Content-Type: multipart/mixed; boundary="-------------0603011710211" This is a multi-part message in MIME format. ---------------0603011710211 Content-Type: text/plain; name="06-42.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-42.keywords" Non equilibrium stationary states, NESS, entropy production, thermal field, fermi golden rule, resonances, quantum resonances, Weyl CCR algebra, von Neumann algebra, dynamical system, KMS, heat reservoirs, heat flow ---------------0603011710211 Content-Type: application/postscript; name="nessII.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nessII.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: nessII.dvi %%Pages: 42 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips nessII.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2006.03.01:1712 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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y Fs(\026)p Fr(\000)p Fv(1)p Fs(=)p Fv(2)p 386 703 193 4 v 386 750 a Fs(\026)p Fv(+1)p Fs(=)p Fv(2)589 722 y Fx(,)k(where)f Fw(\026)c(>)f Fx(1)p Fw(=)p Fx(2)27 b(is)g(giv)n(en)g(in)h(Condition)f(\(A\))i(b)r(elo)n(w\).)37 b(Our)26 b(approac)n(h)g(applies)h(to)h(an)f(arbitrary)180 831 y(\014nite)d(n)n(um)n(b)r(er)f(of)g(b)r(osonic)f(or)h(fermionic)g (reserv)n(oirs;)e(in)j(the)f(latter)g(case,)h(it)f(giv)n(es)f(an)h (extension)g(of)g(the)h(results)180 930 y(of)k([21)o(])g(to)g(the)h (temp)r(erature)e(range)g(men)n(tioned)h(ab)r(o)n(v)n(e.)37 b(Moreo)n(v)n(er,)25 b(w)n(e)j(dev)n(elop)f(a)h(p)r(erturbation)f (theory)h(for)180 1030 y(NESS)g(and)f(use)h(it)g(to)f(pro)n(v)n(e)f (that)i(the)g(en)n(trop)n(y)e(pro)r(duction)h(is)h(strictly)f(p)r (ositiv)n(e.)305 1130 y(An)33 b(iteration)g(of)g(our)g(estimates)g(in)h (the)f(spirit)h(of)f(the)h(sp)r(ectral)f(renormalization)e(group)h(of)h ([5,)g(6,)g(7,)g(8])180 1229 y(giv)n(es)26 b(the)h(ab)r(o)n(v)n(e)f (results)h(for)f Fq(al)t(l)31 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b(The)32 b(op)r(erator)e Fw(K)37 b Fx(do)r(es)31 b(not)h(b)r(elong)f(to)h(a)f(class)g(for)g(whic)n(h)h(the)180 1927 y(ev)n(olution)i(is)g(a)g(priori)f(kno)n(wn)h(to)g(exist)g(\(e.g.) 57 b(classes)34 b(of)g(normal)f(or)h(accretiv)n(e)f(op)r(erators\).)55 b(T)-7 b(o)34 b(o)n(v)n(ercome)180 2026 y(this)25 b(problem)f(w)n(e)g (establish)g(a)g(direct)g(connection)g(b)r(et)n(w)n(een)g(desired)g (ergo)r(dic)g(prop)r(erties)f(of)i Fw(U)9 b Fx(\()p Fw(t)p Fx(\))24 b(and)h(certain)180 2126 y(sp)r(ectral)30 b(prop)r(erties)g (of)g(a)g(complex)g(deformation,)h Fw(K)1948 2138 y Fs(\022)1985 2126 y Fx(,)g Fw(\022)g Fp(2)d Fo(C)2246 2096 y Fv(2)2289 2126 y Fx(,)j(of)g Fw(K)6 b Fx(.)45 b(F)-7 b(or)30 b(tec)n(hnical)g (reasons)f(w)n(e)h(can)g(use)180 2226 y(neither)23 b(the)f(complex)h (deformations)e(in)n(tro)r(duced)h(in)h([18)o(])g(nor)f(those)g(in)n (tro)r(duced)g(in)h([8])f(but)h(w)n(e)g(com)n(bine)f(b)r(oth)180 2325 y(t)n(yp)r(es,)34 b(hence)e Fw(\022)j Fx(is)d(in)h Fo(C)982 2295 y Fv(2)1058 2325 y Fx(rather)e(than)i(in)g Fo(C)15 b Fx(.)57 b(\(Suc)n(h)33 b(a)f(com)n(bination)g(w)n(as)f (already)g(men)n(tioned)i(in)f([8]\).)52 b(In)180 2425 y(order)26 b(to)g(establish)h(the)g(desired)g(sp)r(ectral)f(c)n (haracteristics)f(of)i(the)g(op)r(erator)e(family)i Fw(K)3011 2437 y Fs(\022)3049 2425 y Fx(,)g(w)n(e)f(use)h(the)h(metho)r(d)180 2524 y(of)f(the)g(F)-7 b(esh)n(bac)n(h)26 b(map,)h(including)g(the)g(R) n(G-approac)n(h)d(\(renormalization)h(group\),)h(as)g(dev)n(elop)r(ed)h (in)g([5)o(,)g(6)o(,)g(7].)305 2624 y(Already)37 b(a)g(single)g (application)g(of)h(the)g(F)-7 b(esh)n(bac)n(h)37 b(map)h(considered)f (in)h(this)g(pap)r(er)f(yields)g(the)i(results)180 2724 y(men)n(tioned)30 b(ab)r(o)n(v)n(e)e(under)i(the)g(additional)f (restriction)g(that)h(min)2320 2736 y Fs(j)2355 2724 y Fx(\()p Fw(T)2436 2736 y Fs(j)2471 2724 y Fx(\))d Fp(\035)g Fw(g)2683 2694 y Fv(2+)p Fs(\013)2844 2724 y Fx(\(but)j(with)h(a)e (relaxed)g(infra-)180 2823 y(red)h(condition\).)45 b(Using)30 b(results)f(of)h([5,)g(7,)g(8])g(on)g(the)h(iteration)e(of)h(the)h(F)-7 b(esh)n(bac)n(h)29 b(map)i(\(the)f(R)n(G)h(approac)n(h\),)180 2923 y(this)g(restriction)f(on)g(the)i(temp)r(eratures)e(can)g(b)r(e)i (remo)n(v)n(ed.)45 b(W)-7 b(e)31 b(presen)n(t)f(in)h([26)o(])g(a)g (detailed)f(analysis)g(of)h(the)180 3023 y(R)n(G)d(to)g(the)g(sp)r (eci\014c)h(mo)r(del)f(at)g(hand.)38 b(It)28 b(relies)f(on)h([6,)g(7)o (,)h(8)o(])f(and)g(features)g(some)f(simpli\014cations)h(due)g(to)g (the)180 3122 y(sp)r(eci\014cit)n(y)g(of)f(our)g(problem)g(and)g(some)h (recen)n(t)f(dev)n(elopmen)n(ts)f([5].)305 3222 y(A)g(more)g(detailed)h (outline)f(of)h(our)f(approac)n(h)e(and)j(of)f(the)h(organization)e(of) h(the)h(pap)r(er)f(is)h(giv)n(en)e(in)i(Section)180 3321 y(2.)180 3595 y Fy(2)135 b(Mo)t(del)45 b(and)f(Approac)l(h)180 3777 y Fx(W)-7 b(e)26 b(consider)e(a)g(system)h(consisting)g(of)g(a)g (particle)f(system,)i(describ)r(ed)f(b)n(y)f(a)h(self-adjoin)n(t)g (Hamiltonian)g Fw(H)3629 3789 y Fs(p)3692 3777 y Fx(on)180 3877 y(a)g(Hilb)r(ert)h(space)e Fp(H)820 3889 y Fs(p)859 3877 y Fx(,)i(and)f(t)n(w)n(o)f(b)r(osonic)h(reserv)n(oirs,)e(at)i(in)n (v)n(erse)f(temp)r(eratures)h Fw(\014)2832 3889 y Fv(1)2894 3877 y Fx(and)g Fw(\014)3100 3889 y Fv(2)3138 3877 y Fx(,)h(describ)r(ed)f(b)n(y)g(the)180 3976 y(Hamiltonians)d Fw(H)754 3988 y Fs(r)r Fv(1)846 3976 y Fx(and)f Fw(H)1070 3988 y Fs(r)r Fv(2)1162 3976 y Fx(acting)h(on)g(Hilb)r(ert)g(spaces)f Fp(H)2115 3988 y Fs(r)r Fv(1)2207 3976 y Fx(and)h Fp(H)2433 3988 y Fs(r)r Fv(2)2503 3976 y Fx(,)h(resp)r(ectiv)n(ely)-7 b(.)35 b(The)22 b(full)g(Hamiltonian)180 4076 y(is)1703 4175 y Fw(H)30 b Fx(:=)23 b Fw(H)1982 4187 y Fv(0)2037 4175 y Fx(+)18 b Fw(g)s(v)31 b(;)1352 b Fx(\(2.1\))180 4322 y(acting)27 b(on)g(the)h(tensor)f(pro)r(duct)h(space)f Fp(H)1541 4334 y Fv(0)1601 4322 y Fx(:=)c Fp(H)1782 4334 y Fs(p)1839 4322 y Fp(\012)18 b(H)1992 4334 y Fs(r)r Fv(1)2080 4322 y Fp(\012)g(H)2233 4334 y Fs(r)r Fv(2)2303 4322 y Fx(.)37 b(Here)1118 4500 y Fw(H)1187 4512 y Fv(0)1248 4500 y Fx(:=)22 b Fw(H)1427 4512 y Fs(p)1484 4500 y Fp(\012)c Fn(1)g Fp(\012)g Fn(1)h Fx(+)f Fn(1)g Fp(\012)g Fw(H)2084 4512 y Fs(r)r Fv(1)2172 4500 y Fp(\012)g Fn(1)g Fx(+)g Fn(1)h Fp(\012)f Fn(1)g Fp(\012)g Fw(H)2772 4512 y Fs(r)r Fv(2)3609 4500 y Fx(\(2.2\))180 4678 y(is)31 b(the)g(unp)r(erturb)r(ed) h(Hamiltonian,)g Fw(v)i Fx(is)d(an)f(op)r(erator)g(on)g Fp(H)2208 4690 y Fv(0)2277 4678 y Fx(describing)g(the)h(in)n(teraction) f(and)h Fw(g)g Fp(2)f Fo(R)37 b Fx(is)30 b(a)180 4778 y(coupling)d(constan)n(t.)305 4877 y(The)f(op)r(erators)f Fw(H)910 4889 y Fs(r)r(j)1005 4877 y Fx(describ)r(e)h(free)h(scalar)e (\(or)h(v)n(ector,)g(if)h(wished\))g(quan)n(tum)g(\014elds)g(on)g Fp(H)3250 4889 y Fs(r)r(j)3317 4877 y Fx(,)g(the)h(b)r(osonic)180 4977 y(F)-7 b(o)r(c)n(k)27 b(spaces)g(o)n(v)n(er)f(the)i(one-particle)e (space)h Fw(L)1684 4947 y Fv(2)1720 4977 y Fx(\()p Fo(R)1807 4947 y Fv(3)1850 4977 y Fw(;)14 b(d)1930 4947 y Fv(3)1967 4977 y Fw(k)s Fx(\),)1453 5197 y Fw(H)1522 5209 y Fs(r)r(j)1613 5197 y Fx(=)1700 5084 y Fm(Z)1797 5197 y Fw(!)s Fx(\()p Fw(k)s Fx(\))p Fw(a)2006 5163 y Fr(\003)2006 5218 y Fs(j)2044 5197 y Fx(\()p Fw(k)s Fx(\))p Fw(a)2198 5209 y Fs(j)2233 5197 y Fx(\()p Fw(k)s Fx(\))g Fw(d)2400 5163 y Fv(3)2438 5197 y Fw(k)s(;)1102 b Fx(\(2.3\))1959 5476 y(2)p eop %%Page: 3 3 3 2 bop 180 210 a Fx(where)20 b Fw(a)457 180 y Fr(\003)457 232 y Fs(j)495 210 y Fx(\()p Fw(k)s Fx(\))g(and)g Fw(a)823 222 y Fs(j)858 210 y Fx(\()p Fw(k)s Fx(\))g(are)f(creation)g(and)h (annihilation)g(op)r(erators)e(on)i Fp(H)2576 222 y Fs(r)r(j)2664 210 y Fx(and)g Fw(!)s Fx(\()p Fw(k)s Fx(\))j(=)f Fp(j)p Fw(k)s Fp(j)e Fx(is)g(the)h(disp)r(ersion)180 310 y(relation)27 b(for)g(relativistic)g(massless)f(b)r(osons.)36 b(The)28 b(in)n(teraction)e(op)r(erator)g(is)i(giv)n(en)e(b)n(y)1234 563 y Fw(v)h Fx(=)1432 460 y Fv(2)1388 484 y Fm(X)1391 661 y Fs(j)s Fv(=1)1522 563 y Fw(v)1562 575 y Fs(j)1680 563 y Fx(with)84 b Fw(v)1965 575 y Fs(j)2023 563 y Fx(=)23 b Fw(a)2155 575 y Fs(j)2189 563 y Fx(\()p Fw(G)2286 575 y Fs(j)2322 563 y Fx(\))c(+)f Fw(a)2500 533 y Fr(\003)2500 585 y Fs(j)2538 563 y Fx(\()p Fw(G)2635 575 y Fs(j)2670 563 y Fx(\).)884 b(\(2.4\))180 826 y(Its)19 b(c)n(hoice)g(is)g(motiv)-5 b(ated)20 b(b)n(y)f(standard)f(mo)r(dels)i(of)f(particles)f(in)n (teracting)h(with)h(the)f(quan)n(tized)g(electromagnetic)180 926 y(\014eld)28 b(or)f(with)h(phonons.)305 1026 y(In)e(\(2.4\),)g Fw(G)692 1038 y Fs(j)751 1026 y Fx(:)d Fw(k)j Fp(7!)d Fw(G)1037 1038 y Fs(j)1072 1026 y Fx(\()p Fw(k)s Fx(\))k(is)f(a)f(map)i (from)e Fo(R)1790 996 y Fv(3)1860 1026 y Fx(in)n(to)h Fp(B)s Fx(\()p Fp(H)2187 1038 y Fs(p)2225 1026 y Fx(\),)h(the)f (algebra)f(of)h(b)r(ounded)g(op)r(erators)f(on)g Fp(H)3718 1038 y Fs(p)3757 1026 y Fx(,)180 1125 y(and)827 1247 y Fw(a)871 1259 y Fs(j)906 1247 y Fx(\()p Fw(G)1003 1259 y Fs(j)1039 1247 y Fx(\))e(:=)1205 1134 y Fm(Z)1302 1247 y Fw(G)1367 1259 y Fs(j)1402 1247 y Fx(\()p Fw(k)s Fx(\))1512 1212 y Fr(\003)1569 1247 y Fp(\012)18 b Fw(a)1696 1259 y Fs(j)1731 1247 y Fx(\()p Fw(k)s Fx(\))c Fw(d)1898 1212 y Fv(3)1935 1247 y Fw(k)169 b Fx(and)d Fw(a)2491 1212 y Fr(\003)2491 1267 y Fs(j)2529 1247 y Fx(\()p Fw(G)2626 1259 y Fs(j)2662 1247 y Fx(\))23 b(:=)g Fw(a)2872 1259 y Fs(j)2907 1247 y Fx(\()p Fw(G)3004 1259 y Fs(j)3039 1247 y Fx(\))3071 1212 y Fr(\003)3110 1247 y Fw(:)476 b Fx(\(2.5\))305 1432 y(If)28 b(the)g(coupling)f(op)r(erators)e Fw(G)1295 1444 y Fs(j)1358 1432 y Fx(are)i(suc)n(h)g(that)1015 1656 y Fw(g)1058 1622 y Fv(2)1109 1543 y Fm(Z)1095 1785 y Fl(R)1142 1768 y Fk(3)1206 1589 y Fm(\000)1244 1656 y Fx(1)18 b(+)g Fp(j)p Fw(k)s Fp(j)1479 1622 y Fr(\000)p Fv(1)1568 1589 y Fm(\001)1620 1656 y Fp(k)o Fw(G)1726 1668 y Fs(j)1761 1656 y Fx(\()p Fw(k)s Fx(\))p Fp(k)1913 1615 y Fv(2)1978 1656 y Fw(dk)169 b Fx(is)28 b(su\016cien)n(tly)f (small)p Fw(;)664 b Fx(\(2.6\))180 1944 y(then)28 b(the)g(op)r(erator)e Fw(H)34 b Fx(is)28 b(self-adjoin)n(t)f(\(see)g(e.g.)37 b([8)o(]\).)305 2043 y(No)n(w)28 b(w)n(e)h(set)f(up)i(a)e(mathematical) h(framew)n(ork)e(for)h(non-equilibrium)h(statistical)f(mec)n(hanics.)40 b(Op)r(erators)180 2143 y(on)f(the)h(Hilb)r(ert)h(space)d Fp(H)1064 2155 y Fv(0)1141 2143 y Fx(will)i(b)r(e)g(called)g(observ)-5 b(ables.)71 b(\(Only)40 b(certain)f(self-adjoin)n(t)g(op)r(erators)e (on)j Fp(H)3743 2155 y Fv(0)180 2242 y Fx(are)29 b(actually)h Fq(physic)l(al)j Fx(observ)-5 b(ables.\))44 b(As)30 b(an)h(algebra)d (of)j(observ)-5 b(ables)28 b(describing)i(the)h(system)f(w)n(e)g(tak)n (e)g(the)180 2342 y Fw(C)245 2312 y Fr(\003)283 2342 y Fx(-algebra)1418 2442 y Fp(A)24 b Fx(=)e Fp(B)s Fx(\()p Fp(H)1755 2454 y Fs(p)1793 2442 y Fx(\))d Fp(\012)f Fj(W)p Fx(\()p Fw(L)2103 2407 y Fv(2)2103 2462 y(0)2140 2442 y Fx(\))g Fp(\012)g Fj(W)p Fx(\()p Fw(L)2449 2407 y Fv(2)2449 2462 y(0)2486 2442 y Fx(\))p Fw(;)1068 b Fx(\(2.7\))180 2589 y(where)22 b Fj(W)p Fx(\()p Fw(L)591 2559 y Fv(2)591 2610 y(0)628 2589 y Fx(\))h(denotes)g(the)g(W)-7 b(eyl)23 b(CCR)g(algebra)e(o)n(v)n(er)g Fw(L)2042 2559 y Fv(2)2042 2610 y(0)2102 2589 y Fx(:=)i Fw(L)2270 2559 y Fv(2)2307 2589 y Fx(\()p Fo(R)2393 2559 y Fv(3)2436 2589 y Fw(;)14 b Fx(\(1)9 b(+)g Fp(j)p Fw(k)s Fp(j)2722 2559 y Fr(\000)p Fv(1)2810 2589 y Fx(\))p Fw(d)2885 2559 y Fv(3)2923 2589 y Fw(k)s Fx(\).)36 b(States)22 b(of)h(the)g(system)180 2689 y(are)k(p)r(ositiv)n(e)g(linear)g(\(`exp)r(ectation'\))h (functionals)f Fw( )k Fx(on)c(the)h(algebra)e Fp(A)p Fx(,)i(normalized)f(as)g Fw( )s Fx(\()p Fn(1)p Fx(\))c(=)g(1.)305 2788 y(The)30 b(reason)f(w)n(e)h(c)n(hose)g Fp(A)g Fx(rather)g(than)g Fp(B)s Fx(\()p Fp(H)1793 2800 y Fv(0)1830 2788 y Fx(\))h(is)f(that)h (the)g(algebra)e Fp(A)i Fx(supp)r(orts)f(states)g(in)g(whic)n(h)h(eac)n (h)180 2888 y(reserv)n(oir)24 b(is)j(at)g(a)g(thermal)g(equilibrium)g (at)g(its)g(o)n(wn)f(temp)r(erature.)37 b(More)26 b(precisely)-7 b(,)26 b(consider)g(the)i(ev)n(olution)180 2987 y(for)f(the)h Fw(j)5 b Fx(-th)28 b(reserv)n(oir)d(giv)n(en)h(b)n(y)1515 3087 y Fw(\013)1568 3053 y Fs(t)1568 3108 y(r)r(j)1635 3087 y Fx(\()p Fw(A)p Fx(\))e(:=)f Fw(e)1935 3053 y Fs(iH)2012 3061 y Fi(r)q(j)2073 3053 y Fs(t)2102 3087 y Fw(Ae)2203 3053 y Fr(\000)p Fs(iH)2332 3061 y Fi(r)q(j)2393 3053 y Fs(t)2422 3087 y Fw(:)1164 b Fx(\(2.8\))180 3234 y(Then)25 b(there)f(are)g(stationary)f(states)h(on)h(the)g Fw(j)5 b Fx(-th)24 b(reserv)n(oir)e(algebra)h(of)i(observ)-5 b(ables,)23 b Fj(W)p Fx(\()p Fw(L)3134 3204 y Fv(2)3134 3255 y(0)3171 3234 y Fx(\),)j(whic)n(h)e(describ)r(e)180 3334 y(\(single-phase\))32 b(thermal)g(equilibria.)50 b(These)32 b(states)g(are)g(parametrized)f(b)n(y)h(the)g(in)n(v)n(erse) f(temp)r(erature)h Fw(\014)3649 3346 y Fs(j)3715 3334 y Fx(=)180 3434 y(1)p Fw(=T)313 3446 y Fs(j)374 3434 y Fx(and)c(their)f(generating)f(functional)i(is)g(giv)n(en)e(b)n(y)1032 3675 y Fw(!)1087 3630 y Fv(\()p Fs(\014)1151 3638 y Fi(j)1181 3630 y Fv(\))1084 3698 y Fs(r)r(j)1225 3675 y Fx(\()p Fw(W)1335 3687 y Fs(j)1371 3675 y Fx(\()p Fw(f)9 b Fx(\)\))23 b(=)g(exp)1768 3558 y Fm(\032)1831 3675 y Fp(\000)1906 3619 y Fx(1)p 1906 3656 42 4 v 1906 3732 a(4)1970 3562 y Fm(Z)2017 3751 y Fl(R)2064 3734 y Fk(3)2119 3619 y Fw(e)2158 3589 y Fs(\014)2196 3597 y Fi(j)2226 3589 y Fr(j)p Fs(k)q Fr(j)2325 3619 y Fx(+)18 b(1)p 2119 3656 331 4 v 2119 3735 a Fw(e)2158 3710 y Fs(\014)2196 3718 y Fi(j)2226 3710 y Fr(j)p Fs(k)q Fr(j)2325 3735 y Fp(\000)g Fx(1)2459 3675 y Fp(j)p Fw(f)9 b Fx(\()p Fw(k)s Fx(\))p Fp(j)2665 3641 y Fv(2)2703 3675 y Fw(d)2746 3641 y Fv(3)2783 3675 y Fw(k)2829 3558 y Fm(\033)2905 3675 y Fw(;)681 b Fx(\(2.9\))180 3916 y(where)33 b Fw(W)504 3928 y Fs(j)539 3916 y Fx(\()p Fw(f)9 b Fx(\))34 b(:=)e Fw(e)846 3886 y Fs(i\036)909 3894 y Fi(j)940 3886 y Fv(\()p Fs(f)7 b Fv(\))1035 3916 y Fx(,)35 b(with)f Fw(\036)1337 3928 y Fs(j)1373 3916 y Fx(\()p Fw(f)9 b Fx(\))33 b(:=)1651 3883 y Fv(1)p 1651 3897 34 4 v 1651 3944 a(2)1708 3848 y Fm(\000)1746 3916 y Fw(a)1790 3886 y Fr(\003)1790 3937 y Fs(j)1828 3916 y Fx(\()p Fw(f)9 b Fx(\))18 b(+)g Fw(a)2087 3928 y Fs(j)2122 3916 y Fx(\()p Fw(f)9 b Fx(\))2236 3848 y Fm(\001)2274 3916 y Fx(,)36 b(is)d(the)h(W)-7 b(eyl)34 b(op)r(erator,)f(see)g(e.g.)55 b([9)o(].)g(The)180 4024 y(c)n(hoice)22 b(of)i(the)f(space)g Fw(L)926 3994 y Fv(2)926 4045 y(0)986 4024 y Fx(ab)r(o)n(v)n(e)e(is)i(dictated)h(b)n(y)f(the)g (need)h(to)f(ha)n(v)n(e)f(the)h(r.h.s.)35 b(of)24 b(this)f(functional)g (\014nite.)36 b(These)180 4124 y(states)27 b(are)g(c)n(haracterized)e (b)n(y)j(the)g(KMS)f(condition)g(and)h(are)e(called)i(the)g(\()p Fw(\013)2661 4094 y Fs(t)2661 4146 y(r)r(j)2729 4124 y Fw(;)14 b(\014)2813 4136 y Fs(j)2847 4124 y Fx(\)-KMS)28 b(states.)305 4236 y Fq(R)l(emark.)80 b Fx(It)32 b(is)f(con)n(v)n (enien)n(t)f(to)h(de\014ne)g(states)g Fw( )j Fx(on)d(pro)r(ducts)g Fw(a)2470 4206 y Fv(#)2528 4236 y Fx(\()p Fw(f)2601 4248 y Fv(1)2639 4236 y Fx(\))14 b Fw(:)g(:)g(:)f(a)2839 4206 y Fv(#)2898 4236 y Fx(\()p Fw(f)2971 4248 y Fs(n)3016 4236 y Fx(\))32 b(of)f(the)g(creation)f(and)180 4335 y(annihilation)j(op)r(erators,)f(where)h Fw(a)1334 4305 y Fv(#)1425 4335 y Fx(denotes)g(either)g Fw(a)g Fx(or)f Fw(a)2200 4305 y Fr(\003)2238 4335 y Fx(.)53 b(This)33 b(is)g(done)f(using)h(deriv)-5 b(ativ)n(es)32 b Fw(@)3487 4347 y Fs(s)3518 4356 y Fi(k)3592 4335 y Fx(of)h(its)180 4435 y(v)-5 b(alues)27 b(on)h(the)g(W)-7 b(eyl)27 b(op)r(erators)f Fw(W)12 b Fx(\()p Fw(s)1424 4447 y Fv(1)1461 4435 y Fw(f)1502 4447 y Fv(1)1539 4435 y Fx(\))i Fw(:)g(:)g(:)g(W)e Fx(\()p Fw(s)1857 4447 y Fs(n)1902 4435 y Fw(f)1943 4447 y Fs(n)1988 4435 y Fx(\))28 b(\(see)g([9)o(],)g(Section)g(5.2.3)e(and)h (\(2.15\)\).)305 4535 y(Consider)f(states)h(\(on)h Fp(A)p Fx(\))g(of)g(the)g(form)1530 4730 y Fw(!)1582 4742 y Fv(0)1642 4730 y Fx(:=)23 b Fw(!)1805 4742 y Fs(p)1861 4730 y Fp(\012)18 b Fw(!)1999 4687 y Fv(\()p Fs(\014)2063 4695 y Fk(1)2095 4687 y Fv(\))1996 4752 y Fs(r)r Fv(1)2143 4730 y Fp(\012)g Fw(!)2281 4687 y Fv(\()p Fs(\014)2345 4695 y Fk(2)2377 4687 y Fv(\))2278 4752 y Fs(r)r Fv(2)2407 4730 y Fw(;)1138 b Fx(\(2.10\))180 4928 y(where)29 b Fw(!)474 4940 y Fs(p)542 4928 y Fx(is)h(a)f(state)g(of)h(the)g (particle)f(system)h(and)f Fw(!)1944 4885 y Fv(\()p Fs(\014)s Fv(\))1941 4951 y Fs(r)r(i)2071 4928 y Fx(is)g(the)h(\()p Fw(\013)2386 4898 y Fs(t)2386 4950 y(r)r(i)2447 4928 y Fw(;)14 b(\014)2531 4940 y Fs(i)2558 4928 y Fx(\)-KMS)30 b(state)g(of)f(the)h Fw(i)p Fx(-th)g(reserv)n(oir.)180 5028 y(The)d(set)g(of)f(states)h(whic)n(h)g(are)e(normal)h(w.r.t.)37 b Fw(!)1746 5040 y Fv(0)1810 5028 y Fx(is)26 b(the)h(same)g(for)f(an)n (y)g(c)n(hoice)g(of)h Fw(!)2916 5040 y Fs(p)2954 5028 y Fx(.)36 b(A)28 b(state)e(normal)g(w.r.t.)180 5128 y Fw(!)232 5140 y Fv(0)297 5128 y Fx(will)h(b)r(e)h(called)g(a)f Fw(\014)918 5140 y Fv(1)955 5128 y Fw(\014)1002 5140 y Fv(2)1039 5128 y Fx(-normal)f(state.)305 5227 y(In)h(the)h (particular)f(case)f Fw(!)1162 5239 y Fs(p)1200 5227 y Fx(\()p Fp(\001)p Fx(\))e(=)f(T)-7 b(r)o(\()p Fw(e)1555 5197 y Fr(\000)p Fs(\014)1645 5205 y Fi(p)1680 5197 y Fs(H)1734 5205 y Fi(p)1787 5227 y Fp(\001)p Fx(\))p Fw(=)p Fx(T)g(r)o(\()p Fw(e)2040 5197 y Fr(\000)p Fs(\014)2130 5205 y Fi(p)2165 5197 y Fs(H)2219 5205 y Fi(p)2258 5227 y Fx(\))28 b(w)n(e)f(call)g Fw(!)2644 5239 y Fv(0)2709 5227 y Fx(a)g Fq(r)l(efer)l(enc)l(e)j(state)p Fx(.)1959 5476 y(3)p eop %%Page: 4 4 4 3 bop 305 210 a Fx(The)27 b(Hamiltonian)h Fw(H)34 b Fx(generates)26 b(the)i(dynamics)f(of)h(observ)-5 b(ables)26 b Fw(A)d Fp(2)h(B)s Fx(\()p Fp(H)2787 222 y Fv(0)2823 210 y Fx(\))k(according)e(to)h(the)h(rule)1480 393 y Fw(A)23 b Fp(7!)g Fw(\013)1724 358 y Fs(t)1754 393 y Fx(\()p Fw(A)p Fx(\))h(:=)f Fw(e)2054 358 y Fs(iH)t(t)2165 393 y Fw(Ae)2266 358 y Fr(\000)p Fs(iH)t(t)2457 393 y Fw(:)1088 b Fx(\(2.11\))180 575 y(Eqn)24 b(\(2.11\))g(de\014nes)h(a)g (group)e(of)i(*-automorphisms)e(of)i Fp(B)s Fx(\()p Fp(H)2150 587 y Fv(0)2186 575 y Fx(\).)37 b(Ho)n(w)n(ev)n(er,)23 b Fw(\013)2686 545 y Fs(t)2741 575 y Fx(do)r(es)h(not)h(map)g(the)g (subalgebra)180 675 y Fp(A)38 b(\032)e(B)s Fx(\()p Fp(H)545 687 y Fv(0)582 675 y Fx(\))h(in)n(to)e(itself,)k(so)d(\(2.11\))f(do)r (es)h(not)g(de\014ne)g(a)g(dynamics)g(on)g Fp(A)p Fx(.)62 b(T)-7 b(o)36 b(circum)n(v)n(en)n(t)f(this)i(problem)180 775 y(w)n(e)32 b(de\014ne)h(the)f(in)n(teracting)g(ev)n(olution)f(of)i (a)f(class)f(of)h Fq(states)i(on)g Fp(A)f Fx(b)n(y)f(using)g(the)h (Araki-Dyson)e(expansion.)180 874 y(Namely)-7 b(,)28 b(w)n(e)f(de\014ne)h(the)g(ev)n(olution)e(of)i(a)f(state)h Fw( )i Fx(on)e Fp(A)g Fx(whic)n(h)f(is)h(normal)e(w.r.t.)37 b Fw(!)2912 886 y Fv(0)2977 874 y Fx(b)n(y)899 1119 y Fw( )956 1084 y Fs(t)986 1119 y Fx(\()p Fw(A)p Fx(\))24 b(:=)51 b(lim)1246 1168 y Fs(n)p Fr(!1)1472 1015 y(1)1445 1040 y Fm(X)1433 1216 y Fs(m)p Fv(=0)1576 1119 y Fx(\()p Fw(ig)s Fx(\))1712 1084 y Fs(m)1789 1006 y Fm(Z)1872 1026 y Fs(t)1835 1194 y Fv(0)1915 1119 y Fw(dt)1988 1131 y Fv(1)2039 1119 y Fp(\001)14 b(\001)g(\001)2150 1006 y Fm(Z)2233 1026 y Fs(t)2258 1034 y Fi(m)p Fh(\000)p Fk(1)2196 1194 y Fv(0)2405 1119 y Fw(dt)2478 1131 y Fs(m)2569 1119 y Fw( )2626 1084 y Fs(t;t)2696 1092 y Fk(1)2728 1084 y Fs(;:::)o(;t)2852 1092 y Fi(m)2623 1139 y Fs(n)2911 1119 y Fx(\()p Fw(A)p Fx(\))p Fw(;)508 b Fx(\(2.12\))180 1381 y(where)27 b(the)h(term)g(with)g Fw(m)23 b Fx(=)f(0)27 b(is)h Fw( )s Fx(\()p Fw(\013)1429 1351 y Fs(t)1429 1402 y Fv(0)1467 1381 y Fx(\()p Fw(A)p Fx(\)\),)h(and,)e(for)g Fw(m)c Fp(\025)g Fx(1,)1001 1563 y Fw( )1058 1529 y Fs(t;t)1128 1537 y Fk(1)1160 1529 y Fs(;:::)o(;t)1284 1537 y Fi(m)1055 1584 y Fs(n)1343 1563 y Fx(\()p Fw(A)p Fx(\))h(:=)f Fw( )1675 1496 y Fm(\000)1713 1563 y Fx([)p Fw(\013)1789 1527 y Fs(t)1814 1535 y Fi(m)1789 1586 y Fv(0)1874 1563 y Fx(\()p Fw(v)1946 1575 y Fs(n)1991 1563 y Fx(\))p Fw(:)14 b Fp(\001)g(\001)g (\001)g Fx([)p Fw(\013)2247 1527 y Fs(t)2272 1535 y Fk(1)2247 1586 y Fv(0)2309 1563 y Fx(\()p Fw(v)2381 1575 y Fs(n)2427 1563 y Fx(\))p Fw(;)g(\013)2549 1529 y Fs(t)2549 1584 y Fv(0)2587 1563 y Fx(\()p Fw(A)p Fx(\)])g Fp(\001)g(\001)g(\001)g Fx(])2884 1496 y Fm(\001)2936 1563 y Fw(:)180 1746 y Fx(Here,)27 b Fw(v)439 1758 y Fs(n)508 1746 y Fp(2)c(A)28 b Fx(is)g(an)f(appro)n(ximating)f(sequence)h(for)g(the)h(op)r(erator)e Fw(v)s Fx(,)i(satisfying)f(the)h(relation)1356 1929 y(lim)1327 1978 y Fs(n)p Fr(!1)1514 1929 y Fw(!)1566 1941 y Fv(0)1603 1929 y Fx(\()p Fw(A)1697 1894 y Fr(\003)1736 1929 y Fx(\()p Fw(v)1811 1894 y Fr(\003)1808 1949 y Fs(n)1872 1929 y Fp(\000)18 b Fw(v)1998 1894 y Fr(\003)2036 1929 y Fx(\)\()p Fw(v)2140 1941 y Fs(n)2205 1929 y Fp(\000)g Fw(v)s Fx(\))p Fw(A)p Fx(\))24 b(=)f(0)p Fw(;)934 b Fx(\(2.13\))180 2145 y(for)33 b(all)h Fw(A)f Fx(=)g Fw(W)12 b Fx(\()p Fw(f)d Fx(\))33 b Fp(2)h(A)g Fx(with)g Fw(f)42 b Fp(2)34 b Fw(L)1478 2115 y Fv(2)1478 2166 y(0)1514 2145 y Fx(.)56 b(Suc)n(h)33 b(a)h(sequence)f(is)g(constructed)h(as)f(follo)n(ws.)54 b(Let)34 b Fp(f)p Fw(e)3436 2157 y Fs(m)3498 2145 y Fp(g)f Fx(b)r(e)h(an)180 2245 y(orthonormal)26 b(basis)g(in)i Fw(L)1013 2215 y Fv(2)1013 2266 y(0)1050 2245 y Fx(.)37 b(W)-7 b(e)28 b(de\014ne)g(the)g(appro)n(ximate)e(creation)g(op)r (erators)1393 2498 y Fw(a)1437 2464 y Fr(\003)1437 2518 y Fs(j;n)1529 2498 y Fx(\()p Fw(G)1626 2510 y Fs(j)1662 2498 y Fx(\))d(=)1856 2390 y Fs(\026)1816 2419 y Fm(X)1805 2595 y Fs(m)p Fv(=1)1948 2498 y Fp(h)p Fw(e)2019 2510 y Fs(m)2082 2498 y Fw(;)14 b(G)2184 2510 y Fs(j)2219 2498 y Fp(i)p Fw(b)2287 2464 y Fr(\003)2287 2518 y Fs(j;\025)2377 2498 y Fx(\()p Fw(e)2448 2510 y Fs(m)2511 2498 y Fx(\))p Fw(;)1002 b Fx(\(2.14\))180 2762 y(where)27 b Fw(n)c Fx(=)g(\()p Fw(\025;)14 b(\026)p Fx(\),)28 b(and,)g(for)f(an)n(y)g Fw(f)k Fp(2)24 b Fw(L)1508 2732 y Fv(2)1544 2762 y Fx(\()p Fo(R)1631 2732 y Fv(3)1674 2762 y Fx(\))k(and)f Fw(\025)d(>)e Fx(0,)1086 2988 y Fw(b)1122 2953 y Fr(\003)1122 3008 y Fs(j;\025)1212 2988 y Fx(\()p Fw(f)9 b Fx(\))24 b(:=)1516 2932 y Fw(\025)p 1470 2969 140 4 v 1470 2985 a Fp(p)p 1539 2985 42 4 v 69 x Fx(2)p Fw(i)1633 2988 y Fp(f)p Fw(W)1753 3000 y Fs(j)1788 2988 y Fx(\()p Fw(f)9 b(=\025)p Fx(\))19 b Fp(\000)f Fn(1)g Fp(\000)g Fw(iW)2350 3000 y Fs(j)2385 2988 y Fx(\()p Fw(if)9 b(=\025)p Fx(\))18 b(+)g Fw(i)p Fn(1)p Fp(g)13 b Fw(:)694 b Fx(\(2.15\))180 3218 y(Similarly)24 b(w)n(e)h(de\014ne)g(the)g(appro)n(ximate)e (annihilation)i(op)r(erators)e Fw(a)2372 3230 y Fs(j;n)2464 3218 y Fx(\()p Fw(G)2561 3230 y Fs(j)2596 3218 y Fx(\).)37 b(Via)25 b(the)g(ab)r(o)n(v)n(e)e(construction)h(w)n(e)180 3318 y(obtain)34 b(the)g(family)h(of)f(in)n(teractions)f Fw(v)1453 3330 y Fs(n)1532 3318 y Fx(whic)n(h)h(b)r(elongs)g(to)g Fp(A)g Fx(and,)i(as)e(can)f(b)r(e)i(easily)e(sho)n(wn)g(using)h (\(2.9\),)180 3418 y(satis\014es)27 b(\(2.13\).)305 3517 y(W)-7 b(e)31 b(sho)n(w)g(in)h(App)r(endix)h(A)e(that)h(under)g (condition)f(\(2.13\))g(the)h(in)n(tegrands)e(on)h(the)h(r.h.s.)49 b(of)31 b(\(2.12\))g(are)180 3617 y(con)n(tin)n(uous)c(functions)i(in)g Fw(t)1083 3629 y Fv(1)1120 3617 y Fw(;)14 b(:)g(:)g(:)f(;)h(t)1334 3629 y Fs(m)1397 3617 y Fx(,)29 b(that)g(the)g(series)e(is)h (absolutely)g(con)n(v)n(ergen)n(t)e(and)i(that)h(the)f(limit)h(exists) 180 3716 y(and)e(is)h(indep)r(enden)n(t)g(of)g(the)g(appro)n(ximating)e (sequence)h Fw(v)2071 3728 y Fs(n)2116 3716 y Fx(.)305 3816 y(Our)37 b(goal)g(is)h(to)g(understand)f(stationary)g(states)g(of) h(the)h(in)n(teracting)e(system)h(originating)e(from)i Fw(\014)3631 3828 y Fv(1)3668 3816 y Fw(\014)3715 3828 y Fv(2)3752 3816 y Fx(-)180 3916 y(normal)30 b(states)g(either)g(b)n(y) h(a)f(p)r(erturbation)g(theory)g(or)f(through)h(an)h(ergo)r(dic)e (limit)j(of)e(the)h(full)h(ev)n(olution)d Fw(\013)3727 3886 y Fs(t)3757 3916 y Fx(.)180 4015 y(These)22 b(states)g(are)f(not)h (equilibrium)h(\(KMS\))f(states)g(states.)35 b(They)22 b(will)g(b)r(e)h(called)f Fq(non-e)l(quilibrium)j(stationary)180 4115 y(states)i Fx(or)g(NESS)h(for)f(short.)305 4215 y(Assuming)34 b(certain)g(smo)r(othness)g(and)h(smallness)f(conditions) g(on)g(the)h(coupling)g(op)r(erators)d Fw(g)s(G)3466 4227 y Fs(j)3501 4215 y Fx(\()p Fw(k)s Fx(\))j(and)180 4314 y(assuming)g(that)g(the)h(particle)f(system)g(is)h(e\013ectiv)n (ely)f(coupled)h(to)f(the)h(reserv)n(oirs,)e(w)n(e)i(sho)n(w)e(that,)k (starting)180 4414 y(initially)28 b(in)g(an)n(y)e Fw(\014)787 4426 y Fv(1)825 4414 y Fw(\014)872 4426 y Fv(2)909 4414 y Fx(-normal)g(state)h Fw( )s Fx(,)h(the)g(system)g(con)n(v)n(erges,)d (under)i(the)h(ev)n(olution)f Fw(\013)3134 4384 y Fs(t)3164 4414 y Fx(,)g(to)h(a)f(state)g Fw(\021)s Fx(:)1600 4596 y Fw( )1657 4562 y Fs(t)1709 4596 y Fp(\000)-14 b(!)23 b Fw(\021)86 b Fx(as)27 b Fw(t)c Fp(!)g(1)p Fw(:)1208 b Fx(\(2.16\))180 4779 y(The)25 b(con)n(v)n(ergence)e(\(2.16\))h(is)h (understo)r(o)r(d)g(in)g(the)g(w)n(eak)p Fp(\003)f Fx(sense)h(on)f(the) i(sub-)p Fw(C)2718 4749 y Fr(\003)2756 4779 y Fx(-algebra)d(of)i (\\analytic)f(observ-)180 4879 y(ables")1297 4978 y Fp(A)1363 4990 y Fv(1)1423 4978 y Fx(=)f Fp(B)s Fx(\()p Fp(H)1671 4990 y Fs(p)1709 4978 y Fx(\))18 b Fp(\012)g Fj(W)p Fx(\()p Fw(D)2032 4944 y Fv(anal)2158 4978 y Fx(\))h Fp(\012)f Fj(W)p Fx(\()p Fw(D)2482 4944 y Fv(anal)2608 4978 y Fx(\))p Fw(:)905 b Fx(\(2.17\))180 5128 y(Here,)36 b Fj(W)p Fx(\()p Fw(D)598 5097 y Fv(anal)724 5128 y Fx(\))e(is)g(the)h(W)-7 b(eyl)35 b(CCR)f(algebra)e(o)n(v)n(er)h(the)i(dense)f(set)g Fw(D)2534 5097 y Fv(anal)2694 5128 y Fp(\032)g Fw(L)2850 5097 y Fv(2)2850 5148 y(0)2921 5128 y Fx(whic)n(h)g(w)n(e)g(de\014ne)g (in)h(Ap-)180 5227 y(p)r(endix)h(C.)61 b(Roughly)34 b(sp)r(eaking,)j Fw(D)1388 5197 y Fv(anal)1549 5227 y Fx(consists)e(of)h(v)n(ectors)d (from)j(the)f(space)2840 5165 y Fm(T)2909 5252 y Fs(b)p Fr(\025)p Fv(0)3042 5227 y Fw(e)3081 5197 y Fr(\000)p Fs(b)p Fr(j)p Fs(k)q Fr(j)3242 5227 y Fw(L)3299 5197 y Fv(2)3299 5248 y(0)3371 5227 y Fx(whic)n(h)g(ha)n(v)n(e)1959 5476 y(4)p eop %%Page: 5 5 5 4 bop 180 210 a Fx(some)34 b(analyticit)n(y)h(prop)r(erties)f(in)h Fp(j)p Fw(k)s Fp(j)g Fx(and)g(a)f(certain)g(b)r(eha)n(viour)g(at)h Fw(k)j Fx(=)d(0.)59 b(The)35 b(densit)n(y)f(of)h Fw(D)3425 180 y Fv(anal)3586 210 y Fp(\032)g Fw(L)3743 180 y Fv(2)3743 231 y(0)180 310 y Fx(implies)28 b(that)g Fp(A)708 322 y Fv(1)773 310 y Fx(is)f(strongly)g(dense)g(in)h Fp(A)p Fx(.)305 409 y(As)21 b(men)n(tioned)g(in)h(the)g(in)n(tro)r(duction,)g (so)f(far,)h(w)n(e)f(do)g(not)g(ha)n(v)n(e)f(a)h(simple)g(c)n (haracterization)e(of)i(NESS.)h(Ho)n(w-)180 509 y(ev)n(er,)27 b(there)h(is)g(a)g(k)n(ey)f(ph)n(ysical)h(quan)n(tit)n(y)f(whic)n(h)h (di\013eren)n(tiates)g(b)r(et)n(w)n(een)g(equilibrium)h(and)f (non-equilibrium)180 609 y(stationary)43 b(states)g({)h(the)h(en)n (trop)n(y)e(pro)r(duction.)86 b(Motiv)-5 b(ated)44 b(b)n(y)g(the)h (second)e(la)n(w)h(of)g(thermo)r(dynamics)180 708 y(\()p Fw(dS)28 b Fx(=)422 646 y Fm(P)523 708 y Fw(dQ)632 720 y Fs(j)667 708 y Fw(=T)758 720 y Fs(j)792 708 y Fx(\))g(w)n(e)f(in)n (tro)r(duce)h(the)g(observ)-5 b(able)26 b(of)h(en)n(trop)n(y)g(pro)r (duction)g(\(rate\))g(as)1718 973 y Fw(s)c Fx(:=)1934 869 y Fv(2)1891 894 y Fm(X)1893 1071 y Fs(j)s Fv(=0)2025 973 y Fw(\014)2072 985 y Fs(j)2107 973 y Fw(\036)2156 985 y Fs(j)2219 973 y Fw(;)1326 b Fx(\(2.18\))180 1244 y(where)29 b Fw(\036)471 1256 y Fs(j)506 1244 y Fx(,)i Fw(j)g Fx(=)26 b(1)p Fw(;)14 b Fx(2,)29 b(is)g(the)h(heat)f(\015o)n(w)g (rate)g(\(i.e.)43 b(the)30 b(energy)e(\015o)n(w)h(rate)g(due)h(to)f (thermal)g(con)n(tact\))g(in)n(to)h(the)180 1344 y Fw(j)5 b Fx(-th)28 b(reserv)n(oir:)1598 1482 y Fw(\036)1647 1494 y Fs(j)1705 1482 y Fx(:=)1841 1426 y Fw(@)p 1826 1463 79 4 v 1826 1539 a(@)5 b(t)1914 1411 y Fm(\014)1914 1461 y(\014)1942 1515 y Fs(t)p Fv(=0)2055 1482 y Fw(\013)2108 1448 y Fs(t)2138 1482 y Fx(\()p Fw(H)2239 1494 y Fs(r)r(j)2307 1482 y Fx(\))p Fw(;)1206 b Fx(\(2.19\))180 1655 y(and)27 b Fw(\036)390 1667 y Fv(0)456 1655 y Fx(is)g(the)h(energy)f(\015o)n(w)g (in)n(to)g(the)h(particle)f(system,)1597 1876 y Fw(\036)1646 1888 y Fv(0)1707 1876 y Fx(:=)1843 1820 y Fw(@)p 1828 1857 V 1828 1933 a(@)5 b(t)1916 1805 y Fm(\014)1916 1855 y(\014)1944 1909 y Fs(t)p Fv(=0)2085 1876 y Fw(\013)2138 1842 y Fs(t)2168 1876 y Fx(\()p Fw(H)2269 1888 y Fs(p)2307 1876 y Fx(\);)1206 b(\(2.20\))180 2082 y(for)27 b(notational)g(con)n(v) n(enience,)f(w)n(e)h(write)h Fw(\014)1563 2094 y Fv(0)1623 2082 y Fx(:=)23 b Fw(\014)1781 2094 y Fs(p)1819 2082 y Fx(.)305 2182 y(The)k(en)n(trop)n(y)g(pro)r(duction,)g Fw(E)5 b(P)12 b Fx(\()p Fw(!)s Fx(\),)28 b(in)g(a)f(state)g Fw(!)j Fx(is)e(de\014ned)g(as)f(\(see)g([34)o(,)h(30)o(,)g(31)o(,)g(32) o(,)f(33,)g(19)o(,)h(20)o(,)g(21)o(]\))1695 2364 y Fw(E)5 b(P)12 b Fx(\()p Fw(!)s Fx(\))23 b(=)g Fw(!)s Fx(\()p Fw(s)p Fx(\))28 b Fw(:)1303 b Fx(\(2.21\))180 2547 y(Since)32 b Fw(s)f Fx(is)g(not)g(a)g(b)r(ounded)h(op)r(erator,)f(w)n(e)g(ha)n(v)n (e)f(to)h(use)g(an)g(appro)n(ximation)f(pro)r(cedure)g(similar)h(to)g (the)g(one)180 2646 y(men)n(tioned)k(in)g(the)g(remark)f(after)g (\(2.9\))h(in)g(order)e(to)i(de\014ne)g(the)g(r.h.s.)59 b(of)35 b(\(2.21\))f(for)g(su\016cien)n(tly)h(regular)180 2746 y(states.)305 2846 y(The)h(en)n(trop)n(y)g(pro)r(duction)g Fw(E)5 b(P)12 b Fx(\()p Fw(\021)s Fx(\))38 b(of)e(the)h(NESS)g(is)g (indep)r(enden)n(t)h(of)e(the)h(particle)g(state)f Fw(!)3412 2816 y Fs(p)3487 2846 y Fx(en)n(tering)180 2945 y(De\014nition)f (\(2.10\))f(of)g(the)h(state)f(of)g(the)h(decoupled)f(system,)i(since)e Fw(\021)k Fx(is)c(indep)r(enden)n(t)h(of)f Fw(!)3242 2957 y Fs(p)3280 2945 y Fx(.)57 b(Notice)35 b(that)180 3045 y Fw(\021)s Fx(\()p Fw(\036)305 3057 y Fv(0)343 3045 y Fx(\))43 b(=)g Fw(@)570 3057 y Fs(t)599 2974 y Fm(\014)599 3024 y(\014)627 3078 y Fs(t)p Fv(=0)740 3045 y Fw(\021)784 2978 y Fm(\000)823 3045 y Fw(\013)876 3015 y Fs(t)905 3045 y Fx(\()p Fw(H)1006 3057 y Fs(p)1045 3045 y Fx(\))1077 2978 y Fm(\001)1115 3045 y Fx(,)g(since)c Fw(H)1465 3057 y Fs(p)1547 3045 y Fp(2)k(B)s Fx(\()p Fp(H)1805 3057 y Fs(p)1843 3045 y Fx(\))d(is)f(a)g(prop)r(er)g(observ) -5 b(able)38 b(and)h Fw(\021)k Fx(is)d(a)f(con)n(tin)n(uous)f(and)180 3145 y(stationary)26 b(state.)37 b(Hence)1781 3244 y Fw(\021)s Fx(\()p Fw(\036)1906 3256 y Fv(0)1944 3244 y Fx(\))23 b(=)g(0)k Fw(:)1389 b Fx(\(2.22\))180 3448 y(Therefore,)26 b(writing)907 3369 y Fv(2)879 3385 y Fm(P)866 3521 y Fs(j)s Fv(=1)994 3448 y Fw(\021)s Fx(\()p Fw(\036)1119 3460 y Fs(j)1155 3448 y Fx(\))e(=)1339 3369 y Fv(2)1312 3385 y Fm(P)1298 3521 y Fs(j)s Fv(=0)1427 3448 y Fw(\021)s Fx(\()p Fw(\036)1552 3460 y Fs(j)1588 3448 y Fx(\))f(=)g Fw(\021)s Fx(\()p Fw(@)1851 3460 y Fs(t)1881 3448 y Fp(j)1904 3460 y Fs(t)p Fv(=0)2017 3448 y Fw(\013)2070 3418 y Fs(t)2100 3448 y Fx(\()p Fw(H)2201 3460 y Fv(0)2238 3448 y Fx(\)\))h(=)f Fp(\000)p Fw(@)2523 3460 y Fs(t)2551 3377 y Fm(\014)2551 3427 y(\014)2579 3481 y Fs(t)p Fv(=0)2692 3448 y Fw(\021)s Fx(\()p Fw(\013)2821 3418 y Fs(t)2852 3448 y Fx(\()p Fw(v)s Fx(\)\))h(=)e(0,)28 b(w)n(e)f(obtain)1772 3677 y Fv(2)1729 3702 y Fm(X)1731 3879 y Fs(j)s Fv(=1)1863 3781 y Fw(\021)s Fx(\()p Fw(\036)1988 3793 y Fs(j)2024 3781 y Fx(\))c(=)g(0)p Fw(:)1336 b Fx(\(2.23\))305 4052 y(Observ)n(e)32 b(that)j(the)f(zero)g(total)g(\015o)n(w)f (relation)h(\(2.23\))f(and)h(De\014nition)h(3.1)f(for)g(the)g(en)n (trop)n(y)f(pro)r(duction)180 4152 y(rate)27 b(imply)h(that)1514 4252 y Fw(E)5 b(P)12 b Fx(\()p Fw(\021)s Fx(\))24 b(=)f(\()p Fw(\014)1944 4264 y Fv(1)2000 4252 y Fp(\000)18 b Fw(\014)2130 4264 y Fv(2)2167 4252 y Fx(\))p Fw(\021)s Fx(\()p Fw(\036)2324 4264 y Fv(1)2363 4252 y Fx(\))27 b Fw(:)1123 b Fx(\(2.24\))180 4401 y(Th)n(us,)27 b(the)h(relation)f Fw(E)5 b(P)12 b Fx(\()p Fw(\021)s Fx(\))24 b Fw(>)e Fx(0)28 b(is)f(equiv)-5 b(alen)n(t)27 b(to)1444 4584 y Fw(\021)s Fx(\()p Fw(\036)1569 4596 y Fv(1)1607 4584 y Fx(\))c Fw(>)g Fx(0)k(whenev)n(er)g Fw(T)2233 4596 y Fv(2)2292 4584 y Fw(>)c(T)2429 4596 y Fv(1)2493 4584 y Fw(;)1052 b Fx(\(2.25\))180 4766 y(where)29 b Fw(T)471 4778 y Fs(j)531 4766 y Fx(=)d Fw(\014)673 4731 y Fr(\000)p Fv(1)669 4790 y Fs(j)792 4766 y Fx(is)j(the)h(temp)r (erature)f(of)g(the)h Fw(j)5 b Fx(-th)29 b(reserv)n(oir.)40 b(In)29 b(other)g(w)n(ords,)f(in)i(the)g(state)f Fw(\021)k Fx(the)c(energy)180 4866 y(\015o)n(ws)e(from)g(the)h(hotter)f(to)h(the) g(colder)e(reserv)n(oir.)305 4966 y(A)f(general)f(result)h(due)g(to)g ([19)o(])g(sho)n(ws)f(that)i Fw(E)5 b(P)12 b Fx(\()p Fw(!)s Fx(\))23 b Fp(\025)g Fx(0)h(for)h(an)n(y)g(NESS)g Fw(!)s Fx(.)36 b(W)-7 b(e)25 b(sho)n(w)f(that)i(for)e(the)i(NESS)180 5065 y Fw(\021)s Fx(,)1514 5165 y Fw(E)5 b(P)12 b Fx(\()p Fw(\021)s Fx(\))24 b Fw(>)f Fx(0)82 b(i\013)i Fw(\014)2191 5177 y Fv(1)2251 5165 y Fp(6)p Fx(=)23 b Fw(\014)2386 5177 y Fv(2)2423 5165 y Fw(;)1959 5476 y Fx(5)p eop %%Page: 6 6 6 5 bop 180 210 a Fx(see)38 b(Theorem)f(3.2)g(and)h(Section)g(12)g(for) f(a)h(precise)f(statemen)n(t)i(of)f(this)g(result.)68 b(Moreo)n(v)n(er,)38 b(w)n(e)g(dev)n(elop)f(a)180 310 y(p)r(erturbation)27 b(theory)g(for)g(the)h(NESS)g(and)f(compute)h Fw(E)5 b(P)12 b Fx(\()p Fw(\021)s Fx(\))28 b(in)g(leading)f(order)f(in) i(the)g(coupling)f(constan)n(t)g Fw(g)s Fx(.)305 409 y(Let)42 b(us)g(outline)h(the)f(main)g(steps)h(of)f(our)f(pro)r(of)h (of)g(the)h(con)n(v)n(ergence)d(\(2.16\))h(\(c.f.)82 b([21)o(]\).)f(W)-7 b(e)43 b(pass)180 509 y(to)f(the)h(Araki-W)-7 b(o)r(o)r(ds)42 b(GNS)h(represen)n(tation)e(of)h(\()p Fp(A)p Fw(;)14 b(!)2043 521 y Fv(0)2081 509 y Fx(\),)46 b(with)d Fw(!)2438 521 y Fv(0)2518 509 y Fx(of)f(the)h(form)f(\(2.10\)) g(and)h Fw(!)3480 521 y Fs(p)3518 509 y Fx(\()p Fw(A)p Fx(\))48 b(:=)180 609 y(T)-7 b(r\()p Fw(e)337 579 y Fr(\000)p Fs(\014)427 587 y Fi(p)461 579 y Fs(H)515 587 y Fi(p)554 609 y Fw(A)p Fx(\))p Fw(=)p Fx(T)g(r\()p Fw(e)847 579 y Fr(\000)p Fs(\014)937 587 y Fi(p)971 579 y Fs(H)1025 587 y Fi(p)1065 609 y Fx(\);)1597 708 y(\()p Fp(A)p Fw(;)14 b(!)1784 720 y Fv(0)1822 708 y Fx(\))23 b Fp(!)g Fx(\()p Fp(H)q Fw(;)14 b(\031)s(;)g Fx(\012)2270 720 y Fv(0)2307 708 y Fx(\))p Fw(;)180 858 y Fx(where)27 b Fp(H)q Fx(,)g Fw(\031)j Fx(and)d(\012)839 870 y Fv(0)904 858 y Fx(are)f(a)g(Hilb)r (ert)i(space,)f(a)f(represen)n(tation)g(of)h(the)g(algebra)e Fp(A)j Fx(b)n(y)f(b)r(ounded)g(op)r(erators)e(on)180 957 y Fp(H)q Fx(,)j(and)f(a)g(cyclic)h(elemen)n(t)f(in)h Fp(H)h Fx(\(meaning)e(that)p 1807 885 279 4 v 28 w Fw(\031)s Fx(\()p Fp(A)p Fx(\)\012)2047 969 y Fv(0)2108 957 y Fx(=)c Fp(H)q Fx(\))28 b(s.t.)1562 1140 y Fw(!)1614 1152 y Fv(0)1651 1140 y Fx(\()p Fw(A)p Fx(\))c(=)e Fp(h)q Fx(\012)1981 1152 y Fv(0)2018 1140 y Fw(;)14 b(\031)s Fx(\()p Fw(A)p Fx(\)\012)2291 1152 y Fv(0)2329 1140 y Fp(i)g Fw(:)305 1323 y Fx(The)23 b(GNS)h(represen)n(tation)e(pro)n(vides)g(us)h(with)h (a)f(Hilb)r(ert)h(space)f(framew)n(ork)f(whic)n(h)h(w)n(e)g(use)g(to)h (con)n(v)n(ert)e(the)180 1422 y(dynamical)30 b(problem)g(describ)r(ed)g (ab)r(o)n(v)n(e)f(in)n(to)h(a)g(sp)r(ectral)g(problem)g(for)g(a)g (certain)f(non-self-adjoin)n(t)h(op)r(erator)180 1522 y Fw(K)e Fx(on)22 b(the)h(Hilb)r(ert)g(space)e Fp(H)q Fx(.)36 b(With)23 b(the)g(free)f(ev)n(olution)f Fw(\013)2062 1492 y Fs(t)2062 1543 y Fv(0)2100 1522 y Fx(\()p Fw(A)p Fx(\))i(:=)g Fw(e)2399 1492 y Fs(itH)2501 1500 y Fk(0)2538 1522 y Fw(Ae)2639 1492 y Fr(\000)p Fs(itH)2793 1500 y Fk(0)2853 1522 y Fx(one)f(asso)r(ciates)f(the)i(unitary)180 1622 y(one-parameter)i(group)i Fw(U)1022 1634 y Fv(0)1059 1622 y Fx(\()p Fw(t)p Fx(\))c(=)g Fw(e)1303 1591 y Fs(itL)1397 1599 y Fk(0)1461 1622 y Fx(on)k Fp(H)i Fx(s.t.)1437 1804 y Fw(\031)s Fx(\()p Fw(\013)1572 1770 y Fs(t)1572 1825 y Fv(0)1610 1804 y Fx(\()p Fw(A)p Fx(\)\))24 b(=)f Fw(U)1937 1816 y Fv(0)1974 1804 y Fx(\()p Fw(t)p Fx(\))p Fw(\031)s Fx(\()p Fw(A)p Fx(\))p Fw(U)2301 1816 y Fv(0)2339 1804 y Fx(\()p Fw(t)p Fx(\))2433 1770 y Fr(\000)p Fv(1)3568 1804 y Fx(\(2.26\))180 1987 y(and)40 b Fw(U)411 1999 y Fv(0)448 1987 y Fx(\()p Fw(t)p Fx(\)\012)602 1999 y Fv(0)685 1987 y Fx(=)k(\012)854 1999 y Fv(0)891 1987 y Fx(.)75 b(De\014ne)41 b(the)g(op)r(erator)e Fw(L)1820 1957 y Fv(\()p Fs(`)p Fv(\))1948 1987 y Fx(:=)44 b Fw(L)2137 1999 y Fv(0)2201 1987 y Fx(+)26 b Fw(g)s(\031)s Fx(\()p Fw(v)s Fx(\))42 b(on)e(the)h(dense)f(domain)g(Dom\()p Fw(L)3628 1999 y Fv(0)3665 1987 y Fx(\))28 b Fp(\\)180 2086 y Fx(Dom\()p Fw(\031)s Fx(\()p Fw(v)s Fx(\)\).)77 b(Here)40 b Fw(\031)s Fx(\()p Fw(v)s Fx(\))h(can)f(b)r(e)g(de\014ned)h (either)f(using)g(explicit)h(form)n(ulae)e(for)g Fw(\031)44 b Fx(in)d(the)f(Araki-W)-7 b(o)r(o)r(ds)180 2186 y(represen)n(tation)27 b(giv)n(en)h(b)r(elo)n(w)h(or)f(b)n(y)g(using)h(the)g(appro)n (ximation,)f Fw(v)2380 2198 y Fs(n)2479 2186 y Fp(2)e(A)p Fx(,)k(for)e(the)h(op)r(erator)e Fw(v)33 b Fx(constructed)180 2286 y(ab)r(o)n(v)n(e.)i(By)24 b(the)i(Glimm-Ja\013e-Nelson)e(comm)n (utator)g(theorem)g(the)i(op)r(erator)d Fw(L)2773 2256 y Fv(\()p Fs(`)p Fv(\))2882 2286 y Fx(is)h(essen)n(tially)h (self-adjoin)n(t;)180 2385 y(w)n(e)35 b(denote)g(its)h(self-adjoin)n(t) f(closure)f(again)h(b)n(y)g(the)g(same)g(sym)n(b)r(ol)g Fw(L)2497 2355 y Fv(\()p Fs(`)p Fv(\))2581 2385 y Fx(.)60 b(The)36 b(op)r(erator)d Fw(L)3242 2355 y Fv(\()p Fs(`)p Fv(\))3361 2385 y Fx(induces)j(the)180 2485 y(one-parameter)25 b(group)i Fw(\033)1015 2455 y Fs(t)1072 2485 y Fx(on)g Fw(\031)s Fx(\()p Fp(A)p Fx(\))1367 2455 y Fr(00)1411 2485 y Fx(,)h(the)g(w)n(eak)e(closure)h(of)g Fw(\031)s Fx(\()p Fp(A)p Fx(\),)1537 2688 y Fw(\033)1587 2653 y Fs(t)1617 2688 y Fx(\()p Fw(B)t Fx(\))d(:=)e Fw(e)1921 2653 y Fs(itL)2015 2628 y Fk(\()p Fi(`)p Fk(\))2093 2688 y Fw(B)t(e)2199 2653 y Fr(\000)p Fs(itL)2345 2628 y Fk(\()p Fi(`)p Fk(\))3568 2688 y Fx(\(2.27\))180 2870 y(for)27 b(an)n(y)g Fw(B)g Fp(2)d Fw(\031)s Fx(\()p Fp(A)p Fx(\))813 2840 y Fr(00)856 2870 y Fx(.)37 b(Let)28 b Fw( )j Fx(b)r(e)d(a)f(state) g(on)g(the)h(algebra)e Fp(A)i Fx(normal)f(w.r.t.)37 b Fw(!)2743 2882 y Fv(0)2780 2870 y Fx(,)27 b(i.e.)1648 3053 y Fw( )s Fx(\()p Fw(A)p Fx(\))d(=)e(T)-7 b(r\()p Fw(\032\031)s Fx(\()p Fw(A)p Fx(\)\))1257 b(\(2.28\))180 3235 y(for)27 b(some)f(p)r(ositiv)n(e)h(trace)f(class)g(op)r(erator)g Fw(\032)h Fx(on)g Fp(H)h Fx(of)f(trace)f(one.)37 b(It)27 b(is)g(sho)n(wn)g(in)g(App)r(endix)h(A)f(that)h(for)e Fw( )31 b Fx(as)180 3335 y(ab)r(o)n(v)n(e)26 b(the)i(limit)h(on)e(the)h (r.h.s.)36 b(of)28 b(\(2.12\))f(exists)g(and)g(equals)1549 3518 y Fw( )1606 3483 y Fs(t)1636 3518 y Fx(\()p Fw(A)p Fx(\))d(=)e(T)-7 b(r\()p Fw(\032\033)2084 3483 y Fs(t)2114 3518 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\)\))p Fw(:)1159 b Fx(\(2.29\))180 3700 y(In)28 b(particular,)e(the)i(limit)g(is)g (indep)r(enden)n(t)g(of)g(the)g(c)n(hoice)f(of)g(the)h(appro)n (ximating)e(family)i Fw(v)3155 3712 y Fs(n)3200 3700 y Fx(.)305 3800 y(Due)d(to)f(\(2.29\))g(the)h(dynamics)f(on)g(normal)g (states,)g(de\014ned)h(in)g(\(2.12\),)g(giv)n(es)e(rise)h(to)g(the)h (dynamics)f(on)h(the)180 3900 y(Hilb)r(ert)j(space)f Fp(H)q Fx(,)h(determined)f(b)n(y)h(a)f(one-parameter)e(group)i Fw(U)9 b Fx(\()p Fw(t)p Fx(\),)28 b(satisfying)1219 4082 y(T)-7 b(r)o(\()p Fw(\032U)9 b Fx(\()p Fw(t)p Fx(\))p Fw(\031)s Fx(\()p Fw(A)p Fx(\))p Fw(U)g Fx(\()p Fw(t)p Fx(\))1875 4048 y Fr(\000)p Fv(1)1966 4082 y Fx(\))24 b(=)e Fw( )2166 4048 y Fs(t)2195 4082 y Fx(\()p Fw(A)p Fx(\))p Fw(;)98 b Fp(8)p Fw(A)23 b Fp(2)g(A)p Fw(:)827 b Fx(\(2.30\))180 4265 y(Due)38 b(to)f(fact)h(that)g Fw(\031)s Fx(\()p Fp(A)p Fx(\))1019 4235 y Fr(00)1100 4265 y Fx(has)f(a)g(large)f(comm)n(utan)n(t)h(\(isomorphic)f(to)h (itself,)k(as)c(is)g(kno)n(wn)g(from)g(T)-7 b(omita-)180 4365 y(T)g(ak)n(esaki)30 b(theory\),)i(relation)e(\(2.30\))h(do)r(es)g (not)h(de\014ne)g Fw(U)9 b Fx(\()p Fw(t)p Fx(\))32 b(uniquely;)h(ho)n (w)n(ev)n(er,)e(if)h(w)n(e)f(imp)r(ose)g(in)h(addition)180 4464 y(to)27 b(\(2.30\))g(the)h(in)n(v)-5 b(ariance)27 b(condition)1773 4564 y Fw(U)9 b Fx(\()p Fw(t)p Fx(\)\012)23 b(=)g(\012)p Fw(;)1381 b Fx(\(2.31\))180 4713 y(where)31 b(\012)h(is)g(a)g(\014xed)g(cyclic)f(and)h(separating)f(v)n(ector,)g (then)i Fw(U)9 b Fx(\()p Fw(t)p Fx(\))32 b(is)g(uniquely)g(determined.) 50 b(\(The)32 b(v)n(ector)f(\012)180 4813 y(is)g(called)f(cyclic)h(if)g Fw(\031)s Fx(\()p Fj(A)p Fx(\)\012)h(is)f(dense)f(in)i Fp(H)g Fx(and)e(separating)g(if)h Fw(\031)s Fx(\()p Fj(A)p Fx(\))2424 4783 y Fr(0)2448 4813 y Fx(\012)g(is)f(dense)h(in)g Fp(H)q Fx(,)h(the)g(prime)e(denoting)180 4913 y(the)h(comm)n(uan)n (t.\))45 b(If)30 b(\012)h(w)n(ere)e(the)i(v)n(ector)e(represen)n(ting)g (an)h(equilibrium)g(state)h(then)f Fw(U)9 b Fx(\()p Fw(t)p Fx(\))31 b(satisfying)f(\(2.30\))180 5012 y(and)f(\(2.31\))f(w)n(ould)g (b)r(e)i(a)e(unitary)h(group.)39 b(In)29 b(the)h(non-equilibrium)e (case)g Fw(\014)2655 5024 y Fv(1)2718 5012 y Fp(6)p Fx(=)c Fw(\014)2854 5024 y Fv(2)2892 5012 y Fx(,)29 b(one)g(can)f(see)h(that)g (\(2.31\))180 5112 y(cannot)23 b(b)r(e)g(satis\014ed)g(for)f(a)h (unitary)f Fw(U)9 b Fx(\()p Fw(t)p Fx(\))23 b(implemen)n(ting)h(the)f (dynamics)g(as)f(in)h(\(2.30\).)35 b(F)-7 b(or)22 b(tec)n(hnical)h (reasons,)180 5211 y(w)n(e)k(c)n(ho)r(ose)g Fw(U)9 b Fx(\()p Fw(t)p Fx(\))28 b(to)f(satisfy)g(\(2.31\))g(for)g(a)g(con)n(v)n (enien)n(t)g(v)n(ector)f(\012,)i(rather)e(than)i(to)g(b)r(e)g(unitary) -7 b(.)1959 5476 y(6)p eop %%Page: 7 7 7 6 bop 305 210 a Fx(W)-7 b(e)29 b(will)g(sho)n(w)f(that)h Fw(U)9 b Fx(\()p Fw(t)p Fx(\))30 b(is)f(strongly)e(di\013eren)n(tiable) i(on)g(a)f(dense)h(set)g(of)g(v)n(ectors)e(and)i(w)n(e)g(will)g (calculate)180 310 y(explicitly)h(its)g(generator,)e Fw(K)j Fx(:=)26 b Fp(\000)p Fw(i)1387 277 y Fs(@)p 1374 291 65 4 v 1374 338 a(@)t(t)1449 310 y Fw(U)9 b Fx(\()p Fw(t)p Fx(\))p Fp(j)1632 322 y Fs(t)p Fv(=0)1745 310 y Fx(.)43 b(In)30 b(the)g(non-equilibrium)f(situation)g Fw(K)3091 280 y Fr(\003)3155 310 y Fp(6)p Fx(=)d Fw(K)35 b Fx(\()p Fw(U)9 b Fx(\()p Fw(t)p Fx(\))31 b(is)e(not)180 409 y(unitary!\))62 b(and)36 b(\(2.31\))g(implies)g(that)g Fw(K)6 b Fx(\012)37 b(=)g(0.)62 b(The)36 b(main)g(e\013ort)g(of)g(our)g (analysis)e(is)i(to)g(deriv)n(e)g(enough)180 509 y(sp)r(ectral)27 b(information)g(on)g(the)h(op)r(erator)e Fw(K)33 b Fx(to)28 b(enable)f(us)h(to)f(sho)n(w)g(\(2.16\))g(and)g(to)g(iden)n(tify)i(the) f(NESS)f(with)1602 692 y Fw(\021)s Fx(\()p Fw(A)p Fx(\))d(=)f Fp(h)p Fx(\012)1976 657 y Fr(\003)2014 692 y Fw(;)14 b(\031)s Fx(\()p Fw(A)p Fx(\)\012)q Fp(i)h Fw(;)1210 b Fx(\(2.32\))180 874 y(where)32 b(\012)485 844 y Fr(\003)557 874 y Fx(is)g(a)h(zero)f(resonance)f(of)i(the)h(op)r(erator)d Fw(K)1951 844 y Fr(\003)1988 874 y Fx(:)48 b Fw(K)2136 844 y Fr(\003)2174 874 y Fx(\012)2234 844 y Fr(\003)2304 874 y Fx(=)31 b(0)i(and)g(\012)2702 844 y Fr(\003)2772 874 y Fp(2)f(D)2925 844 y Fr(0)2923 898 y Fv(anal)3049 874 y Fx(,)j(for)d(an)h(appropriate)180 974 y(dense)27 b(set)h Fp(D)600 986 y Fv(anal)749 974 y Fp(\032)23 b(H)q Fx(,)k(and)h Fw(A)g Fx(are)e(suc)n(h)h(that)h Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)d Fp(2)e(D)2118 986 y Fv(anal)2244 974 y Fx(.)305 1074 y(In)32 b(order)e(to)i(obtain)g(rather)e(subtle)j (sp)r(ectral)e(information)g(on)h(the)g(op)r(erator)e Fw(K)6 b Fx(,)33 b(and)f(to)f(giv)n(e)g(a)h(precise)180 1173 y(meaning)k(to)h(expression)e(\(2.32\),)k(w)n(e)d(dev)n(elop)g(a)g (new)h(t)n(yp)r(e)g(of)g(sp)r(ectral)f(deformation,)j Fw(K)k Fp(7!)c Fw(K)3441 1185 y Fs(\022)3478 1173 y Fx(,)g(with)e(a)180 1273 y(sp)r(ectral)23 b(deformation)f(parameter)g Fw(\022)j Fp(2)f Fo(C)1536 1243 y Fv(2)1580 1273 y Fx(,)g(in)f(com)n(bination)g (with)h(an)e(\(iterativ)n(e\))h(application)g(of)g(a)g(F)-7 b(esh)n(bac)n(h)180 1373 y(map)27 b(acting)h(on)f Fw(K)800 1385 y Fs(\022)837 1373 y Fx(.)305 1472 y(In)i(conclusion)f(of)h(this)h (outline)f(w)n(e)g(presen)n(t)g(here)f(the)i(GNS)g(triple)f(pro)n (vided)f(b)n(y)h(the)h(Araki-W)-7 b(o)r(o)r(ds)28 b(con-)180 1572 y(struction,)39 b(whic)n(h)e(forms)f(a)g(mathematical)h(framew)n (ork)e(for)h(our)g(analysis)g(\(see)g([8,)h(18)o(,)g(9)o(])g(for)g (details)f(and)180 1671 y([3,)h(17)o(])h(for)f(original)f(pap)r(ers\).) 66 b(In)38 b(the)g(Araki-W)-7 b(o)r(o)r(ds)36 b(GNS)i(represen)n (tation)e(the)i(\(p)r(ositiv)n(e)f(temp)r(erature\))180 1771 y(Hilb)r(ert)28 b(space)f(is)h(giv)n(en)e(b)n(y)1718 1871 y Fp(H)e Fx(=)f Fp(H)1971 1836 y Fs(p)2028 1871 y Fp(\012)18 b(H)2182 1836 y Fs(r)2218 1871 y Fw(;)1327 b Fx(\(2.33\))180 2020 y(where)27 b Fp(H)491 1990 y Fs(p)552 2020 y Fx(=)c Fp(H)710 2032 y Fs(p)767 2020 y Fp(\012)18 b(H)920 2032 y Fs(p)986 2020 y Fx(and)28 b Fp(H)1219 1990 y Fs(r)1278 2020 y Fx(=)23 b Fp(H)1437 1990 y Fs(r)r Fv(1)1525 2020 y Fp(\012)18 b(H)1679 1990 y Fs(r)r Fv(2)1777 2020 y Fx(with)1655 2203 y Fp(H)1726 2168 y Fs(r)r(j)1817 2203 y Fx(=)23 b Fp(H)1975 2215 y Fs(r)r(j)2061 2203 y Fp(\012)18 b(H)2214 2215 y Fs(r)r(j)2282 2203 y Fw(:)1263 b Fx(\(2.34\))180 2398 y(W)-7 b(e)31 b(denote)g(b)n(y)f Fw(a)759 2358 y Fv(#)759 2423 y Fs(`;j)841 2398 y Fx(\()p Fw(f)9 b Fx(\))31 b(\(resp.,)g Fw(a)1288 2358 y Fv(#)1288 2421 y Fs(r)n(;j)1372 2398 y Fx(\()p Fw(f)9 b Fx(\)\))31 b(the)g(creation)f(and)g(annihilation)g(op)r(erators)f(whic)n(h)i(act)f (on)g(the)h(left)180 2498 y(\(resp.,)i(righ)n(t\))e(factor)h(of)g (\(2.34\).)49 b(They)32 b(are)f(related)g(to)h(the)g(zero)f(temp)r (erature)h(creation)f(and)g(annihilation)180 2607 y(op)r(erators)26 b Fw(a)592 2567 y Fv(#)592 2630 y Fs(j)650 2607 y Fx(\()p Fw(f)9 b Fx(\))28 b(b)n(y)1257 2726 y Fw(\031)s Fx(\()p Fw(a)1383 2738 y Fs(j)1419 2726 y Fx(\()p Fw(f)9 b Fx(\)\))23 b(=)g Fw(a)1720 2738 y Fs(`j)1783 2726 y Fx(\()1815 2656 y Fm(p)p 1898 2656 221 4 v 70 x Fx(1)18 b(+)g Fw(\032)2084 2738 y Fs(j)2133 2726 y Fw(f)9 b Fx(\))18 b(+)g Fw(a)2360 2691 y Fr(\003)2360 2746 y Fs(r)r(j)2428 2726 y Fx(\()2460 2674 y Fp(p)p 2529 2674 78 4 v 52 x Fw(\032)2572 2738 y Fs(j)2639 2704 y Fx(\026)2621 2726 y Fw(f)8 b Fx(\))866 b(\(2.35\))180 2875 y(and)1246 2975 y Fw(\031)1296 2940 y Fr(0)1319 2975 y Fx(\()p Fw(a)1395 2987 y Fs(j)1431 2975 y Fx(\()p Fw(f)9 b Fx(\)\))23 b(=)g Fw(a)1732 2940 y Fr(\003)1732 2995 y Fs(`j)1794 2975 y Fx(\()1826 2923 y Fp(p)p 1896 2923 V 1896 2975 a Fw(\032)1939 2987 y Fs(j)1988 2975 y Fw(f)9 b Fx(\))18 b(+)g Fw(a)2215 2987 y Fs(r)r(j)2282 2975 y Fx(\()2314 2905 y Fm(p)p 2398 2905 221 4 v 2398 2975 a Fx(1)g(+)g Fw(\032)2584 2987 y Fs(j)2650 2953 y Fx(\026)2632 2975 y Fw(f)9 b Fx(\))854 b(\(2.36\))180 3138 y(where)33 b Fw(\032)469 3150 y Fs(j)536 3138 y Fp(\021)e Fw(\032)675 3150 y Fs(j)710 3138 y Fx(\()p Fw(k)s Fx(\))i(=)e(\()p Fw(e)1020 3108 y Fs(\014)1058 3116 y Fi(j)1089 3108 y Fs(!)r Fv(\()p Fs(k)q Fv(\))1248 3138 y Fp(\000)21 b Fx(1\))1408 3108 y Fr(\000)p Fv(1)1530 3138 y Fx(with)34 b Fw(!)s Fx(\()p Fw(k)s Fx(\))e(=)g Fp(j)p Fw(k)s Fp(j)p Fx(.)53 b(Finally)-7 b(,)34 b(w)n(e)f(denote)g (\012)2953 3150 y Fs(r)3022 3138 y Fx(:=)f(\012)3202 3150 y Fs(r)r Fv(1)3294 3138 y Fp(\012)21 b Fx(\012)3440 3150 y Fs(r)r Fv(2)3510 3138 y Fx(,)35 b(where)180 3238 y(\012)240 3250 y Fs(r)r(j)331 3238 y Fx(:=)22 b(\012)501 3250 y Fs(r)r(j;`)631 3238 y Fp(\012)c Fx(\012)774 3250 y Fs(r)r(j;r)918 3238 y Fx(are)27 b(the)h(v)-5 b(acua)27 b(in)h Fp(H)1601 3208 y Fs(r)r(j)1668 3238 y Fx(.)37 b(Th)n(us,)27 b(\012)2021 3250 y Fs(r)2086 3238 y Fx(is)g(the)h(v)-5 b(acuum)28 b(in)g Fp(H)2787 3208 y Fs(r)2824 3238 y Fx(.)305 3338 y(De\014nition)g(\(2.10\))f(and)g(our)g(c)n(hoice)g(of)g Fw(!)1633 3350 y Fs(p)1699 3338 y Fx(made)g(at)h(the)g(b)r(eginning)f (of)h(this)g(section)f(imply)h(that)923 3595 y(\012)983 3607 y Fv(0)1043 3595 y Fx(=)23 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4420 y Fm(Z)1404 4533 y Fw(!)s Fx(\()p Fw(k)s Fx(\))1583 4466 y Fm(\000)1621 4533 y Fw(a)1665 4499 y Fr(\003)1665 4554 y Fs(`;j)1748 4533 y Fx(\()p Fw(k)s Fx(\))p Fw(a)1902 4545 y Fs(`;j)1984 4533 y Fx(\()p Fw(k)s Fx(\))h Fp(\000)f Fw(a)2240 4499 y Fr(\003)2240 4554 y Fs(r)n(;j)2324 4533 y Fx(\()p Fw(k)s Fx(\))p Fw(a)2478 4545 y Fs(r)n(;j)2562 4533 y Fx(\()p Fw(k)s Fx(\))2672 4466 y Fm(\001)2738 4533 y Fw(d)2781 4499 y Fv(3)2818 4533 y Fw(k)s(:)180 4725 y Fx(The)31 b(op)r(erator)f Fw(K)36 b Fx(can)31 b(b)r(e)h(written)f(as)g Fw(K)j Fx(=)29 b Fw(L)1729 4737 y Fv(0)1786 4725 y Fx(+)20 b Fw(g)s Fx(\()p Fw(V)40 b Fp(\000)20 b Fw(W)12 b Fx(\))32 b(with)g Fw(V)47 b Fx(=)29 b Fw(\031)s Fx(\()p Fw(v)s Fx(\))j(and)f Fw(W)41 b Fx(=)29 b Fw(\031)3272 4695 y Fr(0)3295 4725 y Fx(\()p Fw(w)r Fx(\))k(with)f Fw(w)h Fx(a)180 4825 y(non-self-adjoin)n(t)27 b(op)r(erator)e(obtained)j(b)n(y)f(a)g(simple) h(transformation)e(of)h Fw(v)s Fx(.)305 4924 y(A)40 b(standard)e (argumen)n(t)h(sho)n(ws)f(that)i(the)g(sp)r(ectrum)g(of)g(the)g(op)r (erator)e Fw(L)2787 4936 y Fv(0)2863 4924 y Fx(\014lls)i(the)g(axis)f Fo(R)46 b Fx(with)40 b(the)180 5024 y(thresholds)24 b(and)h(eigen)n(v) -5 b(alues)23 b(lo)r(cated)i(at)f Fw(\033)s Fx(\()p Fw(L)1683 5036 y Fs(p)1722 5024 y Fx(\))h(and)g(with)g(0)f(an)h(eigen)n(v)-5 b(alue)23 b(of)i(m)n(ultiplicit)n(y)g(at)g(least)f(dim)15 b Fw(H)3742 5036 y Fs(p)180 5124 y Fx(and)31 b(at)h(most)f(\(dim)14 b Fw(H)911 5136 y Fs(p)950 5124 y Fx(\))982 5093 y Fv(2)1051 5124 y Fx(\(dep)r(ending)32 b(on)f(the)h(degeneracy)d(of)j(the)f(sp)r (ectrum)h(of)f Fw(L)2949 5136 y Fs(p)2987 5124 y Fx(\).)49 b(A)32 b(priori)e(w)n(e)h(do)g(not)180 5223 y(kno)n(w)23 b(an)n(ything)g(ab)r(out)h(the)g(sp)r(ectrum)g(of)g(the)g (non-self-adjoin)n(t)f(op)r(erator)f Fw(K)29 b Fx(b)r(esides)24 b(the)g(fact)g(that)g(it)g(has)f(an)1959 5476 y(7)p eop %%Page: 8 8 8 7 bop 180 210 a Fx(eigen)n(v)-5 b(alue)19 b(0.)34 b(F)-7 b(or)19 b(all)g(w)n(e)g(kno)n(w)g(its)h(sp)r(ectrum)g(migh)n(t)f (\014ll)h(in)g(the)g(en)n(tire)f(complex)g(plane!)34 b(Th)n(us)20 b(understanding)180 310 y(the)28 b(ev)n(olution)f (generated)f(b)n(y)h(the)h(op)r(erator)e Fw(K)33 b Fx(is)28 b(a)f(subtle)h(matter.)305 409 y(This)23 b(pap)r(er)g(is)g(organized)e (as)i(follo)n(ws.)34 b(In)23 b(Section)h(3)e(w)n(e)h(giv)n(e)g(a)f (precise)h(form)n(ulation)f(of)h(our)g(assumptions,)180 509 y(state)35 b(the)g(results)f(and)h(discuss)f(assumptions)g(and)h (results.)58 b(In)35 b(Section)f(4)h(w)n(e)f(presen)n(t)g(the)h(Hilb)r (ert)h(space)180 609 y(framew)n(ork)29 b(and)j(the)f(ev)n(olution)g Fw(U)9 b Fx(\()p Fw(t)p Fx(\))31 b(and)h(in)f(Section)g(5)g(w)n(e)g (describ)r(e)g(the)h(generator)d Fw(K)6 b Fx(.)48 b(In)31 b(Section)h(6)e(w)n(e)180 708 y(in)n(tro)r(duce)d(the)g(complex)f (deformation)g Fw(K)1549 720 y Fs(\022)1614 708 y Fx(of)g Fw(K)33 b Fx(and)27 b(establish)f(the)h(connection)g(b)r(et)n(w)n(een)g (the)g(resolv)n(en)n(ts)e(of)180 808 y Fw(K)35 b Fx(and)29 b Fw(K)520 820 y Fs(\022)558 808 y Fx(.)42 b(In)30 b(Section)g(7)f(w)n (e)g(establish)g(the)h(sp)r(ectral)f(prop)r(erties)g(of)g Fw(K)2580 820 y Fs(\022)2647 808 y Fx(whic)n(h)g(w)n(e)h(then)g(use)f (in)h(Section)f(8)180 908 y(to)h(express)g(the)h(dynamics)f(in)h(terms) f(of)h(an)f(in)n(tegral)g(o)n(v)n(er)e(the)j(resolv)n(en)n(t)e(of)i Fw(K)2796 920 y Fs(\022)2833 908 y Fx(.)46 b(In)31 b(Sections)f(9)g (and)h(10)e(w)n(e)180 1007 y(pro)n(v)n(e)23 b(our)i(\014rst)g(main)g (result,)g(the)h(existence)e(and)h(explicit)h(form)f(of)g(the)g(NESS,)g (and)g(its)h(dynamical)e(stabilit)n(y)-7 b(.)180 1107 y(In)28 b(Section)f(11)g(w)n(e)g(dev)n(elop)g(a)g(p)r(erturbation)g (theory)g(for)g(NESS)h(and)f(in)h(Section)f(12)g(w)n(e)g(pro)n(v)n(e)f (the)i(p)r(ositivit)n(y)180 1206 y(of)g(the)g(en)n(trop)n(y)e(pro)r (duction.)36 b(Finally)-7 b(,)28 b(in)g(App)r(endices)g(A{C)g(w)n(e)f (collect)g(some)g(tec)n(hnical)g(results.)180 1477 y Fy(3)135 b(Assumptions)44 b(and)h(Results)180 1659 y Fx(In)31 b(order)f(to)h(state)g(our)f(assumptions)h(\(A\))h(-)f(\(C\))h (b)r(elo)n(w,)f(it)h(is)f(practical)f(to)h(de\014ne)g(the)h(map)f Fw(\015)i Fx(:)c Fw(L)3469 1629 y Fv(2)3506 1659 y Fx(\()p Fo(R)3592 1629 y Fv(3)3636 1659 y Fx(\))g Fp(!)180 1759 y Fw(L)237 1729 y Fv(2)274 1759 y Fx(\()p Fo(R)24 b Fp(\002)18 b Fw(S)523 1729 y Fv(2)560 1759 y Fx(\),)1222 1911 y(\()p Fw(\015)5 b(f)k Fx(\)\()p Fw(u;)14 b(\033)s Fx(\))24 b(=)1694 1836 y Fm(p)p 1777 1836 94 4 v 75 x Fp(j)p Fw(u)p Fp(j)1912 1793 y Fm(\032)2016 1858 y Fw(f)9 b Fx(\()p Fw(u\033)s Fx(\))p Fw(;)212 b(u)23 b Fp(\025)g Fx(0)p Fw(;)2016 1962 y Fp(\000)p 2081 1894 50 4 v Fw(f)8 b Fx(\()p Fp(\000)p Fw(u\033)s Fx(\))p Fw(;)83 b(u)23 b(<)g Fx(0)p Fw(:)3609 1911 y Fx(\(3.1\))180 2111 y(Let)36 b Fw(j)371 2123 y Fs(\022)409 2111 y Fx(\()p Fw(u)p Fx(\))h(=)g Fw(e)699 2081 y Fs(\016)r Fv(sgn\()p Fs(u)p Fv(\))922 2111 y Fw(u)24 b Fx(+)g Fw(\034)46 b Fx(for)35 b Fw(\022)40 b Fx(=)d(\()p Fw(\016)o(;)14 b(\034)9 b Fx(\))37 b Fp(2)h 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b Fx(or)54 b Fp(j)p Fw(\014)1907 5078 y Fv(1)1963 5066 y Fp(\000)18 b Fw(\014)2093 5078 y Fv(2)2130 5066 y Fp(j)24 b(\025)e Fw(C)34 b Fx(and)28 b(min)2657 5078 y Fs(j)2706 5066 y Fw(\014)2753 5078 y Fs(j)2811 5066 y Fp(\025)22 b Fw(C)34 b(;)595 b Fx(\(3.6\))388 5227 y(where)27 b(0)22 b Fw(<)h(c;)14 b(C)29 b(<)23 b Fp(1)k Fx(are)g(constan)n(ts)g(dep)r(ending)h(only)f(on)g(the)h(in)n (teraction)f Fw(G)2968 5239 y Fv(1)p Fs(;)p Fv(2)3058 5227 y Fx(.)1959 5476 y(8)p eop %%Page: 9 9 9 8 bop 305 210 a Fq(R)l(emarks.)36 b Fx(1\))49 b(The)25 b(map)g(\(3.1\))f(has)h(the)g(follo)n(wing)e(origin.)35 b(In)25 b(the)g(p)r(ositiv)n(e-temp)r(erature)f(represen)n(tation)180 310 y(of)37 b(the)g(CCR)g(\(the)g(Araki-W)-7 b(o)r(o)r(ds)36 b(represen)n(tation)f(on)i(a)f(suitable)h(Hilb)r(ert)g(space,)i(see)d (App)r(endix)i(A\),)g(the)180 409 y(in)n(teraction)27 b(term)g Fw(v)836 421 y Fs(j)899 409 y Fx(is)h(represen)n(ted)e(b)n(y)h Fw(a)1583 421 y Fs(j)1618 409 y Fx(\()q Fm(e)-47 b Fw(\015)1693 421 y Fs(\014)1731 429 y Fi(j)1766 409 y Fw(G)1831 421 y Fs(j)1867 409 y Fx(\))18 b(+)g Fw(a)2044 379 y Fr(\003)2044 431 y Fs(j)2082 409 y Fx(\()q Fm(e)-47 b Fw(\015)2157 421 y Fs(\014)2195 429 y Fi(j)2230 409 y Fw(G)2295 421 y Fs(j)2331 409 y Fx(\),)28 b(where)1611 658 y Fm(e)-47 b Fw(\015)1653 670 y Fs(\014)1721 658 y Fx(:=)1831 540 y Fm(r)p 1914 540 338 4 v 2059 602 a Fw(u)p 1924 639 318 4 v 1924 715 a Fx(1)18 b Fp(\000)g Fw(e)2106 691 y Fr(\000)p Fs(\014)s(u)2280 658 y Fw(\015)5 b(:)1258 b Fx(\(3.7\))305 890 y(2\))32 b(A)i(class)e(of)h(in)n(teractions)e (satisfying)i(Condition)g(\(A\))g(is)g(giv)n(en)f(b)n(y)h Fw(G)2687 902 y Fs(j)2722 890 y Fx(\()p Fw(k)s Fx(\))g(=)e Fw(g)s Fx(\()p Fp(j)p Fw(k)s Fp(j)p Fx(\))p Fw(G)p Fx(,)k(where)e Fw(g)s Fx(\()p Fw(u)p Fx(\))e(=)180 1004 y Fw(u)228 974 y Fs(\013)275 1004 y Fw(e)314 974 y Fr(\000)p Fs(u)405 949 y Fk(2)441 1004 y Fx(,)d(with)g Fw(u)23 b Fp(\025)g Fx(0,)k Fw(\013)d Fx(=)f Fw(n)18 b Fx(+)g(1)p Fw(=)p Fx(2,)26 b Fw(n)d Fx(=)g(0)p Fw(;)14 b Fx(1)p Fw(;)g Fx(2)p Fw(;)g(:)g(:)g(:)n Fx(,)28 b(and)g Fw(G)23 b Fx(=)g Fw(G)2370 974 y Fr(\003)2431 1004 y Fp(2)h(B)s Fx(\()p Fp(H)2670 1016 y Fs(p)2708 1004 y Fx(\).)37 b(A)28 b(straigh)n(tforw)n (ard)d(estimate)180 1103 y(giv)n(es)h(that)i(the)g(norms)f(\(3.4\))g (ha)n(v)n(e)g(the)h(b)r(ound)1318 1311 y Fp(k)p Fw(G)1425 1323 y Fs(j)1460 1311 y Fp(k)1502 1339 y Fs(\016)r(\014)1572 1347 y Fi(j)1603 1339 y Fs(;\026;\022)1743 1311 y Fp(\024)23 b Fw(C)1910 1219 y Fm(\020)1959 1311 y Fx(1)18 b(+)g Fw(e)2141 1277 y Fv(\()p Fs(\016)r(\014)2237 1285 y Fi(j)2268 1277 y Fv(\))2294 1252 y Fk(2)2326 1277 y Fs(=)p Fv(4)2398 1219 y Fm(\021)2461 1311 y Fp(jj)p Fw(G)p Fp(jj)p Fw(;)968 b Fx(\(3.8\))180 1514 y(pro)n(vided)29 b Fw(\026)e(<)g(\013)20 b Fx(+)g(1,)30 b(where)g(the)g(constan)n(t)g Fw(C)36 b Fx(do)r(es)30 b(not)g(dep)r(end)h(on)e(the)i(in)n(v)n(erse)e(temp)r (eratures,)h(nor)f(on)h Fw(\022)180 1614 y Fx(v)-5 b(arying)22 b(in)h(an)n(y)e(compact)i(set)f(\(compare)g(this)h(with)g(the)h(b)r (ound)f(\(4.13\))f(of)g([21)o(]\).)36 b(The)23 b(restriction)f Fw(\013)h Fx(=)g Fw(n)9 b Fx(+)g(1)p Fw(=)p Fx(2)180 1713 y(with)31 b Fw(n)c Fx(=)g(0)p Fw(;)14 b Fx(1)p Fw(;)g Fx(2)p Fw(;)g(:)g(:)g(:)28 b Fx(comes)h(from)h(the)h(requiremen)n(t)e (of)h(translation)f(analyticit)n(y)g(\(the)i Fw(\034)9 b Fx({comp)r(onen)n(t)30 b(of)g Fw(\022)r Fx(\),)180 1813 y(whic)n(h)e(app)r(ears)e(also)g(in)i([21)o(].)305 1912 y(3\))c(The)g(condition)g Fw(\034)967 1924 y Fv(0)1004 1912 y Fw(=)14 b Fx(cos)e Fw(\016)1221 1924 y Fv(0)1282 1912 y Fw(<)22 b Fx(2)p Fw(\031)s(=\014)28 b Fx(after)c(\(3.3\))g (guaran)n(tees)e(that)i(the)g(square)f(ro)r(ot)h(in)g(\(3.7\))g(is)g (analytic)180 2012 y(in)k(translations)e Fw(u)d Fp(7!)g Fw(u)18 b Fx(+)g Fw(\034)9 b Fx(.)305 2112 y(4\))25 b(What)h(w)n(e)f (need)h(in)g(our)e(analysis)h(is)g(that)h(the)g(lev)n(el)f(shift)h(op)r (erator)e(\003)2661 2124 y Fv(0)2698 2112 y Fx(,)i(the)g Fw(N)d Fp(\002)14 b Fw(N)35 b Fx(matrix)25 b(de\014ned)h(in)180 2211 y(\(7.1\),)g(has)f(a)g(sp)r(ectral)g(gap)f(at)i(zero)e(whic)n(h)i (is)f(b)r(ounded)h(b)r(elo)n(w)f(b)n(y)g(a)g(strictly)g(p)r(ositiv)n(e) g(constan)n(t)g(indep)r(enden)n(t)180 2311 y(of)j(the)h(temp)r (eratures.)39 b(Condition)29 b(\(D\))g(ensures)f(this)h(prop)r(ert)n(y) -7 b(.)38 b(If)29 b(one)f(can)h(sho)n(w)e(the)i(desired)f(prop)r(ert)n (y)f(on)180 2411 y(the)h(gap)f(b)n(y)g(other)g(means)g(then)h (Condition)g(\(D\))g(can)g(b)r(e)g(dropp)r(ed.)305 2510 y(Let)1332 2610 y Fw(\033)e Fx(:=)d Fp(f)o(j)p Fw(\025)c Fp(\000)f Fw(\026)p Fp(j)28 b(j)g Fw(\025;)14 b(\026)23 b Fp(2)h Fw(\033)s Fx(\()p Fw(H)2270 2622 y Fs(p)2309 2610 y Fx(\))p Fw(;)14 b(\025)23 b Fp(6)p Fx(=)g Fw(\026)p Fp(g)980 b Fx(\(3.9\))180 2759 y(and)27 b(de\014ne)734 2930 y Fw(g)774 2942 y Fv(0)834 2930 y Fx(:=)22 b Fw(C)6 b(\033)1059 2896 y Fv(1)p Fs(=)p Fv(2)1178 2930 y Fx(sin\()p Fw(\016)1349 2942 y Fv(0)1387 2930 y Fx(\))1433 2788 y Fm(")1481 2930 y Fx(\(1)19 b(+)f Fw(\014)1708 2887 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)1704 2952 y(1)1883 2930 y Fx(+)g Fw(\014)2017 2887 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2013 2952 y(2)2173 2930 y Fx(\))c(max)2281 2982 y Fs(j)2420 2930 y Fx(sup)2388 3003 y Fr(j)p Fs(\022)r Fr(j\024)p Fs(\022)2546 3011 y Fk(0)2590 2930 y Fp(k)p Fw(G)2697 2942 y Fs(j)2732 2930 y Fp(k)2774 2945 y Fs(\016)r(\014)2844 2953 y Fi(j)2874 2945 y Fs(;)p Fv(1)p Fs(=)p Fv(2)p Fs(;\022)3052 2788 y Fm(#)3100 2805 y Fr(\000)p Fv(1)3203 2930 y Fw(;)180 3154 y Fx(where)27 b Fw(C)34 b Fx(is)28 b(a)f(constan)n(t)g(dep)r (ending)h(only)f(on)g(tan)14 b Fw(\016)1868 3166 y Fv(0)1905 3154 y Fx(,)28 b(and)f Fw(\016)s(\014)2204 3166 y Fs(j)2262 3154 y Fx(=)c Fw(\014)g Fp(\000)18 b Fw(\014)2550 3166 y Fs(j)2585 3154 y Fx(,)27 b Fw(\016)s(\014)2722 3166 y Fs(p)2784 3154 y Fx(=)22 b Fp(j)p Fw(\014)h Fp(\000)18 b Fw(\014)3094 3166 y Fs(p)3133 3154 y Fp(j)p Fx(,)27 b(and)h(set)1296 3365 y Fw(g)1336 3377 y Fv(1)1396 3365 y Fx(:=)23 b(min)1659 3272 y Fm(\020)1708 3365 y Fx(\()p Fw(g)1780 3377 y Fv(0)1817 3365 y Fx(\))1849 3330 y Fv(1)p Fs(=\013)1964 3365 y Fw(;)14 b Fx([min\()p Fw(T)2243 3377 y Fv(1)2281 3365 y Fw(;)g(T)2367 3377 y Fv(2)2403 3365 y Fx(\)])2509 3306 y Fk(1)p 2468 3315 109 3 v 2468 3349 a(2+)p Fi(\013)2591 3272 y Fm(\021)2641 3365 y Fw(;)904 b Fx(\(3.10\))180 3594 y(where)27 b Fw(\013)d Fx(=)594 3553 y Fs(\026)p Fr(\000)p Fv(1)p Fs(=)p Fv(2)p 594 3575 193 4 v 594 3622 a Fs(\026)p Fv(+1)p Fs(=)p Fv(2)797 3594 y Fx(,)j(and)h Fw(\026)23 b(>)f Fx(1)p Fw(=)p Fx(2)27 b(is)g(giv)n(en)g(in)h(\(3.4\).)305 3703 y(The)41 b(main)g(results)f (of)i(this)f(pap)r(er)g(are)f(giv)n(en)g(in)i(the)f(follo)n(wing)f (theorems,)k(where)d(b)n(y)g(a)f(\\state")g(on)180 3802 y(a)h(subalgebra)e(\(whic)n(h)j(is)f(not)g(necessarily)f(a)h Fw(C)1810 3772 y Fr(\003)1848 3802 y Fx(-subalgebra\),)i(w)n(e)e(mean)g (a)g(p)r(ositiv)n(e)g(normalized)f(linear)180 3902 y(functional.)180 4085 y Fn(Theorem)30 b(3.1)41 b Fq(Assume)32 b(c)l(onditions)i(\(A\))e ({)i(\(D\))e(ar)l(e)h(ob)l(eye)l(d)i(for)e(some)h Fx(0)28 b Fw(<)g(\014)2833 4097 y Fv(1)2870 4085 y Fw(;)14 b(\014)2954 4097 y Fv(2)3020 4085 y Fw(<)29 b Fp(1)p Fq(,)34 b Fw(\026)29 b(>)f Fx(1)p Fq(,)34 b(and)f(let)180 4184 y Fw(\014)27 b Fx(=)c(max\()p Fw(\014)576 4196 y Fv(1)613 4184 y Fw(;)14 b(\014)697 4196 y Fv(2)734 4184 y Fx(\))p Fq(.)305 4284 y(If)30 b Fx(0)24 b Fw(<)g Fp(j)p Fw(g)s Fp(j)f Fw(<)h(g)788 4296 y Fv(1)855 4284 y Fq(then)30 b(ther)l(e)h(is)f(a)h(stationary)g (state)f Fw(\021)d Fx(=)d Fw(\021)2202 4296 y Fs(\014)2240 4304 y Fk(1)2272 4296 y Fs(\014)2310 4304 y Fk(2)2346 4284 y Fq(,)31 b(de\014ne)l(d)g(on)f(a)h(str)l(ongly)f(dense)h(sub)l (algebr)l(a)180 4384 y Fp(A)246 4396 y Fv(1)313 4384 y Fq(of)g Fp(A)f Fq(\(se)l(e)g(\(2.7\)\),)h(satisfying)1681 4483 y Fw( )1738 4449 y Fs(t)1790 4483 y Fp(!)23 b Fw(\021)s(;)74 b(t)23 b Fp(!)g(1)1289 b Fx(\(3.11\))180 4633 y Fq(for)34 b(any)f Fw(\014)525 4645 y Fv(1)562 4633 y Fw(\014)609 4645 y Fv(2)646 4633 y Fq(-normal)g(initial)h(state)f Fw( )s Fq(.)48 b Fw(\021)36 b Fq(is)d(c)l(ontinuous)f(in)h(the)f(norm)h (of)h Fp(A)p Fq(.)48 b(The)34 b(c)l(onver)l(genc)l(e)f(is)g(in)g(the) 180 4732 y(we)l(ak)p Fp(\003)f Fq(sense)g(\(i.e.,)i(p)l(ointwise)e(for) h(e)l(ach)g Fw(A)27 b Fp(2)f(A)1776 4744 y Fv(1)1814 4732 y Fq(\).)44 b(F)-6 b(or)32 b Fw(A)27 b Fp(2)g(A)2311 4744 y Fv(1)2349 4732 y Fq(,)32 b Fw(\021)s Fx(\()p Fw(A)p Fx(\))h Fq(is)f(analytic)h(in)f Fw(g)s Fq(,)g(and)g Fw(\021)s Fp(j)3450 4744 y Fs(g)r Fv(=0)3600 4732 y Fx(=)26 b Fw(!)3743 4744 y Fv(0)180 4832 y Fq(\(the)k(r)l(efer)l(enc)l(e)g(state)f (\(2.10\))j(wher)l(e)e Fw(!)1438 4844 y Fs(p)1506 4832 y Fq(is)g(the)g(p)l(article)h(Gibbs)g(state)e(at)h(temp)l(er)l(atur)l (e)f Fw(\014)3059 4844 y Fs(p)3097 4832 y Fq(\).)305 5015 y(R)l(emark.)59 b Fx(5\))22 b(Our)f(analysis)f(sho)n(ws)h(that)h (the)g(NESS)f(is)h(actually)f(de\014ned)h(on)f(a)h(bigger)e(\(but)j (somewhat)e(less)180 5114 y(explicit\))27 b(Banac)n(h)e(space)h(of)g (op)r(erators)e Fp(A)1545 5126 y Fv(0)1606 5114 y Fp(\023)e(A)1759 5126 y Fv(1)1823 5114 y Fx(\(see)27 b(\(9.12\)\),)f(and)g(the)h(con)n (v)n(ergence)c(to)k(the)f(NESS,)h(\(3.11\),)180 5214 y(holds)k(on)g Fp(A)586 5226 y Fv(0)624 5214 y Fx(.)48 b(On)31 b Fp(A)903 5226 y Fv(1)972 5214 y Fx(one)g(can)g(in)n(tro)r (duce)g(a)g(\\deformation)f(norm")g Fp(k)20 b(\001)h(k)2649 5226 y Fr(\030)2734 5214 y Fp(\024)29 b(k)20 b(\001)h(k)p Fx(,)32 b(see)f(\(9.10\),)h(suc)n(h)f(that)1959 5476 y(9)p eop %%Page: 10 10 10 9 bop 180 210 a Fx(in)31 b(this)g(norm,)f(the)h(con)n(v)n(ergence)d (in)j(\(3.11\))f(is)h(uniform,)g(sup)2189 230 y Fs(A)p Fr(2A)2337 238 y Fk(1)2373 210 y Fx(\()p Fw( )2462 180 y Fs(t)2492 210 y Fx(\()p Fw(A)p Fx(\))21 b Fp(\000)f Fw(\021)s Fx(\()p Fw(A)p Fx(\)\))p Fw(=)p Fp(k)p Fw(A)p Fp(k)3114 222 y Fr(\030)3198 210 y Fp(!)28 b Fx(0.)46 b(Moreo)n(v)n(er,)180 310 y(on)28 b Fp(A)362 322 y Fv(1)400 310 y Fx(,)h(the)g(con)n(v)n(ergence)d(is)i(exp)r(onen)n(tially)g(fast) h(for)f(initial)h(conditions)f Fw( )j Fx(in)e(a)f(dense)h(set)f(\(in)h (the)g(top)r(ology)180 409 y(of)h(b)r(ounded)h(linear)e(functionals)h (on)g Fp(A)p Fx(\))h({)e(this)i(set)f(is)g(the)h(con)n(v)n(ex)d(h)n (ull)j(of)f(v)n(ector)f(states)g(with)i(deformation)180 509 y(analytic)c(v)n(ectors.)180 692 y Fn(Theorem)j(3.2)41 b Fq(Assume)27 b(that)h(the)g(c)l(onditions)h(of)g(The)l(or)l(em)g(3.1) g(ar)l(e)f(satis\014e)l(d)g(and)h(let)e Fw(\021)3101 704 y Fs(\014)3139 712 y Fk(1)3172 704 y Fs(\014)3210 712 y Fk(2)3274 692 y Fq(and)h Fw(g)j Fq(b)l(e)d(as)g(in)180 791 y(The)l(or)l(em)k(3.1.)45 b(If)32 b Fw(\014)841 803 y Fv(1)904 791 y Fp(6)p Fx(=)26 b Fw(\014)1042 803 y Fv(2)1111 791 y Fq(and)31 b(if)i(either)f Fw(g)i Fq(and)e Fp(j)p Fw(\014)1901 803 y Fv(1)1958 791 y Fp(\000)19 b Fw(\014)2089 803 y Fv(2)2126 791 y Fp(j)32 b Fq(ar)l(e)f (su\016ciently)h(smal)t(l)g(\(indep)l(endently)h(of)f(e)l(ach)180 891 y(other\))e(and)h Fw(G)656 903 y Fv(1)716 891 y Fx(=)23 b Fw(G)869 903 y Fv(2)906 891 y Fq(,)30 b(or)h(if)f Fx(dim)14 b Fp(H)1371 903 y Fs(p)1433 891 y Fx(=)23 b(2)p Fq(,)29 b(then)h Fw(E)5 b(P)12 b Fx(\()p Fw(\021)2006 903 y Fs(\014)2044 911 y Fk(1)2076 903 y Fs(\014)2114 911 y Fk(2)2151 891 y Fx(\))23 b Fw(>)f Fx(0)p Fq(.)305 1074 y Fx(Our)g(analysis)f(giv)n (es)h(a)h(stronger)e(result)h(than)h(the)h(one)e(presen)n(ted)h(in)g (Theorem)f(3.2.)34 b(Namely)-7 b(,)24 b(for)f Fw(\026)g(>)f Fx(3)p Fw(=)p Fx(2,)180 1173 y(w)n(e)28 b(sho)n(w)g(that)h Fw(E)5 b(P)12 b Fx(\()p Fw(\021)895 1185 y Fs(\014)933 1193 y Fk(1)966 1185 y Fs(\014)1004 1193 y Fk(2)1040 1173 y Fx(\))25 b Fw(>)g Fx(0,)k(pro)n(vided)e Fw(o)p Fx(\()p Fw(g)1738 1143 y Fv(0)1776 1173 y Fx(\))p Fw(O)r Fx(\()p Fw(\016)s(\014)t Fx(\))g Fp(\024)d Fw(\021)2188 1143 y Fr(0)2212 1173 y Fx(,)29 b(where)f Fw(\016)s(\014)h Fx(=)c Fp(j)p Fw(\014)2781 1185 y Fv(1)2838 1173 y Fp(\000)18 b Fw(\014)2968 1185 y Fv(2)3006 1173 y Fp(j)28 b Fx(\(see)h(Theorem)f (12.1\).)180 1273 y(Here,)f Fw(\021)443 1243 y Fr(0)494 1273 y Fx(dep)r(ends)i(on)e(the)h(in)n(v)n(erse)e(temp)r(eratures)h (and)g(the)h(coupling)f(functions)h(and)g(is)f(giv)n(en)g(b)n(y)1258 1504 y Fw(\021)1302 1470 y Fr(0)1349 1504 y Fx(=)c(2)p Fw(\031)1542 1425 y Fm(X)1549 1602 y Fs(j)s(>i)1662 1504 y Fx(\()p Fw(\015)1737 1516 y Fs(j)1772 1504 y Fw(e)1811 1470 y Fs(\014)1849 1478 y Fk(1)1881 1470 y Fs(E)1930 1478 y Fi(j)r(i)2006 1504 y Fp(\000)18 b Fw(\015)2132 1516 y Fs(i)2159 1504 y Fx(\))2201 1448 y Fw(E)2262 1460 y Fs(j)s(i)2349 1448 y Fw(g)2389 1460 y Fs(j)s(i)2447 1448 y Fx(\()p Fw(E)2540 1460 y Fs(j)s(i)2599 1448 y Fx(\))2631 1418 y Fv(2)p 2202 1485 468 4 v 2256 1561 a Fw(e)2295 1536 y Fs(\014)2333 1544 y Fk(1)2365 1536 y Fs(E)2414 1544 y Fi(j)r(i)2489 1561 y Fp(\000)g Fx(1)2679 1504 y Fw(;)180 1781 y Fx(where)j Fw(E)475 1793 y Fs(j)s(i)557 1781 y Fx(=)i Fw(E)706 1793 y Fs(j)748 1781 y Fp(\000)7 b Fw(E)881 1793 y Fs(i)908 1781 y Fx(,)23 b Fw(g)994 1793 y Fs(j)s(i)1052 1781 y Fx(\()p Fw(E)5 b Fx(\))1182 1750 y Fv(2)1243 1781 y Fx(=)1331 1714 y Fm(R)1370 1810 y Fl(R)1417 1793 y Fk(3)1462 1781 y Fw(d)1505 1750 y Fv(3)1543 1781 y Fw(k)s Fp(j)14 b(h)p Fw(')1712 1793 y Fs(i)1740 1781 y Fw(;)g(G)1842 1793 y Fv(1)1879 1781 y Fx(\()p Fw(k)s Fx(\))p Fw(')2043 1793 y Fs(j)2079 1781 y Fp(i)g(j)2148 1750 y Fv(2)2185 1781 y Fw(\016)s Fx(\()p Fw(E)2318 1793 y Fs(j)s(i)2384 1781 y Fp(\000)7 b Fw(!)s Fx(\))21 b(\(see)h(Condition)f(\(C\)\).)i(The)f(n)n(um)n(b)r(ers)180 1880 y Fw(\015)223 1892 y Fs(j)281 1880 y Fp(\025)h Fx(0)i(are)g(the)h (co)r(ordinates)e(\(in)j(the)f(basis)f Fp(f)p Fw(')1720 1892 y Fs(j)1770 1880 y Fp(\012)14 b Fw(')1903 1892 y Fs(j)1938 1880 y Fp(g)26 b Fx(of)g(Null\()p Fw(L)2342 1892 y Fs(p)2380 1880 y Fx(\)\))h(of)f(the)g(unique)g(v)n(ector)e Fw(\020)3262 1850 y Fr(\003)3327 1880 y Fx(in)i(the)g(k)n(ernel)180 1980 y(of)34 b(the)g(adjoin)n(t)f(lev)n(el)g(shift)i(op)r(erator)c (\003)1504 1950 y Fr(\003)1504 2000 y Fv(0)1542 1980 y Fx(,)36 b(at)d(the)h(v)-5 b(alue)34 b Fw(\014)2125 1992 y Fs(p)2196 1980 y Fx(=)f(0)g(\(and)h(normalized)f(as)3108 1918 y Fm(P)3195 2005 y Fs(j)3244 1980 y Fw(\015)3287 1992 y Fs(j)3355 1980 y Fx(=)g(1\).)55 b(\(The)180 2079 y(op)r(erator)26 b(\003)573 2091 y Fv(0)637 2079 y Fx(is)i(de\014ned)g (in)g(Section)f(10.\))305 2179 y(By)i(general)g(argumen)n(ts)g(one)h (can)f(pro)n(v)n(e)g(that)h Fw(\021)1895 2149 y Fr(0)1946 2179 y Fp(\025)d Fx(0)i(for)h Fw(\014)2286 2191 y Fv(1)2350 2179 y Fw(>)d(\014)2489 2191 y Fv(2)2526 2179 y Fx(,)k Fw(\021)2624 2149 y Fr(0)2675 2179 y Fp(\024)c Fx(0)i(for)h Fw(\014)3015 2191 y Fv(1)3080 2179 y Fw(<)c(\014)3218 2191 y Fv(2)3256 2179 y Fx(,)k(and)g Fw(\021)3517 2149 y Fr(0)3568 2179 y Fx(=)d(0)j(if)180 2279 y Fw(\014)227 2291 y Fv(1)287 2279 y Fx(=)23 b Fw(\014)422 2291 y Fv(2)459 2279 y Fx(.)36 b(W)-7 b(e)26 b(also)f(sho)n(w)f(that)i Fw(\021)1249 2249 y Fr(0)1296 2279 y Fw(>)c Fx(0)j(for)g Fw(\014)1622 2291 y Fv(1)1682 2279 y Fw(>)e(\014)1817 2291 y Fv(2)1880 2279 y Fx(and)i Fw(\021)2083 2249 y Fr(0)2130 2279 y Fw(<)d Fx(0)j(for)g Fw(\014)2456 2291 y Fv(1)2517 2279 y Fw(<)d(\014)2651 2291 y Fv(2)2688 2279 y Fx(,)27 b(for)d(all)i Fw(\014)3023 2291 y Fv(1)3060 2279 y Fw(;)14 b(\014)3144 2291 y Fv(2)3181 2279 y Fx(,)26 b(except)g(p)r(ossibly)180 2378 y(for)h(\014nitely)h(man)n(y)f(v)-5 b(alues)27 b(in)h(an)n(y)f(compact)g(set,)h(see)f(after)g(\(12.38\).) 305 2478 y(The)e(dep)r(endence)h(of)f Fw(\021)1052 2448 y Fr(0)1101 2478 y Fx(on)g Fw(\016)s(\014)30 b Fx(is)25 b(determined)h(b)n(y)f(the)h(co)r(ordinates)d Fw(\015)2579 2490 y Fs(j)2614 2478 y Fx(.)37 b(W)-7 b(e)25 b(compute)h(those)f(in)g (the)h(cases)180 2578 y(when)g Fw(G)460 2590 y Fv(1)521 2578 y Fx(=)d Fw(G)674 2590 y Fv(2)737 2578 y Fx(and)j(dim)15 b Fp(H)1120 2590 y Fs(p)1181 2578 y Fx(=)23 b(2)j(\(see)g(the)g(pro)r (of)g(of)g(Theorem)f(12.1,)h(and)g(equation)f(\(11.20\),)h(resp)r (ectiv)n(ely\).)305 2677 y Fq(R)l(emarks.)64 b Fx(6\))25 b(Using)g(Araki's)f(theory)h(of)g(p)r(erturbation)g(of)g(KMS)g(states,) h(one)f(sho)n(ws)f(that)i(if)f(the)h(temp)r(er-)180 2777 y(atures)g(of)i(b)r(oth)f(reserv)n(oirs)e(are)h(equal)h(then)h(the)f (limit)h(state)g(is)f(an)g(equilibrium)g(state)g(and)g(has)g(zero)f(en) n(trop)n(y)180 2876 y(pro)r(duction.)37 b(Non-existence)26 b(of)i(equilibrium)g(states)f(for)g Fw(\014)2111 2888 y Fv(1)2171 2876 y Fp(6)p Fx(=)c Fw(\014)2306 2888 y Fv(2)2371 2876 y Fx(has)k(b)r(een)h(sho)n(wn)f(in)g([25].)305 2976 y(7\))f(F)-7 b(or)25 b(a)h(mo)r(del)g(with)h(fermionic)f(reserv)n (oirs,)d(using)j(a)g(su\016cien)n(tly)g(fast)g(con)n(v)n(ergence)e (rate)h(in)i(\(3.11\))e(\(e.g.)180 3076 y Fw(O)r Fx(\()p Fw(t)307 3046 y Fr(\000)p Fs(\013)407 3076 y Fx(\))h(with)g Fw(\013)e(>)e Fx(1)k(su\016ces\))f(and)h(the)g(fact)f(that)h Fw(\021)j Fx(is)d(not)f(a)g(normal)g(state)g(for)g Fw(\014)2853 3088 y Fv(1)2914 3076 y Fp(6)p Fx(=)d Fw(\014)3048 3088 y Fv(1)3086 3076 y Fx(,)k(it)g(has)f(b)r(een)h(sho)n(wn)180 3175 y(b)n(y)31 b(an)f(abstract)g(argumen)n(t)g(that)h Fw(E)5 b(P)12 b Fx(\()p Fw(\021)1506 3187 y Fs(\014)1544 3195 y Fk(1)1576 3187 y Fs(\014)1614 3195 y Fk(2)1650 3175 y Fx(\))29 b Fw(>)f Fx(0,)j(pro)n(vided)f Fp(j)p Fw(\014)2315 3187 y Fv(1)2373 3175 y Fp(\000)20 b Fw(\014)2505 3187 y Fv(2)2542 3175 y Fp(j)29 b(\025)f Fw(C)6 b(g)33 b Fx(for)e(some)f Fw(C)k(>)28 b Fx(0)j(\(see)g([21)o(]\).)180 3275 y(Instead)g(of)f(this)i(indirect)e(deriv)-5 b(ation)31 b(w)n(e)f(compute)h Fw(E)5 b(P)12 b Fx(\()p Fw(\021)2110 3287 y Fs(\014)2148 3295 y Fk(1)2180 3287 y Fs(\014)2218 3295 y Fk(2)2255 3275 y Fx(\))31 b(to)g(the)g(leading)f(order)g(in)h Fw(g)i Fx(and)e(deriv)n(e)f(the)180 3375 y(results)d(stated)h(in)f(the) h(theorem.)305 3474 y(8\))f(The)h(condition)f Fw(G)1006 3486 y Fv(1)1067 3474 y Fx(=)22 b Fw(G)1219 3486 y Fv(2)1284 3474 y Fx(can)28 b(b)r(e)g(relaxed)e(to)i Fw(G)2005 3486 y Fv(1)2061 3474 y Fp(\000)18 b Fw(G)2209 3486 y Fv(2)2274 3474 y Fx(b)r(eing)28 b(small)f(in)h(a)f(suitable)g(sense)305 3574 y(9\))21 b(In)g(this)g(pap)r(er)g(w)n(e)g(pro)n(v)n(e)e(Theorems)h (3.1)g(and)h(3.2)g(under)g(the)g(additional)g(condition)g(that)g(min\() p Fw(T)3500 3586 y Fv(1)3537 3574 y Fw(;)14 b(T)3623 3586 y Fv(2)3660 3574 y Fx(\))23 b Fw(>)180 3687 y(C)6 b(g)288 3657 y Fv(2+)p Fs(\013)419 3687 y Fx(,)32 b(see)e(\(3.10\),)g (with)h Fw(\013)d Fx(=)1252 3646 y Fs(\026)p Fr(\000)p Fv(1)p Fs(=)p Fv(2)p 1252 3668 193 4 v 1252 3715 a Fs(\026)p Fv(+1)p Fs(=)p Fv(2)1455 3687 y Fx(,)j(where)f Fw(\026)e(>)f Fx(1)p Fw(=)p Fx(2)i(is)i(giv)n(en)e(in)i(Condition)f(\(A\).)i(W)-7 b(e)30 b(will)h(remo)n(v)n(e)e(this)180 3800 y(restriction)e(in)g ([26].)180 4075 y Fy(4)135 b(Sp)t(ectral)45 b(Theory)f(of)i(NESS)180 4257 y Fx(In)30 b(this)h(section)f(w)n(e)g(outline)g(a)g(sp)r(ectral)g (theory)f(of)h(NESS)h(applicable)e(to)h(Bose)g(and)g(F)-7 b(ermi)30 b(reserv)n(oirs.)42 b(Our)180 4356 y(approac)n(h)33 b(follo)n(ws)g(the)i(one)f(dev)n(elop)r(ed)g(for)h(the)f(F)-7 b(ermi)35 b(reserv)n(oirs)d(in)i([21].)57 b(Fix)35 b(a)f(state,)j Fw(!)3264 4368 y Fv(0)3300 4356 y Fx(,)g(of)d(the)h(form)180 4456 y(\(2.10\))27 b(with)h Fw(!)661 4468 y Fs(p)699 4456 y Fx(\()p Fw(A)p Fx(\))c(:=)f(T)-7 b(r)o(\()p Fw(e)1116 4426 y Fr(\000)p Fs(\014)1206 4434 y Fi(p)1240 4426 y Fs(H)1294 4434 y Fi(p)1334 4456 y Fw(A)p Fx(\))p Fw(=)p Fx(T)g(r)o(\()p Fw(e)1626 4426 y Fr(\000)p Fs(\014)1716 4434 y Fi(p)1751 4426 y Fs(H)1805 4434 y Fi(p)1844 4456 y Fx(\).)305 4556 y(In)31 b(this)h(and)f(the)h(next)f(section)g(w)n(e)g (use)g(the)h(follo)n(wing)e(conditions)h(on)g(the)h(coupling)f (functions)h(and)f(the)180 4655 y(particle)c(Hamiltonian)1189 4680 y Fm(Z)1286 4793 y Fp(k)p Fw(G)1393 4805 y Fs(j)1427 4793 y Fx(\()p Fw(k)s Fx(\))p Fp(k)1579 4759 y Fv(2)1617 4793 y Fw(e)1656 4759 y Fv(\()p Fs(\014)s Fr(\000)p Fs(\014)1813 4767 y Fi(j)1842 4759 y Fv(\))p Fr(j)p Fs(k)q Fr(j)1949 4793 y Fx([1)18 b(+)g(\()p Fw(\014)2194 4805 y Fs(j)2229 4793 y Fp(j)p Fw(k)s Fp(j)p Fx(\))2353 4759 y Fr(\000)p Fv(1)2442 4793 y Fx(])p Fw(dk)26 b(<)d Fp(1)p Fw(;)838 b Fx(\(4.1\))180 4985 y(where,)27 b(recall,)g Fw(\014)g Fx(=)c(max)o(\()p Fw(\014)1083 4997 y Fv(1)1121 4985 y Fw(;)14 b(\014)1205 4997 y Fv(2)1242 4985 y Fx(\),)28 b(and)1363 5167 y(\()p Fw(H)1464 5179 y Fs(p)1521 5167 y Fx(+)18 b Fw(i)p Fx(\))1665 5133 y Fr(\000)p Fv(1)1782 5167 y Fx(is)27 b(a)h(compact)f(op)r(erator.)1010 b(\(4.2\))1938 5476 y(10)p eop %%Page: 11 11 11 10 bop 180 210 a Fx(\(The)33 b(exp)r(onen)n(tial)f(w)n(eigh)n(t)f (in)i(\(4.1\))f(is)g(used)h(in)f(Section)h(5)e(and)i(could)f(b)r(e)h (omitted)f(in)h(the)g(presen)n(t)e(section)180 310 y(as)c(w)n(ell.\))39 b(These)27 b(conditions)h(are)f(considerably)f(w)n(eak)n(er)g(than)j (Conditions)e(\(A\))i(and)f(\(B\).)h(In)f(particular,)f(the)180 409 y Fw(\022)r Fx(-analyticit)n(y)h(of)i(the)f(coupling)g(functions)h (and)f(the)g(\014niteness)h(of)f(the)h(dimension)f(of)g(the)h(particle) f(space)f(are)180 509 y(not)g(required.)305 609 y(W)-7 b(e)28 b(pass)e(to)i(the)g(Araki-W)-7 b(o)r(o)r(ds)26 b(GNS)i(represen)n(tation)e(for)h(the)h(unp)r(erturb)r(ed)g(system:) 1609 785 y(\()p Fp(A)p Fw(;)14 b(!)1796 797 y Fv(0)1833 785 y Fx(\))23 b Fp(!)h Fx(\()p Fp(H)q Fw(;)14 b(\031)s(;)g Fx(\012)2282 797 y Fv(0)2319 785 y Fx(\))180 961 y(where)27 b Fp(H)q Fx(,)g Fw(\031)j Fx(and)d(\012)839 973 y Fv(0)904 961 y Fx(are)f(a)g(Hilb)r(ert)i(space,)f(a)f(represen)n(tation)g(of)h (the)g(algebra)e Fp(A)j Fx(b)n(y)f(b)r(ounded)g(op)r(erators)e(on)180 1061 y Fp(H)q Fx(,)j(and)f(a)g(cyclic)h(elemen)n(t)f(in)h Fp(H)h Fx(s.t.)1562 1160 y Fw(!)1614 1172 y Fv(0)1651 1160 y Fx(\()p Fw(A)p Fx(\))24 b(=)e Fp(h)q Fx(\012)1981 1172 y Fv(0)2018 1160 y Fw(;)14 b(\031)s Fx(\()p Fw(A)p Fx(\)\012)2291 1172 y Fv(0)2329 1160 y Fp(i)g Fw(:)180 1306 y Fx(There)29 b(is)g(also)f(an)h(an)n(ti-linear)f(represen)n (tation,)g Fw(\031)1811 1276 y Fr(0)1834 1306 y Fx(,)i(of)f(the)h (algebra)d Fp(A)j Fx(in)f(b)r(ounded)h(op)r(erators)d(on)i(the)h(space) 180 1405 y Fp(H)q Fx(,)e(s.t.)37 b Fw(\031)500 1375 y Fr(0)551 1405 y Fx(comm)n(utes)27 b(with)h Fw(\031)j Fx(\(i.e.)38 b([)p Fw(\031)1457 1375 y Fr(0)1480 1405 y Fx(\()p Fw(A)p Fx(\))p Fw(;)14 b(\031)s Fx(\()p Fw(B)t Fx(\)])25 b(=)e(0)k Fp(8)p Fw(A;)14 b(B)26 b Fp(2)e(A)p Fx(\),)p 2492 1333 302 4 v 28 w Fw(\031)2542 1381 y Fr(0)2566 1405 y Fx(\()p Fp(A)p Fx(\)\012)2756 1417 y Fv(0)2817 1405 y Fx(=)e Fp(H)q Fx(.)305 1505 y(The)27 b(full)h(dynamics)g(is)f (implemen)n(ted)h(b)n(y)g(a)f(one-parameter)e(group)i Fw(U)9 b Fx(\()p Fw(t)p Fx(\))28 b(satisfying)1376 1681 y Fw(U)9 b Fx(\()p Fw(t)p Fx(\))p Fw(B)t Fx(\012)23 b(=)g Fw(\033)1824 1647 y Fs(t)1854 1681 y Fx(\()p Fw(B)t Fx(\)\012)p Fw(;)97 b Fp(8)p Fw(B)27 b Fp(2)c Fw(\031)s Fx(\()p Fp(A)p Fx(\))p Fw(;)1026 b Fx(\(4.3\))180 1857 y(where)27 b(\012)h(is)f(a)g (cyclic)h(and)f(separating)f(v)n(ector)g(for)h Fw(\031)s Fx(\()p Fp(A)p Fx(\))i(to)f(b)r(e)g(sp)r(eci\014ed)g(b)r(elo)n(w)f(and) g(where)1537 2053 y Fw(\033)1587 2019 y Fs(t)1617 2053 y Fx(\()p Fw(B)t Fx(\))d(:=)e Fw(e)1921 2019 y Fs(itL)2015 1994 y Fk(\()p Fi(`)p Fk(\))2093 2053 y Fw(B)t(e)2199 2019 y Fr(\000)p Fs(itL)2345 1994 y Fk(\()p Fi(`)p Fk(\))3609 2053 y Fx(\(4.4\))180 2230 y(for)27 b(an)n(y)g Fw(B)g Fp(2)d Fw(\031)s Fx(\()p Fp(A)p Fx(\))k(with)g Fw(L)1087 2199 y Fv(\()p Fs(`)p Fv(\))1194 2230 y Fx(:=)23 b Fw(L)1362 2242 y Fv(0)1417 2230 y Fx(+)18 b Fw(g)s(\031)s Fx(\()p Fw(v)s Fx(\).)37 b(Observ)n(e)26 b(that)i(\(4.3\))g(implies)f(that)1773 2406 y Fw(U)9 b Fx(\()p Fw(t)p Fx(\)\012)23 b(=)g(\012)p Fw(:)1422 b Fx(\(4.5\))180 2582 y(If)29 b(the)g(state)f Fw(!)k Fx(corresp)r(onding)26 b(to)j(\012)f(is)h(stationary)e(\(i.e.)40 b Fw(!)2135 2552 y Fs(t)2188 2582 y Fx(=)24 b Fw(!)s Fx(\))29 b(then)g Fw(U)9 b Fx(\()p Fw(t)p Fx(\))29 b(comes)f(out)g(to)h (b)r(e)g(unitary)-7 b(.)39 b(In)180 2682 y(our)28 b(situation)g(w)n(e)h (exp)r(ect)g(that)g(there)f(is)h(no)f Fw(!)1711 2694 y Fv(0)1748 2682 y Fx(-normal)g(stationary)f(state)h(and)h Fw(U)9 b Fx(\()p Fw(t)p Fx(\))29 b(will)g(b)r(e)g(a)f(non-unitary)180 2781 y(group.)305 2881 y(W)-7 b(e)28 b(pic)n(k)f(the)h(v)n(ector)e (\012)i(as)f(follo)n(ws.)36 b(Let)27 b Fw(\014)h Fx(=)22 b(max)1975 2893 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)2160 2881 y Fw(\014)2207 2893 y Fs(j)2242 2881 y Fx(.)37 b(W)-7 b(e)28 b(de\014ne)1390 3080 y(\012)23 b(:=)g Fw(e)1623 3046 y Fr(\000)p Fs(\014)s(L)1762 3021 y Fk(\()p Fi(`)p Fk(\))1835 3046 y Fs(=)p Fv(2)1906 3080 y Fx(\012)1966 3092 y Fv(0)2003 3080 y Fw(=)p Fp(k)p Fw(e)2126 3046 y Fr(\000)p Fs(\014)s(L)2265 3021 y Fk(\()p Fi(`)p Fk(\))2337 3046 y Fs(=)p Fv(2)2408 3080 y Fx(\012)2468 3092 y Fv(0)2505 3080 y Fp(k)p Fw(:)1039 b Fx(\(4.6\))180 3272 y(The)25 b(facts)g(that)g(\012)g(is)g(w)n(ell)f(de\014ned,)i(i.e.,)g(that)f (\012)1749 3284 y Fv(0)1809 3272 y Fp(2)f Fx(Dom\()p Fw(e)2133 3242 y Fr(\000)p Fs(\014)s(L)2272 3217 y Fk(\()p Fi(`)p Fk(\))2345 3242 y Fs(=)p Fv(2)2416 3272 y Fx(\),)i(and)e(that)i (\012)f(is)f(cyclic)h(and)g(separating,)180 3371 y(are)i(established)g (in)h(Prop)r(osition)e(4.1)g(at)i(the)g(end)g(of)f(this)h(section.)305 3471 y(The)f(family)h Fw(U)9 b Fx(\()p Fw(t)p Fx(\))28 b(is)f(not)h(unitary)f(since)g Fw(!)f Fx(:=)d Fp(h)p Fx(\012)p Fw(;)14 b(\031)s Fx(\()g Fp(\001)g Fx(\)\012)q Fp(i)28 b Fx(is)f(not)h(stationary:)870 3647 y Fp(h)p Fw(U)9 b Fx(\()p Fw(t)p Fx(\))p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)q Fw(;)14 b(U)9 b Fx(\()p Fw(t)p Fx(\))p Fw(\031)s Fx(\()p Fw(B)t Fx(\)\012)r Fp(i)83 b Fx(=)2002 3580 y Fm(\012)2041 3647 y Fx(\012)p Fw(;)14 b(\033)2188 3613 y Fs(t)2217 3647 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)2393 3613 y Fr(\003)2433 3647 y Fw(B)t Fx(\)\)\012)2624 3580 y Fm(\013)2687 3647 y Fx(=)22 b Fw(!)2829 3613 y Fs(t)2858 3647 y Fx(\()p Fw(A)2952 3613 y Fr(\003)2991 3647 y Fw(B)t Fx(\))1854 3772 y Fp(6)p Fx(=)83 b Fw(!)s Fx(\()p Fw(A)2151 3737 y Fr(\003)2189 3772 y Fw(B)t Fx(\))23 b(=)g Fp(h)p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)q Fw(;)14 b(\031)s Fx(\()p Fw(B)t Fx(\)\012)q Fp(i)g Fw(;)593 b Fx(\(4.7\))180 3948 y(for)39 b(some)g Fw(A;)14 b(B)t(;)g(t)p Fx(.)73 b(Let)40 b(no)n(w)f Fw( )k Fx(b)r(e)d(an)f Fw(!)1615 3960 y Fv(0)1652 3948 y Fx(-normal)f(state)i(corresp)r(onding)d(to)j (the)g(v)n(ector)e Fw(Q)p Fx(\012)43 b Fp(2)g(H)e Fx(\(i.e.)180 4048 y Fw( )s Fx(\()p Fw(A)p Fx(\))24 b(=)f Fp(h)p Fw(Q)p Fx(\012)o Fw(;)14 b(\031)s Fx(\()p Fw(A)p Fx(\))p Fw(Q)p Fx(\012)q Fp(i)p Fx(\),)28 b(where)f Fw(Q)c Fp(2)h Fw(\031)1545 4017 y Fr(0)1568 4048 y Fx(\()p Fp(A)p Fx(\))29 b(then)1442 4224 y Fw( )1499 4189 y Fs(t)1528 4224 y Fx(\()p Fw(A)p Fx(\))24 b(=)1766 4156 y Fm(\012)1805 4224 y Fw(Q)p Fx(\012)o Fw(;)14 b(\033)2017 4189 y Fs(t)2047 4224 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\))p Fw(Q)p Fx(\012)2414 4156 y Fm(\013)2495 4224 y Fw(:)1091 b Fx(\(4.8\))180 4400 y(Due)25 b(to)f(Eqns)f(\(2.27\)) h(and)g Fw(\033)1097 4370 y Fs(t)1094 4420 y Fv(0)1131 4400 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\))h(:=)e Fw(\031)s Fx(\()p Fw(\013)1642 4370 y Fs(t)1642 4420 y Fv(0)1680 4400 y Fx(\()p Fw(A)p Fx(\)\))i(w)n(e)f(see)g(that)g(the)h (op)r(erator)d Fw(Q)i Fx(comm)n(utes)g(with)h Fw(\033)3473 4370 y Fs(t)3470 4420 y(n)3515 4400 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\).)180 4499 y(Using)i(this)h(together)f(with)h (\(4.3\))f(and)h(\(4.5\))f(w)n(e)g(arriv)n(e)f(at)1422 4676 y Fw( )1479 4641 y Fs(t)1508 4676 y Fx(\()p Fw(A)p Fx(\))e(=)e Fp(h)q Fw(Q)1844 4641 y Fr(\003)1882 4676 y Fw(Q)p Fx(\012)o Fw(;)14 b(U)9 b Fx(\()p Fw(t)p Fx(\))p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)q Fp(i)42 b Fw(:)1071 b Fx(\(4.9\))180 4852 y(This)25 b(k)n(ey)g(form)n(ula,)g(due)h(to)f ([21)o(],)h(connects)f(the)h(long)f(time)h(b)r(eha)n(viour)e(of)h Fw( )2646 4822 y Fs(t)2675 4852 y Fx(\()p Fw(A)p Fx(\))i(with)f(sp)r (ectral)f(prop)r(erties)f(of)180 4951 y Fw(U)9 b Fx(\()p Fw(t)p Fx(\))28 b(or)f(its)g(generator.)35 b(W)-7 b(e)28 b(explain)f(what)h(this)g(means.)305 5051 y(Assume)f(w)n(e)g(can)h(sho) n(w)e(that,)i(for)f(a)h(certain)e(class)h(of)h Fw(\036)g Fx(and)f(\011,)h(and)f(as)g Fw(t)c Fp(!)g(1)p Fx(,)1580 5227 y Fp(h)q Fw(\036;)14 b(U)9 b Fx(\()p Fw(t)p Fx(\)\011)p Fp(i)23 b(!)g(h)p Fw(\036;)14 b(P)e Fx(\011)p Fp(i)1203 b Fx(\(4.10\))1938 5476 y(11)p eop %%Page: 12 12 12 11 bop 180 210 a Fx(where)23 b Fw(P)36 b Fx(is)24 b(the)g(eigenpro)5 b(jection)23 b(on)h(the)g(\014xed)g(p)r(oin)n(t)g (subspace)f(of)h Fw(U)9 b Fx(\()p Fw(t)p Fx(\))25 b(\(i.e.)36 b Fw(U)9 b Fx(\()p Fw(t)p Fx(\))p Fw(P)35 b Fx(=)22 b Fw(P)12 b(U)d Fx(\()p Fw(t)p Fx(\))24 b(=)e Fw(P)12 b Fx(\),)25 b(whic)n(h)180 310 y(w)n(e)i(assume)g(for)g(a)g(momen)n(t)h (to)f(exist.)37 b(Relations)27 b(\(4.9\))g(and)h(\(4.10\))f(imply)1658 489 y Fw( )1715 454 y Fs(t)1767 489 y Fp(!)c Fw(\021)s(;)97 b(t)23 b Fp(!)g(1)p Fw(;)1266 b Fx(\(4.11\))180 668 y(where)27 b(the)h(state)f Fw(\021)k Fx(is)d(de\014ned)g(\(on)f(an)h(appropriate)e (set)h(of)h(observ)-5 b(ables\))26 b(b)n(y)1493 846 y Fw(\021)s Fx(\()p Fw(A)p Fx(\))e(:=)f Fp(h)p Fw(Q)1896 812 y Fr(\003)1934 846 y Fw(Q)p Fx(\012)o Fw(;)14 b(P)e(\031)s Fx(\()p Fw(A)p Fx(\)\012)q Fp(i)i Fw(:)1101 b Fx(\(4.12\))180 1025 y(W)-7 b(e)33 b(will)f(sho)n(w)g(b)r(elo)n(w)g(that)g(\(4.10\))g (holds)g(for)f(some)h(un)n(b)r(ounded)h(pro)5 b(jection)31 b(op)r(erator)g Fw(P)12 b Fx(.)50 b(T)-7 b(o)32 b(understand)180 1125 y(the)c(structure)f(of)h(this)g(op)r(erator,)d(w)n(e)j(pro)r(ceed) f(as)g(follo)n(ws.)305 1225 y(W)-7 b(e)25 b(will)g(sho)n(w)g(that)g Fw(U)9 b Fx(\()p Fw(t)p Fx(\))26 b(is)f(strongly)e(di\013eren)n(tiable) i(on)g(a)g(dense)g(set)g(and)g(w)n(e)g(will)g(compute)h(its)f (non-self-)180 1324 y(adjoin)n(t)i(generator,)f Fw(K)i Fx(:=)23 b Fp(\000)p Fw(i)1180 1291 y Fs(@)p 1167 1305 65 4 v 1167 1353 a(@)t(t)1241 1324 y Fp(j)1264 1336 y Fs(t)p Fv(=0)1378 1324 y Fw(U)9 b Fx(\()p Fw(t)p Fx(\),)28 b(whic)n(h)f(satis\014es)g(\(see)g(\(4.5\)\))1824 1503 y Fw(K)6 b Fx(\012)23 b(=)f(0)p Fw(:)1432 b Fx(\(4.13\))305 1682 y(The)25 b(op)r(erator)f Fw(P)38 b Fx(is)25 b(the)i(eigenpro)5 b(jection)24 b(on)n(to)h(the)h(eigenspace)e(of)i Fw(K)31 b Fx(asso)r(ciated)24 b(with)j(the)f(eigen)n(v)-5 b(alue)24 b(0)180 1781 y(\(i.e.)37 b Fw(K)6 b(P)35 b Fx(=)22 b Fw(P)12 b(K)29 b Fx(=)22 b(0\).)37 b(W)-7 b(e)28 b(sho)n(w)f(that)h (dim)14 b Fw(P)35 b Fx(=)22 b(1)27 b(and)1751 1960 y Fw(P)35 b Fx(=)22 b Fp(j)q Fx(\012)p Fp(i)14 b(h)p Fx(\012)2148 1926 y Fr(\003)2186 1960 y Fp(j)1359 b Fx(\(4.14\))180 2139 y(for)26 b(some)h(\012)574 2109 y Fr(\003)644 2139 y Fw(=)-51 b Fp(2)23 b(H)q Fx(,)k(satisfying)f Fw(K)1280 2109 y Fr(\003)1318 2139 y Fx(\012)1378 2109 y Fr(\003)1439 2139 y Fx(=)d(0)j(in)h(a)g(w)n(eak)f(sense)g(\(\012)2271 2109 y Fr(\003)2332 2139 y Fp(2)e(D)2477 2109 y Fr(0)2475 2163 y Fv(anal)2601 2139 y Fx(,)j(where)f Fp(D)2954 2151 y Fv(anal)3103 2139 y Fx(=)d Fp([)3246 2151 y Fv(Im)p Fs(\022)r(>)p Fv(0)3447 2139 y Fx(Dom\()p Fw(U)3710 2151 y Fs(\022)3748 2139 y Fx(\))180 2239 y(with)28 b(the)g(family)f Fw(U)822 2251 y Fs(\022)887 2239 y Fx(de\014ned)h(in)g(Section)f(6\).) 37 b(Understanding)27 b(the)h(nature)f(of)g(the)h(v)n(ector)e(\012)3207 2209 y Fr(\003)3245 2239 y Fx(,)i(whic)n(h)f(w)n(e)g(call)180 2338 y(the)h Fq(NESS)h(ve)l(ctor)p Fx(,)f(is)g(a)f(goal)f(of)i(our)e (analysis.)305 2438 y(Substituting)i(\(4.14\))f(in)n(to)g(\(4.12\))g (and)h(using)f(that)h Fp(h)p Fw(Q)2085 2408 y Fr(\003)2123 2438 y Fw(Q)p Fx(\012)o Fw(;)14 b Fx(\012)p Fp(i)24 b Fx(=)e Fp(k)p Fw(Q)p Fx(\012)p Fp(k)2697 2396 y Fv(2)2757 2438 y Fx(=)h Fw( )s Fx(\()p Fn(1)p Fx(\))g(=)g(1,)k(w)n(e)g(obtain) 1602 2617 y Fw(\021)s Fx(\()p Fw(A)p Fx(\))d(=)f Fp(h)p Fx(\012)1976 2583 y Fr(\003)2014 2617 y Fw(;)14 b(\031)s Fx(\()p Fw(A)p Fx(\)\012)q Fp(i)h Fw(:)1210 b Fx(\(4.15\))305 2796 y(Since)27 b(\012)581 2766 y Fr(\003)652 2796 y Fw(=)-52 b Fp(2)24 b(H)k Fx(the)g(state)f Fw(\021)k Fx(is)c(not)h (normal)e(but)i(it)g(is)g(w)n(ell)f(de\014ned)h(for)f(a)g(dense)g(set)g (of)h(observ)-5 b(ables.)35 b(The)180 2895 y(question)27 b(no)n(w)f(is)h(what)g(is)f(\012)1114 2865 y Fr(\003)1153 2895 y Fx(?)36 b(The)27 b(answ)n(er,)f(pro)n(vided)g(in)h(subsequen)n (t)g(sections,)f(is)h(that)g(\012)3212 2865 y Fr(\003)3277 2895 y Fx(is)g(a)g(resonance)180 2995 y(of)h Fw(K)352 2965 y Fr(\003)389 2995 y Fx(.)305 3095 y(In)38 b(the)h(follo)n(wing)e (sections)g(w)n(e)h(construct)g(a)g(mathematical)g(framew)n(ork)e(whic) n(h)i(pro)n(vides)f(meaningful)180 3194 y(expressions)29 b(replacing)h(formal)g(ones,)h(\(4.10\){\(4.15\))e(and)i(with)g(the)g (help)h(of)e(whic)n(h)h(w)n(e)g(can)f(pro)n(v)n(e)g(the)h(con-)180 3294 y(v)n(ergence)26 b(\(4.11\).)305 3394 y Fq(R)l(emark.)35 b Fx(Ev)n(olution)19 b(groups)g(and)i(their)f(generators)f(giv)n(en)g (b)n(y)i(conditions)f(of)h(the)g(t)n(yp)r(e)g(of)f(\(4.5\))h(\(or)f (\(4.13\)\))180 3493 y(w)n(ere)i(in)n(tro)r(duced)g(in)h([21)o(],)g (where)f(the)h(group)e Fw(U)9 b Fx(\()p Fw(t)p Fx(\))23 b(is)g(sp)r(eci\014ed)f(b)n(y)h(the)g(condition)f Fw(U)9 b Fx(\()p Fw(t)p Fx(\)\012)3066 3505 y Fv(0)3126 3493 y Fx(=)23 b(\012)3274 3505 y Fv(0)3311 3493 y Fx(.)35 b(where)22 b(\012)3664 3505 y Fv(0)3724 3493 y Fx(is)180 3593 y(the)g(unp)r(erturb)r(ed)g(v)n(ector)e(\(\\v)-5 b(acuum"\))21 b(in)n(tro)r(duced)g(in)g(\(2.37\))g(ab)r(o)n(v)n(e.)33 b(Ho)n(w)n(ev)n(er,)21 b(an)g(analysis)f(of)h(the)h(op)r(erator)180 3692 y Fw(K)30 b Fx(\(see)25 b(Sections)f(6)h(and)f(7\))h(de\014ned)g (this)g(w)n(a)n(y)f(requires,)g(instead)h(of)f(Condition)h(\(A\),)h (the)f(condition)f(obtained)180 3792 y(from)k(Condition)g(\(A\))h(b)n (y)g(replacing)e(the)h(w)n(eigh)n(t)g Fw(e)1839 3762 y Fs(\016)r(\014)1909 3770 y Fi(j)1940 3762 y Fr(j)p Fs(u)p Fr(j)p Fs(=)p Fv(2)2118 3792 y Fx(b)n(y)g Fw(e)2273 3762 y Fs(\014)2311 3770 y Fi(j)2341 3762 y Fr(j)p Fs(u)p Fr(j)p Fs(=)p Fv(2)2520 3792 y Fx(\(and)g(correp)r(ondingly)-7 b(,)27 b Fw(e)3338 3762 y Fv(\()p Fs(\014)s Fr(\000)p Fs(\014)3495 3770 y Fi(j)3525 3762 y Fv(\))p Fr(j)p Fs(k)q Fr(j)3660 3792 y Fx(has)180 3892 y(to)g(b)r(e)g(replaced)f(b)n(y)h Fw(e)874 3862 y Fs(\014)912 3870 y Fi(j)942 3862 y Fr(j)p Fs(k)q Fr(j)1050 3892 y Fx(in)g(\(4.1\)\).)36 b(This)27 b(leads)g(to)f(an)h(additional)g(restriction)e(on)i(the)g(temp)r 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Fr(\000)p Fs(\014)s(L)2658 4615 y Fk(\()p Fi(`)p Fk(\))2731 4640 y Fs(=)p Fv(2)2802 4670 y Fx(\012)2862 4682 y Fv(0)2899 4670 y Fw(=)p Fp(k)p Fw(e)3022 4640 y Fr(\000)p Fs(\014)s(L)3161 4615 y Fk(\()p Fi(`)p Fk(\))3233 4640 y Fs(=)p Fv(2)3304 4670 y Fx(\012)3364 4682 y Fv(0)3401 4670 y Fp(k)g Fq(is)g(cyclic)180 4770 y(and)30 b(sep)l(ar)l(ating)h (for)f(the)g(von)g(Neumann)f(algebr)l(a)i Fw(\031)s Fx(\()p Fp(A)p Fx(\))2001 4740 y Fr(00)2045 4770 y Fq(.)305 4949 y Fx(W)-7 b(e)32 b(b)r(egin)g(with)h(with)g(some)e(preliminary)h(tec)n (hnical)f(results.)51 b(T)-7 b(o)31 b(do)h(manipulations)g(with)h(un)n (b)r(ounded)180 5048 y(op)r(erators)26 b(w)n(e)h(use)g(the)h(dense)g (subset,)f Fp(D)r Fx(,)h(of)g(our)f(Hilb)r(ert)h(space,)f Fp(H)q Fx(,)h(de\014ned)g(b)n(y)1729 5227 y Fp(D)e Fx(:=)c Fw(\031)s Fx(\()2040 5206 y(~)2011 5227 y Fp(A)q Fx(\)\012)2170 5239 y Fv(0)2208 5227 y Fw(;)1337 b Fx(\(4.17\))1938 5476 y(12)p eop %%Page: 13 13 13 12 bop 180 210 a Fx(where)457 189 y(~)429 210 y Fp(A)38 b Fx(=)f Fp(B)s Fx(\()p Fp(H)795 222 y Fs(p)833 210 y Fx(\))25 b Fp(\012)e(P)1036 222 y Fv(1)1098 210 y Fp(\012)h(P)1245 222 y Fv(2)1282 210 y Fx(.)63 b(Here)36 b Fp(P)1631 222 y Fs(j)1702 210 y Fx(is)g(the)h(p)r(olynomial)f(algebra)e(generated)i (b)n(y)g(the)g(annihilation)180 310 y(and)e(creation)g(op)r(erators,)g Fw(a)1117 322 y Fs(r)r(j)1184 310 y Fx(\()p Fw(f)9 b Fx(\))35 b(and)g Fw(a)1546 280 y Fr(\003)1546 331 y Fs(r)r(j)1613 310 y Fx(\()p Fw(f)9 b Fx(\),)36 b(of)f(the)g Fw(j)5 b Fx(-th)34 b(reserv)n(oir)e(acting)i(on)g Fp(H)3010 322 y Fs(r)r(j)3113 310 y Fx(with)h Fw(f)43 b Fp(2)35 b Fw(C)3548 280 y Fr(1)3542 330 y Fv(0)3618 310 y Fx(\()p Fo(R)3704 280 y Fv(3)3748 310 y Fx(\))180 409 y(satisfying)27 b Fw(f)9 b Fx(\(0\))23 b(=)f(0)27 b(and)h Fp(r)p Fw(f)9 b Fx(\(0\))23 b(=)f(0.)305 509 y(Let)45 b(\001)h(and)g Fw(J)54 b Fx(b)r(e)46 b(the)g(mo)r(dular)f(op)r(erator)f(and)i(mo)r (dular)f(conjugation)g(asso)r(ciated)f(with)j(the)f(pair)180 609 y Fp(f)p Fw(\031)s Fx(\()p Fp(A)p 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Fs(\014)s(L)2799 1780 y Fk(\()p Fi(`)p Fk(\))2872 1805 y Fs(=)p Fv(2)2943 1835 y Fw(e)2982 1805 y Fs(\014)s(L)3069 1813 y Fk(0)3100 1805 y Fs(=)p Fv(2)3200 1835 y Fx(whic)n(h)28 b(is)g(de\014ned)180 1934 y(and)33 b(closable)g(on)g Fp(D)j Fx(\(see)e(the)f(pro)r(of)g(of)h (Lemma)f(4.2)g(b)r(elo)n(w\).)55 b(W)-7 b(e)34 b(denote)f(the)h (closure)e(of)i(\000)f(b)n(y)h(the)g(same)180 2034 y(sym)n(b)r(ol.)i (The)28 b(follo)n(wing)f(results,)g(whic)n(h)g(w)n(e)g(pro)n(v)n(e)f (at)i(the)g(end)g(of)f(this)h(section,)f(are)g(useful.)180 2200 y Fn(Lemma)j(4.2)40 b Fq(L)l(et)29 b Fx(\000)877 2170 y Fv(#)966 2200 y Fq(stand)g(for)i(either)g Fx(\000)1610 2170 y Fr(\003)1677 2200 y Fq(or)f Fx(\000)p Fq(.)39 b(The)31 b(fol)t(lowing)h(statements)d(hold:)1603 2383 y Fw(\031)s Fx(\()p Fp(A)p Fx(\))1783 2348 y Fr(0)1808 2383 y Fp(D)c(\032)e Fx(Dom\(\000)2243 2348 y Fv(#)2302 2383 y Fx(\))p Fw(;)1211 b Fx(\(4.21\))1439 2565 y(\000)p Fw(B)t Fx(\012)1618 2577 y Fv(0)1678 2565 y Fx(=)23 b Fw(B)t Fx(\000\012)1945 2577 y Fv(0)1982 2565 y Fw(;)44 b Fp(8)p Fw(B)26 b Fp(2)e Fw(\031)s Fx(\()p Fp(A)p Fx(\))2444 2531 y Fr(0)2498 2565 y Fw(;)1047 b Fx(\(4.22\))1285 2715 y Fw(J)8 b(A)p Fx(\000\012)1513 2727 y Fv(0)1573 2715 y Fx(=)23 b(\001)1730 2680 y Fv(1)p Fs(=)p Fv(2)1835 2715 y Fx(\000)1887 2680 y Fr(\003)1925 2715 y Fw(A)1987 2680 y Fr(\003)2025 2715 y Fx(\012)2085 2727 y Fv(0)2122 2715 y Fw(;)44 b Fp(8)p Fw(A)22 b Fp(2)i Fw(\031)s Fx(\()p Fp(A)p Fx(\))2579 2680 y Fr(00)2652 2715 y Fw(;)893 b Fx(\(4.23\))1686 2864 y(Null)q(\(\000)1925 2830 y Fv(#)1983 2864 y Fx(\))24 b(=)e Fp(f)p Fx(0)p Fp(g)p Fw(:)1293 b Fx(\(4.24\))305 3030 y Fq(R)l(emark.)67 b Fw(\031)s Fx(\()p Fp(A)p Fx(\))855 3000 y Fr(0)907 3030 y Fx(is)27 b(the)h(v)n(on)f(Neumann)h(algebra)e Fw(\031)2007 3000 y Fr(0)2030 3030 y Fx(\()p Fp(A)p Fx(\))2160 3000 y Fr(00)2204 3030 y Fx(,)h(the)h(w)n(eak)f(closure)f(of)i Fw(\031)3027 3000 y Fr(0)3051 3030 y Fx(\()p Fp(A)p Fx(\).)305 3146 y Fq(Pr)l(o)l(of)33 b(of)g(Pr)l(op)l(osition)g(4.1.)91 b Fx(Since)30 b Fw(e)1556 3115 y Fs(\034)7 b(L)1640 3123 y Fk(0)1675 3146 y Fx(\012)1735 3158 y Fv(0)1800 3146 y Fx(=)27 b(\012)1952 3158 y Fv(0)1989 3146 y Fx(,)k(the)f(prop)r(ert)n (y)f(\012)2590 3158 y Fv(0)2654 3146 y Fp(2)f Fx(Dom\()p Fw(e)2982 3115 y Fr(\000)p Fs(\014)s(L)3121 3090 y Fk(\()p Fi(`)p Fk(\))3194 3115 y Fs(=)p Fv(2)3265 3146 y Fx(\))i(is)g(equiv)-5 b(alen)n(t)180 3245 y(to)27 b(\012)341 3257 y Fv(0)402 3245 y Fp(2)c Fx(Dom\(\000\),)28 b(whic)n(h)g(is)f(pro)n(v)n(en)g(in)g (Lemma)h(4.2,)f(relation)f(\(4.21\).)36 b(Hence)28 b(the)g(v)n(ector)e (\012)i(exists.)305 3345 y(F)-7 b(or)32 b(the)h(cyclicit)n(y)g(it)h(is) f(enough)f(to)h(sho)n(w)f(that)h(if)h Fw(B)i Fp(2)d Fw(\031)s Fx(\()p Fp(A)p Fx(\))2359 3315 y Fr(0)2417 3345 y Fx(and)f Fw(B)t Fx(\012)h(=)e(0)i(then)h Fw(B)i Fx(=)c(0.)53 b(Let)33 b(\012)3637 3315 y Fr(0)3692 3345 y Fx(:=)180 3460 y Fw(e)219 3430 y Fr(\000)p Fs(\014)s(L)358 3405 y Fk(\()p Fi(`)p Fk(\))431 3430 y Fs(=)p Fv(2)502 3460 y Fx(\012)562 3472 y Fv(0)599 3460 y Fx(.)k(By)25 b(\(4.23\))h(and)g(\(4.24\),)g Fw(B)t Fx(\012)1575 3472 y Fv(0)1635 3460 y Fp(2)d Fx(Dom\(\000\))k (and)f(0)d(=)f Fw(B)t Fx(\012)2469 3430 y Fr(0)2516 3460 y Fx(=)g Fw(B)t Fx(\000\012)2782 3472 y Fv(0)2843 3460 y Fx(=)g(\000)p Fw(B)t Fx(\012)3109 3472 y Fv(0)3147 3460 y Fx(.)36 b(Since)26 b(Null)q(\(\000\))d(=)180 3560 y Fp(f)p Fx(0)p Fp(g)p Fx(,)j(this)i(implies)g Fw(B)t Fx(\012)926 3572 y Fv(0)986 3560 y Fx(=)23 b(0.)36 b(Since)28 b(\012)1452 3572 y Fv(0)1517 3560 y Fx(is)g(separating)e(for)h Fw(\031)s Fx(\()p Fp(A)p Fx(\))2310 3530 y Fr(0)2362 3560 y Fx(w)n(e)g(ha)n(v)n(e)f(that)i Fw(B)g Fx(=)22 b(0.)305 3659 y(No)n(w)j(w)n(e)g(sho)n(w)f(that)i(\012)f(is)h (separating)e(for)h Fw(\031)s Fx(\()p Fp(A)p Fx(\))1865 3629 y Fr(00)1908 3659 y Fx(.)37 b(Let)25 b Fw(A)e Fp(2)h Fw(\031)s Fx(\()p Fp(A)p Fx(\))2458 3629 y Fr(00)2527 3659 y Fx(b)r(e)i(suc)n(h)f(that)h Fw(A)p Fx(\012)d(=)g(0.)35 b(The)26 b(relation)180 3759 y(\000\012)292 3771 y Fv(0)352 3759 y Fx(=)d(\012)500 3729 y Fr(0)551 3759 y Fx(and)k(equation)g (\(4.23\))g(imply)h(that)1314 3967 y(0)22 b(=)h Fp(h)p Fw(J)8 b(A)p Fx(\000\012)1726 3979 y Fv(0)1763 3967 y Fw(;)14 b( )s Fp(i)24 b Fx(=)2000 3874 y Fm(D)2051 3967 y Fw(A)2113 3932 y Fr(\003)2151 3967 y Fx(\012)2211 3979 y Fv(0)2276 3967 y Fw(;)14 b Fx(\000\001)2434 3932 y Fv(1)p Fs(=)p Fv(2)2539 3967 y Fw( )2596 3874 y Fm(E)3568 3967 y Fx(\(4.25\))180 4186 y(for)25 b(an)n(y)g Fw( )h Fp(2)e(D)r Fx(.)36 b(Since)26 b(\001)1028 4156 y Fv(1)p Fs(=)p Fv(2)1133 4186 y Fp(D)f Fx(=)e Fp(D)28 b Fx(and)e(\000)p Fp(D)i Fx(is)d(dense)h(\(see)f(at)h(the)g(end)g(of)g(the)g(pro)r(of)f (of)h(Lemma)f(4.2)g(b)r(elo)n(w\))180 4285 y(w)n(e)h(ha)n(v)n(e)e(that) j(\000\001)790 4255 y Fv(1)p Fs(=)p Fv(2)894 4285 y Fp(D)h Fx(is)e(dense)g(and)g(it)g(follo)n(ws)f(from)h(\(4.25\))f(that)h Fw(A)2478 4255 y Fr(\003)2516 4285 y Fx(\012)2576 4297 y Fv(0)2637 4285 y Fx(=)c(0.)36 b(Since)26 b(\012)3100 4297 y Fv(0)3164 4285 y Fx(is)f(separating)g(this)180 4385 y(implies)j(that)g Fw(A)23 b Fx(=)g(0)k(and)g(therefore)g(\012)h (is)f(separating.)1751 b Fg(\004)305 4485 y Fq(Pr)l(o)l(of)31 b(of)g(L)l(emma)g(4.2.)40 b Fx(W)-7 b(e)29 b(consider)e(only)h(the)h (op)r(erator)d(\000;)j(the)g(op)r(erator)d(\000)2860 4454 y Fr(\003)2927 4485 y Fx(is)i(treated)g(similarly)-7 b(.)38 b(W)-7 b(e)180 4584 y(appro)n(ximate)25 b(the)i(op)r(erator)f Fw(v)k Fx(b)n(y)c(UV-cut-o\013)i(op)r(erators)c Fw(v)2139 4596 y Fs(n)2208 4584 y Fp(2)f(A)28 b Fx(b)n(y)e(replacing)g(the)h (coupling)g(functions)g Fw(G)3745 4596 y Fs(j)180 4703 y Fx(b)n(y)34 b(the)g(coupling)f(functions)i Fw(e)1193 4673 y Fr(\000j)p Fs(k)q Fr(j)1321 4648 y Fk(2)1353 4673 y Fs(=n)1432 4703 y Fw(G)1497 4715 y Fs(j)1566 4703 y Fx(in)f(\(2.14\).)55 b(Let)34 b Fw(L)2172 4660 y Fv(\()p Fs(`)p Fv(\))2172 4713 y Fs(n)2289 4703 y Fx(:=)f Fw(L)2467 4715 y Fv(0)2527 4703 y Fx(+)22 b Fw(V)2662 4715 y Fs(n)2708 4703 y Fx(,)35 b(where)f Fw(V)3061 4715 y Fs(n)3140 4703 y Fx(:=)f Fw(\031)s Fx(\()p Fw(v)3383 4715 y Fs(n)3429 4703 y Fx(\),)j(and)e(let)180 4821 y(\000)232 4833 y Fs(n)300 4821 y Fx(:=)23 b Fw(e)450 4791 y Fr(\000)p Fs(\014)s(L)589 4766 y Fk(\()p Fi(`)p Fk(\))589 4808 y Fi(n)661 4791 y Fs(=)p Fv(2)733 4821 y Fw(e)772 4791 y Fs(\014)s(L)859 4799 y Fk(0)890 4791 y Fs(=)p Fv(2)961 4821 y Fx(.)37 b(Note)28 b(that)g(the)g Fw(V)1593 4833 y Fs(n)1666 4821 y Fx(are)f Fw(\033)1852 4833 y Fv(0)1889 4821 y Fx(-en)n(tire.)36 b(W)-7 b(e)28 b(claim)g(that)1712 5004 y Fp(D)d(\032)e Fx(Dom\(\000)2147 5016 y Fs(n)2193 5004 y Fx(\))p Fw(;)1320 b Fx(\(4.26\))1331 5187 y(\000)1383 5199 y Fs(n)1428 5187 y Fw(')55 b Fx(con)n(v)n(erges)25 b(as)i Fw(n)c Fp(!)g(1)p Fw(;)42 b Fp(8)p Fw(')22 b Fp(2)i(D)r Fw(:)939 b Fx(\(4.27\))1938 5476 y(13)p eop %%Page: 14 14 14 13 bop 180 210 a Fx(Indeed,)33 b(due)g(to)f(the)g(UV-cut-o\013,)i (the)e(transformation)e Fw(\033)2102 180 y Fs(i\034)2099 231 y Fv(0)2168 210 y Fx(\()p Fw(V)2248 222 y Fs(n)2294 210 y Fx(\))h(=)f Fw(e)2491 180 y Fr(\000)p Fs(\034)7 b(L)2627 188 y Fk(0)2662 210 y Fw(V)2710 222 y Fs(n)2755 210 y Fw(e)2794 180 y Fs(\034)g(L)2878 188 y Fk(0)2945 210 y Fx(is)32 b(w)n(ell)g(de\014ned.)51 b(Hence,)180 310 y(the)28 b(Dyson)f(series)g(for)g(\000)980 322 y Fs(n)1025 310 y Fw(')h Fx(is)f(w)n(ell)h(de\014ned:)815 553 y(\000)867 565 y Fs(n)912 553 y Fw(')23 b Fx(=)1089 474 y Fm(X)1077 653 y Fs(m)p Fr(\025)p Fv(0)1221 553 y Fx(\()p Fp(\000)p Fw(g)s Fx(\))1393 519 y Fs(m)1469 440 y Fm(Z)1552 461 y Fs(\014)s(=)p Fv(2)1516 629 y(0)1678 553 y Fw(d\034)1757 565 y Fv(1)1809 553 y Fp(\001)14 b(\001)g(\001)1919 440 y Fm(Z)2002 461 y Fs(\034)2033 469 y Fi(m)p Fh(\000)p Fk(1)1966 629 y Fv(0)2179 553 y Fw(d\034)2258 565 y Fs(m)2322 553 y Fw(\033)2372 516 y Fs(i\034)2426 524 y Fi(m)2369 575 y Fv(0)2486 553 y Fx(\()p Fw(V)2566 565 y Fs(n)2612 553 y Fx(\))g Fp(\001)g(\001)g(\001)g Fw(\033)2819 516 y Fs(i\034)2873 524 y Fk(1)2816 575 y Fv(0)2910 553 y Fx(\()p Fw(V)2990 565 y Fs(n)3036 553 y Fx(\))p Fw(';)423 b Fx(\(4.28\))180 821 y(where)38 b(the)g(in)n(tegrands)f(of)h(the)h(r.h.s.)68 b(are)37 b(in)h Fp(H)h Fx(and)f(are)g(strongly)e(con)n(tin)n(uous)h(in)i Fw(\034)3095 833 y Fv(1)3132 821 y Fw(;)14 b(:)g(:)g(:)52 b(;)14 b(\034)3391 833 y Fs(m)3493 821 y Fx(and)38 b(the)180 920 y(in)n(tegrals)26 b(are)h(understo)r(o)r(d)g(as)g(strong)f(limits)i (of)g(Riemann)g(sums.)305 1020 y(Since)23 b(the)h Fw(V)704 1032 y Fs(n)773 1020 y Fx(are)e(b)r(ounded,)j(the)f(series)e(ab)r(o)n (v)n(e)g(con)n(v)n(erges.)33 b(Moreo)n(v)n(er,)21 b(it)j(con)n(v)n (erges)d(uniformly)i(in)h Fw(n)f(>)f Fx(0.)180 1120 y(This)e(can)f(b)r (e)h(seen)f(using)g(a)h(standard)e(argumen)n(t)h(to)g(estimate)h(the)g (norms)f(of)g(the)h(in)n(tegrands)f(\(to)g(k)n(eep)h(notation)180 1219 y(simple)28 b(w)n(e)f(set)h Fw(')23 b Fx(=)g(\012)916 1231 y Fv(0)953 1219 y Fx(\),)416 1402 y Fp(k)g Fw(\033)531 1365 y Fs(i\034)585 1373 y Fi(m)528 1424 y Fv(0)645 1402 y Fx(\()p Fw(V)725 1414 y Fs(n)771 1402 y Fx(\))14 b Fp(\001)g(\001)g(\001)f Fw(\033)977 1365 y Fs(i\034)1031 1373 y Fk(1)974 1424 y Fv(0)1068 1402 y Fx(\()p Fw(V)1148 1414 y Fs(n)1194 1402 y Fx(\)\012)1286 1414 y Fv(0)1347 1402 y Fp(k)p Fx(=)22 b Fw(!)1528 1414 y Fv(0)1565 1402 y Fx(\()p Fw(\013)1650 1365 y Fr(\000)p Fs(i\034)1756 1373 y Fk(1)1650 1424 y Fv(0)1793 1402 y Fx(\()p Fw(v)1865 1414 y Fs(n)1911 1402 y Fx(\))14 b Fp(\001)g(\001)g(\001)g Fw(\013)2121 1365 y Fr(\000)p Fs(i\034)2227 1373 y Fi(m)2121 1424 y Fv(0)2286 1402 y Fx(\()p Fw(v)2358 1414 y Fs(n)2404 1402 y Fx(\))p Fw(\013)2489 1365 y Fs(i\034)2543 1373 y Fi(m)2489 1424 y Fv(0)2603 1402 y Fx(\()p Fw(v)2675 1414 y Fs(n)2721 1402 y Fx(\))g Fp(\001)g(\001)g(\001)g Fw(\013)2931 1365 y Fs(i\034)2985 1373 y Fk(1)2931 1424 y Fv(0)3022 1402 y Fx(\()p Fw(v)3094 1414 y Fs(n)3140 1402 y Fx(\)\))3204 1368 y Fv(1)p Fs(=)p Fv(2)3309 1402 y Fw(;)236 b Fx(\(4.29\))180 1585 y(b)n(y)30 b(using)f(Wic)n(k's)h (theorem,)g(the)g(expression)f(for)g(the)i(imaginary-time)d(t)n(w)n (o-p)r(oin)n(t)h(functions)h(and)g(condition)180 1684 y(\(4.1\),)d(in)h(the)g(same)f(w)n(a)n(y)g(as)g(it)h(is)f(done,)g(for)h (instance,)f(in)h([8)o(],)g(Thm)g(IV.3.)305 1784 y(A)k(similar)g (argumen)n(t)f(sho)n(ws)h(that)h Fp(8)p Fw(')d Fp(2)i(D)r Fw(;)47 b Fp(f)p Fx(\000)1930 1796 y Fs(n)1974 1784 y Fw(')p Fp(g)33 b Fx(is)f(a)g(Cauc)n(h)n(y)f(sequence)h(as)g Fw(n)f Fp(!)h Fx(0)g(and)g(therefore)180 1883 y(\000)232 1895 y Fs(n)277 1883 y Fw(')c Fx(con)n(v)n(erges)d(as)i Fw(n)c Fp(!)g(1)p Fx(,)28 b(whic)n(h)f(completes)h(the)g(pro)r(of)f(of) g(\(4.26\))g(and)g(\(4.27\).)305 1999 y(No)n(w)33 b(w)n(e)h(sho)n(w)f (that)h(\012)1087 2011 y Fv(0)1158 1999 y Fp(2)g Fx(Dom\(\000\))g (\(and)g(\012)1831 2011 y Fv(0)1902 1999 y Fp(2)g Fx(Dom\()p Fw(e)2236 1969 y Fr(\000)p Fs(\014)s(L)2375 1944 y Fk(\()p Fi(`)p Fk(\))2448 1969 y Fs(=)p Fv(2)2519 1999 y Fx(\)\).)57 b(The)34 b(set)g Fp(D)i Fx(giv)n(en)d(in)h(\(4.17\))g(is)f(a)180 2115 y(core)e(for)g(the)h(selfadjoin)n(t)g(op)r(erators)e Fw(L)1471 2071 y Fv(\()p Fs(`)p Fv(\))1471 2124 y Fs(n)1586 2115 y Fx(and)i Fw(L)1809 2084 y Fv(\()p Fs(`)p Fv(\))1893 2115 y Fx(.)49 b(Since)33 b Fw(L)2244 2071 y Fv(\()p Fs(`)p Fv(\))2244 2124 y Fs(n)2357 2115 y Fp(!)d Fw(L)2527 2084 y Fv(\()p Fs(`)p Fv(\))2643 2115 y Fx(on)h Fp(D)r Fx(,)j(the)e(con)n(v)n(ergence,)f(in)h(fact,)180 2214 y(is)h(in)g(the)g(strong)e(resolv)n(en)n(t)g(sense)i(\(see)f(Theorem)g (VI)r(I)r(I.25)h(of)g([28)o(]\).)53 b(T)-7 b(ak)n(e)31 b Fw(')h Fp(2)g(H)i Fx(s.t.)53 b Fw(')32 b Fx(=)f Fw(f)9 b Fx(\()p Fw(L)3478 2184 y Fv(\()p Fs(`)p Fv(\))3561 2214 y Fx(\))p Fw(')34 b Fx(for)180 2330 y(some)28 b Fw(f)34 b Fp(2)26 b Fw(C)610 2300 y Fr(1)604 2351 y Fv(0)680 2330 y Fx(\()p Fo(R)q Fx(\).)47 b(Suc)n(h)29 b Fw(')p Fx('s)g(form)g(a)f(core)g(for)g(the)i(op)r(erator)d Fw(e)2303 2300 y Fr(\000)p Fs(\014)s(L)2442 2275 y Fk(\()p Fi(`)p Fk(\))2514 2300 y Fs(=)p Fv(2)2586 2330 y Fx(.)41 b(Since)29 b Fw(L)2925 2287 y Fv(\()p Fs(`)p Fv(\))2925 2340 y Fs(n)3033 2330 y Fp(!)c Fw(L)3198 2300 y Fv(\()p Fs(`)p Fv(\))3311 2330 y Fx(in)k(the)g(strong)180 2430 y(resolv)n(en)n(t)d(sense,)h(w)n (e)g(ha)n(v)n(e)655 2540 y Fm(D)706 2633 y Fw(e)745 2598 y Fr(\000)p Fs(\014)s(L)884 2573 y Fk(\()p Fi(`)p Fk(\))957 2598 y Fs(=)p Fv(2)1028 2633 y Fw(';)14 b Fx(\012)1179 2645 y Fv(0)1216 2540 y Fm(E)1290 2633 y Fx(=)35 b(lim)1378 2684 y Fs(n)p Fr(!)p Fv(0)1532 2540 y Fm(D)1583 2633 y Fw(e)1622 2598 y Fr(\000)p Fs(\014)s(L)1761 2573 y Fk(\()p Fi(`)p Fk(\))1761 2615 y Fi(n)1833 2598 y Fs(=)p Fv(2)1905 2633 y Fw(f)9 b Fx(\()p Fw(L)2044 2598 y Fv(\()p Fs(`)p Fv(\))2044 2653 y Fs(n)2127 2633 y Fx(\))p Fw(';)14 b Fx(\012)2310 2645 y Fv(0)2348 2540 y Fm(E)2421 2633 y Fx(=)36 b(lim)2509 2684 y Fs(n)p Fr(!)p Fv(0)2663 2540 y Fm(D)2714 2633 y Fw(f)9 b Fx(\()p Fw(L)2853 2598 y Fv(\()p Fs(`)p Fv(\))2853 2653 y Fs(n)2936 2633 y Fx(\))p Fw(')q(;)14 b Fx(\000)3112 2645 y Fs(n)3157 2633 y Fx(\012)3217 2645 y Fv(0)3254 2540 y Fm(E)180 2879 y Fx(and)22 b(therefore,)g(b)n(y) f(\(4.27\),)1070 2787 y Fm(D)1121 2879 y Fw(e)1160 2849 y Fr(\000)p Fs(\014)s(L)1299 2824 y Fk(\()p Fi(`)p Fk(\))1372 2849 y Fs(=)p Fv(2)1443 2879 y Fw(';)14 b Fx(\012)1594 2891 y Fv(0)1631 2787 y Fm(E)1705 2879 y Fx(=)23 b Fp(h)p Fw(';)14 b(\037)p Fp(i)p Fx(,)23 b(where)f Fw(\037)h Fx(:=)f(lim)2582 2891 y Fs(n)p Fr(!)p Fv(0)2740 2879 y Fx(\000)2792 2891 y Fs(n)2837 2879 y Fx(\012)2897 2891 y Fv(0)2934 2879 y Fw(:)g Fx(Since)g Fw(')g Fx(v)-5 b(aries)21 b(through)180 3028 y(a)30 b(core)g(for)g(the)h(op)r(erator)e Fw(e)1084 2997 y Fr(\000)p Fs(\014)s(L)1223 2972 y Fk(\()p Fi(`)p Fk(\))1296 2997 y Fs(=)p Fv(2)1398 3028 y Fx(w)n(e)h(ha)n(v)n(e) f(that)i(\012)1960 3040 y Fv(0)2026 3028 y Fp(2)d Fx(Dom\()p Fw(e)2354 2997 y Fr(\000)p Fs(\014)s(L)2493 2972 y Fk(\()p Fi(`)p Fk(\))2566 2997 y Fs(=)p Fv(2)2638 3028 y Fx(\))j(\(and)f Fw(e)2936 2997 y Fr(\000)p Fs(\014)s(L)3075 2972 y Fk(\()p Fi(`)p Fk(\))3148 2997 y Fs(=)p Fv(2)3219 3028 y Fx(\012)3279 3040 y Fv(0)3345 3028 y Fx(=)d Fw(\037)p Fx(\).)47 b(Since)180 3127 y Fw(e)219 3097 y Fs(\034)7 b(L)303 3105 y Fk(0)338 3127 y Fx(\012)398 3139 y Fv(0)458 3127 y Fx(=)23 b(\012)606 3139 y Fv(0)643 3127 y Fx(,)j(w)n(e)e(conclude)g(that)h(\012)1386 3139 y Fv(0)1446 3127 y Fp(2)f Fx(Dom\(\000\).)36 b(Similarly)24 b(one)h(sho)n(ws)e(that)i Fw(')e Fp(2)h Fx(Dom\(\000\))h(for)f(an)n(y)g Fw(')f Fp(2)h(D)r Fx(,)180 3227 y(i.e.)37 b Fp(D)25 b(\032)e Fx(Dom\(\000\).)305 3326 y(Note)k(that)h Fp(D)d(\032)e Fx(Dom\(\000\),)28 b(\(4.27\),)f(and)h(similar)f(statemen)n(ts)g(for)g (\000)2498 3296 y Fr(\003)2498 3347 y Fs(n)2571 3326 y Fx(and)g(\000)2784 3296 y Fr(\003)2822 3326 y Fx(,)h(imply)g(that) 1365 3509 y(\000)1417 3475 y Fv(#)1417 3530 y Fs(n)1476 3509 y Fw(')23 b Fp(!)g Fx(\000)1711 3475 y Fv(#)1770 3509 y Fw(';)42 b Fx(as)27 b Fw(n)c Fp(!)g(1)p Fx(,)k Fp(8)p Fw(')c Fp(2)g(D)r Fw(:)974 b Fx(\(4.30\))305 3692 y(Since)24 b Fw(\033)568 3662 y Fs(i\034)565 3712 y Fv(0)633 3692 y Fx(\()p Fw(V)713 3704 y Fs(n)759 3692 y Fx(\))f(=)g Fw(\031)s Fx(\()p Fw(\013)1037 3662 y Fs(i\034)1037 3712 y Fv(0)1103 3692 y Fx(\()p Fw(v)1175 3704 y Fs(n)1221 3692 y Fx(\)\))h(w)n(e)g(ha)n(v)n(e)e Fw(\033)1665 3655 y Fs(i\034)1719 3663 y Fi(m)1662 3714 y Fv(0)1779 3692 y Fx(\()p Fw(V)1859 3704 y Fs(n)1905 3692 y Fx(\))14 b Fp(\001)g(\001)g(\001)g Fw(\033)2112 3655 y Fs(i\034)2166 3663 y Fk(1)2109 3714 y Fv(0)2203 3692 y Fx(\()p Fw(V)2283 3704 y Fs(n)2329 3692 y Fx(\))23 b(=)g Fw(\031)s Fx(\()p Fw(\013)2607 3655 y Fs(i\034)2661 3663 y Fi(m)2607 3714 y Fv(0)2721 3692 y Fx(\()p Fw(v)2793 3704 y Fs(n)2839 3692 y Fx(\))14 b Fp(\001)g(\001)g(\001)g Fw(\013)3049 3655 y Fs(i\034)3103 3663 y Fk(1)3049 3714 y Fv(0)3140 3692 y Fx(\()p Fw(v)3212 3704 y Fs(n)3258 3692 y Fx(\)\).)36 b(T)-7 b(runcating)180 3791 y(the)28 b(sum)g(on)f(the)h(r.h.s.)36 b(of)28 b(\(4.28\),)f(and)g(appro)n(ximating)f(the)i(in)n(tegrals)e(b)n (y)h(Riemann)h(sums,)g(one)f(sho)n(ws)f(that)1122 3974 y(\000)1174 3986 y Fs(n)1219 3974 y Fw(')d Fx(=)49 b(lim)1384 4028 y Fs(k)q Fr(!1)1567 3974 y Fx(\000)1619 3986 y Fs(n;k)1720 3974 y Fw(')56 b Fx(with)f(\000)2098 3986 y Fs(n;k)2223 3974 y Fp(2)23 b Fw(\031)s Fx(\()p Fp(A)p Fx(\))p Fw(;)43 b Fp(8)p Fw(')23 b Fp(2)g(D)r Fw(:)730 b Fx(\(4.31\))305 4190 y(Equations)26 b(\(4.30\))h(and)g(\(4.31\))g(imply)h(that)g(for)f (an)n(y)g Fw(';)14 b( )26 b Fp(2)d(D)30 b Fx(and)e(for)f(an)n(y)g Fw(B)g Fp(2)c Fw(\031)s Fx(\()p Fp(A)p Fx(\))3171 4160 y Fr(0)761 4373 y Fp(h)p Fw(B)t( )s(;)14 b Fx(\000)1006 4339 y Fr(\003)1044 4373 y Fw(')p Fp(i)24 b Fx(=)79 b(lim)1241 4427 y Fs(n;k)q Fr(!1)1485 4306 y Fm(\012)1524 4373 y Fw(B)t( )s(;)14 b Fx(\000)1737 4339 y Fr(\003)1737 4393 y Fs(n;k)1839 4373 y Fw(')1893 4306 y Fm(\013)1955 4373 y Fx(=)80 b(lim)2043 4427 y Fs(n;k)q Fr(!1)2287 4373 y Fp(h)p Fw(B)t Fx(\000)2438 4385 y Fs(n;k)2539 4373 y Fw( )t(;)14 b(')p Fp(i)23 b Fx(=)g Fp(h)p Fw(B)t Fx(\000)p Fw( )s(;)14 b(')p Fp(i)g Fw(:)180 4600 y Fx(Hence)33 b(b)n(y)f(an)g(elemen)n(tary)g(abstract)f(argumen)n(t)h Fw(B)t( )i Fp(2)e Fx(Dom\(\000\))h(and)g Fw(B)t Fx(\000)p Fw( )h Fx(=)d(\000)p Fw(B)t( )s Fx(,)j Fp(8)e Fw(B)j Fp(2)d Fw(\031)s Fx(\()p Fp(A)p Fx(\))3519 4570 y Fr(0)3543 4600 y Fx(.)52 b(This)180 4700 y(completes)27 b(the)h(pro)r(of)f(of)h (\(4.21\))f(and)g(sho)n(ws)g(\(4.22\))f(as)h(w)n(ell.)305 4800 y(T)-7 b(o)33 b(pro)n(v)n(e)f(Eqn)h(\(4.23\))h(w)n(e)f(use)h(Eqns) f(\(4.30\))g(and)g(\(4.31\))g(and)h(the)g(de\014ning)g(prop)r(ert)n(y)f (of)h(the)g(op)r(erator)180 4899 y Fw(S)28 b Fx(=)22 b(\001)415 4869 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)572 4899 y Fw(J)36 b Fx(to)27 b(obtain)g Fp(8)p Fw(A)c Fp(2)g Fw(\031)s Fx(\()p Fp(A)p Fx(\))1403 4869 y Fr(00)1474 4899 y Fx(and)28 b Fp(8)p Fw( )d Fp(2)e(D)j Fx(=)d(\001)2087 4869 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2243 4899 y Fp(D)408 5107 y(h)p Fw(J)8 b(A)p Fx(\000\012)668 5119 y Fv(0)705 5107 y Fw(;)14 b( )s Fp(i)24 b Fx(=)80 b(lim)942 5161 y Fs(n;k)q Fr(!1)1186 5107 y Fp(h)p Fw(J)8 b(A)p Fx(\000)1386 5119 y Fs(n;k)1488 5107 y Fx(\012)1548 5119 y Fv(0)1585 5107 y Fw(;)14 b( )s Fp(i)24 b Fx(=)80 b(lim)1822 5161 y Fs(n;k)q Fr(!1)2066 5015 y Fm(D)2117 5107 y Fx(\000)2169 5073 y Fr(\003)2169 5127 y Fs(n;k)2270 5107 y Fw(A)2332 5073 y Fr(\003)2371 5107 y Fx(\012)2431 5119 y Fv(0)2468 5107 y Fw(;)14 b Fx(\001)2574 5073 y Fv(1)p Fs(=)p Fv(2)2678 5107 y Fw( )2735 5015 y Fm(E)2809 5107 y Fx(=)2897 5015 y Fm(D)2948 5107 y Fw(A)3010 5073 y Fr(\003)3048 5107 y Fx(\012)3108 5119 y Fv(0)3145 5107 y Fw(;)g Fx(\000\001)3303 5073 y Fv(1)p Fs(=)p Fv(2)3407 5107 y Fw( )3465 5015 y Fm(E)3529 5107 y Fw(:)1938 5476 y Fx(14)p eop %%Page: 15 15 15 14 bop 305 210 a Fx(Finally)-7 b(,)29 b(to)f(pro)n(v)n(e)g (\(4.24\),)g(it)h(su\016ces)g(to)g(sho)n(w)f(that)h(\000)2108 180 y Fv(#)2166 210 y Fp(D)j Fx(is)c(dense.)41 b(Recall)28 b(that)i Fw(e)3081 180 y Fs(\034)7 b(L)3165 188 y Fk(0)3200 210 y Fp(D)27 b Fx(=)e Fp(D)r Fx(,)30 b Fp(8)p Fw(\034)j Fp(2)26 b Fo(R)p Fx(.)180 335 y(Let)k Fw( )j Fx(b)r(e)d(suc)n(h)g(that) g(0)c(=)1064 267 y Fm(\012)1103 335 y Fw( )s(;)14 b Fx(\000)p Fw(e)1288 305 y Fr(\000)p Fs(\014)s(L)1427 313 y Fk(0)1458 305 y Fs(=)p Fv(2)1530 335 y Fw(')1584 267 y Fm(\013)1650 335 y Fx(=)1741 243 y Fm(D)1792 335 y Fw( )s(;)g(e)1925 305 y Fr(\000)p Fs(\014)s(L)2064 280 y Fk(\()p Fi(`)p Fk(\))2137 305 y Fs(=)p Fv(2)2208 335 y Fw(')2262 243 y Fm(E)2313 335 y Fx(,)30 b Fp(8)p Fw(')d Fp(2)g(D)r Fx(.)44 b(Since)30 b Fp(D)i Fx(is)d(dense,)i(this)f(implies)180 483 y(that)24 b Fw( )j Fp(2)c Fx(Dom\()p Fw(e)760 453 y Fr(\000)p Fs(\014)s(L)899 428 y Fk(\()p Fi(`)p Fk(\))972 453 y Fs(=)p Fv(2)1043 483 y Fx(\))i(whic)n(h,)g(in)f(turn,)h(yields)f (that)g Fw( )j Fx(=)22 b(0.)36 b(Hence)24 b(\000)p Fp(D)i Fx(is)e(dense.)36 b(Similarly)-7 b(,)24 b(one)g(sho)n(ws)180 583 y(that)k(\000)412 553 y Fr(\003)450 583 y Fp(D)i Fx(is)d(dense)h(as)f(w)n(ell.)2595 b Fg(\004)305 683 y Fq(R)l(emark.)75 b Fx(Another)30 b(w)n(a)n(y)e(to)i(demonstrate)f (the)h(prop)r(erties)e(of)i(the)g(op)r(erators)e(\000)2964 652 y Fv(#)3052 683 y Fx(listed)i(in)g(Lemma)g(4.2)180 793 y(is)h(to)g(use)f(the)i(analyticit)n(y)e(of)h(the)g(op)r(erator)e (family)i(\000\()p Fw(z)t Fx(\))e(:=)f Fw(e)2263 763 y Fr(\000)p Fs(z)r(L)2395 738 y Fk(\()p Fi(`)p Fk(\))2472 793 y Fw(e)2511 763 y Fs(z)r(L)2591 771 y Fk(0)2627 793 y Fx(,)k(for)e Fp(j)p Fx(Re)p Fw(z)t Fp(j)e Fw(<)g(\014)t Fx(.)48 b(The)31 b(adv)-5 b(an)n(tage)180 893 y(of)26 b(this)g(approac)n(h)e(is)i(that)h(\000\()p Fw(it)p Fx(\),)f Fw(t)d Fp(2)h Fo(R)p Fx(,)32 b(are)25 b(b)r(ounded)i(op)r(erators)d (and)i(man)n(y)f(of)h(the)h(relations)d(ab)r(o)n(v)n(e)h(can)h(b)r(e) 180 993 y(easily)h(v)n(eri\014ed)g(b)n(y)g(using)g(e.g.)37 b(the)28 b(Kato-T)-7 b(rotter)25 b(form)n(ula.)180 1265 y Fy(5)135 b(Generator)46 b Ff(K)53 b Fy(and)44 b(in)l(terp)t(olating)j (family)f Ff(K)2748 1286 y Fx(\()p Fw(s)p Fx(\))180 1446 y(In)24 b(this)g(section)f(w)n(e)h(\014nd)g(an)f(explicit)h(form)g(and) f(some)g(prop)r(erties)g(of)h(the)g(generator)d Fw(K)30 b Fx(of)23 b(the)h(one-parameter)180 1546 y(group)j Fw(U)9 b Fx(\()p Fw(t)p Fx(\))29 b(in)n(tro)r(duced)f(in)h(the)g(preceding)e (section)h(\(cf.)h([21)o(]\),)g(and)g(of)f(the)h(family)f Fw(K)3001 1561 y Fv(\()p Fs(s)p Fv(\))3117 1546 y Fx(whic)n(h)g(in)n (terp)r(olates)180 1646 y Fw(K)33 b Fx(to)28 b(a)f(selfadjoin)n(t)g(op) r(erator.)305 1745 y(Let)g Fw(!)505 1757 y Fv(0)570 1745 y Fx(b)r(e)h(the)g(state)f(of)h(the)f(algebra)f Fp(A)i Fx(\014xed)g(at)f(the)h(b)r(eginning)g(of)f(the)h(Section)f(4)h(and)f (let)h Fw(J)35 b Fx(and)28 b(\001)g(b)r(e)180 1845 y(the)c(T)-7 b(omita-T)g(ak)n(esaki)21 b(mo)r(dular)i(conjugation)g(and)g(mo)r (dular)g(op)r(erator)f(asso)r(ciated)g(with)i(the)g(couple)f(\()p Fp(A)p Fw(;)14 b(!)3687 1857 y Fv(0)3725 1845 y Fx(\).)180 1945 y(W)-7 b(e)28 b(ha)n(v)n(e)e(the)i(follo)n(wing)f(standard)g (relations:)1670 2113 y Fw(J)8 b(\031)s Fx(\()p Fw(A)p Fx(\))p Fw(J)33 b Fx(=)22 b Fw(\031)2116 2078 y Fr(0)2140 2113 y Fx(\()p Fw(A)p Fx(\))p Fw(;)1320 b Fx(\(5.1\))180 2281 y Fw(J)8 b Fx(\001)303 2251 y Fv(1)p Fs(=)p Fv(2)408 2281 y Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)644 2293 y Fv(0)705 2281 y Fx(=)23 b Fw(\031)s Fx(\()p Fw(A)937 2251 y Fr(\003)976 2281 y Fx(\)\012)1068 2293 y Fv(0)1105 2281 y Fx(,)e(\001)1218 2251 y Fv(1)p Fs(=)p Fv(2)1323 2281 y Fx(\012)1383 2293 y Fv(0)1443 2281 y Fx(=)i(\012)1591 2293 y Fv(0)1648 2281 y Fx(and)c(\001)1870 2251 y Fr(\000)p Fs(it)1975 2281 y Fw(\031)s Fx(\()p Fw(A)p Fx(\)\001)2220 2251 y Fs(it)2297 2281 y Fx(=)k Fw(\031)s Fx(\()p Fw(\024)2515 2251 y Fs(t)2544 2281 y Fx(\()p Fw(A)p Fx(\)\),)g(where)c Fw(\024)h Fx(is)f(the)h(automorphism)180 2381 y(of)28 b(the)g(algebra)d Fp(A)j Fx(de\014ned)g(in)g(\(4.18\).)36 b(The)28 b(last)f(three)h(equations)e(imply)1489 2549 y Fw(J)8 b(\031)s Fx(\()p Fw(A)p Fx(\)\012)1779 2561 y Fv(0)1841 2549 y Fx(=)22 b Fw(\031)s Fx(\()p Fw(\024)2058 2514 y Fs(i=)p Fv(2)2153 2549 y Fx(\()p Fw(A)2247 2514 y Fr(\003)2286 2549 y Fx(\)\)\012)2410 2561 y Fv(0)2448 2549 y Fw(:)1138 b Fx(\(5.2\))305 2717 y(Finally)-7 b(,)41 b(w)n(e)d(recall)g(that)g Fw(\014)46 b Fx(=)41 b(max)o(\()p Fw(\014)1605 2729 y Fv(1)1643 2717 y Fw(;)14 b(\014)1727 2729 y Fv(2)1764 2717 y Fx(\))39 b(and)f(that)h Fw(L)2255 2687 y Fv(\()p Fs(`)p Fv(\))2377 2717 y Fx(is)f(the)h(self-adjoin)n(t)f (op)r(erator)f(de\014ned)i(as)180 2816 y Fw(L)237 2786 y Fv(\()p Fs(`)p Fv(\))343 2816 y Fx(:=)23 b Fw(L)511 2828 y Fv(0)566 2816 y Fx(+)18 b Fw(g)s(V)46 b Fx(where)28 b Fw(V)41 b Fx(:=)23 b Fw(\031)s Fx(\()p Fw(v)s Fx(\).)180 2985 y Fn(Theorem)30 b(5.1)41 b Fq(Assume)35 b(that)h(\(4.1\))h(and)f (\(4.2\))h(hold.)58 b(The)37 b(semigr)l(oup)g Fw(U)9 b Fx(\()p Fw(t)p Fx(\))p Fq(,)38 b(de\014ne)l(d)e(in)g (\(4.3\){\(4.5\),)41 b(is)180 3109 y(di\013er)l(entiable)33 b(on)e(the)h(domain)g Fx(Dom\()p Fw(L)1498 3079 y Fv(\()p Fs(`)p Fv(\))1582 3109 y Fx(\))20 b Fp(\\)g Fw(\031)s Fx(\()p Fp(A)p Fx(\)\012)p Fq(,)33 b(and)f(the)f(gener)l(ator)h Fw(K)f Fx(=)26 b Fp(\000)p Fw(i)2988 3076 y Fs(@)p 2976 3090 65 4 v 2976 3138 a(@)t(t)3050 3014 y Fm(\014)3050 3063 y(\014)3050 3113 y(\014)3077 3167 y Fs(t)p Fv(=0)3191 3109 y Fw(U)9 b Fx(\()p Fw(t)p Fx(\))31 b Fq(is)h(given)g(on)180 3233 y(this)e(domain)h(by)f(the)g(expr)l(ession)1542 3333 y Fw(K)f Fx(=)22 b Fw(L)1786 3345 y Fv(0)1841 3333 y Fx(+)c Fw(g)s Fx(\()p Fw(V)38 b Fp(\000)18 b Fw(V)2234 3299 y Fr(0)2216 3355 y(\000)p Fs(i=)p Fv(2)2363 3333 y Fx(\))p Fw(;)1191 b Fx(\(5.3\))180 3500 y Fq(wher)l(e)30 b Fw(V)42 b Fx(=)22 b Fw(\031)s Fx(\()p Fw(v)s Fx(\))31 b Fq(and)e Fw(V)1006 3469 y Fr(0)987 3520 y Fs(s)1053 3500 y Fx(=)22 b Fw(\031)1190 3469 y Fr(0)1214 3500 y Fx(\()p Fw(\015)p 1294 3436 36 3 v 1294 3469 a Fs(s)1329 3500 y Fx(\()p Fw(v)s Fx(\)\))30 b Fq(with)g Fw(\015)1726 3469 y Fs(s)1784 3500 y Fx(:=)23 b Fw(\013)1948 3460 y Fr(\000)p Fs(\014)s(s)1948 3522 y Fv(0)2093 3500 y Fp(\016)17 b Fw(\024)2200 3469 y Fs(s)2265 3500 y Fq(\(due)29 b(to)g(c)l(ondition)h(\(4.1\))g(the)g(op)l(er)l(ator)g Fw(V)3652 3469 y Fr(0)3633 3526 y(\000)p Fs(i=)p Fv(2)180 3623 y Fq(is)g(wel)t(l)h(de\014ne)l(d\).)39 b(F)-6 b(urthermor)l(e,)30 b(the)g(domain)h Fx(Dom\()p Fw(L)1990 3593 y Fv(\()p Fs(`)p Fv(\))2074 3623 y Fx(\))18 b Fp(\\)h Fw(\031)s Fx(\()p Fp(A)p Fx(\)\012)31 b Fq(is)f(dense)g(in)g Fp(H)q Fq(.)305 3791 y(R)l(emark.)61 b Fx(The)36 b(imaginary)e(part)i(of)f (the)i(generator)c Fw(K)42 b Fx(is)35 b(not)h(semi-b)r(ounded.)61 b(Therefore,)37 b(the)f(group)180 3891 y Fw(U)9 b Fx(\()p Fw(t)p Fx(\),)28 b(densely)f(de\014ned)h(on)g Fw(\031)s Fx(\()p Fp(A)p Fx(\)\012,)h(do)r(es)e(not)g(extend)h(to)g(a)f(group)f (of)i(b)r(ounded)g(op)r(erators.)305 4006 y Fq(Pr)l(o)l(of)33 b(of)g(The)l(or)l(em)g(5.1.)78 b Fx(The)30 b(argumen)n(t)f(whic)n(h)h (w)n(as)f(used)h(to)g(pro)n(v)n(e)f(that)h(\012)2903 4018 y Fv(0)2968 4006 y Fp(2)d Fx(Dom)q(\()p Fw(e)3296 3976 y Fr(\000)p Fs(\014)s(L)3435 3951 y Fk(\()p Fi(`)p Fk(\))3507 3976 y Fs(=)p Fv(2)3579 4006 y Fx(\))j(\(see)180 4121 y(the)f(pro)r(of)g(of)g(Prop)r(osition)e(4.1)h(and)h(Lemma)g (4.2\))f(also)g(sho)n(ws)g(that)h Fw(L)2496 4091 y Fv(\()p Fs(`)p Fv(\))2580 4121 y Fx(\012)2640 4133 y Fv(0)2702 4121 y Fx(=)c Fw(g)s(\031)s Fx(\()p Fw(v)s Fx(\)\012)3052 4133 y Fv(0)3116 4121 y Fp(2)g Fx(Dom\()p Fw(e)3441 4091 y Fr(\000)p Fs(\014)s(L)3580 4066 y Fk(\()p Fi(`)p Fk(\))3653 4091 y Fs(=)p Fv(2)3725 4121 y Fx(\).)180 4237 y(The)k(latter)f(fact)h (implies)g(that)g Fw(e)1247 4207 y Fs(itL)1341 4182 y Fk(\()p Fi(`)p Fk(\))1415 4207 y Fs(=)p Fv(2)1486 4237 y Fx(\012)g(is)f(di\013eren)n(tiable)h(at)f Fw(t)d Fx(=)f(0)29 b(and)f(therefore)g(\012)d Fp(2)g Fx(Dom\()p Fw(L)3416 4207 y Fv(\()p Fs(`)p Fv(\))3500 4237 y Fx(\).)41 b(No)n(w,)180 4336 y(let)29 b Fw(B)f Fp(2)d Fw(\031)s Fx(\()p Fp(A)p Fx(\))30 b(b)r(e)e(suc)n(h)g(that)h Fw(B)t Fx(\012)c Fp(2)f Fx(Dom)q(\()p Fw(L)1659 4306 y Fv(\()p Fs(`)p Fv(\))1742 4336 y Fx(\).)40 b(T)-7 b(aking)27 b(in)n(to)h(accoun)n(t)g (equation)g(\(4.4\))g(w)n(e)g(see)g(that)h Fw(\033)3619 4306 y Fs(t)3648 4336 y Fx(\()p Fw(B)t Fx(\))180 4436 y(and)c(therefore,)f(due)i(to)e(\(4.3\),)i(also)d Fw(U)9 b Fx(\()p Fw(t)p Fx(\))p Fw(B)t Fx(\012,)26 b(is)f(di\013eren)n(tiable) g(in)g Fw(t)p Fx(.)36 b(Di\013eren)n(tiating)25 b(equation)f(\(4.3\))h (w)n(e)g(\014nd)1493 4604 y Fw(K)6 b(B)t Fx(\012)23 b(=)f Fw(L)1864 4570 y Fv(\()p Fs(`)p Fv(\))1948 4604 y Fw(B)t Fx(\012)c Fp(\000)g Fw(B)t(L)2300 4570 y Fv(\()p Fs(`)p Fv(\))2384 4604 y Fx(\012)p Fw(:)1142 b Fx(\(5.4\))180 4772 y(No)n(w)32 b(w)n(e)h(compute)g(the)g(last)g(term)g(on)f(the)h (r.h.s.)53 b(of)32 b(this)i(expression.)51 b(T)-7 b(o)32 b(this)i(end)f(w)n(e)f(use)h(the)g(follo)n(wing)180 4872 y(relations:)1516 4972 y Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)1752 4984 y Fv(0)1813 4972 y Fx(=)23 b Fw(\031)1951 4937 y Fr(0)1975 4972 y Fx(\()p Fw(\024)2055 4937 y Fs(i=)p Fv(2)2150 4972 y Fx(\()p Fw(A)2244 4937 y Fr(\003)2282 4972 y Fx(\)\)\012)2406 4984 y Fv(0)3609 4972 y Fx(\(5.5\))180 5112 y(and,)k(for)h Fw(z)e Fx(=)d Fw(it)p Fx(,)1401 5227 y Fw(e)1440 5193 y Fs(z)r(L)1520 5168 y Fk(\()p Fi(`)p Fk(\))1597 5227 y Fw(\031)1647 5193 y Fr(0)1670 5227 y Fx(\()p Fw(A)p Fx(\))p Fw(e)1835 5193 y Fr(\000)p Fs(z)r(L)1967 5168 y Fk(\()p Fi(`)p Fk(\))2069 5227 y Fx(=)f Fw(\031)2206 5193 y Fr(0)2230 5227 y Fx(\()p Fw(\013)2315 5193 y Fs(i)p 2338 5159 35 3 v(z)2315 5248 y Fv(0)2377 5227 y Fx(\()p Fw(A)p Fx(\)\))p Fw(:)1051 b Fx(\(5.6\))1938 5476 y(15)p eop %%Page: 16 16 16 15 bop 180 210 a Fx(Eqn)27 b(\(5.5\))g(follo)n(ws)g(from)g (relations)f(\(5.1\))i(and)f(\(5.2\).)37 b(Equalit)n(y)26 b(\(5.6\))i(is)f(pro)n(v)n(en)f(b)n(y)h(using)g(the)h(Kato-T)-7 b(rotter)180 310 y(pro)r(duct)28 b(form)n(ula.)305 409 y(No)n(w,)f(w)n(e)g(claim)g(that)h Fp(8)p Fw( )d Fp(2)f(D)793 606 y(h)p Fw( )s(;)14 b(B)t(e)1025 572 y Fr(\000)p Fs(\014)s(L)1164 546 y Fk(\()p Fi(`)p Fk(\))1237 572 y Fs(=)p Fv(2)1309 606 y Fw(\031)s Fx(\()p Fw(v)s Fx(\)\012)1526 618 y Fv(0)1564 606 y Fp(i)23 b Fx(=)g Fp(h)p Fx([)p Fw(\031)1812 572 y Fr(0)1836 606 y Fx(\()p Fw(\013)1921 563 y Fr(\000)p Fs(i\014)s(=)p Fv(2)1921 628 y(0)2127 606 y Fp(\016)18 b Fw(\024)2235 572 y Fs(i=)p Fv(2)2330 606 y Fx(\()p Fw(v)s Fx(\)\)])2492 572 y Fr(\003)2531 606 y Fw( )s(;)c(B)t(e)2731 572 y Fr(\000)p Fs(\014)s(L)2870 546 y Fk(\()p Fi(`)p Fk(\))2943 572 y Fs(=)p Fv(2)3014 606 y Fx(\012)3074 618 y Fv(0)3111 606 y Fp(i)p Fw(:)443 b Fx(\(5.7\))180 787 y(Indeed,)26 b(appro)n(ximate)e(the)h(op)r(erator)f Fw(v)k Fx(as)d(in)h(the)f(pro)r(of)g(of)g(Lemma)g(4.2)g(b)n(y)g (UV-cut-o\013)g(op)r(erators)e Fw(v)3486 799 y Fs(n)3532 787 y Fx(.)36 b(Since)180 886 y(b)r(oth)29 b(sides)g(of)f(Eqn)h (\(5.7\))f(c)n(hange)g(con)n(tin)n(uously)g(under)g(this)h(appro)n (ximation,)f(it)h(su\016ces)f(to)h(pro)n(v)n(e)e(\(5.7\))i(for)180 986 y Fw(v)220 998 y Fs(n)293 986 y Fx(replacing)d Fw(v)s Fx(.)38 b(W)-7 b(e)28 b(pro)n(v)n(e)e(the)i(more)e(general)h(relation) 951 1182 y Fp(h)p Fw( )s(;)14 b(B)t(e)1183 1148 y Fs(z)r(L)1263 1123 y Fk(\()p Fi(`)p Fk(\))1341 1182 y Fw(\031)s Fx(\()p Fw(v)1463 1194 y Fs(n)1509 1182 y Fx(\)\012)1601 1194 y Fv(0)1638 1182 y Fp(i)24 b Fx(=)e Fp(h)p Fx([)p Fw(\031)1886 1148 y Fr(0)1910 1182 y Fx(\()p Fw(\013)1995 1148 y Fs(i)p 2018 1115 35 3 v(z)1995 1203 y Fv(0)2076 1182 y Fp(\016)c Fw(\024)2184 1148 y Fs(i=)p Fv(2)2279 1182 y Fx(\()p Fw(v)2351 1194 y Fs(n)2396 1182 y Fx(\)\)])2483 1148 y Fr(\003)2522 1182 y Fw( )s(;)c(B)t(e)2722 1148 y Fs(z)r(L)2802 1123 y Fk(\()p Fi(`)p Fk(\))2880 1182 y Fx(\012)2940 1194 y Fv(0)2977 1182 y Fp(i)600 b Fx(\(5.8\))180 1379 y(for)34 b Fp(j)p Fx(Re)14 b Fw(z)t Fp(j)33 b Fw(<)g(\014)t Fx(.)57 b(Observ)n(e)33 b(that)h(the)h(v)n(ectors)d Fw(e)1766 1349 y Fs(z)r(L)1846 1324 y Fk(\()p Fi(`)p Fk(\))1924 1379 y Fw(')p Fx(,)k Fw(')e Fp(2)h(D)r Fx(,)h(are)d(analytic)g(in)i Fw(z)i Fx(for)d Fp(j)p Fw(z)t Fp(j)f Fw(<)h(\014)k Fx(as)c(can)g(b)r(e) 180 1479 y(sho)n(wn)26 b(b)n(y)g(using)g(the)h(Araki-Dyson)f(expansion) f(as)h(in)h(the)g(pro)r(of)f(of)g(Lemma)h(4.2,)f(with)h Fw(\014)g Fx(=)c(2)j(and)h(with)g Fw(L)3697 1449 y Fv(\()p Fs(`)p Fv(\))180 1578 y Fx(replaced)h(b)n(y)h Fw(z)t(L)726 1548 y Fv(\()p Fs(`)p Fv(\))808 1578 y Fx(.)41 b(Hence)29 b(b)r(oth)h(sides)e(of)h(Eqn)g(\(5.8\))f(are)g(analytic)h(in)g Fw(z)j Fx(for)c Fp(j)p Fw(z)t Fp(j)d Fw(<)g(\014)t Fx(.)41 b(Therefore)28 b(it)h(su\016ces)180 1678 y(to)k(pro)n(v)n(e)e(\(5.8\))h (for)g Fw(z)j Fx(=)d Fw(it)p Fx(.)52 b(Eqn)32 b(\(5.8\))g(with)i Fw(z)h Fx(=)c Fw(it)h Fx(follo)n(ws)g(from)g(the)i(relations)d(\(5.5\)) i(and)f(\(5.6\))h(pro)n(v)n(en)180 1778 y(ab)r(o)n(v)n(e.)j(Th)n(us)27 b(\(5.7\))g(is)h(demonstrated.)305 1893 y(No)n(w,)21 b(let)g(\012)684 1863 y Fr(0)731 1893 y Fx(:=)h Fw(e)880 1863 y Fr(\000)p Fs(\014)s(L)1019 1838 y Fk(\()p Fi(`)p Fk(\))1092 1863 y Fs(=)p Fv(2)1163 1893 y Fx(\012)1223 1905 y Fv(0)1260 1893 y Fx(.)35 b(Recall)21 b(that)g Fw(L)1792 1863 y Fv(\()p Fs(`)p Fv(\))1875 1893 y Fx(\012)1935 1863 y Fr(0)1981 1893 y Fx(=)i Fw(g)s(e)2151 1863 y Fr(\000)p Fs(\014)s(L)2290 1838 y Fk(\()p Fi(`)p Fk(\))2362 1863 y Fs(=)p Fv(2)2433 1893 y Fw(\031)s Fx(\()p Fw(v)s Fx(\)\012)2650 1905 y Fv(0)2689 1893 y Fx(.)34 b(Then)21 b(\(5.7\))g(and)g(the)g (de\014nition)180 1993 y(of)28 b(the)g(transformation)d Fw(\015)1031 1963 y Fs(s)1094 1993 y Fx(imply)j(that)1216 2189 y Fp(h)p Fw( )s(;)14 b(B)t(L)1466 2155 y Fv(\()p Fs(`)p Fv(\))1550 2189 y Fx(\012)1610 2155 y Fr(0)1633 2189 y Fp(i)84 b Fx(=)e Fw(g)s Fp(h)p Fw( )s(;)14 b(B)t(e)2171 2155 y Fr(\000)p Fs(\014)s(L)2310 2130 y Fk(\()p Fi(`)p Fk(\))2383 2155 y Fs(=)p Fv(2)2454 2189 y Fw(\031)s Fx(\()p Fw(v)s Fx(\)\012)2671 2201 y Fv(0)2710 2189 y Fp(i)1749 2314 y Fx(=)82 b Fw(g)s Fp(h)p Fx([)p Fw(\031)2044 2279 y Fr(0)2068 2314 y Fx(\()p Fw(\015)2148 2279 y Fs(i=)p Fv(2)2243 2314 y Fx(\()p Fw(v)s Fx(\)\)])2405 2279 y Fr(\003)2444 2314 y Fw( )s(;)14 b(B)t Fx(\012)2665 2279 y Fr(0)2688 2314 y Fp(i)p Fw(:)866 b Fx(\(5.9\))180 2495 y(This)28 b(relation)e(sho)n(ws)h(that)h Fw(B)t Fx(\012)1220 2465 y Fr(0)1266 2495 y Fp(2)23 b Fx(Dom)q(\()p Fw(\031)1601 2465 y Fr(0)1624 2495 y Fx(\()p Fw(\015)1704 2465 y Fs(i=)p Fv(2)1799 2495 y Fx(\()p Fw(v)s Fx(\)\))29 b(and)1522 2676 y Fw(B)t(L)1646 2641 y Fv(\()p Fs(`)p Fv(\))1729 2676 y Fx(\012)23 b(=)g Fw(\031)1950 2641 y Fr(0)1973 2676 y Fx(\()p Fw(\015)2053 2641 y Fs(i=)p Fv(2)2148 2676 y Fx(\()p Fw(v)s Fx(\)\))p Fw(B)t Fx(\012)p Fw(:)1131 b Fx(\(5.10\))180 2857 y(Since)41 b(the)f(r.h.s.)75 b(of)40 b(\(5.10\))g(is)g Fw(V)1344 2827 y Fr(0)1325 2883 y(\000)p Fs(i=)p Fv(2)1472 2857 y Fw(B)t Fx(\012,)k(this)d(equation)e(together)h (with)h(\(5.4\))f(implies)g(\(5.3\).)75 b(Finally)-7 b(,)180 2980 y(Dom\()p Fw(L)443 2950 y Fv(\()p Fs(`)p Fv(\))527 2980 y Fx(\))18 b Fp(\\)h Fw(\031)s Fx(\()p Fp(A)p Fx(\)\012)29 b(con)n(tains)e(the)h(set)f Fp(D)j Fx(and)e(is)f(therefore)g(dense)g(in)h Fp(H)q Fx(.)p 3714 2992 59 71 v 305 3080 a(Observ)n(e)d(that)1141 3179 y Fw(\024)1189 3145 y Fs(t)1218 3179 y Fp(j)1241 3191 y Fs(\014)1279 3199 y Fi(j)1310 3191 y Fv(=)p Fs(\014)1399 3199 y Fi(p)1433 3191 y Fv(=)p Fs(\014)1552 3179 y Fx(=)d Fw(\013)1692 3139 y Fs(\014)s(t)1692 3201 y Fv(0)1929 3179 y Fx(and)166 b Fw(\015)2277 3145 y Fs(t)2305 3179 y Fp(j)2328 3191 y Fs(\014)2366 3199 y Fi(j)2397 3191 y Fv(=)p Fs(\014)2486 3199 y Fi(p)2521 3191 y Fv(=)p Fs(\014)2639 3179 y Fx(=)23 b(id)p Fw(:)749 b Fx(\(5.11\))180 3328 y(This)28 b(implies)f(that)1691 3427 y Fw(K)6 b Fp(j)1791 3439 y Fs(\014)1829 3447 y Fi(j)1859 3439 y Fv(=)p Fs(\014)1948 3447 y Fi(p)1983 3439 y Fv(=)p Fs(\014)2102 3427 y Fx(=)22 b Fw(L;)1299 b Fx(\(5.12\))180 3576 y(where)24 b Fw(L)e Fx(:=)h Fw(L)664 3588 y Fv(0)713 3576 y Fx(+)12 b Fw(g)s(V)30 b Fp(\000)12 b Fw(g)s(V)1098 3546 y Fr(0)1121 3576 y Fx(,)25 b(with)g Fw(V)1422 3546 y Fr(0)1468 3576 y Fx(:=)e Fw(\031)1629 3546 y Fr(0)1653 3576 y Fx(\()p Fw(v)s Fx(\),)j(is)e(the)h(standard)e(self-adjoin)n(t)h(Liouville)g(op) r(erator.)34 b(In)25 b(what)180 3675 y(follo)n(ws)i(w)n(e)g(write)g Fw(K)i Fx(=)22 b Fw(L)1031 3687 y Fv(0)1087 3675 y Fx(+)c Fw(g)s(I)7 b Fx(,)27 b(where)1710 3775 y Fw(I)j Fx(=)23 b Fw(V)37 b Fp(\000)18 b Fw(V)2099 3741 y Fr(0)2080 3798 y(\000)p Fs(i=)p Fv(2)2227 3775 y Fw(:)305 3924 y Fx(The)33 b(op)r(erator)e Fw(K)38 b Fx(is)33 b(non-self-adjoin)n(t)f(for)g Fw(\016)s(\014)37 b Fp(6)p Fx(=)31 b(0,)j(and)f(the)h(p)r(erturbation)e Fw(I)40 b Fx(not)33 b(relativ)n(ely)f(b)r(ounded)180 4023 y(w.r.t.)48 b(the)32 b(unp)r(erturb)r(ed)g(op)r(erator)e Fw(L)1449 4035 y Fv(0)1486 4023 y Fx(.)48 b(T)-7 b(o)31 b(study)h(the)g(ev)n(olution)e(generated)g(b)n(y)i Fw(K)k Fx(w)n(e)c(use)f(the)h(family)f(of)180 4123 y(op)r(erators)1528 4222 y Fw(K)1599 4237 y Fv(\()p Fs(s)p Fv(\))1709 4222 y Fx(:=)23 b Fw(L)1877 4234 y Fv(0)1932 4222 y Fx(+)18 b Fw(g)s Fx(\()p Fw(V)37 b Fp(\000)18 b Fw(V)2325 4188 y Fr(0)2306 4243 y Fs(s)2349 4222 y Fx(\))28 b Fw(;)1136 b Fx(\(5.13\))180 4371 y(where,)27 b(recall,)g Fw(V)42 b Fx(=)22 b Fw(\031)s Fx(\()p Fw(v)s Fx(\),)29 b(and)f Fw(V)1303 4341 y Fr(0)1284 4391 y Fs(s)1349 4371 y Fx(:=)23 b Fw(\031)1510 4341 y Fr(0)1533 4371 y Fx(\()p Fw(\015)p 1613 4307 36 3 v 1613 4341 a Fs(s)1649 4371 y Fx(\()p Fw(v)s Fx(\)\).)38 b(This)27 b(family)h(in)n(terp)r(olates)f(b)r(et)n (w)n(een)g(the)h(op)r(erator)e Fw(K)6 b Fx(,)1727 4552 y Fw(K)28 b Fx(=)23 b Fw(K)1985 4567 y Fv(\()p Fr(\000)p Fs(i=)p Fv(2\))2210 4552 y Fw(;)1335 b Fx(\(5.14\))180 4733 y(\(see)26 b(Eqn)f(\(5.3\)\))h(and)g(the)g(self-adjoin)n(t)g(op)r (erators)e Fw(K)1910 4748 y Fv(\()p Fs(s)p Fv(\))2023 4733 y Fx(with)i(real)f Fw(s)p Fx(.)37 b(Under)26 b(condition)f (\(4.1\))h(on)g Fw(v)s Fx(,)g Fw(V)3551 4703 y Fr(0)3532 4760 y Fv(\()p Fs(s)p Fv(\))3646 4733 y Fx(and)180 4856 y Fw(K)251 4871 y Fv(\()p Fs(s)p Fv(\))366 4856 y Fx(are)g(w)n(ell)i (de\014ned)f(on)h(the)g(dense)f(domain)g(Dom)q(\()p Fw(L)2002 4826 y Fv(\()p Fs(`)p Fv(\))2085 4856 y Fx(\))19 b Fp(\\)g Fw(\031)s Fx(\()p Fp(A)p Fx(\)\012)28 b(for)g(all)f Fw(s)g Fx(in)h(the)g(strip)1447 5046 y Fw(S)1498 5058 y Fs(")1556 5046 y Fx(:=)1667 4979 y Fm(\010)1716 5046 y Fw(t)23 b Fp(2)g Fo(C)1901 4976 y Fm(\014)1901 5025 y(\014)1935 5046 y Fp(j)p Fx(Im)14 b Fw(t)p Fp(j)23 b Fw(<)2244 5013 y Fv(1)p 2244 5027 34 4 v 2244 5075 a(2)2306 5046 y Fx(+)18 b Fw(")2428 4979 y Fm(\011)2490 5046 y Fw(;)1055 b Fx(\(5.15\))180 5227 y(for)27 b Fw(")c(>)f Fx(0,)28 b(and)f(are)g(strongly)f(analytic)h (there)g(\(recall)g(that)h Fw(\031)2182 5197 y Fr(0)2233 5227 y Fx(is)g(an)n(ti-linear\).)1938 5476 y(16)p eop %%Page: 17 17 17 16 bop 180 210 a Fy(6)135 b(Sp)t(ectral)45 b(Deformation)i(of)e Ff(K)53 b Fy(and)44 b Ff(K)2358 231 y Fx(\()p Fw(s)p Fx(\))180 392 y(Since)23 b(the)g(op)r(erator)e Fw(K)28 b Fx(is)23 b(not)g(self-adjoin)n(t)f(it)h(is)g(not)g(a)f(simple)h (matter)f(to)h(deriv)n(e)f(long-time)g(prop)r(erties)g(of)h(the)180 492 y(dynamics)29 b Fw(e)588 462 y Fs(iK)t(t)730 492 y Fx(from)g(sp)r(ectral)g(prop)r(erties)f(of)i Fw(K)6 b Fx(.)42 b(As)30 b(a)f(result)g(w)n(e)g(b)n(ypass)f(establishing)h (the)h(connection)f(of)180 591 y(the)22 b(dynamics)g(to)g(the)h(sp)r (ectrum)f(of)g Fw(K)27 b Fx(and)22 b(instead)g(connect)g(it)h(to)f (certain)f(sp)r(ectral)h(prop)r(erties)f(of)h(a)g(complex)180 691 y(deformation,)33 b Fw(K)744 703 y Fs(\022)781 691 y Fx(,)i(of)d(this)h(op)r(erator.)51 b(T)-7 b(o)32 b(do)h(this)g(w)n(e) f(use)h(the)g(in)n(terp)r(olating)f(family)h Fw(K)3156 706 y Fv(\()p Fs(s)p Fv(\))p Fs(\022)3276 691 y Fx(,)h(whic)n(h)f(is)g (the)180 791 y(complex)27 b(deformation)g(of)h(the)h(family)f Fw(K)1533 806 y Fv(\()p Fs(s)p Fv(\))1619 791 y Fx(,)g(\(5.13\).)37 b(In)28 b(this)h(section)e(w)n(e)h(de\014ne)g(complex)f(deformations)g Fw(K)3743 803 y Fs(\022)180 890 y Fx(and)k Fw(K)416 905 y Fv(\()p Fs(s)p Fv(\))p Fs(\022)567 890 y Fx(and)f(in)h(the)g(next)g (section)f(w)n(e)h(establish)f(their)h(sp)r(ectral)f(c)n (haracteristics)e(whic)n(h)j(are)f(relev)-5 b(an)n(t)30 b(for)180 990 y(us.)305 1089 y(In)d(order)e(to)i(carry)e(out)i(the)g (sp)r(ectral)f(analysis)g(of)h(the)g(op)r(erator)e Fw(K)6 b Fx(,)26 b(whic)n(h)h(w)n(e)g(b)r(egin)g(in)g(this)g(section,)f(w)n(e) 180 1189 y(use)k(the)g(sp)r(eci\014cs)g(of)g(the)h(Araki-W)-7 b(o)r(o)r(ds)29 b(represen)n(tation)f(in)i(an)g(essen)n(tial)f(w)n(a)n (y)-7 b(.)43 b(They)30 b(w)n(ere)f(not)h(used)g(in)g(an)180 1289 y(essen)n(tial)d(w)n(a)n(y)f(for)h(the)h(dev)n(elopmen)n(ts)f(up)h (to)f(this)h(section.)305 1388 y(As)38 b(a)g(complex)g(deformation)g(w) n(e)g(c)n(ho)r(ose)g(a)g(com)n(bination)g(of)g(the)h(complex)f (dilation)h(used)f(in)h([8])f(and)180 1488 y(complex)23 b(translation)f(due)h(to)h([18)o(])f(\(see)g([8],)h(Section)f(V.2)h (for)e(a)h(sk)n(etc)n(h)g(of)g(the)h(relev)-5 b(an)n(t)22 b(ideas\).)35 b(This)24 b(complex)180 1588 y(deformation)j(w)n(as)f (used)i(in)g([25)o(])g(in)f(the)h(sp)r(ectral)f(analysis)g(for)g(a)g (general)f(class)h(of)g(Liouville)h(t)n(yp)r(e)f(op)r(erators.)305 1687 y(First)f(w)n(e)g(de\014ne)g(the)h(group)e(of)h(dilations.)36 b(Let)1865 1666 y(^)1851 1687 y Fw(U)1908 1699 y Fs(d;\016)2025 1687 y Fx(b)r(e)27 b(the)f(second)g(quan)n(tization)f(of)i(the)f (one-parameter)180 1787 y(group)1527 1886 y Fw(u)1575 1898 y Fs(d;\016)1688 1886 y Fx(:)d Fw(f)9 b Fx(\()p Fw(k)s Fx(\))23 b Fp(!)g Fw(e)2062 1852 y Fv(3)p Fs(\016)r(=)p Fv(2)2198 1886 y Fw(f)9 b Fx(\()p Fw(e)2319 1852 y Fs(\016)2355 1886 y Fw(k)s Fx(\))180 2036 y(of)30 b(dilations)f(on)h Fw(L)791 2006 y Fv(2)827 2036 y Fx(\()p Fo(R)914 2006 y Fs(n)965 2036 y Fx(\).)44 b(This)29 b(group)g(acts)g(on)h(creation)e (and)i(annihilation)f(op)r(erators)f Fw(a)3145 2006 y Fv(#)3145 2056 y Fs(r)3204 2036 y Fx(\()p Fw(f)9 b Fx(\))30 b(on)f(the)h(F)-7 b(o)r(c)n(k)180 2136 y(space,)27 b Fp(H)495 2148 y Fs(r)532 2136 y Fx(,)h(according)d(to)j(the)g(rule)1132 2297 y(^)1118 2318 y Fw(U)1175 2330 y Fs(d;\016)1265 2318 y Fw(a)1309 2284 y Fv(#)1309 2339 y Fs(r)1368 2318 y Fx(\()p Fw(f)9 b Fx(\))1497 2297 y(^)1482 2318 y Fw(U)1548 2283 y Fr(\000)p Fv(1)1539 2343 y Fs(d;\016)1660 2318 y Fx(=)23 b Fw(a)1792 2284 y Fv(#)1792 2339 y Fs(r)1850 2318 y Fx(\()p Fw(u)1930 2330 y Fs(d;\016)2021 2318 y Fw(f)9 b Fx(\))p Fw(;)2320 2297 y Fx(^)2306 2318 y Fw(U)2363 2330 y Fs(d;\016)2453 2318 y Fx(\012)2513 2330 y Fs(r)r(j)2604 2318 y Fx(=)22 b(\012)2751 2330 y Fs(r)r(j)2819 2318 y Fw(:)767 b Fx(\(6.1\))180 2501 y(W)-7 b(e)28 b(lift)g(this)g(group)f (to)g(the)h(p)r(ositiv)n(e-temp)r(erature)f(Hilb)r(ert)h(space,)f (\(2.33\),)g(according)e(to)j(the)g(form)n(ula)1153 2683 y Fw(U)1210 2695 y Fs(d;\016)1323 2683 y Fx(=)23 b Fn(1)1459 2695 y Fs(p)1516 2683 y Fp(\012)18 b Fn(1)1647 2695 y Fs(p)1703 2683 y Fp(\012)1801 2662 y Fx(^)1786 2683 y Fw(U)1843 2695 y Fs(d;\016)1952 2683 y Fp(\012)2049 2662 y Fx(^)2035 2683 y Fw(U)2092 2695 y Fs(d;)p Fr(\000)p Fs(\016)2253 2683 y Fp(\012)2350 2662 y Fx(^)2336 2683 y Fw(U)2393 2695 y Fs(d;\016)2502 2683 y Fp(\012)2599 2662 y Fx(^)2585 2683 y Fw(U)2642 2695 y Fs(d;)p Fr(\000)p Fs(\016)2784 2683 y Fw(:)802 b Fx(\(6.2\))180 2866 y(Note)28 b(that)h(w)n(e)f(could)g(dilate)g(eac)n(h)g(reserv)n(oir)d(b)n(y)j(a)g (di\013eren)n(t)g(amoun)n(t.)39 b(Ho)n(w)n(ev)n(er,)26 b(this)j(do)r(es)f(not)g(giv)n(e)g(us)g(an)n(y)180 2966 y(adv)-5 b(an)n(tage,)25 b(so)h(to)h(k)n(eep)f(notation)g(simple)h(w)n (e)f(use)h(one)f(dilation)h(parameter)e(for)h(b)r(oth)h(reserv)n(oirs.) 33 b(W)-7 b(e)28 b(record)180 3065 y(for)18 b(future)h(reference)f(ho)n (w)g(the)i(group)d Fw(U)1458 3077 y Fs(d;\016)1568 3065 y Fx(acts)h(on)g(the)h(Liouville)g(op)r(erator)e Fw(L)2685 3077 y Fv(0)2740 3065 y Fx(and)i(the)g(p)r(ositiv)n(e-temp)r(erature) 180 3177 y(photon)27 b(n)n(um)n(b)r(er)h(op)r(erator)e Fw(N)31 b Fx(:=)1308 3114 y Fm(P)1396 3135 y Fv(2)1396 3202 y Fs(j)s Fv(=1)1529 3177 y Fw(N)1596 3189 y Fs(j)1630 3177 y Fx(,)d(where)1165 3417 y Fw(N)1232 3429 y Fs(j)1290 3417 y Fx(:=)1401 3304 y Fm(Z)1497 3350 y(\002)1532 3417 y Fw(a)1576 3383 y Fr(\003)1576 3438 y Fs(`;j)1658 3417 y Fx(\()p Fw(k)s Fx(\))p Fw(a)1812 3429 y Fs(`;j)1895 3417 y Fx(\()p Fw(k)s Fx(\))19 b(+)f Fw(a)2151 3383 y Fr(\003)2151 3438 y Fs(r)n(;j)2235 3417 y Fx(\()p Fw(k)s Fx(\))p Fw(a)2389 3429 y Fs(r)n(;j)2473 3417 y Fx(\()p Fw(k)s Fx(\))2583 3350 y Fm(\003)2645 3417 y Fw(d)2688 3383 y Fv(3)2726 3417 y Fw(k)s(;)814 b Fx(\(6.3\))180 3660 y(and)32 b(the)h(op)r(erators)e Fw(a)911 3620 y Fv(#)911 3688 y Fr(f)p Fs(`;r)r Fr(g)p Fs(;j)1114 3660 y Fx(\()p Fw(k)s Fx(\))i(w)n(ere)e(in)n(tro)r(duced)h(after)h (\(2.34\).)50 b(W)-7 b(e)33 b(ha)n(v)n(e)f(\(b)r(elo)n(w)g(w)n(e)g(do)g (not)h(displa)n(y)f(the)180 3774 y(iden)n(tit)n(y)c(op)r(erators\):) 1284 3873 y Fw(U)1341 3885 y Fs(d;\016)1431 3873 y Fw(L)1488 3885 y Fs(r)r(j)1555 3873 y Fw(U)1621 3838 y Fr(\000)p Fv(1)1612 3898 y Fs(d;\016)1733 3873 y Fx(=)23 b(cosh)o(\()p Fw(\016)s Fx(\))p Fw(L)2139 3885 y Fs(r)r(j)2225 3873 y Fx(+)18 b(sinh\()p Fw(\016)s Fx(\)\003)2618 3885 y Fs(j)2653 3873 y Fw(;)933 b Fx(\(6.4\))180 4023 y(where)27 b(\003)478 4035 y Fs(j)540 4023 y Fx(is)h(the)g(p)r(ositiv)n(e)f(op)r (erator)f(on)h(the)h Fw(j)5 b Fx(th)28 b(reserv)n(oir)d(Hilb)r(ert)j (space)f(giv)n(en)g(b)n(y)1088 4251 y(\003)1146 4263 y Fs(j)1204 4251 y Fx(=)1292 4138 y Fm(Z)1389 4251 y Fw(!)s Fx(\()p Fw(k)s Fx(\))1568 4183 y Fm(\000)1605 4251 y Fw(a)1649 4216 y Fr(\003)1649 4271 y Fs(`;j)1732 4251 y Fx(\()p Fw(k)s Fx(\))p Fw(a)1886 4263 y Fs(`;j)1969 4251 y Fx(\()p Fw(k)s Fx(\))18 b(+)h Fw(a)2225 4216 y Fr(\003)2225 4271 y Fs(r)n(;j)2308 4251 y Fx(\()p Fw(k)s Fx(\))p Fw(a)2462 4263 y Fs(r)n(;j)2546 4251 y Fx(\()p Fw(k)s Fx(\))2656 4183 y Fm(\001)2722 4251 y Fw(d)2765 4216 y Fv(3)2803 4251 y Fw(k)s(;)737 b Fx(\(6.5\))180 4471 y(and)1660 4571 y Fw(U)1717 4583 y Fs(d;\016)1808 4571 y Fw(N)1875 4583 y Fs(j)1909 4571 y Fw(U)1975 4535 y Fr(\000)p Fv(1)1966 4596 y Fs(d;\016)2087 4571 y Fx(=)23 b Fw(N)2242 4583 y Fs(j)2277 4571 y Fw(:)1309 b Fx(\(6.6\))305 4720 y(No)n(w)26 b(w)n(e)g(de\014ne)h(a)g(one-parameter)e(group)g(of)i (translations.)35 b(It)27 b(can)g(b)r(e)g(de\014ned)g(as)g (one-parameter)d(group)180 4820 y(arising)k(from)h(transformations)f (of)h(the)h(underlying)f(ph)n(ysical)f(space)h(similarly)f(to)i(the)g (dilation)f(group.)41 b(This)180 4920 y(is)28 b(done)h(in)f(App)r (endix)i(B.)39 b(Here)29 b(w)n(e)f(de\014ne)h(the)f(translation)g (group)f(b)n(y)h(means)g(of)h(the)g(selfadjoin)n(t)f(generator)180 5019 y Fw(T)34 b Fx(:=)374 4957 y Fm(P)461 4977 y Fv(2)461 5044 y Fs(j)s Fv(=1)594 5019 y Fw(T)643 5031 y Fs(j)678 5019 y Fx(,)28 b(where)1138 5177 y Fw(T)1187 5189 y Fs(j)1245 5177 y Fx(=)1332 5064 y Fm(Z)1429 5109 y(\002)1464 5177 y Fw(a)1508 5142 y Fr(\003)1508 5197 y Fs(`;j)1590 5177 y Fx(\()p Fw(k)s Fx(\))p Fw(\015)5 b(a)1792 5189 y Fs(`;j)1875 5177 y Fx(\()p Fw(k)s Fx(\))18 b(+)h Fw(a)2131 5142 y Fr(\003)2131 5197 y Fs(r)n(;j)2214 5177 y Fx(\()p Fw(k)s Fx(\))p Fw(\015)5 b(a)2416 5189 y Fs(r)n(;j)2500 5177 y Fx(\()p Fw(k)s Fx(\))2610 5109 y Fm(\003)2672 5177 y Fw(d)2715 5142 y Fv(3)2753 5177 y Fw(k)s(:)787 b Fx(\(6.7\))1938 5476 y(17)p eop %%Page: 18 18 18 17 bop 180 210 a Fx(Here,)26 b Fw(\015)h Fx(=)c Fw(i)p Fx(\()619 188 y(^)617 210 y Fw(k)17 b Fp(\001)e(r)g Fx(+)f Fp(r)h(\001)1002 188 y Fx(^)1000 210 y Fw(k)r Fx(\))26 b(with)1293 188 y(^)1291 210 y Fw(k)f Fx(=)e Fw(k)s(=)p Fp(j)p Fw(k)s Fp(j)p Fx(.)36 b(Since)26 b Fw(\015)k Fx(is)c(a)f (self-adjoin)n(t)g(op)r(erator)f(on)h Fw(L)3051 180 y Fv(2)3088 210 y Fx(\()p Fo(R)3174 180 y Fv(3)3218 210 y Fx(\),)h(the)g(op)r(erators)180 310 y Fw(T)229 322 y Fs(j)263 310 y Fw(;)14 b(j)41 b Fx(=)36 b(1)p Fw(;)14 b Fx(2,)36 b(and)f(therefore)f(the)i(op)r(erator)e Fw(T)12 b Fx(,)36 b(are)e(self-adjoin)n(t)h(as)g(w)n(ell.)59 b(W)-7 b(e)36 b(de\014ne)g(the)f(one-parameter)180 409 y(group)26 b(of)i(translations)e(as)1567 509 y Fw(U)1624 521 y Fs(t;\034)1733 509 y Fx(:=)c Fn(1)1891 521 y Fs(p)1948 509 y Fp(\012)c Fn(1)2079 521 y Fs(p)2136 509 y Fp(\012)g Fw(e)2258 475 y Fs(i\034)7 b(T)2370 509 y Fw(:)1216 b Fx(\(6.8\))180 655 y(Eqns.)38 b(\(6.7\))g(-)h(\(6.8\))f(imply)h(the)g (follo)n(wing)f(expressions)f(for)h(the)h(action)f(of)h(this)g(group)e (on)h(the)i(Liouville)180 754 y(op)r(erators:)1516 854 y Fw(U)1573 866 y Fs(t;\034)1659 854 y Fw(L)1716 866 y Fs(r)r(j)1783 854 y Fw(U)1849 818 y Fr(\000)p Fv(1)1840 874 y Fs(t;\034)1961 854 y Fx(=)22 b Fw(L)2105 866 y Fs(r)r(j)2191 854 y Fx(+)c Fw(\034)9 b(N)2386 866 y Fs(j)2421 854 y Fw(:)1165 b Fx(\(6.9\))180 999 y(Observ)n(e)33 b(that)j(neither)f(the)h(dilation)e(nor)h(the)g(translation)f(group)g (a\013ects)h(the)h(particle)e(v)n(ectors,)i(and)f(that)180 1099 y Fw(U)237 1111 y Fs(t;\034)323 1099 y Fw(N)390 1111 y Fs(j)425 1099 y Fw(U)491 1064 y Fr(\000)p Fv(1)482 1119 y Fs(t;\034)603 1099 y Fx(=)22 b Fw(N)757 1111 y Fs(j)792 1099 y Fx(.)305 1199 y(No)n(w)30 b(w)n(e)g(w)n(an)n(t)g(to)g (apply)g(the)h(pro)r(duct)g(of)f(these)h(transformations)e(to)h(the)h (full)g(op)r(erator)e Fw(K)34 b Fx(=)27 b Fw(L)3529 1211 y Fv(0)3586 1199 y Fx(+)20 b Fw(g)s(I)7 b Fx(.)180 1298 y(Since)37 b(the)f(dilation)h(and)f(translation)f(transformations)g(do) h(not)g(comm)n(ute)h(w)n(e)f(ha)n(v)n(e)f(to)h(c)n(ho)r(ose)g(the)g (order)180 1398 y(in)i(whic)n(h)g(w)n(e)g(apply)f(them.)69 b(Since)38 b(the)g(op)r(erator)e(\003)k(=)2110 1336 y Fm(P)2198 1423 y Fs(j)2247 1398 y Fx(\003)2305 1410 y Fs(j)2377 1398 y Fx(is)e(not)g(analytic)f(under)h(the)h(translations) 180 1498 y(while)26 b(the)h(op)r(erator)d Fw(N)35 b Fx(is)26 b(analytic)f(under)h(dilations)f(w)n(e)h(apply)g(\014rst)f(the)i (translation)e(and)g(then)i(the)f(dilation)180 1597 y(transformation.) 35 b(W)-7 b(e)28 b(de\014ne)g(the)g(com)n(bined)f(translation-dilation) f(transformation)g(as)1732 1773 y Fw(U)1789 1785 y Fs(\022)1850 1773 y Fx(=)c Fw(U)1994 1785 y Fs(d;\016)2085 1773 y Fw(U)2142 1785 y Fs(t;\034)3568 1773 y Fx(\(6.10\))180 1949 y(where)33 b Fw(\022)i Fx(=)d(\()p Fw(\016)o(;)14 b(\034)9 b Fx(\).)54 b(In)34 b(what)f(follo)n(ws)f(w)n(e)h(will)h(use)f (the)g(notation)g Fp(j)p Fw(\022)r Fp(j)g Fx(=)f(\()p Fp(j)p Fw(\016)s Fp(j)p Fw(;)14 b Fp(j)p Fw(\034)9 b Fp(j)p Fx(\),)36 b(Im)q Fw(\022)e Fx(=)e(\(Im)q Fw(\016)o(;)14 b Fx(Im)o Fw(\034)9 b Fx(\),)36 b(and)180 2049 y(similarly)27 b(for)g(Re)p Fw(\022)r Fx(,)h(and)1277 2149 y(Im)p Fw(\022)d(>)e Fx(0)128 b Fp(\()-14 b(\))130 b Fx(Im)p Fw(\016)26 b(>)d Fx(0)18 b Fp(^)g Fx(Im)q Fw(\034)32 b(>)23 b Fx(0)p Fw(:)884 b Fx(\(6.11\))305 2294 y(No)n(w)32 b(w)n(e)h(are)g(ready)f(to)h (de\014ne)h(a)f(complex)g(deformation)f(of)i(the)f(op)r(erator)f Fw(K)6 b Fx(.)54 b(On)33 b(the)h(set)f(Dom\(\003\))23 b Fp(\\)180 2394 y Fx(Dom\()p Fw(N)9 b Fx(\))28 b(w)n(e)f(de\014ne)h (for)f Fw(\022)f Fp(2)d Fo(R)1208 2364 y Fv(2)1685 2494 y Fw(K)1756 2506 y Fs(\022)1816 2494 y Fx(:=)g Fw(U)1984 2506 y Fs(\022)2021 2494 y Fw(K)6 b(U)2164 2458 y Fr(\000)p Fv(1)2155 2519 y Fs(\022)2252 2494 y Fw(:)1293 b Fx(\(6.12\))180 2652 y(Recalling)23 b(the)h(decomp)r(osition)f Fw(K)28 b Fx(=)23 b Fw(L)1463 2664 y Fv(0)1510 2652 y Fx(+)10 b Fw(g)s(I)d Fx(,)24 b(where)f Fw(L)2011 2664 y Fv(0)2071 2652 y Fx(:=)g Fw(L)2239 2664 y Fs(p)2287 2652 y Fx(+)10 b Fw(L)2419 2664 y Fs(r)2456 2652 y Fx(,)24 b Fw(L)2560 2664 y Fs(r)2619 2652 y Fx(:=)2730 2590 y Fm(P)2818 2610 y Fv(2)2818 2677 y Fs(j)s Fv(=1)2951 2652 y Fw(L)3008 2664 y Fs(r)r(j)3098 2652 y Fx(and)g Fw(I)30 b Fx(=)23 b Fw(V)29 b Fp(\000)10 b Fw(V)3629 2622 y Fr(0)3610 2679 y(\000)p Fs(i=)p Fv(2)3757 2652 y Fx(,)180 2752 y(w)n(e)27 b(ha)n(v)n(e)1677 2851 y Fw(K)1748 2863 y Fs(\022)1808 2851 y Fx(=)22 b Fw(L)1952 2863 y Fv(0)p Fs(;\022)2061 2851 y Fx(+)c Fw(g)s(I)2223 2863 y Fs(\022)2260 2851 y Fw(;)1285 b Fx(\(6.13\))180 2997 y(where)27 b(the)g(families)g Fw(L)921 3009 y Fv(0)p Fs(;\022)1038 2997 y Fx(and)g Fw(I)1235 3009 y Fs(\022)1300 2997 y Fx(are)f(de\014ned)i(accordingly) -7 b(.)35 b(Due)27 b(to)g(Eqns.)g(\(6.4\),)g(\(6.6\))g(and)g(\(6.9\))f (w)n(e)h(ha)n(v)n(e:)1249 3173 y Fw(L)1306 3185 y Fv(0)p Fs(;\022)1419 3173 y Fx(=)22 b Fw(L)1563 3185 y Fs(p)1620 3173 y Fx(+)c(cosh)o(\()p Fw(\016)s Fx(\))p Fw(L)2021 3185 y Fs(r)2076 3173 y Fx(+)g(sinh\()p Fw(\016)s Fx(\)\003)h(+)f Fw(\034)9 b(N)t(;)858 b Fx(\(6.14\))180 3361 y(where)30 b Fw(\022)g Fx(=)e(\()p Fw(\016)o(;)14 b(\034)9 b Fx(\),)32 b(and)e(\003)e(=)1164 3298 y Fm(P)1252 3319 y Fv(2)1252 3386 y Fs(j)s Fv(=1)1385 3361 y Fx(\003)1443 3373 y Fs(j)1477 3361 y Fx(.)46 b(An)31 b(explicit)g(expression)e(for)h(the)h(family)g Fw(I)2953 3373 y Fs(\022)3021 3361 y Fx(is)g(giv)n(en)e(in)i(App)r (endix)180 3460 y(B.2)c(\(see)h(Eqns)e(\(B.2.5\))i(and)f(\(B.2.7\)\).) 305 3560 y(Similarly)-7 b(,)27 b(w)n(e)g(de\014ne)h(the)g(family)f Fw(K)1503 3575 y Fv(\()p Fs(s)p Fv(\))p Fs(\022)1647 3560 y Fx(:=)c Fw(U)1815 3572 y Fs(\022)1852 3560 y Fw(K)1923 3575 y Fv(\()p Fs(s)p Fv(\))2010 3560 y Fw(U)2076 3524 y Fr(\000)p Fv(1)2067 3585 y Fs(\022)2192 3560 y Fx(\(recall)k (\(5.13\)\).)305 3660 y(Of)i(course)g(the)h(op)r(erator)e(families)i (ab)r(o)n(v)n(e)f(are)f(w)n(ell)i(de\014ned)g(for)g(real)e Fw(\022)r Fx(.)44 b(Our)30 b(task)f(is)h(to)f(de\014ne)h(them)h(as)180 3759 y(analytic)c(families)h(on)f(the)h(strips)1413 3935 y Fw(S)1469 3900 y Fr(\006)1464 3960 y Fs(\022)1496 3968 y Fk(0)1555 3935 y Fx(=)1643 3868 y Fm(\010)1691 3935 y Fw(\022)e Fp(2)d Fo(C)1888 3901 y Fv(2)1931 3935 y Fp(j)p Fx(0)g Fw(<)g Fp(\006)p Fx(Im)o Fw(\022)j(<)c(\022)2461 3947 y Fv(0)2498 3868 y Fm(\011)3568 3935 y Fx(\(6.15\))180 4112 y(where)32 b Fw(\022)464 4124 y Fv(0)532 4112 y Fx(=)e(\()p Fw(\016)696 4124 y Fv(0)734 4112 y Fw(;)14 b(\034)807 4124 y Fv(0)844 4112 y Fx(\))31 b Fw(>)g Fx(0)g(is)i(the)f (same)g(as)g(in)g(Condition)g(\(A\).)h(Recall)f(that)h(the)g(inequalit) n(y)e Fp(\006)p Fx(Im)p Fw(\022)i(<)e(\022)3655 4124 y Fv(0)3724 4112 y Fx(is)180 4211 y(equiv)-5 b(alen)n(t)28 b(to)g(the)h(follo)n(wing)e(inequalities:)37 b Fp(\006)p Fx(Im)p Fw(\016)27 b(<)d(\016)1995 4223 y Fv(0)2060 4211 y Fx(and)k Fp(\006)p Fx(Im)p Fw(\034)34 b(<)23 b(\034)2580 4223 y Fv(0)2618 4211 y Fx(.)38 b(\(The)29 b(fact)f(that)h(analyticit)n (y)e(in)h(a)180 4311 y(neigh)n(b)r(ourho)r(o)r(d)f(of)h(a)g(\014xed)g Fw(\022)f Fp(2)d Fw(S)1315 4275 y Fr(\006)1310 4336 y Fs(\022)1342 4344 y Fk(0)1407 4311 y Fx(implies)k(analyticit)n(y)f(in)i (the)f(corresp)r(onding)f(strip)h(in)g(whic)n(h)g(Re)p Fw(\022)j Fx(is)d(not)180 4410 y(constrain)n(t)22 b(follo)n(ws)f(from)i (the)g(explicit)g(form)n(ulas)f(\(6.14\),)h(\(B.2.5\))f(and)h (\(B.2.7\).\))35 b(The)23 b(analytic)f(con)n(tin)n(uations)180 4510 y(\(if)28 b(they)g(exist\))g(are)f(denoted)g(b)n(y)h(the)f(same)h (sym)n(b)r(ols.)305 4610 y(W)-7 b(e)29 b(de\014ne)h(the)g(family)g Fw(K)1163 4622 y Fs(\022)1229 4610 y Fx(for)f Fw(\022)g Fp(2)d(f)p Fw(\022)i Fp(2)f Fo(C)1752 4580 y Fv(2)1795 4539 y Fm(\014)1795 4589 y(\014)1823 4610 y Fp(j)p Fx(Im)14 b Fw(\022)r Fp(j)27 b Fw(<)e(\022)2179 4622 y Fv(0)2217 4610 y Fp(g)k Fx(b)n(y)g(the)h(explicit)g(expressions)d(\(6.13\),)j (\(6.14\),)180 4709 y(\(B.2.5\))h(and)h(\(B.2.7\).)49 b(Clearly)-7 b(,)32 b(Dom)q(\(\003\))21 b Fp(\\)h Fx(Dom\()p Fw(N)9 b Fx(\))30 b Fp(\032)g Fx(Dom\()p Fw(L)2362 4721 y Fv(0)p Fs(\022)2433 4709 y Fx(\))i(and)f(on)h(this)g(domain)g(the)g (family)g Fw(L)3710 4721 y Fv(0)p Fs(\022)180 4809 y Fx(is)j(manifestly)g(strongly)f(analytic)g(in)h Fw(\022)j Fp(2)e(f)p Fw(\022)h Fp(2)f Fo(C)1867 4779 y Fv(2)1910 4738 y Fm(\014)1910 4788 y(\014)1938 4809 y Fp(j)p Fx(Im)p Fw(\022)r Fp(j)g Fw(<)f(\022)2299 4821 y Fv(0)2336 4809 y Fp(g)p Fx(.)59 b(It)35 b(is)g(sho)n(wn)f(in)h(App)r(endix)h(B)f(that) h(for)180 4920 y Fp(j)p Fx(Im)p Fw(\022)r Fp(j)c Fw(<)f(\022)533 4932 y Fv(0)602 4920 y Fx(w)n(e)h(ha)n(v)n(e)g(Dom\(\003)1190 4890 y Fv(1)p Fs(=)p Fv(2)1294 4920 y Fx(\))g Fp(\032)e Fx(Dom\()p Fw(I)1695 4932 y Fs(\022)1734 4920 y Fx(\))j(and)f Fw(I)2001 4932 y Fs(\022)2039 4920 y Fw(f)41 b Fx(is)33 b(analytic)e Fp(8)p Fw(f)39 b Fp(2)32 b Fx(Dom\(\003)3008 4890 y Fv(1)p Fs(=)p Fv(2)3112 4920 y Fx(\).)52 b(Here)32 b(Condition)180 5020 y(\(A\))39 b(of)g(Section)f(3)g(is)g(used.)69 b(Hence)39 b(the)g(family)f Fw(K)1927 5032 y Fs(\022)2003 5020 y Fx(for)f Fw(\022)44 b Fp(2)d(f)p Fw(\022)i Fp(2)e Fo(C)2593 4989 y Fv(2)2637 4949 y Fm(\014)2637 4999 y(\014)2664 5020 y Fp(j)p Fx(Im)14 b Fw(\022)r Fp(j)42 b Fw(<)e(\022)3050 5032 y Fv(0)3087 5020 y Fp(g)e Fx(is)h(b)r(ounded)f(from)180 5128 y(Dom\(\003\))7 b Fp(\\)g Fx(Dom)r(\()p Fw(N)i Fx(\))22 b(to)g Fp(H)h Fx(\(and)f Fw(K)1331 5140 y Fs(\022)1368 5128 y Fw(f)31 b Fx(is)22 b(analytic)f(in)i Fw(\022)i Fp(2)e(f)p Fw(\022)i Fp(2)f Fo(C)2301 5097 y Fv(2)2344 5057 y Fm(\014)2344 5107 y(\014)2371 5128 y Fp(j)p Fx(Im)15 b Fw(\022)r Fp(j)23 b Fw(<)g(\022)2722 5140 y Fv(0)2759 5128 y Fp(g)p Fx(,)g Fp(8)p Fw(f)30 b Fp(2)24 b Fx(Dom\(\003\))7 b Fp(\\)g Fx(Dom)q(\()p Fw(N)i Fx(\)\).)180 5227 y(Moreo)n(v)n(er,)25 b(for)i Fp(j)p Fx(Im)h Fw(\022)r Fp(j)23 b Fw(>)g Fx(0)k(the)h(op)r (erators)e Fw(K)1671 5239 y Fs(\022)1735 5227 y Fx(are)h(closed)g(on)g (the)h(domain)f(Dom)q(\(\003\))18 b Fp(\\)h Fx(Dom\()p Fw(N)9 b Fx(\).)1938 5476 y(18)p eop %%Page: 19 19 19 18 bop 305 210 a Fx(Ho)n(w)n(ev)n(er,)29 b Fp(f)p Fw(K)779 222 y Fs(\022)845 210 y Fp(j)i Fx(Im)p Fw(\022)f(<)d(\022)1198 222 y Fv(0)1235 210 y Fp(g)j Fx(is)g(not)g(an)g(analytic)g(family)g(in) h(the)g(sense)e(of)h(Kato.)44 b(The)31 b(problem)e(here)h(is)180 310 y(the)j(lac)n(k)f(of)h(co)r(ercivit)n(y)f({)g(the)i(p)r (erturbation)e Fw(I)40 b Fx(is)33 b(not)f(b)r(ounded)i(relativ)n(ely)d (to)i(the)g(unp)r(erturb)r(ed)h(op)r(erator)180 409 y Fw(L)237 421 y Fv(0)274 409 y Fx(.)56 b(T)-7 b(o)34 b(comp)r(ensate)g (for)g(this)g(w)n(e)g(ha)n(v)n(e)f(c)n(hosen)g(the)i(deformation)e Fw(U)2517 421 y Fs(\022)2589 409 y Fx(in)h(suc)n(h)g(a)g(w)n(a)n(y)f (that)h(the)h(op)r(erator)180 509 y Fw(M)261 521 y Fs(\022)336 509 y Fx(:=)i(Im)p Fw(L)617 521 y Fv(0)p Fs(;\022)743 509 y Fx(is)g(co)r(erciv)n(e)d(for)i(Im)p Fw(\022)k(>)d Fx(0)f(,)j(i.e.,)g(the)d(p)r(erturbation)g Fw(I)2567 521 y Fs(\022)2642 509 y Fx(as)f(w)n(ell)h(as)g(Re)p Fw(L)3195 521 y Fv(0)p Fs(;\022)3321 509 y Fx(are)g(b)r(ounded)180 609 y(relativ)n(e)e(to)h(this)g(op)r(erator.)58 b(The)35 b(problem)f(here)h(is)g(that)g Fw(M)2206 621 y Fs(\022)2278 609 y Fp(!)h Fx(0)f(as)f(Im)p Fw(\022)k Fp(!)d Fx(0)g(so)f(w)n(e)h(ha)n (v)n(e)f(to)h(pro)r(ceed)180 708 y(carefully)-7 b(.)305 808 y(Ev)n(erything)26 b(said)h(ab)r(out)g Fw(K)1212 820 y Fs(\022)1277 808 y Fx(applies)g(also)g(to)g(the)h(family)g Fw(K)2290 823 y Fv(\()p Fs(s)p Fv(\))p Fs(\022)2410 808 y Fx(.)305 908 y(The)d(next)g(result)f(is)h(giv)n(es)f(some)g (analyticit)n(y)g(prop)r(erties)g(and)h(some)g(global)e(sp)r(ectral)i (prop)r(erties)f(of)h Fw(K)3637 923 y Fv(\()p Fs(s)p Fv(\))p Fs(\022)3757 908 y Fx(.)180 1069 y Fn(Theorem)30 b(6.1)h(\([25)o(]\))42 b Fq(Assume)25 b(that)h(Condition)h(\(A\))f (holds)h(and)g(let)f Fw(\022)2534 1081 y Fv(0)2594 1069 y Fx(=)d(\()p Fw(\016)2751 1081 y Fv(0)2788 1069 y Fw(;)14 b(\034)2861 1081 y Fv(0)2899 1069 y Fx(\))26 b Fq(b)l(e)g(as)g(in)g (that)g(c)l(ondition.)180 1169 y(T)-6 b(ake)31 b(an)1274 1364 y Fw(a)23 b(>)1552 1308 y(g)1595 1278 y Fv(2)p 1439 1345 306 4 v 1439 1421 a Fx(sin\(Im)p Fw(\016)s Fx(\))1755 1364 y Fw(C)1820 1330 y Fv(2)1814 1384 y(0)1871 1197 y Fm(0)1871 1347 y(@)1987 1260 y Fv(2)1944 1285 y Fm(X)1946 1462 y Fs(j)s Fv(=1)2077 1364 y Fp(k)p Fw(G)2184 1376 y Fs(j)2219 1364 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newpath 0 113 moveto 0 0 lineto 239 0 lineto 239 113 lineto closepath clip newpath -21.3 162.4 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def 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b(\(8.1\))g(is)f(wel)t(l)h(de\014ne)l(d)f(in)g(virtue)g(of)180 5221 y(The)l(or)l(em)c(6.1,)g(\(6.17\),)h(and)e(the)g(estimate)g Fp(j)p Fw(e)1658 5191 y Fs(iz)r(t)1745 5221 y Fp(j)23 b(\024)g Fw(e)1918 5191 y Fr(\000)p Fs(\025\034)2047 5166 y Fh(0)2068 5191 y Fs(=)p Fv(3)p Fs(C)2221 5221 y Fq(on)30 b(the)g(in\014nite)f(br)l(anches)i(of)f Fx(\000)p Fq(.)1938 5476 y Fx(21)p eop %%Page: 22 22 22 21 bop 305 210 a Fq(Pr)l(o)l(of.)68 b Fx(In)28 b(the)g(pro)r(of)f(b) r(elo)n(w,)h(the)g(v)n(ectors)e Fw(\036)i Fx(and)g(\011)f(are)g(as)g (in)h(the)g(theorem.)36 b(T)-7 b(o)28 b(pro)n(v)n(e)e(the)i(equalit)n (y)f(in)180 310 y(\(8.1\))g(w)n(e)h(use)f(the)h(family)g Fw(K)1112 325 y Fv(\()p Fs(s)p Fv(\))1226 310 y Fx(of)g(op)r(erators)d (de\014ned)j(in)g(\(5.13\)-\(5.14\).)35 b(First)28 b(w)n(e)f(sho)n(w) 180 469 y Fn(Lemma)j(8.2)40 b Fx(\012)23 b Fp(2)g Fx(Dom)q(\()p Fw(e)1089 439 y Fs(\027)t(N)1189 469 y Fx(\))30 b Fq(for)g(any)g Fw(\027)f(>)22 b Fx(0)p Fq(.)180 645 y(Pr)l(o)l(of)30 b(of)h(L)l(emma)e(8.2.)68 b Fx(The)27 b(result)g(follo)n(ws)f(from)h (the)g(relation)g(\012)2371 657 y Fv(0)2431 645 y Fp(2)c Fx(Dom\()p Fw(e)2754 614 y Fs(\027)t(N)2854 645 y Fw(e)2893 614 y Fr(\000)p Fs(\014)s(L)3032 589 y Fk(\()p Fi(`)p Fk(\))3105 614 y Fs(=)p Fv(2)3176 645 y Fx(\))28 b(whic)n(h)f(is)g(sho) n(wn)180 760 y(in)h(the)g(same)f(w)n(a)n(y)f(as)h(the)h(relation)f (\012)1406 772 y Fv(0)1466 760 y Fp(2)c Fx(Dom)q(\()p Fw(e)1790 730 y Fr(\000)p Fs(\014)s(L)1929 705 y Fk(\()p Fi(`)p Fk(\))2001 730 y Fs(=)p Fv(2)2073 760 y Fx(\),)28 b(see)f(Prop)r(osition)f(4.1.)848 b Fg(\004)305 860 y Fx(Next,)24 b(w)n(e)f(de\014ne)h(the)f(op)r(erator)f Fw(e)1389 829 y Fs(iK)1468 840 y Fk(\()p Fi(s)p Fk(\))1545 829 y Fs(t)1597 860 y Fx(as)h(follo)n(ws:)34 b Fw(e)2037 829 y Fs(iK)2116 840 y Fk(\()p Fi(s)p Fk(\))2193 829 y Fs(t)2222 860 y Fx(\012)23 b(is)g(giv)n(en)g(b)n(y)g(a)g(Dyson)g (expansion,)g(where)g(the)180 959 y(part)30 b Fp(\000)p Fw(g)s(V)537 929 y Fr(0)519 980 y Fs(s)592 959 y Fx(of)h Fw(K)761 974 y Fv(\()p Fs(s)p Fv(\))878 959 y Fx(is)g(treated)g(as)f(a) h(p)r(erturbation.)46 b(The)31 b(fact)g(that)g(the)g(Dyson)g(series)f (con)n(v)n(erges)e(is)j(easily)180 1059 y(seen)i(from)g(the)g(relation) f(\012)g Fp(2)h Fx(Dom\()p Fw(e)1451 1029 y Fs(\027)t(N)1551 1059 y Fx(\),)i Fw(\027)i(>)32 b Fx(0,)i(sho)n(wn)e(ab)r(o)n(v)n(e.)52 b(Moreo)n(v)n(er,)32 b(it)h(is)g(clear)f(that)i(this)f(series)180 1158 y(de\014nes)c(a)g(v)n(ector)f(whic)n(h)i(is)f(analytic)f(in)i Fw(s)c Fp(2)g Fw(S)1714 1170 y Fs(")1750 1158 y Fx(,)k(i.e.,)g Fw(s)25 b Fp(7!)h Fw(e)2174 1128 y Fs(iK)2253 1139 y Fk(\()p Fi(s)p Fk(\))2330 1128 y Fs(t)2360 1158 y Fx(\012)j(is)g (analytic)g(for)g Fw(s)c Fp(2)i Fw(S)3178 1170 y Fs(")3213 1158 y Fx(.)42 b(W)-7 b(e)30 b(de\014ne)g(the)180 1258 y(action)d(of)h Fw(e)563 1228 y Fs(iK)642 1239 y Fk(\()p Fi(s)p Fk(\))718 1228 y Fs(t)775 1258 y Fx(on)g(v)n(ectors)e Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012,)j Fw(A)23 b Fp(2)g(A)28 b Fx(\(whic)n(h)g(form)f(a)h(dense)f(set\),)h(b)n(y)1380 1433 y Fw(e)1419 1398 y Fs(iK)1498 1409 y Fk(\()p Fi(s)p Fk(\))1575 1398 y Fs(t)1604 1433 y Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)c(=)f Fw(\033)2002 1398 y Fs(t)2031 1433 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\))p Fw(e)2310 1398 y Fs(iK)2389 1409 y Fk(\()p Fi(s)p Fk(\))2468 1398 y Fs(t)2497 1433 y Fx(\012)p Fw(:)1029 b Fx(\(8.3\))180 1607 y(Consequen)n(tly)-7 b(,)34 b(the)g(map)g Fw(s)e Fp(7!)h Fw(e)1284 1577 y Fs(iK)1363 1588 y Fk(\()p Fi(s)p Fk(\))1440 1577 y Fs(t)1469 1607 y Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)i(is)e(analytic)g(for)g Fw(s)g Fp(2)g Fw(S)2495 1619 y Fs(")2531 1607 y Fx(.)55 b(F)-7 b(or)32 b Fw(s)h Fx(=)2943 1574 y Fr(\000)p Fs(i)p 2943 1588 76 4 v 2964 1636 a Fv(2)3062 1607 y Fx(this)g(de\014nition)h(giv)n(es) 180 1707 y Fw(e)219 1676 y Fs(iK)t(t)331 1707 y Fx(.)305 1806 y(Since)27 b Fw(K)592 1821 y Fv(\()p Fs(s)p Fv(\))707 1806 y Fx(is)g(self-adjoin)n(t)g(for)h Fw(s)f Fx(real)g(w)n(e)g(deriv)n (e)g(from)g(Stone's)g(form)n(ula)897 1965 y Fm(\012)936 2032 y Fw(\036;)14 b(e)1061 1998 y Fs(iK)1140 2009 y Fk(\()p Fi(s)p Fk(\))1217 1998 y Fs(t)1246 2032 y Fx(\011)1311 1965 y Fm(\013)1373 2032 y Fx(=)23 b Fp(\000)1575 1976 y Fx(1)p 1536 2013 121 4 v 1536 2089 a(2)p Fw(\031)s(i)1697 1919 y Fm(I)1661 2157 y Fl(R)p Fr(\000)p Fs(i)1811 1965 y Fm(\012)1850 2032 y Fw(\036;)14 b Fx(\()p Fw(K)2039 2047 y Fv(\()p Fs(s)p Fv(\))2145 2032 y Fp(\000)k Fw(z)t Fx(\))2303 1998 y Fr(\000)p Fv(1)2392 2032 y Fx(\011)2456 1965 y Fm(\013)2509 2032 y Fw(e)2548 1998 y Fs(iz)r(t)2635 2032 y Fw(dz)t(;)96 b(s)23 b Fp(2)g Fo(R)p Fw(:)552 b Fx(\(8.4\))180 2329 y(Next,)28 b(using)f Fw(e)662 2299 y Fs(iz)r(t)772 2329 y Fx(=)877 2296 y Fv(1)p 869 2310 49 4 v 869 2357 a Fs(it)1025 2296 y(@)p 938 2310 215 4 v 938 2357 a(@)t Fv(\(Re)11 b Fs(z)r Fv(\))1162 2329 y Fw(e)1201 2299 y Fs(iz)r(t)1315 2329 y Fx(and)28 b(in)n(tegrating)e (b)n(y)h(parts)g(w)n(e)g(can)g(represen)n(t)g(the)h(r.h.s.)36 b(of)28 b(\(8.4\))f(as)1101 2568 y Fw(R)q(H)7 b(S)e Fx(\(8)p Fw(:)p Fx(4\))21 b(=)i Fp(\000)1691 2512 y Fx(1)p 1652 2549 121 4 v 1652 2625 a(2)p Fw(\031)s(i)1813 2455 y Fm(Z)1777 2693 y Fl(R)p Fr(\000)p Fs(i)1927 2501 y Fm(\012)1966 2568 y Fw(\036;)14 b Fx(\()p Fw(K)2155 2583 y Fv(\()p Fs(s)p Fv(\))2261 2568 y Fp(\000)k Fw(z)t Fx(\))2419 2534 y Fr(\000)p Fv(2)2508 2568 y Fx(\011)2572 2501 y Fm(\013)2625 2568 y Fw(e)2664 2534 y Fs(iz)r(t)2751 2568 y Fw(dz)t(:)180 2854 y Fx(No)n(w)27 b(w)n(e)g(p)r(erform)g(the)h(sp)r (ectral)f(deformation,)g(Theorem)g(6.1,)g(\(iv\),)h(to)g(obtain)f(for)g (Im)p Fw(\022)e(>)e Fx(0)1007 3071 y Fw(R)q(H)7 b(S)e Fx(\(8)p Fw(:)p Fx(4\))22 b(=)g Fp(\000)1597 3015 y Fx(1)p 1558 3052 V 1558 3128 a(2)p Fw(\031)s(i)1707 3015 y Fx(1)p 1698 3052 59 4 v 1698 3128 a Fw(it)1798 2958 y Fm(Z)1762 3196 y Fl(R)p Fr(\000)p Fs(i)1912 3004 y Fm(\012)1951 3071 y Fw(\036)p 2000 3048 38 3 v 26 x Fs(\022)2038 3071 y Fw(;)14 b Fx(\()p Fw(K)2178 3086 y Fv(\()p Fs(s)p Fv(\))p Fs(\022)2317 3071 y Fp(\000)k Fw(z)t Fx(\))2475 3037 y Fr(\000)p Fv(2)2564 3071 y Fx(\011)2629 3083 y Fs(\022)2666 3004 y Fm(\013)2719 3071 y Fw(e)2758 3037 y Fs(iz)r(t)2844 3071 y Fw(dz)t(:)656 b Fx(\(8.5\))180 3365 y(The)31 b(in)n(tegral)f (con)n(v)n(erges)f(since)i(due)g(to)h(\(6.17\))e(w)n(e)h(ha)n(v)n(e) 2073 3295 y Fm(\015)2073 3344 y(\015)2119 3365 y Fx(\()p Fw(K)2222 3380 y Fv(\()p Fs(s)p Fv(\))p Fs(\022)2364 3365 y Fp(\000)20 b Fw(z)t Fx(\))2524 3335 y Fr(\000)p Fs(n)2621 3365 y Fp(k)28 b(\024)h Fw(C)2844 3377 y Fs(n)2890 3365 y Fp(h)p Fx(Re)14 b Fw(z)t Fp(i)3109 3335 y Fr(\000)p Fs(n)3237 3365 y Fx(for)30 b Fw(z)j Fp(2)c Fo(R)e Fp(\000)21 b Fw(i")p Fx(,)180 3479 y(where)27 b Fp(h)p Fw(x)p Fp(i)d Fx(:=)666 3410 y Fp(p)p 735 3410 228 4 v 69 x Fx(1)18 b(+)g Fw(x)925 3455 y Fv(2)963 3479 y Fx(.)305 3578 y(In)26 b(\(8.5\))f(w)n(e)h(deform)f(the)h(con)n(tour)f(of)h(in)n(tegration)e (from)i Fo(R)21 b Fp(\000)14 b Fw(i)26 b Fx(to)f(\000)h(whic)n(h)g(is)f 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Fx(\(see)j(Theorem)g(6.1,)f(\(6.17\),)h(and)180 4269 y(after)36 b(equation)f(\(8.3\)\).)62 b(Since)36 b(they)g(are)f(equal)g(for)h(real)e Fw(s)i Fx(they)g(are)f(equal)h(for) f(all)h Fw(s)g Fx(in)g(their)g(domain)f(of)180 4368 y(analyticit)n(y)-7 b(.)36 b(T)-7 b(aking)27 b Fw(s)c Fx(=)g Fp(\000)1124 4336 y Fs(i)p 1119 4350 34 4 v 1119 4397 a Fv(2)1190 4368 y Fx(in)28 b(this)g(form)n(ula)e(giv)n(es)h(\(8.1\).)p 3714 4380 59 71 v 305 4468 a(It)k(is)g(sho)n(wn)g(in)g(Section)g(7)g (that)g(the)h(op)r(erator)d(family)j Fw(K)2205 4480 y Fs(\022)2273 4468 y Fx(has)f(a)f(simple)i(isolated)e(eigen)n(v)-5 b(alue)30 b(at)h(0)g(and)180 4568 y(the)d(rest)f(of)h(its)g(sp)r (ectrum)f(is)h(lo)r(cated)f(in)h(a)f(truncated)h(cone)f(in)h Fp(f)p Fw(z)e Fp(2)d Fo(C)2500 4537 y Fv(+)2562 4568 y Fp(j)p Fx(Im)14 b Fw(z)26 b(>)2861 4535 y Fv(1)p 2861 4549 34 4 v 2861 4596 a(3)2904 4568 y Fw(\034)2949 4537 y Fr(0)2973 4568 y Fp(g)p Fx(,)h(where)g Fw(\034)3350 4537 y Fr(0)3397 4568 y Fx(=)c(Im)14 b Fw(\034)9 b Fx(.)38 b(In)180 4667 y(the)28 b(in)n(tegral)e(on)i(the)g(r.h.s.)36 b(of)28 b(form)n(ula)e(\(8.1\))i(w)n(e)f(deform)g(the)h(con)n(tour)e (of)i(in)n(tegration)e(to)307 4842 y(\000)359 4807 y Fr(0)405 4842 y Fx(:=)d Fp(f)p Fw(z)j Fx(=)c Fw(\025)d Fx(+)870 4809 y Fv(1)p 870 4823 V 870 4870 a(3)913 4842 y Fw(i\034)987 4807 y Fr(0)1011 4842 y Fw(;)14 b Fp(j)p Fw(\025)p Fp(j)23 b(\024)g Fw(C)6 b Fp(g)18 b([)h(f)p Fw(z)26 b Fx(=)c Fw(\025)d Fx(+)f Fw(i\025\034)1918 4807 y Fr(0)1942 4842 y Fw(=)p Fx(3)p Fw(C)q(;)c(\025)23 b Fp(\025)g Fw(C)6 b Fp(g)18 b([)h(f)p Fw(z)26 b Fx(=)d Fw(\025)18 b Fp(\000)g Fw(i\025\034)2947 4807 y Fr(0)2971 4842 y Fw(=)p Fx(3)p Fw(C)q(;)c(\025)24 b Fp(\024)e(\000)p Fw(C)6 b Fp(g)126 b Fx(\(8.7\))180 5016 y(where)30 b Fw(C)37 b Fx(is)31 b(su\016cien)n(tly)f(large.)45 b(Pic)n(king)29 b(up)i(the)g(residue)f(from)h(the)g(simple)f(eigen)n(v)-5 b(alue)30 b(0)g(of)h Fw(K)3370 5028 y Fs(\022)3438 5016 y Fx(w)n(e)f(deriv)n(e)180 5116 y(from)d(\(8.1\))1279 5160 y Fm(\012)1318 5227 y Fw(\036;)14 b(e)1443 5193 y Fs(iK)t(t)1556 5227 y Fx(\011)p Fp(i)23 b Fx(=)f Fp(h)p Fw(\036)p 1844 5204 38 3 v 26 x Fs(\022)1883 5227 y Fw(;)14 b(P)1973 5239 y Fs(\022)2011 5227 y Fx(\011)2076 5239 y Fs(\022)2113 5160 y Fm(\013)2170 5227 y Fx(+)k Fw(O)r Fx(\()p Fw(e)2389 5193 y Fr(\000)p Fs(\034)2479 5168 y Fh(0)2502 5193 y Fs(t=)p Fv(3)2598 5227 y Fx(\))28 b Fw(;)928 b Fx(\(8.8\))1938 5476 y(22)p eop %%Page: 23 23 23 22 bop 180 210 a Fx(where)23 b Fw(P)469 222 y Fs(\022)530 210 y Fx(=)634 178 y Fr(\000)p Fv(1)p 628 192 98 4 v 628 239 a(2)p Fs(\031)r(i)749 143 y Fm(H)805 210 y Fx(\()p Fw(K)908 222 y Fs(\022)956 210 y Fp(\000)11 b Fw(z)t Fx(\))1107 180 y Fr(\000)p Fv(1)1196 210 y Fw(dz)27 b Fx(is)d(the)g(eigenpro)5 b(jection)23 b(of)h Fw(K)2262 222 y Fs(\022)2323 210 y Fx(corresp)r(onding)e(to)i(the)g(simple)g(and) g(isolated)180 310 y(eigen)n(v)-5 b(alue)31 b(0.)49 b(The)31 b(con)n(tour)g(in)n(tegral)f(is)i(o)n(v)n(er)e(a)h(small)h(circle)f (around)f(the)j(origin.)48 b(This)31 b(is)h(the)g(only)g(place)180 409 y(where)26 b(w)n(e)g(use)g(that)h(0)f(is)h(an)f(isolated)g(eigen)n (v)-5 b(alue)25 b(of)h Fw(K)1989 421 y Fs(\022)2053 409 y Fx({)g(the)h(fact)g(w)n(e)f(sho)n(w)f(using)h(complex)h(translation)e (in)180 509 y(addition)i(to)h(complex)f(dilation.)305 609 y Fw(P)358 621 y Fs(\022)423 609 y Fx(is)h(a)f(rank)g(one)g(pro)5 b(jection)27 b(whic)n(h)g(is)h(analytic)f(in)h Fw(\022)d Fp(2)f Fw(S)2248 573 y Fv(+)2243 634 y Fs(\022)2275 642 y Fk(0)2311 609 y Fx(.)37 b(One)27 b(pro)n(v)n(es)f(using)h(a)g (standard)g(argumen)n(t)180 708 y(that)h(it)g(satis\014es)f Fw(K)820 720 y Fs(\022)857 708 y Fw(P)910 720 y Fs(\022)971 708 y Fx(=)22 b Fw(P)1111 720 y Fs(\022)1149 708 y Fw(K)1220 720 y Fs(\022)1280 708 y Fx(=)h(0.)36 b(Hence,)28 b Fw(P)1792 720 y Fs(\022)1858 708 y Fx(can)f(b)r(e)h(written)g(as)1695 873 y Fw(P)1748 885 y Fs(\022)1809 873 y Fx(=)22 b Fp(j)q Fx(\012)1980 885 y Fs(\022)2017 873 y Fp(i)2063 806 y Fm(\012)2102 873 y Fx(\012)2162 839 y Fr(\003)p 2162 858 38 3 v 2162 907 a Fs(\022)2201 803 y Fm(\014)2201 853 y(\014)2242 873 y Fw(;)1344 b Fx(\(8.9\))180 1038 y(where)27 b(\012)480 1050 y Fs(\022)545 1038 y Fx(and)h(\012)767 1008 y Fr(\003)p 767 1027 V 767 1076 a Fs(\022)833 1038 y Fx(are)e(zero)h(eigen)n(v)n(ectors)e(of)i Fw(K)1776 1050 y Fs(\022)1841 1038 y Fx(and)h(its)f(adjoin)n(t)h(op)r(erator)1575 1212 y(\()p Fw(K)1678 1224 y Fs(\022)1716 1212 y Fx(\))1748 1178 y Fr(\003)1809 1212 y Fx(=)23 b(\()p Fw(K)2006 1178 y Fr(\003)2044 1212 y Fx(\))p 2076 1189 V 26 x Fs(\022)2136 1212 y Fx(=:)g Fw(K)2324 1178 y Fr(\003)p 2318 1196 V 2318 1245 a Fs(\022)2362 1212 y Fw(;)1183 b Fx(\(8.10\))180 1377 y(i.e.)37 b(w)n(e)27 b(ha)n(v)n(e)1485 1476 y Fw(K)1556 1488 y Fs(\022)1593 1476 y Fx(\012)1653 1488 y Fs(\022)1714 1476 y Fx(=)22 b(0)55 b(and)g Fw(K)2164 1442 y Fr(\003)p 2158 1461 V 2158 1510 a Fs(\022)2202 1476 y Fx(\012)2262 1442 y Fr(\003)p 2262 1461 V 2262 1510 a Fs(\022)2323 1476 y Fx(=)23 b(0)p Fw(;)1092 b Fx(\(8.11\))180 1629 y(with)29 b(the)h(normalization)d Fp(h)p Fx(\012)1134 1599 y Fr(\003)p 1134 1618 V 1134 1666 a Fs(\022)1172 1629 y Fw(;)14 b Fx(\012)1269 1641 y Fs(\022)1307 1629 y Fp(i)25 b Fx(=)g(1.)41 b(Since)29 b Fw(P)1831 1641 y Fs(\022)1898 1629 y Fx(and)f(\012)2120 1641 y Fs(\022)2187 1629 y Fx(are)g(analytic)g(in)h Fw(\022)f Fp(2)d Fw(S)2945 1593 y Fv(+)2940 1654 y Fs(\022)2972 1662 y Fk(0)3009 1629 y Fx(,)k(then)g(so)g(is)f(\012)3499 1599 y Fr(\003)p 3499 1618 V 3499 1666 a Fs(\022)3567 1629 y Fx(in)h(the)180 1770 y(v)-5 b(ariable)p 495 1703 42 4 v 30 w Fw(\022)32 b Fp(2)d Fw(S)706 1735 y Fr(\000)701 1795 y Fs(\022)733 1803 y Fk(0)769 1770 y Fx(.)49 b(\(The)31 b(p)r(ossibilit)n(y)g(of)g (the)h(normalization,)2245 1678 y Fm(D)2296 1770 y Fx(\012)2356 1740 y Fr(\003)p 2356 1759 38 3 v 2356 1808 a Fs(\022)2394 1770 y Fw(;)14 b Fx(\012)2491 1782 y Fs(\022)2528 1678 y Fm(E)2608 1770 y Fp(6)p Fx(=)29 b(0,)j(follo)n(ws)e(also)g(from)h (results)g(of)180 1907 y(Section)25 b(11.)35 b(Analyticit)n(y)25 b(of)g(\012)1193 1877 y Fr(\003)p 1193 1896 V 1193 1944 a Fs(\022)1256 1907 y Fx(in)p 1351 1839 42 4 v 26 w Fw(\022)g Fp(2)f Fw(S)j Fp(2)d Fw(S)1707 1871 y Fr(\006)1702 1932 y Fs(\022)1734 1940 y Fk(0)1795 1907 y Fx(can)h(also)f(b)r(e)h(sho)n (wn)f(directly)h(b)n(y)g(using)g(the)g(analyticit)n(y)f(and)180 2015 y(sp)r(ectral)j(prop)r(erties)g(of)g Fw(K)1054 1985 y Fr(\003)p 1048 2004 38 3 v 1048 2053 a Fs(\022)1092 2015 y Fx(,)h(see)f(Section)g(7\).)37 b(Equation)27 b(\(8.9\))g (implies)1425 2130 y Fm(\012)1464 2197 y Fw(\036)p 1513 2174 V 26 x Fs(\022)1551 2197 y Fw(;)14 b(P)1641 2209 y Fs(\022)1679 2197 y Fx(\011)1744 2209 y Fs(\022)1781 2130 y Fm(\013)1843 2197 y Fx(=)23 b Fp(h)p Fw(\036;)14 b Fx(\012)p Fp(i)2155 2130 y Fm(\012)2195 2197 y Fx(\012)2255 2163 y Fr(\003)p 2255 2181 V 2255 2230 a Fs(\022)2293 2197 y Fw(;)g Fx(\011)2395 2209 y Fs(\022)2432 2130 y Fm(\013)2512 2197 y Fw(:)1033 b Fx(\(8.12\))305 2362 y Fq(R)l(emark.)75 b Fx(W)-7 b(e)30 b(presen)n(t)f(here)g(another)g (pro)r(of)g(of)g(the)i(relation)d Fw(z)2414 2374 y Fv(0)2478 2362 y Fx(=)e(0,)k(where)f Fw(z)2945 2374 y Fv(0)3011 2362 y Fx(is)h(the)g(simple)g(isolated)180 2461 y(eigen)n(v)-5 b(alue)26 b(of)i Fw(K)742 2473 y Fs(\022)806 2461 y Fx(giv)n(en)f(in)h (Theorem)e(7.4.)36 b(Starting)27 b(with)h(the)g(information)f(on)g(the) g(sp)r(ectrum)h(of)f Fw(K)3526 2473 y Fs(\022)3591 2461 y Fx(giv)n(en)180 2561 y(in)37 b(Theorem)g(7.4)f(and)h(pro)r(ceeding)f (with)i(a)e(con)n(tour)g(deformation)h(as)f(ab)r(o)n(v)n(e)g(w)n(e)g (\014nd)i(that)3303 2494 y Fm(\012)3343 2561 y Fw(\036;)14 b(e)3468 2531 y Fs(iK)t(t)3580 2561 y Fw( )3637 2494 y Fm(\013)3715 2561 y Fx(=)180 2677 y Fw(e)219 2647 y Fs(iz)274 2655 y Fk(0)306 2647 y Fs(t)349 2609 y Fm(\012)389 2677 y Fw(\036)p 438 2654 V 26 x Fs(\022)476 2677 y Fw(;)g(P)566 2689 y Fs(\022)603 2677 y Fw( )657 2689 y Fs(\022)695 2609 y Fm(\013)751 2677 y Fx(+)i Fw(O)r Fx(\()p Fw(e)968 2647 y Fv(\()p Fr(j)p Fv(Im)11 b Fs(z)1136 2655 y Fk(0)1168 2647 y Fr(j\000)p Fs(\034)1278 2621 y Fh(0)1300 2647 y Fs(=)p Fv(3\))p Fs(t)1422 2677 y Fx(\))27 b(instead)g(of)f(\(8.8\).) 37 b(Applying)27 b(this)g(form)n(ula)f(to)g Fw(\036)e Fx(=)e Fw( )k Fx(=)d(\012)k(and)f(using)180 2792 y(\(4.3\))d(w)n(e)g (obtain)g Fp(h)q Fx(\012)p Fw(;)14 b Fx(\012)p Fp(i)23 b Fx(=)g Fw(e)1118 2762 y Fs(iz)1173 2770 y Fk(0)1205 2762 y Fs(t)1248 2792 y Fp(h)p Fx(\012)p Fw(;)14 b Fx(\012)p Fp(i)c Fx(+)g Fw(O)r Fx(\()p Fw(e)1690 2762 y Fv(\()p Fr(j)p Fv(Im)h Fs(z)1858 2770 y Fk(0)1890 2762 y Fr(j\000)p Fs(\034)2000 2737 y Fh(0)2022 2762 y Fs(=)p Fv(3\))p Fs(t)2144 2792 y Fx(\).)36 b(Since)24 b Fp(j)p Fw(z)2510 2804 y Fv(0)2547 2792 y Fp(j)f Fx(=)f Fw(O)r Fx(\()p Fp(j)p Fw(g)s Fp(j)2866 2762 y Fv(2+)p Fs(\013)2998 2792 y Fx(\))i Fw(<)-28 b(<)22 b(\034)3223 2762 y Fr(0)3271 2792 y Fx(the)i(error)d(term)180 2907 y(tends)28 b(to)f(zero)g(as)g Fw(t)c Fp(!)g(1)p Fx(,)28 b(so)f(b)n(y)g(taking)g(lim)1662 2919 y Fs(T)9 b Fr(!1)1861 2841 y Fm(R)1916 2861 y Fs(T)1900 2937 y Fv(0)1982 2907 y Fw(dt)28 b Fx(on)f(b)r(oth)h(sides)g(w)n(e)f (see)g(that)h Fw(z)3069 2919 y Fv(0)3133 2907 y Fx(m)n(ust)g(b)r(e)g (zero.)180 3179 y Fy(9)180 b(Pro)t(of)45 b(of)g(Theorem)g(3.1)180 3361 y Fx(Let)33 b Fw( )i Fx(b)r(e)e(an)n(y)f Fw(!)755 3373 y Fv(0)792 3361 y Fx(-normal)f(state)i(on)f Fp(A)p Fx(.)52 b(F)-7 b(or)32 b(the)h(follo)n(wing)f(reasoning,)g(w)n(e)g(ma)n (y)g(assume)f(without)i(loss)f(of)180 3460 y(generalit)n(y)i(that)j Fw( )s Fx(\()p Fw(A)p Fx(\))h(=)e Fp(h)q Fx(\012)1175 3472 y Fs( )1225 3460 y Fw(;)14 b(\031)s Fx(\()p Fw(A)p Fx(\)\012)1498 3472 y Fs( )1549 3460 y Fp(i)p Fx(,)38 b(for)e(some)f(\012)2054 3472 y Fs( )2141 3460 y Fp(2)j(H)q Fx(.)62 b(Since)36 b(\012)g(is)g(cyclic)g(for)f Fw(\031)3225 3430 y Fr(0)3249 3460 y Fx(\()p Fp(A)p Fx(\),)k Fw( )g Fx(can)d(b)r(e)180 3560 y(appro)n(ximated)26 b(as)h(follo)n(ws.)36 b(F)-7 b(or)27 b(an)n(y)g Fw(\017)22 b(>)h Fx(0)k(there)h(is)f(a)g Fw(Q)c Fp(2)g Fw(\031)2214 3530 y Fr(0)2238 3560 y Fx(\()p Fp(A)p Fx(\))29 b(s.t.,)e(for)g(all)h Fw(A)23 b Fp(2)h(A)p Fx(,)1386 3725 y Fp(j)p Fw( )s Fx(\()p Fw(A)p Fx(\))19 b Fp(\000)f(h)q Fw(Q)p Fx(\012)o Fw(;)c(\031)s Fx(\()p Fw(A)p Fx(\))p Fw(Q)p Fx(\012)q Fp(i)g(j)23 b Fw(<)g(\017)p Fp(k)p Fw(A)p Fp(k)p Fw(:)1034 b Fx(\(9.1\))180 3890 y(T)-7 b(aking)27 b(in)n(to)g(accoun)n(t)g(\(2.30\),)g Fw(U)9 b Fx(\()p Fw(t)p Fx(\))23 b(=)g Fw(e)1505 3860 y Fs(itK)1645 3890 y Fx(and)k Fw(U)9 b Fx(\()p Fw(t)p Fx(\)\012)23 b(=)g(0,)k(w)n(e)g(obtain)1270 4055 y Fp(j)p Fw( )1350 4021 y Fs(t)1379 4055 y Fx(\()p Fw(A)p Fx(\))19 b Fp(\000)1607 3988 y Fm(\012)1647 4055 y Fw(Q)1713 4021 y Fr(\003)1750 4055 y Fw(Q)p Fx(\012)p Fw(;)14 b(e)1952 4021 y Fs(itK)2064 4055 y Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)2301 3988 y Fm(\013)2354 4055 y Fp(j)23 b Fw(<)g(\017)p Fp(k)p Fw(A)p Fp(k)p Fw(;)918 b Fx(\(9.2\))180 4228 y(uniformly)20 b(in)h Fw(t)i Fp(2)g Fo(R)p Fx(.)41 b(In)20 b(order)f(to)h(examine)g (the)h(long)e(time)i(b)r(eha)n(viour)e(of)2563 4161 y Fm(\012)2602 4228 y Fw(Q)2668 4198 y Fr(\003)2706 4228 y Fw(Q)p Fx(\012)p Fw(;)14 b(e)2908 4198 y Fs(itK)3020 4228 y Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)3257 4161 y Fm(\013)3296 4228 y Fx(,)22 b(via)e(\(8.8\),)h(w)n(e)180 4328 y(\014rst)i(appro)n(ximate)e(the)j(v)n(ector)d Fw(Q)1271 4298 y Fr(\003)1309 4328 y Fw(Q)p Fx(\012)i Fp(2)g(H)h Fx(b)n(y)f(a)g(family)g(of)g Fw(U)2202 4340 y Fs(\022)2239 4328 y Fx(-analytic)f(v)n(ectors)g Fw(\037)2908 4340 y Fs(\017)2939 4328 y Fx(,)j(s.t.)35 b Fp(k)p Fw(\037)3227 4340 y Fs(\017)3268 4328 y Fp(\000)9 b Fw(Q)3408 4298 y Fr(\003)3445 4328 y Fw(Q)p Fx(\012)p Fp(k)23 b Fw(<)f(\017)p Fx(.)180 4427 y(W)-7 b(e)28 b(ha)n(v)n(e)715 4527 y Fp(j)752 4460 y Fm(\012)791 4527 y Fw(Q)857 4493 y Fr(\003)895 4527 y Fw(Q)p Fx(\012)18 b Fp(\000)g Fw(\037)1174 4539 y Fs(\017)1206 4527 y Fw(;)c(e)1282 4493 y Fs(itK)1394 4527 y Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)1631 4460 y Fm(\013)1684 4527 y Fp(j)23 b Fw(<)g(\017)p Fp(k)p Fw(e)1933 4493 y Fs(itK)2045 4527 y Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)p Fp(k)g Fx(=)g Fw(\017)p Fp(k)p Fw(\033)2560 4493 y Fs(t)2589 4527 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\)\012)p Fp(k)h Fw(<)f(\017)p Fp(k)p Fw(A)p Fp(k)p Fw(:)363 b Fx(\(9.3\))180 4666 y(It)28 b(follo)n(ws)f(from)g(\(9.2\),)g(\(9.3\),)h (\(8.8\))f(and)g(\(8.12\))g(that)1029 4760 y Fm(\014)1029 4810 y(\014)1056 4831 y Fw( )1113 4797 y Fs(t)1143 4831 y Fx(\()p Fw(A)p Fx(\))19 b Fp(\000)f(h)p Fw(\037)1455 4843 y Fs(\017)1487 4831 y Fw(;)c Fx(\012)p Fp(i)1630 4764 y Fm(\012)1669 4831 y Fx(\012)1729 4797 y Fr(\003)p 1729 4815 V 1729 4864 a Fs(\022)1767 4831 y Fw(;)g Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))2104 4843 y Fs(\022)2143 4764 y Fm(\013)2183 4760 y(\014)2183 4810 y(\014)2233 4831 y Fp(\024)23 b Fx(2)p Fw(\017)p Fp(k)p Fw(A)p Fp(k)17 b Fx(+)h Fp(j)p Fw(R)2729 4843 y Fs(t)2758 4831 y Fx(\()p Fw(A)p Fx(\))p Fp(j)p Fw(;)679 b Fx(\(9.4\))180 4996 y(where)372 5190 y Fp(j)p Fw(R)458 5202 y Fs(t)488 5190 y Fx(\()p Fw(A)p Fx(\))p Fp(j)24 b Fx(=)783 5134 y(1)p 758 5171 92 4 v 758 5247 a(2)p Fw(\031)874 5069 y Fm(\014)874 5119 y(\014)874 5169 y(\014)874 5219 y(\014)901 5077 y(I)948 5265 y Fv(\000)989 5249 y Fh(0)1029 5098 y Fm(D)1080 5190 y Fw(\037)1132 5216 y Fs(\017;)p 1180 5167 34 3 v(\022)1217 5190 y Fw(;)14 b Fx(\()p Fw(K)1357 5202 y Fs(\022)1412 5190 y Fp(\000)k Fw(z)t Fx(\))1570 5156 y Fr(\000)p Fv(1)1659 5190 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))1959 5202 y Fs(\022)1998 5098 y Fm(E)2063 5190 y Fw(e)2102 5156 y Fs(iz)r(t)2188 5190 y Fw(dz)2274 5069 y Fm(\014)2274 5119 y(\014)2274 5169 y(\014)2274 5219 y(\014)2325 5190 y Fp(\024)k Fw(C)6 b Fp(k)p Fw(\037)2571 5216 y Fs(\017;)p 2619 5167 V(\022)2656 5190 y Fp(k)27 b(k)p Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))3067 5202 y Fs(\022)3106 5190 y Fp(k)p Fw(e)3187 5156 y Fr(\000)p Fs(\034)3277 5130 y Fh(0)3298 5156 y Fs(t=)p Fv(3)3394 5190 y Fw(:)192 b Fx(\(9.5\))1938 5476 y(23)p eop %%Page: 24 24 24 23 bop 180 210 a Fx(The)28 b(path)f(\000)596 180 y Fr(0)647 210 y Fx(in)h(\(9.5\))f(is)h(de\014ned)g(in)g(\(8.7\).)305 310 y(Since)e Fp(h)p Fw(Q)618 280 y Fr(\003)656 310 y Fw(Q)p Fx(\012)o Fw(;)14 b Fx(\012)p Fp(i)24 b(!)f Fx(1)i(as)h Fw(\017)d Fp(!)g Fx(0,)j(w)n(e)f(ha)n(v)n(e)g Fp(h)q Fw(\037)1857 322 y Fs(\017)1888 310 y Fw(;)14 b Fx(\012)p Fp(i)24 b Fx(=)e(1)15 b(+)g Fw(o)p Fx(\()p Fw(\017)2371 280 y Fv(0)2409 310 y Fx(\),)26 b(where)g Fw(o)p Fx(\()p Fw(\017)2835 280 y Fv(0)2872 310 y Fx(\))h(denotes)f(a)f(quan)n(tit)n (y)h(that)180 409 y(v)-5 b(anishes)27 b(in)h(the)g(limit)g Fw(\017)23 b Fp(!)g Fx(0.)37 b(Th)n(us)391 574 y Fp(j)p Fw( )471 540 y Fs(t)500 574 y Fx(\()p Fw(A)p Fx(\))19 b Fp(\000)f(h)q Fw(\037)813 586 y Fs(\017)844 574 y Fw(;)c Fx(\012)p Fp(i)987 507 y Fm(\012)1027 574 y Fx(\012)1087 540 y Fr(\003)p 1087 559 38 3 v 1087 607 a Fs(\022)1125 574 y Fw(;)g Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))1462 586 y Fs(\022)1501 507 y Fm(\013)1554 574 y Fp(j)23 b(\025)g(j)p Fw( )1768 540 y Fs(t)1797 574 y Fx(\()p Fw(A)p Fx(\))c Fp(\000)2025 507 y Fm(\012)2065 574 y Fx(\012)2125 540 y Fr(\003)p 2125 559 V 2125 607 a Fs(\022)2163 574 y Fw(;)14 b Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))2500 586 y Fs(\022)2539 507 y Fm(\013)2592 574 y Fp(j)k(\000)g Fw(o)p Fx(\()p Fw(\017)2822 540 y Fv(0)2860 574 y Fx(\))28 b Fp(j)2957 507 y Fm(\012)2996 574 y Fx(\012)3056 540 y Fr(\003)p 3056 559 V 3056 607 a Fs(\022)3094 574 y Fw(;)14 b Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))3431 586 y Fs(\022)3470 507 y Fm(\013)3523 574 y Fp(j)p Fw(:)180 739 y Fx(Com)n(bining)27 b(this)h(estimate)f(with)i(\(9.4\))e(and)g (\(9.5\))h(w)n(e)f(arriv)n(e)f(at)731 903 y Fp(j)p Fw( )811 869 y Fs(t)840 903 y Fx(\()p Fw(A)p Fx(\))19 b Fp(\000)1068 836 y Fm(\012)1107 903 y Fx(\012)1167 869 y Fr(\003)p 1167 888 V 1167 937 a Fs(\022)1206 903 y Fw(;)14 b Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))1543 915 y Fs(\022)1582 836 y Fm(\013)1635 903 y Fp(j)897 1052 y(\024)22 b Fw(o)p Fx(\()p Fw(\017)1090 1018 y Fv(0)1128 1052 y Fx(\))1174 985 y Fm(\000)1212 1052 y Fp(k)p Fw(A)p Fp(k)c Fx(+)g Fp(j)1496 985 y Fm(\012)1535 1052 y Fx(\012)1595 1018 y Fr(\003)p 1595 1037 V 1595 1086 a Fs(\022)1633 1052 y Fw(;)c Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))1970 1064 y Fs(\022)2009 985 y Fm(\013)2062 1052 y Fp(j)2085 985 y Fm(\001)2142 1052 y Fx(+)k Fw(C)6 b Fp(k)p Fw(\037)2384 1078 y Fs(\017;)p 2432 1029 34 3 v(\022)2468 1052 y Fp(k)27 b(k)p Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))2879 1064 y Fs(\022)2918 1052 y Fp(k)p Fw(e)2999 1018 y Fr(\000)p Fs(\034)3089 993 y Fh(0)3110 1018 y Fs(t=)p Fv(3)3206 1052 y Fw(;)380 b Fx(\(9.6\))180 1230 y(where)27 b Fw(o)p Fx(\()p Fw(\017)526 1200 y Fv(0)564 1230 y Fx(\))h(is)f(indep)r(enden)n (t)h(of)g Fw(A)g Fx(and)f Fw(\022)r Fx(.)38 b(In)27 b(particular,)g (taking)g(\014rst)g Fw(t)c Fp(!)g(1)28 b Fx(and)f(then)h Fw(\017)23 b Fp(!)g Fx(0)k(yields)1459 1395 y(lim)1438 1445 y Fs(t)p Fr(!1)1609 1395 y Fw( )1666 1360 y Fs(t)1695 1395 y Fx(\()p Fw(A)p Fx(\))d(=)1933 1327 y Fm(\012)1972 1395 y Fx(\012)2032 1360 y Fr(\003)p 2032 1379 38 3 v 2032 1428 a Fs(\022)2070 1395 y Fw(;)14 b Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))2407 1407 y Fs(\022)2446 1327 y Fm(\013)2499 1395 y Fw(:)1087 b Fx(\(9.7\))180 1622 y(Equation)26 b(\(9.7\))h(sho)n(ws)e(that)j Fp(j)1190 1529 y Fm(D)1240 1622 y Fx(\012)1300 1591 y Fr(\003)p 1300 1610 V 1300 1659 a Fs(\022)1338 1622 y Fw(;)14 b Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))1675 1634 y Fs(\022)1714 1529 y Fm(E)1779 1622 y Fp(j)23 b(\024)g(k)p Fw(A)p Fp(k)p Fx(.)36 b(W)-7 b(e)27 b(re-inject)g(this)g(inequalit)n(y) g(in)n(to)f(r.h.s.)36 b(of)27 b(\(9.6\))180 1739 y(to)g(arriv)n(e)f(at) 1478 1870 y(lim)1457 1919 y Fs(t)p Fr(!1)1640 1870 y Fx(sup)1628 1956 y Fs(A)p Fr(2)1743 1941 y Fe(e)1723 1956 y Fr(A)1786 1813 y Fp(j)p Fw( )1866 1783 y Fs(t)1895 1813 y Fx(\()p Fw(A)p Fx(\))20 b Fp(\000)e Fw(\021)s Fx(\()p Fw(A)p Fx(\))p Fp(j)p 1786 1851 532 4 v 1951 1927 a(k)p Fw(A)p Fp(k)2097 1939 y Fr(\030)2351 1870 y Fx(=)23 b(0)p Fw(;)1105 b Fx(\(9.8\))180 2074 y(where)1117 2153 y Fm(e)1090 2173 y Fp(A)24 b Fx(=)e Fp(f)p Fw(A)h Fp(2)h(A)k(j)f Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)i(is)f Fw(U)2023 2185 y Fs(\022)2060 2173 y Fx(-analytic)f(for)g Fp(j)p Fw(\022)r Fp(j)c Fw(<)g(\022)2768 2185 y Fv(0)2805 2173 y Fp(g)p Fw(;)739 b Fx(\(9.9\))180 2322 y(and)27 b(where)h Fp(k)17 b(\001)i(k)725 2334 y Fr(\030)808 2322 y Fx(is)28 b(the)g(norm)f(on)1393 2301 y Fm(e)1367 2322 y Fp(A)h Fx(de\014ned)g(b)n(y)1152 2497 y Fp(k)p Fw(A)p Fp(k)1298 2509 y Fr(\030)1376 2497 y Fx(=)23 b Fp(k)p Fw(A)p Fp(k)18 b Fx(+)50 b(sup)1711 2571 y Fr(j)p Fs(\022)r Fr(j)p Fs(<\022)1869 2579 y Fk(0)1913 2497 y Fp(k)p Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))2255 2509 y Fs(\022)2294 2497 y Fp(k)p Fw(;)69 b Fx(for)27 b Fw(A)c Fp(2)2745 2476 y Fm(e)2719 2497 y Fp(A)p Fw(:)760 b Fx(\(9.10\))180 2748 y(The)28 b(state)f Fw(\021)k Fx(on)767 2727 y Fm(e)741 2748 y Fp(A)d Fx(in)g(\(9.8\))f(is)h(giv)n(en)f(b)n(y)1533 2913 y Fw(\021)s Fx(\()p Fw(A)p Fx(\))d(:=)1838 2845 y Fm(\012)1877 2913 y Fx(\012)1937 2878 y Fr(\003)p 1937 2897 38 3 v 1937 2946 a Fs(\022)1975 2913 y Fw(;)14 b Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012\))2312 2925 y Fs(\022)2351 2845 y Fm(\013)2404 2913 y Fw(;)1141 b Fx(\(9.11\))180 3077 y(it)31 b(is)g(indep)r(enden)n(t)h(of)f(the)g (deformation)f(parameter)f Fw(\022)r Fx(,)j(if)g(Im)14 b Fw(\022)31 b(>)d Fx(0,)j(and)g(0)d Fw(<)g(g)j(<)d Fp(j)p Fw(g)3041 3089 y Fv(0)3078 3077 y Fp(j)j Fx(\(the)h(upp)r(er)f(b)r (ound)180 3177 y(dep)r(ending)d(on)f(sin\(Im)14 b Fw(\016)s Fx(\),)29 b(see)e(Theorem)g(3.1\).)305 3301 y(Observ)n(e)i(that)i Fn(1)e Fp(2)997 3280 y Fm(e)971 3301 y Fp(A)p Fx(,)j(and)f(that)g(the)g (normalization)2116 3209 y Fm(D)2167 3301 y Fx(\012)2227 3271 y Fr(\003)p 2227 3290 V 2227 3339 a Fs(\022)2265 3301 y Fw(;)14 b Fx(\012)2362 3313 y Fs(\022)2399 3209 y Fm(E)2479 3301 y Fx(=)28 b(1)j(implies)g Fw(\021)s Fx(\()p Fn(1)p Fx(\))e(=)f(1.)3346 3280 y Fm(e)3320 3301 y Fp(A)j Fx(is)g(a)g(linear)180 3421 y(subspace)c(of)g Fp(A)p Fx(,)h(but)h(not)e(an)h(algebra.)305 3521 y(W)-7 b(e)35 b(sho)n(w)f(in)i(App)r(endix)g(C,)h(Prop)r(osition)c(C.1,)k (that)e Fp(A)2173 3533 y Fv(1)2246 3521 y Fx(\(de\014ned)h(in)f (\(2.17\)\))g(is)g(strongly)e(dense)i(in)h Fp(A)180 3620 y Fx(\(de\014ned)h(in)g(\(2.7\)\),)i(and)d(that)h(an)n(y)e Fw(A)j Fp(2)g(A)1661 3632 y Fv(1)1735 3620 y Fx(has)e(the)h(prop)r(ert) n(y)e(that)i Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)h(is)e Fw(U)3004 3632 y Fs(\022)3041 3620 y Fx(-analytic,)i(for)e Fw(\022)j Fx(in)d(a)180 3730 y(neigh)n(b)r(ourho)r(o)r(d)27 b(of)g Fw(\022)e Fx(=)e(0.)37 b(Hence)27 b Fp(A)1408 3742 y Fv(1)1469 3730 y Fp(\022)1583 3709 y Fm(e)1557 3730 y Fp(A)c(\022)g(A)p Fx(,)28 b(and)f(therefore)2387 3709 y Fm(e)2361 3730 y Fp(A)h Fx(is)f(strongly)f(dense)i(in)g Fp(A)p Fx(.)305 3830 y(W)-7 b(e)29 b(ha)n(v)n(e)f(th)n(us)i(sho)n(wn)e (that)i(for)e(an)n(y)h Fw(!)1601 3842 y Fv(0)1638 3830 y Fx(-normal)f(state)h Fw( )s Fx(,)h Fw( )2321 3800 y Fs(t)2376 3830 y Fp(!)c Fw(\021)32 b Fx(as)d Fw(t)c Fp(!)h(1)p Fx(,)k(where)f(the)h(con)n(v)n(ergence)180 3930 y(is)d(understo)r(o)r (d)h(in)g(the)g Fp(k)18 b(\001)g(k)1081 3942 y Fr(\030)1137 3930 y Fx(-top)r(ology)26 b(of)h(linear)g(functionals)g(on)2395 3909 y Fm(e)2369 3930 y Fp(A)p Fx(.)305 4040 y(It)f(is)f(clear)g(from)h (\(9.8\))f(that)h Fp(j)p Fw(\021)s Fx(\()p Fw(A)p Fx(\))p Fp(j)f(\024)d(k)p Fw(A)p Fp(k)p Fx(,)k(for)f Fw(A)e Fp(2)2078 4019 y Fm(e)2052 4040 y Fp(A)p Fx(,)j(hence)g Fw(\021)j Fx(extends)d(to)g(a)f(b)r(ounded)h(p)r(ositiv)n(e)g(linear)180 4139 y(functional)i(on)f(the)h(Banac)n(h)e(space)h(of)h(observ)-5 b(ables)1543 4304 y Fp(A)1609 4316 y Fv(0)1670 4304 y Fx(:=)27 b Fp(k)18 b(\001)g(k)p Fx(-closure)26 b(of)2354 4283 y Fm(e)2328 4304 y Fp(A)p Fx(,)1151 b(\(9.12\))180 4479 y(normalized)29 b(as)h Fw(\021)s Fx(\()p Fn(1)p Fx(\))f(=)e(1.)45 b(Standard)30 b(p)r(erturbation)g(theory)g(sho)n(ws)f (that)i Fw(\021)s Fx(\()p Fw(A)p Fx(\),)i Fw(A)28 b Fp(2)3065 4458 y Fm(e)3039 4479 y Fp(A)p Fx(,)k(is)e(analytic)g(in)h(the)180 4578 y(coupling)c(constan)n(t)g Fw(g)s Fx(.)305 4678 y(Observ)n(e)e(that)j(w)n(e)g(can)f(rewrite)g(the)h(state)f Fw(\021)s Fx(\()p Fw(A)p Fx(\))i(also)d(in)i(the)g(form)1599 4843 y Fw(\021)s Fx(\()p Fw(A)p Fx(\))c(=)e(T)-7 b(r)1980 4775 y Fm(\000)2018 4843 y Fw(\031)s Fx(\()p Fw(A)p Fx(\))2194 4855 y Fs(\022)2232 4843 y Fw(P)2285 4855 y Fs(\022)2323 4775 y Fm(\001)3568 4843 y Fx(\(9.13\))180 5018 y(where)33 b Fw(\031)s Fx(\()p Fw(A)p Fx(\))602 5030 y Fs(\022)673 5018 y Fx(:=)f Fw(U)850 5030 y Fs(\022)887 5018 y Fw(\031)s Fx(\()p Fw(A)p Fx(\))p Fw(U)1129 4982 y Fr(\000)p Fv(1)1120 5043 y Fs(\022)1219 5018 y Fx(.)55 b(F)-7 b(ormally)32 b(one)h(can)g(undo)g(the)h(rotation)e(in)h(\(9.11\))g(to)g(obtain)g (\(4.15\))g(with)180 5117 y(\012)240 5087 y Fr(\003)302 5117 y Fx(:=)24 b Fw(U)480 5082 y Fr(\000)p Fv(1)p 471 5108 V 471 5157 a Fs(\022)569 5117 y Fx(\012)629 5087 y Fr(\003)p 629 5106 V 629 5155 a Fs(\022)667 5117 y Fx(.)39 b(Ho)n(w)n(ev)n(er,)27 b(in)h(the)h(non-equilibrium)f (situation)g(\012)2339 5087 y Fr(\003)2401 5117 y Fp(62)d(H)q Fx(!)39 b(The)28 b(set)h Fp(A)2982 5129 y Fv(1)3048 5117 y Fx(is)f(exactly)f(the)i(set)f(on)180 5227 y(whic)n(h)g(\(4.15\))e (mak)n(es)h(sense.)36 b(Th)n(us)28 b(w)n(e)f(ga)n(v)n(e)f(a)h(rigorous) e(meaning)i(to)h(\(4.15\))e(and)i(the)g(NESS)f(v)n(ector)g(\012)3586 5197 y Fr(\003)3624 5227 y Fx(.)1938 5476 y(24)p eop %%Page: 25 25 25 24 bop 180 210 a Fy(10)135 b(Pro)t(of)45 b(of)g(Theorem)g(7.2)180 392 y Fx(Our)19 b(task)f(is)i(to)f(sho)n(w)f(that)i(the)f(sp)r(ectrum)h (of)f(\(7.1\))g(lies)g(in)h(the)g(upp)r(er)f(complex)g(half)g(plane)g Fp(f)p Fx(Im)14 b Fw(z)26 b(>)d Fx(0)p Fp(g)18 b Fx(if)i Fw(e)j Fp(6)p Fx(=)f(0;)180 492 y(and)32 b(that)g(it)g(has)f(a)g (simple)h(eigen)n(v)-5 b(alue)31 b(at)g(zero)g(and)h(all)f(the)h(other) f(eigen)n(v)-5 b(alues)31 b(lie)h(in)g(the)g(upp)r(er)g(complex)180 591 y(half)25 b(plane)g(if)h Fw(e)d Fx(=)f(0.)36 b(While)26 b(this)f(analysis)f(is)h(standard)f(in)i(the)f(case)f(when)i Fw(I)32 b Fx(is)25 b(a)g(selfadjoin)n(t)g(op)r(erator)e(\(then)180 691 y(the)h(imaginary)e(part)i(of)g(\(7.1\))f(is)h(just)g Fw(P)1460 703 y Fs(e)1496 691 y Fw(I)7 b(\016)s Fx(\()p Fw(L)1668 703 y Fv(0)1716 691 y Fp(\000)k Fw(e)p Fx(\))p Fw(I)c(P)1959 703 y Fs(e)1995 691 y Fx(,)24 b(manifestly)g(a)g (non-negativ)n(e)e(op)r(erator;)h(see)h(e.g.)35 b([8]\),)180 791 y(it)28 b(needs)f(some)h(more)e(though)n(t)i(in)f(our)g(case,)g (where)g(the)h(in)n(teraction)f(is)g(non-selfadjoin)n(t.)36 b(Let)1401 973 y Fw(V)1449 985 y Fs(j)1507 973 y Fx(=)23 b Fw(\031)s Fx(\()p Fw(v)1717 985 y Fs(j)1753 973 y Fx(\))p Fw(;)97 b Fx(and)83 b Fw(V)2189 939 y Fr(0)2170 994 y Fs(j)2235 973 y Fx(=)23 b Fw(\031)2373 939 y Fr(0)2396 973 y Fx(\()p Fw(v)2468 985 y Fs(j)2504 973 y Fx(\))p Fw(:)180 1156 y Fx(The)28 b(main)f(ingredien)n(t)g(of)h(the)g(pro)r(of) f(is)180 1322 y Fn(Prop)s(osition)j(10.1)40 b Fq(We)30 b(have)356 1600 y Fx(\003)414 1612 y Fs(e)472 1600 y Fx(=)560 1507 y Fm(\020)610 1600 y Fw(e)649 1565 y Fr(\000)p Fs(\014)739 1573 y Fi(p)773 1565 y Fs(H)827 1573 y Fi(p)862 1565 y Fs(=)p Fv(2)952 1600 y Fp(\012)18 b Fx(1)-18 b(l)1082 1612 y Fs(p)1119 1507 y Fm(\021)1183 1433 y(2)1183 1582 y(4)1262 1521 y(X)1238 1698 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)1419 1507 y Fm(\020)1469 1600 y Fw(e)1508 1565 y Fs(\014)1546 1573 y Fi(j)1576 1565 y Fs(H)1630 1573 y Fi(p)1666 1565 y Fs(=)p Fv(2)1755 1600 y Fp(\012)18 b Fx(1)-18 b(l)1885 1612 y Fs(p)1923 1507 y Fm(\021)1986 1600 y Fx(\003)2044 1612 y Fs(j)s(e)2124 1507 y Fm(\020)2174 1600 y Fw(e)2213 1565 y Fr(\000)p Fs(\014)2303 1573 y Fi(j)2333 1565 y Fs(H)2387 1573 y Fi(p)2422 1565 y Fs(=)p Fv(2)2512 1600 y Fp(\012)18 b Fx(1)-18 b(l)2642 1612 y Fs(p)2679 1507 y Fm(\021)2729 1433 y(3)2729 1582 y(5)2798 1507 y(\020)2848 1600 y Fw(e)2887 1565 y Fs(\014)2925 1573 y Fi(p)2959 1565 y Fs(H)3013 1573 y Fi(p)3048 1565 y Fs(=)p Fv(2)3138 1600 y Fp(\012)18 b Fx(1)-18 b(l)3268 1612 y Fs(p)3305 1507 y Fm(\021)3369 1600 y Fw(;)176 b Fx(\(10.1\))180 1887 y Fq(wher)l(e,)31 b(setting)e Fp(R)24 b Fx(:=)e(\()p Fw(L)1000 1899 y Fv(0)1056 1887 y Fp(\000)c Fw(e)g Fx(+)g Fw(i)p Fx(0\))1382 1857 y Fr(\000)p Fv(1)1471 1887 y Fq(,)1355 2070 y Fx(\003)1413 2082 y Fs(j)s(e)1502 2070 y Fx(=)23 b Fp(\000)p Fw(P)1708 2082 y Fs(e)1743 2070 y Fx(\()p Fw(V)1823 2082 y Fs(j)1878 2070 y Fp(\000)18 b Fw(V)2027 2036 y Fr(0)2009 2090 y Fs(j)2051 2070 y Fx(\))p Fp(R)p Fx(\()p Fw(V)2233 2082 y Fs(j)2287 2070 y Fp(\000)h Fw(V)2437 2036 y Fr(0)2419 2090 y Fs(j)2461 2070 y Fx(\))p Fw(P)2546 2082 y Fs(e)2582 2070 y Fw(:)963 b Fx(\(10.2\))180 2252 y Fq(Notic)l(e)30 b(that)g(\(10.1\))h(shows)g (that)e(the)h(sp)l(e)l(ctrum)f(of)i Fx(\003)1909 2264 y Fs(e)1974 2252 y Fq(is)f(indep)l(endent)h(of)f Fw(\014)2668 2264 y Fs(p)2706 2252 y Fq(.)305 2418 y Fx(The)25 b(imp)r(ortance)g(of) g(\(10.1\))g(is)g(that)h(it)g(relates)e(\003)1896 2430 y Fs(e)1957 2418 y Fx(to)h(the)h(op)r(erators)d(\003)2620 2430 y Fs(j)s(e)2712 2418 y Fx(whose)i(sp)r(ectral)f(c)n (haracteristics)180 2518 y(are)g(w)n(ell)i(kno)n(wn.)35 b(Indeed,)26 b(\003)1126 2530 y Fs(j)s(e)1218 2518 y Fx(are)e(the)i(lev)n(el)f(shift)h(op)r(erators)d(corresp)r(onding)h(to) h(the)h(reserv)n(oir)d Fw(j)30 b Fx(coupled)25 b(to)180 2618 y(the)j(particle)f(system,)g(studied)h(in)g([8,)g(24)o(].)305 2717 y(Before)e(pro)r(ceeding)h(to)g(the)h(pro)r(of)f(w)n(e)h(examine)f (some)g(consequences)f(of)i(this)g(prop)r(osition.)305 2817 y Fn(The)33 b(case)h Fw(e)24 b Fp(6)p Fx(=)h(0)p Fn(.)73 b Fx(Let)29 b(us)g(assume)f(that)h(the)g(nonzero)f(eigen)n(v)-5 b(alues)27 b(of)i Fw(L)2765 2829 y Fs(p)2832 2817 y Fx(are)f(simple,)h (i.e.)41 b Fw(E)3463 2829 y Fs(i)3510 2817 y Fp(\000)19 b Fw(E)3655 2829 y Fs(j)3715 2817 y Fx(=)180 2917 y Fw(E)241 2929 y Fs(m)326 2917 y Fp(\000)i Fw(E)473 2929 y Fs(n)548 2917 y Fp(,)31 b Fw(i)e Fx(=)h Fw(m;)14 b(j)35 b Fx(=)30 b Fw(n)p Fx(.)50 b(F)-7 b(or)31 b(a)h(treatmen)n(t)f(of)h(the)g(more)f (general)g(case)g(where)g Fw(E)3062 2929 y Fs(i)3112 2917 y Fp(\000)21 b Fw(E)3259 2929 y Fs(j)3324 2917 y Fx(=)30 b Fw(E)3480 2929 y Fs(m)3565 2917 y Fp(\000)20 b Fw(E)3711 2929 y Fs(n)3757 2917 y Fx(,)180 3016 y(for)32 b(\()p Fw(i;)14 b(j)5 b Fx(\))32 b Fp(6)p Fx(=)f(\()p Fw(m;)14 b(n)p Fx(\),)35 b(with)e(simple)g Fw(E)1410 3028 y Fs(j)1445 3016 y Fx(,)h(w)n(e)f(refer)f(to)g([27)o(].)53 b(Since)33 b Fw(P)2413 3028 y Fs(e)2482 3016 y Fx(is)f(of)h(rank)f(one) g(\003)3082 3028 y Fs(e)3150 3016 y Fx(is)h(just)g(a)g(complex)180 3116 y(n)n(um)n(b)r(er,)24 b(namely)g(the)g(sum)g(of)g(\003)1247 3128 y Fv(1)p Fs(e)1326 3116 y Fx(+)11 b(\003)1460 3128 y Fv(2)p Fs(e)1552 3116 y Fx(\(the)24 b(dep)r(endence)h(on)e Fw(\014)2323 3128 y Fv(1)2360 3116 y Fw(;)14 b(\014)2444 3128 y Fv(2)2505 3116 y Fx(disapp)r(ears\).)35 b(Under)24 b(condition)f(\(3.5\),)180 3215 y(one)35 b(has)g(Im)14 b(\003)667 3227 y Fs(e)739 3215 y Fp(\025)36 b Fw(\015)883 3227 y Fv(0)920 3215 y Fx(,)i(where)d Fw(\015)1272 3227 y Fv(0)1345 3215 y Fx(is)g(a)g(strictly)h(p)r(ositiv)n(e)f(constan)n(t) g(whic)n(h)g(is)h(indep)r(enden)n(t)g(of)g(the)g(in)n(v)n(erse)180 3315 y(temp)r(eratures,)27 b(see)g([8]\).)37 b(This)28 b(sho)n(ws)e(\(7.2\))h(and)h(\(7.3\))f(for)g Fw(e)c Fp(6)p Fx(=)g(0.)305 3415 y Fn(The)30 b(case)g Fw(e)23 b Fx(=)f(0)p Fn(.)66 b Fx(Zero)25 b(is)h(necessarily)e(a)i(degenerate)e(eigen)n(v)-5 b(alue)26 b(of)g Fw(L)2691 3427 y Fv(0)2727 3415 y Fx(,)h(so)e(the)i (ab)r(o)n(v)n(e)d(reasoning)g(do)r(es)180 3514 y(not)k(apply)-7 b(.)36 b(In)28 b(particular,)e(\003)1149 3526 y Fv(10)1247 3514 y Fx(and)h(\003)1466 3526 y Fv(20)1564 3514 y Fx(do)g(not)h(comm)n (ute.)37 b(It)28 b(is)f(sho)n(wn)g(in)h([24)o(,)g(27)o(])g(that)1575 3697 y(\003)1633 3709 y Fs(j)s Fv(0)1724 3697 y Fx(=)23 b Fw(i)p Fx(Im)13 b(\003)2011 3709 y Fs(j)s Fv(0)2102 3697 y Fx(=:)23 b Fw(i)p Fx(\000)2294 3709 y Fs(j)s Fv(0)2362 3697 y Fw(;)1183 b Fx(\(10.3\))180 3880 y(where)26 b(Im)14 b(\003)590 3892 y Fs(j)s Fv(0)681 3880 y Fx(:=)813 3847 y Fv(1)p 802 3861 57 4 v 802 3908 a(2)p Fs(i)868 3880 y Fx(\(\003)958 3892 y Fs(j)s Fv(0)1043 3880 y Fp(\000)i Fx(\003)1182 3850 y Fr(\003)1182 3901 y Fs(j)s Fv(0)1250 3880 y Fx(\).)37 b(W)-7 b(e)27 b(use)f(here)g(implicitly)i(Condition)e (\(B\))h(on)g(the)g(non-degeneracy)d(of)j Fw(H)3719 3892 y Fs(p)3757 3880 y Fx(;)180 3979 y(if)f(the)h(small)e(system)h(has)f (degenerate)g(energy)g(lev)n(els)g(then)h(\003)2174 3991 y Fs(j)s Fv(0)2268 3979 y Fx(are)f(not)h(purely)f(imaginary)g([24)o(].) 36 b(One)26 b(sho)n(wn)180 4079 y(as)f(in)i([8)o(,)f(27)o(,)g(24)o(])g (that)h(\000)984 4091 y Fs(j)s Fv(0)1078 4079 y Fx(are)e(real)g (matrices)g(ha)n(ving)g(strictly)h(negativ)n(e)e(o\013-diagonal)g(en)n (tries,)i(\(\000)3400 4091 y Fs(j)s Fv(0)3468 4079 y Fx(\))3500 4091 y Fs(mn)3628 4079 y Fw(<)c Fx(0,)180 4179 y(for)27 b Fw(m)c Fp(6)p Fx(=)g Fw(n)p Fx(,)k(and)h(satisfying) 1740 4278 y(\000)1792 4290 y Fs(j)s Fv(0)1860 4278 y Fx(\012)1920 4244 y Fv(\()p Fs(\014)1984 4252 y Fi(j)2015 4244 y Fv(\))1920 4299 y Fs(p)2068 4278 y Fx(=)22 b(0)p Fw(;)1348 b Fx(\(10.4\))180 4457 y(where)27 b(\012)480 4412 y Fv(\()p Fs(\014)544 4420 y Fi(j)575 4412 y Fv(\))480 4466 y Fs(p)632 4457 y Fx(is)h(the)g(particle)f(Gibbs)h(state)f(at)h (temp)r(erature)f Fw(\014)2225 4469 y Fs(j)2260 4457 y Fx(.)37 b(Hence,)28 b(since)1137 4609 y Fm(\020)1187 4702 y Fw(e)1226 4667 y Fv(\()p Fs(\014)1290 4675 y Fi(p)1324 4667 y Fr(\000)p Fs(\014)1414 4675 y Fi(j)1444 4667 y Fv(\))p Fs(H)1524 4675 y Fi(p)1559 4667 y Fs(=)p Fv(2)1649 4702 y Fp(\012)18 b Fx(1)-18 b(l)1779 4714 y Fs(p)1817 4609 y Fm(\021)1880 4702 y Fx(\012)1940 4667 y Fv(\()p Fs(\014)2004 4675 y Fi(p)2039 4667 y Fv(\))1940 4722 y Fs(p)2092 4702 y Fx(=)2191 4645 y(tr)p Fw(e)2295 4615 y Fr(\000)p Fs(\014)2385 4623 y Fi(j)2415 4615 y Fs(H)2469 4623 y Fi(p)2504 4615 y Fs(=)p Fv(2)p 2189 4683 389 4 v 2189 4761 a Fx(tr)p Fw(e)2293 4736 y Fr(\000)p Fs(\014)2383 4744 y Fi(p)2417 4736 y Fs(H)2471 4744 y Fi(p)2506 4736 y Fs(=)p Fv(2)2615 4702 y Fx(\012)2675 4667 y Fv(\()p Fs(\014)2739 4675 y Fi(j)2770 4667 y Fv(\))2675 4722 y Fs(p)2800 4702 y Fw(;)180 4910 y Fx(w)n(e)27 b(see)g(that)854 5049 y(\000)906 5061 y Fv(0)943 5049 y Fx(\012)1003 5014 y Fv(\()p Fs(\014)1067 5022 y Fi(p)1102 5014 y Fv(\))1003 5069 y Fs(p)1155 5049 y Fx(=)1266 4970 y Fm(X)1242 5147 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)1424 4956 y Fm(\020)1473 5049 y Fw(e)1512 5014 y Fv(\()p Fr(\000)p Fs(\014)1628 5022 y Fi(p)1662 5014 y Fv(+)p Fs(\014)1751 5022 y Fi(j)1782 5014 y Fv(\))p Fs(H)1862 5022 y Fi(p)1897 5014 y Fs(=)p Fv(2)1987 5049 y Fp(\012)18 b Fx(1)-18 b(l)2117 5061 y Fs(p)2154 4956 y Fm(\021)2230 4992 y Fx(tr)p Fw(e)2334 4962 y Fr(\000)p Fs(\014)2424 4970 y Fi(j)2454 4962 y Fs(H)2508 4970 y Fi(p)2543 4962 y Fs(=)p Fv(2)p 2228 5029 V 2228 5108 a Fx(tr)p Fw(e)2332 5083 y Fr(\000)p Fs(\014)2422 5091 y Fi(p)2456 5083 y Fs(H)2510 5091 y Fi(p)2545 5083 y Fs(=)p Fv(2)2626 5049 y Fx(\000)2678 5061 y Fs(j)s Fv(0)2746 5049 y Fx(\012)2806 5014 y Fv(\()p Fs(\014)2870 5022 y Fi(j)2901 5014 y Fv(\))2806 5069 y Fs(p)2954 5049 y Fx(=)22 b(0)p Fw(;)1938 5476 y Fx(25)p eop %%Page: 26 26 26 25 bop 180 216 a Fx(where)27 b(\000)472 228 y Fv(0)532 216 y Fx(:=)c Fp(\000)p Fw(i)p Fx(\003)795 228 y Fv(0)831 216 y Fx(.)37 b(Th)n(us,)27 b(\012)1184 171 y Fv(\()p Fs(\014)1248 179 y Fi(p)1282 171 y Fv(\))1184 225 y Fs(p)1340 216 y Fx(is)g(an)g(eigen)n(v)n(ector)e(of)j(the)f(real)g(matrix)g(\000) 2690 228 y Fv(0)2754 216 y Fx(with)h(eigen)n(v)-5 b(alue)26 b(zero.)36 b(Notice)180 334 y(that)28 b(the)h(v)n(ector)e(\012)814 289 y Fv(\()p Fs(\014)878 297 y Fi(p)912 289 y Fv(\))814 343 y Fs(p)970 334 y Fx(has)h(strictly)g(p)r(ositiv)n(e)g(comp)r(onen)n (ts,)f([tr)p Fw(e)2319 303 y Fr(\000)p Fs(\014)2409 311 y Fi(p)2443 303 y Fs(H)2497 311 y Fi(p)2533 303 y Fs(=)p Fv(2)2604 334 y Fx(])2627 303 y Fr(\000)p Fv(1)2716 334 y Fw(e)2755 303 y Fr(\000)p Fs(\014)2845 311 y Fi(p)2879 303 y Fs(E)2928 311 y Fi(n)2969 303 y Fs(=)p Fv(2)3040 334 y Fx(,)h(in)h(the)f(orthonormal)180 433 y(basis)21 b Fp(f)p Fw(')474 445 y Fs(n)525 433 y Fp(\012)6 b Fw(')650 445 y Fs(n)696 433 y Fp(g)21 b Fx(of)h(Ran)13 b Fw(\037)1062 445 y Fs(L)1108 453 y Fi(p)1143 445 y Fv(=0)1253 433 y Fx(\(where)21 b Fw(H)1588 445 y Fs(p)1626 433 y Fw(')1680 445 y Fs(n)1749 433 y Fx(=)i Fw(E)1898 445 y Fs(n)1943 433 y Fw(')1997 445 y Fs(n)2042 433 y Fw(;)14 b Fp(k)p Fw(')2175 445 y Fs(n)2220 433 y Fp(k)23 b Fx(=)g(1\).)34 b(Moreo)n(v)n(er,)20 b(the)i(o\013-diagonal)e(elemen)n(ts)180 533 y(of)28 b(the)g(real)e(matrix)h(\(whic)n(h)h(is)g(not)f(symmetric)g (for)g Fw(\014)1928 545 y Fv(1)1989 533 y Fp(6)p Fx(=)22 b Fw(\014)2123 545 y Fv(2)2161 533 y Fx(\))28 b(\000)2273 545 y Fv(0)2337 533 y Fx(are)f(giv)n(en)g(b)n(y)868 729 y(\(\000)952 741 y Fv(0)990 729 y Fx(\))1022 741 y Fs(m;n)1169 729 y Fx(=)1281 650 y Fm(X)1257 827 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)1438 637 y Fm(h\020)1527 729 y Fw(e)1566 695 y Fv(\()p Fr(\000)p Fs(\014)1682 703 y Fi(p)1716 695 y Fv(+)p Fs(\014)1805 703 y Fi(j)1836 695 y Fv(\))p Fs(H)1916 703 y Fi(p)1951 695 y Fs(=)p Fv(2)2040 729 y Fp(\012)19 b Fx(1)-18 b(l)2171 741 y Fs(p)2208 637 y Fm(\021)2271 729 y Fx(\000)2323 741 y Fs(j)s Fv(0)2405 637 y Fm(\020)2455 729 y Fw(e)2494 695 y Fr(\000)p Fv(\()p Fr(\000)p Fs(\014)2662 703 y Fi(p)2696 695 y Fv(+)p Fs(\014)2785 703 y Fi(j)2815 695 y Fv(\))p Fs(H)2895 703 y Fi(p)2930 695 y Fs(=)p Fv(2)3020 729 y Fp(\012)18 b Fx(1)-18 b(l)3150 741 y Fs(p)3188 637 y Fm(\021)o(i)3277 787 y Fs(mn)1034 1010 y Fx(=)23 b Fp(\000)p Fw(\031)1275 931 y Fm(X)1251 1108 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)1442 953 y Fw(E)1508 923 y Fv(2)1503 974 y Fs(mn)1608 953 y Fw(e)1647 923 y Fr(\000)p Fv(\()p Fs(\014)1763 931 y Fi(p)1797 923 y Fr(\000)p Fs(\014)1887 931 y Fi(j)1917 923 y Fv(\))p Fs(E)1992 931 y Fi(mn)2084 923 y Fs(=)p Fv(2)p 1442 990 713 4 v 1497 1067 a Fp(j)14 b Fx(sinh\()p Fw(\014)1761 1079 y Fs(j)1796 1067 y Fw(E)1857 1079 y Fs(mn)1962 1067 y Fw(=)p Fx(2\))p Fp(j)2179 897 y Fm(Z)2225 1085 y Fs(S)2269 1069 y Fk(2)2319 1010 y Fw(d\033)31 b Fp(j)p Fx([)p Fw(G)2551 1022 y Fs(j)2587 1010 y Fx(\()p Fw(E)2680 1022 y Fs(mn)2785 1010 y Fw(;)14 b(\033)s Fx(\)])2927 1022 y Fs(nm)3031 1010 y Fp(j)3054 975 y Fv(2)3568 1010 y Fx(\(10.5\))180 1269 y(for)29 b Fw(m)c(>)g(n)k Fx(\(and)h(similarly)e(for)h Fw(m)c(<)g(n)p Fx(,)30 b(see)f(also)f([8)o(,)h(27)o(]\).)42 b(Hence)30 b(Condition)f(\(C\))g(implies)h(that)f(\(\000)3496 1281 y Fv(0)3534 1269 y Fx(\))3566 1281 y Fs(m;n)3715 1269 y Fw(<)180 1369 y Fx(0.)53 b(A)34 b(standard)e Fq(Perr)l(on-F)-6 b(r)l(ob)l(enius)35 b(ar)l(gument)d Fx(sho)n(ws)g(that)h(zero)g(is)g(a) f(simple)i(eigen)n(v)-5 b(alue)32 b(of)h(\000)3366 1381 y Fv(0)3403 1369 y Fx(,)i(and)e(that)180 1468 y Fw(\033)s Fx(\(\000)314 1480 y Fv(0)352 1468 y Fx(\))p Fp(nf)p Fx(0)p Fp(g)28 b(\032)h Fo(C)728 1438 y Fv(+)789 1468 y Fx(.)49 b(This)32 b(sho)n(ws)e(equations)h(\(7.3\))g({)g(\(7.4\))g (for)g Fw(e)f Fx(=)f(0.)48 b(It)32 b(is)g(sho)n(wn)f(in)g([8])h(that)f (the)h(gap)f(at)180 1568 y(the)f(b)r(ottom)f(of)g(the)h(sp)r(ectrum)g (of)f(\000)1373 1580 y Fs(j)s Fv(0)1441 1568 y Fx(,)h Fw(j)g Fx(=)c(1)p Fw(;)14 b Fx(2)28 b(has)h(a)g(lo)n(w)n(er)e(b)r(ound) j(whic)n(h)f(is)h(indep)r(enden)n(t)g(of)f(the)h(in)n(v)n(erse)180 1668 y(temp)r(eratures.)305 1767 y(W)-7 b(e)28 b(no)n(w)g(pro)n(v)n(e)f (the)i(existence)f(of)g Fw(\016)1477 1779 y Fv(0)1514 1767 y Fx(,)h(assuming)e(that)i(Condition)f(\(D\))i(is)e(satis\014ed.) 39 b(If)29 b(dim)14 b Fp(H)3397 1779 y Fs(p)3460 1767 y Fx(=)24 b(2)k(then)180 1867 y(the)35 b(t)n(w)n(o)e(eigen)n(v)-5 b(alues)33 b(of)h(\003)1088 1879 y Fv(0)1160 1867 y Fx(are)f(zero)g (and)h(the)h(trace)e(of)h(\003)2178 1879 y Fv(0)2215 1867 y Fx(.)57 b(Expressions)32 b(\(10.1\))i(and)g(\(10.3\))f(sho)n(w)h (that)180 1967 y(T)-7 b(r\(\003)356 1979 y Fv(0)393 1967 y Fx(\))26 b(=)g Fw(i)p Fx([T)-7 b(r)o(\(\000)763 1979 y Fv(10)834 1967 y Fx(\))20 b(+)f(T)-7 b(r\(\000)1140 1979 y Fv(20)1210 1967 y Fx(\)].)43 b(Th)n(us)30 b(the)g(sp)r(ectral)e (gap)h(of)h(\003)2316 1979 y Fv(0)2382 1967 y Fx(is)g(the)g(sum)f(of)h (the)g(gaps)e(of)i(\000)3372 1979 y Fv(01)3471 1967 y Fx(and)g(\000)3687 1979 y Fv(02)3757 1967 y Fx(,)180 2066 y(whic)n(h)e(ha)n(v)n(e)e(lo)n(w)n(er)g(b)r(ounds)i(uniform)f(in)h (the)g(in)n(v)n(erse)e(temp)r(eratures.)305 2166 y(Next)h(tak)n(e)f (dim)14 b Fp(H)908 2178 y Fs(p)970 2166 y Fp(\025)22 b Fx(3.)37 b(F)-7 b(or)26 b Fw(\016)s(\014)h Fx(=)c Fp(j)p Fw(\014)1579 2178 y Fv(1)1633 2166 y Fp(\000)17 b Fw(\014)1762 2178 y Fv(2)1799 2166 y Fp(j)23 b Fx(=)g(0)j(the)h(matrix)g(\003)2471 2178 y Fv(0)2535 2166 y Fx(has)f(the)h(same)g(sp)r(ectrum)g(as)3495 2104 y Fm(P)3582 2191 y Fs(j)3631 2166 y Fx(\003)3689 2178 y Fs(j)s Fv(0)3757 2166 y Fx(,)180 2265 y(see)e(\(10.1\).)36 b(An)25 b(application)g(of)h(the)g(minimax)f(principle)h(demonstrates)e (that)i(the)g(sp)r(ectral)e(gap)h(of)h(the)f(latter)180 2365 y(op)r(erator)f(has)h(to)h(b)r(e)h(at)e(least)h(as)f(large)f(as)i (the)g(maxim)n(um)g(of)g(the)g(gaps)f(of)h(\000)2659 2377 y Fs(j)s Fv(0)2726 2365 y Fx(,)h Fw(j)h Fx(=)22 b(1)p Fw(;)14 b Fx(2.)36 b(F)-7 b(or)25 b(small)h(v)-5 b(alues)25 b(of)180 2465 y Fw(\016)s(\014)32 b Fx(\(c.f.)38 b(\(3.6\)\),)27 b(the)h(existence)g(of)f Fw(\016)1358 2477 y Fv(0)1423 2465 y Fx(follo)n(ws)g(b)n(y)g(p)r(erturbation)g (theory)-7 b(.)305 2564 y(Finally)26 b(w)n(e)g(consider)f(the)h(case)g (where)f Fw(\016)s(\014)31 b Fx(and)26 b Fw(\014)1908 2576 y Fv(1)1945 2564 y Fw(;)14 b(\014)2029 2576 y Fv(2)2093 2564 y Fx(are)25 b(large)g(\(see)h(\(3.6\)\).)36 b(Let)27 b(us)f(tak)n(e)f Fw(\014)3337 2576 y Fv(2)3398 2564 y Fx(=)d Fw(\014)3532 2576 y Fv(1)3585 2564 y Fx(+)16 b Fw(\016)s(\014)t Fx(.)180 2664 y(As)28 b(is)f(easily)g(seen)g(from)h (\(10.1\))e(w)n(e)i(ha)n(v)n(e)983 2860 y Fw(\033)s Fx(\(\003)1123 2872 y Fv(0)1161 2860 y Fx(\))23 b(=)f Fw(i\033)1396 2768 y Fm(\020h)1485 2860 y Fx(\000)1537 2872 y Fv(10)1626 2860 y Fx(+)c(\()p Fw(e)1780 2826 y Fs(\016)r(\014)s(H)1907 2834 y Fi(p)1942 2826 y Fs(=)p Fv(2)2032 2860 y Fp(\012)g Fx(1)-18 b(l\)\000)2246 2872 y Fv(20)2315 2860 y Fx(\()p Fw(e)2386 2826 y Fr(\000)p Fs(\016)r(\014)s(H)2565 2834 y Fi(p)2600 2826 y Fs(=)p Fv(2)2690 2860 y Fp(\012)18 b Fx(1)-18 b(l\))2852 2768 y Fm(i\021)2954 2860 y Fw(:)591 b Fx(\(10.6\))180 3056 y(Using)36 b(the)g(explicit)h(expression)d(for)i (the)h(matrix)e(elemen)n(ts)h(of)g(\000)2352 3068 y Fv(20)2459 3056 y Fx(in)g(the)h(basis)e Fw(')2982 3068 y Fs(j)3041 3056 y Fp(\012)24 b Fw(')3184 3068 y Fs(j)3256 3056 y Fx(\(whic)n(h)36 b(can)g(b)r(e)180 3156 y(read)31 b(o\013)h(of)f (relation)g(\(10.5\))g(for)h(o\013-diagonal)d(terms,)k(and)f(is)f(easy) g(to)h(obtain)f(for)h(the)g(diagonal)e(ones\),)i(one)180 3255 y(v)n(eri\014es)21 b(that)i(the)f(matrix)g(\()p Fw(e)1100 3225 y Fs(\016)r(\014)s(H)1227 3233 y Fi(p)1262 3225 y Fs(=)p Fv(2)1341 3255 y Fp(\012)8 b Fx(1)-18 b(l\)\000)1545 3267 y Fv(20)1614 3255 y Fx(\()p Fw(e)1685 3225 y Fr(\000)p Fs(\016)r(\014)s(H)1864 3233 y Fi(p)1899 3225 y Fs(=)p Fv(2)1978 3255 y Fp(\012)8 b Fx(1)-18 b(l\))22 b(con)n(v)n(erges)d(to)j 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Fi(p)1765 1273 y Fs(=)p Fv(2)1836 1215 y Fm(\021)1899 1307 y Fw(e)1938 1273 y Fv(\()p Fs(\016)r(\014)2034 1281 y Fk(1)2066 1273 y Fs(L)2112 1281 y Fi(r)q Fk(1)2173 1273 y Fv(+)p Fs(\016)r(\014)2294 1281 y Fk(2)2327 1273 y Fs(L)2373 1281 y Fi(r)q Fk(2)2434 1273 y Fv(\))p Fs(=)p Fv(2)2531 1307 y Fw(V)2598 1273 y Fr(0)2621 1307 y Fw(P)2674 1319 y Fs(e)2710 1307 y Fw(e)2749 1273 y Fr(\000)p Fs(\016)r(\014)2871 1281 y Fi(p)2906 1273 y Fs(e=)p Fv(2)637 1490 y Fx(=)83 b Fw(P)838 1502 y Fs(e)888 1398 y Fm(\020)937 1490 y Fx(1)-18 b(l)984 1502 y Fs(p)1040 1490 y Fp(\012)18 b Fw(e)1162 1455 y Fr(\000)p Fs(\016)r(\014)1284 1463 y Fi(p)1319 1455 y Fs(H)1373 1463 y Fi(p)1408 1455 y Fs(=)p Fv(2)1479 1398 y Fm(\021)1543 1490 y Fw(V)g Fp(R)p Fw(e)1718 1455 y Fv(\()p Fs(\016)r(\014)1814 1463 y Fk(1)1847 1455 y Fs(L)1893 1463 y Fi(r)q Fk(1)1954 1455 y Fv(+)p Fs(\016)r(\014)2075 1463 y Fk(2)2107 1455 y Fs(L)2153 1463 y Fi(r)q Fk(2)2215 1455 y Fv(\))p Fs(=)p Fv(2)2312 1490 y Fw(V)2379 1455 y Fr(0)2416 1398 y Fm(\020)2465 1490 y Fw(e)2504 1455 y Fs(\016)r(\014)2574 1463 y Fi(p)2609 1455 y Fs(H)2663 1463 y Fi(p)2698 1455 y Fs(=)p Fv(2)2788 1490 y Fp(\012)g Fx(1)-18 b(l)2918 1502 y Fs(p)2955 1398 y Fm(\021)3019 1490 y Fw(P)3072 1502 y Fs(e)3108 1490 y Fw(e)3147 1455 y Fr(\000)p Fs(\016)r(\014)3269 1463 y Fi(p)3303 1455 y Fs(e=)p Fv(2)637 1672 y Fx(=)785 1580 y Fm(\020)835 1672 y Fw(e)874 1638 y Fr(\000)p Fs(\016)r(\014)996 1646 y Fi(p)1030 1638 y Fs(H)1084 1646 y Fi(p)1119 1638 y Fs(=)p Fv(2)1209 1672 y Fp(\012)18 b Fx(1)-18 b(l)1339 1684 y Fs(p)1376 1580 y Fm(\021)1440 1672 y Fw(P)1493 1684 y Fs(e)1529 1672 y Fw(V)19 b Fp(R)p Fw(e)1705 1638 y Fv(\()p Fs(\016)r(\014)1801 1646 y Fk(1)1833 1638 y Fs(L)1879 1646 y Fi(r)q Fk(1)1940 1638 y Fv(+)p Fs(\016)r(\014)2061 1646 y Fk(2)2093 1638 y Fs(L)2139 1646 y Fi(r)q Fk(2)2201 1638 y Fv(\))p Fs(=)p Fv(2)2298 1672 y Fw(V)2365 1638 y Fr(0)2388 1672 y Fw(P)2441 1684 y Fs(e)2491 1580 y Fm(\020)2540 1672 y Fw(e)2579 1638 y Fs(\016)r(\014)2649 1646 y Fi(p)2684 1638 y Fs(H)2738 1646 y Fi(p)2773 1638 y Fs(=)p Fv(2)2863 1672 y Fp(\012)f Fx(1)-18 b(l)2993 1684 y Fs(p)3030 1580 y Fm(\021)3094 1672 y Fw(:)180 1880 y Fx(T)-7 b(reating)27 b(the)h(other)f(t)n(w)n(o)f(terms)i(in)g (\(10.7\))e(in)i(a)f(similar)g(w)n(a)n(y)-7 b(,)27 b(w)n(e)g(arriv)n(e) f(at)552 2087 y Fp(\000)p Fx(\003)675 2099 y Fs(e)733 2087 y Fx(=)821 1995 y Fm(\020)870 2087 y Fw(e)909 2053 y Fr(\000)p Fs(\016)r(\014)1031 2061 y Fi(p)1066 2053 y Fs(H)1120 2061 y Fi(p)1155 2053 y Fs(=)p Fv(2)1245 2087 y Fp(\012)18 b Fx(1)-18 b(l)1375 2099 y Fs(p)1412 1995 y Fm(\021)1476 2087 y Fw(P)1529 2099 y Fs(e)1565 2087 y Fx([)p Fw(V)19 b Fp(R)p Fw(V)37 b Fx(+)18 b Fw(V)1960 2053 y Fr(0)1983 2087 y Fp(R)p Fw(V)2121 2053 y Fr(0)718 2270 y Fp(\000)p Fw(V)h Fp(R)p Fw(e)959 2236 y Fv(\()p Fs(\016)r(\014)1055 2244 y Fk(1)1087 2236 y Fs(L)1133 2244 y Fi(r)q Fk(1)1194 2236 y Fv(+)p Fs(\016)r(\014)1315 2244 y Fk(2)1348 2236 y Fs(L)1394 2244 y Fi(r)q Fk(2)1455 2236 y Fv(\))p Fs(=)p Fv(2)1552 2270 y Fw(V)1619 2236 y Fr(0)1661 2270 y Fp(\000)f Fw(V)1811 2236 y Fr(0)1834 2270 y Fp(R)p Fw(e)1943 2236 y Fv(\()p Fr(\000)p Fs(\016)r(\014)2091 2244 y Fk(1)2123 2236 y Fs(L)2169 2244 y Fi(r)q Fk(1)2230 2236 y Fr(\000)p Fs(\016)r(\014)2352 2244 y Fk(2)2385 2236 y Fs(L)2431 2244 y Fi(r)q Fk(2)2492 2236 y Fv(\))p Fs(=)p Fv(2)2589 2270 y Fw(V)h Fx(])p Fw(P)2732 2282 y Fs(e)2782 2178 y Fm(\020)2831 2270 y Fw(e)2870 2236 y Fs(\016)r(\014)2940 2244 y Fi(p)2975 2236 y Fs(H)3029 2244 y Fi(p)3064 2236 y Fs(=)p Fv(2)3154 2270 y Fp(\012)f Fx(1)-18 b(l)3284 2282 y Fs(p)3321 2178 y Fm(\021)3385 2270 y Fw(:)160 b Fx(\(10.9\))180 2478 y(W)-7 b(e)32 b(no)n(w)g(examine)f(the)i(term)f(in)g([)p Fp(\001)14 b(\001)g(\001)g Fx(].)50 b(W)-7 b(rite)33 b Fw(V)49 b Fx(=)30 b Fw(V)1992 2490 y Fv(1)2051 2478 y Fx(+)21 b Fw(V)2185 2490 y Fv(2)2223 2478 y Fx(,)33 b Fw(V)2346 2448 y Fr(0)2400 2478 y Fx(=)d Fw(V)2562 2448 y Fr(0)2543 2498 y Fv(1)2606 2478 y Fx(+)21 b Fw(V)2759 2448 y Fr(0)2740 2498 y Fv(2)2783 2478 y Fx(,)33 b(where,)g(recall,)f Fw(V)3406 2490 y Fs(j)3472 2478 y Fx(=)e Fw(\031)s Fx(\()p Fw(v)3689 2490 y Fs(j)3725 2478 y Fx(\),)180 2577 y Fw(V)247 2547 y Fr(0)228 2599 y Fs(j)295 2577 y Fx(=)25 b Fw(\031)435 2547 y Fr(0)458 2577 y Fx(\()p Fw(v)530 2589 y Fs(j)566 2577 y Fx(\).)41 b(Notice)29 b(that)g(w)n(e)f(ha)n(v)n(e)g(\(1)-18 b(l)1500 2589 y Fs(p)1557 2577 y Fp(\012)19 b Fx(1)-18 b(l)1688 2589 y Fs(p)1744 2577 y Fp(\012)19 b Fw(P)1881 2589 y Fv(\012)1952 2577 y Fp(\012)g Fx(1)-18 b(l)2083 2589 y Fv(1)2120 2577 y Fx(\))p Fw(V)2200 2589 y Fv(1)2238 2577 y Fx(\(1)g(l)2317 2589 y Fs(p)2374 2577 y Fp(\012)18 b Fx(1)-18 b(l)2504 2589 y Fs(p)2561 2577 y Fp(\012)19 b Fw(P)2698 2589 y Fv(\012)2769 2577 y Fp(\012)g Fx(1)-18 b(l)2900 2589 y Fv(2)2936 2577 y Fx(\))26 b(=)e(0,)29 b(and)g(similarly)f(for)180 2677 y Fw(V)228 2689 y Fv(2)266 2677 y Fx(,)f(from)h(whic)n(h)f(it)h(follo)n(ws)f(that)h(the)g (expression)e Fw(P)1882 2689 y Fs(e)1918 2677 y Fx([)p Fp(\001)14 b(\001)g(\001)g Fx(])p Fw(P)2128 2689 y Fs(e)2191 2677 y Fx(in)28 b(\(10.9\))f(splits)h(in)n(to)f(a)g(sum)437 2885 y Fw(P)490 2897 y Fs(e)564 2806 y Fm(X)540 2983 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)722 2792 y Fm(h)761 2885 y Fw(V)809 2897 y Fs(j)844 2885 y Fp(R)p Fw(V)962 2897 y Fs(j)1016 2885 y Fx(+)18 b Fw(V)1166 2850 y Fr(0)1190 2885 y Fp(R)p Fw(V)1327 2850 y Fr(0)1308 2905 y Fs(j)1369 2885 y Fp(\000)g Fw(V)1500 2897 y Fs(j)1535 2885 y Fp(R)p Fw(e)1644 2850 y Fv(\()p Fs(\016)r(\014)1740 2858 y Fk(1)1772 2850 y Fs(L)1818 2858 y Fi(r)q Fk(1)1880 2850 y Fv(+)p Fs(\016)r(\014)2001 2858 y Fk(2)2033 2850 y Fs(L)2079 2858 y Fi(r)q Fk(2)2140 2850 y Fv(\))p Fs(=)p Fv(2)2237 2885 y Fw(V)2304 2850 y Fr(0)2285 2905 y Fs(j)2346 2885 y Fp(\000)g Fw(V)2496 2850 y Fr(0)2477 2905 y Fs(j)2519 2885 y Fp(R)p Fw(e)2628 2850 y Fv(\()p Fr(\000)p Fs(\016)r(\014)2776 2858 y Fk(1)2808 2850 y Fs(L)2854 2858 y Fi(r)q Fk(1)2916 2850 y Fr(\000)p Fs(\016)r(\014)3038 2858 y Fk(2)3070 2850 y Fs(L)3116 2858 y Fi(r)q Fk(2)3177 2850 y Fv(\))p Fs(=)p Fv(2)3274 2885 y Fw(V)3322 2897 y Fs(j)3358 2792 y Fm(i)3411 2885 y Fw(P)3464 2897 y Fs(e)3500 2885 y Fw(:)180 3156 y Fx(W)-7 b(e)27 b(consider)f(the)h Fw(j)h Fx(=)22 b(1)27 b(term.)36 b(Using)26 b(that)h Fw(L)1703 3168 y Fs(r)r Fv(2)1799 3156 y Fx(comm)n(utes)g(with)g Fw(V)2425 3168 y Fv(1)2463 3156 y Fw(;)14 b(V)2566 3126 y Fr(0)2548 3177 y Fv(1)2616 3156 y Fx(and)27 b(that)g Fw(L)3013 3168 y Fs(r)r Fv(2)3082 3156 y Fw(P)3135 3168 y Fs(e)3194 3156 y Fx(=)c(0,)j(w)n(e)h(see)f(that)327 3339 y Fw(P)380 3351 y Fs(e)415 3339 y Fw(V)463 3351 y Fv(1)501 3339 y Fp(R)p Fw(e)610 3304 y Fs(\016)r(\014)680 3312 y Fk(1)712 3304 y Fs(L)758 3312 y Fi(r)q Fk(1)820 3304 y Fv(+)p Fs(\016)r(\014)941 3312 y Fk(2)973 3304 y Fs(L)1019 3312 y Fi(r)q Fk(2)1084 3339 y Fw(V)1151 3304 y Fr(0)1132 3359 y Fv(1)1175 3339 y Fw(P)1228 3351 y Fs(e)1347 3339 y Fx(=)82 b Fw(P)1547 3351 y Fs(e)1583 3339 y Fw(V)1631 3351 y Fv(1)1669 3339 y Fp(R)p Fw(e)1778 3304 y Fs(\016)r(\014)1848 3312 y Fk(1)1880 3304 y Fs(L)1926 3312 y Fi(r)q Fk(1)1987 3304 y Fs(=)p Fv(2)2059 3339 y Fw(V)2126 3304 y Fr(0)2107 3359 y Fv(1)2149 3339 y Fw(P)2202 3351 y Fs(e)2261 3339 y Fx(=)23 b Fw(P)2402 3351 y Fs(e)2437 3339 y Fw(V)2485 3351 y Fv(1)2523 3339 y Fp(R)p Fw(e)2632 3304 y Fs(\016)r(\014)2702 3312 y Fk(1)2734 3304 y Fs(L)2780 3312 y Fk(0)2812 3304 y Fs(=)p Fv(2)2884 3339 y Fw(e)2923 3304 y Fr(\000)p Fs(\016)r(\014)3045 3312 y Fk(1)3077 3304 y Fs(L)3123 3312 y Fi(p)3157 3304 y Fs(=)p Fv(2)3228 3339 y Fw(V)3295 3304 y Fr(0)3276 3359 y Fv(1)3318 3339 y Fw(P)3371 3351 y Fs(e)1347 3488 y Fx(=)82 b Fw(P)1547 3500 y Fs(e)1597 3396 y Fm(\020)1647 3488 y Fx(1)-18 b(l)1694 3500 y Fs(p)1750 3488 y Fp(\012)18 b Fw(e)1872 3453 y Fs(\016)r(\014)1942 3461 y Fk(1)1974 3453 y Fs(H)2028 3461 y Fi(p)2063 3453 y Fs(=)p Fv(2)2134 3396 y Fm(\021)2197 3488 y Fw(V)2245 3500 y Fv(1)2283 3488 y Fp(R)p Fw(e)2392 3453 y Fs(\016)r(\014)2462 3461 y Fk(1)2494 3453 y Fs(L)2540 3461 y Fk(0)2572 3453 y Fs(=)p Fv(2)2644 3488 y Fw(V)2711 3453 y Fr(0)2692 3508 y Fv(1)2748 3396 y Fm(\020)2797 3488 y Fw(e)2836 3453 y Fr(\000)p Fs(\016)r(\014)2958 3461 y Fi(p)2993 3453 y Fs(H)3047 3461 y Fi(p)3082 3453 y Fs(=)p Fv(2)3172 3488 y Fp(\012)g Fx(1)-18 b(l)3302 3500 y Fs(p)3339 3396 y Fm(\021)3402 3488 y Fw(P)3455 3500 y Fs(e)1347 3670 y Fx(=)1494 3578 y Fm(\020)1544 3670 y Fw(e)1583 3636 y Fs(\016)r(\014)1653 3644 y Fk(1)1685 3636 y Fs(H)1739 3644 y Fi(p)1774 3636 y Fs(=)p Fv(2)1864 3670 y Fp(\012)18 b Fx(1)-18 b(l)1994 3682 y Fs(p)2031 3578 y Fm(\021)2095 3670 y Fw(P)2148 3682 y Fs(e)2184 3670 y Fw(V)2232 3682 y Fv(1)2269 3670 y Fp(R)p Fw(e)2378 3636 y Fs(\016)r(\014)2448 3644 y Fk(1)2481 3636 y Fv(\()p Fs(L)2553 3644 y Fk(0)2585 3636 y Fr(\000)p Fs(e)p Fv(\))p Fs(=)p Fv(2)2765 3670 y Fw(V)2832 3636 y Fr(0)2813 3691 y Fv(1)2855 3670 y Fw(P)2908 3682 y Fs(e)2958 3578 y Fm(\020)3008 3670 y Fw(e)3047 3636 y Fr(\000)p Fs(\016)r(\014)3169 3644 y Fk(1)3201 3636 y Fs(H)3255 3644 y Fi(p)3290 3636 y Fs(=)p Fv(2)3379 3670 y Fp(\012)18 b Fx(1)-18 b(l)3509 3682 y Fs(p)3547 3578 y Fm(\021)3610 3670 y Fw(:)180 3878 y Fx(All)34 b(other)f(terms)g(in)g(\(10.9\))g(for)f Fw(j)38 b Fx(=)32 b(1,)j(as)d(w)n(ell)h(as)g(the)h(terms)f(for)g Fw(j)k Fx(=)32 b(2,)j(are)d(treated)h(similarly)f(and)h(one)180 3978 y(arriv)n(es)25 b(at)451 4160 y Fp(\000)p Fx(\003)574 4172 y Fs(e)632 4160 y Fx(=)719 4068 y Fm(\020)769 4160 y Fw(e)808 4126 y Fr(\000)p Fs(\014)898 4134 y Fi(p)932 4126 y Fs(H)986 4134 y Fi(p)1021 4126 y Fs(=)p Fv(2)1111 4160 y Fp(\012)18 b Fx(1)-18 b(l)1241 4172 y Fs(p)1279 4068 y Fm(\021)1366 4081 y(X)1342 4258 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)1524 4068 y Fm(\020)1573 4160 y Fw(e)1612 4126 y Fs(\014)1650 4134 y Fi(j)1680 4126 y Fs(H)1734 4134 y Fi(p)1770 4126 y Fs(=)p Fv(2)1859 4160 y Fp(\012)18 b Fx(1)-18 b(l)1989 4068 y Fm(\021)2052 4160 y Fw(P)2105 4172 y Fs(e)2141 4160 y Fx([)p Fw(V)2212 4172 y Fs(j)2247 4160 y Fp(R)p Fw(V)2365 4172 y Fs(j)2420 4160 y Fx(+)18 b Fw(V)2569 4126 y Fr(0)2551 4181 y Fs(j)2593 4160 y Fp(R)p Fw(V)2730 4126 y Fr(0)2711 4181 y Fs(j)617 4406 y Fp(\000)p Fw(V)730 4418 y Fs(j)765 4406 y Fp(R)p Fw(e)874 4372 y Fs(\016)r(\014)944 4380 y Fi(j)975 4372 y Fv(\()p Fs(L)1047 4380 y Fk(0)1079 4372 y Fr(\000)p Fs(e)p Fv(\))p Fs(=)p Fv(2)1259 4406 y Fw(V)1326 4372 y Fr(0)1307 4427 y Fs(j)1368 4406 y Fp(\000)g Fw(V)1518 4372 y Fr(0)1499 4427 y Fs(j)1541 4406 y Fp(R)p Fw(e)1650 4372 y Fr(\000)p Fs(\016)r(\014)1772 4380 y Fi(j)1803 4372 y Fv(\()p Fs(L)1875 4380 y Fk(0)1907 4372 y Fr(\000)p Fs(e)p Fv(\))p Fs(=)p Fv(2)2088 4406 y Fw(V)2136 4418 y Fs(j)2171 4406 y Fx(])p Fw(P)2247 4418 y Fs(e)2297 4314 y Fm(\020)2346 4406 y Fw(e)2385 4372 y Fr(\000)p Fs(\014)2475 4380 y Fi(j)2506 4372 y Fs(H)2560 4380 y Fi(p)2595 4372 y Fs(=)p Fv(2)2685 4406 y Fp(\012)g Fx(1)-18 b(l)2815 4418 y Fs(p)2852 4314 y Fm(\021)13 b(\020)2965 4406 y Fw(e)3004 4372 y Fs(\014)3042 4380 y Fi(p)3076 4372 y Fs(H)3130 4380 y Fi(p)3165 4372 y Fs(=)p Fv(2)3255 4406 y Fp(\012)18 b Fx(1)-18 b(l)3385 4418 y Fs(p)3423 4314 y Fm(\021)3486 4406 y Fw(;)180 4622 y Fx(where)25 b(w)n(e)f(used)i Fw(\016)s(\014)812 4634 y Fs(p)863 4622 y Fp(\000)14 b Fw(\016)s(\014)1029 4634 y Fs(j)1087 4622 y Fx(=)22 b Fw(\014)1221 4634 y Fs(p)1273 4622 y Fp(\000)14 b Fw(\014)1399 4634 y Fs(j)1433 4622 y Fx(.)36 b(Hence,)26 b Fp(\000)p Fx(\003)1883 4634 y Fs(e)1941 4622 y Fx(=)d Fw(R)q(H)7 b(S)e Fx(\(10)p Fw(:)p Fx(1\))14 b(+)g(\()p Fw(e)2602 4592 y Fr(\000)p Fs(\016)2684 4600 y Fi(p)2716 4592 y Fs(H)2770 4600 y Fi(p)2822 4622 y Fp(\012)g Fx(1)-18 b(l)2948 4634 y Fs(p)2985 4622 y Fx(\))p Fw(R)q Fx(\()p Fw(e)3152 4592 y Fs(\016)3182 4600 y Fi(p)3217 4592 y Fs(H)3271 4600 y Fi(p)3324 4622 y Fp(\012)14 b Fx(1)-18 b(l)3450 4634 y Fs(p)3487 4622 y Fx(\),)26 b(where)1017 4833 y Fw(R)e Fx(=)1215 4754 y Fm(X)1192 4931 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)1373 4741 y Fm(\020)1423 4833 y Fw(e)1462 4799 y Fs(\014)1500 4807 y Fi(j)1530 4799 y Fs(H)1584 4807 y Fi(p)1619 4799 y Fs(=)p Fv(2)1709 4833 y Fp(\012)18 b Fx(1)-18 b(l)1839 4741 y Fm(\021)1901 4833 y Fw(P)1954 4845 y Fs(e)1990 4833 y Fx([)p Fw(V)2061 4845 y Fs(j)2097 4833 y Fp(R)p Fx(\(1)19 b Fp(\000)f Fw(e)2382 4799 y Fs(\016)r(\014)2452 4807 y Fi(j)2482 4799 y Fv(\()p Fs(L)2554 4807 y Fk(0)2586 4799 y Fr(\000)p Fs(e)p Fv(\))p Fs(=)p Fv(2)2767 4833 y Fx(\))p Fw(V)2866 4799 y Fr(0)2847 4853 y Fs(j)1183 5079 y Fx(+)p Fw(V)1315 5045 y Fr(0)1296 5100 y Fs(j)1338 5079 y Fp(R)p Fx(\(1)h Fp(\000)f Fw(e)1623 5045 y Fr(\000)p Fs(\016)r(\014)1745 5053 y Fi(j)1775 5045 y Fv(\()p Fs(L)1847 5053 y Fk(0)1879 5045 y Fr(\000)p Fs(e)p Fv(\))p Fs(=)p Fv(2)2060 5079 y Fx(\))p Fw(V)2140 5091 y Fs(j)2175 5079 y Fx(])p Fw(P)2251 5091 y Fs(e)2301 4987 y Fm(\020)2351 5079 y Fw(e)2390 5045 y Fr(\000)p Fs(\014)2480 5053 y Fi(j)2510 5045 y Fs(H)2564 5053 y Fi(p)2599 5045 y Fs(=)p Fv(2)2689 5079 y Fp(\012)g Fx(1)-18 b(l)2819 5091 y Fs(p)2856 4987 y Fm(\021)2920 5079 y Fw(:)1938 5476 y Fx(27)p eop %%Page: 28 28 28 27 bop 180 210 a Fx(Since)30 b Fw(L)456 222 y Fv(0)523 210 y Fx(implemen)n(ts)g(the)g(free)g(dynamics,)g(w)n(e)g(ha)n(v)n(e)e (that)j Fw(e)2206 180 y Fs(iz)r(L)2309 188 y Fk(0)2345 210 y Fw(V)2393 222 y Fs(j)2428 210 y Fw(e)2467 180 y Fr(\000)p Fs(iz)r(L)2622 188 y Fk(0)2688 210 y Fx(comm)n(utes)f(with)g Fw(V)3339 180 y Fr(0)3320 232 y Fs(j)3363 210 y Fx(,)g(for)g Fw(z)g Fp(2)d Fo(C)15 b Fx(.)180 343 y(Using)27 b(this,)h(writing)894 310 y Fd(1)p Fr(\000)p Fs(e)1015 283 y Fi(\016)q(\014)1076 296 y(j)1108 283 y Fk(\()p Fi(L)1170 295 y Fk(0)1202 283 y Fh(\000)p Fi(e)p Fk(\))p Fi(=)p Fk(2)p 894 324 467 4 v 993 372 a Fs(L)1039 380 y Fk(0)1071 372 y Fr(\000)p Fs(e)p Fr(\006)p Fs(i)p Fv(0)1394 343 y Fx(=)23 b Fp(\000)1561 276 y Fm(R)1615 297 y Fs(\016)r(\014)1685 305 y Fi(j)1716 297 y Fs(=)p Fv(2)1599 372 y(0)1829 343 y Fw(ds)14 b(e)1964 313 y Fs(s)p Fv(\()p Fs(L)2067 321 y Fk(0)2099 313 y Fr(\000)p Fs(e)p Fv(\))2240 343 y Fx(and)27 b(using)h(that)g Fw(P)2852 355 y Fs(e)2888 343 y Fw(L)2945 355 y Fv(0)3004 343 y Fx(=)23 b Fw(eP)3184 355 y Fs(e)3220 343 y Fx(,)k(w)n(e)h(see)f (that)910 599 y Fw(P)963 611 y Fs(e)999 599 y Fw(V)1047 611 y Fs(j)1093 543 y Fn(1)18 b Fp(\000)g Fw(e)1281 512 y Fs(\016)r(\014)1351 520 y Fi(j)1381 512 y Fv(\()p Fs(L)1453 520 y Fk(0)1485 512 y Fr(\000)p Fs(e)p Fv(\))p Fs(=)p Fv(2)p 1093 580 574 4 v 1176 656 a Fw(L)1233 668 y Fv(0)1289 656 y Fp(\000)g Fw(e)g Fp(\000)g Fw(i)p Fx(0)1676 599 y Fw(V)1743 564 y Fr(0)1724 619 y Fs(j)1766 599 y Fw(P)1819 611 y Fs(e)1878 599 y Fx(=)23 b Fp(\000)p Fw(P)2084 611 y Fs(e)2119 599 y Fw(V)2186 564 y Fr(0)2167 619 y Fs(j)2219 543 y Fn(1)18 b Fp(\000)g Fw(e)2407 512 y Fr(\000)p Fs(\016)r(\014)2529 520 y Fi(j)2560 512 y Fv(\()p Fs(L)2632 520 y Fk(0)2664 512 y Fr(\000)p Fs(e)p Fv(\))p Fs(=)p Fv(2)p 2219 580 626 4 v 2329 656 a Fw(L)2386 668 y Fv(0)2441 656 y Fp(\000)g Fw(e)g Fx(+)g Fw(i)p Fx(0)2854 599 y Fw(V)2902 611 y Fs(j)2938 599 y Fw(P)2991 611 y Fs(e)3027 599 y Fw(:)180 815 y Fx(Consequen)n(tly)-7 b(,)27 b Fw(R)d Fx(=)e(0.)37 b(This)27 b(concludes)g(the)h(pro)r(of)f(of)h(Prop)r(osition)e(10.1.)p 3714 827 59 71 v 180 1089 a Fy(11)135 b(P)l(erturbation)46 b(Theory)f(for)g(NESS)180 1298 y Fx(In)25 b(this)h(section)f(w)n(e)g (dev)n(elop)f(a)h(p)r(erturbation)g(theory)f(for)h(the)g(NESS)h Fw(\021)s Fx(\()p Fw(A)p Fx(\))e(:=)2749 1206 y Fm(D)2800 1298 y Fx(\012)2860 1268 y Fr(\003)p 2860 1287 38 3 v 2860 1336 a Fs(\022)2898 1298 y Fw(;)14 b(\031)s Fx(\()p Fw(A)p Fx(\))3111 1310 y Fs(\022)3150 1298 y Fx(\012)3210 1310 y Fs(\022)3247 1206 y Fm(E)3298 1298 y Fx(.)36 b(The)26 b(v)n(ectors)180 1423 y(\012)240 1393 y Fr(\003)p 240 1412 V 240 1461 a Fs(\022)305 1423 y Fx(and)g(\012)525 1435 y Fs(\022)589 1423 y Fx(are)g(the)h(zero)e(eigen)n(v)n(ectors)f (of)j(the)g(op)r(erators)d Fw(K)2185 1393 y Fr(\003)p 2179 1412 V 2179 1461 a Fs(\022)2249 1423 y Fx(and)j Fw(K)2481 1435 y Fs(\022)2544 1423 y Fx(resp)r(ectiv)n(ely)-7 b(,)26 b(see)h(\(8.10\))e(and)i(\(8.11\).)180 1531 y(W)-7 b(e)25 b(deriv)n(e)f(p)r(erturbation)h(expansions)f(for)g(\012)1647 1501 y Fr(\003)p 1647 1520 V 1647 1569 a Fs(\022)1710 1531 y Fx(and)h(\012)1929 1543 y Fs(\022)1967 1531 y Fx(,)g(see)g(\(11.10\))f(and)h(\(11.15\))e(b)r(elo)n(w)i(using)g(the)g (F)-7 b(esh)n(bac)n(h)180 1639 y(maps)33 b(in)n(tro)r(duced)g(in)h([6)o (,)g(7)o(],)h(and)f(extended)f(in)h([5].)54 b(W)-7 b(e)34 b(review)e(the)i(de\014nitions)g(and)f(some)g(prop)r(erties)f(of)180 1739 y(these)27 b(maps)g(referring)f(the)i(reader)e(to)h([7,)h(5)o(])g (for)e(more)h(detail.)37 b(F)-7 b(or)27 b(simplicit)n(y)g(w)n(e)g (presen)n(t)g(here)g(the)h(original)180 1839 y(v)n(ersion,)22 b([6)o(,)h(7)o(],)h(though)e(the)g(re\014ned)g(one,)h([5)o(],)h(the)e (smo)r(oth)g(F)-7 b(esh)n(bac)n(h)22 b(map,)h(is)f(easier)f(to)h(use)g (from)g(a)f(tec)n(hnical)180 1938 y(p)r(oin)n(t)28 b(of)f(view.)305 2038 y(Let)22 b Fw(X)28 b Fx(b)r(e)22 b(a)g(Banac)n(h)f(space)g(and)h (let)g Fw(P)34 b Fx(b)r(e)22 b(a)g(pro)5 b(jection)21 b(on)g Fw(X)7 b Fx(.)35 b(De\014ne)p 2634 1971 65 4 v 22 w Fw(P)g Fx(:=)23 b Fn(1)7 b Fp(\000)g Fw(P)33 b Fx(and)22 b(let)g Fw(H)p 3385 2015 56 3 v 25 x Fs(P)3464 2038 y Fx(:=)p 3575 1971 65 4 v 23 w Fw(P)11 b(H)p 3715 1971 V 7 w(P)180 2138 y Fx(and)22 b Fw(R)p 399 2115 56 3 v 25 x Fs(P)454 2138 y Fx(\()p Fw(H)7 b Fx(\))24 b(:=)p 729 2071 65 4 v 23 w Fw(P)11 b(H)869 2102 y Fr(\000)p Fv(1)p 862 2128 56 3 v 862 2176 a Fs(P)p 958 2071 65 4 v 958 2138 a Fw(P)34 b Fx(if)23 b Fw(H)p 1185 2115 56 3 v 25 x Fs(P)1263 2138 y Fx(is)f(in)n(v)n(ertible)f(on)h(Ran)p 1961 2071 65 4 v Fw(P)12 b Fx(.)35 b(W)-7 b(e)23 b(de\014ne)f(the)h(F) -7 b(esh)n(bac)n(h)21 b(map)h Fw(F)3177 2150 y Fs(P)3255 2138 y Fx(b)n(y)g(the)h(relation)180 2247 y Fw(F)233 2259 y Fs(P)289 2247 y Fx(\()p Fw(H)7 b Fx(\))23 b(:=)g Fw(P)j Fx(\()p Fw(H)f Fp(\000)18 b Fw(H)7 b(R)p 990 2224 56 3 v 25 x Fs(P)1045 2247 y Fx(\()p Fw(H)g Fx(\))p Fw(H)g Fx(\))14 b Fw(P)40 b Fx(on)27 b(the)h(domain)537 2427 y(Dom\()p Fw(F)796 2439 y Fs(P)852 2427 y Fx(\))23 b(=)620 2551 y Fp(f)p Fw(H)29 b Fx(:)23 b Fw(X)30 b Fp(!)23 b Fw(X)7 b Fp(j)p Fw(H)p 1179 2528 V 26 x Fs(P)1261 2551 y Fx(is)28 b(in)n(v)n(ertible)o Fw(;)14 b Fx(Ran)o Fw(P)35 b Fp(\022)23 b Fx(Dom\()p Fw(H)7 b Fx(\))p Fw(;)14 b Fx(Ran)p Fw(R)p 2608 2528 V 26 x Fs(P)2663 2551 y Fx(\()p Fw(H)7 b Fx(\))24 b Fp(\022)f Fx(Dom\()p Fw(P)12 b(H)p 3262 2485 65 4 v 7 w(P)f Fx(\))p Fp(g)p Fw(:)180 2732 y Fx(A)28 b(k)n(ey)f(prop)r(ert)n(y)f(of)i(the)g(maps)f Fw(F)1267 2744 y Fs(P)1351 2732 y Fx(is)g(giv)n(en)g(in)h(the)g(follo)n (wing)e(statemen)n(t)i(pro)n(v)n(en)e(in)i([7)o(]:)180 2912 y Fn(Theorem)i(11.1)h(\(Isosp)s(ectralit)m(y)h(Theorem\))113 b Fq(\(i\))42 b Fx(0)22 b Fp(2)i Fw(\033)s Fx(\()p Fw(H)7 b Fx(\))46 b Fp(\()-14 b(\))47 b Fx(0)22 b Fp(2)i Fw(\033)s Fx(\()p Fw(F)2934 2924 y Fs(P)2990 2912 y Fx(\()p Fw(H)7 b Fx(\)\))p Fq(,)227 3077 y(\(ii\))43 b Fw(H)7 b( )25 b Fx(=)e(0)46 b Fp(\()-14 b(\))46 b Fw(F)970 3089 y Fs(P)1026 3077 y Fx(\()p Fw(H)7 b Fx(\))p Fw(')23 b Fx(=)g(0)29 b Fq(with)i 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y Fx(is)h(the)f(sp)r(ectral)g(pro)5 b(jection)27 b(for)g(the)h(self-adjoin)n(t)f(op)r(erator)e Fw(M)2653 3617 y Fs(\022)2718 3605 y Fx(corresp)r(onding)g(to)j(the)g (sp)r(ectral)180 3704 y(in)n(terv)-5 b(al)27 b([0)p Fw(;)14 b(\032)p Fx(])27 b(\(remem)n(b)r(er)g(that)h Fw(M)1356 3716 y Fs(\022)1421 3704 y Fx(is)f(a)h(p)r(ositiv)n(e)f(op)r(erator\).) 305 3804 y(The)g(follo)n(wing)g(result)g(is)h(pro)n(v)n(en)e(in)h ([25],)g(Lemma)h(6.3.)180 3968 y Fn(Lemma)i(11.2)40 b Fq(T)-6 b(ake)27 b Fw(\032)971 3980 y Fv(0)1031 3968 y Fp(2)d Fx(\(0)p Fw(;)14 b(\033)s(=)p Fx(2\))25 b Fq(and)i(let)f Fp(j)p Fw(g)s Fp(j)d Fw(<)1883 3916 y Fp(p)p 1952 3916 81 4 v 52 x Fw(\032)1995 3980 y Fv(0)2046 3968 y Fw(g)2086 3980 y Fv(0)2123 3968 y Fq(.)37 b(If)27 b Fw(z)f Fp(2)e Fw(S)2464 3980 y Fs(e)2526 3968 y Fq(then)i Fw(K)2778 3980 y Fs(\022)r(z)2872 3968 y Fx(:=)d Fw(K)3054 3980 y Fs(\022)3102 3968 y Fp(\000)11 b Fw(z)26 b Fp(2)d Fx(Dom\()p Fw(F)3580 3980 y Fs(P)3622 3988 y Fi(e\032)3680 4000 y Fk(0)3722 3968 y Fx(\))p Fq(,)180 4096 y(and)30 b(the)g(op)l(er)l (ator)h Fw(K)883 4053 y Fv(\(1\))877 4121 y Fs(\022)r(z)995 4096 y Fx(:=)22 b Fw(F)1158 4108 y Fs(P)1200 4116 y Fi(e\032)1258 4128 y Fk(0)1300 4096 y Fx(\()p Fw(K)1403 4108 y Fs(\022)r(z)1474 4096 y Fx(\))30 b Fq(acting)h(on)e Fx(Ran)p Fw(P)2105 4108 y Fs(e\032)2170 4116 y Fk(0)2238 4096 y Fq(is)h(of)g(the)g(form) 1187 4305 y Fw(K)1264 4262 y Fv(\(1\))1258 4330 y Fs(\022)r(z)1375 4305 y Fx(=)23 b(\()p Fw(e)18 b Fp(\000)h Fw(z)t Fx(\))p Fn(1)e Fx(+)h Fw(L)1916 4317 y Fs(r)r(\022)2004 4305 y Fx(+)g Fw(g)2130 4271 y Fv(2)2167 4305 y Fx(\003)2225 4317 y Fs(e)2279 4305 y Fx(+)g Fw(O)r Fx(\()p Fw(\017)p Fx(\()p Fw(g)s(;)c(\032)2648 4317 y Fv(0)2686 4305 y Fx(\)\))p Fw(:)795 b Fx(\(11.2\))180 4485 y Fq(The)61 b(r)l(emainder)h(is)e(estimate)l(d)h(in)g(op)l(er)l(ator)g(norm,)69 b Fp(k)p Fw(O)r Fx(\()p Fw(\017)p Fx(\()p Fw(g)s(;)14 b(\032)2431 4497 y Fv(0)2468 4485 y Fx(\)\))p Fp(k)78 b(\024)g Fw(C)6 b(\017)p Fx(\()p Fw(g)2956 4507 y Fi(\016)q(\014)3028 4485 y Fw(;)14 b(\032)3108 4497 y Fv(0)3145 4485 y Fx(\))p Fq(,)69 b(wher)l(e)61 b Fw(g)3566 4507 y Fi(\016)q(\014)3715 4485 y Fx(=)180 4585 y Fp(j)p Fw(g)s Fp(j)14 b Fx(max)437 4597 y Fs(j)486 4585 y Fx(sup)611 4605 y Fr(j)p Fs(\022)r Fr(j)p Fs(<\022)769 4613 y Fk(0)818 4585 y Fp(k)p Fw(G)925 4597 y Fs(j)959 4585 y Fp(k)1001 4600 y Fs(\016)r(\014)1071 4608 y Fi(j)1102 4600 y Fs(;)p Fv(1)p Fs(=)p Fv(2)p Fs(;\022)1279 4585 y Fq(,)30 b(with)h(a)f Fw(C)36 b Fq(indep)l(endent)30 b(of)h Fw(\016)s(\014)2327 4597 y Fs(j)2362 4585 y Fq(,)f Fw(\022)r Fq(,)g(and)g(wher)l(e)h(we)f(have)h(set)1273 4786 y Fw(\017)p Fx(\()p Fw(g)s(;)14 b(\032)p Fx(\))23 b(:=)g Fp(j)p Fw(g)s Fp(j)p Fw(\032)1760 4752 y Fs(\026)1822 4786 y Fx(+)18 b Fp(j)p Fw(g)s Fp(j)1994 4752 y Fv(3)2031 4786 y Fw(\032)2074 4752 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2249 4786 y Fx(+)g Fp(j)p Fw(g)s Fp(j)2421 4752 y Fv(2)2458 4786 y Fw(\032)2501 4752 y Fv(2)p Fs(\026)p Fr(\000)p Fv(1)2664 4786 y Fw(:)881 b Fx(\(11.3\))180 4976 y(T)-7 b(o)30 b(unify)h(the)g(follo)n(wing)e(analysis)g(w)n(e)h(write)g(\012) 1734 4937 y Fv(#)1734 5007 y Fs(\022)1768 4990 y Fk(#)1851 4976 y Fx(for)g(either)g(\012)2279 4988 y Fs(\022)2347 4976 y Fx(or)g(\012)2512 4946 y Fr(\003)p 2512 4965 38 3 v 2512 5014 a Fs(\022)2550 4976 y Fx(.)45 b(Corresp)r(ondingly)-7 b(,)29 b Fw(I)3304 4937 y Fv(#)3297 5007 y Fs(\022)3331 4990 y Fk(#)3385 4976 y Fx(,)i Fw(L)3496 4993 y Fv(0)p Fs(\022)3563 4977 y Fk(#)3646 4976 y Fx(and)180 5103 y Fw(K)257 5063 y Fv(#)251 5133 y Fs(\022)285 5116 y Fk(#)367 5103 y Fx(stand)e(for)f(either)h Fw(I)996 5115 y Fs(\022)1063 5103 y Fx(or)f Fw(I)1209 5072 y Fr(\003)p 1202 5091 V 1202 5140 a Fs(\022)1247 5103 y Fx(,)h(for)g(either)f Fw(L)1721 5115 y Fv(0)p Fs(\022)1821 5103 y Fx(or)g Fw(L)1981 5128 y Fv(0)p 2014 5080 34 3 v Fs(\022)2051 5103 y Fx(,)h(and)g(for)f (either)h Fw(K)2702 5115 y Fs(\022)2768 5103 y Fx(or)f Fw(K)2948 5072 y Fr(\003)p 2942 5091 38 3 v 2942 5140 a Fs(\022)2986 5103 y Fx(,)h(resp)r(ectiv)n(ely)-7 b(.)40 b(W)-7 b(e)29 b(use)180 5222 y(the)f(shorthand)f Fw(P)769 5234 y Fv(0)829 5222 y Fp(\021)c Fw(P)970 5234 y Fv(0)p Fs(\032)1037 5242 y Fk(0)1102 5222 y Fx(and)p 1263 5155 64 4 v 27 w Fw(R)1327 5234 y Fv(0)1364 5222 y Fx(\()p Fw(A)p Fx(\))h(:=)p 1625 5155 65 4 v 23 w Fw(P)1690 5234 y Fv(0)1727 5222 y Fw(A)1789 5186 y Fr(\000)p Fv(1)p 1789 5212 52 3 v 1789 5260 a Fs(P)1841 5268 y Fk(0)p 1878 5155 65 4 v 1878 5222 a Fw(P)1943 5234 y Fv(0)1981 5222 y Fx(,)j(where)g Fw(A)2333 5234 y Fs(P)2412 5222 y Fx(:=)c Fw(P)12 b(AP)g Fx(.)1938 5476 y(28)p eop %%Page: 29 29 29 28 bop 305 210 a Fx(Theorem)26 b(11.1)h(and)g(Lemma)g(11.2)g(imply)h (that)f Fw(K)1953 170 y Fv(#)1947 240 y Fs(\022)1981 224 y Fk(#)2057 210 y Fp(2)d Fx(Dom\()p Fw(F)2395 222 y Fs(P)2437 230 y Fk(0)2474 210 y Fx(\),)k(that)1596 395 y Fw(F)1649 407 y Fs(P)1691 415 y Fk(0)1728 395 y Fx(\()p Fw(K)1837 355 y Fv(#)1831 425 y Fs(\022)1865 408 y Fk(#)1918 395 y Fx(\))p Fw(P)2003 407 y Fv(0)2041 395 y Fx(\012)2101 355 y Fv(#)2101 425 y Fs(\022)2135 408 y Fk(#)2212 395 y Fx(=)22 b(0)p Fw(;)1204 b Fx(\(11.4\))180 579 y(and)27 b(that)h(the)g(original)e(eigen)n(v)n(ector)g(\012)1456 539 y Fv(#)1456 609 y Fs(\022)1490 593 y Fk(#)1571 579 y Fx(can)h(b)r(e)h(reconstructed)e(as)1362 779 y(\012)1422 739 y Fv(#)1422 809 y Fs(\022)1456 793 y Fk(#)1532 779 y Fx(=)1620 687 y Fm(h)1659 779 y Fn(1)18 b Fp(\000)g Fw(g)p 1851 712 64 4 v 3 w(R)1915 791 y Fv(0)1952 779 y Fx(\()p Fw(K)2061 739 y Fv(#)2055 809 y Fs(\022)2089 793 y Fk(#)2142 779 y Fx(\))p Fw(I)2217 739 y Fv(#)2210 809 y Fs(\022)2244 793 y Fk(#)2298 687 y Fm(i)2337 779 y Fw(P)2390 791 y Fv(0)2428 779 y Fx(\012)2488 739 y Fv(#)2488 809 y Fs(\022)2522 793 y Fk(#)2575 779 y Fw(:)970 b Fx(\(11.5\))180 988 y(W)-7 b(e)28 b(expand)p 611 921 V 27 w Fw(R)675 1000 y Fv(0)712 988 y Fx(\()p Fw(K)821 948 y Fv(#)815 1018 y Fs(\022)849 1001 y Fk(#)902 988 y Fx(\))g(in)g(this)g(expression)e(in)n(to)h(a)h(Neumann)g(series,)917 1244 y(\012)977 1204 y Fv(#)977 1274 y Fs(\022)1011 1257 y Fk(#)1087 1244 y Fx(=)1175 1140 y Fs(N)6 b Fr(\000)p Fv(1)1187 1165 y Fm(X)1184 1341 y Fs(n)p Fv(=0)1332 1244 y Fw(g)1375 1209 y Fs(n)1420 1244 y Fx(\()p Fp(\000)p 1517 1177 V Fw(R)1581 1256 y Fv(0)1618 1244 y Fx(\()p Fw(L)1707 1261 y Fv(0)p Fs(\022)1774 1244 y Fk(#)h Fx(\))p Fw(I)1902 1204 y Fv(#)1895 1274 y Fs(\022)1929 1257 y Fk(#)1983 1244 y Fx(\))2015 1209 y Fs(n)2061 1244 y Fw(P)2114 1256 y Fv(0)2151 1244 y Fx(\012)2211 1204 y Fv(#)2211 1274 y Fs(\022)2245 1257 y Fk(#)2317 1244 y Fx(+)18 b Fw(O)2479 1152 y Fm(\020)2529 1244 y Fx(\()p Fw(g)2591 1265 y Fi(\016)q(\014)2662 1244 y Fw(\032)2705 1201 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2705 1266 y(0)2861 1244 y Fx(\))2893 1209 y Fs(N)2957 1152 y Fm(\021)3020 1244 y Fw(;)525 b Fx(\(11.6\))180 1510 y(for)30 b(an)n(y)f Fw(N)36 b Fp(\025)27 b Fx(1,)j(pro)n(vided)f(that)h Fw(O)r Fx(\(\()p Fw(g)1444 1531 y Fi(\016)q(\014)1517 1510 y Fw(\032)1560 1467 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)1560 1532 y(0)1716 1510 y Fx(\)\))e(=)e Fw(o)1939 1522 y Fs(g)1978 1510 y Fx(\(1\).)45 b(The)30 b(remainder)f(term)h(in)g(\(11.6\))g(is)g (obtained)f(b)n(y)180 1610 y(using)e(a)g(standard)g(estimate)h(on)f (the)h Fw(N)9 b Fx(th)28 b(term)f(of)h(the)g(con)n(v)n(ergen)n(t)d (Neumann)j(series.)36 b(Indeed,)28 b(writing)488 1722 y Fm(h)p 527 1748 V 93 x Fw(R)591 1827 y Fv(0)628 1815 y Fx(\()p Fw(L)717 1831 y Fv(0)p Fs(\022)784 1815 y Fk(#)7 b Fx(\))p Fw(I)912 1775 y Fv(#)905 1845 y Fs(\022)939 1828 y Fk(#)993 1722 y Fm(i)1032 1740 y Fs(N)3568 1815 y Fx(\(11.7\))571 2041 y(=)82 b(\()p Fw(M)831 2053 y Fs(\022)887 2041 y Fx(+)18 b Fw(\032)1013 2053 y Fv(0)1050 2041 y Fx(\))1082 2007 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)1253 1924 y Fm(\024)1306 1985 y Fw(M)1387 1997 y Fs(\022)1443 1985 y Fx(+)g Fw(\032)1569 1997 y Fv(0)p 1306 2022 300 4 v 1317 2105 a Fw(L)1374 2122 y Fv(0)p Fs(\022)1441 2106 y Fk(#)p 1494 2039 65 4 v 1494 2105 a Fw(P)1559 2117 y Fv(0)p 1616 1975 V 1616 2041 a Fw(P)1681 2053 y Fv(0)1718 2041 y Fx(\()p Fw(M)1831 2053 y Fs(\022)1887 2041 y Fx(+)g Fw(\032)2013 2053 y Fv(0)2050 2041 y Fx(\))2082 2007 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2239 2041 y Fw(I)2282 2001 y Fv(#)2275 2071 y Fs(\022)2309 2055 y Fk(#)2363 2041 y Fx(\()p Fw(M)2476 2053 y Fs(\022)2531 2041 y Fx(+)g Fw(\032)2657 2053 y Fv(0)2695 2041 y Fx(\))2727 2007 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2883 1924 y Fm(\025)2927 1941 y Fs(N)3004 2041 y Fx(\()p Fw(M)3117 2053 y Fs(\022)3173 2041 y Fx(+)g Fw(\032)3299 2053 y Fv(0)3336 2041 y Fx(\))3368 2007 y Fv(1)p Fs(=)p Fv(2)180 2254 y Fx(and)27 b(using)h(the)g (estimates)1095 2440 y Fp(j)p Fw(g)s Fp(j)1198 2345 y Fm(\015)1198 2394 y(\015)1198 2444 y(\015)1243 2440 y Fx(\()p Fw(M)1356 2452 y Fs(\022)1412 2440 y Fx(+)18 b Fw(\032)1538 2452 y Fv(0)1575 2440 y Fx(\))1607 2406 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)1764 2440 y Fw(I)1807 2400 y Fv(#)1800 2470 y Fs(\022)1834 2454 y Fk(#)1888 2440 y Fx(\()p Fw(M)2001 2452 y Fs(\022)2056 2440 y Fx(+)g Fw(\032)2182 2452 y Fv(0)2220 2440 y Fx(\))2252 2406 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2408 2345 y Fm(\015)2408 2394 y(\015)2408 2444 y(\015)2477 2440 y Fp(\024)23 b Fw(g)2595 2461 y Fi(\016)q(\014)2666 2440 y Fw(\032)2709 2397 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2709 2462 y(0)3568 2440 y Fx(\(11.8\))180 2664 y(\(see)28 b(also)e(Lemma)h(5.3)g(of)h([25) o(]\))g(and)1388 2568 y Fm(\015)1388 2618 y(\015)1388 2668 y(\015)1458 2627 y Fs(M)1521 2635 y Fk(0)1554 2627 y Fv(+)p Fs(\032)1639 2635 y Fk(0)p 1444 2645 241 4 v 1444 2702 a Fs(L)1490 2723 y Fk(0)p Fi(\022)1547 2710 y Fk(#)p 1601 2654 52 3 v 1601 2702 a Fs(P)1653 2710 y Fk(0)p 1695 2597 65 4 v 1695 2664 a Fw(P)1760 2676 y Fv(0)1797 2568 y Fm(\015)1797 2618 y(\015)1797 2668 y(\015)1866 2664 y Fp(\024)23 b Fw(C)6 b Fx(,)28 b(w)n(e)f(obtain)954 2918 y Fp(j)p Fw(g)s Fp(j)1043 2883 y Fs(N)1119 2797 y Fm(\015)1119 2847 y(\015)1119 2897 y(\015)1119 2947 y(\015)1165 2826 y(h)p 1205 2851 64 4 v 1205 2918 a Fw(R)1268 2930 y Fv(0)1305 2918 y Fx(\()p Fw(L)1394 2934 y Fv(0)p Fs(\022)1461 2918 y Fk(#)8 b Fx(\))p Fw(I)1590 2878 y Fv(#)1583 2948 y Fs(\022)1617 2931 y Fk(#)1671 2826 y Fm(i)1710 2843 y Fs(N)1787 2918 y Fw(P)1840 2930 y Fv(0)1877 2918 y Fx(\012)1937 2878 y Fv(#)1937 2948 y Fs(\022)1971 2931 y Fk(#)2024 2797 y Fm(\015)2024 2847 y(\015)2024 2897 y(\015)2024 2947 y(\015)2094 2918 y Fp(\024)22 b Fw(\032)2224 2875 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2224 2940 y(0)2380 2918 y Fx(\()p Fw(g)2442 2939 y Fi(\016)q(\014)2514 2918 y Fw(\032)2557 2875 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)2557 2940 y(0)2713 2918 y Fx(\))2745 2883 y Fs(N)2808 2918 y Fw(\032)2851 2875 y Fv(1)p Fs(=)p Fv(2)2851 2940 y(0)2983 2918 y Fw(:)562 b Fx(\(11.9\))180 3144 y(Observ)n(e)26 b(that)i(since)f Fw(g)912 3165 y Fi(\016)q(\014)983 3144 y Fw(\032)1026 3108 y Fr(\000)p Fv(1)1026 3166 y(0)1139 3144 y Fx(=)22 b Fw(o)1266 3156 y Fs(g)1305 3144 y Fx(\(1\))28 b(w)n(e)f(ha)n(v)n(e)1136 3334 y(\012)1196 3295 y Fv(#)1196 3365 y Fs(\022)1230 3348 y Fk(#)1306 3334 y Fx(=)1423 3256 y Fm(X)1394 3431 y Fs(n)p Fv(=0)p Fs(;)p Fv(1)1586 3334 y Fw(g)1629 3300 y Fs(n)1674 3334 y Fx(\()p Fp(\000)p 1771 3268 V Fw(R)1834 3346 y Fv(0)1871 3334 y Fx(\()p Fw(L)1960 3351 y Fv(0)p Fs(\022)2027 3335 y Fk(#)2081 3334 y Fx(\))p Fw(I)2156 3295 y Fv(#)2149 3365 y Fs(\022)2183 3348 y Fk(#)2237 3334 y Fx(\))2269 3300 y Fs(n)2314 3334 y Fw(P)2367 3346 y Fv(0)2405 3334 y Fx(\012)2465 3295 y Fv(#)2465 3365 y Fs(\022)2499 3348 y Fk(#)2570 3334 y Fx(+)18 b Fw(o)p Fx(\()p Fw(g)s Fx(\))p Fw(:)703 b Fx(\(11.10\))305 3605 y(Let)27 b Fw(K)530 3575 y Fv(#\(1\))696 3605 y Fx(:=)c Fw(F)860 3617 y Fs(\032)894 3625 y Fk(0)932 3605 y Fx(\()p Fw(K)1041 3566 y Fv(#)1035 3636 y Fs(\022)1069 3619 y Fk(#)1122 3605 y Fx(\).)37 b(As)28 b(in)f(\(11.2\))g(it)h(can)f (b)r(e)h(written)g(as)1568 3796 y Fw(K)1645 3762 y Fv(#\(1\))1811 3796 y Fx(=)23 b Fw(K)1976 3753 y Fv(#\(1\))1970 3819 y(0)2137 3796 y Fx(+)18 b Fw(W)2310 3762 y Fv(#)2369 3796 y Fw(;)180 3980 y Fx(where)28 b Fw(K)498 3936 y Fv(#\(1\))492 4002 y(0)667 3980 y Fx(:=)c Fw(L)836 3996 y Fs(r)r(\022)903 3980 y Fk(#)i Fx(+)19 b Fw(g)1102 3949 y Fv(2)1139 3980 y Fx(\003)1197 3949 y Fv(#)1284 3980 y Fx(with)29 b(\003)1532 3949 y Fv(#)1620 3980 y Fx(either)f(\003)1914 3992 y Fv(0)1980 3980 y Fx(or)g(\003)2141 3949 y Fr(\003)2141 4000 y Fv(0)2179 3980 y Fx(,)h(where)g(\003)2531 3992 y Fv(0)2597 3980 y Fx(is)f(giv)n(en)g(b)n(y)h(\(7.1\))g(with)g Fw(e)c Fx(=)f(0.)41 b(W)-7 b(e)180 4079 y(assume)31 b(that)g Fw(\016)s(\014)36 b Fx(v)-5 b(aries)31 b(in)g(the)h(set)g Fp(j)p Fw(\016)s(\014)t Fp(j)e(\024)f Fw(c)p Fx(,)j(for)f(some)g Fw(c)e(>)g Fx(0,)j(so)f(that)h Fw(g)2667 4100 y Fi(\016)q(\014)2769 4079 y Fx(can)f(b)r(e)h(replaced)f(b)n(y)g Fw(g)s Fx(.)48 b(T)-7 b(ak)n(e)180 4201 y Fw(\032)223 4213 y Fv(0)283 4201 y Fx(=)23 b Fw(g)414 4171 y Fv(2)p Fr(\000)p Fv(2)p Fs(\013)607 4201 y Fx(with)28 b Fw(\013)23 b Fx(=)970 4160 y Fs(\026)p Fr(\000)p Fv(1)p Fs(=)p Fv(2)p 970 4182 193 4 v 970 4229 a Fs(\026)p Fv(+1)p Fs(=)p Fv(2)1172 4201 y Fx(,)28 b(then)g(Lemma)f(11.2)g(giv)n(es)1687 4393 y Fw(W)1777 4358 y Fv(#)1858 4393 y Fx(=)c Fw(O)r Fx(\()p Fw(g)2086 4358 y Fv(2+)p Fs(\013)2218 4393 y Fx(\))p Fw(:)1253 b Fx(\(11.11\))180 4558 y(By)27 b(Theorem)g(11.1,) 1663 4658 y Fw(K)1740 4624 y Fv(#\(1\))1884 4658 y Fw(P)1937 4670 y Fv(0)1974 4658 y Fx(\012)2034 4618 y Fv(#)2034 4688 y Fs(\022)2068 4671 y Fk(#)2144 4658 y Fx(=)c(0)p Fw(:)1229 b Fx(\(11.12\))180 4816 y(Let)28 b Fw(Q)395 4776 y Fv(#)395 4838 y(0)476 4816 y Fx(:=)23 b Fw(\037)639 4849 y Fs(K)699 4819 y Fk(#\(1\))695 4867 y(0)822 4849 y Fv(=0)933 4816 y Fx(=)g Fw(\037)1073 4833 y Fv(\003)1118 4816 y Fk(#)1168 4833 y Fv(=0)1275 4816 y Fp(\012)18 b Fw(\037)1410 4828 y Fs(L)1456 4836 y Fi(r)1489 4828 y Fv(=0)1577 4816 y Fx(,)28 b(where)f Fw(\037)1920 4833 y Fv(\003)1965 4816 y Fk(#)2015 4833 y Fv(=0)2131 4816 y Fx(is)h(the)g(Riesz)g(pro)5 b(jection)26 b(on)n(to)h(the)h(k)n(ernel) f(of)h(\003)3699 4786 y Fv(#)3757 4816 y Fx(.)180 4962 y(Since)i(the)g(eigen)n(v)-5 b(alue)28 b(zero)h(of)h(the)g(op)r(erator) d Fw(K)1775 4932 y Fv(#\(1\))1948 4962 y Fx(\(and)j(of)g(the)g(op)r (erator)d Fw(K)2799 4919 y Fv(#\(1\))2793 4984 y(0)2943 4962 y Fx(\))j(is)f(separated)f(from)i(the)180 5062 y(rest)d(of)h(its)f (sp)r(ectrum)h(b)n(y)g(a)f(gap)g(of)g(order)1623 5227 y(min\()p Fw(\034)1838 5193 y Fr(0)1862 5227 y Fw(;)14 b(g)1942 5193 y Fv(2)1979 5227 y Fx(\))23 b Fp(\035)g Fw(g)2183 5193 y Fv(2+)p Fs(\013)2314 5227 y Fw(;)1189 b Fx(\(11.13\))1938 5476 y(29)p eop %%Page: 30 30 30 29 bop 180 210 a Fx(as)35 b(is)h(seen)g(b)n(y)g(a)f(standard)g (argumen)n(t,)i(w)n(e)f(conclude)g(that)g Fw(K)2278 180 y Fv(#\(1\))2458 210 y Fp(2)i Fx(Dom\()p Fw(F)2810 239 y Fs(Q)2862 210 y Fk(#)2862 257 y(0)2917 210 y Fx(\).)62 b(Therefore,)37 b(b)n(y)f(Theo-)180 318 y(rem)27 b(11.1,)1226 438 y Fw(P)1279 450 y Fv(0)1317 438 y Fx(\012)1377 398 y Fv(#)1377 468 y Fs(\022)1411 452 y Fk(#)1487 438 y Fx(=)1575 346 y Fm(\020)1624 438 y Fn(1)18 b Fp(\000)g Fw(R)p 1836 415 107 3 v 43 x Fs(Q)1888 453 y Fk(#)1888 499 y(0)1943 438 y Fx(\()p Fw(K)2052 404 y Fv(#\(1\))2195 438 y Fx(\))p Fw(W)2317 404 y Fv(#)2376 346 y Fm(\021)2439 438 y Fw(Q)2505 398 y Fv(#)2505 460 y(0)2564 438 y Fx(\012)2624 398 y Fv(#)2624 468 y Fs(\022)2658 452 y Fk(#)2711 438 y Fw(:)792 b Fx(\(11.14\))180 615 y(Since)28 b Fp(k)p Fw(R)p 502 592 V 42 x Fs(Q)554 629 y Fk(#)554 675 y(0)607 615 y Fx(\()p Fw(K)716 585 y Fv(#\(1\))860 615 y Fx(\))p Fp(k)23 b(\024)f Fw(C)6 b Fx([min)q(\()p Fw(\034)1348 585 y Fr(0)1372 615 y Fw(;)14 b(g)1452 585 y Fv(2)1489 615 y Fx(\)])1544 585 y Fr(\000)p Fv(1)1661 615 y Fx(w)n(e)27 b(ha)n(v)n(e)g(the)h(absolutely)f(con)n(v)n(ergen)n(t)e(p)r (erturbation)i(expansion)1094 878 y Fw(P)1147 890 y Fv(0)1184 878 y Fx(\012)1244 838 y Fv(#)1244 908 y Fs(\022)1278 891 y Fk(#)1354 878 y Fx(=)c Fw(C)1507 844 y Fv(#)1609 774 y Fr(1)1582 799 y Fm(X)1580 975 y Fs(n)p Fv(=0)1719 786 y Fm(\020)1768 878 y Fp(\000)p Fw(R)p 1896 855 V 42 x Fs(Q)1948 892 y Fk(#)1948 938 y(0)2002 878 y Fx(\()p Fw(K)2111 835 y Fv(#\(1\))2105 900 y(0)2255 878 y Fx(\))p Fw(W)2377 844 y Fv(#)2435 786 y Fm(\021)2485 803 y Fs(n)2544 878 y Fw(\020)2586 844 y Fv(#)2664 878 y Fp(\012)18 b Fx(\012)2807 890 y Fs(r)2843 878 y Fw(:)660 b Fx(\(11.15\))180 1119 y(Here,)27 b Fw(\020)441 1089 y Fv(#)524 1119 y Fp(2)c Fx(Ran)14 b Fw(\037)817 1131 y Fs(L)863 1139 y Fi(p)897 1131 y Fv(=0)1013 1119 y Fx(is)27 b(the)h(unique)g(v)n(ector)e (in)i(the)g(k)n(ernel)f(of)g(\003)2395 1089 y Fv(#)2454 1119 y Fx(,)g(normalized)g(as)1392 1288 y(\003)1450 1254 y Fv(#)1508 1288 y Fw(\020)1550 1254 y Fv(#)1632 1288 y Fx(=)c(0)p Fw(;)41 b Fp(k)p Fw(\020)6 b Fp(k)23 b Fx(=)f(1)p Fw(;)41 b Fp(h)q Fw(\020)2243 1254 y Fr(\003)2281 1288 y Fw(;)14 b(\020)6 b Fp(i)24 b Fx(=)f(1)p Fw(:)957 b Fx(\(11.16\))180 1444 y(Letting)33 b Fw(\020)519 1414 y Fv(##)665 1444 y Fx(equal)f Fw(\020)39 b Fx(if)32 b(#)g(=)e Fp(\003)i Fx(and)h(letting)f(it)h(equal)f Fw(\020)2102 1414 y Fr(\003)2173 1444 y Fx(otherwise,)h(the)g(constan)n(t)e(in)i (\(11.15\))e(tak)n(es)h(the)180 1569 y(form)27 b Fw(C)441 1539 y Fv(#)523 1569 y Fx(=)611 1477 y Fm(D)662 1569 y Fw(\020)704 1539 y Fv(##)836 1569 y Fp(\012)18 b Fx(\012)979 1581 y Fs(r)1016 1569 y Fw(;)c Fx(\012)1113 1529 y Fv(#)1113 1599 y Fs(\022)1147 1583 y Fk(#)1200 1477 y Fm(E)1250 1569 y Fx(.)305 1702 y(The)29 b(o)n(v)n(erlap)d Fp(h)q Fw(\020)841 1672 y Fr(\003)879 1702 y Fw(;)14 b(\020)6 b Fp(i)30 b Fx(can)f(b)r(e)g(c)n(hosen)f(strictly)h(p)r(ositiv)n(e)f (since)h(the)g Fw(\020)2540 1672 y Fv(#)2628 1702 y Fx(are)f(the)i(P)n (erron-F)-7 b(rob)r(enius)26 b(eigen-)180 1801 y(v)n(ectors)35 b(of)h(\003)632 1771 y Fv(#)727 1801 y Fx(\(i.e.,)i(their)f(comp)r (onen)n(ts)e(can)h(b)r(e)h(c)n(hosen)e(non-negativ)n(e\),)i(and)f(ev)n (ery)f(comp)r(onen)n(t)h(of)g Fw(\020)43 b Fx(is)180 1901 y(strictly)34 b(p)r(ositiv)n(e)f(\(see)h(b)r(elo)n(w\).)56 b(The)34 b(last)f(relation)g(in)h(\(11.16\))f(is)h(then)g(ac)n(hiev)n (ed)f(b)n(y)g(scaling)g Fw(\020)3386 1871 y Fr(\003)3459 1901 y Fx(prop)r(erly)-7 b(.)180 2001 y(The)28 b(normalization)e Fp(h)p Fx(\012)969 1970 y Fr(\003)969 2024 y Fs(\022)1007 2001 y Fw(;)14 b Fx(\012)1104 2013 y Fs(\022)1142 2001 y Fp(i)23 b Fx(=)g(1)k(together)f(with)j(\(11.6\),)d(\(11.11\))h (\(11.15\))f(and)i(\(11.16\))e(implies)i(that)p 1684 2091 104 4 v 1684 2157 a Fw(C)1749 2133 y Fr(\003)1787 2157 y Fw(C)h Fx(=)23 b(1)18 b(+)g Fw(o)p Fx(\()p Fw(g)s Fx(\))p Fw(:)1250 b Fx(\(11.17\))305 2314 y(If)35 b(the)h(condition)g (\(11.13\),)g Fw(\034)1278 2284 y Fr(0)1338 2314 y Fp(\035)h Fw(g)1501 2284 y Fv(2+)p Fs(\013)1632 2314 y Fx(,)g(do)r(es)e(not)h (hold)f(then)h(w)n(e)g(ha)n(v)n(e)e(to)h(apply)h(the)f(F)-7 b(esh)n(bac)n(h)35 b(map)180 2414 y(iterativ)n(ely)c(and)g(use)h(a)f (corresp)r(onding)f(no)n(v)n(el)g(p)r(erturbation)h(theory)g(for)g (eigen)n(v)n(ectors.)47 b(W)-7 b(e)32 b(omit)g(here)f(this)180 2513 y(analysis)23 b(and)h(refer)f(the)h(reader)f(to)h([6)o(,)h(7)o(,)f (5])g(for)g(general)e(references)h(on)h(suc)n(h)g(a)f(R)n(G)h(p)r (erturbation)g(theory)f(and)180 2613 y([26)o(,)28 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Fv(1)2174 4346 y Fx(+)g Fw(')2311 4358 y Fv(2)2367 4346 y Fp(\012)g Fw(')2504 4358 y Fv(2)3526 4346 y Fx(\(11.20\))388 4479 y(where)1367 4633 y Fw(\013)23 b Fx(=)1531 4491 y Fm(")1579 4633 y Fx(1)18 b(+)1836 4502 y Fm(P)1924 4589 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)2110 4564 y Fw(g)2150 4576 y Fs(j)2184 4564 y Fx(\()p Fw(E)5 b Fx(\))p 1732 4614 688 4 v 1732 4628 a Fm(P)1820 4715 y Fs(j)s Fv(=1)p Fs(;)p Fv(2)2006 4690 y Fw(g)2046 4702 y Fs(j)2080 4690 y Fx(\()p Fw(E)g Fx(\))p Fw(\032)2253 4702 y Fs(j)2289 4690 y Fx(\()p Fw(E)g Fx(\))2429 4491 y Fm(#)2492 4633 y Fw(e)2531 4599 y Fr(\000)p Fs(\014)2621 4607 y Fi(p)2655 4599 y Fs(E)s(=)p Fv(2)2778 4633 y Fw(:)725 b Fx(\(11.21\))388 4849 y(Here,)27 b(w)n(e)g(use)h(the)g(notation)f Fw(E)h Fx(:=)22 b Fw(E)1608 4861 y Fv(2)1664 4849 y Fp(\000)c Fw(E)1808 4861 y Fv(1)1846 4849 y Fx(,)28 b Fw(\032)1940 4861 y Fs(j)1975 4849 y Fx(\()p Fw(E)5 b Fx(\))23 b(:=)2347 4817 y Fv(1)p 2249 4831 229 4 v 2249 4890 a Fs(e)2280 4862 y Fi(\014)2313 4875 y(j)2344 4862 y(E)2392 4890 y Fr(\000)p Fv(1)2515 4849 y Fx(and)1265 5075 y Fw(g)1305 5087 y Fs(j)1339 5075 y Fx(\()p Fw(E)5 b Fx(\))24 b(:=)1671 4962 y Fm(Z)1586 5204 y Fr(j)p Fs(k)q Fr(j)p Fv(=)p Fr(j)p Fs(E)s Fr(j)1836 5008 y Fm(\002)1871 5075 y Fp(j)p Fw(G)1959 5087 y Fs(j)1994 5075 y Fx(\()p Fw(k)s Fx(\))2104 5087 y Fv(12)2175 5075 y Fp(j)2198 5041 y Fv(2)2253 5075 y Fx(+)18 b Fp(j)p Fw(G)2424 5087 y Fs(j)2460 5075 y Fx(\()p Fw(k)s Fx(\))2570 5087 y Fv(21)2640 5075 y Fp(j)2663 5041 y Fv(2)2701 5008 y Fm(\003)2763 5075 y Fw(dk)31 b(:)1938 5476 y Fx(30)p eop %%Page: 31 31 31 30 bop 180 210 a Fx(Equation)35 b(\(11.18\))h(follo)n(ws)f(from)h (\(10.1\),)i(\(10.3\))e(and)g(\(10.4\).)63 b(The)36 b(expressions)f (\(11.20\))g({)h(\(11.21\))g(come)180 310 y(simply)22 b(from)g(solving)f(a)h(t)n(w)n(o-dimensional)e(eigen)n(v)-5 b(alue)21 b(problem.)35 b(Equation)21 b(\(11.19\))g(follo)n(ws)g(from)h (a)g(straigh)n(t-)180 409 y(forw)n(ard)k(p)r(erturbation)h(theory)g(in) h Fw(\014)1377 421 y Fv(2)1414 409 y Fx(.)305 509 y(Moreo)n(v)n(er,)k (expressions)g(\(10.1\){\(10.2\))g(for)i(\003)1831 521 y Fv(0)1901 509 y Fx(implies)g(the)h(follo)n(wing)e(relations)f(among)h (v)n(ectors)f Fw(\020)3643 479 y Fr(\003)3715 509 y Fp(\021)180 609 y Fw(\020)222 579 y Fr(\003)261 609 y Fx(\()p Fw(\014)340 621 y Fs(p)378 609 y Fx(\))c(corresp)r(onding)e(to)h(di\013eren)n(t)h (particle)f(temp)r(eratures)1430 802 y Fw(\020)1472 767 y Fr(\003)1510 802 y Fx(\()p Fw(\014)1589 814 y Fs(p)1628 802 y Fx(\))c(=)g(\()p Fw(e)1842 767 y Fs(\014)1880 775 y Fi(p)1914 767 y Fs(H)1968 775 y Fi(p)2003 767 y Fs(=)p Fv(2)2093 802 y Fp(\012)18 b Fx(1)-18 b(l)2223 814 y Fs(p)2261 802 y Fx(\))p Fw(\020)2335 767 y Fr(\003)2374 802 y Fx(\(0\))27 b Fw(:)996 b Fx(\(11.22\))180 1076 y Fy(12)135 b(En)l(trop)l(y)46 b(pro)t(duction)e(rate)i(for)f Ff(\021)180 1258 y Fx(In)21 b(this)g(section)f(w)n(e)g(pro)n(v)n(e)f (Theorem)h(3.2.)34 b(Recall)20 b(that)h(the)g(stationary)e(state)i Fw(\021)j Fx(men)n(tioned)c(in)h(this)g(theorem)f(is,)180 1358 y(in)25 b(fact,)h(giv)n(en)e(in)h(\(9.11\).)35 b(T)-7 b(o)25 b(analyze)f(the)h(en)n(trop)n(y)f(pro)r(duction,)h Fw(E)5 b(P)12 b Fx(\()p Fw(\021)s Fx(\),)26 b(in)f(this)g(state)g Fw(\021)j Fx(w)n(e)d(use)g(expression)180 1457 y(\(2.24\),)856 1557 y Fw(E)5 b(P)12 b Fx(\()p Fw(\021)s Fx(\))24 b(=)e(\()p Fw(\014)1285 1569 y Fv(1)1341 1557 y Fp(\000)c Fw(\014)1471 1569 y Fv(2)1508 1557 y Fx(\))p Fw(\021)s Fx(\()p Fw(\036)1665 1569 y Fv(1)1704 1557 y Fx(\))23 b(=)g Fw(ig)s Fx(\()p Fw(\014)1998 1569 y Fv(1)2053 1557 y Fp(\000)18 b Fw(\014)2183 1569 y Fv(2)2220 1557 y Fx(\))p Fw(\021)2296 1490 y Fm(\000)2335 1557 y Fw(a)2379 1569 y Fv(1)2416 1557 y Fx(\()p Fw(!)s(G)2568 1569 y Fv(1)2605 1557 y Fx(\))h Fp(\000)f Fw(a)2783 1523 y Fr(\003)2783 1577 y Fv(1)2821 1557 y Fx(\()p Fw(!)s(G)2973 1569 y Fv(1)3011 1557 y Fx(\))3043 1490 y Fm(\001)3081 1557 y Fw(;)464 b Fx(\(12.1\))180 1706 y(whic)n(h)28 b(relates)e(it)i(to)g(the)g(heat)f(\015o)n(w,)g Fw(\021)s Fx(\()p Fw(\036)1514 1718 y Fv(1)1552 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Fw(L)1776 3196 y Fs(r)r Fv(1)1845 3184 y Fw(i)p Fx(\()p Fw(L)1963 3196 y Fv(0)2018 3184 y Fx(+)19 b Fw(i)p Fx(0\))2205 3150 y Fr(\000)p Fv(1)2293 3184 y Fw(e)2332 3150 y Fr(\000)p Fs(\014)s(L)2471 3158 y Fk(0)2502 3150 y Fs(=)p Fv(2)2573 3184 y Fw(\031)s Fx(\()p Fw(v)2695 3196 y Fv(1)2734 3184 y Fx(\)\012)2826 3196 y Fv(0)2863 3092 y Fm(E)2928 3184 y Fw(:)617 b Fx(\(12.5\))180 3354 y Fq(Mor)l(e)l(over,)32 b(we)e(have)h(the)e(explicit)i(expr)l(ession)1257 3585 y Fw(\021)1301 3551 y Fr(0)1348 3585 y Fx(=)23 b(2)p Fw(\031)1541 3506 y Fm(X)1548 3683 y Fs(j)s(>i)1661 3585 y Fx(\()p Fw(\015)1736 3597 y Fs(j)1771 3585 y Fw(e)1810 3551 y Fs(\014)1848 3559 y Fk(1)1880 3551 y Fs(E)1929 3559 y Fi(j)r(i)2005 3585 y Fp(\000)18 b Fw(\015)2131 3597 y Fs(i)2158 3585 y Fx(\))2200 3529 y Fw(E)2261 3541 y Fs(j)s(i)2350 3529 y Fw(g)2390 3541 y Fs(j)s(i)2448 3529 y Fx(\()p Fw(E)2541 3541 y Fs(j)s(i)2600 3529 y Fx(\))2632 3499 y Fv(2)p 2200 3566 470 4 v 2256 3642 a Fw(e)2295 3617 y Fs(\014)2333 3625 y Fk(1)2365 3617 y Fs(E)2414 3625 y Fi(j)r(i)2489 3642 y Fp(\000)g Fx(1)2680 3585 y Fw(;)865 b Fx(\(12.6\))180 3862 y Fq(wher)l(e)24 b Fw(E)469 3874 y Fs(j)s(i)551 3862 y Fx(=)f Fw(E)700 3874 y Fs(j)741 3862 y Fp(\000)5 b Fw(E)872 3874 y Fs(i)900 3862 y Fq(,)25 b Fw(g)990 3874 y Fs(j)s(i)1048 3862 y Fx(\()p Fw(E)5 b Fx(\))1178 3831 y Fv(2)1239 3862 y Fx(=)1327 3795 y Fm(R)1366 3891 y Fl(R)1413 3874 y Fk(3)1458 3862 y Fw(d)1501 3831 y Fv(3)1539 3862 y Fw(k)s Fp(j)14 b(h)p Fw(')1708 3874 y Fs(j)1743 3862 y Fw(;)g(G)1845 3874 y Fv(1)1882 3862 y Fx(\()p Fw(k)s Fx(\))p Fw(')2046 3874 y Fs(i)2075 3862 y Fp(i)g(j)2144 3831 y Fv(2)2181 3862 y Fw(\016)s Fx(\()p Fw(E)2314 3874 y Fs(j)s(i)2379 3862 y Fp(\000)5 b Fw(!)s Fx(\))24 b Fq(\(se)l(e)f(Condition)j(\(C\)\).)e (The)h(numb)l(ers)180 3961 y Fw(\015)223 3973 y Fs(j)281 3961 y Fp(\025)e Fx(0)29 b Fq(ar)l(e)h(the)g(c)l(omp)l(onents)g(of)g (the)g(ve)l(ctor)g Fw(\020)1689 3931 y Fr(\003)1728 3961 y Fq(,)g(normalize)l(d)h(as)f(in)g(\(11.16\),)i(at)e Fw(\014)2877 3973 y Fs(p)2938 3961 y Fx(=)23 b(0)p Fq(:)1535 4163 y Fw(\020)1577 4129 y Fr(\003)1616 4163 y Fp(j)1639 4175 y Fs(\014)1677 4183 y Fi(p)1711 4175 y Fv(=0)1822 4163 y Fx(=)1910 4085 y Fm(X)1955 4261 y Fs(j)2044 4163 y Fw(\015)2087 4175 y Fs(j)2122 4163 y Fw(')2176 4175 y Fs(j)2230 4163 y Fp(\012)18 b Fw(')2367 4175 y Fs(j)2402 4163 y Fw(:)1143 b Fx(\(12.7\))180 4435 y Fq(Observe)30 b(that)g(by)g(\(11.18\),)i Fw(\021)1135 4405 y Fr(0)1182 4435 y Fx(=)22 b(0)30 b Fq(for)g Fw(\014)1520 4447 y Fv(1)1581 4435 y Fx(=)22 b Fw(\014)1715 4447 y Fv(2)1752 4435 y Fq(.)305 4617 y Fx(The)27 b(follo)n(wing)g(result)g(sho)n(ws)f (that)i Fw(\021)1519 4587 y Fr(0)1571 4617 y Fx(is)f(strictly)g(p)r (ositiv)n(e)h(for)f(small)g(nonzero)f(temp)r(erature)h(di\013erences.) 180 4800 y Fn(Theorem)j(12.2)41 b Fq(If)29 b Fw(\016)s(\014)f Fx(=)22 b Fw(\014)1132 4812 y Fv(1)1186 4800 y Fp(\000)17 b Fw(\014)1315 4812 y Fv(2)1375 4800 y Fw(>)23 b Fx(0)28 b Fq(is)h(smal)t(l)h(and)g(the)f(c)l(oupling)h(functions)f(\(2.4\))h (satisfy)g Fw(G)24 b Fx(=)e Fw(G)3529 4812 y Fv(1)3590 4800 y Fx(=)g Fw(G)3742 4812 y Fv(2)180 4900 y Fq(then)30 b(the)f(line)l(ar)i(term)e(of)i Fw(\021)1077 4870 y Fr(0)1130 4900 y Fq(in)f Fw(\016)s(\014)k Fq(is)1242 5086 y Fw(\016)s(\014)p 1242 5123 92 4 v 1267 5199 a Fx(2)1487 5086 y Fw(Z)1544 5098 y Fs(p)1582 5086 y Fx(\()p Fw(\014)1661 5098 y Fs(p)1700 5086 y Fx(\))p 1353 5123 514 4 v 1353 5199 a Fw(Z)1410 5211 y Fs(p)1448 5199 y Fx(\()p Fw(\014)1527 5211 y Fv(1)1583 5199 y Fx(+)18 b Fw(\014)1713 5211 y Fs(p)1751 5199 y Fw(=)p Fx(2\))1891 5063 y Fm(X)1891 5242 y Fs(j)s(>k)2034 5075 y Fw(E)2100 5045 y Fv(2)2095 5098 y Fs(j)s(k)2197 5075 y Fw(g)2237 5087 y Fs(j)s(k)2308 5075 y Fx(\()p Fw(E)2401 5087 y Fs(j)s(k)2473 5075 y Fx(\))2505 5045 y Fv(2)p 2034 5123 509 4 v 2042 5199 a Fw(e)2081 5174 y Fs(\014)2119 5182 y Fk(1)2151 5174 y Fs(E)2200 5182 y Fi(j)2254 5199 y Fp(\000)g Fw(e)2376 5175 y Fs(\014)2414 5183 y Fk(1)2445 5175 y Fs(E)2494 5184 y Fi(k)2576 5142 y Fw(>)23 b Fx(0)p Fw(;)839 b Fx(\(12.8\))1938 5476 y(31)p eop %%Page: 32 32 32 31 bop 180 210 a Fq(wher)l(e)30 b Fw(Z)471 222 y Fs(p)509 210 y Fx(\()p Fw(\014)t Fx(\))24 b(=)f(tr)14 b Fw(e)854 180 y Fr(\000)p Fs(\014)s(H)1001 188 y Fi(p)1069 210 y Fq(is)30 b(the)g(p)l(article)h(p)l(artition)g(function.)305 310 y(R)l(emark.)73 b Fx(Using)29 b(\(11.20\))f(for)h(the)g(t)n(w)n (o-dimensional)f(case)g(it)i(is)f(easy)f(to)h(v)n(erify)f(that)i(the)g (linear)e(term)h(of)180 409 y Fw(\021)224 379 y Fr(0)279 409 y Fx(in)i Fw(\016)s(\014)36 b Fx(is)c(strictly)f(p)r(ositiv)n(e)g (\(without)h(the)f(assumption)g Fw(G)2188 421 y Fv(1)2255 409 y Fx(=)e Fw(G)2414 421 y Fv(2)2451 409 y Fx(\).)49 b(Also,)32 b(in)f(the)h(general)e(case,)h(if)h Fw(G)3655 421 y Fv(1)3724 409 y Fx(is)180 509 y(close)27 b(to)g Fw(G)545 521 y Fv(2)610 509 y Fx(one)h(deduces)f(strict)h(p)r(ositivit) n(y)f(of)h Fw(\021)1798 479 y Fr(0)1849 509 y Fx(in)g(the)g(linear)e (term)i(in)g Fw(\016)s(\014)k Fx(b)n(y)27 b(a)g(p)r(erturbation)h (argumen)n(t.)305 708 y Fn(Pro)s(of)j(of)h(Theorem)e(12.1.)67 b Fx(W)-7 b(e)28 b(\014rst)g(sho)n(w)e(1.)37 b(T)-7 b(o)27 b(simplify)h(the)g(exp)r(osition)f(w)n(e)h(restrict)f(ourselv)n(es)e (to)180 817 y(the)31 b(case)f Fw(\034)550 787 y Fr(0)601 817 y Fp(\035)e Fw(g)755 787 y Fv(2+)p Fs(\013)917 817 y Fx(for)i Fw(\013)e Fx(=)1231 776 y Fs(\026)p Fr(\000)p Fv(1)p Fs(=)p Fv(2)p 1231 798 193 4 v 1231 845 a Fs(\026)p Fv(+1)p Fs(=)p Fv(2)1433 817 y Fx(,)j(and)g Fw(\026)d(>)f Fx(3)p Fw(=)p Fx(2.)45 b(Pic)n(k)29 b Fw(\032)2247 829 y Fv(0)2312 817 y Fx(=)f Fw(g)2448 787 y Fv(2)p Fr(\000)p Fv(2)p Fs(\013)2613 817 y Fx(.)45 b(First)31 b(w)n(e)f(pro)n(v)n(e)f (an)h(estimate)g(on)180 930 y Fw(\021)s Fx(\()p Fw(\036)305 942 y Fv(1)343 930 y Fx(\))e(whic)n(h)g(is)f(rougher)f(than)i(\(12.4\)) f(and)g(then)h(w)n(e)f(explain)h(ho)n(w)f(to)g(obtain)g(\(12.4\).)37 b(Recall)27 b(that)1533 1109 y Fw(\021)s Fx(\()p Fw(\036)1658 1121 y Fv(1)1697 1109 y Fx(\))c(=)1840 1042 y Fm(\012)1879 1109 y Fx(\012)1939 1075 y Fr(\003)p 1939 1094 38 3 v 1939 1143 a Fs(\022)1977 1109 y Fw(;)14 b(\031)s Fx(\()p Fw(\036)2145 1121 y Fv(1)2183 1109 y Fx(\))2215 1121 y Fs(\022)2253 1109 y Fx(\012)2313 1121 y Fs(\022)2350 1042 y Fm(\013)2404 1109 y Fw(:)1141 b Fx(\(12.9\))180 1311 y(Let)26 b(\012)387 1271 y Fv(#)387 1333 y(0)469 1311 y Fx(=)d Fw(\020)599 1280 y Fv(#)673 1311 y Fp(\012)15 b Fx(\012)813 1323 y Fs(r)876 1311 y Fx(and)26 b(tak)n(e)f(a)h(n)n(um)n (b)r(er)f Fw(M)35 b Fx(so)26 b(large)e(that)i(\()p Fw(g)2253 1280 y Fv(2+)p Fs(\013)2385 1311 y Fx([min\()p Fw(\034)2623 1280 y Fr(0)2647 1311 y Fw(;)14 b(g)2727 1280 y Fv(2)2764 1311 y Fx(\)])2819 1280 y Fr(\000)p Fv(1)2909 1311 y Fx(\))2941 1280 y Fs(M)3038 1311 y Fx(=)22 b Fw(o)p Fx(\()p Fw(g)s Fx(\).)37 b(Substituting)180 1428 y(expansions)31 b(\(11.6\))g(and)h(\(11.15\))f(in)n(to)g(the)i(r.h.s.)49 b(of)32 b(this)h(expression,)e(using)h(that)g(\()p Fw(g)3011 1449 y Fi(\016)q(\014)3083 1428 y Fw(\032)3126 1385 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)3126 1450 y(0)3282 1428 y Fx(\))3314 1398 y Fv(2)3382 1428 y Fx(=)e Fw(o)p Fx(\()p Fw(g)s Fx(\),)k(for)180 1527 y Fw(\032)223 1539 y Fv(0)283 1527 y Fx(=)23 b Fw(g)414 1497 y Fv(2)p Fr(\000)p Fv(2)p Fs(\013)579 1527 y Fx(,)28 b Fw(\013)23 b(>)804 1495 y Fv(1)p 804 1509 34 4 v 804 1556 a(2)847 1527 y Fx(,)k(and)h(using)f (that)h Fw(\036)1505 1539 y Fv(1)1570 1527 y Fx(is)g(prop)r(ortional)e (to)h Fw(g)s Fx(,)g(w)n(e)h(\014nd)1409 1715 y Fw(\021)s Fx(\()p Fw(\036)1534 1727 y Fv(1)1572 1715 y Fx(\))23 b(=)g Fw(\021)1756 1727 y Fv(0)1812 1715 y Fx(+)18 b Fw(\021)1936 1727 y Fv(1)1992 1715 y Fx(+)g(Rem)g(+)h Fw(o)p Fx(\()p Fw(g)2459 1681 y Fv(2)2496 1715 y Fx(\))p Fw(;)975 b Fx(\(12.10\))180 1894 y(where)884 2073 y Fw(\021)925 2085 y Fv(0)1045 2073 y Fx(:=)83 b Fp(h)p Fx(\012)1308 2039 y Fr(\003)1308 2093 y Fv(0)1346 2073 y Fw(;)14 b(\031)s Fx(\()p Fw(\036)1514 2085 y Fv(1)1553 2073 y Fx(\)\012)1645 2085 y Fv(0)1682 2073 y Fp(i)g Fw(;)1775 b Fx(\(12.11\))884 2197 y Fw(\021)925 2209 y Fv(1)1045 2197 y Fx(:=)83 b Fp(\000)p Fw(g)1337 2130 y Fm(\012)1376 2197 y Fx(\012)1436 2163 y Fr(\003)1436 2218 y Fv(0)1474 2197 y Fw(;)1511 2130 y Fm(\002)1546 2197 y Fw(I)1582 2209 y Fs(\022)p 1620 2131 64 4 v 1620 2197 a Fw(R)1684 2209 y Fv(0)1721 2197 y Fx(\()p Fw(L)1810 2209 y Fv(0)p Fs(\022)1880 2197 y Fx(\))p Fw(\031)s Fx(\()p Fw(\036)2043 2209 y Fv(1)2082 2197 y Fx(\))2114 2209 y Fs(\022)2170 2197 y Fx(+)18 b Fw(\031)s Fx(\()p Fw(\036)2384 2209 y Fv(1)2422 2197 y Fx(\))2454 2209 y Fs(\022)p 2492 2131 V 2492 2197 a Fw(R)2556 2209 y Fv(0)2593 2197 y Fx(\()p Fw(L)2682 2209 y Fv(0)p Fs(\022)2753 2197 y Fx(\))p Fw(I)2821 2209 y Fs(\022)2859 2197 y Fx(\))2891 2130 y Fm(\003)2940 2197 y Fx(\012)3000 2209 y Fv(0)3037 2130 y Fm(\013)3526 2197 y Fx(\(12.12\))180 2376 y(and)746 2616 y(Rem)83 b(=)1227 2512 y Fs(M)1202 2537 y Fm(X)1144 2715 y Fs(m)p Fv(+)p Fs(n)p Fr(\025)p Fv(1)1458 2512 y(1)1415 2537 y Fm(X)1394 2716 y Fs(k)q(;l)p Fv(=0)1569 2616 y Fw(g)1612 2581 y Fs(k)q Fv(+)p Fs(l)1725 2524 y Fm(D)1776 2616 y Fx(\()p Fp(\000)p 1873 2549 V Fw(R)1936 2628 y Fv(0)1974 2616 y Fx(\()p Fw(L)2063 2642 y Fv(0)p 2096 2593 34 3 v Fs(\022)2133 2616 y Fx(\))p Fw(I)2208 2581 y Fr(\003)p 2201 2600 38 3 v 2201 2649 a Fs(\022)2246 2616 y Fx(\))2278 2581 y Fs(k)2320 2616 y Fx(\()p Fp(\000)p Fw(R)p 2480 2593 92 3 v 25 x Fs(Q)2532 2621 y Fh(\003)2532 2659 y Fk(0)2571 2616 y Fx(\()p Fw(K)2680 2573 y Fv(#\(1\))2674 2638 y(0)2823 2616 y Fx(\))p Fw(W)2945 2581 y Fr(\003)2983 2616 y Fx(\))3015 2581 y Fs(m)3079 2616 y Fx(\012)3139 2581 y Fr(\003)3139 2636 y Fv(0)3177 2616 y Fw(;)1144 2864 y(\031)s Fx(\()p Fw(\036)1275 2876 y Fv(1)1313 2864 y Fx(\))1345 2876 y Fs(\022)1383 2864 y Fx(\()p Fp(\000)p 1480 2797 64 4 v Fw(R)1544 2876 y Fv(0)1581 2864 y Fx(\()p Fw(L)1670 2876 y Fv(0)p Fs(\022)1740 2864 y Fx(\))p Fw(I)1808 2876 y Fs(\022)1847 2864 y Fx(\))1879 2830 y Fs(l)1904 2864 y Fx(\()p Fp(\000)p Fw(R)p 2064 2841 53 3 v 25 x Fs(Q)2117 2905 y Fk(0)2153 2864 y Fx(\()p Fw(K)2262 2821 y Fv(\(1\))2256 2886 y(0)2351 2864 y Fx(\))p Fw(W)12 b Fx(\))2505 2830 y Fs(n)2551 2864 y Fx(\012)2611 2876 y Fv(0)2648 2772 y Fm(E)2699 2864 y Fw(:)804 b Fx(\(12.13\))180 3079 y(Here)30 b(w)n(e)g(replaced)f(the)i(factor)p 1221 3012 104 4 v 30 w Fw(C)1286 3055 y Fr(\003)1324 3079 y Fw(C)j Fx(=)27 b(1)20 b(+)g Fw(o)p Fx(\()p Fw(g)s Fx(\))31 b(b)n(y)f(1)g(\(see)g (\(11.15\))f(and)h(\(11.17\)\))g(in)g(the)h(r.h.s.)45 b(Using)30 b(the)180 3178 y(pull-through)d(pro)r(cedure)g(and)g(elemen) n(tary)g(estimates)g(of)g(the)h(resulting)f(in)n(tegrals)f(w)n(e)i (obtain)f(that)1737 3357 y(Rem)c(=)g Fw(o)p Fx(\()p Fw(g)2130 3323 y Fv(2)2167 3357 y Fx(\))p Fw(:)1304 b Fx(\(12.14\))180 3536 y(Since)28 b Fw(\036)446 3548 y Fv(1)511 3536 y Fx(is)g(linear)e(in)i(creation)f(and)g(annihilation)h(op)r(erators,)d (see)i(\(12.1\),)g(w)n(e)h(ha)n(v)n(e)1853 3715 y Fw(\021)1894 3727 y Fv(0)1955 3715 y Fx(=)22 b(0)p Fw(:)1419 b Fx(\(12.15\))305 3894 y(It)27 b(remains)g(to)h(compute)f Fw(\021)1183 3906 y Fv(1)1221 3894 y Fx(.)37 b(Using)1523 4073 y(\()p 1555 4006 65 4 v Fw(P)1621 4085 y Fs(\032)1655 4093 y Fk(0)1710 4073 y Fp(\000)18 b Fn(1)p Fx(\))p Fw(I)1916 4033 y Fv(#)1909 4103 y Fs(\022)1943 4086 y Fk(#)1997 4073 y Fx(\012)2057 4033 y Fv(#)2057 4095 y(0)2139 4073 y Fx(=)k Fw(O)r Fx(\()p Fw(\032)2366 4085 y Fv(0)2404 4073 y Fx(\))1090 b(\(12.16\))180 4252 y(\([25)o(])33 b(Lemma)g(5.3\),)h(remo)n(ving)d(the)j(sp)r(ectral)e(deformation)g(and) h(using)f Fw(\031)s Fx(\()p Fw(\036)2692 4264 y Fv(1)2731 4252 y Fx(\))g(=)f Fw(g)s Fx([)p Fw(\031)s Fx(\()p Fw(v)3079 4264 y Fv(1)3117 4252 y Fx(\))p Fw(;)14 b(iL)3272 4264 y Fs(r)r Fv(1)3341 4252 y Fx(],)35 b(w)n(e)d(obtain)180 4351 y Fw(\021)221 4363 y Fv(1)282 4351 y Fx(=)22 b Fw(g)412 4321 y Fv(2)449 4351 y Fw(\021)493 4321 y Fr(0)535 4351 y Fx(+)c Fw(o)p Fx(\()p Fw(g)733 4321 y Fv(2)770 4351 y Fx(\),)28 b(where)776 4530 y Fw(\021)820 4496 y Fr(0)867 4530 y Fx(=)23 b Fp(\000)1034 4463 y Fm(\012)1072 4530 y Fx(\012)1132 4496 y Fr(\003)1132 4551 y Fv(0)1170 4530 y Fw(;)1207 4463 y Fm(\002)1242 4530 y Fw(\031)s Fx(\()p Fw(v)1364 4542 y Fv(1)1402 4530 y Fx(\))p Fw(iL)1520 4542 y Fs(r)r Fv(1)1589 4530 y Fx(\()p Fw(L)1678 4542 y Fv(0)1734 4530 y Fx(+)18 b Fw(i)p Fx(0\))1920 4496 y Fr(\000)p Fv(1)2008 4530 y Fw(I)26 b Fp(\000)18 b Fw(I)7 b Fx(\()p Fw(L)2285 4542 y Fv(0)2340 4530 y Fx(+)18 b Fw(i)p Fx(0\))2526 4496 y Fr(\000)p Fv(1)2615 4530 y Fw(iL)2701 4542 y Fs(r)r Fv(1)2770 4530 y Fw(\031)s Fx(\()p Fw(v)2892 4542 y Fv(1)2930 4530 y Fx(\))2962 4463 y Fm(\003)3010 4530 y Fx(\012)3070 4542 y Fv(0)3108 4463 y Fm(\013)3161 4530 y Fw(:)342 b Fx(\(12.17\))180 4709 y(Next,)23 b(note)e(that)g(the) g(con)n(tribution)f(of)h(the)h Fw(v)1619 4721 y Fv(2)1656 4709 y Fx(-part)e(of)h Fw(I)28 b Fx(to)21 b Fw(\021)2148 4679 y Fr(0)2192 4709 y Fx(is)g(zero)f(since)h(the)g(resulting)g (expression)e(is)i(linear)180 4809 y(in)g(creation)f(and)g (annihilation)h(op)r(erators)e(for)h(the)h(\014rst)g(and)g(second)f (reserv)n(oirs)e(separately)-7 b(.)33 b(The)21 b(con)n(tribution)180 4908 y(of)28 b(the)g Fw(\031)s Fx(\()p Fw(v)540 4920 y Fv(1)578 4908 y Fx(\)-part)f(of)g Fw(I)35 b Fx(is)27 b(also)g(zero)f(b)n(y)i(the)g(symmetry)f(of)g(\(12.17\).)36 b(Hence)28 b(w)n(e)f(ha)n(v)n(e)f Fw(\021)3067 4878 y Fr(0)3114 4908 y Fx(=)d Fw(A)18 b Fp(\000)g Fw(B)t Fx(,)28 b(where)1069 5087 y Fw(A)83 b Fx(=)g Fp(h)p Fx(\012)1454 5053 y Fr(\003)1454 5108 y Fv(0)1492 5087 y Fw(;)14 b(\031)s Fx(\()p Fw(v)1651 5099 y Fv(1)1689 5087 y Fx(\))p Fw(iL)1807 5099 y Fs(r)r Fv(1)1876 5087 y Fx(\()p Fw(L)1965 5099 y Fv(0)2021 5087 y Fx(+)k Fw(i)p Fx(0\))2207 5053 y Fr(\000)p Fv(1)2295 5087 y Fw(\031)2345 5053 y Fr(0)2369 5020 y Fm(\000)2407 5087 y Fw(\015)2455 5053 y Fs(i=)p Fv(2)2549 5087 y Fx(\()p Fw(v)2621 5099 y Fv(1)2659 5087 y Fx(\))2691 5020 y Fm(\001)2730 5087 y Fx(\012)2790 5099 y Fv(0)2827 5087 y Fp(i)p Fw(;)1064 5227 y(B)87 b Fx(=)c Fp(h)p Fx(\012)1454 5193 y Fr(\003)1454 5248 y Fv(0)1492 5227 y Fw(;)14 b(\031)1579 5193 y Fr(0)1603 5160 y Fm(\000)1641 5227 y Fw(\015)1689 5193 y Fs(i=)p Fv(2)1783 5227 y Fx(\()p Fw(v)1855 5239 y Fv(1)1893 5227 y Fx(\))1925 5160 y Fm(\001)1963 5227 y Fw(iL)2049 5239 y Fs(r)r Fv(1)2118 5227 y Fx(\()p Fw(L)2207 5239 y Fv(0)2263 5227 y Fx(+)k Fw(i)p Fx(0\))2449 5193 y Fr(\000)p Fv(1)2537 5227 y Fw(\031)s Fx(\()p Fw(v)2659 5239 y Fv(1)2697 5227 y Fx(\)\012)2789 5239 y Fv(0)2827 5227 y Fp(i)p Fw(:)1938 5476 y Fx(32)p eop %%Page: 33 33 33 32 bop 180 211 a Fx(Using)32 b(that)g Fw(\031)652 181 y Fr(0)676 144 y Fm(\000)714 211 y Fw(\015)762 181 y Fs(i=)p Fv(2)856 211 y Fx(\()p Fw(v)928 223 y Fv(1)966 211 y Fx(\))998 144 y Fm(\001)1036 211 y Fx(\012)1096 223 y Fv(0)1164 211 y Fx(=)d Fw(J)8 b(e)1351 181 y Fv(\()p Fs(\014)s(L)1464 189 y Fk(0)1496 181 y Fr(\000)1556 166 y Fv(~)1548 181 y Fs(L)o Fv(\))p Fs(=)p Fv(2)1690 211 y Fw(\031)s Fx(\()p Fw(v)1812 223 y Fv(1)1850 211 y Fx(\)\012)1942 223 y Fv(0)1980 211 y Fx(,)33 b(where)2290 190 y(~)2280 211 y Fw(L)f Fx(is)g(giv)n(en)f(after)g(\(4.19\),)i(and)e(the)i(fact)f (that)180 332 y Fw(J)8 b(e)273 302 y Fr(\000)333 287 y Fv(~)325 302 y Fs(L)o(=)p Fv(2)441 332 y Fw(\031)s Fx(\()p Fw(v)563 344 y Fv(1)601 332 y Fx(\)\012)693 344 y Fv(0)754 332 y Fx(=)23 b Fw(\031)s Fx(\()p Fw(v)967 302 y Fr(\003)964 353 y Fv(1)1006 332 y Fx(\)\012)1098 344 y Fv(0)1158 332 y Fx(=)g Fw(\031)s Fx(\()p Fw(v)1368 344 y Fv(1)1406 332 y Fx(\)\012)1498 344 y Fv(0)1535 332 y Fx(,)28 b(w)n(e)f(transform)1100 499 y Fw(A)d Fx(=)e Fp(h)p Fx(\012)1365 465 y Fr(\003)1365 519 y Fv(0)1404 499 y Fw(;)14 b(\031)s Fx(\()p Fw(v)1563 511 y Fv(1)1601 499 y Fx(\))p Fw(iL)1719 511 y Fs(r)r Fv(1)1788 499 y Fx(\()p Fw(L)1877 511 y Fv(0)1932 499 y Fx(+)k Fw(i)p Fx(0\))2118 465 y Fr(\000)p Fv(1)2207 499 y Fw(e)2246 465 y Fr(\000)p Fs(\014)s(L)2385 473 y Fk(0)2416 465 y Fs(=)p Fv(2)2487 499 y Fw(\031)s Fx(\()p Fw(v)2609 511 y Fv(1)2647 499 y Fx(\)\012)2739 511 y Fv(0)2777 499 y Fp(i)28 b Fw(:)180 679 y Fx(W)-7 b(e)28 b(use)f(the)h(relations)f Fw(\031)997 649 y Fr(0)1020 612 y Fm(\000)1058 679 y Fw(\015)1106 649 y Fs(i=)p Fv(2)1201 679 y Fx(\()p Fw(v)1273 691 y Fv(1)1311 679 y Fx(\))1343 612 y Fm(\001)1404 679 y Fx(=)c Fw(J)8 b(e)1585 649 y Fv(\()p Fs(\014)s(L)1698 657 y Fk(0)1729 649 y Fr(\000)1789 634 y Fv(~)1781 649 y Fs(L)o Fv(\))p Fs(=)p Fv(2)1923 679 y Fw(\031)s Fx(\()p Fw(v)2045 691 y Fv(1)2084 679 y Fx(\))p Fw(e)2155 649 y Fr(\000)p Fv(\()p Fs(\014)s(L)2320 657 y Fk(0)2351 649 y Fr(\000)2411 634 y Fv(~)2403 649 y Fs(L)o Fv(\))p Fs(=)p Fv(2)2546 679 y Fw(J)35 b Fx(and)873 871 y Fw(J)8 b(iL)1013 883 y Fs(r)r Fv(1)1082 871 y Fx(\()p Fw(L)1171 883 y Fv(0)1226 871 y Fx(+)18 b Fw(i)p Fx(0\))1412 837 y Fr(\000)p Fv(1)1501 871 y Fw(\031)s Fx(\()p Fw(v)1623 883 y Fv(1)1661 871 y Fx(\)\012)1753 883 y Fv(0)1813 871 y Fx(=)23 b Fp(\000)p Fw(iL)2052 883 y Fs(r)r Fv(1)2121 871 y Fx(\()p Fw(L)2210 883 y Fv(0)2265 871 y Fx(+)18 b Fw(i)p Fx(0\))2451 837 y Fr(\000)p Fv(1)2540 871 y Fw(e)2579 837 y Fr(\000)2639 822 y Fv(~)2631 837 y Fs(L)o(=)p Fv(2)2747 871 y Fw(\031)s Fx(\()p Fw(v)2869 883 y Fv(1)2907 871 y Fx(\)\012)2999 883 y Fv(0)3064 871 y Fw(;)180 1038 y Fx(see)27 b(also)g(\(5.1\),)g(to)h(\014nd)g(that)850 1222 y Fw(B)f Fx(=)c Fp(\000h)p Fx(\012)1185 1188 y Fr(\003)1185 1243 y Fv(0)1223 1222 y Fw(;)14 b(J)8 b(e)1353 1188 y Fv(\()p Fs(\014)s(L)1466 1196 y Fk(0)1497 1188 y Fv(+)1556 1173 y(~)1548 1188 y Fs(L)o Fv(\))p Fs(=)p Fv(2)1691 1222 y Fw(\031)s Fx(\()p Fw(v)1813 1234 y Fv(1)1851 1222 y Fx(\))p Fw(iL)1969 1234 y Fs(r)r Fv(1)2038 1222 y Fx(\()p Fw(L)2127 1234 y Fv(0)2182 1222 y Fx(+)18 b Fw(i)p Fx(0\))2368 1188 y Fr(\000)p Fv(1)2457 1222 y Fw(e)2496 1188 y Fr(\000)p Fs(\014)s(L)2635 1196 y Fk(0)2666 1188 y Fs(=)p Fv(2)2737 1222 y Fw(\031)s Fx(\()p Fw(v)2859 1234 y Fv(1)2897 1222 y Fx(\)\012)2989 1234 y Fv(0)3027 1222 y Fp(i)28 b Fw(:)180 1389 y Fx(Fingally)-7 b(,)27 b(since)h Fp(h)p Fw(J)8 b(u;)14 b(J)8 b(v)s Fp(i)23 b Fx(=)p 1135 1316 193 4 v 23 w Fp(h)p Fw(u;)14 b(v)s Fp(i)p Fx(,)27 b Fw(J)8 b Fx(\012)1491 1359 y Fr(\003)1491 1409 y Fv(0)1553 1389 y Fx(=)22 b(\012)1700 1359 y Fr(\003)1700 1409 y Fv(0)1738 1389 y Fx(,)28 b Fw(L)1846 1401 y Fs(p)1884 1389 y Fx(\012)1944 1359 y Fr(\003)1944 1409 y Fv(0)2005 1389 y Fx(=)23 b(0)k(and)h(\()p Fw(\014)t(L)2464 1401 y Fv(0)2519 1389 y Fp(\000)p 2602 1322 57 4 v 18 w Fw(L)p Fx(\)\012)2751 1359 y Fr(\003)2751 1409 y Fv(0)2812 1389 y Fx(=)23 b(0)k(w)n(e)g(obtain)p 961 1499 68 4 v 961 1566 a Fw(B)g Fx(=)22 b Fp(\000h)p Fx(\012)1295 1531 y Fr(\003)1295 1586 y Fv(0)1333 1566 y Fw(;)14 b(\031)s Fx(\()p Fw(v)1492 1578 y Fv(1)1530 1566 y Fx(\))p Fw(iL)1648 1578 y Fs(r)r Fv(1)1718 1566 y Fx(\()p Fw(L)1807 1578 y Fv(0)1862 1566 y Fx(+)k Fw(i)p Fx(0\))2048 1531 y Fr(\000)p Fv(1)2137 1566 y Fw(e)2176 1531 y Fr(\000)p Fs(\014)s(L)2315 1539 y Fk(0)2346 1531 y Fs(=)p Fv(2)2417 1566 y Fw(\031)s Fx(\()p Fw(v)2539 1578 y Fv(1)2577 1566 y Fx(\)\012)2669 1578 y Fv(0)2707 1566 y Fp(i)23 b Fx(=)g Fp(\000)p Fw(A:)180 1732 y Fx(Since)28 b Fw(\021)441 1702 y Fr(0)487 1732 y Fx(=)23 b Fw(A)c Fp(\000)f Fw(B)32 b Fx(this)27 b(giv)n(es)1021 1899 y Fw(\021)1065 1865 y Fr(0)1111 1899 y Fx(=)c(2Re)13 b Fp(h)p Fx(\012)1444 1865 y Fr(\003)1444 1919 y Fv(0)1483 1899 y Fw(;)h(\031)s Fx(\()p Fw(v)1642 1911 y Fv(1)1680 1899 y Fx(\))p Fw(iL)1798 1911 y Fs(r)r Fv(1)1867 1899 y Fx(\()p Fw(L)1956 1911 y Fv(0)2012 1899 y Fx(+)k Fw(i)p Fx(0\))2198 1865 y Fr(\000)p Fv(1)2286 1899 y Fw(e)2325 1865 y Fr(\000)p Fs(\014)s(L)2464 1873 y Fk(0)2495 1865 y Fs(=)p Fv(2)2567 1899 y Fw(\031)s Fx(\()p Fw(v)2689 1911 y Fv(1)2727 1899 y Fx(\)\012)2819 1911 y Fv(0)2856 1899 y Fp(i)28 b Fw(:)587 b Fx(\(12.18\))305 2065 y(Collecting)27 b(estimates)g(\(12.10\),)f(\(12.14\),)h(\(12.15\))f(and)h Fw(\021)2154 2077 y Fv(1)2215 2065 y Fx(=)c Fw(g)2346 2035 y Fv(2)2382 2065 y Fw(\021)2426 2035 y Fr(0)2468 2065 y Fx(+)18 b Fw(o)p Fx(\()p Fw(g)2666 2035 y Fv(2)2704 2065 y Fx(\))28 b(w)n(e)f(\014nd)h(that)1610 2232 y Fw(\021)s Fx(\()p Fw(\036)1735 2244 y Fv(1)1773 2232 y Fx(\))c(=)e Fw(g)1959 2198 y Fv(2)1996 2232 y Fw(\021)2040 2198 y Fr(0)2082 2232 y Fx(+)c Fw(o)p Fx(\()p Fw(g)2280 2198 y Fv(2)2318 2232 y Fx(\))1176 b(\(12.19\))180 2399 y(where)27 b Fw(\021)464 2369 y Fr(0)515 2399 y Fx(is)h(giv)n(en)f(b)n(y)g (\(12.5\).)36 b(This)28 b(pro)n(v)n(es)d(the)j(rough)f(v)n(ersion)f(of) h(\(12.4\){\(12.5\).)305 2498 y(Before)j(w)n(e)h(re\014ne)f(estimate)i (\(12.19\))e(let)h(us)g(sho)n(w)g(\(12.6\).)47 b(W)-7 b(e)31 b(expand)g(the)h(v)n(ectors)d(\012)3201 2468 y Fr(\003)3201 2519 y Fv(0)3271 2498 y Fx(and)i(\012)3496 2510 y Fv(0)3564 2498 y Fx(in)h(the)180 2598 y(basis)27 b Fw(')438 2610 y Fs(j)s(j)522 2598 y Fp(\012)18 b Fx(\012)665 2610 y Fs(r)702 2598 y Fx(,)28 b Fw(')807 2610 y Fs(j)s(j)896 2598 y Fx(=)23 b Fw(')1038 2610 y Fs(j)1091 2598 y Fp(\012)c Fw(')1229 2610 y Fs(j)1264 2598 y Fx(,)1177 2834 y(\012)1237 2846 y Fv(0)1297 2834 y Fx(=)1424 2730 y Fs(n)1384 2755 y Fm(X)1387 2932 y Fs(j)s Fv(=1)1518 2834 y Fw(\013)1571 2846 y Fs(j)1606 2834 y Fw(')1660 2846 y Fs(j)s(j)1745 2834 y Fp(\012)f Fx(\012)1888 2846 y Fs(r)1925 2834 y Fw(;)97 b Fx(\012)2105 2800 y Fr(\003)2105 2855 y Fv(0)2166 2834 y Fx(=)2293 2730 y Fs(n)2253 2755 y Fm(X)2256 2932 y Fs(j)s Fv(=1)2390 2834 y Fx(~)-45 b Fw(\015)2430 2846 y Fs(j)2465 2834 y Fw(')2519 2846 y Fs(j)s(j)2604 2834 y Fp(\012)18 b Fx(\012)2747 2846 y Fs(r)3526 2834 y Fx(\(12.20\))180 3089 y(with)30 b Fw(\013)424 3101 y Fs(j)485 3089 y Fp(\025)c Fx(0,)670 3027 y Fm(P)758 3114 y Fs(j)807 3089 y Fw(\013)860 3101 y Fs(j)921 3089 y Fx(=)g(1)j(and)j(~)-45 b Fw(\015)1289 3101 y Fs(j)1350 3089 y Fp(\025)26 b Fx(0,)1536 3027 y Fm(P)1623 3114 y Fs(j)1675 3089 y Fx(~)-45 b Fw(\015)1715 3101 y Fs(j)1776 3089 y Fx(=)26 b(1.)42 b(Plugging)28 b(the)i(expressions)d(in)j(\(12.20\))e(in)n(to)i(the)f(r.h.s.)180 3214 y(of)e(\(12.5\),)g(using)g(\(2.35\))g(in)g(order)g(to)g(express)f Fw(\031)s Fx(\()p Fw(v)1820 3226 y Fv(1)1858 3214 y Fx(\))i(in)g(terms) f(of)g(creation)f(and)i(annihilation)f(op)r(erators,)e Fw(a)3715 3174 y Fv(#)3715 3239 y Fs(`)p Fv(1)180 3328 y Fx(and)i Fw(a)385 3288 y Fv(#)385 3350 y Fs(r)r Fv(1)455 3328 y Fx(,)1103 3438 y Fw(\031)s Fx(\()p Fw(v)1225 3450 y Fv(1)1263 3438 y Fx(\))c(=)g Fw(a)1450 3450 y Fs(`)p Fv(1)1515 3438 y Fx(\()1547 3365 y Fm(p)p 1630 3365 224 4 v 73 x Fx(1)18 b(+)g Fw(\032)1816 3450 y Fv(1)1867 3438 y Fw(G)1932 3450 y Fv(1)p Fs(`)1997 3438 y Fx(\))h(+)f Fw(a)2175 3450 y Fs(r)r Fv(1)2245 3438 y Fx(\()2277 3382 y Fp(p)p 2346 3382 81 4 v 56 x Fw(\032)2389 3450 y Fv(1)2440 3438 y Fw(G)2505 3450 y Fv(1)p Fs(`)2571 3438 y Fx(\))g(+)g Fw(h:c:;)180 3577 y Fx(where)35 b Fw(\032)471 3589 y Fv(1)543 3577 y Fx(=)g(\()p Fw(e)714 3547 y Fs(\014)752 3555 y Fk(1)784 3547 y Fs(!)856 3577 y Fp(\000)23 b Fx(1\))1018 3547 y Fr(\000)p Fv(1)1107 3577 y Fx(,)37 b(pulling)e(through)f(the)i (annihilation)e(op)r(erators)f(to)i(the)h(righ)n(t)e(and)h(using)g (that)180 3677 y Fw(\031)s Fx(\()p Fw(v)302 3689 y Fv(1)340 3677 y Fx(\))28 b(\(or)f Fw(G)599 3689 y Fv(1)p Fs(`)664 3677 y Fx(\))h(acts)f(only)h(on)f(the)h(\014rst)f(\(left\))i(factor)e (in)h Fw(')2106 3689 y Fs(j)s(j)2195 3677 y Fx(=)23 b Fw(')2337 3689 y Fs(j)2390 3677 y Fp(\012)18 b Fw(')2527 3689 y Fs(j)2563 3677 y Fx(,)27 b(w)n(e)h(obtain)364 3889 y Fw(\021)408 3855 y Fr(0)514 3889 y Fx(=)83 b Fp(\000)p Fx(2)783 3810 y Fm(X)826 3987 y Fs(j)915 3889 y Fw(\013)968 3901 y Fs(j)1007 3889 y Fx(~)-45 b Fw(\015)1047 3901 y Fs(j)1082 3889 y Fx(Im)1195 3776 y Fm(Z)1291 3797 y(n)1347 3889 y Fx(\(1)18 b(+)g Fw(\032)1565 3901 y Fv(1)1602 3889 y Fx(\))p Fw(!)1703 3797 y Fm(D)1754 3889 y Fw(')1808 3901 y Fs(j)1843 3889 y Fw(;)c(G)1945 3855 y Fr(\003)1945 3910 y Fv(1)1983 3889 y Fx(\()p Fw(H)2084 3901 y Fs(p)2141 3889 y Fp(\000)k Fw(E)2285 3901 y Fs(j)2339 3889 y Fx(+)g Fw(!)j Fx(+)d Fw(i)p Fx(0\))2681 3855 y Fr(\000)p Fv(1)2770 3889 y Fw(e)2809 3855 y Fr(\000)p Fs(\014)s Fv(\()p Fs(H)2982 3863 y Fi(p)3016 3855 y Fr(\000)p Fs(E)3117 3863 y Fi(j)3148 3855 y Fv(+)p Fs(!)r Fv(\))p Fs(=)p Fv(2)3340 3889 y Fw(G)3405 3901 y Fv(1)3442 3889 y Fw(')3496 3901 y Fs(j)3532 3797 y Fm(E)662 4135 y Fp(\000)p Fw(\032)770 4147 y Fv(1)807 4135 y Fw(!)875 4043 y Fm(D)926 4135 y Fw(')980 4147 y Fs(j)1015 4135 y Fw(;)c(G)1117 4101 y Fr(\003)1117 4156 y Fv(1)1155 4135 y Fx(\()p Fw(H)1256 4147 y Fs(p)1314 4135 y Fp(\000)k Fw(E)1458 4147 y Fs(j)1511 4135 y Fp(\000)g Fw(!)j Fx(+)d Fw(i)p Fx(0\))1853 4101 y Fr(\000)p Fv(1)1942 4135 y Fw(e)1981 4101 y Fr(\000)p Fs(\014)s Fv(\()p Fs(H)2154 4109 y Fi(p)2188 4101 y Fr(\000)p Fs(E)2289 4109 y Fi(j)2320 4101 y Fr(\000)p Fs(!)r Fv(\))p Fs(=)p Fv(2)2513 4135 y Fw(G)2578 4101 y Fr(\003)2578 4156 y Fv(1)2616 4135 y Fw(')2670 4147 y Fs(j)2706 4043 y Fm(E)13 b(o)2826 4135 y Fw(d)2869 4101 y Fv(3)2906 4135 y Fw(k)s(:)180 4327 y Fx(Inserting)34 b(the)g(partition)g(of)g(unit)n(y)h Fn(1)e Fx(=)1543 4265 y Fm(P)1631 4352 y Fs(j)1680 4327 y Fp(j)p Fw(')1757 4339 y Fs(j)1792 4327 y Fp(i)14 b(h)p Fw(')1924 4339 y Fs(j)1960 4327 y Fp(j)34 b Fx(in)n(to)g(the)h(inner)f (pro)r(ducts)g(on)g(the)g(r.h.s.)57 b(w)n(e)33 b(obtain)180 4426 y(furthermore)624 4615 y Fw(\021)668 4581 y Fr(0)715 4615 y Fx(=)23 b(2)p Fw(\031)908 4536 y Fm(X)931 4713 y Fs(i;j)1042 4615 y Fw(\013)1095 4627 y Fs(i)1126 4615 y Fx(~)-45 b Fw(\015)1166 4627 y Fs(i)1207 4502 y Fm(Z)1304 4615 y Fw(!)s Fp(j)14 b(h)p Fw(')1482 4627 y Fs(i)1510 4615 y Fw(;)g(G)1612 4627 y Fv(1)1649 4615 y Fw(')1703 4627 y Fs(j)1739 4615 y Fp(i)g(j)1808 4581 y Fv(2)1859 4615 y Fp(f)o Fx(\(1)19 b(+)f Fw(\032)2119 4627 y Fv(1)2156 4615 y Fx(\))p Fw(\016)s Fx(\()p Fw(E)2321 4627 y Fs(j)s(i)2399 4615 y Fx(+)g Fw(!)s Fx(\))g Fp(\000)g Fw(\032)2713 4627 y Fv(1)2750 4615 y Fw(\016)s Fx(\()p Fw(E)2883 4627 y Fs(j)s(i)2961 4615 y Fp(\000)g Fw(!)s Fx(\))p Fp(g)13 b Fw(d)3229 4581 y Fv(3)3267 4615 y Fw(k)s(:)180 4865 y Fx(In)n(terc)n(hanging)30 b(the)i(lab)r(els)f(in)h(the)g(sum)f(of)h (the)g(\014rst)f(term)h(and)f(noticing)g(that)h(in)g(the)g(resulting)f (expression)180 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Fx(\(12.21\))180 1173 y(where)1681 1252 y(~)1671 1273 y Fw(I)30 b Fx(=)d(~)-46 b Fw(\031)s Fx(\()p Fw(v)s Fx(\))19 b Fp(\000)2100 1252 y Fx(~)2084 1273 y Fw(\031)2134 1249 y Fr(0)2158 1273 y Fx(\()p Fw(v)s Fx(\))p Fw(:)1238 b Fx(\(12.22\))180 1459 y(De\014ne)444 1438 y Fm(e)437 1459 y Fx(\012)24 b(=)653 1425 y Fs(e)684 1400 y Fh(\000)p Fi(\014)772 1390 y Fc(e)764 1400 y Fi(L)804 1383 y Fk(\()p Fi(l)p Fk(\))873 1400 y Fi(=)p Fk(2)936 1425 y Fv(\012)983 1433 y Fk(0)p 619 1440 431 4 v 619 1506 a Fr(k)p Fs(e)684 1489 y Fh(\000)p Fi(\014)772 1479 y Fc(e)764 1489 y Fi(L)804 1474 y Fk(\()p Fi(l)p Fk(\))873 1489 y Fi(=)p Fk(2)936 1506 y Fv(\012)983 1514 y Fk(0)1016 1506 y Fr(k)1060 1459 y Fw(;)k Fx(where)1359 1438 y Fm(e)1351 1459 y Fw(L)1408 1429 y Fv(\()p Fs(l)p Fv(\))1508 1459 y Fx(=)c Fw(L)1654 1471 y Fv(0)1709 1459 y Fx(+)c Fm(e)-48 b Fw(\031)s Fx(\()p Fw(v)s Fx(\))29 b(and)h Fm(e)-48 b Fw(\031)27 b Fx(=)c Fw(\031)s Fp(j)2375 1471 y Fs(\014)2413 1479 y Fk(1)2445 1471 y Fv(=)p Fs(\014)2534 1479 y Fk(2)2566 1471 y Fv(=)p Fs(\014)2662 1459 y Fx(.)38 b(Since)28 b(the)g(en)n(trop)n(y)e(pro)r (duction)180 1591 y(do)r(es)f(not)f(dep)r(end)i(on)f Fw(\014)955 1603 y Fs(p)1018 1591 y Fx(w)n(e)f(put)i(from)e(no)n(w)h (on)f Fw(\014)1810 1603 y Fs(p)1872 1591 y Fx(=)e Fw(\014)2006 1603 y Fv(1)2043 1591 y Fx(.)36 b(The)25 b(op)r(erator)e Fw(K)6 b Fp(j)2702 1603 y Fs(\014)2740 1611 y Fi(p)2775 1603 y Fv(=)p Fs(\014)2864 1611 y Fk(1)2896 1603 y Fv(=)p Fs(\014)2985 1611 y Fk(2)3017 1603 y Fv(=)p Fs(\014)3135 1591 y Fx(=)23 b Fw(L)h Fx(is)h(selfadjoin)n(t,)180 1709 y(\(5.12\),)i(and)g(hence)842 1688 y Fm(e)835 1709 y Fx(\012)895 1679 y Fr(\003)p 895 1698 38 3 v 895 1746 a Fs(\022)956 1709 y Fx(=)1051 1688 y Fm(e)1044 1709 y Fx(\012)p 1104 1686 V 26 x Fs(\022)1141 1709 y Fx(.)37 b(F)-7 b(rom)27 b(\(12.9\))g(w)n(e)g(obtain)873 1933 y Fw(\021)s Fx(\()p Fw(\036)998 1945 y Fv(1)1036 1933 y Fx(\))83 b(=)1299 1841 y Fm(D)1350 1933 y Fx(\(\012)1442 1899 y Fr(\003)p 1442 1918 V 1442 1967 a Fs(\022)1499 1933 y Fp(\000)1589 1912 y Fm(e)1582 1933 y Fx(\012)p 1642 1910 V 26 x Fs(\022)1679 1933 y Fx(\))p Fw(;)14 b(\031)s Fx(\()p Fw(\036)1879 1945 y Fv(1)1918 1933 y Fx(\))1950 1945 y Fs(\022)1987 1933 y Fx(\012)2047 1945 y Fs(\022)2085 1841 y Fm(E)2154 1933 y Fx(+)2237 1841 y Fm(D)2295 1912 y(e)2288 1933 y Fx(\012)p 2348 1910 V 26 x Fs(\022)2385 1933 y Fw(;)g(\031)s Fx(\()p Fw(\036)2553 1945 y Fv(1)2592 1933 y Fx(\))2624 1945 y Fs(\022)2661 1933 y Fx(\(\012)2753 1945 y Fs(\022)2810 1933 y Fp(\000)2900 1912 y Fm(e)2893 1933 y Fx(\012)2953 1945 y Fs(\022)2990 1933 y Fx(\))3023 1841 y Fm(E)1299 2116 y Fx(+)1378 2024 y Fm(D)1435 2095 y(e)1428 2116 y Fx(\012)p 1488 2093 V 26 x Fs(\022)1526 2116 y Fw(;)g(\031)s Fx(\()p Fw(\036)1694 2128 y Fv(1)1732 2116 y Fx(\))1764 2128 y Fs(\022)1809 2095 y Fm(e)1802 2116 y Fx(\012)1862 2128 y Fs(\022)1899 2024 y Fm(E)1964 2116 y Fw(:)1539 b Fx(\(12.23\))180 2323 y(W)-7 b(e)19 b(consider)f(the)i(last)e(term)h(\014rst.)34 b(Recall)19 b(that)g Fw(\036)1765 2335 y Fv(1)1826 2323 y Fx(=)j Fw(g)s Fx([)p Fw(v)s(;)14 b(iH)2157 2335 y Fs(r)r Fv(1)2227 2323 y Fx(])23 b(=)f Fw(g)s Fx([)p Fw(v)2466 2335 y Fv(1)2504 2323 y Fw(;)14 b(iH)2639 2335 y Fs(r)r Fv(1)2708 2323 y Fx(])19 b(and)g(therefore)f Fw(\031)s Fx(\()p Fw(\036)3374 2335 y Fv(1)3412 2323 y Fx(\))24 b(=)g Fm(e)-48 b Fw(\031)t Fx(\()p Fw(\036)3687 2335 y Fv(1)3725 2323 y Fx(\).)180 2423 y(It)28 b(follo)n(ws)f(that)1111 2523 y Fp(h)1150 2502 y Fm(e)1143 2523 y Fx(\012)p 1203 2500 V 26 x Fs(\022)1240 2523 y Fw(;)14 b(\031)s Fx(\()p Fw(\036)1408 2535 y Fv(1)1447 2523 y Fx(\))1479 2535 y Fs(\022)1523 2502 y Fm(e)1516 2523 y Fx(\012)1576 2535 y Fs(\022)1614 2523 y Fp(i)23 b Fx(=)g Fp(h)1796 2502 y Fm(e)1789 2523 y Fx(\012)p 1849 2500 V 26 x Fs(\022)1887 2523 y Fw(;)16 b Fm(e)-48 b Fw(\031)s Fx(\()p Fw(\036)2055 2535 y Fv(1)2093 2523 y Fx(\))2125 2535 y Fs(\022)2170 2502 y Fm(e)2163 2523 y Fx(\012)2223 2535 y Fs(\022)2260 2523 y Fp(i)24 b Fx(=)e Fp(h)2442 2502 y Fm(e)2435 2523 y Fx(\012)q Fw(;)16 b Fm(e)-48 b Fw(\031)s Fx(\()p Fw(\036)2664 2535 y Fv(1)2702 2523 y Fx(\))2741 2502 y Fm(e)2734 2523 y Fx(\012)p Fp(i)p Fw(:)677 b Fx(\(12.24\))180 2672 y(Note)32 b(that)g(the)h(r.h.s.)49 b(of)32 b(\(12.24\))f(describ)r(es)g(the)i (heat)f(\015o)n(w)f(in)n(to)h(reserv)n(oir)d Fw(r)r Fx(1)j(for)g(the)g (equal)f(temp)r(erature)180 2772 y(system.)37 b(Since)28 b(the)h(heat)e(\015o)n(ws)g(v)-5 b(anish)28 b(individually)g(in)g(the)g (equal)f(temp)r(erature)h(case)f(w)n(e)g(ha)n(v)n(e)g(sho)n(wn)g(that) 180 2871 y(the)h(last)f(term)h(in)g(\(12.23\))e(v)-5 b(anishes.)305 2971 y(T)e(o)28 b(estimate)h(the)g(\014rst)g(t)n(w)n(o)f (terms)h(on)f(the)i(r.h.s.)40 b(w)n(e)29 b(use)f(as)h(b)r(efore)f (expansions)g(\(11.6\))g(and)h(\(11.15\))f(for)180 3080 y(\012)240 3040 y Fv(#)240 3110 y Fs(\022)274 3094 y Fk(#)357 3080 y Fx(=)h(\012)511 3092 y Fs(\022)549 3080 y Fw(;)14 b Fx(\012)646 3050 y Fr(\003)p 646 3069 V 646 3118 a Fs(\022)715 3080 y Fx(and)32 b(similar)f(expansions)f (\(obtained)i(b)n(y)f(setting)g Fw(\014)2400 3092 y Fv(2)2467 3080 y Fx(=)f Fw(\014)2609 3092 y Fv(1)2676 3080 y Fx(=)f Fw(\014)36 b Fx(in)c(\(11.6\))f(and)g(\(11.15\)\))g(for)187 3186 y Fm(e)180 3207 y Fx(\012)240 3219 y Fs(\022)278 3207 y Fx(.)57 b(As)35 b(a)f(result)g(w)n(e)g(obtain)g(an)g(expression) f(for)h(\(12.23\))f(of)i(the)g(t)n(yp)r(e)f(\(12.10\))f({)h(\(12.13\))g (but)h(with)g(some)180 3317 y(of)c(the)f(p)r(o)n(w)n(ers)g(in)g(Rem,)i (\(12.13\),)e(replaced)g(b)n(y)g(the)h(di\013erences,)g(e.g.)45 b(\()p Fp(\000)p 2624 3250 64 4 v Fw(R)2688 3329 y Fv(0)2725 3317 y Fx(\()p Fw(L)2814 3329 y Fv(0)p Fs(\022)2884 3317 y Fx(\))p Fw(I)2952 3329 y Fs(\022)2991 3317 y Fx(\))3023 3287 y Fs(l)3069 3317 y Fp(\000)20 b Fx(\()p Fp(\000)p 3251 3250 V Fw(R)3315 3329 y Fv(0)3352 3317 y Fx(\()p Fw(L)3441 3329 y Fv(0)p Fs(\022)3511 3317 y Fx(\))3551 3296 y Fm(e)3543 3317 y Fw(I)3579 3329 y Fs(\022)3618 3317 y Fx(\))3650 3287 y Fs(l)3706 3317 y Fx(or)180 3433 y(\()p Fw(R)p 275 3410 53 3 v 25 x Fs(Q)328 3474 y Fk(0)364 3433 y Fx(\()p Fw(K)473 3390 y Fv(\(1\))467 3455 y(0)562 3433 y Fx(\))p Fw(W)12 b Fx(\))716 3403 y Fs(n)783 3433 y Fp(\000)22 b Fx(\()p Fw(R)p 965 3410 V 25 x Fs(Q)1017 3474 y Fk(0)1054 3433 y Fx(\()p Fw(K)1163 3390 y Fv(\(1\))1157 3455 y(0)1252 3433 y Fx(\))p Fw(W)12 b Fx(\))1406 3403 y Fs(n)1451 3433 y Fp(j)1474 3445 y Fs(\014)1512 3453 y Fk(2)1545 3445 y Fv(=)p Fs(\014)1634 3453 y Fk(1)1666 3445 y Fv(=)p Fs(\014)1761 3433 y Fx(.)52 b(These)32 b(di\013erences)g(are)g(estimated)g(b)n(y)g(using)h(a)f(telescopic)180 3533 y(expansion,)27 b(e.g.,)815 3715 y(\()p Fp(\000)p 912 3648 64 4 v Fw(R)976 3727 y Fv(0)1013 3715 y Fx(\()p Fw(L)1102 3727 y Fv(0)p Fs(\022)1173 3715 y Fx(\))p Fw(I)1241 3727 y Fs(\022)1279 3715 y Fx(\))1311 3681 y Fs(l)1355 3715 y Fp(\000)18 b Fx(\()p Fp(\000)p 1535 3648 V Fw(R)1599 3727 y Fv(0)1636 3715 y Fx(\()p Fw(L)1725 3727 y Fv(0)p Fs(\022)1795 3715 y Fx(\))1835 3694 y Fm(e)1827 3715 y Fw(I)1863 3727 y Fs(\022)1902 3715 y Fx(\))1934 3681 y Fs(l)898 3922 y Fx(=)1095 3818 y Fs(l)1046 3843 y Fm(X)1048 4020 y Fs(j)s Fv(=1)1166 3922 y Fx(\()p Fp(\000)p 1263 3855 V Fw(R)1326 3934 y Fv(0)1364 3922 y Fx(\()p Fw(L)1453 3934 y Fv(0)p Fs(\022)1523 3922 y Fx(\))p Fw(I)1591 3934 y Fs(\022)1629 3922 y Fx(\))1661 3887 y Fs(j)s Fr(\000)p Fv(1)1782 3922 y Fx(\()p Fp(\000)p 1879 3855 V Fw(R)1942 3934 y Fv(0)1979 3922 y Fx(\()p Fw(L)2068 3934 y Fv(0)p Fs(\022)2139 3922 y Fx(\)\)\()p Fw(I)2271 3934 y Fs(\022)2328 3922 y Fp(\000)2419 3901 y Fm(e)2411 3922 y Fw(I)2447 3934 y Fs(\022)2485 3922 y Fx(\)\()p Fp(\000)p 2614 3855 V Fw(R)2678 3934 y Fv(0)2715 3922 y Fx(\()p Fw(L)2804 3934 y Fv(0)p Fs(\022)2875 3922 y Fx(\))2915 3901 y Fm(e)2907 3922 y Fw(I)2943 3934 y Fs(\022)2981 3922 y Fx(\))3013 3887 y Fs(l)p Fr(\000)p Fs(j)3122 3922 y Fw(;)381 b Fx(\(12.25\))180 4188 y(and)31 b(then)h(estimating)e(the)i(\014rst)f(t)n(yp)r(e)g(of)g (the)h(di\013erences)f(in)g(norm)f(while)i(for)e(the)i(second)e(t)n(yp) r(e)i(w)n(e)e(do)h(\014rst)180 4288 y(the)h(pull-through)f(and)h(con)n (traction)e(pro)r(cedure)h(and)h(then)g(estimate)g(the)g(resulting)f (in)n(tegrals.)48 b(As)32 b(a)f(result)180 4387 y(w)n(e)c(ha)n(v)n(e) 1034 4570 y Fw(\021)s Fx(\()p Fw(\036)1159 4582 y Fv(1)1197 4570 y Fx(\))83 b(=)g Fp(\000)p Fw(g)1581 4503 y Fm(\012)1620 4570 y Fx(\012)1680 4536 y Fr(\003)1680 4591 y Fv(0)1737 4570 y Fp(\000)18 b Fx(\012)1880 4582 y Fv(0)1917 4570 y Fw(;)c(\031)s Fx(\()p Fw(\036)2085 4582 y Fv(1)2123 4570 y Fx(\))2155 4582 y Fs(\022)p 2193 4503 V 2193 4570 a Fw(R)2257 4582 y Fv(0)2294 4570 y Fx(\()p Fw(L)2383 4582 y Fv(0)p Fs(;\022)2473 4570 y Fx(\))p Fw(I)2541 4582 y Fs(\022)2579 4570 y Fx(\012)2639 4582 y Fv(0)2677 4503 y Fm(\013)1460 4728 y Fp(\000)p Fw(g)1581 4636 y Fm(D)p 1632 4661 V 92 x Fw(R)1695 4740 y Fv(0)1733 4728 y Fx(\()p Fw(L)1822 4754 y Fv(0)p Fs(;)p 1875 4705 34 3 v(\022)1912 4728 y Fx(\))1958 4636 y Fm(h)1997 4728 y Fw(I)2040 4693 y Fr(\003)p 2033 4712 38 3 v 2033 4761 a Fs(\022)2078 4728 y Fx(\012)2138 4693 y Fr(\003)2138 4748 y Fv(0)2195 4728 y Fp(\000)2286 4707 y Fm(e)2278 4728 y Fw(I)p 2314 4705 V 26 x Fs(\022)2352 4728 y Fx(\012)2412 4740 y Fv(0)2449 4636 y Fm(i)2488 4728 y Fw(;)g(\031)s Fx(\()p Fw(\036)2656 4740 y Fv(1)2695 4728 y Fx(\))2727 4740 y Fs(\022)2764 4728 y Fx(\012)2824 4740 y Fv(0)2862 4636 y Fm(E)1460 4910 y Fp(\000)p Fw(g)1581 4818 y Fm(D)1632 4910 y Fx(\012)1692 4922 y Fv(0)1729 4910 y Fw(;)g(\031)s Fx(\()p Fw(\036)1897 4922 y Fv(1)1935 4910 y Fx(\))1967 4922 y Fs(\022)p 2005 4844 64 4 v 2005 4910 a Fw(R)2069 4922 y Fv(0)2106 4910 y Fx(\()p Fw(L)2195 4922 y Fv(0)p Fs(;\022)2285 4910 y Fx(\))2331 4818 y Fm(h)2370 4910 y Fw(I)2406 4922 y Fs(\022)2463 4910 y Fp(\000)2554 4889 y Fm(e)2546 4910 y Fw(I)2582 4922 y Fs(\022)2620 4818 y Fm(i)2673 4910 y Fx(\012)2733 4922 y Fv(0)2770 4818 y Fm(E)1460 5069 y Fx(+)p Fw(o)p Fx(\()p Fw(g)1640 5035 y Fv(2)1677 5069 y Fx(\))p Fw(O)r Fx(\()p Fw(\016)s(\014)t Fx(\))1312 5194 y(=)83 b Fw(\021)1501 5206 y Fv(1)1557 5194 y Fx(+)18 b Fw(\021)1681 5206 y Fv(2)1737 5194 y Fx(+)g Fw(o)p Fx(\()p Fw(g)1935 5160 y Fv(2)1972 5194 y Fx(\))p Fw(O)r Fx(\()p Fw(\016)s(\014)t Fx(\))1302 b(\(12.26\))1938 5476 y(34)p eop %%Page: 35 35 35 34 bop 180 210 a Fx(where)27 b Fw(\021)461 222 y Fv(1)526 210 y Fx(is)h(giv)n(en)e(in)i(\(12.12\))f(and)950 415 y Fw(\021)991 427 y Fv(2)1051 415 y Fx(=)c Fw(g)1196 323 y Fm(D)1246 415 y Fx(\012)1306 427 y Fv(0)1343 415 y Fw(;)1380 323 y Fm(h)1427 394 y(e)1420 415 y Fw(I)1456 427 y Fs(\022)p 1494 348 64 4 v 1494 415 a Fw(R)1557 427 y Fv(0)1595 415 y Fx(\()p Fw(L)1684 427 y Fv(0)p Fs(;\022)1774 415 y Fx(\))p Fw(\031)s Fx(\()p Fw(\036)1937 427 y Fv(1)1975 415 y Fx(\))2007 427 y Fs(\022)2064 415 y Fx(+)18 b Fw(\031)s Fx(\()p Fw(\036)2278 427 y Fv(1)2316 415 y Fx(\))2348 427 y Fs(\022)p 2386 348 V 2386 415 a Fw(R)2449 427 y Fv(0)2487 415 y Fx(\()p Fw(L)2576 427 y Fv(0)p Fs(;\022)2666 415 y Fx(\))2706 394 y Fm(e)2698 415 y Fw(I)2734 427 y Fs(\022)2772 323 y Fm(i)2825 415 y Fx(\012)2885 427 y Fv(0)2922 323 y Fm(E)2987 415 y Fw(:)516 b Fx(\(12.27\))180 639 y(Since)27 b(the)h(con)n(tribution)f (of)g(the)h Fw(v)1289 651 y Fv(2)1326 639 y Fx(-comp)r(onen)n(t)f(of) 1880 618 y Fm(e)1872 639 y Fw(I)1908 651 y Fs(\022)1974 639 y Fx(is)g(zero)f(w)n(e)h(can)g(omit)g(the)h(tilde)g(\()p Fp(\030)p Fx(\))f(in)h(\(12.27\).)35 b(Th)n(us)180 738 y(the)f(expression)f(for)g Fw(\021)910 750 y Fv(2)981 738 y Fx(coincides)g(up)i(to)e(the)h(sign)g(and)f(the)i(substitution)f (\012)2744 708 y Fr(\003)2744 759 y Fv(0)2816 738 y Fp(!)f Fx(\012)2992 750 y Fv(0)3063 738 y Fx(with)h(the)h(expression)180 838 y(\(12.12\))26 b(for)h Fw(\021)629 850 y Fv(1)667 838 y Fx(,)h(i.e.)744 1018 y Fw(\021)785 1030 y Fv(1)841 1018 y Fx(+)18 b Fw(\021)965 1030 y Fv(2)1025 1018 y Fx(=)23 b Fp(\000)p Fw(g)1234 951 y Fm(\012)1273 1018 y Fx(\012)1333 984 y Fr(\003)1333 1038 y Fv(0)1390 1018 y Fp(\000)18 b Fx(\012)1533 1030 y Fv(0)1570 1018 y Fw(;)1607 951 y Fm(\002)1642 1018 y Fw(I)1678 1030 y Fs(\022)p 1716 951 V 1716 1018 a Fw(R)1779 1030 y Fv(0)1817 1018 y Fx(\()p Fw(L)1906 1030 y Fv(0)p Fs(;\022)1996 1018 y Fx(\))p Fw(\031)s Fx(\()p Fw(\036)2159 1030 y Fv(1)2197 1018 y Fx(\))2229 1030 y Fs(\022)2286 1018 y Fx(+)g Fw(\031)s Fx(\()p Fw(\036)2500 1030 y Fv(1)2538 1018 y Fx(\))2570 1030 y Fs(\022)p 2608 951 V 2608 1018 a Fw(R)2671 1030 y Fv(0)2709 1018 y Fx(\()p Fw(L)2798 1030 y Fv(0)p Fs(;\022)2888 1018 y Fx(\))p Fw(I)2956 1030 y Fs(\022)2994 951 y Fm(\003)3043 1018 y Fx(\012)3103 1030 y Fv(0)3140 951 y Fm(\013)3193 1018 y Fw(:)310 b Fx(\(12.28\))180 1198 y(W)-7 b(e)30 b(pro)r(ceed)e(with)i(the)g(r.h.s.)41 b(of)29 b(\(12.28\))f(exactly)h (as)f(w)n(e)h(did)h(ab)r(o)n(v)n(e)e(with)h Fw(\021)2698 1210 y Fv(1)2765 1198 y Fx(alone)g(in)g(equations)f(\(12.16\))g({)180 1298 y(\(12.18\),)e(and)i(w)n(e)f(arriv)n(e)f(at)559 1503 y Fw(\021)s Fx(\()p Fw(\036)684 1515 y Fv(1)722 1503 y Fx(\))e(=)e(2)p Fw(g)950 1468 y Fv(2)987 1503 y Fx(Re)1099 1410 y Fm(D)1149 1503 y Fx(\012)1209 1468 y Fr(\003)1209 1523 y Fv(0)1266 1503 y Fp(\000)c Fx(\012)1409 1515 y Fv(0)1446 1503 y Fw(;)c(\031)s Fx(\()p Fw(v)1605 1515 y Fv(1)1643 1503 y Fx(\))p Fw(iL)1761 1515 y Fs(r)r Fv(1)1830 1503 y Fx(\()p Fw(L)1919 1515 y Fv(0)1975 1503 y Fx(+)k Fw(i)p Fx(0\))2161 1468 y Fr(\000)p Fv(1)2249 1503 y Fw(e)2288 1468 y Fr(\000)p Fs(\014)s(L)2427 1476 y Fk(0)2458 1468 y Fs(=)p Fv(2)2530 1503 y Fw(\031)s Fx(\()p Fw(v)2652 1515 y Fv(1)2690 1503 y Fx(\)\012)2782 1515 y Fv(0)2819 1410 y Fm(E)2888 1503 y Fx(+)h Fw(o)p Fx(\()p Fw(g)3087 1468 y Fv(2)3124 1503 y Fx(\))p Fw(O)r Fx(\()p Fw(\016)s(\014)t Fx(\))p Fw(:)180 1707 y Fx(\(12.4\))27 b(follo)n(ws)f(b)n(y)i(noticing)f(that)958 1912 y(Re)1070 1820 y Fm(D)1120 1912 y Fx(\012)1180 1924 y Fv(0)1218 1912 y Fw(;)14 b(\031)s Fx(\()p Fw(v)1377 1924 y Fv(1)1415 1912 y Fx(\))p Fw(iL)1533 1924 y Fs(r)r Fv(1)1602 1912 y Fx(\()p Fw(L)1691 1924 y Fv(0)1746 1912 y Fx(+)k Fw(i)p Fx(0\))1932 1878 y Fr(\000)p Fv(1)2021 1912 y Fw(e)2060 1878 y Fr(\000)p Fs(\014)s(L)2199 1886 y Fk(0)2230 1878 y Fs(=)p Fv(2)2301 1912 y Fw(\031)s Fx(\()p Fw(v)2423 1924 y Fv(1)2461 1912 y Fx(\)\012)2553 1924 y Fv(0)2591 1820 y Fm(E)2665 1912 y Fx(=)k(0)p Fw(:)p 2936 1924 59 71 v 305 2117 a Fn(Pro)s(of)39 b(of)g(Theorem)f(12.2.)93 b Fx(W)-7 b(e)35 b(w)n(an)n(t)e(to)h(con)n(trol)f(the)h(comp)r(onen)n (ts)g Fw(\015)2805 2129 y Fs(j)2840 2117 y Fx(,)i(\(12.7\),)f(app)r (earing)d(in)j(the)180 2217 y(expression)30 b(\(12.6\))g(for)g Fw(\021)1001 2187 y Fr(0)1025 2217 y Fx(.)47 b(T)-7 b(o)31 b(this)g(end)g(w)n(e)g(emplo)n(y)f(basic)h(analytic)f(p)r(erturbation)h (theory)f(\(in)h Fw(\016)s(\014)t Fx(\))h(for)f(the)180 2316 y(matrix)24 b(family)h Fw(M)9 b Fx(\()p Fw(\016)s(\014)t Fx(\))24 b(:=)f(\003)1136 2286 y Fr(\003)1136 2337 y Fv(0)1173 2316 y Fx(\()p Fw(\016)s(\014)t Fx(\))p Fp(j)1351 2328 y Fs(\014)1389 2336 y Fi(p)1425 2328 y Fv(=0)1513 2316 y Fx(,)j(where)e(w)n(e)h(consider)e Fw(\014)2286 2328 y Fv(1)2348 2316 y Fx(to)i(b)r(e)g(\014xed.)37 b(W)-7 b(rite)25 b(\000)3070 2328 y Fs(j)3129 2316 y Fx(instead)g(of)g(\000) 3557 2328 y Fs(j)s Fv(0)3625 2316 y Fx(,)g(see)180 2416 y(\(10.3\).)36 b(According)27 b(to)g(Prop)r(osition)f(10.1)g(w)n(e)i (ha)n(v)n(e)e Fw(M)9 b Fx(\()p Fw(\016)s(\014)t Fx(\))24 b(=)e Fw(M)2321 2428 y Fv(0)2377 2416 y Fx(+)c Fw(\016)s(\014)t(M)2632 2428 y Fv(1)2687 2416 y Fx(+)g Fw(O)r Fx(\()p Fw(\016)s(\014)2958 2386 y Fv(2)2997 2416 y Fx(\),)28 b(where)507 2596 y Fw(M)588 2608 y Fv(0)707 2596 y Fx(=)83 b Fp(\000)p Fw(i)p Fx(\()p Fw(e)1020 2562 y Fr(\000)p Fs(\014)1110 2570 y Fk(1)1141 2562 y Fs(H)1195 2570 y Fi(p)1230 2562 y Fs(=)p Fv(2)1320 2596 y Fp(\012)18 b Fx(1)-18 b(l\))14 b Fp(f)o Fx(\000)1589 2608 y Fv(1)1626 2596 y Fx(\()p Fw(\014)1705 2608 y Fv(1)1742 2596 y Fx(\))19 b(+)f(\000)1928 2608 y Fv(2)1965 2596 y Fx(\()p Fw(\014)2044 2608 y Fv(1)2082 2596 y Fx(\))p Fp(g)13 b Fx(\()p Fw(e)2240 2562 y Fs(\014)2278 2570 y Fk(1)2310 2562 y Fs(H)2364 2570 y Fi(p)2400 2562 y Fs(=)p Fv(2)2489 2596 y Fp(\012)18 b Fx(1)-18 b(l\))p Fw(;)852 b Fx(\(12.29\))507 2721 y Fw(M)588 2733 y Fv(1)707 2721 y Fx(=)83 b Fp(\000)p Fw(i)p Fx(\()p Fw(e)1020 2686 y Fr(\000)p Fs(\014)1110 2694 y Fk(1)1141 2686 y Fs(H)1195 2694 y Fi(p)1230 2686 y Fs(=)p Fv(2)1320 2721 y Fp(\012)18 b Fx(1)-18 b(l\))1496 2653 y Fm(\010)1554 2688 y Fv(1)p 1554 2702 34 4 v 1554 2749 a(2)1597 2721 y Fx([)p Fw(H)1689 2733 y Fs(p)1746 2721 y Fp(\012)18 b Fx(1)-18 b(l)p Fw(;)14 b Fx(\000)1965 2733 y Fv(2)2001 2721 y Fx(\()p Fw(\014)2080 2733 y Fv(1)2117 2721 y Fx(\)])19 b Fp(\000)f Fx(\()p Fw(@)2350 2733 y Fs(\014)2395 2721 y Fx(\000)2447 2733 y Fv(2)2484 2721 y Fx(\)\()p Fw(\014)2595 2733 y Fv(1)2633 2721 y Fx(\))2665 2653 y Fm(\011)2728 2721 y Fx(\()p Fw(e)2799 2686 y Fs(\014)2837 2694 y Fk(1)2869 2686 y Fs(H)2923 2694 y Fi(p)2958 2686 y Fs(=)p Fv(2)3048 2721 y Fp(\012)g Fx(1)-18 b(l\))p Fw(:)293 b Fx(\(12.30\))180 2901 y(Let)24 b Fw(\020)367 2871 y Fr(\003)429 2901 y Fx(=)f Fw(\020)559 2871 y Fr(\003)553 2921 y Fv(0)609 2901 y Fx(+)12 b Fw(\016)s(\014)t(\020)819 2871 y Fr(\003)813 2921 y Fv(1)870 2901 y Fx(+)g Fw(O)r Fx(\()p Fw(\016)s(\014)1135 2871 y Fv(2)1173 2901 y Fx(\).)36 b(The)24 b(normalization)f Fp(h)p Fw(\020)2028 2871 y Fr(\003)2067 2901 y Fw(;)14 b(\020)6 b Fp(i)24 b Fx(=)e(1)i(\(where)g Fw(\020)30 b Fx(=)22 b(\012)2837 2913 y Fs(p)2876 2901 y Fx(\()p Fw(\014)2955 2913 y Fs(p)2993 2901 y Fx(\))j(is)f(the)h(particle)f (Gibbs)180 3000 y(state\))k(implies)g(that)h Fp(h)p Fw(\020)953 2970 y Fr(\003)947 3021 y Fv(0)992 3000 y Fw(;)14 b(\020)6 b Fp(i)24 b Fx(=)f(1)28 b(and)1447 2933 y Fm(\012)1486 3000 y Fw(\020)1528 2970 y Fr(\003)1522 3022 y Fs(j)1567 3000 y Fw(;)14 b(\020)1646 2933 y Fm(\013)1709 3000 y Fx(=)24 b(0,)j(for)h Fw(j)h Fp(\025)23 b Fx(1.)38 b(Solving)27 b(the)i(zero-o)r(der)d(eigen)n(v)-5 b(alue)27 b(equation)180 3100 y Fw(M)261 3112 y Fv(0)298 3100 y Fw(\020)340 3070 y Fr(\003)334 3121 y Fv(0)401 3100 y Fx(=)c(0)k(giv)n(es)1025 3247 y Fw(\020)1067 3213 y Fr(\003)1061 3268 y Fv(0)1129 3247 y Fx(=)1216 3168 y Fm(X)1261 3345 y Fs(j)1350 3247 y Fw(\015)1398 3204 y Fv(\(0\))1393 3270 y Fs(j)1487 3247 y Fw(')1541 3259 y Fs(j)1594 3247 y Fp(\012)18 b Fw(')1731 3259 y Fs(j)1767 3247 y Fw(;)97 b(\015)1935 3204 y Fv(\(0\))1930 3270 y Fs(j)2046 3247 y Fx(=)2278 3191 y Fw(Z)2335 3203 y Fs(p)2373 3191 y Fx(\()p Fw(\014)2452 3203 y Fs(p)2491 3191 y Fx(\))p 2144 3228 514 4 v 2144 3304 a Fw(Z)2201 3316 y Fs(p)2239 3304 y Fx(\()p Fw(\014)2318 3316 y Fv(1)2374 3304 y Fx(+)18 b Fw(\014)2504 3316 y Fs(p)2542 3304 y Fw(=)p Fx(2\))2668 3247 y Fw(e)2707 3213 y Fr(\000)p Fs(\014)2797 3221 y Fk(1)2828 3213 y Fs(E)2877 3221 y Fi(j)2912 3247 y Fw(:)591 b Fx(\(12.31\))180 3479 y(The)28 b(\014rst-order)d(eigen)n(v)-5 b(alue)27 b(equation)g(reads)f Fw(M)1773 3491 y Fv(1)1810 3479 y Fw(\020)1852 3449 y Fr(\003)1846 3500 y Fv(0)1909 3479 y Fx(+)18 b Fw(M)2073 3491 y Fv(0)2110 3479 y Fw(\020)2152 3449 y Fr(\003)2146 3500 y Fv(1)2214 3479 y Fx(=)k(0,)28 b(whic)n(h)f(implies)859 3659 y([\000)934 3671 y Fv(1)971 3659 y Fx(\()p 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Fs(p)2247 3839 y Fx(.)53 b(Let)33 b(\011\()p Fw(\014)t Fx(\))f(:=)2809 3777 y Fm(P)2897 3864 y Fs(j)2945 3839 y Fw(e)2984 3809 y Fr(\000)p Fs(\014)s(E)3126 3817 y Fi(j)3156 3809 y Fs(=)p Fv(2)3228 3839 y Fw(')3282 3851 y Fs(j)3339 3839 y Fp(\012)22 b Fw(')3480 3851 y Fs(j)3515 3839 y Fx(.)53 b(Since)180 3939 y(\000)232 3951 y Fv(2)269 3939 y Fx(\()p Fw(\014)t Fx(\)\011\()p Fw(\014)t Fx(\))25 b(=)d(0)27 b(w)n(e)h(ha)n(v)n(e)e(\()p Fw(@)1135 3951 y Fs(\014)1180 3939 y Fx(\000)1232 3951 y Fv(2)1269 3939 y Fx(\)\()p Fw(\014)t Fx(\)\011\()p Fw(\014)t Fx(\))f(=)e Fp(\000)p Fx(\000)1826 3951 y Fv(2)1862 3939 y Fx(\()p Fw(\014)t Fx(\)\()p Fw(@)2053 3951 y Fs(\014)2099 3939 y Fx(\011\)\()p Fw(\014)t Fx(\),)29 b(so)896 4129 y Fw(@)940 4141 y Fs(\014)985 4129 y Fp(j)1008 4141 y Fs(\014)1046 4149 y Fk(1)1083 4129 y Fx(\000)1135 4141 y Fv(2)1172 4129 y Fx(\()p Fw(\014)t Fx(\)\()p Fw(e)1358 4095 y Fs(\014)s(H)1453 4103 y Fi(p)1488 4095 y Fs(=)p Fv(2)1578 4129 y Fp(\012)18 b Fx(1)-18 b(l\))p Fw(\020)1782 4095 y Fr(\003)1776 4149 y Fv(0)1903 4129 y Fx(=)83 b Fw(C)6 b Fx(\()p Fw(\014)2195 4141 y Fv(1)2232 4129 y Fx(\))p Fw(@)2308 4141 y Fs(\014)2354 4129 y Fp(j)2377 4141 y Fs(\014)2415 4149 y Fk(1)2451 4129 y Fx(\000)2503 4141 y Fv(2)2540 4129 y Fx(\()p Fw(\014)t Fx(\)\011\(2)p Fw(\014)2841 4141 y Fv(1)2897 4129 y Fp(\000)18 b Fw(\014)t Fx(\))1903 4253 y(=)83 b Fw(C)6 b Fx(\()p Fw(\014)2195 4265 y Fv(1)2232 4253 y Fx(\)\000)2316 4265 y Fv(2)2354 4253 y Fx(\()p Fw(\014)2433 4265 y Fv(1)2470 4253 y Fx(\)\()p Fw(H)2603 4265 y Fs(p)2661 4253 y Fp(\012)18 b Fx(1)-18 b(l\)\011\()p Fw(\014)2967 4265 y Fv(1)3003 4253 y Fx(\))p Fw(;)468 b Fx(\(12.33\))180 4433 y(where)22 b Fw(C)6 b Fx(\()p Fw(\014)559 4445 y Fv(1)597 4433 y Fx(\))23 b(=)g Fw(Z)797 4445 y Fs(p)835 4433 y Fx(\()p Fw(\014)914 4445 y Fs(p)952 4433 y Fx(\))p Fw(=)-5 b(Z)1078 4445 y Fs(p)1116 4433 y Fx(\()p Fw(\014)1195 4445 y Fv(1)1241 4433 y Fx(+)8 b Fw(\014)1361 4445 y Fs(p)1399 4433 y Fw(=)p Fx(2\).)34 b(The)23 b(r.h.s.)34 b(of)23 b(\(12.33\))e(is)h(a)g (v)n(ector)f(in)h(the)h(orthogonal)d(complemen)n(t)180 4533 y(of)28 b(k)n(er\000)439 4545 y Fv(2)475 4533 y Fx(\()p Fw(\014)554 4545 y Fv(1)591 4533 y Fx(\))23 b(=)g Fo(C)15 b Fx(\011\()p Fw(\014)932 4545 y Fv(1)975 4533 y Fx(\).)37 b(Using)28 b(this)g(fact)f(and)h(\(12.33\))e(w)n(e)h(solv)n (e)g(\(12.32\))f(for)h Fw(\020)2845 4503 y Fr(\003)2839 4554 y Fv(1)2884 4533 y Fx(:)709 4713 y Fw(\020)751 4679 y Fr(\003)745 4734 y Fv(1)873 4713 y Fx(=)82 b Fw(C)6 b Fx(\()p Fw(\014)1164 4725 y Fv(1)1202 4713 y Fx(\)\()p Fw(e)1305 4679 y Fr(\000)p Fs(\014)1395 4687 y Fk(1)1427 4679 y Fs(H)1481 4687 y Fi(p)1516 4679 y Fs(=)p Fv(2)1606 4713 y Fp(\012)18 b Fx(1)-18 b(l\)[\000)1843 4725 y Fv(1)1880 4713 y Fx(\()p Fw(\014)1959 4725 y Fv(1)1996 4713 y Fx(\))19 b(+)f(\000)2182 4725 y Fv(2)2219 4713 y Fx(\()p Fw(\014)2298 4725 y Fv(1)2336 4713 y Fx(\)])2391 4679 y Fr(\000)p Fv(1)2480 4713 y Fx(\000)2532 4725 y Fv(2)2569 4713 y Fx(\()p Fw(\014)2648 4725 y Fv(1)2686 4713 y Fx(\)\()p Fw(H)2819 4725 y Fs(p)2876 4713 y Fp(\012)g Fx(1)-18 b(l\)\011\()p Fw(\014)3182 4725 y Fv(1)3219 4713 y Fx(\))275 b(\(12.34\))1020 4848 y(+)p Fw(C)1150 4814 y Fr(0)1173 4848 y Fx(\()p Fw(e)1244 4814 y Fr(\000)p Fs(\014)1334 4822 y Fk(1)1366 4814 y Fs(H)1420 4822 y Fi(p)1456 4814 y Fs(=)p Fv(2)1545 4848 y Fp(\012)18 b Fx(1)-18 b(l\)\011\()p Fw(\014)1851 4860 y Fv(1)1888 4848 y Fx(\))p Fw(;)1583 b Fx(\(12.35\))180 5028 y(where)31 b(the)h(constan)n(t)f Fw(C)975 4998 y Fr(0)1031 5028 y Fx(is)h(determined)g(b)n(y)f(the)h (normalization)e(condition)i Fp(h)p Fw(\020)2796 4998 y Fr(\003)2790 5049 y Fv(1)2835 5028 y Fw(;)14 b(\020)6 b Fp(i)31 b Fx(=)e(0.)49 b(F)-7 b(rom)32 b(expression)180 5128 y(\(12.6\))g(it)i(is)e(clear)g(that)h(the)h(term)f(\(12.35\))e(do) r(es)i(not)g(con)n(tribute)f(to)h(the)h(v)-5 b(alue)32 b(of)h Fw(\021)2999 5097 y Fr(0)3056 5128 y Fx(\(this)g(is)g(the)g (same)g(as)180 5227 y(sa)n(ying)26 b(that)i Fw(\021)658 5197 y Fr(0)705 5227 y Fx(=)22 b(0)27 b(for)g Fw(\016)s(\014)h Fx(=)23 b(0\).)1938 5476 y(35)p eop %%Page: 36 36 36 35 bop 305 210 a Fx(Under)24 b(the)h(assumption)f Fw(G)1194 222 y Fv(1)1254 210 y Fx(=)f Fw(G)1407 222 y Fv(2)1467 210 y Fx(=)g Fw(G)i Fx(w)n(e)f(ha)n(v)n(e)f(\000)2004 222 y Fv(1)2041 210 y Fx(\()p Fw(\014)2120 222 y Fv(1)2158 210 y Fx(\))g(=)g(\000)2353 222 y Fv(2)2390 210 y Fx(\()p Fw(\014)2469 222 y Fv(1)2506 210 y Fx(\))i(and)f(the)h(r.h.s.)35 b(of)25 b(\(12.34\))e(simpli\014es)180 310 y(to)k(an)h(easy)e (expression,)h(whic)n(h,)g(when)h(used)g(in)f(\(12.6\),)g(yields)h (\(12.8\).)p 3714 322 59 71 v 305 509 a Fq(Pr)l(o)l(of)43 b(of)g(The)l(or)l(em)h(3.2.)122 b Fx(No)n(w)41 b(w)n(e)g(are)f(ready)h (to)g(pro)n(v)n(e)f(\(12.3\).)77 b(By)41 b(an)g(abstract)f(result)h(of) h([19)o(],)180 609 y Fw(E)5 b(P)12 b Fx(\()p Fw(\021)s Fx(\))24 b Fp(\025)g Fx(0.)37 b(Therefore,)27 b(due)h(to)g(\(12.1\),)f Fw(\021)s Fx(\()p Fw(\036)1681 621 y Fv(1)1719 609 y Fx(\))d Fp(\025)f Fx(0)28 b(for)f Fw(\014)2107 621 y Fv(1)2168 609 y Fp(\025)c Fw(\014)2303 621 y Fv(2)2341 609 y Fx(.)38 b(Hence,)28 b(due)g(to)g(\(12.4\),)f(for)h Fw(g)i Fx(su\016cien)n(tly)180 708 y(small)d(\(dep)r(ending)h(on)g Fw(\016)s(\014)k Fx(in)c(general\),)1586 879 y Fw(\021)1630 845 y Fr(0)1677 879 y Fp(\025)23 b Fx(0)82 b(for)g Fw(\014)2118 891 y Fv(1)2179 879 y Fp(\025)22 b Fw(\014)2313 891 y Fv(2)2351 879 y Fw(:)1152 b Fx(\(12.36\))180 1051 y(Next,)28 b(expression)e(\(12.6\))h(sho)n(ws)f(that)1506 1222 y Fw(\021)1550 1188 y Fr(0)1601 1222 y Fx(is)h(analytic)g(in)h Fw(\014)2144 1234 y Fv(1)2209 1222 y Fx(and)f Fw(\014)2417 1234 y Fv(2)3526 1222 y Fx(\(12.37\))180 1393 y(\(the)h Fw(\015)398 1405 y Fs(j)461 1393 y Fx(are)f(analytic)f(in)i Fw(\014)1059 1405 y Fv(1)1097 1393 y Fw(;)14 b(\014)1181 1405 y Fv(2)1218 1393 y Fx(\).)37 b(Equations)26 b(\(11.19\))h(and)g (\(12.6\))g(imply)h(that)1100 1564 y Fw(\021)1144 1530 y Fr(0)1191 1564 y Fw(>)22 b Fx(0)83 b(if)28 b Fw(\014)1526 1576 y Fv(1)1591 1564 y Fx(is)f(\014xed)h(and)f Fw(\014)2083 1576 y Fv(2)2148 1564 y Fx(is)h(su\016cien)n(tly)f(small)p Fw(:)666 b Fx(\(12.38\))180 1735 y(Relations)32 b(\(12.36\))f({)i (\(12.38\))e(imply)i(\(12.3\))f(for)g(almost)g(all)g(v)-5 b(alues)32 b(of)h(\()p Fw(\014)2637 1747 y Fv(1)2674 1735 y Fw(;)14 b(\014)2758 1747 y Fv(2)2795 1735 y Fx(\))32 b Fp(2)g Fx(\(0)p Fw(;)14 b Fp(1)p Fx(\))22 b Fp(\002)f Fx(\(0)p Fw(;)14 b Fp(1)p Fx(\),)34 b(in)f(the)180 1835 y(sense)f(that)i(for)e(\014xed)h Fw(\014)969 1847 y Fv(1)1039 1835 y Fp(2)f Fx(\(0)p Fw(;)14 b Fp(1)p Fx(\),)34 b Fw(E)5 b(P)12 b Fx(\()p Fw(\021)1613 1847 y Fs(\014)1651 1855 y Fk(1)1684 1847 y Fs(\014)1722 1855 y Fk(2)1758 1835 y Fx(\))33 b(can)g(v)-5 b(anish)33 b(only)f(for)h(\014nitely)g(man)n(y) g(v)-5 b(alues)32 b(of)h Fw(\014)3478 1847 y Fv(2)3549 1835 y Fx(in)g(an)n(y)180 1935 y(b)r(ounded)28 b(subset)g(of)f(\(0)p Fw(;)14 b Fp(1)p Fx(\).)37 b(\(The)28 b(same)f(holds)g(for)g Fw(\014)1956 1947 y Fv(1)2021 1935 y Fx(and)h Fw(\014)2230 1947 y Fv(2)2295 1935 y Fx(in)n(terc)n(hanged\).)305 2034 y(If)d Fw(G)450 2046 y Fv(1)510 2034 y Fx(=)e Fw(G)663 2046 y Fv(2)725 2034 y Fx(or)h(if)h(the)g(dimension)f(of)h(the)g (particle)f(system)g(is)g(2)h(then)g(w)n(e)f(ha)n(v)n(e)f Fw(\021)2911 2004 y Fr(0)2958 2034 y Fx(=)p 3046 1988 45 4 v 23 w Fw(\021)s(\016)s(\014)17 b Fx(+)12 b Fw(O)r Fx(\(\()p Fw(\016)s(\014)t Fx(\))3523 2004 y Fv(2)3562 2034 y Fx(\))25 b(with)p 180 2088 V 180 2134 a Fw(\021)31 b(>)d Fx(0)i(indep)r(enden)n(t)i(of)e Fw(\016)s(\014)t Fx(.)46 b(Hence)31 b(for)f Fw(g)k Fx(and)c Fw(\016)s(\014)35 b Fx(b)r(oth)c(small,)g(but)g(indep)r(enden)n(t)h(of)e(eac)n(h)g (other,)h(w)n(e)f(ha)n(v)n(e)180 2233 y Fw(\021)224 2203 y Fr(0)271 2233 y Fw(>)22 b Fx(0.)37 b(This)27 b(follo)n(ws)g(from)g (Theorem)g(12.2)f(and)i(for)f(the)h(t)n(w)n(o-dimensional)d(case)i (from)g(\(11.20\),)g(\(11.21\).)305 2333 y(The)g(ab)r(o)n(v)n(e)f (results)h(together)g(with)h(\(12.4\))f(imply)h(the)g(statemen)n(t)g (in)f(Theorem)g(3.2)g(for)g Fw(\014)3248 2345 y Fv(1)3308 2333 y Fp(6)p Fx(=)c Fw(\014)3443 2345 y Fv(2)3480 2333 y Fx(.)212 b Fg(\004)305 2523 y Fn(Ac)m(kno)m(wledgemen)m(ts.)32 b Fx(The)20 b(authors)f(are)g(grateful)h(to)g(V.)g(Bac)n(h,)h(G.)f (Elliott,)i(J.)e(F)-7 b(r\177)-42 b(ohlic)n(h,)21 b(and)f(esp)r (ecially)180 2623 y(C.-A.)30 b(Pillet)g(for)g(useful)g(discussions)f (and)h(to)g(V.)g(Jak)l(\025)-37 b(si)n(\023)e(c)27 b(and)j(C.-A.)g (Pillet,)g(for)g(dra)n(wing)e(their)i(atten)n(tion)g(to)180 2722 y(pap)r(er)k([34)o(])g(whic)n(h)g(discusses)f(the)h(en)n(trop)n(y) f(pro)r(duction)h(in)g(a)f(form)h(close)f(to)h(the)h(one)e(used)h(in)g (the)h(presen)n(t)180 2822 y(pap)r(er.)h(P)n(art)26 b(of)g(this)i(w)n (ork)d(w)n(as)h(done)g(while)h(the)h(\014rst)e(author)g(w)n(as)g (visiting)g(the)i(Univ)n(ersit)n(y)e(of)g(T)-7 b(oron)n(to,)26 b(the)180 2922 y(third)31 b(author,)g(ETH,)g(Z)r(\177)-44 b(uric)n(h)30 b(and)g(the)i(second)e(and)g(third)h(authors)f(visited)h (ESI)g(Vienna.)46 b(During)31 b(w)n(ork)e(on)180 3021 y(this)e(pap)r(er)g(the)h(second)e(author)h(w)n(as)f(at)h(the)h(Univ)n (ersit)n(y)e(of)h(T)-7 b(oron)n(to)25 b(on)i(a)g(D)n(AAD)h(fello)n (wship.)37 b(The)27 b(authors)180 3121 y(are)g(grateful)g(to)g(these)h (places)f(for)g(hospitalit)n(y)-7 b(.)180 3393 y Fy(A)134 b(Pro)t(of)45 b(of)g(existence)h(of)g(dynamics)180 3575 y Fx(In)33 b(this)g(app)r(endix)h(w)n(e)e(pro)n(v)n(e)f(the)j (existence)f(of)f(dynamics)h(\(2.12\).)52 b(Recall)33 b(the)g(de\014nition)h(of)e(the)i(op)r(erator)180 3691 y Fw(L)237 3660 y Fv(\()p Fs(`)p Fv(\))343 3691 y Fx(:=)23 b Fw(L)511 3703 y Fv(0)566 3691 y Fx(+)18 b Fw(g)s(\031)s Fx(\()p Fw(v)s Fx(\))29 b(and)e(the)h(one)f(parameter)f(group)h Fw(\033)2017 3660 y Fs(t)2047 3691 y Fx(\()p Fw(B)t Fx(\))c(:=)g Fw(e)2351 3660 y Fs(itL)2445 3635 y Fk(\()p Fi(`)p Fk(\))2523 3691 y Fw(B)t(e)2629 3660 y Fr(\000)p Fs(itL)2775 3635 y Fk(\()p Fi(`)p Fk(\))2853 3691 y Fx(,)k Fw(B)h Fp(2)23 b Fw(\031)s Fx(\()p Fp(A)p Fx(\))3252 3660 y Fr(00)3296 3691 y Fx(.)180 3847 y Fn(Prop)s(osition)30 b(A.1)41 b Fq(Assume)33 b(the)h(op)l(er)l(ators)g Fw(v)1744 3859 y Fs(n)1820 3847 y Fp(2)c(A)k Fq(satisfy)h(\(2.13\).)52 b(Then)34 b(the)g(inte)l(gr)l(ands)g(on)f(the)h(r.h.s.)180 3947 y(of)d(\(2.12\))g(ar)l(e)f(c)l(ontinuous)f(functions,)h(the)g (series)h(is)f(absolutely)g(c)l(onver)l(gent,)g(the)g(limit)h(exists)e (and)h(e)l(quals)1561 4118 y Fw( )1618 4084 y Fs(t)1647 4118 y Fx(\()p Fw(A)p Fx(\))24 b(=)f(T)-7 b(r)o(\()p Fw(\032\033)2095 4084 y Fs(t)2125 4118 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\)\))1192 b(\(A.1\))180 4289 y Fq(and,)31 b(c)l(onse)l(quently,)f(is)g(indep)l(endent)g(of)h(the)f(appr)l (oximating)h(op)l(er)l(ators.)305 4446 y Fn(Pro)s(of.)59 b Fx(Let)22 b Fw(v)806 4458 y Fs(n)874 4446 y Fp(2)i(A)e Fx(b)r(e)g(an)g(appro)n(ximating)e(sequence)h(for)g(the)h(op)r(erator)e Fw(v)25 b Fx(satisfying)c(\(2.13\).)35 b(W)-7 b(e)22 b(de\014ne)180 4562 y(the)32 b(selfadjoin)n(t)f(op)r(erators)f Fw(L)1159 4519 y Fv(\()p Fs(`)p Fv(\))1159 4572 y Fs(n)1271 4562 y Fx(:=)g Fw(L)1446 4574 y Fv(0)1503 4562 y Fx(+)21 b Fw(g)s(\031)s Fx(\()p Fw(v)1754 4574 y Fs(n)1800 4562 y Fx(\))31 b(on)h(the)f(dense)h(domain)f Fp(D)r Fx(\()p Fw(L)2814 4574 y Fv(0)2851 4562 y Fx(\).)49 b(Let)32 b(the)f(one)h(parameter)180 4662 y(group)26 b Fw(\033)465 4632 y Fs(t)462 4688 y Fv(\()p Fs(n)p Fv(\))588 4662 y Fx(on)h Fw(\031)s Fx(\()p Fp(A)p Fx(\))i(b)r(e)f(giv)n(en)f(b)n(y) 1493 4873 y Fw(\033)1543 4839 y Fs(t)1540 4896 y Fv(\()p Fs(n)p Fv(\))1638 4873 y Fx(\()p Fw(B)t Fx(\))c(:=)g Fw(e)1942 4839 y Fs(itL)2036 4814 y Fk(\()p Fi(`)p Fk(\))2036 4856 y Fi(n)2114 4873 y Fw(B)t(e)2220 4839 y Fr(\000)p Fs(itL)2366 4814 y Fk(\()p Fi(`)p Fk(\))2366 4856 y Fi(n)2444 4873 y Fw(:)180 5056 y Fx(Set)28 b Fw(\033)373 5026 y Fs(t)370 5077 y Fv(0)408 5056 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\))c(:=)f Fw(\031)s Fx(\()p Fw(\013)918 5026 y Fs(t)918 5077 y Fv(0)956 5056 y Fx(\()p Fw(A)p Fx(\)\))29 b(and)e(let)h Fw( )j Fx(b)r(e)d(an)f Fw(!)1789 5068 y Fv(0)1826 5056 y Fx(-normal)f(state)i(on)f Fp(A)p Fx(,)h(i.e.)1648 5227 y Fw( )s Fx(\()p Fw(A)p Fx(\))c(=)e(T)-7 b(r\()p Fw(\032\031)s Fx(\()p Fw(A)p Fx(\)\))1278 b(\(A.2\))1938 5476 y(36)p eop %%Page: 37 37 37 36 bop 180 210 a Fx(for)29 b(some)f(p)r(ositiv)n(e,)i(trace)e(class) g(op)r(erator)g Fw(\032)h Fx(on)g Fp(H)h Fx(of)f(trace)f(1.)42 b(Then)29 b(using)g(the)h(de\014nition)f Fw(V)3294 222 y Fs(n)3366 210 y Fx(=)c Fw(\031)s Fx(\()p Fw(v)3578 222 y Fs(n)3624 210 y Fx(\))k(w)n(e)180 310 y(\014nd)582 409 y Fw( )s Fx(\([)p Fw(\013)747 373 y Fs(t)772 381 y Fi(m)747 432 y Fv(0)832 409 y Fx(\()p Fw(v)904 421 y Fs(n)950 409 y Fx(\))p Fw(;)14 b Fp(\001)g(\001)g(\001)g Fx([)p Fw(\013)1206 373 y Fs(t)1231 381 y Fk(1)1206 432 y Fv(0)1268 409 y Fx(\()p Fw(v)1340 421 y Fs(n)1386 409 y Fx(\))p Fw(;)g(\013)1508 375 y Fs(t)1508 430 y Fv(0)1545 409 y Fx(\()p Fw(A)p Fx(\)])g Fp(\001)g(\001)g(\001)h Fx(]\))23 b(=)g(T)-7 b(r)o(\()p Fw(\032)p Fx([)p Fw(\033)2219 373 y Fs(t)2244 381 y Fi(m)2216 432 y Fv(0)2304 409 y Fx(\()p Fw(V)2384 421 y Fs(n)2430 409 y Fx(\))p Fw(;)14 b Fp(\001)g(\001)g(\001)g Fx([)p Fw(\033)2683 373 y Fs(t)2708 381 y Fk(1)2680 432 y Fv(0)2745 409 y Fx(\()p Fw(V)2825 421 y Fs(n)2871 409 y Fx(\))p Fw(;)g(\033)2990 375 y Fs(t)2987 430 y Fv(0)3025 409 y Fx(\()p Fw(A)p Fx(\)])g Fp(\001)g(\001)g(\001)g Fx(]\))p Fw(:)212 b Fx(\(A.3\))180 550 y(Clearly)31 b(the)i(r.h.s.)50 b(is)32 b(con)n(tin)n(uous)f(in)i Fw(t)1492 562 y Fv(1)1529 550 y Fw(;)14 b Fp(\001)g(\001)g(\001)28 b Fw(;)14 b(t)1758 562 y Fs(m)1853 550 y Fx(and)32 b(therefore)f(the)i (in)n(tegrals)e(in)h(\(2.12\))f(are)h(w)n(ell)g(de\014ned)180 650 y(and,)g(b)n(y)g(a)f(standard)f(estimate,)j(the)f(series)e(on)h (the)h(r.h.s.)48 b(of)32 b(\(2.12\))e(con)n(v)n(erges)f(absolutely)-7 b(.)48 b(In)32 b(fact,)g(using)180 750 y(the)c(Araki-Dyson)e(series)819 980 y Fw(\033)869 946 y Fs(t)866 1003 y Fv(\()p Fs(n)p Fv(\))963 980 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\))85 b(=)1474 876 y Fr(1)1447 901 y Fm(X)1436 1077 y Fs(m)p Fv(=0)1579 980 y Fx(\()p Fw(ig)s Fx(\))1715 946 y Fs(m)1791 867 y Fm(Z)1874 888 y Fs(t)1837 1056 y Fv(0)1917 980 y Fw(dt)1990 992 y Fv(1)2042 980 y Fp(\001)14 b(\001)g(\001)2152 867 y Fm(Z)2235 888 y Fs(t)2260 896 y Fi(m)p Fh(\000)p Fk(1)2199 1056 y Fv(0)2407 980 y Fw(dt)2480 992 y Fs(m)2571 980 y Fx([)p Fw(\033)2644 943 y Fs(t)2669 951 y Fi(m)2641 1002 y Fv(0)2729 980 y Fx(\()p Fw(\031)s Fx(\()p Fw(v)2883 992 y Fs(n)2929 980 y Fx(\)\))p Fw(;)g Fp(\001)g(\001)g(\001)1436 1191 y Fx([)p Fw(\033)1509 1154 y Fs(t)1534 1162 y Fk(1)1506 1213 y Fv(0)1571 1191 y Fx(\()p Fw(\031)s Fx(\()p Fw(v)1725 1203 y Fs(n)1771 1191 y Fx(\)\))p Fw(;)g(\033)1922 1157 y Fs(t)1919 1212 y Fv(0)1957 1191 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\)])g Fp(\001)g(\001)g(\001)i Fx(])p Fw(;)1196 b Fx(\(A.4\))180 1360 y(one)46 b(can)g(easily)f(see)h(that)g (this)h(series)e(is)h(nothing)g(but)g(the)h(Araki-Dyson)e(expansion)g (of)h(the)h(function)180 1459 y(T)-7 b(r\()p Fw(\032\033)391 1429 y Fs(t)388 1486 y Fv(\()p Fs(n)p Fv(\))485 1459 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\)\).)39 b(Th)n(us)27 b(w)n(e)g(ha)n(v)n(e)f(sho)n(wn)h(that)h(the)f(r.h.s.)37 b(of)27 b(\(2.12\))g(is)g(equal)g(to)g(lim)2986 1471 y Fs(n)p Fr(!1)3178 1459 y Fx(T)-7 b(r)o(\()p Fw(\032\033)3388 1429 y Fs(t)3385 1486 y Fv(\()p Fs(n)p Fv(\))3483 1459 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\)\).)305 1576 y(No)n(w,)36 b Fw(V)574 1588 y Fs(n)654 1576 y Fx(con)n(v)n(erges)d(to) h Fw(V)54 b Fx(strongly)34 b(on)g(the)i(dense)e(set)h(Span)p Fp(f)p Fw(\031)s Fx(\()p Fw(W)12 b Fx(\()p Fw(f)d Fx(\)\)\012)2816 1588 y Fv(0)2854 1576 y Fp(j)p Fw(f)44 b Fp(2)35 b Fw(L)3109 1546 y Fv(2)3109 1597 y(0)3146 1576 y Fp(g)f Fx(as)h(follo)n(ws)e(from) 180 1676 y(\(2.13\))27 b(and)g(the)h(relation)1081 1845 y Fp(k)p Fx(\()p Fw(V)1203 1857 y Fs(n)1267 1845 y Fp(\000)18 b Fw(V)g Fx(\))p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\012)1684 1857 y Fv(0)1723 1845 y Fp(k)1765 1810 y Fv(2)1825 1845 y Fx(=)23 b Fw(!)1965 1857 y Fv(0)2001 1845 y Fx(\()p Fw(A)2095 1810 y Fr(\003)2134 1845 y Fx(\()p Fw(v)2209 1810 y Fr(\003)2206 1865 y Fs(n)2270 1845 y Fp(\000)18 b Fw(v)2396 1810 y Fr(\003)2435 1845 y Fx(\)\()p Fw(v)2539 1857 y Fs(n)2603 1845 y Fp(\000)g Fw(v)s Fx(\))p Fw(A)p Fx(\))p Fw(:)711 b Fx(\(A.5\))180 2029 y(Hence)30 b Fw(L)486 1986 y Fv(\()p Fs(`)p Fv(\))486 2039 y Fs(n)598 2029 y Fx(con)n(v)n(erges)d(to)i Fw(L)1132 1999 y Fv(\()p Fs(`)p Fv(\))1245 2029 y Fx(strongly)f(on)h(the)h(same)f(set.)42 b(Since)29 b(this)h(set)f(is)h(a)f(core)f(for)h Fw(L)3240 1986 y Fv(\()p Fs(`)p Fv(\))3240 2039 y Fs(n)3352 2029 y Fx(and)h Fw(L)3573 1999 y Fv(\()p Fs(`)p Fv(\))3685 2029 y Fx(w)n(e)180 2140 y(conclude)g(that)g Fw(L)763 2097 y Fv(\()p Fs(`)p Fv(\))763 2150 y Fs(n)877 2140 y Fx(con)n(v)n(erge)e(to)i Fw(L)1380 2110 y Fv(\()p Fs(`)p Fv(\))1493 2140 y Fx(in)g(the)h(strong)e(resolv)n(en)n(t)f(sense)i(as)f Fw(n)e Fp(\000)-14 b(!)27 b(1)p Fx(.)45 b(Hence)30 b(the)h(functions) 180 2240 y(T)-7 b(r\()p Fw(\032\033)391 2210 y Fs(t)388 2260 y(n)433 2240 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\))30 b(con)n(v)n(erge)25 b(to)j(T)-7 b(r)o(\()p Fw(\032\033)1354 2210 y Fs(t)1384 2240 y Fx(\()p Fw(\031)s Fx(\()p Fw(A)p Fx(\)\)\))30 b(whic)n(h,)d(in)h(particular,)e(sho)n(ws)h (\(A.1\).)p 3714 2252 59 71 v 180 2512 a Fy(B)134 b(P)l(ositiv)l(e)47 b(T)-11 b(emp)t(erature)45 b(Represen)l(tation)180 2710 y Fb(B.1)112 b(Jak)-6 b(\025)-50 b(si)m(\023)d(c-Pillet)34 b(Gluing)180 2864 y Fx(In)27 b(this)h(app)r(endix,)f(w)n(e)g(represen)n (t)f(the)i(Hilb)r(ert)g(space)e Fp(H)i Fx(in)g(a)f(form)f(whic)n(h)i (is)f(w)n(ell)g(suited)g(for)g(a)g(de\014nition)g(of)180 2963 y(the)h(translation)e(transformation.)36 b(This)27 b(represen)n(tation)f(is)i(due)f(to)h([18)o(].)305 3063 y(Consider)e(the)i(F)-7 b(o)r(c)n(k)27 b(space)1264 3231 y Fp(F)k Fx(:=)22 b Fp(F)8 b Fx(\()p Fw(L)1622 3197 y Fv(2)1659 3231 y Fx(\()p Fw(X)25 b Fp(\002)18 b(f)p Fx(1)p Fw(;)c Fx(2)p Fp(g)p Fx(\)\))p Fw(;)96 b(X)29 b Fx(=)23 b Fo(R)h Fp(\002)18 b Fw(S)2659 3197 y Fv(2)3527 3231 y Fx(\(B.1.1\))180 3400 y(and)25 b(denote)g Fw(x)f Fx(=)e(\()p Fw(u;)14 b(\033)s Fx(\))24 b Fp(2)f Fw(X)7 b Fx(.)36 b(The)25 b(v)-5 b(acuum)25 b(in)h Fp(F)33 b Fx(is)25 b(denoted)h(b)n(y)2373 3379 y(~)2364 3400 y(\012)2424 3412 y Fs(r)2460 3400 y Fx(.)37 b(The)25 b(smeared-out)f(creation)g(op) r(erator)180 3500 y Fw(a)224 3469 y Fr(\003)262 3500 y Fx(\()p Fw(F)12 b Fx(\),)28 b Fw(F)35 b Fp(2)24 b Fw(L)666 3469 y Fv(2)702 3500 y Fx(\()p Fw(X)h Fp(\002)18 b(f)p Fx(1)p Fw(;)c Fx(2)p Fp(g)p Fx(\))26 b(is)i(giv)n(en)f(b)n(y)1415 3711 y Fw(a)1459 3676 y Fr(\003)1497 3711 y Fx(\()p Fw(F)12 b Fx(\))23 b(=)1737 3632 y Fm(X)1775 3806 y Fs(\013)1871 3597 y Fm(Z)1917 3786 y Fs(X)1994 3711 y Fw(F)12 b Fx(\()p Fw(x;)i(\013)p Fx(\))p Fw(a)2304 3676 y Fr(\003)2343 3711 y Fx(\()p Fw(x;)g(\013)p Fx(\))180 3949 y(and)27 b(analogously)f(for)h(annihilation)g(op)r(erators.)35 b(The)27 b(CCR)h(read)1336 4118 y([)p Fw(a)p Fx(\()p Fw(x;)14 b(\013)p Fx(\))p Fw(;)g(a)1685 4083 y Fr(\003)1724 4118 y Fx(\()p Fw(x)1803 4083 y Fr(0)1827 4118 y Fw(;)g(\013)1917 4083 y Fr(0)1941 4118 y Fx(\)])23 b(=)g Fw(\016)2144 4130 y Fs(\013;\013)2250 4113 y Fh(0)2277 4118 y Fw(\016)s Fx(\()p Fw(x)c Fp(\000)f Fw(x)2545 4083 y Fr(0)2569 4118 y Fx(\))p Fw(:)180 4286 y Fx(F)-7 b(ollo)n(wing)26 b([18],)h(w)n(e)h (in)n(tro)r(duce)f(the)h(unitary)f(map)389 4455 y Fw(U)32 b Fx(:)524 4387 y Fm(\002)559 4455 y Fp(F)8 b Fx(\()p Fw(L)716 4420 y Fv(2)752 4455 y Fx(\()p Fo(R)839 4420 y Fv(3)882 4455 y Fx(\)\))19 b Fp(\012)f(F)8 b Fx(\()p Fw(L)1205 4420 y Fv(2)1242 4455 y Fx(\()p Fo(R)1328 4420 y Fv(3)1371 4455 y Fx(\)\))1435 4387 y Fm(\003)1489 4455 y Fp(\012)1572 4387 y Fm(\002)1607 4455 y Fp(F)g Fx(\()p Fw(L)1764 4420 y Fv(2)1801 4455 y Fx(\()p Fo(R)1887 4420 y Fv(3)1930 4455 y Fx(\)\))19 b Fp(\012)f(F)8 b Fx(\()p Fw(L)2253 4420 y Fv(2)2290 4455 y Fx(\()p Fo(R)2376 4420 y Fv(3)2420 4455 y Fx(\)\))2484 4387 y Fm(\003)2542 4455 y Fp(!)23 b(F)8 b Fx(\()p Fw(L)2805 4420 y Fv(2)2842 4455 y Fx(\()p Fw(X)25 b Fp(\002)18 b(f)p Fx(1)p Fw(;)c Fx(2)p Fp(g)p Fx(\)\))207 b(\(B.1.2\))180 4623 y(de\014ned)28 b(b)n(y)1335 4723 y Fw(U)22 b Fx(\()q([\012)1530 4735 y Fs(r)r Fv(1)1618 4723 y Fp(\012)c Fx(\012)1761 4735 y Fs(r)r Fv(1)1831 4723 y Fx(])g Fp(\012)g Fx([\012)2038 4735 y Fs(r)r Fv(2)2127 4723 y Fp(\012)g Fx(\012)2270 4735 y Fs(r)r Fv(2)2339 4723 y Fx(])q(\))23 b(:=)2538 4702 y(~)2529 4723 y(\012)2589 4735 y Fs(r)3527 4723 y Fx(\(B.1.3\))180 4864 y(and)876 5032 y Fw(U)942 4940 y Fm(\020)1005 5032 y Fx([)p Fw(a)1072 4998 y Fr(\003)1110 5032 y Fx(\()p Fw(f)1183 5044 y Fv(1)1221 5032 y Fx(\))18 b Fp(\012)g Fn(1)h Fx(+)f Fn(1)g Fp(\012)g Fw(a)1697 4998 y Fr(\003)1735 5032 y Fx(\()p Fw(g)1807 5044 y Fv(1)1844 5032 y Fx(\)])h Fp(\012)f Fn(1)g Fp(\012)g Fn(1)992 5215 y Fx(+)p Fn(1)f Fp(\012)h Fn(1)g Fp(\012)h Fx([)p Fw(a)1422 5180 y Fr(\003)1460 5215 y Fx(\()p Fw(f)1533 5227 y Fv(2)1570 5215 y Fx(\))g Fp(\012)f Fn(1)g Fx(+)g Fn(1)g Fp(\012)g Fw(a)2046 5180 y Fr(\003)2084 5215 y Fx(\()p Fw(g)2156 5227 y Fv(2)2193 5215 y Fx(\)])2262 5123 y Fm(\021)2312 5215 y Fw(U)2378 5180 y Fr(\000)p Fv(1)2550 5215 y Fx(:=)82 b Fw(a)2764 5180 y Fr(\003)2803 5215 y Fx(\()p Fw(f)27 b Fp(\010)18 b Fw(g)s Fx(\))p Fw(;)443 b Fx(\(B.1.4\))1938 5476 y(37)p eop %%Page: 38 38 38 37 bop 180 210 a Fx(where,)27 b(for)g Fw(x)d Fx(=)e(\()p Fw(u;)14 b(\033)s Fx(\))24 b Fp(2)f Fw(X)7 b Fx(,)1253 462 y([)p Fw(f)27 b Fp(\010)18 b Fw(g)s Fx(])c(\()p Fw(u;)g(\033)n(;)g (\013)p Fx(\))24 b(:=)1926 320 y Fm(\()1993 406 y Fw(u)14 b(f)2096 418 y Fs(\013)2142 406 y Fx(\()p Fw(u\033)s Fx(\))p Fw(;)147 b(u)23 b Fp(\025)f Fx(0)p Fw(;)1993 525 y(u)14 b(g)2095 537 y Fs(\013)2141 525 y Fx(\()p Fp(\000)p Fw(u\033)s Fx(\))p Fw(;)83 b(u)23 b(<)f Fx(0)p Fw(:)3527 462 y Fx(\(B.1.5\))180 710 y(This)35 b(map)f(is)h(extended)g (to)f(the)h(Hilb)r(ert)g(space)f Fp(H)i Fx(=)e Fp(H)2075 680 y Fs(p)2137 710 y Fp(\012)23 b(F)42 b Fx(in)35 b(the)g(ob)n(vious)f (w)n(a)n(y)-7 b(.)56 b(W)-7 b(e)35 b(k)n(eep)g(the)g(same)180 810 y(notation)27 b(for)g(its)h(extension.)305 909 y(The)23 b(op)r(erators)e Fw(L)891 921 y Fs(r)r Fv(1)971 909 y Fp(\012)10 b Fn(1)1094 921 y Fs(r)r Fv(2)1173 909 y Fx(+)g Fn(1)1296 921 y Fs(r)r Fv(1)1375 909 y Fp(\012)g Fw(L)1507 921 y Fs(r)r Fv(2)1599 909 y Fx(and)24 b Fw(N)1824 921 y Fs(r)r Fv(1)1903 909 y Fp(\012)10 b Fn(1)2026 921 y Fs(r)r Fv(2)2105 909 y Fx(+)g Fn(1)2228 921 y Fs(r)r Fv(1)2308 909 y Fp(\012)g Fw(N)2450 921 y Fs(r)r Fv(2)2542 909 y Fx(are)23 b(mapp)r(ed)g(under)h Fw(U)32 b Fx(to)23 b(the)h(\(total\))180 1009 y(free)j(\014eld)h(Liouvillian)f(and)h(n)n (um)n(b)r(er)f(op)r(erator)f(giv)n(en)h(b)n(y)1219 1224 y Fw(L)1276 1236 y Fs(f)1401 1224 y Fx(=)83 b(d\000\()p Fw(u)p Fx(\))23 b(=)1870 1145 y Fm(X)1908 1320 y Fs(\013)2003 1111 y Fm(Z)2049 1300 y Fs(X)2126 1224 y Fw(a)2170 1190 y Fr(\003)2208 1224 y Fx(\()p Fw(x;)14 b(\013)p Fx(\))p Fw(ua)p Fx(\()p Fw(x;)g(\013)p Fx(\))p Fw(;)1242 1474 y(N)92 b Fx(=)83 b(d\000\(1)-18 b(l\))23 b(=)1868 1395 y Fm(X)1907 1570 y Fs(\013)2002 1361 y Fm(Z)2048 1550 y Fs(X)2125 1474 y Fw(a)2169 1440 y Fr(\003)2207 1474 y Fx(\()p Fw(x;)14 b(\013)p Fx(\))p Fw(a)p Fx(\()p Fw(x;)g(\013)p Fx(\))p Fw(:)180 1722 y Fx(Moreo)n(v)n(er,)25 b(the)j(in)n(teraction)f (tak)n(es)f(the)i(form)1512 1899 y Fw(U)9 b(I)e(U)1687 1865 y Fr(\000)p Fv(1)1799 1899 y Fx(=)23 b Fw(a)1931 1865 y Fr(\003)1969 1899 y Fx(\()p Fw(F)2054 1911 y Fv(1)2092 1899 y Fx(\))18 b(+)g Fw(a)p Fx(\()p Fw(F)2354 1911 y Fv(2)2392 1899 y Fx(\))p Fw(;)1080 b Fx(\(B.1.6\))180 2077 y(where)27 b Fw(F)473 2089 y Fv(1)p Fs(;)p Fv(2)587 2077 y Fp(2)c Fw(L)722 2047 y Fv(2)759 2077 y Fx(\()p Fw(X)i Fp(\002)18 b(f)p Fx(1)p Fw(;)c Fx(2)p Fp(g)p Fw(;)g Fp(B)s Fx(\()p Fp(H)1370 2089 y Fs(p)1424 2077 y Fp(\012)k(H)1577 2089 y Fs(p)1615 2077 y Fx(\)\))29 b(are)d(explicitly)i(giv)n(en)f(b)n (y)g(\()p Fw(x)d Fx(=)f(\()p Fw(u;)14 b(\033)s Fx(\))23 b Fp(2)g Fw(X)30 b Fx(=)22 b Fo(R)j Fp(\002)18 b Fw(S)3433 2047 y Fv(2)3470 2077 y Fx(\))410 2311 y Fw(F)463 2323 y Fv(1)500 2311 y Fx(\()p 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388 y(~)1190 409 y Fw(U)1247 421 y Fs(\022)1312 409 y Fx(is)h(related)e(to)i(the)g(family)g Fw(U)2228 421 y Fs(\022)2293 409 y Fx(in)n(tro)r(duced)f(in)h(Section)f(6)g(as)1709 587 y Fw(U)1766 599 y Fs(\022)1826 587 y Fx(=)c Fw(U)1994 566 y Fx(~)1980 587 y Fw(U)2037 599 y Fs(\022)2073 587 y Fw(U)2139 553 y Fr(\000)p Fv(1)2228 587 y Fw(:)305 764 y Fx(The)k(op)r(erator)832 743 y(~)810 764 y Fw(K)33 b Fx(b)r(ecomes)27 b(after)h(sp)r(ectral)f(deformation)1481 921 y(~)1458 942 y Fw(K)1529 954 y Fs(\022)1590 942 y Fx(:=)1715 921 y(~)1700 942 y Fw(U)1757 954 y Fs(\022)1795 942 y Fw(K)1885 921 y Fx(~)1872 942 y Fw(U)1938 906 y Fr(\000)p Fv(1)1929 967 y Fs(\022)2049 942 y Fx(=)2146 921 y(~)2137 942 y Fw(L)2194 954 y Fv(0)p Fs(;\022)2302 942 y Fx(+)18 b Fw(g)2438 921 y Fx(~)2428 942 y Fw(I)2464 954 y Fs(\022)3527 942 y Fx(\(B.2.3\))180 1119 y(where)1079 1276 y(~)1069 1297 y Fw(L)1126 1309 y Fv(0)p Fs(;\022)1299 1297 y Fx(=)83 b Fw(L)1504 1309 y Fs(p)1560 1297 y Fx(+)18 b(cosh)13 b Fw(\016)31 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Fx(as)1412 2035 y Fw(K)1483 2047 y Fs(\022)1543 2035 y Fx(:=)23 b Fw(U)1720 2001 y Fr(\000)p Fv(1)1831 2014 y Fx(~)1809 2035 y Fw(K)1880 2047 y Fs(\022)1917 2035 y Fw(U)1983 2001 y Fr(\000)p Fv(1)2095 2035 y Fx(=)g Fw(L)2240 2047 y Fv(0)p Fs(;\022)2348 2035 y Fx(+)18 b Fw(g)s(I)2510 2047 y Fs(\022)3527 2035 y Fx(\(B.2.6\))180 2221 y(where)27 b Fw(L)477 2233 y Fv(0)p Fs(;\022)590 2221 y Fx(:=)c Fw(U)767 2191 y Fr(\000)p Fv(1)865 2200 y Fx(~)856 2221 y Fw(L)913 2233 y Fv(0)p Fs(;\022)1002 2221 y Fw(U)37 b Fx(is)27 b(giv)n(en)g(b)n(y)g(\(6.14\))g (and)1733 2407 y Fw(I)1769 2419 y Fs(\022)1830 2407 y Fx(=)c Fw(U)1984 2373 y Fr(\000)p Fv(1)2083 2386 y Fx(~)2073 2407 y Fw(I)2109 2419 y Fs(\022)2147 2407 y Fw(U:)1300 b Fx(\(B.2.7\))180 2681 y Fy(C)135 b(The)44 b Ff(C)779 2638 y Fp(\003)824 2681 y Fy(-algebra)i Fa(A)1435 2699 y Fx(1)180 2863 y Fn(Prop)s(osition)30 b(C.1)41 b Fq(L)l(et)26 b(the)i(algebr)l(as)g Fp(A)1535 2875 y Fv(1)1599 2863 y Fq(and)g Fp(A)f 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y Fs(\014)398 3720 y Fx(is)28 b(b)r(ounded)h(and)f(in)n(v)n (ertible)f(\(on)h Fw(R)c Fx(:=)h Fm(e)-47 b Fw(\015)1741 3732 y Fs(\014)1786 3720 y Fw(L)1843 3690 y Fv(2)1843 3740 y(1)1903 3720 y Fp(\032)24 b Fw(L)2049 3690 y Fv(2)2085 3720 y Fx(\),)29 b(and)f Fm(e)-46 b Fw(\015)2379 3684 y Fr(\000)p Fv(1)2374 3745 y Fs(\014)2496 3720 y Fx(is)28 b(a)f(\(real)h(linear\))f(b)r(ounded)i(map.)38 b(W)-7 b(e)180 3819 y(ha)n(v)n(e)26 b(the)i(equiv)-5 b(alence)1018 3997 y Fw(g)26 b Fp(2)d Fw(R)84 b Fp(,)f Fw(g)s Fx(\()p Fw(u;)14 b(\033)s Fx(\))23 b(=)g Fp(\000)p Fw(e)1932 3963 y Fs(\014)s(u=)p Fv(2)p 2082 3951 43 4 v 2082 3997 a Fw(g)s Fx(\()p Fp(\000)p Fw(u;)14 b(\033)s Fx(\))55 b(for)27 b(a.e.)37 b Fw(u)22 b Fp(2)i Fo(R)1475 4164 y Fx(and)1706 4051 y Fm(Z)1708 4289 y Fl(R)1803 4164 y Fw(du)1908 4051 y Fm(Z)1893 4292 y Fs(S)1937 4276 y Fk(2)2004 4164 y Fw(d\033)s Fp(j)p Fw(g)s Fx(\()p Fw(u;)14 b(\033)s Fx(\))p Fp(j)2385 4129 y Fv(2)2446 4164 y Fw(<)23 b Fp(1)p Fw(:)951 b Fx(\(C.2\))180 4469 y(Let)25 b Fw(R)389 4481 y Fv(0)449 4469 y Fx(:=)e Fp(f)p Fw(g)i Fp(2)f Fw(R)q(;)38 b(e)910 4439 y Fs(bu)978 4414 y Fk(2)1015 4469 y Fw(g)25 b Fp(2)f Fw(L)1216 4439 y Fv(2)1277 4469 y Fx(for)h(some)f Fw(b)e Fp(\025)h Fx(0)p Fp(g)f(\032)h Fw(R)q Fx(.)36 b(The)24 b(set)h Fw(R)2427 4481 y Fv(0)2489 4469 y Fx(is)g(dense)g(in)g Fw(R)g Fx(and)g(\(C.2\))g(implies)g(that)726 4662 y Fw(R)789 4674 y Fv(0)849 4662 y Fx(=)e Fp(f)p Fw(e)1018 4627 y Fs(\014)s(u=)p Fv(4)1168 4662 y Fw(h)p Fp(j)g Fx(:)h Fw(e)1348 4627 y Fs(bu)1416 4602 y Fk(2)1452 4662 y Fw(h)f Fp(2)g Fw(L)1658 4627 y Fv(2)1723 4662 y Fx(for)k(some)g Fw(b)c(>)f Fx(0)28 b(and)f Fw(h)p Fx(\()p Fw(u;)14 b(\033)s Fx(\))23 b(=)g Fp(\000)p 2858 4594 48 4 v Fw(h)o Fx(\()p Fp(\000)p Fw(u;)14 b(\033)s Fx(\))p Fp(g)p Fw(:)357 b Fx(\(C.3\))180 4839 y(Giv)n(en)24 b Fw(g)i Fx(=)c Fw(e)609 4809 y Fs(\014)s(u=)p Fv(4)760 4839 y Fw(h)h Fp(2)h Fw(R)973 4851 y Fv(0)1010 4839 y Fx(,)h(de\014ne)g Fw(h)1343 4851 y Fs(\017)1397 4839 y Fx(:=)e Fw(G)1573 4851 y Fs(\017)1617 4839 y Fp(\003)12 b Fw(h)p Fx(,)25 b(the)f(con)n(v)n(olution)f(in)i (the)g(v)-5 b(ariable)23 b Fw(u)h Fx(of)g Fw(h)g Fx(with)h(the)g (Gaussian)180 4939 y Fw(G)245 4951 y Fs(\017)277 4939 y Fx(\()p Fw(u)p Fx(\))39 b(:=)f Fw(\017)588 4909 y Fr(\000)p Fv(1)677 4939 y Fw(G)p Fx(\()p Fw(u=\017)p Fx(\),)h(where)d Fw(G)p Fx(\()p Fw(u)p Fx(\))j(=)f Fw(\031)1610 4909 y Fr(\000)p Fv(1)p Fs(=)p Fv(2)1766 4939 y Fw(e)1805 4909 y Fr(\000)p Fs(u)1896 4884 y Fk(2)1970 4939 y Fx(and)e Fw(\017)i(>)h Fx(0.)64 b Fw(h)2493 4951 y Fs(\017)2561 4939 y Fx(is)37 b(con)n(tin)n(uous)f(\(actually)g(analytic\),)180 5050 y(satis\014es)27 b Fw(e)525 5020 y Fs(bu)593 4995 y Fk(2)629 5050 y Fw(h)677 5062 y Fs(\017)732 5050 y Fp(2)d Fw(L)868 5020 y Fv(2)904 5050 y Fx(,)k(and)f(since)h Fw(G)1385 5062 y Fs(\017)1417 5050 y Fx(\()p Fp(\001)p Fx(\))g(is)g(real)e(v)-5 b(alued)28 b(and)f(o)r(dd:)1569 5227 y Fw(h)1617 5239 y Fs(\017)1649 5227 y Fx(\()p Fw(u;)14 b(\033)s Fx(\))24 b(=)e Fp(\000)p 2024 5160 V Fw(h)2072 5239 y Fs(\017)2103 5227 y Fx(\()p Fp(\000)p Fw(u;)14 b(\033)s Fx(\))p Fw(:)1938 5476 y Fx(39)p eop %%Page: 40 40 40 39 bop 180 210 a Fx(Therefore,)23 b Fw(g)616 222 y Fs(\017)670 210 y Fx(:=)g Fw(e)820 180 y Fs(\014)s(u=)p Fv(4)971 210 y Fw(h)1019 222 y Fs(\017)1074 210 y Fp(2)g Fw(R)1215 222 y Fv(0)1252 210 y Fx(.)36 b(Since)23 b Fw(h)1571 222 y Fs(\017)1626 210 y Fp(!)g Fw(h)g Fx(in)h Fw(L)1953 180 y Fv(2)1989 210 y Fx(,)h(w)n(e)d(conclude)h(that)h Fw(g)2707 222 y Fs(\017)2761 210 y Fp(!)f Fw(g)j Fx(in)e Fw(L)3083 180 y Fv(2)3143 210 y Fx(as)e Fw(\017)h Fp(!)g Fx(0.)35 b(Clearly)-7 b(,)180 310 y Fw(g)220 322 y Fs(\017)279 310 y Fx(extends)28 b(to)f(an)g(en)n(tire)h(function)g Fw(z)e Fp(7!)d Fw(g)1568 322 y Fs(\017)1600 310 y Fx(\()p Fw(z)t(;)14 b(\033)s Fx(\).)37 b(De\014ne)28 b(the)g(set)718 492 y Fw(R)782 458 y Fv(anal)931 492 y Fx(:=)22 b Fp(f)p Fw(e)1122 458 y Fs(\014)s(u=)p Fv(4)1273 492 y Fw(h)1321 504 y Fs(\017)1352 492 y Fp(j)28 b Fw(h)f Fx(satis\014es)g(the)h (conditions)f(on)h(r.h.s.)36 b(of)28 b(\(C.3\),)g Fw(\017)22 b(>)h Fx(0)p Fp(g)p Fw(:)180 675 y(R)244 645 y Fv(anal)399 675 y Fx(is)28 b(a)h(subset)g(of)g Fw(R)h Fx(that)f(is)g(dense)f(in)i Fw(R)q Fx(.)40 b(Since)30 b Fm(e)-46 b Fw(\015)1985 640 y Fr(\000)p Fv(1)1980 700 y Fs(\014)2102 675 y Fx(is)29 b(b)r(ounded,)h(then)f Fw(D)2812 645 y Fv(anal)2964 675 y Fx(:=)c Fm(e)-46 b Fw(\015)3125 640 y Fr(\000)p Fv(1)3120 700 y Fs(\014)3213 675 y Fx(\()p Fw(R)3309 645 y Fv(anal)3435 675 y Fx(\))29 b(is)g(dense)180 790 y(in)f Fw(L)334 760 y Fv(2)334 811 y(1)371 790 y Fx(.)37 b(Since)27 b Fw(L)704 760 y Fv(2)704 811 y(1)769 790 y Fx(is)g(dense)h(in)g Fw(L)1233 760 y Fv(2)1269 790 y Fx(,)g(w)n(e)f(conclude)h(that)f Fw(D)2034 760 y Fv(anal)2188 790 y Fx(is)g(also)g(dense)g(in)h Fw(L)2818 760 y Fv(2)2855 790 y Fx(.)305 890 y(De\014ne)h Fp(A)629 902 y Fv(1)696 890 y Fx(as)g(in)g(\(2.17\))g(with)g Fp(D)1396 860 y Fv(anal)1552 890 y Fx(giv)n(en)f(ab)r(o)n(v)n(e.)40 b(Since)30 b Fp(D)2327 860 y Fv(anal)2482 890 y Fx(is)f(dense)g(in)h Fw(L)2951 860 y Fv(2)2988 890 y Fx(,)f Fp(A)3106 902 y Fv(1)3173 890 y Fx(is)g(strongly)f(dense)180 990 y(in)g Fp(A)g Fx(\(de\014ned)g(b)n(y)f(\(2.7\)\).)37 b(Next)28 b(w)n(e)f(ha)n(v)n(e)g(for)g(real)g Fw(\022)1386 1172 y(U)1443 1184 y Fs(\022)1480 1172 y Fw(\031)1530 1105 y Fm(\000)1568 1172 y Fw(A)1630 1184 y Fs(p)1687 1172 y Fp(\012)19 b Fw(W)12 b Fx(\()p Fw(f)1934 1184 y Fv(1)1971 1172 y Fx(\))18 b Fp(\012)g Fw(W)12 b Fx(\()p Fw(f)2267 1184 y Fv(2)2304 1172 y Fx(\))2336 1105 y Fm(\001)2375 1172 y Fx(\012)1238 1305 y(=)83 b Fw(\031)1433 1317 y Fs(p)1471 1305 y Fx(\()p Fw(A)1565 1317 y Fs(p)1604 1305 y Fx(\))19 b Fp(\012)f Fw(W)1828 1238 y Fm(\000)1866 1305 y Fx(\()q Fm(e)-47 b Fw(\015)1941 1317 y Fs(\014)1986 1305 y Fw(f)2027 1317 y Fv(1)2064 1305 y Fx(\))2096 1317 y Fs(\022)2134 1238 y Fm(\001)2190 1305 y Fp(\012)18 b Fw(W)2363 1238 y Fm(\000)2401 1305 y Fx(\()q Fm(e)-47 b Fw(\015)2476 1317 y Fs(\014)2522 1305 y Fw(f)2563 1317 y Fv(2)2599 1305 y Fx(\))2631 1317 y Fs(\022)2669 1238 y Fm(\001)2707 1305 y Fx(\012)2767 1317 y Fs(\022)3591 1305 y Fx(\(C.4\))180 1488 y(where)31 b(the)h(map)g Fw(g)g Fp(!)e Fw(g)985 1500 y Fs(\022)1053 1488 y Fx(is)i(de\014ned)g(b)n(y)f Fw(g)1590 1500 y Fs(\022)1627 1488 y Fx(\()p Fw(u;)14 b(\033)s Fx(\))30 b(:=)g Fw(g)s Fx(\()p Fw(j)2083 1500 y Fs(\022)2120 1488 y Fx(\()p Fw(u)p Fx(\))p Fw(;)14 b(\033)s Fx(\))33 b(with)f(the)g(function)g Fw(j)3087 1500 y Fs(\022)3156 1488 y Fx(de\014ned)g(in)g(\(A2.2\))180 1587 y(and)i(where)f(w)n(e)h(understand)f(the)i(W)-7 b(eyl)34 b(op)r(erators)e(on)h(the)i(r.h.s.)55 b(as)33 b(acting)h(on)f(the)h(\(Jak)l(\024)-37 b(si)n(\023)e(c-Pillet)31 b(glued\))180 1687 y(GNS)40 b(space)e(and)h(\012)859 1699 y Fs(\022)935 1687 y Fx(giv)n(en)f(in)i(the)f(same)f(represen)n (tation.)69 b(Using)39 b(that)g Fw(\022)45 b Fp(!)d Fx(\()q Fm(e)-47 b Fw(\015)2974 1699 y Fs(\014)3019 1687 y Fw(f)3060 1699 y Fs(j)3094 1687 y Fx(\))3126 1699 y Fs(\022)3203 1687 y Fx(are)38 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(b)r(ourho)r(o)r(d)1066 2140 y Fm(\010)1115 2207 y Fw(\022)d Fp(2)f Fo(C)1312 2173 y Fv(2)1355 2136 y Fm(\014)1355 2186 y(\014)1383 2207 y Fp(j)14 b Fx(tan)o(\(Im)h Fw(\016)s Fx(\))p Fp(j)23 b Fw(<)1972 2151 y(b)p 1901 2188 179 4 v 1901 2264 a Fx(1)18 b(+)g Fw(b)2172 2207 y Fx(and)83 b Fp(j)p Fx(Im)14 b Fw(\034)9 b Fp(j)24 b Fw(<)f(\014)2756 2173 y Fr(\000)p Fv(1)2845 2140 y Fm(\011)180 2420 y Fx(of)28 b Fo(R)p Fx(.)p 3714 2432 59 71 v 180 2695 a Fy(References)222 2877 y Fx([1])41 b(W.)30 b(K.)g(Ab)r(ou)h(Salem.)44 b(On)30 b(the)g(quasi-static)f(ev)n(olution)g(of)h(non-equilibrium)g (steady)f(states.)44 b(Preprin)n(t,)351 2976 y(mp-arc)26 b(05-341)222 3142 y([2])41 b(W.)29 b(K.)f(Ab)r(ou)g(Salem)h(and)f(J.)g (F)-7 b(r\177)-42 b(ohlic)n(h.)39 b(Cyclic)28 b(thermo)r(dynamic)g(pro) r(cesses)f(and)h(en)n(trop)n(y)f(pro)r(duction.)351 3242 y(Preprin)n(t,)f(mp-arc)h(05-436)222 3408 y([3])41 b(H.)27 b(Araki)g(and)g(E.J.)g(W)-7 b(o)r(o)r(ds.)36 b(Represen)n(tation)26 b(of)h(the)h(canonical)e(comm)n(utation)h(relations)f(describing)g(a) 351 3508 y(nonrelativistic)g(in\014nite)j(free)e(b)r(ose)g(gas.)36 b Fq(J.)30 b(Math.)h(Phys.)p Fx(,)e(4:637{662,)24 b(1963.)222 3674 y([4])41 b(W.)33 b(Asc)n(h)n(bac)n(her)d(and)j(C.-A.)f(Pillet.)52 b(Non-equilibrium)32 b(steady)g(states)g(of)h(the)g Fw(X)7 b(Y)50 b Fx(c)n(hain.)h(J.)33 b(Statist.)351 3773 y(Ph)n(ys.)26 b(112)g(\(2003\),)h(no.)g(5-6,)g(1153{1175.)222 3939 y([5])41 b(V.)35 b(Bac)n(h,)h(T.)g(Chen,)h(J.)e(F)-7 b(r\177)-42 b(ohlic)n(h,)36 b(and)f(I.M.)g(Sigal.)59 b(Smo)r(oth)35 b(F)-7 b(esh)n(bac)n(h)34 b(map)h(and)g(op)r(erator)f (theoric)351 4039 y(renormalization)25 b(group)i(metho)r(d.)37 b Fq(J.)30 b(F)-6 b(unc.)29 b(A)n(nalysis)f Fx(203,)f(2003.)222 4205 y([6])41 b(V.)c(Bac)n(h,)h(J.)f(F)-7 b(r\177)-42 b(ohlic)n(h,)38 b(and)f(I.M.)g(Sigal.)64 b(Quan)n(tum)36 b(electro)r(dynamics)g(of)h(con\014ned)f(non-relativistic)351 4305 y(particles.)g Fq(A)l(dv.)30 b(in)g(Math.)p Fx(,)f(137:299{395,)23 b(1998.)222 4471 y([7])41 b(V.)32 b(Bac)n(h,)g(J.)g(F)-7 b(r\177)-42 b(ohlic)n(h,)32 b(and)g(I.M.)g(Sigal.)49 b(Renormalization)30 b(group)h(analysis)g(of)g(sp)r(ectral)h(problems)f (in)351 4570 y(quan)n(tum)c(\014eld)h(theory)-7 b(.)36 b Fq(A)l(dv.)30 b(in)g(Math.)p Fx(,)f(137:205{298,)23 b(1998.)222 4736 y([8])41 b(V.)34 b(Bac)n(h,)i(J.)e(F)-7 b(r\177)-42 b(ohlic)n(h,)36 b(and)e(I.M.)h(Sigal.)56 b(Return)35 b(to)f(equilibrium.)57 b Fq(J.)36 b(Math.)i(Phys.)p Fx(,)f(41:3985{4060,)351 4836 y(2000.)222 5002 y([9])k(O.)19 b(Bratteli)g(and)h(D.W.)h(Robinson.)i Fq(Op)l(er)l(ator)g(A)n(lgebr)l (as)g(and)g(Quantum)e(Statistic)l(al)i(Me)l(chanics)p Fx(,)g(v)n(olumes)351 5102 y(1)k(and)g(2.)37 b(Springer)26 b(V)-7 b(erlag,)27 b(New)h(Y)-7 b(ork,)27 b(Heidelb)r(erg,)g(Berlin,)g (1979.)1938 5476 y(40)p eop %%Page: 41 41 41 40 bop 180 210 a Fx([10])41 b(J.)27 b(Derezi)r(\023)-44 b(nski)27 b(and)g(V.)h(Jak)l(\024)-37 b(si)n(\023)e(c.)33 b(Sp)r(ectral)28 b(theory)e(of)i(P)n(auli-Fierz)d(op)r(erators.)35 b Fq(J.)30 b(F)-6 b(unc.)29 b(A)n(nal.)p Fx(,)f(180:243,)351 310 y(2001.)180 476 y([11])41 b(J.)31 b(Derezi)r(\023)-44 b(nski,)33 b(V.)f(Jak)l(\024)-37 b(si)n(\023)e(c,)30 b(and)h(C.-A.)h(Pillet.)50 b(P)n(erturbation)30 b(theory)h(of)h Fw(W)2886 446 y Fr(\003)2924 476 y Fx(-dynamics,)g(Liouvilleans)351 575 y(and)27 b(KMS-states.)36 b(Rev.)27 b(Math.)h(Ph)n(ys.)f(15)g (\(2003\),)f(no.)h(5,)h(447{489)180 742 y([12])41 b(S.)f(Dirren.)72 b(Dobrushin's)40 b(Uniqueness)g(Theorem,)i(High)d(T)-7 b(emp)r(erature)40 b(Expansions.)71 b Fq(ETH)41 b(Z)q(\177)-43 b(urich)351 841 y(Diploma)31 b(Thesis,)g(written)f(under)f(sup)l (ervision)i(of)g(J.)e(F)-6 b(r\177)-42 b(ohlich)32 b(and)e(G.M.)i(Gr)l (af)p Fx(,)c(1999.)180 1007 y([13])41 b(J.)25 b(F)-7 b(r\177)-42 b(ohlic)n(h)25 b(and)h(M.)g(Merkli.)33 b(Thermal)25 b(ionization.)33 b(Mathematical)25 b(Ph)n(ysics,)g(Analysis)g(and)h (Geometry)351 1107 y(7,)h(no.)g(3,)h(239-287)c(\(2004\).)180 1273 y([14])41 b(J.)23 b(F)-7 b(r\177)-42 b(ohlic)n(h)23 b(and)g(M.)h(Merkli.)29 b(Another)23 b(Return)h(of)f(\\Return)h(to)f (Equilibrium".)29 b(Comm)n(un.)23 b(Math.)h(Ph)n(ys.)351 1373 y Fn(251)p Fx(,)j(235{262)d(\(2004\))180 1539 y([15])41 b(J.)24 b(F)-7 b(r\177)-42 b(ohlic)n(h,)24 b(M.)h(Merkli,)f(and)h(D.)g (Ueltsc)n(hi.)31 b(Dissipativ)n(e)24 b(transp)r(ort:)34 b(Thermal)24 b(con)n(tacts)f(and)i(tunneling)351 1638 y(junctions.)37 b(Annales)27 b(Henri)h(P)n(oincar)n(\023)-39 b(e)24 b(4,)j(no.5,)g(897{945)e(\(2004\))180 1804 y([16])41 b(R.)28 b(Haag.)35 b(Lo)r(cal)27 b(Quan)n(tum)g(Ph)n(ysics.)36 b(T)-7 b(exts)27 b(and)h(Monographs)d(in)j(Ph)n(ysics,)e(Springer)h(V) -7 b(erlag,)26 b(1992)180 1970 y([17])41 b(R.)35 b(Haag,)h(N.)g (Hugenholz,)g(and)f(M.)h(Winnink.)60 b(On)35 b(the)g(equilibrium)g (states)g(in)g(quan)n(tum)h(statistical)351 2070 y(mec)n(hanics.)g Fq(Comm.)30 b(Math.)i(Phys.)p Fx(,)d(5:215{236,)24 b(1967.)180 2236 y([18])41 b(V.)20 b(Jak)l(\024)-37 b(si)n(\023)e(c)17 b(and)i(C.-A.)h(Pillet.)25 b(On)19 b(a)h(mo)r(del)g(for)f(quan)n(tum)h (friction)g(I)r(I.)g(F)-7 b(ermi's)20 b(golden)f(rule)h(and)g(dynamics) 351 2336 y(at)27 b(p)r(ositiv)n(e)g(temp)r(erature.)37 b Fq(Comm.)31 b(Math.)g(Phys.)p Fx(,)e(176:619{644,)23 b(1996.)180 2502 y([19])41 b(V.)31 b(Jak)l(\024)-37 b(si)n(\023)e(c)27 b(and)k(C.-A.)f(Pillet.)46 b(On)31 b(en)n(trop)n(y)e(pro)r(duction)h (in)h(quan)n(tum)g(statistical)f(mec)n(hanics.)45 b Fq(Comm.)351 2601 y(Math.)31 b(Phys.)p Fx(,)e(217:285{293,)23 b(2001.)180 2767 y([20])41 b(V.)34 b(Jak)l(\024)-37 b(si)n(\023)e(c)30 b(and)k(C.-A.)g(Pillet.)54 b(Mathematical)34 b(theory)f(of)g (non-equilibrium)g(quan)n(tum)h(statistical)f(me-)351 2867 y(c)n(hanics.)j Fq(J.)29 b(Stat.)h(Phys.)p Fx(,)f(108:787{829,)23 b(2002.)180 3033 y([21])41 b(V.)25 b(Jak)l(\024)-37 b(si)n(\023)e(c)22 b(and)k(C.-A.)f(Pillet.)33 b(Non-equilibrium)25 b(steady)g(states)g(of) g(\014nite)h(quan)n(tum)f(systems)g(coupled)g(to)351 3133 y(thermal)i(reserv)n(oirs.)34 b Fq(Comm.)d(Math.)g(Phys.)p Fx(,)e(226:131{162,)23 b(2002.)180 3299 y([22])41 b(M.)21 b(Merkli.)27 b(P)n(ositiv)n(e)19 b(comm)n(utators)h(in)i (non-equilibrium)f(statistical)g(mec)n(hanics.)26 b Fq(Comm.)f(Math.)g (Phys.)p Fx(,)351 3398 y(223:327-362,)e(2001)180 3564 y([23])41 b(M.)23 b(Merkli.)53 b(Stabilit)n(y)23 b(of)g(Equilibria)f (with)i(a)f(Condensate.)29 b Fq(Comm.)e(Math.)g(Phys.)p Fx(,)e(257,)e(no.)g(3:)34 b(621{640)351 3664 y(\(2005\))180 3830 y([24])41 b(M.)27 b(Merkli.)65 b(Lev)n(el)26 b(shift)j(op)r (erators)c(for)i(op)r(en)h(quan)n(tum)g(systems.)36 b(mp-arc)26 b(preprin)n(t)h(no.)h(06-5,)e(2006)180 3996 y([25])41 b(M.)27 b(Merkli,)g(M.)g(M)r(\177)-44 b(uc)n(k,)27 b(I.M.)g(Sigal.)35 b(Instabilit)n(y)27 b(of)g(Equilibrium)g(States)g(for)f(Coupled)h(Heat) g(Reserv)n(oirs)351 4096 y(at)g(Di\013eren)n(t)h(T)-7 b(emp)r(eratures,)36 b(mp-arc)27 b(preprin)n(t)g(no.)g(05-239,)e(2005) 180 4262 y([26])41 b(M.)24 b(Merkli,)g(M.)f(M)r(\177)-44 b(uc)n(k,)25 b(I.M.)f(Sigal.)29 b(Renormalization)22 b(group)h(approac)n(h)e(in)j(non-equilibrium)g(statistical)351 4361 y(quan)n(tum)j(mec)n(hanics.)36 b Fq(in)30 b(pr)l(ep)l(ar)l(ation) p Fx(.)180 4527 y([27])41 b(M.)31 b(M)r(\177)-44 b(uc)n(k.)48 b(Thermal)30 b(Relaxation)h(for)f(P)n(article)g(Systems)h(in)h(In)n (teraction)e(with)i(Sev)n(eral)e(Bosonic)g(Heat)351 4627 y(Reserv)n(oirs.)44 b(Ph.D.)31 b(Dissertation,)g(Departmen)n(t)g(of)g (Mathematics,)h(Johannes-Guten)n(b)r(erg)d(Univ)n(ersit)n(y)-7 b(,)351 4727 y(Mainz,)27 b(July)g(2004,)f(ISBN)i(3-8334-1866-4)180 4893 y([28])41 b(M.)21 b(Reed,)i(B.)f(Simon.)f Fq(Metho)l(ds)k(of)g(Mo) l(dern)g(Mathematic)l(al)h(Physics)g(I,)f(F)-6 b(unctional)24 b(A)n(nalysis)p Fx(,)f(Academic)351 4992 y(Press,)j(1980)180 5158 y([29])41 b(D.)28 b(Ruelle.)37 b Fq(Statistic)l(al)30 b(Me)l(chanics.)i(R)n(igor)l(ous)e(R)l(esults)p Fx(.)35 b(Benjamin,)28 b(New)g(Y)-7 b(ork,)27 b(1969.)1938 5476 y(41)p eop %%Page: 42 42 42 41 bop 180 210 a Fx([30])41 b(D.)35 b(Ruelle.)59 b(Natural)35 b(non-equilibrium)f(states)h(in)g(quan)n(tum)g(statistical)g(mec)n (hanics.)58 b Fq(J.)36 b(Stat.)g(Phys.)p Fx(,)351 310 y(98:57,)26 b(2000.)180 476 y([31])41 b(D.)28 b(Ruelle.)37 b(En)n(trop)n(y)26 b(pro)r(duction)h(in)h(quan)n(tum)f(spin)h(systems.) 36 b Fq(Comm.)31 b(Math.)g(Phys.)p Fx(,)f(224:3,)25 b(2001.)180 642 y([32])41 b(D.)19 b(Ruelle.)j(Ho)n(w)c(should)g(one)g(de\014ne)g (en)n(trop)n(y)g(pro)r(duction)g(for)g(non-equilibrium)f(quan)n(tum)i (spin)f(systems?)351 742 y Fq(Pr)l(eprint)p Fx(,)28 b(2001.)180 908 y([33])41 b(D.)28 b(Ruelle.)37 b(T)-7 b(opics)27 b(in)h(quan)n(tum)f(statistical)g(mec)n(hanics)g(and)h(op)r(erator)e (algebras.)34 b Fq(Pr)l(eprint)p Fx(,)28 b(2001.)180 1074 y([34])41 b(H.)32 b(Sp)r(ohn,)h(J.)f(L.)g(Leb)r(o)n(witz.)49 b(Irrev)n(ersible)29 b(thermo)r(dynamics)i(for)h(quan)n(tum)f(systems)h (w)n(eakly)e(coupled)351 1173 y(to)d(thermal)g(reserv)n(oirs.)34 b Fq(A)l(dvanc)l(es)c(in)g(Chemic)l(al)i(Physics)d Fx(38,)e(109-142,)d (1978.)1938 5476 y(42)p eop 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