Content-Type: multipart/mixed; boundary="-------------0306130919579" This is a multi-part message in MIME format. ---------------0306130919579 Content-Type: text/plain; name="03-280.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-280.keywords" thermal ionization, stationary states, thermal field, KMS state, equilibrium state, black body radiation, C* algebra, von Neumann algebra, standard form, Liouville operator, positive commutator, virial theorem, CCR algebra, Fermi Golden Rule, radiation field, return to equilibrium ---------------0306130919579 Content-Type: application/postscript; name="fms-ti.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fms-ti.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: fms-ti.dvi %%Pages: 53 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips fms-ti -o %DVIPSParameters: dpi=600, 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b(stationary)h(state)g(disin)m(tegrates)f(after)h(coupling)d(to) j(the)f(radiation)f(\014eld)572 3344 y(then)25 b(it)h(is)f(unstable,)h (pro)m(vided)e(the)i(coupling)f(constan)m(t)i(is)e(su\016cien)m(tly)f (small)572 3457 y(\(dep)s(ending)k(on)i(the)h(temp)s(erature\).)708 3570 y(The)43 b(result)g(is)f(pro)m(v)m(en)i(b)m(y)f(analyzing)g(the)h (sp)s(ectrum)e(of)i(the)g(thermal)572 3683 y(Hamiltonian)27 b(\(Liouvillian\))e(of)j(the)h(system)g(within)d(the)i(framew)m(ork)h (of)g Fy(W)3253 3650 y Fx(\003)3292 3683 y Fz(-)572 3796 y(dynamical)43 b(systems.)85 b(A)45 b(k)m(ey)h(elemen)m(t)f(of)g(our)g (sp)s(ectral)f(analysis)g(is)g(the)572 3909 y(p)s(ositiv)m(e)29 b(comm)m(utator)j(metho)s(d.)p 328 4273 1296 4 v 439 4335 a Fw(\003)477 4365 y Fv(Institut)41 b(f)r(\177)-44 b(ur)39 b(Theoretisc)n(he)f(Ph)n(ysik,)j(ETH)e(H\177)-42 b(onggerb)r(erg,)40 b(CH-8093)d(Z)r(\177)-44 b(uric)n(h,)41 b(Switzerland)328 4464 y(\(juerg@ph)n(ys.ethz.c)n(h\))443 4535 y Fw(y)477 4565 y Fv(Institut)g(f)r(\177)-44 b(ur)39 b(Theoretisc)n(he)f(Ph)n(ysik,)j(ETH)e(H\177)-42 b(onggerb)r(erg,)40 b(CH-8093)d(Z)r(\177)-44 b(uric)n(h,)41 b(Switzerland)328 4664 y(\(merkli@ph)n(ys.ethz.c)n(h\))443 4735 y Fw(z)477 4765 y Fv(Departmen)n(t)30 b(of)g(Mathematics,)g(Univ)n(ersit)n(y)e(of) i(T)-7 b(oron)n(to,)28 b(T)-7 b(oron)n(to,)29 b(ON,)h(M5S)f(3G3)g (Canada)328 4864 y(\(w)n(ebpage:)36 b(www.math.toron)n(to.edu/sigal\)) 443 4935 y Fw(x)477 4965 y Fv(Supp)r(orted)28 b(b)n(y)f(NSER)n(C)h (under)f(gran)n(t)g(NA7901.)p eop %%Page: 2 2 2 1 bop 328 631 a Fu(Con)l(ten)l(ts)328 850 y Ft(1)90 b(In)m(tro)s(duction)2418 b(3)328 1068 y(2)90 b(De\014nition)36 b(of)i(the)f(mo)s(del)g(and)h(main)f(results)1038 b(6)474 1188 y Fs(2.1)100 b(De\014nition)30 b(of)j(the)g(mo)s(del)46 b(.)k(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)f(.)h(.)g(.)138 b(6)699 1309 y(2.1.1)111 b(Kinematical)29 b(algebra)j Fr(A)p Fs(,)h(and)f(regularized)g(dynamics)g Fq(\013)3213 1258 y Fp(\()p Fo(\017)p Fp(\))3212 1336 y Fo(t;\025)3352 1309 y Fs(.)138 b(7)699 1440 y(2.1.2)111 b(Reference)34 b(state)f Fq(!)1758 1404 y Fp(ref)1891 1440 y Fs(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)f(.)h(.)g(.)89 b(13)699 1560 y(2.1.3)111 b Fq(W)1117 1524 y Fx(\003)1156 1560 y Fs(-dynamical)30 b(system)k(\()p Fr(M)2124 1575 y Fo(\014)2170 1560 y Fq(;)17 b(\033)2269 1575 y Fo(t;\025)2360 1560 y Fs(\))31 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)f(.)h(.)g(.)89 b(16)699 1681 y(2.1.4)111 b(Characterization) 29 b(of)h(the)g Fq(\033)2080 1696 y Fo(t;\025)2171 1681 y Fs(-in)m(v)-5 b(arian)m(t)29 b(normal)f(states)j(on)1011 1801 y Fr(M)1116 1816 y Fo(\014)1199 1801 y Fs(.)50 b(.)g(.)g(.)f(.)h (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)f(.)h(.)g(.)89 b(17)699 1921 y(2.1.5)111 b(A)32 b(quic)m(k-reference)j(list)53 b(.)d(.)g(.)g(.)g(.)f(.)h(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)89 b(18)474 2042 y(2.2)100 b(Main)32 b(results)48 b(.)i(.)g(.)f(.)h(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.) h(.)g(.)89 b(19)328 2260 y Ft(3)h(Virial)35 b(theorems)i(and)h(the)g(p) s(ositiv)m(e)e(comm)m(utator)f(metho)s(d)348 b(20)474 2380 y Fs(3.1)100 b(Tw)m(o)33 b(abstract)g(virial)d(theorems)j(.)50 b(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g (.)89 b(21)474 2501 y(3.2)100 b(Outline)41 b(of)h(the)h(pro)s(ofs)f(of) g(Theorems)i(2.1)e(and)g(2.3;)48 b(the)43 b(p)s(ositiv)m(e)699 2621 y(comm)m(utator)31 b(metho)s(d)88 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)89 b(23)474 2741 y(3.3)100 b(Applications)30 b(of)j(the)g(virial)d (theorems)94 b(.)49 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h (.)g(.)89 b(26)699 2862 y(3.3.1)111 b(Setting)32 b(for)g(Theorem)h(2.1) 89 b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g (.)89 b(28)699 2982 y(3.3.2)111 b(Setting)32 b(for)g(Theorem)h(2.3)89 b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.) 89 b(29)328 3200 y Ft(4)h(Pro)s(of)37 b(of)h(Theorem)f(2.1)1925 b(31)328 3418 y(5)90 b(Pro)s(of)37 b(of)h(Theorem)f(2.3)1925 b(35)474 3538 y Fs(5.1)100 b(Reduction)32 b(of)g(the)h(Liouvillian)65 b(.)50 b(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.) h(.)g(.)89 b(35)474 3659 y(5.2)100 b(The)33 b(k)m(ernel)g(of)f Fq(L)1363 3674 y Fo(\025)1437 3659 y Fn(\026)27 b Fs(Ran)16 b Fq(P)54 b Fs(.)c(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)f(.)h(.)g(.)89 b(36)328 3877 y Ft(A)61 b(App)s(endix)2502 b(41)474 3997 y Fs(A.1)76 b(In)m(v)-5 b(ariance)32 b(of)g(domains,)g(comm)m(utator)f(expansion)76 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)89 b(41)474 4117 y(A.2)76 b(Pro)s(of)32 b(of)g(Prop)s(osition)e(3.3)88 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)f(.)h(.)g(.)89 b(43)474 4238 y(A.3)76 b(Pro)s(of)32 b(of)g(Prop)s(osition)e(3.4)88 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)89 b(47)474 4358 y(A.4)76 b(Pro)s(of)32 b(of)g(Prop)s(osition)e(3.2)88 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)f(.)h(.)g(.)89 b(50)1922 5214 y(2)p eop %%Page: 3 3 3 2 bop 328 631 a Fu(1)161 b(In)l(tro)t(duction)328 850 y Fs(In)31 b(this)f(pap)s(er,)h(w)m(e)g(study)h(a)e(quan)m(tum)h(mec)m (hanical)e(mo)s(del)g(of)g(an)i(atom)e(in)m(teracting)328 970 y(with)k(blac)m(k-b)s(o)s(dy)h(radiation.)45 b(The)35 b(atom)e(is)g(describ)s(ed)i(b)m(y)g(an)f(electron)g(mo)m(ving)f(in)328 1091 y(a)f(p)s(oten)m(tial,)f(e.g.)44 b(the)33 b(Coulom)m(b)e(p)s(oten) m(tial)g(of)h(a)h(static)f(n)m(ucleus.)474 1211 y(Our)47 b(goal)f(is)g(to)h(sho)m(w)h(that)f(when)h(the)f(atom)f(is)h(coupled)g (to)f(the)i(quan)m(tized)328 1332 y(radiation)34 b(\014eld)i(in)f(the)i (state)f(of)g(blac)m(k-b)s(o)s(dy)g(radiation)e(at)h(su\016cien)m(tly)i (high)f(tem-)328 1452 y(p)s(erature,)h(it)e(is)g(ionized.)52 b(W)-8 b(e)36 b(describ)s(e)h(this)e(ionization)e(pro)s(cess)k(b)m(y)g (sho)m(wing)f(that)328 1572 y(stationary)h(states)i(of)e(the)i(system)f (b)s(ecome)g(unstable)g(when)h(the)g(atom)d(is)i(coupled)328 1693 y(to)h(the)g(radiation)e(\014eld.)63 b(Eac)m(h)40 b(b)s(ound)f(state)g(of)g(the)g(atom)f(leads)h(to)g(a)f(stationary)328 1813 y(state)30 b(of)f(the)h(uncoupled)g(system,)i(where)f(the)f (\014eld)f(is)g(in)g(an)h(equilibrium)c(state.)43 b(W)-8 b(e)328 1933 y(consider)34 b(a)f(class)g(of)g(small)e(in)m(teractions)i (lo)s(calized)e(in)h(space)j(whic)m(h)f(couple)f FB(\014nitely)328 2054 y Fs(man)m(y)43 b(b)s(oundstates)g(of)g(the)g(atom)e(to)h(the)h (\014eld.)74 b(W)-8 b(e)43 b(sho)m(w)h(that)e(the)h(stationary)328 2174 y(states)35 b(for)f(the)h(coupled)g(system)g(corresp)s(ond)g(to)f (those)h(eigen)m(v)-5 b(alues)35 b(of)f(the)h(atomic)328 2295 y(Hamiltonian)21 b(whic)m(h)26 b(are)f(not)g(coupled)h(to)e(the)i (\014eld.)41 b(In)25 b(other)h(w)m(ords,)h(all)c(stationary)328 2415 y(states)29 b(arising)e(from)h(atomic)e(b)s(ound)j(states)h(whic)m (h)f(are)f(coupled)h(to)f(the)h(\014eld)g(b)m(y)g(the)328 2535 y(in)m(teraction)i(are)i(unstable,)g(pro)m(vided)g(the)g(coupling) e(constan)m(t)j(is)e(small)e(enough.)474 2656 y(This)k(instabilit)m(y)e (is)h(explained)h(b)m(y)h(the)f(follo)m(wing)e(mec)m(hanism.)47 b(If)33 b(the)i(electron)328 2776 y(is)g(in)f(a)h(b)s(ound)g(state)h (with)e(energy)j Fq(E)h(<)32 b Fs(0,)j(it)f(will)f(b)s(e)j(hit,)f (after)f(some)h(time,)g(b)m(y)h(a)328 2896 y(quan)m(tum)c(\(photon\))g (of)f(energy)i Fq(!)e Fm(\025)d(\000)p Fq(E)6 b Fs(,)32 b(and)g(hence)i(will)29 b(mak)m(e)j(a)f(transition)g(to)g(a)328 3017 y(scattering)h(state)h(of)f(energy)i Fq(E)28 b Fs(+)22 b Fq(!)t Fs(.)43 b(In)33 b(other)g(w)m(ords,)g(the)g(atom)f(is)g (ionized.)474 3137 y(The)27 b(a)m(v)m(erage)h(time,)e Fq(t)1291 3152 y Fo(E)1351 3137 y Fs(,)h(it)f(tak)m(es)h(for)f(an)g (atom)f(in)h(a)g(b)s(ound)g(state)h(of)f(energy)h Fq(E)32 b Fs(to)328 3258 y(b)s(e)c(ionized)f(is)g(giv)m(en,)i(to)e(second)j (order)d(in)h(the)g(p)s(erturbation,)g(b)m(y)g(the)h(F)-8 b(ermi)26 b(Golden)328 3378 y(Rule.)43 b(Heuristically)-8 b(,)30 b(for)i(an)h(in)m(v)m(erse)h(temp)s(erature)e Fq(\014)38 b Fs(of)33 b(the)g(\014eld,)f(it)g(satis\014es)1550 3569 y Fq(t)1585 3584 y Fo(E)1672 3569 y Fm(/)c Fq(e)1822 3528 y Fo(\014)s Fx(j)p Fo(E)t Fx(j)1964 3569 y Fm(j)p Fq(p)p Fs(\()p Fq(E)6 b Fs(\))p Fm(j)2223 3528 y Fx(\000)p Fp(2)2317 3569 y Fq(;)1021 b Fs(\(1.1\))328 3761 y(where)40 b Fm(j)p Fq(p)p Fs(\()p Fq(E)6 b Fs(\))p Fm(j)875 3725 y Fp(2)953 3761 y Fs(is)38 b(the)i(probabilit)m(y)d(for)h(the)i (electron)f(to)g(mak)m(e)g(a)g(transition)e(from)328 3881 y(the)d(b)s(ound)g(state)g(to)f(a)h(scattering)f(state)h(b)m(y)h (absorbing)e(a)h(photon)f(of)g(energy)i Fq(>)30 b(E)6 b Fs(.)328 4002 y(The)38 b(factor)e Fq(e)861 3966 y Fo(\014)s Fx(j)p Fo(E)t Fx(j)1039 4002 y Fs(in)g(\(1.1\))g(can)h(b)s(e)g (explained)f(b)m(y)i(Planc)m(k's)f(la)m(w,)h(whic)m(h)f(sa)m(ys)h(that) 328 4122 y(the)32 b(probabilit)m(y)d(densit)m(y)k(for)e(a)g(photon)h (to)f(ha)m(v)m(e)i(energy)f Fq(!)j Fs(is)2826 4083 y Fp(1)p 2741 4099 206 4 v 2741 4157 a Fo(e)2774 4138 y Fl(\014)s(!)2856 4157 y Fx(\000)p Fp(1)2956 4122 y Fs(.)43 b(A)m(t)32 b(zero)g(tem-)328 4242 y(p)s(erature,)40 b Fq(\014)i Fs(=)37 b Fm(1)p Fs(,)i(one)f(\014nds)h(that)f Fq(t)1806 4257 y Fo(E)1902 4242 y Fs(=)f Fm(1)p Fs(,)i(and)f(thermal)e (ionization)f(do)s(es)k(not)328 4363 y(o)s(ccur.)52 b(F)-8 b(or)34 b(a)h(giv)m(en)g(strength)h(of)f(the)g(in)m(teraction,)g(as)h (measured)f(b)m(y)h(the)g(size)f(of)g(a)328 4483 y(coupling)e(constan)m (t)j Fq(\025)31 b Fm(2)g Fk(R)5 b Fs(,)41 b(w)m(e)36 b(are)e(able)g(to)g(sho)m(w)i(that)f(thermal)e(ionization)e(tak)m(es) 328 4604 y(place,)37 b(pro)m(vided)g(0)d Fq(<)f Fm(j)p Fq(\025)p Fm(j)h Fq(<)g(k)s(e)1562 4567 y Fx(\000)p Fp(2)p Fo(\014)s Fx(j)p Fo(E)1767 4576 y Fj(0)1801 4567 y Fx(j)1825 4604 y Fs(,)j(where)h Fq(E)2247 4619 y Fp(0)2320 4604 y Fq(<)c Fs(0)i(is)g(the)h(minimal)32 b(energy)37 b(of)328 4724 y(the)i(electron)f(in)f(a)h(b)s(ound)g(state)h(coupled)f(to)g(the) g(radiation)e(\014eld,)k(and)e Fq(k)j Fs(is)d(some)328 4844 y(constan)m(t.)56 b(This)37 b(restriction)e(is)h(of)g(tec)m (hnical)g(nature;)j(ph)m(ysically)-8 b(,)37 b(thermal)e(ioniza-)328 4965 y(tion)e(is)h(exp)s(ected)i(to)e(b)s(e)g(observ)m(ed)i(for)e (arbitrarily)d(small)h(temp)s(eratures,)j(pro)m(vided)1922 5214 y(3)p eop %%Page: 4 4 4 3 bop 328 631 a Fs(the)33 b(coupling)e(constan)m(t)j(is)e(small)e (enough)j FB(indep)-5 b(endently)31 b Fs(of)h Fq(\014)6 b Fs(.)474 872 y(Next,)37 b(w)m(e)f(describ)s(e)f(the)h(system)g(and)f (our)f(main)g(results)h(in)f(some)h(more)f(detail.)328 992 y(The)45 b(atomic)e(Hamiltonian)e(is)j(a)h(Sc)m(hr\177)-49 b(odinger)45 b(op)s(erator)e Fq(H)2680 1007 y Fo(p)2768 992 y Fs(=)48 b Fm(\000)p Fs(\001)31 b(+)f Fq(v)49 b Fs(on)44 b(the)328 1112 y(Hilb)s(ert)30 b(space)j Fm(H)1007 1127 y Fo(p)1074 1112 y Fs(=)28 b Fq(L)1244 1076 y Fp(2)1284 1112 y Fs(\()p Fk(R)1387 1076 y Fp(3)1433 1112 y Fq(;)17 b(d)1528 1076 y Fp(3)1567 1112 y Fq(x)p Fs(\),)32 b(where)h Fq(v)i Fs(b)s(elongs)c(to)h(a)f(certain)g(class)h(of)f(p)s(oten-)328 1233 y(tials)25 b(including)g(the)j(Coulom)m(b)e(p)s(oten)m(tial)f (regularized)h(at)h(the)g(origin.)39 b(The)28 b(op)s(erator)328 1353 y Fq(H)409 1368 y Fo(p)481 1353 y Fs(generates)34 b(the)f(Heisen)m(b)s(erg)g(dynamics)1499 1561 y Fq(\013)1562 1513 y Fo(p)1561 1583 y(t)1601 1561 y Fs(\()p Fq(A)p Fs(\))28 b(=)f Fq(e)1926 1519 y Fo(itH)2033 1527 y Fl(p)2074 1561 y Fq(Ae)2192 1519 y Fx(\000)p Fo(itH)2354 1527 y Fl(p)328 1768 y Fs(on)32 b(the)h(v)m(on)h(Neumann)e(algebra)g Fr(A)1670 1783 y Fo(p)1738 1768 y Fs(=)27 b Fm(B)s Fs(\()p Fm(H)2031 1783 y Fo(p)2071 1768 y Fs(\))33 b(of)f(b)s(ounded)h(op)s (erators)g(on)f Fm(H)3302 1783 y Fo(p)3342 1768 y Fs(.)474 1888 y(The)43 b(\014eld)e(is)f(con)m(v)m(enien)m(tly)j(describ)s(ed)f (in)f(terms)g(of)g(a)g Fq(C)2710 1852 y Fx(\003)2749 1888 y Fs(-algebra)f Fr(A)3203 1903 y Fo(f)3249 1888 y Fs(,)j(whic)m(h)328 2009 y(can)32 b(b)s(e)f(view)m(ed)i(as)e(a)g (time-a)m(v)m(eraged)g(W)-8 b(eyl)31 b(algebra.)42 b(The)32 b(dynamics)f(is)g(giv)m(en)h(b)m(y)g(a)328 2129 y Fm(\003)p Fs(automorphism)e(group)j(of)f Fr(A)1482 2144 y Fo(f)1560 2129 y Fs(describing)g(free)h(massless)g(b)s(osons.)474 2249 y(The)h(com)m(binded)e(system)i(is)e(describ)s(ed)h(in)f(terms)h (of)f(the)h(algebra)1657 2457 y Fr(A)28 b Fs(=)g Fr(A)1931 2472 y Fo(p)1993 2457 y Fm(\012)23 b Fr(A)2164 2472 y Fo(f)2210 2457 y Fq(;)328 2664 y Fs(and)33 b(the)g(uncoupled)g (dynamics)f(is)g(giv)m(en)h(b)m(y)g(the)g(automorphisms)1628 2871 y Fq(\013)1690 2886 y Fo(t;)p Fp(0)1803 2871 y Fs(=)27 b Fq(\013)1969 2824 y Fo(p)1968 2894 y(t)2031 2871 y Fm(\012)22 b Fq(\013)2193 2824 y Fo(f)2192 2894 y(t)2238 2871 y Fq(:)328 3078 y Fs(T)-8 b(o)34 b(de\014ne)h(the)g(coupled)f (dynamics,)g(w)m(e)h(sp)s(ecify)g(a)e(\(regularized\))g(in)m(teraction) g(term)328 3199 y Fq(\025V)463 3163 y Fp(\()p Fo(\017)p Fp(\))551 3199 y Fs(,)51 b(whose)e(form)d(is)h(motiv)-5 b(ated)46 b(b)m(y)j(standard)f(mo)s(dels)e(of)h(atoms)g(in)m(teracting) 328 3319 y(with)35 b(the)g(radiation)e(\014eld.)52 b(The)36 b(regularization)c(is)j(in)m(tro)s(duced)g(to)g(guaran)m(tee)h(that)328 3440 y Fq(V)406 3403 y Fp(\()p Fo(\017)p Fp(\))522 3440 y Fm(2)28 b Fr(A)p Fs(,)j(for)e(all)f Fq(\017)g Fm(6)p Fs(=)g(0.)42 b(The)31 b(in)m(teracting)e(dynamics,)h Fq(\013)2518 3389 y Fp(\()p Fo(\017)p Fp(\))2517 3467 y Fo(t;\025)2607 3440 y Fs(,)h(is)e(then)i(de\014ned)g(as)f(the)328 3560 y Fm(\003)p Fs(automorphism)g(group)j(of)f Fr(A)h Fs(obtained)f(b)m(y)h(the)g(Sc)m(h)m(winger-Dyson)h(series.)474 3680 y(A)m(t)22 b(zero)h(temp)s(erature,)h(the)e(dynamics)g(of)f(the)i (mo)s(del)d(is)h(generated)i(b)m(y)g(the)f(formal)328 3801 y(Hamiltonian)1520 3921 y Fq(H)36 b Fs(=)27 b Fq(H)1821 3936 y Fo(p)1883 3921 y Fs(+)22 b Fq(H)2062 3936 y Fo(f)2129 3921 y Fs(+)g Fq(\025V)5 b(;)328 4090 y Fs(where)36 b Fq(H)693 4105 y Fo(f)771 4090 y Fs(=)c(d\000\()p Fm(j)p Fq(k)s Fm(j)p Fs(\))i(is)h(the)h(free-\014eld)f(Hamiltonian,)d(i.e.,)k (the)f(second)i(quan)m(tized)328 4210 y(m)m(ultiplication)19 b(op)s(erator)k Fm(j)p Fq(k)s Fm(j)p Fs(,)i(acting)e(on)h(b)s(osonic)f (F)-8 b(o)s(c)m(k)24 b(space)h Fm(F)10 b Fs(\()p Fq(L)2904 4174 y Fp(2)2943 4210 y Fs(\()p Fk(R)3047 4174 y Fp(3)3092 4210 y Fq(;)17 b(d)3187 4174 y Fp(3)3226 4210 y Fq(k)s Fs(\)\),)26 b(and)328 4331 y(the)33 b(in)m(teraction)e(term)h Fq(V)54 b Fs(is)33 b(giv)m(en)f(b)m(y)1275 4550 y Fq(V)49 b Fs(=)1485 4455 y Fi(X)1534 4664 y Fo(\013)1645 4550 y Fq(G)1722 4565 y Fo(\013)1794 4550 y Fm(\012)22 b Fs(\()p Fq(a)p Fs(\()p Fq(g)2067 4565 y Fo(\013)2117 4550 y Fs(\))g(+)g Fq(a)2326 4509 y Fx(\003)2365 4550 y Fs(\()p Fq(g)2450 4565 y Fo(\013)2499 4550 y Fs(\)\))17 b Fq(:)328 4844 y Fs(The)31 b(sum)e(is)g(o)m(v)m(er)i(a)e(\014nite)h(set,)h Fq(G)1609 4859 y Fo(\013)1688 4844 y Fs(are)f(b)s(ounded)g(selfadjoin)m (t)e(op)s(erators)i(on)g Fm(B)s Fs(\()p Fm(H)3461 4859 y Fo(p)3501 4844 y Fs(\),)328 4965 y(and)j(the)g(form)e(factors)i Fq(g)1281 4980 y Fo(\013)1362 4965 y Fs(are)g(in)f Fq(L)1705 4929 y Fp(2)1744 4965 y Fs(\()p Fk(R)1848 4929 y Fp(3)1894 4965 y Fq(;)17 b(d)1989 4929 y Fp(3)2028 4965 y Fq(k)s Fs(\).)1922 5214 y(4)p eop %%Page: 5 5 5 4 bop 474 631 a Fs(W)-8 b(e)33 b(in)m(tro)s(duce)g(a)f(reference)i (state)1636 851 y Fq(!)1701 810 y Fp(ref)1818 851 y Fs(=)28 b Fq(!)1987 810 y Fo(p)2048 851 y Fm(\012)23 b Fq(!)2213 804 y Fo(f)2209 879 y(\014)328 1093 y Fs(on)30 b Fr(A)p Fs(,)h(where)h Fq(!)935 1057 y Fo(p)1004 1093 y Fs(is)e(giv)m(en)g(b)m (y)i(a)e(\(strictly)f(p)s(ositiv)m(e\))h(densit)m(y)h(matrix,)e(and)i Fq(!)3260 1046 y Fo(f)3256 1121 y(\014)3335 1093 y Fs(is)e(the)328 1235 y(\()p Fq(\014)6 b(;)17 b(\013)534 1188 y Fo(f)533 1258 y(t)578 1235 y Fs(\)-KMS)42 b(state)g(of)g Fr(A)1350 1250 y Fo(f)1396 1235 y Fs(,)i(i.e.,)g(the)e(state)h(of)e(blac)m(k-b)s (o)s(dy)h(radiation)e(at)h(in)m(v)m(erse)328 1356 y(temp)s(erature)30 b Fq(\014)6 b Fs(.)42 b(W)-8 b(e)31 b(are)f(in)m(terested)i(in)d(the)i (time)e(ev)m(olution)g(of)h(states)h(on)f Fr(A)h Fs(whic)m(h)328 1476 y(are)44 b(close)g(to)g(\(normal)e(w.r.t.\))78 b Fq(!)1666 1440 y Fp(ref)1755 1476 y Fs(,)47 b(i.e.,)g(whic)m(h)d(are)g (represen)m(ted)j(b)m(y)e(a)f(densit)m(y)328 1596 y(matrix)39 b(on)h(the)g(GNS)g(Hilb)s(ert)f(space)i Fm(H)g Fs(of)f(\()p Fr(A)p Fq(;)17 b(!)2289 1560 y Fp(ref)2378 1596 y Fs(\).)66 b(The)41 b(GNS)f(represen)m(tation)328 1717 y(pro)m(vides)30 b(us)g(with)e(a)h(represen)m(tation)h(map)e Fq(\031)2030 1732 y Fo(\014)2106 1717 y Fs(:)f Fr(A)i Fm(!)e(B)s Fs(\()p Fm(H)q Fs(\))i(and)h(a)e(v)m(ector)j(\012)3269 1681 y Fp(ref)3387 1717 y Fm(2)d(H)328 1854 y Fs(s.t.)66 b Fq(!)589 1818 y Fp(ref)678 1854 y Fs(\()p Fq(A)p Fs(\))40 b(=)983 1773 y Fi(\012)1030 1854 y Fs(\012)1100 1818 y Fp(ref)1190 1854 y Fq(;)17 b(\031)1289 1869 y Fo(\014)1336 1854 y Fs(\()p Fq(A)p Fs(\)\012)1555 1818 y Fp(ref)1645 1773 y Fi(\013)1692 1854 y Fs(.)66 b(There)41 b(is)e(a)h(selfadjoin)m(t)e (op)s(erator)h Fq(L)3210 1803 y Fp(\()p Fo(\017)p Fp(\))3210 1881 y Fo(\025)3338 1854 y Fs(on)h Fm(H)328 1974 y Fs(generating)32 b(the)h(coupled)g(time)e(ev)m(olution)g(in)h(the)h(represen)m(tation)h Fq(\031)2950 1989 y Fo(\014)2997 1974 y Fs(,)1250 2224 y Fq(\031)1305 2239 y Fo(\014)1352 2224 y Fs(\()p Fq(\013)1453 2173 y Fp(\()p Fo(\017)p Fp(\))1452 2251 y Fo(t;\025)1542 2224 y Fs(\()p Fq(A)p Fs(\)\))28 b(=)g Fq(e)1906 2183 y Fo(itL)2003 2147 y Fj(\()p Fl(\017)p Fj(\))2003 2206 y Fl(\025)2086 2224 y Fq(\031)2141 2239 y Fo(\014)2188 2224 y Fs(\()p Fq(A)p Fs(\))p Fq(e)2382 2183 y Fx(\000)p Fo(itL)2534 2147 y Fj(\()p Fl(\017)p Fj(\))2534 2206 y Fl(\025)2617 2224 y Fq(;)328 2444 y Fs(for)k(all)e Fq(A)e Fm(2)g Fr(A)33 b Fs(and)g Fq(t)28 b Fm(2)g Fk(R)5 b Fs(.)49 b(W)-8 b(e)33 b(will)e(sho)m(w)i(that)1533 2664 y(s)p Fm(\000)17 b Fs(lim)1666 2723 y Fo(\017)p Fx(!)p Fp(0)1817 2664 y Fq(e)1862 2623 y Fo(itL)1959 2587 y Fj(\()p Fl(\017)p Fj(\))1959 2646 y Fl(\025)2070 2664 y Fs(=)27 b Fq(e)2218 2623 y Fo(itL)2315 2635 y Fl(\025)328 2911 y Fs(exists,)33 b(for)f(all)f Fq(t)p Fs(,)i(and)f(de\014nes)j(a)d Fm(\003)p Fs(automorphism)e(group)1482 3131 y Fq(\033)1537 3146 y Fo(t;\025)1628 3131 y Fs(\()p Fq(A)p Fs(\))e(=)f Fq(e)1953 3089 y Fo(itL)2050 3101 y Fl(\025)2096 3131 y Fq(Ae)2214 3089 y Fx(\000)p Fo(itL)2366 3101 y Fl(\025)328 3351 y Fs(of)32 b(the)h(v)m(on)g(Neumann)g(algebra) 1465 3571 y Fr(M)1570 3586 y Fo(\014)1644 3571 y Fs(=)28 b Fq(\031)1803 3586 y Fo(\014)1850 3571 y Fs(\()p Fr(A)p Fs(\))1997 3529 y Fx(00)2067 3571 y Fm(\032)h(B)s Fs(\()p Fm(H)q Fs(\))p Fq(:)328 3791 y Fs(The)d(pair)d(\()p Fr(M)856 3806 y Fo(\014)903 3791 y Fq(;)17 b(\033)1002 3806 y Fo(t;\025)1093 3791 y Fs(\))24 b(is)h(called)e(a)i Fq(W)1694 3754 y Fx(\003)1733 3791 y Fs(-dynamical)d(system.)42 b(Our)25 b(results)g(concern)h(the)328 3911 y(structure)34 b(of)e(the)g(set)i(of)e(normal)e(\()p Fq(\033)t Fs(-w)m(eakly)j(con)m (tin)m(uous\))g(time-translation)c(\()p Fq(\033)3405 3926 y Fo(t;\025)3495 3911 y Fs(-\))328 4031 y(in)m(v)-5 b(arian)m(t)31 b(states)j(on)e Fr(M)1255 4046 y Fo(\014)1302 4031 y Fs(.)474 4152 y(The)37 b(general)d(theory)i(of)f(v)m(on)h (Neumann)f(algebras)g(sho)m(ws)i(that)e(there)h(is)f(a)g(one-)328 4272 y(to-one)d(corresp)s(ondence)k(b)s(et)m(w)m(een)f(normal)c Fq(\033)2071 4287 y Fo(t;\025)2162 4272 y Fs(-in)m(v)-5 b(arian)m(t)31 b(states)j(on)f Fr(M)3123 4287 y Fo(\014)3202 4272 y Fs(and)g(nor-)328 4392 y(malized)e(v)m(ectors)j(in)e(the)h(set) 1710 4513 y Fm(P)e(\\)22 b Fs(k)m(er)c Fq(L)2111 4528 y Fo(\025)2157 4513 y Fq(;)1181 b Fs(\(1.2\))328 4687 y(where)47 b Fm(P)55 b Fs(is)46 b(a)g(certain)g(cone)h(in)e Fm(H)q Fs(,)50 b(the)d(so-called)e(natural)g(cone)i(asso)s(ciated)f(to) 328 4808 y(\()p Fr(M)471 4823 y Fo(\014)518 4808 y Fq(;)17 b Fs(\012)632 4771 y Fp(ref)722 4808 y Fs(\),)34 b(pro)m(vided)g(w)m(e) h(c)m(ho)s(ose)g(the)g(thermal)d(Hamiltonian)e(\(Liouvillian\))g Fq(L)3405 4823 y Fo(\025)3485 4808 y Fs(in)328 4928 y(suc)m(h)40 b(a)f(w)m(a)m(y)h(that)e(the)h(unitary)g(one-parameter)f(group)g Fm(f)p Fq(e)2616 4892 y Fo(itL)2713 4904 y Fl(\025)2798 4928 y Fm(j)g Fq(t)g Fm(2)h Fk(R)5 b Fm(g)44 b Fs(lea)m(v)m(es)c Fm(P)1922 5214 y Fs(5)p eop %%Page: 6 6 6 5 bop 328 631 a Fs(in)m(v)-5 b(arian)m(t.)41 b(Let)28 b Fm(M)g Fs(b)s(e)g(a)g(lab)s(elling)c(of)k(the)g(eigen)m(v)-5 b(alues)28 b(of)g Fq(H)2643 646 y Fo(p)2682 631 y Fs(,)h(including)e(m) m(ultiplic-)328 751 y(ities.)42 b(An)30 b(elemen)m(t)h Fq(m)d Fm(2)g(M)i Fs(is)g(called)f(a)h(mo)s(de)g(and)g(the)h(corresp)s (onding)f(eigen)m(v)-5 b(alue)328 872 y(of)32 b Fq(H)520 887 y Fo(p)592 872 y Fs(is)g(denoted)i(b)m(y)f Fq(E)6 b Fs(\()p Fq(m)p Fs(\).)44 b(W)-8 b(e)33 b(will)d(see)k(that)418 1122 y Fm(P)d(\\)22 b Fs(k)m(er)c Fq(L)819 1137 y Fp(0)887 1122 y Fs(=)27 b Fm(P)k(\\)55 b Fs(span)1406 1012 y Fi(n)1473 1122 y Fq(')1537 1137 y Fo(m)1626 1122 y Fm(\012)22 b Fq(')1789 1137 y Fo(n)1858 1122 y Fm(\012)h Fs(\012)2061 1037 y Fi(\014)2061 1097 y(\014)2126 1122 y Fq(m;)17 b(n)28 b Fm(2)g(M)p Fq(;)17 b(E)6 b Fs(\()p Fq(m)p Fs(\))28 b(=)f Fq(E)6 b Fs(\()p Fq(n)p Fs(\))3181 1012 y Fi(o)3248 1122 y Fq(;)90 b Fs(\(1.3\))328 1364 y(where)39 b Fq(')679 1379 y Fo(m)783 1364 y Fs(is)f(the)g(eigen)m(v)m(ector)h(of)f Fq(H)1770 1379 y Fo(p)1847 1364 y Fs(corresp)s(onding)g(to)g(the)g(mo)s (de)f Fq(m)p Fs(,)j(and)e(\012)g(is)328 1493 y(the)f(v)m(ector)i (represen)m(tativ)m(e)g(of)d Fq(!)1609 1446 y Fo(f)1605 1521 y(\014)1654 1493 y Fs(.)57 b(Our)37 b(main)f(result,)i(Theorem)f (2.3,)h(sho)m(ws)h(that)328 1614 y(\(1.2\))29 b(is)f(the)i(subset)h(of) e(\(1.3\))f(with)h Fq(m;)17 b(n)30 b Fs(ranging)e(o)m(v)m(er)i(those)g (mo)s(des)f(whic)m(h)h(are)f(not)328 1734 y(coupled)k(to)f(the)h (\014eld.)474 1855 y(While)42 b(this)g(result)h(holds)g(for)f(a)h(sp)s (eci\014c)g(class)g(of)f(p)s(oten)m(tials)g(\(see)i(\(2.2\)\),)h(w)m(e) 328 1975 y(pro)m(v)m(e)35 b(in)d(Theorem)i(2.2)f(a)g(result)g(whic)m(h) h(holds)f(for)g(a)g(v)m(ery)i(general)e(class)g(of)g(p)s(oten-)328 2095 y(tials:)57 b(F)-8 b(or)39 b(an)m(y)i Fq(\033)1022 2110 y Fo(t;)p Fp(0)1107 2095 y Fs(-in)m(v)-5 b(arian)m(t)38 b(normal)g(state)i Fq(!)2205 2059 y Fp(0)2284 2095 y Fs(and)g(an)m(y)h Fq(\033)2728 2110 y Fo(t;\025)2819 2095 y Fs(-in)m(v)-5 b(arian)m(t)38 b(normal)328 2216 y(state)e Fq(!)635 2180 y Fo(\025)715 2216 y Fs(on)f Fr(M)958 2231 y Fo(\014)1040 2216 y Fs(w)m(e)i(pro)m(v)m(e)g(that)e Fm(k)p Fq(!)1782 2180 y Fp(0)1845 2216 y Fm(\000)24 b Fq(!)2011 2180 y Fo(\025)2056 2216 y Fm(k)32 b(\025)i Fq(k)h(>)e Fs(0,)j(pro)m(vided)g Fq(\025)c Fm(6)p Fs(=)h(0)i(is)g (small)328 2336 y(enough,)j(for)d(a)h(constan)m(t)h Fq(k)i Fs(indep)s(enden)m(t)f(of)e Fq(\025)p Fs(.)54 b(Here)37 b Fm(k)24 b(\001)g(k)36 b Fs(denotes)i(the)e(norm)g(on)328 2457 y(the)i(space)h(of)f(linear)e(functionals)h(on)g Fr(M)1907 2472 y Fo(\014)1954 2457 y Fs(.)59 b(Theorem)39 b(2.2)e(is)g(pro)m(v)m(en)j(for)d(b)s(ounded)328 2577 y(p)s(oten)m(tials)32 b Fq(v)38 b Fs(suc)m(h)d(that)e Fm(\000)p Fs(\001)24 b(+)e Fq(v)38 b Fs(has)c(only)f(\014nitely)g(man)m (y)g(eigen)m(v)-5 b(alues)34 b(b)s(elo)m(w)f(the)328 2697 y(threshold)k(of)f(the)h(con)m(tin)m(uous)g(sp)s(ectrum,)h(all)d (of)h(whic)m(h)h(are)g(coupled)g(to)f(the)h(\014eld.)328 2818 y(Alternativ)m(ely)-8 b(,)27 b(w)m(e)i(could)e(relax)g(this)g (\014niteness)i(condition)d(but)h(couple)h(only)e(\014nitely)328 2938 y(man)m(y)32 b(mo)s(des)h(to)f(the)h(\014eld.)328 3391 y Fu(2)161 b(De\014nition)53 b(of)h(the)f(mo)t(del)h(and)f(main)h (results)328 3610 y Fs(In)40 b(Section)f(2.1)f(w)m(e)j(in)m(tro)s(duce) e(the)h(mo)s(del)d(and)j(sho)m(w)g(in)f(whic)m(h)g(w)m(a)m(y)i(it)d (de\014nes)j(a)328 3731 y Fq(W)434 3694 y Fx(\003)473 3731 y Fs(-dynamical)31 b(system)j(\()p Fr(M)1442 3746 y Fo(\014)1488 3731 y Fq(;)17 b(\033)1587 3746 y Fo(t;\025)1678 3731 y Fs(\).)44 b(Our)33 b(main)e(results)j(are)f(presen)m(ted)i(in)d (Section)328 3851 y(2.2.)328 4140 y Fh(2.1)135 b(De\014nition)46 b(of)f(the)g(mo)t(del)328 4325 y Fs(Starting)27 b(with)g(an)h(algebra)e Fr(A)j Fs(describing)e(the)h(join)m(t)f(system)i(atom-\014eld)d(and)i (a)f(\(reg-)328 4459 y(ularized\))e(dynamics)h Fq(\013)1216 4408 y Fp(\()p Fo(\017)p Fp(\))1215 4486 y Fo(t;\025)1331 4459 y Fs(on)g(it,)h(w)m(e)g(in)m(tro)s(duce)f(a)g(reference)h(state)g Fq(!)2919 4422 y Fp(ref)3008 4459 y Fs(,)g(describing)f(a)328 4579 y(b)s(oundstate)34 b(of)f(the)h(atom)e(and)i(blac)m(k-b)s(o)s(dy)f (radiation)e(at)i(in)m(v)m(erse)i(temp)s(erature)e Fq(\014)6 b Fs(.)328 4699 y(W)-8 b(e)39 b(then)f(consider)h(the)f(induced)h (\(regularized\))e(dynamics)h Fq(\033)2733 4648 y Fp(\()p Fo(\017)p Fp(\))2729 4727 y Fo(t;\025)2859 4699 y Fs(on)g Fq(\031)3055 4714 y Fo(\014)3103 4699 y Fs(\()p Fr(A)p Fs(\),)i(where)328 4831 y(\()p Fm(H)q Fq(;)17 b(\031)550 4846 y Fo(\014)597 4831 y Fq(;)g Fs(\012)711 4794 y Fp(ref)801 4831 y Fs(\))35 b(denotes)i(the)e(GNS)g(represen)m(tation)h(corresp)s (onding)g(to)f(\()p Fr(A)p Fq(;)17 b(!)3249 4794 y Fp(ref)3338 4831 y Fs(\).)51 b(As)1922 5214 y(6)p eop %%Page: 7 7 7 6 bop 328 632 a Fq(\017)36 b Fm(!)g Fs(0,)i Fq(\033)712 581 y Fp(\()p Fo(\017)p Fp(\))708 660 y Fo(t;\025)837 632 y Fs(tends)g(to)f(a)g Fm(\003)p Fs(automorphism)e(group,)k Fq(\033)2377 647 y Fo(t;\025)2468 632 y Fs(,)f(of)f(the)h(v)m(on)g (Neumann)f(al-)328 752 y(gebra)c Fr(M)699 767 y Fo(\014)745 752 y Fs(,)g(de\014ned)h(as)f(the)g(w)m(eak)h(closure)f(of)f Fq(\031)2165 767 y Fo(\014)2213 752 y Fs(\()p Fr(A)p Fs(\))h(in)f Fm(B)s Fs(\()p Fm(H)q Fs(\).)44 b(W)-8 b(e)33 b(determine)f(the)328 873 y(generator,)47 b Fq(L)869 888 y Fo(\025)915 873 y Fs(,)f(of)e(the)g(unitary)g(group,)j Fq(e)2009 837 y Fo(itL)2106 849 y Fl(\025)2151 873 y Fs(,)g(on)d Fm(H)h Fs(implemen)m(ting)c Fq(\033)3177 888 y Fo(t;\025)3268 873 y Fs(;)50 b Fq(L)3411 888 y Fo(\025)3500 873 y Fs(is)328 993 y(called)33 b(a)i FB(Liouvil)5 b(lian)p Fs(.)48 b(W)-8 b(e)35 b(explain)f(the)h(relation)e(b)s(et)m(w) m(een)k(eigen)m(v)-5 b(alues)34 b(of)g Fq(L)3328 1008 y Fo(\025)3409 993 y Fs(and)328 1114 y(in)m(v)-5 b(arian)m(t)31 b(normal)g(states)i(on)g Fr(M)1586 1129 y Fo(\014)1632 1114 y Fs(.)328 1390 y Ft(2.1.1)112 b(Kinematical)35 b(algebra)j Fr(A)p Ft(,)g(and)g(regularized)e(dynamics)i Fq(\013)3183 1339 y Fp(\()p Fo(\017)p Fp(\))3182 1418 y Fo(t;\025)328 1575 y Fs(W)-8 b(e)34 b(consider)g(a)g(system)g (consisting)g(of)f(a)g(quan)m(tum)h(mec)m(hanical)f(particle)f(\(an)i (elec-)328 1695 y(tron)e(in)g(the)h(p)s(oten)m(tial)e(of)h(a)g(static)g (n)m(ucleus\))i(in)m(teracting)e(with)g(a)g(quan)m(tized)h(\014eld.)474 1815 y(Pure)43 b(states)f(of)f(the)i(particle)d(system)i(are)g(describ) s(ed)h(b)m(y)f(unit)f(v)m(ectors)i(in)e(the)328 1936 y(Hilb)s(ert)31 b(space)j Fm(H)1009 1951 y Fo(p)1076 1936 y Fs(=)28 b Fq(L)1246 1900 y Fp(2)1286 1936 y Fs(\()p Fk(R)1390 1900 y Fp(3)1435 1936 y Fq(;)17 b(d)1530 1900 y Fp(3)1569 1936 y Fq(x)p Fs(\),)33 b(their)f(dynamics)g(is)g (determined)h(b)m(y)g(the)h(Sc)m(hr\177)-49 b(o-)328 2056 y(dinger)32 b(equation)g(with)h(Hamiltonian)1643 2276 y Fq(H)1724 2291 y Fo(p)1791 2276 y Fs(=)27 b Fm(\000)p Fs(\001)c(+)f Fq(v)t(;)1114 b Fs(\(2.1\))328 2496 y(where)35 b(the)f(p)s(oten)m(tial)e Fq(v)37 b Fs(is)d(b)s(ounded)g(and)g (satis\014es)g(one)g(of)f(the)i(follo)m(wing)c(t)m(w)m(o)j(con-)328 2617 y(ditions.)473 2820 y Fm(\017)49 b FB(Condition)30 b Fs(C)1094 2835 y Fp(A)1152 2820 y FB(.)74 b Fs(The)30 b(p)s(oten)m(tial)d Fq(v)33 b Fs(is)c(s.t.)43 b(the)29 b(sp)s(ectrum)h(of)f Fq(H)2985 2835 y Fo(p)3053 2820 y Fs(consists)h(of)f(a)572 2940 y(\014nite)37 b(n)m(um)m(b)s(er)g Fq(d)g Fs(of)f(eigen)m(v)-5 b(alues)38 b(\(coun)m(ting)e(m)m (ultiplicit)m(y\))e(lying)h(b)s(elo)m(w)i(the)572 3061 y(con)m(tin)m(uous)32 b(sp)s(ectrum)h(whic)m(h)f(co)m(v)m(ers)i([0)p Fq(;)17 b Fm(1)p Fs(\).)42 b(W)-8 b(e)32 b(set)h Fq(E)2771 3076 y Fp(0)2838 3061 y Fs(:=)28 b(inf)22 b Fq(\033)t Fs(\()p Fq(H)3281 3076 y Fo(p)3321 3061 y Fs(\))27 b Fq(<)h Fs(0.)473 3264 y Fm(\017)49 b FB(Condition)34 b Fs(C)1098 3279 y Fp(B)1152 3264 y FB(.)78 b Fs(The)34 b(p)s(oten)m(tial)d Fq(v)36 b Fs(is)c(giv)m(en)h(b)m(y)1390 3534 y Fq(v)t Fs(\()p Fq(x)p Fs(\))28 b(=)g Fm(\000)1796 3466 y Fq(\032)p Fs(\()p Fm(j)p Fq(x)p Fm(j)p Fs(\))p 1791 3511 248 4 v 1791 3602 a Fm(j)p Fq(x)p Fm(j)1902 3573 y Fp(1+)p Fo(\026)2049 3534 y Fq(;)146 b Fm(\000)p Fs(1)28 b Fq(<)g(\026)f Fm(\024)h Fs(1)p Fq(;)618 b Fs(\(2.2\))572 3809 y(where)28 b Fq(\032)p Fs(\()p Fm(j)p Fq(x)p Fm(j)p Fs(\))e(is)g(a)g(smo)s(oth,)h(non-negativ)m(e)g(function)f(that)g(has)h (a)g(zero)g(of)f(order)572 3930 y(1)21 b(+)g Fq(\026)31 b Fs(at)h(the)g(origin,)e(and)j(increases)g(to)e(a)h(constan)m(t)h(v)-5 b(alue)31 b Fq(\032)i Fs(as)f Fm(j)p Fq(x)p Fm(j)27 b(!)h(1)p Fs(,)k(in)572 4050 y(suc)m(h)i(a)e(w)m(a)m(y)i(that)e Fq(v)37 b Fs(is)32 b(smo)s(oth,)f(and)1882 4270 y(\()p Fq(x)23 b Fm(\001)e(r)p Fs(\))2168 4229 y Fo(j)2205 4270 y Fq(v)1113 b Fs(\(2.3\))572 4490 y(are)30 b(b)s(ounded,)h(for)f Fq(j)j Fs(=)28 b(0)p Fq(;)17 b(:)g(:)g(:)e(;)i Fs(3.)42 b(Notice)30 b(that)g(the)g(eigen)m(v)-5 b(alues)30 b(of)f Fq(H)3233 4505 y Fo(p)3303 4490 y Fs(are)h(all)572 4611 y(negativ)m(e)j(and)f(can)h(accum)m(ulate)f(only)g(at)h(the)g (threshold)f(0.)328 4814 y FB(R)-5 b(emark.)94 b Fs(In)37 b(Condition)e(C)1434 4829 y Fp(B)1526 4814 y Fs(w)m(e)i(admit)e(p)s (oten)m(tials)h(suc)m(h)i(that)f Fq(H)2935 4829 y Fo(p)3011 4814 y Fs(has)g(in\014nitely)328 4934 y(man)m(y)g(eigen)m(v)-5 b(alues)37 b(b)s(elo)m(w)g(zero,)h(but)f(couple)g(only)f(\014nitely)h (man)m(y)g(of)f(them)h(to)f(the)1922 5214 y(7)p eop %%Page: 8 8 8 7 bop 328 631 a Fs(\014eld,)32 b(as)h(w)m(e)h(explain)d(b)s(elo)m(w.) 474 872 y(The)25 b(\014eld)f(is)g(a)g(scalar)f(massless)h(free)h(b)s (osonic)e(\014eld.)41 b(\(It)24 b(w)m(ould)g(b)s(e)g(more)g(in)m (terest-)328 992 y(ing,)j(ph)m(ysically)-8 b(,)28 b(to)e(consider)h (the)h(quan)m(tized)f(electromagnetic)f(\014eld.)41 b(Our)27 b(metho)s(ds)328 1112 y(can)38 b(b)s(e)g(applied)f(to)g(the)i (resulting)e(mo)s(del)f(at)h(the)i(price)f(of)f(sligh)m(tly)f(more)h (compli-)328 1233 y(cated)f(notations.\))53 b(The)37 b(scalar)e(free)h(\014eld)g(is)f(con)m(v)m(enien)m(tly)j(describ)s(ed)e (in)f(terms)h(of)328 1353 y(a)42 b(\\time-a)m(v)m(eraged")f(W)-8 b(eyl)42 b(algebra,)i Fr(A)1860 1368 y Fo(f)1906 1353 y Fs(,)g(whic)m(h)f(is)f(the)h Fq(C)2629 1317 y Fx(\003)2668 1353 y Fs(-algebra)e(\(of)g(\\observ-)328 1474 y(ables"\),)48 b(de\014ned)f(as)f(follo)m(ws.)81 b(Let)46 b Fr(W)g Fs(b)s(e)g(the)g(W) -8 b(eyl)45 b(algebra)g(o)m(v)m(er)h(the)g(Hilb)s(ert)328 1594 y(space)1177 1714 y Fq(L)1243 1673 y Fp(2)1243 1739 y(0)1310 1714 y Fs(=)28 b Fq(L)1480 1673 y Fp(2)1519 1714 y Fs(\()p Fk(R)1623 1673 y Fp(3)1669 1714 y Fq(;)17 b(d)1764 1673 y Fp(3)1803 1714 y Fq(k)s Fs(\))22 b Fm(\\)g Fq(L)2071 1673 y Fp(2)2111 1714 y Fs(\()p Fk(R)2215 1673 y Fp(3)2260 1714 y Fq(;)17 b Fm(j)p Fq(k)s Fm(j)2414 1673 y Fx(\000)p Fp(1)2508 1714 y Fq(d)2559 1673 y Fp(3)2598 1714 y Fq(k)s Fs(\))p Fq(;)648 b Fs(\(2.4\))328 1889 y(i.e.,)33 b Fr(W)g Fs(is)f(the)i Fq(C)994 1853 y Fx(\003)1033 1889 y Fs(-algebra)e(generated)h(b)m(y)h(W)-8 b(eyl)33 b(op)s(erators,)g Fq(W)14 b Fs(\()p Fq(f)d Fs(\),)33 b Fq(f)39 b Fm(2)29 b Fq(L)3243 1853 y Fp(2)3243 1913 y(0)3282 1889 y Fs(,)34 b(satis-)328 2009 y(fying)e(the)h(W)-8 b(eyl)32 b(relations)1216 2229 y Fq(W)14 b Fs(\()p Fq(f)d Fs(\))p Fq(W)j Fs(\()p Fq(g)t Fs(\))25 b(=)j Fq(e)1864 2188 y Fx(\000)p Fo(i)p Fp(Im)o Fx(h)q Fo(f)s(;g)r Fx(i)2179 2229 y Fq(W)14 b Fs(\()p Fq(g)t Fs(\))p Fq(W)g Fs(\()p Fq(f)d Fs(\))p Fq(:)685 b Fs(\(2.5\))328 2449 y(The)33 b(free)g(\014eld)g(dynamics)f(on)h Fr(W)f Fs(is)g(giv)m(en)h(b)m(y)h (the)f Fm(\003)p Fs(automorphism)d(group)1250 2669 y Fq(W)14 b Fs(\()p Fq(f)d Fs(\))27 b Fm(7!)g Fq(\013)1708 2628 y Fg(W)1707 2694 y Fo(t)1784 2669 y Fs(\()p Fq(W)14 b Fs(\()p Fq(f)d Fs(\)\))27 b(=)g Fq(W)14 b Fs(\()p Fq(e)2420 2628 y Fo(i!)r(t)2520 2669 y Fq(f)d Fs(\))p Fq(;)721 b Fs(\(2.6\))328 2889 y(where)36 b Fq(!)t Fs(\()p Fq(k)s Fs(\))31 b(=)h Fm(j)p Fq(k)s Fm(j)j Fs(is)f(the)i(energy)g(of)f(a)f (single)g(b)s(oson.)51 b(F)-8 b(or)35 b(functions)g Fq(f)42 b Fm(2)33 b Fq(L)3328 2853 y Fp(2)3328 2914 y(0)3368 2889 y Fs(,)i(the)328 3010 y(exp)s(ectation)e(functional)1478 3169 y Fq(f)39 b Fm(7!)27 b Fq(e)1737 3114 y Fx(\000)1802 3086 y Fj(1)p 1802 3098 31 3 v 1802 3140 a(4)1843 3040 y Ff(D)1886 3114 y Fo(f)s(;)1943 3040 y Ff(\020)1985 3114 y Fp(1+)2165 3086 y Fj(2)p 2085 3098 190 3 v 2085 3151 a Fl(e)2114 3136 y(\014)s(!)2196 3151 y Fe(\000)p Fj(1)2285 3040 y Ff(\021)2327 3114 y Fo(f)2368 3040 y Ff(E)3365 3169 y Fs(\(2.7\))328 3343 y(is)33 b(w)m(ell)f(de\014ned)j (and)e(determines)h(a)f(\()p Fq(\014)6 b(;)17 b(\013)1935 3307 y Fg(W)1934 3368 y Fo(t)2010 3343 y Fs(\)-KMS)33 b(state)h(on)f Fr(W)p Fs(.)46 b(It)33 b(is)g(w)m(ell)f(kno)m(wn)328 3463 y(that)37 b(the)h Fm(\003)p Fs(automorphism)e(group)h Fq(\013)1762 3427 y Fg(W)1761 3488 y Fo(t)1875 3463 y Fs(of)h Fr(W)f Fs(is)g(not)h(norm-con)m(tin)m(uous)f(\(i.e.,)h Fk(R)47 b Fm(3)328 3584 y Fq(t)d Fm(7!)f Fq(\013)613 3548 y Fg(W)612 3609 y Fo(t)689 3584 y Fs(\()p Fq(W)14 b Fs(\()p Fq(f)d Fs(\)\))41 b(is)h(not)g(con)m(tin)m(uous)h(in)e(the)i (norm)e(of)g Fr(W)p Fs(\).)72 b(The)43 b(time-a)m(v)m(eraged)328 3704 y Fq(C)405 3668 y Fx(\003)444 3704 y Fs(-algebra)31 b Fr(A)889 3719 y Fo(f)968 3704 y Fs(is)h(generated)h(b)m(y)h(elemen)m (ts)e(of)h(the)g(form)1427 3976 y Fq(a)p Fs(\()p Fq(h)p Fs(\))28 b(=)1742 3840 y Fi(Z)1797 4066 y Fd(R)1866 3976 y Fq(ds)k(h)p Fs(\()p Fq(s)p Fs(\))p Fq(\013)2236 3935 y Fg(W)2235 4000 y Fo(s)2312 3976 y Fs(\()p Fq(a)p Fs(\))p Fq(;)899 b Fs(\(2.8\))328 4243 y(where)35 b Fq(a)29 b Fm(2)h Fr(W)k Fs(and)g Fq(h)29 b Fs(:)h Fk(R)40 b Fm(!)30 b Fk(C)59 b Fs(are)34 b(functions)f(whose)i(F)-8 b(ourier)33 b(transforms)g(satisfy)326 4337 y Fi(b)328 4363 y Fq(h)28 b Fm(2)g Fq(C)583 4327 y Fx(1)576 4388 y Fp(0)683 4363 y Fs(\(this)d(is)h(a)f(con)m(v)m(enien)m(t)i(class)f(of)f(functions)h (whic)m(h)g(allo)m(ws)f(us)h(to)f(de\014ne)i(KMS)328 4483 y(states)33 b(on)g Fr(A)812 4498 y Fo(f)858 4483 y Fs(,)f(see)i([FM]\).)f(The)g(free)g(\014eld)g(dynamics)f(on)h Fr(A)2617 4498 y Fo(f)2695 4483 y Fs(is)f(de\014ned)i(b)m(y)1071 4755 y Fq(\013)1134 4708 y Fo(f)1133 4777 y(t)1179 4755 y Fs(\()p Fq(a)p Fs(\()p Fq(h)p Fs(\)\))28 b(=)1570 4619 y Fi(Z)1625 4845 y Fd(R)1694 4755 y Fq(ds)k(h)p Fs(\()p Fq(s)22 b Fm(\000)h Fq(t)p Fs(\))p Fq(\013)2221 4714 y Fg(W)2220 4779 y Fo(s)2297 4755 y Fs(\()p Fq(a)p Fs(\))28 b(=:)f Fq(a)p Fs(\()p Fq(h)2727 4770 y Fo(t)2757 4755 y Fs(\))p Fq(:)543 b Fs(\(2.9\))1922 5214 y(8)p eop %%Page: 9 9 9 8 bop 328 631 a Fs(It)34 b(is)f(a)g(norm-con)m(tin)m(uous)g Fm(\003)p Fs(-automorphism)e(group)i(on)h Fr(A)2574 646 y Fo(f)2620 631 y Fs(.)46 b(W)-8 b(e)34 b(refer)g(to)f([FM])h(for)328 751 y(more)e(details)f(on)i(the)g(construction)f(and)h(the)g(prop)s (erties)g(of)f Fr(A)2742 766 y Fo(f)2788 751 y Fs(.)474 992 y(The)26 b(join)m(t)e(system)i(describing)e(the)h(particle)f(and)h (the)g(\014eld)g(is)f(describ)s(ed)i(in)e(terms)328 1112 y(of)32 b(the)h Fq(C)684 1076 y Fx(\003)723 1112 y Fs(-algebra)1657 1233 y Fr(A)28 b Fs(=)g Fr(A)1931 1248 y Fo(p)1993 1233 y Fm(\012)23 b Fr(A)2164 1248 y Fo(f)2210 1233 y Fq(;)1080 b Fs(\(2.10\))328 1407 y(where)54 b Fr(A)701 1422 y Fo(p)802 1407 y Fs(=)62 b Fm(B)s Fs(\()p Fm(H)1130 1422 y Fo(p)1170 1407 y Fs(\))52 b(is)g(the)h(v)m(on)g(Neumann)g(algebra)e(of)h(all)e(b) s(ounded)k(op)s(era-)328 1528 y(tors)49 b(on)f(the)i(Hilb)s(ert)d (space)i Fm(H)1588 1543 y Fo(p)1628 1528 y Fs(.)92 b(The)50 b(uncoupled)f(dynamics)g(is)f(giv)m(en)h(b)m(y)g(the)328 1648 y Fm(\003)p Fs(automorphism)30 b(group)1642 1768 y Fq(\013)1704 1783 y Fo(t;)p Fp(0)1816 1768 y Fs(=)e Fq(\013)1983 1721 y Fo(p)1982 1791 y(t)2044 1768 y Fm(\012)23 b Fq(\013)2207 1721 y Fo(f)2206 1791 y(t)3317 1768 y Fs(\(2.11\))328 1943 y(of)j Fr(A)p Fs(,)i(where)g Fq(\013)898 1895 y Fo(p)897 1965 y(t)937 1943 y Fs(\()p Fm(\001)p Fs(\))f(=)h Fq(e)1217 1907 y Fo(itH)1324 1915 y Fl(p)1375 1943 y Fm(\001)10 b Fq(e)1458 1907 y Fx(\000)p Fo(itH)1620 1915 y Fl(p)1660 1943 y Fs(.)41 b(In)27 b(order)g(to)f(de\014ne)h(the)g (dynamics)g(of)f(the)g(in)m(ter-)328 2063 y(acting)h(system)h(in)f(a)h (represen)m(tation)g(indep)s(enden)m(t)h(w)m(a)m(y)g(\(i.e.,)g(as)f(a)f Fm(\003)p Fs(automorphism)328 2183 y(group)j(on)f Fr(A)p Fs(\),)i(w)m(e)g(need)g(to)f(in)m(tro)s(duce)g(a)f(regularized)g(in)m (teraction)g(term.)42 b(F)-8 b(or)29 b Fq(\017)f Fm(6)p Fs(=)g(0,)328 2304 y(this)k(term)g(is)g(giv)m(en)h(b)m(y)g(giv)m(en)g (b)m(y)629 2563 y Fq(V)707 2512 y Fp(\()p Fo(\017)p Fp(\))686 2591 y(#)822 2563 y Fs(=)926 2468 y Fi(X)975 2677 y Fo(\013)1086 2563 y Fq(G)1163 2578 y Fo(\013;)p Fp(#)1313 2563 y Fm(\012)1459 2495 y Fs(1)p 1423 2540 122 4 v 1423 2631 a(2)p Fq(i\017)1554 2482 y Fi(\010)1613 2563 y Fq(W)14 b Fs(\()p Fq(\017g)1843 2578 y Fo(\013)1892 2563 y Fs(\)\()p Fq(h)2024 2578 y Fo(\017)2056 2563 y Fs(\))22 b Fm(\000)h Fq(W)14 b Fs(\()p Fq(\017g)2446 2578 y Fo(\013)2495 2563 y Fs(\)\()p Fq(h)2627 2578 y Fo(\017)2660 2563 y Fs(\))2698 2522 y Fx(\003)2737 2482 y Fi(\011)2823 2563 y Fm(2)28 b Fr(A)p Fq(;)302 b Fs(\(2.12\))328 2870 y(where)37 b(the)g(sum)f(is)f(o)m(v)m(er)i (\014nitely)f(man)m(y)f(indices)h Fq(\013)q Fs(,)h(with)e Fq(G)2661 2885 y Fo(\013;)p Fp(#)2823 2870 y Fs(=)e Fq(G)3009 2834 y Fx(\003)3009 2896 y Fo(\013;)p Fp(#)3171 2870 y Fm(2)h(B)s Fs(\()p Fm(H)3461 2885 y Fo(p)3501 2870 y Fs(\),)328 2990 y Fq(g)375 3005 y Fo(\013)452 2990 y Fm(2)28 b Fq(L)612 2954 y Fp(2)612 3015 y(0)652 2990 y Fs(,)g(for)e(all)f Fq(\013)q Fs(,)j(and)f(where)h Fq(h)1614 3005 y Fo(\017)1674 2990 y Fs(is)e(an)h(appro)m(ximation)d(of)j(the)g (Dirac)f(distribution)328 3111 y(lo)s(calized)36 b(at)i(zero.)61 b(T)-8 b(o)38 b(b)s(e)h(sp)s(eci\014c)g(w)m(e)g(can)g(tak)m(e)g Fq(h)2358 3126 y Fo(\017)2391 3111 y Fs(\()p Fq(t)p Fs(\))e(=)2663 3071 y Fp(1)p 2663 3088 36 4 v 2666 3145 a Fo(\017)2708 3111 y Fq(e)2753 3075 y Fx(\000)p Fo(t)2833 3051 y Fj(2)2868 3075 y Fo(=\017)2932 3051 y Fj(2)2971 3111 y Fs(.)60 b(The)39 b(sym)m(b)s(ol)328 3254 y Fq(G)405 3269 y Fo(\013;)p Fp(#)559 3254 y Fs(\(and)27 b(similarly)c Fq(V)1252 3203 y Fp(\()p Fo(\017)p Fp(\))1231 3282 y(#)1340 3254 y Fs(\))j(stands)i (for)e(either)h Fq(G)2194 3269 y Fo(\013)2270 3254 y Fs(or)f Fq(G)2460 3269 y Fo(\013;J)2574 3254 y Fs(,)i(where)f Fq(J)36 b Fs(is)26 b(some)h(cuto\013)328 3374 y(determining)37 b(whic)m(h)i(mo)s(des)f(of)g(the)h(particle)e(are)i(coupled)f(to)h(the) g(\014eld.)61 b(In)39 b(order)328 3495 y(to)34 b(describ)s(e)h(this)f (more)f(precisely)-8 b(,)35 b(w)m(e)h(in)m(tro)s(duce)e(the)h(follo)m (wing)c(terminology)-8 b(.)47 b(Let)328 3615 y Fm(M)41 b Fs(b)s(e)h(the)f(index)h(set)g(of)f(the)h(discrete)g(\\mo)s(des")f (of)g Fq(H)2513 3630 y Fo(p)2552 3615 y Fs(,)j(i.e.,)f(a)f(lab)s (elling)37 b(of)k(the)328 3736 y(eigen)m(v)-5 b(alues)37 b(of)f Fq(H)1033 3751 y Fo(p)1110 3736 y Fs(including)f(m)m(ultiplicit) m(y)-8 b(.)53 b(Giv)m(en)37 b Fq(m)e Fm(2)h(M)p Fs(,)i Fq(E)6 b Fs(\()p Fq(m)p Fs(\))37 b(denotes)h(the)328 3856 y(corresp)s(onding)24 b(eigen)m(v)-5 b(alue)24 b(of)g Fq(H)1588 3871 y Fo(p)1627 3856 y Fs(.)41 b(An)25 b(eigen)m(v)-5 b(alue)24 b Fq(E)30 b Fs(of)24 b Fq(H)2591 3871 y Fo(p)2655 3856 y Fs(is)g(simple)e(if)i(and)g(only)g(if)328 3976 y(there)29 b(is)e(a)h(unique)g Fq(m)g Fm(2)g(M)g Fs(s.t.)42 b Fq(E)34 b Fs(=)27 b Fq(E)6 b Fs(\()p Fq(m)p Fs(\).)42 b(W)-8 b(e)29 b(denote)f(the)h(rank-one)f(pro)5 b(jection)328 4097 y(corresp)s(onding)32 b(to)h(the)g(mo)s(de)e Fq(m)d Fm(2)h(M)j Fs(b)m(y)h Fq(p)2048 4112 y Fo(m)2115 4097 y Fs(.)474 4217 y(Let)40 b Fq(J)710 4232 y Fo(d)790 4217 y Fm(\032)h(M)e Fs(b)s(e)h(a)f(set)h(of)g(\014nitely)f(man)m(y)g (discrete)i(mo)s(des)e(of)g Fq(H)3053 4232 y Fo(p)3132 4217 y Fs(and)h(let)f Fq(J)3531 4232 y Fo(c)328 4338 y Fs(b)s(e)32 b(an)g(op)s(en)g(in)m(terv)-5 b(al)30 b(in)i(the)g(con)m (tin)m(uous)h(sp)s(ectrum)f Fk(R)2441 4353 y Fp(+)2538 4338 y Fs(of)f Fq(H)2729 4353 y Fo(p)2800 4338 y Fs(\(w)m(e)i(ma)m(y)f (also)f(tak)m(e)328 4458 y(a)36 b(\014nite)g(union)g(of)g(disjoin)m(t)f (in)m(terv)-5 b(als\),)37 b(s.t.)55 b Fq(J)2107 4473 y Fo(c)2176 4458 y Fm(\032)35 b Fs([)p Fq(r)m(;)17 b(R)q Fs(],)37 b(for)f(some)g Fq(r)m(;)17 b(R)38 b Fs(satisfying)328 4578 y(0)27 b Fq(<)h(r)i(<)e(R)h(<)e Fm(1)p Fs(.)43 b(The)34 b(set)1689 4699 y Fq(J)j Fs(:=)28 b Fq(J)1965 4714 y Fo(d)2027 4699 y Fm([)23 b Fq(J)2170 4714 y Fo(c)1922 5214 y Fs(9)p eop %%Page: 10 10 10 9 bop 328 631 a Fs(determines)25 b(the)h(mo)s(des)f(of)f(the)i (particle)e(whic)m(h)h(are)g(coupled)h(to)e(the)i(\014eld,)g(according) 328 751 y(to)32 b(the)h(in)m(teraction)1115 931 y Fq(G)1192 946 y Fo(\013;J)1333 931 y Fs(=)1437 850 y Fi(\000)1482 931 y Fq(p)1531 946 y Fo(J)1570 958 y Fl(d)1632 931 y Fs(+)22 b Fq(\026)p Fs(\()p Fq(H)1908 946 y Fo(p)1948 931 y Fs(\))1986 850 y Fi(\001)2031 931 y Fq(G)2108 946 y Fo(\013)2158 850 y Fi(\000)2203 931 y Fq(p)2252 946 y Fo(J)2291 958 y Fl(d)2353 931 y Fs(+)g Fq(\026)p Fs(\()p Fq(H)2629 946 y Fo(p)2668 931 y Fs(\))2706 850 y Fi(\001)2752 931 y Fq(;)538 b Fs(\(2.13\))328 1110 y(where)34 b Fq(G)687 1125 y Fo(\013)769 1110 y Fs(is)e(a)g(b)s(ounded,)h(selfadjoin)m(t)f (op)s(erator)g(on)g Fm(H)2455 1125 y Fo(p)2495 1110 y Fs(,)h(and)1646 1310 y Fq(p)1695 1325 y Fo(J)1734 1337 y Fl(d)1801 1310 y Fs(=)1925 1216 y Fi(X)1905 1427 y Fo(m)p Fx(2)p Fo(J)2053 1439 y Fl(d)2105 1310 y Fq(p)2154 1325 y Fo(m)2221 1310 y Fq(;)1069 b Fs(\(2.14\))328 1591 y Fq(\026)27 b Fm(2)h Fq(C)585 1555 y Fx(1)578 1616 y Fp(0)660 1591 y Fs(\()p Fq(J)752 1606 y Fo(c)787 1591 y Fs(\))h(is)h(a)f(smo)s(oth)g(v)m(ersion)i(of)e(the)i(indicator)d (function)h(with)h(supp)s(ort)g(in)f Fq(J)3504 1606 y Fo(c)3539 1591 y Fs(,)328 1712 y(and)k Fq(\026)p Fs(\()p Fq(H)696 1727 y Fo(p)735 1712 y Fs(\))f(is)g(de\014ned)i(via)e(the)h(F) -8 b(ourier)32 b(transform)1492 1942 y Fq(\026)p Fs(\()p Fq(H)1670 1957 y Fo(p)1709 1942 y Fs(\))27 b(=)1878 1807 y Fi(Z)1999 1942 y(b)-60 b Fq(\026)p Fs(\()p Fq(s)p Fs(\))p Fq(e)2220 1901 y Fo(isH)2335 1909 y Fl(p)2375 1942 y Fq(:)328 2165 y Fs(Clearly)-8 b(,)27 b Fq(G)758 2180 y Fo(\013;J)900 2165 y Fs(tends)h(to)f Fq(G)1346 2180 y Fo(\013)1395 2165 y Fs(,)i(in)d(the)i(strong)g(sense)h(as)e Fq(\026)g Fs(increases)i(to)e(the)g(c)m(haracter-)328 2285 y(istic)33 b(function)h(of)g Fk(R)1099 2300 y Fp(+)1199 2285 y Fs(\(i.e.)49 b Fq(J)1465 2300 y Fo(c)1530 2285 y Fm(")31 b Fk(R)1677 2300 y Fp(+)1742 2285 y Fs(\))j(and)h Fq(J)2060 2300 y Fo(d)2135 2285 y Fs(increases)g(to)f(the)h(set)h(of)e (all)e(discrete)328 2419 y(mo)s(des)f(of)f Fq(H)818 2434 y Fo(p)857 2419 y Fs(.)43 b(Th)m(us,)33 b Fq(V)1279 2368 y Fp(\()p Fo(\017)p Fp(\))1257 2446 y Fo(J)1397 2419 y Fs(can)e(b)s(e)g(view)m(ed)h(as)f(an)g(appro)m(ximation)d(of)j Fq(V)3111 2383 y Fp(\()p Fo(\017)p Fp(\))3226 2419 y Fs(=)d Fq(V)3409 2368 y Fp(\()p Fo(\017)p Fp(\))3387 2447 y Fo(J)6 b Fp(=)p Fd(R)3539 2419 y Fs(.)474 2660 y(The)28 b(in)m(teraction)e(term)h(\(2.12\))f(determines)h(a)g Fm(\003)p Fs(automorphism)d(group)j Fq(\013)3245 2609 y Fp(\()p Fo(\017)p Fp(\))3244 2688 y Fo(t;\025)3362 2660 y Fs(of)f Fr(A)p Fs(,)328 2780 y(the)33 b(coupled)g(dynamics,)f (via)g(the)h(norm-con)m(v)m(ergen)m(t)g(Dyson)g(series)498 3017 y Fq(\013)561 2967 y Fp(\()p Fo(\017)p Fp(\))560 3045 y Fo(t;\025)650 3017 y Fs(\()p Fq(A)p Fs(\))83 b(:=)g Fq(\013)1130 3032 y Fo(t;)p Fp(0)1215 3017 y Fs(\()p Fq(A)p Fs(\))22 b(+)1484 2923 y Fi(X)1489 3133 y Fo(n)p Fx(\025)p Fp(1)1628 3017 y Fs(\()p Fq(i\025)p Fs(\))1794 2976 y Fo(n)1857 2882 y Fi(Z)1957 2908 y Fo(t)1913 3107 y Fp(0)2003 3017 y Fq(dt)2089 3032 y Fp(1)2145 3017 y Fm(\001)17 b(\001)g(\001)2278 2882 y Fi(Z)2378 2908 y Fo(t)2403 2917 y Fl(n)p Fe(\000)p Fj(1)2333 3107 y Fp(0)2545 3017 y Fq(dt)2631 3032 y Fo(n)2678 2907 y Fi(h)2725 3017 y Fq(\013)2787 3032 y Fo(t)2812 3040 y Fl(n)2855 3032 y Fo(;)p Fp(0)2914 3017 y Fs(\()p Fq(V)3030 2967 y Fp(\()p Fo(\017)p Fp(\))3009 3045 y(#)3118 3017 y Fs(\))p Fq(;)3200 2907 y Fi(h)3263 3017 y Fm(\001)g(\001)g(\001)1393 3302 y(\001)g(\001)g(\001)1526 3192 y Fi(h)1573 3302 y Fq(\013)1635 3317 y Fo(t)1660 3326 y Fj(1)1695 3317 y Fo(;)p Fp(0)1754 3302 y Fs(\()p Fq(V)1871 3251 y Fp(\()p Fo(\017)p Fp(\))1849 3330 y(#)1958 3302 y Fs(\))p Fq(;)g(\013)2102 3317 y Fo(t;)p Fp(0)2187 3302 y Fs(\()p Fq(A)p Fs(\))2336 3192 y Fi(i)2399 3302 y Fm(\001)g(\001)g(\001)2532 3192 y Fi(ii)2626 3302 y Fq(;)664 b Fs(\(2.15\))328 3504 y(where)36 b Fq(A)c Fm(2)h Fr(A)p Fs(,)j(and)f Fq(\025)d Fm(2)h Fk(R)46 b Fs(is)34 b(the)i(coupling)d(constan)m(t.)52 b(The)36 b(m)m(ultiple)d(in)m(tegral)h(in)328 3624 y(\(2.15\))22 b(is)h(understo)s(o)s(d)h(in)e(the)i(pro)s(duct)g(top)s(ology)d(coming) h(from)g(the)i(strong)f(top)s(ology)328 3744 y(of)32 b Fm(B)s Fs(\()p Fm(H)629 3759 y Fo(p)669 3744 y Fs(\))h(and)f(the)h (norm)f(top)s(ology)f(of)h Fr(A)1935 3759 y Fo(f)1981 3744 y Fs(.)474 3881 y(One)37 b(ma)m(y)g(view)g Fq(\013)1192 3830 y Fp(\()p Fo(\017)p Fp(\))1191 3909 y Fo(t;\025)1319 3881 y Fs(as)g(a)f FB(r)-5 b(e)g(gularize)g(d)38 b(dynamics)p Fs(,)f(in)f(the)i(sense)g(that)f(it)f(has)h(a)328 4002 y(limit,)i(as)i Fq(\017)i Fm(!)e Fs(0,)i(in)d(suitably)g(c)m(hosen)i (represen)m(tations)h(of)d Fr(A)p Fs(;)45 b(\(this)c(is)f(sho)m(wn)j (in)328 4122 y([FM])33 b(and)f(explained)h(b)s(elo)m(w\).)474 4242 y(The)39 b(functions)e Fq(g)1152 4257 y Fo(\013)1237 4242 y Fm(2)g Fq(L)1406 4206 y Fp(2)1406 4267 y(0)1483 4242 y Fs(are)g(called)g FB(form)h(factors)p Fs(.)58 b(Using)37 b(p)s(olar)f(co)s(ordinates)328 4363 y(in)c Fk(R)508 4327 y Fp(3)553 4363 y Fs(,)h(w)m(e)g(often)g(write)f Fq(g)1299 4378 y Fo(\013)1376 4363 y Fs(=)c Fq(g)1527 4378 y Fo(\013)1576 4363 y Fs(\()p Fq(!)t(;)17 b Fs(\006\),)32 b(where)i(\()p Fq(!)t(;)17 b Fs(\006\))27 b Fm(2)h Fk(R)2614 4378 y Fp(+)2701 4363 y Fm(\002)22 b Fq(S)2866 4327 y Fp(2)2906 4363 y Fs(.)474 4604 y(W)-8 b(e)33 b(no)m(w)g(sp)s(ecify)g(t) m(w)m(o)h(sets)f(of)g(assumptions)f(on)g(the)h(in)m(teractions.)474 4844 y FB(Condition)38 b Fq(A)p FB(.)97 b Fs(The)38 b(p)s(oten)m(tial)d Fq(v)41 b Fs(satis\014es)d(condition)e(C)2710 4859 y Fp(A)2767 4844 y Fs(,)j(the)e(in)m(teraction)f(is)328 4965 y(giv)m(en)d(b)m(y)g Fq(V)797 4929 y Fp(\()p Fo(\017)p Fp(\))884 4965 y Fs(,)g(and)g(the)g(follo)m(wing)c(prop)s(erties)k (hold.)1898 5214 y(10)p eop %%Page: 11 11 11 10 bop 473 631 a Fm(\017)49 b FB(Infr)-5 b(ar)g(e)g(d)32 b(and)g(ultr)-5 b(aviolet)33 b(b)-5 b(ehaviour)32 b(of)h(the)g(form)f (factors)p Fs(.)43 b(F)-8 b(or)29 b(an)m(y)j(\014xed)f(\006,)572 751 y Fq(g)619 766 y Fo(\013)668 751 y Fs(\()p Fm(\001)p Fq(;)17 b Fs(\006\))30 b Fm(2)h Fq(C)1090 715 y Fp(4)1129 751 y Fs(\()p Fk(R)1233 766 y Fp(+)1298 751 y Fs(\),)k(and)f(there)h (are)f(t)m(w)m(o)h(constan)m(ts)g(0)30 b Fq(<)g(k)2859 766 y Fp(1)2899 751 y Fq(;)17 b(k)2994 766 y Fp(2)3063 751 y Fq(<)30 b Fm(1)p Fs(,)k(s.t.)49 b(if)572 872 y Fq(!)31 b(<)c(k)818 887 y Fp(1)858 872 y Fs(,)32 b(then)1213 1071 y Fm(j)p Fq(@)1297 1030 y Fo(j)1292 1096 y(!)1343 1071 y Fq(g)1390 1086 y Fo(\013)1439 1071 y Fs(\()p Fq(!)t(;)17 b Fs(\006\))p Fm(j)27 b Fq(<)g(k)1903 1086 y Fp(2)1942 1071 y Fq(!)2007 1030 y Fo(p)p Fx(\000)p Fo(j)2134 1071 y Fq(;)114 b Fs(for)32 b(some)g Fq(p)c(>)g Fs(2)o Fq(;)393 b Fs(\(2.16\))572 1270 y(uniformly)43 b(in)h Fq(\013)q Fs(,)49 b Fq(j)55 b Fs(=)49 b(0)p Fq(;)17 b(:)g(:)g(:)f(;)h Fs(4)44 b(and)i(\006)k Fm(2)f Fq(S)2379 1234 y Fp(2)2419 1270 y Fs(.)81 b(Similarly)-8 b(,)45 b(there)h(are)f(t)m(w)m(o)572 1391 y(constan)m(t)33 b(0)28 b Fq(<)f(K)1228 1406 y Fp(1)1268 1391 y Fq(;)17 b(K)1395 1406 y Fp(2)1461 1391 y Fq(<)28 b Fm(1)p Fs(,)k(s.t.)44 b(if)32 b Fq(!)f(>)c(K)2266 1406 y Fp(1)2306 1391 y Fs(,)32 b(then)1172 1590 y Fm(j)p Fq(@)1256 1549 y Fo(j)1251 1614 y(!)1301 1590 y Fq(g)1348 1605 y Fo(\013)1397 1590 y Fs(\()p Fq(!)t(;)17 b Fs(\006\))p Fm(j)27 b Fq(<)h(K)1894 1605 y Fp(2)1933 1590 y Fq(!)1998 1549 y Fx(\000)p Fo(q)r Fx(\000)p Fo(j)2177 1590 y Fq(;)115 b Fs(for)32 b(some)g Fq(q)g(>)27 b Fs(3)p Fq(:)351 b Fs(\(2.17\))473 1828 y Fm(\017)49 b FB(R)-5 b(elative)46 b(b)-5 b(ound)46 b(on)h Fs([)p Fq(G)1505 1843 y Fo(\013)1554 1828 y Fq(;)17 b(H)1679 1843 y Fo(p)1718 1828 y Fs(].)83 b(De\014ne)46 b(the)f(comm)m(utator)f([)p Fq(G)3017 1843 y Fo(\013)3067 1828 y Fq(;)17 b(H)3192 1843 y Fo(p)3231 1828 y Fs(])46 b(in)e(the)572 1948 y(w)m(eak)34 b(sense)g(on)f Fq(C)1279 1912 y Fx(1)1272 1973 y Fp(0)1375 1948 y Fm(\002)23 b Fq(C)1552 1912 y Fx(1)1545 1973 y Fp(0)1659 1948 y Fs(b)m(y)1123 2147 y Fm(h)p Fq( )t(;)17 b Fs([)p Fq(G)1377 2162 y Fo(\013)1426 2147 y Fq(;)g(H)1551 2162 y Fo(p)1590 2147 y Fs(])p Fq(')p Fm(i)28 b Fs(=)f Fm(h)p Fq(G)1967 2162 y Fo(\013)2016 2147 y Fq( )t(;)17 b(H)2208 2162 y Fo(p)2247 2147 y Fq(')p Fm(i)22 b(\000)h(h)p Fq(H)2592 2162 y Fo(p)2631 2147 y Fq( )t(;)17 b(G)2819 2162 y Fo(\013)2868 2147 y Fq(')p Fm(i)g Fq(:)572 2347 y Fs(Then)24 b([)p Fq(G)921 2362 y Fo(\013)970 2347 y Fq(;)17 b(H)1095 2362 y Fo(p)1135 2347 y Fs(])23 b(extends)h(to)f(a)g(relativ)m(ely)e(\()p Fq(H)2246 2362 y Fo(p)2288 2347 y Fm(\000)r Fq(E)2439 2362 y Fp(0)2482 2347 y Fs(+)r(1\))2647 2310 y Fp(1)p Fo(=)p Fp(2)2757 2347 y Fs(-b)s(ounded)i(op)s(erator,)572 2467 y(i.e.)43 b(there)33 b(is)f(a)g Fq(k)f(<)d Fm(1)k Fs(s.t.)44 b(for)32 b(an)m(y)h Fq( )f Fm(2)c Fq(C)2259 2431 y Fx(1)2252 2492 y Fp(0)2334 2467 y Fs(,)1246 2666 y Fm(k)p Fs([)p Fq(G)1400 2681 y Fo(\013)1450 2666 y Fq(;)17 b(H)1575 2681 y Fo(p)1614 2666 y Fs(])p Fq( )t Fm(k)27 b(\024)i Fq(k)1961 2581 y Fi(\015)1961 2641 y(\015)2017 2666 y Fs(\()p Fq(H)2136 2681 y Fo(p)2197 2666 y Fm(\000)23 b Fq(E)2369 2681 y Fp(0)2431 2666 y Fs(+)f(1\))2616 2625 y Fp(1)p Fo(=)p Fp(2)2725 2666 y Fq( )2792 2581 y Fi(\015)2792 2641 y(\015)2864 2666 y Fq(;)426 b Fs(\(2.18\))572 2865 y(where)34 b Fq(E)926 2880 y Fp(0)993 2865 y Fs(=)27 b(inf)c Fq(\033)t Fs(\()p Fq(H)1409 2880 y Fo(p)1449 2865 y Fs(\))k Fq(<)h Fs(0.)473 3063 y Fm(\017)49 b FB(The)d(F)-7 b(ermi)45 b(Golden)i(R)n(ule)f(Condition)p Fs(.)81 b(W)-8 b(e)46 b(de\014ne)h(a)e(family)e(of)i(b)s(ounded)572 3183 y(op)s(erators)c(on)g Fm(H)1240 3198 y Fo(p)1322 3183 y Fs(b)m(y)h Fq(F)14 b Fs(\()p Fq(!)t(;)j Fs(\006\))42 b(=)1959 3109 y Fi(P)2064 3212 y Fo(\013)2130 3183 y Fq(g)2177 3198 y Fo(\013)2226 3183 y Fs(\()p Fq(!)t(;)17 b Fs(\006\))p Fq(G)2558 3198 y Fo(\013)2648 3183 y Fs(and)42 b(let,)h(for)e(arbitrary)572 3304 y Fq(\017)28 b(>)f Fs(0,)763 3546 y Fq(T)820 3561 y Fo(\017)852 3546 y Fs(\()p Fq(!)t(;)17 b(E)6 b Fs(\))27 b(=)1246 3410 y Fi(Z)1301 3636 y Fo(S)1348 3617 y Fj(2)1403 3546 y Fq(d)p Fs(\006)33 b Fq(F)14 b Fs(\()p Fq(!)t(;)j Fs(\006\))2230 3478 y Fq(p)2279 3493 y Fo(c)2346 3478 y Fq(\017)p 1899 3523 821 4 v 1899 3614 a Fs(\()p Fq(H)2018 3629 y Fo(p)2078 3614 y Fm(\000)23 b Fq(E)28 b Fm(\000)23 b Fq(!)t Fs(\))2481 3585 y Fp(2)2541 3614 y Fs(+)f Fq(\017)2678 3585 y Fp(2)2728 3546 y Fq(F)14 b Fs(\()p Fq(!)t(;)j Fs(\006\))3060 3505 y Fx(\003)3099 3546 y Fq(;)191 b Fs(\(2.19\))572 3798 y(where)24 b Fq(E)29 b Fs(is)23 b(an)g(eigen)m(v)-5 b(alue)23 b(of)f Fq(H)1799 3813 y Fo(p)1862 3798 y Fs(and)h Fq(p)2091 3813 y Fo(c)2149 3798 y Fs(is)g(the)g(pro)5 b(jection)23 b(on)m(to)g(the)h(con)m(tin)m(u-)572 3919 y(ous)h(subspace)i(of)e Fq(H)1323 3934 y Fo(p)1362 3919 y Fs(.)41 b(Let)25 b Fq(p)p Fs(\()p Fq(E)6 b Fs(\))25 b(denote)h(the)g(pro)5 b(jection)24 b(on)m(to)h(the)h(eigenspace)572 4039 y(corresp)s(onding) 48 b(to)g Fq(E)6 b Fs(.)90 b(W)-8 b(e)49 b(assume)f(that)g(there)h(is)f (an)g Fq(\017)2876 4054 y Fp(0)2970 4039 y Fq(>)54 b Fs(0,)e(s.t.)91 b(for)572 4159 y(0)27 b Fq(<)h(\017)g(<)f(\017)961 4174 y Fp(0)1001 4159 y Fs(,)1087 4277 y Fi(Z)1187 4303 y Fx(1)1143 4502 y(\000)p Fo(E)1278 4412 y Fq(d!)1538 4345 y(!)1603 4309 y Fp(2)p 1436 4389 310 4 v 1436 4481 a Fq(e)1481 4452 y Fo(\014)s(!)1596 4481 y Fm(\000)c Fs(1)1755 4412 y Fq(p)p Fs(\()p Fq(E)6 b Fs(\))p Fq(T)2015 4427 y Fo(\017)2048 4412 y Fs(\()p Fq(!)t(;)17 b(E)6 b Fs(\))p Fq(p)p Fs(\()p Fq(E)g Fs(\))26 b Fm(\025)i Fq(\015)2696 4427 y Fo(E)2788 4412 y Fq(p)p Fs(\()p Fq(E)6 b Fs(\))p Fq(;)299 b Fs(\(2.20\))572 4671 y(for)32 b(an)m(y)h Fq(E)h Fm(2)28 b Fq(\033)1160 4686 y Fo(p)1200 4671 y Fs(\()p Fq(H)1319 4686 y Fo(p)1358 4671 y Fs(\),)33 b(where)h Fq(\015)1789 4686 y Fo(E)1881 4671 y Fs(is)e(a)g(strictly)g(p)s(ositiv) m(e)g(constan)m(t.)44 b(W)-8 b(e)33 b(set)1631 4870 y Fq(\015)f Fs(:=)111 b(min)1845 4937 y Fo(E)t Fx(2)p Fo(\033)1988 4945 y Fl(p)2024 4937 y Fp(\()p Fo(H)2109 4945 y Fl(p)2146 4937 y Fp(\))2190 4870 y Fq(\015)2241 4885 y Fo(E)2327 4870 y Fq(>)28 b Fs(0)p Fq(:)810 b Fs(\(2.21\))1898 5214 y(11)p eop %%Page: 12 12 12 11 bop 474 631 a FB(R)-5 b(emarks.)114 b Fs(1\))42 b(All)e(requiremen)m(ts)j(in)e(Condition)g Fq(A)h Fs(are)g(indep)s (enden)m(t)h(of)e(the)328 751 y(regularization)30 b(of)i(the)h(in)m (teraction.)474 872 y(2\))d(F)-8 b(or)28 b(the)j(ph)m(ysical)e(mo)s (del)f(of)h(an)h(atom)e(in)m(teracting)h(with)g(the)h(radiation)e (\014eld,)328 992 y(the)37 b(v)-5 b(alue)35 b(of)h(the)h(constan)m(t)g Fq(p)g Fs(in)e(\(2.16\))h(is)f Fq(p)g Fs(=)e Fm(\000)p Fs(1)p Fq(=)p Fs(2)k(\(or)e Fq(p)g Fs(=)e(1)p Fq(=)p Fs(2)j(in)g(the)g(dip)s(ole)328 1112 y(appro)m(ximation\),)f(see)i (e.g.)55 b([BFS].)37 b(Although)e Fq(p)f(>)g Fs(2)i(is)g(quite)g(far)g (from)f(the)i(ph)m(ys-)328 1233 y(ical)g(range,)k(w)m(e)f(do)f(not)g (attempt)f(here)i(to)e(optimize)f(condition)h(\(2.16\).)62 b(This)39 b(will)328 1353 y(b)s(e)i(the)g(aim)e(of)h(subsequen)m(t)k(w) m(ork.)69 b(Su\016ce)42 b(it)d(to)i(note)g(that)f(the)h(discrete)h(v)-5 b(alues)328 1474 y Fq(p)28 b Fs(=)f Fm(\000)p Fs(1)p Fq(=)p Fs(2)p Fq(;)17 b Fs(1)p Fq(=)p Fs(2)31 b(are)i(also)f (admissible)e(in)i(our)h(analysis.)474 1594 y(3\))d(The)h(op)s(erator)e Fq(T)1236 1609 y Fo(\017)1269 1594 y Fs(\()p Fq(!)t(;)17 b(E)6 b Fs(\))29 b(is)g(just)i(a)e(\(non-negativ)m(e\))h(n)m(um)m(b)s (er)g(if)f Fq(E)36 b Fs(is)30 b(a)f(simple)328 1714 y(eigen)m(v)-5 b(alue.)43 b(F)-8 b(or)30 b Fq(\017)i Fs(small,)e(it)g(represen)m(ts)k (the)e(probabilit)m(y)e(that)h(the)h(particle)e(mak)m(es)328 1835 y(a)f(transition)f(from)g(the)h(b)s(ound)h(state)g(corresp)s (onding)f(to)g(the)h(energy)g Fq(E)35 b Fs(in)m(to)29 b(a)g(scat-)328 1955 y(tering)i(state)h(with)g(energy)h Fq(E)26 b Fs(+)21 b Fq(!)31 b Fm(\025)d Fs(0)k(b)m(y)g(absorbing)g(a)f (photon)h(of)f(energy)i Fq(!)t Fs(.)43 b(The)328 2076 y(probabilit)m(y)34 b(densit)m(y)j(for)e(a)h(photon)g(to)f(ha)m(v)m(e)j (energy)f Fq(!)i Fs(is)c(giv)m(en)h(b)m(y)h(Planc)m(k's)g(la)m(w,)328 2196 y(i.e.,)h(b)m(y)g(\()p Fq(e)741 2160 y Fo(\014)s(!)859 2196 y Fm(\000)26 b Fs(1\))1049 2160 y Fx(\000)p Fp(1)1143 2196 y Fs(.)56 b(Hence)39 b Fq(\015)j Fs(is)36 b(a)h(p)s(erturbativ)m (e)g(b)s(ound)g(on)g(the)g(probabilit)m(y)e(of)328 2316 y(an)f(ionization)d(pro)s(cess;)36 b(it)d(dep)s(ends)j(on)e(the)g(in)m (v)m(erse)i(temp)s(erature)e Fq(\014)39 b Fs(as)34 b Fq(\015)h Fm(\030)c Fq(e)3405 2280 y Fo(\014)s(E)3500 2289 y Fj(0)3539 2316 y Fs(,)328 2437 y(where)h Fq(E)680 2452 y Fp(0)747 2437 y Fq(<)c Fs(0)i(is)g(the)h(ground)f(state)h (energy)h(of)e Fq(H)2257 2452 y Fo(p)2296 2437 y Fs(.)43 b(More)31 b(precisely)-8 b(,)31 b(if)f(w)m(e)h(assume)328 2557 y(that,)j(for)f(0)c Fq(<)h(\017)g(<)f(\017)1115 2572 y Fp(0)1155 2557 y Fs(,)34 b Fq(!)f Fm(7!)c Fq(p)p Fs(\()p Fq(E)6 b Fs(\))p Fq(T)1699 2572 y Fo(\017)1732 2557 y Fs(\()p Fq(!)t(;)17 b(E)6 b Fs(\))p Fq(p)p Fs(\()p Fq(E)g Fs(\))32 b(is)i(con)m(tin)m(uous,)g(and)g(that)g(there)g(is)328 2677 y(a)f(constan)m(t)h Fq(t)29 b(>)f Fs(0)33 b(s.t.)46 b Fq(p)p Fs(\()p Fq(E)6 b Fs(\))p Fq(T)1490 2692 y Fo(\017)1523 2677 y Fs(\()p Fq(!)t(;)17 b(E)6 b Fs(\))p Fq(p)p Fs(\()p Fq(E)g Fs(\))27 b Fm(\025)j Fq(t)22 b Fm(\001)h Fq(p)p Fs(\()p Fq(E)6 b Fs(\))33 b(at)f Fq(!)g Fs(=)d Fm(\000)p Fq(E)6 b Fs(,)34 b(then)g(one)f(sees)328 2808 y(that)f Fq(k)609 2769 y Fo(e)642 2745 y Fl(\014)s(E)p 603 2785 134 4 v 603 2842 a Fp(1+)p Fo(\014)774 2808 y Fm(\024)c Fq(\015)930 2823 y Fo(E)1017 2808 y Fm(\024)g Fq(k)s(e)1221 2772 y Fo(\014)s(E)1324 2808 y Fs(,)33 b(for)f(some)g Fq(k)k Fs(whic)m(h)d(do)s(es)g(not)f(dep)s(end)i(on)f Fq(\014)6 b Fs(.)474 3049 y FB(Condition)38 b Fq(B)5 b FB(.)95 b Fs(The)38 b(p)s(oten)m(tial)e Fq(v)k Fs(satis\014es)e (condition)d(C)2713 3064 y Fp(B)2768 3049 y Fs(,)j(the)f(in)m (teraction)f(is)328 3169 y(giv)m(en)d(b)m(y)g Fq(V)797 3118 y Fp(\()p Fo(\017)p Fp(\))775 3196 y Fo(J)884 3169 y Fs(,)g(and)g(the)g(follo)m(wing)c(prop)s(erties)k(hold.)473 3329 y Fm(\017)49 b Fs(The)44 b(infra-red)e(and)i(ultra-violet)d(b)s (eha)m(viour)i(of)g(the)h(form)e(factors)i(is)f(as)h(in)572 3449 y(\(2.16\),)32 b(\(2.17\).)473 3638 y Fm(\017)49 b Fs(Spatial)30 b(deca)m(y)k(of)e Fq(G)1361 3653 y Fo(\013)1411 3638 y Fs(.)43 b(There)34 b(is)e(a)h(constan)m(t)g Fq(k)e(<)c Fm(1)32 b Fs(s.t.)1182 3807 y Fm(kh)p Fq(x)p Fm(i)1365 3765 y Fo(n)1408 3774 y Fj(1)1447 3807 y Fq(G)1524 3822 y Fo(\013)1573 3807 y Fm(h)p Fq(x)p Fm(i)1706 3765 y Fo(n)1749 3774 y Fj(2)1787 3807 y Fm(k)c(\024)g Fq(k)s(;)114 b(n)2223 3822 y Fp(1)2285 3807 y Fs(+)22 b Fq(n)2441 3822 y Fp(2)2509 3807 y Fs(=)27 b(0)p Fq(;)17 b(:)g(:)g(:)f(;)h Fs(5)p Fq(;)361 b Fs(\(2.22\))572 3975 y(where)47 b(w)m(e)h(set)f Fm(h)p Fq(x)p Fm(i)k Fs(=)g(\()p Fq(x)1595 3939 y Fp(2)1666 3975 y Fs(+)31 b(1\))1860 3939 y Fp(1)p Fo(=)p Fp(2)1970 3975 y Fs(,)50 b(for)45 b Fq(x)52 b Fm(2)f Fk(R)2499 3939 y Fp(3)2545 3975 y Fs(.)84 b(Notice)46 b(that)g(this)g(is)g(a)572 4096 y(condition)31 b(on)h Fq(G)1212 4111 y Fo(\013)1294 4096 y Fs(not)h(dep)s(ending)g(on)f(the)h(regularization.)473 4284 y Fm(\017)49 b Fs(The)44 b(F)-8 b(ermi)41 b(Golden)h(Rule)h (Condition.)74 b(F)-8 b(or)42 b(all)f(eigen)m(v)-5 b(alues)43 b Fq(E)49 b Fs(of)43 b Fq(H)3353 4299 y Fo(p)3435 4284 y Fs(s.t.)572 4405 y Fq(E)37 b Fs(=)30 b Fq(E)6 b Fs(\()p Fq(m)p Fs(\))35 b(for)e(some)i Fq(m)c Fm(2)g Fq(J)1725 4420 y Fo(d)1765 4405 y Fs(,)k(let)f Fq(T)2027 4420 y Fo(\017)2059 4405 y Fs(\()p Fq(!)t(;)17 b(E)6 b Fs(\))34 b(b)s(e)g(de\014ned)i(as)e(in)g(\(2.19\),)g(with)572 4525 y Fq(p)621 4540 y Fo(c)702 4525 y Fs(replaced)46 b(b)m(y)h Fq(\026)p Fs(\()p Fq(H)1427 4540 y Fo(p)1467 4525 y Fs(\))1505 4489 y Fp(2)1544 4525 y Fs(,)j(and)c(let)g Fq(p)2028 4540 y Fo(J)2067 4552 y Fl(d)2107 4525 y Fs(\()p Fq(E)6 b Fs(\))51 b(=)2438 4450 y Fi(P)2593 4525 y Fl(m)p Fe(2)p Fl(J)2723 4542 y(d)2554 4589 y(E)s Fj(\()p Fl(m)p Fj(\)=)p Fl(E)2828 4525 y Fq(p)2877 4540 y Fo(m)2944 4525 y Fs(.)84 b(There)48 b(is)e(an)572 4679 y Fq(\017)611 4694 y Fp(0)678 4679 y Fq(>)28 b Fs(0)k(s.t.,)h(for)f(0)c Fq(<)f(\017)h(<)g(\017)1592 4694 y Fp(0)1632 4679 y Fs(,)840 4765 y Fi(Z)939 4792 y Fx(1)895 4991 y(\000)p Fo(E)1031 4901 y Fq(d!)1295 4834 y(!)1360 4798 y Fp(2)p 1188 4878 319 4 v 1188 4969 a Fq(e)1233 4941 y Fo(\014)s(E)1358 4969 y Fm(\000)23 b Fs(1)1549 4901 y Fq(p)1598 4916 y Fo(J)1637 4928 y Fl(d)1677 4901 y Fs(\()p Fq(E)6 b Fs(\))p Fq(T)1888 4916 y Fo(\017)1921 4901 y Fs(\()p Fq(!)t(;)17 b(E)6 b Fs(\))p Fq(p)2233 4916 y Fo(J)2272 4928 y Fl(d)2310 4901 y Fs(\()p Fq(E)g Fs(\))28 b Fm(\025)g Fq(\015)2648 4916 y Fo(E)2740 4901 y Fq(p)2789 4916 y Fo(J)2828 4928 y Fl(d)2868 4901 y Fs(\()p Fq(E)6 b Fs(\))p Fq(;)268 b Fs(\(2.23\))1898 5214 y(12)p eop %%Page: 13 13 13 12 bop 572 631 a Fs(for)32 b(some)g(strictly)g(p)s(ositiv)m(e)g (constan)m(t)h Fq(\015)2104 646 y Fo(E)2164 631 y Fs(.)43 b(W)-8 b(e)33 b(set)689 851 y Fq(\015)f Fs(:=)c(min)o Fm(f)p Fq(\015)1167 866 y Fo(E)1258 851 y Fm(j)k Fq(E)i Fm(2)28 b Fq(\033)1573 866 y Fo(p)1613 851 y Fs(\()p Fq(H)1732 866 y Fo(p)1772 851 y Fs(\))k(s.t.)44 b Fq(E)34 b Fs(=)27 b Fq(E)6 b Fs(\()p Fq(m)p Fs(\))33 b(for)f(some)h Fq(m)28 b Fm(2)g Fq(J)3152 866 y Fo(d)3192 851 y Fm(g)g Fq(>)f Fs(0)p Fq(:)3317 971 y Fs(\(2.24\))474 1175 y FB(R)-5 b(emarks.)112 b Fs(1\))40 b Fq(\015)47 b Fs(is)40 b(exp)s(onen)m(tially)h(small)e(in)h Fq(\014)6 b Fs(,)43 b(as)f(observ)m(ed)h(in)d(Remark)h(3\))328 1295 y(after)32 b(\(2.21\).)474 1416 y(2\))g(The)h(op)s(erator)f Fq(T)1243 1431 y Fo(\017)1276 1416 y Fs(\()p Fq(!)t(;)17 b(E)6 b Fs(\))31 b(is)g(a)h(decreasing)h(function)f(of)f Fq(r)s Fs(,)h(and)g(an)h(increasing)328 1536 y(function)42 b(of)h Fq(R)q Fs(.)74 b(Th)m(us,)48 b(w)m(e)c(ma)m(y)e(assume)i(without)e (loss)h(of)f(generalit)m(y)h(that)f Fq(\015)48 b Fs(is)328 1656 y(indep)s(enden)m(t)34 b(of)e Fq(r)e Fm(\024)e Fs(1)p Fq(;)17 b(R)29 b Fm(\025)f Fs(2.)328 2036 y Ft(2.1.2)112 b(Reference)38 b(state)f Fq(!)1527 2000 y Fp(ref)328 2221 y Fs(The)c FB(r)-5 b(efer)g(enc)g(e)34 b(state)f Fs(of)f(the)h(system)h(is)e(giv)m(en)h(b)m(y)g(the)g(pro)s(duct)g (state)1623 2441 y Fq(!)1688 2400 y Fp(ref)1805 2441 y Fs(=)27 b Fq(!)1973 2400 y Fo(p)2034 2441 y Fm(\012)c Fq(!)2199 2394 y Fo(f)2195 2469 y(\014)2244 2441 y Fq(;)1046 b Fs(\(2.25\))328 2661 y(where)26 b Fq(!)667 2625 y Fo(p)730 2661 y Fs(is)e(a)g(state)h(on)f Fm(B)s Fs(\()p Fm(H)1441 2676 y Fo(p)1482 2661 y Fs(\),)i(determined)e(b)m(y)h(a)f(strictly)g(p) s(ositiv)m(e)g(densit)m(y)h(matrix)328 2782 y Fq(\032)378 2797 y Fo(p)446 2782 y Fq(>)i Fs(0,)33 b(i.e.)1586 2902 y Fq(!)1647 2917 y Fo(p)1686 2902 y Fs(\()p Fq(A)p Fs(\))28 b(=)f(tr\()p Fq(\032)2130 2917 y Fo(p)2170 2902 y Fq(A)p Fs(\))p Fq(;)1009 b Fs(\(2.26\))328 3085 y(for)45 b(an)m(y)i Fq(A)j Fm(2)h(B)s Fs(\()p Fm(H)1118 3100 y Fo(p)1158 3085 y Fs(\).)84 b(The)47 b(state)f Fq(!)1838 3038 y Fo(f)1834 3113 y(\014)1929 3085 y Fs(is)f(the)h Fq(\014)6 b Fs(-KMS)46 b(state)g(of)g Fr(A)3028 3100 y Fo(f)3119 3085 y Fs(w.r.t.)84 b(the)328 3206 y(free)36 b(\014eld)f(dynamics)g (\(2.9\))g(determined)g(b)m(y)i(the)f(exp)s(ectation)f(functional)f (\(2.7\).)52 b(It)328 3326 y(describ)s(es)34 b FB(black)g(b)-5 b(o)g(dy)34 b(r)-5 b(adiation)32 b Fs(of)g(the)h(\014eld)g(at)f(temp)s (erature)g(1)p Fq(=\014)6 b Fs(.)474 3447 y(Let)25 b(\()p Fm(H)q Fq(;)17 b(\031)863 3462 y Fo(\014)911 3447 y Fq(;)g Fs(\012)1025 3410 y Fp(ref)1115 3447 y Fs(\))24 b(b)s(e)h(the)h(GNS)e (represen)m(tation)i(of)e(\()p Fr(A)p Fq(;)17 b(!)2642 3410 y Fp(ref)2732 3447 y Fs(\),)26 b(i.e.)40 b Fm(H)26 b Fs(is)f(a)f(Hilb)s(ert)328 3567 y(space,)k Fq(\031)665 3582 y Fo(\014)738 3567 y Fs(is)e(a)f Fm(\003)p Fs(-morphism)f Fr(A)k Fm(!)f(B)s Fs(\()p Fm(H)q Fs(\),)h(and)e(\012)2199 3531 y Fp(ref)2315 3567 y Fs(is)f(a)g(v)m(ector)i(in)e Fm(H)i Fs(s.t.)42 b Fq(\031)3211 3582 y Fo(\014)3258 3567 y Fs(\()p Fr(A)p Fs(\)\012)3475 3531 y Fp(ref)328 3687 y Fs(is)32 b(dense)i(in)e Fm(H)q Fs(,)h(and)1149 3907 y Fq(!)1214 3866 y Fp(ref)1303 3907 y Fs(\()p Fq(A)p Fs(\))28 b(=)1583 3827 y Fi(\012)1631 3907 y Fs(\012)1701 3866 y Fp(ref)1791 3907 y Fq(;)17 b(\031)1890 3922 y Fo(\014)1937 3907 y Fs(\()p Fq(A)p Fs(\)\012)2156 3866 y Fp(ref)2246 3827 y Fi(\013)2310 3907 y Fq(;)114 b(A)28 b Fm(2)g Fr(A)p Fq(:)328 4127 y Fs(An)48 b(explicit)f(realization)e(of) j(the)g(GNS)g(represen)m(tation)h(is)e(w)m(ell)h(kno)m(wn.)91 b(It)48 b(w)m(as)328 4248 y(\014rst)33 b(constructed)i(b)m(y)e(Araki)f (and)h(W)-8 b(o)s(o)s(ds,)33 b([A)-11 b(W],)34 b(and)e(has)i(b)s(een)f (used)h(recen)m(tly)g(b)m(y)328 4368 y(sev)m(eral)j(authors.)56 b(Here)38 b(w)m(e)f(just)g(recall)e(the)i(explicit)f(form)m(ulas)f (that)h(are)h(useful)g(in)328 4489 y(the)c(presen)m(t)h(pap)s(er)f(and) g(refer)f(to)h([JP],)g([FM])g(for)f(a)g(more)g(detailed)f(discussion.) 474 4609 y(The)j(represen)m(tation)f(Hilb)s(ert)e(space)j(is)1539 4829 y Fm(H)29 b Fs(=)e Fm(H)1839 4844 y Fo(p)1901 4829 y Fm(\012)c(H)2085 4844 y Fo(p)2147 4829 y Fm(\012)f(F)10 b Fq(;)962 b Fs(\(2.27\))1898 5214 y(13)p eop %%Page: 14 14 14 13 bop 328 631 a Fs(where)34 b Fm(F)42 b Fs(is)32 b(a)g(shorthand)h(for)f(the)h(F)-8 b(o)s(c)m(k)33 b(space)1252 820 y Fm(F)k Fs(=)28 b Fm(F)1563 739 y Fi(\000)1608 820 y Fs(\()p Fq(L)1712 779 y Fp(2)1752 820 y Fs(\()p Fk(R)33 b Fm(\002)23 b Fq(S)2050 779 y Fp(2)2089 820 y Fq(;)49 b(du)22 b Fm(\002)g Fq(d)p Fs(\006\))2552 739 y Fi(\001)2615 820 y Fq(;)675 b Fs(\(2.28\))328 1009 y Fq(du)37 b Fs(b)s(eing)h(the)g (Leb)s(esgue)i(measure)e(on)h Fk(R)5 b Fs(,)45 b(and)38 b Fq(d)p Fs(\006)h(the)f(uniform)f(measure)h(on)g Fq(S)3499 973 y Fp(2)3539 1009 y Fs(.)328 1129 y(Here)30 b Fm(F)10 b Fs(\()p Fq(X)e Fs(\))29 b(denotes)i(the)f(Bosonic)g(F)-8 b(o)s(c)m(k)30 b(space)h(o)m(v)m(er)f(a)g(\(normed)f(v)m(ector\))i (space)g Fq(X)8 b Fs(,)1367 1336 y Fm(F)i Fs(\()p Fq(X)e Fs(\))27 b(:=)h Fk(C)48 b Fm(\010)1966 1241 y Fi(M)1974 1452 y Fo(n)p Fx(\025)p Fp(1)2133 1255 y Fi(\000)2178 1336 y Fm(S)7 b Fq(X)2334 1295 y Fx(\012)p Fo(n)2437 1255 y Fi(\001)2499 1336 y Fq(;)791 b Fs(\(2.29\))328 1619 y(where)27 b Fm(S)34 b Fs(is)26 b(the)h(pro)5 b(jection)26 b(on)m(to)g(the)h(symmetric)e(subspace)j(of)e(the)h(tensor)f(pro)s (duct.)328 1739 y(W)-8 b(e)30 b(use)g(standard)g(notation,)e(e.g.)43 b(\012)30 b(is)e(the)i(v)-5 b(acuum)29 b(v)m(ector,)i([)p Fq( )t Fs(])2819 1754 y Fo(n)2896 1739 y Fs(is)e(the)g Fq(n)p Fs(-particle)328 1860 y(comp)s(onen)m(t)36 b(of)g Fq( )j Fm(2)34 b(F)10 b Fs(\()p Fq(X)e Fs(\),)37 b(d\000\()p Fq(A)p Fs(\))f(is)g(the)h(second)h(quan)m(tization)e(of)g(the)g(op)s (erator)328 1980 y Fq(A)d Fs(on)f Fq(X)8 b Fs(,)32 b Fq(N)39 b Fs(=)27 b(d\000\(1)-22 b(l\))31 b(is)i(the)g(n)m(um)m(b)s(er) g(op)s(erator.)474 2100 y(The)h(represen)m(tation)f(map)f Fq(\031)1584 2115 y Fo(\014)1659 2100 y Fs(:)c Fr(A)g Fm(!)f(B)s Fs(\()p Fm(H)q Fs(\))33 b(is)f(the)h(pro)s(duct)1655 2298 y Fq(\031)1710 2313 y Fo(\014)1785 2298 y Fs(=)28 b Fq(\031)1944 2313 y Fo(p)2006 2298 y Fm(\012)23 b Fq(\031)2165 2251 y Fo(\014)2161 2326 y(f)2212 2298 y Fq(;)328 2487 y Fs(where)34 b(the)f Fm(\003)p Fs(homomorphism)c Fq(\031)1571 2502 y Fo(p)1639 2487 y Fs(:)f Fr(A)1765 2502 y Fo(p)1833 2487 y Fm(!)f(B)s Fs(\()p Fm(H)2150 2502 y Fo(p)2212 2487 y Fm(\012)c(H)2396 2502 y Fo(p)2436 2487 y Fs(\))32 b(is)g(giv)m(en)h(b)m(y)1602 2676 y Fq(\031)1657 2691 y Fo(p)1697 2676 y Fs(\()p Fq(A)p Fs(\))27 b(=)h Fq(A)22 b Fm(\012)h Fs(1)-22 b(l)2227 2691 y Fo(p)2265 2676 y Fq(:)1025 b Fs(\(2.30\))328 2874 y(The)33 b(represen)m(tation)h(map)e Fq(\031)1442 2827 y Fo(\014)1438 2902 y(f)1516 2874 y Fs(:)c Fr(A)1642 2889 y Fo(f)1716 2874 y Fm(!)f(B)s Fs(\()p Fm(F)10 b Fs(\))32 b(is)g(determined)h(b)m(y)1227 3128 y Fq(\031)1286 3080 y Fo(\014)1282 3155 y(f)1333 3128 y Fs(\()p Fq(a)p Fs(\()p Fq(h)p Fs(\)\))28 b(=)1724 2992 y Fi(Z)1779 3218 y Fd(R)1848 3128 y Fq(dt)k(h)p Fs(\()p Fq(t)p Fs(\))h Fq(\031)2225 3080 y Fo(\014)2221 3155 y Fg(W)2298 3128 y Fs(\()p Fq(\013)2399 3087 y Fg(W)2398 3152 y Fo(t)2474 3128 y Fs(\()p Fq(a)p Fs(\)\))p Fq(;)651 b Fs(\(2.31\))328 3388 y(where)34 b Fq(\031)669 3341 y Fo(\014)665 3416 y Fg(W)769 3388 y Fs(:)28 b Fr(W)g Fm(!)f(B)s Fs(\()p Fm(F)10 b Fs(\))32 b(is)g(a)h(represen)m(tation)g (of)f(the)h(W)-8 b(eyl)33 b(algebra)e(giv)m(en)i(b)m(y)1602 3586 y Fq(\031)1661 3539 y Fo(\014)1657 3613 y Fg(W)1762 3586 y Fs(=)27 b Fq(\031)1920 3601 y Fp(F)-6 b(o)r(c)n(k)2091 3586 y Fm(\016)22 b(T)2217 3601 y Fo(\014)2264 3586 y Fq(:)328 3775 y Fs(Here,)k Fm(T)632 3790 y Fo(\014)702 3775 y Fs(is)d(the)g(Bogoliub)s(o)m(v)e(transformation,)i(mapping)f Fr(W)p Fs(\()p Fq(L)2734 3739 y Fp(2)2734 3799 y(0)2774 3775 y Fs(\))g(to)h Fr(W)p Fs(\()p Fq(L)3152 3739 y Fp(2)3192 3775 y Fs(\()p Fk(R)8 b Fm(\002)s Fq(S)3445 3739 y Fp(2)3490 3775 y Fs(\)\))328 3895 y(de\014ned)34 b(b)m(y)f Fq(W)14 b Fs(\()p Fq(f)d Fs(\))27 b Fm(7!)h Fq(W)14 b Fs(\()p Fq(\034)1381 3910 y Fo(\014)1428 3895 y Fq(f)d Fs(\),)32 b(with)g Fq(\034)1848 3910 y Fo(\014)1923 3895 y Fs(:)c Fq(L)2044 3859 y Fp(2)2084 3895 y Fs(\()p Fk(R)2188 3910 y Fp(+)2275 3895 y Fm(\002)23 b Fq(S)2441 3859 y Fp(2)2480 3895 y Fs(\))28 b Fm(!)f Fq(L)2739 3859 y Fp(2)2779 3895 y Fs(\()p Fk(R)33 b Fm(\002)22 b Fq(S)3076 3859 y Fp(2)3116 3895 y Fs(\))32 b(giv)m(en)h(b)m(y)650 4150 y(\()p Fq(\034)730 4165 y Fo(\014)778 4150 y Fq(f)11 b Fs(\)\()p Fq(u;)17 b Fs(\006\))27 b(=)1251 4008 y Fi(r)p 1351 4008 379 4 v 1512 4082 a Fq(u)p 1361 4127 359 4 v 1361 4218 a Fs(1)22 b Fm(\000)g Fq(e)1576 4189 y Fx(\000)p Fo(\014)s(u)1746 4009 y Fi(\032)1862 4014 y Fm(p)p 1945 4014 56 4 v 72 x Fq(u)32 b(f)11 b Fs(\()p Fq(u;)17 b Fs(\006\))p Fq(;)315 b(u)27 b(>)h Fs(0)p Fq(;)1862 4211 y Fm(\000)1939 4136 y(p)p 2023 4136 134 4 v 2023 4211 a(\000)p Fq(u)p 2188 4130 59 4 v 32 w(f)11 b Fs(\()p Fm(\000)p Fq(u;)17 b Fs(\006\))p Fq(;)83 b(u)27 b(<)h Fs(0)p Fq(:)3317 4150 y Fs(\(2.32\))328 4399 y FB(R)-5 b(emarks.)39 b Fs(1\))21 b(It)h(is)f(easily)g(v)m(eri\014ed)i(that)e(Im)16 b Fm(h)p Fq(\034)2057 4414 y Fo(\014)2105 4399 y Fq(f)10 b(;)17 b(\034)2249 4414 y Fo(\014)2297 4399 y Fq(g)s Fm(i)2386 4431 y Fo(L)2434 4412 y Fj(2)2468 4431 y Fp(\()p Fd(R)p Fx(\002)p Fo(S)2645 4412 y Fj(2)2680 4431 y Fp(\))2739 4399 y Fs(=)28 b(Im)16 b Fm(h)o Fq(f)5 b(;)17 b(g)t Fm(i)3200 4428 y Fo(L)3248 4409 y Fj(2)3283 4428 y Fp(\()p Fd(R)3358 4437 y Fj(+)3409 4428 y Fx(\002)p Fo(S)3511 4409 y Fj(2)3545 4428 y Fp(\))3577 4399 y Fs(,)328 4535 y(for)32 b(all)e Fq(f)5 b(;)17 b(g)31 b Fm(2)e Fq(L)948 4499 y Fp(2)948 4560 y(0)987 4535 y Fs(,)k(so)g(the)g(CCR)g(\(2.5\))f(are)h(preserv)m (ed)i(under)e(the)g(map)f Fq(\034)3113 4550 y Fo(\014)3160 4535 y Fs(.)474 4655 y(2\))g(In)h(the)g(limit)c Fq(\014)34 b Fm(!)27 b(1)p Fs(,)32 b(the)h(r.h.s.)44 b(of)33 b(\(2.32\))e(tends)j (to)1480 4762 y Fi(\032)1597 4841 y Fq(u)e(f)11 b Fs(\()p Fq(u;)17 b Fs(\006\))p Fq(;)82 b(u)27 b(>)h Fs(0)p Fq(;)1597 4962 y Fs(0)p Fq(;)426 b(u)27 b(<)h Fs(0)p Fq(:)3317 4902 y Fs(\(2.33\))1898 5214 y(14)p eop %%Page: 15 15 15 14 bop 328 631 a Fs(Notice)27 b(that)g Fq(L)901 595 y Fp(2)941 631 y Fs(\()p Fk(R)1045 646 y Fp(+)1121 631 y Fm(\002)11 b Fq(S)1275 595 y Fp(2)1315 631 y Fs(\))g Fm(\010)g Fq(L)1518 595 y Fp(2)1559 631 y Fs(\()p Fk(R)1663 646 y Fp(+)1740 631 y Fm(\002)g Fq(S)1894 595 y Fp(2)1934 631 y Fs(\))27 b(is)g(isometrically)d(isomorphic)h(to)i Fq(L)3327 595 y Fp(2)3367 631 y Fs(\()p Fk(R)16 b Fm(\002)328 751 y Fq(S)394 715 y Fp(2)433 751 y Fs(\))33 b(via)e(the)i(map)958 1020 y(\()p Fq(f)5 b(;)17 b(g)t Fs(\))27 b Fm(7!)g Fq(h;)82 b(h)p Fs(\()p Fq(u;)17 b Fs(\006\))27 b(=)1934 879 y Fi(\032)2050 959 y Fq(u)32 b(f)11 b Fs(\()p Fq(u;)17 b Fs(\006\))p Fq(;)152 b(u)27 b(>)g Fs(0)p Fq(;)2050 1079 y(u)32 b(g)t Fs(\()p Fm(\000)p Fq(u;)17 b Fs(\006\))p Fq(;)83 b(u)27 b(<)g Fs(0)p Fq(;)3317 1020 y Fs(\(2.34\))328 1292 y(so)33 b(\(2.33\))g(can)h(b)s(e)f(iden)m(ti\014ed,)h(via)e (\(2.34\),)h(with)g Fq(f)40 b Fm(2)29 b Fq(L)2440 1256 y Fp(2)2440 1317 y(0)2480 1292 y Fs(.)45 b(Th)m(us,)36 b Fm(T)2882 1307 y Fo(\014)2962 1292 y Fs(reduces)f(to)e(the)328 1421 y(iden)m(tit)m(y)i(\(an)g(im)m(b)s(edding\),)g Fq(\031)1472 1374 y Fo(\014)1468 1449 y Fg(W)1580 1421 y Fs(b)s(ecomes)g(the)h(F)-8 b(o)s(c)m(k)36 b(represen)m(tation)g(of)f Fr(W)p Fs(\()p Fq(L)3338 1385 y Fp(2)3338 1446 y(0)3378 1421 y Fs(\),)h(as)328 1542 y Fq(\014)d Fm(!)28 b(1)p Fs(,)k(and)g(w)m(e)i(reco)m(v)m(er)g (the)f(zero)g(temp)s(erature)g(situation.)474 1662 y(It)e(is)g(useful)g (to)g(in)m(tro)s(duce)g(the)h(follo)m(wing)c(notation.)42 b(W)-8 b(e)31 b(de\014ne)i(unitary)e(op)s(era-)328 1783 y(tors)527 1757 y Fi(c)524 1783 y Fq(W)13 b Fs(\()p Fq(f)e Fs(\))32 b(on)h(the)g(Hilb)s(ert)e(space)i(\(2.27\))f(b)m(y)1258 1980 y Fi(c)1255 2005 y Fq(W)14 b Fs(\()p Fq(f)d Fs(\))27 b(=)g Fq(e)1671 1964 y Fo(i')p Fp(\()p Fo(f)7 b Fp(\))1842 2005 y Fq(;)82 b(f)38 b Fm(2)28 b Fq(L)2197 1964 y Fp(2)2237 2005 y Fs(\()p Fk(R)33 b Fm(\002)23 b Fq(S)2535 1964 y Fp(2)2574 2005 y Fs(\))p Fq(;)328 2215 y Fs(where)34 b Fq(')p Fs(\()p Fq(f)11 b Fs(\))32 b(is)g(the)h(selfadjoin)m(t)e(op)s (erator)h(on)h Fm(F)42 b Fs(giv)m(en)32 b(b)m(y)1493 2481 y Fq(')p Fs(\()p Fq(f)11 b Fs(\))27 b(=)1833 2413 y Fq(a)1884 2377 y Fx(\003)1923 2413 y Fs(\()p Fq(f)11 b Fs(\))22 b(+)g Fq(a)p Fs(\()p Fq(f)11 b Fs(\))p 1833 2458 532 4 v 2032 2478 a Fm(p)p 2115 2478 49 4 v 82 x Fs(2)2374 2481 y Fq(;)916 b Fs(\(2.35\))328 2746 y(and)34 b Fq(a)570 2710 y Fx(\003)610 2746 y Fs(\()p Fq(f)11 b Fs(\),)35 b Fq(a)p Fs(\()p Fq(f)11 b Fs(\))34 b(are)g(the)h (creation-)e(and)i(annihilation)30 b(op)s(erators)35 b(on)f Fm(F)10 b Fs(,)34 b(smeared)328 2866 y(out)e(with)h Fq(f)11 b Fs(.)43 b(One)33 b(easily)e(v)m(eri\014es)j(that)1479 3076 y Fq(\031)1538 3029 y Fo(\014)1534 3104 y Fg(W)1611 3076 y Fs(\()p Fq(W)14 b Fs(\()p Fq(f)d Fs(\)\))26 b(=)2061 3051 y Fi(c)2058 3076 y Fq(W)14 b Fs(\()p Fq(\034)2244 3091 y Fo(\014)2291 3076 y Fq(f)d Fs(\))p Fq(:)474 3286 y Fs(The)34 b(cyclic)e(GNS)g(v)m(ector)i(is)e(giv)m(en)h(b)m(y)1636 3496 y(\012)1706 3455 y Fp(ref)1824 3496 y Fs(=)28 b(\012)1998 3511 y Fo(p)2060 3496 y Fm(\012)23 b Fs(\012)p Fq(;)328 3707 y Fs(where)34 b(\012)f(is)f(the)h(v)-5 b(acuum)32 b(in)g Fm(F)10 b Fs(,)32 b(and)1220 3929 y(\012)1290 3944 y Fo(p)1358 3929 y Fs(=)1461 3834 y Fi(X)1467 4044 y Fo(n)p Fx(\025)p Fp(0)1622 3929 y Fq(k)1673 3944 y Fo(n)1720 3929 y Fq(')1784 3944 y Fo(n)1853 3929 y Fm(\012)22 b(C)2004 3944 y Fo(p)2044 3929 y Fq(')2108 3944 y Fo(n)2183 3929 y Fm(2)28 b(H)2361 3944 y Fo(p)2423 3929 y Fm(\012)23 b(H)2607 3944 y Fo(p)2646 3929 y Fq(:)644 b Fs(\(2.36\))328 4242 y(Here,)42 b Fm(f)p Fq(k)698 4206 y Fp(2)695 4267 y Fo(n)742 4242 y Fm(g)792 4206 y Fx(1)792 4267 y Fo(n)p Fp(=0)968 4242 y Fs(is)e(the)g(sp)s(ectrum)g(of)f Fq(\032)1850 4257 y Fo(p)1890 4242 y Fs(,)j Fm(f)p Fq(')2073 4257 y Fo(n)2119 4242 y Fm(g)e Fs(is)f(an)h(orthogonal)e(basis)h(of)h (eigen-)328 4363 y(v)m(ectors)46 b(of)f Fq(\032)845 4378 y Fo(p)885 4363 y Fs(,)j(and)d Fm(C)1214 4378 y Fo(p)1299 4363 y Fs(is)f(an)h(an)m(tilinear)e(in)m(v)m(olution)g(on)i Fm(H)2708 4378 y Fo(p)2748 4363 y Fs(.)81 b(The)45 b(op)s(erator)g Fm(C)3526 4378 y Fo(p)328 4483 y Fs(comes)d(from)e(the)i(iden)m (ti\014cation)e(of)h Fq(l)1789 4447 y Fp(2)1829 4483 y Fs(\()p Fm(H)1951 4498 y Fo(p)1991 4483 y Fs(\))g(\(Hilb)s(ert-Sc)m (hmitt)e(op)s(erators)j(on)f Fm(H)3488 4498 y Fo(p)3528 4483 y Fs(\))328 4604 y(with)34 b Fm(H)636 4619 y Fo(p)700 4604 y Fm(\012)24 b(H)885 4619 y Fo(p)925 4604 y Fs(,)35 b(via)f Fm(j)p Fq(')p Fm(ih)p Fq( )t Fm(j)d(7!)g Fq(')23 b Fm(\012)i(C)1817 4619 y Fo(p)1857 4604 y Fq( )t Fs(.)50 b(W)-8 b(e)35 b(\014x)h(a)e(con)m(v)m(enien)m(t)j(c)m(hoice)e(for)f Fm(C)3375 4619 y Fo(p)3415 4604 y Fs(.)51 b(It)328 4724 y(is)37 b(the)h(an)m(tilinear)e(in)m(v)m(olution)g(on)h Fm(H)1733 4739 y Fo(p)1811 4724 y Fs(corresp)s(onding)g(to)h(complex)f (conjugation)f(of)328 4844 y(comp)s(onen)m(ts)e(of)e(v)m(ectors)j(in)d (the)i(basis)f(in)f(whic)m(h)i(the)f(Hamiltonian)d Fq(H)3037 4859 y Fo(p)3109 4844 y Fs(is)j(diagonal)328 4965 y(\(i.e.)43 b(it)31 b(is)i(the)g(the)g(time)e(rev)m(ersal)i(op)s(erator\).)43 b(Then)33 b Fm(C)2421 4980 y Fo(p)2462 4965 y Fq(H)2543 4980 y Fo(p)2582 4965 y Fm(C)2634 4980 y Fo(p)2702 4965 y Fs(=)27 b Fq(H)2886 4980 y Fo(p)2926 4965 y Fq(:)1898 5214 y Fs(15)p eop %%Page: 16 16 16 15 bop 328 631 a Ft(2.1.3)112 b Fq(W)776 595 y Fx(\003)815 631 y Ft(-dynamical)37 b(system)g Fs(\()p Fr(M)1904 646 y Fo(\014)1951 631 y Fq(;)17 b(\033)2050 646 y Fo(t;\025)2141 631 y Fs(\))328 816 y(Let)29 b Fr(M)604 831 y Fo(\014)680 816 y Fs(b)s(e)h(the)g(v)m(on)g(Neumann)f(algebra)f(obtained)h(b)m(y)h (taking)e(the)i(w)m(eak)h(closure)e(of)328 936 y Fq(\031)383 951 y Fo(\014)430 936 y Fs(\()p Fr(A)p Fs(\))k(in)f Fm(B)s Fs(\()p Fm(H)q Fs(\),)1138 1152 y Fr(M)1243 1167 y Fo(\014)1317 1152 y Fs(=)27 b Fm(B)s Fs(\()p Fm(H)1610 1167 y Fo(p)1651 1152 y Fs(\))22 b Fm(\012)g Fs(1)-22 b(l)1865 1167 y Fo(p)1926 1152 y Fm(\012)23 b Fq(\031)2085 1105 y Fo(\014)2081 1180 y(f)2132 1152 y Fs(\()p Fr(A)2241 1167 y Fo(f)2286 1152 y Fs(\))2324 1111 y Fx(00)2394 1152 y Fm(\032)29 b(B)s Fs(\()p Fm(H)q Fs(\))p Fq(:)561 b Fs(\(2.37\))328 1359 y(Since)39 b(the)h(densit)m(y)g(matrix)e Fq(\032)1481 1374 y Fo(p)1560 1359 y Fs(is)g(strictly)h(p)s(ositiv)m(e,)h(\012)2471 1374 y Fo(p)2550 1359 y Fs(is)f(cyclic)g(and)g(separating)328 1480 y(for)e(the)g(v)m(on)h(Neumann)f(algebra)f Fq(\031)1690 1495 y Fo(p)1730 1480 y Fs(\()p Fr(A)1839 1495 y Fo(p)1879 1480 y Fs(\))1917 1444 y Fx(00)1995 1480 y Fs(=)g Fm(B)s Fs(\()p Fm(H)2297 1495 y Fo(p)2337 1480 y Fs(\))25 b Fm(\012)h Fs(1)-22 b(l)2558 1495 y Fo(p)2596 1480 y Fs(.)57 b(Similarly)-8 b(,)35 b(\012)i(is)g(cyclic)328 1609 y(and)f(separating) f(for)h Fq(\031)1208 1562 y Fo(\014)1204 1637 y(f)1255 1609 y Fs(\()p Fr(A)1364 1624 y Fo(f)1410 1609 y Fs(\))1448 1573 y Fx(00)1490 1609 y Fs(,)h(since)f(it)f(is)h(the)g(GNS)g(v)m (ector)h(of)e(a)h(KMS)g(state)h(\(see)328 1741 y(e.g.)61 b([BRI)s(I]\).)39 b(Consequen)m(tly)-8 b(,)42 b(\012)1608 1704 y Fp(ref)1737 1741 y Fs(is)c(cyclic)g(and)h(separating)e(for)h Fr(M)3048 1756 y Fo(\014)3095 1741 y Fs(.)61 b(Let)39 b Fq(J)48 b Fs(b)s(e)328 1861 y(the)33 b(mo)s(dular)d(conjugation)i(op) s(erator)g(asso)s(ciated)g(to)g(\()p Fr(M)2531 1876 y Fo(\014)2578 1861 y Fq(;)17 b Fs(\012)2692 1825 y Fp(ref)2782 1861 y Fs(\).)43 b(It)33 b(is)f(giv)m(en)h(b)m(y)1679 2068 y Fq(J)k Fs(=)27 b Fq(J)1927 2083 y Fo(p)1989 2068 y Fm(\012)c Fq(J)2143 2083 y Fo(f)2188 2068 y Fq(;)1102 b Fs(\(2.38\))328 2275 y(where,)34 b(for)e Fq(';)17 b( )31 b Fm(2)d(H)1166 2290 y Fo(p)1206 2275 y Fs(,)33 b Fq(J)1320 2290 y Fo(p)1376 2275 y Fs(\()p Fq(')22 b Fm(\012)g(C)1651 2290 y Fo(p)1692 2275 y Fq( )t Fs(\))27 b(=)h Fq( )e Fm(\012)d(C)2169 2290 y Fo(p)2209 2275 y Fq(';)32 b Fs(and,)h(for)f Fq( )g Fs(=)27 b Fm(f)p Fs([)p Fq( )t Fs(])3067 2290 y Fo(n)3114 2275 y Fm(g)3164 2290 y Fo(n)p Fx(\025)p Fp(0)3329 2275 y Fm(2)h(F)10 b Fs(,)832 2493 y([)p Fq(J)913 2508 y Fo(f)958 2493 y Fq( )t Fs(])1052 2508 y Fo(n)1099 2493 y Fs(\()p Fq(u)1193 2508 y Fp(1)1232 2493 y Fq(;)17 b(:)g(:)g(:)f(;)h(u)1507 2508 y Fo(n)1553 2493 y Fs(\))28 b(=)p 1722 2406 815 4 v 27 w([)p Fq( )t Fs(])1843 2508 y Fo(n)1890 2493 y Fs(\()p Fm(\000)p Fq(u)2061 2508 y Fp(1)2101 2493 y Fq(;)17 b(:)g(:)g(:)f(;)h Fm(\000)p Fq(u)2453 2508 y Fo(n)2499 2493 y Fs(\))p Fq(;)82 b Fs(for)32 b Fq(n)c Fm(\025)g Fs(1)p Fq(;)328 2710 y Fs(and)33 b([)p Fq(J)599 2725 y Fo(f)644 2710 y Fq( )t Fs(])738 2725 y Fp(0)805 2710 y Fs(=)p 909 2624 260 4 v 28 w([)p Fq(J)990 2725 y Fo(f)1035 2710 y Fq( )t Fs(])1129 2725 y Fp(0)1196 2710 y Fm(2)28 b Fk(C)20 b Fs(.)50 b(Clearly)-8 b(,)32 b Fq(J)9 b Fs(\012)1924 2674 y Fp(ref)2042 2710 y Fs(=)27 b(\012)2215 2674 y Fp(ref)2305 2710 y Fs(,)33 b(and)g(one)g(v)m (eri\014es)g(that)916 2917 y Fq(J)970 2932 y Fo(p)1010 2917 y Fq(\031)1065 2932 y Fo(p)1105 2917 y Fs(\()p Fq(A)p Fs(\))p Fq(J)1308 2932 y Fo(p)1430 2917 y Fs(=)83 b(1)-22 b(l)1644 2932 y Fo(p)1705 2917 y Fm(\012)22 b(C)1856 2932 y Fo(p)1896 2917 y Fq(A)p Fm(C)2021 2932 y Fo(p)2062 2917 y Fq(;)1228 b Fs(\(2.39\))701 3078 y Fq(J)755 3093 y Fo(f)800 3078 y Fq(\031)859 3031 y Fo(\014)855 3106 y Fg(W)932 3078 y Fs(\()p Fq(W)14 b Fs(\()p Fq(f)d Fs(\)\))p Fq(J)1303 3093 y Fo(f)1430 3078 y Fs(=)1592 3053 y Fi(c)1589 3078 y Fq(W)j Fs(\()p Fm(\000)p Fq(e)1855 3037 y Fx(\000)p Fo(\014)s(u=)p Fp(2)2069 3078 y Fq(\034)2111 3093 y Fo(\014)2158 3078 y Fs(\()p Fq(f)d Fs(\)\))28 b(=)2465 3053 y Fi(c)2462 3078 y Fq(W)14 b Fs(\()p Fq(e)2651 3037 y Fx(\000)p Fo(\014)s(u=)p Fp(2)2864 3078 y Fq(\034)2906 3093 y Fo(\014)2954 3078 y Fs(\()p Fq(f)d Fs(\)\))3127 3037 y Fx(\003)3166 3078 y Fq(:)124 b Fs(\(2.40\))328 3285 y(It)33 b(is)f(not)g(di\016cult)g(to) g(see)i(\([FM]\))e(that)983 3493 y Fq(\033)1038 3508 y Fo(t;)p Fp(0)1123 3493 y Fs(\()p Fq(\031)1216 3508 y Fo(\014)1264 3493 y Fs(\()p Fq(A)p Fs(\)\))27 b(:=)h Fq(\031)1664 3508 y Fo(\014)1711 3493 y Fs(\()p Fq(\013)1811 3508 y Fo(t;)p Fp(0)1896 3493 y Fs(\()p Fq(A)p Fs(\)\))g(=)f Fq(e)2259 3451 y Fo(itL)2356 3460 y Fj(0)2396 3493 y Fq(\031)2451 3508 y Fo(\014)2498 3493 y Fs(\()p Fq(A)p Fs(\))p Fq(e)2692 3451 y Fx(\000)p Fo(itL)2844 3460 y Fj(0)2883 3493 y Fq(;)407 b Fs(\(2.41\))328 3700 y(for)32 b(all)e Fq(A)e Fm(2)g Fr(A)p Fs(,)33 b(where)h Fq(L)1286 3715 y Fp(0)1359 3700 y Fs(is)e(the)h(selfadjoin)m(t)e(op)s(erator)h (on)g Fm(H)q Fs(,)h(giv)m(en)g(b)m(y)1235 3907 y Fq(L)1301 3922 y Fp(0)1368 3907 y Fs(=)28 b Fq(H)1553 3922 y Fo(p)1615 3907 y Fm(\012)22 b Fs(1)-22 b(l)1769 3922 y Fo(p)1830 3907 y Fm(\000)22 b Fs(1)-22 b(l)1984 3922 y Fo(p)2045 3907 y Fm(\012)23 b Fq(H)2226 3922 y Fo(p)2287 3907 y Fs(+)f(d\000\()p Fq(u)p Fs(\))p Fq(;)658 b Fs(\(2.42\))328 4114 y(commonly)27 b(called)g(the)i(\(non-in)m(teracting,)f(standard\)) h(Liouvillian.)38 b(One)29 b(easily)f(sees)328 4248 y(that)k Fq(\013)602 4197 y Fp(\()p Fo(\017)p Fp(\))601 4276 y Fo(t;\025)724 4248 y Fs(is)g(unitarily)f(implemen)m(ted)g(in)h(the)h (represen)m(tation)g Fq(\031)2772 4263 y Fo(\014)2852 4248 y Fs(as)933 4497 y Fq(\031)988 4512 y Fo(\014)1036 4497 y Fs(\()p Fq(\013)1137 4446 y Fp(\()p Fo(\017)p Fp(\))1136 4525 y Fo(t;\025)1226 4497 y Fs(\()p Fq(A)p Fs(\)\))28 b(=)f Fq(e)1589 4456 y Fo(itL)1686 4421 y Fj(\()p Fl(\017)p Fj(\))1686 4480 y Fl(\025)1770 4497 y Fq(\031)1825 4512 y Fo(\014)1872 4497 y Fs(\()p Fq(A)p Fs(\))p Fq(e)2066 4456 y Fx(\000)p Fo(itL)2218 4421 y Fj(\()p Fl(\017)p Fj(\))2218 4480 y Fl(\025)2329 4497 y Fs(=:)g Fq(\033)2518 4446 y Fp(\()p Fo(\017)p Fp(\))2514 4525 y Fo(t;\025)2606 4497 y Fs(\()p Fq(\031)2699 4512 y Fo(\014)2746 4497 y Fs(\()p Fq(A)p Fs(\)\))p Fq(;)328 4734 y Fs(where)34 b(the)f FB(r)-5 b(e)g(gularize)g(d)34 b(Liouvil)5 b(lian)32 b Fq(L)1817 4683 y Fp(\()p Fo(\017)p Fp(\))1817 4761 y Fo(\025)1937 4734 y Fs(is)g(giv)m(en)h(b)m(y)1158 4957 y Fq(L)1224 4906 y Fp(\()p Fo(\017)p Fp(\))1224 4985 y Fo(\025)1339 4957 y Fs(=)28 b Fq(L)1509 4972 y Fp(0)1571 4957 y Fs(+)22 b Fq(\025\031)1781 4972 y Fo(\014)1828 4957 y Fs(\()p Fq(V)1945 4906 y Fp(\()p Fo(\017)p Fp(\))1923 4985 y(#)2032 4957 y Fs(\))g Fm(\000)h Fq(\025J)9 b(\031)2367 4972 y Fo(\014)2415 4957 y Fs(\()p Fq(V)2531 4906 y Fp(\()p Fo(\017)p Fp(\))2510 4985 y(#)2619 4957 y Fs(\))p Fq(J)n(:)1898 5214 y Fs(16)p eop %%Page: 17 17 17 16 bop 328 631 a Fs(An)47 b(application)e(of)h(the)i (Glimm-Ja\013e-Nelson)42 b(Theorem)47 b(\(Theorem)h(3.1\))e(sho)m(ws) 328 765 y(that)32 b Fq(L)605 714 y Fp(\()p Fo(\017)p Fp(\))605 793 y Fo(\025)726 765 y Fs(is)g(essen)m(tially)g(selfadjoin)m (t)f(on)955 985 y Fm(D)f Fs(=)e Fq(C)1243 944 y Fx(1)1236 1010 y Fp(0)1339 985 y Fm(\012)23 b Fq(C)1516 944 y Fx(1)1509 1010 y Fp(0)1613 985 y Fm(\012)f Fs(\()p Fm(F)10 b Fs(\()p Fq(C)1947 944 y Fx(1)1940 1010 y Fp(0)2022 985 y Fs(\()p Fk(R)33 b Fm(\002)22 b Fq(S)2319 944 y Fp(2)2358 985 y Fs(\)\))g Fm(\\)h(F)2617 1000 y Fp(0)2656 985 y Fs(\))28 b Fm(\032)g(H)q Fq(;)378 b Fs(\(2.43\))328 1205 y(where)31 b Fm(F)679 1220 y Fp(0)747 1205 y Fs(is)e(the)h(\014nite-particle)d (subspace)32 b(\(see)e([FM]\).)g(Moreo)m(v)m(er,)i(from)c(the)i(theo-) 328 1325 y(rem)i(on)g(in)m(v)-5 b(ariance)32 b(of)f(domains,)h(Theorem) g(A.1,)h(and)f(the)h(Duhamel)e(form)m(ula,)f(one)328 1446 y(easily)i(sees)i(that)1585 1587 y(lim)1586 1647 y Fo(\017)p Fx(!)p Fp(0)1737 1587 y Fq(e)1782 1546 y Fo(itL)1879 1511 y Fj(\()p Fl(\017)p Fj(\))1879 1570 y Fl(\025)1990 1587 y Fs(=)28 b Fq(e)2139 1546 y Fo(itL)2236 1558 y Fl(\025)2281 1587 y Fq(;)1009 b Fs(\(2.44\))328 1788 y(in)32 b(the)h(strong)f(sense)j(on)d Fm(H)q Fs(,)h(where)h(the)f (Liouvillian)28 b Fq(L)2441 1803 y Fo(\025)2519 1788 y Fs(is)k(giv)m(en)h(b)m(y)451 2008 y Fq(L)517 2023 y Fo(\025)645 2008 y Fs(=)83 b Fq(L)870 2023 y Fp(0)932 2008 y Fs(+)22 b Fq(\025I)8 b(;)2152 b Fs(\(2.45\))512 2171 y Fq(I)90 b Fs(=)804 2076 y Fi(X)854 2286 y Fo(\013)965 2171 y Fq(G)1042 2186 y Fo(\013;)p Fp(#)1192 2171 y Fm(\012)22 b Fs(1)-22 b(l)1346 2186 y Fo(p)1407 2171 y Fm(\012)23 b Fq(')p Fs(\()p Fq(\034)1651 2186 y Fo(\014)1698 2171 y Fs(\()p Fq(g)1783 2186 y Fo(\013)1832 2171 y Fs(\)\))f Fm(\000)h Fs(1)-22 b(l)2085 2186 y Fo(p)2145 2171 y Fm(\012)23 b(C)2297 2186 y Fo(p)2337 2171 y Fq(G)2414 2186 y Fo(\013;)p Fp(#)2542 2171 y Fm(C)2594 2186 y Fo(p)2656 2171 y Fm(\012)g Fq(')p Fs(\()p Fq(e)2903 2130 y Fx(\000)p Fo(\014)s(u=)p Fp(2)3116 2171 y Fq(\034)3158 2186 y Fo(\014)3206 2171 y Fs(\()p Fq(g)3291 2186 y Fo(\013)3340 2171 y Fs(\)\))p Fq(:)328 2478 y Fs(The)34 b(op)s(erator)e Fq(L)988 2493 y Fo(\025)1067 2478 y Fs(is)g(essen)m(tially)h(selfadjoin)m(t)e(on)i (the)g(domain)f Fm(D)j Fs(de\014ned)f(in)f(\(2.43\))328 2599 y(and)g(de\014nes)h(a)e Fm(\003)p Fs(automorphism)e(group)j(on)f Fr(M)2131 2614 y Fo(\014)2211 2599 y Fs(giv)m(en)g(b)m(y)1225 2819 y Fq(\033)1280 2834 y Fo(t;\025)1371 2819 y Fs(\()p Fq(A)p Fs(\))27 b(=)h Fq(e)1696 2778 y Fo(itL)1793 2790 y Fl(\025)1838 2819 y Fq(Ae)1956 2778 y Fx(\000)p Fo(itL)2108 2790 y Fl(\025)2154 2819 y Fq(;)114 b(A)28 b Fm(2)g Fr(M)2595 2834 y Fo(\014)2642 2819 y Fq(:)648 b Fs(\(2.46\))328 3039 y(An)33 b(imp)s(ortan)m(t)d(prop)s(ert)m(y)k(of)e Fq(L)1524 3054 y Fo(\025)1602 3039 y Fs(is)g(that)1311 3259 y Fq(e)1356 3218 y Fo(itL)1453 3230 y Fl(\025)1499 3259 y Fq(J)37 b Fs(=)27 b Fq(J)9 b(e)1801 3218 y Fo(itL)1898 3230 y Fl(\025)1944 3259 y Fq(;)49 b Fs(for)32 b(all)f Fq(\025)c Fm(2)h Fk(R)11 b Fq(:)735 b Fs(\(2.47\))328 3518 y Ft(2.1.4)112 b(Characterization)36 b(of)i(the)f Fq(\033)1901 3533 y Fo(t;\025)1992 3518 y Ft(-in)m(v)-6 b(arian)m(t)37 b(normal)f(states)i(on)g Fr(M)3459 3533 y Fo(\014)328 3703 y Fs(A)i(state)g Fq(!)k Fs(on)39 b(a)h(v)m(on)h (Neumann)f(algebra)e Fr(M)i Fm(\032)h(B)s Fs(\()p Fm(H)q Fs(\))f(is)g(called)e(normal)g(i\013)h(it)g(is)328 3824 y(giv)m(en)f(b)m(y)h(a)f(densit)m(y)h(matrix)d Fq(\032)h Fm(2)h(B)s Fs(\()p Fm(H)q Fs(\),)h(i.e.)60 b Fq(!)t Fs(\()p Fq(A)p Fs(\))36 b(=)h(tr)16 b Fq(\032A)p Fs(,)40 b Fq(A)d Fm(2)g Fr(M)p Fs(.)60 b(If)38 b Fq(\034)3346 3839 y Fo(t)3414 3824 y Fs(is)f(a)328 3944 y(group)e(of)f(homomorphisms)f(of)h Fr(M)h Fs(the)g(state)g(is)g(called)f Fq(\034)2536 3959 y Fo(t)2566 3944 y Fs(-in)m(v)-5 b(arian)m(t)33 b(i\013)h Fq(!)27 b Fm(\016)c Fq(\034)3332 3959 y Fo(t)3394 3944 y Fs(=)31 b Fq(!)328 4064 y Fs(for)e(all)f Fq(t)g Fm(2)g Fk(R)5 b Fs(.)48 b(In)30 b(order)g(to)g(c)m(haracterize)g(the)g Fq(\033)2149 4079 y Fo(t;\025)2240 4064 y Fs(-in)m(v)-5 b(arian)m(t)28 b(normal)g(states)i(on)g Fr(M)3519 4079 y Fo(\014)328 4185 y Fs(it)35 b(is)h(useful)h(to)f(in)m(tro)s(duce)g (the)h(natural)e(cone)i Fm(P)45 b Fs(asso)s(ciated)36 b(to)g(\()p Fr(M)2966 4200 y Fo(\014)3013 4185 y Fq(;)17 b Fs(\012)3127 4149 y Fp(ref)3217 4185 y Fs(\),)37 b(whic)m(h)328 4305 y(is)32 b(de\014ned)i(b)m(y)1266 4426 y Fm(P)i Fs(=)p 1474 4339 909 4 v 27 w Fm(f)p Fq(AJ)9 b(A)p Fs(\012)1803 4397 y Fp(ref)1926 4426 y Fm(j)33 b Fq(A)27 b Fm(2)h Fr(M)2286 4441 y Fo(\014)2333 4426 y Fm(g)g(\032)g(H)q Fq(;)689 b Fs(\(2.48\))328 4600 y(where)28 b(the)f(bar)f(denotes)i(the) f(closure)g(in)f(the)h(norm)f(of)g Fm(H)q Fs(.)41 b(The)28 b(follo)m(wing)c(prop)s(erties)328 4720 y(of)29 b(the)i(natural)e(cone) i(are)f(the)g(con)m(ten)m(ts)i(of)d(the)i FB(A)n(r)-5 b(aki-Connes-Haagerup)28 b Fs(theorem,)328 4841 y(a)k(deep)i(result)e (in)g(the)h(theory)g(of)f(v)m(on)i(Neumann)e(algebras)g(\(see)i(e.g.)44 b([BRI]\).)474 4961 y(Giv)m(en)f(an)m(y)g(normal)d(state)j Fq(!)i Fs(on)e Fr(M)1907 4976 y Fo(\014)1953 4961 y Fs(,)i(there)e(is)f (a)g(unique)h(v)m(ector)h Fq(\030)49 b Fm(2)44 b(P)51 b Fs(s.t.)1898 5214 y(17)p eop %%Page: 18 18 18 17 bop 328 631 a Fq(!)t Fs(\()p Fq(A)p Fs(\))27 b(=)g Fm(h)p Fq(\030)5 b(;)17 b(A\030)t Fm(i)p Fs(,)30 b(for)f(all)f Fq(A)g Fm(2)g Fr(M)1598 646 y Fo(\014)1645 631 y Fs(.)42 b(Moreo)m(v)m(er,)32 b(if)d Fq(!)2315 646 y Fp(1)2384 631 y Fs(and)h Fq(!)2632 646 y Fp(2)2701 631 y Fs(are)f(normal)f (states)j(on)328 751 y Fr(M)433 766 y Fo(\014)512 751 y Fs(with)h(corresp)s(onding)h(v)m(ectors)h Fq(\030)1735 766 y Fp(1)1774 751 y Fs(,)f Fq(\030)1877 766 y Fp(2)1948 751 y Fs(in)f Fm(P)41 b Fs(then)976 971 y Fm(k)p Fq(\030)1069 986 y Fp(1)1130 971 y Fm(\000)23 b Fq(\030)1273 986 y Fp(2)1312 971 y Fm(k)1362 930 y Fp(2)1429 971 y Fm(\024)28 b(k)p Fq(!)1645 986 y Fp(1)1706 971 y Fm(\000)23 b Fq(!)1867 986 y Fp(2)1906 971 y Fm(k)k(\024)i(k)p Fq(\030)2182 986 y Fp(1)2243 971 y Fm(\000)22 b Fq(\030)2385 986 y Fp(2)2424 971 y Fm(k)33 b(k)p Fq(\030)2600 986 y Fp(1)2661 971 y Fs(+)22 b Fq(\030)2802 986 y Fp(2)2841 971 y Fm(k)p Fq(:)399 b Fs(\(2.49\))328 1191 y(The)33 b(norm)f(of)g(a)h(state)g Fq(!)i Fs(of)d Fr(M)1527 1206 y Fo(\014)1607 1191 y Fs(is)g(giv)m(en)h (b)m(y)g Fm(k)p Fq(!)t Fm(k)27 b Fs(=)g(sup)2537 1215 y Fo(A)p Fx(2)p Fg(M)2709 1227 y Fl(\014)2772 1191 y Fm(j)p Fq(!)t Fs(\()p Fq(A)p Fs(\))p Fm(j)p Fq(=)p Fm(k)p Fq(A)p Fm(k)p Fs(.)474 1326 y(It)k(is)f(not)h(di\016cult)e(to)i(see)h (that)e(\(2.47\))g(implies)e(that)j Fq(e)2548 1290 y Fo(itL)2645 1302 y Fl(\025)2690 1326 y Fm(P)36 b Fs(=)28 b Fm(P)8 b Fs(,)31 b(for)g(all)d Fq(\025)g Fm(2)g Fk(R)328 1447 y Fs(and)35 b Fq(t)e Fm(2)f Fk(R)5 b Fs(.)58 b(F)-8 b(rom)34 b(the)h(uniqueness)j(of)c(the)i(v)m(ector)h(represen)m(tativ)m (e)g(in)d(the)i(natural)328 1567 y(cone)f(it)f(follo)m(ws)g(that)h(the) g Fq(\033)1414 1582 y Fo(t;\025)1505 1567 y Fs(-in)m(v)-5 b(arian)m(t)33 b(normal)g(states)j(are)f(in)f(one-to-one)g(corre-)328 1687 y(sp)s(ondence)39 b(with)e(the)h(unit)f(v)m(ectors)h(in)f(the)h (set)g Fm(P)c(\\)26 b Fs(k)m(er)17 b Fq(L)2604 1702 y Fo(\025)2650 1687 y Fs(,)38 b(whic)m(h,)i(for)c Fq(\025)g Fs(=)g(0,)i(is)328 1808 y(giv)m(en)33 b(b)m(y)397 2050 y Fm(P)d(\\)23 b Fs(k)m(er)17 b Fq(L)797 2065 y Fp(0)865 2050 y Fs(=)28 b Fm(P)i(\\)55 b Fs(span)1385 1939 y Fi(n)1451 2050 y Fq(')1515 2065 y Fo(m)1604 2050 y Fm(\012)23 b Fq(')1768 2065 y Fo(n)1837 2050 y Fm(\012)f Fs(\012)33 b Fm(j)f Fq(m;)17 b(n)28 b Fm(2)g(M)p Fq(;)17 b(E)6 b Fs(\()p Fq(m)p Fs(\))28 b(=)f Fq(E)6 b Fs(\()p Fq(n)p Fs(\))3154 1939 y Fi(o)3221 2050 y Fq(:)69 b Fs(\(2.50\))328 2291 y(W)-8 b(e)28 b(will)e(sho)m(w)j(in)e(Theorem)h(2.3)g(that,)h(for) e Fq(\025)g Fm(6)p Fs(=)h(0,)h(the)f Fq(\033)2467 2306 y Fo(t;\025)2558 2291 y Fs(-in)m(v)-5 b(arian)m(t)26 b(normal)g(states)328 2412 y(are)g(giv)m(en)h(b)m(y)g(the)g(subset)h (of)e(\(2.50\))g(determined)g(b)m(y)i(the)e(mo)s(des)h Fq(m;)17 b(n)28 b Fm(2)g(Mn)p Fq(J)3321 2427 y Fo(d)3387 2412 y Fs(that)328 2532 y(do)k(not)h(in)m(teract)f(with)g(the)h (\014eld.)328 2792 y Ft(2.1.5)112 b(A)37 b(quic)m(k-reference)h(list) 328 2977 y Fs(F)-8 b(or)27 b(the)i(con)m(v)m(enience)h(of)e(the)h (reader)g(and)f(for)g(future)g(reference,)j(w)m(e)e(collect)e(the)i (def-)328 3097 y(initions)h(of)h(some)h(imp)s(ortan)m(t)f(op)s(erators) h(in)f(a)h(list.)42 b(Generally)-8 b(,)31 b(if)g Fq(\031)k Fs(is)d(a)g(pro)5 b(jection)328 3217 y(then)33 b(w)m(e)h(set)p 846 3163 59 4 v 33 w Fq(\031)e Fs(=)27 b(1)-22 b(l)21 b Fm(\000)i Fq(\031)t Fs(.)782 3430 y Fq(p)831 3445 y Fo(d)974 3430 y Fs(pro)5 b(jection)32 b(on)m(to)h(the)g(discrete)g (subspace)i(of)d Fq(H)2783 3445 y Fo(p)785 3551 y Fq(p)834 3566 y Fo(c)974 3551 y Fs(pro)5 b(jection)32 b(on)m(to)h(the)g(con)m (tin)m(uous)g(subspace)i(of)d Fq(H)2913 3566 y Fo(p)769 3671 y Fq(p)818 3686 y Fo(m)974 3671 y Fs(one-dimensional)e(pro)5 b(jection)32 b(on)m(to)h(the)g(mo)s(de)f Fq(m)c Fm(2)g(M)763 3792 y Fq(p)812 3807 y Fo(J)851 3819 y Fl(d)974 3792 y Fs(=)1078 3717 y Fi(P)1183 3821 y Fo(m)p Fx(2)p Fo(J)1331 3833 y Fl(d)1388 3792 y Fq(p)1437 3807 y Fo(m)765 3917 y Fq(p)814 3932 y Fo(J)853 3940 y Fl(c)974 3917 y Fs(sp)s(ectral)k(pro) 5 b(jection)33 b(of)f Fq(H)1996 3932 y Fo(p)2068 3917 y Fs(on)m(to)g(the)h(in)m(terv)-5 b(al)32 b Fq(J)2862 3932 y Fo(c)803 4037 y Fq(p)122 b Fs(=)28 b Fq(p)1127 4052 y Fo(J)1166 4064 y Fl(d)1228 4037 y Fs(+)22 b Fq(p)1375 4052 y Fo(J)1414 4060 y Fl(c)789 4157 y Fq(P)122 b Fs(=)28 b Fq(p)22 b Fm(\012)g Fq(p)g Fm(\012)h Fs(1)-22 b(l)1474 4172 y Fo(f)1551 4157 y Fs(is)32 b(a)g(pro)5 b(jection)32 b(on)h Fm(H)2413 4172 y Fo(p)2475 4157 y Fm(\012)23 b(H)2659 4172 y Fo(p)2720 4157 y Fm(\012)g(F)776 4278 y Fq(P)853 4242 y Fo(l)974 4278 y Fs(=)28 b Fq(p)22 b Fm(\012)p 1248 4223 49 4 v 22 w Fq(p)g Fm(\012)h Fs(1)-22 b(l)1474 4293 y Fo(f)770 4398 y Fq(P)847 4362 y Fo(r)974 4398 y Fs(=)p 1078 4343 V 28 w Fq(p)22 b Fm(\012)g Fq(p)g Fm(\012)h Fs(1)-22 b(l)1474 4413 y Fo(f)769 4518 y Fq(P)846 4482 y Fp(0)974 4518 y Fs(=)p 1078 4464 V 28 w Fq(p)22 b Fm(\012)p 1248 4464 V 22 w Fq(p)g Fm(\012)h Fs(1)-22 b(l)1474 4533 y Fo(f)776 4639 y Fq(P)839 4654 y Fp(0)974 4639 y Fs(pro)5 b(jection)32 b(on)m(to)h(k)m(er)18 b Fq(L)1870 4654 y Fo(p)791 4759 y Fs(\005)110 b(=)28 b Fq(P)1141 4774 y Fp(0)1202 4759 y Fm(\012)23 b Fq(P)1365 4774 y Fp(\012)1452 4759 y Fs(is)32 b(the)h(pro)5 b(jection)33 b(on)m(to)f(k)m(er)18 b Fq(L)2614 4774 y Fp(0)1898 5214 y Fs(18)p eop %%Page: 19 19 19 18 bop 328 631 a Fh(2.2)135 b(Main)45 b(results)328 816 y Fs(Our)24 b(main)e(results)j(concern)g(the)g(dynamical)d(system)j (\()p Fr(M)2484 831 y Fo(\014)2531 816 y Fq(;)17 b(\033)2630 831 y Fo(t;\025)2720 816 y Fs(\),)26 b(where)f(w)m(e)g(ha)m(v)m(e)h (de-)328 936 y(\014ned)g(the)f(v)m(on)g(Neumann)g(algebra)f Fr(M)1768 951 y Fo(\014)1840 936 y Fs(in)g(\(2.37\),)h(and)g(where)h Fq(\033)2759 951 y Fo(t;\025)2875 936 y Fs(is)f(the)g Fm(\003)p Fs(automor-)328 1056 y(phism)32 b(group)g(\(2.46\))g(of)g Fr(M)1390 1071 y Fo(\014)1469 1056 y Fs(generated)i(b)m(y)f(the)g (Liouvillian)c(\(2.45\).)474 1177 y(The)45 b(follo)m(wing)c(theorem)j (describ)s(es)h(some)f(prop)s(erties)g(of)f(eigen)m(v)m(ectors)j(of)d Fq(L)3520 1192 y Fo(\025)328 1297 y Fs(whic)m(h,)i(as)c(w)m(e)i(ha)m(v) m(e)g(seen)g(in)e(Subsection)h(2.1.4,)i(pla)m(y)d(an)h(imp)s(ortan)m(t) e(role)g(in)h(the)328 1418 y(c)m(haracterization)32 b(of)g(in)m(v)-5 b(arian)m(t)31 b(normal)f(states.)328 1613 y Ft(Theorem)37 b(2.1)49 b(\(Bounds)38 b(on)f(eigen)m(v)m(ectors\).)433 1791 y FB(1\))49 b(Assume)31 b(that)h(either)f(Condition)f(A)i(or)f (Condition)g(B)f(of)h(Se)-5 b(ction)31 b(2.1.1)g(holds.)572 1912 y(L)-5 b(et)40 b Fq(N)48 b Fs(=)37 b(d\000\(1)-22 b(l\))38 b FB(denote)i(the)g(numb)-5 b(er)39 b(op)-5 b(er)g(ator)40 b(on)g(F)-7 b(o)i(ck)38 b(sp)-5 b(ac)g(e)39 b Fm(F)10 b Fs(\()p Fq(L)3313 1876 y Fp(2)3353 1912 y Fs(\()p Fk(R)36 b Fm(\002)572 2032 y Fq(S)638 1996 y Fp(2)677 2032 y Fs(\)\))p FB(.)55 b(A)n(ny)38 b(eigenve)-5 b(ctor)37 b Fq( )42 b FB(of)c Fq(L)1847 2047 y Fo(\025)1931 2032 y FB(satis\014es)f Fq( )h Fm(2)c(D)s Fs(\()p Fq(N)2707 1996 y Fp(1)p Fo(=)p Fp(2)2817 2032 y Fs(\))p FB(,)39 b(for)f(any)g Fq(\025)c Fm(2)g Fk(R)5 b FB(,)572 2152 y(and)34 b(ther)-5 b(e)35 b(is)g(a)f(c)-5 b(onstant)35 b Fq(k)c(<)c Fm(1)35 b FB(s.t.)1631 2343 y Fm(k)p Fq(N)1769 2302 y Fp(1)p Fo(=)p Fp(2)1880 2343 y Fq( )t Fm(k)27 b(\024)h Fq(k)s Fm(j)p Fq(\025)p Fm(j)17 b(k)p Fq( )t Fm(k)p Fq(:)810 b Fs(\(2.51\))572 2534 y FB(The)48 b(c)-5 b(onstant)49 b Fq(k)j FB(satis\014es)c Fq(k)57 b(<)c(k)1962 2498 y Fx(0)1986 2534 y Fs(\(1)32 b(+)g(1)p Fq(=\014)6 b Fs(\))p FB(,)52 b(wher)-5 b(e)48 b Fq(k)2835 2498 y Fx(0)2908 2534 y FB(dep)-5 b(ends)47 b(on)i(the)572 2655 y(inter)-5 b(action,)34 b(but)h(not)g(on)g Fq(\014)6 b FB(.)433 2850 y(2\))49 b(Assume)g(Condition)g(A.)g(Given)g(any)h Fs(0)k Fq(<)h(\014)60 b(<)54 b Fm(1)p FB(,)f(ther)-5 b(e)49 b(ar)-5 b(e)49 b(c)-5 b(onstants)572 2970 y Fq(\025)629 2985 y Fp(0)668 2970 y Fs(\()p Fq(\014)6 b Fs(\))27 b Fq(>)h Fs(0)p FB(,)33 b Fq(k)s Fs(\()p Fq(\014)6 b Fs(\))27 b Fq(>)h Fs(0)p FB(,)33 b(s.t.)45 b(if)33 b Fs(0)27 b Fq(<)h Fm(j)p Fq(\025)p Fm(j)f Fq(<)g(\025)2232 2985 y Fp(0)2272 2970 y Fs(\()p Fq(\014)6 b Fs(\))p FB(,)33 b(and)f(if)h Fq( )38 b FB(is)33 b(an)g(eigenve)-5 b(ctor)572 3091 y(of)34 b Fq(L)752 3106 y Fo(\025)798 3091 y FB(,)h(then)1549 3211 y Fm(k)p 1599 3131 77 4 v Fq(P)1675 3226 y Fp(0)1737 3211 y Fm(\012)23 b Fq(P)1900 3226 y Fp(\012)1955 3211 y Fq( )t Fm(k)k(\025)h Fq(k)s Fs(\()p Fq(\014)6 b Fs(\))p Fm(k)p Fq( )t Fm(k)p Fq(;)728 b Fs(\(2.52\))572 3373 y FB(wher)-5 b(e)31 b Fq(P)907 3388 y Fp(0)979 3373 y FB(is)h(the)g(pr)-5 b(oje)g(ction)32 b(onto)g(the)g(zer)-5 b(o)32 b(eigensp)-5 b(ac)g(e)31 b(of)h Fq(L)2920 3388 y Fo(p)2960 3373 y FB(,)p 3023 3293 V 33 w Fq(P)3099 3388 y Fp(0)3166 3373 y Fs(=)c(1)-22 b(l)16 b Fm(\000)g Fq(P)3497 3388 y Fp(0)3536 3373 y FB(,)572 3493 y(and)25 b Fq(P)815 3508 y Fp(\012)897 3493 y FB(is)h(the)g(pr)-5 b(oje)g(ction)25 b(onto)h(the)h(vacuum)f(se)-5 b(ctor)26 b(in)g Fm(F)10 b FB(.)41 b(We)26 b(have)g Fq(\025)3285 3508 y Fp(0)3324 3493 y Fs(\()p Fq(\014)6 b Fs(\))27 b Fm(\025)572 3614 y Fq(k)s(\015)43 b FB(\(se)-5 b(e)38 b(\(2.21\))f(and)g(r)-5 b(emark)38 b(2\))g(ther)-5 b(e)g(after\))38 b(and)f Fq(k)s Fs(\()p Fq(\014)6 b Fs(\))34 b Fm(\025)g Fq(k)s(\015)2992 3578 y Fp(2)3032 3614 y FB(,)k(for)g(some)g Fq(k)572 3734 y FB(indep)-5 b(endent)36 b(of)h Fq(\014)43 b FB(and)37 b Fq(\025)g FB(\(i.e.,)h(b)-5 b(oth)37 b(c)-5 b(onstants)37 b(de)-5 b(c)g(ay)37 b(exp)-5 b(onential)5 b(ly)36 b(in)h Fq(\014)6 b FB(,)572 3855 y(for)34 b(lar)-5 b(ge)35 b Fq(\014)6 b FB(\).)328 4050 y Fs(The)47 b(pro)s(of)e(of)g (Theorem)h(2.1)g(is)f(giv)m(en)h(in)f(Section)h(4.)83 b(Here)47 b(w)m(e)g(sho)m(w)g(that)e(the)328 4170 y(b)s(ounds)32 b(\(2.51\))f(and)h(\(2.52\))e(imply)g(that)h(bifurcations)g(of)g (stationary)g(states)h(for)f(the)328 4291 y(in)m(teracting)k(dynamics)h (generated)h(b)m(y)h Fq(L)1911 4306 y Fo(\025)1993 4291 y Fs(from)d(an)m(y)i(stationary)e(states)i(for)f Fq(\025)e Fs(=)g(0)328 4411 y(cannot)f(o)s(ccur.)474 4531 y(Let)27 b(\005)h(=)g Fq(P)911 4546 y Fp(0)961 4531 y Fm(\012)11 b Fq(P)1112 4546 y Fp(\012)1194 4531 y Fs(denote)27 b(the)h(pro)5 b(jection)26 b(on)m(to)h(the)g(zero)h(eigenspace)g(of)e Fq(L)3342 4546 y Fp(0)3409 4531 y Fs(and)328 4652 y(set)p 474 4572 74 4 v 27 w(\005)h(:=)h(1)-22 b(l)9 b Fm(\000)g Fs(\005.)40 b(Assume)27 b(that)f Fq( )k Fs(is)25 b(an)h(eigen)m(v)m (ector)h(of)f Fq(L)2539 4667 y Fo(\025)2584 4652 y Fs(,)i(for)d(some)h (0)h Fq(<)h Fm(j)p Fq(\025)p Fm(j)f Fq(<)g(\025)3499 4667 y Fp(0)3539 4652 y Fs(.)328 4772 y(Using)i(the)i(decomp)s(osition) p 1405 4692 V 28 w(\005)c(=)p 1609 4692 77 4 v 28 w Fq(P)1685 4787 y Fp(0)1741 4772 y Fm(\012)17 b Fq(P)1898 4787 y Fp(\012)1970 4772 y Fs(+)p 2063 4692 V 17 w Fq(P)2139 4787 y Fp(\012)2224 4772 y Fs(and)30 b(\(2.51\),)f(\(2.52\),)h(w)m(e)h (ha)m(v)m(e)g(that)832 4878 y Fi(\015)832 4938 y(\015)p 888 4883 74 4 v 888 4963 a Fs(\005)p Fq( )1028 4878 y Fi(\015)1028 4938 y(\015)1111 4963 y Fm(\025)1216 4878 y Fi(\015)1216 4938 y(\015)p 1271 4883 77 4 v 25 x Fq(P)1348 4978 y Fp(0)1409 4963 y Fm(\012)23 b Fq(P)1572 4978 y Fp(\012)1627 4963 y Fq( )1694 4878 y Fi(\015)1694 4938 y(\015)1771 4963 y Fm(\000)1871 4878 y Fi(\015)1871 4938 y(\015)p 1926 4883 V 25 x Fq(P)2003 4978 y Fp(\012)2058 4963 y Fq( )2125 4878 y Fi(\015)2125 4938 y(\015)2208 4963 y Fm(\025)28 b Fs(\()p Fq(k)s Fs(\()p Fq(\014)6 b Fs(\))22 b Fm(\000)g Fq(k)s Fm(j)p Fq(\025)p Fm(j)p Fs(\))p Fm(k)p Fq( )t Fm(k)p Fq(:)1898 5214 y Fs(19)p eop %%Page: 20 20 20 19 bop 328 631 a Fs(Let)33 b Fq( )566 646 y Fp(0)638 631 y Fs(b)s(e)g(an)f(arbitrary)g(elemen)m(t)g(of)g(k)m(er)18 b Fq(L)2005 646 y Fp(0)2045 631 y Fs(.)43 b(Then)804 835 y Fm(k)p Fq( )917 850 y Fp(0)979 835 y Fm(\000)23 b Fq( )t Fm(k)k(\025)1329 751 y Fi(\015)1329 811 y(\015)p 1384 755 74 4 v 24 x Fs(\005)p Fq( )1524 751 y Fi(\015)1524 811 y(\015)1602 835 y Fm(\000)22 b(k)p Fq( )1814 850 y Fp(0)1876 835 y Fm(\000)h Fs(\005)p Fq( )t Fm(k)k(\025)2298 751 y Fi(\015)2298 811 y(\015)p 2354 755 V 2354 835 a Fs(\005)p Fq( )2494 751 y Fi(\015)2494 811 y(\015)2571 835 y Fm(\000)c(k)p Fq( )2784 850 y Fp(0)2846 835 y Fm(\000)f Fq( )t Fm(k)p Fq(;)328 1040 y Fs(so)1276 1180 y Fm(k)p Fq( )1389 1195 y Fp(0)1451 1180 y Fm(\000)h Fq( )t Fm(k)k(\025)1810 1112 y Fs(1)p 1810 1157 49 4 v 1810 1248 a(2)1869 1180 y(\()p Fq(k)s Fs(\()p Fq(\014)6 b Fs(\))22 b Fm(\000)g Fq(k)s Fm(j)p Fq(\025)p Fm(j)p Fs(\))p Fm(k)p Fq( )t Fm(k)p Fq(:)699 b Fs(\(2.53\))328 1401 y(This)26 b(sho)m(ws)i(that)e (for)f(0)j Fq(<)f Fm(j)p Fq(\025)p Fm(j)g Fq(<)h Fs(min)o(\()p Fq(\025)1847 1416 y Fp(0)1886 1401 y Fq(;)1939 1362 y Fp(1)p 1939 1378 36 4 v 1939 1435 a(2)1995 1354 y Fo(k)r Fp(\()p Fo(\014)s Fp(\))p 1995 1378 137 4 v 2044 1435 a Fo(k)2141 1401 y Fs(\),)f(the)g(distance)f(b)s(et)m(w)m(een)j(an)m(y) e(eigen-)328 1521 y(v)m(ector)39 b(of)f Fq(L)809 1536 y Fo(\025)893 1521 y Fs(and)h(an)m(y)g(eigen)m(v)m(ector)h(of)d Fq(L)1974 1536 y Fp(0)2053 1521 y Fs(is)h(greater)g(than)g Fq(k)s Fs(\()p Fq(\014)6 b Fs(\))p Fq(=)p Fs(4.)60 b(Com)m(bining)328 1642 y(\(2.53\))32 b(with)g(\(2.49\))g(yields)g(the)h(follo)m(wing)d (result.)328 1852 y Ft(Theorem)37 b(2.2)49 b(\(No)29 b(bifurcation\).)71 b FB(Assume)29 b(that)g(the)g(Condition)f Fq(A)h FB(of)f(Se)-5 b(ction)328 1983 y(2.1.1)44 b(holds,)h(and)f(that) h Fs(0)g Fq(<)g Fm(j)p Fq(\025)p Fm(j)g Fq(<)g Fs(min)n(\()p Fq(\025)2031 1998 y Fp(0)2071 1983 y Fq(;)2124 1944 y Fp(1)p 2124 1960 36 4 v 2124 2017 a(2)2179 1935 y Fo(k)r Fp(\()p Fo(\014)s Fp(\))p 2179 1960 137 4 v 2228 2017 a Fo(k)2326 1983 y Fs(\))p FB(,)h(wher)-5 b(e)44 b Fq(\025)2782 1998 y Fp(0)2821 1983 y FB(,)j Fq(k)s Fs(\()p Fq(\014)6 b Fs(\))p FB(,)46 b Fq(k)i FB(ar)-5 b(e)44 b(the)328 2103 y(c)-5 b(onstants)31 b(in)g(The)-5 b(or)g(em)30 b(2.1,)i(2\).)43 b(F)-7 b(or)31 b(any)g(normal)g Fq(\033)2374 2118 y Fo(t;)p Fp(0)2459 2103 y FB(-invariant)f(state)i Fq(!)3208 2067 y Fp(0)3278 2103 y FB(on)f Fr(M)3519 2118 y Fo(\014)328 2224 y FB(and)j(any)h(normal)f Fq(\033)1094 2239 y Fo(t;\025)1185 2224 y FB(-invariant)g(state)h Fq(!)1941 2188 y Fo(\025)2021 2224 y FB(on)f Fr(M)2265 2239 y Fo(\014)2312 2224 y FB(,)1456 2344 y Fi(\015)1456 2403 y(\015)1511 2428 y Fq(!)1576 2387 y Fp(0)1637 2428 y Fm(\000)23 b Fq(!)1802 2387 y Fo(\025)1846 2344 y Fi(\015)1846 2403 y(\015)1929 2428 y Fm(\025)29 b Fq(k)s Fs(\()p Fq(\014)6 b Fs(\))2226 2387 y Fp(2)2264 2428 y Fq(=)p Fs(16)p Fq(:)879 b Fs(\(2.54\))472 2639 y(Our)36 b(next)h(result)g(sho)m(ws)g(that)f (the)h(mo)s(des)f(of)f(the)i(particle)e(whic)m(h)h(are)h(coupled)328 2759 y(to)29 b(the)h(\014eld)f(do)g(not)h(giv)m(e)f(rise)g(to)g(in)m(v) -5 b(arian)m(t)28 b(states;)k(\(compare)d(with)g(Section)g(2.1.4\).)328 2949 y Ft(Theorem)37 b(2.3)49 b(\(Instabilit)m(y)40 b(of)j(normal)f(in) m(v)-6 b(arian)m(t)42 b(states\).)100 b FB(Assume)39 b(that)328 3070 y(Condition)e Fq(B)42 b FB(of)c(Se)-5 b(ction)37 b(2.1.1)g(holds.)52 b(Given)38 b(any)g Fs(0)32 b Fq(<)h(\014)39 b(<)32 b Fm(1)38 b FB(and)f(any)h Fq(r)d(>)e Fs(0)p FB(,)328 3190 y(ther)-5 b(e)37 b(is)h(a)f Fq(\025)825 3205 y Fp(0)864 3190 y Fs(\()p Fq(\014)6 b(;)17 b(r)s Fs(\))31 b Fq(>)i Fs(0)k FB(s.t.)52 b(for)38 b Fs(0)32 b Fq(<)g Fm(j)p Fq(\025)p Fm(j)g(\024)h Fq(\025)2162 3205 y Fp(0)2201 3190 y Fs(\()p Fq(\014)6 b(;)17 b(r)s Fs(\))p FB(,)37 b(the)g Fq(\033)2715 3205 y Fo(t;\025)2806 3190 y FB(-invariant)g(normal)328 3310 y(states)i(on)f Fr(M)854 3325 y Fo(\014)939 3310 y FB(ar)-5 b(e)39 b(in)f(one-to-one)f (c)-5 b(orr)g(esp)g(ondenc)g(e)37 b(with)i(the)f(unit)h(ve)-5 b(ctors)38 b(in)h(the)328 3431 y(set)591 3635 y Fm(P)30 b(\\)55 b Fs(span)1007 3525 y Fi(n)1073 3635 y Fq(')1137 3650 y Fo(m)1226 3635 y Fm(\012)23 b Fq(')1390 3650 y Fo(n)1459 3635 y Fm(\012)f Fs(\012)36 b Fm(j)e Fq(m;)17 b(n)28 b Fm(2)g(Mn)p Fq(J)2259 3650 y Fo(d)2299 3635 y Fq(;)52 b(E)6 b Fs(\()p Fq(m)p Fs(\))28 b(=)f Fq(E)6 b Fs(\()p Fq(n)p Fs(\))2960 3525 y Fi(o)3027 3635 y Fq(:)263 b Fs(\(2.55\))328 3871 y FB(We)33 b(have)f Fq(\025)782 3886 y Fp(0)822 3871 y Fs(\()p Fq(\014)6 b(;)17 b(r)s Fs(\))26 b Fm(\025)i Fq(k)s(\015)1291 3835 y Fp(2)1331 3871 y Fq(r)s FB(,)33 b(for)f(some)g Fq(k)k FB(which)c(is)h(indep)-5 b(endent)32 b(of)g Fq(\014)6 b(;)17 b(r)35 b FB(and)d(wher)-5 b(e)328 3992 y Fq(\015)40 b FB(is)34 b(given)g(in)h(\(2.24\).)328 4322 y Fu(3)161 b(Virial)47 b(theorems)41 b(and)j(the)f(p)t(ositiv)l(e) h(comm)l(uta-)570 4505 y(tor)53 b(metho)t(d)328 4724 y Fs(Our)38 b(pro)s(ofs)f(of)g(Theorems)i(2.1)e(and)h(2.3)g(are)f (based)i(on)f(the)g(p)s(ositiv)m(e)f(comm)m(utator)328 4844 y(metho)s(d,)31 b(whic)m(h)h(w)m(e)g(explain)f(in)f(Section)i (3.2.)42 b(In)32 b(the)g(next)g(section)f(w)m(e)i(describ)s(e)f(an)328 4965 y(essen)m(tial)h(ingredien)m(t)e(of)i(this)f(metho)s(d,)g(the)h (virial)d(theorem.)1898 5214 y(20)p eop %%Page: 21 21 21 20 bop 328 631 a Fh(3.1)135 b(Tw)l(o)45 b(abstract)h(virial)g (theorems)328 816 y Fs(Let)38 b Fm(H)h Fs(b)s(e)f(a)f(Hilb)s(ert)g (space,)j Fm(D)f(\032)f(H)g Fs(a)g(core)g(for)g(a)f(selfadjoin)m(t)g (op)s(erator)g Fq(Y)58 b Fm(\025)37 b Fs(1)-22 b(l,)328 936 y(and)33 b Fq(X)42 b Fs(a)33 b(symmetric)f(op)s(erator)h(on)g Fm(D)s Fs(.)46 b(W)-8 b(e)33 b(sa)m(y)i(the)f(triple)e(\()p Fq(X)r(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))32 b(satis\014es)i(the)328 1056 y FB(GJN)41 b(\(Glimm-Ja\013e-Nelson\))d(Condition)p Fs(,)h(or)f(that)g(\()p Fq(X)r(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))38 b(is)f(a)h FB(GJN-triple)p Fs(,)j(if)328 1177 y(there)33 b(is)f(a)h(constan)m(t)g Fq(k)e(<)c Fm(1)p Fs(,)33 b(s.t.)44 b(for)32 b(all)e Fq( )i Fm(2)c(D)s Fs(:)1942 1393 y Fm(k)p Fq(X)8 b( )t Fm(k)83 b(\024)g Fq(k)s Fm(k)p Fq(Y)22 b( )t Fm(k)624 b Fs(\(3.1\))1005 1538 y Fm(\006)p Fq(i)17 b Fm(fh)o Fq(X)8 b( )t(;)17 b(Y)k( )t Fm(i)h(\000)h(h)o Fq(Y)f( )t(;)17 b(X)8 b( )s Fm(ig)83 b(\024)g Fq(k)20 b Fm(h)p Fq( )t(;)d(Y)k( )t Fm(i)16 b Fq(:)476 b Fs(\(3.2\))328 1755 y(Notice)40 b(that)h(if)f(\()p Fq(X)1079 1770 y Fp(1)1118 1755 y Fq(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))40 b(and)h(\()p Fq(X)1743 1770 y Fp(2)1783 1755 y Fq(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))40 b(are)h(GJN)f(triples,)i(then)g(so)f(is)g(\()p Fq(X)3423 1770 y Fp(1)3490 1755 y Fs(+)328 1875 y Fq(X)409 1890 y Fp(2)448 1875 y Fq(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\).)43 b(Since)32 b Fq(Y)49 b Fm(\025)29 b Fs(1)-22 b(l,)31 b(inequalit)m(y)g(\(3.1\))h(is)h(equiv)-5 b(alen)m(t)32 b(to)1383 2091 y Fm(k)p Fq(X)8 b( )t Fm(k)27 b(\024)h Fq(k)1822 2106 y Fp(1)1862 2091 y Fm(k)p Fq(Y)21 b( )t Fm(k)h Fs(+)g Fq(k)2278 2106 y Fp(2)2317 2091 y Fm(k)p Fq( )t Fm(k)p Fq(;)328 2308 y Fs(for)33 b(some)g Fq(k)774 2323 y Fp(1)813 2308 y Fq(;)17 b(k)908 2323 y Fp(2)977 2308 y Fq(<)29 b Fm(1)p Fs(.)46 b(F)-8 b(or)32 b(a)i(more)e(detailed)h (exp)s(osition)f(of)i(the)f(follo)m(wing)e(results)328 2428 y(\(and)i(pro)s(ofs)f(of)g(Theorems)h(3.2)f(and)h(3.3\))f(w)m(e)i (refer)f(to)f([FM].)328 2652 y Ft(Theorem)37 b(3.1)49 b(\(GJN)44 b(comm)m(utator)f(theorem\))125 b FB(If)40 b Fs(\()p Fq(X)r(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))40 b FB(satis\014es)g(the)328 2772 y(GJN)35 b(Condition,)e(then)g Fq(X)42 b FB(determines)33 b(a)g(selfadjoint)g(op)-5 b(er)g(ator)34 b(\(again)e(denote)-5 b(d)34 b(by)328 2893 y Fq(X)8 b FB(\),)38 b(s.t.)54 b Fm(D)s Fs(\()p Fq(X)8 b Fs(\))32 b Fm(\033)i(D)s Fs(\()p Fq(Y)21 b Fs(\))p FB(.)54 b(Mor)-5 b(e)g(over,)38 b Fq(X)45 b FB(is)38 b(essential)5 b(ly)37 b(selfadjoint)g(on)g(any)h(c)-5 b(or)g(e)328 3013 y(for)35 b Fq(Y)21 b FB(,)35 b(and)f(\(3.1\))g(is)h (valid)f(for)g(al)5 b(l)35 b Fq( )d Fm(2)c(D)s Fs(\()p Fq(Y)20 b Fs(\))p FB(.)474 3237 y Fs(Based)31 b(on)e(the)h(GJN)f(comm)m (utator)g(theorem,)h(w)m(e)g(next)h(describ)s(e)f(the)g(setting)f(for) 328 3358 y(a)35 b(general)g FB(virial)j(the)-5 b(or)g(em)p Fs(.)52 b(Supp)s(ose)36 b(one)g(is)f(giv)m(en)h(a)g(selfadjoin)m(t)e (op)s(erator)h(\003)e Fm(\025)g Fs(1)-22 b(l)328 3478 y(with)33 b(core)i Fm(D)d(\032)f(H)q Fs(,)j(and)g(op)s(erators)g Fq(L;)17 b(A;)g(N)5 b(;)17 b(D)s(;)g(C)2298 3493 y Fo(n)2344 3478 y Fs(,)34 b Fq(n)c Fs(=)g(0)p Fq(;)17 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g Fs(3,)33 b(all)f(symmetric)328 3598 y(on)g Fm(D)s Fs(,)h(and)f(satisfying)719 3815 y Fm(h)p Fq(';)17 b(D)s( )s Fm(i)83 b Fs(=)g Fq(i)17 b Fm(f)o(h)p Fq(L';)g(N)10 b( )5 b Fm(i)21 b(\000)i(h)p Fq(N)10 b(';)17 b(L )t Fm(ig)983 b Fs(\(3.3\))946 3960 y Fq(C)1016 3975 y Fp(0)1138 3960 y Fs(=)83 b Fq(L)686 4105 y Fm(h)p Fq(';)17 b(C)903 4120 y Fo(n)949 4105 y Fq( )t Fm(i)83 b Fs(=)g Fq(i)17 b Fm(f)o(h)p Fq(C)1505 4120 y Fo(n)p Fx(\000)p Fp(1)1642 4105 y Fq(';)g(A )t Fm(i)22 b(\000)g(h)p Fq(A';)17 b(C)2340 4120 y Fo(n)p Fx(\000)p Fp(1)2477 4105 y Fq( )t Fm(i)o(g)g Fq(;)82 b(n)27 b Fs(=)h(1)p Fq(;)17 b Fs(2)p Fq(;)g Fs(3)p Fq(;)156 b Fs(\(3.4\))328 4321 y(where)34 b Fq(';)17 b( )31 b Fm(2)d(D)s Fs(.)43 b(W)-8 b(e)33 b(assume)g(that)473 4522 y Fm(\017)49 b Fs(\()p Fq(X)r(;)17 b Fs(\003)p Fq(;)g Fm(D)s Fs(\))41 b(satis\014es)h(the)g(GJN)g(Condition,)h(for)e Fq(X)51 b Fs(=)44 b Fq(L;)17 b(N)5 b(;)17 b(D)s(;)g(C)3134 4537 y Fo(n)3180 4522 y Fs(.)71 b(Conse-)572 4642 y(quen)m(tly)-8 b(,)39 b(all)34 b(these)k(op)s(erators)f(determine)f(selfadjoin)m(t)g (op)s(erators,)h(whic)m(h)h(w)m(e)572 4762 y(denote)33 b(b)m(y)h(the)f(same)f(letters.)473 4965 y Fm(\017)49 b Fq(A)32 b Fs(is)h(selfadjoin)m(t,)e Fm(D)f(\032)e(D)s Fs(\()p Fq(A)p Fs(\),)33 b(and)f Fq(e)2007 4929 y Fo(itA)2147 4965 y Fs(lea)m(v)m(es)h Fm(D)s Fs(\(\003\))f(in)m(v)-5 b(arian)m(t.)1898 5214 y(21)p eop %%Page: 22 22 22 21 bop 328 631 a FB(R)-5 b(emarks.)70 b Fs(1\))26 b(F)-8 b(rom)25 b(the)h(in)m(v)-5 b(ariance)26 b(condition)f Fq(e)2241 595 y Fo(itA)2348 631 y Fm(D)s Fs(\(\003\))h Fm(\032)j(D)s Fs(\(\003\),)d(it)g(follo)m(ws)f(that)328 751 y(for)32 b(some)g(0)c Fm(\024)g Fq(k)s(;)17 b(k)1055 715 y Fx(0)1106 751 y Fq(<)27 b Fm(1)p Fs(,)33 b(and)f(all)f Fq( )g Fm(2)e(D)s Fs(\(\003\),)1442 966 y Fm(k)p Fs(\003)p Fq(e)1605 925 y Fo(itA)1711 966 y Fq( )t Fm(k)f(\024)g Fq(k)s(e)2060 925 y Fo(k)2099 901 y Fe(0)2121 925 y Fx(j)p Fo(t)p Fx(j)2190 966 y Fm(k)p Fs(\003)p Fq( )t Fm(k)p Fq(:)913 b Fs(\(3.5\))328 1181 y(A)33 b(pro)s(of)e(of)h(this)h(can)f(b) s(e)h(found)g(in)f([ABG],)g(Prop)s(ositions)g(3.2.2)g(and)g(3.2.5.)328 1301 y(2\))f(Condition)g(\(3.1\))g(is)g(phrased)i(equiv)-5 b(alen)m(tly)31 b(as)h(\\)p Fq(X)j Fm(\024)28 b Fq(k)s(Y)22 b Fs(,)32 b(in)f(the)h(sense)h(of)e(Kato)328 1421 y(on)h Fm(D)s Fs(".)328 1542 y(3\))38 b(One)h(can)f(sho)m(w)h(that)g(if)e(\()p Fq(A;)17 b Fs(\003)p Fq(;)g Fm(D)s Fs(\))37 b(satis\014es)i(conditions) e(\(3.1\),)i(\(3.2\),)g(then)g(the)328 1662 y(ab)s(o)m(v)m(e)33 b(assumption)f(on)h Fq(A)f Fs(holds;)h(see)g(Theorem)g(A.1.)328 1884 y Ft(Theorem)k(3.2)49 b(\()p Fs(1)1088 1848 y Fp(st)1185 1884 y Ft(virial)36 b(theorem\))72 b FB(Assume)35 b Fq(N)46 b FB(and)35 b Fq(e)2719 1848 y Fo(itA)2861 1884 y FB(c)-5 b(ommute,)35 b(for)g(al)5 b(l)328 2005 y Fq(t)28 b Fm(2)g Fk(R)5 b FB(,)41 b(in)34 b(the)h(str)-5 b(ong)35 b(sense)f(on)g Fm(D)s FB(,)h(and)f(that)1183 2220 y Fq(D)85 b Fm(\024)f Fq(k)s(N)1652 2178 y Fp(1)p Fo(=)p Fp(2)1763 2220 y Fq(;)1575 b Fs(\(3.6\))1157 2365 y Fq(C)1227 2380 y Fp(1)1349 2365 y Fm(\024)84 b Fq(k)s(N)1652 2324 y Fo(p)1692 2365 y Fq(;)122 b FB(for)34 b(some)g Fs(0)28 b Fm(\024)g Fq(p)f(<)h Fm(1)p FB(,)628 b Fs(\(3.7\))1157 2510 y Fq(C)1227 2525 y Fp(3)1349 2510 y Fm(\024)84 b Fq(k)s(N)1652 2469 y Fp(1)p Fo(=)p Fp(2)3365 2510 y Fs(\(3.8\))328 2725 y FB(in)45 b(the)h(sense)f(of)g(Kato)h(on)f Fm(D)s FB(,)j(for)e(some)f Fq(k)51 b(<)c Fm(1)p FB(.)77 b(Then,)48 b(if)d Fq( )52 b Fm(2)c(D)s Fs(\()p Fq(L)p Fs(\))e FB(is)f(an)328 2845 y(eigenve)-5 b(ctor)45 b(of)i Fq(L)p FB(,)i(ther)-5 b(e)47 b(is)f(a)h(family)f(of)g(appr)-5 b(oximating)45 b(eigenve)-5 b(ctors)45 b Fm(f)p Fq( )3339 2860 y Fo(\013)3389 2845 y Fm(g)k(\032)328 2966 y(D)s Fs(\()p Fq(L)p Fs(\))22 b Fm(\\)g(D)s Fs(\()p Fq(C)848 2981 y Fp(1)887 2966 y Fs(\))p FB(,)35 b Fq(\013)28 b(>)g Fs(0)p FB(,)34 b(such)h(that)g Fq( )1781 2981 y Fo(\013)1859 2966 y Fm(!)27 b Fq( )39 b FB(\(in)34 b Fm(H)q FB(\))h(as)g Fq(\013)28 b Fm(!)f Fs(0)p FB(,)35 b(and)1539 3180 y Fs(lim)1531 3240 y Fo(\013)p Fx(!)p Fp(0)1699 3180 y Fm(h)p Fq( )1801 3195 y Fo(\013)1851 3180 y Fq(;)17 b(C)1965 3195 y Fp(1)2004 3180 y Fq( )2067 3195 y Fo(\013)2117 3180 y Fm(i)27 b Fs(=)h(0)p Fq(:)1002 b Fs(\(3.9\))474 3442 y FB(R)-5 b(emarks.)69 b Fs(1\))34 b(It)g(is)f(not)h(necessary)i(that)d Fq(N)45 b Fs(and)34 b Fq(e)2446 3406 y Fo(itA)2586 3442 y Fs(comm)m(ute)f(for)h(the)g (result)328 3562 y(to)27 b(hold,)h(but)g(this)f(will)f(b)s(e)i(the)g (case)g(in)f(all)f(our)h(applications)f(\(see)j([FM])f(for)f(the)h (case)328 3682 y(where)34 b Fq(N)43 b Fs(and)32 b Fq(A)h Fs(do)g(not)f(comm)m(ute\).)328 3803 y(2\))d(In)g(a)g(heuristic)g(w)m (a)m(y)-8 b(,)31 b(w)m(e)g(understand)f Fq(C)1964 3818 y Fp(1)2033 3803 y Fs(as)f(the)h(comm)m(utator)d Fq(i)p Fs([)p Fq(L;)17 b(A)p Fs(])29 b(=)e Fq(i)p Fs(\()p Fq(LA)15 b Fm(\000)328 3923 y Fq(AL)p Fs(\),)29 b(and)e(\(3.9\))g(as)h Fm(h)p Fq( )t(;)17 b(i)p Fs([)p Fq(L;)g(A)p Fs(])p Fq( )t Fm(i)28 b Fs(=)f(0,)h(whic)m(h)g(is)f(the)h(usual)f(statemen)m(t)h(of)f (the)h(virial)328 4044 y(theorem;)g(see)f(e.g.)41 b([ABG],)26 b(and)g([GG])f(for)g(a)g(comparison)g(\(and)h(correction\))f(of)g (virial)328 4164 y(theorems)33 b(encoun)m(tered)h(in)e(the)h (literature.)474 4405 y(The)j(Virial)31 b(Theorem)k(is)f(still)e(v)-5 b(alid)32 b(if)i(w)m(e)h(add)g(to)f(the)h(op)s(erator)e Fq(A)i Fs(a)f(b)s(ounded)328 4525 y(p)s(erturbation)e Fq(A)973 4540 y Fp(0)1045 4525 y Fs(lea)m(ving)f(the)i(domain)e(of)h Fq(L)h Fs(in)m(v)-5 b(arian)m(t.)328 4724 y Ft(Theorem)37 b(3.3)49 b(\()p Fs(2)1088 4688 y Fp(nd)1209 4724 y Ft(virial)37 b(theorem\))75 b FB(Supp)-5 b(ose)35 b(that)i(we)f(ar)-5 b(e)36 b(in)g(the)h(situation)328 4844 y(of)44 b(The)-5 b(or)g(em)44 b(3.2,)i(and)e(that)h Fq(A)1554 4859 y Fp(0)1638 4844 y FB(is)g(a)f(b)-5 b(ounde)g(d)44 b(op)-5 b(er)g(ator)44 b(on)h Fm(H)q FB(,)i(s.t.)74 b Fs(Ran)16 b Fq(A)3403 4859 y Fp(0)3488 4844 y Fm(\022)328 4965 y(D)s Fs(\()p Fq(L)p Fs(\))30 b Fm(\\)h Fs(Ran)17 b Fq(P)d Fs(\()p Fq(N)58 b Fm(\024)49 b Fq(n)1304 4980 y Fp(0)1344 4965 y Fs(\))p FB(,)g(for)d(some)f Fq(n)1946 4980 y Fp(0)2035 4965 y Fq(<)j Fm(1)p FB(.)79 b(The)45 b(c)-5 b(ommutator)46 b Fq(i)p Fs([)p Fq(L;)17 b(A)3374 4980 y Fp(0)3414 4965 y Fs(])49 b(=)1898 5214 y(22)p eop %%Page: 23 23 23 22 bop 328 631 a Fq(i)p Fs(\()p Fq(LA)538 646 y Fp(0)608 631 y Fm(\000)30 b Fq(A)788 646 y Fp(0)827 631 y Fq(L)p Fs(\))45 b FB(is)g(wel)5 b(l)44 b(de\014ne)-5 b(d)44 b(in)g(the)h(str)-5 b(ong)45 b(sense)f(on)g Fm(D)s Fs(\()p Fq(L)p Fs(\))p FB(.)75 b(F)-7 b(or)44 b(the)h(same)328 751 y(family)h(of)h(appr)-5 b(oximating)46 b(eigenve)-5 b(ctors)46 b(as)g(in)h(the)g(pr)-5 b(evious)46 b(the)-5 b(or)g(em,)50 b(we)d(have)328 872 y(that)1286 992 y Fs(lim)1278 1052 y Fo(\013)p Fx(!)p Fp(0)1446 992 y Fm(h)p Fq( )1548 1007 y Fo(\013)1598 992 y Fq(;)17 b Fs(\()p Fq(C)1750 1007 y Fp(1)1811 992 y Fs(+)22 b Fq(i)p Fs([)p Fq(L;)17 b(A)2152 1007 y Fp(0)2192 992 y Fs(]\))p Fq( )2320 1007 y Fo(\013)2370 992 y Fm(i)27 b Fs(=)h(0)p Fq(:)701 b Fs(\(3.10\))328 1293 y Fh(3.2)135 b(Outline)41 b(of)g(the)f(pro)t(ofs)h (of)f(Theorems)g(2.1)h(and)f(2.3;)j(the)634 1442 y(p)t(ositiv)l(e)k (comm)l(utator)f(metho)t(d)328 1627 y Fs(The)34 b(p)s(ositiv)m(e)e (comm)m(utator)f(metho)s(d)h(giv)m(es)h(a)f(conceptually)h(easy)g(pro)s (of)f(of)g(the)h(ab-)328 1747 y(sence)40 b(of)f(p)s(oin)m(t)f(sp)s (ectrum)h(of)f Fq(L)p Fs(.)63 b(W)-8 b(e)39 b(outline)e(a)i(v)m(ersion) g(that)f(is)h(adapted)g(to)f(the)328 1868 y(pro)s(ofs)c(of)g(Theorems)h (2.1)f(and)g(2.3.)48 b(The)35 b(full)e(pro)s(ofs)h(are)g(giv)m(en)g(in) g(Sections)h(4)f(and)328 1988 y(5.)41 b(In)28 b(the)g(presen)m(t)h (section)e(w)m(e)h(use)g(the)g(notation)e(of)h(Section)g(3.1)g(and)g (write)g Fq(i)p Fs([)p Fq(L)3376 2003 y Fo(\025)3422 1988 y Fq(;)17 b(A)p Fs(])328 2108 y(for)32 b Fq(C)547 2123 y Fp(1)586 2108 y Fs(.)474 2349 y FB(Outline)47 b(of)g(the)g(pr)-5 b(o)g(of)46 b(of)h(The)-5 b(or)g(em)46 b(2.3.)130 b Fs(According)45 b(to)h(the)g(discussion)g(of)328 2469 y(Section)29 b(2.1.4)g(w)m(e)i(ha)m(v)m(e)g(to)e(sho)m(w)i(that)e Fm(P)24 b(\\)16 b Fs(k)m(er)i Fq(L)2210 2484 y Fo(\025)2286 2469 y Fs(is)29 b(giv)m(en)g(b)m(y)i(the)f(set)g(\(2.55\).)42 b(The)328 2590 y(Liouvillian)28 b Fq(L)880 2605 y Fo(\025)959 2590 y Fs(is)k(reduced)i(b)m(y)f(the)g(decomp)s(osition)1055 2810 y Fm(H)c Fs(=)f(Ran)16 b Fq(P)1540 2769 y Fp(0)1601 2810 y Fm(\010)23 b Fs(Ran)16 b Fq(P)35 b Fm(\010)23 b Fs(Ran)16 b Fq(P)2358 2769 y Fo(l)2406 2810 y Fm(\010)23 b Fs(Ran)16 b Fq(P)2774 2769 y Fo(r)2811 2810 y Fq(;)328 3030 y Fs(where)38 b(the)f(v)-5 b(arious)37 b(pro)5 b(jections)37 b(are)g(de\014ned)h(in)e(Section)h(2.1.5.)56 b(It)37 b(is)f(easy)i(to)f(see)328 3150 y(that)c Fm(P)e(\\)24 b Fs(Ran)16 b Fq(P)998 3114 y Fo(l)1053 3150 y Fs(=)29 b Fm(P)i(\\)23 b Fs(Ran)17 b Fq(P)1616 3114 y Fo(r)1682 3150 y Fs(=)29 b Fm(f)p Fs(0)p Fm(g)k Fs(and)h(that)f Fq(L)2438 3165 y Fo(\025)2513 3150 y Fn(\026)c Fs(Ran)16 b Fq(P)2852 3114 y Fp(0)2920 3150 y Fs(=)30 b Fq(L)3092 3165 y Fp(0)3161 3150 y Fn(\026)e Fs(Ran)17 b Fq(P)3500 3114 y Fp(0)3539 3150 y Fs(.)328 3271 y(Consequen)m(tly)-8 b(,)39 b Fm(P)32 b(\\)25 b Fs(k)m(er)18 b Fq(L)1361 3286 y Fo(\025)1439 3271 y Fs(=)33 b Fm(P)g(\\)25 b Fs(\(k)m(er)17 b Fq(L)1991 3286 y Fp(0)2059 3271 y Fn(\026)27 b Fs(Ran)16 b Fq(P)2396 3234 y Fp(0)2457 3271 y Fm([)23 b Fs(k)m(er)18 b Fq(L)2759 3286 y Fo(\025)2832 3271 y Fn(\026)27 b Fs(Ran)17 b Fq(P)d Fs(\))i Fq(;)35 b Fs(and)h(to)328 3391 y(pro)m(v)m(e)e(the)f (theorem,)f(it)g(is)g(enough)h(to)f(sho)m(w)i(that)1482 3611 y(k)m(er)18 b Fq(L)1695 3626 y Fo(\025)1768 3611 y Fn(\026)27 b Fs(Ran)17 b Fq(P)41 b Fs(=)27 b Fm(f)p Fs(0)p Fm(g)p Fq(:)905 b Fs(\(3.11\))328 3831 y(W)-8 b(e)35 b(construct)g(selfadjoin)m(t)e(op)s(erators)h(\003)p Fq(;)17 b(A;)g(A)2134 3846 y Fp(0)2207 3831 y Fs(suc)m(h)36 b(that)e(the)g(conditions)g(of)f(Sec-)328 3951 y(tion)f(3.1)h(are)g (ful\014lled,)e(with)i Fq(L)c Fs(=)g Fq(L)1716 3966 y Fo(\025)1795 3951 y Fs(and)k Fq(N)39 b Fs(=)29 b(d\000\(1)-22 b(l\).)44 b(The)34 b(op)s(erators)f Fq(A)g Fs(and)g Fq(A)3526 3966 y Fp(0)328 4072 y Fs(ha)m(v)m(e)h(the)f(prop)s(erties)f(that)1223 4292 y Fq(i)p Fs([)p Fq(L)1349 4307 y Fo(\025)1396 4292 y Fq(;)17 b(A)p Fs(])22 b(+)g Fq(i)p Fs([)p Fq(L;)17 b(A)1903 4307 y Fp(0)1943 4292 y Fs(])27 b Fm(\025)i Fq(P)14 b(M)2274 4307 y Fp(0)2313 4292 y Fq(P)35 b Fs(+)22 b Fq(M)2603 4307 y Fp(1)2643 4292 y Fq(;)328 4512 y Fs(where)34 b(the)f(b)s(ounded)g(op)s(erators)f Fq(M)1701 4527 y Fp(0)1774 4512 y Fs(and)g Fq(M)2057 4527 y Fp(1)2130 4512 y Fs(satisfy)1593 4732 y Fq(P)14 b(M)1764 4747 y Fp(1)1803 4732 y Fq(P)97 b Fs(=)84 b(0)p Fq(;)1491 4877 y Fm(h)o Fq( )t(;)17 b(M)1734 4892 y Fp(0)1774 4877 y Fq( )t Fm(i)83 b(\025)g Fq(\016)t Fm(k)p Fq( )t Fm(k)2337 4836 y Fp(2)2376 4877 y Fq(;)914 b Fs(\(3.12\))1898 5214 y(23)p eop %%Page: 24 24 24 23 bop 328 631 a Fs(for)33 b(an)m(y)h Fq( )f Fm(2)d Fs(k)m(er)17 b Fq(L)1067 646 y Fo(\025)1142 631 y Fn(\026)29 b Fs(Ran)16 b Fq(P)e Fs(,)33 b(and)h(where)g Fq(\016)k Fs(is)32 b(strictly)h(p)s(ositiv)m(e.)46 b(F)-8 b(rom)31 b(Theorem)328 751 y(3.3)h(w)m(e)i(obtain)959 928 y(0)27 b(=)h(lim)1184 988 y Fo(\013)1291 928 y Fm(h)p Fq( )1393 943 y Fo(\013)1443 928 y Fq(;)17 b Fs(\()o Fq(i)p Fs([)p Fq(L)1650 943 y Fo(\025)1696 928 y Fq(;)g(A)p Fs(])22 b(+)g Fq(i)p Fs([)p Fq(L)2086 943 y Fo(\025)2132 928 y Fq(;)17 b(A)2249 943 y Fp(0)2289 928 y Fs(]\))f Fq( )2433 943 y Fo(\013)2483 928 y Fm(i)28 b(\025)g Fq(\016)t Fm(k)p Fq( )t Fm(k)2869 887 y Fp(2)2908 928 y Fq(:)328 1137 y Fs(Because)34 b Fq(\016)e(>)27 b Fs(0,)33 b(w)m(e)g(ha)m(v)m(e)h (that)e(k)m(er)18 b Fq(L)1780 1152 y Fo(\025)1854 1137 y Fn(\026)27 b Fs(Ran)16 b Fq(P)41 b Fs(=)28 b Fm(f)p Fs(0)p Fm(g)p Fs(.)474 1257 y(W)-8 b(e)36 b(no)m(w)f(explain)g(ho)m(w)g (to)g(arriv)m(e)g(at)g(the)g(k)m(ey)i(inequalit)m(y)d(\(3.12\).)50 b(The)36 b(form)e(of)328 1378 y(the)f(non-in)m(teracting)e (Liouvillian,)e Fq(L)1756 1393 y Fp(0)1795 1378 y Fs(,)k(giv)m(en)g(in) e(\(2.42\))h(suggests)i(to)f(consider)664 1554 y Fq(A)27 b Fs(=)h Fq(\037A)1002 1569 y Fo(p)1042 1554 y Fq(\037)22 b Fm(\012)h Fs(1)-22 b(l)1280 1569 y Fo(p)1340 1554 y Fm(\012)23 b Fs(1)-22 b(l)1495 1569 y Fo(f)1561 1554 y Fm(\000)23 b Fs(1)-22 b(l)1716 1569 y Fo(p)1776 1554 y Fm(\012)23 b Fq(\037A)2010 1569 y Fo(p)2050 1554 y Fq(\037)f Fm(\012)h Fs(1)-22 b(l)2288 1569 y Fo(f)2354 1554 y Fs(+)22 b(1)-22 b(l)2507 1569 y Fo(p)2567 1554 y Fm(\012)23 b Fs(1)-22 b(l)2722 1569 y Fo(p)2783 1554 y Fm(\012)22 b Fs(d\000\()p Fq(i@)3119 1569 y Fo(u)3165 1554 y Fs(\))p Fq(;)328 1731 y Fs(where)47 b Fq(A)696 1746 y Fo(p)782 1731 y Fs(is)f(the)g(dilation)d(generator,)50 b(see)d(\(3.40\),)i(and)d(d\000\()p Fq(i@)2861 1746 y Fo(u)2907 1731 y Fs(\))f(is)h(the)g(second)328 1851 y(quan)m(tization) 34 b(of)h(the)g(translation)e(generator)j(in)e(the)h(radial)e(v)-5 b(ariable)34 b Fq(u)g Fs(of)h Fq(L)3315 1815 y Fp(2)3355 1851 y Fs(\()p Fk(R)f Fm(\002)328 1971 y Fq(S)394 1935 y Fp(2)433 1971 y Fq(;)17 b(du)29 b Fm(\002)h Fq(d)p Fs(\006\),)47 b(see)e(\(2.28\).)76 b(Here,)47 b Fq(\037)d Fs(is)f(a)h(function)f(of)g Fq(H)2649 1986 y Fo(p)2732 1971 y Fs(with)h(supp)s(ort)g(in)f(an)328 2092 y(in)m(terv)-5 b(al)33 b(in)g Fk(R)863 2107 y Fp(+)928 2092 y Fs(,)h(con)m(taining)e Fq(J)1518 2107 y Fo(c)1587 2092 y Fs(but)i(not)g Fm(f)p Fs(0)p Fm(g)p Fs(,)f(and)h(suc)m(h)i(that)d Fq(\037)p Fm(j)2865 2107 y Fo(J)2904 2115 y Fl(c)2970 2092 y Fs(=)d(1.)47 b(Then)35 b(w)m(e)328 2212 y(ha)m(v)m(e)660 2389 y Fq(i)p Fs([)p Fq(L)786 2404 y Fo(\025)832 2389 y Fq(;)17 b(A)p Fs(])28 b(=)f Fq(\037)1168 2348 y Fp(2)1208 2389 y Fq(H)1289 2404 y Fo(p)1350 2389 y Fm(\012)c Fs(1)-22 b(l)1505 2404 y Fo(p)1566 2389 y Fm(\012)22 b Fs(1)-22 b(l)1720 2404 y Fo(f)1786 2389 y Fs(+)22 b(1)-22 b(l)1939 2404 y Fo(p)2000 2389 y Fm(\012)23 b Fq(\037)2161 2348 y Fp(2)2200 2389 y Fq(H)2281 2404 y Fo(p)2343 2389 y Fm(\012)f Fs(1)-22 b(l)2497 2404 y Fo(f)2564 2389 y Fs(+)22 b Fq(N)32 b Fs(+)22 b Fq(U)33 b Fs(+)22 b Fq(\025I)3167 2404 y Fp(1)3207 2389 y Fq(;)328 2565 y Fs(where)39 b Fq(N)49 b Fs(is)38 b(the)h(n)m(um)m(b)s(er)f(op)s(erator,)i Fq(U)48 b Fs(is)38 b(a)g(non-negativ)m(e)g(op)s(erator,)i(b)s(ecause)f(of)328 2686 y(the)30 b(c)m(hoice)g(of)f(the)h(parameter)f Fq(\026)g Fs(in)g(the)h(p)s(oten)m(tial)e Fq(v)33 b Fs(of)c(equation)g(\(2.2\),)h (and)g(where)328 2806 y Fq(I)371 2821 y Fp(1)438 2806 y Fs(=)e Fq(i)p Fs([)p Fq(I)8 b(;)17 b(A)p Fs(])32 b(is)g (in\014nitesimally)d(small)i(w.r.t.)44 b Fq(N)10 b Fs(,)1548 3042 y Fm(\006)p Fq(\025I)1725 3057 y Fp(1)1792 3042 y Fm(\024)29 b Fq(cN)j Fs(+)2158 2974 y Fq(\025)2215 2938 y Fp(2)p 2158 3019 97 4 v 2185 3110 a Fq(c)2265 3042 y(k)s(;)971 b Fs(\(3.13\))328 3244 y(for)34 b(an)m(y)h Fq(c)c(>)g Fs(0)k(and)f(some)h Fq(k)f(<)d Fm(1)p Fs(.)49 b(The)36 b(role)d(of)h Fq(\037)h Fs(is)f(to)g(pro)5 b(ject)36 b(out)e(the)h(discrete)328 3364 y(mo)s(des)e(in)g Fq(J)799 3379 y Fo(d)839 3364 y Fs(.)47 b(Using)33 b(that)h Fq(P)42 b Fs(=)30 b Fq(p)23 b Fm(\012)g Fq(p)g Fm(\012)g Fs(1)-22 b(l)2011 3379 y Fo(f)2055 3364 y Fs(,)34 b Fq(p\037)2226 3328 y Fp(2)2295 3364 y Fs(=)29 b Fq(p)2449 3379 y Fo(J)2488 3387 y Fl(c)2524 3364 y Fs(,)34 b(and)f(that)h Fq(p)3037 3379 y Fo(J)3076 3387 y Fl(c)3112 3364 y Fq(H)3193 3379 y Fo(p)3261 3364 y Fm(\025)c Fq(r)s(p)3464 3379 y Fo(J)3503 3387 y Fl(c)3539 3364 y Fs(,)328 3485 y(since)37 b(the)g(in)m(terv)-5 b(al)36 b Fq(J)1154 3500 y Fo(c)1225 3485 y Fs(is)g(a)m(w)m(a)m(y)j (from)c(the)i(origin)e(b)m(y)i(a)g(distance)g(of)f(at)h(least)f Fq(r)s Fs(,)i(w)m(e)328 3605 y(obtain)443 3781 y Fq(P)14 b(i)p Fs([)p Fq(L)646 3796 y Fo(\025)691 3781 y Fq(;)j(A)p Fs(])p Fq(P)526 3984 y Fm(\025)83 b Fq(P)779 3844 y Fi(\022)852 3984 y Fq(p)901 3999 y Fo(J)940 4007 y Fl(c)976 3984 y Fq(H)1057 3999 y Fo(p)1119 3984 y Fm(\012)22 b Fs(1)-22 b(l)1273 3999 y Fo(p)1334 3984 y Fm(\012)23 b Fq(P)1497 3999 y Fp(\012)1574 3984 y Fs(+)f(1)-22 b(l)1727 3999 y Fo(p)1787 3984 y Fm(\012)23 b Fq(p)1936 3999 y Fo(J)1975 4007 y Fl(c)2011 3984 y Fq(H)2092 3999 y Fo(p)2153 3984 y Fm(\012)g Fq(P)2316 3999 y Fp(\012)2393 3984 y Fs(+)2501 3917 y(1)p 2501 3961 49 4 v 2501 4052 a(2)p 2560 3904 77 4 v 2560 3984 a Fq(P)2636 3999 y Fp(\012)2691 3844 y Fi(\023)2781 3984 y Fq(P)36 b Fm(\000)22 b Fq(k)s(\025)3090 3943 y Fp(2)3130 3984 y Fq(P)526 4256 y Fm(\025)83 b Fs(min)o(\()p Fq(r)m(;)17 b Fs(1)p Fq(=)p Fs(2\))p Fq(P)1249 4116 y Fi(\022)1322 4256 y Fq(p)1371 4271 y Fo(J)1410 4279 y Fl(c)1468 4256 y Fm(\012)23 b Fs(1)-22 b(l)1623 4271 y Fo(p)1683 4256 y Fm(\012)23 b Fq(P)1846 4271 y Fp(\012)1923 4256 y Fs(+)f(1)-22 b(l)2076 4271 y Fo(p)2136 4256 y Fm(\012)23 b Fq(p)2285 4271 y Fo(J)2324 4279 y Fl(c)2382 4256 y Fm(\012)g Fq(P)2545 4271 y Fp(\012)2622 4256 y Fs(+)2730 4189 y(1)p 2730 4233 49 4 v 2730 4325 a(2)p 2788 4176 77 4 v 2788 4256 a Fq(P)2865 4271 y Fp(\012)2920 4116 y Fi(\023)3010 4256 y Fq(P)35 b Fm(\000)23 b Fq(k)s(\025)3319 4215 y Fp(2)3358 4256 y Fq(P)s(;)328 4490 y Fs(where)40 b(w)m(e)g(c)m(ho)s(ose)f Fq(c)f Fs(=)g(1)p Fq(=)p Fs(2)g(in)g (\(3.13\).)61 b(The)40 b(\014rst)f(term)f(on)h(the)g(r.h.s.)62 b(is)38 b(strictly)328 4611 y(p)s(ositiv)m(e)29 b(except)i(on)e(the)h (subspace)h(Ran)16 b Fq(p)1930 4626 y Fo(J)1969 4638 y Fl(d)2025 4611 y Fm(\012)g Fq(p)2167 4626 y Fo(J)2206 4638 y Fl(d)2261 4611 y Fm(\012)g Fq(P)2417 4626 y Fp(\012)2500 4611 y Fm(\032)28 b Fs(Ran)17 b Fq(P)d Fs(.)42 b(W)-8 b(e)29 b(decomp)s(ose)954 4806 y Fq(p)1003 4821 y Fo(J)1042 4833 y Fl(d)1104 4806 y Fm(\012)23 b Fq(p)1253 4821 y Fo(J)1292 4833 y Fl(d)1354 4806 y Fm(\012)g Fq(P)1517 4821 y Fp(\012)1599 4806 y Fs(=)28 b Fq(P)14 b Fs(\005)21 b(+)2076 4712 y Fi(X)2036 4912 y Fl(m;n)p Fe(2)p Fl(J)2223 4929 y(d)1982 4976 y(E)s Fj(\()p Fl(m)p Fj(\))p Fe(6)p Fj(=)p Fl(E)s Fj(\()p Fl(n)p Fj(\))2339 4806 y Fq(p)2388 4821 y Fo(m)2477 4806 y Fm(\012)i Fq(p)2626 4821 y Fo(n)2695 4806 y Fm(\012)f Fq(P)2857 4821 y Fp(\012)2912 4806 y Fq(;)378 b Fs(\(3.14\))1898 5214 y(24)p eop %%Page: 25 25 25 24 bop 328 631 a Fs(where)45 b(\005)f(is)g(the)g(pro)5 b(jection)44 b(on)m(to)g(the)g(k)m(ernel)h(of)e Fq(L)2399 646 y Fp(0)2439 631 y Fs(.)78 b(Note)44 b(that)g Fq(P)14 b Fs(\005)43 b(is)h(\014nite-)328 751 y(dimensional.)53 b(On)36 b(the)h(range)g(of)f(the)h(second)h(pro)5 b(jection)36 b(on)g(the)h(r.h.s.)56 b(of)36 b(\(3.14\),)328 872 y(the)d(free)g (Liouvillian)28 b(satis\014es)537 1066 y Fm(j)p Fq(L)631 1081 y Fp(0)671 1066 y Fm(j)f(\025)h Fs(min)1011 956 y Fi(n)1077 1066 y Fm(j)p Fq(E)6 b Fs(\()p Fq(m)p Fs(\))22 b Fm(\000)h Fq(E)6 b Fs(\()p Fq(n)p Fs(\))p Fm(j)1738 982 y Fi(\014)1738 1041 y(\014)1804 1066 y Fq(m;)17 b(n)28 b Fm(2)g Fq(J)2167 1081 y Fo(d)2207 1066 y Fq(;)17 b(E)6 b Fs(\()p Fq(m)p Fs(\))28 b Fm(6)p Fs(=)f Fq(E)6 b Fs(\()p Fq(n)p Fs(\))2833 956 y Fi(o)2928 1066 y Fq(>)27 b Fs(0)p Fq(:)210 b Fs(\(3.15\))328 1288 y(Let)32 b(\001)g(b)s(e)f(an)h(in)m (terv)-5 b(al)30 b(around)i(zero)g(whose)h(size)f Fm(j)p Fs(\001)p Fm(j)f Fs(is)g(smaller)f(than)h(the)h(r.h.s.)44 b(of)328 1409 y(\(3.14\),)39 b(and)f(let)f Fq(E)1063 1373 y Fp(0)1057 1434 y(\001)1158 1409 y Fs(b)s(e)i(the)f(sp)s(ectral)g (pro)5 b(jection)38 b(of)f Fq(L)2493 1424 y Fp(0)2571 1409 y Fs(on)m(to)h(\001.)60 b(Then)39 b(w)m(e)g(ha)m(v)m(e)328 1529 y(that)32 b Fq(E)617 1493 y Fp(0)611 1554 y(\001)707 1529 y Fq(p)756 1544 y Fo(J)795 1556 y Fl(d)857 1529 y Fm(\012)23 b Fq(p)1006 1544 y Fo(J)1045 1556 y Fl(d)1107 1529 y Fm(\012)f Fq(P)1269 1544 y Fp(\012)1352 1529 y Fs(=)27 b Fq(E)1533 1493 y Fp(0)1527 1554 y(\001)1591 1529 y Fq(P)14 b Fs(\005)27 b(=)g Fq(P)14 b Fs(\005.)43 b(Consider)33 b(the)g(decomp)s(osition)1195 1724 y(Ran)17 b Fq(P)d(E)1542 1683 y Fp(0)1536 1748 y(\001)1626 1724 y Fs(=)27 b(Ran)17 b Fq(P)d(E)2076 1683 y Fp(0)2070 1748 y(\001)p 2132 1644 77 4 v 2132 1724 a Fq(P)36 b Fm(\010)23 b Fs(Ran)16 b Fq(P)e Fs(\005)p Fq(:)618 b Fs(\(3.16\))328 1918 y(F)-8 b(rom)49 b(the)i(ab)s(o)m(v)m(e)g(discussion)g(it)e(is)h (apparen)m(t)h(that)g(on)f(the)h(blo)s(c)m(k)f(Ran)17 b Fq(P)d(E)3406 1882 y Fp(0)3400 1944 y(\001)p 3462 1838 V 3462 1918 a Fq(P)g Fs(,)328 2039 y Fq(i)p Fs([)p Fq(L)454 2054 y Fo(\025)500 2039 y Fq(;)j(A)p Fs(])45 b(is)f(bigger)g(than)i Fq(r)33 b Fm(\000)e Fq(k)s(\025)1641 2003 y Fp(2)1729 2039 y Fm(\025)49 b Fq(r)s(=)p Fs(2)c(\(w)m(e)h(require)f Fm(j)p Fq(\025)p Fm(j)j(\024)h Fq(k)2920 1967 y Fm(p)p 3003 1967 47 4 v 72 x Fq(r)s Fs(\),)f(while)c(the)328 2159 y(comm)m(utator)38 b(is)h(zero)h(on)f(the)h(blo)s(c)m(k)f(Ran)17 b Fq(P)d Fs(\005.)63 b(This)40 b(is)f(where)h(w)m(e)h(in)m(tro)s(duce)e (the)328 2280 y(op)s(erator)32 b Fq(A)794 2295 y Fp(0)833 2280 y Fs(.)474 2400 y(One)25 b(can)g(c)m(ho)s(ose)g Fq(A)1217 2415 y Fp(0)1281 2400 y Fs(s.t.)41 b Fq(P)14 b Fs(\005)p Fq(i)p Fs([)p Fq(L)1728 2415 y Fo(\025)1774 2400 y Fq(;)j(A)1891 2415 y Fp(0)1930 2400 y Fs(]\005)p Fq(P)38 b Fs(is)24 b(strictly)g(p)s(ositiv)m(e)f(pro)m(vided)i(the)g (in-)328 2520 y(teraction)c(satis\014es)h(the)h(F)-8 b(ermi)19 b(Golden)i(Rule)g(Condition.)39 b(Moreo)m(v)m(er,)26 b(on)21 b(Ran)c Fq(P)d(E)3453 2484 y Fp(0)3447 2546 y(\001)p 3509 2440 74 4 v 3509 2520 a Fs(\005,)328 2641 y Fq(i)p Fs([)p Fq(L)454 2656 y Fo(\025)500 2641 y Fq(;)j(A)617 2656 y Fp(0)656 2641 y Fs(])30 b(is)f(small)f(relativ)m(e)h(to)g Fq(r)s Fs(.)42 b(The)31 b(construction)f(of)f Fq(A)2572 2656 y Fp(0)2641 2641 y Fs(has)h(b)s(een)h(giv)m(en,)f(in)f(the)328 2761 y(con)m(text)38 b(of)f(zero)h(temp)s(erature)f(systems,)j(in)c ([BFSS],)i(and)f(has)g(b)s(een)h(mo)s(di\014ed)e(for)328 2882 y(p)s(ositiv)m(e)c(temp)s(erature)g(systems)i(in)e([M].)474 3002 y(The)49 b(ab)s(o)m(v)m(e)f(discussion)g(sho)m(ws)h(that)e(the)h (op)s(erator)f Fq(i)p Fs([)p Fq(L)2691 3017 y Fo(\025)2737 3002 y Fq(;)17 b(A)p Fs(])32 b(+)g Fq(i)p Fs([)p Fq(L)3147 3017 y Fo(\025)3194 3002 y Fq(;)17 b(A)3311 3017 y Fp(0)3350 3002 y Fs(])47 b(has)328 3122 y(strictly)30 b(p)s(ositiv)m(e)f (diagonal)f(blo)s(c)m(ks)j(in)e(the)i(decomp)s(osition)e(\(3.16\).)42 b(An)30 b(application)328 3243 y(of)i(the)h(F)-8 b(esh)m(bac)m(h)34 b(metho)s(d)e(then)h(sho)m(ws)i(that)927 3437 y Fq(E)1005 3396 y Fp(0)999 3462 y(\001)1062 3437 y Fq(P)30 b Fs(\()p Fq(i)p Fs([)p Fq(L)1319 3452 y Fo(\025)1365 3437 y Fq(;)17 b(A)p Fs(])23 b(+)f Fq(i)p Fs([)p Fq(L)1756 3452 y Fo(\025)1802 3437 y Fq(;)17 b(A)1919 3452 y Fp(0)1958 3437 y Fs(]\))g Fq(P)d(E)2195 3396 y Fp(0)2189 3462 y(\001)2279 3437 y Fs(=:)27 b Fq(E)2487 3396 y Fp(0)2481 3462 y(\001)2545 3437 y Fq(P)14 b(M)2716 3452 y Fp(0)2755 3437 y Fq(P)g(E)2910 3396 y Fp(0)2904 3462 y(\001)3317 3437 y Fs(\(3.17\))328 3632 y(is)30 b(strictly)g(p)s(ositiv)m(e.)42 b(Since)31 b Fm(k)p Fq(E)1535 3596 y Fp(0)1529 3657 y(\001)1592 3632 y Fq( )22 b Fm(\000)c Fq( )t Fm(k)28 b(\024)g Fq(k)s Fm(j)p Fq(\025)p Fm(j)i(k)p Fq( )t Fm(k)p Fs(,)g(for)g(an)m(y)i Fq( )f Fm(2)d Fs(k)m(er)18 b Fq(L)3174 3647 y Fo(\025)3220 3632 y Fs(,)31 b(w)m(e)g(can)328 3752 y(pass)i(from)f(\(3.17\))g(to)g (estimate)f(\(3.12\).)474 3873 y(In)e(this)g(pro)s(of,)f(the)h (coupling)f(constan)m(t)h(cannot)g(b)s(e)g(c)m(hosen)h(idep)s(enden)m (tly)f(of)f(the)328 3993 y(in)m(v)m(erse)36 b(temp)s(erature)e Fq(\014)6 b Fs(.)48 b(This)35 b(is)f(due)h(to)f(the)g(fact)h(that)f (the)g(constan)m(t)i Fq(\016)i Fs(in)c(\(3.12\))328 4114 y(is)c(prop)s(ortional)e(to)i Fq(\015)5 b Fs(,)31 b(see)h(\(2.24\),)e (whic)m(h)h(in)f(turn)h(deca)m(ys)i(exp)s(onen)m(tially)c(in)h Fq(\014)6 b Fs(.)42 b(W)-8 b(e)328 4234 y(ha)m(v)m(e)36 b(to)f(require)g(that)g(certain)f(error)h(terms)g(whic)m(h)h(dep)s(end) g(on)f Fq(\025)g Fs(are)g(small)d(w.r.t.)328 4354 y Fq(\015)5 b Fs(,)33 b(hence)h(the)f Fq(\014)6 b Fs(-dep)s(enden)m(t)33 b(smallness)f(condition)f(on)i Fq(\025)p Fs(.)474 4595 y FB(Outline)38 b(of)f(the)h(pr)-5 b(o)g(of)38 b(of)f(The)-5 b(or)g(em)37 b(2.1)g Fs(.)53 b(P)m(art)36 b(1\))g(is)f(an)h(easy)g (consequence)j(of)328 4715 y(the)33 b(virial)d(theorem)i(com)m(bined)h (with)f(the)h(b)s(ound)1494 4930 y Fq(i)p Fs([)p Fq(L)1620 4945 y Fo(\025)1666 4930 y Fq(;)17 b(A)p Fs(])28 b Fm(\025)1953 4862 y Fs(1)p 1953 4907 49 4 v 1953 4998 a(2)2012 4930 y Fq(N)k Fm(\000)23 b Fq(k)s(\025)2333 4888 y Fp(2)2372 4930 y Fq(;)1898 5214 y Fs(25)p eop %%Page: 26 26 26 25 bop 328 631 a Fs(for)32 b Fq(A)d Fs(=)g(d\000\()p Fq(i@)921 646 y Fo(u)967 631 y Fs(\).)44 b(The)35 b(pro)s(of)d(of)g (part)h(2\))g(pro)s(ceeds)h(as)g(follo)m(ws.)44 b(W)-8 b(e)33 b(construct)h Fq(A)3526 646 y Fp(0)328 751 y Fs(\(the)f(same)f (as)h(for)f(the)h(pro)s(of)f(of)g(Theorem)h(2.1\))f(s.t.)1100 924 y Fq(i)p Fs([)p Fq(L)1226 939 y Fo(\025)1272 924 y Fq(;)17 b(A)p Fs(])22 b(+)g Fq(i)p Fs([)p Fq(L)1662 939 y Fo(\025)1708 924 y Fq(;)17 b(A)1825 939 y Fp(0)1865 924 y Fs(])28 b Fm(\025)g Fq(\024)2081 939 y Fp(1)2120 924 y Fs(\005)23 b Fm(\000)f Fq(\024)2371 939 y Fp(2)p 2411 843 77 4 v 2411 924 a Fq(P)2487 939 y Fp(0)2549 924 y Fm(\012)g Fq(P)2711 939 y Fp(\012)2766 924 y Fq(;)524 b Fs(\(3.18\))328 1096 y(for)52 b(some)g Fq(\024)817 1111 y Fp(1)857 1096 y Fq(;)17 b(\024)957 1111 y Fp(2)1057 1096 y Fq(>)62 b Fs(0,)57 b(and)52 b(where)i(\005)e(is)g(the)h(pro)5 b(jection)52 b(on)m(to)g(the)h(k)m(ernel)g(of)328 1216 y Fq(L)394 1231 y Fp(0)434 1216 y Fs(.)84 b(If)46 b Fq( )51 b Fs(is)45 b(an)i(eigen)m(v)m(ector)g(of)f Fq(L)1742 1231 y Fo(\025)1834 1216 y Fs(then)h(b)m(y)g(part)f(1\))g(w)m(e)h(ha)m (v)m(e)h(that)e Fm(k)p Fs(\005)p Fq( )t Fm(k)k(\025)328 1336 y Fs(\(1)20 b Fm(\000)h Fq(k)s Fm(j)p Fq(\025)p Fm(j)p Fs(\))p Fm(k)p Fq( )t Fm(k)e(\000)i(k)p 1072 1256 V Fq(P)1148 1351 y Fp(0)1208 1336 y Fm(\012)g Fq(P)1369 1351 y Fp(\012)1424 1336 y Fq( )t Fm(k)p Fs(.)43 b(Inserting)32 b(this)f(b)s(ound)h(in)m(to)f(\(3.18\))g(and)h(using)f(the)328 1457 y(virial)f(theorem)i(yields)h(the)g(b)s(ound)f(\(2.52\).)474 1698 y(This)h(outline)e(indicates)h(that)g(the)h(pro)s(ofs)f(of)g (Theorems)i(2.1)e(and)g(2.3)g(consist)h(of)328 1818 y(t)m(w)m(o)e (steps.)44 b(First)29 b(w)m(e)j(v)m(erify)e(that)g(the)h(virial)d (theorems)i(are)h(applicable)d(and)i(second)328 1938 y(establish)f(a)f(p)s(ositiv)m(e)h(comm)m(utator)e(estimate)h(in)h(the) g(ab)s(o)m(v)m(e)h(sense.)44 b(The)30 b(latter)e(task)328 2059 y(is)k(carried)g(out)h(in)e(Sections)i(4)g(and)f(5.)328 2341 y Fh(3.3)135 b(Applications)46 b(of)f(the)g(virial)h(theorems)328 2525 y Fs(Corresp)s(onding)27 b(to)g(the)h(di\013eren)m(t)g(h)m(yp)s (otheses)h(of)e(Theorems)h(2.1)f(and)h(2.3,)g(w)m(e)g(in)m(tro-)328 2646 y(duce)38 b(t)m(w)m(o)g(sets)g(of)e(op)s(erators)h(\003)p Fq(;)17 b(L;)g(A;)g(A)1907 2661 y Fp(0)1946 2646 y Fq(;)g(N)10 b Fs(,)39 b(and)e(v)m(erify)-8 b(,)38 b(in)e(eac)m(h)i(case,)i(that)c (the)328 2766 y(virial)25 b(theorems)k(are)f(applicable.)40 b(The)29 b(follo)m(wing)c(ob)5 b(jects)30 b(app)s(ear)e(in)f(b)s(oth)h (applica-)328 2886 y(tions:)44 b(the)33 b(Hilb)s(ert)f(space)i(is)f (the)g(GNS)g(space)h(giv)m(en)f(in)g(\(2.27\);)f(the)i(dense)g(domain) 328 3007 y Fm(D)h Fs(is)d(c)m(hosen)i(to)f(b)s(e)1306 3127 y Fm(D)d Fs(=)e Fq(C)1594 3086 y Fx(1)1587 3152 y Fp(0)1669 3127 y Fs(\()p Fk(R)1772 3086 y Fp(3)1818 3127 y Fs(\))22 b Fm(\012)h Fq(C)2055 3086 y Fx(1)2048 3152 y Fp(0)2129 3127 y Fs(\()p Fk(R)2233 3086 y Fp(3)2279 3127 y Fs(\))f Fm(\012)g(D)2515 3142 y Fo(f)2561 3127 y Fq(;)729 b Fs(\(3.19\))328 3281 y(where)1338 3401 y Fm(D)1415 3416 y Fo(f)1488 3401 y Fs(=)27 b Fm(F)1689 3321 y Fi(\000)1735 3401 y Fq(C)1812 3360 y Fx(1)1805 3426 y Fp(0)1887 3401 y Fs(\()p Fk(R)33 b Fm(\002)22 b Fq(S)2184 3360 y Fp(2)2224 3401 y Fs(\))2262 3321 y Fi(\001)2329 3401 y Fm(\\)h(F)2490 3416 y Fp(0)2529 3401 y Fq(;)328 3555 y Fs(where)28 b(the)e(F)-8 b(o)s(c)m(k)27 b(space)h Fm(F)35 b Fs(has)27 b(b)s(een)g(de\014ned)h(in)e(\(2.29\),)h (and)g Fm(F)2743 3570 y Fp(0)2808 3555 y Fs(denotes)h(the)f(\014nite-) 328 3676 y(particle)d(subspace.)44 b(The)26 b(op)s(erator)f Fq(L)h Fs(is)f(the)i(in)m(teracting)d(Liouvillian)e(in)m(tro)s(duced)k (in)328 3796 y(\(2.45\),)f(and)e Fq(N)38 b Fs(=)28 b(d\000\(1)-22 b(l\))22 b(is)h(the)h(particle)e(n)m(um)m(b)s(er)i(op)s(erator)f(in)g Fm(F)37 b(\021)28 b(F)10 b Fs(\()p Fq(L)3124 3760 y Fp(2)3163 3796 y Fs(\()p Fk(R)f Fm(\002)t Fq(S)3418 3760 y Fp(2)3463 3796 y Fs(\)\).)328 3916 y(Clearly)-8 b(,)34 b Fq(X)39 b Fs(=)32 b Fq(L;)17 b(N)45 b Fs(are)35 b(symmetric)f(op)s(erators)g (on)h Fm(D)s Fs(.)50 b(The)35 b(op)s(erator)f Fq(D)s Fs(,)h(de\014ned)328 4037 y(in)d(\(3.3\),)g(is)g(giv)m(en)h(b)m(y)390 4226 y Fq(D)85 b Fs(=)e Fq(i\025)822 4132 y Fi(X)871 4341 y Fo(\013)983 4146 y Fi(\010)1041 4226 y Fq(G)1118 4241 y Fo(\013;)p Fp(#)1268 4226 y Fm(\012)22 b Fs(1)-22 b(l)1422 4241 y Fo(p)1483 4226 y Fm(\012)23 b Fs(\()p Fm(\000)p Fq(a)1749 4185 y Fx(\003)1789 4226 y Fs(\()p Fq(\034)1869 4241 y Fo(\014)1916 4226 y Fs(\()p Fq(g)2001 4241 y Fo(\013)2050 4226 y Fs(\)\))f(+)g Fq(a)p Fs(\()p Fq(\034)2377 4241 y Fo(\014)2425 4226 y Fs(\()p Fq(g)2510 4241 y Fo(\013)2559 4226 y Fs(\)\)\))715 4475 y Fm(\000)p Fs(1)-22 b(l)847 4490 y Fo(p)908 4475 y Fm(\012)23 b(C)1060 4490 y Fo(p)1100 4475 y Fq(G)1177 4490 y Fo(\013;)p Fp(#)1305 4475 y Fm(C)1357 4490 y Fo(p)1419 4475 y Fm(\012)1519 4394 y Fi(\000)1565 4475 y Fm(\000)p Fq(a)1693 4434 y Fx(\003)1733 4475 y Fs(\()p Fq(e)1816 4434 y Fx(\000)p Fo(\014)s(u=)p Fp(2)2029 4475 y Fq(\034)2071 4490 y Fo(\014)2119 4475 y Fs(\()p Fq(g)2204 4490 y Fo(\013)2253 4475 y Fs(\)\))f(+)g Fq(a)p Fs(\()p Fq(e)2583 4434 y Fx(\000)p Fo(\014)s(u=)p Fp(2)2797 4475 y Fq(\034)2839 4490 y Fo(\014)2886 4475 y Fs(\()p Fq(g)2971 4490 y Fo(\013)3020 4475 y Fs(\)\))3096 4394 y Fi(\001)17 b(\011)3217 4475 y Fq(:)73 b Fs(\(3.20\))474 4647 y(W)-8 b(e)33 b(de\014ne)h(a)e(b)s(ounded,)i(selfadjoin)m(t)d(op)s (erator)h Fq(A)2366 4662 y Fp(0)2438 4647 y Fs(on)h Fm(H)g Fs(b)m(y)1269 4819 y Fq(A)1342 4834 y Fp(0)1465 4819 y Fs(=)83 b Fq(i\022)s(\025)p Fs(\(\005)p Fq(I)8 b(R)1999 4778 y Fp(2)1998 4844 y Fo(\017)p 2038 4739 74 4 v 2038 4819 a Fs(\005)23 b Fm(\000)p 2233 4739 V 22 w Fs(\005)p Fq(R)2381 4778 y Fp(2)2380 4844 y Fo(\017)2421 4819 y Fq(I)8 b Fs(\005\))p Fq(;)707 b Fs(\(3.21\))1267 4965 y Fq(R)1342 4924 y Fp(2)1341 4989 y Fo(\017)1465 4965 y Fs(=)83 b(\()p Fq(L)1728 4924 y Fp(2)1728 4989 y(0)1789 4965 y Fs(+)23 b Fq(\017)1927 4924 y Fp(2)1966 4965 y Fs(\))2004 4924 y Fx(\000)p Fp(1)2099 4965 y Fq(:)1191 b Fs(\(3.22\))1898 5214 y(26)p eop %%Page: 27 27 27 26 bop 328 631 a Fs(Here,)33 b Fq(\022)j Fs(and)d Fq(\017)g Fs(are)f(p)s(ositiv)m(e)g(parameters,)h(and)f(\005)h(is)f (the)h(pro)5 b(jection)1563 851 y(\005)83 b(=)g Fq(P)1941 866 y Fp(0)2002 851 y Fm(\012)23 b Fq(P)2165 866 y Fp(\012)2220 851 y Fq(;)1070 b Fs(\(3.23\))1534 996 y Fq(P)1597 1011 y Fp(0)1719 996 y Fs(=)83 b Fq(P)14 b Fs(\()p Fq(L)2059 1011 y Fo(p)2126 996 y Fs(=)28 b(0\))p Fq(;)973 b Fs(\(3.24\))p 1563 1062 74 4 v 1563 1142 a(\005)83 b(=)g(1)-22 b(l)21 b Fm(\000)h Fs(\005)p Fq(:)1164 b Fs(\(3.25\))328 1362 y(W)-8 b(e)33 b(also)f(in)m(tro)s(duce)g(the)h(notation)p 1740 1502 59 4 v 1723 1582 a Fq(R)1799 1597 y Fo(\017)1859 1582 y Fs(=)p 1963 1502 74 4 v 28 w(\005)p Fq(R)2110 1597 y Fo(\017)2143 1582 y Fq(:)474 1802 y Fs(Notice)d(that)f(the)h(op) s(erator)f Fq(A)1614 1817 y Fp(0)1684 1802 y Fs(satis\014es)h(the)g (conditions)f(giv)m(en)h(in)f(Theorem)h(3.3)328 1922 y(with)i Fq(n)608 1937 y Fp(0)675 1922 y Fs(=)c(1.)43 b(Moreo)m(v)m(er,)34 b([)p Fq(L)1446 1937 y Fo(;)1470 1922 y Fq(A)1543 1937 y Fp(0)1583 1922 y Fs(])27 b(=)h Fq(LA)1880 1937 y Fp(0)1942 1922 y Fm(\000)22 b Fq(A)2114 1937 y Fp(0)2153 1922 y Fq(L)33 b Fs(extends)h(to)e(a)g(b)s(ounded)h (op)s(erator)328 2042 y(on)f(the)h(en)m(tire)g(Hilb)s(ert)e(space,)j (and)1329 2316 y Fm(k)p Fs([)p Fq(L;)17 b(A)1589 2331 y Fp(0)1629 2316 y Fs(])p Fm(k)27 b(\024)i Fq(k)1909 2176 y Fi(\022)1992 2249 y Fq(\022)s Fm(j)p Fq(\025)p Fm(j)p 1992 2293 161 4 v 2053 2384 a Fq(\017)2185 2316 y Fs(+)2293 2249 y Fq(\022)s(\025)2398 2212 y Fp(2)p 2293 2293 145 4 v 2326 2384 a Fq(\017)2365 2356 y Fp(2)2447 2176 y Fi(\023)2537 2316 y Fq(:)753 b Fs(\(3.26\))474 2593 y(This)30 b(c)m(hoice)g(for)e(the)i(op)s(erator)f Fq(A)1754 2608 y Fp(0)1823 2593 y Fs(w)m(as)h(initially)c(in)m(tro)s (duced)j(in)g([BFSS])g(for)g(the)328 2714 y(sp)s(ectral)46 b(analysis)g(of)h(P)m(auli-Fierz)d(Hamiltonians)g(\(zero)j(temp)s (erature)f(systems\),)328 2834 y(and)31 b(w)m(as)g(adopted)g(in)f([M])i (to)e(sho)m(w)i(return)f(to)f(equilibrium)e(\(p)s(ositiv)m(e)i(temp)s (erature)328 2954 y(systems\).)45 b(The)33 b(k)m(ey)h(feature)f(of)f Fq(A)1654 2969 y Fp(0)1727 2954 y Fs(is)g(that)1375 3192 y Fq(i)p Fs(\005[)p Fq(L;)17 b(A)1691 3207 y Fp(0)1731 3192 y Fs(]\005)28 b(=)g(2)p Fq(\022)s(\025)2117 3151 y Fp(2)2156 3192 y Fs(\005)p Fq(I)p 2297 3112 59 4 v 8 w(R)2355 3130 y Fp(2)2355 3216 y Fo(\017)2395 3192 y Fq(I)8 b Fs(\005)328 3412 y(is)42 b(a)g(non-negativ)m(e)g(op)s (erator.)73 b(The)43 b(F)-8 b(ermi)41 b(Golden)g(Rule)h(Condition,)h (\(2.20\))f(\(or)328 3532 y(\(2.23\)\),)32 b(sa)m(ys)i(that)e(it)g(is)g (a)g FB(strictly)k(p)-5 b(ositive)34 b(op)-5 b(er)g(ator)32 b Fs(on)h(Ran)16 b(\005.)328 3736 y Ft(Prop)s(osition)35 b(3.2)49 b FB(Assume)35 b(\(2.20\))f(and)g(let)h Fs(0)28 b Fq(<)f(\017)i(<)e(\017)2499 3751 y Fp(0)2539 3736 y FB(.)45 b(Then)1587 4000 y Fs(\005)p Fq(I)p 1727 3920 V 7 w(R)1786 3938 y Fp(2)1786 4025 y Fo(\017)1825 4000 y Fq(I)8 b Fs(\005)28 b Fm(\025)2092 3933 y Fs(1)p 2092 3977 49 4 v 2097 4068 a Fq(\017)2151 4000 y(\015)5 b Fs(\005)p Fq(;)1010 b Fs(\(3.27\))328 4251 y FB(wher)-5 b(e)46 b Fq(\015)51 b FB(is)46 b(given)g(by)g(\(2.21\).)78 b(Assuming)46 b(c)-5 b(ondition)46 b(\(2.23\))f(inste)-5 b(ad)46 b(of)g(\(2.20\),)328 4371 y(the)37 b(same)g(lower)g(b)-5 b(ound)37 b(holds)f(\(with)h Fq(\015)43 b FB(given)36 b(in)h(\(2.24\)\))g(if)g(we)g(r)-5 b(eplac)g(e)36 b Fs(\005)i FB(by)f Fs(\005)p Fq(P)328 4492 y FB(\()p Fq(P)k Fs(=)27 b Fq(p)c Fm(\012)f Fq(p)p FB(,)35 b Fq(p)28 b Fs(=)f Fq(p)1089 4507 y Fo(J)1128 4519 y Fl(d)1190 4492 y Fs(+)22 b Fq(p)1337 4507 y Fo(J)1376 4515 y Fl(c)1412 4492 y FB(\))34 b(and)h Fq(I)42 b FB(by)35 b(the)g(r)-5 b(e)g(gularize)g(d)35 b(inter)-5 b(action;)34 b(se)-5 b(e)34 b(\(2.13\).)474 4695 y Fs(The)g(pro)s(of)d(of)i(Prop)s(osition)d(3.2)j(is)f(giv)m(en)g (in)g(Section)h(A.4.)328 4816 y(Next,)45 b(w)m(e)e(de\014ne)g(the)f(op) s(erators)g(\003)f(and)h Fq(A)g Fs(and)g(v)m(erify)g(the)g(h)m(yp)s (otheses)j(used)e(in)328 4936 y(Section)32 b(3.1.)1898 5214 y(27)p eop %%Page: 28 28 28 27 bop 328 631 a Ft(3.3.1)112 b(Setting)37 b(for)g(Theorem)g(2.1)328 816 y Fs(W)-8 b(e)33 b(de\014ne)858 1036 y(\003)82 b(=)h(\003)1235 1051 y Fo(p)1297 1036 y Fm(\012)22 b Fs(1)-22 b(l)1451 1051 y Fo(p)1512 1036 y Fm(\012)23 b Fs(1)-22 b(l)1667 1051 y Fo(f)1733 1036 y Fs(+)22 b(1)-22 b(l)1886 1051 y Fo(p)1946 1036 y Fm(\012)23 b Fs(\003)2114 1051 y Fo(p)2175 1036 y Fm(\012)g Fs(1)-22 b(l)2330 1051 y Fo(f)2396 1036 y Fs(+)22 b(1)-22 b(l)2549 1051 y Fo(p)2610 1036 y Fm(\012)23 b Fs(1)-22 b(l)2765 1051 y Fo(p)2825 1036 y Fm(\012)23 b Fs(\003)2993 1051 y Fo(f)3038 1036 y Fq(;)252 b Fs(\(3.28\))818 1181 y(\003)886 1196 y Fo(p)1008 1181 y Fs(=)83 b Fq(H)1248 1196 y Fo(p)1310 1181 y Fm(\000)22 b Fq(E)1481 1196 y Fp(0)1543 1181 y Fs(+)g(1)p Fq(;)1600 b Fs(\(3.29\))812 1326 y(\003)880 1341 y Fo(f)1008 1326 y Fs(=)83 b(d\000\()p Fq(u)1376 1285 y Fp(2)1437 1326 y Fs(+)22 b(1\))p Fq(;)1668 b Fs(\(3.30\))328 1546 y(where,)35 b(w)m(e)g(recall,)d Fq(E)1143 1561 y Fp(0)1212 1546 y Fs(=)e(inf)22 b Fq(\033)t Fs(\()p Fq(H)1630 1561 y Fo(p)1670 1546 y Fs(\))29 b Fq(<)h Fs(0.)46 b(Clearly)-8 b(,)33 b(\003)g(is)g(essen)m(tially)g (selfadjoin)m(t)g(on)328 1667 y(the)40 b(domain)d Fm(D)42 b Fs(de\014ned)f(in)d(\(3.19\),)i(and)g(\003)2020 1682 y Fo(p)2098 1667 y Fm(\025)f Fs(1)-22 b(l,)40 b(\003)2404 1682 y Fo(f)2488 1667 y Fm(\025)g Fs(1)-22 b(l.)62 b(In)39 b(what)h(follo)m(ws)e(w)m(e)328 1787 y(shall)28 b(often)h(use)i(the)e (standard)h(fact)f(that)g(if)g Fq(f)38 b Fm(2)28 b Fq(L)2260 1751 y Fp(2)2300 1787 y Fs(\()p Fk(R)21 b Fm(\002)16 b Fq(S)2579 1751 y Fp(2)2623 1787 y Fq(;)h(du)f Fm(\002)g Fq(d)p Fs(\006\),)29 b(then)h Fq(a)3368 1751 y Fp(#)3431 1787 y Fs(\()p Fq(f)11 b Fs(\))328 1907 y(is)38 b(relativ)m(ely)g Fq(N)952 1871 y Fp(1)p Fo(=)p Fp(2)1102 1907 y Fs(b)s(ounded)h(in)g (the)g(sense)i(of)d(Kato.)62 b(This)39 b(implies)e(immediately)328 2028 y(that)32 b Fq(a)590 1992 y Fp(#)653 2028 y Fs(\()p Fq(f)11 b Fs(\))33 b(is)f(relativ)m(ely)f(\003)1412 1977 y Fp(1)p Fo(=)p Fp(2)1412 2056 y Fo(f)1554 2028 y Fs(b)s(ounded.)474 2148 y(W)-8 b(e)40 b(v)m(erify)f(that)g(\()p Fq(L;)17 b Fs(\003)p Fq(;)g Fm(D)s Fs(\))38 b(is)g(a)h(GJN)g(triple.)61 b(The)40 b(b)s(ound)f(\(3.1\))g(is)f(trivial)f(b)m(y)328 2269 y(the)28 b(ab)s(o)m(v)m(e)h(observ)-5 b(ation,)28 b(and)g(the)g(fact)g(that)f Fq(\034)2088 2284 y Fo(\014)2136 2269 y Fs(\()p Fq(g)2221 2284 y Fo(\013)2270 2269 y Fs(\))g Fm(2)i Fq(L)2496 2232 y Fp(2)2535 2269 y Fs(\()p Fk(R)18 b Fm(\002)12 b Fq(S)2807 2232 y Fp(2)2853 2269 y Fs(\).)41 b(Next,)30 b(the)e(only)328 2389 y(con)m(tribution)f(to)h(the)h(comm)m (utator)e(of)h Fq(L)h Fs(with)f(\003)g(comes)h(from)e(the)i(in)m (teraction,)g(and)328 2509 y(a)f(t)m(ypical)f(term)h(to)g(estimate)f (is)h(of)g(the)h(form)e([)p Fq(G)2145 2524 y Fo(\013)2194 2509 y Fq(;)17 b(H)2319 2524 y Fo(p)2358 2509 y Fs(])c Fm(\012)g Fs(1)-22 b(l)13 b Fm(\012)g Fq(')p Fs(\()p Fq(\034)2790 2524 y Fo(\014)2839 2509 y Fs(\()p Fq(g)2924 2524 y Fo(\013)2973 2509 y Fs(\)\))g(+)g Fq(G)3228 2524 y Fo(\013)3291 2509 y Fm(\012)g Fs(1)-22 b(l)3436 2524 y Fo(p)3488 2509 y Fm(\012)328 2630 y Fs([)p Fq(')p Fs(\()p Fq(\034)499 2645 y Fo(\014)546 2630 y Fs(\()p Fq(g)631 2645 y Fo(\013)681 2630 y Fs(\)\))p Fq(;)17 b Fs(\003)869 2645 y Fo(f)913 2630 y Fs(].)44 b(Using)32 b(the)h(b)s(ound)g (\(2.18\),)f(w)m(e)h(obtain)f(for)g(the)h(\014rst)g(term)710 2850 y Fm(j)o(h)p Fq( )t(;)17 b Fs([)p Fq(G)991 2865 y Fo(\013)1040 2850 y Fq(;)g(H)1165 2865 y Fo(p)1204 2850 y Fs(])23 b Fm(\012)f Fs(1)-22 b(l)1408 2865 y Fo(p)1469 2850 y Fm(\012)22 b Fq(')p Fs(\()p Fq(\034)1712 2865 y Fo(\014)1760 2850 y Fs(\()p Fq(g)1845 2865 y Fo(\013)1894 2850 y Fs(\)\))p Fq( )t Fm(ij)793 3012 y(\024)83 b Fq(k)s Fm(k)p Fs([)p Fq(G)1161 3027 y Fo(\013)1211 3012 y Fq(;)17 b(H)1336 3027 y Fo(p)1375 3012 y Fs(]\()p Fq(H)1521 3027 y Fo(p)1582 3012 y Fm(\000)23 b Fq(E)1754 3027 y Fp(0)1816 3012 y Fs(+)f(1\))2001 2970 y Fx(\000)p Fp(1)p Fo(=)p Fp(2)2165 3012 y Fm(k)33 b(k)p Fs(\003)2366 2970 y Fp(1)p Fo(=)p Fp(2)2366 3036 y Fo(p)2497 3012 y Fm(\012)23 b Fs(1)-22 b(l)2652 3027 y Fo(p)2690 3012 y Fq( )t Fm(k)33 b(k)p Fs(\003)2958 2961 y Fp(1)p Fo(=)p Fp(2)2958 3039 y Fo(f)3067 3012 y Fq( )t Fm(k)793 3157 y(\024)83 b Fq(k)20 b Fm(h)p Fq( )t(;)d Fs(\003)p Fq( )s Fm(i)f Fq(:)1927 b Fs(\(3.31\))328 3377 y(Next,)328 3597 y Fm(jh)p Fq( )t(;)17 b(G)583 3612 y Fo(\013)654 3597 y Fm(\012)23 b Fs(1)-22 b(l)809 3612 y Fo(p)869 3597 y Fm(\012)23 b Fs([)p Fq(')p Fs(\()p Fq(\034)1140 3612 y Fo(\014)1187 3597 y Fs(\()p Fq(g)1272 3612 y Fo(\013)1322 3597 y Fs(\)\))p Fq(;)17 b Fs(\003)1510 3612 y Fo(f)1554 3597 y Fs(])p Fq( )t Fm(ij)83 b(\024)g Fq(k)s Fm(k)p Fs(\()p Fq(u)2156 3556 y Fp(2)2217 3597 y Fs(+)22 b(1\))2402 3556 y Fp(1)p Fo(=)p Fp(2)2512 3597 y Fq(\034)2554 3612 y Fo(\014)2601 3597 y Fs(\()p Fq(g)2686 3612 y Fo(\013)2736 3597 y Fs(\))p Fm(k)2824 3614 y Fo(L)2872 3595 y Fj(2)2906 3614 y Fp(\()p Fd(R)p Fx(\002)p Fo(S)3083 3595 y Fj(2)3117 3614 y Fp(\))3181 3597 y Fm(k)p Fs(\003)3299 3556 y Fp(1)p Fo(=)p Fp(2)3409 3597 y Fq( )t Fm(k)3526 3556 y Fp(2)1798 3742 y Fm(\024)83 b Fq(k)20 b Fm(h)p Fq( )t(;)d Fs(\003)p Fq( )s Fm(i)f Fq(;)922 b Fs(\(3.32\))328 3962 y(where)23 b(w)m(e)g(ha)m(v)m(e)g(used) g(that)e([)p Fq(a)1436 3926 y Fx(\003)1476 3962 y Fs(\()p Fq(\034)1556 3977 y Fo(\014)1603 3962 y Fs(\()p Fq(g)1688 3977 y Fo(\013)1737 3962 y Fs(\)\))p Fq(;)c Fs(\003)1925 3977 y Fo(f)1970 3962 y Fs(])28 b(=)f Fq(a)2179 3926 y Fx(\003)2236 3962 y Fs(\(\()p Fq(u)2368 3926 y Fp(2)2429 3962 y Fs(+)22 b(1\))p Fq(\034)2656 3977 y Fo(\014)2703 3962 y Fs(\()p Fq(g)2788 3977 y Fo(\013)2837 3962 y Fs(\)\))g(and)f([)q Fq(a)p Fs(\()p Fq(\034)3272 3977 y Fo(\014)3319 3962 y Fs(\()p Fq(g)3404 3977 y Fo(\013)3453 3962 y Fs(\)\))p Fq(;)c Fs(\003)3641 3977 y Fo(f)3686 3962 y Fs(])28 b(=)328 4083 y Fm(\000)p Fq(a)17 b Fs(\(\()p Fq(u)605 4046 y Fp(2)666 4083 y Fs(+)22 b(1\))p Fq(\034)893 4098 y Fo(\014)940 4083 y Fs(\()p Fq(g)1025 4098 y Fo(\013)1075 4083 y Fs(\)\))o(,)27 b(so)e(that)g([)p Fq(')p Fs(\()p Fq(\034)1691 4098 y Fo(\014)1738 4083 y Fs(\()p Fq(g)1823 4098 y Fo(\013)1872 4083 y Fs(\)\))p Fq(;)17 b Fs(\003)2060 4098 y Fo(f)2105 4083 y Fs(])25 b(is)g(still)d Fq(N)2518 4046 y Fp(1)p Fo(=)p Fp(2)2654 4083 y Fs(b)s(ounded,)27 b(since)e Fq(\034)3346 4098 y Fo(\014)3394 4083 y Fs(\()p Fq(g)3479 4098 y Fo(\013)3528 4083 y Fs(\))328 4203 y(has)33 b(the)h(deca)m(y)g(prop)s(ert)m(y)g (\(2.17\).)44 b(The)34 b(form)e(b)s(ound)h(\(3.2\))g(follo)m(ws)e(from) h(these)i(ob-)328 4323 y(serv)-5 b(ations.)43 b(In)33 b(a)g(similar)c(w)m(a)m(y)-8 b(,)34 b(one)f(sho)m(ws)h(that)e(\()p Fq(D)s(;)17 b Fs(\003)p Fq(;)g Fm(D)s Fs(\))31 b(is)h(a)g(GJN)h (triple.)474 4564 y(Next,)41 b(w)m(e)e(de\014ne)g(the)g(op)s(erator)e Fq(A)g Fm(\021)h Fq(A)2053 4579 y Fo(f)2137 4564 y Fs(to)f(b)s(e)i(the) f(selfadjoin)m(t)f(generator)h(of)328 4684 y(the)33 b(translation)e (group)h(acting)g(on)g(the)h(radial)e(v)-5 b(ariable)30 b(of)i(elemen)m(ts)h(in)f Fm(F)42 b Fs(b)m(y)488 4904 y([)p Fq(e)560 4863 y Fo(itA)662 4875 y Fl(f)706 4904 y Fq( )t Fs(])800 4919 y Fo(n)847 4904 y Fs(\()p Fq(u)941 4919 y Fp(1)980 4904 y Fq(;)17 b Fs(\006)1094 4919 y Fp(1)1134 4904 y Fq(;)g(:)g(:)g(:)f(;)h(u)1409 4919 y Fo(n)1455 4904 y Fq(;)g Fs(\006)1569 4919 y Fo(n)1616 4904 y Fs(\))28 b(=)f([)p Fq( )t Fs(])1906 4919 y Fo(n)1953 4904 y Fs(\()p Fq(u)2047 4919 y Fp(1)2108 4904 y Fm(\000)c Fq(t;)17 b Fs(\006)2357 4919 y Fp(1)2397 4904 y Fq(;)g(:)g(:)g(:)f(;)h (u)2672 4919 y Fo(n)2740 4904 y Fm(\000)23 b Fq(t;)17 b Fs(\006)2989 4919 y Fo(n)3036 4904 y Fs(\))p Fq(;)49 b(t)28 b Fm(2)g Fk(R)5 b Fq(:)1898 5214 y Fs(28)p eop %%Page: 29 29 29 28 bop 328 631 a Fs(In)37 b(what)g(follo)m(ws,)g(w)m(e)h(will)c (often)j(not)g(displa)m(y)f(the)i(angular)d(v)-5 b(ariables)36 b(\006)3163 646 y Fp(1)3203 631 y Fq(;)17 b(:)g(:)g(:)f(;)h Fs(\006)3492 646 y Fo(n)3539 631 y Fs(.)328 751 y(W)-8 b(e)31 b(set)h Fq(e)690 715 y Fo(itA)792 727 y Fl(f)837 751 y Fs(\012)c(:=)g(\012.)43 b(Clearly)-8 b(,)31 b Fm(D)f(\032)e(D)s Fs(\()p Fq(A)1966 766 y Fo(f)2011 751 y Fs(\),)j Fm(D)j Fs(is)d(in)m(v)-5 b(arian)m(t)30 b(under)i Fq(e)3043 715 y Fo(itA)3145 727 y Fl(f)3189 751 y Fs(,)f(hence)i(a)328 872 y(core)g(for)f Fq(A)756 887 y Fo(f)801 872 y Fs(,)h(and)g Fq(A)1124 887 y Fo(f)1202 872 y Fs(acts)g(on)f Fm(D)j Fs(as)1648 1084 y Fq(A)1721 1099 y Fo(f)1794 1084 y Fs(=)28 b(d\000\()p Fq(i@)2135 1099 y Fo(u)2181 1084 y Fs(\))p Fq(:)1071 b Fs(\(3.33\))328 1295 y(An)33 b(easy)g(calculation)e(sho)m (ws)j(that,)e(on)h Fm(D)s Fs(,)1223 1507 y(\003)p Fq(e)1336 1466 y Fo(itA)1438 1478 y Fl(f)1510 1507 y Fs(=)27 b Fq(e)1658 1466 y Fo(itA)1760 1478 y Fl(f)1821 1427 y Fi(\000)1867 1507 y Fs(\003)22 b(+)g(d\000\(2)p Fq(ut)f Fm(\000)i Fq(t)2504 1466 y Fp(2)2544 1507 y Fs(\))2582 1427 y Fi(\001)2644 1507 y Fq(;)646 b Fs(\(3.34\))328 1719 y(so)38 b(estimate)g(\(3.5\),)h(with)f Fq(k)1399 1683 y Fx(0)1459 1719 y Fs(=)g(0,)h(is)f(satis\014ed)h(for)e(all)f Fq( )42 b Fm(2)37 b(D)s Fs(,)j(hence)f(for)f(all)e Fq( )41 b Fm(2)328 1839 y(D)s Fs(\(\003\).)i(F)-8 b(or)31 b Fq(A)d Fs(:=)g Fq(A)1101 1854 y Fo(f)1146 1839 y Fs(,)33 b(w)m(e)h(\014nd)f (that)1575 2051 y Fq(C)1645 2066 y Fp(1)1768 2051 y Fs(=)83 b Fq(N)32 b Fs(+)22 b Fq(\025I)2235 2066 y Fp(1)2275 2051 y Fq(;)1015 b Fs(\(3.35\))1575 2197 y Fq(C)1645 2212 y Fp(2)1768 2197 y Fs(=)83 b Fq(\025I)2027 2212 y Fp(2)2066 2197 y Fq(;)1224 b Fs(\(3.36\))1575 2342 y Fq(C)1645 2357 y Fp(3)1768 2342 y Fs(=)83 b Fq(\025I)2027 2357 y Fp(3)2066 2342 y Fq(;)1224 b Fs(\(3.37\))328 2554 y(where)964 2766 y Fq(I)1007 2781 y Fo(n)1081 2766 y Fs(=)28 b Fq(i)1218 2724 y Fo(n)1282 2671 y Fi(X)1331 2880 y Fo(\013)1442 2685 y Fi(\010)1500 2766 y Fq(G)1577 2781 y Fo(\013)1649 2766 y Fm(\012)22 b Fs(1)-22 b(l)1803 2781 y Fo(p)1864 2766 y Fm(\012)23 b Fq(')p Fs(\(\()p Fm(\000)p Fq(i@)2265 2781 y Fo(u)2311 2766 y Fs(\))2349 2724 y Fo(n)2396 2766 y Fq(\034)2438 2781 y Fo(\014)2485 2766 y Fs(\()p Fq(g)2570 2781 y Fo(\013)2619 2766 y Fs(\)\))1130 3014 y Fm(\000)p Fs(1)-22 b(l)1262 3029 y Fo(p)1323 3014 y Fm(\012)22 b(C)1474 3029 y Fo(p)1514 3014 y Fq(G)1591 3029 y Fo(\013)1641 3014 y Fm(C)1693 3029 y Fo(p)1755 3014 y Fm(\012)h Fq(')p Fs(\(\()p Fm(\000)p Fq(i@)2156 3029 y Fo(u)2202 3014 y Fs(\))2240 2973 y Fo(n)2287 3014 y Fq(e)2332 2973 y Fx(\000)p Fo(\014)s(u=)p Fp(2)2545 3014 y Fq(\034)2587 3029 y Fo(\014)2635 3014 y Fs(\()p Fq(g)2720 3029 y Fo(\013)2769 3014 y Fs(\)\))2845 2934 y Fi(\011)2903 3014 y Fq(:)387 b Fs(\(3.38\))474 3226 y(W)-8 b(e)31 b(no)m(w)g(sho)m(w)g(that)f(\()p Fq(C)1397 3241 y Fo(n)1443 3226 y Fq(;)17 b Fs(\003)p Fq(;)g Fm(D)s Fs(\))29 b(are)h(GJN)g(triples,)g(for)g Fq(n)e Fs(=)f(1)p Fq(;)17 b Fs(2)p Fq(;)g Fs(3.)41 b(The)32 b(op)s(era-)328 3347 y(tors)d Fq(I)563 3362 y Fo(n)638 3347 y Fs(are)g Fq(N)885 3310 y Fp(1)p Fo(=)p Fp(2)995 3347 y Fs(-b)s(ounded,)h(since)f Fq(\034)1727 3362 y Fo(\014)1774 3347 y Fs(\()p Fq(g)1859 3362 y Fo(\013)1908 3347 y Fs(\))g(and)g Fq(e)2206 3310 y Fx(\000)p Fo(\014)s(u=)p Fp(2)2419 3347 y Fq(\034)2461 3362 y Fo(\014)2509 3347 y Fs(\()p Fq(g)2594 3362 y Fo(\013)2643 3347 y Fs(\))f(are)h(in)f(in)f(the)i(domain)328 3467 y(of)k(the)i(op)s(erators)e(\()p Fq(i@)1164 3482 y Fo(u)1210 3467 y Fs(\))1248 3431 y Fo(n)1295 3467 y Fs(,)h Fq(n)c Fs(=)g(1)p Fq(;)17 b Fs(2)p Fq(;)g Fs(3)32 b(\(see)j(also)e (\(2.16\)\),)g(hence)i(\(3.1\))f(holds.)47 b(Note)328 3587 y(that)28 b(this)g(also)f(yields)h(\(3.7\).)42 b(Next,)30 b(w)m(e)g(need)f(to)f(calculate)f(the)i(comm)m(utators)e(of)h Fq(C)3519 3602 y Fo(n)328 3708 y Fs(with)38 b(\003.)59 b(The)39 b(estimates)f(on)g(the)g(comm)m(utators)f(of)h Fq(I)2419 3723 y Fo(n)2504 3708 y Fs(with)f(\003,)j(for)d Fq(n)g Fs(=)g(2)p Fq(;)17 b Fs(3,)39 b(are)328 3828 y(similar)27 b(to)j(the)g(ones)h(for)f Fq(n)e Fs(=)f(1.)42 b(The)32 b(latter)d(has)h(b)s(een)h(outlined)e(ab)s(o)m(v)m(e;)j(it)d(requires) 328 3948 y(that)i(\()p Fq(u)632 3912 y Fp(2)691 3948 y Fs(+)20 b(1\)\()p Fm(\000)p Fq(i@)1073 3963 y Fo(u)1119 3948 y Fs(\))1157 3912 y Fo(n)1204 3948 y Fq(\034)1246 3963 y Fo(\014)1293 3948 y Fs(\()p Fq(g)1378 3963 y Fo(\013)1427 3948 y Fs(\))28 b Fm(2)g Fq(L)1653 3912 y Fp(2)1693 3948 y Fs(\()p Fk(R)j Fm(\002)20 b Fq(S)1986 3912 y Fp(2)2025 3948 y Fs(\),)32 b(whic)m(h)g(is)f(guaran)m(teed)h(b)m(y)h(conditions) 328 4069 y(\(2.16\))f(and)g(\(2.17\).)474 4189 y(This)39 b(discussion)h(sho)m(ws)g(that)f(w)m(e)h(are)f(in)g(the)g(situation)f (describ)s(ed)i(in)e(Section)328 4310 y(3.1,)32 b(and)h(Theorems)g (3.2,)f(3.3)h(apply)-8 b(.)328 4568 y Ft(3.3.2)112 b(Setting)37 b(for)g(Theorem)g(2.3)328 4753 y Fs(W)-8 b(e)33 b(de\014ne)h(the)f(op)s (erator)f(\003)g(as)h(in)e(\(3.28\),)h(but)h(where)h(\003)2510 4768 y Fo(p)2582 4753 y Fs(is)e(no)m(w)h(giv)m(en)g(b)m(y)1627 4965 y(\003)1695 4980 y Fo(p)1762 4965 y Fs(=)28 b Fm(\000)p Fs(\001)23 b(+)f Fq(x)2200 4924 y Fp(2)2240 4965 y Fq(:)1050 b Fs(\(3.39\))1898 5214 y(29)p eop %%Page: 30 30 30 29 bop 328 631 a Fs(\003)32 b(is)g(essen)m(tially)g(selfadjoin)m(t)g (on)g Fm(D)j Fs(\(see)f(\(3.19\)\),)e(\003)2321 646 y Fo(p)2388 631 y Fm(\025)c Fs(1)-22 b(l,)31 b(\003)2674 646 y Fo(f)2747 631 y Fm(\025)d Fs(1)-22 b(l.)474 751 y(V)-8 b(erifying)36 b(that)h(\()p Fq(L)1216 766 y Fo(J)1265 751 y Fq(;)17 b Fs(\003)p Fq(;)g Fm(D)s Fs(\))35 b(is)i(a)f(GJN)h (triple)e(is)i(done)g(as)g(in)g(Subsection)g(3.3.1,)328 872 y(using)k(that)f([)p Fq(G)915 887 y Fo(\013;J)1029 872 y Fq(;)17 b Fs(\003)1141 887 y Fo(p)1180 872 y Fs(])41 b(is)g(b)s(ounded;)46 b(see)c(Lemma)e(A.5.)68 b(It)41 b(is)g(also)f(easy)i(to)e(c)m(hec)m(k)328 992 y(that)32 b(\()p Fq(D)s(;)17 b Fs(\003)p Fq(;)g Fm(D)s Fs(\))31 b(is)h(a)h(GJN)f(triple.)474 1233 y(Next,)52 b(w)m(e)d(de\014ne)g(an)e (op)s(erator)g Fq(A)h Fs(di\013ering)e(substan)m(tially)g(from)g(the)i (c)m(hoice)328 1353 y Fq(A)36 b Fs(=)g Fq(A)622 1368 y Fo(f)704 1353 y Fs(\(see)j(\(3.33\)\))d(in)h(Subsection)h(3.3.1:)52 b(W)-8 b(e)38 b(add)g(a)f(\(regularized\))f(dilatation)328 1474 y(on)c(the)h(particle)f(space)h(to)g Fq(A)1437 1489 y Fo(f)1482 1474 y Fs(.)474 1594 y(Let)i Fq(\037)30 b Fm(2)h Fq(C)916 1558 y Fx(1)909 1619 y Fp(0)990 1594 y Fs(\()p Fk(R)1094 1609 y Fp(+)1159 1594 y Fs(\))j(b)s(e)h(a)f(smo)s (oth)f(c)m(haracteristic)h(function)f(of)h(the)h(set)f Fq(J)3269 1609 y Fo(c)3338 1594 y Fs(\(with)328 1714 y(the)f(prop)s(ert)m(y)g Fq(\037)p Fm(j)983 1729 y Fo(J)1022 1737 y Fl(c)1085 1714 y Fs(=)28 b(1\),)k(whic)m(h)h(has)g(compact)f (supp)s(ort)h(not)f(con)m(taining)f(zero.)44 b(W)-8 b(e)328 1835 y(de\014ne)39 b Fq(\037)p Fs(\()p Fq(H)795 1850 y Fo(p)835 1835 y Fs(\))e(=)1023 1755 y Fi(R)1117 1835 y Fs(^)-60 b Fq(\037)p Fs(\()p Fq(s)p Fs(\))p Fq(e)1334 1799 y Fo(isH)1449 1807 y Fl(p)1489 1835 y Fs(,)39 b(where)51 b(^)-60 b Fq(\037)38 b Fs(is)g(the)g(F)-8 b(ourier)37 b(transform)g(of)h Fq(\037)p Fs(,)h(and)g(w)m(e)328 1955 y(abbreviate)31 b Fq(\037)p Fs(\()p Fq(H)986 1970 y Fo(p)1026 1955 y Fs(\))g(b)m(y)h Fq(\037)p Fs(.)43 b(Let)31 b Fq(A)1606 1970 y Fo(p)1677 1955 y Fs(b)s(e)g(the)h(symmetric)e(op)s(erator)h(on)g Fq(C)3051 1919 y Fx(1)3044 1980 y Fp(0)3125 1955 y Fs(\()p Fk(R)3229 1919 y Fp(3)3275 1955 y Fs(\))g(giv)m(en)328 2076 y(b)m(y)1430 2236 y Fq(A)1503 2251 y Fo(p)1570 2236 y Fs(=)d Fm(\000)1769 2169 y Fq(i)p 1761 2213 49 4 v 1761 2304 a Fs(4)1820 2236 y(\()p Fq(x)23 b Fm(\001)e(r)h Fs(+)g Fm(r)h(\001)e Fq(x)p Fs(\))p Fq(:)854 b Fs(\(3.40\))328 2443 y(Notice)32 b(that)g(\()p Fq(A)956 2458 y Fo(p)996 2443 y Fq(;)17 b Fs(\003)1108 2458 y Fo(p)1147 2443 y Fq(;)g(C)1268 2407 y Fx(1)1261 2468 y Fp(0)1343 2443 y Fs(\()p Fk(R)1446 2407 y Fp(3)1492 2443 y Fs(\)\))32 b(is)g(a)g(GJN)h(triple,)e(so)i Fq(A)2492 2458 y Fo(p)2564 2443 y Fs(is)f(essen)m(tially)g(selfadjoin)m(t)328 2563 y(on)g Fq(C)540 2527 y Fx(1)533 2588 y Fp(0)615 2563 y Fs(\()p Fk(R)719 2527 y Fp(3)764 2563 y Fs(\).)44 b(W)-8 b(e)33 b(denote)g(the)g(selfadjoin)m(t)e(closure)i(again)e(b)m(y)j Fq(A)2787 2578 y Fo(p)2827 2563 y Fs(.)474 2684 y FB(R)-5 b(emark.)93 b Fs(T)-8 b(o)37 b(sho)m(w)h(that)e Fq(A)1606 2699 y Fo(p)1682 2684 y Fs(is)g(essen)m(tially)h(selfadjoin)m(t)e(on)h Fq(C)2945 2648 y Fx(1)2938 2708 y Fp(0)3020 2684 y Fs(\()p Fk(R)3124 2648 y Fp(3)3169 2684 y Fs(\),)i(w)m(e)f(can)328 2804 y(also)d(use)i(the)g(fact)f(that)g(the)h(dense)h(set)f Fq(C)1949 2768 y Fx(1)1942 2829 y Fp(0)2023 2804 y Fs(\()p Fk(R)2127 2768 y Fp(3)2173 2804 y Fs(\))f(is)g(in)m(v)-5 b(arian)m(t)33 b(under)j(the)g(group)f(of)328 2925 y(dilatations)e(on)i Fq(L)1018 2888 y Fp(2)1058 2925 y Fs(\()p Fk(R)1161 2888 y Fp(3)1207 2925 y Fq(;)17 b(d)1302 2888 y Fp(3)1341 2925 y Fq(x)p Fs(\),)36 b(hence)h(a)e(core)g(for)g(the)h(selfadjoin)m (t)e(generator)h(of)g(this)328 3045 y(group.)43 b(The)34 b(generator)e(acts)h(on)g Fq(C)1690 3009 y Fx(1)1683 3070 y Fp(0)1797 3045 y Fs(as)g(in)f(\(3.40\).)328 3244 y Ft(Prop)s(osition)j(3.3)49 b Fs(\()p Fq(\037A)1297 3259 y Fo(p)1337 3244 y Fq(\037;)17 b Fs(\003)1510 3259 y Fo(p)1550 3244 y Fq(;)g(C)1671 3207 y Fx(1)1664 3268 y Fp(0)1745 3244 y Fs(\()p Fk(R)1849 3207 y Fp(3)1894 3244 y Fs(\)\))35 b FB(is)g(a)g(GJN)h(triple.)45 b(In)35 b(p)-5 b(articular,)35 b Fq(\037A)3465 3259 y Fo(p)3505 3244 y Fq(\037)328 3364 y FB(is)c(wel)5 b(l)31 b(de\014ne)-5 b(d)30 b(and)h(symmetric)g(on)g Fq(C)1820 3328 y Fx(1)1813 3389 y Fp(0)1894 3364 y Fs(\()p Fk(R)1998 3328 y Fp(3)2044 3364 y Fs(\))p FB(,)h(and)e(it)i(is)f(essential)5 b(ly)31 b(selfadjoint)f(on)328 3484 y Fq(C)405 3448 y Fx(1)398 3509 y Fp(0)480 3484 y Fs(\()p Fk(R)583 3448 y Fp(3)629 3484 y Fs(\))p FB(.)45 b(We)35 b(denote)f(the)h(selfadjoint)f(closur)-5 b(e)34 b(again)g(by)h Fq(\037A)2715 3499 y Fo(p)2755 3484 y Fq(\037)p FB(.)328 3683 y Fs(W)-8 b(e)33 b(giv)m(e)f(the)h(pro)s (of)f(in)g(Section)g(A.2.)44 b(Let)33 b(us)g(no)m(w)g(de\014ne)h(the)f (op)s(erator)1201 3897 y Fq(A)28 b Fs(=)f Fq(\037A)1539 3912 y Fo(p)1579 3897 y Fq(\037)22 b Fm(\012)h Fs(1)-22 b(l)1817 3912 y Fo(p)1877 3897 y Fm(\000)23 b Fs(1)-22 b(l)2032 3912 y Fo(p)2093 3897 y Fm(\012)22 b Fq(\037A)2326 3912 y Fo(p)2366 3897 y Fq(\037)g Fs(+)g Fq(A)2620 3912 y Fo(f)2666 3897 y Fq(;)624 b Fs(\(3.41\))328 4112 y(whic)m(h)26 b(is)f(essen)m(tially)g(selfadjoin)m(t)g(on)g Fm(D)s Fs(.)41 b(It)25 b(follo)m(ws)g(immediately)d(from)i(Prop)s(osition)328 4232 y(3.3,)38 b(Theorem)f(A.1,)h(and)f(relation)e(\(3.34\),)i(that)g Fq(e)2279 4196 y Fo(itA)2423 4232 y Fs(lea)m(v)m(es)h Fm(D)s Fs(\(\003\))e(in)m(v)-5 b(arian)m(t,)37 b(and)328 4353 y(that)32 b(the)h(estimate)f(\(3.5\))g(holds)g(true.)44 b(W)-8 b(e)33 b(calculate)f(explicitly)494 4581 y Fq(C)564 4596 y Fo(n)639 4581 y Fs(=)27 b Fq(\016)785 4596 y Fo(n;)p Fp(1)887 4581 y Fq(N)33 b Fs(+)22 b(ad)1199 4530 y Fp(\()p Fo(n)p Fp(\))1199 4608 y Fo(\037A)1296 4616 y Fl(p)1332 4608 y Fo(\037)1380 4581 y Fs(\()p Fq(H)1499 4596 y Fo(p)1538 4581 y Fs(\))g Fm(\012)h Fs(1)-22 b(l)1753 4596 y Fo(p)1814 4581 y Fs(+)22 b(\()p Fm(\000)p Fs(1\))2114 4539 y Fo(n)2161 4581 y Fs(1)-22 b(l)2216 4596 y Fo(p)2276 4581 y Fm(\012)23 b Fs(ad)2479 4530 y Fp(\()p Fo(n)p Fp(\))2479 4608 y Fo(\037A)2576 4616 y Fl(p)2612 4608 y Fo(\037)2660 4581 y Fs(\()p Fq(H)2779 4596 y Fo(p)2818 4581 y Fs(\))f(+)g Fq(\025I)3076 4596 y Fo(n)3123 4581 y Fq(;)167 b Fs(\(3.42\))328 4829 y(for)42 b Fq(n)j Fs(=)g(1)p Fq(;)17 b Fs(2)p Fq(;)g Fs(3,)44 b(where)f(w)m(e)h(de\014ne)g(the)f(m)m(ultiple)d(comm)m (utators)i(ad)3026 4778 y Fp(\(0\))3026 4856 y Fo(Y)3120 4829 y Fs(\()p Fq(X)8 b Fs(\))44 b(=)h Fq(X)8 b Fs(,)328 4965 y(and)34 b(for)f Fq(n)d Fm(\025)g Fs(1,)k(ad)1077 4914 y Fp(\()p Fo(n)p Fp(\))1077 4992 y Fo(Y)1179 4965 y Fs(\()p Fq(X)8 b Fs(\))29 b(=)h Fq(i)p Fs([ad)1642 4914 y Fp(\()p Fo(n)p Fx(\000)p Fp(1\))1642 4992 y Fo(Y)1834 4965 y Fs(\()p Fq(X)8 b Fs(\))p Fq(;)17 b(Y)k Fs(],)34 b(in)f(the)h(w)m(eak)h(sense)h(on)d Fq(C)3203 4929 y Fx(1)3196 4989 y Fp(0)3278 4965 y Fs(\()p Fk(R)3382 4929 y Fp(3)3427 4965 y Fs(\))23 b Fm(\002)1898 5214 y Fs(30)p eop %%Page: 31 31 31 30 bop 328 631 a Fq(C)405 595 y Fx(1)398 656 y Fp(0)480 631 y Fs(\()p Fk(R)583 595 y Fp(3)629 631 y Fs(\).)43 b(F)-8 b(or)32 b Fq(n)c Fs(=)g(1)p Fq(;)17 b Fs(2)p Fq(;)g Fs(3,)31 b(w)m(e)i(ha)m(v)m(e)h(de\014ned)328 897 y Fq(I)371 912 y Fo(n)501 897 y Fs(=)710 773 y Fo(n)660 803 y Fi(X)667 1015 y Fo(k)r Fp(=0)820 757 y Fi(\022)894 830 y Fq(n)896 966 y(k)952 757 y Fi(\023)1025 897 y Fs(2)1074 856 y Fx(\000)p Fp(\()p Fo(n)p Fx(\000)p Fo(k)r Fp(\))1341 803 y Fi(X)1390 1012 y Fo(\013)1501 787 y Fi(n)1568 897 y Fs(ad)1671 846 y Fp(\()p Fo(n)p Fx(\000)p Fo(k)r Fp(\))1671 924 y Fo(\037A)1768 932 y Fl(p)1804 924 y Fo(\037)1866 897 y Fs(\()p Fq(G)1981 912 y Fo(\013)2030 897 y Fs(\))22 b Fm(\012)h Fs(1)-22 b(l)2245 912 y Fo(p)2306 897 y Fm(\012)22 b Fq(')2486 817 y Fi(\000)2531 897 y Fs(\()p Fm(\000)p Fq(i@)2730 912 y Fo(u)2777 897 y Fs(\))2815 856 y Fo(k)2857 897 y Fq(\034)2899 912 y Fo(\014)2947 897 y Fs(\()p Fq(g)3032 912 y Fo(\013)3081 897 y Fs(\))3119 817 y Fi(\001)923 1173 y Fs(+\()p Fm(\000)p Fs(1\))1201 1131 y Fo(n)p Fx(\000)p Fo(k)1341 1173 y Fs(1)-22 b(l)1396 1188 y Fo(p)1457 1173 y Fm(\012)22 b Fs(ad)1659 1122 y Fp(\()p Fo(n)p Fx(\000)p Fo(k)r Fp(\))1659 1200 y Fo(\037A)1756 1208 y Fl(p)1792 1200 y Fo(\037)1854 1173 y Fs(\()p Fq(G)1969 1188 y Fo(\013)2019 1173 y Fs(\))g Fm(\012)g Fq(')2259 1092 y Fi(\000)2305 1173 y Fs(\()p Fm(\000)p Fq(i@)2504 1188 y Fo(u)2550 1173 y Fs(\))2588 1131 y Fo(k)2630 1173 y Fq(e)2675 1131 y Fo(\014)s(u=)p Fp(2)2834 1173 y Fq(\034)2876 1188 y Fo(\014)2924 1173 y Fs(\()p Fq(g)3009 1188 y Fo(\013)3058 1173 y Fs(\))3096 1092 y Fi(\001)3141 1062 y(o)3225 1173 y Fq(:)75 b Fs(\(3.43\))328 1385 y(Note)33 b(that)1355 1505 y(ad)1458 1454 y Fp(\(1\))1458 1532 y Fo(\037A)1555 1540 y Fl(p)1591 1532 y Fo(\037)1639 1505 y Fs(\()p Fq(H)1758 1520 y Fo(p)1798 1505 y Fs(\))27 b(=)h Fq(\037)p Fs(\()p Fq(H)2147 1520 y Fo(p)2208 1505 y Fs(+)22 b Fq(W)14 b Fs(\))p Fq(\037;)779 b Fs(\(3.44\))328 1667 y(with)654 1890 y Fq(W)41 b Fs(=)28 b Fm(\000)978 1822 y Fs(1)p 978 1867 49 4 v 978 1958 a(2)1054 1890 y(\()p Fq(x)22 b Fm(\001)g(r)p Fq(v)k Fs(+)c(2)p Fq(v)t Fs(\))27 b(=)1752 1822 y(1)p 1752 1867 V 1752 1958 a(2)1827 1749 y Fi(\022)1910 1822 y Fq(\032)1960 1786 y Fx(0)1984 1822 y Fs(\()p Fm(j)p Fq(x)p Fm(j)p Fs(\))p 1910 1867 261 4 v 1962 1958 a Fm(j)p Fq(x)p Fm(j)2073 1929 y Fo(\026)2203 1890 y Fs(+)22 b(\(1)g Fm(\000)g Fq(\026)p Fs(\))2621 1822 y Fq(\032)p Fs(\()p Fm(j)p Fq(x)p Fm(j)p Fs(\))p 2616 1867 248 4 v 2616 1958 a Fm(j)p Fq(x)p Fm(j)2727 1929 y Fp(1+)p Fo(\026)2873 1749 y Fi(\023)2963 1890 y Fq(;)327 b Fs(\(3.45\))328 2138 y(and)33 b(the)g(c)m(hoice)g(of)f Fq(\026)p Fs(,)g Fq(\032)h Fs(giv)m(en)g(in)e(\(2.2\))h(implies)e(that)1790 2328 y Fq(W)41 b Fm(\025)28 b Fs(0)p Fq(:)1213 b Fs(\(3.46\))328 2519 y(In)33 b(App)s(endix)g(A.3)f(w)m(e)i(pro)m(v)m(e)g(the)f(follo)m (wing)d(Prop)s(osition.)328 2713 y Ft(Prop)s(osition)35 b(3.4)49 b Fs(\()p Fq(C)1233 2728 y Fo(n)1280 2713 y Fq(;)17 b Fs(\003)p Fq(;)g Fm(D)s Fs(\))40 b FB(ar)-5 b(e)41 b(GJN)g(triples,)i(for)d Fq(n)g Fs(=)e(1)p Fq(;)17 b Fs(2)p Fq(;)g Fs(3)p FB(,)42 b(and)e(the)h(esti-)328 2834 y(mates)34 b(\(3.6\))g(-\(3.8\))g(ar)-5 b(e)35 b(satis\014e)-5 b(d.)328 3028 y Fs(This)29 b(sho)m(ws)h(that)f(with)f(the)i(c)m(hoice)f (of)f(op)s(erators)h(in)m(tro)s(duced)g(in)f(this)g(section,)i(The-)328 3148 y(orems)i(3.2)g(and)h(3.3)f(apply)-8 b(.)328 3477 y Fu(4)161 b(Pro)t(of)54 b(of)g(Theorem)e(2.1)328 3696 y Fs(1\))78 b(Set)676 3671 y Fi(e)668 3696 y Fq(I)711 3711 y Fp(1)789 3696 y Fs(=)39 b Fq(i)p Fs([)p Fq(I)8 b(;)17 b(A)1132 3711 y Fo(f)1177 3696 y Fs(],)41 b(where)g Fq(A)1634 3711 y Fo(f)1718 3696 y Fs(and)e Fq(I)47 b Fs(are)40 b(giv)m(en)f(in)f(\(3.33\),)j(\(2.45\).)62 b(F)-8 b(or)38 b Fq( )43 b Fm(2)328 3816 y(D)s Fs(\()p Fq(N)534 3780 y Fp(1)p Fo(=)p Fp(2)644 3816 y Fs(\),)33 b(w)m(e)g(ha)m(v)m(e)328 3920 y Fi(\014)328 3979 y(\014)328 4039 y(\014)361 3924 y(D)422 4034 y Fq( )t(;)541 4009 y Fi(e)533 4034 y Fq(I)576 4049 y Fp(1)615 4034 y Fq( )682 3924 y Fi(E)743 3920 y(\014)743 3979 y(\014)743 4039 y(\014)804 4034 y Fm(\024)909 3920 y Fi(\014)909 3979 y(\014)909 4039 y(\014)942 3924 y(D)1003 4034 y Fq( )t(;)1122 4009 y Fi(e)1114 4034 y Fq(I)1157 4049 y Fp(1)p 1196 3954 77 4 v 1196 4034 a Fq(P)1273 4049 y Fp(\012)1328 4034 y Fq( )1395 3924 y Fi(E)1456 3920 y(\014)1456 3979 y(\014)1456 4039 y(\014)1489 4034 y Fs(+)1565 3920 y Fi(\014)1565 3979 y(\014)1565 4039 y(\014)1598 3924 y(D)1659 4034 y Fq( )t(;)p 1770 3954 V 17 w(P)1846 4049 y Fp(\012)1910 4009 y Fi(e)1901 4034 y Fq(I)1944 4049 y Fp(1)1984 4034 y Fq(P)2047 4049 y Fp(\012)2102 4034 y Fq( )2169 3924 y Fi(E)2230 3920 y(\014)2230 3979 y(\014)2230 4039 y(\014)2290 4034 y Fm(\024)c Fs(2)p Fm(k)2504 4009 y Fi(e)2495 4034 y Fq(I)2538 4049 y Fp(1)2577 4034 y Fq(N)2665 3993 y Fx(\000)p Fp(1)p Fo(=)p Fp(2)p 2830 3954 V 2830 4034 a Fq(P)2906 4049 y Fp(\012)2962 4034 y Fm(kk)p Fq( )t Fm(k)21 b(k)p Fq(N)3338 3993 y Fp(1)p Fo(=)p Fp(2)3448 4034 y Fq( )t Fm(k)p Fq(:)328 4275 y Fs(This)31 b(sho)m(ws)h(that)f(in) f(the)h(sense)h(of)f(quadratic)f(forms)g(on)h Fm(D)s Fs(\()p Fq(N)10 b Fs(\),)31 b Fq(\025)2876 4249 y Fi(e)2867 4275 y Fq(I)2910 4290 y Fp(1)2977 4275 y Fm(\025)d(\000)p Fq(cN)i Fm(\000)3414 4235 y Fo(\025)3455 4212 y Fj(2)3490 4235 y Fo(k)p 3414 4252 115 4 v 3456 4309 a(c)3539 4275 y Fq(;)328 4414 y Fs(for)e(an)m(y)h Fq(c)e(>)h Fs(0,)h(where)h Fq(k)g Fs(=)e Fm(k)1453 4389 y Fi(e)1444 4414 y Fq(I)1487 4429 y Fp(1)1526 4414 y Fq(N)1614 4378 y Fx(\000)p Fp(1)p Fo(=)p Fp(2)p 1780 4334 77 4 v 1780 4414 a Fq(P)1856 4429 y Fp(\012)1911 4414 y Fm(k)1961 4378 y Fp(2)2028 4414 y Fm(\024)g Fq(k)2187 4378 y Fx(0)2227 4340 y Fi(P)2332 4443 y Fo(\013)2398 4414 y Fm(k)p Fq(@)2499 4429 y Fo(u)2545 4414 y Fq(\034)2587 4429 y Fo(\014)2634 4414 y Fs(\()p Fq(g)2719 4429 y Fo(\013)2768 4414 y Fs(\))p Fm(k)2856 4378 y Fp(2)2856 4445 y Fo(L)2904 4426 y Fj(2)2971 4414 y Fm(\024)g Fq(k)3130 4378 y Fx(0)3153 4414 y Fs(\(1)13 b(+)g(1)p Fq(=\014)6 b Fs(\).)328 4551 y(Let)532 4525 y Fi(e)514 4551 y Fq(C)584 4566 y Fp(1)669 4551 y Fs(=)46 b Fq(i)p Fs([)p Fq(L)917 4566 y Fo(\025)963 4551 y Fq(;)17 b(A)1080 4566 y Fo(f)1125 4551 y Fs(])46 b(=)g Fq(N)41 b Fs(+)29 b Fq(\025)1610 4525 y Fi(e)1601 4551 y Fq(I)1644 4566 y Fp(1)1727 4551 y Fs(and)43 b(c)m(ho)s(ose)h Fq(c)j Fs(=)f(1)p Fq(=)p Fs(2.)75 b(Then)45 b(w)m(e)f(\014nd)3352 4525 y Fi(e)3333 4551 y Fq(C)3403 4566 y Fp(1)3488 4551 y Fm(\025)338 4645 y Fp(1)p 338 4661 36 4 v 338 4718 a(2)383 4684 y Fq(N)33 b Fm(\000)22 b Fq(k)s(\025)704 4648 y Fp(2)744 4684 y Fs(,)32 b(in)g(the)h(sense)i(of)d(forms)g(on)g Fm(D)s Fs(\()p Fq(N)10 b Fs(\))28 b Fm(\032)g(D)s Fs(\()2364 4659 y Fi(e)2346 4684 y Fq(C)2416 4699 y Fp(1)2455 4684 y Fs(\),)k(and)h(from)e(Theorem)i(3.2)852 4928 y(0)27 b(=)36 b(lim)1032 4988 y Fo(\013)p Fx(!)p Fp(0)1200 4818 y Fi(D)1261 4928 y Fq( )1324 4943 y Fo(\013)1373 4928 y Fq(;)1436 4903 y Fi(e)1417 4928 y Fq(C)1487 4943 y Fp(1)1526 4928 y Fq( )1589 4943 y Fo(\013)1639 4818 y Fi(E)1728 4928 y Fm(\025)1843 4861 y Fs(1)p 1843 4905 49 4 v 1843 4997 a(2)1926 4928 y(lim)1918 4988 y Fo(\013)p Fx(!)p Fp(0)2086 4928 y Fm(k)p Fq(N)2224 4887 y Fp(1)p Fo(=)p Fp(2)2334 4928 y Fq( )2397 4943 y Fo(\013)2447 4928 y Fm(k)2497 4887 y Fp(2)2559 4928 y Fm(\000)22 b Fq(k)s(\025)2769 4887 y Fp(2)2809 4928 y Fm(k)p Fq( )t Fm(k)2976 4887 y Fp(2)3015 4928 y Fq(;)1898 5214 y Fs(31)p eop %%Page: 32 32 32 31 bop 328 631 a Fs(where)34 b Fq( )i Fs(is)c(an)h(eigen)m(v)m (ector)h(of)e Fq(L)1627 646 y Fo(\025)1672 631 y Fs(,)h(and)g Fq( )1985 646 y Fo(\013)2067 631 y Fs(its)f(regularization.)41 b(It)32 b(follo)m(ws)g(that)1387 851 y(lim)1380 911 y Fo(\013)p Fx(!)p Fp(0)1547 851 y Fm(k)p Fq(N)1685 810 y Fp(1)p Fo(=)p Fp(2)1796 851 y Fq( )1859 866 y Fo(\013)1909 851 y Fm(k)1959 810 y Fp(2)2025 851 y Fm(\024)d Fq(k)s(\025)2242 810 y Fp(2)2281 851 y Fm(k)p Fq( )t Fm(k)2448 810 y Fp(2)2487 851 y Fq(;)328 1115 y Fs(whic)m(h)k(tells)f(us)h(that)f Fq( )g Fm(2)c(D)s Fs(\()p Fq(N)1545 1078 y Fp(1)p Fo(=)p Fp(2)1655 1115 y Fs(\),)k(and)h(that)f Fm(k)p Fq(N)2291 1078 y Fp(1)p Fo(=)p Fp(2)2402 1115 y Fq( )t Fm(k)27 b(\024)h Fq(k)s Fm(j)p Fq(\025)p Fm(j)17 b(k)p Fq( )t Fm(k)p Fs(.)474 1235 y(2\))31 b(In)h(what)f(follo)m(ws,)g(the)g (constan)m(ts)i Fq(k)s(;)17 b(k)2048 1250 y Fp(1)2087 1235 y Fs(,)32 b Fq(\025)2203 1250 y Fp(1)2242 1235 y Fs(,)g Fq(\025)2358 1250 y Fp(2)2428 1235 y Fs(are)f(indep)s(enden)m(t) i(of)e Fq(\014)i Fm(\025)28 b Fq(\014)3499 1250 y Fp(0)3539 1235 y Fs(,)328 1355 y(where)h Fq(\014)660 1370 y Fp(0)727 1355 y Fq(>)f Fs(0)f(is)h(arbitrary)f(but)h(\014xed.)43 b(In)28 b(the)h(course)g(of)e(the)h(pro)s(of)f(w)m(e)i(will)d(imp)s (ose)328 1476 y(sev)m(eral)k(conditions)e(on)i(the)f(parameters)h Fq(\017;)17 b(\025;)g(\022)32 b Fs(whic)m(h)e(are)f(collected)g(in)g (\(4.14\).)41 b(W)-8 b(e)328 1596 y(adopt)32 b(the)h(notation)f(of)g (Subsection)h(3.3.1.)474 1717 y(On)g Fm(D)s Fs(\()p Fq(N)10 b Fs(\))28 b Fm(\032)g(D)s Fs(\()p Fq(C)1202 1732 y Fp(1)1240 1717 y Fs(\),)33 b(w)m(e)h(de\014ne)f(the)g(op)s(erator)1536 1937 y Fq(B)g Fs(=)27 b Fq(C)1816 1952 y Fp(1)1877 1937 y Fs(+)22 b Fq(i)p Fs([)p Fq(L)2101 1952 y Fo(\025)2148 1937 y Fq(;)17 b(A)2265 1952 y Fp(0)2304 1937 y Fs(])p Fq(:)328 2157 y Fs(Recall)48 b(that)i Fq(A)941 2172 y Fp(0)1031 2157 y Fs(is)g(de\014ned)h(in)f(\(3.21\),)j(that)d(w)m(e)i (write)p 2636 2077 59 4 v 49 w Fq(R)2694 2172 y Fo(\017)2785 2157 y Fs(=)p 2918 2077 74 4 v 57 w(\005)p Fq(R)3065 2172 y Fo(\017)3098 2157 y Fs(,)j(and)50 b(that)328 2277 y Fq(P)391 2292 y Fp(\012)446 2277 y Fq(I)8 b(P)560 2292 y Fp(\012)642 2277 y Fs(=)28 b Fq(P)809 2292 y Fp(\012)864 2277 y Fq(I)907 2292 y Fp(1)946 2277 y Fq(P)1009 2292 y Fp(\012)1092 2277 y Fs(=)g(0.)43 b(Using)32 b(that)g Fq(P)1863 2292 y Fp(\012)1946 2277 y Fs(=)c(\005)22 b(+)p 2243 2197 77 4 v 22 w Fq(P)2319 2292 y Fp(0)2381 2277 y Fm(\012)h Fq(P)2544 2292 y Fp(\012)2599 2277 y Fs(,)32 b(one)h(\014nds)h(that)388 2497 y Fq(P)451 2512 y Fp(\012)506 2497 y Fq(B)5 b(P)648 2512 y Fp(\012)786 2497 y Fs(=)83 b(\005)p Fq(B)5 b Fs(\005)22 b(+)p 1290 2417 V 22 w Fq(P)1367 2512 y Fp(0)1428 2497 y Fm(\012)h Fq(P)1591 2512 y Fp(\012)1646 2497 y Fq(B)5 b Fs(\005)22 b(+)g(\005)p Fq(B)p 2070 2417 V 5 w(P)2147 2512 y Fp(0)2209 2497 y Fm(\012)g Fq(P)2371 2512 y Fp(\012)2448 2497 y Fs(+)p 2546 2417 V 22 w Fq(P)2623 2512 y Fp(0)2684 2497 y Fm(\012)h Fq(P)2847 2512 y Fp(\012)2902 2497 y Fq(B)p 2981 2417 V 5 w(P)3058 2512 y Fp(0)3119 2497 y Fm(\012)g Fq(P)3282 2512 y Fp(\012)786 2660 y Fs(=)83 b(2)p Fq(\022)s(\025)1099 2619 y Fp(2)1138 2660 y Fs(\005)p Fq(I)p 1279 2580 59 4 v 8 w(R)1337 2598 y Fp(2)1337 2684 y Fo(\017)1377 2660 y Fq(I)8 b Fs(\005)22 b(+)g Fq(\022)s(\025)1726 2619 y Fp(2)1782 2549 y Fi(\020)p 1841 2580 77 4 v 111 x Fq(P)1918 2675 y Fp(0)1979 2660 y Fm(\012)h Fq(P)2142 2675 y Fp(\012)2197 2660 y Fq(I)p 2264 2580 59 4 v 7 w(R)2323 2598 y Fp(2)2323 2684 y Fo(\017)2362 2660 y Fq(I)8 b Fs(\005)22 b(+)g(\005)p Fq(I)p 2747 2580 V 8 w(R)2805 2598 y Fp(2)2805 2684 y Fo(\017)2845 2660 y Fq(I)p 2896 2580 77 4 v 8 w(P)2972 2675 y Fp(0)3034 2660 y Fm(\012)g Fq(P)3196 2675 y Fp(\012)3251 2549 y Fi(\021)3294 2660 y Fq(;)44 b Fs(\(4.1\))374 2862 y Fq(P)437 2877 y Fp(\012)492 2862 y Fq(B)p 571 2782 V 5 w(P)648 2877 y Fp(\012)786 2862 y Fs(=)83 b Fq(\025P)1065 2877 y Fp(\012)1120 2862 y Fq(I)1163 2877 y Fp(1)p 1202 2782 V 1202 2862 a Fq(P)1279 2877 y Fp(\012)1356 2862 y Fs(+)22 b Fq(\022)s(\025)p Fs(\005)p Fq(I)p 1699 2782 59 4 v 7 w(R)1758 2800 y Fp(2)1758 2887 y Fo(\017)1797 2862 y Fq(L)1863 2877 y Fp(0)p 1903 2782 77 4 v 1903 2862 a Fq(P)1980 2877 y Fp(\012)2057 2862 y Fs(+)g Fq(\022)s(\025)2260 2821 y Fp(2)2299 2862 y Fs(\005)p Fq(I)p 2440 2782 59 4 v 8 w(R)2498 2800 y Fp(2)2498 2887 y Fo(\017)2538 2862 y Fq(I)p 2589 2782 77 4 v 8 w(P)2665 2877 y Fp(\012)2720 2862 y Fq(;)618 b Fs(\(4.2\))p 374 2945 V 374 3025 a Fq(P)451 3040 y Fp(\012)506 3025 y Fq(B)5 b(P)648 3040 y Fp(\012)786 3025 y Fs(=)83 b Fq(\025)p 1002 2945 V(P)1078 3040 y Fp(\012)1133 3025 y Fq(I)1176 3040 y Fp(1)1216 3025 y Fq(P)1279 3040 y Fp(\012)1356 3025 y Fs(+)22 b Fq(\022)s(\025)p 1559 2945 V(P)1635 3040 y Fp(\012)1690 3025 y Fq(L)1756 3040 y Fp(0)p 1813 2945 59 4 v 1796 3025 a Fq(R)1871 2963 y Fp(2)1871 3049 y Fo(\017)1911 3025 y Fq(I)8 b Fs(\005)22 b(+)g Fq(\022)s(\025)2260 2984 y Fp(2)p 2299 2945 77 4 v 2299 3025 a Fq(P)2376 3040 y Fp(\012)2431 3025 y Fq(I)p 2498 2945 59 4 v 7 w(R)2557 2963 y Fp(2)2557 3049 y Fo(\017)2596 3025 y Fq(I)8 b Fs(\005)p Fq(;)618 b Fs(\(4.3\))p 361 3117 77 4 v 361 3197 a Fq(P)437 3212 y Fp(\012)492 3197 y Fq(B)p 571 3117 V 5 w(P)648 3212 y Fp(\012)786 3197 y Fs(=)p 945 3117 V 83 w Fq(P)1021 3212 y Fp(\012)1076 3197 y Fq(N)33 b Fs(+)p 1285 3117 V 22 w Fq(P)1361 3212 y Fp(\012)1433 3087 y Fi(\020)1493 3197 y Fq(\025I)1593 3212 y Fp(1)1654 3197 y Fm(\000)23 b Fq(\022)s(\025)1859 3156 y Fp(2)1898 3197 y Fs(\()p Fq(I)8 b Fs(\005)p Fq(I)p 2128 3117 59 4 v 8 w(R)2186 3136 y Fp(2)2186 3222 y Fo(\017)2248 3197 y Fs(+)p 2362 3117 V 21 w Fq(R)2421 3136 y Fp(2)2421 3222 y Fo(\017)2460 3197 y Fq(I)g Fs(\005)p Fq(I)g Fs(\))2673 3087 y Fi(\021)p 2749 3117 77 4 v 2749 3197 a Fq(P)2826 3212 y Fp(\012)2881 3197 y Fq(:)328 3475 y Fs(F)-8 b(rom)32 b(the)i(estimates)f Fm(k)p 1234 3395 V Fq(P)1310 3490 y Fp(\012)1365 3475 y Fq(N)1453 3438 y Fx(\000)p Fp(1)p Fo(=)p Fp(2)1618 3475 y Fq(I)1661 3490 y Fp(1)1701 3475 y Fq(N)1789 3438 y Fx(\000)p Fp(1)p Fo(=)p Fp(2)p 1954 3395 V 1954 3475 a Fq(P)2031 3490 y Fp(\012)2086 3475 y Fm(k)c(\024)g Fq(k)s Fs(,)34 b Fm(k)p Fq(I)8 b Fs(\005)p Fq(I)g Fm(k)28 b(\024)i Fq(k)s Fs(,)k Fm(k)p 2978 3395 59 4 v Fq(R)3036 3413 y Fp(2)3036 3499 y Fo(\017)3075 3475 y Fm(k)29 b(\024)g Fq(\017)3299 3438 y Fx(\000)p Fp(2)3394 3475 y Fs(,)34 b(w)m(e)328 3595 y(see)g(that)e(there)h(is)f (some)h(constan)m(t)g Fq(\025)1739 3610 y Fp(1)1806 3595 y Fq(<)28 b Fm(1)k Fs(\(indep)s(enden)m(t)i(of)e Fq(\025;)17 b(\017;)g(\022)s Fs(\),)32 b(s.t.)p 1596 3779 77 4 v 1596 3859 a Fq(P)1672 3874 y Fp(\012)1727 3859 y Fq(B)p 1806 3779 V 5 w(P)1883 3874 y Fp(\012)1966 3859 y Fm(\025)2081 3792 y Fs(1)p 2081 3837 49 4 v 2081 3928 a(2)p 2139 3779 77 4 v 2139 3859 a Fq(P)2216 3874 y Fp(\012)2271 3859 y Fq(;)1067 b Fs(\(4.4\))328 4105 y(pro)m(vided)1643 4279 y Fm(j)p Fq(\025)p Fm(j)p Fq(;)1842 4212 y(\022)s(\025)1947 4175 y Fp(2)p 1842 4256 145 4 v 1874 4347 a Fq(\017)1913 4319 y Fp(2)2024 4279 y Fq(<)27 b(\025)2184 4294 y Fp(1)2224 4279 y Fq(;)1114 b Fs(\(4.5\))328 4484 y(see)34 b(also)d(\(4.14\).)43 b(Using)32 b(the)h(estimates)803 4722 y Fm(k)p Fq(I)p 920 4642 59 4 v 7 w(R)979 4660 y Fp(2)979 4746 y Fo(\017)1018 4722 y Fq(I)8 b Fs(\005)p Fm(k)p Fq(;)17 b Fm(k)p Fs(\005)p Fq(I)p 1426 4642 V 7 w(R)1485 4660 y Fp(2)1485 4746 y Fo(\017)1524 4722 y Fq(I)8 b Fm(k)27 b(\024)i Fq(\017)1797 4685 y Fx(\000)p Fp(2)1891 4722 y Fq(k)68 b Fs(and)e Fm(k)p Fq(P)2346 4737 y Fp(\012)2400 4722 y Fq(I)2443 4737 y Fp(1)2483 4722 y Fm(k)p Fq(;)17 b Fm(k)p Fq(I)2670 4737 y Fp(1)2709 4722 y Fq(P)2772 4737 y Fp(\012)2827 4722 y Fm(k)27 b(\024)h Fq(k)s Fs(,)1898 5214 y(32)p eop %%Page: 33 33 33 32 bop 328 631 a Fs(where)38 b Fq(k)j Fs(is)36 b(indep)s(enden)m(t)j (of)e(the)g(parameters)g Fq(\025;)17 b(\022)s(;)g(\017)p Fs(,)39 b(w)m(e)f(arriv)m(e)f(at)g(the)g(follo)m(wing)328 751 y(lo)m(w)m(er)c(b)s(ound.)43 b(F)-8 b(or)32 b(an)m(y)h Fq(\036)28 b Fm(2)g(D)s Fs(\()p Fq(N)10 b Fs(\))32 b(and)h(some)g Fq(k)2222 766 y Fp(1)2288 751 y Fq(<)28 b Fm(1)693 994 y(h)p Fq(B)5 b Fm(i)850 1023 y Fo(\036)979 994 y Fm(\025)83 b Fs(2)p Fq(\022)s(\025)1293 953 y Fp(2)1349 884 y Fi(D)1410 994 y Fs(\005)p Fq(I)p 1551 914 59 4 v 8 w(R)1609 932 y Fp(2)1609 1019 y Fo(\017)1648 994 y Fq(I)8 b Fs(\005)1772 884 y Fi(E)1833 1063 y Fo(\036)1902 994 y Fs(+)2010 927 y(1)p 2010 971 49 4 v 2010 1062 a(2)2068 994 y Fm(k)p 2118 914 77 4 v Fq(P)2195 1009 y Fp(\012)2250 994 y Fq(\036)p Fm(k)2358 953 y Fp(2)1139 1256 y Fm(\000)p Fq(k)1267 1271 y Fp(1)1317 1188 y Fq(\022)s(\025)1422 1152 y Fp(2)p 1317 1233 145 4 v 1350 1324 a Fq(\017)1389 1295 y Fp(2)1471 1256 y Fm(k)p Fs(\005)p Fq(\036)p Fm(k)32 b(k)p 1784 1175 77 4 v Fq(P)1861 1271 y Fp(0)1922 1256 y Fm(\012)23 b Fq(P)2085 1271 y Fp(\012)2140 1256 y Fq(\036)p Fm(k)e(\000)i Fq(k)2420 1271 y Fp(1)2459 1256 y Fm(j)p Fq(\025)p Fm(j)32 b(k)p Fq(P)2717 1271 y Fp(\012)2772 1256 y Fq(\036)p Fm(k)g(k)p 2962 1175 V Fq(P)3038 1271 y Fp(\012)3093 1256 y Fq(\036)p Fm(k)1139 1503 y(\000)1233 1363 y Fi(\022)1307 1503 y Fs(2)p Fq(\022)s Fm(j)p Fq(\025)p Fm(j)21 b Fs(+)h(2)p Fq(k)1736 1518 y Fp(1)1785 1436 y Fq(\022)s(\025)1890 1400 y Fp(2)p 1785 1480 145 4 v 1837 1572 a Fq(\017)1939 1363 y Fi(\023)2029 1503 y Fm(k)p 2096 1423 59 4 v Fq(R)2154 1518 y Fo(\017)2187 1503 y Fq(I)8 b Fs(\005)p Fq(\036)p Fm(k)32 b(k)p 2501 1423 77 4 v Fq(P)2577 1518 y Fp(\012)2632 1503 y Fq(\036)p Fm(k)p Fq(:)598 b Fs(\(4.6\))328 1799 y(Clearly)-8 b(,)50 b Fm(k)p 771 1719 59 4 v Fq(R)829 1814 y Fo(\017)862 1799 y Fq(I)8 b Fs(\005)p Fq(\036)p Fm(k)47 b(k)p 1191 1719 77 4 v Fq(P)1267 1814 y Fp(\012)1323 1799 y Fq(\036)p Fm(k)52 b(\024)i Fq(\016)1678 1688 y Fi(D)1739 1799 y Fs(\005)p Fq(I)p 1879 1719 59 4 v 7 w(R)1938 1737 y Fp(2)1938 1824 y Fo(\017)1977 1799 y Fq(I)8 b Fs(\005)2101 1688 y Fi(E)2162 1868 y Fo(\036)2241 1799 y Fs(+)32 b Fq(\016)2396 1763 y Fx(\000)p Fp(1)2490 1799 y Fm(k)p 2540 1719 77 4 v Fq(P)2616 1814 y Fp(\012)2671 1799 y Fq(\036)p Fm(k)2779 1763 y Fp(2)2818 1799 y Fs(,)52 b(for)47 b(an)m(y)h Fq(\016)58 b(>)53 b Fs(0.)328 1964 y(Cho)s(osing)30 b(appropriate)g(v)-5 b(alues)30 b(of)h Fq(\016)t Fs(,)g(w)m(e)g(b)s(ound)g(the)g(last)f(line)f(in)h(\(4.6\))h (from)e(b)s(elo)m(w)328 2084 y(b)m(y)996 2258 y Fm(\000)p Fq(\022)s(\025)1178 2217 y Fp(2)1235 2148 y Fi(D)1296 2258 y Fs(\005)p Fq(I)p 1436 2178 59 4 v 7 w(R)1495 2196 y Fp(2)1495 2283 y Fo(\017)1534 2258 y Fq(I)8 b Fs(\005)1658 2148 y Fi(E)1719 2327 y Fo(\036)1787 2258 y Fm(\000)23 b Fs(4)1953 2118 y Fi(\022)2026 2258 y Fq(\022)i Fs(+)d Fq(k)2248 2217 y Fp(2)2245 2283 y(1)2298 2191 y Fq(\022)s(\025)2403 2155 y Fp(2)p 2298 2235 145 4 v 2330 2326 a Fq(\017)2369 2298 y Fp(2)2452 2118 y Fi(\023)2542 2258 y Fm(k)p 2592 2178 77 4 v Fq(P)2668 2273 y Fp(\012)2723 2258 y Fq(\036)p Fm(k)2831 2217 y Fp(2)2870 2258 y Fq(;)328 2475 y Fs(and)33 b(it)e(follo)m(ws)g(that)679 2708 y Fm(h)p Fq(B)5 b Fm(i)836 2737 y Fo(\036)965 2708 y Fm(\025)1136 2640 y Fq(\022)s(\025)1241 2604 y Fp(2)p 1136 2685 145 4 v 1188 2776 a Fq(\017)1290 2708 y(\015)g Fm(k)p Fs(\005)p Fq(\036)p Fm(k)1577 2666 y Fp(2)1638 2708 y Fs(+)1736 2567 y Fi(\022)1819 2640 y Fs(1)p 1819 2685 49 4 v 1819 2776 a(2)1900 2708 y Fm(\000)23 b Fs(4)p Fq(\022)i Fm(\000)e Fs(4)p Fq(k)2322 2666 y Fp(2)2319 2732 y(1)2371 2640 y Fq(\022)s(\025)2476 2604 y Fp(2)p 2371 2685 145 4 v 2404 2776 a Fq(\017)2443 2747 y Fp(2)2525 2567 y Fi(\023)2615 2708 y Fm(k)p 2665 2628 77 4 v Fq(P)2742 2723 y Fp(\012)2797 2708 y Fq(\036)p Fm(k)2905 2666 y Fp(2)1126 2982 y Fm(\000)p Fq(k)1254 2997 y Fp(1)1303 2914 y Fq(\022)s(\025)1408 2878 y Fp(2)p 1303 2959 145 4 v 1336 3050 a Fq(\017)1375 3021 y Fp(2)1458 2982 y Fm(k)p Fs(\005)p Fq(\036)p Fm(k)32 b(k)p 1771 2902 77 4 v Fq(P)1847 2997 y Fp(0)1909 2982 y Fm(\012)22 b Fq(P)2071 2997 y Fp(\012)2126 2982 y Fq(\036)p Fm(k)g(\000)h Fq(k)2407 2997 y Fp(1)2446 2982 y Fm(j)p Fq(\025)p Fm(j)31 b(k)p Fq(P)2703 2997 y Fp(\012)2758 2982 y Fq(\036)p Fm(k)h(k)p 2948 2902 V Fq(P)3025 2997 y Fp(\012)3080 2982 y Fq(\036)p Fm(k)p Fq(;)150 b Fs(\(4.7\))328 3211 y(where)37 b(w)m(e)g(ha)m(v)m(e)g(used)g(\(3.27\))f(and)g(hence)h (assumed)g(that)e(0)f Fq(<)f(\017)h(<)f(\017)2989 3226 y Fp(0)3029 3211 y Fs(.)53 b(Using)36 b(that)328 3331 y Fm(k)p Fq(P)441 3346 y Fp(\012)496 3331 y Fq(\036)p Fm(k)27 b(\024)h(k)p Fs(\005)p Fq(\036)p Fm(k)21 b Fs(+)h Fm(k)p 1136 3251 V Fq(P)1212 3346 y Fp(0)1273 3331 y Fm(\012)g Fq(P)1435 3346 y Fp(\012)1490 3331 y Fq(\036)p Fm(k)p Fs(,)32 b(w)m(e)h(estimate)e(the)i(t)m(w)m(o)g(terms)f(in)g(the) g(last)g(line)f(on)328 3452 y(the)i(r.h.s.)44 b(of)32 b(\(4.7\))g(as)527 3695 y Fm(\000)p Fq(k)655 3710 y Fp(1)694 3695 y Fm(j)p Fq(\025)p Fm(j)g(k)p Fq(P)952 3710 y Fp(\012)1007 3695 y Fq(\036)p Fm(k)g(k)p 1197 3615 V Fq(P)1273 3710 y Fp(\012)1328 3695 y Fq(\036)p Fm(k)83 b(\025)h(\000)1767 3627 y Fs(1)p 1767 3672 49 4 v 1767 3763 a(4)1826 3695 y Fm(k)p 1876 3615 77 4 v Fq(P)1952 3710 y Fp(\012)2007 3695 y Fq(\036)p Fm(k)2115 3654 y Fp(2)2176 3695 y Fm(\000)23 b Fs(8)p Fq(\025)2382 3654 y Fp(2)2421 3695 y Fq(k)2475 3654 y Fp(2)2472 3719 y(1)2531 3614 y Fi(\000)2577 3695 y Fm(k)p Fs(\005)p Fq(\036)p Fm(k)2808 3654 y Fp(2)2869 3695 y Fs(+)f Fm(k)p 3017 3615 V Fq(P)3093 3710 y Fp(0)3155 3695 y Fm(\012)g Fq(P)3317 3710 y Fp(\012)3372 3695 y Fq(\036)p Fm(k)3480 3654 y Fp(2)3519 3614 y Fi(\001)3581 3695 y Fq(;)328 3943 y Fm(\000)p Fq(k)456 3958 y Fp(1)506 3875 y Fq(\022)s(\025)611 3839 y Fp(2)p 506 3920 145 4 v 538 4011 a Fq(\017)577 3982 y Fp(2)660 3943 y Fm(k)p Fs(\005)p Fq(\036)p Fm(k)32 b(k)p 973 3863 77 4 v Fq(P)1049 3958 y Fp(0)1111 3943 y Fm(\012)22 b Fq(P)1273 3958 y Fp(\012)1328 3943 y Fq(\036)p Fm(k)83 b(\025)h(\000)1767 3875 y Fs(1)p 1767 3920 49 4 v 1767 4011 a(2)1836 3875 y Fq(\022)s(\025)1941 3839 y Fp(2)p 1836 3920 145 4 v 1888 4011 a Fq(\017)1990 3943 y(\015)5 b Fm(k)p Fs(\005)p Fq(\036)p Fm(k)2277 3901 y Fp(2)2338 3943 y Fm(\000)23 b Fs(2)p Fq(k)2541 3901 y Fp(2)2538 3967 y(1)2590 3875 y Fq(\022)s(\025)2695 3839 y Fp(2)p 2590 3920 V 2623 4011 a Fq(\017)2662 3982 y Fp(3)2745 3943 y Fq(\015)5 b Fm(k)p 2851 3863 77 4 v Fq(P)2927 3958 y Fp(0)2988 3943 y Fm(\012)23 b Fq(P)3151 3958 y Fp(\012)3206 3943 y Fq(\036)p Fm(k)3314 3901 y Fp(2)3353 3943 y Fq(:)328 4172 y Fs(Using)32 b(these)i(t)m(w)m(o)f(estimates)f(in)g(\(4.7\),)g(w) m(e)i(arriv)m(e)e(at)456 4428 y Fm(h)o Fq(B)5 b Fm(i)612 4457 y Fo(\036)686 4428 y Fm(\025)28 b Fq(\025)848 4386 y Fp(2)904 4287 y Fi(\022)987 4360 y Fs(1)p 987 4405 49 4 v 987 4496 a(2)1056 4360 y Fq(\022)p 1056 4405 V 1060 4496 a(\017)1114 4428 y(\015)f Fm(\000)c Fs(8)p Fq(k)1395 4386 y Fp(2)1392 4452 y(1)1434 4287 y Fi(\023)1524 4428 y Fm(k)p Fs(\005)p Fq(\036)p Fm(k)1755 4386 y Fp(2)1816 4428 y Fm(\000)g Fs(2)p Fq(\025)2022 4386 y Fp(2)2061 4428 y Fq(k)2115 4386 y Fp(2)2112 4452 y(1)2171 4287 y Fi(\022)2244 4428 y Fs(4)f(+)2467 4360 y Fq(\022)p 2423 4405 135 4 v 2423 4496 a(\017)2462 4467 y Fp(3)2502 4496 y Fq(\015)2568 4287 y Fi(\023)2658 4428 y Fm(k)p 2708 4347 77 4 v Fq(P)2784 4443 y Fp(0)2846 4428 y Fm(\012)h Fq(P)3009 4443 y Fp(\012)3064 4428 y Fq(\036)p Fm(k)3172 4386 y Fp(2)3211 4428 y Fq(;)127 b Fs(\(4.8\))328 4678 y(where)34 b(w)m(e)f(require)g(the)g(condition)1495 4862 y(1)p 1495 4907 49 4 v 1495 4998 a(4)1575 4930 y Fm(\000)23 b Fs(4)p Fq(\022)i Fm(\000)e Fs(4)p Fq(k)1997 4888 y Fp(2)1994 4954 y(1)2046 4862 y Fq(\022)s(\025)2151 4826 y Fp(2)p 2046 4907 145 4 v 2079 4998 a Fq(\017)2118 4969 y Fp(2)2228 4930 y Fm(\025)28 b Fs(0)p Fq(;)956 b Fs(\(4.9\))1898 5214 y(33)p eop %%Page: 34 34 34 33 bop 328 631 a Fs(whic)m(h)32 b(guaran)m(tees)g(that)f(the)h(con)m (tribution)e(of)h(the)h(term)f(in)f(\(4.7\))h(whic)m(h)h(is)f(prop)s (or-)328 751 y(tional)23 b(to)h Fm(k)p 758 671 77 4 v Fq(P)835 766 y Fp(\012)890 751 y Fq(\036)p Fm(k)998 715 y Fp(2)1062 751 y Fs(is)g(non-negativ)m(e,)j(and)e(can)g(hence)i(b)s(e) e(dropp)s(ed.)41 b(\(4.9\))25 b(is)f(satis\014ed)328 872 y(if)33 b(\(4.14\))g(holds.)47 b(Let)35 b Fq(\036)29 b Fs(=)h Fq( )1432 887 y Fo(\013)1512 872 y Fm(2)h(D)s Fs(\()p Fq(N)10 b Fs(\))34 b(b)s(e)g(the)g(regularization)e(of)h(the)i (eigen)m(v)m(ector)328 992 y Fq( )h Fs(as)d(de\014ned)h(in)e(Theorem)h (3.2.)43 b(Then)34 b(it)d(follo)m(ws)h(from)f(\(4.8\))h(that)1000 1212 y(0)27 b(=)36 b(lim)1180 1272 y Fo(\013)p Fx(!)p Fp(0)1348 1212 y Fm(h)o Fq(B)5 b Fm(i)1504 1241 y Fo( )1550 1249 y Fl(\013)1626 1212 y Fm(\025)29 b Fq(\024)1788 1227 y Fp(1)1827 1212 y Fm(k)p Fs(\005)p Fq(\036)p Fm(k)2058 1171 y Fp(2)2119 1212 y Fm(\000)23 b Fq(\024)2275 1227 y Fp(2)2314 1212 y Fm(k)p 2364 1132 V Fq(P)2441 1227 y Fp(0)2502 1212 y Fm(\012)g Fq(P)2665 1227 y Fp(\012)2720 1212 y Fq(\036)p Fm(k)2828 1171 y Fp(2)2867 1212 y Fq(;)423 b Fs(\(4.10\))328 1459 y(where)1360 1703 y Fq(\024)1416 1718 y Fp(1)1539 1703 y Fs(=)1707 1636 y(1)p 1707 1681 49 4 v 1707 1772 a(2)1776 1636 y Fq(\022)p 1776 1681 V 1780 1772 a(\017)1834 1703 y(\015)27 b Fm(\000)c Fs(8)p Fq(k)2115 1662 y Fp(2)2112 1728 y(1)2182 1703 y Fq(>)28 b(k)2340 1662 y Fp(2)2337 1728 y(1)2379 1703 y Fq(;)911 b Fs(\(4.11\))1360 1949 y Fq(\024)1416 1964 y Fp(2)1539 1949 y Fs(=)82 b(2)p Fq(k)1800 1908 y Fp(2)1797 1974 y(1)1856 1809 y Fi(\022)1930 1949 y Fs(4)22 b(+)2152 1882 y Fq(\022)p 2109 1927 135 4 v 2109 2018 a(\015)5 b(\017)2204 1989 y Fp(3)2253 1809 y Fi(\023)2354 1949 y Fq(>)28 b Fs(0)p Fq(:)783 b Fs(\(4.12\))328 2242 y(The)33 b(lo)m(w)m(er)g(b)s(ound)g(\(4.11\))f(is)g(a)g(consequence)k(of)c Fq(\017)c(<)2422 2198 y Fo(\022)r(\015)p 2387 2219 144 4 v 2387 2278 a Fp(18)p Fo(k)2496 2255 y Fj(2)2494 2299 y(1)2541 2242 y Fs(,)33 b(see)g(\(4.14\).)474 2382 y(F)-8 b(rom)31 b(\(2.51\),)h(w)m(e)i(\014nd)f(that)454 2602 y Fm(k)p Fs(\005)p Fq( )t Fm(k)27 b(\025)h(k)p Fq( )t Fm(k)22 b(\000)h(k)p 1165 2522 77 4 v Fq(P)1241 2617 y Fp(\012)1296 2602 y Fq( )t Fm(k)f(\000)h(k)p 1585 2522 V Fq(P)1661 2617 y Fp(0)1722 2602 y Fm(\012)g Fq(P)1885 2617 y Fp(\012)1940 2602 y Fq( )t Fm(k)k(\025)i Fs(\(1)21 b Fm(\000)i Fq(k)s Fm(j)p Fq(\025)p Fm(j)p Fs(\))p Fm(k)p Fq( )t Fm(k)e(\000)i(k)p 2941 2522 V Fq(P)3017 2617 y Fp(0)3079 2602 y Fm(\012)f Fq(P)3241 2617 y Fp(\012)3296 2602 y Fq( )t Fm(k)p Fq(:)328 2822 y Fs(Th)m(us)i(there)g(is)f(a)f(p)s (ositiv)m(e)h(constan)m(t)h Fq(\025)1758 2837 y Fp(2)1820 2822 y Fs(\(indep)s(enden)m(t)g(of)f Fq(\017;)17 b(\025;)g(\022)s Fs(\))22 b(s.t.)41 b(if)22 b(0)27 b Fq(<)h Fm(j)p Fq(\025)p Fm(j)f Fq(<)g(\025)3526 2837 y Fp(2)328 2943 y Fs(then)33 b Fm(k)p Fs(\005)p Fq( )t Fm(k)28 b(\025)933 2903 y Fp(1)p 933 2920 36 4 v 933 2977 a(2)978 2943 y Fm(k)p Fq( )t Fm(k)22 b(\000)g(k)p 1316 2863 77 4 v Fq(P)1393 2958 y Fp(0)1454 2943 y Fm(\012)h Fq(P)1617 2958 y Fp(\012)1672 2943 y Fq( )t Fm(k)p Fs(.)43 b(Th)m(us)34 b(w)m(e)g(get)e(from)g (\(4.10\))1181 3217 y Fm(k)p 1231 3136 V Fq(P)1307 3232 y Fp(0)1369 3217 y Fm(\012)22 b Fq(P)1531 3232 y Fp(\012)1586 3217 y Fq( )t Fm(k)28 b(\025)1846 3149 y Fs(1)p 1846 3194 49 4 v 1846 3285 a(2)2188 3149 y(1)p 1915 3194 595 4 v 1915 3285 a(1)21 b(+)h(\()p Fq(\024)2177 3300 y Fp(2)2217 3285 y Fq(=\024)2322 3300 y Fp(1)2361 3285 y Fs(\))2399 3256 y Fp(1)p Fo(=)p Fp(2)2519 3217 y Fm(k)p Fq( )t Fm(k)p Fq(:)604 b Fs(\(4.13\))328 3487 y(Consequen)m(tly)-8 b(,)35 b(under)e(the)g(conditions)f(that)328 3759 y(0)27 b Fq(<)h Fm(j)p Fq(\025)p Fm(j)f Fq(<)g Fs(min)931 3618 y Fi(\032)1005 3759 y Fq(\025)1062 3774 y Fp(1)1102 3759 y Fq(;)17 b(\025)1203 3774 y Fp(2)1242 3759 y Fq(;)1341 3691 y(\017)p 1295 3736 132 4 v 1295 3756 a Fm(p)p 1378 3756 49 4 v 85 x Fq(\022)1453 3618 y Fi(\022)1526 3666 y(p)p 1626 3666 97 4 v 93 x Fq(\025)1683 3774 y Fp(1)1745 3759 y Fs(+)1964 3691 y(1)p 1853 3736 271 4 v 1853 3838 a(4)1902 3756 y Fm(p)p 1984 3756 49 4 v 1984 3838 a Fs(2)p Fq(k)2084 3853 y Fp(1)2133 3618 y Fi(\023)q(\033)2298 3759 y Fq(;)38 b(\022)31 b(<)2577 3691 y Fs(1)p 2553 3736 98 4 v 2553 3827 a(32)2660 3759 y Fq(;)39 b(\017)28 b(<)f Fs(min)3075 3618 y Fi(\032)3203 3691 y Fq(\022)s(\015)p 3160 3736 192 4 v 3160 3827 a Fs(18)p Fq(k)3312 3793 y Fp(2)3309 3851 y(1)3361 3759 y Fq(;)17 b(\017)3444 3774 y Fp(0)3484 3618 y Fi(\033)3575 3759 y Fq(;)3317 3936 y Fs(\(4.14\))328 4057 y(w)m(e)34 b(obtain)1232 4197 y Fm(k)p 1282 4117 77 4 v Fq(P)1359 4212 y Fp(0)1420 4197 y Fm(\012)23 b Fq(P)1583 4212 y Fp(\012)1638 4197 y Fq( )t Fm(k)k(\025)1897 4129 y Fs(1)p 1897 4174 49 4 v 1897 4265 a(2)2284 4129 y(1)p 1966 4174 686 4 v 1966 4303 a(1)22 b(+)2135 4194 y Fi(q)p 2235 4194 417 4 v 109 x Fs(2\(4)f(+)2534 4264 y Fo(\022)p 2500 4280 104 4 v 2500 4337 a(\015)t(\017)2569 4318 y Fj(3)2614 4303 y Fs(\))3317 4197 y(\(4.15\))328 4506 y(Cho)s(ose)37 b(for)e(instance)i Fq(\022)g Fs(=)d(1)p Fq(=)p Fs(100,)i Fq(\017)e Fs(=)g(min)n Fm(f)2199 4462 y Fo(\015)p 2112 4483 215 4 v 2112 4542 a Fp(2000)p Fo(k)2291 4519 y Fj(2)2289 4563 y(1)2336 4506 y Fq(;)17 b(\017)2419 4521 y Fp(0)2459 4506 y Fm(g)p Fs(.)54 b(Then)37 b(\(4.14\))f(holds)f(pro-)328 4646 y(vided)d(0)c Fq(<)f Fm(j)p Fq(\025)p Fm(j)g Fq(<)h(k)s(\015)5 b Fs(,)32 b(for)g(some)g Fq(k)j Fs(indep)s(enden)m(t)e(of)f Fq(\014)37 b Fs(pro)m(vided)c(that)f Fq(\014)h Fm(\025)28 b Fq(\014)3277 4661 y Fp(0)3317 4646 y Fs(,)k(with)328 4767 y Fq(\014)383 4782 y Fp(0)454 4767 y Fq(>)g Fs(0)i(arbitrary)g (but)i(\014xed.)51 b(F)-8 b(or)35 b(large)e Fq(\014)6 b Fs(,)36 b(the)f(r.h.s.)51 b(of)35 b(\(4.15\))f(b)s(eha)m(v)m(es)j (lik)m(e)d Fq(\015)3499 4730 y Fp(2)3539 4767 y Fs(.)328 4887 y Fn(\004)1898 5214 y Fs(34)p eop %%Page: 35 35 35 34 bop 328 631 a Fu(5)161 b(Pro)t(of)54 b(of)g(Theorem)e(2.3)328 850 y Fs(The)47 b Fq(\033)597 865 y Fo(t;\025)688 850 y Fs(-in)m(v)-5 b(arian)m(t)44 b(normal)g(states)j(on)f Fr(M)2032 865 y Fo(\014)2124 850 y Fs(are)g(in)f(one-to-one)h(corresp)s (ondence)328 970 y(with)41 b(the)i(normalized)d(v)m(ectors)j(in)e(the)i (span)f(of)f Fm(P)c(\\)29 b Fs(k)m(er)18 b Fq(L)2657 985 y Fo(\025)2745 970 y Fs(\(see)42 b(Section)g(2.1.4\).)328 1091 y(Our)32 b(task)i(is)e(to)g(sho)m(w)i(that)e Fm(P)f(\\)22 b Fs(k)m(er)c Fq(L)1809 1106 y Fo(\025)1887 1091 y Fs(equals)33 b(the)g(set)g(\(2.55\).)474 1211 y(In)25 b(this)f(section,)i(w)m(e)g (will)c(alw)m(a)m(ys)j(deal)e(with)h(the)h(cuto\013)g(in)m(teraction)e (\(determined)328 1332 y(b)m(y)33 b Fq(G)540 1347 y Fo(\013;J)654 1332 y Fs(\))g(but)g(w)m(e)g(shall)e(drop)i(the)g(subscript)2089 1347 y Fo(J)2171 1332 y Fs(in)f(the)h(notation.)328 1620 y Fh(5.1)135 b(Reduction)46 b(of)f(the)g(Liouvillian)328 1805 y Fs(W)-8 b(e)43 b(de\014ne)g(the)g(pro)5 b(jection)42 b Fq(p)i Fs(=)g Fq(p)1710 1820 y Fo(J)1749 1832 y Fl(d)1818 1805 y Fs(+)28 b Fq(p)1971 1820 y Fo(J)2010 1828 y Fl(c)2046 1805 y Fs(,)45 b(where)e Fq(p)2458 1820 y Fo(J)2497 1832 y Fl(d)2537 1805 y Fq(;)17 b(p)2630 1820 y Fo(J)2669 1828 y Fl(c)2746 1805 y Fs(are)43 b(the)f(pro)5 b(jections)328 1925 y(corresp)s(onding)42 b(to)h(the)g(discrete)g(and)f(con)m(tin)m (uous)i(mo)s(des)e(in)g Fq(J)2827 1940 y Fo(d)2910 1925 y Fs(and)g Fq(J)3163 1940 y Fo(c)3198 1925 y Fs(,)j(resp)s(ec-)328 2046 y(tiv)m(ely;)33 b(\(see)h(also)e(\(2.14\)\).)44 b(Setting)p 1728 1991 49 4 v 49 w Fq(p)i Fs(=)28 b(1)-22 b(l)1982 2061 y Fo(p)2043 2046 y Fm(\000)23 b Fq(p)p Fs(,)33 b Fq(P)42 b Fs(=)28 b Fq(p)23 b Fm(\012)g Fq(p)f Fm(\012)h Fs(1)-22 b(l)2859 2061 y Fo(f)2903 2046 y Fs(,)33 b(w)m(e)h(decomp)s(ose)p 328 2086 77 4 v 328 2166 a Fq(P)41 b Fs(=)28 b(1)-22 b(l)20 b Fm(\000)j Fq(P)46 b Fs(as)p 940 2086 V 33 w Fq(P)41 b Fs(=)27 b Fq(P)1224 2130 y Fo(l)1272 2166 y Fs(+)22 b Fq(P)1447 2130 y Fo(r)1507 2166 y Fs(+)g Fq(P)1682 2130 y Fp(0)1720 2166 y Fs(,)33 b(where)623 2386 y Fq(P)700 2345 y Fo(l)753 2386 y Fs(=)28 b Fq(p)22 b Fm(\012)p 1044 2331 49 4 v 39 w Fq(p)39 b Fm(\012)23 b Fs(1)-22 b(l)1287 2401 y Fo(f)1428 2386 y Fq(P)1505 2345 y Fo(r)1570 2386 y Fs(=)p 1690 2331 V 44 w Fq(p)39 b Fm(\012)23 b Fq(p)f Fm(\012)g Fs(1)-22 b(l)2103 2401 y Fo(f)2148 2386 y Fq(;)114 b(P)2366 2345 y Fp(0)2432 2386 y Fs(=)p 2553 2331 V 45 w Fq(p)38 b Fm(\012)p 2756 2331 V 39 w Fq(p)h Fm(\012)23 b Fs(1)-22 b(l)2999 2401 y Fo(f)3043 2386 y Fq(:)295 b Fs(\(5.1\))328 2606 y(It)32 b(is)g(easy)h(to)e(v)m(erify)i(that)f(the)g (\(regularized\))f(Liouvillian)d Fq(L)2638 2621 y Fo(\025)2684 2606 y Fs(,)k(de\014ned)h(in)f(\(2.45\),)f(is)328 2727 y(reduced)j(b)m(y)g(the)f(decomp)s(osition)1055 2947 y Fm(H)c Fs(=)f(Ran)16 b Fq(P)35 b Fm(\010)23 b Fs(Ran)16 b Fq(P)1929 2906 y Fo(l)1977 2947 y Fm(\010)23 b Fs(Ran)16 b Fq(P)2345 2906 y Fo(r)2405 2947 y Fm(\010)22 b Fs(Ran)17 b Fq(P)2773 2906 y Fp(0)2811 2947 y Fq(;)328 3167 y Fs(and)33 b(that)1355 3287 y Fq(L)1421 3302 y Fo(\025)1494 3287 y Fn(\026)27 b Fs(Ran)17 b Fq(P)1832 3246 y Fp(0)1898 3287 y Fs(=)28 b Fq(L)2068 3302 y Fp(0)2135 3287 y Fn(\026)g Fs(Ran)16 b Fq(P)2473 3246 y Fp(0)2512 3287 y Fq(:)826 b Fs(\(5.2\))328 3461 y(F)-8 b(rom)31 b(the)i(de\014nition,)f (\(2.38\),)g(of)g(the)h(mo)s(dular)d(conjugation)i Fq(J)9 b Fs(,)32 b(it)g(follo)m(ws)f(that)1702 3681 y Fq(J)9 b(P)1842 3640 y Fo(l)1895 3681 y Fs(=)27 b Fq(P)2075 3640 y Fo(r)2113 3681 y Fq(J)n(:)1173 b Fs(\(5.3\))328 3901 y(Because)45 b(ev)m(ery)h Fq( )k Fm(2)d(P)52 b Fs(satis\014es)44 b Fq(J)9 b( )51 b Fs(=)46 b Fq( )t Fs(,)h(\(5.3\))c(implies)e(that)i Fm(P)c(\\)30 b Fs(Ran)16 b Fq(P)3418 3865 y Fo(l)3490 3901 y Fs(=)328 4022 y Fm(P)31 b(\\)22 b Fs(Ran)16 b Fq(P)784 3986 y Fo(r)850 4022 y Fs(=)27 b Fm(f)p Fs(0)p Fm(g)p Fs(,)32 b(and,)h(consequen)m(tly)-8 b(,)35 b(w)m(e)e(ha)m(v)m(e) h(that)798 4242 y Fm(P)d(\\)22 b Fs(k)m(er)c Fq(L)1199 4257 y Fo(\025)1273 4242 y Fs(=)27 b Fm(P)k(\\)1564 4161 y Fi(\000)1610 4242 y Fs(k)m(er)18 b Fq(L)1823 4257 y Fo(\025)1896 4242 y Fn(\026)27 b Fs(Ran)16 b Fq(P)2233 4201 y Fp(0)2294 4242 y Fm([)23 b Fs(k)m(er)18 b Fq(L)2596 4257 y Fo(\025)2669 4242 y Fn(\026)27 b Fs(Ran)17 b Fq(P)3007 4161 y Fi(\001)3068 4242 y Fq(:)270 b Fs(\(5.4\))328 4462 y(W)-8 b(e)42 b(pro)m(v)m(e)h(in)e(the)h(next)g(section)g(that)g (k)m(er)17 b Fq(L)2073 4477 y Fo(\025)2162 4462 y Fn(\026)43 b Fs(Ran)16 b Fq(P)57 b Fs(=)43 b Fm(f)p Fs(0)p Fm(g)p Fs(,)g(whic)m(h,)i(together)328 4582 y(with)32 b(\(5.4\))g(and)g (\(5.2\),)g(sho)m(ws)i(that)e Fm(P)e(\\)22 b Fs(k)m(er)17 b Fq(L)2122 4597 y Fo(\025)2200 4582 y Fs(is)32 b(giv)m(en)g(b)m(y)h (the)g(subspace)h(de\014ned)328 4703 y(in)e(\(2.55\).)1898 5214 y(35)p eop %%Page: 36 36 36 35 bop 328 631 a Fh(5.2)135 b(The)45 b(k)l(ernel)h(of)f Fc(L)1544 649 y Fb(\025)1629 631 y Fa(\026)33 b FD(Ran)20 b Fc(P)328 816 y Ft(Theorem)37 b(5.1)49 b FB(Given)35 b(any)h Fs(0)29 b Fq(<)g(\014)35 b(<)29 b Fm(1)36 b FB(and)f(any)g Fq(r)d(>)d Fs(0)36 b FB(ther)-5 b(e)36 b(is)f(a)g Fq(\025)3116 831 y Fp(0)3156 816 y Fs(\()p Fq(\014)6 b(;)17 b(r)s Fs(\))28 b Fq(>)h Fs(0)328 936 y FB(s.t.)50 b(if)36 b Fs(0)30 b Fq(<)h Fm(j)p Fq(\025)p Fm(j)f Fq(<)g(\025)1098 951 y Fp(0)1137 936 y Fs(\()p Fq(\014)6 b(;)17 b(r)s Fs(\))36 b FB(then)g Fs(k)m(er)18 b Fq(L)1832 951 y Fo(\025)1908 936 y Fn(\026)30 b Fs(Ran)16 b Fq(P)44 b Fs(=)31 b Fm(f)p Fs(0)p Fm(g)p FB(.)49 b(Her)-5 b(e,)37 b Fq(\025)2935 951 y Fp(0)2974 936 y Fs(\()p Fq(\014)6 b(;)17 b(r)s Fs(\))30 b Fm(\025)h Fq(k)s(\015)3450 900 y Fp(2)3489 936 y Fq(r)s FB(,)328 1056 y(for)k(some)f Fq(k)k FB(indep)-5 b(endent)33 b(of)i Fq(\014)6 b(;)17 b(r)s FB(.)43 b(The)35 b(c)-5 b(onstant)34 b Fq(\015)40 b FB(is)35 b(given)f(in)g(\(2.24\).) 474 1285 y(Pr)-5 b(o)g(of.)69 b Fs(W)-8 b(e)26 b(use)g(the)g(notation)e (of)h(Subsection)i(3.3.2)d(and)i(write)f Fq(L)h Fs(for)f Fq(L)3202 1300 y Fo(\025)3247 1285 y Fs(.)42 b(In)25 b(the)328 1405 y(spirit)i(of)h(the)h(p)s(ositiv)m(e)e(comm)m(utator)g (metho)s(d)h(outlined)f(in)g(Section)i(3.2,)g(w)m(e)g(w)m(an)m(t)g(to) 328 1526 y(establish)38 b(a)g(lo)m(w)m(er)g(b)s(ound)h(on)f(the)h(exp)s (ectation)f(v)-5 b(alue)38 b Fm(h)p Fq( )t(;)17 b Fs(\()p Fq(C)2752 1541 y Fp(1)2813 1526 y Fs(+)22 b Fq(i)p Fs([)p Fq(L;)17 b(A)3154 1541 y Fp(0)3194 1526 y Fs(]\))p Fq( )t Fm(i)38 b Fs(\(see)328 1646 y(also)32 b(\(3.42\)\),)f(where)j Fq( )j Fs(is)32 b(a)g(\(h)m(yp)s(othetical\))g(eigen)m(v)m(ector)i(of)e Fq(L)h Fs(in)e(Ran)17 b Fq(P)d Fs(.)474 1766 y(Using)32 b(the)h(relativ)m(e)f(b)s(ound)1326 2040 y Fm(\006)p Fq(\025I)1503 2055 y Fp(1)1571 2040 y Fm(\025)c(\000)p Fq(cN)33 b Fm(\000)2016 1973 y Fq(\025)2073 1936 y Fp(2)p 2016 2017 97 4 v 2043 2108 a Fq(c)2122 2040 y(;)114 b Fm(8)p Fq(c)28 b(>)g Fs(0)p Fq(;)797 b Fs(\(5.5\))328 2291 y(with)35 b Fq(c)d Fs(=)g(1)p Fq(=)p Fs(10,)j(and)g(\(3.46\),)g(w) m(e)i(obtain)d(a)h(lo)m(w)m(er)g(b)s(ound)g(\(alw)m(a)m(ys)h(in)f(the)g (sense)i(of)328 2411 y(quadratic)32 b(forms)g(on)g Fm(D)s Fs(\))391 2631 y Fq(C)461 2646 y Fp(1)522 2631 y Fs(+)22 b Fq(i)p Fs([)p Fq(L;)17 b(A)863 2646 y Fp(0)903 2631 y Fs(])486 2836 y Fm(\025)97 b Fq(\037)721 2795 y Fp(2)760 2836 y Fq(H)841 2851 y Fo(p)903 2836 y Fm(\012)22 b Fs(1)-22 b(l)1057 2851 y Fo(p)1118 2836 y Fs(+)22 b(1)-22 b(l)1271 2851 y Fo(p)1332 2836 y Fm(\012)22 b Fq(\037)1492 2795 y Fp(2)1532 2836 y Fq(H)1613 2851 y Fo(p)1674 2836 y Fs(+)1807 2768 y(9)p 1782 2813 98 4 v 1782 2904 a(10)1890 2836 y Fq(N)32 b Fs(+)23 b Fq(i)p Fs([)p Fq(L;)17 b(A)2342 2851 y Fp(0)2382 2836 y Fs(])22 b Fm(\000)2541 2768 y Fq(\025)2598 2732 y Fp(2)p 2540 2813 V 2540 2904 a Fs(10)486 3084 y Fm(\025)97 b Fq(P)753 2943 y Fi(\022)826 3084 y Fq(\037)887 3043 y Fp(2)927 3084 y Fq(H)1008 3099 y Fo(p)1069 3084 y Fm(\012)23 b Fs(1)-22 b(l)1224 3099 y Fo(p)1284 3084 y Fm(\012)23 b Fq(P)1447 3099 y Fp(\012)1524 3084 y Fs(+)f(1)-22 b(l)1677 3099 y Fo(p)1738 3084 y Fm(\012)22 b Fq(\037)1898 3043 y Fp(2)1938 3084 y Fq(H)2019 3099 y Fo(p)2080 3084 y Fm(\012)h Fq(P)2243 3099 y Fp(\012)2320 3084 y Fs(+)2453 3016 y(9)p 2428 3061 V 2428 3152 a(10)p 2536 3004 77 4 v 2536 3084 a Fq(P)2612 3099 y Fp(\012)2689 3084 y Fs(+)f Fq(i)p Fs([)p Fq(L;)17 b(A)3030 3099 y Fp(0)3070 3084 y Fs(])23 b Fm(\000)3230 3016 y Fq(\025)3287 2980 y Fp(2)p 3229 3061 98 4 v 3229 3152 a Fs(10)3337 2943 y Fi(\023)3427 3084 y Fq(P)660 3358 y Fs(+)p 736 3278 77 4 v Fq(P)13 b(i)p Fs([)p Fq(L;)k(A)1055 3373 y Fp(0)1095 3358 y Fs(])p Fq(P)36 b Fs(+)22 b Fq(P)14 b(i)p Fs([)p Fq(L;)j(A)1639 3373 y Fp(0)1678 3358 y Fs(])p 1705 3278 V Fq(P)36 b Fs(+)p 1902 3278 V 22 w Fq(P)13 b(i)p Fs([)p Fq(L;)k(A)2221 3373 y Fp(0)2261 3358 y Fs(])p 2288 3278 V Fq(P)36 b Fm(\000)2497 3291 y Fq(\025)2554 3254 y Fp(2)p 2496 3335 98 4 v 2496 3426 a Fs(10)p 2604 3278 77 4 v 2604 3358 a Fq(P)474 3534 y Fs(=:)83 b Fq(P)14 b(M)831 3549 y Fp(0)870 3534 y Fq(P)35 b Fs(+)22 b Fq(M)1160 3549 y Fp(1)1200 3534 y Fq(;)2138 b Fs(\(5.6\))328 3754 y(where)34 b(the)f(b)s(ounded)g(op)s(erators)f Fq(M)1701 3769 y Fp(0)1774 3754 y Fs(and)g Fq(M)2057 3769 y Fp(1)2130 3754 y Fs(are)g(giv)m(en)h(b)m(y)759 4013 y Fq(M)853 4028 y Fp(0)976 4013 y Fs(:=)83 b Fq(p)1211 4028 y Fo(J)1250 4036 y Fl(c)1286 4013 y Fq(H)1367 4028 y Fo(p)1429 4013 y Fm(\012)22 b Fs(1)-22 b(l)1583 4028 y Fo(p)1644 4013 y Fm(\012)22 b Fq(P)1806 4028 y Fp(\012)1884 4013 y Fs(+)g(1)-22 b(l)2037 4028 y Fo(p)2097 4013 y Fm(\012)23 b Fq(p)2246 4028 y Fo(J)2285 4036 y Fl(c)2320 4013 y Fq(H)2401 4028 y Fo(p)2463 4013 y Fm(\012)g Fq(P)2626 4028 y Fp(\012)2703 4013 y Fs(+)2835 3946 y(9)p 2811 3990 98 4 v 2811 4081 a(10)p 2918 3933 77 4 v 2918 4013 a Fq(P)2995 4028 y Fp(\012)1260 4261 y Fs(+)p Fq(i)p Fs([)p Fq(L;)17 b(A)1579 4276 y Fp(0)1619 4261 y Fs(])22 b Fm(\000)1778 4194 y Fq(\025)1835 4157 y Fp(2)p 1778 4238 98 4 v 1778 4329 a Fs(10)1885 4261 y Fq(;)1453 b Fs(\(5.7\))759 4509 y Fq(M)853 4524 y Fp(1)976 4509 y Fs(:=)p 1162 4429 77 4 v 83 w Fq(P)14 b(i)p Fs([)p Fq(L;)j(A)1482 4524 y Fp(0)1522 4509 y Fs(])p Fq(P)35 b Fs(+)22 b Fq(P)14 b(i)p Fs([)p Fq(L;)j(A)2065 4524 y Fp(0)2105 4509 y Fs(])p 2132 4429 V Fq(P)36 b Fs(+)p 2329 4429 V 22 w Fq(P)13 b(i)p Fs([)p Fq(L;)k(A)2648 4524 y Fp(0)2688 4509 y Fs(])p 2715 4429 V Fq(P)36 b Fm(\000)2924 4441 y Fq(\025)2981 4405 y Fp(2)p 2923 4486 98 4 v 2923 4577 a Fs(10)p 3031 4429 77 4 v 3031 4509 a Fq(P)13 b(:)231 b Fs(\(5.8\))474 4754 y(The)27 b(di\016cult)d(part)h(of)g(the)h(pro)s(of)f(of)g(Theorem)g(2.3)g(is)g (con)m(tained)h(in)f(the)g(follo)m(wing)328 4875 y(t)m(w)m(o)33 b(prop)s(ositions.)1898 5214 y(36)p eop %%Page: 37 37 37 36 bop 328 631 a Ft(Prop)s(osition)35 b(5.1)49 b FB(Supp)-5 b(ose)26 b(Pr)-5 b(op)g(osition)26 b(3.2)g(holds)f(and)h(that)h(the)f (p)-5 b(ar)g(ameters)26 b(sat-)328 751 y(isfy)403 1023 y Fs(0)h Fq(<)h Fm(j)p Fq(\025)p Fm(j)f Fq(<)h Fs(min)1006 882 y Fi(\022)1079 1023 y Fs(1)p Fq(;)1172 946 y Fm(p)p 1255 946 47 4 v 77 x Fq(r)r(;)1452 955 y(\017)p 1355 1000 235 4 v 1355 1104 a Fs(3)1404 1020 y Fm(p)p 1487 1020 49 4 v 84 x Fq(\022)s(k)1599 1023 y(;)1701 955 y(\017)p 1653 1000 138 4 v 1653 1020 a Fm(p)p 1736 1020 55 4 v 84 x Fq(k)1800 882 y Fi(\023)1890 1023 y Fq(;)51 b Fs(0)28 b Fq(<)f(\017)h(<)g Fs(min)n(\(5)p Fq(\022)s(\015)5 b(;)17 b(\017)2755 1038 y Fp(0)2795 1023 y Fs(\))p Fq(;)51 b Fs(0)28 b Fq(<)f(\022)k(<)d(r)s(=)p Fs(32)p Fq(;)3365 1200 y Fs(\(5.9\))328 1321 y FB(wher)-5 b(e)42 b Fq(k)k FB(is)c(a)h(c)-5 b(onstant)43 b(dep)-5 b(ending)41 b(on)h(the)h(inter) -5 b(action,)44 b(but)f(not)g(on)g(any)f(of)h(the)328 1441 y(p)-5 b(ar)g(ameters)38 b Fq(\017;)17 b(\025;)g(\022)s FB(,)40 b(nor)e(on)h Fq(\014)44 b FB(\(for)39 b Fq(\014)h Fm(\025)c Fq(\014)2022 1456 y Fp(0)2062 1441 y FB(,)j(with)g Fq(\014)2402 1456 y Fp(0)2477 1441 y Fq(>)c Fs(0)j FB(\014xe)-5 b(d\).)56 b(Then)38 b(ther)-5 b(e)39 b(is)328 1561 y(an)34 b(interval)h Fs(\001)g FB(ar)-5 b(ound)35 b(zer)-5 b(o)34 b(such)h(that)1378 1815 y Fq(P)14 b(E)1533 1774 y Fp(0)1527 1840 y(\001)1589 1815 y Fq(M)1683 1830 y Fp(0)1723 1815 y Fq(E)1801 1774 y Fp(0)1795 1840 y(\001)1858 1815 y Fq(P)41 b Fm(\025)2078 1748 y Fq(\022)s(\025)2183 1711 y Fp(2)p 2078 1792 145 4 v 2130 1883 a Fq(\017)2232 1815 y(\015)5 b(E)2366 1774 y Fp(0)2360 1840 y(\001)2423 1815 y Fq(P)s(;)801 b Fs(\(5.10\))328 2070 y FB(wher)-5 b(e)34 b Fq(E)681 2033 y Fp(0)675 2095 y(\001)766 2070 y Fs(=)28 b Fq(E)942 2085 y Fp(\001)1005 2070 y Fs(\()p Fq(L)1109 2085 y Fp(0)1148 2070 y Fs(\))35 b FB(is)g(the)g(sp)-5 b(e)g(ctr)g(al)34 b(pr)-5 b(oje)g(ction)34 b(of)h Fq(L)2468 2085 y Fp(0)2543 2070 y FB(onto)f Fs(\001)p FB(.)328 2297 y Ft(Prop)s(osition)h(5.2)49 b FB(Assume)29 b(that)g(the)g(c)-5 b(onditions)27 b(of)i(Pr)-5 b(op)g(osition)28 b(5.1)g(ar)-5 b(e)29 b(satis\014e)-5 b(d)328 2418 y(and)34 b(that)1474 2538 y Fm(j)p Fq(\025)p Fm(j)26 b Fq(<)1727 2471 y(\015)p 1727 2515 57 4 v 1728 2606 a(k)1810 2538 y Fs(min)15 b(\(1)p Fq(;)i(\017;)g(\022)s(=\017)p Fs(\))f Fq(:)897 b Fs(\(5.11\))328 2738 y FB(If)34 b Fq( )e Fm(2)c Fs(Ran)16 b Fq(P)49 b FB(is)34 b(an)h(eigenve)-5 b(ctor)34 b(of)g Fq(L)1853 2753 y Fo(\025)1934 2738 y FB(then)1425 3011 y Fm(h)p Fq( )t(;)17 b(M)1669 3026 y Fp(0)1708 3011 y Fq( )t Fm(i)27 b(\025)1957 2943 y Fs(1)p 1957 2988 49 4 v 1957 3079 a(2)2025 2943 y Fq(\022)s(\025)2130 2907 y Fp(2)p 2025 2988 145 4 v 2078 3079 a Fq(\017)2180 3011 y(\015)5 b Fm(k)p Fq( )t Fm(k)2403 2969 y Fp(2)2442 3011 y Fq(:)848 b Fs(\(5.12\))517 3293 y(W)-8 b(e)43 b(ma)m(y)f(c)m(ho)s (ose)h(the)g(parameters)g(as)f Fq(\025)j Fs(=)2280 3266 y Fi(e)2279 3293 y Fq(\025\015)2392 3257 y Fp(2)2431 3293 y Fs(,)g Fq(\017)g Fs(=)d Fi(e)-52 b Fq(\017\015)5 b Fs(,)45 b(with)3108 3266 y Fi(e)3107 3293 y Fq(\025)p Fs(,)e Fi(e)-53 b Fq(\017)43 b Fs(and)g Fq(\022)328 3413 y Fs(indep)s(enden)m(t)c(of)f(the)h(in)m(v)m(erse)h(temp)s(erature)e Fq(\014)6 b Fs(.)60 b(Conditions)37 b(\(5.9\))h(and)g(\(5.11\))g(are) 328 3545 y(satis\014ed)33 b(pro)m(vided)g(0)27 b Fq(<)h Fm(j)1313 3518 y Fi(e)1312 3545 y Fq(\025)p Fm(j)f Fq(<)g(k)s(r)s Fs(,)33 b(for)f(some)g Fq(k)k Fs(indep)s(enden)m(t)e(of)e Fq(\014)38 b Fs(and)33 b Fq(r)s Fs(.)474 3665 y(Theorem)43 b(5.1)f(is)g(no)m(w)h(pro)m(v)m(en)h(as)f(follo)m(ws.)72 b(Assume)44 b(that)e Fq( )49 b Fm(2)c Fs(Ran)16 b Fq(P)56 b Fs(is)42 b(an)328 3785 y(eigen)m(v)m(ector)28 b(of)e Fq(L)1000 3800 y Fo(\025)1046 3785 y Fs(,)i(and)f(let)f Fq( )1483 3800 y Fo(\013)1560 3785 y Fs(b)s(e)h(the)g(family)d(of)i (appro)m(ximate)g(eigen)m(v)m(ectors)j(giv)m(en)328 3906 y(in)i(Theorem)h(3.3.)43 b(F)-8 b(rom)30 b(\(5.6\),)i(\(5.12\),)f(and)h (using)f(that)h Fm(h)p Fq( )t(;)17 b(M)2768 3921 y Fp(1)2807 3906 y Fq( )t Fm(i)28 b Fs(=)f(0,)32 b(w)m(e)h(obtain)972 4184 y(0)28 b(=)f(lim)1197 4244 y Fo(\013)1304 4184 y Fm(h)p Fq( )1406 4199 y Fo(\013)1456 4184 y Fq(;)17 b Fs(\()p Fq(C)1608 4199 y Fp(1)1669 4184 y Fs(+)22 b Fq(i)p Fs([)p Fq(L;)17 b(A)2010 4199 y Fp(0)2050 4184 y Fs(]\))p Fq( )2178 4199 y Fo(\013)2228 4184 y Fm(i)27 b(\025)2409 4117 y Fs(1)p 2409 4161 49 4 v 2409 4253 a(2)2478 4117 y Fq(\022)s(\025)2583 4081 y Fp(2)p 2478 4161 145 4 v 2531 4253 a Fq(\017)2632 4184 y(\015)5 b Fm(k)p Fq( )t Fm(k)2855 4143 y Fp(2)2895 4184 y Fq(;)328 4431 y Fs(whic)m(h)33 b(is)f(a)g(con)m(tradiction,)g(b)s(ecause)i(the)f(r.h.s.)44 b(is)32 b(strictly)g(p)s(ositiv)m(e.)474 4671 y FB(Pr)-5 b(o)g(of)35 b(of)f(Pr)-5 b(op)g(osition)35 b(5.1.)77 b Fs(Pic)m(k)33 b(\001)g(suc)m(h)h(that)598 4930 y Fm(j)p Fs(\001)p Fm(j)27 b Fq(<)876 4862 y Fs(1)p 876 4907 49 4 v 876 4998 a(2)951 4930 y(min)1130 4819 y Fi(n)1197 4930 y Fm(j)p Fq(E)6 b Fs(\()p Fq(m)p Fs(\))22 b Fm(\000)g Fq(E)6 b Fs(\()p Fq(n)p Fs(\))p Fm(j)1858 4845 y Fi(\014)1858 4905 y(\014)1923 4930 y Fq(m;)17 b(n)28 b Fm(2)g Fq(J)2286 4945 y Fo(d)2327 4930 y Fq(;)17 b(E)6 b Fs(\()p Fq(m)p Fs(\))27 b Fm(6)p Fs(=)h Fq(E)6 b Fs(\()p Fq(n)p Fs(\))2953 4819 y Fi(o)3020 4930 y Fq(:)270 b Fs(\(5.13\))1898 5214 y(37)p eop %%Page: 38 38 38 37 bop 328 631 a Fs(The)33 b(Hilb)s(ert)e(space)j(Ran)16 b Fq(E)1394 595 y Fp(0)1388 656 y(\001)1452 631 y Fq(P)46 b Fs(has)33 b(the)g(decomp)s(osition)1197 844 y(Ran)16 b Fq(E)1466 803 y Fp(0)1460 869 y(\001)1523 844 y Fq(P)41 b Fs(=)28 b(Ran)16 b Fq(P)e Fs(\005)22 b Fm(\010)h Fs(Ran)16 b Fq(E)2463 803 y Fp(0)2457 869 y(\001)2520 844 y Fq(P)p 2597 764 74 4 v 14 w Fs(\005)p Fq(;)620 b Fs(\(5.14\))328 1058 y(where,)40 b(w)m(e)e(recall,)f(\005)f(=)g Fq(P)1368 1073 y Fp(0)1433 1058 y Fm(\012)26 b Fq(P)1599 1073 y Fp(\012)1691 1058 y Fs(is)37 b(the)h(pro)5 b(jection)37 b(on)m(to)g(the)h(k)m(ernel)g(of)f Fq(L)3306 1073 y Fp(0)3346 1058 y Fs(.)57 b(W)-8 b(e)328 1178 y(analyze)40 b(the)g(sp)s(ectrum)g (of)f(the)h(op)s(erator)f Fq(P)14 b(E)2141 1142 y Fp(0)2135 1203 y(\001)2198 1178 y Fq(M)2292 1193 y Fp(0)2332 1178 y Fq(E)2410 1142 y Fp(0)2404 1203 y(\001)2467 1178 y Fq(P)53 b Fs(on)39 b(Ran)17 b Fq(E)2995 1142 y Fp(0)2989 1203 y(\001)3052 1178 y Fq(P)53 b Fs(using)39 b(the)328 1298 y(F)-8 b(esh)m(bac)m(h)40 b(metho)s(d.)61 b(F)-8 b(or)38 b(details)g(on)g(the)i(F)-8 b(esh)m(bac)m(h)40 b(metho)s(d,)g(w)m(e)f(refer)g(to)g([BFS].)328 1419 y(Let)i Fq(m)g Fs(b)s(e)g(a)f(n)m(um)m(b)s(er)h(in)f(the)h(resolv)m(en)m(t)h (set)f(of)p 2225 1339 V 40 w(\005)p Fq(P)14 b(E)2453 1383 y Fp(0)2447 1444 y(\001)2510 1419 y Fq(M)2604 1434 y Fp(0)2644 1419 y Fq(E)2722 1383 y Fp(0)2716 1444 y(\001)2779 1419 y Fq(P)p 2856 1339 V 14 w Fs(\005)40 b(\(view)m(ed)i(as)f(an)328 1539 y(op)s(erator)c(on)h(Ran)16 b Fq(E)1136 1503 y Fp(0)1130 1564 y(\001)1194 1539 y Fq(P)p 1271 1459 V 14 w Fs(\005)o(\).)60 b(The)39 b(F)-8 b(esh)m(bac)m(h)39 b(map)e Fq(F)2383 1554 y Fp(\005)p Fo(;m)2561 1539 y Fs(applied)f(to)i(the)g(op)s(erator) 328 1660 y Fq(P)14 b(E)483 1623 y Fp(0)477 1685 y(\001)540 1660 y Fq(M)634 1675 y Fp(0)673 1660 y Fq(E)751 1623 y Fp(0)745 1685 y(\001)809 1660 y Fq(P)45 b Fs(is)33 b(de\014ned)h(as)399 1873 y Fq(F)462 1888 y Fp(\005)p Fo(;m)602 1873 y Fs(\()p Fq(P)14 b(E)795 1832 y Fp(0)789 1898 y(\001)851 1873 y Fq(M)945 1888 y Fp(0)985 1873 y Fq(E)1063 1832 y Fp(0)1057 1898 y(\001)1120 1873 y Fq(P)g Fs(\))2082 b(\(5.15\))482 2049 y(=)83 b(\005)731 1938 y Fi(\020)791 2049 y Fq(P)14 b(M)962 2064 y Fp(0)1001 2049 y Fq(P)35 b Fm(\000)23 b Fq(P)14 b(M)1370 2064 y Fp(0)p 1409 1969 V 1409 2049 a Fs(\005)p Fq(P)g(E)1637 2008 y Fp(0)1631 2073 y(\001)1710 1968 y Fi(\000)p 1756 1969 V 81 x Fs(\005)p Fq(P)g(E)1984 2008 y Fp(0)1978 2073 y(\001)2041 2049 y Fq(M)2135 2064 y Fp(0)2175 2049 y Fq(E)2253 2008 y Fp(0)2247 2073 y(\001)2310 2049 y Fq(P)p 2387 1969 V 14 w Fs(\005)22 b Fm(\000)g Fq(m)2666 1968 y Fi(\001)2712 1987 y Fx(\000)p Fp(1)2823 2049 y Fq(E)2901 2008 y Fp(0)2895 2073 y(\001)2958 2049 y Fq(P)p 3035 1969 V 14 w Fs(\005)p Fq(M)3202 2064 y Fp(0)3242 2049 y Fq(P)3319 1938 y Fi(\021)3394 2049 y Fs(\005)p Fq(;)328 2284 y Fs(and)47 b(has)g(the)g(follo)m(wing)d(prop)s(ert)m(y)k (of)e FB(isosp)-5 b(e)g(ctr)g(ality)p Fs(:)71 b(Let)47 b Fq(\033)k Fs(and)c Fq(\032)g Fs(denote)g(the)328 2404 y(sp)s(ectrum)33 b(and)g(the)g(resolv)m(en)m(t)g(set)h(of)e(an)g(op)s (erator.)43 b(Then)704 2618 y Fq(z)33 b Fm(2)28 b Fq(\033)t Fs(\()p Fq(P)14 b(E)1128 2577 y Fp(0)1122 2642 y(\001)1184 2618 y Fq(M)1278 2633 y Fp(0)1318 2618 y Fq(E)1396 2577 y Fp(0)1390 2642 y(\001)1453 2618 y Fq(P)g Fs(\))22 b Fm(\\)g Fq(\032)p Fs(\()p 1766 2538 V(\005)q Fq(P)14 b(E)1995 2577 y Fp(0)1989 2642 y(\001)2051 2618 y Fq(M)2145 2633 y Fp(0)2185 2618 y Fq(E)2263 2577 y Fp(0)2257 2642 y(\001)2320 2618 y Fq(P)p 2397 2538 V 14 w Fs(\005\))809 b(\(5.16\))870 2763 y Fm(\()-17 b(\))28 b Fq(z)k Fm(2)c Fq(\033)t Fs(\()p Fq(F)1412 2778 y Fp(\005)p Fo(;m)1551 2763 y Fs(\()p Fq(P)14 b(E)1744 2722 y Fp(0)1738 2788 y(\001)1801 2763 y Fq(M)1895 2778 y Fp(0)1935 2763 y Fq(E)2013 2722 y Fp(0)2007 2788 y(\001)2070 2763 y Fq(P)g Fs(\)\))21 b Fm(\\)i Fq(\032)p Fs(\()p 2421 2683 V(\005)p Fq(P)14 b(E)2649 2722 y Fp(0)2643 2788 y(\001)2706 2763 y Fq(M)2800 2778 y Fp(0)2840 2763 y Fq(E)2918 2722 y Fp(0)2912 2788 y(\001)2975 2763 y Fq(P)p 3052 2683 V 14 w Fs(\005)o(\))p Fq(:)328 2976 y Fs(The)43 b(p)s(oin)m(t)e(of)h(the) h(F)-8 b(esh)m(bac)m(h)44 b(metho)s(d)d(is)h(that)g(it)f(can)i(b)s(e)f (easier)g(to)g(analyze)g(the)328 3097 y(sp)s(ectrum)c(of)g(the)g(op)s (erator)f(\(5.15\))g(than)h(the)g(one)g(of)g Fq(P)14 b(E)2596 3061 y Fp(0)2590 3122 y(\001)2652 3097 y Fq(M)2746 3112 y Fp(0)2786 3097 y Fq(E)2864 3061 y Fp(0)2858 3122 y(\001)2921 3097 y Fq(P)g Fs(,)39 b(b)s(ecause)g(the)328 3217 y(op)s(erator)32 b(\(5.15\))g(acts)h(on)f(the)h(smaller)e(space)i (Ran)17 b Fq(P)d Fs(\005.)474 3337 y(Let)32 b(us)g(examine)f(the)h (diagonal)d(blo)s(c)m(ks)j(of)f Fq(P)14 b(E)2269 3301 y Fp(0)2263 3363 y(\001)2326 3337 y Fq(M)2420 3352 y Fp(0)2460 3337 y Fq(E)2538 3301 y Fp(0)2532 3363 y(\001)2595 3337 y Fq(P)45 b Fs(in)31 b(the)h(decomp)s(ositon)328 3458 y(\(5.14\).)43 b(It)32 b(is)g(readily)g(v)m(eri\014ed)h(that)1103 3730 y(\005)p Fq(P)14 b(M)1347 3745 y Fp(0)1386 3730 y Fq(P)g Fs(\005)27 b(=)h(2)p Fq(\022)s(\025)1821 3689 y Fp(2)1860 3730 y Fs(\005)p Fq(P)14 b(I)p 2077 3650 59 4 v 7 w(R)2135 3669 y Fp(2)2135 3755 y Fo(\017)2175 3730 y Fq(I)8 b(P)14 b Fs(\005)21 b Fm(\000)2508 3663 y Fq(\025)2565 3627 y Fp(2)p 2507 3708 98 4 v 2507 3799 a Fs(10)2615 3730 y Fq(P)14 b Fs(\005)p Fq(:)525 b Fs(\(5.17\))328 3988 y(Because)34 b(of)e(\(5.13\),)g Fq(E)1199 3952 y Fp(0)1193 4013 y(\001)p 1256 3908 77 4 v 1256 3988 a Fq(P)1333 4003 y Fp(0)1405 3988 y Fq(p)1454 4003 y Fo(J)1493 4015 y Fl(d)1555 3988 y Fm(\012)22 b Fq(p)1703 4003 y Fo(J)1742 4015 y Fl(d)1804 3988 y Fm(\012)h Fq(P)1967 4003 y Fp(\012)2049 3988 y Fs(=)28 b(0.)43 b(Hence)544 4201 y Fq(E)622 4160 y Fp(0)616 4226 y(\001)679 4201 y Fq(P)p 756 4121 74 4 v 14 w Fs(\005)27 b(=)h Fq(E)1038 4160 y Fp(0)1032 4226 y(\001)1095 4201 y Fq(P)p 1172 4121 77 4 v 14 w(P)1248 4216 y Fp(\012)1325 4201 y Fs(+)22 b Fq(E)1501 4160 y Fp(0)1495 4226 y(\001)1558 4201 y Fq(P)30 b Fs(\()p Fq(p)1738 4216 y Fo(J)1777 4228 y Fl(d)1839 4201 y Fm(\012)23 b Fq(p)1988 4216 y Fo(J)2027 4224 y Fl(c)2085 4201 y Fs(+)f Fq(p)2232 4216 y Fo(J)2271 4224 y Fl(c)2328 4201 y Fm(\012)h Fq(p)2477 4216 y Fo(J)2516 4228 y Fl(d)2578 4201 y Fs(+)f Fq(p)2725 4216 y Fo(J)2764 4224 y Fl(c)2822 4201 y Fm(\012)g Fq(p)2970 4216 y Fo(J)3009 4224 y Fl(c)3045 4201 y Fs(\))g Fm(\012)h Fq(P)3268 4216 y Fp(\012)3323 4201 y Fq(:)328 4415 y Fs(Set)47 b Fq(Q)587 4430 y Fp(1)679 4415 y Fs(=)k Fq(E)884 4379 y Fp(0)878 4440 y(\001)941 4415 y Fq(P)p 1018 4335 V 14 w(P)1094 4430 y Fp(\012)1196 4415 y Fs(and)c(let)f Fq(Q)1632 4430 y Fp(2)1718 4415 y Fs(b)s(e)h(the)g(other)g(pro)5 b(jection)47 b(on)f(the)i(righ)m(t)e(side.)328 4535 y Fq(P)14 b(E)483 4499 y Fp(0)477 4560 y(\001)540 4535 y Fq(M)634 4550 y Fp(0)673 4535 y Fq(E)751 4499 y Fp(0)745 4560 y(\001)809 4535 y Fq(P)48 b Fs(is)34 b(diagonal)f(in)h(this)g(decomp)s(osition)f (of)i(Ran)16 b Fq(E)2749 4499 y Fp(0)2743 4560 y(\001)2806 4535 y Fq(P)p 2883 4455 74 4 v 14 w Fs(\005.)50 b(W)-8 b(e)35 b(ha)m(v)m(e)h(the)328 4656 y(estimates)607 4903 y Fq(Q)684 4918 y Fp(1)724 4903 y Fq(M)818 4918 y Fp(0)858 4903 y Fq(Q)935 4918 y Fp(1)1002 4903 y Fm(\025)1108 4763 y Fi(\022)1215 4836 y Fs(9)p 1191 4880 98 4 v 1191 4972 a(10)1320 4903 y Fm(\000)1431 4836 y Fq(\025)1488 4800 y Fp(2)p 1430 4880 V 1430 4972 a Fs(10)1560 4903 y Fm(\000)22 b Fq(k)1723 4836 y(\022)s(\025)1828 4800 y Fp(2)p 1723 4880 145 4 v 1756 4972 a Fq(\017)1795 4943 y Fp(2)1878 4763 y Fi(\023)1968 4903 y Fq(Q)2045 4918 y Fp(1)2084 4903 y Fq(;)82 b(Q)2270 4918 y Fp(2)2310 4903 y Fq(M)2404 4918 y Fp(0)2444 4903 y Fq(Q)2521 4918 y Fp(2)2588 4903 y Fm(\025)2693 4763 y Fi(\022)2767 4903 y Fq(r)24 b Fm(\000)2946 4836 y Fq(\025)3003 4800 y Fp(2)p 2945 4880 98 4 v 2945 4972 a Fs(10)3053 4763 y Fi(\023)3142 4903 y Fq(Q)3219 4918 y Fp(2)3259 4903 y Fq(;)1898 5214 y Fs(38)p eop %%Page: 39 39 39 38 bop 328 631 a Fs(where)33 b Fq(k)d Fs(=)e(2)p Fm(k)p Fq(I)8 b Fs(\005)p Fq(I)g Fm(k)31 b Fs(and)g(w)m(e)i(used)g(that)e Fq(p)1961 646 y Fo(J)2000 654 y Fl(c)2036 631 y Fq(H)2117 646 y Fo(p)2184 631 y Fm(\025)d Fq(r)s(p)2385 646 y Fo(J)2424 654 y Fl(c)2459 631 y Fs(.)43 b(Consequen)m(tly)-8 b(,)34 b(w)m(e)f(obtain)328 751 y(the)g(lo)m(w)m(er)g(b)s(ound)451 981 y Fq(P)p 528 901 74 4 v 14 w Fs(\005)o Fq(E)678 940 y Fp(0)672 1005 y(\001)735 981 y Fq(M)829 996 y Fp(0)869 981 y Fq(E)947 940 y Fp(0)941 1005 y(\001)p 1004 901 V 1004 981 a Fs(\005)p Fq(P)42 b Fm(\025)28 b Fs(min)1466 840 y Fi(\022)1574 913 y Fs(9)p 1549 958 98 4 v 1549 1049 a(10)1679 981 y Fm(\000)1789 913 y Fq(\025)1846 877 y Fp(2)p 1788 958 V 1788 1049 a Fs(10)1918 981 y Fm(\000)23 b Fq(k)2082 913 y(\022)s(\025)2187 877 y Fp(2)p 2082 958 145 4 v 2115 1049 a Fq(\017)2154 1020 y Fp(2)2236 981 y Fq(;)17 b(r)25 b Fm(\000)2459 913 y Fq(\025)2516 877 y Fp(2)p 2458 958 98 4 v 2458 1049 a Fs(10)2566 840 y Fi(\023)2656 981 y Fq(E)2734 940 y Fp(0)2728 1005 y(\001)p 2791 901 74 4 v 2791 981 a Fs(\005)p Fq(P)41 b Fm(\025)3084 913 y Fq(r)p 3083 958 49 4 v 3083 1049 a Fs(2)3142 981 y Fq(E)3220 940 y Fp(0)3214 1005 y(\001)p 3277 901 74 4 v 3277 981 a Fs(\005)p Fq(P)s(;)3317 1158 y Fs(\(5.18\))328 1279 y(due)47 b(to)e(\(5.9\).)83 b(It)46 b(follo)m(ws)f(that)h(an)m(y)h Fq(m)j(<)h(r)s(=)p Fs(4)45 b(is)g(in)h(the)g(resolv)m(en)m(t)h(set)g (of)e(the)328 1399 y(op)s(erator)32 b Fq(P)p 798 1319 V 14 w Fs(\005)o Fq(E)948 1363 y Fp(0)942 1424 y(\001)1006 1399 y Fq(M)1100 1414 y Fp(0)1139 1399 y Fq(E)1217 1363 y Fp(0)1211 1424 y(\001)p 1275 1319 V 1275 1399 a Fs(\005)p Fq(P)46 b Fs(and)1194 1507 y Fi(\015)1194 1567 y(\015)1194 1627 y(\015)1250 1541 y(\000)1295 1622 y Fq(P)p 1372 1542 V 14 w Fs(\005)p Fq(E)1523 1581 y Fp(0)1517 1647 y(\001)1580 1622 y Fq(M)1674 1637 y Fp(0)1714 1622 y Fq(E)1792 1581 y Fp(0)1786 1647 y(\001)p 1849 1542 V 1849 1622 a Fs(\005)p Fq(P)36 b Fm(\000)22 b Fq(m)2205 1541 y Fi(\001)2251 1560 y Fx(\000)p Fp(1)2345 1507 y Fi(\015)2345 1567 y(\015)2345 1627 y(\015)2429 1622 y Fm(\024)28 b Fs(4)p Fq(=r)m(:)617 b Fs(\(5.19\))328 1839 y(W)-8 b(e)33 b(sho)m(w)h(no)m(w)f(that)1223 2068 y Fq(F)1286 2083 y Fp(\005)p Fo(;m)1425 2068 y Fs(\()p Fq(P)14 b(E)1618 2027 y Fp(0)1612 2093 y(\001)1674 2068 y Fq(M)1768 2083 y Fp(0)1808 2068 y Fq(E)1886 2027 y Fp(0)1880 2093 y(\001)1943 2068 y Fq(P)g Fs(\))27 b Fm(\025)2200 2001 y Fq(\022)s(\025)2305 1965 y Fp(2)p 2200 2045 145 4 v 2253 2137 a Fq(\017)2355 2068 y(\015)37 b(E)2521 2027 y Fp(0)2515 2093 y(\001)2579 2068 y Fq(P)s(;)645 b Fs(\(5.20\))328 2294 y(uniformly)37 b(in)i Fq(m)h(<)g(r)s(=)p Fs(4,)h(pro)m(vided)f(\(5.9\))f(is)g (satis\014ed,)j(and)e(where)h(the)f(F)-8 b(esh)m(bac)m(h)328 2415 y(map)27 b(w)m(as)h(in)m(tro)s(duced)g(in)f(\(5.15\).)41 b(The)29 b(b)s(ound)f(\(5.20\))f(and)g(the)i(isosp)s(ectralit)m(y)d (prop-)328 2535 y(ert)m(y)33 b(\(5.16\))f(of)g(the)h(F)-8 b(esh)m(bac)m(h)35 b(map)c(imply)g(\(5.10\).)474 2655 y(W)-8 b(e)30 b(complete)e(the)i(pro)s(of)e(of)h(the)h(prop)s(osition)d (b)m(y)j(sho)m(wing)f(\(5.20\).)42 b(F)-8 b(rom)28 b(\(5.19\))328 2776 y(it)j(follo)m(ws)h(that)g(for)g(an)m(y)h Fq( )k Fs(and)c Fq(m)28 b(<)f(r)s(=)p Fs(4)986 2888 y Fi(D)1047 2998 y Fq( )t(;)17 b Fs(\005)p Fq(P)d(M)1402 3013 y Fp(0)p 1441 2918 74 4 v 1441 2998 a Fs(\005)p Fq(P)g(E)1669 2957 y Fp(0)1663 3023 y(\001)1742 2918 y Fi(\000)p 1788 2918 V 80 x Fs(\005)p Fq(P)g(E)2016 2957 y Fp(0)2010 3023 y(\001)2073 2998 y Fq(M)2167 3013 y Fp(0)2206 2998 y Fq(E)2284 2957 y Fp(0)2278 3023 y(\001)2342 2998 y Fq(P)p 2419 2918 V 14 w Fs(\005)21 b Fm(\000)i Fq(m)2698 2918 y Fi(\001)2744 2937 y Fx(\000)p Fp(1)2855 2998 y Fq(E)2933 2957 y Fp(0)2927 3023 y(\001)2990 2998 y Fq(P)p 3067 2918 V 14 w Fs(\005)p Fq(M)3234 3013 y Fp(0)3273 2998 y Fq(P)14 b Fs(\005)p Fq( )3490 2888 y Fi(E)1152 3228 y Fm(\024)1267 3160 y Fs(4)p 1267 3205 49 4 v 1268 3296 a Fq(r)1326 3228 y Fm(k)p 1376 3148 74 4 v Fs(\005)p Fq(P)g(E)1604 3187 y Fp(0)1598 3253 y(\001)1660 3228 y Fq(M)1754 3243 y Fp(0)1794 3228 y Fq(P)g Fs(\005)p Fq( )t Fm(k)2061 3187 y Fp(2)1152 3476 y Fm(\024)1267 3408 y Fs(8)p Fq(\022)1364 3372 y Fp(2)1404 3408 y Fq(\025)1461 3372 y Fp(2)p 1267 3453 233 4 v 1360 3544 a Fq(r)1510 3476 y Fs(\(1)22 b(+)g Fq(k)s(\025)1828 3435 y Fp(2)1867 3476 y Fq(=\017)1955 3435 y Fp(2)1995 3476 y Fs(\))2050 3365 y Fi(D)2110 3476 y Fq( )t(;)17 b(P)d Fs(\005)p Fq(I)p 2438 3396 59 4 v 7 w(R)2496 3414 y Fp(2)2496 3500 y Fo(\017)2536 3476 y Fq(I)8 b Fs(\005)p Fq(P)14 b( )2803 3365 y Fi(E)2881 3476 y Fq(;)409 b Fs(\(5.21\))328 3732 y(where)34 b(w)m(e)f(estimate)p 1144 3652 74 4 v 32 w(\005)p Fq(P)14 b(E)1372 3695 y Fp(0)1366 3757 y(\001)1429 3732 y Fq(M)1523 3747 y Fp(0)1563 3732 y Fs(\005)p Fq(P)41 b Fs(=)27 b Fq(\022)s(\025)p 1948 3652 V Fs(\005)q Fq(P)14 b(E)2177 3695 y Fp(0)2171 3757 y(\001)2233 3732 y Fq(L)p 2316 3652 59 4 v(R)2375 3670 y Fp(2)2375 3756 y Fo(\017)2414 3732 y Fq(I)8 b Fs(\005)p Fq(P)46 b Fs(as)932 3927 y Fm(k)p 982 3847 74 4 v Fs(\005)p Fq(P)14 b(E)1210 3886 y Fp(0)1204 3951 y(\001)1267 3927 y Fq(M)1361 3942 y Fp(0)1401 3927 y Fs(\005)p Fq(P)g( )t Fm(k)27 b(\024)h Fq(\022)s Fm(j)p Fq(\025)p Fm(j)p Fs(\(1)21 b(+)h Fq(k)s Fm(j)p Fq(\025)p Fm(j)p Fq(=\017)p Fs(\))p Fm(k)p 2527 3847 59 4 v Fq(R)2584 3942 y Fo(\017)2617 3927 y Fq(I)8 b Fs(\005)p Fq(P)14 b( )t Fm(k)p Fq(;)328 4122 y Fs(with)33 b Fq(k)f Fs(=)c Fm(k)p Fq(I)8 b(P)14 b Fs(\()p Fq(N)39 b Fm(\024)29 b Fs(2\))p Fm(k)p Fs(.)45 b(T)-8 b(aking)33 b(in)m(to)f(accoun)m(t)i (\(5.17\))f(and)g(Prop)s(osition)f(3.2,)h(w)m(e)328 4242 y(obtain)e(the)i(estimate)415 4472 y Fq(F)478 4487 y Fp(\005)p Fo(;m)617 4472 y Fs(\()p Fq(P)14 b(E)810 4430 y Fp(0)804 4496 y(\001)867 4472 y Fq(M)961 4487 y Fp(0)1001 4472 y Fq(E)1079 4430 y Fp(0)1073 4496 y(\001)1136 4472 y Fq(P)g Fs(\))27 b Fm(\025)h Fs(2)1442 4404 y Fq(\022)s(\025)1547 4368 y Fp(2)p 1442 4449 145 4 v 1494 4540 a Fq(\017)1596 4472 y(\015)1701 4331 y Fi(\022)1775 4472 y Fs(1)22 b Fm(\000)1955 4404 y Fs(4)p Fq(\022)p 1955 4449 97 4 v 1980 4540 a(r)2062 4472 y Fs(\(1)g(+)g Fq(k)s(\025)2380 4430 y Fp(2)2419 4472 y Fq(=\017)2507 4430 y Fp(2)2547 4472 y Fs(\))2585 4331 y Fi(\023)2675 4472 y Fs(\005)p Fq(P)35 b Fm(\000)2957 4404 y Fq(\025)3014 4368 y Fp(2)p 2956 4449 98 4 v 2956 4540 a Fs(10)3064 4472 y(\005)p Fq(P)s(:)87 b Fs(\(5.22\))328 4724 y(The)33 b(b)s(ound)f(\(5.20\))f (follo)m(ws)g(from)g(\(5.22\))g(and)h(conditions)f(\(5.9\).)43 b(This)32 b(\014nishes)h(the)328 4844 y(pro)s(of)f(of)g(Prop)s(osition) f(5.1.)1898 5214 y(39)p eop %%Page: 40 40 40 39 bop 474 631 a FB(Pr)-5 b(o)g(of)45 b(of)f(Pr)-5 b(op)g(osition)44 b(5.2.)120 b Fs(Let)43 b(0)j Fm(\024)g Fq(g)j Fm(\024)e Fs(1)c(b)s(e)g(a)g(smo)s(oth)f(function)h(with)328 751 y(supp)s(ort)31 b(in)e(the)i(in)m(terv)-5 b(al)29 b(\001,)i(s.t.)43 b Fq(g)31 b Fs(=)d(1)i(on)g(the)g(in)m(terv)-5 b(al)29 b(\()p Fm(\000j)p Fs(\001)p Fm(j)p Fq(=)p Fs(4)p Fq(;)17 b Fm(j)p Fs(\001)p Fm(j)p Fq(=)p Fs(4\),)30 b(and)g(set)328 872 y Fq(g)375 887 y Fp(0)446 872 y Fs(=)j Fq(g)t Fs(\()p Fq(L)710 887 y Fp(0)749 872 y Fs(\).)51 b(Since)35 b(the)h(in)m (teraction)e Fq(I)43 b Fs(is)35 b(relativ)m(ely)f Fq(N)2489 836 y Fp(1)p Fo(=)p Fp(2)2600 872 y Fs(-b)s(ounded,)i(it)f(follo)m(ws)f (in)328 992 y(a)c(standard)g(w)m(a)m(y)h(\(b)m(y)g(using)f(e.g.)43 b(the)30 b(functional)e(calculus)i(presen)m(ted)i(in)d(App)s(endix)328 1112 y(A\))k(that)966 1333 y Fm(k)p Fs(\(1)22 b Fm(\000)g Fq(g)1271 1348 y Fp(0)1310 1333 y Fs(\))p Fq( )t Fm(k)83 b(\024)h Fq(k)s Fm(j)p Fq(\025)p Fm(j)31 b(k)p Fs(\()p Fq(N)i Fs(+)22 b(1\))2291 1291 y Fp(1)p Fo(=)p Fp(2)2401 1333 y Fq( )t Fm(k)27 b(\024)h Fq(k)s Fm(j)p Fq(\025)p Fm(j)k(k)p Fq( )t Fm(k)p Fq(;)274 b Fs(\(5.23\))834 1478 y Fm(k)p Fs(\(1)22 b Fm(\000)h Fq(g)1140 1493 y Fp(0)1179 1478 y Fs(\))p 1217 1398 77 4 v Fq(P)1293 1493 y Fp(\012)1348 1478 y Fq( )t Fm(k)83 b(\024)h Fq(k)s Fm(j)p Fq(\025)p Fm(j)31 b(k)p Fq(N)2045 1437 y Fp(1)p Fo(=)p Fp(2)p 2156 1398 V 2156 1478 a Fq(P)2232 1493 y Fp(\012)2287 1478 y Fq( )t Fm(k)d(\024)g Fq(k)s Fm(j)p Fq(\025)p Fm(j)2704 1437 y Fp(2)2775 1478 y Fm(k)p Fq( )t Fm(k)p Fq(;)348 b Fs(\(5.24\))328 1698 y(where)29 b(w)m(e)g(used)f(\(2.51\))f(in)g(the) h(last)f(line.)41 b(Notice)27 b(also)g(that)g(\005\(1)12 b Fm(\000)g Fq(g)2917 1713 y Fp(0)2957 1698 y Fs(\))27 b(=)h(0)f(and)h(that)328 1818 y(on)39 b(Ran)16 b(\(1)26 b Fm(\000)h Fq(g)925 1833 y Fp(0)965 1818 y Fs(\))39 b(w)m(e)h(ha)m(v)m(e)g Fm(j)p Fq(L)1517 1833 y Fp(0)1556 1818 y Fm(j)f(\025)g(j)p Fs(\001)p Fm(j)p Fq(=)p Fs(4)f(hence)i Fm(k)p Fs(\(1)26 b Fm(\000)h Fq(g)2603 1833 y Fp(0)2643 1818 y Fs(\))p Fq(R)2755 1833 y Fo(\017)2787 1818 y Fm(k)39 b(\024)g Fs(4)p Fq(=)p Fm(j)p Fs(\001)p Fm(j)p Fs(.)62 b(These)328 1939 y(estimates)32 b(are)h(used)g(b)s(elo)m(w)g(without)f (explicit)f(men)m(tion.)43 b(W)-8 b(e)33 b(decomp)s(ose)1155 2159 y Fq(P)14 b(M)1326 2174 y Fp(0)1365 2159 y Fq(P)97 b Fs(=)83 b Fq(g)1731 2174 y Fp(0)1770 2159 y Fq(P)14 b(M)1941 2174 y Fp(0)1980 2159 y Fq(P)g(g)2104 2174 y Fp(0)1684 2304 y Fs(+2Re\(1)21 b Fm(\000)i Fq(g)2179 2319 y Fp(0)2218 2304 y Fs(\))p Fq(P)14 b(M)2427 2319 y Fp(0)2466 2304 y Fq(P)g(g)2590 2319 y Fp(0)3317 2304 y Fs(\(5.25\))1684 2449 y(+\(1)22 b Fm(\000)g Fq(g)2015 2464 y Fp(0)2054 2449 y Fs(\))p Fq(P)14 b(M)2263 2464 y Fp(0)2302 2449 y Fq(P)g Fs(\(1)22 b Fm(\000)g Fq(g)2634 2464 y Fp(0)2673 2449 y Fs(\))p Fq(:)579 b Fs(\(5.26\))328 2669 y(Prop)s(osition)31 b(5.1)h(yields)g(the)h(b)s(ound)591 2943 y Fm(h)p Fq( )t(;)17 b(g)788 2958 y Fp(0)827 2943 y Fq(P)d(M)998 2958 y Fp(0)1037 2943 y Fq(P)g(g)1161 2958 y Fp(0)1200 2943 y Fq( )t Fm(i)27 b(\025)1448 2875 y Fq(\022)s(\025)1553 2839 y Fp(2)p 1448 2920 145 4 v 1501 3011 a Fq(\017)1603 2943 y(\015)37 b Fm(k)p Fq(g)1788 2958 y Fp(0)1827 2943 y Fq( )t Fm(k)1944 2902 y Fp(2)2011 2943 y Fs(=)2125 2875 y Fq(\022)s(\025)2230 2839 y Fp(2)p 2125 2920 V 2177 3011 a Fq(\017)2279 2943 y(\015)h Fs(\(1)22 b Fm(\000)g Fq(k)s Fm(j)p Fq(\025)p Fm(j)p Fs(\))2781 2902 y Fp(2)2820 2943 y Fm(k)p Fq( )t Fm(k)2987 2902 y Fp(2)3026 2943 y Fq(:)264 b Fs(\(5.27\))328 3187 y(W)-8 b(e)33 b(estimate)434 3407 y(2Re\(1)21 b Fm(\000)i Fq(g)853 3422 y Fp(0)892 3407 y Fs(\))p Fq(P)14 b(M)1101 3422 y Fp(0)1140 3407 y Fq(P)g(g)1264 3422 y Fp(0)1331 3407 y Fm(\025)28 b Fs(2Re\(1)22 b Fm(\000)g Fq(g)1855 3422 y Fp(0)1894 3407 y Fs(\))p Fq(P)14 b(i)p Fs([)p Fq(L;)j(A)2252 3422 y Fp(0)2292 3407 y Fs(])p Fq(P)d(g)2443 3422 y Fp(0)2503 3407 y Fm(\000)23 b Fq(\025)2660 3366 y Fp(2)2699 3407 y Fs(\(1)f Fm(\000)h Fq(g)2955 3422 y Fp(0)2994 3407 y Fs(\))p Fq(g)3079 3422 y Fp(0)3118 3407 y Fq(P)s(;)106 b Fs(\(5.28\))328 3627 y(and)33 b(since)520 3847 y(\(1)22 b Fm(\000)g Fq(g)775 3862 y Fp(0)814 3847 y Fs(\))p Fq(i)p Fs([)p Fq(L;)17 b(A)1095 3862 y Fp(0)1135 3847 y Fs(])p Fq(g)1209 3862 y Fp(0)1276 3847 y Fs(=)28 b Fm(\000)p Fq(\022)s(\025)p Fs(\(1)22 b Fm(\000)h Fq(g)1818 3862 y Fp(0)1857 3847 y Fs(\))1912 3737 y Fi(\020)1971 3847 y Fq(\025I)8 b Fs(\005)p Fq(I)p 2219 3767 59 4 v 7 w(R)2278 3785 y Fp(2)2278 3872 y Fo(\017)2339 3847 y Fm(\000)23 b Fq(L)p 2522 3767 V(R)2580 3785 y Fp(2)2580 3872 y Fo(\017)2620 3847 y Fq(I)8 b Fs(\005)22 b Fm(\000)h Fq(\025)p 2940 3767 V(R)2998 3785 y Fp(2)2998 3872 y Fo(\017)3037 3847 y Fq(I)8 b Fs(\005)p Fq(I)3212 3737 y Fi(\021)3288 3847 y Fq(g)3335 3862 y Fp(0)328 4089 y Fs(w)m(e)34 b(conclude)f(that)619 4343 y(2Re)16 b Fm(h)p Fq( )t(;)h Fs(\(1)22 b Fm(\000)g Fq(g)1204 4358 y Fp(0)1243 4343 y Fs(\))p Fq(P)14 b(M)1452 4358 y Fp(0)1492 4343 y Fq(P)g(g)1616 4358 y Fp(0)1654 4343 y Fq( )t Fm(i)28 b(\025)g(\000)p Fq(k)2034 4276 y(\022)s(\025)2139 4240 y Fp(2)p 2034 4320 145 4 v 2087 4412 a Fq(\017)2205 4203 y Fi(\022)2289 4276 y Fm(j)p Fq(\025)p Fm(j)p 2289 4320 113 4 v 2325 4412 a Fq(\017)2433 4343 y Fs(+)2541 4276 y Fm(j)p Fq(\025)p Fm(j)p Fq(\017)p 2541 4320 152 4 v 2593 4412 a(\022)2703 4203 y Fi(\023)2792 4343 y Fm(k)p Fq( )t Fm(k)2959 4302 y Fp(2)2999 4343 y Fq(:)291 b Fs(\(5.29\))328 4615 y(Next,)33 b(w)m(e)h(ha)m(v)m(e)g (that)328 4889 y(\(1)16 b Fm(\000)g Fq(g)571 4904 y Fp(0)611 4889 y Fs(\))p Fq(M)743 4904 y Fp(0)783 4889 y Fs(\(1)g Fm(\000)g Fq(g)1026 4904 y Fp(0)1065 4889 y Fs(\))28 b Fm(\025)g(\000)p Fq(\022)s(\025)1418 4848 y Fp(2)1458 4889 y Fs(\(1)16 b Fm(\000)g Fq(g)1701 4904 y Fp(0)1741 4889 y Fs(\))1796 4778 y Fi(\020)1855 4889 y Fq(I)8 b Fs(\005)p Fq(I)p 2047 4809 59 4 v 8 w(R)2105 4827 y Fp(2)2105 4913 y Fo(\017)2167 4889 y Fs(+)p 2281 4809 V 21 w Fq(R)2340 4827 y Fp(2)2340 4913 y Fo(\017)2379 4889 y Fq(I)g Fs(\005)p Fq(I)2554 4778 y Fi(\021)2630 4889 y Fs(\(1)16 b Fm(\000)g Fq(g)2873 4904 y Fp(0)2913 4889 y Fs(\))g Fm(\000)3072 4821 y Fq(\025)3129 4785 y Fp(2)p 3071 4866 98 4 v 3071 4957 a Fs(10)3178 4889 y(\(1)g Fm(\000)g Fq(g)3421 4904 y Fp(0)3461 4889 y Fs(\))3499 4848 y Fp(2)3539 4889 y Fq(;)1898 5214 y Fs(40)p eop %%Page: 41 41 41 40 bop 328 631 a Fs(from)31 b(whic)m(h)i(it)f(follo)m(ws)f(that)592 885 y Fm(h)p Fq( )t(;)17 b Fs(\(1)k Fm(\000)i Fq(g)997 900 y Fp(0)1036 885 y Fs(\))p Fq(P)14 b(M)1245 900 y Fp(0)1284 885 y Fq(P)g Fs(\(1)22 b Fm(\000)g Fq(g)1616 900 y Fp(0)1655 885 y Fs(\))p Fq( )t Fm(i)28 b(\025)g(\000)p Fq(k)2073 818 y(\022)s(\025)2178 782 y Fp(2)p 2073 862 145 4 v 2126 954 a Fq(\017)2244 745 y Fi(\022)2318 885 y Fq(\025)2375 844 y Fp(4)2414 885 y Fq(\017)23 b Fs(+)2584 818 y Fq(\025)2641 782 y Fp(2)2680 818 y Fq(\017)p 2584 862 136 4 v 2627 954 a(\022)2729 745 y Fi(\023)2819 885 y Fm(k)p Fq( )t Fm(k)2986 844 y Fp(2)3025 885 y Fq(:)265 b Fs(\(5.30\))328 1163 y(Collecting)31 b(the)i(b)s(ounds)g(\(5.27\),)f (\(5.29\))g(and)g(\(5.30\))g(w)m(e)i(obtain)716 1442 y Fm(h)o Fq( )t(;)17 b(P)d(M)1036 1457 y Fp(0)1075 1442 y Fq(P)g( )t Fm(i)27 b(\025)1400 1374 y Fq(\022)s(\025)1505 1338 y Fp(2)p 1400 1419 145 4 v 1453 1510 a Fq(\017)1555 1442 y(\015)1660 1301 y Fi(\022)1733 1442 y Fs(1)22 b Fm(\000)h Fq(k)s Fm(j)p Fq(\025)p Fm(j)e(\000)2203 1374 y Fq(k)p 2202 1419 57 4 v 2202 1510 a(\015)2284 1301 y Fi(\022)2368 1374 y Fm(j)p Fq(\025)p Fm(j)p 2368 1419 113 4 v 2404 1510 a Fq(\017)2512 1442 y Fs(+)2620 1374 y Fm(j)p Fq(\025)p Fm(j)p Fq(\017)p 2620 1419 152 4 v 2672 1510 a(\022)2782 1301 y Fi(\023\023)2945 1442 y Fm(k)p Fq( )t Fm(k)3112 1401 y Fp(2)3151 1442 y Fq(;)328 1719 y Fs(and)39 b(\(5.12\))g(follo)m(ws)f(from)g(the)i(conditions)e (\(5.11\).)63 b(This)39 b(completes)g(the)h(pro)s(of)e(of)328 1839 y(Prop)s(osition)31 b(5.2)h(and)h(of)f(Theorem)h(5.1.)1613 b Fn(\004)328 2172 y Fu(A)161 b(App)t(endix)328 2420 y Fh(A.1)135 b(In)l(v)-7 b(ariance)45 b(of)g(domains,)h(comm)l(utator)g (expansion)328 2605 y Fs(The)33 b(follo)m(wing)d(t)m(w)m(o)k(theorems)e (are)h(useful)g(in)e(our)i(analysis.)328 2808 y Ft(Theorem)k(A.1)49 b(\(in)m(v)-6 b(ariance)42 b(of)h(domain,)i([F]\))83 b FB(Supp)-5 b(ose)39 b Fs(\()p Fq(X)r(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))39 b FB(satis\014es)328 2929 y(the)i(GJN)h(Condition,)g (\(3.1\),)g(\(3.2\).)63 b(Then)41 b(the)g(unitary)h(gr)-5 b(oup)41 b(gener)-5 b(ate)g(d)40 b(by)i(the)328 3049 y(selfadjoint)34 b Fq(X)8 b FB(,)34 b Fq(e)997 3013 y Fo(itX)1114 3049 y FB(,)h(le)-5 b(aves)34 b Fm(D)s Fs(\()p Fq(Y)21 b Fs(\))35 b FB(invariant,)f(and)g(we)g(have)h(the)g(estimate) 1465 3269 y Fm(k)p Fq(Y)21 b(e)1638 3228 y Fo(itX)1755 3269 y Fq( )t Fm(k)27 b(\024)i Fq(e)2050 3228 y Fo(k)r Fx(j)p Fo(t)p Fx(j)2157 3269 y Fm(k)p Fq(Y)21 b( )t Fm(k)p Fq(;)912 b Fs(\(A.1\))328 3489 y FB(for)35 b(some)f Fq(k)c Fm(\025)f Fs(0)p FB(,)34 b(and)g(al)5 b(l)35 b Fq( )d Fm(2)c(D)s Fs(\()p Fq(Y)20 b Fs(\))p FB(.)328 3718 y Ft(Theorem)37 b(A.2)49 b(\(comm)m(utator)43 b(expansion,)48 b([F]\))87 b FB(Supp)-5 b(ose)41 b Fm(D)j FB(is)d(a)g(c)-5 b(or)g(e)41 b(for)328 3838 y(the)29 b(selfadjoint)g Fq(Y)49 b Fm(\025)28 b Fs(1)-22 b(l)p FB(.)42 b(L)-5 b(et)29 b Fq(X)r(;)17 b(Z)37 b FB(b)-5 b(e)29 b(two)h(symmetric)f(op)-5 b(er)g(ators)29 b(on)g Fm(D)s FB(,)h(and)f(de\014ne)328 3958 y(the)35 b(symmetric)f(op)-5 b(er)g(ators)35 b Fs(ad)1491 3907 y Fp(\()p Fo(n)p Fp(\))1491 3985 y Fo(X)1593 3958 y Fs(\()p Fq(Z)7 b Fs(\))34 b FB(on)h Fm(D)i FB(by)799 4194 y Fs(ad)902 4143 y Fp(\(0\))902 4221 y Fo(X)996 4194 y Fs(\()p Fq(Z)7 b Fs(\))83 b(=)g Fq(Z)r(;)492 4258 y Fi(D)553 4369 y Fq( )t(;)17 b Fs(ad)766 4318 y Fp(\()p Fo(n)p Fp(\))766 4396 y Fo(X)868 4369 y Fs(\()p Fq(Z)7 b Fs(\))p Fq( )1085 4258 y Fi(E)1229 4369 y Fs(=)83 b Fq(i)1438 4258 y Fi(nD)1565 4369 y Fs(ad)1668 4318 y Fp(\()p Fo(n)p Fx(\000)p Fp(1\))1668 4396 y Fo(X)1860 4369 y Fs(\()p Fq(Z)7 b Fs(\))p Fq( )t(;)17 b(X)8 b( )2276 4258 y Fi(E)2359 4369 y Fm(\000)2459 4258 y Fi(D)2520 4369 y Fq(X)g( )t(;)17 b Fs(ad)2822 4318 y Fp(\()p Fo(n)p Fx(\000)p Fp(1\))2822 4396 y Fo(X)3014 4369 y Fs(\()p Fq(Z)7 b Fs(\))p Fq( )3231 4258 y Fi(Eo)3375 4369 y Fq(;)328 4642 y FB(for)40 b(al)5 b(l)39 b Fq( )i Fm(2)c(D)s FB(,)j Fq(n)e Fs(=)e(1)p Fq(;)17 b(:)g(:)g(:)f(;)h(M)10 b FB(.)60 b(We)40 b(supp)-5 b(ose)39 b(that)h(the)g(triples)f Fs(\(ad)3017 4591 y Fp(\()p Fo(n)p Fp(\))3017 4669 y Fo(X)3119 4642 y Fs(\()p Fq(Z)7 b Fs(\))p Fq(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))p FB(,)328 4763 y Fq(n)31 b Fs(=)f(0)p Fq(;)17 b Fs(1)p Fq(;)g(:)g(:)g(:)e(;)i(M)10 b FB(,)38 b(satisfy)e(the)h(GJN)g (Condition)e(\(3.1\),)h(\(3.2\),)g(and)g(that)h Fq(X)44 b FB(is)36 b(self-)328 4883 y(adjoint,)41 b Fm(D)f(\032)f(D)s Fs(\()p Fq(X)8 b Fs(\))p FB(,)41 b Fq(e)1290 4847 y Fo(itX)1447 4883 y FB(le)-5 b(aves)40 b Fm(D)s Fs(\()p Fq(Y)20 b Fs(\))41 b FB(invariant,)g(and)e(that\(A.1\))i(holds.)60 b(The)1898 5214 y Fs(41)p eop %%Page: 42 42 42 41 bop 328 631 a FB(we)34 b(have)h(on)f Fm(D)s Fs(\()p Fq(Y)21 b Fs(\))650 928 y Fq(e)695 887 y Fo(itX)812 928 y Fq(Z)7 b(e)931 887 y Fx(\000)p Fo(itX)1186 928 y Fs(=)83 b Fq(Z)29 b Fm(\000)1541 804 y Fo(M)7 b Fx(\000)p Fp(1)1551 833 y Fi(X)1557 1043 y Fo(n)p Fp(=1)1734 861 y Fq(t)1769 825 y Fo(n)p 1732 905 86 4 v 1732 996 a Fq(n)p Fs(!)1827 928 y(ad)1930 877 y Fp(\()p Fo(n)p Fp(\))1930 955 y Fo(X)2032 928 y Fs(\()p Fq(Z)g Fs(\))1345 1236 y Fm(\000)1439 1100 y Fi(Z)1539 1127 y Fo(t)1494 1326 y Fp(0)1585 1236 y Fq(dt)1671 1251 y Fp(1)1727 1236 y Fm(\001)17 b(\001)g(\001)1860 1100 y Fi(Z)1959 1127 y Fo(t)1984 1138 y Fl(M)5 b Fe(\000)p Fj(1)1915 1326 y Fp(0)2151 1236 y Fq(dt)2237 1251 y Fo(M)2316 1236 y Fq(e)2361 1195 y Fo(it)2410 1206 y Fl(M)2478 1195 y Fo(X)2546 1236 y Fs(ad)2649 1185 y Fp(\()p Fo(M)i Fp(\))2649 1263 y Fo(X)2783 1236 y Fs(\()p Fq(Z)g Fs(\))p Fq(e)2978 1195 y Fx(\000)p Fo(it)3082 1206 y Fl(M)3149 1195 y Fo(X)3217 1236 y Fq(:)97 b Fs(\(A.2\))474 1494 y(The)34 b(follo)m(wing)c(Lemma)h (is)h(a)g(consequence)k(of)c(the)h(ab)s(o)m(v)m(e)g(t)m(w)m(o)h (theorems.)328 1704 y Ft(Lemma)j(A.1)49 b FB(Supp)-5 b(ose)46 b Fs(\()p Fq(X)r(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))47 b FB(and)f Fs(\(ad)2107 1653 y Fp(\()p Fo(n)p Fp(\))2107 1731 y Fo(X)2209 1704 y Fs(\()p Fq(Y)21 b Fs(\))p Fq(;)c(Y)5 b(;)17 b Fm(D)s Fs(\))47 b FB(ar)-5 b(e)47 b(GJN)h(triples,)i(for)328 1825 y Fq(n)41 b Fs(=)g(1)p Fq(;)17 b(:)g(:)g(:)e(;)i(M)10 b FB(,)44 b(some)d Fq(M)52 b Fm(\025)41 b Fs(1)p FB(.)66 b(Mor)-5 b(e)g(over,)43 b(assume)e(that)i(in)e(the)h(sense)f(of)h(Kato)328 1945 y(on)j Fm(D)s Fs(\()p Fq(Y)20 b Fs(\))p FB(:)65 b Fm(\006)p Fs(ad)987 1894 y Fp(\()p Fo(M)7 b Fp(\))987 1972 y Fo(X)1121 1945 y Fs(\()p Fq(Y)21 b Fs(\))46 b Fm(\024)h Fq(k)s(X)8 b FB(,)48 b(for)c(some)h Fq(k)k Fm(\025)e Fs(0)p FB(.)75 b(F)-7 b(or)44 b Fq(\037)j Fm(2)g(S)7 b Fs(\()p Fk(R)e Fs(\))p FB(,)54 b(a)44 b(smo)-5 b(oth)328 2065 y(function)33 b(of)g(r)-5 b(apid)33 b(de)-5 b(cr)g(e)g(ase,)33 b(de\014ne)f Fq(\037)p Fs(\()p Fq(X)8 b Fs(\))27 b(=)2120 1985 y Fi(R)2215 2065 y Fs(^)-61 b Fq(\037)p Fs(\()p Fq(s)p Fs(\))p Fq(e)2431 2029 y Fo(isX)2555 2065 y FB(,)34 b(wher)-5 b(e)44 b Fs(^)-60 b Fq(\037)33 b FB(is)g(the)h(F)-7 b(ourier)328 2186 y(tr)i(ansform)34 b(of)h Fq(\037)p FB(.)45 b(Then)34 b Fq(\037)p Fs(\()p Fq(X)8 b Fs(\))34 b FB(le)-5 b(aves)34 b Fm(D)s Fs(\()p Fq(Y)21 b Fs(\))35 b FB(invariant.)474 2408 y(Pr)-5 b(o)g(of.)123 b Fs(F)-8 b(or)43 b Fq(R)49 b(>)e Fs(0,)g(set)d Fq(\037)1633 2423 y Fo(R)1691 2408 y Fs(\()p Fq(X)8 b Fs(\))47 b(=)2026 2328 y Fi(R)2093 2354 y Fo(R)2073 2443 y Fx(\000)p Fo(R)2214 2408 y Fs(^)-60 b Fq(\037)p Fs(\()p Fq(s)p Fs(\))p Fq(e)2431 2372 y Fo(isX)2555 2408 y Fs(,)47 b(then)d Fq(\037)2923 2423 y Fo(R)2981 2408 y Fs(\()p Fq(X)8 b Fs(\))47 b Fm(!)g Fq(\037)p Fs(\()p Fq(X)8 b Fs(\))328 2529 y(in)45 b(op)s(erator)h(norm,)j(as)d Fq(R)52 b Fm(!)e(1)p Fs(.)84 b(F)-8 b(rom)45 b(the)h(in)m(v)-5 b(ariance)46 b(of)f(domain)g(theorem,)328 2649 y(w)m(e)d(see)h(that)e Fq(\037)928 2664 y Fo(R)986 2649 y Fs(\()p Fq(X)8 b Fs(\))41 b(lea)m(v)m(es)h Fm(D)s Fs(\()p Fq(Y)21 b Fs(\))41 b(in)m(v)-5 b(arian)m(t.)69 b(Let)41 b Fq( )47 b Fm(2)c(D)s Fs(\()p Fq(Y)21 b Fs(\),)43 b(then)f(using)f(the)328 2770 y(comm)m(utator)31 b(expansion)i(theorem)f(ab)s(o)m(v)m(e,)i(w)m(e)g(ha)m(v)m(e)640 2974 y Fq(Y)21 b(\037)779 2989 y Fo(R)837 2974 y Fs(\()p Fq(X)8 b Fs(\))p Fq( )723 3187 y Fs(=)83 b Fq(\037)943 3202 y Fo(R)1000 3187 y Fs(\()p Fq(X)8 b Fs(\))p Fq(Y)21 b( )26 b Fs(+)1430 3051 y Fi(Z)1530 3078 y Fo(R)1486 3277 y Fx(\000)p Fo(R)1627 3187 y Fs(^)-61 b Fq(\037)p Fs(\()p Fq(s)p Fs(\))p Fq(e)1843 3146 y Fo(isX)1984 3106 y Fi(\000)2029 3187 y Fq(e)2074 3146 y Fx(\000)p Fo(isX)2254 3187 y Fq(Y)21 b(e)2377 3146 y Fo(isX)2523 3187 y Fm(\000)i Fq(Y)2701 3106 y Fi(\001)2763 3187 y Fq( )723 3497 y Fs(=)83 b Fq(\037)943 3512 y Fo(R)1000 3497 y Fs(\()p Fq(X)8 b Fs(\))p Fq(Y)21 b( )26 b Fm(\000)1432 3361 y Fi(Z)1532 3387 y Fo(R)1487 3587 y Fx(\000)p Fo(R)1628 3497 y Fs(^)-60 b Fq(\037)p Fs(\()p Fq(s)p Fs(\))p Fq(e)1845 3456 y Fo(isX)1985 3326 y Fi( )2064 3372 y Fo(M)7 b Fx(\000)p Fp(1)2075 3402 y Fi(X)2080 3611 y Fo(n)p Fp(=1)2256 3429 y Fs(\()p Fm(\000)p Fq(s)p Fs(\))2455 3393 y Fo(n)p 2256 3474 247 4 v 2336 3565 a Fq(n)p Fs(!)2512 3497 y(ad)2615 3446 y Fp(\()p Fo(n)p Fp(\))2615 3524 y Fo(X)2717 3497 y Fs(\()p Fq(Y)21 b Fs(\))891 3799 y(+\()p Fm(\000)p Fs(1\))1169 3758 y Fo(M)1265 3663 y Fi(Z)1365 3690 y Fo(s)1320 3889 y Fp(0)1418 3799 y Fq(ds)1515 3814 y Fp(1)1571 3799 y Fm(\001)c(\001)g(\001)1704 3663 y Fi(Z)1803 3690 y Fo(s)1836 3701 y Fl(M)5 b Fe(\000)p Fj(1)1759 3889 y Fp(0)2002 3799 y Fq(ds)2099 3814 y Fo(M)2178 3799 y Fq(e)2223 3758 y Fx(\000)p Fo(is)2335 3769 y Fl(M)2402 3758 y Fo(X)2470 3799 y Fs(ad)2573 3748 y Fp(\()p Fo(M)i Fp(\))2573 3826 y Fo(X)2706 3799 y Fs(\()p Fq(Y)22 b Fs(\))p Fq(e)2906 3758 y Fo(is)2963 3769 y Fl(M)3030 3758 y Fo(X)3097 3658 y Fi(\023)3187 3799 y Fq( )91 b Fs(\(A.3\))328 4061 y(The)33 b(in)m(tegrand)g(of)f(the)h Fq(s)p Fs(-in)m(tegration)d(in)i(\(A.3\))g(is)g(b)s(ounded)i(in)e(norm) f(b)m(y)909 4265 y Fq(k)s Fs(\()p Fm(j)p Fq(s)p Fm(j)1103 4224 y Fo(M)1203 4265 y Fs(+)22 b(1\))17 b(\()p Fm(k)p Fq(Y)k( )t Fm(k)h Fs(+)g Fm(k)p Fq(X)8 b( )t Fm(k)p Fs(\))27 b Fm(\024)h Fq(k)s Fs(\()p Fm(j)p Fq(s)p Fm(j)2428 4224 y Fo(M)2528 4265 y Fs(+)22 b(1\))p Fm(k)p Fq(Y)f( )t Fm(k)p Fq(;)328 4483 y Fs(where)39 b(w)m(e)g(used)g(that)f Fm(k)p Fs(ad)1361 4432 y Fp(\()p Fo(M)7 b Fp(\))1361 4510 y Fo(X)1495 4483 y Fs(\()p Fq(Y)21 b Fs(\))p Fq(e)1694 4447 y Fo(is)1751 4458 y Fl(M)1819 4447 y Fo(X)1886 4483 y Fq( )t Fm(k)36 b(\024)i Fq(k)s Fm(k)p Fq(X)8 b(e)2392 4447 y Fo(is)2449 4458 y Fl(M)2516 4447 y Fo(X)2583 4483 y Fq( )t Fm(k)36 b Fs(=)h Fq(k)s Fm(k)p Fq(X)8 b( )t Fm(k)p Fs(.)59 b(Since)49 b(^)-60 b Fq(\037)328 4604 y Fs(is)40 b(of)f(rapid)h(decrease,)k(it)39 b(can)h(b)s(e)h(in)m (tegrated)f(against)f(an)m(y)i(p)s(o)m(w)m(er)g(of)f Fm(j)p Fq(s)p Fm(j)p Fs(,)h(and)g(w)m(e)328 4724 y(conclude)34 b(that)g(the)g(r.h.s.)47 b(of)34 b(\(A.3\))f(has)h(a)g(limit)c(as)k Fq(R)c Fm(!)g(1)p Fs(.)46 b(Since)34 b Fq(Y)55 b Fs(is)33 b(a)h(closed)328 4844 y(op)s(erator)e(it)f(follo)m(ws)h(that)g Fq(\037)p Fs(\()p Fq(X)8 b Fs(\))p Fq( )31 b Fm(2)e(D)s Fs(\()p Fq(Y)20 b Fs(\).)1463 b Fn(\004)1898 5214 y Fs(42)p eop %%Page: 43 43 43 42 bop 328 631 a Ft(Lemma)37 b(A.2)49 b FB(L)-5 b(et)44 b Fq(\037)i Fm(2)g Fq(C)1419 595 y Fx(1)1412 656 y Fp(0)1493 631 y Fs(\()p Fk(R)1597 595 y Fp(3)1643 631 y Fs(\))p FB(,)g Fq(\037)g Fs(=)f Fq(F)2062 595 y Fp(2)2146 631 y Fm(\025)h Fs(0)p FB(.)73 b(Supp)-5 b(ose)44 b Fs(\()p Fq(X)r(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))44 b FB(satis\014es)328 751 y(the)d(GJN)h(c)-5 b(ondition)40 b(and)h(de\014ne)f Fq(F)14 b Fs(\()p Fq(X)8 b Fs(\))p Fq(;)17 b(\037)p Fs(\()p Fq(X)8 b Fs(\))40 b FB(via)g(the)i(F)-7 b(ourier)40 b(tr)-5 b(ansform)40 b(as)h(in)328 872 y(L)-5 b(emma)45 b(A.1.)79 b(Supp)-5 b(ose)45 b Fq(F)14 b Fs(\()p Fq(X)8 b Fs(\))46 b FB(le)-5 b(aves)45 b Fm(D)s Fs(\()p Fq(Y)21 b Fs(\))46 b FB(invariant.)78 b(L)-5 b(et)46 b Fq(Z)53 b FB(b)-5 b(e)46 b(a)g(symmet-)328 992 y(ric)c(op)-5 b(er)g(ator)42 b(on)g Fm(D)s FB(,)h(s.t.)67 b(for)42 b(some)f Fq(M)52 b Fm(\025)42 b Fs(1)p FB(,)h(and)f Fq(n)f Fs(=)g(0)p Fq(;)17 b Fs(1)p Fq(;)g(:)g(:)g(:)e(;)i(M)10 b FB(,)45 b(the)d(triples)328 1112 y Fs(\(ad)469 1062 y Fp(\()p Fo(n)p Fp(\))469 1140 y Fo(X)571 1112 y Fs(\()p Fq(Z)7 b Fs(\))p Fq(;)17 b(Y)5 b(;)17 b Fm(D)s Fs(\))29 b FB(satisfy)h(the)h (GJN)g(c)-5 b(ondition.)42 b(Mor)-5 b(e)g(over,)31 b(assume)f(that)g (the)h(multi-)328 1233 y(c)-5 b(ommutators)35 b(for)g Fq(n)28 b Fs(=)g(1)p Fq(;)17 b(:)g(:)g(:)f(;)h(M)33 b Fm(\000)22 b Fs(1)35 b FB(ar)-5 b(e)35 b(b)-5 b(ounde)g(d,)35 b(and)f(the)h Fq(M)10 b FB(-th)36 b(multic)-5 b(ommu-)328 1353 y(tator)39 b(is)g(r)-5 b(elatively)39 b Fq(X)8 b FB(-b)-5 b(ounde)g(d)38 b(in)g(the)h(sense)f(of)h(Kato)g(on)f Fm(D)s FB(:)53 b(ther)-5 b(e)38 b(is)h(a)g Fq(k)f(<)d Fm(1)328 1474 y FB(s.t.)45 b Fm(8)p Fq( )32 b Fm(2)c(D)s FB(:)991 1661 y Fm(k)p Fs(ad)1143 1610 y Fp(\()p Fo(n)p Fp(\))1143 1688 y Fo(X)1245 1661 y Fs(\()p Fq(Z)7 b Fs(\))p Fq( )t Fm(k)83 b(\024)g Fq(k)s Fm(k)p Fq( )t Fm(k)p Fq(;)156 b(n)28 b Fs(=)g(1)p Fq(;)17 b(:)g(:)g(:)e(;)i(M)33 b Fm(\000)23 b Fs(1)p Fq(;)959 1822 y Fm(k)p Fs(ad)1111 1771 y Fp(\()p Fo(M)7 b Fp(\))1111 1849 y Fo(X)1245 1822 y Fs(\()p Fq(Z)g Fs(\))p Fq( )t Fm(k)83 b(\024)g Fq(k)s Fm(k)p Fq(X)8 b( )t Fm(k)p Fq(:)328 2010 y FB(Then)31 b(the)g(c)-5 b(ommutator)31 b Fs([)p Fq(\037)p Fs(\()p Fq(X)8 b Fs(\))p Fq(;)17 b(Z)7 b Fs(])28 b(=)f Fq(\037)p Fs(\()p Fq(X)8 b Fs(\))p Fq(Z)22 b Fm(\000)15 b Fq(Z)7 b(\037)p Fs(\()p Fq(X)h Fs(\))31 b FB(is)g(wel)5 b(l)31 b(de\014ne)-5 b(d)30 b(on)h Fm(D)j FB(and)328 2130 y(b)-5 b(ounde)g(d:)44 b(ther)-5 b(e)35 b(is)f(a)h Fq(k)c(<)c Fm(1)35 b FB(s.t.)45 b Fm(k)p Fs([)p Fq(\037)p Fs(\()p Fq(X)8 b Fs(\))p Fq(;)17 b(Z)7 b Fs(])p Fq( )t Fm(k)27 b(\024)h Fq(k)s Fm(k)p Fq( )t Fm(k)p FB(,)34 b Fm(8)p Fq( )e Fm(2)c(D)s FB(.)474 2442 y(Pr)-5 b(o)g(of.)139 b Fs(W)-8 b(e)48 b(write)g Fq(F)s(;)17 b(\037)47 b Fs(instead)h(of)g Fq(F)14 b Fs(\()p Fq(X)8 b Fs(\))p Fq(;)17 b(\037)p Fs(\()p Fq(X)8 b Fs(\).)88 b(Since)48 b Fq(F)62 b Fs(lea)m(v)m(es)49 b Fm(D)s Fs(\()p Fq(Y)21 b Fs(\))328 2562 y(in)m(v)-5 b(arian)m(t)31 b(w)m(e)j(ha)m(v)m(e)g(in)d(the)i(strong)g(sense)h(on)f Fm(D)s Fs(\()p Fq(Y)21 b Fs(\):)1382 2750 y([)p Fq(\037;)c(Z)7 b Fs(])28 b(=)f Fq(F)14 b Fs([)p Fq(F)s(;)j(Z)7 b Fs(])22 b(+)g([)p Fq(F)s(;)17 b(Z)7 b Fs(])p Fq(F)s(:)328 2938 y Fs(W)-8 b(e)33 b(expand)h(the)f(comm)m(utator)634 3168 y([)p Fq(F)s(;)17 b(Z)7 b Fs(])83 b(=)1114 3033 y Fi(Z)1252 3143 y Fs(^)1230 3168 y Fq(F)13 b Fs(\()p Fq(s)p Fs(\))p Fq(e)1473 3127 y Fo(isX)1614 3088 y Fi(\000)1660 3168 y Fq(Z)29 b Fm(\000)23 b Fq(e)1901 3127 y Fx(\000)p Fo(isX)2080 3168 y Fq(Z)7 b(e)2199 3127 y Fo(isX)2323 3088 y Fi(\001)955 3467 y Fs(=)1114 3332 y Fi(Z)1252 3442 y Fs(^)1230 3467 y Fq(F)13 b Fs(\()p Fq(s)p Fs(\))p Fq(e)1473 3426 y Fo(isX)1614 3297 y Fi(\()1694 3343 y Fo(M)7 b Fx(\000)p Fp(1)1705 3373 y Fi(X)1710 3582 y Fo(n)p Fp(=1)1886 3400 y Fq(s)1932 3364 y Fo(n)p 1886 3444 93 4 v 1890 3536 a Fq(n)p Fs(!)1989 3467 y(ad)2092 3416 y Fp(\()p Fo(n)p Fp(\))2092 3494 y Fo(X)2194 3467 y Fs(\()p Fq(Z)g Fs(\))1221 3769 y(+)1314 3634 y Fi(Z)1413 3660 y Fo(s)1369 3859 y Fp(0)1467 3769 y Fq(ds)1564 3784 y Fp(1)1619 3769 y Fm(\001)17 b(\001)g(\001)1752 3634 y Fi(Z)1852 3660 y Fo(s)1885 3671 y Fl(M)5 b Fe(\000)p Fj(1)1807 3859 y Fp(0)2051 3769 y Fq(ds)2148 3784 y Fo(M)2227 3769 y Fq(e)2272 3728 y Fx(\000)p Fo(is)2384 3739 y Fl(M)2451 3728 y Fo(X)2518 3769 y Fs(ad)2621 3719 y Fp(\()p Fo(M)i Fp(\))2621 3796 y Fo(X)2755 3769 y Fs(\()p Fq(Z)g Fs(\))p Fq(e)2950 3728 y Fo(is)3007 3739 y Fl(M)3074 3728 y Fo(X)3141 3629 y Fi(\033)3233 3769 y Fq(:)328 4014 y Fs(Multiplying)34 b(this)i(from)f(the)i(left)e(with)h Fq(F)50 b Fs(\(and)37 b(noticing)e(that)h Fq(F)50 b Fs(comm)m(utes)36 b(with)328 4135 y Fq(e)373 4099 y Fo(is)430 4110 y Fl(M)497 4099 y Fo(X)565 4135 y Fs(\),)i(w)m(e)h(see)g(immediately)34 b(that)k([)p Fq(F)s(;)17 b(Z)7 b Fs(])p Fq(F)51 b Fs(is)37 b(b)s(ounded.)59 b(Next,)40 b(for)d(an)m(y)i Fq(\036;)17 b( )41 b Fs(in)328 4255 y(the)h(dense)h(set)f Fm(D)s Fs(,)h(w)m(e)f(ha)m(v)m(e)h(the)f(estimate)e Fm(jh)p Fq(\036;)17 b(F)d Fs([)p Fq(F)s(;)j(Z)7 b Fs(])p Fq( )s Fm(i)o(j)43 b Fs(=)f Fm(jh)o Fs([)p Fq(F)s(;)17 b(Z)7 b Fs(])p Fq(F)14 b(\036;)j( )t Fm(i)o(j)42 b(\024)328 4376 y Fq(k)s Fm(k)p Fq(\036)p Fm(k)32 b(k)p Fq( )t Fm(k)p Fs(,)g(hence)i Fm(k)p Fq(F)14 b Fs([)p Fq(F)s(;)j(Z)7 b Fs(])p Fq( )t Fm(k)27 b Fs(=)g(sup)1828 4399 y Fp(0)p Fx(6)p Fp(=)p Fo(\036)p Fx(2D)2085 4376 y Fm(k)p Fq(\036)p Fm(k)2243 4339 y Fx(\000)p Fp(1)2336 4376 y Fm(j)17 b(h)o Fq(\036;)g(F)d Fs([)p Fq(F)s(;)j(Z)7 b Fs(])p Fq( )s Fm(i)16 b(j)28 b(\024)g Fq(k)s Fm(k)p Fq( )t Fm(k)p Fs(.)122 b Fn(\004)328 4660 y Fh(A.2)135 b(Pro)t(of)45 b(of)g(Prop)t(osition)g (3.3)328 4844 y Fs(Before)31 b(pro)m(ving)f(Prop)s(osition)f(3.3,)i(w)m (e)g(sho)m(w)h(certain)e(triples)g(satisfy)h(the)g(GJN)f(con-)328 4965 y(ditions.)1898 5214 y(43)p eop %%Page: 44 44 44 43 bop 328 631 a Ft(Lemma)37 b(A.3)49 b FB(The)34 b(fol)5 b(lowing)33 b(triples)i(satisfy)g(the)g(GJN)h(Condition)d (\(3.1\),)h(\(3.2\):)1295 851 y Fs(\()p Fq(H)1414 866 y Fo(p)1454 851 y Fq(;)17 b Fs(\003)1566 866 y Fo(p)1605 851 y Fq(;)g(C)1726 810 y Fx(1)1719 876 y Fp(0)1800 851 y Fs(\()p Fk(R)1904 810 y Fp(3)1950 851 y Fs(\)\))1315 b(\(A.4\))1295 996 y(\()p Fq(A)1406 1011 y Fo(p)1446 996 y Fq(;)17 b Fs(\003)1558 1011 y Fo(p)1597 996 y Fq(;)g(C)1718 955 y Fx(1)1711 1021 y Fp(0)1793 996 y Fs(\()p Fk(R)1897 955 y Fp(3)1942 996 y Fs(\)\))1323 b(\(A.5\))1295 1158 y(\(ad)1436 1107 y Fp(\()p Fo(n)p Fp(\))1436 1185 y Fo(H)1494 1193 y Fl(p)1538 1158 y Fs(\(\003)1644 1173 y Fo(p)1684 1158 y Fs(\))p Fq(;)17 b Fs(\003)1834 1173 y Fo(p)1873 1158 y Fq(;)g(C)1994 1117 y Fx(1)1987 1183 y Fp(0)2068 1158 y Fs(\()p Fk(R)2172 1117 y Fp(3)2217 1158 y Fs(\)\))p Fq(;)87 b(n)27 b Fs(=)h(1)p Fq(;)17 b Fs(2)p Fq(:)576 b Fs(\(A.6\))474 1398 y FB(Pr)-5 b(o)g(of.)97 b Fs(F)-8 b(or)36 b Fq( )k Fm(2)d Fq(C)1290 1362 y Fx(1)1283 1423 y Fp(0)1364 1398 y Fs(\()p Fk(R)1468 1362 y Fp(3)1514 1398 y Fs(\))g(w)m(e)h(ha)m(v)m(e)h Fm(k)p Fq(H)2098 1413 y Fo(p)2137 1398 y Fq( )t Fm(k)2254 1362 y Fp(2)2329 1398 y Fm(\024)e Fs(2)p Fm(k)p Fs(\()p Fm(\000)p Fs(\001\))p Fq( )t Fm(k)2893 1362 y Fp(2)2958 1398 y Fs(+)25 b(2)p Fm(k)p Fq(v)t Fm(k)3259 1362 y Fp(2)3259 1423 y Fx(1)3333 1398 y Fm(k)p Fq( )t Fm(k)3500 1362 y Fp(2)3539 1398 y Fq(:)328 1519 y Fs(The)e(\014rst)f(term)f(on)h(the)g(r.h.s.)40 b(is)22 b(b)s(ounded)g(from)f(ab)s(o)m(v)m(e)h(b)m(y)h(2)p Fm(k)p Fs(\003)2715 1534 y Fo(p)2754 1519 y Fq( )t Fm(k)2871 1482 y Fp(2)2910 1519 y Fs(+4Re)17 b Fm(h)o Fq( )t(;)g Fs(\001)p Fq(x)3452 1482 y Fp(2)3492 1519 y Fq( )t Fm(i)328 1639 y Fs(and)26 b(w)m(e)h(ha)m(v)m(e)g(the)f(estimate)f(Re)17 b Fm(h)o Fq( )t(;)g Fs(\001)p Fq(x)1828 1603 y Fp(2)1868 1639 y Fq( )t Fm(i)28 b Fs(=)2105 1564 y Fi(P)2210 1591 y Fp(3)2210 1668 y Fo(j)t Fp(=1)2337 1639 y Fs(\()p Fm(h)p Fq( )t(;)17 b(x)2580 1654 y Fo(j)2616 1639 y Fs(\001)p Fq(x)2752 1654 y Fo(j)2790 1639 y Fq( )t Fm(i)7 b Fs(+)h Fm(h)p Fq( )t(;)17 b(@)3188 1654 y Fo(j)3225 1639 y Fq(x)3280 1654 y Fo(j)3317 1639 y Fq( )t Fm(i)p Fs(\))27 b Fm(\024)328 1706 y Fi(P)433 1732 y Fp(3)433 1810 y Fo(j)t Fp(=1)576 1781 y Fm(k)p Fq(@)677 1796 y Fo(j)714 1781 y Fq( )t Fm(k)17 b(k)p Fq(x)953 1796 y Fo(j)989 1781 y Fq( )t Fm(k)40 b(\024)h(h)p Fq( )t(;)17 b Fs(\()p Fm(\000)p Fs(\001)22 b(+)g Fq(x)1785 1744 y Fp(2)1825 1781 y Fs(\))p Fq( )t Fm(i)p Fs(.)65 b(Therefore,)43 b(since)d(\003)2853 1796 y Fo(p)2933 1781 y Fm(\025)h Fs(1)-22 b(l,)40 b(it)f(follo)m(ws) 328 1910 y(that)h(that)f Fm(k)p Fq(H)896 1925 y Fo(p)936 1910 y Fq( )t Fm(k)1053 1874 y Fp(2)1132 1910 y Fm(\024)i Fq(k)s Fm(k)p Fs(\003)1422 1925 y Fo(p)1461 1910 y Fq( )t Fm(k)1578 1874 y Fp(2)1617 1910 y Fs(.)66 b(In)40 b(a)g(similar)c(w)m (a)m(y)41 b(it)e(is)h(simple)e(to)i(v)m(erify)g(that)328 2047 y Fm(\006)p Fs(ad)508 1996 y Fp(\(1\))508 2074 y(\003)557 2082 y Fl(p)603 2047 y Fs(\()p Fq(H)722 2062 y Fo(p)761 2047 y Fs(\))28 b Fm(\024)g Fq(k)s Fs(\003)1054 2062 y Fo(p)1093 2047 y Fs(.)44 b(This)32 b(sho)m(ws)i(that)e(\(A.4\))g (satis\014es)i(the)e(GJN)g(conditions.)43 b(The)328 2167 y(pro)s(of)32 b(for)g(\(A.5\))g(is)g(similarly)d(easy)-8 b(.)474 2301 y(F)g(rom)41 b(the)i(ab)s(o)m(v)m(e)g(calculations,)g(and) f(ad)2077 2250 y Fp(\(1\))2077 2328 y Fo(H)2135 2336 y Fl(p)2176 2301 y Fs(\(\003)2282 2316 y Fo(p)2321 2301 y Fs(\))i(=)g Fm(\000)p Fs(ad)2704 2250 y Fp(\(1\))2704 2328 y(\003)2753 2336 y Fl(p)2798 2301 y Fs(\()p Fq(H)2917 2316 y Fo(p)2956 2301 y Fs(\),)h(w)m(e)f(see)f(that)328 2455 y Fm(k)p Fs(ad)481 2404 y Fp(\(1\))481 2482 y Fo(H)539 2490 y Fl(p)579 2455 y Fs(\(\003)685 2470 y Fo(p)724 2455 y Fs(\))p Fq( )t Fm(k)28 b(\024)g Fq(k)s Fm(k)p Fs(\003)1184 2404 y Fp(1)p Fo(=)p Fp(2)1184 2465 y Fo(p)1293 2455 y Fq( )t Fm(k)p Fs(.)44 b(Moreo)m(v)m(er,)34 b(w)m(e)g(calculate) 668 2722 y(ad)771 2672 y Fp(\(1\))771 2750 y(\003)820 2758 y Fl(p)882 2612 y Fi(\020)942 2722 y Fs(ad)1044 2672 y Fp(\(1\))1044 2750 y Fo(H)1102 2758 y Fl(p)1143 2722 y Fs(\(\003)1249 2737 y Fo(p)1288 2722 y Fs(\))1326 2612 y Fi(\021)834 2905 y Fs(=)28 b Fm(\000)1032 2824 y Fi(\000)1077 2905 y Fs(2)p Fq(x)23 b Fm(\001)f(r)p Fq(v)k Fs(+)c(2)p Fq(@)1608 2920 y Fo(m)1674 2905 y Fq(v)1721 2920 y Fo(mn)1831 2905 y Fq(@)1882 2920 y Fo(n)1952 2905 y Fs(+)g Fq(@)2101 2920 y Fo(n)2148 2905 y Fq(v)2195 2920 y Fo(nmm)2389 2905 y Fs(+)g(4\001)g(+)g(4)p Fq(x)2841 2864 y Fp(2)2881 2824 y Fi(\001)2949 2905 y Fs(+)g(h)p Fq(:)p Fs(c)p Fq(:)q(;)328 3125 y Fs(where)53 b Fq(v)676 3140 y Fo(k)r(l)q(m)864 3125 y Fs(=)61 b Fq(@)1052 3140 y Fo(k)1095 3125 y Fq(@)1146 3140 y Fo(l)1173 3125 y Fq(@)1224 3140 y Fo(m)1291 3125 y Fq(v)t Fs(,)56 b(etc.)103 b(Since)52 b(all)e(the)j(deriv)-5 b(ativ)m(es)52 b(of)f Fq(v)56 b Fs(in)m(v)m(olv)m(ed)d(are)328 3246 y(b)s(ounded,)d(the)45 b(r.h.s.)84 b(is)45 b(again)f(relativ)m(ely)g(\003)2135 3261 y Fo(p)2174 3246 y Fs(-b)s(ounded,)50 b(in)44 b(the)i(form)f (sense)i(on)328 3366 y Fq(C)405 3330 y Fx(1)398 3391 y Fp(0)480 3366 y Fs(\()p Fk(R)583 3330 y Fp(3)629 3366 y Fs(\).)c(This)33 b(sho)m(ws)h(\(A.6\))f(for)f Fq(n)c Fs(=)f(1.)43 b(F)-8 b(or)32 b Fq(n)c Fs(=)g(2,)k(w)m(e)i(calculate)378 3599 y(ad)481 3549 y Fp(\(2\))481 3626 y Fo(H)539 3634 y Fl(p)579 3599 y Fs(\(\003)685 3614 y Fo(p)724 3599 y Fs(\))28 b(=)f Fm(\000)17 b Fs(\()p Fm(r)p Fq(v)26 b Fm(\001)c(r)p Fq(v)k Fs(+)c(2)p Fq(x)g Fm(\001)g(r)p Fq(v)k Fs(+)c(2)p Fq(@)2015 3614 y Fo(m)2082 3599 y Fq(v)2129 3614 y Fo(nm)2239 3599 y Fq(@)2290 3614 y Fo(n)2359 3599 y Fs(+)g Fq(@)2508 3614 y Fo(m)2575 3599 y Fq(v)2622 3614 y Fo(mnn)2797 3599 y Fs(+)g(4\001\)+h)p Fq(:)p Fs(c)p Fq(:)51 b Fs(\(A.7\))328 3853 y(Notice)29 b(that)f(ad)941 3802 y Fp(\(2\))941 3880 y Fo(H)999 3888 y Fl(p)1040 3853 y Fs(\(\003)1146 3868 y Fo(p)1185 3853 y Fs(\))h(is)f(relativ)m (ely)g Fq(H)1849 3868 y Fo(p)1889 3853 y Fs(-b)s(ounded,)i(hence)g (relativ)m(ely)e(\003)3101 3868 y Fo(p)3140 3853 y Fs(-b)s(ounded,)328 3985 y(in)35 b(the)h(sense)h(of)e(Kato)g(on)h Fq(C)1446 3949 y Fx(1)1439 4010 y Fp(0)1520 3985 y Fs(\()p Fk(R)1624 3949 y Fp(3)1670 3985 y Fs(\),)g(b)s(ecause)h Fm(r)p Fq(v)28 b Fm(\001)c(r)p Fq(v)39 b Fs(and)d Fq(x)24 b Fm(\001)g(r)p Fq(v)39 b Fs(are)d(b)s(ounded.)328 4133 y(Moreo)m(v)m(er,)50 b(one)c(can)g(calculate)e(ad)1705 4082 y Fp(\(1\))1705 4160 y(\003)1754 4168 y Fl(p)1816 4023 y Fi(\020)1876 4133 y Fs(ad)1979 4082 y Fp(\(2\))1979 4160 y Fo(H)2037 4168 y Fl(p)2077 4133 y Fs(\(\003)2183 4148 y Fo(p)2222 4133 y Fs(\))2260 4023 y Fi(\021)2320 4133 y Fs(,)49 b(and)c(see)i(that)e(it)g(is)g(\003)3283 4148 y Fo(p)3368 4133 y Fs(form)328 4285 y(b)s(ounded)33 b(on)g Fq(C)939 4249 y Fx(1)932 4310 y Fp(0)1013 4285 y Fs(\()p Fk(R)1117 4249 y Fp(3)1163 4285 y Fs(\).)43 b(This)33 b(sho)m(ws)h(\(A.6\))e(for)g Fq(n)c Fs(=)g(2.)1042 b Fn(\004)474 4526 y FB(Pr)-5 b(o)g(of)49 b(of)f(Pr)-5 b(op)g(osition)48 b(3.3.)135 b Fs(W)-8 b(e)48 b(start)g(b)m(y)g(sho)m (wing)g(that)f Fq(\037)g Fs(lea)m(v)m(es)i Fm(D)s Fs(\(\003)3489 4541 y Fo(p)3528 4526 y Fs(\))328 4646 y(in)m(v)-5 b(arian)m(t.)42 b(This)33 b(follo)m(ws)e(from)g(Lemma)g(A.1)i(and)f(the)h(follo)m(wing) d(t)m(w)m(o)j(facts:)44 b(\014rstly)-8 b(,)328 4767 y(\()p Fq(H)447 4782 y Fo(p)486 4767 y Fq(;)17 b Fs(\003)598 4782 y Fo(p)637 4767 y Fq(;)g(C)758 4731 y Fx(1)751 4791 y Fp(0)833 4767 y Fs(\()p Fk(R)937 4731 y Fp(3)982 4767 y Fs(\)\))25 b(and)g(\(ad)1406 4716 y Fp(\()p Fo(n)p Fp(\))1406 4794 y Fo(H)1464 4802 y Fl(p)1508 4767 y Fq(;)17 b Fs(\003)1620 4782 y Fo(p)1659 4767 y Fq(;)g(C)1780 4731 y Fx(1)1773 4791 y Fp(0)1854 4767 y Fs(\()p Fk(R)1958 4731 y Fp(3)2004 4767 y Fs(\)\))24 b(satisfy)h(the)h(GJN)e(Condition,)i (for)e Fq(n)k Fs(=)328 4921 y(1)p Fq(;)17 b Fs(2)31 b(\(see)h(Lemma)e (A.3\),)i(and)f(secondly)-8 b(,)33 b Fm(\006)p Fs(ad)2068 4870 y Fp(\(2\))2068 4948 y Fo(H)2126 4956 y Fl(p)2166 4921 y Fs(\(\003)2272 4936 y Fo(p)2312 4921 y Fs(\))27 b Fm(\024)i Fq(k)s(H)2618 4936 y Fo(p)2657 4921 y Fs(,)j(in)e(the)i (sense)i(of)c(Kato)1898 5214 y(44)p eop %%Page: 45 45 45 44 bop 328 631 a Fs(on)23 b Fm(D)s Fs(\(\003)640 646 y Fo(p)679 631 y Fs(\))h(\(see)g(\(A.7\)\).)40 b(This)24 b(sho)m(ws)h(that)e Fq(\037)28 b Fs(:)g Fm(D)s Fs(\(\003)2274 646 y Fo(p)2313 631 y Fs(\))g Fm(!)f(D)s Fs(\(\003)2692 646 y Fo(p)2731 631 y Fs(\),)e(and)f(consequen)m(tly)-8 b(,)328 751 y Fq(\037A)462 766 y Fo(p)502 751 y Fq(\037)32 b Fs(is)h(w)m(ell)e(de\014ned)j(on)f Fm(D)s Fs(\(\003)1549 766 y Fo(p)1588 751 y Fs(\).)474 872 y(W)-8 b(e)33 b(ha)m(v)m(e)h(in)e (the)h(strong)g(sense)h(on)e Fm(D)s Fs(\(\003)2019 887 y Fo(p)2058 872 y Fs(\))1417 1092 y Fq(\037A)1551 1107 y Fo(p)1591 1092 y Fq(\037)c Fs(=)f Fq(\037)1844 1051 y Fp(2)1884 1092 y Fq(A)1957 1107 y Fo(p)2019 1092 y Fs(+)22 b Fq(\037)p Fs([)p Fq(A)2278 1107 y Fo(p)2318 1092 y Fq(;)17 b(\037)p Fs(])p Fq(:)864 b Fs(\(A.8\))328 1312 y(Let)33 b(us)g(estimate)f(eac)m(h)h(term)f(on)h(the)g(r.h.s.)44 b(separately)-8 b(.)44 b(F)-8 b(or)31 b Fq( )h Fm(2)c Fq(C)2964 1276 y Fx(1)2957 1336 y Fp(0)3039 1312 y Fs(\()p Fk(R)3143 1276 y Fp(3)3188 1312 y Fs(\),)560 1576 y Fm(k)p Fq(\037)671 1535 y Fp(2)711 1576 y Fq(A)784 1591 y Fo(p)824 1576 y Fq( )t Fm(k)83 b Fs(=)h Fm(k)p Fq(\037)1295 1535 y Fp(2)1334 1576 y Fs(\()p Fm(r)22 b(\001)g Fq(x)h Fs(+)f(3)p Fq(=)p Fs(2\))p Fq( )t Fm(k)k(\024)2137 1482 y Fi(X)2187 1691 y Fo(n)2297 1576 y Fm(k)p Fq(\037)2408 1535 y Fp(2)2447 1576 y Fq(@)2498 1591 y Fo(n)2546 1576 y Fm(k)32 b(k)p Fq(x)2733 1591 y Fo(n)2780 1576 y Fq( )t Fm(k)22 b Fs(+)3027 1509 y(3)p 3027 1553 49 4 v 3027 1645 a(2)3086 1576 y Fm(k)p Fq( )t Fm(k)1023 1866 y(\024)84 b Fq(k)1255 1771 y Fi(X)1305 1980 y Fo(n)1415 1785 y Fi(\012)1462 1866 y Fq( )t(;)17 b(x)1628 1824 y Fp(2)1628 1890 y Fo(n)1675 1866 y Fq( )1742 1785 y Fi(\013)1789 1807 y Fp(1)p Fo(=)p Fp(2)1921 1866 y Fs(+)2029 1798 y(3)p 2029 1843 V 2029 1934 a(2)2088 1866 y Fm(k)p Fq( )t Fm(k)27 b(\024)i Fq(k)2458 1785 y Fi(\012)2505 1866 y Fq( )t(;)17 b Fs(\()p Fm(\000)p Fs(\001)23 b(+)f Fq(x)2988 1824 y Fp(2)3028 1866 y Fs(\))p Fq( )3133 1785 y Fi(\013)3180 1807 y Fp(1)p Fo(=)p Fp(2)3306 1866 y Fq(;)328 2177 y Fs(where)34 b(w)m(e)f(used)h(that)e Fm(k)p Fq(\037)1298 2141 y Fp(2)1338 2177 y Fq(@)1389 2192 y Fo(n)1436 2177 y Fm(k)c Fq(<)f Fm(1)p Fs(.)43 b(Consequen)m(tly)-8 b(,)1478 2397 y Fm(k)p Fq(\037)1589 2356 y Fp(2)1628 2397 y Fq(A)1701 2412 y Fo(p)1741 2397 y Fq( )t Fm(k)27 b(\024)i Fq(k)s Fm(k)p Fs(\003)2163 2356 y Fp(1)p Fo(=)p Fp(2)2163 2422 y Fo(p)2272 2397 y Fq( )t Fm(k)p Fq(:)925 b Fs(\(A.9\))328 2617 y(Next)33 b(w)m(e)h(ha)m(v)m(e)g(on)e Fm(D)s Fs(\(\003)1256 2632 y Fo(p)1295 2617 y Fs(\))798 2891 y Fq(\037)p Fs([)p Fq(A)959 2906 y Fo(p)999 2891 y Fq(;)17 b(\037)p Fs(])27 b(=)h Fm(\000)p Fq(\037)1417 2755 y Fi(Z)1533 2891 y Fq(ds)44 b Fs(^)-61 b Fq(\037)p Fs(\()p Fq(s)p Fs(\))1862 2755 y Fi(Z)1962 2781 y Fo(s)1917 2981 y Fp(0)2015 2891 y Fq(ds)2112 2906 y Fp(1)2151 2891 y Fq(e)2196 2850 y Fo(is)2253 2859 y Fj(1)2287 2850 y Fo(H)2345 2858 y Fl(p)2386 2891 y Fs(ad)2489 2840 y Fp(\(1\))2489 2918 y Fo(A)2542 2926 y Fl(p)2583 2891 y Fs(\()p Fq(H)2702 2906 y Fo(p)2741 2891 y Fs(\))p Fq(e)2824 2850 y Fx(\000)p Fo(is)2936 2859 y Fj(1)2971 2850 y Fo(H)3029 2858 y Fl(p)3069 2891 y Fq(;)328 3189 y Fs(where)30 b(ad)709 3138 y Fp(\(1\))709 3216 y Fo(A)762 3224 y Fl(p)803 3189 y Fs(\()p Fq(H)922 3204 y Fo(p)962 3189 y Fs(\))d(=)h Fq(H)1212 3204 y Fo(p)1266 3189 y Fs(+)15 b Fq(W)f Fs(,)29 b(see)h(\(3.44\),)f(and)g(since)g Fq(\037)g Fs(comm)m(utes)g(with)f Fq(e)3208 3153 y Fo(is)3265 3162 y Fj(1)3300 3153 y Fo(H)3358 3161 y Fl(p)3398 3189 y Fs(,)i(w)m(e)328 3310 y(obtain)h(the)i(estimate)883 3553 y Fm(k)p Fq(\037)p Fs([)p Fq(A)1094 3568 y Fo(p)1134 3553 y Fq(;)17 b(\037)p Fs(])p Fm(k)28 b(\024)1449 3418 y Fi(Z)1565 3553 y Fq(ds)k Fm(j)12 b Fs(^)-61 b Fq(\037)p Fs(\()p Fq(s)p Fs(\))p Fm(j)p Fq(s)17 b Fs(\()o Fm(k)p Fq(\037H)2225 3568 y Fo(p)2264 3553 y Fm(k)22 b Fs(+)g Fm(k)p Fq(W)14 b Fm(k)2640 3568 y Fx(1)2714 3553 y Fs(\))28 b Fq(<)g Fm(1)p Fq(:)281 b Fs(\(A.10\))328 3822 y(It)33 b(follo)m(ws,)e(together)i(with)f(with)g(\(A.8\))h(and)f(\(A.9\),)h (that)980 4042 y Fm(k)p Fq(\037A)1164 4057 y Fo(p)1203 4042 y Fq(\037 )t Fm(k)28 b(\024)g Fq(k)1585 3961 y Fi(\000)1630 4042 y Fm(k)p Fs(\003)1748 4000 y Fp(1)p Fo(=)p Fp(2)1748 4066 y Fo(p)1858 4042 y Fq( )t Fm(k)22 b Fs(+)g Fm(k)p Fq( )t Fm(k)2262 3961 y Fi(\001)2335 4042 y Fm(\024)28 b Fs(2)p Fq(k)s Fm(k)p Fs(\003)2661 4000 y Fp(1)p Fo(=)p Fp(2)2661 4066 y Fo(p)2770 4042 y Fq( )t Fm(k)p Fq(:)328 4262 y Fs(This)45 b(sho)m(ws)i(that)e(the)g(\014rst)h(GJN)f(condition,) i(\(3.1\),)g(is)e(satis\014ed)h(for)e(our)h(triple.)328 4382 y(Next,)33 b(w)m(e)h(write)1153 4502 y Fm(h)p Fq(\037A)1326 4517 y Fo(p)1366 4502 y Fq(\037 )t(;)17 b Fs(\003)1606 4517 y Fo(p)1645 4502 y Fq( )t Fm(i)27 b Fs(=)h Fm(h)p Fq(A)1994 4517 y Fo(p)2034 4502 y Fq(\037 )t(;)17 b Fs(\003)2274 4517 y Fo(p)2313 4502 y Fq(\037 )t Fm(i)22 b Fs(+)g Fq(R)2674 4517 y Fp(1)2713 4502 y Fq(;)552 b Fs(\(A.11\))328 4677 y(where)1455 4797 y Fq(R)1529 4812 y Fp(1)1596 4797 y Fs(=)28 b Fm(h)o Fq(A)1811 4812 y Fo(p)1851 4797 y Fq(\037 )t(;)17 b Fs([)p Fq(\037;)g Fs(\003)2223 4812 y Fo(p)2262 4797 y Fs(])p Fq( )t Fm(i)g Fq(:)853 b Fs(\(A.12\))1898 5214 y(45)p eop %%Page: 46 46 46 45 bop 328 631 a Fs(Since)35 b Fq(\037 )g Fm(2)c(D)s Fs(\(\003)1027 646 y Fo(p)1066 631 y Fs(\),)k(and)f(\003)1425 646 y Fo(p)1499 631 y Fs(is)g(essen)m(tially)g(selfadjoin)m(t)f(on)i Fq(C)2754 595 y Fx(1)2747 656 y Fp(0)2829 631 y Fs(\()p Fk(R)2932 595 y Fp(3)2978 631 y Fs(\),)g(there)g(exists)328 751 y(a)f(sequence)i Fm(f)p Fq(')930 766 y Fo(n)977 751 y Fm(g)30 b(\032)h Fq(C)1242 715 y Fx(1)1235 776 y Fp(0)1316 751 y Fs(\()p Fk(R)1420 715 y Fp(3)1466 751 y Fs(\))j(s.t.)48 b Fq(')1780 766 y Fo(n)1857 751 y Fm(!)30 b Fq(\037 )38 b Fs(and)c(\003)2408 766 y Fo(p)2448 751 y Fq(')2512 766 y Fo(n)2589 751 y Fm(!)29 b Fs(\003)2786 766 y Fo(p)2826 751 y Fq(\037 )t Fs(.)48 b(Moreo)m(v)m(er)35 b(\003)3526 766 y Fo(p)328 872 y Fs(lea)m(v)m(es)f Fq(C)685 836 y Fx(1)678 896 y Fp(0)759 872 y Fs(\()p Fk(R)863 836 y Fp(3)909 872 y Fs(\))e(in)m(v)-5 b(arian)m(t,)31 b(so)i(w)m(e)h(ha)m(v) m(e)690 1086 y Fm(h)p Fq(A)802 1101 y Fo(p)842 1086 y Fq(\037 )t(;)17 b Fs(\003)1082 1101 y Fo(p)1121 1086 y Fq(\037 )t Fm(i)83 b Fs(=)g(lim)1576 1146 y Fo(n)1682 1086 y Fm(h)o Fq(A)1793 1101 y Fo(p)1833 1086 y Fq(\037 )t(;)17 b Fs(\003)2073 1101 y Fo(p)2112 1086 y Fq(')2176 1101 y Fo(n)2223 1086 y Fm(i)1371 1264 y Fs(=)83 b(lim)1576 1324 y Fo(n)1682 1264 y Fm(f)o(h)p Fs(\003)1838 1279 y Fo(p)1878 1264 y Fq(\037 )t(;)17 b(A)2123 1279 y Fo(p)2162 1264 y Fq(')2226 1279 y Fo(n)2273 1264 y Fm(i)22 b Fs(+)g Fm(h)p Fq(\037 )t(;)17 b Fs([)p Fq(A)2743 1279 y Fo(p)2782 1264 y Fq(;)g Fs(\003)2894 1279 y Fo(p)2933 1264 y Fs(])p Fq(')3024 1279 y Fo(n)3071 1264 y Fm(ig)g Fq(;)88 b Fs(\(A.13\))328 1515 y(and)32 b(w)m(e)g(calculate)f(\(strongly)g(on)h Fq(C)1691 1479 y Fx(1)1684 1539 y Fp(0)1765 1515 y Fs(\()p Fk(R)1869 1479 y Fp(3)1915 1515 y Fs(\)\):)42 b Fq(i)17 b Fs([)p Fq(A)2210 1530 y Fo(p)2250 1515 y Fq(;)g Fs(\003)2362 1530 y Fo(p)2401 1515 y Fs(])28 b(=)g Fq(H)2641 1530 y Fo(p)2700 1515 y Fs(+)20 b Fq(W)34 b Fm(\000)21 b Fq(x)g Fm(\001)e(r)p Fq(v)24 b Fm(\000)d Fs(2)p Fq(x)3499 1479 y Fp(2)3539 1515 y Fs(.)328 1635 y(Since)42 b Fq(\037)p Fs(\()p Fq(H)772 1650 y Fo(p)841 1635 y Fs(+)28 b Fq(W)43 b Fm(\000)29 b Fq(x)g Fm(\001)g(r)p Fq(v)t Fs(\))41 b(is)h(b)s(ounded,) k(and)c Fq(\037 )49 b Fm(2)44 b(D)s Fs(\()p Fq(x)2741 1599 y Fp(2)2781 1635 y Fs(\))e(\(b)s(ecause)h(it)f(is)g(in)328 1756 y Fm(D)s Fs(\(\003)514 1771 y Fo(p)553 1756 y Fs(\)\),)32 b(w)m(e)i(get)1248 1876 y(\()p Fq(A:)p Fs(13\))27 b(=)h(lim)1699 1936 y Fo(n)1805 1876 y Fm(h)o Fs(\003)1911 1891 y Fo(p)1951 1876 y Fq(\037 )t(;)17 b(A)2196 1891 y Fo(p)2235 1876 y Fq(')2299 1891 y Fo(n)2346 1876 y Fm(i)22 b Fs(+)g Fq(R)2579 1891 y Fp(2)2619 1876 y Fq(;)646 b Fs(\(A.14\))328 2075 y(where)34 b(w)m(e)f(de\014ned)1070 2289 y Fq(R)1144 2304 y Fp(2)1212 2289 y Fs(=)1315 2208 y Fi(\012)1362 2289 y Fq( )t(;)17 b(\037)p Fs(\()p Fq(H)1653 2304 y Fo(p)1715 2289 y Fs(+)22 b Fq(W)35 b Fm(\000)23 b Fq(x)g Fm(\001)e(r)p Fq(v)26 b Fm(\000)d Fs(2)p Fq(x)2527 2248 y Fp(2)2567 2289 y Fs(\))p Fq(\037 )2733 2208 y Fi(\013)2796 2289 y Fq(:)469 b Fs(\(A.15\))328 2504 y(Next,)26 b(it)d(is)g(not)g (di\016cult)f(to)h(see)i(that)e(if)g Fq( )31 b Fm(2)d Fq(C)2083 2467 y Fx(1)2076 2528 y Fp(0)2181 2504 y Fs(then)c Fq(\037 )32 b Fm(2)c(D)s Fs(\(\003)2830 2467 y Fp(2)2830 2528 y Fo(p)2869 2504 y Fs(\).)41 b(Consequen)m(tly)-8 b(,)328 2624 y(w)m(e)31 b(can)f(mo)m(v)m(e)g Fq(A)967 2639 y Fo(p)1037 2624 y Fs(in)f(\(A.14\))h(to)f(the)h(left)f(factor)h (in)f(the)h(scalar)f(pro)s(duct)i(\(recall)d(that)328 2744 y Fm(D)s Fs(\()p Fq(A)519 2759 y Fo(p)558 2744 y Fs(\))36 b Fm(\033)g(D)s Fs(\(\003)931 2759 y Fo(p)970 2744 y Fs(\)\),)i(p)s(erform)e(the)i(limit,)c(and)k(mo)m(v)m(e)f Fq(A)2446 2759 y Fo(p)2523 2744 y Fs(bac)m(k)h(to)f(the)h(righ)m(t)e (factor.)328 2865 y(W)-8 b(e)33 b(then)g(obtain)417 3079 y Fm(h)p Fq(A)529 3094 y Fo(p)568 3079 y Fq(\037 )5 b(;)17 b Fs(\003)809 3094 y Fo(p)848 3079 y Fq(\037 )t Fm(i)27 b Fs(=)h Fm(h)o Fs(\003)1252 3094 y Fo(p)1292 3079 y Fq(\037 )t(;)17 b(A)1537 3094 y Fo(p)1576 3079 y Fq(\037 )5 b Fm(i)21 b Fs(+)h Fq(R)1937 3094 y Fp(2)2005 3079 y Fs(=)27 b Fm(h)p Fs(\003)2215 3094 y Fo(p)2254 3079 y Fq( )t(;)17 b(\037A)2499 3094 y Fo(p)2539 3079 y Fq(\037 )t Fm(i)22 b Fs(+)g Fq(R)2900 3094 y Fp(2)2962 3079 y Fm(\000)p 3062 2999 76 4 v 23 w Fq(R)3137 3094 y Fp(1)3176 3079 y Fq(;)89 b Fs(\(A.16\))328 3294 y(where)34 b(the)f(bar)f(denotes)i (complex)e(conjugate.)44 b(T)-8 b(ogether)33 b(with)f(\(A.11\))g(this)g (giv)m(es)917 3508 y Fm(h)p Fq(\037A)1090 3523 y Fo(p)1130 3508 y Fq(\037 )t(;)17 b Fs(\003)1370 3523 y Fo(p)1409 3508 y Fq( )t Fm(i)22 b(\000)h(h)o Fs(\003)1743 3523 y Fo(p)1783 3508 y Fq( )t(;)17 b(\037A)2028 3523 y Fo(p)2067 3508 y Fq(\037 )5 b Fm(i)27 b Fs(=)g Fq(R)2439 3523 y Fp(1)2501 3508 y Fs(+)22 b Fq(R)2673 3523 y Fp(2)2735 3508 y Fm(\000)p 2835 3428 V 23 w Fq(R)2910 3523 y Fp(1)2949 3508 y Fq(:)316 b Fs(\(A.17\))328 3722 y(Let)33 b(us)g(\014rst)g (consider)g Fq(R)1283 3737 y Fp(1)1323 3722 y Fs(.)43 b(W)-8 b(e)33 b(estimate)616 3937 y Fm(j)p Fq(R)718 3952 y Fp(1)757 3937 y Fm(j)27 b(\024)h(k)p Fq(A)1040 3952 y Fo(p)1080 3937 y Fq(\037 )t Fm(k)k(k)p Fs([)p Fq(\037;)17 b Fs(\003)1540 3952 y Fo(p)1580 3937 y Fs(])p Fq( )t Fm(k)27 b(\024)1856 3856 y Fi(\000)1902 3937 y Fm(k)p Fq(A)2025 3952 y Fo(p)2065 3937 y Fq( )t Fm(k)21 b Fs(+)i Fm(k)p Fs([)p Fq(A)2452 3952 y Fo(p)2491 3937 y Fq(;)17 b(\037)p Fs(])p Fq( )t Fm(k)2740 3856 y Fi(\001)2818 3937 y Fm(k)p Fs([)p Fq(\037;)g Fs(\003)3068 3952 y Fo(p)3107 3937 y Fs(])p Fq( )t Fm(k)p Fq(:)328 4175 y Fs(It)40 b(is)f(clear)g(that)h Fm(k)p Fq(A)1128 4190 y Fo(p)1167 4175 y Fq( )t Fm(k)g(\024)g Fq(k)s Fm(k)p Fs(\003)1613 4124 y Fp(1)p Fo(=)p Fp(2)1613 4185 y Fo(p)1723 4175 y Fq( )t Fm(k)p Fs(,)h(and)f(b)m(y)g(the)h(same)e(argumen)m(t)g(as)h (the)g(one)328 4295 y(leading)35 b(to)h(\(A.10\),)h(that)f Fm(k)p Fs([)p Fq(A)1494 4310 y Fo(p)1533 4295 y Fq(;)17 b(\037)p Fs(])p Fq( )t Fm(k)34 b(\024)g Fq(k)s Fm(k)p Fq( )t Fm(k)i Fs(\(use)i(that)e Fq(\037)e Fs(=)g Fq(F)2892 4259 y Fp(2)2965 4295 y Fm(\025)g Fs(0,)j(as)g(in)e(the)328 4416 y(pro)s(of)d(of)g(Lemma)f(A.2\),)h(so)h(that)549 4630 y Fm(j)p Fq(R)651 4645 y Fp(1)690 4630 y Fm(j)27 b(\024)i Fq(k)s Fm(k)p Fs(\003)1023 4589 y Fp(1)p Fo(=)p Fp(2)1023 4655 y Fo(p)1132 4630 y Fq( )t Fm(k)j(k)p Fs([)p Fq(\037;)17 b Fs(\003)1531 4645 y Fo(p)1571 4630 y Fs(])p Fq( )t Fm(k)27 b(\024)h Fq(k)s Fm(k)p Fs(\003)2019 4589 y Fp(1)p Fo(=)p Fp(2)2019 4655 y Fo(p)2129 4630 y Fq( )t Fm(k)2246 4549 y Fi(\000)2291 4630 y Fm(k)p Fq( )t Fm(k)22 b Fs(+)g Fm(k)p Fs([)p Fq(\037;)17 b(x)2815 4589 y Fp(2)2855 4630 y Fs(])p Fq( )t Fm(k)2999 4549 y Fi(\001)3044 4630 y Fq(;)221 b Fs(\(A.18\))328 4844 y(where)28 b(w)m(e)f(used)h(in)e(the) h(second)h(step)g(that)e([)p Fq(\037;)17 b Fm(\000)p Fs(\001])27 b(is)g(b)s(ounded.)42 b(Next,)29 b(Lemma)c(A.2)328 4965 y(\(with)k Fq(X)35 b Fs(=)28 b Fq(H)886 4980 y Fo(p)925 4965 y Fq(;)17 b(Z)35 b Fs(=)27 b Fq(x)1229 4980 y Fo(n)1277 4965 y Fq(;)17 b(Y)48 b Fs(=)28 b(\003)1598 4980 y Fo(p)1637 4965 y Fq(;)17 b(M)38 b Fs(=)28 b(1\))h(sho)m(ws)i(that)f([)p Fq(\037;)17 b(x)2706 4980 y Fo(n)2753 4965 y Fs(])29 b(is)g(b)s(ounded,)i(hence)1898 5214 y(46)p eop %%Page: 47 47 47 46 bop 328 632 a Fm(k)p Fs([)p Fq(\037;)17 b(x)565 596 y Fp(2)605 632 y Fs(])p Fq( )t Fm(k)46 b(\024)919 557 y Fi(P)1024 661 y Fo(n)1071 632 y Fs(\()p Fq(k)s Fm(k)p Fq(x)1268 647 y Fo(n)1315 632 y Fq( )t Fm(k)30 b Fs(+)f Fm(k)p Fq(x)1672 647 y Fo(n)1719 632 y Fs([)p Fq(\037;)17 b(x)1906 647 y Fo(n)1954 632 y Fs(])p Fq( )t Fm(k)p Fs(\))46 b Fm(\024)h Fq(k)s Fm(k)p Fs(\003)2478 581 y Fp(1)p Fo(=)p Fp(2)2478 642 y Fo(p)2588 632 y Fq( )t Fm(k)29 b Fs(+)2840 557 y Fi(P)2945 661 y Fo(n)3008 632 y Fm(k)p Fq(x)3113 647 y Fo(n)3160 632 y Fs([)p Fq(\037;)17 b(x)3347 647 y Fo(n)3395 632 y Fs(])p Fq( )t Fm(k)p Fs(.)328 752 y(W)-8 b(e)35 b(write)f Fq(x)804 767 y Fo(n)851 752 y Fs([)p Fq(\037;)17 b(x)1038 767 y Fo(n)1086 752 y Fs(])p Fq( )34 b Fs(=)d([)p Fq(\037;)17 b(x)1504 767 y Fo(n)1551 752 y Fs(])p Fq(x)1633 767 y Fo(n)1681 752 y Fq( )27 b Fs(+)d([)p Fq(x)1953 767 y Fo(n)2000 752 y Fq(;)17 b Fs([)p Fq(\037;)g(x)2231 767 y Fo(n)2278 752 y Fs(]])p Fq( )t Fs(.)49 b(As)35 b(ab)s(o)m(v)m(e,)h Fm(k)p Fs([)p Fq(\037;)17 b(x)3164 767 y Fo(n)3211 752 y Fs(])p Fq(x)3293 767 y Fo(n)3341 752 y Fq( )t Fm(k)30 b(\024)328 886 y Fq(k)s Fm(k)p Fs(\003)500 835 y Fp(1)p Fo(=)p Fp(2)500 897 y Fo(p)610 886 y Fq( )t Fm(k)p Fs(,)i(and)h(using)f Fq(\037)1292 850 y Fp(2)1359 886 y Fs(=)c Fq(F)14 b Fs(,)32 b(w)m(e)i(write)648 1106 y([)p Fq(x)730 1121 y Fo(n)778 1106 y Fq(;)17 b Fs([)p Fq(\037;)g(x)1009 1121 y Fo(n)1056 1106 y Fs(]])27 b(=)h Fm(\000)p Fs(2[)p Fq(F)s(;)17 b(x)1559 1121 y Fo(n)1606 1106 y Fs(])1633 1065 y Fp(2)1695 1106 y Fs(+)22 b Fq(F)14 b Fs([)p Fq(x)1952 1121 y Fo(n)1999 1106 y Fs([)p Fq(F)s(;)j(x)2191 1121 y Fo(n)2238 1106 y Fs(]])23 b(+)f([)p Fq(x)2495 1121 y Fo(n)2542 1106 y Fq(;)17 b Fs([)p Fq(F)s(;)g(x)2778 1121 y Fo(n)2825 1106 y Fs(]])p Fq(F)s(:)320 b Fs(\(A.19\))328 1326 y(As)43 b(ab)s(o)m(v)m(e,)j([)p Fq(F)s(;)17 b(x)990 1341 y Fo(n)1037 1326 y Fs(])43 b(is)f(b)s(ounded,)47 b(and)42 b(a)h(b)m(y)h(no)m(w)f (standard)g(comm)m(utator)e(expan-)328 1447 y(sion)j(sho)m(ws)i(that)e (so)h(are)f(the)h(other)g(t)m(w)m(o)g(terms)g(in)e(\(A.19\).)79 b(W)-8 b(e)45 b(conclude)g(that)328 1581 y Fm(k)p Fs([)p Fq(\037;)17 b(x)565 1545 y Fp(2)605 1581 y Fs(])p Fq( )t Fm(k)44 b(\024)h Fq(k)s Fs(\()p Fm(k)p Fs(\003)1125 1530 y Fp(1)p Fo(=)p Fp(2)1125 1591 y Fo(p)1234 1581 y Fq( )t Fm(k)29 b Fs(+)g Fm(k)p Fq( )t Fm(k)p Fs(\))44 b Fm(\024)h Fq(k)s Fm(k)p Fs(\003)2028 1530 y Fp(1)p Fo(=)p Fp(2)2028 1591 y Fo(p)2137 1581 y Fq( )t Fm(k)p Fs(.)73 b(Com)m(bining)41 b(this)h(with)g(\(A.18\))328 1701 y(yields)1563 1822 y Fm(j)p Fq(R)1665 1837 y Fp(1)1705 1822 y Fm(j)27 b(\024)h Fq(k)s Fm(k)p Fs(\003)2037 1780 y Fp(1)p Fo(=)p Fp(2)2037 1846 y Fo(p)2147 1822 y Fq( )t Fm(k)2264 1780 y Fp(2)2303 1822 y Fq(:)962 b Fs(\(A.20\))328 1996 y(Finally)-8 b(,)25 b(let)h(us)i(obtain)e(the)h(same)f(upp)s(er)i(b)s(ound)f(for)f Fq(R)2413 2011 y Fp(2)2453 1996 y Fs(.)41 b(Since)27 b Fq(\037)p Fs(\()p Fq(H)2950 2011 y Fo(p)3000 1996 y Fs(+)10 b Fq(W)24 b Fm(\000)10 b Fq(x)g Fm(\001)g(r)p Fq(v)t Fs(\))328 2133 y(is)24 b(b)s(ounded)i(w)m(e)g(only)f(need)h(to)e (sho)m(w)i(that)f Fm(j)17 b(h)o Fq(\037 )t(;)g(x)2231 2097 y Fp(2)2271 2133 y Fq(\037 )t Fm(i)f(j)27 b(\024)i Fq(k)s Fm(k)p Fs(\003)2787 2082 y Fp(1)p Fo(=)p Fp(2)2787 2143 y Fo(p)2896 2133 y Fq( )t Fm(k)3013 2097 y Fp(2)3052 2133 y Fs(.)41 b(Using)25 b(that)328 2253 y([)p Fq(\037;)17 b(x)515 2268 y Fo(n)562 2253 y Fs(])33 b(is)f(b)s(ounded,)h(w)m(e)h (arriv)m(e)e(at)502 2491 y Fm(j)547 2410 y Fi(\012)594 2491 y Fq(\037 )t(;)17 b(x)821 2449 y Fp(2)860 2491 y Fq(\037 )988 2410 y Fi(\013)1052 2491 y Fm(j)27 b Fs(=)1211 2396 y Fi(X)1261 2605 y Fo(n)1371 2491 y Fm(k)p Fq(x)1476 2506 y Fo(n)1524 2491 y Fq(\037 )t Fm(k)1702 2449 y Fp(2)1769 2491 y Fm(\024)h Fq(k)s Fm(k)p Fq( )t Fm(k)2095 2449 y Fp(2)2156 2491 y Fs(+)22 b Fq(k)2325 2396 y Fi(X)2375 2605 y Fo(n)2485 2491 y Fm(k)p Fq(x)2590 2506 y Fo(n)2637 2491 y Fq( )t Fm(k)2754 2449 y Fp(2)2821 2491 y Fm(\024)29 b Fq(k)s Fm(k)p Fs(\003)3099 2449 y Fp(1)p Fo(=)p Fp(2)3099 2515 y Fo(p)3208 2491 y Fq( )t Fm(k)3325 2449 y Fp(2)3364 2491 y Fq(;)328 2792 y Fs(whic)m(h)k(sho)m(ws)h(that)1563 2913 y Fm(j)p Fq(R)1665 2928 y Fp(2)1705 2913 y Fm(j)27 b(\024)h Fq(k)s Fm(k)p Fs(\003)2037 2871 y Fp(1)p Fo(=)p Fp(2)2037 2937 y Fo(p)2147 2913 y Fq( )t Fm(k)2264 2871 y Fp(2)2303 2913 y Fq(:)962 b Fs(\(A.21\))328 3087 y(Com)m(bining)33 b(\(A.17\),)h(\(A.20\))g(and)h(\(A.21\))e(sho)m(ws)j(that)f(\()p Fq(\037A)2636 3102 y Fo(p)2675 3087 y Fq(\037;)17 b Fs(\003)2848 3102 y Fo(p)2888 3087 y Fq(;)g(C)3009 3051 y Fx(1)3002 3112 y Fp(0)3083 3087 y Fs(\()p Fk(R)3187 3051 y Fp(3)3232 3087 y Fs(\)\))35 b(satis-)328 3207 y(\014es)e(the)g(second)h(GJN)f (condition,)e(\(3.2\).)1593 b Fn(\004)328 3617 y Fh(A.3)135 b(Pro)t(of)45 b(of)g(Prop)t(osition)g(3.4)328 3801 y Fs(W)-8 b(e)35 b(giv)m(e)g(the)h(follo)m(wing)c(lemma)h(without)h(a)h (pro)s(of,)g(whic)m(h)g(is)g(not)f(di\016cult)g(to)h(\014nd,)328 3922 y(e.g.)44 b(b)m(y)33 b(using)f(the)h(results)g(of)f(App)s(endix)i (A.1.)328 4150 y Ft(Lemma)j(A.4)49 b FB(L)-5 b(et)26 b Fq(v)32 b Fm(2)c Fq(C)1355 4114 y Fo(p)p Fx(\000)p Fp(1)1485 4150 y Fs(\()p Fk(R)1588 4114 y Fo(d)1635 4150 y Fs(\))e FB(b)-5 b(e)26 b(s.t.)42 b Fq(x)2039 4114 y Fo(\013)2089 4150 y Fq(@)2145 4114 y Fo(\013)2195 4150 y Fq(v)30 b FB(is)c(b)-5 b(ounde)g(d,)27 b(for)f(any)h(multi-index)328 4270 y Fm(j)p Fq(\013)q Fm(j)34 b(\024)h Fq(p)25 b Fm(\000)g Fs(1)p FB(.)56 b(L)-5 b(et)39 b Fq(H)j Fs(=)35 b Fm(\000)p Fs(\001)26 b(+)f Fq(v)t FB(,)39 b(which)f(is)g(essential)5 b(ly)38 b(selfadjoint)g(on)g Fq(C)3273 4234 y Fx(1)3266 4295 y Fp(0)3348 4270 y Fs(\()p Fk(R)3452 4234 y Fo(d)3498 4270 y Fs(\))p FB(.)328 4391 y(Given)d(a)f(function)h Fq(\026)27 b Fm(2)h Fq(C)1338 4355 y Fx(1)1331 4415 y Fp(0)1413 4391 y Fs(\()p Fk(R)5 b Fs(\))p FB(,)40 b(de\014ne)34 b Fq(\026)p Fs(\()p Fq(H)8 b Fs(\))27 b(=)2264 4310 y Fi(R)2354 4391 y Fs(^)-56 b Fq(\026)o Fs(\()p Fq(s)p Fs(\))p Fq(e)2572 4355 y Fo(isH)2696 4391 y FB(.)45 b(Then)34 b(we)h(have)939 4611 y Fm(kh)p Fq(x)p Fm(i)1122 4570 y Fx(\007)p Fo(n)1223 4611 y Fq(\026)p Fs(\()p Fq(H)8 b Fs(\))p Fm(h)p Fq(x)p Fm(i)1580 4570 y Fx(\006)p Fo(n)1681 4611 y Fm(k)28 b(\024)g Fq(K)7 b Fs(\()p Fq(p;)17 b(\026)p Fs(\))p Fq(;)121 b(n)28 b Fs(=)f(0)p Fq(;)17 b Fs(1)p Fq(;)g(:)g(:)g(:)e(;)i(p;)337 b Fs(\(A.22\))328 4831 y FB(wher)-5 b(e)34 b Fq(K)7 b Fs(\()p Fq(p;)17 b(\026)p Fs(\))34 b FB(is)h(some)f(\014nite)h(c)-5 b(onstant.)1898 5214 y Fs(47)p eop %%Page: 48 48 48 47 bop 328 631 a Ft(Lemma)37 b(A.5)49 b FB(The)30 b(r)-5 b(e)g(gularize)g(d)30 b Fq(G)1697 646 y Fo(\013;J)1839 631 y Fs(=)e(\()p Fq(p)2030 646 y Fo(J)2069 658 y Fl(d)2122 631 y Fs(+)14 b Fq(\026)p Fs(\))p Fq(G)2386 646 y Fo(\013)2435 631 y Fs(\()p Fq(p)2522 646 y Fo(J)2561 658 y Fl(d)2614 631 y Fs(+)g Fq(\026)p Fs(\))30 b FB(satis\014es)h(the)g(same)328 768 y(b)-5 b(ounds)34 b(\(2.22\))g(as)h Fq(G)1145 783 y Fo(\013)1194 768 y FB(.)45 b(Mor)-5 b(e)g(over,)34 b Fs(ad)1830 717 y Fp(\()p Fo(n)p Fp(\))1830 795 y Fo(\037A)1927 803 y Fl(p)1963 795 y Fo(\037)2011 768 y Fs(\()p Fq(G)2126 783 y Fo(\013;J)2240 768 y Fs(\))h FB(is)f(b)-5 b(ounde)g(d,)34 b Fq(n)28 b Fs(=)g(0)p Fq(;)17 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g Fs(3)p FB(.)474 955 y(Pr)-5 b(o)g(of.)73 b Fs(The)29 b(\014rst)g(assertion)f(follo)m(ws)f(easily)g(from)g(the)i(fact)f(that) g Fm(h)p Fq(x)p Fm(i)3098 918 y Fo(m)3164 955 y Fq(p)3213 970 y Fo(J)3252 982 y Fl(d)3292 955 y Fm(h)p Fq(x)p Fm(i)3425 918 y Fo(n)3500 955 y Fs(is)328 1075 y(b)s(ounded,)33 b(for)f(all)f Fq(m;)17 b(n)p Fs(,)33 b(and)f(from)g(Lemma)f(A.4)h (\(use)i Fm(h)p Fq(x)p Fm(i)2574 1039 y Fo(n)2621 1075 y Fq(\026)27 b Fs(=)h Fm(h)p Fq(x)p Fm(i)2944 1039 y Fo(n)2991 1075 y Fq(\026)p Fm(h)p Fq(x)p Fm(i)3183 1039 y Fx(\000)p Fo(n)3284 1075 y Fm(h)p Fq(x)p Fm(i)3417 1039 y Fo(n)3464 1075 y Fs(\).)474 1195 y(In)k(order)g(to)g(sho)m(w)h (b)s(oundedness)h(of)d(the)i(m)m(ulti-comm)m(utators,)28 b(w)m(e)33 b(treat)f(a)f(t)m(yp-)328 1316 y(ical)g(term)h(app)s(earing) f(in)h(ad)1409 1265 y Fp(\(3\))1409 1343 y Fo(\037A)1506 1351 y Fl(p)1542 1343 y Fo(\037)1590 1316 y Fs(\()p Fq(G)1705 1331 y Fo(\013;J)1819 1316 y Fs(\):)655 1499 y Fq(\037A)789 1514 y Fo(p)829 1499 y Fq(\037G)967 1514 y Fo(\013;J)1081 1499 y Fq(\037A)1215 1514 y Fo(p)1255 1499 y Fq(\037\037A)1450 1514 y Fo(p)1490 1499 y Fq(\037)738 1645 y Fs(=)83 b Fq(\037A)1031 1660 y Fo(p)1071 1645 y Fm(h)p Fq(x)p Fm(i)1204 1604 y Fx(\000)p Fp(1)1298 1645 y Fm(h)p Fq(x)p Fm(i)p Fq(\037)p Fm(h)p Fq(x)p Fm(i)1625 1604 y Fx(\000)p Fp(1)1719 1645 y Fm(h)p Fq(x)p Fm(i)p Fq(G)1929 1660 y Fo(\013;J)2043 1645 y Fm(h)p Fq(x)p Fm(i)2176 1604 y Fp(2)2215 1645 y Fm(h)p Fq(x)p Fm(i)2348 1604 y Fx(\000)p Fp(2)2443 1645 y Fq(\037A)2577 1660 y Fo(p)2617 1645 y Fq(\037)p Fm(h)p Fq(x)p Fm(ih)p Fq(x)p Fm(i)2944 1604 y Fx(\000)p Fp(1)3038 1645 y Fq(A)3111 1660 y Fo(p)3151 1645 y Fq(\037:)328 1828 y Fs(Since)25 b Fq(\037A)709 1843 y Fo(p)749 1828 y Fm(h)p Fq(x)p Fm(i)882 1792 y Fx(\000)p Fp(1)1002 1828 y Fs(and)g Fm(h)p Fq(x)p Fm(i)1317 1792 y Fx(\000)p Fp(1)1411 1828 y Fq(A)1484 1843 y Fo(p)1524 1828 y Fq(\037)g Fs(are)g(b)s (ounded,)j(w)m(e)e(see)h(from)d(Lemmas)g(A.4)h(and)g(the)328 1949 y(b)s(ound)d(\(2.22\))f(\(for)g Fq(G)1142 1964 y Fo(\013;J)1256 1949 y Fs(\),)j(that)d(the)h(r.h.s.)41 b(is)21 b(b)s(ounded,)k(pro)m(vided)d Fm(kh)p Fq(x)p Fm(i)3031 1912 y Fx(\000)p Fp(2)3125 1949 y Fq(\037A)3259 1964 y Fo(p)3299 1949 y Fq(\037)p Fm(h)p Fq(x)p Fm(ik)28 b Fq(<)328 2069 y Fm(1)p Fs(.)48 b(T)-8 b(o)34 b(obtain)f(the)i(latter) e(b)s(ound,)i(it)e(is)h(enough)h(\(due)g(to)e(due)i(to)f(Lemma)f(A.4\)) h(to)328 2189 y(sho)m(w)g Fm(kh)p Fq(x)p Fm(i)753 2153 y Fx(\000)p Fp(2)847 2189 y Fq(\037A)981 2204 y Fo(p)1021 2189 y Fm(h)p Fq(x)p Fm(ik)27 b Fq(<)h Fm(1)p Fs(,)k(whic)m(h)h(in)f (turn)g(is)h(pro)m(v)m(ed)h(b)m(y)f(writing)1004 2422 y Fm(h)p Fq(x)p Fm(i)1137 2380 y Fx(\000)p Fp(2)1231 2422 y Fq(\037A)1365 2437 y Fo(p)1405 2422 y Fm(h)p Fq(x)p Fm(i)28 b Fs(=)1687 2354 y Fq(i)p 1679 2399 49 4 v 1679 2490 a Fs(2)1738 2422 y Fm(h)p Fq(x)p Fm(i)1871 2380 y Fx(\000)p Fp(2)1965 2422 y Fq(\037)2043 2327 y Fi(X)2093 2536 y Fo(n)2187 2422 y Fs(\()p Fq(x)2280 2437 y Fo(n)2327 2422 y Fm(h)p Fq(x)p Fm(i)p Fq(@)2511 2437 y Fo(n)2581 2422 y Fs(+)22 b(1)p Fq(=)p Fs(2\))p Fq(;)328 2692 y Fs(and)j(pro)s(ceeding)g(as)g(in)g(the)g(pro)s(of)f(of)h(Lemma)f (\(A.4\),)i(b)m(y)g(comm)m(uting)d Fq(x)3024 2707 y Fo(n)3072 2692 y Fm(h)p Fq(x)p Fm(i)i Fs(through)328 2813 y Fq(\037)33 b Fs(to)f(the)h(left.)2613 b Fn(\004)474 3053 y FB(Pr)-5 b(o)g(of)38 b(of)g(Pr)-5 b(op)g(osition)38 b(3.4)p Fs(.)72 b(The)38 b(op)s(erator)d Fq(\037)p Fs(\()p Fq(H)2389 3068 y Fo(p)2454 3053 y Fs(+)24 b Fq(W)14 b Fs(\))p Fq(\037)36 b Fs(is)g(b)s(ounded,)i(hence)328 3174 y(relativ)m(ely)f(\003)827 3189 y Fo(p)866 3174 y Fs(-b)s(ounded.)61 b(W)-8 b(e)38 b(will)e(use)j(b)s(elo)m(w)f(the)h(fact)e(that)h([)p Fq(\037;)17 b(A)2964 3189 y Fo(p)3004 3174 y Fs(])38 b(is)g(b)s(ounded,)328 3294 y(whic)m(h)31 b(follo)m(ws)f(from)f(Lemma)g (A.2,)j(with)e Fq(X)35 b Fs(=)28 b Fq(H)2227 3309 y Fo(p)2266 3294 y Fs(,)k Fq(Z)i Fs(=)28 b Fq(A)2603 3309 y Fo(p)2643 3294 y Fs(,)j Fq(Y)49 b Fs(=)27 b(\003)2978 3309 y Fo(p)3018 3294 y Fs(,)k Fq(M)38 b Fs(=)28 b(1.)42 b(W)-8 b(e)328 3415 y(ha)m(v)m(e,)34 b(in)e(the)h(strong)f(sense)j(on)d Fm(D)s Fs(\(\003)1732 3430 y Fo(p)1771 3415 y Fs(\),)1010 3615 y(ad)1113 3564 y Fp(\(2\))1113 3642 y Fo(\037A)1210 3650 y Fl(p)1246 3642 y Fo(\037)1294 3615 y Fs(\()p Fq(H)1413 3630 y Fo(p)1452 3615 y Fs(\))c(=)g Fq(\037)p Fs([)p Fq(\037)1771 3574 y Fp(2)1810 3615 y Fq(H)1891 3630 y Fo(p)1931 3615 y Fq(;)17 b(A)2048 3630 y Fo(p)2087 3615 y Fs(])p Fq(\037)23 b Fs(+)f([)p Fq(\037W)14 b(\037;)j(\037A)2729 3630 y Fo(p)2768 3615 y Fq(\037)p Fs(])p Fq(;)328 3798 y Fs(where)44 b(the)f(comm)m(utator)f(in)g(the)h(\014rst)g(term)f(is)h (b)s(ounded,)j(and)d(the)g(second)h(term)328 3919 y(equals)37 b Fq(\037)p Fs(\()p Fq(W)14 b(\037)894 3883 y Fp(2)933 3919 y Fq(A)1006 3934 y Fo(p)1070 3919 y Fm(\000)26 b Fq(A)1246 3934 y Fo(p)1285 3919 y Fq(\037)1346 3883 y Fp(2)1386 3919 y Fq(W)14 b Fs(\))p Fq(\037)p Fs(,)37 b(whic)m(h)g(is)f(easily)f(seen)j(to)e(b)s(e)g(b)s(ounded,)i(to)s(o)e (\(use)328 4039 y(Lemma)47 b(A.4)i(together)g(with)f(the)h(fact)g(that) f Fq(\037A)2283 4054 y Fo(p)2323 4039 y Fm(h)p Fq(x)p Fm(i)2456 4003 y Fx(\000)p Fp(1)2599 4039 y Fs(is)g(b)s(ounded,)54 b(and)48 b(so)h(is)328 4160 y Fq(W)14 b Fm(h)p Fq(x)p Fm(i)p Fs(\).)43 b(Next,)33 b(w)m(e)h(sho)m(w)g(that)554 4357 y(ad)657 4306 y Fp(\(3\))657 4384 y Fo(\037A)754 4392 y Fl(p)790 4384 y Fo(\037)838 4357 y Fs(\()p Fq(H)957 4372 y Fo(p)996 4357 y Fs(\))28 b(=)f([)p Fq(\037)p Fs([)p Fq(\037)1341 4316 y Fp(2)1381 4357 y Fq(H)1462 4372 y Fo(p)1502 4357 y Fq(;)17 b(A)1619 4372 y Fo(p)1658 4357 y Fs(])p Fq(\037;)g(\037A)1924 4372 y Fo(p)1964 4357 y Fq(\037)p Fs(])22 b(+)g([[)p Fq(\037W)14 b(\037;)j(\037A)2632 4372 y Fo(p)2672 4357 y Fq(\037)p Fs(])p Fq(;)g(\037A)2938 4372 y Fo(p)2978 4357 y Fq(\037)p Fs(])226 b(\(A.23\))328 4540 y(is)28 b(b)s(ounded.)43 b(The)29 b(\014rst)f(term)g(on)g(the)h (r.h.s.)43 b(is)27 b(the)i(sum)f(of)g(t)m(w)m(o)h(b)s(ounded)g(op)s (erators)328 4661 y(plus)35 b Fq(\037)p Fs([)17 b([)p Fq(\037)730 4625 y Fp(2)770 4661 y Fq(H)851 4676 y Fo(p)890 4661 y Fq(;)g(A)1007 4676 y Fo(p)1047 4661 y Fs(])p Fq(;)g(\037A)1252 4676 y Fo(p)1291 4661 y Fq(\037)p Fs(])p Fq(\037)p Fs(,)37 b(whic)m(h)f(can)g(b)s(e)f(written)g(\(b)m(y)i(comm)m(uting)c Fq(\037)j Fs(through)328 4781 y Fq(A)401 4796 y Fo(p)441 4781 y Fs(\))c(as)h(a)f(b)s(ounded)i(op)s(erator)d(plus)1492 4965 y Fq(\037)p Fs([)17 b([)p Fq(\037)1685 4924 y Fp(2)1725 4965 y Fq(H)1806 4980 y Fo(p)1845 4965 y Fq(;)g(A)1962 4980 y Fo(p)2002 4965 y Fs(])p Fq(;)g(A)2146 4980 y Fo(p)2185 4965 y Fq(\037)2246 4924 y Fp(2)2286 4965 y Fs(])p Fq(\037:)891 b Fs(\(A.24\))1898 5214 y(48)p eop %%Page: 49 49 49 48 bop 328 631 a Fs(Setting)29 b Fq(\037)722 646 y Fp(1)789 631 y Fs(=)f Fq(\037H)1035 646 y Fo(p)1074 631 y Fs(,)j(w)m(e)f(expand)h([)p Fq(\037)1696 646 y Fp(1)1736 631 y Fq(;)17 b(A)1853 646 y Fo(p)1892 631 y Fs(])28 b(=)f Fq(\037)2111 595 y Fx(0)2111 656 y Fp(1)2151 631 y Fq(H)2232 646 y Fo(p)2287 631 y Fs(+)2379 551 y Fi(R)2474 631 y Fs(^)-61 b Fq(\037)2523 646 y Fp(1)2563 631 y Fs(\()p Fq(s)p Fs(\))2702 551 y Fi(R)2768 577 y Fo(s)2748 666 y Fp(0)2821 631 y Fq(e)2866 595 y Fo(i)p Fp(\()p Fo(s)p Fx(\000)p Fo(s)3038 604 y Fj(1)3073 595 y Fp(\))p Fo(H)3158 603 y Fl(p)3198 631 y Fq(W)14 b(e)3349 595 y Fo(is)3406 604 y Fj(1)3440 595 y Fo(H)3498 603 y Fl(p)3539 631 y Fs(,)328 751 y(and)40 b(\(A.24\))g(splits)f(in)m(to)g(t)m(w)m(o)h (terms,)i(the)f(\014rst)f(one,)i Fq(\037)p Fs([)p Fq(\037)2554 715 y Fx(0)2554 776 y Fp(1)2594 751 y Fq(H)2675 766 y Fo(p)2714 751 y Fq(;)17 b(A)2831 766 y Fo(p)2871 751 y Fs(])p Fq(\037)2959 715 y Fp(4)2999 751 y Fs(,)41 b(is)f(b)s(ounded.) 328 872 y(T)-8 b(o)33 b(see)g(b)s(oundedness)i(of)d(the)h(second)h (term,)e(write)g(it)g(as)463 996 y Fi(Z)590 1132 y Fs(^)-60 b Fq(\037)640 1147 y Fp(1)679 1132 y Fs(\()p Fq(s)p Fs(\))818 996 y Fi(Z)917 1022 y Fo(s)873 1221 y Fp(0)971 1132 y Fq(\037)1049 1051 y Fi(\010)1107 1132 y Fs([)p Fq(e)1179 1090 y Fo(i)p Fp(\()p Fo(s)p Fx(\000)p Fo(s)1351 1099 y Fj(1)1385 1090 y Fp(\))p Fo(H)1470 1098 y Fl(p)1511 1132 y Fq(W)14 b(e)1662 1090 y Fo(is)1719 1099 y Fj(1)1753 1090 y Fo(H)1811 1098 y Fl(p)1851 1132 y Fq(;)j(A)1968 1147 y Fo(p)2008 1132 y Fs(])p Fq(\037)2096 1090 y Fp(2)2158 1132 y Fs(+)22 b Fq(A)2329 1147 y Fo(p)2368 1132 y Fq(e)2413 1090 y Fo(i)p Fp(\()p Fo(s)p Fx(\000)p Fo(s)2585 1099 y Fj(1)2620 1090 y Fp(\))p Fo(H)2705 1098 y Fl(p)2746 1132 y Fs([)p Fq(W)m(;)17 b(\037)2967 1090 y Fp(2)3006 1132 y Fs(])p Fq(e)3078 1090 y Fo(is)3135 1099 y Fj(1)3170 1090 y Fo(H)3228 1098 y Fl(p)3268 1051 y Fi(\011)3343 1132 y Fq(\037;)328 1399 y Fs(and)29 b(use)h(that)f Fm(h)p Fq(x)p Fm(i)p Fq(W)43 b Fs(is)28 b(b)s(ounded.)44 b(The)30 b(second)g(term)f(in)f(\(A.23\))h(can)g(b)s(e)h(written)f(as)328 1519 y(a)f(b)s(ounded)h(op)s(erator)e(plus)h Fq(\037)p Fs([)p Fq(W)14 b(\037)1644 1483 y Fp(2)1683 1519 y Fq(A)1756 1534 y Fo(p)1809 1519 y Fm(\000)f Fq(A)1972 1534 y Fo(p)2013 1519 y Fq(\037)2074 1483 y Fp(2)2113 1519 y Fq(W)m(;)k(\037)2307 1483 y Fp(2)2347 1519 y Fq(A)2420 1534 y Fo(p)2460 1519 y Fs(])p Fq(\037)p Fs(.)42 b(The)29 b(latter)e(term)g(equals)328 1639 y(again)h(a)h(b)s(ounded)i(op)s(erator)e(plus)g(2Re[)p Fq(W)14 b(\037)2010 1603 y Fp(2)2049 1639 y Fq(A)2122 1654 y Fo(p)2162 1639 y Fq(;)j(\037)2267 1603 y Fp(2)2306 1639 y Fq(A)2379 1654 y Fo(p)2419 1639 y Fs(],)30 b(whic)m(h)h(is)e (easily)f(seen)j(to)f(b)s(e)328 1760 y(b)s(ounded,)38 b(b)m(y)f(again)e(noticing)g(that)h Fm(h)p Fq(x)p Fm(i)p Fq(W)14 b Fs(,)36 b(and)h(deriv)-5 b(ativ)m(es)36 b(of)g Fq(W)50 b Fs(m)m(ultiplied)33 b(b)m(y)328 1880 y Fm(h)p Fq(x)p Fm(i)461 1844 y Fp(2)533 1880 y Fs(are)f(b)s(ounded.)474 2001 y(W)-8 b(e)32 b(ha)m(v)m(e)g(th)m(us)g(sho)m(wn)g(that)f(the)h(m)m (ulti-comm)m(utators)27 b(app)s(earing)j(in)g(\(3.42\))h(are)328 2121 y(b)s(ounded)26 b(\(hence)g(\003)1088 2136 y Fo(p)1127 2121 y Fs(-b)s(ounded)f(in)f(the)i(sense)g(of)f(Kato\).)40 b(Clearly)-8 b(,)25 b Fq(N)36 b Fs(is)24 b(\003)3103 2136 y Fo(f)3173 2121 y Fs(b)s(ounded,)328 2241 y(and)35 b(Lemma)e(A.5,)j(together)f(with)g(the)g(fact)g(that)f Fq(')p Fs(\(\()p Fm(\000)p Fq(i@)2573 2256 y Fo(u)2620 2241 y Fs(\))2658 2205 y Fo(k)2700 2241 y Fq(\034)2742 2256 y Fo(\014)2790 2241 y Fs(\()p Fq(g)2875 2256 y Fo(\013)2924 2241 y Fs(\)\)\))g(is)h(relativ)m(ely)328 2362 y Fq(N)416 2326 y Fp(1)p Fo(=)p Fp(2)526 2362 y Fs(-b)s(ounded,)44 b(sho)m(ws)f(that)e Fq(I)1547 2377 y Fo(n)1635 2362 y Fs(is)f(\003)1809 2377 y Fo(f)1854 2362 y Fs(-b)s(ounded)i(in)e(the)h (sense)i(of)e(Kato.)68 b(Conse-)328 2482 y(quen)m(tly)-8 b(,)33 b(condition)f(\(3.1\))g(is)g(satis\014ed)h(for)f Fq(X)j Fs(=)28 b Fq(C)2270 2497 y Fo(n)2317 2482 y Fs(.)474 2603 y(Next,)35 b(w)m(e)g(v)m(erify)f(that)f(\(3.2\))g(is)g (satis\014ed.)47 b(Let)34 b(us)g(start)g(with)f(the)h(comm)m(utator)328 2723 y(of)27 b Fq(C)504 2738 y Fp(1)572 2723 y Fs(with)g(\003.)42 b(W)-8 b(e)28 b(need)i(to)d(sho)m(w)i(relativ)m(e)f(b)s(oundedness)i (of)e([)p Fq(\037)p Fs(\()p Fq(H)20 b Fs(+)13 b Fq(W)h Fs(\))p Fq(\037;)j Fs(\003)3315 2738 y Fo(p)3354 2723 y Fs(])28 b(and)328 2843 y([)p Fq(I)398 2858 y Fp(1)437 2843 y Fq(;)17 b Fs(\003])34 b(\(relativ)m(e)f(to)h(\003)1185 2858 y Fo(p)1259 2843 y Fs(and)g(\003)1518 2858 y Fo(f)1563 2843 y Fs(,)h(resp)s(ectiv)m(ely)-8 b(,)35 b(in)e(the)i(sense)h(of)e (quadratic)f(forms\).)328 2964 y(Noticing)k(that)i(\003)1013 2979 y Fo(p)1092 2964 y Fs(=)f Fq(H)1287 2979 y Fo(p)1353 2964 y Fm(\000)28 b Fq(v)i Fs(+)c Fq(x)1692 2928 y Fp(2)1732 2964 y Fs(,)41 b(w)m(e)f(write)f([)p Fq(\037)p Fs(\()p Fq(H)c Fs(+)26 b Fq(W)14 b Fs(\))p Fq(\037;)j Fs(\003)2867 2979 y Fo(p)2906 2964 y Fs(])39 b(as)g(a)g(sum)g(of)g(a)328 3084 y(b)s(ounded)33 b(op)s(erator)f(plus)1456 3299 y([)p Fq(\037)1544 3258 y Fp(2)1584 3299 y Fq(H)1665 3314 y Fo(p)1704 3299 y Fq(;)17 b(x)1803 3258 y Fp(2)1843 3299 y Fs(])22 b(+)g([)p Fq(\037W)14 b(\037;)j(x)2344 3258 y Fp(2)2384 3299 y Fs(])p Fq(:)854 b Fs(\(A.25\))328 3513 y(No)m(w)46 b(setting)f Fq(\037)957 3528 y Fp(1)1047 3513 y Fs(=)50 b Fq(\037)1234 3477 y Fp(2)1273 3513 y Fq(H)1354 3528 y Fo(p)1394 3513 y Fs(,)e(the)f(\014rst)f(term)f(in)f (\(A.25\))i(equals)2866 3439 y Fi(P)2971 3542 y Fo(n)3018 3513 y Fs(\()p Fq(x)3111 3528 y Fo(n)3158 3513 y Fs([)p Fq(\037)3246 3528 y Fp(1)3286 3513 y Fq(;)17 b(x)3385 3528 y Fo(n)3432 3513 y Fs(])31 b(+)328 3634 y([)p Fq(\037)416 3649 y Fp(1)456 3634 y Fq(;)17 b(x)555 3649 y Fo(n)602 3634 y Fs(])p Fq(x)684 3649 y Fo(n)731 3634 y Fs(\),)33 b(so)g(for)f(an)m(y)h Fq( )e Fm(2)e Fq(C)1548 3597 y Fx(1)1541 3658 y Fp(0)1622 3634 y Fs(,)921 3866 y Fm(j)966 3785 y Fi(\012)1013 3866 y Fq( )t(;)17 b Fs([)p Fq(\037)1212 3881 y Fp(1)1251 3866 y Fq(;)g(x)1350 3825 y Fp(2)1389 3866 y Fs(])p Fq( )1484 3785 y Fi(\013)1558 3866 y Fm(\024)28 b Fq(k)1734 3771 y Fi(X)1785 3980 y Fo(n)1895 3866 y Fm(k)p Fq(x)2000 3881 y Fo(n)2047 3866 y Fq( )t Fm(k)k(k)p Fq( )t Fm(k)27 b(\024)i Fq(k)19 b Fm(h)p Fq( )t(;)e Fs(\003)2784 3881 y Fo(p)2823 3866 y Fq( )t Fm(i)f Fq(:)320 b Fs(\(A.26\))328 4161 y(Next,)403 4375 y Fm(j)448 4295 y Fi(\012)494 4375 y Fq( )t(;)17 b Fs([)p Fq(\037W)d(\037;)j(x)959 4334 y Fp(2)999 4375 y Fs(])p Fq( )1093 4295 y Fi(\013)1156 4375 y Fm(j)2108 b Fs(\(A.27\))569 4548 y Fm(\024)28 b Fs(2)p Fm(j)768 4467 y Fi(\012)814 4548 y Fq( )t(;)17 b(\037W)d Fs([)p Fq(\037;)j(x)1279 4507 y Fp(2)1319 4548 y Fs(])p Fq( )1413 4467 y Fi(\013)1476 4548 y Fm(j)28 b(\024)g Fs(2)1703 4453 y Fi(X)1753 4663 y Fo(n)1863 4548 y Fm(j)17 b(h)o Fq( )t(;)g Fs(\()p Fq(\037W)d(x)2317 4563 y Fo(n)2364 4548 y Fs([)p Fq(\037;)j(x)2551 4563 y Fo(n)2598 4548 y Fs(])22 b(+)g Fq(\037W)14 b Fs([)p Fq(\037;)j(x)3099 4563 y Fo(n)3146 4548 y Fs(])p Fq(x)3228 4563 y Fo(n)3276 4548 y Fs(\))p Fq( )t Fm(i)f(j)p Fq(:)328 4844 y Fs(Comm)m(uting)22 b Fq(x)908 4859 y Fo(n)979 4844 y Fs(in)i(the)g(\014rst)g(term)g(in)f(the)h(sum)g(through)g Fq(\037)g Fs(to)g(the)g(left,)h(one)f(sees)i(that)328 4965 y Fm(j)17 b(h)o Fq( )t(;)g(\037W)d(x)744 4980 y Fo(n)791 4965 y Fs([)p Fq(\037;)j(x)978 4980 y Fo(n)1025 4965 y Fs(])p Fq( )t Fm(i)g(j)27 b(\024)h Fq(k)s Fs(\()p Fm(k)p Fq( )t Fm(k)1594 4929 y Fp(2)1655 4965 y Fs(+)21 b Fm(k)p Fq(x)1857 4980 y Fo(n)1904 4965 y Fq( )t Fm(k)32 b(k)p Fq( )t Fm(k)p Fs(\),)g(whic)m(h)h(is)f(b)s(ounded)h(from)e(ab)s (o)m(v)m(e)1898 5214 y(49)p eop %%Page: 50 50 50 49 bop 328 631 a Fs(b)m(y)33 b Fq(k)20 b Fm(h)p Fq( )t(;)d Fs(\003)752 646 y Fo(p)791 631 y Fq( )t Fm(i)32 b Fs(\(pro)s(ceed)h(as) g(in)e(\(A.26\)\).)43 b(The)34 b(second)g(term)e(in)f(the)i(sum)f(in)g (\(A.27\))328 751 y(is)g(estimated)g(in)g(the)h(same)f(w)m(a)m(y)-8 b(.)45 b(This)33 b(sho)m(ws)h(that)e(\(A.25\))g(is)g(\003)2811 766 y Fo(p)2851 751 y Fs(-form-b)s(ounded.)474 872 y(Next,)g(in)e (order)g(to)g(sho)m(w)i(the)f(relativ)m(e)e(b)s(ound)i(on)f([)p Fq(I)2471 887 y Fo(n)2518 872 y Fq(;)17 b Fs(\003],)31 b(it)e(is)h(enough)h(to)f(sho)m(w)328 1006 y(that)i([ad)669 955 y Fp(\()p Fo(n)p Fp(\))669 1033 y Fo(\037A)766 1041 y Fl(p)802 1033 y Fo(\037)850 1006 y Fs(\()p Fq(G)965 1021 y Fo(\013)1015 1006 y Fs(\))p Fq(;)17 b Fs(\003)1165 1021 y Fo(p)1204 1006 y Fs(])32 b(is)g(relativ)m(ely)g(\003)1855 1021 y Fo(p)1894 1006 y Fs(-form-b)s(ounded,)f(and)i(that)1495 1177 y([)p Fq(')p Fs(\(\()p Fm(\000)p Fq(i@)1823 1192 y Fo(u)1869 1177 y Fs(\))1907 1136 y Fo(n)1954 1177 y Fq(\034)1996 1192 y Fo(\014)2043 1177 y Fs(\()p Fq(g)2128 1192 y Fo(\013)2177 1177 y Fs(\))p Fq(;)17 b Fs(\003)2327 1192 y Fo(f)2372 1177 y Fs(])328 1348 y(is)43 b(relativ)m(ely)g(\003) 942 1363 y Fo(f)987 1348 y Fs(-form-b)s(ounded.)76 b(The)45 b(former)d(b)s(ound)i(is)g(easily)f(obtained)g(from)328 1469 y(\(2.22\),)34 b(and)h(the)g(latter)e(has)i(b)s(een)g(treated)g (in)e(subsection)j(3.3.1.)48 b(This)35 b(sho)m(ws)h(that)328 1589 y Fq(I)371 1604 y Fo(n)457 1589 y Fs(are)i(relativ)m(ely)g (\003-form-b)s(ounded,)g(hence)j(also)c(completing)g(the)i(pro)s(of)f (that)h Fq(C)3527 1604 y Fp(1)328 1710 y Fs(satis\014es)33 b(condition)e(\(3.2\).)474 1830 y(Next,)g(w)m(e)e(consider)g(the)g (comm)m(utator)f(of)g Fq(C)2140 1845 y Fp(2)2208 1830 y Fs(with)g(\003.)42 b(The)29 b(only)f(thing)g(to)g(c)m(hec)m(k)328 1950 y(is)33 b(that)h([ad)770 1965 y Fo(\037A)867 1973 y Fl(p)903 1965 y Fo(\037)951 1950 y Fs(\()p Fq(H)1070 1965 y Fo(p)1110 1950 y Fs(\))p Fq(;)17 b Fs(\003)1260 1965 y Fo(p)1299 1950 y Fs(])34 b(is)f(\003)1527 1965 y Fo(p)1566 1950 y Fs(-form-b)s(ounded.)47 b(This)34 b(comm)m(utator)e(can)j(b)s(e)f(writ-)328 2092 y(ten)43 b(as)f(a)g(b)s(ounded)h(op)s(erator)f(plus)g([ad)1883 2041 y Fp(\(2\))1883 2119 y Fo(\037A)1980 2127 y Fl(p)2016 2119 y Fo(\037)2064 2092 y Fs(\()p Fq(H)2183 2107 y Fo(p)2223 2092 y Fs(\))p Fq(;)17 b(x)2360 2056 y Fp(2)2399 2092 y Fs(],)45 b(hence)f(it)d(su\016ces)k(to)d(sho)m(w)328 2246 y(that)f Fq(x)603 2261 y Fo(n)651 2246 y Fs([ad)781 2195 y Fp(\(2\))781 2273 y Fo(\037A)878 2281 y Fl(p)913 2273 y Fo(\037)961 2246 y Fs(\()p Fq(H)1080 2261 y Fo(p)1120 2246 y Fs(\))p Fq(;)17 b(x)1257 2261 y Fo(n)1304 2246 y Fs(])41 b(is)g(\003)1547 2261 y Fo(p)1587 2246 y Fs(-form-b)s(ounded) f(\()p Fq(n)j Fs(=)f(1)p Fq(;)17 b Fs(2)p Fq(;)g Fs(3\).)69 b(One)42 b(sho)m(ws)h(that)328 2400 y([ad)458 2349 y Fp(\(2\))458 2427 y Fo(\037A)555 2435 y Fl(p)591 2427 y Fo(\037)639 2400 y Fs(\()p Fq(H)758 2415 y Fo(p)797 2400 y Fs(\))p Fq(;)17 b(x)934 2415 y Fo(n)981 2400 y Fs(])34 b(is)g(b)s(ounded,)h(b)m(y)g(simple)e(estimates)h(as)g(ab)s(o)m (v)m(e.)49 b(Relativ)m(e)33 b(b)s(ound-)328 2554 y(edness)45 b(of)e Fq(x)820 2569 y Fo(n)867 2554 y Fs([ad)997 2504 y Fp(\(2\))997 2581 y Fo(\037A)1094 2589 y Fl(p)1130 2581 y Fo(\037)1178 2554 y Fs(\()p Fq(H)1297 2569 y Fo(p)1336 2554 y Fs(\))p Fq(;)17 b(x)1473 2569 y Fo(n)1520 2554 y Fs(])44 b(then)f(follo)m(ws)f(easily)h(\(pro)s(ceeding)f(as)i(in)e (\(A.26\)\).)328 2675 y(Consequen)m(tly)-8 b(,)35 b(\(3.2\))d(is)g (satis\014ed)h(for)f Fq(C)1877 2690 y Fp(2)1916 2675 y Fs(.)474 2795 y(W)-8 b(e)31 b(no)m(w)g(consider)g(the)g(comm)m (utator)e(of)h Fq(C)2112 2810 y Fp(3)2181 2795 y Fs(with)g(\003,)h(and) f(it)g(is)g(enough)h(to)f(sho)m(w)328 2916 y(that)37 b([ad)674 2865 y Fp(\(3\))674 2943 y Fo(\037A)771 2951 y Fl(p)807 2943 y Fo(\037)855 2916 y Fs(\()p Fq(H)974 2931 y Fo(p)1013 2916 y Fs(\))p Fq(;)17 b(x)1150 2879 y Fp(2)1190 2916 y Fs(])37 b(is)f(relativ)m(ely)g(\003)1854 2931 y Fo(p)1894 2916 y Fs(-form-b)s(ounded.)55 b(W)-8 b(e)38 b(write)e(this)h(comm)m(u-)328 3070 y(tator)32 b(as)h(2Re)16 b([ad)1002 3019 y Fp(\(2\))1002 3097 y Fo(\037A)1099 3105 y Fl(p)1135 3097 y Fo(\037)1183 3070 y Fs(\()p Fq(H)1302 3085 y Fo(p)1341 3070 y Fs(\))p Fq(\037A)1513 3085 y Fo(p)1553 3070 y Fq(\037;)h(x)1713 3033 y Fp(2)1753 3070 y Fs(].)43 b(No)m(w)34 b(w)m(e)f(ha)m(v)m(e)434 3275 y([ad)564 3224 y Fp(\(2\))564 3302 y Fo(\037A)661 3310 y Fl(p)697 3302 y Fo(\037)745 3275 y Fs(\()p Fq(H)864 3290 y Fo(p)904 3275 y Fs(\))p Fq(\037A)1076 3290 y Fo(p)1116 3275 y Fq(\037;)17 b(x)1276 3290 y Fo(n)1323 3275 y Fs(])28 b(=)f([ad)1611 3224 y Fp(\(2\))1611 3302 y Fo(\037A)1708 3310 y Fl(p)1744 3302 y Fo(\037)1792 3275 y Fs(\()p Fq(H)1911 3290 y Fo(p)1950 3275 y Fs(\))p Fq(;)17 b(x)2087 3290 y Fo(n)2134 3275 y Fs(])p Fq(\037A)2295 3290 y Fo(p)2335 3275 y Fq(\037)23 b Fs(+)f(ad)2620 3224 y Fp(\(2\))2620 3302 y Fo(\037A)2717 3310 y Fl(p)2753 3302 y Fo(\037)2801 3275 y Fs(\()p Fq(H)2920 3290 y Fo(p)2959 3275 y Fs(\)[)p Fq(\037A)3158 3290 y Fo(p)3198 3275 y Fq(\037;)17 b(x)3358 3290 y Fo(n)3405 3275 y Fs(])p Fq(;)328 3480 y Fs(and)33 b(it)e(is)h(clear)g(that)h([ad)1288 3429 y Fp(\(2\))1288 3507 y Fo(\037A)1385 3515 y Fl(p)1421 3507 y Fo(\037)1469 3480 y Fs(\()p Fq(H)1588 3495 y Fo(p)1627 3480 y Fs(\))p Fq(\037A)1799 3495 y Fo(p)1839 3480 y Fq(\037;)17 b(x)1999 3495 y Fo(n)2046 3480 y Fs(])p Fm(h)p Fq(x)p Fm(i)2206 3444 y Fx(\000)p Fp(1)2333 3480 y Fs(is)32 b(b)s(ounded.)44 b(Consequen)m(tly)-8 b(,)651 3584 y Fi(\014)651 3644 y(\014)651 3704 y(\014)684 3588 y(D)745 3699 y Fq( )t(;)17 b Fs([ad)986 3648 y Fp(\(3\))986 3726 y Fo(\037A)1083 3734 y Fl(p)1118 3726 y Fo(\037)1166 3699 y Fs(\()p Fq(H)1285 3714 y Fo(p)1325 3699 y Fs(\))p Fq(;)g(x)1462 3657 y Fp(2)1501 3699 y Fs(])p Fq( )1595 3588 y Fi(E)1656 3584 y(\014)1656 3644 y(\014)1656 3704 y(\014)1717 3699 y Fm(\024)28 b Fq(k)s Fs(\()p Fm(k)p Fq( )t Fm(k)2081 3657 y Fp(2)2143 3699 y Fs(+)2241 3618 y Fi(\012)2288 3699 y Fq( )t(;)17 b(x)2454 3657 y Fp(2)2493 3699 y Fq( 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4637 y Fp(0)911 4622 y Fs(=)27 b(1)-22 b(l)21 b Fm(\000)i Fq(P)1253 4637 y Fp(0)1325 4622 y Fs(is)32 b(giv)m(en)g(b)m(y)p 703 4734 V 703 4814 a Fq(P)780 4829 y Fp(0)847 4814 y Fs(=)1054 4719 y Fi(X)1013 4919 y Fl(m;n)p Fe(2M)960 4976 y Fl(E)s Fj(\()p Fl(m)p Fj(\))p Fe(6)p Fj(=)p Fl(E)s Fj(\()p Fl(n)p Fj(\))1317 4814 y Fq(p)1366 4829 y Fo(m)1455 4814 y Fm(\012)22 b Fq(p)1603 4829 y Fo(n)1673 4814 y Fs(+)g Fq(p)1820 4829 y Fo(d)1882 4814 y Fm(\012)h Fq(p)2031 4829 y Fo(c)2088 4814 y Fs(+)f Fq(p)2235 4829 y Fo(c)2291 4814 y Fm(\012)h Fq(p)2440 4829 y Fo(d)2503 4814 y Fs(+)f Fq(p)2650 4829 y Fo(c)2706 4814 y Fm(\012)h Fq(p)2855 4829 y Fo(c)2890 4814 y Fq(;)375 b Fs(\(A.28\))1898 5214 y(50)p eop %%Page: 51 51 51 50 bop 328 631 a Fs(where)41 b Fq(p)666 646 y Fo(d)747 631 y Fs(and)g Fq(p)994 646 y Fo(c)1068 631 y Fs(are)g(the)g(pro)5 b(jections)40 b(on)m(to)g(the)h(discrete)g(and)f(con)m(tin)m(uous)h (sub-)328 751 y(spaces)34 b(corresp)s(onding)f(to)f Fq(H)1454 766 y Fo(p)1493 751 y Fs(.)44 b(One)33 b(can)g(see)g(that)834 992 y Fq(\017)p Fs(\005)p Fq(I)8 b(R)1072 951 y Fp(2)1071 1016 y Fo(\017)1112 992 y Fq(P)1175 1007 y Fp(0)1214 992 y Fq(I)g Fs(\005)28 b Fm(!)f Fs(0)p Fq(;)82 b(\017)p Fs(\005)p Fq(I)8 b(R)1889 951 y Fp(2)1888 1016 y Fo(\017)2049 897 y Fi(X)2008 1097 y Fl(m;n)p Fe(2M)1955 1154 y Fl(E)s Fj(\()p Fl(m)p Fj(\)=)p Fl(E)s Fj(\()p Fl(n)p Fj(\))2296 992 y Fs(\()p Fq(p)2383 1007 y Fo(m)2471 992 y Fm(\012)23 b Fq(p)2620 1007 y Fo(n)2667 992 y Fs(\))p Fq(I)8 b Fs(\005)27 b Fm(!)h Fs(0)p Fq(;)328 1363 y Fs(as)f Fq(\017)h Fm(!)f Fs(0,)h(that)e(\005)p Fq(I)8 b(R)1144 1327 y Fp(2)1143 1388 y Fo(\017)1184 1363 y Fs(\()p Fq(p)1271 1378 y Fo(c)1315 1363 y Fm(\012)i Fq(p)1451 1378 y Fo(c)1486 1363 y Fs(\))p Fq(I)e Fs(\005)28 b(=)g(0,)f(and)g(that)f(\005)p Fq(I)8 b(R)2471 1327 y Fp(2)2470 1388 y Fo(\017)2511 1363 y Fs(\()p Fq(p)2598 1378 y Fo(c)2642 1363 y Fm(\012)i Fq(p)2778 1378 y Fo(d)2819 1363 y Fs(\))p Fq(I)e Fs(\005)28 b(=)f(\005)p Fq(I)8 b(R)3311 1327 y Fp(2)3310 1388 y Fo(\017)3351 1363 y Fs(\()p Fq(p)3438 1378 y Fo(d)3488 1363 y Fm(\012)328 1483 y Fq(p)377 1498 y Fo(c)412 1483 y Fs(\))p Fq(I)g Fs(\005.)43 b(F)-8 b(rom)31 b(form)m(ula)g(\(2.45\))h(for)g(the)h(in)m (teraction,)e(w)m(e)j(obtain)d(the)i(b)s(ound)454 1721 y(\005)p Fq(I)p 594 1641 59 4 v 7 w(R)653 1659 y Fp(2)653 1745 y Fo(\017)692 1721 y Fq(I)8 b Fs(\005)83 b Fm(\025)h Fs(\005)p Fq(I)8 b(R)1259 1680 y Fp(2)1258 1745 y Fo(\017)1298 1721 y Fs(\()p Fq(p)1385 1736 y Fo(c)1442 1721 y Fm(\012)22 b Fq(p)1590 1736 y Fo(d)1631 1721 y Fs(\))p Fq(I)8 b Fs(\005)900 1883 y(=)1272 1789 y Fi(X)1184 1997 y Fl(m;n;m)1368 1983 y Fe(0)1392 1997 y(2M)1070 2057 y Fl(E)s Fj(\()p Fl(m)p Fj(\)=)p Fl(E)s Fj(\()p Fl(n)p Fj(\)=)p Fl(E)s Fj(\()p Fl(m)1574 2043 y Fe(0)1595 2057 y Fj(\))1645 1789 y Fi(X)1651 2001 y Fo(\013;\013)1761 1982 y Fe(0)1789 1883 y Fs(\()p Fq(p)1876 1898 y Fo(m)1965 1883 y Fm(\012)22 b Fq(p)2113 1898 y Fo(n)2183 1883 y Fm(\012)g Fq(P)2345 1898 y Fp(\012)2400 1883 y Fs(\))17 b Fm(f)p Fq(G)2582 1898 y Fo(\013)2653 1883 y Fm(\012)23 b Fs(1)-22 b(l)2808 1898 y Fo(p)2868 1883 y Fm(\012)23 b Fq(a)p Fs(\()p Fq(\034)3099 1898 y Fo(\014)3147 1883 y Fs(\()p Fq(g)3232 1898 y Fo(\013)3281 1883 y Fs(\)\))p Fm(g)1060 2250 y(\002)1147 2182 y Fq(p)1196 2197 y Fo(c)1253 2182 y Fm(\012)g Fq(p)1402 2197 y Fo(d)1464 2182 y Fm(\012)g Fs(1)-22 b(l)1619 2197 y Fo(f)p 1147 2227 516 4 v 1253 2318 a Fq(L)1319 2284 y Fp(2)1319 2342 y(0)1380 2318 y Fs(+)22 b Fq(\017)1517 2289 y Fp(2)1689 2250 y Fm(f)p Fq(G)1816 2265 y Fo(\013)1861 2246 y Fe(0)1910 2250 y Fm(\012)h Fs(1)-22 b(l)2065 2265 y Fo(p)2125 2250 y Fm(\012)23 b Fq(a)2276 2208 y Fx(\003)2316 2250 y Fs(\()p Fq(\034)2396 2265 y Fo(\014)2443 2250 y Fs(\()p Fq(g)2528 2265 y Fo(\013)2573 2246 y Fe(0)2600 2250 y Fs(\)\))p Fm(g)16 b Fs(\()p Fq(p)2829 2265 y Fo(m)2891 2246 y Fe(0)2940 2250 y Fm(\012)22 b Fq(p)3088 2265 y Fo(n)3157 2250 y Fm(\012)h Fq(P)3320 2265 y Fp(\012)3375 2250 y Fs(\))p Fq(:)328 2545 y Fs(W)-8 b(e)48 b(write)f Fq(a)p Fs(\()p Fq(\034)906 2560 y Fo(\014)954 2545 y Fs(\()p Fq(g)1039 2560 y Fo(\013)1088 2545 y Fs(\)\))53 b(=)1346 2465 y Fi(R)1393 2580 y Fd(R)p Fx(\002)p Fo(S)1543 2561 y Fj(2)p 1598 2458 262 4 v 1598 2545 a Fq(\034)1640 2560 y Fo(\014)1687 2545 y Fs(\()p Fq(g)1772 2560 y Fo(\013)1821 2545 y Fs(\)\()p Fq(u;)17 b Fs(\006\))p Fq(a)p Fs(\()p Fq(u;)g Fs(\006\))47 b(and)g(use)i(the)f(pull)e(through)328 2665 y(form)m(ula)31 b Fq(a)p Fs(\()p Fq(u;)17 b Fs(\006\))p Fq(L)1049 2680 y Fp(0)1116 2665 y Fs(=)27 b(\()p Fq(L)1323 2680 y Fp(0)1385 2665 y Fs(+)22 b Fq(u)p Fs(\))p Fq(a)p Fs(\()p Fq(u;)17 b Fs(\006\))32 b(and)h(obtain)362 2957 y(\005)p Fq(I)p 503 2877 59 4 v 8 w(R)561 2896 y Fp(2)561 2982 y Fo(\017)600 2957 y Fq(I)8 b Fs(\005)83 b Fm(\025)1064 2863 y Fi(X)976 3071 y Fl(m;n;m)1160 3057 y Fe(0)1184 3071 y(2M)862 3131 y Fl(E)s Fj(\()p Fl(m)p Fj(\)=)p Fl(E)s Fj(\()p Fl(n)p Fj(\)=)p Fl(E)s Fj(\()p Fl(m)1366 3117 y Fe(0)1387 3131 y Fj(\))1437 2863 y Fi(X)1443 3075 y Fo(\013;\013)1553 3056 y Fe(0)1598 2822 y Fi(Z)1698 2848 y Fo(E)t Fp(\()p Fo(m)p Fp(\))1653 3047 y Fx(\0001)1891 2957 y Fq(du)2015 2822 y Fi(Z)2069 3047 y Fo(S)2116 3028 y Fj(2)2171 2957 y Fq(d)p Fs(\006)2434 2890 y Fq(u)2490 2854 y Fp(2)p 2302 2934 359 4 v 2302 3026 a Fq(e)2347 2997 y Fx(\000)p Fo(\014)s(u)2513 3026 y Fm(\000)22 b Fs(1)2671 2957 y Fq(g)2718 2972 y Fo(\013)2767 2957 y Fs(\()p Fm(\000)p Fq(u;)17 b Fs(\006\))p 3090 2902 119 4 v Fq(g)3137 2972 y Fo(\013)3182 2953 y Fe(0)3209 2957 y Fs(\()p Fm(\000)p Fq(u;)g Fs(\006\))968 3331 y Fm(\002)1062 3191 y Fi(\022)1135 3331 y Fq(p)1184 3346 y Fo(m)1251 3331 y Fq(G)1328 3346 y Fo(\013)1831 3264 y Fq(p)1880 3279 y Fo(c)p 1387 3308 972 4 v 1387 3400 a Fs(\()p Fq(H)1506 3415 y Fo(p)1568 3400 y Fm(\000)23 b Fq(E)6 b Fs(\()p Fq(m)p Fs(\))22 b(+)g Fq(u)p Fs(\))2121 3371 y Fp(2)2182 3400 y Fs(+)g Fq(\017)2319 3371 y Fp(2)2369 3331 y Fq(G)2446 3346 y Fo(\013)2491 3327 y Fe(0)2518 3331 y Fq(p)2567 3346 y Fo(m)2629 3327 y Fe(0)2655 3191 y Fi(\023)2751 3331 y Fm(\012)h Fq(P)2914 3346 y Fo(n)2983 3331 y Fm(\012)f Fq(P)3145 3346 y Fp(\012)3200 3331 y Fq(;)65 b Fs(\(A.29\))328 3610 y(where)28 b(w)m(e)g(recall)e(that)h Fq(E)6 b Fs(\()p Fq(m)p Fs(\))27 b(is)g(the)g(eigen)m(v)-5 b(alue)27 b(of)g Fq(H)2372 3625 y Fo(p)2438 3610 y Fs(corresp)s(onding)g(to)g(the)g(mo)s (de)328 3731 y Fq(m)p Fs(.)80 b(W)-8 b(e)45 b(ha)m(v)m(e)h(dropp)s(ed)g (the)f(in)m(tegration)e(o)m(v)m(er)j(the)f(v)-5 b(alues)45 b Fq(u)j Fm(\025)g Fq(E)6 b Fs(\()p Fq(m)p Fs(\))45 b(b)s(ecause)328 3851 y Fq(\017)p Fs(\(\()p Fq(H)524 3866 y Fo(p)594 3851 y Fm(\000)31 b Fq(E)6 b Fs(\()p Fq(m)p Fs(\))31 b(+)f Fq(u)p Fs(\))1172 3815 y Fp(2)1241 3851 y Fs(+)g Fq(\017)1386 3815 y Fp(2)1426 3851 y Fs(\))1464 3815 y Fx(\000)p Fp(1)1607 3851 y Fm(!)48 b Fq(\016)t Fs(\()p Fq(H)1921 3866 y Fo(p)1990 3851 y Fm(\000)31 b Fq(E)6 b Fs(\()p Fq(m)p Fs(\))31 b(+)f Fq(u)p Fs(\))44 b(as)h Fq(\017)k Fm(!)f Fs(0,)f(hence)f Fq(u)i Fs(=)328 3971 y Fm(\000)p Fq(H)486 3986 y Fo(p)553 3971 y Fs(+)27 b Fq(E)6 b Fs(\()p Fq(m)p Fs(\))41 b Fm(\024)h Fq(E)6 b Fs(\()p Fq(m)p Fs(\).)66 b(Recalling)38 b(the)j(de\014nition)e (of)h Fq(F)14 b Fs(,)41 b(see)h(b)s(efore)e(2.19,)i(and)328 4092 y(making)31 b(the)i(c)m(hange)h(of)e(v)-5 b(ariable)30 b Fq(u)e Fm(7!)f(\000)p Fq(u)32 b Fs(in)g(the)h(in)m(tegral,)e(w)m(e)j (arriv)m(e)e(at)328 4365 y(\005)p Fq(I)p 469 4285 59 4 v 8 w(R)527 4304 y Fp(2)527 4390 y Fo(\017)566 4365 y Fq(I)8 b Fs(\005)83 b Fm(\025)1147 4271 y Fi(X)1058 4479 y Fl(m;n;m)1242 4465 y Fe(0)1266 4479 y(2M)944 4539 y Fl(E)s Fj(\()p Fl(m)p Fj(\)=)p Fl(E)s Fj(\()p Fl(n)p Fj(\)=)p Fl(E)s Fj(\()p Fl(m)1448 4525 y Fe(0)1469 4539 y Fj(\))1520 4230 y Fi(Z)1619 4256 y Fx(1)1575 4455 y(\000)p Fo(E)t Fp(\()p Fo(m)p Fp(\))1823 4365 y Fq(du)1947 4230 y Fi(Z)2002 4455 y Fo(S)2049 4436 y Fj(2)2104 4365 y Fq(d)p Fs(\006)2339 4298 y Fq(u)2395 4262 y Fp(2)p 2235 4342 304 4 v 2235 4434 a Fq(e)2280 4405 y Fo(\014)s(u)2390 4434 y Fm(\000)23 b Fs(1)3316 4365 y(\(A.30\))934 4739 y Fm(\002)1028 4599 y Fi(\022)1101 4739 y Fq(p)1150 4754 y Fo(m)1217 4739 y Fq(F)14 b Fs(\()p Fq(u;)j Fs(\006\))1994 4672 y Fq(p)2043 4687 y Fo(c)p 1550 4717 974 4 v 1550 4808 a Fs(\()p Fq(H)1669 4823 y Fo(p)1730 4808 y Fm(\000)22 b Fq(E)6 b Fs(\()p Fq(m)p Fs(\))23 b Fm(\000)f Fq(u)p Fs(\))2284 4779 y Fp(2)2345 4808 y Fs(+)g Fq(\017)2482 4779 y Fp(2)2532 4739 y Fq(F)14 b Fs(\()p Fq(u;)j Fs(\006\))2855 4698 y Fx(\003)2894 4739 y Fq(p)2943 4754 y Fo(m)3005 4735 y Fe(0)3032 4599 y Fi(\023)3127 4739 y Fm(\012)23 b Fq(p)3276 4754 y Fo(n)3345 4739 y Fm(\012)g Fq(P)3508 4754 y Fp(\012)3563 4739 y Fq(:)1898 5214 y Fs(51)p eop %%Page: 52 52 52 51 bop 328 631 a Fs(The)38 b(pro)5 b(jection)37 b Fq(p)p Fs(\()p Fq(E)6 b Fs(\))38 b(on)m(to)f(the)h(eigenspace)g (corresp)s(onding)f(to)h(an)f(eigen)m(v)-5 b(alue)37 b Fq(E)328 751 y Fs(of)32 b Fq(H)520 766 y Fo(p)592 751 y Fs(is)g(giv)m(en)h(b)m(y)1080 677 y Fi(P)1186 780 y Fo(m)p Fx(2M)p Fp(:)f Fo(E)t Fp(\()p Fo(m)p Fp(\)=)p Fo(E)1735 751 y Fq(p)1784 766 y Fo(m)1883 751 y Fs(and)h(w)m(e)g(use) 1221 915 y Fi(X)1133 1123 y Fl(m;n;m)1317 1109 y Fe(0)1341 1123 y(2M)1019 1184 y Fl(E)s Fj(\()p Fl(m)p Fj(\)=)p Fl(E)s Fj(\()p Fl(n)p Fj(\)=)p Fl(E)s Fj(\()p Fl(m)1523 1170 y Fe(0)1544 1184 y Fj(\))1606 1010 y Fs(=)1801 915 y Fi(X)1709 1131 y Fo(E)t Fx(2)p Fo(\033)1852 1139 y Fl(p)1888 1131 y Fp(\()p Fo(H)1973 1139 y Fl(p)2010 1131 y Fp(\))2114 915 y Fi(X)2102 1115 y Fl(m)p Fe(2M)2064 1171 y Fl(E)s Fj(\()p Fl(m)p Fj(\)=)p Fl(E)2386 915 y Fi(X)2383 1115 y Fl(n)p Fe(2M)2344 1171 y Fl(E)s Fj(\()p Fl(n)p Fj(\)=)p Fl(E)2670 915 y Fi(X)2647 1123 y Fl(m)2701 1109 y Fe(0)2724 1123 y(2M)2609 1179 y Fl(E)s Fj(\()p Fl(m)2735 1165 y Fe(0)2757 1179 y Fj(\)=)p Fl(E)328 1380 y Fs(to)f(arriv)m(e)h(at)534 1634 y(\005)p Fq(I)p 675 1554 59 4 v 8 w(R)733 1573 y Fp(2)733 1659 y Fo(\017)772 1634 y Fq(I)8 b Fs(\005)28 b Fm(\025)1121 1540 y Fi(X)1029 1756 y Fo(E)t Fx(2)p Fo(\033)1172 1764 y Fl(p)1208 1756 y Fp(\()p Fo(H)1293 1764 y Fl(p)1330 1756 y Fp(\))1374 1499 y Fi(Z)1474 1525 y Fx(1)1429 1724 y(\000)p Fo(E)1565 1634 y Fq(du)1689 1499 y Fi(Z)1743 1724 y Fo(S)1790 1705 y Fj(2)1845 1634 y Fq(d)p Fs(\006)2081 1567 y Fq(u)2137 1531 y Fp(2)p 1976 1611 304 4 v 1976 1703 a Fq(e)2021 1674 y Fo(\014)s(u)2131 1703 y Fm(\000)23 b Fs(1)3292 1634 y(\(A.31\))700 1821 y Fi(\022)773 1961 y Fq(p)p Fs(\()p Fq(E)6 b Fs(\))p Fq(F)14 b Fs(\()p Fq(u;)j Fs(\006\))1673 1894 y Fq(p)1722 1909 y Fo(c)p 1309 1938 812 4 v 1309 2029 a Fs(\()p Fq(H)1428 2044 y Fo(p)1489 2029 y Fm(\000)23 b Fq(E)28 b Fm(\000)22 b Fq(u)p Fs(\))1882 2001 y Fp(2)1944 2029 y Fs(+)g Fq(\017)2081 2001 y Fp(2)2130 1961 y Fq(F)14 b Fs(\()p Fq(u;)j Fs(\006\))2453 1920 y Fx(\003)2492 1961 y Fq(p)p Fs(\()p Fq(E)6 b Fs(\))2695 1821 y Fi(\023)2790 1961 y Fm(\012)23 b Fq(p)p Fs(\()p Fq(E)6 b Fs(\))22 b Fm(\012)h Fq(P)3278 1976 y Fp(\012)3333 1961 y Fq(:)328 2240 y Fs(The)33 b(desired)h(b)s(ound)e(\(3.27\))g(no)m(w)h(follo)m(ws) f(from)f(\(2.20\))h(and)h(\(2.21\).)474 2360 y(The)h(case)f(when)h (\(2.23\))e(holds)g(and)h Fq(\015)k Fs(is)32 b(giv)m(en)h(b)m(y)h (\(2.24\))d(is)h(done)h(similarly)-8 b(.)47 b Fn(\004)328 2693 y Fu(References)328 2912 y Fs([ABG])i(Amrein,)27 b(W.,)h(Boutet)f(de)h(Mon)m(v)m(el,)h(A.,)f(Georgescu,)h(V.:)41 b Fq(C)2894 2927 y Fp(0)2933 2912 y FB(-Gr)-5 b(oups,)31 b(Com-)528 3033 y(mutator)42 b(Metho)-5 b(ds)42 b(and)e(Sp)-5 b(e)g(ctr)g(al)41 b(The)-5 b(ory)41 b(of)g Fq(N)10 b FB(-b)-5 b(o)g(dy)42 b(Hamiltonians.)c Fs(Basel-)528 3153 y(Boston-Berlin:)k(Birkh\177)-49 b(auser,)34 b(1996)328 3356 y([A)-11 b(W])49 b(Araki,)38 b(H.,)h(W)-8 b(o)s(o)s(ds,)38 b(E.:)53 b FB(R)-5 b(epr)g(esentations)38 b(of)h(the)g(c)-5 b(anonic)g(al)37 b(c)-5 b(ommutation)528 3477 y(r)g(elations)49 b(describing)f(a)h(non-r)-5 b(elativistic)48 b(in\014nite)h(fr)-5 b(e)g(e)49 b(b)-5 b(ose)48 b(gas.)g Fs(J.)g(Math.)528 3597 y(Ph)m(ys.)35 b Ft(4)p Fs(,)e(637-662)d(\(1963\))328 3801 y([BFS])49 b(Bac)m(h,)37 b(V.,)g(F)-8 b(r\177)-49 b(ohlic)m(h,)35 b(J.,)i(Sigal,)e(I.M.:)50 b FB(Quantum)38 b(ele)-5 b(ctr)g(o)g(dynamics)37 b(of)g(c)-5 b(on-)528 3921 y(\014ne)g(d)35 b(nonr)-5 b(elativistic)33 b(p)-5 b(articles.)32 b Fs(Adv.)i(Math.)f Ft(137)p Fs(,)f(no.2,)h(299-395)d (\(1995\))328 4124 y([BFSS])49 b(Bac)m(h,)33 b(V.,)f(F)-8 b(r\177)-49 b(ohlic)m(h,)31 b(J.,)h(Sigal,)e(I.M.,)j(So\013er,)f(A.:)43 b FB(Positive)34 b(Commutators)528 4245 y(and)40 b(the)g(sp)-5 b(e)g(ctrum)40 b(of)g(Pauli-Fierz)g(hamiltonians)f(of)g(atoms)h(and)g (mole)-5 b(cules.)528 4365 y Fs(Comm)m(un.)33 b(Math.)f(Ph)m(ys.)j Ft(207)p Fs(,)e(no.)f(3,)h(557-587)d(\(1999\))328 4568 y([BR])49 b(Bratteli,)28 b(O.,)i(Robinson,)g(D.:)41 b FB(Op)-5 b(er)g(ator)32 b(A)n(lgebr)-5 b(as)31 b(and)g(Quantum)h (Statistic)-5 b(al)528 4689 y(Me)g(chanics)43 b(I,)h(II.)d Fs(T)-8 b(exts)44 b(and)f(Monographs)g(in)f(Ph)m(ysics,)47 b(Berlin:)62 b(Springer-)528 4809 y(V)-8 b(erlag,)32 b(2nd)h(edition,)e(1987)1898 5214 y(52)p eop %%Page: 53 53 53 52 bop 328 631 a Fs([F])82 b(F)-8 b(r\177)-49 b(ohlic)m(h,)29 b(J.:)43 b FB(Applic)-5 b(ation)32 b(of)g(Commutator)g(The)-5 b(or)g(ems)31 b(to)i(the)f(Inte)-5 b(gr)g(ation)32 b(of)528 751 y(R)-5 b(epr)g(esentations)32 b(of)g(Lie)h(A)n(lgebr)-5 b(as)31 b(and)h(Commutation)g(R)-5 b(elations.)29 b Fs(Comm)m(un.)528 872 y(Math.)k(Ph)m(ys.)i Ft(54)p Fs(,)d(135-150)f(\(1977\))328 1075 y([FM])49 b(F)-8 b(r\177)-49 b(ohlic)m(h,)31 b(J.,)i(Merkli,)f (M.:)44 b FB(Thermal)34 b(Ionization)p Fs(,)d(submitted,)h(2003)328 1279 y([GG])48 b(Georgescu,)41 b(V.,)g(G)m(\023)-46 b(erard,)40 b(C.:)56 b FB(On)40 b(the)h(Virial)f(The)-5 b(or)g(em)40 b(in)g(Quantum)h(Me-)528 1399 y(chanics.)32 b Fs(Comm)m(un.)g(Math.)h (Ph)m(ys.)h Ft(208)f Fs(275-281)d(\(1999\))328 1602 y([JP])49 b(Jak)-5 b(\025)-44 b(si)m(\023)e(c,)30 b(V.,)f(Pillet,)f(C.-A.:)41 b FB(On)31 b(a)g(Mo)-5 b(del)30 b(for)h(Quantum)g(F)-7 b(riction)30 b(III.)f(Er)-5 b(go)g(dic)528 1723 y(Pr)g(op)g(erties)35 b(of)g(the)g(Spin-Boson)e(System.)g Fs(Comm)m(un.)f(Math.)h(Ph)m(ys.)i Ft(178)p Fs(,)e(627-)528 1843 y(651)f(\(1996\))328 2046 y([M])57 b(Merkli,)43 b(M.:)61 b FB(Positive)42 b(Commutators)g(in)g (Non-Equilibrium)g(Quantum)h(Sta-)528 2167 y(tistic)-5 b(al)35 b(Me)-5 b(chanics.)32 b Fs(Comm)m(un.)g(Math.)h(Ph)m(ys.)h Ft(223)p Fs(,)f(327-362)e(\(2001\))1898 5214 y(53)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0306130919579--