1401 2968 y Fm(\017)1455 2953 y Fp(\000)g Fq(h\025V)1747 2917 y Fo(\(1\))1725 2977 y Fm(\017)1841 2953 y Ft(\))p Fq(G)1956 2968 y Fm(\017)1989 2953 y Ft(\()p Fq(z)t Ft(\))0 3171 y(F)-8 b(or)32 b Fp(j)p Fq(h)p Fp(j)27 b(\024)432 3132 y Fm(\017)p 429 3148 V 429 3206 a Fo(2)474 3171 y Ft(,)33 b(w)m(e)h(ha)m(v)m(e)778 3460 y Fp(k)p Ft(\()p Fq(V)923 3475 y Fm(\017)p Fo(+)p Fm(h)1073 3460 y Fp(\000)22 b Fq(V)1229 3475 y Fm(\017)1284 3460 y Fp(\000)h Fq(hV)1518 3419 y Fo(\(1\))1497 3485 y Fm(\017)1613 3460 y Ft(\))p Fq(N)1739 3419 y Fl(\000)p Fo(1)1833 3460 y Fp(k)28 b(\024)g Fq(C)2103 3393 y Fp(j)p Fq(h)p Fp(j)2215 3357 y Fo(2)p 2103 3437 151 4 v 2139 3528 a Fq(\017)2178 3500 y Fo(2)2264 3460 y Fp(j)p Fq(\025)p Fp(jk)p Fq(\013)q Fp(k)17 b Ft(sup)2616 3534 y Fm(t)2718 3460 y Fp(j)p Fq(\030)2794 3419 y Fl(00)2836 3460 y Ft(\()p Fq(t)p Ft(\))p Fp(j)p Fq(:)479 b Ft(\(7.111\))146 3729 y(Using)35 b(\(7.108\),)g(\(7.109\))f(and)i(\(7.111\))e(w)m(e)i(see)h (that)e(I)g(and)g(I)s(I)h(are)f(less)h(than)f Fq(C)7 b(h)3246 3693 y Fo(2)3285 3729 y Ft(.)52 b(This)35 b(ends)0 3850 y(the)e(pro)s(of)f(of)g(the)h(lemma)d(for)i Fq(n)c Ft(=)g(1.)43 b Fe(2)146 4020 y Ft(W)-8 b(e)23 b(pro)s(ceed)g(to)e (deriv)m(e)i(an)f(alternativ)m(e)f(expression)i(for)2263 3981 y Fo(d)p 2249 3997 68 4 v 2249 4054 a(d)p Fm(\017)2326 4020 y Fq(G)2403 4035 y Fm(\017)2436 4020 y Ft(\()p Fq(z)t Ft(\))f(whic)m(h)h(will)d(pla)m(y)h(an)h(imp)s(ortan)m(t)0 4140 y(role)32 b(in)f(the)i(sequel.)146 4261 y(The)h(comm)m(utator)d([) p Fq(S;)17 b(H)1109 4276 y Fm(\017)1142 4261 y Ft(],)32 b(de\014ned)i(as)f(a)f(quadratic)h(form)e(on)h Fp(D)s Ft(\()p Fq(S)6 b Ft(\))22 b Fp(\\)g(D)s Ft(\()p Fq(H)3098 4276 y Fm(\017)3130 4261 y Ft(\))33 b(is)f(equal)g(to)786 4530 y([)p Fq(S;)17 b(H)998 4545 y Fm(\017)1031 4530 y Ft(])28 b(=)f Fp(\000)p Ft(i)p Fq(N)32 b Ft(+)1550 4463 y Fq(\025)p 1512 4507 132 4 v 1512 4525 a Fp(p)p 1595 4525 49 4 v 82 x Ft(2)1671 4530 y(\()p Fp(\000)p Fq(a)p Ft(\()p Fq(s\030)5 b Ft(\()p Fp(\000)p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\))22 b(+)g Fq(a)2479 4489 y Fl(\003)2519 4530 y Ft(\()p Fq(s\030)5 b Ft(\()p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\)\))15 b Fq(:)488 b Ft(\(7.112\))0 4814 y(If)41 b(w)m(e)h(assume)g(S\(0\),)h(then)f(it)f(is)f(easy)j(to)d(sho)m (w)j(that)2155 4775 y Fm(\025)p 2129 4790 95 4 v 2129 4800 a Fl(p)p 2188 4800 36 3 v 55 x Fo(2)2250 4814 y Ft(\()o Fp(\000)p Fq(a)p Ft(\()p Fq(s\030)5 b Ft(\()p Fp(\000)p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\))23 b(+)f Fq(a)3058 4778 y Fl(\003)3098 4814 y Ft(\()p Fq(s\030)5 b Ft(\()p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\)\))40 b(is)h(an)0 4946 y(in\014nitesimal)33 b(p)s(erturbation)h(of)i Fp(\000)p Ft(i)p Fq(N)10 b Ft(.)53 b(Hence)37 b(the)f(righ)m(t)f(hand)h(side)g (of)f(\(7.112\))g(de\014nes)j(a)d(closed)0 5067 y(op)s(erator)41 b(with)g(domain)e Fp(D)s Ft(\()p Fq(N)10 b Ft(\).)70 b(By)42 b(a)f(sligh)m(t)f(abuse)i(of)f(notation,)h(this)f(op)s(erator)g (will)e(b)s(e)i(also)0 5187 y(denoted)34 b(b)m(y)f([)p Fq(S;)17 b(H)716 5202 y Fm(\017)749 5187 y Ft(].)1841 5753 y(49)p eop %%Page: 50 50 50 49 bop 0 407 a Fr(Lemma)37 b(7.5)49 b Fi(Assume)d(that)g(Hyp)-5 b(othesis)45 b(S\(0\))g(holds.)76 b(L)-5 b(et)46 b Fq(\017)i Fp(6)p Ft(=)g(0)p Fi(,)g Fq(z)k Fp(62)c Fq(\033)t Ft(\()p Fq(H)3127 422 y Fm(\017)3159 407 y Ft(\))e Fi(and)f Fq(m)j Fp(\025)g Ft(1)p Fi(.)0 527 y(Then,)29 b Ft([)p Fq(S;)17 b(G)487 491 y Fm(m)487 552 y(\017)553 527 y Ft(\()p Fq(z)t Ft(\)])p Fi(,)30 b(de\014ne)-5 b(d)27 b(as)h(a)g(quadr)-5 b(atic)28 b(form)g(on)g Fp(D)s Ft(\()p Fq(S)6 b Ft(\))p Fi(,)28 b(extends)g(by)g(c)-5 b(ontinuity)29 b(to)f(a)g(b)-5 b(ounde)g(d)0 648 y(op)g(er)g(ator)35 b(on)f Fp(H)i Fi(e)-5 b(qual)35 b(to)1027 893 y Ft([)p Fq(S;)17 b(G)1235 852 y Fm(m)1235 918 y(\017)1302 893 y Ft(\()p Fq(z)t Ft(\)])29 b(=)1586 785 y Fm(m)p Fl(\000)p Fo(1)1602 810 y Fj(X)1598 995 y Fm(k)r Fo(=0)1755 893 y Fq(G)1832 852 y Fm(k)r Fo(+1)1832 918 y Fm(\017)1965 893 y Ft(\()p Fq(z)t Ft(\)[)p Fq(S;)17 b(H)2302 908 y Fm(\017)2335 893 y Ft(])p Fq(G)2439 852 y Fm(m)p Fl(\000)p Fm(k)2439 918 y(\017)2599 893 y Ft(\()p Fq(z)t Ft(\))p Fq(:)146 1255 y Fr(Remark.)64 b Ft(In)40 b(this)f(section,)i(w)m(e)g(will)c(use)j(Lemmas)f(7.5)g(and) g(7.6)g(with)g Fq(m)h Ft(=)f(1.)64 b(The)40 b(cases)0 1375 y Fq(m)28 b(>)g Ft(1)k(will)e(b)s(e)j(used)h(in)d(the)i(next)h (section.)0 1616 y Fr(Pro)s(of.)65 b Ft(F)-8 b(or)32 b Fq(t)g Ft(real)g(w)m(e)i(de\014ne)693 1767 y Fq(H)774 1782 y Fm(\017;t)935 1767 y Ft(:=)28 b(e)1109 1731 y Fo(i)p Fm(tS)1205 1767 y Fq(H)1286 1782 y Fm(\017)1318 1767 y Ft(e)1361 1731 y Fl(\000)p Fo(i)p Fm(tS)935 1960 y Ft(=)g Fq(H)1120 1975 y Fm(\017;)p Fo(fr)1243 1960 y Ft(+)22 b Fq(tN)33 b Ft(+)1621 1920 y Fm(\025)p 1595 1936 95 4 v 1595 1946 a Fl(p)p 1654 1946 36 3 v 55 x Fo(2)1699 1960 y Ft(\()p Fq(a)p Ft(\(e)1869 1923 y Fo(i)p Fm(ts)1951 1960 y Fq(\030)5 b Ft(\()p Fp(\000)p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\))22 b(+)g Fq(a)2509 1923 y Fl(\003)2549 1960 y Ft(\(e)2630 1923 y Fo(i)p Fm(ts)2712 1960 y Fq(\030)5 b Ft(\()p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\)\))p Fq(:)0 2144 y Ft(Let)1311 2237 y Fq(G)1388 2252 y Fm(\017;t)1549 2237 y Ft(:=)28 b(e)1723 2201 y Fo(i)p Fm(tS)1819 2237 y Ft(\()p Fq(z)f Fp(\000)22 b Fq(H)2109 2252 y Fm(\017)2142 2237 y Ft(\))2180 2201 y Fl(\000)p Fo(1)2274 2237 y Ft(e)2317 2201 y Fl(\000)p Fo(i)p Fm(tS)1549 2428 y Ft(=)28 b(\()p Fq(z)e Fp(\000)d Fq(H)1943 2443 y Fm(\017;t)2021 2428 y Ft(\))2059 2392 y Fl(\000)p Fo(1)2153 2428 y Fq(:)0 2578 y Ft(Arguing)32 b(as)g(in)g(the)h(pro)s(of)f(of)g(Lemma)f(7.4,)h (one)h(sho)m(ws)h(that)f(the)g(function)1615 2758 y Fr(R)27 b Fp(3)h Fq(t)g Fp(7!)f Fq(G)2087 2773 y Fm(\017;t)0 2937 y Ft(is)32 b(di\013eren)m(tiable)f(and)i(that)1248 3074 y(d)p 1230 3118 90 4 v 1230 3209 a(d)p Fq(t)1329 3141 y(G)1406 3156 y Fm(\017;t)1484 3141 y Fp(j)1512 3170 y Fm(t)p Fo(=0)1659 3141 y Ft(=)28 b Fq(G)1840 3156 y Fm(\017)1873 3141 y Ft(\()p Fq(z)t Ft(\)i[)p Fq(S;)17 b(H)2238 3156 y Fm(\017)2270 3141 y Ft(])p Fq(G)2374 3156 y Fm(\017)2407 3141 y Ft(\()p Fq(z)t Ft(\))p Fq(:)0 3346 y Ft(It)33 b(follo)m(ws)e(that)1049 3446 y(d)p 1032 3490 V 1032 3581 a(d)p Fq(t)1131 3513 y(G)1208 3472 y Fm(m)1208 3538 y(\017;t)1286 3414 y Fj(\014)1286 3463 y(\014)1286 3513 y(\014)1314 3567 y Fm(t)p Fo(=0)1461 3513 y Ft(=)1565 3405 y Fm(m)p Fl(\000)p Fo(1)1581 3430 y Fj(X)1577 3615 y Fm(k)r Fo(=0)1734 3513 y Fq(G)1811 3472 y Fm(k)r Fo(+1)1811 3538 y Fm(\017)1944 3513 y Ft(\()p Fq(z)t Ft(\)i[)p Fq(S;)17 b(H)2309 3528 y Fm(\017)2341 3513 y Ft(])p Fq(G)2445 3472 y Fm(m)p Fl(\000)p Fm(k)2445 3538 y(\017)2605 3513 y Ft(\()p Fq(z)t Ft(\))p Fq(:)724 b Ft(\(7.113\))0 3743 y(On)33 b(the)g(other)f(hand,)h(in)f(the)h (quadratic)f(form)g(sense)i(on)f Fp(D)s Ft(\()p Fq(S)6 b Ft(\),)1405 3904 y(d)p 1388 3948 V 1388 4040 a(d)p Fq(t)1487 3971 y(G)1564 3930 y Fm(m)1564 3996 y(\017;t)1642 3872 y Fj(\014)1642 3921 y(\014)1642 3971 y(\014)1669 4025 y Fm(t)p Fo(=0)1817 3971 y Ft(=)27 b(i[)p Fq(S;)17 b(G)2156 3930 y Fm(m)2156 3996 y(\017)2222 3971 y Ft(\()p Fq(z)t Ft(\)])p Fq(:)1080 b Ft(\(7.114\))0 4182 y(Com)m(bining)31 b(\(7.113\))g(and)i(\(7.114\))f(w)m(e)h(deriv)m(e)h(the)f(statemen)m (t.)44 b Fe(2)146 4340 y Ft(Let)33 b Fq(\020)40 b Ft(b)s(e)33 b(as)f(in)g(\(7.99\))g(and)h(let)1217 4520 y Fq(\021)t Ft(\()p Fq(t)p Ft(\))28 b(:=)f(e)1581 4478 y Fm(t)1611 4520 y Fq(t\020)1697 4478 y Fl(0)1720 4520 y Ft(\()p Fq(t)p Ft(\))h(=)f Fq(t)p Ft(\()p Fq(\030)2083 4478 y Fl(0)2106 4520 y Ft(\()p Fq(t)p Ft(\))22 b Fp(\000)h Fq(\030)5 b Ft(\()p Fq(t)p Ft(\)\))p Fq(:)918 b Ft(\(7.115\))0 4699 y(W)-8 b(e)33 b(set)1058 4854 y Fq(K)1141 4869 y Fm(\017)1202 4854 y Ft(:=)1342 4786 y Fq(\025\017)1438 4750 y Fl(\000)p Fo(1)p 1342 4830 191 4 v 1372 4848 a Fp(p)p 1455 4848 49 4 v 82 x Ft(2)1559 4854 y(\()p Fq(a)p Ft(\()p Fq(\021)t Ft(\()p Fp(\000)p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\))23 b(+)f Fq(a)2249 4812 y Fl(\003)2288 4854 y Ft(\()p Fq(\021)t Ft(\()p Fq(\017s)p Ft(\))p Fq(\013)q Ft(\)\))16 b Fq(:)760 b Ft(\(7.116\))0 5064 y(Note)33 b(that)f(0)c Fp(62)g Ft(supp)17 b Fq(\021)36 b Ft(\(this)d(fact)f(will) e(pla)m(y)j(an)f(imp)s(ortan)m(t)f(role)g(in)h(the)h(sequel\))h(and)e (that)1465 5243 y Fq(H)1554 5202 y Fo(\(1\))1546 5268 y Fm(\017)1670 5243 y Fp(\000)22 b Ft([)p Fq(S;)17 b(H)1981 5258 y Fm(\017)2014 5243 y Ft(])28 b(=)f Fq(K)2255 5258 y Fm(\017)2288 5243 y Fq(:)1166 b Ft(\(7.117\))1841 5753 y(50)p eop %%Page: 51 51 51 50 bop 0 407 a Fr(Lemma)37 b(7.6)49 b Fi(Assume)41 b(that)g(Hyp)-5 b(othesis)41 b(S\(0\))g(holds)f(and)g(let)h Fq(z)j Fp(62)c Fq(\033)t Ft(\()p Fq(H)2848 422 y Fm(\017)2880 407 y Ft(\))h Fi(b)-5 b(e)41 b(\014xe)-5 b(d.)62 b(Then,)42 b(for)0 527 y(any)35 b Fq(m)28 b Fp(\025)g Ft(1)p Fi(,)884 646 y Ft(d)p 864 691 94 4 v 864 782 a(d)p Fq(\017)968 714 y(G)1045 673 y Fm(m)1045 739 y(\017)1111 714 y Ft(\()p Fq(z)t Ft(\))g(=)g([)p Fq(S;)17 b(G)1576 673 y Fm(m)1576 739 y(\017)1643 714 y Ft(\()p Fq(z)t Ft(\)])23 b(+)1916 606 y Fm(m)p Fl(\000)p Fo(1)1932 631 y Fj(X)1927 815 y Fm(k)r Fo(=0)2085 714 y Fq(G)2162 673 y Fm(k)r Fo(+1)2162 739 y Fm(\017)2295 714 y Ft(\()p Fq(z)t Ft(\))p Fq(K)2503 729 y Fm(\017)2536 714 y Fq(G)2613 673 y Fm(m)p Fl(\000)p Fm(k)2613 739 y(\017)2773 714 y Ft(\()p Fq(z)t Ft(\))p Fq(:)556 b Ft(\(7.118\))0 1117 y Fr(Pro)s(of.)65 b Ft(Note)33 b(that)935 1285 y Fo(d)p 921 1301 68 4 v 921 1358 a(d)p Fm(\017)999 1324 y Fq(G)1076 1288 y Fm(m)1076 1349 y(\017)1142 1324 y Ft(\()p Fq(z)t Ft(\))84 b(=)1454 1258 y Fj(P)1542 1283 y Fm(m)p Fl(\000)p Fo(1)1542 1349 y Fm(k)r Fo(=0)1715 1324 y Fq(G)1792 1288 y Fm(k)1792 1349 y(\017)1835 1324 y Ft(\()p Fq(z)t Ft(\))1977 1228 y Fj(\020)2051 1285 y Fo(d)p 2037 1301 V 2037 1358 a(d)p Fm(\017)2114 1324 y Fq(G)2191 1339 y Fm(\017)2224 1324 y Ft(\()p Fq(z)t Ft(\))2349 1228 y Fj(\021)2416 1324 y Fq(G)2493 1288 y Fm(m)p Fl(\000)p Fo(1)p Fl(\000)p Fm(k)2493 1349 y(\017)2743 1324 y Ft(\()p Fq(z)t Ft(\))1351 1517 y(=)1454 1451 y Fj(P)1542 1476 y Fm(m)p Fl(\000)p Fo(1)1542 1542 y Fm(k)r Fo(=0)1715 1517 y Fq(G)1792 1481 y Fm(k)r Fo(+1)1792 1542 y Fm(\017)1925 1517 y Ft(\()p Fq(z)t Ft(\))p Fq(H)2139 1481 y Fo(\(1\))2131 1542 y Fm(\017)2234 1517 y Fq(G)2311 1481 y Fm(m)p Fl(\000)p Fm(k)2311 1542 y(\017)2470 1517 y Ft(\()p Fq(z)t Ft(\))p Fq(:)0 1736 y Ft(The)34 b(result)e(follo)m(ws) f(from)h(the)h(iden)m(tit)m(y)f(\(7.117\))g(and)g(Lemma)f(7.5.)44 b Fe(2)146 1906 y Ft(F)-8 b(rom)25 b(no)m(w)i(on)f(w)m(e)h(strengthen)g (the)f(h)m(yp)s(othesis)i(on)e(the)g(in)m(teraction)f(and)h(w)m(e)h (will)d(assume)j(S\(1\).)0 2026 y(Note)33 b(that)f(the)h(follo)m(wing)d (inequalit)m(y)h(pla)m(ys)i(the)g(role)f(of)g(the)h(Mourre)g(estimate.) 0 2255 y Fr(Lemma)k(7.7)49 b Fi(Assume)35 b(that)g(Hyp)-5 b(othesis)35 b(S\(1\))f(holds)g(and)h(let)g Fq(\017)28 b(>)f Ft(0)p Fi(.)45 b(Then,)34 b(for)g(any)h Ft(\011)28 b Fp(2)g Fh(D)p Fi(,)712 2519 y Fp(\000)833 2452 y Ft(1)p 799 2496 116 4 v 799 2587 a(2i)p Fq(\017)925 2519 y Ft(\(\011)p Fp(j)p Ft(\()p Fq(H)1186 2534 y Fm(\017)1240 2519 y Fp(\000)23 b Fq(H)1429 2478 y Fl(\003)1421 2544 y Fm(\017)1468 2519 y Ft(\)\011\))k Fp(\025)h Ft(\(1)22 b Fp(\000)1961 2432 y(p)p 2044 2432 49 4 v 87 x Ft(2)p Fp(j)p Fq(\025)p Fp(jk)p Fq(\030)2304 2478 y Fl(0)2325 2519 y Fp(k)2375 2534 y Fl(1)2450 2519 y Fp(k)p Fq(s\013)q Fp(k)p Ft(\)\(\011)p Fp(j)p Fq(N)10 b Ft(\011\))p Fq(:)0 2967 y Fr(Remark.)47 b Ft(Since)34 b Fq(\030)770 2931 y Fl(0)793 2967 y Ft(\()p Fq(t)p Ft(\))c(=)g(e)1083 2931 y Fm(t)1146 2967 y Ft(around)k(0,)g(w)m (e)h(ha)m(v)m(e)h(that)d Fp(k)p Fq(\030)2270 2931 y Fl(0)2293 2967 y Fp(k)2343 2982 y Fl(1)2447 2967 y Fp(\025)e Ft(1.)47 b(On)34 b(the)g(other)g(hand,)h(b)m(y)g(an)0 3088 y(appropriate)d(c)m (hoice)h(of)f(the)h(function)f Fq(\020)8 b Ft(,)32 b(w)m(e)h(can)g(mak) m(e)g Fp(k)p Fq(\030)2262 3052 y Fl(0)2284 3088 y Fp(k)2334 3103 y Fl(1)2441 3088 y Ft(as)g(close)g(to)f(1)g(as)h(w)m(e)h(wish.)0 3329 y Fr(Pro)s(of.)65 b Ft(Let)1352 3469 y Fq(\030)1395 3484 y Fo(1)1434 3469 y Ft(\()p Fq(t)p Ft(\))28 b(=)1718 3401 y(1)p 1687 3445 112 4 v 1687 3537 a(2i)p Fq(t)1808 3469 y Ft(\()p Fq(\030)5 b Ft(\()p Fp(\000)p Fq(t)p Ft(\))22 b Fp(\000)h Fq(\030)5 b Ft(\()p Fq(t)p Ft(\)\))p Fq(:)0 3668 y Ft(Then)34 b(on)e Fh(D)p Ft(,)707 3881 y Fp(\000)819 3842 y Fo(1)p 794 3858 84 4 v 794 3915 a(2i)p Fm(\017)888 3881 y Ft(\()p Fq(H)1007 3896 y Fm(\017)1061 3881 y Fp(\000)23 b Fq(H)1250 3845 y Fl(\003)1242 3906 y Fm(\017)1289 3881 y Ft(\))83 b(=)28 b Fq(N)k Ft(+)1753 3842 y Fm(\025)p 1732 3858 V 1732 3915 a Fo(2i)p Fm(\017)1826 3881 y Ft(\()p Fq(V)1921 3896 y Fl(\000)p Fm(\017)2030 3881 y Fp(\000)23 b Fq(V)2187 3896 y Fm(\017)2219 3881 y Ft(\))1410 4072 y(=)28 b Fq(N)k Ft(+)22 b Fq(\025')p Ft(\()p Fq(\030)1924 4087 y Fo(1)1963 4072 y Ft(\()p Fq(\017s)p Ft(\))p Fq(s\013)q Ft(\))1410 4279 y(=)28 b Fq(N)1612 4215 y Fk(1)p 1612 4227 31 4 v 1612 4268 a(2)1673 4183 y Fj(\020)1723 4279 y Ft(1)22 b(+)g Fq(\025N)2037 4242 y Fl(\000)2102 4215 y Fk(1)p 2102 4227 V 2102 4268 a(2)2147 4279 y Fq(')p Ft(\()p Fq(\030)2292 4294 y Fo(1)2330 4279 y Ft(\()p Fq(\017s)p Ft(\))p Fq(s\013)q Ft(\))p Fq(N)2726 4242 y Fl(\000)2791 4215 y Fk(1)p 2792 4227 V 2792 4268 a(2)2836 4183 y Fj(\021)2902 4279 y Fq(N)3001 4215 y Fk(1)p 3001 4227 V 3001 4268 a(2)3045 4279 y Fq(:)0 4506 y Ft(The)34 b(result)e(follo)m(ws)f(from)h(this)g(iden)m(tit)m(y)g(and)h(the)g (elemen)m(tary)g(estimate)631 4729 y Fp(k)p Fq(N)769 4692 y Fl(\000)834 4665 y Fk(1)p 835 4677 V 835 4718 a(2)879 4729 y Fq(')p Ft(\()p Fq(\030)1024 4744 y Fo(1)1063 4729 y Ft(\()p Fq(\017s)p Ft(\))p Fq(s\013)q Ft(\))p Fq(N)1459 4692 y Fl(\000)1524 4665 y Fk(1)p 1524 4677 V 1524 4718 a(2)1569 4729 y Fp(k)83 b(\024)1807 4647 y(p)p 1890 4647 49 4 v 82 x Ft(2)o Fp(k)p Fq(\030)2031 4744 y Fo(1)2070 4729 y Ft(\()p Fq(\017s)p Ft(\))p Fq(s\013)q Fp(k)1702 4926 y(\024)1807 4844 y(p)p 1890 4844 V 82 x Ft(2)o Fp(k)p Fq(\030)2031 4941 y Fo(1)2070 4926 y Fp(k)2120 4941 y Fl(1)2195 4926 y Fp(k)p Fq(s\013)q Fp(k)27 b(\024)2536 4844 y(p)p 2619 4844 V 82 x Ft(2)p Fp(k)p Fq(\030)2766 4890 y Fl(0)2788 4926 y Fp(k)2838 4941 y Fl(1)2913 4926 y Fp(k)p Fq(s\013)q Fp(k)p Fq(:)0 5114 y Fe(2)1841 5753 y Ft(51)p eop %%Page: 52 52 52 51 bop 146 407 a Ft(In)33 b(the)g(sequel)h(w)m(e)f(c)m(ho)s(ose)h (\003)1248 422 y Fo(1)1315 407 y Fq(>)27 b Ft(0)32 b(and)h Fq(C)1759 422 y Fo(0)1826 407 y Fq(>)28 b Ft(0)k(suc)m(h)i(that)1422 581 y(\003)1490 596 y Fo(1)1612 581 y Fq(<)27 b Ft(\()1753 499 y Fp(p)p 1836 499 49 4 v 82 x Ft(2)p Fp(k)p Fq(s\013)q Fp(k)p Ft(\))2132 545 y Fl(\000)p Fo(1)2225 581 y Fq(;)1419 778 y(C)1489 793 y Fo(0)1612 778 y Fq(<)g Ft(1)22 b Fp(\000)1886 696 y(p)p 1969 696 V 82 x Ft(2\003)2086 793 y Fo(1)2125 778 y Fp(k)p Fq(s\013)q Fp(k)p Fq(:)3481 678 y Ft(\(7.119\))0 954 y(It)30 b(follo)m(ws)e(from)g(Lemma)g(7.7)h(that)h(w)m(e)g(can)g(c) m(ho)s(ose)g Fq(\020)37 b Ft(in)29 b(\(7.99\))f(so)i(that)f(for)g Fp(j)p Fq(\025)p Fp(j)e(\024)i Ft(\003)3231 969 y Fo(1)3299 954 y Ft(and)h(\011)d Fp(2)h Fh(D)p Ft(,)1130 1176 y Fp(\000)1250 1109 y Ft(1)p 1217 1153 116 4 v 1217 1244 a(2i)p Fq(\017)1342 1176 y Ft(\(\011)p Fp(j)p Ft(\()p Fq(H)1603 1191 y Fm(\017)1657 1176 y Fp(\000)23 b Fq(H)1846 1135 y Fl(\003)1838 1201 y Fm(\017)1885 1176 y Ft(\)\011\))k Fp(\025)i Fq(C)2240 1191 y Fo(0)2279 1176 y Ft(\(\011)p Fp(j)p Fq(N)10 b Ft(\011\))p Fq(:)831 b Ft(\(7.120\))0 1384 y(All)31 b(the)i(results)g(in)f(the)h(sequel)g(will)d(hold)i(for)g (real)g Fq(\025)g Ft(suc)m(h)i(that)f Fp(j)p Fq(\025)p Fp(j)27 b(\024)h Ft(\003)2784 1399 y Fo(1)2823 1384 y Ft(,)33 b(uniformly)d(in)i Fq(\025)p Ft(.)0 1564 y Fr(Lemma)37 b(7.8)49 b Fi(Assume)d(that)g(Hyp)-5 b(othesis)46 b(S\(1\))f(holds)f (and)i(let)f Fq(\017)k(>)e Ft(0)p Fi(.)77 b(If)45 b Ft(Im)p Fq(z)52 b(>)c Fp(\000)p Fq(C)3473 1579 y Fo(0)3513 1564 y Fq(\017)e Fi(then)0 1684 y Fq(z)32 b Fp(62)d Fq(\033)t Ft(\()p Fq(H)350 1699 y Fm(\017)382 1684 y Ft(\))35 b Fi(and)1415 1849 y Fp(k)p Fq(G)1542 1864 y Fm(\017)1574 1849 y Ft(\()p Fq(z)t Ft(\))p Fp(k)28 b(\024)2086 1781 y Ft(1)p 1893 1826 436 4 v 1893 1917 a(Im)o Fq(z)f Ft(+)22 b Fq(C)2249 1932 y Fo(0)2288 1917 y Fq(\017)2338 1849 y(:)0 2184 y Fr(Pro)s(of.)73 b Ft(It)36 b(follo)m(ws)f(from)g(Relation) g(\(7.120\))g(that)h(the)h(n)m(umerical)e(range)i(of)f(the)g(op)s (erator)g Fq(H)3645 2199 y Fm(\017)3714 2184 y Ft(is)0 2305 y(con)m(tained)e(in)g(the)g(region)g(Im)o Fq(z)h Fp(\024)c(\000)p Fq(C)1474 2320 y Fo(0)1514 2305 y Fq(\017)p Ft(.)48 b(Since)35 b Fp(D)s Ft(\()p Fq(H)2084 2320 y Fm(\017)2116 2305 y Ft(\))30 b(=)g Fp(D)s Ft(\()p Fq(H)2497 2268 y Fl(\003)2489 2329 y Fm(\017)2536 2305 y Ft(\),)35 b(the)f(statemen)m(t)h(follo)m(ws)e(from)0 2425 y(Prop)s(osition)e (2.9.)43 b Fe(2)146 2583 y Ft(Before)33 b(w)m(e)h(mak)m(e)e(use)i(of)e (the)h(last)f(result,)h(w)m(e)g(need)h(one)f(additional)c(lemma.)0 2750 y Fr(Lemma)37 b(7.9)49 b Fi(Assume)c(that)h(Hyp)-5 b(othesis)45 b(S\(1\))f(holds)g(and)h(let)g Fq(\017)i(>)g Ft(0)e Fi(and)f Fq(z)52 b Fp(2)47 b Fr(C)3297 2765 y Fo(+)3401 2750 y Fi(b)-5 b(e)44 b(given.)0 2870 y(Then,)34 b(for)h(any)f Ft(\011)28 b Fp(2)g(H)q Fi(,)1041 3051 y Fp(k)p Fq(N)1190 2987 y Fk(1)p 1190 2999 31 4 v 1190 3040 a(2)1234 3051 y Fq(G)1311 3066 y Fm(\017)1344 3051 y Ft(\()p Fq(z)t Ft(\)\011)p Fp(k)83 b(\024)28 b Ft(\()p Fq(C)1891 3066 y Fo(0)1931 3051 y Fq(\017)p Ft(\))2008 3014 y Fl(\000)2073 2987 y Fk(1)p 2073 2999 V 2073 3040 a(2)2117 3051 y Fp(j)p Ft(\(\011)p Fp(j)p Fq(G)2364 3066 y Fm(\017)2396 3051 y Ft(\()p Fq(z)t Ft(\)\011\))p Fp(j)2673 2987 y Fk(1)p 2673 2999 V 2673 3040 a(2)2718 3051 y Fq(;)1035 3254 y Fp(k)p Fq(N)1183 3190 y Fk(1)p 1183 3202 V 1183 3243 a(2)1228 3254 y Fq(G)1305 3218 y Fl(\003)1305 3278 y Fm(\017)1344 3254 y Ft(\()p Fq(z)t Ft(\)\011)p Fp(k)83 b(\024)28 b Ft(\()p Fq(C)1891 3269 y Fo(0)1931 3254 y Fq(\017)p Ft(\))2008 3217 y Fl(\000)2073 3190 y Fk(1)p 2073 3202 V 2073 3243 a(2)2117 3254 y Fp(j)p Ft(\(\011)p Fp(j)p Fq(G)2364 3269 y Fm(\017)2396 3254 y Ft(\()p Fq(z)t Ft(\)\011\))p Fp(j)2673 3190 y Fk(1)p 2673 3202 V 2673 3243 a(2)2718 3254 y Fq(:)0 3522 y Fr(Pro)s(of.)65 b Ft(W)-8 b(e)33 b(pro)m(v)m(e)h(the)f(\014rst)g(relation.)41 b(A)33 b(similar)c(argumen)m(t)k(yields)f(the)h(second.)45 b(W)-8 b(e)33 b(ha)m(v)m(e)677 3702 y Fp(k)p Fq(N)826 3638 y Fk(1)p 826 3650 V 826 3692 a(2)870 3702 y Fq(G)947 3717 y Fm(\017)980 3702 y Ft(\()p Fq(z)t Ft(\)\011)p Fp(k)1231 3666 y Fo(2)1354 3702 y Ft(=)27 b(\(\011)p Fp(j)p Fq(G)1676 3666 y Fl(\003)1676 3727 y Fm(\017)1715 3702 y Ft(\()p Fq(z)t Ft(\))p Fq(N)10 b(G)2005 3717 y Fm(\017)2039 3702 y Ft(\()p Fq(z)t Ft(\)\011\))p Fp(j)1354 3893 y(\024)28 b Ft(\()p Fq(C)1567 3908 y Fo(0)1606 3893 y Fq(\017)p Ft(\))1683 3857 y Fl(\000)p Fo(1)1777 3893 y Ft(\(\011)p Fp(j)p Fq(G)1996 3857 y Fl(\003)1996 3918 y Fm(\017)2035 3893 y Ft(\()p Fq(z)t Ft(\)\(Im)p Fq(H)2396 3908 y Fm(\017)2451 3893 y Ft(+)22 b(Im)p Fq(z)t Ft(\))p Fq(G)2830 3908 y Fm(\017)2863 3893 y Ft(\()p Fq(z)t Ft(\)\011\))1354 4085 y(=)27 b(\(i2)p Fq(C)1642 4100 y Fo(0)1680 4085 y Fq(\017)p Ft(\))1757 4049 y Fl(\000)p Fo(1)1852 4085 y Ft(\(\011)p Fp(j)p Ft(\()p Fq(G)2109 4049 y Fl(\003)2109 4109 y Fm(\017)2148 4085 y Ft(\()p Fq(z)t Ft(\))22 b Fp(\000)h Fq(G)2472 4100 y Fm(\017)2505 4085 y Ft(\()p Fq(z)t Ft(\)\)\011\))1354 4276 y Fp(\024)28 b Ft(\()p Fq(C)1567 4291 y Fo(0)1606 4276 y Fq(\017)p Ft(\))1683 4240 y Fl(\000)p Fo(1)1777 4276 y Fp(j)p Ft(\(\011)p Fp(j)p Fq(G)2024 4291 y Fm(\017)2056 4276 y Ft(\()p Fq(z)t Ft(\)\011\))p Fp(j)p Fq(;)0 4452 y Ft(where)34 b(in)e(the)h(\014rst)g (estimate)e(w)m(e)j(used)g(\(7.120\))d Fe(2)146 4610 y Ft(F)-8 b(rom)31 b(no)m(w)j(on)e Fq(\032)h Ft(will)d(denote)k(a)e(Sc) m(h)m(w)m(artz)i(function.)43 b(Set)1301 4779 y Fp(h)p Fq(S)6 b Fp(i)1445 4743 y Fl(\000)p Fm(\026)1445 4803 y(\017;\032)1629 4779 y Ft(:=)27 b Fp(h)p Fq(S)6 b Fp(i)1903 4743 y Fl(\000)p Fm(\026)2004 4779 y Fq(\032)p Ft(\()p Fq(\017S)g Ft(\))p Fq(;)1268 4970 y(F)1331 4985 y Fm(\017;\032)1420 4970 y Ft(\()p Fq(z)t Ft(\))84 b(:=)27 b Fp(h)p Fq(S)6 b Fp(i)1903 4934 y Fl(\000)p Fm(\026)1903 4995 y(\017;\032)2004 4970 y Fq(G)2081 4985 y Fm(\017)2114 4970 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)g Fp(i)2383 4934 y Fl(\000)p Fm(\026)2383 4995 y(\017;\032)2484 4970 y Fq(:)3481 4876 y Ft(\(7.121\))0 5147 y(Note)27 b(that)f(Lemma)g(7.4)g(yields)g(that)h(the)g(function)f Fr(R)2025 5162 y Fo(+)2112 5147 y Fp(3)i Fq(\017)g Fp(7!)f Fq(F)2463 5162 y Fm(\017;\032)2552 5147 y Ft(\()p Fq(z)t Ft(\))g(is)f(in\014nitely)g(di\013eren)m(tiable.)0 5268 y(W)-8 b(e)33 b(are)g(no)m(w)g(ready)g(to)f(pro)m(v)m(e)i(one)f(of)f (the)h(k)m(ey)h(tec)m(hnical)e(results)h(of)f(this)h(section.)1841 5753 y(52)p eop %%Page: 53 53 53 52 bop 0 407 a Fr(Lemma)37 b(7.10)49 b Fi(Assume)36 b(that)g(Hyp)-5 b(othesis)36 b Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))36 b Fi(holds)f(with)h Fq(\027)g Fp(\025)30 b Ft(1)35 b Fi(and)h(let)g Fq(\026)29 b(>)g Ft(0)p Fi(.)48 b(Set)36 b Fq(\015)5 b Ft(\()p Fq(\026)p Ft(\))29 b(=)0 527 y(min)o(\()p Fq(\026;)17 b Ft(1\))p Fi(.)43 b(Then,)34 b(for)h(any)g Fq(z)d Fp(2)c Fr(C)1343 542 y Fo(+)1402 527 y Fi(,)654 633 y Fj(\015)654 683 y(\015)654 733 y(\015)654 783 y(\015)654 833 y(\015)730 715 y Ft(d)p 710 760 94 4 v 710 851 a(d)p Fq(\017)814 783 y(F)877 798 y Fm(\017;\032)966 783 y Ft(\()p Fq(z)t Ft(\))1091 633 y Fj(\015)1091 683 y(\015)1091 733 y(\015)1091 783 y(\015)1091 833 y(\015)1165 783 y Fp(\024)g Fq(C)1340 798 y Fo(1)1379 783 y Fq(\017)1418 742 y Fl(\000)1483 714 y Fk(3)p 1484 726 31 4 v 1484 768 a(2)1524 742 y Fo(+)p Fm(\015)t Fo(\()p Fm(\026)p Fo(\))1721 783 y Fp(k)p Fq(F)1834 798 y Fm(\017;\032)1922 783 y Ft(\()p Fq(z)t Ft(\))p Fp(k)2107 714 y Fk(1)p 2107 726 V 2107 768 a(2)2174 783 y Ft(+)22 b Fp(j)p Fq(\025)p Fp(j)p Fq(C)2455 798 y Fo(2)2494 783 y Fq(\017)2533 742 y Fl(\000)p Fo(2+)p Fm(\027)2721 783 y Fp(k)p Fq(F)2834 798 y Fm(\017;\032)2923 783 y Ft(\()p Fq(z)t Ft(\))p Fp(k)p Fq(:)356 b Ft(\(7.122\))0 1157 y Fr(Pro)s(of.)65 b Ft(It)33 b(follo)m(ws)e(from)g(Lemma)g(7.6)i(that)1415 1307 y(d)p 1395 1351 94 4 v 1395 1443 a(d)p Fq(\017)1499 1374 y(F)1562 1389 y Fm(\017;\032)1651 1374 y Ft(\()p Fq(z)t Ft(\))28 b(=)f(I)c(+)f(I)s(I)g(+)g(I)s(I)s(I)p Fq(;)0 1593 y Ft(where)793 1701 y(I)83 b(:=)1042 1605 y Fj(\020)1116 1662 y Fo(d)p 1101 1678 68 4 v 1101 1735 a(d)p Fm(\017)1179 1701 y Fp(h)p Fq(S)6 b Fp(i)1323 1665 y Fl(\000)p Fm(\026)1323 1726 y(\017;\032)1424 1605 y Fj(\021)1490 1701 y Fq(G)1567 1716 y Fm(\017)1600 1701 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)g Fp(i)1869 1665 y Fl(\000)p Fm(\026)1869 1726 y(\017;\032)1992 1701 y Ft(+)22 b Fp(h)p Fq(S)6 b Fp(i)2234 1665 y Fl(\000)p Fm(\026)2234 1726 y(\017;\032)2335 1701 y Fq(G)2412 1716 y Fm(\017)2445 1701 y Ft(\()p Fq(z)t Ft(\))2587 1605 y Fj(\020)2661 1662 y Fo(d)p 2646 1678 V 2646 1735 a(d)p Fm(\017)2724 1701 y Fp(h)p Fq(S)g Fp(i)2868 1665 y Fl(\000)p Fm(\026)2868 1726 y(\017;\032)2969 1605 y Fj(\021)3035 1701 y Fq(;)755 1892 y Ft(I)s(I)83 b(:=)28 b Fp(h)p Fq(S)6 b Fp(i)1186 1856 y Fl(\000)p Fm(\026)1186 1917 y(\017;\032)1287 1892 y Ft([)p Fq(S;)17 b(G)1495 1907 y Fm(\017)1528 1892 y Ft(\()p Fq(z)t Ft(\)])p Fp(h)p Fq(S)6 b Fp(i)1824 1856 y Fl(\000)p Fm(\026)1824 1917 y(\017;\032)1925 1892 y Fq(;)717 2084 y Ft(I)s(I)s(I)83 b(:=)28 b Fp(h)p Fq(S)6 b Fp(i)1186 2047 y Fl(\000)p Fm(\026)1186 2108 y(\017;\032)1287 2084 y Fq(G)1364 2099 y Fm(\017)1396 2084 y Ft(\()p Fq(z)t Ft(\))p Fq(K)1604 2099 y Fm(\017)1638 2084 y Fq(G)1715 2099 y Fm(\017)1747 2084 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)g Fp(i)2016 2047 y Fl(\000)p Fm(\026)2016 2108 y(\017;\032)2118 2084 y Fq(:)0 2241 y Ft(Note)33 b(that)1297 2249 y Fj(\015)1297 2299 y(\015)1297 2349 y(\015)1297 2399 y(\015)1297 2449 y(\015)1373 2332 y Ft(d)p 1354 2376 94 4 v 1354 2467 a(d)p Fq(\017)1457 2399 y Fp(h)p Fq(S)6 b Fp(i)1601 2358 y Fl(\000)p Fm(\026)1601 2424 y(\017;\032)1702 2249 y Fj(\015)1702 2299 y(\015)1702 2349 y(\015)1702 2399 y(\015)1702 2449 y(\015)1776 2399 y Fp(\024)28 b Fq(\017)1920 2358 y Fl(\000)p Fo(1+)p Fm(\015)t Fo(\()p Fm(\026)p Fo(\))2207 2399 y Fp(k)p Fq(\032)2307 2358 y Fl(0)2330 2399 y Fp(k)2380 2414 y Fl(1)2455 2399 y Fq(:)0 2619 y Ft(Note)33 b(also)e(that)i(Lemma) e(7.9)h(yields)g(the)h(estimates)675 2815 y Fp(k)p Fq(G)802 2779 y Fl(\003)802 2840 y Fm(\017)841 2815 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)6 b Fp(i)1110 2779 y Fl(\000)p Fm(\026)1110 2840 y(\017;\032)1212 2815 y Fp(k)83 b(\024)28 b(k)p Fq(N)1598 2751 y Fk(1)p 1598 2763 31 4 v 1598 2804 a(2)1643 2815 y Fq(G)1720 2779 y Fl(\003)1720 2840 y Fm(\017)1759 2815 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)6 b Fp(i)2028 2779 y Fl(\000)p Fm(\026)2028 2840 y(\017;\032)2129 2815 y Fp(k)28 b(\024)g Ft(\()p Fq(C)2420 2830 y Fo(0)2459 2815 y Fq(\017)p Ft(\))2536 2778 y Fl(\000)2601 2751 y Fk(1)p 2601 2763 V 2601 2804 a(2)2646 2815 y Fp(k)p Fq(F)2759 2830 y Fm(\017;\032)2847 2815 y Ft(\()p Fq(z)t Ft(\))p Fp(k)3032 2751 y Fk(1)p 3033 2763 V 3033 2804 a(2)3077 2815 y Fq(;)682 3018 y Fp(k)p Fq(G)809 3033 y Fm(\017)841 3018 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)6 b Fp(i)1110 2982 y Fl(\000)p Fm(\026)1110 3043 y(\017;\032)1212 3018 y Fp(k)83 b(\024)28 b(k)p Fq(N)1598 2954 y Fk(1)p 1598 2966 V 1598 3007 a(2)1643 3018 y Fq(G)1720 3033 y Fm(\017)1752 3018 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)6 b Fp(i)2021 2982 y Fl(\000)p Fm(\026)2021 3043 y(\017;\032)2123 3018 y Fp(k)27 b(\024)h Ft(\()p Fq(C)2413 3033 y Fo(0)2452 3018 y Fq(\017)p Ft(\))2529 2981 y Fl(\000)2594 2954 y Fk(1)p 2595 2966 V 2595 3007 a(2)2639 3018 y Fp(k)p Fq(F)2752 3033 y Fm(\017;\032)2841 3018 y Ft(\()p Fq(z)t Ft(\))p Fp(k)3026 2954 y Fk(1)p 3026 2966 V 3026 3007 a(2)3071 3018 y Fq(:)3481 2912 y Ft(\(7.123\))0 3206 y(Th)m(us,)34 b(the)f(term)f(I)h(is)f(estimated)g(as)h(follo)m(ws:)740 3406 y Fp(k)p Ft(I)p Fp(k)83 b(\024)28 b Fq(\017)1102 3370 y Fl(\000)p Fo(1+)p Fm(\015)t Fo(\()p Fm(\026)p Fo(\))1389 3406 y Fp(k)p Fq(\032)1489 3370 y Fl(0)1513 3406 y Fp(k)1563 3421 y Fl(1)1654 3310 y Fj(\020)1704 3406 y Fp(k)p Fq(G)1831 3421 y Fm(\017)1863 3406 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)6 b Fp(i)2132 3370 y Fl(\000)p Fm(\026)2132 3431 y(\017;\032)2233 3406 y Fp(k)22 b Ft(+)g Fp(k)p Fq(G)2530 3370 y Fl(\003)2530 3431 y Fm(\017)2570 3406 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)6 b Fp(i)2839 3370 y Fl(\000)p Fm(\026)2839 3431 y(\017;\032)2940 3406 y Fp(k)2990 3310 y Fj(\021)958 3633 y Fp(\024)28 b Ft(2)p Fq(C)1189 3573 y Fl(\000)1254 3545 y Fk(1)p 1254 3557 V 1254 3599 a(2)1182 3655 y Fo(0)1298 3633 y Fp(k)p Fq(\032)1398 3597 y Fl(0)1422 3633 y Fp(k)1472 3648 y Fl(1)1546 3633 y Fq(\017)1585 3596 y Fl(\000)1650 3569 y Fk(3)p 1651 3581 V 1651 3622 a(2)1691 3596 y Fo(+)p Fm(\015)t Fo(\()p Fm(\026)p Fo(\))1888 3633 y Fp(k)p Fq(F)2001 3648 y Fm(\017;\032)2089 3633 y Ft(\()p Fq(z)t Ft(\))p Fp(k)2274 3569 y Fk(1)p 2274 3581 V 2274 3622 a(2)2319 3633 y Fq(:)3481 3513 y Ft(\(7.124\))0 3820 y(Since)33 b(for)f(an)m(y)h Fq(\017)28 b(>)g Ft(0,)1059 3940 y Fp(k)p Fq(S)6 b Fp(h)p Fq(S)g Fp(i)1319 3899 y Fl(\000)p Fm(\026)1319 3965 y(\017;\032)1419 3940 y Fp(k)27 b Ft(=)h Fp(kh)p Fq(S)6 b Fp(i)1794 3899 y Fl(\000)p Fm(\026)1794 3965 y(\017;\032)1895 3940 y Fq(S)g Fp(k)27 b(\024)h Fq(\017)2182 3899 y Fl(\000)p Fo(1+)p Fm(\015)t Fo(\()p Fm(\026)p Fo(\))2469 3940 y Fp(k)p Fq(\032)p Fp(k)2619 3955 y Fl(1)2694 3940 y Fq(;)0 4103 y Ft(the)33 b(estimates)f(\(7.123\))g(yield)733 4309 y Fp(k)p Ft(I)s(I)p Fp(k)83 b(\024)28 b Fq(\017)1133 4272 y Fl(\000)p Fo(1+)p Fm(\015)t Fo(\()p Fm(\026)p Fo(\))1420 4309 y Fp(k)p Fq(\032)p Fp(k)1570 4324 y Fl(1)1661 4212 y Fj(\020)1711 4309 y Fp(k)p Fq(G)1838 4324 y Fm(\017)1870 4309 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)6 b Fp(i)2139 4272 y Fl(\000)p Fm(\026)2139 4333 y(\017;\032)2241 4309 y Fp(k)22 b Ft(+)g Fp(k)p Fq(G)2538 4272 y Fl(\003)2538 4333 y Fm(\017)2577 4309 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)6 b Fp(i)2846 4272 y Fl(\000)p Fm(\026)2846 4333 y(\017;\032)2947 4309 y Fp(k)2997 4212 y Fj(\021)989 4535 y Fp(\024)28 b Ft(2)p Fq(C)1220 4475 y Fl(\000)1285 4448 y Fk(1)p 1284 4460 V 1284 4501 a(2)1213 4557 y Fo(0)1329 4535 y Fp(k)p Fq(\032)p Fp(k)1479 4550 y Fl(1)1554 4535 y Fq(\017)1593 4498 y Fl(\000)1658 4471 y Fk(3)p 1658 4483 V 1658 4524 a(2)1698 4498 y Fo(+)p Fm(\015)t Fo(\()p Fm(\026)p Fo(\))1895 4535 y Fp(k)p Fq(F)2008 4550 y Fm(\017;\032)2096 4535 y Ft(\()p Fq(z)t Ft(\))p Fp(k)2281 4471 y Fk(1)p 2282 4483 V 2282 4524 a(2)2326 4535 y Fq(:)3481 4415 y Ft(\(7.125\))0 4722 y(The)34 b(term)e(I)s(I)s(I)g(is)g(estimated)g (as)659 4899 y Fp(k)p Ft(I)s(I)s(I)p Fp(k)82 b(\024)29 b(k)p Fq(N)1206 4835 y Fk(1)p 1206 4847 V 1206 4888 a(2)1250 4899 y Fq(G)1327 4863 y Fl(\003)1327 4924 y Fm(\017)1367 4899 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)6 b Fp(i)1636 4863 y Fl(\000)p Fm(\026)1636 4924 y(\017;\032)1737 4899 y Fp(kk)p Fq(N)1935 4835 y Fk(1)p 1935 4847 V 1935 4888 a(2)1980 4899 y Fq(G)2057 4914 y Fm(\017)2089 4899 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)g Fp(i)2358 4863 y Fl(\000)p Fm(\026)2358 4924 y(\017;\032)2460 4899 y Fp(kk)p Fq(N)2648 4862 y Fl(\000)2713 4835 y Fk(1)p 2713 4847 V 2713 4888 a(2)2757 4899 y Fq(K)2840 4914 y Fm(\017)2873 4899 y Fq(N)2961 4862 y Fl(\000)3026 4835 y Fk(1)p 3026 4847 V 3026 4888 a(2)3071 4899 y Fp(k)952 5090 y(\024)29 b Fq(C)1135 5049 y Fl(\000)p Fo(1)1128 5112 y(0)1229 5090 y Fp(j)p Fq(\025)p Fp(j)p Fq(\017)1381 5054 y Fl(\000)p Fo(2)1475 5090 y Ft(\()p Fp(k)p Fq(\021)t Ft(\()p Fp(\000)p Fq(\017s)p Ft(\))p Fq(\013)q Fp(k)22 b Ft(+)g Fp(k)p Fq(\021)t Ft(\()p Fq(\017s)p Ft(\))p Fq(\013)q Fp(k)p Ft(\))p Fp(k)p Fq(F)2613 5105 y Fm(\017;\032)2700 5090 y Ft(\()p Fq(z)t Ft(\))p Fp(k)952 5282 y(\024)29 b Ft(2)p Fq(C)1184 5240 y Fl(\000)p Fo(1)1177 5303 y(0)1278 5282 y Fp(j)p Fq(\025)p Fp(jk)p Fq(s)1487 5245 y Fm(\027)1529 5282 y Fq(\013)q Fp(k)17 b Ft(sup)1805 5305 y Fm(t)1851 5282 y Fp(j)p Fq(t)1914 5245 y Fl(\000)p Fm(\027)2012 5282 y Fq(\021)t Ft(\()p Fq(t)p Ft(\))p Fp(j)p Fq(\017)2242 5245 y Fl(\000)p Fo(2+)p Fm(\027)2430 5282 y Fp(k)p Fq(F)2543 5297 y Fm(\017;\032)2631 5282 y Ft(\()p Fq(z)t Ft(\))p Fp(k)p Fq(:)3481 5085 y Ft(\(7.126\))1841 5753 y(53)p eop %%Page: 54 54 54 53 bop 0 407 a Ft(Note)33 b(that)f(since)h(0)27 b Fp(62)h Ft(supp)18 b Fq(\021)t Ft(,)32 b(sup)1332 430 y Fm(t)1378 407 y Fp(j)p Fq(t)1441 371 y Fl(\000)p Fm(\027)1539 407 y Fq(\021)t Ft(\()p Fq(t)p Ft(\))p Fp(j)27 b Fq(<)h Fp(1)p Ft(.)43 b(Com)m(bining)31 b(the)i(estimates)f(\(7.124\),)f (\(7.125\))0 527 y(and)i(\(7.126\))e(w)m(e)j(deriv)m(e)f(that)f (Relation)f(\(7.122\))h(holds)g(with)1212 772 y Fq(C)1282 787 y Fo(1)1404 772 y Ft(=)c(2)p Fq(C)1634 712 y Fl(\000)1699 685 y Fk(1)p 1698 697 31 4 v 1698 738 a(2)1627 794 y Fo(0)1743 772 y Ft(\()p Fp(k)p Fq(\032)p Fp(k)1931 787 y Fl(1)2028 772 y Ft(+)22 b Fp(k)p Fq(\032)2226 736 y Fl(0)2249 772 y Fp(k)2299 787 y Fl(1)2374 772 y Ft(\))p Fq(;)1212 964 y(C)1282 979 y Fo(2)1404 964 y Ft(=)28 b(2)p Fq(C)1634 922 y Fl(\000)p Fo(1)1627 985 y(0)1728 964 y Fp(k)p Fq(s)1824 927 y Fm(\027)1867 964 y Fq(\013)q Fp(k)17 b Ft(sup)2143 987 y Fm(t)2189 964 y Fp(j)p Fq(t)2252 927 y Fl(\000)p Fm(\027)2350 964 y Fq(\021)t Ft(\()p Fq(t)p Ft(\))p Fp(j)p Fq(:)3481 851 y Ft(\(7.127\))0 1145 y Fe(2)0 1412 y Fr(Lemma)37 b(7.11)49 b Fi(Assume)35 b(that)g(Hyp)-5 b(othesis)35 b Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))35 b Fi(holds)f(for)h(some)f Fq(\027)g(>)28 b Ft(1)34 b Fi(and)h(let)g Fq(\026)27 b(>)3374 1373 y Fo(1)p 3374 1389 36 4 v 3374 1447 a(2)3420 1412 y Fi(.)44 b(Then,)0 1533 y Ft(\(i\))34 b(sup)285 1556 y Fo(\()p Fm(\017;z)s Fo(\))p Fl(2)p Fd(R)532 1565 y Fk(+)582 1556 y Fl(\002)p Fd(C)696 1565 y Fk(+)767 1533 y Fp(k)p Fq(F)880 1548 y Fm(\017;\032)968 1533 y Ft(\()p Fq(z)t Ft(\))p Fp(k)29 b(\024)f Fq(C)7 b Fi(.)0 1653 y Ft(\(ii\))33 b Fi(F)-7 b(or)34 b(any)h Fq(z)d Fp(2)c Fr(C)786 1668 y Fo(+)845 1653 y Fi(,)35 b(the)g(norm-limit)f Ft(lim)1699 1668 y Fm(\017)p Fl(#)p Fo(0)1819 1653 y Fq(F)1882 1668 y Fm(\017;\032)1971 1653 y Ft(\()p Fq(z)t Ft(\))h Fi(exist.)0 1994 y Fr(Pro)s(of.)65 b Ft(Since)1321 2114 y Fp(k)p Fq(F)1434 2129 y Fm(\017;\032)1523 2114 y Ft(\()p Fq(z)t Ft(\))p Fp(k)1708 2046 y Fk(1)p 1708 2058 31 4 v 1708 2099 a(2)1780 2114 y Fp(\024)28 b Ft(1)22 b(+)g Fp(k)p Fq(F)2167 2129 y Fm(\017;\032)2256 2114 y Ft(\()p Fq(z)t Ft(\))p Fp(k)p Fq(;)0 2285 y Ft(it)32 b(follo)m(ws)f(from)g(\(7.122\))h (that)1241 2411 y Fj(\015)1241 2461 y(\015)1241 2510 y(\015)1241 2560 y(\015)1241 2610 y(\015)1317 2493 y Ft(d)p 1297 2537 94 4 v 1297 2629 a(d)p Fq(\017)1400 2560 y(F)1463 2575 y Fm(\017;\032)1552 2560 y Ft(\()p Fq(z)t Ft(\))1677 2411 y Fj(\015)1677 2461 y(\015)1677 2510 y(\015)1677 2560 y(\015)1677 2610 y(\015)1751 2560 y Fp(\024)d Fq(a)1908 2575 y Fm(\017)1940 2560 y Fp(k)p Fq(F)2053 2575 y Fm(\017;\032)2142 2560 y Ft(\()p Fq(z)t Ft(\))p Fp(k)22 b Ft(+)g Fq(b)2478 2575 y Fm(\017)2512 2560 y Fq(;)942 b Ft(\(7.128\))0 2830 y(where)1218 2934 y Fq(a)1269 2949 y Fm(\017)1384 2934 y Ft(:=)28 b Fq(C)1585 2949 y Fo(1)1624 2934 y Fq(\017)1663 2897 y Fl(\000)1728 2870 y Fk(3)p 1729 2882 31 4 v 1729 2923 a(2)1769 2897 y Fo(+)p Fm(\015)t Fo(\()p Fm(\026)p Fo(\))1988 2934 y Ft(+)22 b Fp(j)p Fq(\025)p Fp(j)p Fq(C)2269 2949 y Fo(2)2307 2934 y Fq(\017)2346 2898 y Fl(\000)p Fo(2+)p Fm(\027)2535 2934 y Fq(;)1227 3137 y(b)1268 3152 y Fm(\017)1384 3137 y Ft(:=)28 b Fq(C)1585 3152 y Fo(1)1624 3137 y Fq(\017)1663 3100 y Fl(\000)1728 3073 y Fk(3)p 1729 3085 V 1729 3126 a(2)1769 3100 y Fo(+)p Fm(\015)t Fo(\()p Fm(\026)p Fo(\))1966 3137 y Fq(:)0 3314 y Ft(Since)33 b Fq(\026)27 b(>)455 3275 y Fo(1)p 455 3291 36 4 v 455 3349 a(2)500 3314 y Ft(,)32 b Fq(\027)j(>)27 b Ft(1,)33 b(w)m(e)g(ha)m(v)m(e)1222 3466 y Fj(Z)1305 3492 y Fo(1)1268 3655 y(0)1361 3583 y Fq(a)1412 3598 y Fm(\034)1456 3583 y Ft(d)p Fq(\034)39 b(<)28 b Fp(1)p Fq(;)1968 3466 y Fj(Z)2051 3492 y Fo(1)2014 3655 y(0)2107 3583 y Fq(b)2148 3598 y Fm(\034)2192 3583 y Ft(d)p Fq(\034)39 b(<)28 b Fp(1)p Fq(:)0 3839 y Ft(Note)k(also)e (that)i(Lemma)e(7.8)h(yields)g(that)g(for)g Fq(\017)d Fp(\025)h Ft(1,)i Fp(k)p Fq(F)2167 3854 y Fm(\017;\032)2255 3839 y Ft(\()p Fq(z)t Ft(\))p Fp(k)e(\024)f Fq(C)2641 3798 y Fl(\000)p Fo(1)2634 3861 y(0)2766 3839 y Ft(for)j(all)f Fq(z)i Fp(2)c Fr(C)3301 3854 y Fo(+)3360 3839 y Ft(.)43 b(Th)m(us)34 b(b)m(y)0 3960 y(the)h(Gron)m(w)m(all)e(inequalit)m(y)h (\(see)h(e.g.)50 b([DG)o(],)35 b(Prop)s(osition)e(A.1.1\))i(w)m(e)g (obtain)f(for)g(all)e Fq(z)k Fp(2)c Fr(C)3529 3975 y Fo(+)3622 3960 y Ft(and)0 4080 y Fq(\017)c(>)g Ft(0,)1586 4200 y Fp(k)p Fq(F)1699 4215 y Fm(\017;\032)1787 4200 y Ft(\()p Fq(z)t Ft(\))p Fp(k)g(\024)g Fq(C)r(;)1287 b Ft(\(7.129\))0 4372 y(where)880 4514 y Fq(C)34 b Ft(:=)28 b(exp)1281 4392 y Fj(\022)1342 4396 y(Z)1425 4423 y Fo(1)1388 4585 y(0)1481 4514 y Fq(a)1532 4529 y Fm(\034)1575 4514 y Fq(d\034)1679 4392 y Fj(\023)17 b(\022)1818 4514 y Fq(C)1895 4472 y Fl(\000)p Fo(1)1888 4538 y(0)2006 4396 y Fj(Z)2089 4423 y Fo(1)2052 4585 y(0)2145 4514 y Fq(a)2196 4529 y Fm(\034)2239 4514 y Ft(d)p Fq(\034)34 b Ft(+)2467 4396 y Fj(Z)2550 4423 y Fo(1)2513 4585 y(0)2606 4514 y Fq(b)2647 4529 y Fm(\034)2691 4514 y Fq(d\034)2795 4392 y Fj(\023)2873 4514 y Fq(:)581 b Ft(\(7.130\))0 4722 y(This)33 b(yields)f(P)m(art)h(\(i\))e(of)h(the)h(lemma.)146 4843 y(T)-8 b(o)35 b(establish)f(P)m(art)i(\(ii\),)d(note)i(that)g (Relations)e(\(7.128\))h(and)g(\(7.129\))g(yield)g(that)h(for)f(an)m(y) h(0)d Fq(<)0 4963 y(\017)39 4978 y Fo(1)106 4963 y Fq(<)c(\017)249 4978 y Fo(2)289 4963 y Ft(,)1004 5090 y Fp(k)p Fq(F)1117 5105 y Fm(\017)1146 5114 y Fk(2)1180 5105 y Fm(;\032)1240 5090 y Ft(\()p Fq(z)t Ft(\))23 b Fp(\000)g Fq(F)1551 5105 y Fm(\017)1580 5114 y Fk(1)1614 5105 y Fm(;\032)1674 5090 y Ft(\()p Fq(z)t Ft(\))p Fp(k)83 b(\024)2038 5019 y Fj(R)2093 5046 y Fm(\017)2122 5055 y Fk(2)2077 5116 y Fm(\017)2106 5125 y Fk(1)2177 4991 y Fj(\015)2177 5040 y(\015)2177 5090 y(\015)2252 5051 y Fo(d)p 2233 5067 79 4 v 2233 5125 a(d)p Fm(\034)2321 5090 y Fq(F)2384 5105 y Fm(\034)t(;\032)2479 5090 y Ft(\()p Fq(z)t Ft(\))2604 4991 y Fj(\015)2604 5040 y(\015)2604 5090 y(\015)17 b Ft(d)p Fq(\034)1932 5282 y Fp(\024)2038 5210 y Fj(R)2093 5237 y Fm(\017)2122 5246 y Fk(2)2077 5307 y Fm(\017)2106 5316 y Fk(1)2160 5282 y Ft(\()p Fq(C)7 b(a)2326 5297 y Fm(\034)2392 5282 y Ft(+)22 b Fq(b)2531 5297 y Fm(\034)2574 5282 y Ft(\)d)p Fq(\034)6 b(:)1841 5753 y Ft(54)p eop %%Page: 55 55 55 54 bop 0 407 a Ft(Th)m(us,)34 b(if)e Fq(\017)403 422 y Fm(n)478 407 y Fp(!)27 b Ft(0)32 b(then)h(the)g(sequence)j Fq(F)1544 422 y Fm(\017)1573 430 y Fg(n)1615 422 y Fm(;\032)1675 407 y Ft(\()p Fq(z)t Ft(\))d(is)f(Cauc)m(h)m(y)j(and)d(this)h(yields)f (the)h(statemen)m(t.)44 b Fe(2)146 572 y Ft(Assume)34 b(no)m(w)f(that)f Fq(\032)p Ft(\(0\))c(=)g(1.)43 b(Clearly)-8 b(,)1209 776 y Fq(F)1272 791 y Fo(0)p Fm(;\032)1367 776 y Ft(\()p Fq(z)t Ft(\))28 b(=)g Fp(h)p Fq(S)6 b Fp(i)1768 734 y Fl(\000)p Fm(\026)1869 776 y Ft(\()p Fq(z)27 b Fp(\000)22 b Fq(H)8 b Ft(\))2205 734 y Fl(\000)p Fo(1)2299 776 y Fp(h)p Fq(S)e Fp(i)2443 734 y Fl(\000)p Fm(\026)2544 776 y Fq(:)0 979 y Ft(Th)m(us,)34 b(to)e(\014nish)h(the)g(pro)s(of)f (of)g(Theorem)h(7.1)f(it)g(su\016ces)i(to)e(sho)m(w)i(that)1454 1182 y(lim)1472 1242 y Fm(\017)p Fl(#)p Fo(0)1606 1182 y Fq(F)1669 1197 y Fm(\017;\032)1758 1182 y Ft(\()p Fq(z)t Ft(\))28 b(=)f Fq(F)2077 1197 y Fo(0)p Fm(;\032)2173 1182 y Ft(\()p Fq(z)t Ft(\))p Fq(:)1156 b Ft(\(7.131\))0 1429 y(\(Note)24 b(that)g(P)m(art)h(\(ii\))d(of)i(Lemma)f(7.11)h (guaran)m(tees)h(the)g(existence)g(of)f(the)h(limit)c(in)i(\(7.131\))g (but)i(sa)m(ys)0 1550 y(nothing)32 b(ab)s(out)h(its)f(v)-5 b(alue\).)45 b(The)34 b(pro)s(of)e(of)g(Relation)f(\(7.131\))i(resolv)m (es)h(the)f(infrared)g(problem)e(w)m(e)0 1670 y(ha)m(v)m(e)j(discussed) g(in)d(the)i(in)m(tro)s(duction.)42 b(In)32 b(the)h(ph)m(ysics)h (language,)d(there)i(is)f(no)g(infrared)g(problem)0 1791 y(as)e(long)e(as)i Fq(\017)e(>)g Ft(0)h({)g(this)h(constan)m(t)g(pla)m (ys)g(a)f(role)g(of)g(a)h(\\complex)f(b)s(oson)g(mass".)43 b(In)29 b(our)h(approac)m(h,)0 1911 y(the)40 b(infrared)f(problem)f (app)s(ears)j(in)d(the)j(limit)36 b Fq(\017)k Fp(#)f Ft(0)h(and)g(is)f(due)h(to)f(the)i(fact)e(that)h(domains)e(of)0 2031 y Fq(H)48 b Ft(and)40 b Fq(H)407 2046 y Fm(\017)480 2031 y Ft(with)g Fq(\017)i(>)e Ft(0)g(are)h(di\013eren)m(t.)67 b(This)41 b(di\016cult)m(y)f(is)g(resolv)m(ed)h(b)s(elo)m(w.)67 b(W)-8 b(e)41 b(remark)f(that)0 2152 y(an)f(argumen)m(t)g(similar)d(to) i(ours)i(has)f(b)s(een)h(used)g(in)f([JP1].)63 b(The)40 b(reader)g(ma)m(y)e(consult)i([JP3])f(for)0 2272 y(additional)30 b(discussion)j(of)f(this)g(p)s(oin)m(t.)146 2392 y(The)26 b(follo)m(wing)c(tec)m(hnical)j(result)g(pla)m(ys)g(a)f(k)m(ey)j(role)d (in)g(the)h(resolution)f(of)g(the)h(infrared)g(problem.)0 2513 y(Recall)31 b(that)h Fq(G)582 2528 y Fo(0)622 2513 y Ft(\()p Fq(z)t Ft(\))c(=)g(\()p Fq(z)e Fp(\000)d Fq(H)8 b Ft(\))1215 2477 y Fl(\000)p Fo(1)1309 2513 y Ft(.)0 2722 y Fr(Lemma)37 b(7.12)49 b Fi(Assume)35 b(that)g(Hyp)-5 b(othesis)35 b Fq(S)6 b Ft(\(1\))35 b Fi(holds)f(and)g(let)i Fq(z)c Fp(2)c Fr(C)2748 2737 y Fo(+)2842 2722 y Fi(b)-5 b(e)35 b(given.)44 b(L)-5 b(et)36 b Fq(\014)d Ft(=)28 b Fp(\006)3704 2683 y Fo(1)p 3704 2699 36 4 v 3704 2756 a(2)3750 2722 y Fi(.)0 2842 y(Then,)0 2963 y Ft(\(i\))34 b Fi(F)-7 b(or)34 b(any)h Fq(\017)28 b(>)f Ft(0)p Fi(,)1006 3153 y Fp(k)p Fq(N)1144 3112 y Fl(\000)p Fm(\014)1247 3153 y Fq(G)1324 3168 y Fm(\017)1356 3153 y Ft(\()p Fq(z)t Ft(\))p Fq(N)1569 3112 y Fm(\014)1618 3153 y Fp(k)g(\024)1869 3086 y Ft(2)p 1810 3130 167 4 v 1810 3222 a(Im)p Fq(z)2003 3007 y Fj( )2069 3153 y Ft(1)22 b(+)2248 3004 y Fp(p)p 2331 3004 49 4 v 82 x Ft(2)o Fp(k)p Fq(\013)q Fp(kj)p Fq(\025)p Fp(j)p 2248 3130 407 4 v 2368 3222 a Ft(Im)o Fq(z)2664 3007 y Fj(!)2746 3153 y Fq(:)708 b Ft(\(7.132\))0 3383 y(\(ii\))26 b Fi(Assume)i(in)f(addition)g(that)h(Hyp)-5 b(othesis)28 b Ft(A)g Fi(is)f(satis\014e)-5 b(d.)42 b(Then,)28 b(\(7.132\))f(is)g(true)i(also)e(for)g Fq(\017)h Ft(=)g(0)p Fi(.)0 3713 y Fr(Pro)s(of.)65 b Ft(Let)33 b Fq(N)621 3728 y Fm(\016)691 3713 y Ft(b)s(e)g(as)g(in)f(Lemma)f(4.2.)43 b(F)-8 b(or)32 b(an)m(y)h Fq(\017)28 b Fp(\025)g Ft(0)k(consider)h(the) g(op)s(erator)1197 3916 y Fq(H)1278 3931 y Fm(\017;\016)o(;\014)1450 3916 y Ft(:=)28 b Fq(H)1662 3931 y Fm(\017)1716 3916 y Ft(+)22 b Fq(\025N)1959 3869 y Fl(\000)p Fm(\014)1949 3941 y(\016)2062 3916 y Fq(V)2119 3931 y Fm(\017)2152 3916 y Fq(N)2240 3869 y Fm(\014)2230 3941 y(\016)2309 3916 y Fp(\000)h Fq(\025V)2523 3931 y Fm(\017)2555 3916 y Fq(:)0 4119 y Ft(It)33 b(follo)m(ws)e(from)g(Lemma)g(4.2)i(that)1343 4323 y Fp(k)p Fq(N)1481 4275 y Fl(\000)p Fm(\014)1471 4348 y(\016)1583 4323 y Fq(V)1640 4338 y Fm(\017)1673 4323 y Fq(N)1761 4275 y Fm(\014)1751 4348 y(\016)1831 4323 y Fp(\000)22 b Fq(V)1987 4338 y Fm(\017)2020 4323 y Fp(k)27 b(\024)i Fq(C)2280 4233 y Fp(p)p 2363 4233 47 4 v 90 x Fq(\016)s(;)0 4526 y Ft(where)35 b(w)m(e)g(use)h(the)e (shorthand)h Fq(C)i Ft(:=)1471 4444 y Fp(p)p 1554 4444 49 4 v 82 x Ft(2)p Fp(k)p Fq(\013)q Fp(k)p Ft(.)47 b(Therefore,)36 b(if)d Fq(\017)d(>)h Ft(0,)j Fq(H)2769 4541 y Fm(\017;\016)o(;\014)2947 4526 y Ft(is)g(a)g(closed)g(op)s(erator)0 4646 y(on)g Fh(D)p Ft(.)48 b(If)34 b Fq(\017)d Ft(=)f(0,)35 b(Hyp)s(othesis)g(A)f (yields)g(that)f Fq(H)1847 4661 y Fm(\017;\016)o(;\014)2026 4646 y Ft(is)h(essen)m(tially)f(self-adjoin)m(t)g(on)h Fh(D)p Ft(.)48 b(Note)34 b(also)0 4767 y(that)e(for)g(an)m(y)i Fq(\017)28 b Fp(\025)g Ft(0,)1286 4887 y Fh(N)p Ft(\()p Fq(H)1474 4902 y Fm(\017)1506 4887 y Ft(\))g Fp(\032)g(f)p Fq(z)k Ft(:)61 b(Im)o Fq(z)33 b Fp(\024)28 b(\000)p Fq(C)2338 4902 y Fo(0)2377 4887 y Fq(\017)p Fp(g)p Fq(;)0 5054 y Ft(\(recall)j(the)i(estimate)f(\(7.120\)\),)f(therefore)1010 5268 y Fh(N)p Ft(\()p Fq(H)1198 5283 y Fm(\017;\016)o(;\014)1343 5268 y Ft(\))c Fp(\032)h(f)p Fq(z)33 b Ft(:)60 b(Im)o Fq(z)33 b Fp(\024)28 b(\000)p Fq(C)2174 5283 y Fo(0)2214 5268 y Fq(\017)22 b Ft(+)g Fq(C)7 b Fp(j)p Fq(\025)p Fp(j)2563 5178 y(p)p 2646 5178 47 4 v 90 x Fq(\016)s Fp(g)p Fq(:)1841 5753 y Ft(55)p eop %%Page: 56 56 56 55 bop 0 407 a Ft(Th)m(us,)34 b(if)e(Im)o Fq(z)h(>)27 b Fp(\000)p Fq(C)808 422 y Fo(0)848 407 y Fq(\017)22 b Ft(+)g Fq(C)7 b Fp(j)p Fq(\025)p Fp(j)1197 322 y(p)p 1280 322 47 4 v 85 x Fq(\016)36 b Ft(then)d(the)g(op)s(erator)f Fq(z)27 b Fp(\000)c Fq(H)2395 422 y Fm(\017;\016)o(;\014)2572 407 y Ft(is)32 b(in)m(v)m(ertible)g(and)933 659 y Fp(k)p Ft(\()p Fq(z)27 b Fp(\000)c Fq(H)1274 674 y Fm(\017;\016)o(;\014)1418 659 y Ft(\))1456 618 y Fl(\000)p Fo(1)1550 659 y Fp(k)28 b(\024)1733 562 y Fj(\020)1783 659 y Ft(Im)o Fq(z)f Ft(+)22 b Fq(C)2139 674 y Fo(0)2178 659 y Fq(\017)h Fp(\000)f Fq(C)7 b Fp(j)p Fq(\025)p Fp(j)2529 569 y(p)p 2612 569 V 90 x Fq(\016)2659 562 y Fj(\021)2708 585 y Fl(\000)p Fo(1)2819 659 y Fq(:)0 879 y Ft(F)-8 b(rom)31 b(no)m(w)i(on)g(w)m(e)g (\014x)h Fq(z)e Fp(2)c Fr(C)1129 894 y Fo(+)1221 879 y Ft(and)k(c)m(ho)s(ose)i Fq(\016)i Ft(suc)m(h)e(that)1568 1099 y Fq(C)7 b Fp(j)p Fq(\025)p Fp(j)1758 1009 y(p)p 1840 1009 V 1840 1099 a Fq(\016)32 b(<)27 b Ft(Im)p Fq(z)t(:)0 1319 y Ft(Let)1422 1439 y Fh(D)1491 1454 y Fo(\014n)1601 1439 y Ft(:=)h Fh(D)22 b Fp(\\)h Ft(\()p Fp(K)g(\012)g Ft(\000)2210 1454 y Fo(\014n)2292 1439 y Ft(\))p Fq(:)0 1613 y Ft(F)-8 b(or)32 b(an)m(y)h(\011)28 b Fp(2)g Fh(D)626 1628 y Fo(\014n)741 1613 y Ft(w)m(e)33 b(ha)m(v)m(e)1389 1734 y Fq(H)1470 1749 y Fm(\017;)p Fo(fr)1572 1734 y Ft(\011)27 b(=)h Fq(N)1867 1687 y Fl(\000)p Fm(\014)1857 1759 y(\016)1969 1734 y Fq(H)2050 1749 y Fm(\017;)p Fo(fr)2151 1734 y Fq(N)2239 1687 y Fm(\014)2229 1759 y(\016)2287 1734 y Ft(\011)p Fq(:)0 1908 y Ft(Similarly)-8 b(,)29 b(for)j(\011)27 b Fp(2)h Fh(D)849 1923 y Fo(\014n)964 1908 y Ft(w)m(e)34 b(ha)m(v)m(e)1032 2128 y(\()p Fq(N)1158 2081 y Fl(\000)p Fm(\014)1148 2153 y(\016)1260 2128 y Fq(V)1317 2143 y Fm(\017)1350 2128 y Fq(N)1438 2081 y Fm(\014)1428 2153 y(\016)1508 2128 y Fp(\000)22 b Fq(V)1664 2143 y Fm(\017)1697 2128 y Ft(\)\011)27 b(=)h Fq(N)2030 2081 y Fl(\000)p Fm(\014)2020 2153 y(\016)2132 2128 y Fq(V)2189 2143 y Fm(\017)2222 2128 y Fq(N)2310 2081 y Fm(\014)2300 2153 y(\016)2358 2128 y Ft(\011)22 b Fp(\000)g Fq(V)2612 2143 y Fm(\017)2645 2128 y Ft(\011)p Fq(:)0 2348 y Ft(Th)m(us,)34 b(on)f Fh(D)479 2363 y Fo(\014n)561 2348 y Ft(,)1325 2469 y Fq(H)1406 2484 y Fm(\017;\016)o(;\014)1633 2469 y Ft(=)28 b Fq(H)1818 2484 y Fm(\017;)p Fo(fr)1941 2469 y Ft(+)22 b Fq(N)2127 2422 y Fl(\000)p Fm(\014)2117 2494 y(\016)2230 2469 y Fq(V)2287 2484 y Fm(\017)2319 2469 y Fq(N)2407 2422 y Fm(\014)2397 2494 y(\016)1633 2670 y Ft(=)28 b Fq(N)1825 2622 y Fl(\000)p Fm(\014)1815 2695 y(\016)1927 2670 y Fq(H)2008 2685 y Fm(\017)2041 2670 y Fq(N)2129 2622 y Fm(\014)2119 2695 y(\016)2177 2670 y Fq(:)3481 2566 y Ft(\(7.133\))0 2837 y(One)34 b(easily)e(sho)m(ws)j(that)f Fh(D)1039 2852 y Fo(\014n)1154 2837 y Ft(is)f(a)g(core)h(for)f Fq(H)1773 2852 y Fm(\017;)p Fo(fr)1874 2837 y Ft(.)46 b(If)33 b Fq(\017)d(>)f Ft(0,)k Fq(V)2385 2852 y Fm(\017)2451 2837 y Ft(is)g(an)g(in\014nitesimal)d(p)s (erturbation)0 2958 y(of)35 b Fq(H)195 2973 y Fm(\017;)p Fo(fr)296 2958 y Ft(,)h(and)f(therefore)h Fh(D)1032 2973 y Fo(\014n)1150 2958 y Ft(is)e(also)h(a)f(core)i(for)f Fq(H)1974 2973 y Fm(\017)2041 2958 y Ft(and)h Fq(H)2315 2973 y Fm(\017;\016)o(;\014)2459 2958 y Ft(.)51 b(It)36 b(follo)m(ws)e(from)f(Hyp)s(othesis)j(A)0 3078 y(that)c Fh(D)280 3093 y Fo(\014n)395 3078 y Ft(is)h(a)f(core)h(of)f Fq(H)973 3093 y Fo(0)1012 3078 y Ft(.)44 b(Therefore,)33 b(for)f Fq(\017)c Fp(\025)h Ft(0,)1419 3283 y(~)1409 3298 y Fh(D)1478 3313 y Fo(\014n)1588 3298 y Ft(:=)f(\()p Fq(z)f Fp(\000)22 b Fq(H)2009 3313 y Fm(\017;\016)o(;\014)2154 3298 y Ft(\))p Fh(D)2261 3313 y Fo(\014n)2343 3298 y Fq(;)1111 b Ft(\(7.134\))0 3518 y(is)32 b(dense)i(in)e Fp(H)i Ft(and)e(on)931 3503 y(~)921 3518 y Fh(D)990 3533 y Fo(\014n)1072 3518 y Ft(,)1136 3738 y(\()p Fq(z)27 b Fp(\000)c Fq(H)1427 3753 y Fm(\017;\016)o(;\014)1571 3738 y Ft(\))1609 3697 y Fl(\000)p Fo(1)1731 3738 y Ft(=)28 b Fq(N)1923 3691 y Fl(\000)p Fm(\014)1913 3763 y(\016)2025 3738 y Ft(\()p Fq(z)f Fp(\000)c Fq(H)2316 3753 y Fm(\017)2348 3738 y Ft(\))2386 3697 y Fl(\000)p Fo(1)2480 3738 y Fq(N)2568 3691 y Fm(\014)2558 3763 y(\016)2616 3738 y Fq(:)0 3958 y Ft(Next,)34 b(w)m(e)f(note)g(that)844 4235 y Fp(k)p Fq(N)982 4187 y Fm(\014)972 4260 y(\016)1029 4235 y Fq(N)1117 4194 y Fl(\000)p Fm(\014)1220 4235 y Fp(k)28 b Ft(=)f Fp(k)p Fq(N)1539 4194 y Fl(\000)p Fm(\014)1641 4235 y Fq(N)1729 4187 y Fm(\014)1719 4260 y(\016)1777 4235 y Fp(k)h(\024)1960 4089 y Fj(\()2068 4174 y Ft(\(1)22 b(+)g Fq(\016)t Ft(\))2360 4138 y Fm(\014)2407 4174 y Fq(;)181 b(\014)33 b(>)28 b Ft(0)o Fq(;)2068 4294 y(\016)2115 4258 y Fm(\014)2162 4294 y Fq(;)426 b(\014)33 b Fp(\024)28 b Ft(0.)0 4523 y(Therefore,)34 b(for)e(an)m(y)h Fq(\017)28 b Fp(\025)g Ft(0)33 b(and)f(\011)c Fp(2)1453 4508 y Ft(~)1443 4523 y Fh(D)1512 4538 y Fo(\014n)1595 4523 y Ft(,)465 4823 y Fp(k)p Fq(N)603 4782 y Fl(\000)p Fm(\014)706 4823 y Ft(\()p Fq(z)f Fp(\000)22 b Fq(H)996 4838 y Fm(\017)1029 4823 y Ft(\))1067 4782 y Fl(\000)p Fo(1)1161 4823 y Fq(N)1249 4782 y Fm(\014)1297 4823 y Ft(\011)p Fp(k)27 b(\024)1555 4677 y Fj( )1631 4756 y Ft(1)22 b(+)g Fq(\016)p 1631 4800 216 4 v 1715 4891 a(\016)1857 4677 y Fj(!)1922 4700 y Fl(j)p Fm(\014)s Fl(j)2025 4727 y Fj(\020)2075 4823 y Ft(Im)p Fq(z)k Ft(+)c Fq(C)2431 4838 y Fo(0)2471 4823 y Fq(\017)g Fp(\000)h Fq(C)7 b Fp(j)p Fq(\025)p Fp(j)2822 4734 y(p)p 2904 4734 47 4 v 2904 4823 a Fq(\016)2951 4727 y Fj(\021)3001 4750 y Fl(\000)p Fo(1)3111 4823 y Fp(k)p Ft(\011)p Fp(k)p Fq(:)0 5147 y Ft(T)-8 b(aking)32 b Fq(\016)g Ft(=)503 5051 y Fj(\020)588 5108 y Fo(Im)o Fm(z)p 563 5124 171 4 v 563 5181 a Fo(2)p Fm(C)5 b Fl(j)p Fm(\025)p Fl(j)743 5051 y Fj(\021)793 5074 y Fo(2)849 5147 y Fq(;)33 b Ft(w)m(e)g(deriv)m(e)h(the)f(statemen)m(ts)g(of)f(the) h(lemma.)41 b Fe(2)1841 5753 y Ft(56)p eop %%Page: 57 57 57 56 bop 0 407 a Fr(Lemma)37 b(7.13)49 b Fi(Assume)35 b(that)g(Hyp)-5 b(otheses)35 b Ft(A)g Fi(and)f Fq(S)6 b Ft(\(1\))35 b Fi(hold.)44 b(Then)1006 640 y Ft(lim)1024 700 y Fm(\017)p Fl(#)p Fo(0)1158 640 y Fq(N)1246 599 y Fl(\000)1311 572 y Fk(1)p 1311 584 31 4 v 1311 625 a(2)1356 640 y Fq(G)1433 655 y Fm(\017)1466 640 y Ft(\()p Fq(z)t Ft(\))p Fq(N)1679 599 y Fl(\000)1744 572 y Fk(1)p 1745 584 V 1745 625 a(2)1817 640 y Ft(=)27 b Fq(N)2008 599 y Fl(\000)2073 572 y Fk(1)p 2074 584 V 2074 625 a(2)2118 640 y Ft(\()p Fq(z)g Fp(\000)c Fq(H)8 b Ft(\))2455 599 y Fl(\000)p Fo(1)2549 640 y Fq(N)2637 599 y Fl(\000)2702 572 y Fk(1)p 2702 584 V 2702 625 a(2)2746 640 y Fq(:)708 b Ft(\(7.135\))0 1088 y Fr(Pro)s(of.)65 b Ft(Let)33 b Fq(\017)28 b(>)f Ft(0.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)652 1308 y Fq(N)740 1267 y Fl(\000)805 1240 y Fk(1)p 806 1252 V 806 1293 a(2)850 1308 y Ft(\()p Fq(G)965 1323 y Fm(\017)998 1308 y Ft(\()p Fq(z)t Ft(\))23 b Fp(\000)f Fq(G)1322 1323 y Fo(0)1362 1308 y Ft(\()p Fq(z)t Ft(\)\))p Fq(N)1613 1267 y Fl(\000)1678 1240 y Fk(1)p 1679 1252 V 1679 1293 a(2)1751 1308 y Ft(=)27 b Fq(N)1942 1267 y Fl(\000)2007 1240 y Fk(1)p 2008 1252 V 2008 1293 a(2)2052 1308 y Fq(G)2129 1323 y Fo(0)2169 1308 y Ft(\()p Fq(z)t Ft(\)\()p Fq(H)2413 1323 y Fm(\017)2468 1308 y Fp(\000)c Fq(H)8 b Ft(\))p Fq(G)2772 1323 y Fm(\017)2804 1308 y Ft(\()p Fq(z)t Ft(\))p Fq(N)3017 1267 y Fl(\000)3082 1240 y Fk(1)p 3083 1252 V 3083 1293 a(2)3481 1308 y Ft(\(7.136\))637 1542 y(=)28 b Fq(N)829 1500 y Fl(\000)894 1473 y Fk(1)p 894 1485 V 894 1526 a(2)939 1542 y Fq(G)1016 1557 y Fo(0)1055 1542 y Ft(\()p Fq(z)t Ft(\))p Fq(N)1279 1473 y Fk(1)p 1279 1485 V 1279 1526 a(2)1340 1445 y Fj(\020)1390 1542 y Fp(\000)p Ft(i)p Fq(\017)22 b Ft(+)g Fq(\025N)1799 1500 y Fl(\000)1864 1473 y Fk(1)p 1864 1485 V 1864 1526 a(2)1909 1542 y Ft(\()p Fq(V)2004 1557 y Fm(\017)2059 1542 y Fp(\000)g Fq(V)g Ft(\))p Fq(N)2363 1500 y Fl(\000)2428 1473 y Fk(1)p 2428 1485 V 2428 1526 a(2)2473 1445 y Fj(\021)2539 1542 y Fq(N)2637 1473 y Fk(1)p 2637 1485 V 2637 1526 a(2)2682 1542 y Fq(G)2759 1557 y Fm(\017)2792 1542 y Ft(\()p Fq(z)t Ft(\))p Fq(N)3005 1500 y Fl(\000)3070 1473 y Fk(1)p 3071 1485 V 3071 1526 a(2)3115 1542 y Fq(:)339 b Ft(\(7.137\))0 1716 y(This)33 b(iden)m(tit)m(y)f(and)h(Lemma)e(7.12)h (yield)g(that)306 1993 y Fp(k)p Fq(N)444 1952 y Fl(\000)509 1924 y Fk(1)p 509 1936 V 509 1978 a(2)554 1993 y Ft(\()p Fq(G)669 2008 y Fm(\017)701 1993 y Ft(\()p Fq(z)t Ft(\))23 b Fp(\000)g Fq(G)1026 2008 y Fo(0)1065 1993 y Ft(\()p Fq(z)t Ft(\)\))p Fq(N)1316 1952 y Fl(\000)1381 1924 y Fk(1)p 1382 1936 V 1382 1978 a(2)1427 1993 y Fp(k)k(\024)h Fq(C)1686 1952 y Fo(2)1679 2017 y Fm(z)1726 1993 y Fq(\017)1782 1847 y Fj( )1847 1993 y Ft(1)22 b(+)g(2)p Fp(j)p Fq(\025)p Fp(jk)p Fq(\013)q Fp(k)17 b Ft(sup)2360 2067 y Fm(t)2385 2076 y Fk(1)2420 2067 y Fm(;t)2465 2076 y Fk(2)2520 1993 y Fp(j)p Fq(t)2583 2008 y Fo(1)2622 1993 y Fq(\030)2670 1952 y Fl(0)2693 1993 y Ft(\()p Fq(t)2766 2008 y Fo(1)2828 1993 y Ft(+)22 b Fq(t)2961 2008 y Fo(2)3000 1993 y Ft(\))p Fp(j)3066 1847 y Fj(!)3148 1993 y Fq(;)306 b Ft(\(7.138\))0 2275 y(where)37 b Fq(C)355 2290 y Fm(z)431 2275 y Ft(is)e(the)h (constan)m(t)h(on)f(the)g(righ)m(t-hand)f(side)h(in)g(\(7.132\).)52 b(Clearly)-8 b(,)36 b(this)g(estimate)f(yields)0 2395 y(\(7.135\).)43 b Fe(2)146 2566 y Ft(W)-8 b(e)33 b(are)g(no)m(w)g (ready)g(to)g(\014nish)0 2686 y Fr(Pro)s(of)g(of)h(Theorem)g(7.1.)43 b Ft(As)29 b(w)m(e)i(ha)m(v)m(e)g(already)e(remark)m(ed,)h(it)f(follo)m (ws)f(from)g(Lemma)g(7.11)h(that)0 2806 y(to)j(pro)m(v)m(e)i(Theorem)f (7.1)f(it)g(su\016ces)i(to)f(sho)m(w)g(that)g(for)f(an)m(y)h Fq(z)f Fp(2)c Fr(C)2538 2821 y Fo(+)2597 2806 y Ft(,)1454 3026 y(lim)1472 3086 y Fm(\017)p Fl(#)p Fo(0)1606 3026 y Fq(F)1669 3041 y Fm(\017;\032)1758 3026 y Ft(\()p Fq(z)t Ft(\))g(=)f Fq(F)2077 3041 y Fo(0)p Fm(;\032)2173 3026 y Ft(\()p Fq(z)t Ft(\))p Fq(:)1156 b Ft(\(7.139\))0 3285 y(Since)33 b(w)m(e)g(kno)m(w)h(that)e(the)h(limit)c(on)k(the)g(righ)m (t)f(hand)h(side)f(exists,)i(it)d(su\016ces)k(to)d(sho)m(w)i(that)1371 3505 y(w)q Fp(\000)18 b Ft(lim)1555 3565 y Fm(\017)p Fl(#)p Fo(0)1689 3505 y Fq(F)1752 3520 y Fm(\017;\032)1841 3505 y Ft(\()p Fq(z)t Ft(\))28 b(=)f Fq(F)2160 3520 y Fo(0)p Fm(;\032)2256 3505 y Ft(\()p Fq(z)t Ft(\))p Fq(:)1073 b Ft(\(7.140\))0 3768 y(This)33 b(relation)d(follo)m(ws)i(from)f (\(7.135\).)43 b Fe(2)0 4107 y Fn(7.2)135 b(H\177)-67 b(older)46 b(con)l(tin)l(uit)l(y)0 4292 y Ft(This)33 b(section)f(is)h(dev)m(oted)h(to)e(the)h(pro)s(of)f(of)g(the)h(follo)m (wing)d(theorem:)0 4495 y Fr(Theorem)74 b(7.14)49 b Fi(Assume)e(that)g (Hyp)-5 b(otheses)47 b Ft(A)g Fi(and)f Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))47 b Fi(hold)f(with)h Fq(\027)56 b(>)50 b Ft(1)p Fi(.)80 b(L)-5 b(et)47 b Fq(n)j Fp(2)g Fr(N)p Fi(,)0 4616 y Ft(0)34 b Fq(<)g(\022)k Fp(\024)c Ft(1)p Fi(,)39 b(and)f Fq(\026)c Fp(\025)913 4576 y Fo(1)p 913 4592 36 4 v 913 4650 a(2)996 4616 y Fi(b)-5 b(e)38 b(such)h(that)g Fq(\027)h Fp(\025)35 b Fq(\026)24 b Ft(+)1942 4576 y Fo(1)p 1942 4592 V 1942 4650 a(2)2022 4616 y Ft(=)34 b(1)24 b(+)h Fq(n)g Ft(+)f Fq(\022)s Fi(.)56 b(L)-5 b(et)39 b Ft(0)34 b Fq(<)g Ft(\003)3056 4631 y Fo(1)3129 4616 y Fq(<)g Ft(\()3277 4534 y Fp(p)p 3360 4534 49 4 v 82 x Ft(2)p Fp(k)p Fq(s\013)q Fp(k)p Ft(\))3656 4579 y Fl(\000)p Fo(1)3750 4616 y Fi(.)0 4736 y(Then,)g(for)h Fp(j)p Fq(\025)p Fp(j)27 b(\024)h Ft(\003)753 4751 y Fo(1)792 4736 y Fi(,)35 b(the)f(function)1007 4956 y Fr(C)1088 4971 y Fo(+)1175 4956 y Fp(3)28 b Fq(z)33 b Fp(7!)27 b(h)p Fq(S)p 1579 4876 39 4 v 1579 4915 a Fo(v)p 1617 4876 V 1 w(v)1659 4956 y Fp(i)1698 4915 y Fl(\000)p Fm(\026)1799 4956 y Ft(\()p Fq(z)t Fr(1)p 1942 4876 V -41 x Fo(v)p 1981 4876 V 2 w(v)2046 4956 y Fp(\000)22 b Fq(H)p 2234 4876 V 2234 4915 a Fo(v)p 2272 4876 V 1 w(v)2314 4956 y Ft(\))2352 4915 y Fl(\000)p Fo(1)2447 4956 y Fp(h)p Fq(S)p 2552 4876 V 2552 4915 a Fo(v)p 2590 4876 V 1 w(v)2632 4956 y Fp(i)2671 4915 y Fl(\000)p Fm(\026)3481 4956 y Ft(\(7.141\))0 5176 y Fi(extends)34 b(by)h(c)-5 b(ontinuity)36 b(to)p 1049 5098 81 4 v 35 w Fr(C)1130 5191 y Fo(+)1223 5176 y Fi(and)f(is)f(of)h(the)g(class)f Fq(C)2105 5140 y Fm(n;\022)2098 5201 y Fo(u)2206 5176 y Ft(\()p 2244 5098 V Fr(C)2325 5191 y Fo(+)2384 5176 y Ft(\))h Fi(uniformly)g(in)f Fq(\025)p Fi(.)1841 5753 y Ft(57)p eop %%Page: 58 58 58 57 bop 146 407 a Ft(The)39 b(rest)f(of)f(this)h(section)g(is)f(dev)m (oted)i(to)e(the)h(pro)s(of)f(of)g(Theorem)h(7.14.)58 b(W)-8 b(e)39 b(will)c(freely)j(use)0 527 y(the)f(results)f(and)h (notation)d(of)i(Section)g(7.1.)54 b(In)37 b(particular,)e(w)m(e)i (drop)g(sup)s(erscripts)p 3268 474 53 4 v 37 w(v)p 3321 474 V 2 w(v)h(un)m(til)d(the)0 648 y(end)e(of)g(this)f(section.)44 b(W)-8 b(e)33 b(\014x)h(\003)1225 663 y Fo(1)1292 648 y Fq(>)27 b Ft(0,)33 b Fq(C)1574 663 y Fo(0)1641 648 y Fq(>)28 b Ft(0)k(and)h Fq(\020)40 b Ft(suc)m(h)34 b(that)f(Relations) e(\(7.119\))h(and)g(\(7.120\))0 768 y(hold.)43 b(All)31 b(the)i(results)g(in)e(the)i(sequel)h(will)c(hold)i(for)g(real)g Fq(\025)g Ft(suc)m(h)i(that)e Fp(j)p Fq(\025)p Fp(j)27 b(\024)h Ft(\003)3038 783 y Fo(1)3078 768 y Ft(,)k(uniformly)f(in)g Fq(\025)p Ft(.)146 888 y(F)-8 b(or)32 b(an)m(y)h Fq(\017)28 b Fp(\025)h Ft(0,)j(the)h(function)1525 1009 y Fr(C)1606 1024 y Fo(+)1693 1009 y Fp(3)28 b Fq(z)33 b Fp(7!)27 b Fq(G)2069 1024 y Fm(\017)2101 1009 y Ft(\()p Fq(z)t Ft(\))p Fq(;)0 1182 y Ft(is)32 b(analytic)f(and)1345 1302 y Fq(@)1401 1261 y Fm(l)1396 1327 y(z)1436 1302 y Fq(G)1513 1317 y Fm(\017)1546 1302 y Ft(\()p Fq(z)t Ft(\))d(=)f(\()p Fp(\000)p Ft(1\))2004 1261 y Fm(l)2031 1302 y Fq(l)r Ft(!)p Fq(G)2166 1261 y Fm(l)q Fo(+1)2166 1327 y Fm(\017)2282 1302 y Ft(\()p Fq(z)t Ft(\))p Fq(:)0 1476 y Ft(It)33 b(follo)m(ws)e(from)g(Lemma)g(7.4)i(that)f(for)g(an)m (y)h Fq(l)i Ft(and)e Fq(m)g Ft(the)g(mixed)e(deriv)-5 b(ativ)m(es)1665 1693 y Fq(@)1721 1652 y Fm(l)1716 1718 y(z)1756 1693 y Fq(@)1812 1652 y Fm(m)1807 1718 y(\017)1880 1693 y Fq(G)1957 1708 y Fm(\017)1989 1693 y Ft(\()p Fq(z)t Ft(\))0 1910 y(exists)33 b(on)g Fr(C)486 1925 y Fo(+)567 1910 y Fp(\002)23 b Fr(R)751 1925 y Fo(+)809 1910 y Ft(,)33 b(and)g(that)f(they)h(are)g(linear)e(com)m(binations)g(of)h(the)h (terms)989 2128 y Fq(G)1066 2087 y Fm(l)1087 2096 y Fk(1)1066 2152 y Fm(\017)1126 2128 y Ft(\()p Fq(z)t Ft(\))p Fq(H)1340 2087 y Fo(\()p Fm(m)1429 2096 y Fk(1)1465 2087 y Fo(\))1332 2152 y Fm(\017)1496 2128 y Fq(G)1573 2087 y Fm(l)1594 2096 y Fk(2)1573 2152 y Fm(\017)1633 2128 y Ft(\()p Fq(z)t Ft(\))17 b Fq(:)g(:)g(:)f(G)1983 2087 y Fm(l)2004 2099 y Fg(k)1983 2152 y Fm(\017)2047 2128 y Ft(\()p Fq(z)t Ft(\))p Fq(H)2261 2087 y Fo(\()p Fm(m)2350 2099 y Fg(k)2389 2087 y Fo(\))2253 2152 y Fm(\017)2420 2128 y Fq(G)2497 2087 y Fm(l)2518 2099 y Fg(k)q Fk(+1)2497 2152 y Fm(\017)2638 2128 y Ft(\()p Fq(z)t Ft(\))p Fq(;)691 b Ft(\(7.142\))0 2345 y(where)34 b Fq(l)311 2360 y Fm(k)354 2345 y Ft('s)f(and)f Fq(m)726 2360 y Fm(k)769 2345 y Ft('s)h(are)g(p)s(ositiv)m(e)f(in)m (tegers)h(suc)m(h)h(that)1202 2524 y Fm(k)1161 2549 y Fj(X)1160 2731 y Fm(j)t Fo(=1)1299 2632 y Fq(m)1384 2647 y Fm(j)1448 2632 y Ft(=)28 b Fq(m;)1876 2524 y Fm(k)r Fo(+1)1881 2549 y Fj(X)1879 2731 y Fm(j)t Fo(=1)2021 2632 y Fq(l)2050 2647 y Fm(j)2115 2632 y Ft(=)f Fq(l)e Ft(+)d Fq(k)j Ft(+)d(1)p Fq(:)861 b Ft(\(7.143\))0 2927 y(W)-8 b(e)33 b(pro)s(ceed)g(to)g(study)g(in)f(some)h(detail)e(these)i (mixed)f(deriv)-5 b(ativ)m(es.)0 3153 y Fr(Lemma)37 b(7.15)49 b Fi(Assume)35 b(that)g(Hyp)-5 b(othesis)35 b Fq(S)6 b 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3535 y Fk(1)p 1366 3547 V 1366 3588 a(2)1427 3502 y Fj(\020)1476 3598 y Fq(@)1532 3562 y Fm(l)1527 3623 y(z)1568 3598 y Fq(@)1624 3562 y Fm(m)1619 3623 y(\017)1691 3598 y Fq(G)1768 3613 y Fm(\017)1801 3598 y Ft(\()p Fq(z)t Ft(\))1926 3502 y Fj(\021)1992 3598 y Fp(h)p Fq(S)f Fp(i)2136 3562 y Fl(\000)p Fm(\026)2136 3623 y(\017;\032)2237 3499 y Fj(\015)2237 3549 y(\015)2237 3598 y(\015)83 b Fp(\024)29 b Fq(C)7 b(\017)2588 3562 y Fl(\000)2653 3535 y Fk(1)p 2653 3547 V 2653 3588 a(2)2693 3562 y Fl(\000)p Fm(l)q Fl(\000)p Fm(m)2891 3598 y Fq(:)3481 3495 y Ft(\(7.144\))0 3979 y Fr(Pro)s(of.)63 b Ft(W)-8 b(e)33 b(will)c(pro)m(v)m(e)k(the)g(\014rst)f(relation.)41 b(A)32 b(similar)d(argumen)m(t)i(yields)h(the)g(second.)45 b(W)-8 b(e)32 b(write)0 4099 y Fq(@)56 4063 y Fm(l)51 4124 y(z)91 4099 y Fq(@)147 4063 y Fm(m)142 4124 y(\017)215 4099 y Fq(G)292 4114 y Fm(\017)324 4099 y Ft(\()p Fq(z)t Ft(\))j(as)f(a)f(linear)f(com)m(binations)h(of)g(the)h(terms)g (\(7.142\).)46 b(After)34 b(inserting)f Fp(h)p Fq(S)6 b Fp(i)3284 4063 y Fl(\000)p Fm(\026)3284 4124 y(\017;\032)3418 4099 y Ft(and)34 b Fq(N)3708 4035 y Fk(1)p 3708 4047 V 3708 4089 a(2)3752 4099 y Ft(,)0 4220 y(w)m(e)g(estimate)d(the)i (norm)f(of)g(eac)m(h)i(term)e(b)m(y)367 4437 y Fp(kh)p Fq(S)6 b Fp(i)561 4396 y Fl(\000)p Fm(\026)561 4462 y(\017;\032)662 4437 y Fq(G)739 4452 y Fm(\017)771 4437 y Ft(\()p Fq(z)t Ft(\))p Fq(N)995 4369 y Fk(1)p 995 4381 V 995 4422 a(2)1040 4437 y Fp(kk)p Fq(N)1238 4369 y Fk(1)p 1238 4381 V 1238 4422 a(2)1283 4437 y Fq(G)1360 4452 y Fm(\017)1392 4437 y Ft(\()p Fq(z)t Ft(\))p Fq(N)1616 4369 y Fk(1)p 1616 4381 V 1616 4422 a(2)1661 4437 y Fp(k)1711 4396 y Fm(l)1732 4405 y Fk(1)1766 4396 y Fl(\000)p Fo(1)1861 4437 y Fp(k)p Fq(N)1999 4396 y Fl(\000)2064 4369 y Fk(1)p 2064 4381 V 2064 4422 a(2)2108 4437 y Fq(H)2197 4396 y Fo(\()p Fm(m)2286 4405 y Fk(1)2321 4396 y Fo(\))2189 4462 y Fm(\017)2353 4437 y Fq(N)2441 4396 y Fl(\000)2506 4369 y Fk(1)p 2506 4381 V 2506 4422 a(2)2551 4437 y Fp(kk)p Fq(N)2749 4369 y Fk(1)p 2749 4381 V 2749 4422 a(2)2794 4437 y Fq(G)2871 4452 y Fm(\017)2903 4437 y Ft(\()p Fq(z)t Ft(\))p Fq(N)3127 4369 y Fk(1)p 3127 4381 V 3127 4422 a(2)3172 4437 y Fp(k)3222 4396 y Fm(l)3243 4405 y Fk(2)3298 4437 y Fq(:)17 b(:)g(:)716 4680 y(:)g(:)g(:)f Fp(k)p Fq(N)995 4612 y Fk(1)p 995 4624 V 995 4665 a(2)1040 4680 y Fq(G)1117 4695 y Fm(\017)1150 4680 y Ft(\()p Fq(z)t Ft(\))p Fq(N)1374 4612 y Fk(1)p 1374 4624 V 1374 4665 a(2)1418 4680 y Fp(k)1468 4639 y Fm(l)1489 4651 y Fg(k)1531 4680 y Fp(k)p Fq(N)1669 4639 y Fl(\000)1734 4612 y Fk(1)p 1735 4624 V 1735 4665 a(2)1779 4680 y Fq(H)1868 4639 y Fo(\()p Fm(m)1957 4651 y Fg(k)1996 4639 y Fo(\))1860 4705 y Fm(\017)2027 4680 y Fq(N)2115 4639 y Fl(\000)2180 4612 y Fk(1)p 2181 4624 V 2181 4665 a(2)2225 4680 y Fp(kk)p Fq(N)2423 4612 y Fk(1)p 2423 4624 V 2423 4665 a(2)2468 4680 y Fq(G)2545 4695 y Fm(\017)2578 4680 y Ft(\()p Fq(z)t Ft(\))p Fq(N)2802 4612 y Fk(1)p 2802 4624 V 2802 4665 a(2)2846 4680 y Fp(k)2896 4639 y Fm(l)2917 4651 y Fg(k)q Fk(+1)3036 4680 y Fq(:)418 b Ft(\(7.145\))0 4854 y(It)33 b(follo)m(ws)e(from)g(Lemmas)h(7.9)g(and) h(7.11)f(that)793 5079 y Fp(kh)p Fq(S)6 b Fp(i)987 5042 y Fl(\000)p Fm(\026)987 5103 y(\017;\032)1087 5079 y Fq(G)1164 5094 y Fm(\017)1197 5079 y Ft(\()p Fq(z)t Ft(\))p Fq(N)1421 5015 y Fk(1)p 1421 5027 V 1421 5068 a(2)1465 5079 y Fp(k)83 b(\024)28 b Fq(C)1780 5018 y Fl(\000)1845 4991 y Fk(1)p 1845 5003 V 1845 5044 a(2)1773 5100 y Fo(0)1890 5079 y Fq(\017)1929 5042 y Fl(\000)1994 5015 y Fk(1)p 1994 5027 V 1994 5068 a(2)2039 5079 y Fp(k)p Fq(F)2152 5094 y Fm(\017;\032)2240 5079 y Ft(\()p Fq(z)t Ft(\))p Fp(k)g(\024)g Fq(C)2635 4991 y Fk(1)p 2635 5003 V 2635 5044 a(2)2618 5100 y Fo(0)2680 5079 y Fq(\017)2719 5042 y Fl(\000)2784 5015 y Fk(1)p 2784 5027 V 2784 5068 a(2)2828 5079 y Fq(C)2915 5015 y Fk(1)p 2915 5027 V 2915 5068 a(2)2960 5079 y Fq(;)894 5282 y Fp(k)p Fq(N)1043 5218 y Fk(1)p 1043 5230 V 1043 5271 a(2)1087 5282 y Fq(G)1164 5297 y Fm(\017)1197 5282 y Ft(\()p Fq(z)t Ft(\))p Fq(N)1421 5218 y Fk(1)p 1421 5230 V 1421 5271 a(2)1465 5282 y Fp(k)83 b(\024)28 b Fq(C)1780 5240 y Fl(\000)p Fo(1)1773 5303 y(0)1875 5282 y Fq(\017)1914 5245 y Fl(\000)p Fo(1)2008 5282 y Fq(:)3481 5163 y Ft(\(7.146\))1841 5753 y(58)p eop %%Page: 59 59 59 58 bop 0 407 a Ft(F)-8 b(urthermore,)32 b(for)g Fq(m)826 422 y Fm(j)891 407 y Fp(\025)c Ft(1,)1256 627 y Fp(k)p Fq(N)1394 586 y Fl(\000)1459 559 y Fk(1)p 1459 571 31 4 v 1459 612 a(2)1504 627 y Fq(H)1593 586 y Fo(\()p Fm(m)1682 596 y Fg(j)1715 586 y Fo(\))1585 652 y Fm(\017)1746 627 y Fq(N)1834 586 y Fl(\000)1899 559 y Fk(1)p 1900 571 V 1900 612 a(2)1944 627 y Fp(k)g(\024)g Fq(c)2169 642 y Fm(m)2231 652 y Fg(j)2268 627 y Fq(\017)2307 586 y Fo(1)p Fl(\000)p Fm(m)2459 596 y Fg(j)2496 627 y Fq(;)958 b Ft(\(7.147\))0 832 y(where)1020 952 y Fq(c)1062 967 y Fm(m)1124 977 y Fg(j)1189 952 y Ft(=)28 b Fq(\016)1336 967 y Fo(1)p Fm(m)1433 977 y Fg(j)1492 952 y Ft(+)22 b(2)p Fp(j)p Fq(\025)p Fp(j)17 b Ft(sup)1829 1026 y Fm(t)1931 952 y Fp(j)p Fq(t)1994 911 y Fm(m)2056 921 y Fg(j)2089 911 y Fl(\000)p Fo(1)2184 952 y Fq(\030)2232 911 y Fo(\()p Fm(m)2321 921 y Fg(j)2353 911 y Fo(\))2385 952 y Ft(\()p Fq(t)p Ft(\))p Fp(jk)p Fq(s\013)q Fp(k)p Fq(:)721 b Ft(\(7.148\))0 1165 y(Com)m(bining)31 b(these)j(estimates)e(and)h(using)f(\(7.143\))f (w)m(e)j(b)s(ound)f(\(7.145\))e(with)1348 1401 y Fq(\017)1387 1360 y Fl(\000)1452 1333 y Fk(1)p 1452 1345 V 1452 1386 a(2)1493 1360 y Fl(\000)p Fm(l)q Fl(\000)p Fm(m)1691 1401 y Fq(C)1778 1314 y Fk(1)p 1778 1326 V 1778 1367 a(2)1818 1341 y Fl(\000)p Fm(l)q Fl(\000)p Fm(k)1761 1423 y Fo(0)1993 1401 y Fq(C)2079 1333 y Fk(1)p 2079 1345 V 2079 1386 a(2)2141 1318 y Fj(Y)2263 1401 y Fq(c)2305 1416 y Fm(m)2367 1426 y Fg(j)2404 1401 y Fq(;)0 1606 y Ft(Summing)g(o)m(v)m(er)i(the)g(terms)g(\(7.142\))e(w)m(e)j(deriv)m (e)f(the)g(Relation)e(\(7.144\).)42 b Fe(2)0 1861 y Fr(Lemma)37 b(7.16)49 b Fi(Assume)42 b(that)g(Hyp)-5 b(othesis)42 b Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))41 b Fi(holds)g(with)h Fq(\027)47 b(>)40 b Ft(1)h Fi(and)h(let)f Fq(\026)f(>)3294 1822 y Fo(1)p 3294 1838 36 4 v 3294 1896 a(2)3339 1861 y Fi(.)65 b(L)-5 b(et)42 b Fq(k)i Ft(=)0 1982 y Fq(k)51 1997 y Fo(1)112 1982 y Ft(+)22 b Fq(k)261 1997 y Fo(2)300 1982 y Fi(.)45 b(Then)363 2186 y Ft(sup)340 2265 y Fm(z)s Fl(2)p Fd(C)482 2274 y Fk(+)549 2086 y Fj(\015)549 2136 y(\015)549 2186 y(\015)p Fp(h)p Fq(S)6 b Fp(i)739 2145 y Fl(\000)p Fm(\026)739 2211 y(\017;\032)824 2220 y Fk(1)861 2186 y Fq(S)927 2145 y Fm(k)964 2154 y Fk(1)1019 2090 y Fj(\020)1069 2186 y Fq(@)1125 2145 y Fm(l)1120 2211 y(z)1160 2186 y Fq(@)1216 2145 y Fm(m)1211 2211 y(\017)1283 2186 y Fq(G)1360 2201 y Fm(\017)1393 2186 y Ft(\()p Fq(z)t Ft(\))p Fq(K)1601 2201 y Fm(\017)1634 2186 y Fq(G)1711 2201 y Fm(\017)1744 2186 y Ft(\()p Fq(z)t Ft(\))1869 2090 y Fj(\021)1936 2186 y Fq(S)2002 2145 y Fm(k)2039 2154 y Fk(2)2077 2186 y Fp(h)p Fq(S)g Fp(i)2221 2145 y Fl(\000)p Fm(\026)2221 2211 y(\017;\032)2306 2220 y Fk(2)2343 2086 y Fj(\015)2343 2136 y(\015)2343 2186 y(\015)28 b Fp(\024)g Fq(C)7 b(\017)2638 2145 y Fl(\000)p Fm(l)q Fl(\000)p Fm(m)p Fl(\000)p Fm(k)r Fl(\000)p Fo(2+)p Fm(\027)3114 2186 y Fq(:)340 b Ft(\(7.149\))0 2601 y Fr(Pro)s(of.)65 b Ft(Note)33 b(that)1489 2721 y Fq(@)1545 2680 y Fm(l)1540 2746 y(z)1581 2721 y Fq(@)1637 2680 y Fm(m)1632 2746 y(\017)1704 2721 y Fq(G)1781 2736 y Fm(\017)1814 2721 y Ft(\()p Fq(z)t Ft(\))p Fq(K)2022 2736 y Fm(\017)2055 2721 y Fq(G)2132 2736 y Fm(\017)2165 2721 y Ft(\()p Fq(z)t Ft(\))0 2889 y(is)f(a)g(linear)f(com)m(bination)g(of)h(terms)1145 3093 y(\()p Fq(@)1239 3052 y Fm(l)1260 3061 y Fk(1)1234 3118 y Fm(z)1299 3093 y Fq(@)1355 3052 y Fm(m)1417 3061 y Fk(1)1350 3118 y Fm(\017)1457 3093 y Fq(G)1534 3108 y Fm(\017)1567 3093 y Ft(\()p Fq(z)t Ft(\)\))q(\()p Fq(@)1825 3052 y Fm(k)1820 3118 y(\017)1868 3093 y Fq(K)1951 3108 y Fm(\017)1984 3093 y Ft(\)\()p Fq(@)2116 3052 y Fm(l)2137 3061 y Fk(2)2111 3118 y Fm(z)2176 3093 y Fq(@)2232 3052 y Fm(m)2294 3061 y Fk(2)2227 3118 y Fm(\017)2335 3093 y Fq(G)2412 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Fk(2)2398 4763 y Fp(k)27 b(\024)h Fq(C)7 b(\017)2696 4722 y Fl(\000)p Fm(l)q Fl(\000)p Fm(m)p Fl(\000)p Fo(2+)p Fm(\027)3079 4763 y Fq(:)375 b Ft(\(7.152\))146 5027 y(Next)35 b(note)f(that)f(w)m (e)i(can)f(\014nd)g(Sc)m(h)m(w)m(artz)i(functions)e Fq(\015)2226 5042 y Fm(i)2253 5027 y Ft(,)k(~)-53 b Fq(\015)2365 5042 y Fm(i)2427 5027 y Ft(suc)m(h)35 b(that)f Fq(\032)2911 5042 y Fm(i)2969 5027 y Ft(=)29 b Fq(\015)3125 5042 y Fm(i)3156 5027 y Ft(~)-52 b Fq(\015)3204 5042 y Fm(i)3232 5027 y Ft(,)34 b(for)f Fq(i)d Ft(=)f(1)p Fq(;)17 b Ft(2.)0 5147 y(Clearly)1199 5268 y Fp(h)p Fq(S)6 b Fp(i)1343 5227 y Fl(\000)p Fm(\026)1343 5292 y(\017;\032)1428 5302 y Fg(i)1485 5268 y Ft(=)31 b(~)-53 b Fq(\015)1639 5283 y Fm(i)1667 5268 y Ft(\()p Fq(\017S)6 b Ft(\))p Fp(h)p Fq(S)g Fp(i)1992 5227 y Fl(\000)p Fm(\026)1992 5292 y(\017;\015)2077 5302 y Fg(i)2106 5268 y Fq(;)115 b(i)28 b Ft(=)f(1)p Fq(;)17 b Ft(2)p Fq(:)1841 5753 y Ft(59)p eop %%Page: 60 60 60 59 bop 0 407 a Ft(Therefore,)34 b(w)m(e)f(can)g(estimate)f(the)h (left-hand)f(side)g(of)g(\(7.149\))g(b)m(y)568 615 y Fp(k)t Ft(~)-53 b Fq(\015)669 630 y Fo(1)708 615 y Ft(\()p Fq(\017S)6 b Ft(\))p Fq(S)955 574 y Fm(k)992 583 y Fk(1)1031 615 y Fp(k)1098 515 y Fj(\015)1097 565 y(\015)1097 615 y(\015)p Fp(h)p Fq(S)g Fp(i)1287 574 y Fl(\000)p Fm(\026)1287 640 y(\017;\015)1372 649 y Fk(2)1426 519 y Fj(\020)1476 615 y Fq(@)1532 574 y Fm(l)1527 640 y(z)1567 615 y Fq(@)1623 574 y Fm(m)1618 640 y(\017)1691 615 y Fq(G)1768 630 y Fm(\017)1800 615 y Ft(\()p Fq(z)t Ft(\))p Fq(K)2008 630 y Fm(\017)2041 615 y Fq(G)2118 630 y Fm(\017)2151 615 y Ft(\()p Fq(z)t Ft(\))2276 519 y Fj(\021)2343 615 y Fp(h)p Fq(S)g Fp(i)2487 574 y Fl(\000)p Fm(\026)2487 640 y(\017;\015)2572 649 y Fk(2)2609 515 y Fj(\015)2609 565 y(\015)2609 615 y(\015)17 b Fp(k)p Fq(S)2788 574 y Fm(k)2825 583 y Fk(2)2867 615 y Ft(~)-53 b Fq(\015)2914 630 y Fo(2)2953 615 y Ft(\()p Fq(\017S)6 b Ft(\))p Fp(k)p Fq(:)0 823 y Ft(Th)m(us,)34 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Ft(sup)838 1853 y Fm(z)s Fl(2)p Fd(C)980 1862 y Fk(+)1047 1775 y Fp(k)p Fq(@)1153 1734 y Fm(l)1148 1799 y(z)1188 1775 y Fp(h)p Fq(S)6 b Fp(i)1332 1734 y Fl(\000)p Fm(\026)1332 1799 y(\017;\032)1417 1808 y Fk(1)1454 1775 y Fq(S)1520 1734 y Fm(k)1557 1743 y Fk(1)1596 1775 y Fq(G)1673 1790 y Fm(\017)1705 1775 y Ft(\()p Fq(z)t Ft(\))p Fq(S)1896 1734 y Fm(k)1933 1743 y Fk(2)1972 1775 y Fp(h)p Fq(S)g Fp(i)2116 1734 y Fl(\000)p Fm(\026)2116 1799 y(\017;\032)2201 1808 y Fk(2)2238 1775 y Fp(k)28 b(\024)g Fq(C)7 b(\017)2537 1734 y Fl(\000)p Fm(k)r Fl(\000)p Fm(l)q Fl(\000)2773 1706 y Fk(1)p 2772 1718 31 4 v 2772 1760 a(2)2813 1734 y Fo(+)p Fm(\026)2914 1775 y Fq(:)540 b Ft(\(7.154\))0 2171 y Fr(Pro)s(of.)58 b Ft(W)-8 b(e)30 b(use)g(the)f(splitting)e Fq(\032)1287 2186 y Fo(2)1354 2171 y Ft(=)32 b(~)-53 b Fq(\015)1509 2186 y Fo(2)1548 2171 y Fq(\015)1599 2186 y Fo(2)1638 2171 y Ft(,)30 b(as)f(in)g(the)g(pro)s(of)f(of)h(the)h (previous)f(lemma.)40 b(If)29 b Fq(k)3490 2186 y Fo(1)3557 2171 y Fp(\025)f Fq(k)3713 2186 y Fo(2)3752 2171 y Ft(,)0 2292 y(then)33 b Fq(k)273 2307 y Fo(1)340 2292 y Fp(\025)28 b Fq(\026)k Ft(and)h(w)m(e)h(use)f(the)g(estimates)1151 2479 y Fp(kh)p Fq(S)6 b Fp(i)1345 2443 y Fl(\000)p Fm(\026)1345 2504 y(\017;\032)1430 2513 y Fk(1)1467 2479 y Fq(S)1533 2443 y Fm(k)1570 2452 y Fk(1)1608 2479 y Fp(k)83 b(\024)28 b Fq(\017)1885 2443 y Fl(\000)p Fm(k)1977 2452 y Fk(1)2012 2443 y Fo(+)p Fm(\026)2130 2479 y Ft(sup)2277 2502 y Fm(t)2323 2479 y Fp(j)p Fq(t)2386 2443 y Fm(k)2423 2452 y Fk(1)2457 2443 y Fl(\000)p Fm(\026)2559 2479 y Fq(\032)2609 2494 y Fo(1)2648 2479 y Ft(\()p Fq(t)p Ft(\))p Fp(j)p Fq(;)1146 2670 y Fp(k)p Fq(S)1262 2634 y Fm(k)1299 2643 y Fk(2)1341 2670 y Ft(~)-53 b Fq(\015)1388 2685 y Fo(2)1427 2670 y Ft(\()p Fq(\017S)6 b Ft(\))p Fp(k)83 b(\024)28 b Fq(\017)1885 2634 y Fl(\000)p Fm(k)1977 2643 y Fk(2)2032 2670 y Ft(sup)2179 2694 y Fm(t)2226 2670 y Fp(j)p Fq(t)2289 2634 y Fm(k)2326 2643 y Fk(2)2368 2670 y Ft(~)-53 b Fq(\015)2415 2685 y Fo(2)2454 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Fp(\\)g(D)s Ft(\()p Fq(A)p Ft(\).)39 b(If)25 b Fq(A)g Ft(is)g(b)s(ounded,)i(one)e(can)g(de\014ne)i(the)e(m)m (ultiple)0 3859 y(comm)m(utators)32 b Fr(S)651 3823 y Fm(p)690 3859 y Fq(A)h Ft(for)f(an)m(y)h(p)s(ositiv)m(e)f(in)m(teger)h Fq(p)f Ft(as)h(quadratic)f(forms)g(on)h Fp(D)s Ft(\()p Fq(S)3039 3823 y Fm(p)3078 3859 y Ft(\).)146 3980 y(In)g(the)g(follo)m (wing)d(lemma)g(it)i(will)e(b)s(e)j(con)m(v)m(enien)m(t)h(to)f(use)g (the)g(follo)m(wing)d(function:)1068 4336 y Fq(!)t Ft(\()p Fq(\027)q(;)17 b(\017)p Ft(\))27 b(:=)1499 4086 y Fj(8)1499 4161 y(>)1499 4186 y(>)1499 4211 y(>)1499 4236 y(<)1499 4385 y(>)1499 4410 y(>)1499 4435 y(>)1499 4460 y(:)1614 4167 y Fq(\017)1653 4131 y Fm(\027)1697 4167 y Fq(;)674 b(\027)34 b(<)28 b Ft(0)p Fq(;)1614 4335 y Ft(1)22 b(+)g(log\(1)g(+)g Fq(\017)2155 4299 y Fl(\000)p Fo(1)2250 4335 y Ft(\))p Fq(;)83 b(\027)34 b Ft(=)28 b(0)p Fq(;)1614 4502 y Ft(1)p Fq(;)708 b(\027)34 b(>)28 b Ft(0)p Fq(:)3481 4336 y Ft(\(7.155\))0 4691 y(Let)33 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Fi(.)55 b(Then,)39 b(for)0 527 y(some)34 b(c)-5 b(onstants)34 b Fq(C)747 542 y Fo(1)822 527 y Fi(and)g Fq(C)1081 542 y Fo(2)1120 527 y Fi(,)799 729 y Ft(sup)946 752 y Fm(z)s Fl(2)p Fd(C)1088 761 y Fk(+)1158 729 y Fp(k)p Fq(@)1264 693 y Fm(l)1259 753 y(z)1300 729 y Fq(@)1356 693 y Fm(k)1351 753 y(\017)1399 729 y Fp(h)p Fq(S)6 b Fp(i)1543 693 y Fl(\000)p Fm(\026)1543 753 y(\017;\032)1644 729 y Fq(G)1721 744 y Fm(\017)1754 729 y Ft(\()p Fq(z)t Ft(\))p Fp(h)p Fq(S)g Fp(i)2023 693 y Fl(\000)p Fm(\026)2023 753 y(\017;\032)2124 729 y Fp(k)799 920 y(\024)28 b Fq(C)974 935 y Fo(1)1013 920 y Fq(!)t Ft(\()p Fp(\000)p Fq(k)d Fp(\000)e Fq(l)h Fp(\000)1532 881 y Fo(1)p 1532 897 V 1532 954 a(2)1599 920 y Ft(+)e Fq(\026;)17 b(\017)p Ft(\))22 b(+)g Fq(C)2067 935 y Fo(2)2106 920 y Fq(!)t Ft(\()p Fp(\000)p Fq(k)j Fp(\000)e Fq(l)h Fp(\000)f Ft(1)f(+)g Fq(\027)q(;)17 b(\017)p Ft(\))p Fq(:)3481 825 y Ft(\(7.157\))0 1249 y Fr(Pro)s(of.)65 b Ft(Note)33 b(that)1464 1369 y Fq(@)1520 1328 y Fm(k)1515 1394 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4984 V 2709 5023 a Fo(v)p 2746 4984 V(v)2789 5064 y Fp(i)2828 5023 y Fm(\027)2871 5064 y Fq(')p Ft(\()p Fq(\013)q Ft(\))p 3074 4984 V -41 x Fo(vv)3154 5064 y Fq(;)300 b Ft(\(7.165\))0 5233 y(w)m(e)34 b(deriv)m(e)f(the)g (result)f(from)g(Theorem)h(7.19.)42 b Fe(2)1841 5753 y Ft(62)p eop %%Page: 63 63 63 62 bop 0 407 a Fn(7.4)135 b(Comparison)46 b(with)f(the)h(free)f (resolv)l(en)l(t)0 592 y Ft(In)27 b(this)g(section)g(w)m(e)h(estimate)e (the)i(di\013erence)f(of)g(the)g(free)h(and)f(the)g(full)e(resolv)m(en) m(t)j(on)f(the)h(radiation)0 712 y(sector.)0 910 y Fr(Theorem)74 b(7.20)49 b Fi(Assume)40 b(that)h(Hyp)-5 b(otheses)40 b Ft(A)g Fi(and)f Fq(S)6 b Ft(\()p Fq(\027)g Ft(\))40 b Fi(hold)g(with)g Fq(\027)j(>)37 b Ft(1)p Fi(.)60 b(L)-5 b(et)41 b Fq(\026)36 b Ft(=)h Fq(\027)c Fp(\000)3734 871 y Fo(1)p 3734 887 36 4 v 3734 944 a(2)0 1043 y Fi(and)h Ft(0)28 b Fq(<)f Ft(\003)437 1058 y Fo(1)504 1043 y Fq(<)h Ft(\()646 961 y Fp(p)p 729 961 49 4 v 82 x Ft(2)o Fp(k)p Fq(s\013)q Fp(k)p Ft(\))1024 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Fo(2)p Fl(\000)p Fm(\024)1651 2207 y Fp(k)p Fq(W)1793 2222 y Fm(k)1835 2207 y Ft(\()p Fq(z)t Ft(\))23 b Fp(\000)g Fq(\025)2140 2166 y Fo(2)2179 2207 y Fq(w)2249 2222 y Fm(k)2292 2207 y Fp(k)k Fq(<)h(\014)2528 2222 y Fo(2)2567 2207 y Fq(;)887 b Ft(\(8.188\))0 2413 y Fi(wher)-5 b(e)34 b(the)h(supr)-5 b(emum)35 b(is)f(taken)h(over)f Fq(z)f Fp(2)p 1660 2334 81 4 v 28 w Fr(C)1741 2428 y Fo(+)1822 2413 y Fp(\\)p 1910 2335 80 4 v 22 w Fq(B)6 b Ft(\()p Fq(k)s(;)17 b(\025)2183 2376 y Fo(2)2222 2413 y Fq(\014)2277 2428 y Fo(1)2316 2413 y Ft(\))p Fi(,)35 b Fp(j)p Fq(\025)p Fp(j)27 b(\024)h Ft(\003)2732 2428 y Fo(2)2771 2413 y Fi(.)0 2624 y Fr(Pro)s(of.)65 b Ft(It)33 b(follo)m(ws)e(from)g(\(8.187\))h(that)382 2841 y Fq(W)474 2856 y Fm(k)517 2841 y Ft(\()p Fq(z)t Ft(\))84 b(=)27 b Fq(H)918 2805 y Fm(k)p 957 2749 39 4 v 2 w(k)999 2841 y Ft(\()p Fq(z)t Fr(1)p 1142 2767 V -36 x Fo(v)p 1181 2767 V 2 w(v)1246 2841 y Fp(\000)22 b Fq(H)p 1434 2767 V 1434 2805 a Fo(v)p 1472 2767 V 1 w(v)1514 2841 y Ft(\))1552 2805 y Fl(\000)p Fo(1)1647 2841 y Fq(H)p 1736 2749 V 1736 2805 a Fm(k)q(k)726 3047 y Ft(+)p Fq(H)891 3011 y Fm(k)p 930 2955 V 2 w(k)971 3047 y Ft(\()p Fq(z)t Fr(1)p 1114 2973 V -36 x Fo(v)p 1153 2973 V 2 w(v)1218 3047 y Fp(\000)h Fq(H)p 1407 2973 V 1407 3011 a Fo(v)p 1444 2973 V(v)1487 3047 y Ft(\))1525 3011 y Fl(\000)p Fo(1)1619 3047 y Fq(H)p 1708 2973 V 1708 3011 a Fo(v)q Fm(k)p 1746 3024 V 1788 3047 a Fq(G)1865 3006 y Fl(\000)p Fo(1)1865 3072 y Fm(k)p 1865 3085 43 4 v 1960 3047 a Ft(\()p Fq(z)t Ft(\))p Fq(H)2174 3011 y Fm(k)p 2174 3024 39 4 v 2212 2973 V 1 w Fo(v)2255 3047 y Ft(\()p Fq(z)t Fr(1)p 2398 2973 V -36 x Fo(v)p 2437 2973 V 2 w(v)2501 3047 y Fp(\000)g Fq(H)p 2690 2973 V 2690 3011 a Fo(v)p 2728 2973 V 1 w(v)2770 3047 y Ft(\))2808 3011 y Fl(\000)p Fo(1)2902 3047 y Fq(H)p 2991 2955 V 2991 3011 a Fm(k)q(k)3072 3047 y Fq(:)3481 2941 y Ft(\(8.189\))0 3267 y(Since)34 b Fq(\014)311 3282 y Fo(1)350 3267 y Ft(\003)418 3231 y Fo(2)418 3292 y(2)487 3267 y Fq(<)c(\016)636 3282 y Fo(1)676 3267 y Ft(,)k(Lemma)e(8.4)h(yields)h(that)f(for)h Fq(z)g Fp(2)c Fr(C)2139 3282 y Fo(+)2221 3267 y Fp(\\)p 2310 3189 80 4 v 23 w Fq(B)6 b Ft(\()p Fq(k)s(;)17 b(\025)2583 3231 y Fo(2)2622 3267 y Fq(\014)2677 3282 y Fo(1)2716 3267 y Ft(\))34 b(the)g(second)h(term)e(on)h(the)0 3387 y(righ)m(t-hand)e(side)g(is)g Fq(O)s Ft(\()p Fq(\025)946 3351 y Fo(4)985 3387 y Ft(\).)43 b(It)33 b(follo)m(ws)e(from)h(Theorem) g(7.20)g(that)h(the)g(\014rst)g(term)f(equals)536 3599 y Fq(H)625 3563 y Fm(k)p 664 3507 39 4 v 2 w(k)705 3599 y Ft(\()p Fq(z)t Fr(1)p 848 3525 V -36 x Fo(v)p 887 3525 V 2 w(v)952 3599 y Fp(\000)23 b Fq(H)p 1141 3525 V 1141 3563 a Fo(v)p 1178 3525 V(v)1221 3599 y Ft(\))1259 3563 y Fl(\000)p Fo(1)1353 3599 y Fq(H)p 1442 3507 V 1442 3563 a Fm(k)q(k)1606 3599 y Ft(=)k Fq(H)1798 3563 y Fm(k)p 1837 3507 V 2 w(k)1879 3599 y Ft(\()p Fq(z)t Fr(1)p 2022 3525 V -36 x Fo(v)p 2061 3525 V 2 w(v)2125 3599 y Fp(\000)c Fq(H)p 2314 3525 V 2314 3563 a Fo(v)p 2352 3525 V 1 w(v)2306 3624 y(fr)2394 3599 y Ft(\))2432 3563 y Fl(\000)p Fo(1)2526 3599 y Fq(H)p 2615 3507 V 2615 3563 a Fm(k)q(k)2718 3599 y Ft(+)f Fq(O)s Ft(\()p Fp(j)p Fq(\025)p Fp(j)3045 3563 y Fo(2+)p Fm(\024)3179 3599 y Ft(\))p Fq(:)146 3798 y Ft(Recall)36 b(that)g(in)g(\(2.34\))g(w)m(e)i(de\014ned)g(the)f (function)g Fq(`)2152 3813 y Fm(\022)2191 3798 y Ft(\()p Fq(\034)11 b Ft(\).)56 b(W)-8 b(e)37 b(extend)i(the)e(de\014nition)f (of)g(this)0 3919 y(function)c(for)g Fq(\022)f Fp(2)d Ft([0)p Fq(;)17 b Fp(1)p Ft([)32 b(as)h(follo)m(ws:)920 4243 y Fq(`)961 4258 y Fm(\022)1000 4243 y Ft(\()p Fq(\034)11 b Ft(\))28 b(:=)1288 4043 y Fj(8)1288 4118 y(>)1288 4143 y(<)1288 4292 y(>)1288 4317 y(:)1403 4121 y Fq(\034)1456 4085 y Fm(\022)1496 4121 y Fq(;)850 b Ft(0)27 b Fq(<)h(\022)i(<)e Ft(1)p Fq(;)1403 4242 y(\034)g Ft(\(1)22 b(+)g(ln)o(\(1)g(+)g Fq(\034)2021 4206 y Fl(\000)p Fo(1)2116 4242 y Ft(\)\))181 b Fq(\022)30 b Ft(=)e(1,)1403 4362 y Fq(\034)6 b(;)862 b(\022)31 b(>)c Ft(1)p Fq(:)0 4561 y Ft(Then,)34 b(b)m(y)f(Theorem)g (7.19,)778 4777 y Fq(H)867 4736 y Fm(k)p 906 4680 V 2 w(k)948 4777 y Ft(\()p Fq(z)t Fr(1)p 1091 4698 V -41 x Fo(v)p 1130 4698 V 2 w(v)1195 4777 y Fp(\000)22 b Fq(H)p 1383 4698 V 1383 4736 a Fo(v)p 1421 4698 V 1 w(v)1375 4802 y(fr)1463 4777 y Ft(\))1501 4736 y Fl(\000)p Fo(1)1596 4777 y Fq(H)p 1685 4680 V 1685 4736 a Fm(k)q(k)1793 4777 y Ft(=)27 b Fq(\025)1953 4736 y Fo(2)1993 4777 y Fq(w)2066 4736 y Fm(k)r(k)2146 4777 y Ft(\()p Fq(z)t Ft(\))i(=)e Fq(\025)2460 4736 y Fo(2)2500 4777 y Fq(w)2573 4736 y Fm(k)r(k)2570 4802 y(k)2675 4777 y Ft(+)22 b Fq(R)q Ft(\()p Fq(z)t Ft(\))p Fq(;)0 4983 y Ft(where)1056 5075 y Fp(k)p Fq(R)q Ft(\()p Fq(z)t Ft(\))p Fp(k)84 b(\024)28 b Fq(C)1615 5090 y Fo(1)1654 5075 y Fq(\025)1711 5039 y Fo(2)1751 5075 y Fq(`)1792 5100 y Fm(\027)t Fl(\000)1896 5073 y Fk(1)p 1895 5085 31 4 v 1895 5126 a(2)1940 5075 y Ft(\()p Fp(j)p Fq(z)e Fp(\000)d Fq(k)s Fp(j)p Ft(\))1440 5267 y Fp(\024)28 b Fq(C)1615 5282 y Fo(2)1654 5267 y Fq(\025)1711 5230 y Fo(2)1751 5267 y Fq(`)1792 5292 y Fm(\027)t Fl(\000)1896 5264 y Fk(1)p 1895 5276 V 1895 5318 a(2)1940 5267 y Ft(\()p Fq(\014)2033 5282 y Fo(1)2072 5267 y Fq(\025)2129 5230 y Fo(2)2169 5267 y Ft(\))f Fp(\024)h Fq(C)2409 5282 y Fo(3)2449 5267 y Fp(j)p Fq(\025)p Fp(j)2562 5230 y Fo(2+)p Fm(\024)2696 5267 y Fq(;)1841 5753 y Ft(68)p eop %%Page: 69 69 69 68 bop 0 407 a Ft(where)34 b(w)m(e)f(used)h(2)p Fq(\027)29 b Fp(\000)22 b Ft(1)28 b Fq(>)f(\024)33 b Ft(and)g(2)27 b Fq(>)h(\024)k Ft(in)g(the)h(last)f(step.)44 b Fe(2)146 569 y Ft(The)34 b(op)s(erator)e Fq(w)810 584 y Fm(k)885 569 y Ft(is)g(dissipativ)m(e)g(and)g(b)m(y)i(the)f(assumption,)1292 762 y Fp(T)1346 777 y Fm(k)1417 762 y Ft(=)27 b Fq(\033)t Ft(\()p Fq(w)1687 777 y Fm(k)1729 762 y Ft(\))c Fp(\\)f Fr(R)27 b Fp(\032)i Fq(\033)2150 777 y Fo(disc)2272 762 y Ft(\()p Fq(w)2380 777 y Fm(k)2422 762 y Ft(\))p Fq(:)0 955 y Ft(Therefore,)34 b(Prop)s(ositions)d(3.3)h(and)h(3.4)f(can)h(b)s (e)g(applied.)42 b(Let)1129 1148 y Fq(c)28 b Ft(:=)50 b(sup)1330 1243 y Fm(z)s Fl(2)p 1413 1188 59 4 v Fd(C)1471 1252 y Fk(+)1538 1148 y Fp(k)p Ft(\()p Fq(z)27 b Fp(\000)c Fq(w)1868 1163 y Fm(k)1910 1148 y Ft(\))1948 1107 y Fl(\000)p Fo(1)2042 1148 y Fr(1)2098 1163 y Fm(\033)r Fo(\()p Fm(w)2218 1175 y Fg(k)2257 1163 y Fo(\))p Fl(n)p Fd(R)2385 1148 y Ft(\()p Fq(w)2493 1163 y Fm(k)2535 1148 y Ft(\))p Fp(k)p Fq(;)831 b Ft(\(8.190\))0 1423 y(\(whic)m(h)34 b(is)f(the)h(constan)m (t)h Fq(c)e Ft(in)g(Prop)s(osition)f(3.3)h(applied)f(to)i Fq(w)2388 1438 y Fm(k)2430 1423 y Ft(\).)46 b(In)34 b(addition)e(to)h (\(8.184\),)g(in)g(the)0 1543 y(sequel)g(w)m(e)h(assume)f(that)g(\003) 1051 1558 y Fo(2)1122 1543 y Ft(satis\014es)1662 1664 y Fq(\014)1717 1679 y Fo(2)1756 1664 y Ft(\003)1824 1622 y Fm(\024)1824 1688 y Fo(2)1869 1664 y Fq(c)27 b(<)h Ft(1)p Fq(:)1363 b Ft(\(8.191\))0 1861 y Fr(Lemma)37 b(8.7)49 b Fi(The)34 b(op)-5 b(er)g(ator)35 b Fq(G)1249 1876 y Fm(k)1292 1861 y Ft(\()p Fq(z)t Ft(\))g Fi(is)g(invertible)f (for)884 2066 y Fq(z)f Fp(2)p 1056 1988 81 4 v 28 w Fr(C)1137 2081 y Fo(+)1218 2066 y Fp(\\)1307 1970 y Fj(\020)p 1356 1988 80 4 v 1356 2066 a Fq(B)5 b Ft(\()p Fq(k)s(;)17 b(\025)1628 2025 y Fo(2)1667 2066 y Fq(\014)1722 2081 y Fo(1)1762 2066 y Ft(\))22 b Fp(n)g Fq(B)5 b Ft(\()p Fq(k)25 b Ft(+)d Fq(\025)2242 2025 y Fo(2)2282 2066 y Fp(T)2336 2081 y Fm(k)2378 2066 y Fq(;)17 b Fp(j)p Fq(\025)p Fp(j)2535 2025 y Fo(2+)p Fm(\024)2669 2066 y Fq(\014)2724 2081 y Fo(2)2764 2066 y Ft(\))2802 1970 y Fj(\021)2868 2066 y Fq(;)0 2281 y Fi(and)34 b(the)h(function)g Fq(G)812 2239 y Fl(\000)p Fo(1)812 2306 y Fm(k)906 2281 y Ft(\()p Fq(z)t Ft(\))h Fi(is)e(in)h(the)g(class)f Fq(C)1764 2245 y Fm(n;\022)1757 2305 y Fo(u)1900 2281 y Fi(of)h(this)f(set.)0 2478 y Fr(Pro)s(of.)51 b Ft(It)26 b(follo)m(ws)e(from)h(the)h (de\014nition)f(of)g Fq(G)1759 2493 y Fm(k)1802 2478 y Ft(\()p Fq(z)t Ft(\))h(and)g(Lemma)e(8.6)h(that)h(if)f Fq(z)32 b Fp(2)p 3086 2400 81 4 v 28 w Fr(C)3167 2493 y Fo(+)3234 2478 y Fp(\\)p 3308 2400 80 4 v 8 w Fq(B)6 b Ft(\()p Fq(k)s(;)17 b(\025)3581 2442 y Fo(2)3620 2478 y Fq(\014)3675 2493 y Fo(1)3714 2478 y Ft(\),)0 2598 y(then)898 2691 y Fq(G)975 2706 y Fm(k)1017 2691 y Ft(\()p Fq(z)t Ft(\))84 b(=)28 b(\()p Fq(z)f Fp(\000)22 b Fq(k)s Ft(\))p Fr(1)1687 2655 y Fm(k)r(k)1791 2691 y Fp(\000)g Fq(W)1982 2706 y Fm(k)2025 2691 y Ft(\()p Fq(z)t Ft(\))1226 2882 y(=)28 b(\()p Fq(z)f Fp(\000)22 b Fq(k)s Ft(\))p Fr(1)1687 2846 y Fm(k)r(k)1791 2882 y Fp(\000)g Fq(\025)1947 2846 y Fo(2)1986 2882 y Fq(w)2056 2897 y Fm(k)2121 2882 y Ft(+)g Fq(R)q Ft(\()p Fq(z)t Ft(\))1226 3089 y(=)28 b Fq(\025)1387 3053 y Fo(2)1443 2992 y Fj(\020)1492 3089 y Fq(\025)1549 3053 y Fl(\000)p Fo(2)1643 3089 y Ft(\()p Fq(z)f Fp(\000)c Fq(k)s Ft(\))p Fr(1)2001 3053 y Fm(k)r(k)2104 3089 y Fp(\000)g Fq(w)2274 3104 y Fm(k)2338 3089 y Ft(+)f Fq(\025)2493 3053 y Fl(\000)p Fo(2)2588 3089 y Fq(R)q Ft(\()p Fq(z)t Ft(\))2788 2992 y Fj(\021)2855 3089 y Fq(;)3481 2898 y Ft(\(8.192\))0 3268 y(where)34 b Fp(k)p Fq(R)q Ft(\()p Fq(z)t Ft(\))p Fp(k)28 b Fq(<)f(\014)768 3283 y Fo(2)808 3268 y Fp(j)p Fq(\025)p Fp(j)921 3232 y Fo(2+)p Fm(\024)1055 3268 y Ft(.)44 b(If)32 b Fq(c)h Ft(is)f(de\014ned)i(b)m(y)f(\(8.190\),)f(then)h Fp(k)p Fq(\025)2554 3232 y Fl(\000)p Fo(2)2648 3268 y Fq(R)q Ft(\()p Fq(z)t Ft(\))p Fp(k)p Fq(c)28 b(<)g Ft(1)k(and)980 3461 y(sup)18 b(dist\()p Fq(\033)t Ft(\()p Fq(\025)1494 3420 y Fl(\000)p Fo(2)1588 3461 y Fq(k)25 b Ft(+)d Fq(w)1832 3476 y Fm(k)1874 3461 y Ft(\))g Fp(\\)h Fr(R)p Fq(;)17 b(\025)2208 3420 y Fl(\000)p Fo(2)2302 3461 y Fq(z)t Ft(\))28 b Fq(>)f Fp(j)p Fq(\025)p Fp(j)2633 3420 y Fm(\024)2677 3461 y Fq(\014)2732 3476 y Fo(2)2772 3461 y Fq(;)0 3654 y Ft(where)38 b(the)f(suprem)m(um)f(is)g(tak)m(en)i(o)m(v)m(er)g Fq(z)32 b Fp(2)p 1692 3576 81 4 v 28 w Fr(C)1773 3669 y Fo(+)1854 3654 y Fp(n)22 b Fq(B)5 b Ft(\()p Fq(k)26 b Ft(+)c Fq(\025)2275 3618 y Fo(2)2314 3654 y Fp(T)2368 3669 y Fm(k)2411 3654 y Fq(;)17 b Fp(j)p Fq(\025)p Fp(j)2568 3618 y Fo(2+)p Fm(\024)2702 3654 y Fq(\014)2757 3669 y Fo(2)2796 3654 y Ft(\).)56 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Fq(R)2679 1805 y Fm(m)p 2679 1818 V(m)p 2741 1818 V 2808 1841 a Ft(\()p Fq(z)t Ft(\)\))17 b Fq(;)0 2064 y Ft(where)34 b Fq(R)q Ft(\()p Fq(z)t Ft(\))28 b(=)g Fq(W)706 2079 y Fm(k)748 2064 y Ft(\()p Fq(z)t Ft(\))23 b Fp(\000)g Fq(\025)1053 2028 y Fo(2)1092 2064 y Fq(w)1162 2079 y Fm(k)1237 2064 y Ft(is)32 b(the)h(same)g(as)f(in)g(\(8.192\).)43 b(By)33 b(Lemma)e(8.6,)1208 2284 y Fp(k)p Fq(R)1333 2243 y Fm(m)p 1333 2256 V(m)p 1395 2256 V 1462 2284 a Ft(\()p Fq(z)t Ft(\))p Fp(k)d(\024)g(k)p Fq(R)q Ft(\()p Fq(z)t Ft(\))p Fp(k)g(\024)g Fq(\014)2258 2299 y Fo(2)2298 2284 y Fp(j)p Fq(\025)p Fp(j)2411 2243 y Fo(2+)p Fm(\024)2545 2284 y Fq(;)0 2504 y Ft(if)j Fq(z)i Fp(2)p 261 2426 81 4 v 28 w Fr(C)342 2519 y Fo(+)423 2504 y Fp(\\)p 512 2426 80 4 v 23 w Fq(B)5 b Ft(\()p Fq(k)s(;)17 b(\025)784 2468 y Fo(2)823 2504 y Fq(\014)878 2519 y Fo(1)917 2504 y Ft(\).)44 b(Moreo)m(v)m(er,)1040 2724 y Fq(w)1113 2673 y Fm(m)p 1113 2686 63 4 v(m)p 1175 2686 V 1110 2749 a(k)1241 2724 y Fr(1)1297 2751 y Fm(\033)r 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b(the)h(same)g(as)g(in)f(\(8.190\).)43 b(Th)m(us,)34 b(w)m(e)g(can)f(apply)f(Prop)s(osition)f(3.4,)h(whic)m(h) i(implies)c(that)i Fq(G)3587 3345 y Fm(m)p 3587 3358 67 4 v 3654 3330 a Ft(\()p Fq(z)t Ft(\))0 3450 y(is)g(in)m(v)m(ertible) g(and)h(satis\014es)g(the)g(b)s(ound)g(\(8.198\))e(with)h Fq(j)i Ft(=)28 b(0.)146 3571 y(The)h(regularit)m(y)e(prop)s(erties)g (of)h Fq(G)1419 3535 y Fl(\000)p Fo(1)1419 3595 y Fm(m)p 1419 3608 V 1513 3571 a Ft(\()p Fq(z)t Ft(\))g(are)g(inferred)g(from)e (the)j(form)m(ula)c(analogous)i(to)g(\(8.176\))0 3691 y(and)33 b(Lemma)e(8.5.)43 b(The)33 b(b)s(ound)g(\(8.198\))f(for)g Fq(j)i Ft(=)27 b(0)32 b(and)h(induction)f(yield)f(\(8.198\))h(for)g (all)e Fq(j)6 b Ft(.)44 b Fe(2)146 3861 y Ft(F)-8 b(or)32 b Fq(z)h Fp(2)28 b Fr(C)574 3876 y Fo(+)665 3861 y Ft(w)m(e)34 b(set)1065 4075 y Fq(W)1157 4090 y Fm(m)1224 4075 y Ft(\()p Fq(z)t Ft(\))28 b(:=)g Fq(H)1597 4039 y Fm(m)p 1659 4001 63 4 v(m)1725 4075 y Ft(\()p Fq(z)t Fr(1)p 1868 4001 V -36 x Fm(m)p 1932 4001 V 2 w(m)2020 4075 y Fp(\000)23 b Fq(H)p 2209 4001 V 2209 4039 a Fm(m)p 2271 4001 V(m)2337 4075 y Ft(\))2375 4039 y Fl(\000)p Fo(1)2470 4075 y Fq(H)p 2559 4001 V 2559 4039 a Fm(mm)2687 4075 y Fq(;)1065 4267 y(G)1142 4282 y Fm(m)1209 4267 y Ft(\()p Fq(z)t Ft(\))28 b(:=)g Fq(z)t Fr(1)1598 4230 y Fm(mm)1750 4267 y Fp(\000)22 b Fq(H)1938 4230 y Fm(mm)2089 4267 y Fp(\000)h Fq(W)2281 4282 y Fm(m)2347 4267 y Ft(\()p Fq(z)t Ft(\))p Fq(:)0 4488 y Fr(Lemma)37 b(8.10)49 b Fi(The)34 b(function)1026 4708 y Fr(C)1107 4723 y Fo(+)1194 4708 y Fp(3)28 b Fq(z)k Fp(7!)c(h)p Fq(S)6 b Fp(i)1637 4667 y Fl(\000)p Fm(\026)1737 4708 y Ft(\()p Fq(z)t Fr(1)p 1880 4628 V -41 x Fm(m)p 1943 4628 V 1 w(m)2032 4708 y Fp(\000)23 b Fq(H)p 2221 4628 V 2221 4667 a Fm(m)p 2283 4628 V(m)2349 4708 y Ft(\))2387 4667 y Fl(\000)p Fo(1)2482 4708 y Fp(h)p Fq(S)6 b Fp(i)2626 4667 y Fl(\000)p Fm(\026)2726 4708 y Fq(;)0 4928 y Fi(extends)40 b(by)h(c)-5 b(ontinuity)42 b(to)f Fq(z)i Fp(2)p 1267 4850 81 4 v 40 w Fr(C)1348 4943 y Fo(+)1433 4928 y Fp(\\)p 1526 4850 80 4 v 27 w Fq(B)6 b Ft(\()p Fq(k)s(;)17 b(\025)1799 4892 y Fo(2)1838 4928 y Fq(\014)1893 4943 y Fo(1)1932 4928 y Ft(\))27 b Fp(\\)p 2090 4850 V 27 w Fq(B)5 b Ft(\()p Fq(k)30 b Ft(+)c Fq(\025)2447 4892 y Fo(2)2486 4928 y Fq(m;)17 b(\025)2672 4892 y Fo(2+)p Fm(\024)2807 4928 y Fq(\014)2862 4943 y Fo(2)2902 4928 y Ft(\))41 b Fi(and)f(is)h(in)f (the)h(class)0 5049 y Fq(C)77 5012 y Fm(n;\022)70 5073 y Fo(u)213 5049 y Fi(of)35 b(this)g(set.)44 b(The)35 b(same)f(r)-5 b(esult)35 b(holds)f(for)h(the)g(function)f Fq(W)2462 5064 y Fm(m)2529 5049 y Ft(\()p Fq(z)t Ft(\))p Fi(.)1841 5753 y Ft(71)p eop %%Page: 72 72 72 71 bop 0 407 a Fr(Pro)s(of.)65 b Ft(The)33 b(F)-8 b(esh)m(bac)m(h)35 b(form)m(ula)30 b(yields)42 617 y(\()p Fq(z)t Fr(1)p 185 542 63 4 v -36 x Fm(m)p 248 542 V 1 w(m)336 617 y Fp(\000)23 b Fq(H)p 525 542 V 525 581 a Fm(m)p 587 542 V(m)654 617 y Ft(\))692 581 y Fl(\000)p Fo(1)869 617 y Ft(=)k(\()p Fq(z)t Fr(1)p 1115 525 39 4 v -36 x Fm(k)p 1155 525 V 3 w(k)1219 617 y Fp(\000)c Fq(H)p 1408 525 V 1408 581 a Fm(k)p 1446 525 V 1 w(k)1489 617 y Ft(\))1527 581 y Fl(\000)p Fo(1)869 823 y Ft(+\()p Fr(1)1039 787 y Fm(m)p 1039 800 63 4 v(m)p 1101 800 V 1190 823 a Ft(+)f(\()p Fq(z)t Fr(1)p 1431 731 39 4 v -36 x Fm(k)p 1470 731 V 2 w(k)1535 823 y Fp(\000)g Fq(H)p 1723 731 V 1723 787 a Fm(k)p 1761 731 V 1 w(k)1804 823 y Ft(\))1842 787 y Fl(\000)p Fo(1)1936 823 y Fq(H)p 2025 731 V 2025 787 a Fm(k)q(m)p 2063 800 63 4 v 2130 823 a Ft(\))p Fq(G)2245 787 y Fl(\000)p Fo(1)2245 847 y Fm(m)p 2245 860 67 4 v 2339 823 a Ft(\()p Fq(z)t Ft(\)\()p Fq(z)t Fr(1)2607 787 y Fm(m)p 2607 800 63 4 v 2 w(m)p 2671 800 V 2760 823 a Ft(+)g Fq(H)2947 787 y Fm(m)p 2947 800 V 3009 731 39 4 v(k)3051 823 y Ft(\()p Fq(z)t Fr(1)p 3194 731 V -36 x Fm(k)p 3234 731 V 3 w(k)3298 823 y Fp(\000)h Fq(H)p 3487 731 V 3487 787 a Fm(k)p 3525 731 V 1 w(k)3568 823 y Ft(\))3606 787 y Fl(\000)p Fo(1)3700 823 y Ft(\))p Fq(;)0 1025 y Ft(and)33 b(the)g(result)f(follo)m(ws)f(from)h(Lemmas)f 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Fp(\024)28 b Fq(\015)5 b Fp(j)p Fq(\025)p Fp(j)2255 1626 y Fo(2)p Fm(\024)2334 1667 y Fq(:)1120 b Ft(\(8.199\))0 2081 y Fr(Pro)s(of.)65 b Ft(It)33 b(follo)m(ws)e(from)g(the)i(F)-8 b(esh)m(bac)m(h)34 b(form)m(ula)d(that)297 2272 y Fq(W)389 2287 y Fm(m)456 2272 y Ft(\()p Fq(z)t Ft(\))84 b(=)27 b Fq(H)857 2235 y Fm(m)p 919 2197 63 4 v(m)986 2272 y Ft(\()p Fq(z)t Fr(1)p 1129 2179 39 4 v -37 x Fm(k)p 1168 2179 V 2 w(k)1233 2272 y Fp(\000)22 b Fq(H)p 1421 2179 V 1421 2235 a Fm(k)p 1459 2179 V 1 w(k)1502 2272 y Ft(\))1540 2235 y Fl(\000)p Fo(1)1634 2272 y Fq(H)p 1723 2197 63 4 v 1723 2235 a Fm(mm)665 2477 y Ft(+)p Fq(H)830 2441 y Fm(m)p 892 2403 V(m)958 2477 y Ft(\()p Fq(z)t Fr(1)p 1101 2385 39 4 v -36 x Fm(k)p 1140 2385 V 2 w(k)1205 2477 y Fp(\000)h Fq(H)p 1394 2385 V 1394 2441 a Fm(k)p 1432 2385 V 1 w(k)1474 2477 y Ft(\))1512 2441 y Fl(\000)p Fo(1)1607 2477 y Fq(H)p 1696 2385 V 1696 2441 a Fm(k)q(m)p 1734 2454 63 4 v 1800 2477 a Fq(G)1877 2441 y Fl(\000)p Fo(1)1877 2502 y Fm(m)p 1877 2515 67 4 v 1972 2477 a Ft(\()p Fq(z)t Ft(\))p Fq(H)2186 2441 y Fm(m)p 2186 2454 63 4 v 2248 2385 39 4 v(k)2291 2477 y Ft(\()p Fq(z)t Fr(1)p 2434 2385 V -36 x Fm(k)p 2473 2385 V 2 w(k)2538 2477 y Fp(\000)g Fq(H)p 2727 2385 V 2727 2441 a Fm(k)p 2765 2385 V 1 w(k)2807 2477 y Ft(\))2845 2441 y Fl(\000)p Fo(1)2939 2477 y Fq(H)p 3028 2403 63 4 v 3028 2441 a Fm(mm)3157 2477 y Fq(:)3481 2371 y Ft(\(8.200\))0 2689 y(The)32 b(deriv)-5 b(ativ)m(e)31 b(of)f(the)i(\014rst)f(term)g(in)f (\(8.200\))g(is)g Fq(O)s Ft(\()p Fq(\025)2062 2653 y Fo(2)2101 2689 y Ft(\))h(b)m(y)h(Lemma)d(8.8.)43 b(When)32 b(w)m(e)g(di\013eren)m(tiate)0 2822 y(the)h(second)h(term)e(and)h(the)g (deriv)-5 b(ativ)m(e)32 b(hits)g(\()p Fq(z)t Fr(1)p 1854 2729 39 4 v -37 x Fm(k)p 1894 2729 V 3 w(k)1958 2822 y Fp(\000)23 b Fq(H)p 2147 2729 V 2147 2785 a Fm(k)p 2185 2729 V 1 w(k)2228 2822 y Ft(\))2266 2785 y Fl(\000)p Fo(1)2360 2822 y Ft(,)33 b(then)g(w)m(e)h(get)e(b)m(y)i(Lemmas)e(8.8,)g (8.9)0 2942 y(and)h(\(8.198\))1133 3062 y Fq(O)s Ft(\()p Fq(\025)1306 3021 y Fo(2)1345 3062 y Ft(\))p Fq(O)s Ft(\()p Fq(\025)1556 3021 y Fl(\000)p Fo(2)1649 3062 y Ft(\))p Fq(O)s Ft(\()p Fp(j)p Fq(\025)p Fp(j)1916 3021 y Fo(2+)p Fm(\024)2050 3062 y Ft(\))27 b(=)h Fq(O)s Ft(\()p Fp(j)p Fq(\025)p Fp(j)2448 3021 y Fo(2+)p Fm(\024)2581 3062 y Ft(\))p Fq(:)0 3230 y Ft(When)33 b(the)g(deriv)-5 b(ativ)m(e)33 b(hits)f Fq(G)1167 3194 y Fl(\000)p Fo(1)1167 3254 y Fm(m)p 1167 3267 67 4 v 1261 3230 a Ft(\()p Fq(z)t Ft(\),)h(then)h(w)m (e)f(get)g(b)m(y)g(the)g(same)g(lemmas)d(and)j(\(8.198\))1085 3447 y Fq(O)s Ft(\()p Fp(j)p Fq(\025)p Fp(j)1314 3406 y Fo(2+)p Fm(\024)1448 3447 y Ft(\))p Fq(O)s Ft(\()p Fq(\025)1659 3406 y Fl(\000)p Fo(4)1752 3447 y Ft(\))p Fq(O)s Ft(\()p Fp(j)p Fq(\025)p Fp(j)2019 3406 y Fo(2+)p Fm(\024)2152 3447 y Ft(\))28 b(=)g Fq(O)s Ft(\()p Fp(j)p Fq(\025)p Fp(j)2551 3406 y Fo(2)p Fm(\024)2629 3447 y Ft(\))p Fq(:)0 3651 y Ft(Then)34 b(w)m(e)f(use)h(2)p Fq(\024)27 b Fp(\024)i Ft(2)22 b(+)g Fq(\024)p Ft(.)43 b Fe(2)146 3816 y 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Fp(\010)c(H)p 1505 4989 V 1505 5028 a Fm(m)1571 5064 y Ft(.)44 b(By)33 b(Corollary)e(3.13,)1167 5268 y(dim)16 b Fr(1)1403 5221 y Fo(pp)1403 5297 y(\002\()p Fm(k)r(;m)p Fo(\))1665 5268 y Fp(\024)28 b Ft(dim)15 b Fp(H)2034 5227 y Fm(m)2129 5268 y Ft(=)27 b(dim)15 b Fq(p)2460 5283 y Fm(k)r(;m)2585 5268 y Fq(:)1841 5753 y Ft(72)p eop %%Page: 73 73 73 72 bop 0 407 a Ft(This)33 b(completes)f(the)h(pro)s(of)f(of)g(P)m (art)h(\(i\).)146 527 y(Since)e(\(iii\))c Fp(\))j Ft(\(ii\),)f(it)g (remains)g(to)h(pro)m(v)m(e)i(\(iii\).)39 b(First)30 b(note)g(that,)h(b)m(y)g(Lemma)e(8.10,)h(\002\()p Fq(k)s(;)17 b(m)p Ft(\))g Fp(\\)0 648 y Fq(\033)55 663 y Fo(pp)138 648 y Ft(\()p Fq(H)p 265 573 63 4 v 265 611 a Fm(m)p 327 573 V(m)393 648 y Ft(\))32 b(=)g Fp(;)p Ft(.)50 b(Therefore,)37 b(b)m(y)e(Prop)s(osition)f(3.7)g(and)h(Theorem)g(3.8,)h Fq(\033)2843 663 y Fo(pp)2926 648 y Ft(\()p Fq(H)8 b Ft(\))23 b Fp(\\)h Ft(\002\()p Fq(k)s(;)17 b(m)p Ft(\))35 b(coin-)0 768 y(cides)g(with)f Fp(f)p Fq(x)d Fp(2)g Ft(\002\()p Fq(k)s(;)17 b(m)p Ft(\))65 b(:)h(0)30 b Fp(2)i Fq(\033)t Ft(\()p Fq(G)1542 783 y Fm(m)1608 768 y Ft(\()p Fq(x)24 b Ft(+)f(i0\)\))p Fp(g)p Ft(.)48 b(Let)35 b Fq(\017)c(>)g Ft(0.)48 b(Since)35 b Fq(G)2914 783 y Fm(m)2981 768 y Ft(\()p Fq(z)t Ft(\))g(is)f(a)g(con)m(tin)m(uous)0 888 y(function)e(and)p 572 810 81 4 v 33 w Fr(C)653 903 y Fo(+)734 888 y Fp(\\)p 822 810 80 4 v 22 w Fq(B)6 b Ft(\()p Fq(k)s(;)17 b(\025)1095 852 y Fo(2)1134 888 y Fq(\014)1189 903 y Fo(1)1228 888 y Ft(\))22 b Fp(\\)p 1377 810 V 23 w Fq(B)5 b Ft(\()p Fq(k)25 b Ft(+)d Fq(\025)1725 852 y Fo(2)1764 888 y Fq(m;)17 b(\014)1948 903 y Fo(2)1988 888 y Fq(\025)2045 852 y Fo(2+)p Fm(\024)2180 888 y Ft(\))22 b Fp(n)g Fq(B)5 b Ft(\()p Fq(\033)2484 903 y Fo(pp)2567 888 y Ft(\()p Fq(H)j Ft(\))p Fq(;)17 b(\017)p Ft(\))32 b(is)g(compact,)1576 1108 y Fp(k)p Fq(G)1703 1067 y Fl(\000)p Fo(1)1703 1133 y Fm(m)1797 1108 y Ft(\()p Fq(z)t Ft(\))p Fp(k)c(\024)g Fq(C)r(:)0 1328 y Ft(on)33 b(this)f(set.)44 b(A)33 b(form)m(ula)e(analogous)g(to)h(\(8.176\))g(and)h(Lemma)e(8.10)h (yield)g(that)g Fq(G)3148 1292 y Fl(\000)p Fo(1)3148 1353 y Fm(m)3242 1328 y Ft(\()p Fq(z)t Ft(\))d Fp(2)f Fq(C)3567 1292 y Fm(n;\022)3560 1353 y Fo(u)3701 1328 y Ft(of)0 1449 y(this)k(set.)146 1569 y(The)i(F)-8 b(esh)m(bac)m(h)34 b(form)m(ula)d(yields)h(that)g(for)g Fq(z)h Fp(2)28 b Fr(C)2011 1584 y Fo(+)2070 1569 y Ft(,)90 1783 y(\()p Fq(z)f Fp(\000)c Fq(H)8 b Ft(\))427 1747 y Fl(\000)p Fo(1)604 1783 y Ft(=)27 b(\()p Fq(z)t Fr(1)p 850 1708 63 4 v -36 x Fm(m)p 913 1708 V 1 w(m)1002 1783 y Fp(\000)c Fq(H)p 1191 1708 V 1191 1747 a Fm(m)p 1253 1708 V(m)1319 1783 y Ft(\))1357 1747 y Fl(\000)p Fo(1)604 1974 y Ft(+\()p Fr(1)774 1938 y Fm(mm)925 1974 y Ft(+)f(\()p Fq(z)t Fr(1)p 1166 1900 V -36 x Fm(m)p 1229 1900 V 1 w(m)1317 1974 y Fp(\000)h Fq(H)p 1506 1900 V 1506 1938 a Fm(m)p 1568 1900 V(m)1635 1974 y Ft(\))1673 1938 y Fl(\000)p Fo(1)1767 1974 y Fq(H)p 1856 1900 V 1856 1938 a Fm(mm)1984 1974 y Ft(\))p Fq(G)2099 1938 y Fl(\000)p Fo(1)2099 1999 y Fm(m)2194 1974 y Ft(\()p Fq(z)t Ft(\)\()p Fr(1)2413 1938 y Fm(mm)2564 1974 y Ft(+)f Fq(H)2751 1938 y Fm(m)p 2813 1900 V(m)2880 1974 y Ft(\()p Fq(z)t Fr(1)p 3023 1900 V -36 x Fm(m)p 3086 1900 V 1 w(m)3175 1974 y Fp(\000)g Fq(H)p 3363 1900 V 3363 1938 a Fm(m)p 3425 1900 V(m)3492 1974 y Ft(\))3530 1938 y Fl(\000)p Fo(1)3624 1974 y Ft(\))p Fq(:)0 2195 y Ft(Sandwic)m(hing)30 b(this)f(form)m(ula)f(with)h Fp(h)p Fq(S)6 b Fp(i)1469 2159 y Fl(\000)p Fm(\026)1570 2195 y Ft(,)30 b(w)m(e)h(deriv)m(e)g(\(iii\))c(from)h(Lemma)g(8.10)i (and)f(the)i(regularit)m(y)0 2315 y(prop)s(erties)h(of)h Fq(G)647 2279 y Fl(\000)p Fo(1)647 2340 y Fm(m)741 2315 y Ft(\()p Fq(z)t Ft(\))g(pro)m(v)m(en)h(ab)s(o)m(v)m(e.)44 b(The)34 b(pro)s(of)e(of)g(Theorem)h(6.4)f(is)g(complete.)43 b Fe(2)0 2698 y Fs(References)0 2917 y Ft([AHS])93 b(Agmon,)29 b(S.,)i(Herbst,)g(I.,)f(Skibsted,)h(E.:)43 b(P)m(erturbation)29 b(of)g(em)m(b)s(edded)i(eigen)m(v)-5 b(alues)29 b(in)g(the)347 3037 y(generalized)j(N-b)s(o)s(dy)g(problem,)g(Comm)m(un.)g(Math.)h(Ph) m(ys.)h Fr(122)p Ft(,)f(411)f(\(1989\).)0 3241 y([A)m(C])153 b(Aguilar,)65 b(J.,)i(Com)m(b)s(es)61 b(J.M.:)99 b(A)60 b(class)h(of)e(analytic)g(p)s(erturbations)h(for)g(one-b)s(o)s(dy)347 3361 y(Sc)m(hr\177)-49 b(odinger)33 b(Hamiltonians,)c(Comm)m(un.)j (Math.)h(Ph)m(ys.)i Fr(22)p Ft(,)d(269)g(\(1971\).)0 3564 y([ABG])74 b(Amrein,)49 b(W.,)i(Boutet)d(de)f(Mon)m(v)m(el,)52 b(A.,)e(Georgescu,)i(V.:)72 b Fq(C)2775 3579 y Fo(0)2815 3564 y Ft(-Groups,)50 b(Comm)m(utator)347 3685 y(Metho)s(ds)39 b(and)f(Sp)s(ectral)g(Theory)h(of)f Fq(N)10 b Ft(-b)s(o)s(dy)38 b(Hamiltonians,)e(Progress)j(in)f(Math.)g(Ser.)347 3805 y(135,)32 b(Birkh\177)-49 b(auser,)33 b(Basel)f(\(1996\).)0 4009 y([Ar])182 b(Arai,)38 b(A:)g(On)g(a)f(mo)s(del)f(of)h(a)h (harmonic)e(oscillator)f(coupled)j(to)g(a)f(quan)m(tized,)j(massless,) 347 4129 y(scalar)29 b(\014eld,)i(I.)f(J.)h(Math.)f(Ph)m(ys.)i Fr(22)p Ft(,)f(2539)e(\(1981\).)g(On)h(a)g(mo)s(del)f(of)g(harmonic)g (oscillator)347 4249 y(coupled)i(to)f(a)g(quan)m(tized,)i(massless,)g (scalar)e(\014eld,)h(I)s(I.)f(J.)h(Math.)g(Ph)m(ys.)i Fr(22)p Ft(,)e(2549)e(\(1981\).)0 4453 y([AH])147 b(Arai,)33 b(A,)h(Hirok)-5 b(a)m(w)m(a,)34 b(M.:)46 b(On)34 b(the)g(existence)i (and)e(uniqueness)i(of)d(ground)h(states)h(of)e(the)347 4573 y(spin-b)s(oson)f(Hamiltonian,)d(J.)k(F)-8 b(unc.)33 b(Anal.)e Fr(151)p Ft(,)i(455)f(\(1997\).)0 4776 y([A)-11 b(W])131 b(Araki,)33 b(H.,)i(W)-8 b(o)s(o)s(ds,)33 b(E.J.:)47 b(Represen)m(tations)35 b(of)e(the)h(canonical)e(comm)m(utation)g (relations)347 4897 y(describing)g(a)g(nonrelativistic)f(in\014nite)g (free)i(Bose)g(gas,)g(J.)g(Math.)g(Ph)m(ys.)h Fr(4)p Ft(,)f(637)f(\(1963\).)0 5100 y([BFS1])57 b(Bac)m(h,)f(V.,)f(F)-8 b(r\177)-49 b(ohlic)m(h,)54 b(J.,)i(Sigal,)d(I.:)80 b(Quan)m(tum)51 b(electro)s(dynamics)f(of)g(con\014ned)i(non-)347 5221 y(relativistic)30 b(particles,)i(Adv.)h(Math.)g Fr(137)p Ft(,)g(299)f(\(1998\).)1841 5753 y(73)p eop %%Page: 74 74 74 73 bop 0 407 a Ft([BFS2])57 b(Bac)m(h)45 b(V.,)j(F)-8 b(r\177)-49 b(ohlic)m(h)43 b(J.,)48 b(Sigal,)d(I.:)68 b(Con)m(v)m(ergen)m(t)47 b(renormalization)41 b(group)j(analysis)g(for) 347 527 y(non-selfadjoin)m(t)31 b(op)s(erators)h(on)h(F)-8 b(o)s(c)m(k)32 b(space,)i(Adv.)g(Math.)e Fr(137)p Ft(,)h(205)f (\(1998\).)0 731 y([BFS3])57 b(Bac)m(h,)e(V.,)f(F)-8 b(r\177)-49 b(ohlic)m(h,)52 b(J.,)j(Sigal,)d(I.:)78 b(Sp)s(ectral)49 b(analysis)g(for)g(systems)i(of)e(atoms)g(and)347 851 y(molecules)36 b(coupled)h(to)g(the)h(quan)m(tized)g(radiation)d (\014eld,)k(Comm)m(un.)e(Math.)g(Ph)m(ys.)i Fr(207)p Ft(,)347 971 y(249)32 b(\(1999\).)0 1175 y([BFS4])57 b(Bac)m(h,)33 b(V.,)g(F)-8 b(r\177)-49 b(ohlic)m(h,)31 b(J.,)i(Sigal,)d(I.:)44 b(Return)33 b(to)f(equilibrium,)e(preprin)m(t)j (1999)0 1378 y([BFSS])52 b(Bac)m(h,)25 b(V.,)f(F)-8 b(r\177)-49 b(ohlic)m(h,)22 b(J.,)j(Sigal,)d(I.,)i(So\013er,)g(A.:)39 b(P)m(ositiv)m(e)22 b(comm)m(utators)f(and)h(the)g(sp)s(ectrum)347 1499 y(of)32 b(P)m(auli-Fierz)f(Hamiltonian)e(of)k(atoms)f(and)h (molecules,)g(Comm)m(un.)f(Math.)h(Ph)m(ys.)i Fr(207)p Ft(,)347 1619 y(557)d(\(1999\).)0 1822 y([BSZ])110 b(Baez,)29 b(J.C.,)h(Segal,)f(I.E.,)g(Zhou,)g(Z.:)41 b Fi(Intr)-5 b(o)g(duction)30 b(to)h(algebr)-5 b(aic)30 b(and)g(c)-5 b(onstructive)31 b(quan-)347 1943 y(tum)k(\014eld)f(the)-5 b(ory)p Ft(,)33 b(Princeton)g(NJ,)g(Princeton)g(Univ)m(ersit)m(y)g (Press)h(\(1991\).)0 2146 y([BC])154 b(Balslev,)32 b(E.,)i(Com)m(b)s (es,)g(J.-M.:)44 b(Sp)s(ectral)33 b(prop)s(erties)g(of)f(man)m(y-b)s(o) s(dy)h(Sc)m(hr\177)-49 b(odinger)33 b(op)s(er-)347 2267 y(ators)f(with)g(dilation)e(analytic)h(in)m(teractions,)h(Comm.)g (Math.)h(Ph)m(ys.)h Fr(22)p Ft(,)f(280)f(\(1971\).)0 2470 y([Bl])196 b(Blanc)m(hard,)33 b(P)-8 b(.:)44 b(Discussion)32 b(mathematique)f(du)i(mo)s(d)m(\022)-46 b(ele)32 b(de)h(P)m(auli)f(et)g (Fierz)h(relatif)d(\022)-49 b(a)33 b(la)347 2590 y(catastrophe)g (infrarouge,)f(Comm.)f(Math.)i(Ph)m(ys.)h Fr(15)p Ft(,)f(156)f (\(1969\).)0 2794 y([BG])147 b(Boutet)32 b(de)g(Mon)m(v)m(el,)h(A.,)f (Georgescu,)h(V.:)44 b(Boundary)32 b(v)-5 b(alues)32 b(of)f(the)h(resolv)m(en)m(t)h(of)f(a)f(self-)347 2914 y(adjoin)m(t)i(op)s(erator:)45 b(Higher)33 b(order)h(estimates,)g(in:) 46 b(Boutet)34 b(de)g(Mon)m(v)m(el,)h(A.,)f(Marc)m(henk)m(o,)347 3034 y(V.,)26 b(eds,)h(Algebraic)c(and)i(Geometric)e(Metho)s(ds)i(in)f (Mathematical)e(Ph)m(ysics,)28 b(9-52,)d(Klu)m(w)m(er)347 3155 y(Academic)32 b(Publishers)h(1996.)0 3358 y([BGS])93 b(Boutet)33 b(de)g(Mon)m(v)m(el,)g(A.,)g(Georgescu,)g(V.,)g(Sah)m (bani,)g(J.:)43 b(Higher)32 b(order)h(estimates)f(in)g(the)347 3479 y(conjugate)h(op)s(erator)e(theory)-8 b(,)34 b(preprin)m(t.)0 3682 y([BS])170 b(Boutet)32 b(de)g(Mon)m(v)m(el,)g(A.,)g(Sah)m(bani,)g (J.:)43 b(On)31 b(the)h(sp)s(ectral)g(prop)s(erties)f(of)g(the)h (Spin-Boson)347 3802 y(Hamiltonians,)d(Lett.)k(Math.)g(Ph)m(ys.)h Fr(44)p Ft(,)f(23)f(\(1998\).)0 4006 y([BR])152 b(Brattelli,)39 b(O,)h(Robinson)g(D.)g(W.:)59 b Fi(Op)-5 b(er)g(ator)41 b(A)n(lgebr)-5 b(as)41 b(and)h(Quantum)g(Statistic)-5 b(al)41 b(Me-)347 4126 y(chanics)34 b(2)p Ft(,)e(Springer,)g(Berlin)g (\(1981\).)0 4330 y([CT])153 b(Cohen-T)-8 b(annoudji)30 b(C.,)h(Dup)s(on)m(t-Ro)s(c,)e(J.,)i(Gryn)m(b)s(erg,)g(G.:)41 b Fi(Photons)32 b(and)g(A)n(toms{)g(Intr)-5 b(o-)347 4450 y(duction)35 b(to)g(Quantum)f(Ele)-5 b(ctr)g(o)g(dynamics)p Ft(,)32 b(John)h(Wiley)-8 b(,)32 b(New)h(Y)-8 b(ork)33 b(\(1991\).)0 4653 y([CFKS])49 b(Cycon,)36 b(H.,)f(F)-8 b(ro)s(ese,)34 b(R.,)h(Kirsc)m(h,)g(W.,)g(Simon,)e(B.:)46 b Fi(Schr\177)-50 b(odinger)35 b(Op)-5 b(er)g(ators)p Ft(,)34 b(Springer-)347 4774 y(V)-8 b(erlag,)31 b(Berlin)h(\(1987\).)0 4977 y([De])175 b(Derezi)s(\023)-51 b(nski,)53 b(J.:)79 b(Asymptotic)50 b(completeness)g(in)g(quan)m(tum)g(\014eld)g(theory)-8 b(.)51 b(A)f(class)g(of)347 5098 y(Gallilei-co)m(v)-5 b(arian)m(t)27 b(mo)s(dels,)k(Rev.)i(Math.)g(Ph)m(ys.)i Fr(10)p Ft(,)d(191)g(\(1997\).)1841 5753 y(74)p eop %%Page: 75 75 75 74 bop 0 407 a Ft([DG])141 b(Derezi)s(\023)-51 b(nski,)46 b(J.,)i(G)m(\023)-46 b(erard,)47 b(C.:)67 b(Asymptotic)44 b(completeness)h(in)f(quan)m(tum)h(\014eld)f(theory)-8 b(.)347 527 y(Massiv)m(e)34 b(P)m(auli-Fierz)c(Hamiltonians,)f(Rev.)34 b(Math.)f(Ph)m(ys.)h Fr(11)p Ft(,)f(383)f(\(1999\).)0 731 y([DJP])102 b(Derezi)s(\023)-51 b(nski,)31 b(J.,)i(Jak)-5 b(\024)-44 b(s)q(i)m(\023)e(c,)32 b(V.,)h(Pillet,)d(C.-A.:)44 b(In)33 b(preparation.)0 934 y([Di])190 b(Dirac,)30 b(P)-8 b(.A.M.:)44 b(The)32 b(quan)m(tum)f(theory)h(of)e(the)i(emission)e(and) h(absorption)f(of)h(radiation.)347 1054 y(Pro)s(c.)i(Ro)m(y)m(al)f(So)s (c.)g(London,)h(Series)g(A)g Fr(114)p Ft(,)f(243)g(\(1927\).)0 1258 y([DuSp])56 b(D)s(\177)-51 b(umc)m(k)m(e,)27 b(R.,)h(Sp)s(ohn,)g (H.:)41 b(Quan)m(tum)26 b(tunneling)g(with)g(dissipation)f(and)i(the)g (Ising)f(mo)s(del)347 1378 y(o)m(v)m(er)33 b Fr(R)p Ft(,)g(J.)f(Stat.)h (Ph)m(ys.)h Fr(41)p Ft(,)f(389)f(\(1985\).)0 1582 y([FNV])83 b(F)-8 b(annes,)38 b(M.,)f(Nac)m(h)m(tergaele,)h(B.,)g(V)-8 b(erb)s(eure,)38 b(A.:)51 b(The)37 b(equilibrium)c(states)38 b(of)d(the)i(spin-)347 1702 y(b)s(oson)32 b(mo)s(del,)g(Comm)m(un.)g (Math.)h(Ph)m(ys.)h Fr(114)p Ft(,)f(537)f(\(1988\).)0 1905 y([F)-8 b(r])199 b(F)-8 b(riedric)m(hs,)49 b(K.)d(O.:)70 b Fi(Perturb)-5 b(ation)48 b(of)e(Sp)-5 b(e)g(ctr)g(a)47 b(in)g(Hilb)-5 b(ert)48 b(Sp)-5 b(ac)g(e)p Ft(,)48 b(AMS,)f(Pro)m (vidence)347 2026 y(\(1965\).)0 2229 y([F)-8 b(ri])171 b(F)-8 b(rigerio,)28 b(A.:)43 b(Quan)m(tum)31 b(dynamical)e(semigroups) h(and)h(approac)m(h)g(to)f(equilibrium,)e(Lett.)347 2350 y(Math.)33 b(Ph)m(ys.)h Fr(2)p Ft(,)f(79)f(\(1997\).)0 2553 y([FH])156 b(F)-8 b(ro)s(ese,)38 b(R.,)h(Herbst,)g(I.:)53 b(A)37 b(new)h(pro)s(of)e(of)h(the)g(Mourre)h(estimate,)g(Duk)m(e)g (Math.)f(J.)g Fr(49)p Ft(,)347 2673 y(1075)31 b(\(1982\).)0 2877 y([GJ])166 b(Glimm.)40 b(J.,)46 b(Ja\013e,)g(A.:)66 b Fi(Quantum)44 b(Physics.)h(A)g(F)-7 b(unctional)43 b(Inte)-5 b(gr)g(al)44 b(Point)h(of)g(View,)347 2997 y Ft(second)34 b(edition,)d(Springer,)h(New)i(Y)-8 b(ork)32 b(\(1987\).)0 3200 y([GGK])63 b(Goh)m(b)s(erg,)31 b(I.,)g(Goldb)s(erg,) f(S.,)i(Kaasho)s(ek,)f(M.)h(A.:)42 b Fi(Classes)32 b(of)h(Line)-5 b(ar)33 b(Op)-5 b(er)g(ators)p Ft(,)31 b(V)-8 b(ol.)30 b(2,)347 3321 y(Birkh\177)-49 b(auser)33 b(1993.)0 3524 y([Ha])171 b(Haag,)32 b(R.:)43 b Fi(L)-5 b(o)g(c)g(al)35 b(Quantum)g(Physics)p Ft(,)d(New)h(Y)-8 b(ork,)33 b(Springer)f (\(1993\).)0 3728 y([He])177 b(Heitler,)27 b(W.:)41 b Fi(The)29 b(Quantum)g(The)-5 b(ory)30 b(of)f(R)-5 b(adiation,)27 b Ft(Oxford,)h(Oxford)f(Univ)m(ersit)m(y)h(Press)347 3848 y(\(1954\).)0 4051 y([Ho)m(w])104 b(Ho)m(wland,)41 b(J.:)56 b(The)40 b(Livsic)f(matrix)e(in)i(p)s(erturbation)f(theory)-8 b(,)41 b(Jour.)f(Math.)f(Anal.)f(and)347 4172 y(Appl.)32 b Fr(50)p Ft(,)h(415)f(\(1975\).)0 4375 y([HHW])49 b(Haag,)32 b(R.,)g(Hugenholtz,)g(N.M.,)g(Winnik,)f(M.:)44 b(On)32 b(the)g(equilibrium)d(states)j(in)f(quan)m(tum)347 4496 y(statistical)f(mec)m(hanics,)j(Comm)m(un.)f(Math.)h(Ph)m(ys.)i Fr(5)p Ft(,)d(215)g(\(1967\).)0 4699 y([HuSp1])49 b(H)s(\177)-51 b(ubner,)47 b(M.,)f(Sp)s(ohn,)g(H.:)66 b(Radiativ)m(e)42 b(deca)m(y:)66 b(non-p)s(erturbativ)m(e)43 b(approac)m(hes,)48 b(Rev.)347 4819 y(Math.)33 b(Ph)m(ys.)h Fr(7)p Ft(,)f(363)f(\(1995\).)0 5023 y([HuSp2])49 b(H)s(\177)-51 b(ubner,)27 b(M.,)h(Sp)s(ohn,)f(H.:)40 b(Sp)s(ectral)25 b(prop)s(erties)h(of)f(the)h(spin-b)s(oson)f (Hamiltonian,)e(Ann.)347 5143 y(Inst.)33 b(H.)g(P)m(oincare)g Fr(62)p Ft(,)f(289)g(\(1995\).)1841 5753 y(75)p eop %%Page: 76 76 76 75 bop 0 407 a Ft([JP1])128 b(Jak)-5 b(\024)-44 b(s)q(i)m(\023)e(c,) 31 b(V.,)g(Pillet,)f(C.-A.:)43 b(On)31 b(a)g(mo)s(del)f(for)g(quan)m (tum)i(friction)d(I)s(I:)i(F)-8 b(ermi's)30 b(golden)h(rule)347 527 y(and)i(dynamics)f(at)g(p)s(ositiv)m(e)g(temp)s(erature,)h(Comm)m (un.)f(Math.)h(Ph)m(ys.)h Fr(176)p Ft(,)f(619)f(\(1996\).)0 731 y([JP2])128 b(Jak)-5 b(\024)-44 b(s)q(i)m(\023)e(c,)30 b(V.,)h(Pillet,)d(C.-A.:)43 b(On)30 b(a)g(mo)s(del)e(for)i(quan)m(tum)g (friction)e(I)s(I)s(I:)i(Ergo)s(dic)g(prop)s(erties)347 851 y(of)i(the)h(spin-b)s(oson)f(system,)i(Comm)m(un.)e(Math.)h(Ph)m (ys.)h Fr(178)p Ft(,)f(627)f(\(1996\).)0 1054 y([JP3])128 b(Jak)-5 b(\024)-44 b(s)q(i)m(\023)e(c,)40 b(V.,)g(Pillet,)f(C.-A.:)56 b(Sp)s(ectral)39 b(theory)g(of)g(thermal)e(relaxation,)i(J.)g(Math.)h (Ph)m(ys.)347 1175 y Fr(38)p Ft(,)33 b(1757)e(\(1997\).)0 1378 y([JMP])88 b(Jensen,)37 b(A.,)e(Mourre,)h(E.,)f(P)m(erry)-8 b(,)37 b(P)-8 b(.:)47 b(Multiple)33 b(comm)m(utator)g(estimates)i(and)f (resolv)m(en)m(t)347 1499 y(smo)s(othness)c(in)e(quan)m(tum)i (scattering)f(theory)-8 b(,)30 b(Ann.)g(Inst.)g(H.)g(P)m(oincare)f Fr(41)p Ft(,)h(207)e(\(1984\).)0 1702 y([Kato])81 b(Kato,)43 b(T.:)62 b Fi(Perturb)-5 b(ation)44 b(The)-5 b(ory)43 b(for)g(Line)-5 b(ar)42 b(Op)-5 b(er)g(ators)p Ft(,)44 b(second)f(edition,)f(Springer-)347 1822 y(V)-8 b(erlag,)31 b(Berlin)h(\(1976\).)0 2026 y([LCD])87 b(Legget)33 b(A.J.,)g(Chakra)m (v)-5 b(art)m(y)35 b(S.,)e(Dorsey)h(A.T.,)f(Fisher)g(M.P)-8 b(.A.,)34 b(Garg)e(A.,)h(Zw)m(erger)h(W.,)347 2146 y(Dynamics)e(of)g (the)h(dissipativ)m(e)f(t)m(w)m(o-state)h(system,)h(Rev.)f(Mo)s(d.)f (Ph)m(ys.)j Fr(59)p Ft(,)e(1)f(\(1987\).)0 2350 y([MeMo])50 b(Mennic)m(k)m(en,)37 b(R.,)f(Moto)m(vilo)m(v,)e(A.)h(K.:)48 b(Op)s(erator)34 b(in)m(terpretation)g(of)h(resonances)h(arising)347 2470 y(in)31 b(sp)s(ectral)g(problems)g(for)g(2)20 b Fp(\002)g Ft(2)32 b(op)s(erator)f(matrices,)g(Theoret.)h(and)g(Math.)g (Ph)m(ys.)h Fr(116)p Ft(,)347 2590 y(867)f(\(1998\).)0 2794 y([Mo])155 b(Mourre,)34 b(E.:)46 b(Absence)35 b(of)e(singular)f (con)m(tin)m(uous)i(sp)s(ectrum)g(for)e(certain)h(self-adjoin)m(t)f (op-)347 2914 y(erators,)h(Comm)m(un.)f(Math.)h(Ph)m(ys.)h Fr(78)p Ft(,)f(391)f(\(1981\).)0 3117 y([O)m(Y])147 b(Ok)-5 b(amoto)29 b(T.,)j(Y)-8 b(a)5 b(jima,)30 b(K:)g(Complex)h(scaling)f (tec)m(hnique)i(in)e(non-relativistic)f(qed,)j(Ann.)347 3238 y(Inst.)h(H.)g(P)m(oincare)g Fr(42)p Ft(,)f(311)g(\(1985\).)0 3441 y([PF])163 b(P)m(auli)31 b(W.,)i(Fierz,)f(M.:)44 b(Nuo)m(v)m(o)34 b(Cimen)m(to)d Fr(15)p Ft(,)i(167)f(\(1938\).)0 3645 y([PSS])119 b(P)m(erry)-8 b(,)35 b(P)-8 b(.)34 b(Sigal,)e(I.)h(M.) h(Simon,)f(B.:)45 b(Sp)s(ectral)33 b(analysis)g(of)g(N-b)s(o)s(dy)g(Sc) m(hr\177)-49 b(odinger)34 b(op)s(era-)347 3765 y(tors,)f(Ann.)g(Math.)g Fr(114)p Ft(,)f(519)g(\(1981\).)0 3968 y([Ro1])123 b(Robinson,)32 b(D.W.:)43 b(Return)33 b(to)f(equilibrium,)e(Comm)m(un.)i(Math.)h(Ph)m (ys.)i Fr(31)p Ft(,)d(171)g(\(1973\).)0 4172 y([Ro2])123 b(Robinson,)43 b(D.W.:)61 b Fq(C)1209 4136 y Fl(\003)1249 4172 y Ft(-algebras)40 b(in)h(quan)m(tum)h(statistical)d(mec)m(hanics,) 45 b(in)40 b Fq(C)3370 4136 y Fl(\003)3410 4172 y Fi(-algebr)-5 b(as)347 4292 y(and)38 b(their)h(Applic)-5 b(ations)39 b(to)g(Statistic)-5 b(al)39 b(Me)-5 b(chanics)38 b(and)h(Quantum)g (Field)f(The)-5 b(ory)p Ft(,)38 b(\(D.)347 4413 y(Kastler)32 b(editor\),)g(Amsterdam,)g(North-Holand)f(\(1976\).)0 4616 y([RS1])118 b(Reed,)44 b(M.,)f(Simon,)f(B.:)60 b Fi(Metho)-5 b(ds)43 b(of)f(Mo)-5 b(dern)43 b(Mathematic)-5 b(al)42 b(Physics,)i(I.)e(F)-7 b(unctional)347 4736 y(A)n(nalysis)p Ft(,)32 b(London,)h(Academic)f(Press)i(\(1980\).)0 4940 y([RS2])118 b(Reed,)27 b(M.,)f(Simon,)f(B.:)39 b Fi(Metho)-5 b(ds)27 b(of)g(Mo)-5 b(dern)27 b(Mathematic)-5 b(al)27 b(Physics,)h(II.)f(F)-7 b(ourier)26 b(A)n(nal-)347 5060 y(ysis,)35 b(Self-A)-5 b(djointness)p Ft(,)31 b(London,)h(Academic)h (Press)h(\(1975\).)1841 5753 y(76)p eop %%Page: 77 77 77 76 bop 0 407 a Ft([RS3])118 b(Reed,)38 b(M.,)g(Simon,)f(B.:)51 b Fi(Metho)-5 b(ds)39 b(of)f(Mo)-5 b(dern)39 b(Mathematic)-5 b(al)38 b(Physics,)h(III.)e(Sc)-5 b(attering)347 527 y(The)g(ory)p Ft(,)32 b(London,)h(Academic)f(Press)i(\(1978\).)0 731 y([RS4])118 b(Reed,)33 b(M.,)h(Simon,)d(B.:)44 b Fi(Metho)-5 b(ds)35 b(of)g(Mo)-5 b(dern)35 b(Mathematic)-5 b(al)35 b(Physics,)f(IV.)h(A)n(nalysis)g(of)347 851 y(Op)-5 b(er)g(ators)p Ft(,)32 b(London,)h(Academic)f(Press)i(\(1978\).)0 1054 y([RZ])161 b(Raggio,)24 b(G.)g(A.,)i(Zivi,)f(S.)f(H.:)40 b(Semiclassical)22 b(description)i(of)g Fq(N)10 b Ft(-lev)m(el)24 b(systems)i(in)m(teracting)347 1175 y(with)36 b(radiation)e(\014elds,)k (in)e Fi(Quantum)j(pr)-5 b(ob)g(ability)38 b(II,)f(Heidelb)-5 b(er)g(g)38 b(1984)p Ft(,)f(Lecture)h(Notes)347 1295 y(in)32 b(Math.)h(1136,)e(Springer-V)-8 b(erlag)31 b(\(1985\).)0 1499 y([Si])211 b(Simon,)27 b(B.:)42 b(Resonances)30 b(in)d(N-b)s(o)s(dy)h(quan)m(tum)g(systems)i(with)e(dilation)d (analytic)i(p)s(oten-)347 1619 y(tial)i(and)j(foundations)f(of)g (time-dep)s(enden)m(t)g(p)s(erturbation)g(theory)-8 b(,)32 b(Ann.)g(Math.)g Fr(97)p Ft(,)g(247)347 1739 y(\(1973\).)0 1943 y([Sk])188 b(Skibsted,)37 b(E.:)49 b(Sp)s(ectral)35 b(analysis)f(of)h Fq(N)10 b Ft(-b)s(o)s(dy)35 b(systems)i(coupled)e(to) g(a)g(b)s(osonic)f(system,)347 2063 y(Rev.)f(Math.)g(Ph)m(ys.)h Fr(10)p Ft(,)f(989)f(\(1998\).)0 2267 y([Sp1])136 b(Sp)s(ohn,)32 b(H.:)43 b(Ground)32 b(states)g(of)g(the)g(spin-b)s(oson)f (hamiltonian,)d(Comm)m(un.)k(Math.)g(Ph)m(ys.)347 2387 y Fr(123)p Ft(,)h(277)e(\(1989\).)0 2590 y([Sp2])136 b(Sp)s(ohn,)42 b(H.:)59 b(Ground)40 b(state)g(of)g(a)g(quan)m(tum)g (particle)f(coupled)h(to)g(a)f(scalar)h(Bose)h(\014eld,)347 2711 y(Lett.)33 b(Math.)g(Ph)m(ys.)h Fr(44)p Ft(,)f(9)f(\(1998\).)0 2914 y([Sp3])136 b(Sp)s(ohn,)43 b(H.:)61 b(An)41 b(algebraic)e (condition)g(for)i(the)g(approac)m(h)h(to)e(equilibrium)e(of)i(an)h(op) s(en)347 3034 y Fq(N)10 b Ft(-lev)m(el)32 b(system,)i(Lett.)f(Math.)f (Ph)m(ys.)j Fr(2)p Ft(,)e(33)f(\(1977\).)0 3238 y([W)-8 b(e])158 b(W)-8 b(ein)m(b)s(erger,)40 b(H.)f(F.:)56 b(V)-8 b(ariational)35 b(metho)s(ds)k(fror)f(eigen)m(v)-5 b(alue)38 b(appro)m(ximation,)g(CBMS,)347 3358 y(v)m(ol.)32 b(15,)g(SIAM,)i (Philadelphia)c(\(1974\).)1841 5753 y(77)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF