Content-Type: multipart/mixed; boundary="-------------0107090905164" This is a multi-part message in MIME format. ---------------0107090905164 Content-Type: text/plain; name="01-252.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-252.keywords" Non-equilibrium quantum statistical mechanics, steady-states, Liouvillean, entropy production, spectral theory ---------------0107090905164 Content-Type: application/postscript; name="ness_us.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ness_us.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ness.dvi %%Pages: 36 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Times-Roman Times-Italic Times-Bold %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -t letter -o ness_us.ps ness %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.07.09:1532 %%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: 8r.enc % @@psencodingfile@{ % author = "S. 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3400 y(Ha].)43 b(F)o(or)28 b(the)h(reader)h(con)l(v)o (enience)i(and)d(notational)j(purposes,)g(in)d(this)g(section)h(we)e (re)n(vie)n(w)h(some)244 3513 y(basic)24 b(notions)i(of)d(this)h(frame) n(w)o(ork.)380 3626 y(In)39 b(the)g(algebraic)j(formalism)d(a)g (physical)i(system)e(is)g(described)j(either)e(by)f(a)f Fu(C)3146 3593 y Ft(\003)3185 3626 y Fy(-)h(or)f Fu(W)3466 3593 y Ft(\003)3505 3626 y Fy(-)244 3738 y(dynamical)26 b(system.)31 b(The)24 b(adv)n(antage)i(of)e(Fermi)g(reserv)n(oirs)j(is) d(that)g(we)f(can)i(deal)g(with)f Fu(C)3179 3705 y Ft(\003)3217 3738 y Fy(-systems)244 3851 y(which)e(are)h(conceptually)i(simpler)-5 b(.)29 b(A)21 b Fu(C)1582 3818 y Ft(\003)1621 3851 y Fy(-dynamical)j(system)f(is)f(a)g(pair)g Fx(\()p Fw(O)s Fu(;)15 b(\034)10 b Fx(\))p Fy(,)23 b(where)f Fw(O)i Fy(is)e(a)f Fu(C)3466 3818 y Ft(\003)3505 3851 y Fy(-)244 3964 y(algebra)26 b(with)f(identity)h(and)f Fu(\034)34 b Fy(is)24 b(a)g(strongly)j(continuous)h(group)e(of)e(automorphisms)k (of)c Fw(O)j Fy(\(that)e(is,)244 4077 y(the)f(map)f Fp(R)35 b Fw(3)25 b Fu(t)g Fw(7!)h Fu(\034)960 4044 y Fq(t)989 4077 y Fx(\()p Fu(A)p Fx(\))e Fy(is)g(norm)f(continuous)k(for)d(each)h Fu(A)g Fw(2)g(O)s Fy(\).)k(The)23 b(elements)i(of)f Fw(O)i Fy(describe)244 4190 y(observ)n(ables)f(of)e(the)f(physical)i(system)f (under)h(consideration)i(and)d(the)g(group)h Fu(\034)31 b Fy(speci\002es)24 b(their)f(time)244 4303 y(e)n(v)n(olution.)52 b(A)29 b(physical)j(state)f(is)f(described)j(by)d(a)g(mathematical)j (state)e(on)f Fw(O)s Fy(,)h(that)g(is,)g(a)f(positi)n(v)o(e)244 4416 y(linear)i(functional)i Fu(!)f Fy(such)f(that)g Fu(!)s Fx(\()p Fo(1)p Fx(\))40 b(=)f(1)p Fy(.)51 b(The)30 b(set)i Fu(E)5 b Fx(\()p Fw(O)s Fx(\))31 b Fy(of)g(all)g(states)h(is)f (a)g(con)l(v)o(e)o(x,)j(weak-)p Fw(\003)244 4529 y Fy(compact)28 b(subset)f(of)g(the)g(dual)g Fw(O)1326 4496 y Ft(\003)1365 4529 y Fy(.)37 b(A)25 b(state)j Fu(!)g Fy(is)e(called)i(f)o(aithful)h (if)d Fu(!)s Fx(\()p Fu(A)2660 4496 y Ft(\003)2700 4529 y Fu(A)p 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b Fw(j)15 b Fx(0)36 b Fu(<)e Fx(sign)o(\()p Fu(\014)5 b Fx(\)Im)q Fu(z)39 b(<)34 b Fw(j)p Fu(\014)5 b Fw(jg)p Fy(,)30 b(bounded)h(and)e(continuous)j(on)d (its)g(closure,)i(and)e(satisfying)p eop %%Page: 3 3 3 2 bop 3730 100 a FA(3)244 407 y Fy(the)24 b(KMS)d(boundary)27 b(conditions)831 611 y Fu(F)889 625 y Fq(A;B)1022 611 y Fx(\()p Fu(t)p Fx(\))f(=)f Fu(!)s Fx(\()p Fu(A\034)1460 574 y Fq(t)1490 611 y Fx(\()p Fu(B)5 b Fx(\)\))p Fu(;)198 b(F)1950 625 y Fq(A;B)2084 611 y Fx(\()p Fu(t)20 b Fx(+)g(i)p Fu(\014)5 b Fx(\))25 b(=)g Fu(!)s Fx(\()p Fu(\034)2646 574 y Fq(t)2676 611 y Fx(\()p Fu(B)5 b Fx(\))p Fu(A)p Fx(\))p Fu(:)244 815 y Fy(A)22 b Fx(\()p Fu(\034)5 b(;)15 b(\014)5 b Fx(\))p Fy(-KMS)24 b(state)g(is)f(f)o(aithful)j(and)e Fu(\034)10 b Fy(-in)l(v)n(ariant.)380 928 y(Let)28 b Fx(\()p Fw(O)s Fu(;)15 b(\034)10 b Fx(\))28 b Fy(be)g(a)f Fu(C)1044 895 y Ft(\003)1083 928 y Fy(-dynamical)j(system)f(and)g(let)f Fu(\016)j Fy(be)d(the)g(generator)i(of)e Fu(\034)37 b Fy(\()p Fu(\034)3028 895 y Fq(t)3092 928 y Fx(=)c(e)3236 895 y Fq(t\016)3300 928 y Fy(\).)41 b(The)244 1041 y(operator)26 b Fu(\016)g Fy(is)d(a)h Fw(\003)p Fy(-deri)n(v)n(ation:)32 b(Its)23 b(domain)i Fw(D)s Fx(\()p Fu(\016)s Fx(\))f Fy(is)f(a)g Fw(\003)p Fy(-subalgebra)28 b(of)23 b Fw(O)j Fy(and)e(for)f Fu(A;)15 b(B)31 b Fw(2)25 b(D)s Fx(\()p Fu(\016)s Fx(\))p Fy(,)983 1245 y Fu(\016)s Fx(\()p Fu(A)p Fx(\))1164 1208 y Ft(\003)1230 1245 y Fx(=)g Fu(\016)s Fx(\()p Fu(A)1472 1208 y Ft(\003)1513 1245 y Fx(\))p Fu(;)197 b(\016)s Fx(\()p Fu(AB)5 b Fx(\))27 b(=)e Fu(\016)s Fx(\()p Fu(A)p Fx(\))p Fu(B)h Fx(+)20 b Fu(A\016)s Fx(\()p Fu(B)5 b Fx(\))p Fu(:)244 1450 y Fy(Let)24 b Fu(V)49 b Fx(=)28 b Fu(V)663 1417 y Ft(\003)731 1450 y Fw(2)f(O)h Fy(be)d(a)f(perturbation)29 b(\(such)d(perturbations)j(are)d(called)g (local\).)35 b(The)24 b(generator)k(of)244 1562 y(the)c(perturbed)j (dynamics)f(is)f Fu(\016)1247 1576 y Fq(V)1308 1562 y Fx(\()p Fu(A)p Fx(\))i(=)g Fu(\016)s Fx(\()p Fu(A)p Fx(\))22 b(+)f(i[)p Fu(V)5 b(;)15 b(A)p Fx(])p Fy(.)31 b(The)24 b(operator)i Fu(\016)2693 1576 y Fq(V)2778 1562 y Fy(is)e(also)h(a)f Fw(\003)p Fy(-deri)n(v)n(ation)244 1675 y(and)g Fw(D)s Fx(\()p Fu(\016)546 1689 y Fq(V)607 1675 y Fx(\))i(=)f Fw(D)s Fx(\()p Fu(\016)s Fx(\))p Fy(.)k(The)23 b(perturbed)j(dynamics)f (is)f(described)i(by)321 1881 y Fu(\034)371 1844 y Fq(t)361 1904 y(V)422 1881 y Fx(\()p Fu(A)p Fx(\))g(:=)f(e)747 1844 y Fq(t\016)803 1855 y Fn(V)862 1881 y Fx(\()p Fu(A)p Fx(\))586 2158 y(=)g Fu(\034)732 2120 y Fq(t)761 2158 y Fx(\()p Fu(A)p Fx(\))d(+)1012 2071 y Fm(X)1011 2267 y Fq(n)p Ft(\025)p Fv(1)1160 2158 y Fx(i)1186 2120 y Fq(n)1247 2034 y Fm(Z)1338 2060 y Fq(t)1297 2240 y Fv(0)1383 2158 y Fx(d)p Fu(t)1467 2172 y Fv(1)1521 2034 y Fm(Z)1611 2060 y Fq(t)1636 2069 y Fl(1)1571 2240 y Fv(0)1691 2158 y Fx(d)p Fu(t)1775 2172 y Fv(2)1829 2158 y Fw(\001)15 b(\001)g(\001)1950 2034 y Fm(Z)2041 2060 y Fq(t)2066 2069 y Fn(n)p Fk(\000)p Fl(1)2000 2240 y Fv(0)2207 2158 y Fx(d)p Fu(t)2291 2172 y Fq(n)2337 2158 y Fx([)p 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3920 y Fy(whose)24 b(elements)h(are)f Fu(\034)1019 3934 y Fq(V)1079 3920 y Fy(-in)l(v)n(ariant.)380 4033 y(One)32 b(of)g(the)h(k)o(e)o(y)f(concept)i(of)e(non-equilibrium)k (thermodynamics)g(is)c(the)g(notion)i(of)e(entrop)o(y)244 4146 y(production.)49 b(W)l(ithin)30 b(the)g(frame)n(w)o(ork)g(of)f (algebraic)j(quantum)e(statistical)i(mechanics)f(this)f(notion)244 4259 y(has)f(been)h(precisely)h(de\002ned)e(in)g(the)g(recent)h(w)o (orks)f([Ru2,)h(JP3],)g(see)f(also)g([Sp1)q(,)f(O1,)f(O2,)h(OHI])g(.) 244 4372 y(W)-7 b(e)23 b(recall)h(the)g(de\002nitions)i(and)e(the)f (results)i(we)e(will)g(need.)380 4485 y(F)o(or)g(positi)n(v)o(e)i (linear)g(functionals)i Fu(\021)s(;)15 b(\030)30 b Fw(2)25 b(O)1811 4452 y Ft(\003)1850 4485 y Fy(,)e(let)g Fx(En)m(t)q(\()p Fu(\021)c Fw(j)c Fu(\030)t Fx(\))23 b Fy(be)h(the)g(relati)n(v)o(e)g (entrop)o(y)i(of)e(Araki)244 4598 y(\(we)d(use)g(the)h(ordering)h(and)f (the)g(sign)g(con)l(v)o(ention)i(of)d(Bratelli-Robinson)k([BR2,)c (Don]\).)28 b(F)o(or)20 b(de\002ni-)244 4711 y(tion)25 b(and)g(properties)i(of)d(Araki')-5 b(s)25 b(relati)n(v)o(e)h(entrop)o (y)g(we)d(refer)i(the)g(reader)h(to)e([Ar1)q(,)f(Ar2,)g(BR2,)g(Don)q(,) 244 4823 y(OP)o(].)380 4936 y(W)-7 b(e)23 b(mak)o(e)h(the)f(follo)n (wing)i(assumption.)244 5149 y Fz(E\(1\))e Fy(There)h(e)o(xists)g(a)g Fu(C)1032 5116 y Ft(\003)1070 5149 y Fy(-dynamics)i Fu(\033)1524 5163 y Fq(!)1597 5149 y Fy(such)e(that)g Fu(!)i Fy(is)d(a)g Fx(\()p Fu(\033)2261 5163 y Fq(!)2312 5149 y Fu(;)15 b Fw(\000)p Fx(1\))p Fy(-KMS)23 b(state.)p eop %%Page: 4 4 4 3 bop 3730 100 a FA(4)380 407 y Fy(The)23 b(choice)h(of)f(reference)i (temperature)g Fu(\014)31 b Fx(=)25 b Fw(\000)p Fx(1)d Fy(is)h(made)g(for)g(mathematical)i(con)l(v)o(enience.)32 b(If)244 520 y(\(E1\))25 b(holds,)i(then)f(for)f(an)o(y)h Fu(\014)33 b Fw(6)p Fx(=)c(0)c Fy(there)h(is)f(a)g Fu(C)1823 487 y Ft(\003)1862 520 y Fy(-dynamics)i Fu(\033)2317 535 y Fq(!)r(;\014)2455 520 y Fy(such)f(that)g Fu(!)h Fy(is)e Fx(\()p Fu(\033)3063 535 y Fq(!)r(;\014)3176 520 y Fu(;)15 b(\014)5 b Fx(\))p Fy(-KMS)244 646 y(state)24 b(\(set)g Fu(\033)642 613 y Fq(t)639 674 y(!)r(;\014)777 646 y Fx(=)h Fu(\033)928 602 y Ft(\000)p Fq(\014)s(t)925 658 y(!)1056 646 y Fx(\))p Fy(.)380 759 y(Let)h Fu(\016)567 773 y Fq(!)643 759 y Fy(be)g(the)g(generator)j(of)d Fu(\033)1412 773 y Fq(!)1462 759 y Fy(.)36 b(Our)25 b(second)j(assumption)h (concerns)f(the)e(local)h(perturbation)244 872 y Fu(V)20 b Fy(.)244 1084 y Fz(\(E2\))j Fu(V)46 b Fw(2)25 b(D)s Fx(\()p Fu(\016)766 1098 y Fq(!)816 1084 y Fx(\))p Fy(.)380 1297 y(Until)h(the)h(end)f(of)g(this)h(section)h(we)d(assume)i(that)f (\(E1\))g(and)h(\(E2\))f(hold.)37 b(W)-7 b(e)25 b(set)h Fu(\033)3084 1311 y Fq(V)3175 1297 y Fx(:=)k Fu(\016)3341 1311 y Fq(!)3392 1297 y Fx(\()p Fu(V)20 b Fx(\))244 1410 y Fy(and)30 b(call)g Fx(Ep\()p Fu(\021)s Fx(\))37 b(=)g Fu(\021)s Fx(\()p Fu(\033)1075 1424 y Fq(V)1137 1410 y Fx(\))29 b Fy(the)h(entrop)o(y)i(production)g(\(w)-6 b(.r)h(.t.)47 b(the)30 b(reference)i(state)e Fu(!)s Fy(\))f(of)h(the)g (per)n(-)244 1523 y(turbed)25 b(system)f Fx(\()p Fw(O)s Fu(;)15 b(\034)964 1537 y Fq(V)1025 1523 y Fx(\))24 b Fy(in)f(the)h(state)g Fu(\021)29 b Fw(2)c Fu(E)5 b Fx(\()p Fw(O)s Fx(\))p Fy(.)380 1636 y(The)23 b(follo)n(wing)i(identity)h(w)o (as)d(pro)o(v)o(en)h(in)g([JP3)q(]:)1163 1893 y Fx(En)m(t\()p Fu(!)g Fw(\016)c Fu(\034)1539 1856 y Fq(t)1529 1916 y(V)1605 1893 y Fw(j)15 b Fu(!)s Fx(\))26 b(=)f Fw(\000)1948 1770 y Fm(Z)2039 1796 y Fq(t)1998 1976 y Fv(0)2083 1893 y Fu(!)s Fx(\()p Fu(\034)2228 1856 y Fq(s)2218 1916 y(V)2279 1893 y Fx(\()p Fu(\033)2366 1907 y Fq(V)2428 1893 y Fx(\)\)d)p Fu(s:)744 b Fy(\(1.2\))244 2137 y(This)33 b(identity)j(moti)n(v)n(ates) f(the)f(de\002nition)i(of)e(entrop)o(y)h(production)i(and)d(is)g(the)g (starting)i(point)f(for)244 2250 y(study)g(of)g(this)g(notion)g([JP3)q (,)e(JP4)q(].)60 b(In)34 b(particular)l(,)40 b(since)35 b(the)g(relati)n(v)o(e)h(entrop)o(y)g(is)e(non-positi)n(v)o(e,)244 2363 y(Relation)25 b(\(1.2\))f(yields)h(that)f(for)f(an)o(y)h Fu(!)1512 2325 y Fv(+)1509 2392 y Fq(V)1596 2363 y Fw(2)h Fx(\006)1748 2325 y Fv(+)1748 2392 y Fq(V)1809 2363 y Fx(\()p Fu(!)s Fx(\))p Fy(,)e Fx(Ep)o(\()p Fu(!)2192 2325 y Fv(+)2189 2392 y Fq(V)2251 2363 y Fx(\))j Fw(\025)f Fx(0)p Fy(.)380 2476 y(The)31 b(NESS)f Fu(!)866 2438 y Fv(+)863 2505 y Fq(V)955 2476 y Fy(is)i(thermodynamically)k(non-tri)n (vial)e(if)d Fx(Ep\()p Fu(!)2491 2438 y Fv(+)2488 2505 y Fq(V)2550 2476 y Fx(\))41 b Fu(>)f Fx(0)p Fy(.)52 b(One)32 b(of)f(the)h(central)244 2589 y(problems)h(of)f(mathematical)i(theory)g (of)e(non-equilibrium)k(quantum)d(statistical)i(mechanics)f(is)e(to)244 2702 y(sho)n(w)f(that)i(the)f(NESS)d(of)i(concrete)j(physically)g(rele) n(v)n(ant)g(models)e(are)g(thermodynamically)k(non-)244 2815 y(tri)n(vial.)47 b(W)-7 b(e)29 b(describe)i(belo)n(w)f(one)g (simple)g(criterion)h(which)f(ensures)h(strict)g(positi)n(vity)h(of)d (entrop)o(y)244 2928 y(production)e(and)d(which)f(will)h(be)f(used)h (in)g(this)g(paper)-5 b(.)380 3041 y(Let)33 b Fx(\()p Fw(H)646 3055 y Fq(!)697 3041 y Fu(;)15 b(\031)789 3055 y Fq(!)839 3041 y Fu(;)g Fx(\012)945 3055 y Fq(!)996 3041 y Fx(\))33 b Fy(be)g(the)h(GNS-representation)i(of)d(the)h (algebra)h Fw(O)h Fy(associated)g(to)d Fu(!)s Fy(.)57 b(The)244 3154 y(states)24 b(in)e Fw(O)635 3121 y Ft(\003)696 3154 y Fy(which)h(are)g(represented)i(by)e(density)h(matrices)g(on)f Fw(H)2430 3168 y Fq(!)2502 3154 y Fy(are)g(called)g Fu(!)s Fy(-normal.)30 b(The)22 b(set)244 3266 y(of)k(all)g Fu(!)s Fy(-normal)h(states)h(is)e(a)g(norm)g(closed)i(subset)f(of)f Fu(E)5 b Fx(\()p Fw(O)s Fx(\))27 b Fy(which)f(we)g(denote)h(by)g Fw(N)3124 3280 y Fq(!)3174 3266 y Fy(.)36 b(One)25 b(can)244 3379 y(sho)n(w)e(that)h(the)g(entrop)o(y)i(production)g(of)e Fu(!)s Fy(-normal)g(NESS)d(is)j(zero,)f(see)h([JP4)q(].)244 3592 y Fz(Theor)n(em)46 b(1.1)k Fj(Assume)23 b(that)h(NESS)f Fu(!)1558 3554 y Fv(+)1555 3621 y Fq(V)1640 3592 y Fj(satis\002es)i (the)f(following:)244 3705 y Fy(\(a\))g Fu(!)428 3667 y Fv(+)425 3733 y Fq(V)512 3705 y Fw(62)h(N)673 3719 y Fq(!)723 3705 y Fj(.)244 3846 y Fy(\(b\))f Fx(sup)509 3868 y Fq(T)10 b(>)p Fv(0)670 3742 y Fm(\014)670 3796 y(\014)670 3851 y(\014)700 3773 y(R)761 3799 y Fq(T)743 3878 y Fv(0)816 3846 y Fx(\()p Fu(!)s Fx(\()p Fu(\034)996 3813 y Fq(t)986 3873 y(V)1047 3846 y Fx(\()p Fu(\033)1134 3860 y Fq(V)1195 3846 y Fx(\)\))21 b Fw(\000)f Fu(!)1437 3808 y Fv(+)1434 3875 y Fq(V)1496 3846 y Fx(\()p Fu(\033)1583 3860 y Fq(V)1644 3846 y Fx(\)\)d)p Fu(t)1798 3742 y Fm(\014)1798 3796 y(\014)1798 3851 y(\014)1854 3846 y Fu(<)25 b Fw(1)p Fj(.)244 3992 y(Then)e Fx(Ep\()p Fu(!)656 3954 y Fv(+)653 4021 y Fq(V)715 3992 y Fx(\))j Fu(>)f Fx(0)p Fj(.)244 4171 y Fy(W)-7 b(e)23 b(will)g(pro)o(v)o(e)h(this)g(theorem)g(in)g (Section)g(5.)380 4284 y(One)k(of)g(the)g(main)g(results)i(of)e(this)h (paper)g(is)f(that)h(the)f(class)h(of)f(systems)h(we)e(study)j(has)e (strictly)244 4397 y(positi)n(v)o(e)d(entrop)o(y)g(production.)380 4510 y(F)o(or)i(additional)k(information)f(about)f(NESS)d(and)i(entrop) o(y)i(production)h(we)c(refer)i(the)f(reader)h(to)244 4623 y([JP4)q(].)244 4905 y Fr(1.2)119 b(The)30 b(model)g(and)h(the)f (r)n(esults)244 5080 y Fy(W)-7 b(e)23 b(no)n(w)g(describe)i(the)f (speci\002c)g(model)g(we)f(will)g(study)i(in)e(this)i(paper)-5 b(.)p eop %%Page: 5 5 5 4 bop 3730 100 a FA(5)380 407 y Fy(The)31 b Fu(C)624 374 y Ft(\003)663 407 y Fy(-algebra)j(of)e(observ)n(ables)i(of)e(the)g (system)g Fw(S)38 b Fy(is)32 b Fw(O)2345 421 y Fv(s)2417 407 y Fw(\021)40 b Fu(M)10 b Fx(\()p Fp(C)2721 374 y Fv(2)2767 407 y Fx(\))p Fy(,)33 b(the)f(matrix)g(algebra)244 520 y(on)26 b Fi(H)425 534 y Fv(s)488 520 y Fw(\021)31 b Fp(C)649 487 y Fv(2)695 520 y Fy(.)36 b(Let)26 b Fu(\033)953 534 y Fq(x)997 520 y Fu(;)15 b(\033)1089 534 y Fq(y)1130 520 y Fu(;)g(\033)1222 534 y Fq(z)1288 520 y Fy(be)27 b(the)f(usual)i(P)o(auli)e(matrices.)38 b(The)26 b(dynamics)i(is)e (speci\002ed)i(by)f(the)244 633 y(automorphisms)1470 746 y Fu(\034)1520 708 y Fq(t)1510 768 y Fv(s)1549 746 y Fx(\()p Fu(A)p Fx(\))f(=)f(e)1849 708 y Fv(i)p Fq(tH)1952 716 y Fl(s)1985 746 y Fu(A)p Fx(e)2093 708 y Ft(\000)p Fv(i)p Fq(tH)2251 716 y Fl(s)2285 746 y Fu(;)1051 b Fy(\(1.3\))244 912 y(where)24 b Fu(H)565 926 y Fv(s)621 912 y Fw(\021)h Fu(\033)769 926 y Fq(z)832 912 y Fy(is)e(the)h(Hamiltonian)h(of)f(the)g (system)g Fw(S)7 b Fy(.)380 1025 y(Let)33 b Fi(h)f Fy(be)i(the)f (Hilbert)h(space)g(of)f(a)g(single)h(fermion)h(and)e Fu(h)g Fy(its)g(ener)n(gy)i(operator)-5 b(.)60 b(Let)32 b Fi(H)3390 1040 y Fv(f)3465 1025 y Fw(\021)244 1138 y Fx(\000)301 1152 y Ft(\000)360 1138 y Fx(\()p Fi(h)p Fx(\))19 b Fy(be)f(the)h(Fermi)f(F)o(ock)g(space)i(and)f Fu(a)p Fx(\()p Fu(f)10 b Fx(\))p Fy(,)19 b Fu(a)1793 1105 y Ft(\003)1832 1138 y Fx(\()p Fu(f)10 b Fx(\))18 b Fy(the)h(corresponding)k(annihilation)f(and)d(creation)244 1251 y(operators)24 b(on)e Fi(H)780 1266 y Fv(f)812 1251 y Fy(.)27 b(In)22 b(the)g(sequel)h Fu(a)1392 1218 y Fv(#)1476 1251 y 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Fy(The)23 b(Hamiltonian)i(and)f(the)g(dynamics)h(are)f(speci\002ed)h(by)e Fu(H)2213 1968 y Fv(f)2269 1953 y Fx(=)i(d\000\()p Fu(h)p Fx(\))e Fy(and)1316 2159 y Fu(\034)1366 2121 y Fq(t)1356 2182 y Fv(f)1395 2159 y Fx(\()p Fu(a)1478 2121 y Fv(#)1542 2159 y Fx(\()p Fu(f)10 b Fx(\)\))25 b(=)g(e)1863 2121 y Fv(i)p Fq(tH)1966 2133 y Fl(f)1998 2159 y Fu(a)2046 2121 y Fv(#)2109 2159 y Fx(\()p Fu(f)10 b Fx(\)e)2274 2121 y Ft(\000)p Fv(i)p Fq(tH)2432 2133 y Fl(f)1727 2381 y Fx(=)25 b Fu(a)1871 2343 y Fv(#)1934 2381 y Fx(\(e)2009 2343 y Fv(i)p Fq(th)2100 2381 y Fu(f)10 b Fx(\))p Fu(:)244 2582 y Fy(The)28 b(pair)g Fx(\()p Fw(O)688 2597 y Fv(f)720 2582 y Fu(;)15 b(\034)800 2597 y Fv(f)831 2582 y Fx(\))28 b Fy(is)g(a)g Fu(C)1122 2549 y Ft(\003)1161 2582 y Fy(-dynamical)i (system)f(describing)i(a)d(free)g(Fermi)g(gas.)42 b(F)o(or)28 b(each)h Fu(\014)39 b(>)33 b Fx(0)244 2695 y Fy(there)f(e)o(xists)g(a)e (unique)j Fx(\()p Fu(\034)1115 2710 y Fv(f)1146 2695 y Fu(;)15 b(\014)5 b Fx(\))p Fy(-KMS)31 b(state)h Fu(!)1790 2710 y Fv(f)5 b Fq(;\014)1913 2695 y Fy(on)31 b Fw(O)2106 2710 y Fv(f)2138 2695 y Fy(.)50 b Fu(!)2268 2710 y Fv(f)5 b Fq(;\014)2392 2695 y Fy(is)31 b(a)f(quasi-free,)35 b(gauge-in)l(v)n(ariant)244 2808 y(state)24 b(uniquely)i(determined)g (by)d(the)h(tw)o(o)f(point)i(function)1181 3012 y Fu(!)1238 3027 y Fv(f)5 b Fq(;\014)1331 3012 y Fx(\()p Fu(a)1414 2975 y Ft(\003)1454 3012 y Fx(\()p Fu(f)10 b Fx(\))p Fu(a)p Fx(\()p Fu(f)g Fx(\)\))26 b(=)f(\()p Fu(f)5 b(;)15 b Fx(\(e)2109 2975 y Fq(\014)s(h)2217 3012 y Fx(+)20 b(1\))2388 2975 y Ft(\000)p Fv(1)2484 3012 y Fu(f)10 b Fx(\))p Fu(:)244 3316 y Fz(Notation.)29 b Fy(In)23 b(the)g(sequel,)h(whene)n(v)o(er)g(the)f(meaning)h(is)f(clear)h(within) f(the)h(conte)o(xt,)g(we)e(denote)i(by)f Fu(A)244 3429 y Fy(the)h(operators)i Fu(A)20 b Fw(\012)g Fo(1)p Fy(,)j Fo(1)d Fw(\012)g Fu(A)p Fy(.)380 3654 y(W)-7 b(e)34 b(consider)j(no)n (w)d(tw)o(o)h(identical)i(reserv)n(oirs)g Fx(\()p Fw(O)2080 3606 y Fv(\()p Fq(i)p Fv(\))2077 3684 y(f)2164 3654 y Fu(;)15 b(\034)2254 3606 y Fv(\()p Fq(i)p Fv(\))2244 3684 y(f)2337 3654 y Fx(\))p Fy(,)37 b Fu(i)46 b Fx(=)g(1)p Fu(;)15 b Fx(2)p Fy(.)63 b(The)34 b Fu(C)3089 3621 y Ft(\003)3128 3654 y Fy(-algebra)j(of)244 3767 y(observ)n(ables)27 b(of)c(the)h(combined)h(system)f Fw(S)j Fx(+)20 b Fw(R)1832 3781 y Fv(1)1892 3767 y Fx(+)g Fw(R)2060 3781 y Fv(2)2122 3767 y Fy(is)1447 3984 y Fw(O)28 b(\021)d(O)1715 3998 y Fv(s)1767 3984 y Fw(\012)20 b(O)1933 3936 y Fv(\(1\))1930 4014 y(f)2048 3984 y Fw(\012)f(O)2213 3936 y Fv(\(2\))2210 4014 y(f)2308 3984 y Fu(;)1028 b Fy(\(1.4\))244 4189 y(the)25 b(tensor)g(product)i(algebra)f(of)e(operators)j(on)e Fi(H)i Fw(\021)f Fi(H)2047 4203 y Fv(s)2101 4189 y Fw(\012)20 b Fi(H)2257 4204 y Fv(f)2310 4189 y Fw(\012)g Fi(H)2466 4204 y Fv(f)2498 4189 y Fy(.)30 b(The)24 b(free)h(dynamics)h(is)e(gi)n (v)o(en)244 4314 y(by)j(the)g(group)i(of)e(automorphisms)j Fu(\034)41 b Fx(=)32 b Fu(\034)1650 4328 y Fv(s)1705 4314 y Fw(\012)22 b Fu(\034)1848 4266 y Fv(\(1\))1838 4344 y(f)1965 4314 y Fw(\012)h Fu(\034)2109 4266 y Fv(\(2\))2099 4344 y(f)2203 4314 y Fy(.)39 b(The)26 b(pair)i Fx(\()p Fw(O)s Fu(;)15 b(\034)10 b Fx(\))27 b Fy(is)g(a)g Fu(C)3088 4281 y Ft(\003)3127 4314 y Fy(-dynamical)244 4427 y(system)d (describing)j(the)c(combined)j(system)e(in)f(absence)j(of)d (interaction.)32 b(Note)23 b(that)1493 4631 y Fu(\034)1543 4594 y Fq(t)1573 4631 y Fx(\()p Fu(A)p Fx(\))j(=)f(e)1873 4594 y Fv(i)p Fq(tH)1985 4631 y Fu(A)p Fx(e)2093 4594 y Ft(\000)p Fv(i)p Fq(tH)2261 4631 y Fu(;)244 4836 y Fy(where)1433 4949 y Fu(H)32 b Fx(=)25 b Fu(H)1713 4963 y Fv(s)1765 4949 y Fx(+)20 b Fu(H)1939 4901 y Fv(\(1\))1932 4978 y(f)2053 4949 y Fx(+)g Fu(H)2227 4901 y Fv(\(2\))2220 4978 y(f)2321 4949 y Fu(:)p eop %%Page: 6 6 6 5 bop 3730 100 a FA(6)380 407 y Fy(W)-7 b(e)24 b(no)n(w)g(describe)j (the)d(interaction)k(of)c Fw(S)31 b Fy(with)24 b(the)h(reserv)n(oirs.) 35 b(Choose)25 b(form-f)o(actors)i Fu(\013)3322 421 y Fq(i)3378 407 y Fw(2)g Fi(h)p Fy(,)244 520 y 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a(2)2595 1513 y Fx(e)2635 1480 y Ft(\000)p Fq(k)2729 1456 y Fl(4)2768 1513 y Fy(,)29 b(then)h(\(A2\))f(and)g(\(A3\))244 1626 y(hold)24 b(\(in)g(this)g(e)o(xample)g(\(A2\))g(holds)g(for)g(all)g Fu(\016)i Fy(and)e Fu(\014)1963 1640 y Fq(i)1992 1626 y Fy(\).)380 1738 y(Our)f(\002rst)h(result)g(is:)244 1951 y Fz(Theor)n(em)46 b(1.2)k Fj(Assume)23 b(that)h Fy(\(A1\))p Fj(-)p Fy(\(A3\))g Fj(hold.)29 b(Then,)23 b(for)h(some)f Fx(\003)j Fu(>)f Fx(0)e Fj(and)g Fx(0)j Fu(<)f Fw(j)p Fu(\025)p Fw(j)h Fu(<)f Fx(\003)p Fj(,)d(ther)m(e)244 2064 y(is)h(a)g(state)i Fu(!)650 2026 y Fv(+)647 2093 y Fq(\025)731 2064 y Fj(on)f Fw(O)i Fj(so)d(that)h(the)g(following)i (hold:)244 2177 y Fy(\(i\))d Fj(F)-10 b(or)24 b(all)g Fu(\021)k Fw(2)d(N)36 b Fj(and)24 b Fu(A)h Fw(2)g(O)s Fj(,)1445 2381 y Fx(lim)1425 2437 y Fq(t)p Ft(!1)1607 2381 y Fu(\021)s Fx(\()p Fu(\034)1740 2343 y Fq(t)1730 2404 y(\025)1776 2381 y Fx(\()p Fu(A)p Fx(\)\))h(=)f Fu(!)2131 2343 y Fv(+)2128 2410 y Fq(\025)2190 2381 y Fx(\()p Fu(A)p Fx(\))p 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Fx(\()p Fu(\025)2213 4836 y Fv(4)2253 4874 y Fx(\))p Fu(;)244 5078 y Fy(where)1260 5191 y Fu(\015)1307 5205 y Fv(0)1372 5191 y Fx(=)1478 5129 y Fu(\031)p 1478 5170 56 4 v 1483 5253 a Fx(2)1558 5117 y Fm(\000)1600 5191 y Fw(k)p Fu(\013)1703 5205 y Fv(1)1743 5191 y Fx(\(2\))p Fw(k)1903 5153 y Fv(2)1903 5214 y Fg(G)1983 5191 y Fx(+)g Fw(k)p Fu(\013)2177 5205 y Fv(2)2217 5191 y Fx(\(2\))p Fw(k)2377 5153 y Fv(2)2377 5214 y Fg(G)2437 5117 y Fm(\001)2494 5191 y Fu(:)p eop %%Page: 8 8 8 7 bop 3730 100 a FA(8)244 407 y Fz(Remark)18 b(5.)26 b Fy(Re)o(garding)20 b(\(iii\),)f(it)g(follo)n(ws)g(from)f(our)h(ar)n (guments)h(that)f(there)h(e)o(xists)f(linear)h(functionals)244 520 y Fu(!)304 481 y Fv(+)301 549 y Fq(k)388 520 y Fx(:)26 b Fw(O)511 534 y Fv(0)576 520 y Fw(7!)f Fp(C)17 b Fy(,)29 b Fu(k)f Fw(\025)d Fx(0)p Fy(,)e(such)i(that)f(for)f Fu(A)j Fw(2)f(O)1794 534 y Fv(0)1833 520 y Fy(,)1438 787 y Fu(!)1498 749 y Fv(+)1495 817 y Fq(\025)1557 787 y Fx(\()p Fu(A)p Fx(\))h(=)1847 674 y Ft(1)1817 701 y Fm(X)1818 899 y Fq(k)r Fv(=0)1963 787 y Fu(\025)2016 750 y Fq(k)2059 787 y Fu(!)2119 749 y Fv(+)2116 817 y Fq(k)2178 787 y Fx(\()p Fu(A)p Fx(\))p Fu(:)975 b Fy(\(1.10\))244 1058 y(The)31 b(\002rst)g(term)g Fu(!)842 1072 y Fv(0)912 1058 y Fy(is)h(computed)h(from)e(a)g(linear)i(eigen)l(v)n(alue)i (problem)d(on)g Fi(H)2848 1072 y Fv(s)2881 1058 y Fy(.)51 b(This)32 b(eigen)l(v)n(alue)244 1171 y(problem)27 b(is)f(determined)i (by)e(the)h(second)g(order)g(correction)i(\(Fermi')-5 b(s)26 b(Golden)h(Rule\))f(for)h(the)f(reso-)244 1284 y(nances)i(of)f(a)g(suitable)i(non-self-adjoint)j(operator)d(\()p Fu(C)7 b Fy(-Liouvillean\).)41 b(Although)29 b(formulas)f(for)g(the)244 1396 y(higher)d(order)g(terms)e(become)i(quickly)g(v)o(ery)f (complicated,)i(in)e(principle)i(it)d(is)g(possible)j(to)e(compute)244 1509 y(all)g(terms)f(in)h(the)g(e)o(xpansion)h(\(1.10\).)30 b(W)-7 b(e)23 b(will)g(discuss)i(this)f(point)h(at)e(the)h(end)g(of)g (Section)g(4.)380 1735 y(Theorem)19 b(1.2)f(establishes)k(the)c(basic)i (thermodynamical)h(property)g(of)d(the)h(system)g Fw(S)8 b Fx(+)q Fw(R)3285 1749 y Fv(1)3325 1735 y Fx(+)q Fw(R)3474 1749 y Fv(2)3513 1735 y Fy(,)244 1848 y(namely)29 b(that)g(the)f(set)h (of)f(initial)i(states)f Fw(N)40 b Fy(belongs)31 b(to)d(the)g(basin)i (of)e(attraction)j(of)d(a)g(single)h(NESS)244 1961 y Fu(!)304 1923 y Fv(+)301 1990 y Fq(\025)363 1961 y Fy(.)f(W)-7 b(e)22 b(no)n(w)h(discuss)j(the)d(other)i(thermodynamical)i(properties) f(of)e(this)g(system.)380 2074 y(The)f(\002rst)h(question)h(is)f (whether)g Fu(!)1498 2036 y Fv(+)1495 2103 y Fq(\025)1580 2074 y Fy(belongs)h(to)f(the)g(set)f Fw(N)36 b Fy(of)24 b(normal)g(states.)244 2280 y Fz(Theor)n(em)46 b(1.3)k Fj(Assume)24 b(that)h Fx(\(A1\))d Fw(\000)e Fx(\(A3\))25 b Fj(hold)g(and)g(that)g Fu(\014)2335 2294 y Fv(1)2402 2280 y Fw(6)p Fx(=)h Fu(\014)2550 2294 y Fv(2)2590 2280 y Fj(.)k(Then)24 b(ther)m(e)h(is)f Fu(`)j(>)f Fx(0)e Fj(suc)o(h)244 2393 y(that)32 b(for)f Fx(0)40 b Fu(<)f Fw(j)p Fu(\025)p Fw(j)h Fu(<)f(`)30 b Fj(ther)m(e)i(ar)m(e)f(no)g Fu(\034)1591 2408 y Fq(\025)1636 2393 y Fj(-in)l(variant)k(states)d(in) f Fw(N)13 b Fj(.)51 b(In)31 b(particular)-10 b(,)35 b(if)c Fx(0)40 b Fu(<)f Fw(j)p Fu(\025)p Fw(j)h Fu(<)244 2506 y Fx(min)n(\(\003)p Fu(;)15 b(`)p Fx(\))p Fj(,)24 b(then)g Fu(!)892 2468 y Fv(+)889 2535 y Fq(\025)976 2506 y Fw(62)h(N)13 b Fj(.)244 2682 y Fz(Remark)28 b(1.)44 b Fy(This)29 b(result)h(can)f (be)g(pro)o(v)o(en)g(under)h(more)f(general)i(condition)g(then)e (\(A2\),)h(see)f([DJ1)q(,)244 2795 y(DJ2].)244 2908 y Fz(Remark)c(2.)34 b Fy(The)25 b(constant)j Fu(`)d Fy(dif)n(fers)i(from) e(the)h(constant)i Fx(\003)d Fy(in)g(Theorem)h(1.2.)35 b(In)25 b(contrast)j(to)e Fx(\003)p Fy(,)f Fu(`)244 3021 y Fy(can)h(be)f(chosen)i(independently)j(of)25 b(the)h(size)f(of)h Fu(\014)1875 3035 y Fq(i)1903 3021 y Fy(')-5 b(s)25 b(as)h Fu(\014)2140 3035 y Fq(i)2197 3021 y Fw(!)i(1)d Fy(\(see)h([DJ2)q(])e (for)i(details\).)36 b(On)25 b(the)244 3134 y(other)h(hand,)g Fu(`)f Fy(depends)i(on)e Fu(d)k Fx(=)f Fw(j)p Fu(\014)1430 3148 y Fv(1)1492 3134 y Fw(\000)21 b Fu(\014)1635 3148 y Fv(2)1675 3134 y Fw(j)k Fy(and)g Fu(`)k Fw(#)f Fx(0)d Fy(as)g Fu(d)k Fw(#)g Fx(0)p Fy(.)34 b(The)24 b(constant)k Fx(\003)c Fy(can)i(be)f(chosen)244 3247 y(independently)j(of)23 b Fu(d)g Fy(as)h(long)g(as)f Fx(0)j Fw(\024)f Fu(d)h Fw(\024)f Fx(const)p Fu(:)380 3360 y Fy(Recall)i(that)g(the)f(entrop)o (y)i(production)i(depends)e(on)e(the)h(choice)g(of)g(the)f(initial)i (state)f Fu(!)s Fy(.)35 b(Let)3455 3337 y Fm(b)3429 3360 y Fw(N)3504 3374 y Fv(s)244 3473 y Fy(be)24 b(the)g(set)g(of)g(states)h (in)f Fw(N)1104 3487 y Fv(s)1158 3473 y Fy(with)g(the)g(property)i (that)f Fu(!)2025 3487 y Fv(s)2082 3473 y Fu(>)h Fx(0)e Fy(and)g(is)g Fu(\034)2526 3487 y Fv(s)2557 3473 y Fy(-in)l(v)n (ariant.)33 b(The)23 b(assumption)244 3585 y(\(E1\))h(of)g(Section)h (1.1)f(hold)g(for)h(all)f Fu(!)29 b Fw(2)1588 3562 y Fm(b)1562 3585 y Fw(N)1637 3599 y Fv(s)1668 3585 y Fy(.)h(If)24 b(\(A2\))g(holds,)h(then)g(\(E2\))e(holds)j(for)e(the)g(perturbation) 244 3698 y Fu(V)c Fy(.)244 3905 y Fz(Theor)n(em)46 b(1.4)k Fj(Under)24 b(the)f(assumptions)k(of)c(Theor)m(em)h(1.3,)f(for)h(any)g (initial)h(state)f Fu(!)k Fw(2)3174 3882 y Fm(b)3148 3905 y Fw(N)3223 3919 y Fv(s)3255 3905 y Fj(,)1344 4096 y Fx(Ep)o(\()p Fu(!)1551 4058 y Fv(+)1548 4126 y Fq(\025)1610 4096 y Fx(\))e(=)f Fu(!)1827 4058 y Fv(+)1824 4126 y Fq(\025)1886 4096 y Fx(\()p Fu(\016)1961 4110 y Fq(!)2012 4096 y Fx(\()p Fu(\025V)20 b Fx(\)\))26 b Fu(>)f Fx(0)p Fu(;)244 4288 y Fj(for)i Fx(0)33 b Fu(<)e Fw(j)p Fu(\025)p Fw(j)i Fu(<)f Fx(min)n(\(\003)p Fu(;)15 b(`)p Fx(\))p Fj(.)40 b(Mor)m(eo)o(ver)-10 b(,)29 b Fx(Ep\()p Fu(!)1822 4250 y Fv(+)1819 4317 y Fq(\025)1881 4288 y Fx(\))d Fj(does)i(not)g (depend)h(on)e(the)h(c)o(hoice)g(of)f(the)g(initial)244 4414 y(state)d Fu(!)k Fw(2)636 4391 y Fm(b)609 4414 y Fw(N)684 4428 y Fv(s)716 4414 y Fj(.)244 4590 y Fz(Remark.)36 b Fy(This)27 b(theorem)g(can)g(be)f(pro)o(v)o(en)h(in)g(tw)o(o)f(dif)n (ferent)i(w)o(ays.)37 b(The)26 b(short)h(proof)h(\(the)f(one)g(we)244 4703 y(will)f(gi)n(v)o(e)h(in)g(this)h(paper\))g(is)f(based)h(on)f (Theorem)g(1.1.)39 b(This)27 b(proof)h(gi)n(v)o(es)f(no)g(estimate)h (on)f(the)g(size)244 4816 y(of)22 b(entrop)o(y)i(production.)32 b(The)21 b(second)j(proof)f(is)g(based)g(on)f(the)h(perturbati)n(v)o(e) i(e)o(xpansion)f(of)f(the)f(state)244 4929 y Fu(!)304 4891 y Fv(+)301 4958 y Fq(\025)363 4929 y Fy(.)48 b(Although)32 b(computationally)i(tedious,)g(this)d(proof)g(has)g(the)f(adv)n(antage) j(of)d(sho)n(wing)i(that)f(the)244 5042 y(entrop)o(y)i(production)i(is) c(strictly)j(positi)n(v)o(e)f(to)e(the)h(lo)n(west)g(non-tri)n(vial)i (order)f(\(the)f(\002rst)f(non-tri)n(vial)244 5155 y(term)g(can)g(be)g (also)h(computed)h(using)f(the)f(v)n(an)h(Ho)o(v)o(e)e(weak)h(coupling) j(limit,)e(see)g([LS)o(]\).)51 b(W)-7 b(e)30 b(will)244 5268 y(discuss)25 b(the)f(perturbati)n(v)o(e)i(proof)f(of)e(Theorem)h (1.4)g(in)f([JP4)q(].)p eop %%Page: 9 9 9 8 bop 3730 100 a FA(9)380 407 y Fy(W)-7 b(e)24 b(\002nish)h(this)g (section)h(with)e(a)g(brief)i(discussion)h(of)d(the)h(heat)g(\003ux)o (es.)32 b(Let)24 b Fu(\016)2896 421 y Fq(i)2948 407 y Fy(be)h(the)f(generator)244 533 y(of)33 b Fu(\034)402 485 y Fv(\()p Fq(i)p Fv(\))392 562 y(f)485 533 y Fy(.)55 b(\(A2\))33 b(implies)h(that)f Fu(V)1288 547 y Fq(i)1359 533 y Fw(2)42 b(D)s Fx(\()p Fu(\016)1610 547 y Fq(i)1638 533 y Fx(\))p Fy(.)56 b(The)33 b(observ)n(able)i(describing)h(the)d (heat)g(\003ux)g(\(ener)n(gy)244 645 y(transfer\))26 b(from)d(the)h(rest)g(of)f(the)h(system)g(into)g(the)g Fu(i)p Fy(-th)g(reserv)n(oir)i(is)e Fx(\010)2516 659 y Fq(i)2569 645 y Fx(:=)h Fu(\016)2730 659 y Fq(i)2759 645 y Fx(\()p Fu(\025V)2900 659 y Fq(i)2928 645 y Fx(\))p Fy(.)244 858 y Fz(Theor)n(em)46 b(1.5)k Fj(Assume)37 b(that)h Fy(\(A1\)-\(A3\))h Fj(hold)f(and)g(that)g Fu(\014)2307 872 y Fv(1)2398 858 y Fw(6)p Fx(=)51 b Fu(\014)2571 872 y Fv(2)2611 858 y Fj(.)70 b(Then,)41 b(for)d Fx(0)51 b Fu(<)g Fw(j)p Fu(\025)p Fw(j)h Fu(<)244 971 y Fx(min)n(\(\003)p Fu(;)15 b(`)p Fx(\))p Fj(,)24 b(the)g(following)h(r)m(elations)h(hold:) 1287 1172 y Fu(!)1347 1134 y Fv(+)1344 1202 y Fq(\025)1406 1172 y Fx(\(\010)1507 1186 y Fv(1)1547 1172 y Fx(\))20 b(+)g Fu(!)1753 1134 y Fv(+)1750 1202 y Fq(\025)1812 1172 y Fx(\(\010)1913 1186 y Fv(2)1953 1172 y Fx(\))25 b(=)g(0)p Fu(;)1106 1381 y(\014)1157 1395 y Fv(1)1197 1381 y Fu(!)1257 1343 y Fv(+)1254 1410 y Fq(\025)1316 1381 y Fx(\(\010)1417 1395 y Fv(1)1456 1381 y Fx(\))c(+)e Fu(\014)1653 1395 y Fv(2)1693 1381 y Fu(!)1753 1343 y Fv(+)1750 1410 y Fq(\025)1812 1381 y Fx(\(\010)1913 1395 y Fv(2)1953 1381 y Fx(\))25 b(=)g Fw(\000)p Fx(Ep)o(\()p Fu(!)2387 1343 y Fv(+)2384 1410 y Fq(\025)2446 1381 y Fx(\))h Fu(<)f Fx(0)p Fu(;)3316 1277 y Fy(\(1.11\))244 1577 y Fj(wher)m(e)j(in)g(the)g(second)i(r)m(elation)g(the)e(entr)l (opy)i(pr)l(oduction)i(is)27 b(computed)j(w)-7 b(.r)d(.t.)42 b(any)28 b(initial)i(state)f(in)270 1667 y Fm(b)244 1690 y Fw(N)319 1704 y Fv(s)350 1690 y Fj(.)244 1869 y Fz(Remark)17 b(1.)27 b Fy(Relations)20 b(\(1.11\))f(are)g(respecti)n(v)o(ely)i(the)e (\002rst)f(and)g(the)h(second)h(la)o(w)d(of)i(thermodynamics)244 1982 y(for)24 b(the)f(model)h Fw(S)j Fx(+)20 b Fw(R)1006 1996 y Fv(1)1066 1982 y Fx(+)g Fw(R)1234 1996 y Fv(2)1273 1982 y Fy(.)244 2095 y Fz(Remark)30 b(2.)49 b Fy(If)30 b Fu(\014)850 2109 y Fv(1)928 2095 y Fu(>)38 b(\014)1088 2109 y Fv(2)1128 2095 y Fy(,)31 b(then)g Fu(!)1428 2057 y Fv(+)1425 2124 y Fq(\025)1487 2095 y Fx(\(\010)1588 2109 y Fv(1)1628 2095 y Fx(\))38 b Fu(>)g Fx(0)p Fy(.)49 b(Thus,)32 b(in)f(NESS)d Fu(!)2576 2057 y Fv(+)2573 2124 y Fq(\025)2664 2095 y Fy(there)k(is)e(a)g(constant)j(non-)244 2208 y(v)n(anishing)26 b(heat)e(\003o)n(w)e(from)i(the)f(hotter)i(to)f (the)f(colder)i(reserv)n(oir)h(across)f(the)f(system)g Fw(S)7 b Fy(.)244 2321 y Fz(Remark)20 b(3.)27 b Fy(Except)21 b(for)g(the)g(strict)g(positi)n(vity)i(of)e(entrop)o(y)i(production,)g (the)e(relations)i(\(1.11\))f(follo)n(w)244 2433 y(only)32 b(from)g(a)f(fe)n(w)f(structural)k(properties)g(of)e(the)g(model)f Fw(S)i Fx(+)26 b Fw(R)2415 2447 y Fv(1)2481 2433 y Fx(+)f Fw(R)2654 2447 y Fv(2)2694 2433 y Fy(,)32 b(and)g(can)g(be)g(pro)o(v)o (en)g(in)244 2546 y(considerable)27 b(generality)-6 b(,)26 b(see)e([JP4)q(])f(for)g(details.)244 2829 y Fr(1.3)119 b(Remarks)244 3003 y Fy(Although)27 b(in)e(this)h(paper)h(we)e(ha)n(v)o (e)h(chosen)h(to)e(study)i(the)e(simplest)i(non-tri)n(vial)h(model,)e (our)g(results)244 3116 y(can)34 b(be)h(easily)g(e)o(xtended)h(to)e (the)h(case)g(where)f Fw(S)41 b Fy(is)34 b(an)g Fu(N)10 b Fy(-le)n(v)o(el)35 b(atom,)h(there)f(are)g Fu(M)10 b Fy(-reserv)n(oirs)244 3229 y(instead)25 b(of)f(tw)o(o,)f(and)h Fu(V)1011 3243 y Fq(i)1061 3229 y Fy(is)g(a)f(\002nite)h(sum)f(of)g (terms)h(of)f(the)h(form)1282 3433 y Fu(Q)1354 3447 y Fq(i)1403 3433 y Fw(\012)c Fu(')p Fx(\()p Fu(\013)1646 3447 y Fq(i)p Fv(1)1710 3433 y Fx(\))15 b Fu(:)g(:)g(:)i(')p Fx(\()p 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Fy(-systems\).)30 b(It)23 b(is)f(an)h(important)h (open)244 4879 y(problem)h(to)e(pro)o(v)o(e)h(the)g(analog)h(of)e (Theorem)h(1.2)g(for)f(Bose)h(reserv)n(oirs.)380 4992 y(Among)30 b(the)g(w)o(orks)g(related)h(to)f(ours,)i(we)d(mention)h (the)g(one)h(of)e(Da)n(vies)i([Da],)f(where)g(the)g(dy-)244 5105 y(namics)c(of)g(the)g(system)g Fw(S)j Fx(+)1217 5037 y Fm(P)1313 5132 y Fq(i)1356 5105 y Fw(R)1433 5119 y Fq(i)1486 5105 y Fy(is)d(studied)h(in)f(the)g(v)n(an)g(Ho)o(v)o(e)f (weak-coupling)k(limit)p 3226 5038 33 4 v 26 w Fu(t)g Fx(=)g Fu(\025)3441 5072 y Fv(2)3480 5105 y Fu(t)p Fy(,)244 5218 y Fu(\025)c Fw(#)h Fx(0)p Fy(,)c Fu(t)j Fw(")h(1)p Fy(.)h(In)22 b(particular)l(,)j(Da)n(vies)e(pro)o(v)o(es)g(the)g(e)o (xistence)h(and)f(uniqueness)i(of)d(NESS)e(in)i(the)h(v)n(an)p eop %%Page: 10 10 10 9 bop 3680 100 a FA(10)244 407 y Fy(Ho)o(v)o(e)21 b(limit)h(\(this)h(state)g(coincides)i(with)c Fu(!)1627 421 y Fv(0)1688 407 y Fy(in)h(the)g(e)o(xpansion)i(\(1.10\)\).)30 b(Lebo)n(witz)22 b(and)g(Spohn)h([LS)o(])244 520 y(ha)n(v)o(e)h(used)g (Da)n(vies)h(results)f(to)g(study)g(the)g(thermodynamics)j(of)c(the)h (system)g Fw(S)j Fx(+)2912 452 y Fm(P)3008 547 y Fq(i)3052 520 y Fw(R)3129 534 y Fq(i)3180 520 y Fy(in)c(the)h(v)n(an)244 633 y(Ho)o(v)o(e)29 b(limit)h(steady)h(state)g Fu(!)1182 647 y Fv(0)1221 633 y Fy(.)47 b(There)30 b(is)g(a)g(substantial)j (literature)f(on)e(the)g(use)h(of)e(v)n(an)i(Ho)o(v)o(e)e(limit)244 746 y(and)j(Mark)o(o)o(vian)h(master)f(equations)j(in)c(statistical)j (mechanics,)i(see)31 b([GFV)o(,)g(Hak])g(for)h(references)244 858 y(and)f(additional)i(information.)52 b(The)29 b(results)j(be)o (yond)g(v)n(an)f(Ho)o(v)o(e)e(limit)i(are)f(scarce.)51 b(In)30 b([JP1)q(,)f(JP2])244 971 y(Theorem)f(1.2)g(w)o(as)f(pro)o(v)o (en)i(in)f(thermal)h(equilibrium)h(case)f(where)f Fu(\014)2522 985 y Fv(1)2595 971 y Fx(=)33 b Fu(\014)2750 985 y Fv(2)2790 971 y Fy(.)41 b(The)27 b(method)i(of)f(the)244 1084 y(proof)d(w)o(as)e 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b(This)22 b(hypothesis)j(is)d(dif)n(\002cult)h(to)f(v) o(erify)i(in)e(concrete)244 1875 y(models,)37 b(and)e(in)f(particular)i (it)e(is)g(not)g(kno)n(wn)g(whether)i(it)d(holds)i(for)g(the)f(model)g (studied)i(in)e(this)244 1988 y(paper)-5 b(.)380 2100 y(W)e(e)21 b(w)o(ould)h(lik)o(e)g(to)f(add)h(the)g(follo)n(wing)h (general)g(remark.)29 b(It)21 b(is)h(kno)n(wn)f(that)h(the)g(er)n (godic)i(proper)n(-)244 2213 y(ties)f(of)f Fu(C)561 2180 y Ft(\003)600 2213 y Fy(-dynamical)j(systems)e(in)f(thermal)i (equilibrium)h(are)d(encoded)j(in)d(the)h(spectrum)h(of)e(a)g(suit-)244 2326 y(able)30 b(self-adjoint)j(operator)l(,)g(the)d(quantum)h(K)m (oopman)f(operator)i(or)d(Liouvillean,)34 b(see)29 b(e.g.)47 b([JP2)q(].)244 2439 y(In)25 b(non-equilibrium)k(situations,)e(the)f (quantum)g(K)m(oopmanism)g(is)f(not)g(applicable,)j(and)d(it)g(has)g (been)244 2552 y(generally)k(belie)n(v)o(ed)g(that)f(the)f (understanding)32 b(of)27 b(NESS)d(requires)29 b(the)f(de)n(v)o (elopment)h(of)e(scattering)244 2665 y(theory)-6 b(.)38 b(In)26 b(the)h(models)g(of)f(physical)i(interest)f(this)g(is)f(a)g (dif)n(\002cult)h(task,)g(and)g(the)f(progress)i(has)f(been)244 2778 y(slo)n(w)g(\(see)h(ho)n(we)n(v)o(er)f([DG1,)f(DG2,)g(FGS)o(]\).) 40 b(A)26 b(perhaps)j(surprising)h(aspect)f(of)e(our)h(method)g(is)f (that)244 2891 y(at)22 b(least)i(in)e(some)h(situations,)i(the)e (spectral)i(approach)g(to)d(NESS)e(is)j(possible,)h(and)g(that)f(the)g (structure)244 3004 y(of)30 b(NESS)e(is)i(encoded)i(in)f(the)f (spectral)j(resonances)g(of)d(a)g(suitable)i(non-self)o(adjoint)j (operator)l(,)g(the)244 3117 y Fu(C)7 b Fy(-Liouvillean.)380 3230 y(The)23 b(paper)i(is)e(or)n(ganized)j(as)e(follo)n(ws.)380 3342 y(The)36 b(method)g(of)g(the)g(proof)h(is)f(described)i(in)e(the)g (abstract)h(setting)h(in)d(Section)i(2)f(where)g(we)244 3455 y(introduce)g(the)d(concept)i(of)e Fu(C)7 b Fy(-Liouvillean,)37 b Fu(L)p Fy(,)d(and)g(sho)n(w)f(ho)n(w)f(the)i(NESS)c(of)j(an)g (abstract)i Fu(C)3466 3422 y Ft(\003)3505 3455 y Fy(-)244 3568 y(dynamical)d(system)f(are)f(related)h(to)f(the)g(resonances)j(of) d Fu(L)2185 3535 y Ft(\003)2225 3568 y Fy(.)47 b(The)30 b(results)h(of)f(Section)h(2)f(are)g(quite)244 3681 y(general)d(and,)f (we)f(belie)n(v)o(e,)i(shed)f(some)f(light)i(on)e(the)h(structure)i(of) d(non-equilibrium)30 b(quantum)c(sta-)244 3794 y(tistical)g(mechanics.) 33 b(In)25 b(Sections)g(3)g(and)f(4)h(we)e(apply)j(the)f(abstract)h (formalism)f(of)g(Section)g(2)f(to)h(the)244 3907 y(speci\002c)k(model) g Fw(S)h Fx(+)23 b Fw(R)1060 3921 y Fv(1)1124 3907 y Fx(+)g Fw(R)1295 3921 y Fv(2)1362 3907 y Fy(\226)28 b(in)g(Section)h(3) f(we)f(e)o(xplicitly)j(compute)g(the)e(modular)i(structure)244 4020 y(and)f Fu(C)7 b Fy(-Liouvillean)32 b Fu(L)p Fy(,)d(and)g(in)g (Section)h(4)f(we)f(study)i(the)g(resonances)i(of)d Fu(L)2792 3987 y Ft(\003)2859 4020 y Fy(using)i(the)e(comple)o(x)244 4133 y(deformation)d(technique)g(pre)n(viously)g(de)n(v)o(eloped)g(in)e ([JP1].)244 4345 y Fz(Ackno)o(wledgments.)k Fy(W)-7 b(e)23 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y Ft(\003)1099 1291 y Fy(-dynamical)i(system)e(and)h Fu(!)h Fy(a)d(gi)n(v)o(en)h(f)o(aithful)i(state.)55 b(Let)31 b Fx(\()p Fw(H)q Fu(;)15 b(\031)s(;)g Fx(\012\))32 b Fy(be)244 1404 y(the)f(GNS-representation)k(of)c(the)g(algebra)i Fw(O)g Fy(associated)h(to)d Fu(!)j Fy(\(for)d(simplicity)-6 b(,)35 b(we)30 b(write)i Fw(H)f Fy(for)244 1517 y Fw(H)321 1531 y Fq(!)371 1517 y Fy(,)h(etc\).)51 b(Since)31 b Fu(!)i Fy(is)e(f)o(aithful,)j Fu(\031)f Fy(is)e(an)g(injection)i(and)f (we)e(can)h(identify)i Fw(O)g Fy(and)e Fu(\031)s Fx(\()p Fw(O)s Fx(\))g Fy(\(with)g(a)244 1630 y(slight)25 b(ab)n(use)g(of)e (notation,)j(we)c(write)i Fu(A)f Fy(for)h Fu(\031)s Fx(\()p Fu(A)p Fx(\))p Fy(\).)29 b(W)-7 b(e)23 b(set)g Fi(M)j Fx(=)f Fu(\031)s Fx(\()p Fw(O)s Fx(\))2654 1597 y Ft(00)2720 1630 y Fy(and)f(assume)g(that)g Fx(\012)f Fy(is)g(a)244 1743 y(separating)j(v)o(ector)f(for)f(the)f(v)n(on)i(Neumann)f(algebra) h Fi(M)e Fy(\()p Fu(A)j Fw(2)f Fi(M)p Fu(;)15 b(A)p Fx(\012)26 b(=)f(0)g Fw(\))g Fu(A)h 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Fu(A)p Fx(\012)15 b Fw(j)g Fu(A)25 b Fw(2)g(D)s Fx(\()p Fu(\016)2273 5198 y Fq(V)2335 5184 y Fx(\))p Fw(g)16 b Fu(;)p eop %%Page: 13 13 13 12 bop 3680 100 a FA(13)244 407 y Fy(and)1523 520 y Fx(i)p Fu(L)1611 534 y Fq(V)1671 520 y Fu(A)p Fx(\012)25 b(=)g Fu(\016)1966 534 y Fq(V)2027 520 y Fx(\()p Fu(A)p Fx(\)\012)p Fu(:)244 683 y Fy(W)-7 b(e)23 b(proceed)i(to)e(compute)i (the)f(operator)i Fu(L)1647 697 y Fq(V)1730 683 y Fy(in)e(terms)f(of)h (the)g(modular)g(structure.)380 796 y(Let)f Fu(A)j Fw(2)e(D)s Fx(\()p Fu(\016)851 810 y Fq(V)913 796 y Fx(\))h(=)g Fw(D)s Fx(\()p Fu(\016)s Fx(\))f Fy(be)g(gi)n(v)o(en.)29 b(Dif)n(ferentiating)e(the)d(relation)1255 994 y Fx(e)1295 957 y Fv(i)p Fq(tL)1388 968 y Fn(V)1446 994 y Fu(A)p Fx(\012)h(=)g(e)1741 957 y Fv(i)p Fq(t)p Fv(\()p Ft(L)p Fv(+)p Fq(V)16 b Fv(\))2006 994 y Fu(A)p Fx(e)2114 957 y Ft(\000)p Fv(i)p Fq(t)p Fv(\()p Ft(L)p Fv(+)p Fq(V)g Fv(\))2434 994 y Fx(\012)p Fu(;)244 1192 y Fy(and)24 b(setting)h Fu(t)g Fx(=)g(0)e 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Ft(\000)p Fv(i)p Fq(=)p Fv(2)2284 2837 y Fu(J)o(;)1421 3055 y(L)1483 3017 y Ft(\003)1483 3077 y Fq(V)1569 3055 y Fx(=)25 b Fw(L)20 b Fx(+)g Fu(V)40 b Fw(\000)20 b Fu(J)9 b(V)2135 3073 y Fv(i)p Fq(=)p Fv(2)2230 3055 y Fu(J)o(:)244 3258 y Fy(Moreo)o(v)o(er)l(,)22 b(if)e(\(R3\))g(holds,)i(then)f(one)g(can)g (tak)o(e)g Fu(a)k Fx(=)g Fw(k)p Fu(V)2050 3276 y Fv(i)p Fq(=)p Fv(2)2145 3258 y Fw(k)g Fx(=)g Fw(k)p Fu(V)2409 3276 y Ft(\000)p Fv(i)p Fq(=)p Fv(2)2559 3258 y Fw(k)20 b Fy(and)h Fu(M)35 b Fx(=)25 b(1)20 b Fy(in)g(\(R1\)-\(R2\).)380 3371 y(If)25 b Fu(!)i Fy(is)e(a)g Fx(\()p Fu(\034)5 b(;)15 b(\014)5 b Fx(\))p Fy(-KMS)24 b(state,)i(there)g(is)f(an)g(important)i (relation)g(between)f(standard)h(Liouvillean)244 3484 y Fw(L)307 3498 y Fq(V)390 3484 y Fy(and)d Fu(C)7 b Fy(-Liouvillean)26 b Fu(L)1150 3498 y Fq(V)1211 3484 y Fy(.)i(A)22 b(simple)i(computation) i(sho)n(ws)e(that)g(for)g Fu(t)h Fw(2)g Fp(R)s Fy(,)1100 3688 y Fw(L)20 b Fx(+)g Fu(V)40 b Fw(\000)20 b Fu(J)9 b(V)1570 3702 y Fq(t)1600 3688 y Fu(J)34 b Fx(=)25 b(e)1820 3650 y Ft(\000)p Fv(i)p Fq(\014)s(t)p Fv(\()p Ft(L)p Fv(+)p Fq(V)16 b Fv(\))2183 3688 y Fw(L)2246 3702 y Fq(V)2307 3688 y Fx(e)2347 3650 y Fv(i)p Fq(\014)s(t)p Fv(\()p Ft(L)p Fv(+)p Fq(V)g Fv(\))2654 3688 y Fu(:)244 3892 y Fy(If)23 b(\(R3\))h(holds,)g(then)g(by)g(analytic)h(continuation)j (the)23 b(relation)1293 4096 y Fu(L)1355 4110 y Fq(V)1441 4096 y Fx(=)i(e)1577 4059 y Fq(\014)s Fv(\()p Ft(L)p Fv(+)p Fq(V)16 b Fv(\))p Fq(=)p Fv(2)1910 4096 y Fw(L)1973 4110 y Fq(V)2034 4096 y Fx(e)2074 4059 y Ft(\000)p Fq(\014)s Fv(\()p Ft(L)p Fv(+)p Fq(V)f Fv(\))p Fq(=)p Fv(2)2462 4096 y Fu(;)829 b Fy(\(2.18\))244 4300 y(holds)28 b(in)f(quadratic)i (form)e(sense)h(on)f(a)f(domain)i Fw(D)s Fx(\(e)1984 4267 y Ft(\000)p Fq(\014)s Fv(\()p Ft(L)p Fv(+)p Fq(V)16 b Fv(\))p Fq(=)p Fv(2)2372 4300 y Fx(\))23 b Fw(\\)f(D)s Fx(\(e)2661 4267 y Fq(\014)s Fv(\()p Ft(L)p Fv(+)p Fq(V)16 b Fv(\))p Fq(=)p Fv(2)2994 4300 y Fx(\))p Fy(.)38 b(The)27 b(identity)244 4413 y(\(2.18\))g(leads)h(to)e(a)g(simpler)i(proof)g(of) e(some)h(fundamental)i(results)f(of)f(Araki')-5 b(s)27 b(theory)h(of)e(perturba-)244 4526 y(tions)33 b(of)e Fu(W)657 4493 y Ft(\003)696 4526 y Fy(-dynamical)j(systems)f(\(see)f ([DJP])f(for)h(details\).)55 b(It)31 b(can)i(also)f(be)g(used)g(to)g (relate)h(the)244 4639 y(method)27 b(of)f(the)h(proof)g(of)f(Theorem)g (1.2,)h(restricted)h(to)f(thermal)g(equilibrium)h(case)f Fu(\014)3064 4653 y Fv(1)3134 4639 y Fx(=)j Fu(\014)3286 4653 y Fv(2)3356 4639 y Fx(=)g Fu(\014)5 b Fy(,)244 4752 y(to)23 b(the)h(method)h(of)e([JP1)q(,)f(JP2)q(].)28 b(F)o(or)22 b(reasons)k(of)d(space)i(we)d(omit)i(the)g(details.)380 4865 y(If)g Fu(!)h Fy(is)e(not)h(a)f(KMS-state,)h(then)g(there)h(is)e (no)h(direct)g(relation)i(between)e Fu(L)2769 4879 y Fq(V)2853 4865 y Fy(and)g Fw(L)3070 4879 y Fq(V)3130 4865 y Fy(.)p eop %%Page: 15 15 15 14 bop 3680 100 a FA(15)244 407 y Fr(2.3)119 b(Spectral)31 b(theory)f(of)f(NESS)244 581 y Fy(Our)h(goal)i(is)f(to)g(study)h(NESS)c (using)k(spectral)h(theory)f(of)f Fu(C)7 b Fy(-Liouvilleans.)54 b(F)o(or)30 b(this)h(reason)h(it)f(is)244 694 y(more)25 b(con)l(v)o(enient)j(to)c(deal)i(with)e(NESS)f(de\002ned)i(using)h (Abelian)g(limits.)33 b(The)24 b(weak-*)i(limit)e(points)244 807 y(of)f(the)h(set)g(of)f(states)1519 942 y Fu(\017)1571 819 y Fm(Z)1662 845 y Ft(1)1622 1025 y Fv(0)1752 942 y Fx(e)1792 905 y Ft(\000)p Fq(\017t)1905 942 y Fu(!)g Fw(\016)e Fu(\034)2101 905 y Fq(t)2091 965 y(V)2152 942 y Fx(d)p Fu(t;)244 1161 y Fy(as)i Fu(\017)i Fw(#)h Fx(0)d Fy(we)g(denote)i(by)e Fx(\006)1116 1123 y Fv(+)1116 1190 y Fq(V)t(;)p Fv(Ab)1277 1161 y Fx(\()p Fu(!)s Fx(\))p Fy(.)28 b(The)23 b(set)g Fx(\006)1811 1123 y Fv(+)1811 1190 y Fq(V)t(;)p Fv(Ab)1972 1161 y Fx(\()p Fu(!)s Fx(\))g Fy(is)g(a)g(non-empty)j(weak-*)e(compact)g(subset)244 1274 y(of)f Fu(E)5 b Fx(\()p Fw(O)s Fx(\))24 b Fy(whose)g(elements)h (are)f Fu(\034)1358 1288 y Fq(V)1418 1274 y Fy(-in)l(v)n(ariant.)32 b(Moreo)o(v)o(er:)244 1482 y Fz(Pr)n(oposition)25 b(2.3)49 b Fj(If)23 b(either)i Fx(\006)1256 1443 y Fv(+)1256 1511 y Fq(V)t(;)p Fv(Ab)1416 1482 y Fx(\()p Fu(!)s Fx(\))f Fj(or)f Fx(\006)1739 1443 y Fv(+)1739 1510 y Fq(V)1800 1482 y Fx(\()p Fu(!)s Fx(\))g Fj(consists)j(of)d(a)g(single)i(state)o (,)g(then)1509 1693 y Fx(\006)1575 1655 y Fv(+)1575 1722 y Fq(V)t(;)p Fv(Ab)1736 1693 y Fx(\()p Fu(!)s Fx(\))h(=)f(\006)2054 1655 y Fv(+)2054 1721 y Fq(V)2114 1693 y Fx(\()p Fu(!)s Fx(\))p Fu(:)244 1888 y Fy(The)e(proof)i(of)e(this)h(proposition)j (follo)n(ws)d(from)g(standard)h(Abelian)g(and)f(T)-7 b(auberian)25 b(theorems)g([Si].)380 2001 y(W)l(ith)20 b(a)g(slight)h(ab)n(use)g(of)e(terminology)k(we)c(will)g(also)h(call)h (the)f(elements)h(of)e Fx(\006)2873 1963 y Fv(+)2873 2030 y Fq(V)t(;)p Fv(Ab)3034 2001 y Fx(\()p Fu(!)s Fx(\))g Fy(the)h(NESS)244 2114 y(of)j Fx(\()p Fw(O)s Fu(;)15 b(\034)532 2128 y Fq(V)594 2114 y Fx(\))23 b Fy(associated)j(to)e(the)f (initial)i(state)g Fu(!)s Fy(.)380 2227 y(In)f(what)f(follo)n(ws)h(we)f (assume)h(that)g(the)g(assumptions)j(\(R1\))c(and)h(\(R2\))f(hold.)380 2340 y(Our)18 b(goal)h(is)f(to)g(characterize)j(NESS)15 b(in)j Fx(\006)1728 2301 y Fv(+)1728 2369 y Fq(V)t(;)p Fv(Ab)1889 2340 y Fx(\()p Fu(!)s Fx(\))g Fy(in)g(terms)g(of)g(the)g (corresponding)23 b(C-Liouvillean.)244 2466 y(T)-7 b(o)25 b(moti)n(v)n(ate)j(this)f(characterization,)32 b(for)27 b Fx(Im)o Fu(z)36 b(>)30 b(a)c Fy(let)h Fx(\012)2166 2480 y Fq(z)2237 2466 y Fx(:=)k(\()p Fu(z)c Fw(\000)22 b Fu(L)2623 2433 y Ft(\003)2623 2493 y Fq(V)2684 2466 y Fx(\))2719 2433 y Ft(\000)p Fv(1)2813 2466 y Fx(\012)p Fy(,)k(and)i(let)e Fu(!)3259 2480 y Fq(z)3330 2466 y Fw(2)k(O)3496 2433 y Ft(\003)244 2579 y Fy(be)23 b(de\002ned)i(by)e Fu(!)813 2593 y Fq(z)853 2579 y Fx(\()p Fu(A)p Fx(\))j(=)f(\(\012)p Fu(;)15 b(A)p Fx(\012)1388 2593 y Fq(z)1428 2579 y Fx(\))p Fy(.)28 b(Then,)23 b(since)1279 2818 y Fu(!)1336 2832 y Fq(z)1375 2818 y Fx(\()p Fu(A)p Fx(\))j(=)1645 2757 y(1)p 1645 2797 46 4 v 1655 2881 a(i)1716 2694 y Fm(Z)1807 2721 y Ft(1)1766 2901 y Fv(0)1897 2818 y Fx(e)1937 2781 y Fv(i)p Fq(z)s(t)2022 2818 y Fu(!)s Fx(\()p Fu(\034)2167 2781 y Fq(t)2157 2841 y(V)2218 2818 y Fx(\()p Fu(A)p Fx(\)\)d)p Fu(t;)244 3070 y Fy(the)c(functionals)i Fu(!)857 3084 y Fq(z)918 3070 y Fy(ha)n(v)o(e)e(weak-)p Fw(\003)g Fy(analytic)i(e)o(xtension)g(to)d(the)h(half-plane)i Fx(Im)p Fu(z)29 b(>)c Fx(0)d Fy(and)g Fx(\006)3245 3031 y Fv(+)3245 3099 y Fq(V)t(;)p Fv(Ab)3405 3070 y Fx(\()p Fu(!)s Fx(\))244 3182 y Fy(is)30 b(the)h(weak-)p Fw(\003)g Fy(limit)g(point)h(set)e(of)h(the)g(set)f(of)h(states)h Fw(f)p Fx(i)p Fu(\017!)2201 3196 y Fv(i)p Fq(\017)2267 3182 y Fw(j)15 b Fu(\017)39 b(>)f Fx(0)p Fw(g)30 b Fy(as)h Fu(\017)38 b Fw(#)h Fx(0)p Fy(.)49 b(W)-7 b(e)30 b(wish)g(to)h(go)244 3295 y(further)22 b(along)g(these)g(lines)g(and)g(characterize)i Fx(\006)1809 3257 y Fv(+)1809 3325 y Fq(V)t(;)p Fv(Ab)1969 3295 y Fx(\()p Fu(!)s Fx(\))d Fy(directly)i(in)d(terms)i(of)e(the)i(v)o (ectors)g Fx(\012)3305 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Fy(be)g(the)h(norm)g(closure)h(of) f Fw(O)1460 421 y Fq(D)1524 407 y Fy(.)k(Our)23 b(ne)o(xt)h(tw)o(o)f (assumptions)j(are:)244 710 y Fz(\(DL2\))d Fw(O)574 677 y Fv(cl)571 737 y Fq(D)661 710 y Fx(=)h Fw(O)s Fy(.)244 1012 y Fz(\(DL3\))f Fy(The)g(set)h Fw(f)p Fu(D)910 979 y Ft(\000)p Fv(1)1004 1012 y Fu(A)1072 979 y Ft(\003)1112 1012 y Fx(\012)15 b Fw(j)g Fu(A)26 b Fw(2)e(O)1484 1026 y Fq(D)1549 1012 y Fw(g)f Fy(is)g(dense)i(in)e Fw(H)q Fy(.)380 1315 y(Let)31 b Fi(W)627 1277 y Fv(+)627 1344 y Fq(V)719 1315 y Fy(be)g(the)h(weak)g(limit)f(point)i(set)f(of)f Fx(i)p Fu(\017)p Fw(R)1998 1329 y Fq(D)2061 1315 y Fx(\(i)p Fu(\017)p Fx(\)\012)g Fy(as)g Fu(\017)40 b Fw(#)h Fx(0)p Fy(.)52 b(Since)32 b(the)g(unit)g(ball)g(in)f(a)244 1428 y(Hilbert)24 b(space)h(is)e(weakly)h(compact,)h(\(2.19\))f(implies)h (that)f Fi(W)2258 1390 y Fv(+)2258 1457 y Fq(V)2342 1428 y Fy(is)f(non-empty)-6 b(.)244 1637 y Fz(Pr)n(oposition)25 b(2.4)49 b Fj(Assume)23 b(that)i Fx(\(DL1\))f Fj(and)g Fx(\(DL2\))g Fj(hold.)29 b(Then)24 b(ther)m(e)g(is)g(an)f(injection) 1329 1834 y Fi(W)1424 1796 y Fv(+)1424 1863 y Fq(V)1510 1834 y Fw(3)i Fx(\012)1662 1796 y Fv(+)1662 1863 y Fq(V)1748 1834 y Fw(7!)g Fu(!)1924 1796 y Fv(+)1921 1863 y Fq(V)2008 1834 y Fw(2)g Fx(\006)2160 1796 y Fv(+)2160 1864 y Fq(V)t(;)p Fv(Ab)2320 1834 y Fx(\()p Fu(!)s Fx(\))866 b Fy(\(2.20\))244 2031 y Fj(suc)o(h)24 b(that)g(for)g Fu(A)h Fw(2)g(O)976 2045 y Fq(D)1041 2031 y Fj(,)1396 2144 y Fu(!)1456 2106 y Fv(+)1453 2173 y Fq(V)1515 2144 y Fx(\()p Fu(A)p Fx(\))h(=)f(\()p Fu(D)1888 2107 y Ft(\000)p Fv(1)1983 2144 y Fu(A)2051 2107 y Ft(\003)2090 2144 y Fx(\012)p Fu(;)15 b Fx(\012)2262 2106 y Fv(+)2262 2173 y Fq(V)2323 2144 y Fx(\))p Fu(:)933 b Fy(\(2.21\))267 2308 y Fj(If)23 b(in)g(addition)j Fy(\(DL3\))d Fj(holds,)i(then)f(the)g(map)f(\(2.20\))h(is)g(a)f(bijection.)244 2485 y Fz(Remark.)39 b Fy(The)26 b(v)o(ectors)j(in)e Fi(W)1271 2447 y Fv(+)1271 2514 y Fq(V)1358 2485 y Fy(are)h(naturally)h (interpreted)h(as)d(the)h(zero-resonance)j(eigen)l(v)o(ectors)244 2598 y(associated)k(to)e(the)f(triple)i Fx(\()p Fu(L)1215 2565 y Ft(\003)1215 2625 y Fq(V)1276 2598 y Fu(;)15 b(D)s(;)g Fx(\012\))p Fy(,)35 b(and)e(in)f(this)h(sense)h(Theorem)f(2.4)f (identi\002es)i(NESS)c(with)244 2711 y(zero)24 b(resonance)i(eigen)l(v) o(ectors)h(of)d Fu(L)1447 2678 y Ft(\003)1447 2738 y Fq(V)1508 2711 y Fy(.)244 2824 y Fz(Pr)n(oof)o(.)46 b Fy(Proposition)26 b(2.2)e(yields)g(that)g(for)g Fu(A)i Fw(2)f(O)1884 2838 y Fq(D)975 3062 y Fu(\017)1012 2939 y Fm(Z)1103 2965 y Ft(1)1063 3145 y Fv(0)1193 3062 y Fx(e)1233 3025 y Ft(\000)p Fq(\017t)1346 3062 y Fu(!)s Fx(\()p Fu(\034)1491 3025 y Fq(t)1481 3085 y(V)1542 3062 y Fx(\()p Fu(A)p Fx(\)\)d)p Fu(t)h Fx(=)f(i)p Fu(\017)p Fx(\()p Fu(D)2097 3025 y Ft(\000)p Fv(1)2191 3062 y Fu(A)2259 3025 y Ft(\003)2298 3062 y Fx(\012)p Fu(;)15 b Fw(R)2481 3076 y Fq(D)2545 3062 y Fx(\(i)p Fu(\017)p Fx(\)\012\))p Fu(:)512 b Fy(\(2.22\))244 3316 y(Since)22 b Fw(O)542 3283 y Fv(cl)539 3343 y Fq(D)629 3316 y Fx(=)j Fw(O)s Fy(,)c(from)h(this)h(relation)h(it)e(follo)n(ws)h(that)f(each)h Fx(\012)2269 3278 y Fv(+)2269 3345 y Fq(V)2355 3316 y Fw(2)i Fi(W)2536 3278 y Fv(+)2536 3345 y Fq(V)2618 3316 y Fy(determines)g(a)c(unique)j(state)244 3429 y Fu(!)304 3391 y Fv(+)301 3458 y Fq(V)388 3429 y Fw(2)h Fx(\006)540 3391 y Fv(+)540 3459 y Fq(V)t(;)p Fv(Ab)700 3429 y Fx(\()p Fu(!)s Fx(\))f Fy(and)g(that)g(\(2.21\))g(holds)h(for)f Fu(A)h Fw(2)g(O)2009 3443 y Fq(D)2073 3429 y Fy(.)380 3542 y(If)31 b(in)h(addition)h(\(DL3\))e(holds,)k(then)d(Relation)h (\(2.22\))f(and)g(the)g(uniform)g(bound)h(\(2.19\))f(imply)244 3655 y(that)24 b(each)g Fu(!)652 3617 y Fv(+)649 3684 y Fq(V)736 3655 y Fw(2)h Fx(\006)888 3617 y Fv(+)888 3685 y Fq(V)t(;)p Fv(Ab)1049 3655 y Fx(\()p Fu(!)s Fx(\))e Fy(determines)j(a)d(unique)i Fx(\012)2023 3617 y Fv(+)2023 3684 y Fq(V)2109 3655 y Fw(2)g Fi(W)2290 3617 y Fv(+)2290 3684 y Fq(V)2351 3655 y Fy(.)j Fe(2)380 3816 y Fy(An)34 b(immediate)h(consequence)i(of)d(Proposition)j(2.4)d(is)f(that)i(under) g(the)f(assumptions)j(\(DL1\)-)244 3928 y(\(DL3\),)23 b Fx(\006)583 3890 y Fv(+)583 3958 y Fq(V)t(;)p Fv(Ab)743 3928 y Fx(\()p Fu(!)s Fx(\))g Fy(consists)j(of)d(a)g(single)i(state)g Fu(!)1858 3890 y Fv(+)1855 3957 y Fq(V)1939 3928 y Fy(if)n(f)1393 4142 y Fx(w)c Fw(\000)f Fx(lim)1584 4203 y Fq(\017)p Ft(#)p Fv(0)1712 4142 y Fx(i)p Fu(\017)p Fw(R)1852 4156 y Fq(D)1915 4142 y Fx(\(i)p Fu(\017)p Fx(\)\012)25 b(=)g(\012)2301 4103 y Fv(+)2301 4170 y Fq(V)2362 4142 y Fu(;)244 4380 y Fy(and)f(in)f(this)h(case)h(for)e(all)h Fu(A)h Fw(2)g(O)h Fy(we)d(ha)n(v)o(e)708 4619 y Fx(lim)675 4679 y Fq(T)10 b Ft(!1)902 4557 y Fx(1)p 892 4598 66 4 v 892 4681 a Fu(T)983 4495 y Fm(Z)1074 4521 y Fq(T)1034 4701 y Fv(0)1144 4619 y Fu(!)s Fx(\()p Fu(\034)1289 4581 y Fq(t)1279 4641 y(V)1340 4619 y Fx(\()p Fu(A)p Fx(\)\)d)p Fu(t)26 b Fx(=)f(lim)1733 4680 y Fq(\017)p Ft(#)p Fv(0)1860 4619 y Fu(\017)1912 4495 y Fm(Z)2003 4521 y Ft(1)1963 4701 y Fv(0)2093 4619 y Fx(e)2133 4581 y Ft(\000)p Fq(\017t)2247 4619 y Fu(!)s Fx(\()p Fu(\034)2392 4581 y Fq(t)2382 4641 y(V)2443 4619 y Fx(\()p Fu(A)p Fx(\)\)d)p Fu(t)g Fx(=)g Fu(!)2881 4580 y Fv(+)2878 4647 y Fq(V)2940 4619 y Fx(\()p Fu(A)p Fx(\))p Fu(:)380 4856 y Fy(T)-7 b(o)23 b(re\002ne)g(the)h(abo)o(v)o(e)g (result,)h(we)e(need)h(additional)i(assumptions.)32 b(Let)1158 5054 y Fi(M)1253 5016 y Ft(0)1253 5076 y Fq(D)1343 5054 y Fx(=)1439 4980 y Fm(\010)1492 5054 y Fu(C)f Fw(2)25 b Fi(M)1769 5016 y Ft(0)1808 4976 y Fm(\014)1808 5031 y(\014)1854 5054 y Fu(C)1926 5016 y Ft(\003)1964 5054 y Fu(C)7 b Fx(\012)25 b Fw(2)g(D)s Fx(\()p Fu(D)2399 5016 y Ft(\000)p Fv(1)2493 5054 y Fx(\))2528 4980 y Fm(\011)2596 5054 y Fu(;)244 5268 y Fy(and)f(let)g Fx(\()p Fi(M)642 5235 y Ft(0)642 5295 y Fq(D)706 5268 y Fx(\012\))807 5224 y Fv(cl)885 5268 y Fy(be)g(the)f(closure)j(of)d Fi(M)1606 5235 y Ft(0)1606 5295 y Fq(D)1671 5268 y Fx(\012)f Fy(in)h Fw(H)q Fy(.)p eop %%Page: 17 17 17 16 bop 3680 100 a FA(17)244 407 y Fz(\(DL4\))23 b Fx(\()p Fi(M)629 374 y Ft(0)629 434 y Fq(D)694 407 y Fx(\012\))795 363 y Fv(cl)875 407 y Fx(=)i Fw(H)q Fy(.)244 719 y(Note)e(that)h(since)h Fx(\012)d Fy(is)i(a)f(separating)j(v)o (ector)e(for)g Fi(M)p Fy(,)f Fx(\()p Fi(M)2091 686 y Ft(0)2115 719 y Fx(\012\))2216 686 y Fv(cl)2296 719 y Fx(=)i Fw(H)q Fy(.)j(W)-7 b(e)23 b(denote)i(by)e Fw(N)3116 733 y Fq(D)3202 719 y Fy(the)h(set)g(of)244 832 y(v)o(ector)g(states) 1509 945 y Fu(\021)s Fx(\()15 b Fw(\001)g Fx(\))27 b(=)e(\()p Fu(C)7 b Fx(\012)p Fu(;)30 b Fw(\001)15 b Fu(C)7 b Fx(\012\))p Fu(;)244 1112 y Fy(where)24 b Fu(C)31 b Fw(2)25 b Fi(M)766 1079 y Ft(0)766 1139 y Fq(D)853 1112 y Fy(and)f Fw(k)p Fu(C)7 b Fx(\012)p Fw(k)26 b Fx(=)f(1)p Fy(.)j(\(DL4\))23 b(implies)h(that)g Fw(N)2227 1126 y Fq(D)2314 1112 y Fy(is)f(norm-dense)j(in)e Fw(N)13 b Fy(.)380 1225 y(W)-7 b(e)23 b(will)g(replace)i(assumption)h(\(DL1\))d(with)244 1537 y Fz(\(DL5\))k Fy(The)f(operator)n(-v)n(alued)32 b(function)e Fu(z)36 b Fw(7!)c(R)1876 1551 y Fq(D)1940 1537 y Fx(\()p Fu(z)t Fx(\))p Fy(,)d(originally)h(de\002ned)e(for)f Fx(Im)p Fu(z)36 b(>)c(a)p Fy(,)27 b(has)h(an)244 1650 y(analytic)d(continuation)j(to)23 b(the)h(re)o(gion)h Fx(Im)o Fu(z)30 b(>)25 b Fx(0)e Fy(and)h(there)g(is)g(a)f(bounded)j (operator)f Fu(P)3094 1611 y Fv(+)3081 1678 y Fq(V)3176 1650 y Fy(such)f(that)1424 1854 y Fx(w)d Fw(\000)f Fx(lim)1615 1915 y Fq(\017)p Ft(#)p Fv(0)1743 1854 y Fx(i)p Fu(\017)p Fw(R)1883 1868 y Fq(D)1946 1854 y Fx(\(i)p Fu(\017)p Fx(\))26 b(=)e Fu(P)2271 1816 y Fv(+)2258 1882 y Fq(V)2331 1854 y Fu(:)244 2404 y Fz(Pr)n(oposition)h(2.5)49 b Fj(Assume)38 b(that)h(the)f(assumptions)j Fx(\(DL2\))p Fu(;)15 b Fx(\(DL4\))40 b Fj(and)f Fx(\(DL5\))g Fj(hold)g(and)f(that)244 2517 y Fx(dim)14 b(Ran)o Fu(P)644 2479 y Fv(+)631 2545 y Fq(V)729 2517 y Fx(=)25 b(1)p Fj(.)j(Then,)23 b(for)h(all)g Fu(\021)k Fw(2)d(N)13 b Fj(,)1300 2721 y Fx(\006)1366 2683 y Fv(+)1366 2750 y Fq(V)t(;)p Fv(Ab)1526 2721 y Fx(\()p Fu(\021)s Fx(\))27 b(=)e(\006)1833 2683 y Fv(+)1833 2750 y Fq(V)t(;)p Fv(Ab)1993 2721 y Fx(\()p Fu(!)s Fx(\))h(=)f Fw(f)p Fu(!)2350 2683 y Fv(+)2347 2750 y Fq(V)2409 2721 y Fw(g)p Fu(:)244 2941 y Fz(Pr)n(oof)o(.)52 b Fy(Note)26 b(that)h(since)h Fu(P)1180 2903 y Fv(+)p Ft(\003)1167 2970 y Fq(V)1274 2941 y Fx(\012)i(=)g(\012)c Fy(and)h Fx(dim)13 b(Ran)p Fu(P)2120 2903 y Fv(+)2107 2970 y Fq(V)2210 2941 y Fx(=)30 b(1)p Fy(,)c Fu(P)2476 2903 y Fv(+)2463 2970 y Fq(V)2536 2941 y Fx(\()15 b Fw(\001)g Fx(\))31 b(=)g(\(\012)p Fu(;)f Fw(\001)15 b Fx(\)\012)3091 2903 y Fv(+)3091 2970 y Fq(V)3153 2941 y Fy(.)36 b(T)-7 b(o)25 b(pro)o(v)o(e)244 3054 y(the)f (proposition)j(it)c(suf)n(\002ces)h(to)f(sho)n(w)h(that)g(for)g Fu(\021)k Fw(2)d(N)2014 3068 y Fq(D)2101 3054 y Fy(and)f Fu(A)h Fw(2)g(O)2506 3068 y Fq(D)2570 3054 y Fy(,)1048 3300 y Fx(lim)1061 3361 y Fq(\017)p Ft(#)p Fv(0)1189 3300 y Fu(\017)1226 3176 y Fm(Z)1317 3202 y Ft(1)1277 3382 y Fv(0)1407 3300 y Fx(e)1447 3262 y Ft(\000)p Fq(\017t)1560 3300 y Fu(\021)s Fx(\()p Fu(\034)1693 3262 y Fq(t)1683 3322 y(V)1745 3300 y Fx(\()p Fu(A)p Fx(\)\)d)p Fu(t)h Fx(=)e(\()p Fu(D)2236 3262 y Ft(\000)p Fv(1)2331 3300 y Fu(A)2399 3262 y Ft(\003)2439 3300 y Fx(\012)p Fu(;)15 b Fx(\012)2611 3261 y Fv(+)2611 3328 y Fq(V)2671 3300 y Fx(\))p Fu(:)585 b Fy(\(2.23\))244 3552 y(Let)23 b Fu(\021)30 b Fw(2)d(N)626 3566 y Fq(D)713 3552 y Fy(and)d Fu(A)j Fw(2)f(O)1121 3566 y Fq(D)1209 3552 y Fy(be)e(gi)n(v)o(en.)32 b(Let)23 b Fu(C)33 b Fw(2)26 b Fi(M)1994 3519 y Ft(0)1994 3579 y Fq(D)2082 3552 y Fy(be)e(such)h(that)g Fu(\021)s Fx(\()15 b Fw(\001)g Fx(\))28 b(=)f(\()p Fu(C)7 b Fx(\012)p Fu(;)30 b Fw(\001)15 b Fu(C)7 b Fx(\012\))p Fy(.)30 b(Since)244 3665 y Fx([)p Fu(C)q(;)15 b(\034)425 3632 y Fq(t)415 3692 y(V)477 3665 y Fx(\()p Fu(A)p Fx(\)])26 b(=)f(0)p Fy(,)e(we)g(deri)n(v)o(e)h(from)f(Proposition)j(2.2)e(that)841 3794 y Fm(Z)931 3820 y Ft(1)891 4000 y Fv(0)1021 3918 y Fx(e)1061 3880 y Ft(\000)p Fq(\017t)1175 3918 y Fu(\021)s Fx(\()p Fu(\034)1308 3880 y Fq(t)1298 3940 y(V)1359 3918 y Fx(\()p Fu(A)p Fx(\)\)d)p Fu(t)i Fx(=)1738 3794 y Fm(Z)1829 3820 y Ft(1)1788 4000 y Fv(0)1919 3918 y Fx(e)1959 3880 y Ft(\000)p Fq(\017t)2072 3918 y Fx(\()p Fu(C)7 b Fx(\012)p Fu(;)15 b(\034)2335 3880 y Fq(t)2325 3940 y(V)2386 3918 y Fx(\()p Fu(A)p Fx(\))p Fu(C)7 b Fx(\012\)d)p Fu(t)1642 4185 y Fx(=)25 b(i\()p Fu(D)1877 4147 y Ft(\000)p Fv(1)1971 4185 y Fu(A)2039 4147 y Ft(\003)2078 4185 y Fx(\012)p Fu(;)15 b Fw(R)2261 4199 y Fq(D)2325 4185 y Fx(\(i)p Fu(\017)p Fx(\))p Fu(D)2536 4147 y Ft(\000)p Fv(1)2630 4185 y Fu(C)2702 4147 y Ft(\003)2741 4185 y Fu(C)7 b Fx(\012\))p Fu(:)244 4380 y Fy(Therefore)836 4516 y Fx(lim)849 4577 y Fq(\017)p Ft(#)p Fv(0)977 4516 y Fu(\017)1014 4392 y Fm(Z)1105 4418 y Ft(1)1065 4598 y Fv(0)1195 4516 y Fx(e)1235 4478 y Ft(\000)p Fq(\017t)1348 4516 y Fu(\021)s Fx(\()p Fu(\034)1481 4478 y Fq(t)1471 4538 y(V)1533 4516 y Fx(\()p Fu(A)p Fx(\)\)d)p Fu(t)26 b Fx(=)f(\()p Fu(D)2025 4478 y Ft(\000)p Fv(1)2119 4516 y Fu(A)2187 4478 y Ft(\003)2227 4516 y Fx(\012)p Fu(;)15 b(P)2404 4477 y Fv(+)2391 4544 y Fq(V)2463 4516 y Fu(D)2541 4478 y Ft(\000)p Fv(1)2635 4516 y Fu(C)2707 4478 y Ft(\003)2746 4516 y Fu(C)7 b Fx(\012\))p Fu(:)372 b Fy(\(2.24\))244 4739 y(Relations)25 b Fu(P)681 4701 y Fv(+)668 4768 y Fq(V)740 4739 y Fx(\()15 b Fw(\001)g Fx(\))27 b(=)e(\(\012)p Fu(;)30 b Fw(\001)15 b Fx(\)\012)1285 4701 y Fv(+)1285 4768 y Fq(V)1346 4739 y Fy(,)23 b Fu(D)1470 4706 y Ft(\000)p Fv(1)1564 4739 y Fx(\012)i(=)g(\012)p Fy(,)d(and)i Fw(k)p Fu(C)7 b Fx(\012)p Fw(k)25 b Fx(=)g(1)e Fy(yield)1478 4944 y Fu(P)1549 4905 y Fv(+)1536 4972 y Fq(V)1608 4944 y Fu(D)1686 4906 y Ft(\000)p Fv(1)1780 4944 y Fu(C)1852 4906 y Ft(\003)1891 4944 y Fu(C)7 b Fx(\012)25 b(=)g(\012)2216 4905 y Fv(+)2216 4972 y Fq(V)2276 4944 y Fu(;)1015 b Fy(\(2.25\))244 5148 y(and)24 b(\(2.23\))g(follo)n(ws)g(from)g(\(2.24\))g(and)g(\(2.25\).)30 b Fe(2)p eop %%Page: 18 18 18 17 bop 3680 100 a FA(18)380 407 y Fy(The)34 b(last)g(result)h(we)e (wish)g(to)h(discuss)i(concerns)g(conditions)h(under)e(which)f(the)g (approach)i(to)244 520 y(NESS)29 b(is)j(e)o(xponentially)k(f)o(ast.)55 b(F)o(or)31 b Fu(\026)41 b Fw(2)g Fp(R)f Fy(let)32 b Fi(P)p Fx(\()p Fu(\026)p Fx(\))g Fy(be)g(the)g(half-plane)j Fw(f)p Fu(z)20 b Fw(j)15 b Fx(Im)p Fu(z)45 b(>)c(\026)p Fw(g)p Fy(.)54 b(W)-7 b(e)244 633 y(replace)25 b(\(DL5\))e(with:)244 943 y Fz(\(DL6\))28 b Fy(The)f(operator)n(-v)n(alued)33 b(function)e Fu(z)38 b Fw(7!)c(R)1884 957 y Fq(D)1948 943 y Fx(\()p Fu(z)t Fx(\))p Fy(,)c(originally)h(de\002ned)e(for)f Fu(z)39 b Fw(2)33 b Fi(P)p Fx(\()p Fu(a)p Fx(\))p Fy(,)d(has)e(a)244 1056 y(meromorphic)d(continuation)j(to)23 b(a)g(half-plane)j Fi(P)p Fx(\()p Fu(\026)p Fx(\))e Fy(for)g(some)f Fu(\026)i(<)g Fx(0)p Fy(.)380 1366 y(Since)h Fx(\(\012)p Fu(;)15 b Fw(R)825 1380 y Fq(D)890 1366 y Fx(\()p Fu(z)t Fx(\)\012\))31 b(=)e(1)p Fu(=z)t Fy(,)e(zero)f(is)g(al)o(w)o(ays)h(a)f(pole)g(of)g Fw(R)2392 1380 y Fq(D)2456 1366 y Fx(\()p Fu(z)t Fx(\))p Fy(.)36 b(It)26 b(is)g(not)g(dif)n(\002cult)h(to)f(sho)n(w)244 1479 y(that)f(if)f(in)g(addition)j(\(DL3\))d(holds,)h(then)h(zero)f(is) f(a)g(simple)h(pole)g(of)g Fw(R)2525 1493 y Fq(D)2589 1479 y Fx(\()p Fu(z)t Fx(\))f Fy(and)h(all)g(other)g(poles)h(are)244 1592 y(in)g(the)g(half-plane)i Fx(Im)p Fu(z)34 b Fw(\024)29 b Fx(0)p Fy(.)35 b(In)26 b(particular)l(,)j(\(DL3\))c Fw(^)g Fy(\(DL6\))g Fw(\))g Fy(\(DL5\).)35 b(W)-7 b(e)25 b(will)g(not)h(mak)o(e)g(use)244 1705 y(of)d(assumption)j(\(DL3\))d (belo)n(w)-6 b(.)380 1818 y(Assume)36 b(in)g(addition)i(to)d(\(DL6\))h (that)g(the)g(function)i Fw(R)2247 1832 y Fq(D)2311 1818 y Fx(\()p Fu(z)t Fx(\))e Fy(has)g(only)h(\002nitely)f(man)o(y)g(poles) 244 1931 y Fw(f)p Fu(z)331 1945 y Fv(0)371 1931 y Fu(;)15 b(z)453 1945 y Fv(1)493 1931 y Fu(;)g(:)g(:)g(:)i(;)e(z)737 1945 y Fq(n)784 1931 y Fw(g)30 b Fy(\()p Fu(z)931 1945 y Fv(0)1008 1931 y Fx(=)37 b(0)p Fy(\))30 b(in)g(the)g(half-plane)j Fi(P)p Fx(\()p Fu(\026)p Fx(\))d Fy(and)g(let)g Fu(m)2449 1946 y Fq(k)2521 1931 y Fy(be)g(the)g(order)h(or)f(the)g(pole)h Fu(z)3470 1946 y Fq(k)3513 1931 y Fy(.)244 2044 y(Then)23 b(we)g(can)h(decompose)i Fw(R)1240 2058 y Fq(D)1304 2044 y Fx(\()p Fu(z)t Fx(\))e Fy(as)1374 2247 y Fw(R)1451 2261 y Fq(D)1515 2247 y Fx(\()p Fu(z)t Fx(\))i(=)f Fw(R)1830 2209 y Fv(a)1830 2269 y Fq(D)1894 2247 y Fx(\()p Fu(z)t Fx(\))c(+)f Fw(R)2199 2209 y Fv(s)2199 2269 y Fq(D)2263 2247 y Fx(\()p Fu(z)t Fx(\))p Fu(;)912 b Fy(\(2.26\))244 2449 y(where)24 b Fw(R)566 2416 y Fv(a)566 2476 y Fq(D)630 2449 y Fx(\()p Fu(z)t Fx(\))f Fy(is)h(an)f(analytic)j(operator)n(-v)n (alued)i(function)e(in)d(the)h(half-plane)i Fi(P)p Fx(\()p Fu(\026)p Fx(\))e Fy(and)1434 2726 y Fw(R)1511 2688 y Fv(s)1511 2748 y Fq(D)1575 2726 y Fx(\()p Fu(z)t Fx(\))i(=)1857 2612 y Fq(n)1813 2640 y Fm(X)1814 2837 y Fq(k)r Fv(=0)1959 2726 y Fu(S)2015 2741 y Fq(k)2058 2726 y Fx(\()p Fu(z)t Fx(\))1476 3105 y Fu(S)1532 3120 y Fq(k)1575 3105 y Fx(\()p Fu(z)t Fx(\))g(=)1828 2988 y Fq(m)1890 3000 y Fn(k)1813 3019 y Fm(X)1822 3214 y Fq(i)p Fv(=1)2079 3044 y Fu(S)2135 3059 y Fq(k)r(i)p 1969 3084 342 4 v 1969 3168 a Fx(\()p Fu(z)f Fw(\000)20 b Fu(z)2204 3183 y Fq(k)2247 3168 y Fx(\))2282 3141 y Fq(i)2321 3105 y Fu(:)3316 2922 y Fy(\(2.27\))380 3400 y(Let)j Fu(P)595 3362 y Fv(+)582 3429 y Fq(V)677 3400 y Fy(be)g(the)h(residue)i(of)d Fw(R)1380 3414 y Fq(D)1444 3400 y Fx(\()p Fu(z)t Fx(\))h Fy(at)f Fu(z)30 b Fx(=)25 b(0)p Fy(.)j(Then)1313 3651 y Fu(P)1384 3612 y Fv(+)1371 3679 y Fq(V)1468 3651 y Fx(=)1614 3589 y(1)p 1574 3630 126 4 v 1574 3713 a(2)p Fu(\031)s Fx(i)1725 3527 y Fm(I)1775 3733 y Fq(\015)1835 3651 y Fw(R)1912 3665 y Fq(D)1976 3651 y Fx(\()p Fu(z)t Fx(\)d)p Fu(z)i Fx(=)25 b Fu(S)2367 3665 y Fv(01)2441 3651 y Fu(;)244 3906 y Fy(where)35 b Fu(\015)40 b Fy(is)34 b(a)h(small)g(circle)h (around)h(zero)f(such)g(that)f(inside)i Fu(\015)i Fy(zero)d(is)f(the)g (only)h(singularity)i(of)244 4019 y Fw(R)321 4033 y Fq(D)385 4019 y Fx(\()p Fu(z)t Fx(\))p Fy(.)244 4231 y Fz(Theor)n(em)46 b(2.6)k Fj(Assume)23 b(the)h(following:)244 4344 y Fy(\(a\))g Fj(Assumptions)h Fx(\(DL2\))p Fu(;)15 b Fx(\(DL4\))26 b Fj(and)e Fx(\(DL6\))g Fj(hold.)244 4457 y Fy(\(b\))19 b Fj(The)e(function)k Fw(R)914 4471 y Fq(D)978 4457 y Fx(\()p Fu(z)t Fx(\))e Fj(has)g(only)g(\002nitely)h(many)e (singularities)23 b Fw(f)p Fu(z)2470 4471 y Fv(0)2510 4457 y Fu(;)15 b(z)2592 4471 y Fv(1)2632 4457 y Fu(;)g(:)g(:)g(:)i(;)e (z)2876 4471 y Fq(n)2923 4457 y Fw(g)j Fj(in)h Fi(P)p Fx(\()p Fu(\026)p Fx(\))p Fj(,)g(wher)m(e)244 4570 y Fu(z)286 4584 y Fv(0)351 4570 y Fx(=)25 b(0)e Fj(and)h Fx(Im)p Fu(z)825 4585 y Fq(k)893 4570 y Fu(<)h Fx(0)e Fj(for)h Fu(k)k Fw(\025)d Fx(1)p Fj(.)244 4683 y Fy(\(c\))f Fx(dim)13 b(Ran)p Fu(P)768 4645 y Fv(+)755 4711 y Fq(V)852 4683 y Fx(=)25 b(1)p Fj(.)244 4808 y Fy(\(d\))f Fj(F)-10 b(or)23 b(all)h Fx(\011)h Fw(2)g(H)e Fj(and)h Fu(j)31 b Fx(=)25 b(0)p Fu(;)15 b Fx(1)p Fj(,)24 b Fx(sup)1560 4830 y Fq(y)r(>\026)1714 4735 y Fm(R)1756 4840 y Fh(R)1824 4808 y Fw(j)p Fu(@)1902 4764 y Fq(j)1897 4820 y(x)1941 4808 y Fx(\(\011)p Fu(;)15 b Fw(R)2164 4775 y Fv(a)2164 4835 y Fq(D)2229 4808 y Fx(\()p Fu(x)20 b Fx(+)g(i)p Fu(y)s Fx(\)\011\))p Fw(j)2667 4775 y Fv(2)p Ft(\000)p Fq(j)2794 4808 y Fx(d)p Fu(x)k(<)h Fw(1)p Fj(.)244 4921 y(Then,)244 5034 y Fy(\(i\))e Fj(F)-10 b(or)24 b(all)g Fu(\021)k Fw(2)d(N)13 b Fj(,)23 b Fx(\006)980 4995 y Fv(+)980 5063 y Fq(V)t(;)p Fv(Ab)1140 5034 y Fx(\()p Fu(\021)s Fx(\))j(=)f(\006)1446 4995 y Fv(+)1446 5063 y Fq(V)t(;)p Fv(Ab)1607 5034 y Fx(\()p Fu(!)s Fx(\))h(=)f Fw(f)p Fu(!)1964 4995 y Fv(+)1961 5062 y Fq(V)2023 5034 y Fw(g)p Fj(.)j(Mor)m(eo)o(ver)-10 b(,)25 b(for)f(all)f Fu(A)j Fw(2)f(O)s Fj(,)1438 5253 y Fx(lim)1417 5309 y Fq(t)p Ft(!1)1599 5253 y Fu(\021)s Fx(\()p Fu(\034)1732 5215 y Fq(t)1722 5275 y(V)1784 5253 y Fx(\()p Fu(A)p Fx(\)\))h(=)f Fu(!)2139 5214 y Fv(+)2136 5281 y Fq(V)2198 5253 y Fx(\()p Fu(A)p Fx(\))p Fu(:)955 b Fy(\(2.28\))p eop %%Page: 19 19 19 18 bop 3680 100 a FA(19)244 407 y Fy(\(ii\))24 b Fj(F)-10 b(or)23 b(all)h Fu(\021)29 b Fw(2)24 b(N)880 421 y Fq(D)944 407 y Fj(,)f Fu(A)i Fw(2)g(O)1241 421 y Fq(D)1305 407 y Fj(,)e(and)h Fu(t)h(>)g Fx(0)p Fj(,)1134 526 y Fm(\014)1134 581 y(\014)1165 604 y Fu(\021)s Fx(\()p Fu(\034)1298 566 y Fq(t)1288 626 y(V)1349 604 y Fx(\()p Fu(A)p Fx(\)\))d Fw(\000)d Fu(!)1694 565 y Fv(+)1691 632 y Fq(V)1753 604 y Fx(\()p Fu(A)p Fx(\))1891 526 y Fm(\014)1891 581 y(\014)1948 604 y Fw(\024)25 b Fu(C)2109 618 y Fq(\021)r(;A)2223 604 y Fx(e)2263 566 y Ft(\000)p Fq(\015)t(t)2388 604 y Fw(h)p Fu(t)p Fw(i)2491 566 y Fq(r)r Ft(\000)p Fv(1)2620 604 y Fu(;)244 800 y Fj(wher)m(e)h Fu(\015)36 b Fw(\021)31 b Fx(min)824 815 y Fv(1)p Ft(\024)p Fq(k)r Ft(\024)p Fq(n)1070 800 y Fw(j)p Fx(Im)p Fu(z)1246 815 y Fq(k)1289 800 y Fw(j)25 b Fj(and)j Fu(r)g Fj(is)e(the)h(maximum)f(or)m(der)h(of)g (dominant)h(poles)f(\(the)g(poles)h(in)244 913 y Fw(f)p Fu(z)331 927 y Fv(1)371 913 y Fu(;)15 b Fw(\001)g(\001)g(\001)32 b Fu(;)15 b(z)630 927 y Fq(n)677 913 y Fw(g)24 b Fj(closest)h(to)e(the) h(r)m(eal)g(axis\).)244 1091 y Fz(Pr)n(oof)o(.)46 b Fy(Since)24 b(\(ii\))g Fw(\))e Fy(\(i\),)i(we)e(ha)n(v)o(e)j(to)e(pro)o(v)o(e)h (\(ii\))g(only)-6 b(.)380 1204 y(Fix)23 b Fu(\021)29 b Fw(2)c(N)759 1218 y Fq(D)822 1204 y Fy(,)e Fu(\021)s Fx(\()15 b Fw(\001)g Fx(\))27 b(=)e(\()p Fu(C)7 b Fx(\012)p Fu(;)30 b Fw(\001)15 b Fu(C)7 b Fx(\012\))p Fy(,)22 b(and)i Fu(A)h Fw(2)g(O)2055 1218 y Fq(D)2120 1204 y Fy(.)j(Then,)724 1321 y Fm(Z)815 1347 y Ft(1)775 1527 y Fv(0)905 1445 y Fx(e)945 1407 y Fv(i)p Fq(z)s(t)1031 1445 y Fu(\021)s Fx(\()p Fu(\034)1164 1407 y Fq(t)1154 1467 y(V)1215 1445 y Fx(\()p Fu(A)p Fx(\)\)d)p Fu(t)e Fx(=)f(i\()p Fu(D)1733 1407 y Ft(\000)p Fv(1)1826 1445 y Fu(A)1894 1407 y Ft(\003)1934 1445 y Fx(\012)p Fu(;)15 b Fw(R)2117 1459 y Fq(D)2181 1445 y Fx(\()p Fu(z)t Fx(\))p Fu(D)2375 1407 y Ft(\000)p Fv(1)2470 1445 y Fu(C)2542 1407 y Ft(\003)2581 1445 y Fu(C)7 b Fx(\012\))25 b Fw(\021)g Fu(`)p Fx(\()p Fu(z)t Fx(\))p Fu(:)244 1689 y Fy(Fix)19 b Fu(\016)29 b(>)c Fx(0)20 b Fy(and)g Fu(\026)819 1656 y Ft(0)862 1689 y Fy(such)h(that)f Fu(\026)25 b(<)g(\026)1434 1656 y Ft(0)1482 1689 y Fu(<)g Fw(\000)p Fu(\015)5 b Fy(.)27 b(Let)19 b Fu(\013)26 b(>)f Fx(0)20 b Fy(be)g(a)f(lar)n(ge)i(number)g(and)g Fx(\000)3004 1703 y Fq(\013)3072 1689 y Fy(the)f(rectangle)244 1802 y(with)j(v)o(ertices)i Fw(f\006)p Fu(\013)c Fx(+)f(i)p Fu(\016)n(;)15 b Fw(\006)p Fu(\013)21 b Fx(+)f(i)p Fu(\026)1444 1769 y Ft(0)1466 1802 y Fw(g)p Fy(.)29 b(Then,)23 b(for)h(an)o(y)f Fu(\017)i(>)g Fx(0)p Fy(,)349 1982 y Fx(1)p 321 2022 101 4 v 321 2105 a(2)p Fu(\031)447 1919 y Fm(Z)497 2126 y Fv(\000)541 2134 y Fn(\013)605 2043 y Fx(e)645 2006 y Ft(\000)p Fv(i)p Fq(tz)785 2043 y Fu(`)p Fx(\()p Fu(z)t Fx(\)d)p Fu(z)30 b Fx(=)25 b Fw(\000)1266 1982 y Fx(1)p 1239 2022 V 1239 2105 a(2)p Fu(\031)1364 1919 y Fm(Z)1455 1946 y Fq(\013)1415 2126 y Ft(\000)p Fq(\013)1534 2043 y Fx(e)1574 2006 y Ft(\000)p Fv(i)p Fq(t)p Fv(\()p Fq(x)p Fv(+i)p Fq(\016)r Fv(\))1882 2043 y Fu(`)p Fx(\()p Fu(x)20 b Fx(+)g 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Fy(can)e(be)f(chosen)i(independently)i(of)c Fu(\016)j Fy(as)d(long)h(as)f Fu(\016)30 b(<)c Fx(const)q Fy(.)j(Pick)24 b(a)244 3359 y(subsequence)j Fu(\013)784 3373 y Fq(n)827 3385 y Fn(k)869 3359 y Fy(such)e(that)975 3603 y Fx(lim)948 3664 y Fq(k)r Ft(!1)1181 3541 y Fx(1)p 1153 3582 101 4 v 1153 3665 a(2)p Fu(\031)1279 3479 y Fm(Z)1370 3506 y Fq(\013)1415 3514 y Fn(n)1453 3531 y(k)1329 3685 y Ft(\000)p Fq(\013)1429 3693 y Fn(n)1467 3711 y(k)1529 3603 y Fx(e)1569 3565 y Ft(\000)p Fv(i)p Fq(t)p Fv(\()p Fq(x)p Fv(+i)p Fq(\016)r Fv(\))1877 3603 y Fu(`)p Fx(\()p Fu(x)20 b Fx(+)g Fu(i\016)s Fx(\)d)p Fu(x)26 b Fx(=)f Fu(\021)s Fx(\()p Fu(\034)2580 3565 y Fq(t)2570 3625 y(V)2632 3603 y Fx(\()p Fu(A)p Fx(\)\))p Fu(;)244 3866 y Fy(for)h(Lebesgue)h(a.e.)34 b Fu(t)29 b(>)g Fx(0)c Fy(and)h(set)g Fu(\013)j Fx(=)g Fu(\013)1667 3880 y Fq(n)1710 3892 y Fn(k)1777 3866 y Fy(in)c(\(2.29\).)36 b(T)-7 b(aking)26 b Fu(k)32 b Fw(!)d(1)c Fy(we)f(deri)n(v)o(e)j(that)f(for)f(a.e.)244 3979 y Fu(t)g(>)g Fx(0)p 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Fx(\()p Fw(O)s Fu(;)15 b(\034)10 b Fx(\))29 b Fy(and)h(the)g(states)g(in)f Fw(N)2060 5169 y Fv(s)2092 5155 y Fy(.)45 b(W)-7 b(e)29 b(then)h(use)f(these)i (results)f(to)g(compute)244 5268 y(the)24 b(standard)h(and)f(the)g Fu(C)7 b Fy(-Liouvillean)26 b(of)d(the)h(locally)h(perturbed)h(system.) p eop %%Page: 20 20 20 19 bop 3680 100 a FA(20)380 407 y Fy(Since)26 b(the)f(results)i(of)e (this)h(section)h(are)e(either)h(well-kno)n(wn)h(or)e(follo)n(w)g(from) g(simple)h(computa-)244 520 y(tions)e(we)f(will)g(omit)h(the)g(proofs.) 244 732 y Fz(Notation.)29 b Fy(If)23 b Fu(A)g Fy(is)h(a)f(linear)i (operator)g(on)f Fi(H)1688 746 y Fv(s)1720 732 y Fy(,)f(we)f(denote)k (by)p 2273 659 69 4 v 23 w Fu(A)d Fy(the)h(linear)h(operator)p 1686 878 V 1686 951 a Fu(A )k Fx(=)p 1938 859 131 4 v 25 w Fu(A)p 2006 877 63 4 v( )s(;)244 1155 y Fy(where)24 b(on)f(the)h(right-hand)j(side)p 1307 1104 56 4 v 39 w Fw(\001)38 b Fy(is)23 b(the)h(usual)h(comple)o(x)f(conjugation)j(on)c Fi(H)2781 1169 y Fv(s)2839 1155 y Fx(=)i Fp(C)2995 1122 y Fv(2)3040 1155 y Fy(.)380 1368 y(W)-7 b(e)23 b(be)o(gin)h(by)g (computing)i(the)d(modular)i(structure)h(associated)g(to)e(the)f(small) h(system)g Fw(S)7 b Fy(.)28 b(Set)1683 1566 y Fw(H)1760 1580 y Fv(s)1817 1566 y Fx(=)d Fi(H)1978 1580 y Fv(s)2031 1566 y Fw(\012)20 b Fi(H)2187 1580 y Fv(s)2219 1566 y Fu(;)1569 1775 y(\031)1621 1789 y Fv(s)1653 1775 y Fx(\()p Fu(A)p Fx(\))26 b(=)f Fu(A)c Fw(\012)e Fo(1)p Fu(;)1535 1996 y(\031)1590 1959 y Fv(#)1587 2019 y(s)1653 1996 y Fx(\()p Fu(A)p Fx(\))26 b(=)f Fo(1)c Fw(\012)p 2077 1923 69 4 v 20 w Fu(A:)244 2198 y Fy(Let)i Fu(!)445 2212 y Fv(s)499 2198 y Fy(be)g(a)g(state)h(on)g Fw(O)1045 2212 y Fv(s)1077 2198 y Fy(.)k(Then)c(there)g(is)f(a)g(density)i (matrix)f Fu(\032)2280 2212 y Fv(s)2335 2198 y Fy(such)g(that)g Fu(!)2740 2212 y Fv(s)2772 2198 y Fx(\()p Fu(A)p Fx(\))i(=)f(T)-8 b(r\()p Fu(\032)3208 2212 y Fv(s)3240 2198 y Fu(A)p Fx(\))p Fy(.)28 b(The)244 2311 y(state)g Fu(!)494 2325 y Fv(s)552 2311 y Fy(is)f(f)o(aithful)i(if)n(f)e Fu(\032)1089 2325 y Fv(s)1153 2311 y Fu(>)k Fx(0)c Fy(and)h Fu(\034)1525 2325 y Fv(s)1557 2311 y Fy(-in)l(v)n(ariant)h(if)n(f)e Fx([)p Fu(H)2141 2325 y Fv(s)2173 2311 y Fu(;)15 b(\032)2260 2325 y Fv(s)2293 2311 y Fx(])32 b(=)f(0)p Fy(.)40 b(If)27 b Fu(\032)2694 2325 y Fv(s)2726 2311 y Fx(\()15 b Fw(\001)g Fx(\))33 b(=)2986 2243 y Fm(P)3098 2311 y Fu(p)3144 2325 y Fq(i)3171 2311 y Fx(\()p Fu( )3265 2325 y Fq(i)3294 2311 y Fu(;)e Fw(\001)15 b Fx(\))p Fu( )3484 2325 y Fq(i)3513 2311 y Fy(,)244 2424 y(let)1487 2550 y Fx(\012)1553 2564 y Fv(s)1610 2550 y Fx(:=)1731 2464 y Fm(X)1878 2550 y Fu(p)1934 2465 y Fl(1)p 1933 2477 31 3 v 1933 2519 a(2)1924 2577 y Fq(i)1978 2550 y Fu( )2037 2564 y Fq(i)2086 2550 y Fw(\012)p 2176 2476 63 4 v 19 w Fu( )2239 2572 y Fq(i)2267 2550 y Fu(:)380 2717 y Fy(Recall)24 b(that)g(the)g(dynamics)h(of)f Fw(S)29 b Fy(is)24 b(speci\002ed)h(by)e(automorphisms)k(\(1.3\).)i(Let) 1443 2921 y Fw(L)1506 2935 y Fv(s)1563 2921 y Fw(\021)c Fu(H)1735 2935 y Fv(s)1786 2921 y Fw(\012)20 b Fo(1)h Fw(\000)f Fo(1)g Fw(\012)g Fu(H)2280 2935 y Fv(s)2312 2921 y Fu(:)244 3134 y Fz(Pr)n(oposition)25 b(3.1)49 b Fj(The)25 b(triple)i Fx(\()p Fw(H)1372 3148 y Fv(s)1405 3134 y Fu(;)15 b(\031)1497 3148 y Fv(s)1529 3134 y Fu(;)g Fx(\012)1635 3148 y Fv(s)1667 3134 y Fx(\))25 b Fj(is)h(the)g(GNS)f(r)m (epr)m(esentation)k(of)d Fw(O)2862 3148 y Fv(s)2920 3134 y Fj(associated)j(to)c Fu(!)3481 3148 y Fv(s)3513 3134 y Fj(.)244 3246 y(If)30 b Fu(!)386 3260 y Fv(s)447 3246 y Fj(is)g Fu(\034)577 3260 y Fv(s)609 3246 y Fj(-in)l(variant,)35 b(then)c Fw(L)1272 3260 y Fv(s)1334 3246 y Fj(is)f(the)h(corr)m (esponding)j(standar)m(d)f(Liouvillean.)52 b(If)30 b Fu(!)3099 3260 y Fv(s)3160 3246 y Fj(is)g(faithful,)244 3359 y(consider)c(the)e(pair)g Fx(\()p Fu(\031)975 3373 y Fv(s)1007 3359 y Fx(\()p Fw(O)1114 3373 y Fv(s)1147 3359 y Fx(\))p Fu(;)15 b Fx(\012)1288 3373 y Fv(s)1320 3359 y Fx(\))p Fj(.)244 3472 y Fy(\(i\))23 b Fj(Its)h(modular)h(oper)o (ator)g(is)1581 3610 y Fx(\001)1657 3624 y Fv(s)1714 3610 y Fx(=)g Fu(\032)1857 3624 y Fv(s)1909 3610 y Fw(\012)p 2000 3560 48 4 v 20 w Fu(\032)2047 3632 y Fv(s)2079 3573 y Ft(\000)p Fv(1)2173 3610 y Fu(:)244 3777 y Fy(\(ii\))f Fj(Its)f(modular)i(conjugation)i(is)d Fu(J)1413 3791 y Fv(s)1445 3777 y Fx(\()p Fu(\036)d Fw(\012)f Fu( )s Fx(\))26 b(=)p 1865 3703 63 4 v 25 w Fu( )d Fw(\012)p 2038 3703 55 4 v 20 w Fu(\036)p Fj(.)244 3898 y Fy(\(iii\))h Fu(J)453 3912 y Fv(s)485 3898 y Fu(\031)537 3912 y Fv(s)569 3898 y Fx(\()p Fu(A)p Fx(\))p Fu(J)757 3912 y Fv(s)816 3898 y Fx(=)h Fu(\031)967 3854 y Fv(#)964 3910 y(s)1030 3898 y Fx(\()p Fu(A)p Fx(\))p Fj(.)380 4078 y Fy(W)-7 b(e)29 b(no)n(w)g(discuss)i(the)f(modular)h(structure)h(associated)h (to)c(a)g(free)h(Fermi)f(reserv)n(oir)j(in)e(thermal)244 4191 y(equilibrium)e(at)e(in)l(v)o(erse)h(temperature)h Fu(\014)5 b Fy(.)35 b(W)-7 b(e)25 b(\002x)g(a)g(comple)o(x)h (conjugation)j(\(an)d(anti-unitary)j(in)l(v)n(o-)244 4304 y(lution\))24 b Fu(f)35 b Fw(7!)p 704 4229 V 25 w Fu(f)d Fy(which)23 b(commutes)h(with)f(the)g(single)h(particle)h (Hamiltonian)g Fu(h)p Fy(.)j(Let)3005 4281 y Fx(^)2995 4304 y(\012)3061 4319 y Fv(f)3114 4304 y Fy(be)23 b(the)g(F)o(ock)244 4416 y(v)n(acuum)h(on)g Fi(H)731 4431 y Fv(f)762 4416 y Fy(,)f Fu(N)32 b Fy(the)24 b(number)h(operator)l(,)1473 4621 y Fu(#)g Fw(\021)g Fx(\000\()p Fw(\000)p Fo(1)p Fx(\))h(=)f(\()p Fw(\000)p Fo(1)p Fx(\))2213 4583 y Fq(N)2281 4621 y Fu(;)244 4825 y Fy(and)1518 4955 y Fu(\045)1565 4970 y Fq(\014)1637 4955 y Fw(\021)1733 4854 y Fm(\020)1788 4955 y Fx(e)1828 4918 y Fq(\014)s(h)1936 4955 y Fx(+)20 b(1)2072 4854 y Fm(\021)2127 4877 y Ft(\000)p Fv(1)2236 4955 y Fu(:)p eop %%Page: 21 21 21 20 bop 3680 100 a FA(21)244 407 y Fy(The)25 b(comple)o(x)h (conjugation)p 1199 355 56 4 v 43 w Fw(\001)41 b Fy(on)25 b Fi(h)g Fy(naturally)j(e)o(xtends)f(to)e(a)g(comple)o(x)h(conjugation) j(on)c Fw(H)3258 422 y Fv(f)3313 407 y Fy(which)244 520 y(we)e(denote)i(by)e(the)h(same)g(symbol,)g(i.e.)k Fx(\011)d Fw(7!)p 1759 447 71 4 v 25 w Fx(\011)p Fy(.)j(Let)1610 695 y Fw(H)1687 710 y Fv(f)1743 695 y Fw(\021)d Fi(H)1904 710 y Fv(f)1956 695 y Fw(\012)20 b Fi(H)2112 710 y Fv(f)2143 695 y Fu(;)1622 919 y Fx(\012)1688 934 y Fv(f)1743 919 y Fx(=)1850 896 y(^)1839 919 y(\012)1905 934 y Fv(f)1956 919 y Fw(\012)2057 896 y Fx(^)2047 919 y(\012)2113 934 y Fv(f)2144 919 y Fu(:)244 1092 y Fy(The)j(Araki-W)-7 b(yss)25 b(representation)j Fu(\031)1453 1107 y Fq(\014)1523 1092 y Fy(of)23 b Fw(O)1693 1107 y Fv(f)1747 1092 y Fy(on)h Fw(H)1938 1107 y Fv(f)1991 1092 y Fy(is)g(de\002ned)g(by)1010 1314 y Fu(\031)1062 1329 y Fq(\014)1109 1314 y Fx(\()p Fu(a)p Fx(\()p Fu(f)10 b Fx(\)\))26 b(=)f Fu(a)p Fx(\(\()p Fo(1)c Fw(\000)f Fu(\045)1803 1329 y Fq(\014)1851 1314 y Fx(\))1896 1249 y Fl(1)p 1896 1261 31 3 v 1896 1302 a(2)1940 1314 y Fu(f)10 b Fx(\))20 b Fw(\012)g Fo(1)h Fx(+)f Fu(#)g Fw(\012)f Fu(a)2517 1276 y Ft(\003)2557 1314 y Fx(\()p Fu(\045)2649 1229 y Fl(1)p 2649 1241 V 2649 1282 a(2)2639 1343 y Fq(\014)p 2694 1240 55 4 v 2694 1314 a Fu(f)9 b Fx(\))p Fu(;)971 1577 y(\031)1023 1592 y Fq(\014)1070 1577 y Fx(\()p Fu(a)1153 1539 y Ft(\003)1193 1577 y Fx(\()p Fu(f)h Fx(\)\))25 b(=)g Fu(a)1522 1539 y Ft(\003)1562 1577 y Fx(\(\()p Fo(1)c Fw(\000)f Fu(\045)1843 1592 y Fq(\014)1890 1577 y Fx(\))1935 1512 y Fl(1)p 1935 1524 31 3 v 1935 1565 a(2)1980 1577 y Fu(f)10 b Fx(\))20 b Fw(\012)g Fo(1)g Fx(+)g Fu(#)g Fw(\012)g Fu(a)p Fx(\()p Fu(\045)2649 1492 y Fl(1)p 2649 1504 V 2649 1545 a(2)2639 1606 y Fq(\014)p 2694 1503 55 4 v 2694 1577 a Fu(f)9 b Fx(\))p Fu(:)244 1786 y Fy(The)23 b(dual)h(representation)k Fu(\031)1180 1742 y Fv(#)1177 1815 y Fq(\014)1266 1786 y Fy(is)23 b(de\002ned)h(by)911 2022 y Fu(\031)966 1978 y Fv(#)963 2051 y Fq(\014)1029 2022 y Fx(\()p Fu(a)1112 1984 y Ft(\003)1152 2022 y Fx(\()p Fu(f)10 b Fx(\)\))25 b(=)g Fu(#a)p Fx(\()p Fu(\045)1627 1937 y Fl(1)p 1628 1949 31 3 v 1628 1991 a(2)1617 2051 y Fq(\014)1672 2022 y Fu(f)10 b Fx(\))20 b Fw(\012)g Fu(#)g Fx(+)g Fo(1)g Fw(\012)g Fu(a)2249 1984 y Ft(\003)2289 2022 y Fx(\(\(1)h Fw(\000)f Fu(\045)2563 2037 y Fq(\014)2610 2022 y Fx(\))2655 1957 y Fl(1)p 2655 1969 V 2655 2010 a(2)p 2700 1948 55 4 v 2700 2022 a Fu(f)9 b Fx(\))p Fu(#;)950 2285 y(\031)1005 2241 y Fv(#)1002 2314 y Fq(\014)1068 2285 y Fx(\()p Fu(a)p Fx(\()p Fu(f)h Fx(\)\))26 b(=)f Fu(a)1481 2247 y Ft(\003)1521 2285 y Fx(\()p Fu(\045)1613 2200 y Fl(1)p 1613 2212 31 3 v 1613 2254 a(2)1603 2314 y Fq(\014)1658 2285 y Fu(f)10 b Fx(\))p Fu(#)19 b Fw(\012)h Fu(#)g Fx(+)g Fo(1)h Fw(\012)f Fu(#a)p Fx(\(\(1)h Fw(\000)f Fu(\045)2617 2300 y Fq(\014)2664 2285 y Fx(\))2709 2220 y Fl(1)p 2709 2232 V 2709 2273 a(2)p 2754 2211 55 4 v 2754 2285 a Fu(f)9 b Fx(\))p Fu(:)244 2494 y Fy(The)19 b(representations)k Fu(\031)1024 2509 y Fq(\014)1090 2494 y Fy(and)c Fu(\031)1294 2450 y Fv(#)1291 2524 y Fq(\014)1376 2494 y Fy(were)g(introduced)j(for)d(the)h(\002rst)f (time)g(in)g([A)-8 b(W)o(])19 b(\(see)h(also)f(Example)244 2607 y(5.2.20)24 b(in)f([BR2]\).)29 b(Let)1444 2720 y Fw(L)1507 2735 y Fv(f)1563 2720 y Fw(\021)c Fu(H)1735 2735 y Fv(f)1786 2720 y Fw(\012)20 b Fo(1)g Fw(\000)g Fo(1)h Fw(\012)f Fu(H)2280 2735 y Fv(f)2310 2720 y Fu(:)244 2921 y Fz(Pr)n(oposition)25 b(3.2)49 b Fj(The)24 b(triple)i Fx(\()p Fw(H)1370 2936 y Fv(f)1401 2921 y Fu(;)15 b(\031)1493 2936 y Fq(\014)1541 2921 y Fu(;)g Fx(\012)1647 2936 y Fv(f)1678 2921 y Fx(\))24 b Fj(is)h(the)g(GNS)e(r)m(epr)m(esentation)29 b(of)c Fw(O)2867 2936 y Fv(f)2922 2921 y Fj(associated)j(to)d(the)244 3034 y(KMS-state)j Fu(!)711 3049 y Fv(f)5 b Fq(;\014)830 3034 y Fj(and)28 b Fw(L)1056 3049 y Fv(f)1113 3034 y Fj(is)f(the)g(corr)m(esponding)32 b(standar)m(d)d(Liouvillean.)42 b(The)26 b(vector)j Fx(\012)3195 3049 y Fv(f)3252 3034 y Fj(is)e(sepa-)244 3147 y(r)o(ating)e(for)e(the)h(en)l(veloping)j(von) d(Neumann)g(alg)o(ebr)o(a)1564 3329 y Fi(M)1659 3344 y Fv(f)5 b Fq(;\014)1779 3329 y Fw(\021)25 b Fu(\031)1927 3344 y Fq(\014)1973 3329 y Fx(\()p Fw(O)2080 3344 y Fv(f)2112 3329 y Fx(\))2147 3291 y Ft(00)2190 3329 y Fu(:)244 3510 y Fj(Consider)g(the)f(pair)g Fx(\()p Fi(M)1038 3525 y Fv(f)5 b Fq(;\014)1132 3510 y Fu(;)15 b Fx(\012)1238 3525 y Fv(f)1269 3510 y Fx(\))p Fj(.)244 3623 y Fy(\(i\))23 b Fj(Its)h(modular)h(oper)o(ator)g(is)f Fx(\001)1296 3638 y Fv(f)1352 3623 y Fx(=)h(e)1488 3590 y Ft(\000)p Fq(\014)s Ft(L)1635 3602 y Fl(f)1666 3623 y Fj(.)244 3736 y Fy(\(ii\))f Fj(Its)f(modular)i(conjugation)i(is)1441 3918 y Fu(J)1491 3933 y Fv(f)1522 3918 y Fx(\(\010)21 b Fw(\012)e Fx(\011\))26 b(=)f Fu(u)p 2014 3845 71 4 v Fx(\011)20 b Fw(\012)g Fu(u)p 2248 3845 66 4 v Fx(\010)o Fu(;)244 4099 y Fj(wher)m(e)j Fu(u)j Fw(\021)f Fx(\()p Fw(\000)p Fo(1)p Fx(\))852 4066 y Fq(N)7 b Fv(\()p Fq(N)g Ft(\000)p Fv(1\))p Fq(=)p Fv(2)1198 4099 y Fj(.)244 4221 y Fy(\(iii\))24 b Fu(J)453 4236 y Fv(f)484 4221 y Fu(\031)536 4236 y Fq(\014)583 4221 y Fx(\()p Fu(A)p Fx(\))p Fu(J)771 4236 y Fv(f)829 4221 y Fx(=)h Fu(\031)980 4176 y Fv(#)977 4250 y Fq(\014)1043 4221 y Fx(\()p Fu(A)p Fx(\))p Fj(.)380 4394 y Fy(If)18 b(\(A1\))h(holds,)h(then)f(the)g(GNS)d(representation) 23 b(and)c(modular)g(structure)i(of)d(a)g(free)h(Fermi)f(gas)g(can)244 4507 y(be)24 b(described)i(in)e(a)g(some)n(what)h(dif)n(ferent)g(form)f (which)h(is)f(more)g(suitable)i(for)e(the)g(spectral)i(analysis.)244 4620 y(In)e(what)f(follo)n(ws)i(we)e(assume)h(that)h(\(A1\))e(holds.)31 b(Let)2003 4596 y Fx(~)2002 4620 y Fi(h)26 b Fx(=)f Fu(L)2233 4587 y Fv(2)2273 4620 y Fx(\()p Fp(R)s Fu(;)15 b Fi(G)p Fx(\))q Fy(.)34 b(T)-7 b(o)23 b(an)o(y)h Fu(f)35 b Fw(2)25 b Fi(h)f Fy(we)e(associate)244 4742 y(a)h(pair)h(of)g(functions)i Fu(f)976 4757 y Fq(\014)1022 4742 y Fu(;)15 b(f)1117 4697 y Fv(#)1107 4771 y Fq(\014)1205 4742 y Fw(2)1292 4718 y Fx(~)1291 4742 y Fi(h)23 b Fy(by)1273 4998 y Fu(f)1318 5013 y Fq(\014)1365 4998 y Fx(\()p Fu(s)p Fx(\))i(=)1599 4897 y Fm(\020)1653 4998 y Fx(e)1693 4961 y Ft(\000)p Fq(\014)s(s)1849 4998 y Fx(+)20 b(1)1985 4897 y Fm(\021)2039 4920 y Ft(\000)2104 4893 y Fl(1)p 2104 4905 31 3 v 2104 4946 a(2)2183 4974 y Fx(~)2164 4998 y Fu(f)9 b Fx(\()p Fu(s)p Fx(\))p Fu(;)1247 5275 y(f)1302 5231 y Fv(#)1292 5305 y Fq(\014)1365 5275 y Fx(\()p Fu(s)p Fx(\))25 b(=)g(ie)1665 5238 y Ft(\000)p Fq(\014)s(s=)p Fv(2)1870 5275 y Fu(f)1915 5290 y Fq(\014)1962 5275 y Fx(\()p Fu(s)p Fx(\))g(=)g(i)p 2222 5201 55 4 v Fu(f)2276 5297 y Fq(\014)2323 5275 y Fx(\()p Fw(\000)p Fu(s)p Fx(\))p Fu(;)p eop %%Page: 22 22 22 21 bop 3680 100 a FA(22)244 407 y Fy(\()294 383 y Fx(~)274 407 y Fu(f)39 b Fy(is)30 b(de\002ned)h(by)g(\(1.7\)\).)50 b(F)o(or)29 b(latter)j(purposes)g(we)e(mak)o(e)g(the)h(follo)n(wing)h (remark.)50 b(Assume)30 b(that)263 496 y Fx(~)244 520 y Fu(f)k Fw(2)25 b Fu(H)492 487 y Fv(2)532 520 y Fx(\()p Fu(\016)s Fx(\))f Fy(for)g(some)f Fx(0)j Fu(<)f(\016)k(<)c(\031)s (=\014)5 b Fy(.)29 b(Then)1165 695 y Fu(C)7 b Fx(\()p Fu(\016)n(;)15 b(\014)5 b Fx(\))27 b Fw(\021)80 b Fx(sup)1564 777 y Ft(j)p Fv(Im)o Fq(z)s Ft(j)p Fq(<\016)1827 695 y Fw(j)p Fx(1)21 b(+)f(e)2049 657 y Ft(\000)p Fq(\014)s(z)2187 695 y Fw(j)2212 657 y Ft(\000)p Fv(1)p Fq(=)p Fv(2)2402 695 y Fu(<)25 b Fw(1)p Fu(:)244 963 y Fy(It)e(then)h(follo)n(ws)h(that) f Fu(f)999 978 y Fq(\014)1045 963 y Fu(;)15 b(f)1140 919 y Fv(#)1130 993 y Fq(\014)1228 963 y Fw(2)25 b Fu(H)1397 930 y Fv(2)1437 963 y Fx(\()p Fu(\016)s Fx(\))p Fy(,)706 1166 y Fw(k)p Fu(f)796 1181 y Fq(\014)843 1166 y Fw(k)888 1186 y Fq(H)951 1167 y Fl(2)986 1186 y Fv(\()p Fq(\016)r Fv(\))1104 1166 y Fx(=)g Fw(k)p Fu(f)1300 1122 y Fv(#)1290 1196 y Fq(\014)1362 1166 y Fw(k)1407 1186 y Fq(H)1470 1167 y Fl(2)1506 1186 y Fv(\()p Fq(\016)r Fv(\))1624 1166 y Fx(=)g Fw(k)p Fx(e)1805 1129 y Fq(\014)s(s=)p Fv(2)1956 1166 y Fu(f)2011 1122 y Fv(#)2001 1196 y Fq(\014)2073 1166 y Fw(k)2118 1186 y Fq(H)2181 1167 y Fl(2)2216 1186 y Fv(\()p Fq(\016)r Fv(\))2334 1166 y Fx(=)g Fw(k)p Fx(e)2515 1129 y Ft(\000)p Fq(\014)s(s=)p Fv(2)2721 1166 y Fu(f)2766 1181 y Fq(\014)2813 1166 y Fw(k)2858 1186 y Fq(H)2921 1167 y Fl(2)2956 1186 y Fv(\()p Fq(\016)r Fv(\))3049 1166 y Fu(;)1323 1364 y Fw(k)p Fu(f)1413 1379 y Fq(\014)1461 1364 y Fw(k)1506 1383 y Fq(H)1569 1364 y Fl(2)1604 1383 y Fv(\()p Fq(\016)r Fv(\))1722 1364 y Fw(\024)g Fu(C)7 b Fx(\()p Fu(\016)n(;)15 b(\014)5 b Fx(\))p Fw(k)2160 1340 y Fx(~)2139 1364 y Fu(f)11 b Fw(k)2240 1383 y Fq(H)2303 1364 y Fl(2)2338 1383 y Fv(\()p Fq(\016)r Fv(\))2431 1364 y Fu(;)860 b Fy(\(3.31\))1065 1538 y Fw(k)p Fx(e)1150 1500 y Ft(\000)p Fq(\014)s(s=)p Fv(2)1356 1538 y Fu(f)1411 1494 y Fv(#)1401 1567 y Fq(\014)1473 1538 y Fw(k)1518 1557 y Fq(H)1581 1539 y Fl(2)1616 1557 y Fv(\()p Fq(\016)r Fv(\))1734 1538 y Fw(\024)25 b Fu(C)7 b Fx(\()p Fu(\016)n(;)15 b(\014)5 b Fx(\))p Fw(k)p Fx(e)2191 1500 y Ft(\000)p Fq(\014)s(s=)p Fv(2)2419 1514 y Fx(~)2399 1538 y Fu(f)k Fw(k)2498 1557 y Fq(H)2561 1539 y Fl(2)2597 1557 y Fv(\()p Fq(\016)r Fv(\))2689 1538 y Fu(:)380 1712 y Fy(W)-7 b(e)19 b(denote)i(by)f Fu(s)f Fy(the)h(operator)i(of)d(multiplication)k(by)d Fu(s)25 b Fw(2)g Fp(R)s Fy(.)32 b(Let)2555 1689 y Fx(~)2545 1712 y(\012)19 b Fy(be)h(the)g(v)n(acuum)g(on)g Fx(\000)3336 1726 y Ft(\000)3395 1712 y Fx(\()3431 1688 y(~)3430 1712 y Fi(h)q Fx(\))p Fy(.)244 1910 y Fz(Theor)n(em)46 b(3.3)k Fj(Ther)m(e)23 b(e)n(xists)i(a)e(unitary)i(map)1549 2085 y Fu(U)35 b Fx(:)26 b Fw(H)1774 2100 y Fv(f)1830 2085 y Fw(7!)f Fx(d\000)2054 2099 y Ft(\000)2112 2085 y Fx(\()2148 2061 y(~)2147 2085 y Fi(h)q Fx(\))244 2260 y Fj(suc)o(h)f(that)1692 2415 y Fu(U)10 b Fx(\012)1830 2430 y Fv(f)1886 2415 y Fx(=)1992 2392 y(~)1982 2415 y(\012)2048 2430 y Fv(f)1529 2633 y Fu(U)g Fw(L)1664 2648 y Fv(f)1694 2633 y Fu(U)1766 2595 y Ft(\000)p Fv(1)1886 2633 y Fx(=)25 b(d\000\()p Fu(s)p Fx(\))1268 2851 y Fu(U)10 b(\031)1392 2866 y Fq(\014)1439 2851 y Fx(\()p Fu(')p Fx(\()p Fu(f)g Fx(\)\))p Fu(U)1765 2814 y Ft(\000)p Fv(1)1886 2851 y Fx(=)25 b Fu(')p Fx(\()p Fu(f)2121 2866 y Fq(\014)2168 2851 y Fx(\))1249 3080 y Fu(U)10 b(\031)1376 3035 y Fv(#)1373 3109 y Fq(\014)1439 3080 y Fx(\()p Fu(')p Fx(\()p Fu(f)g Fx(\)\))p Fu(U)1765 3042 y Ft(\000)p Fv(1)1886 3080 y Fx(=)25 b(i\000\()p Fw(\000)p Fo(1)p Fx(\))p Fu(')p Fx(\()p Fu(f)2407 3035 y Fv(#)2397 3109 y Fq(\014)2470 3080 y Fx(\))p Fu(:)244 3364 y Fz(Pr)n(oof)o(.)54 b Fy(This)27 b(result)h(follo)n(ws)g(from)g (the)f(identi\002cation)k Fi(h)23 b Fw(\010)g Fi(h)32 b Fx(=)g Fu(L)2503 3331 y Fv(2)2543 3364 y Fx(\()p Fp(R)s Fu(;)15 b Fi(G)q Fx(\))32 b Fy(and)c(the)g(e)o(xponential)244 3477 y(la)o(w)23 b(for)g(fermionic)i(systems)g(\(see)f(Theorem)g(3.2)f (in)h([BSZ)o(]\).)k Fe(2)380 3630 y Fy(In)h(what)f(follo)n(ws)i(we)e (will)g(w)o(ork)h(e)o(xclusi)n(v)o(ely)i(in)d(the)h(representation)k (gi)n(v)o(en)c(by)g(Theorem)g(3.3)244 3743 y(and)h(we)f(identify)i(the) f(quantities)j(related)e(by)f Fu(U)38 b Fy(\()p Fw(H)1976 3758 y Fv(f)2036 3743 y Fy(no)n(w)29 b(stands)i(for)f Fx(\000)2667 3757 y Ft(\000)2726 3743 y Fx(\()2762 3719 y(~)2761 3743 y Fi(h)q Fx(\))p Fy(,)g Fx(\012)2963 3758 y Fv(f)3023 3743 y Fy(for)3168 3720 y Fx(~)3157 3743 y(\012)3223 3758 y Fv(f)3254 3743 y Fy(,)g Fw(L)3370 3758 y Fv(f)3430 3743 y Fy(for)244 3856 y Fx(d\000\()p Fu(s)p Fx(\))22 b Fy(etc.\).)380 3985 y(Consider)i(no)n(w)e(tw)o(o)g (identical)i(reserv)n(oirs)h Fx(\()p Fw(O)1885 3937 y Fv(\()p Fq(i)p Fv(\))1882 4014 y(f)1968 3985 y Fu(;)15 b(\034)2048 3999 y Fq(i)2077 3985 y Fx(\))22 b Fy(and)g(let)h Fw(O)h Fy(be)e(gi)n(v)o(en)h(by)f(\(1.4\).)29 b(Let)22 b Fu(!)3359 4000 y Fq(\014)3399 4010 y Fn(i)3450 3985 y Fy(be)244 4120 y Fx(\()p Fu(\034)319 4134 y Fq(i)347 4120 y Fu(;)15 b(\014)438 4134 y Fq(i)467 4120 y Fx(\))p Fy(-KMS)22 b(on)i Fw(O)941 4072 y Fv(\()p Fq(i)p Fv(\))938 4149 y(f)1047 4120 y Fy(for)g(some)f Fu(\014)1441 4134 y Fq(i)1495 4120 y Fu(>)i Fx(0)p Fy(.)j(Set)1473 4317 y Fw(H)e Fx(=)f Fw(H)1749 4331 y Fv(s)1801 4317 y Fw(\012)20 b(H)1970 4269 y Fv(\(1\))1969 4347 y(f)2084 4317 y Fw(\012)g(H)2253 4269 y Fv(\(2\))2252 4347 y(f)1485 4553 y Fx(\012)25 b(=)g(\012)1738 4567 y Fv(s)1790 4553 y Fw(\012)20 b Fx(\012)1947 4505 y Fv(\(1\))1947 4582 y(f)2061 4553 y Fw(\012)g Fx(\012)2218 4505 y Fv(\(2\))2218 4582 y(f)2312 4553 y Fu(;)1496 4762 y(\031)28 b Fx(=)d Fu(\031)1724 4776 y Fv(s)1776 4762 y Fw(\012)20 b Fu(\031)1919 4777 y Fq(\014)1959 4786 y Fl(1)2017 4762 y Fw(\012)g Fu(\031)2160 4777 y Fq(\014)2200 4786 y Fl(2)2238 4762 y Fu(;)1433 4990 y(\031)1488 4952 y Fv(#)1576 4990 y Fx(=)25 b Fu(\031)1727 4952 y Fv(#)1724 5012 y(s)1810 4990 y Fw(\012)20 b Fu(\031)1956 4946 y Fv(#)1953 5019 y Fq(\014)1993 5028 y Fl(1)2051 4990 y Fw(\012)g Fu(\031)2197 4946 y Fv(#)2194 5019 y Fq(\014)2234 5028 y Fl(2)2273 4990 y Fu(;)1488 5234 y Fw(L)25 b Fx(=)g Fw(L)1735 5248 y Fv(s)1787 5234 y Fx(+)20 b Fw(L)1941 5186 y Fv(\(1\))1941 5263 y(f)2055 5234 y Fx(+)g Fw(L)2209 5186 y Fv(\(2\))2209 5263 y(f)2303 5234 y Fu(:)p eop %%Page: 23 23 23 22 bop 3680 100 a FA(23)244 407 y Fz(Pr)n(oposition)25 b(3.4)49 b Fj(The)24 b(GNS)f(r)m(epr)m(esentation)28 b(of)c Fw(O)i Fj(associated)i(to)c Fu(!)2530 421 y Fv(s)2582 407 y Fw(\012)c Fu(!)2730 422 y Fq(\014)2770 431 y Fl(1)2829 407 y Fw(\012)h Fu(!)2978 422 y Fq(\014)3018 431 y Fl(2)3079 407 y Fj(is)j Fx(\()p Fw(H)q Fu(;)15 b(\031)s(;)g Fx(\012\))p Fj(.)244 520 y(If)21 b Fu(!)377 534 y Fv(s)428 520 y Fj(is)g Fu(\034)549 534 y Fv(s)581 520 y Fj(-in)l(variant,)j(then)e Fw(L)e Fj(is)h(the)g(corr)m(esponding)k(standar)m(d)e(Liouvillean.)31 b(If)20 b Fu(!)2940 534 y Fv(s)2992 520 y Fj(is)h(faithful,)i(then)244 633 y Fx(\012)f Fj(is)i(a)f(separ)o(ating)j(vector)f(for)f(the)g(en)l (veloping)j(von)d(Neumann)g(alg)o(ebr)o(a)1126 791 y Fi(M)i Fw(\021)f Fu(\031)s Fx(\()p Fw(O)s Fx(\))1543 754 y Ft(00)1611 791 y Fx(=)g Fu(\031)1759 805 y Fv(s)1791 791 y Fx(\()p Fw(O)1898 805 y Fv(s)1930 791 y Fx(\))c Fw(\012)f Fi(M)2172 806 y Fv(f)5 b Fq(;\014)2259 815 y Fl(1)2317 791 y Fw(\012)20 b Fi(M)2503 806 y Fv(f)5 b Fq(;\014)2590 815 y Fl(2)2628 791 y Fu(:)244 950 y Fj(F)-10 b(or)23 b Fu(!)450 964 y Fv(s)505 950 y Fj(faithful,)i (consider)h(the)e(pair)g Fx(\()p Fi(M)p Fu(;)15 b Fx(\012\))p Fj(.)244 1079 y Fy(\(i\))23 b Fj(Its)h(modular)h(oper)o(ator)g(is)f Fx(\001)h(=)g(\001)1493 1093 y Fv(s)1545 1079 y Fw(\012)20 b Fx(\001)1712 1031 y Fv(\(1\))1712 1108 y(f)1826 1079 y Fw(\012)g Fx(\001)1993 1031 y Fv(\(2\))1993 1108 y(f)2087 1079 y Fj(.)244 1192 y Fy(\(ii\))k Fj(Its)f(modular)i(conjugation)i(is) d Fu(J)34 b Fx(=)25 b Fu(J)1593 1206 y Fv(s)1646 1192 y Fw(\012)19 b Fu(J)1786 1207 y Fv(f)1838 1192 y Fw(\012)h Fu(J)1979 1207 y Fv(f)2010 1192 y Fj(.)244 1305 y Fy(\(iii\))k Fu(J)9 b(\031)s Fx(\()p Fu(A)p Fx(\))p Fu(J)35 b Fx(=)25 b Fu(\031)891 1272 y Fv(#)954 1305 y Fx(\()p Fu(A)p Fx(\))p Fj(.)380 1473 y Fy(Let)30 b(no)n(w)h Fu(V)50 b Fy(be)31 b(the)g(perturbation)k(\(1.5\).)51 b(The)30 b(standard)j(Liouvillean)g Fw(L)2819 1487 y Fq(V)2910 1473 y Fy(for)e(the)g(perturbed)244 1586 y(dynamics)e(is)e(no)n(w)g(easily)h(computed)h(in)e(the)h (representation)j Fu(\031)s Fy(.)40 b(W)l(ith)27 b(a)g(slight)i(ab)n (use)f(of)g(notation)244 1711 y(we)d(identify)j Fu(V)46 b Fy(and)26 b Fu(\031)s Fx(\()p Fu(V)21 b Fx(\))p Fy(.)36 b(Moreo)o(v)o(er)l(,)28 b(we)d(denote)j(the)e(\002eld)g(and)h(number)g (operators)h(on)f Fw(H)3337 1664 y Fv(\()p Fq(i)p Fv(\))3336 1741 y(f)3445 1711 y Fy(by)244 1824 y Fu(')303 1791 y Fv(\()p Fq(i)p Fv(\))409 1824 y Fy(and)d Fu(N)636 1838 y Fq(i)664 1824 y Fy(.)k(Then,)887 1983 y Fu(V)46 b Fx(=)25 b(\()p Fu(\033)1169 1997 y Fq(x)1233 1983 y Fw(\012)20 b Fo(1)p Fx(\))h Fw(\012)f Fu(')1582 1946 y Fv(\(1\))1676 1983 y Fx(\()p Fu(\013)1769 1998 y Fv(1)p Fq(\014)1844 2007 y Fl(1)1883 1983 y Fx(\))h(+)f(\()p Fu(\033)2117 1997 y Fq(x)2181 1983 y Fw(\012)g Fo(1)p Fx(\))h Fw(\012)f Fu(')2530 1946 y Fv(\(2\))2625 1983 y Fx(\()p Fu(\013)2718 1998 y Fv(2)p Fq(\014)2793 2007 y Fl(2)2832 1983 y Fx(\))p Fu(;)398 2170 y(J)9 b(V)20 b(J)35 b Fx(=)25 b(\()p Fo(1)c Fw(\012)f Fu(\033)962 2184 y Fq(x)1005 2170 y Fx(\))h Fw(\012)1152 2069 y Fm(\020)1206 2170 y Fx(i\()p Fw(\000)p Fo(1)p Fx(\))1425 2132 y Fq(N)1481 2141 y Fl(1)1520 2170 y Fu(')1579 2132 y Fv(\(1\))1674 2170 y Fx(\()p Fu(\013)1767 2125 y Fv(#)1767 2199 y(1)p Fq(\014)1842 2208 y Fl(1)1881 2170 y Fx(\))1916 2069 y Fm(\021)1991 2170 y Fx(+)e(\()p Fo(1)i Fw(\012)f Fu(\033)2332 2184 y Fq(x)2376 2170 y Fx(\))h Fw(\012)2523 2069 y Fm(\020)2577 2170 y Fx(i\()p Fw(\000)p Fo(1)p Fx(\))2796 2132 y Fq(N)2852 2141 y Fl(2)2891 2170 y Fu(')2950 2132 y Fv(\(2\))3044 2170 y Fx(\()p Fu(\013)3137 2125 y Fv(#)3137 2199 y(2)p Fq(\014)3212 2208 y Fl(2)3251 2170 y Fx(\))3286 2069 y Fm(\021)3356 2170 y Fu(:)244 2382 y Fz(Pr)n(oposition)k(3.5)49 b Fj(The)21 b(standar)m(d)j(Liouvillean)h(of)d(the)g(perturbed)i(system)f Fx(\()p Fw(O)s Fu(;)15 b(\034)2868 2397 y Fq(\025)2914 2382 y Fx(\))21 b Fj(in)h(the)g(r)m(epr)m(esen-)244 2495 y(tation)j Fu(\031)h Fj(is)1440 2608 y Fw(L)1503 2623 y Fq(\025)1573 2608 y Fx(=)f Fw(L)20 b Fx(+)g Fu(\025V)40 b Fw(\000)20 b Fu(\025J)9 b(V)21 b(J)o(:)380 2920 y Fy(Assume)j(no)n(w) f(that)h(\(A2\))f(holds.)30 b(Then,)23 b(the)h(assumption)i(\(R3\))e (of)f(the)h(Section)g(2.2)g(holds)g(and)244 3114 y Fu(J)9 b(V)356 3133 y Ft(\000)p Fv(i)p Fq(=)p Fv(2)505 3114 y Fu(J)35 b Fx(=)25 b(i)727 3028 y Fm(X)780 3223 y Fq(i)857 3114 y Fx(\()p Fo(1)c Fw(\012)p 1056 3064 48 4 v 20 w Fu(\032)1103 3077 y Fv(1)p Fq(=)p Fv(2)1103 3137 y(s)1213 3114 y Fu(\033)1265 3128 y Fq(x)p 1309 3064 V 1309 3114 a Fu(\032)1356 3136 y Fv(s)1388 3077 y Ft(\000)p Fv(1)p Fq(=)p Fv(2)1553 3114 y Fx(\))f Fw(\012)1747 3053 y Fx(1)p 1709 3093 122 4 v 1709 3111 a Fw(p)p 1785 3111 46 4 v 76 x Fx(2)1840 3114 y(\()p Fw(\000)p Fo(1)q Fx(\))2034 3077 y Fq(N)2090 3087 y Fn(i)2136 3013 y Fm(\020)2190 3114 y Fu(a)2238 3077 y Fv(\()p Fq(i)p Fv(\))2321 3114 y Fx(\(e)2396 3077 y Ft(\000)p Fq(\014)2491 3087 y Fn(i)2518 3077 y Fq(s=)p Fv(2)2625 3114 y Fu(\013)2683 3070 y Fv(#)2683 3143 y Fq(i\014)2747 3153 y Fn(i)2777 3114 y Fx(\))h(+)f Fu(a)2972 3077 y Fv(\()p Fq(i)p Fv(\))p Ft(\003)3090 3114 y Fx(\(e)3165 3077 y Fq(\014)3205 3087 y Fn(i)3232 3077 y Fq(s=)p Fv(2)3339 3114 y Fu(\013)3397 3070 y Fv(#)3397 3143 y Fq(i\014)3461 3153 y Fn(i)3492 3114 y Fx(\))3527 3013 y Fm(\021)3596 3114 y Fu(;)299 3456 y(J)9 b(V)411 3475 y Fv(i)p Fq(=)p Fv(2)505 3456 y Fu(J)35 b Fx(=)25 b(i)727 3370 y Fm(X)780 3565 y Fq(i)857 3456 y Fx(\()p Fo(1)c Fw(\012)p 1056 3406 48 4 v 20 w Fu(\032)1103 3478 y Fv(s)1135 3419 y Ft(\000)p Fv(1)p Fq(=)p Fv(2)1300 3456 y Fu(\033)1352 3470 y Fq(x)p 1396 3406 V 1396 3456 a Fu(\032)1443 3419 y Fv(1)p Fq(=)p Fv(2)1443 3479 y(s)1553 3456 y Fx(\))f Fw(\012)1747 3395 y Fx(1)p 1709 3435 122 4 v 1709 3454 a Fw(p)p 1785 3454 46 4 v 75 x Fx(2)1840 3456 y(\()p Fw(\000)p Fo(1)p Fx(\))2033 3419 y Fq(N)2089 3429 y Fn(i)2136 3355 y Fm(\020)2190 3456 y Fu(a)2238 3419 y Fv(\()p Fq(i)p Fv(\))2321 3456 y Fx(\(e)2396 3419 y Fq(\014)2436 3429 y Fn(i)2463 3419 y Fq(s=)p Fv(2)2570 3456 y Fu(\013)2628 3412 y Fv(#)2628 3486 y Fq(i\014)2692 3496 y Fn(i)2722 3456 y Fx(\))h(+)f Fu(a)2917 3419 y Fv(\()p Fq(i)p Fv(\))p Ft(\003)3035 3456 y Fx(\(e)3110 3419 y Ft(\000)p Fq(\014)3205 3429 y Fn(i)3232 3419 y Fq(s=)p Fv(2)3339 3456 y Fu(\013)3397 3412 y Fv(#)3397 3486 y Fq(i\014)3461 3496 y Fn(i)3492 3456 y Fx(\))3527 3355 y Fm(\021)3596 3456 y Fu(:)244 3722 y Fz(Pr)n(oposition)25 b(3.6)49 b Fj(If)30 b Fu(!)1019 3736 y Fv(s)1079 3722 y Fj(is)g(faithful)i(and)e(Hypothesis)i Fy(\(A2\))e Fj(holds,)i(then)f (the)f(Hypothesis)i Fy(\(R3\))e Fj(of)244 3835 y(Section)25 b(2.2)f(holds)g(for)g(the)g(perturbation)j Fu(V)43 b Fj(and)24 b(the)g Fu(C)7 b Fj(-Liouvillean)26 b(is)1376 3994 y Fu(L)1438 4009 y Fq(\025)1508 3994 y Fx(=)f Fw(L)20 b Fx(+)g Fu(\025V)40 b Fw(\000)20 b Fu(\025J)9 b(V)2180 4013 y Ft(\000)p Fv(i)p Fq(=)p Fv(2)2330 3994 y Fu(J)o(:)244 4153 y Fj(The)23 b(adjoint)i(of)f Fu(L)839 4168 y Fq(\025)907 4153 y Fj(is)1403 4266 y Fu(L)1465 4228 y Ft(\003)1465 4289 y Fq(\025)1535 4266 y Fx(=)h Fw(L)20 b Fx(+)g Fu(\025V)41 b Fw(\000)20 b Fu(\025J)9 b(V)2208 4284 y Fv(i)p Fq(=)p Fv(2)2302 4266 y Fu(J)o(:)244 4434 y Fy(Although)27 b(the)f(standard)i (Liouvillean)g(does)f(not)f(depend)h(on)f(the)g(choice)h(of)e(the)h (initial)h(state)g(of)f(the)244 4547 y(small)h(system,)h(the)f Fu(C)7 b Fy(-Liouvillean)29 b(does)e(through)i(the)e(term)f Fu(J)9 b(V)2395 4565 y Ft(\000)p Fv(i)p Fq(=)p Fv(2)2545 4547 y Fu(J)g Fy(.)37 b(It)27 b(is)f(often)i(con)l(v)o(enient)i(to)244 4660 y(tak)o(e)24 b(a)f(simple)h(choice)h(for)f(the)g(initial)h(state)f Fu(!)1749 4674 y Fv(s)1781 4660 y Fy(,)e(namely)1541 4818 y Fu(!)1598 4832 y Fv(s)1630 4818 y Fx(\()p Fu(A)p Fx(\))k(=)f(T)-8 b(r)o(\()p Fu(A)p Fx(\))p Fu(=)p Fx(2)p Fu(;)1080 b Fy(\(3.32\))244 4977 y(whose)24 b(density)h(matrix)f(is)g Fu(\032)1175 4991 y Fv(s)1232 4977 y Fx(=)h Fo(1)p Fu(=)p Fx(2)p Fy(.)k(In)23 b(this)i(case)f Fu(L)2016 4992 y Fq(\025)2084 4977 y Fy(tak)o(es)g(a)f(slightly)j(simpler)e(form)g(and) 1371 5150 y Fw(k)p Fu(V)1469 5168 y Ft(\006)p Fv(i)p Fq(=)p Fv(2)1619 5150 y Fw(k)h(\024)g Fx(2)1845 5063 y Fm(X)1900 5259 y Fq(i)1992 5150 y Fw(k)p Fx(e)2077 5112 y Fq(\014)2117 5122 y Fn(i)2144 5112 y Fq(s=)p Fv(2)2252 5150 y Fu(\013)2310 5164 y Fq(i)2338 5150 y Fw(k)p Fu(:)p eop %%Page: 24 24 24 23 bop 3680 100 a FA(24)244 407 y Fs(4)143 b(Spectral)34 b(analysis)244 614 y Fy(The)26 b(spectral)i(analysis)g(of)e(the)g (operators)j Fw(L)1701 629 y Fq(\025)1771 614 y Fy(and)e Fu(L)1990 581 y Ft(\003)1990 641 y Fq(\025)2060 614 y Fy(follo)n(ws)g(closely)h([JP1].)36 b(In)26 b(this)h(section)h(we)244 727 y(will)23 b(state)h(the)f(main)h(results)g(of)g(this)f(analysis)j (and)e(discuss)h(some)e(of)g(their)h(consequences.)33 b(W)-7 b(e)22 b(will)244 840 y(only)h(indicate)h(the)f(main)f(steps)i (of)e(the)g(proofs)i(and)f(the)g(interested)i(reader)e(should)h (consult)g([JP1)q(])e(for)244 953 y(details.)30 b(Throughout)c(this)e (section)h(we)e(assume)i(that)f(the)f(assumptions)k(\(A1\))c(and)h (\(A2\))f(hold.)380 1065 y(Recall)36 b(that)818 1042 y Fx(~)817 1065 y Fi(h)47 b Fx(=)f Fu(L)1090 1032 y Fv(2)1129 1065 y Fx(\()p Fp(R)s Fu(;)16 b Fi(G)p Fx(\))41 b Fy(and)35 b Fw(H)1654 1080 y Fv(f)1731 1065 y Fx(=)47 b(\000)1906 1079 y Ft(\000)1964 1065 y Fx(\()2000 1042 y(~)1999 1065 y Fi(h)q Fx(\))p Fy(.)63 b(Let)34 b Fu(p)46 b Fw(\021)g Fx(i)p Fu(@)2606 1079 y Fv(s)2672 1065 y Fy(be)35 b(the)g(generator)j (of)d(the)244 1178 y(group)24 b(of)g(translations)i(on)1133 1155 y Fx(~)1132 1178 y Fi(h)d Fy(and)g Fu(P)39 b Fx(=)25 b(d\000\()p Fu(p)p Fx(\))d Fy(its)h(second)i(quantization.)33 b(W)-7 b(e)22 b(adopt)j(the)e(shorthand)244 1304 y Fw(h)p Fu(P)13 b Fw(i)26 b Fx(=)f(\(1)c(+)f Fu(P)770 1271 y Fv(2)809 1304 y Fx(\))854 1241 y Fl(1)p 854 1253 31 3 v 854 1295 a(2)899 1304 y Fy(.)28 b(Let)23 b Fu(\016)29 b(>)c Fx(0)e Fy(be)h(as)f(in)h(\(A2\).)k(In)c(what)f(follo)n(ws)h(we)f (\002x)g Fu(\024)g Fy(such)h(that)1342 1490 y Fx(0)i Fu(<)f(\024)g(<)g Fx(min)o(\()p Fu(\031)s(=\014)2020 1504 y Fv(1)2061 1490 y Fu(;)15 b(\031)s(=\014)2252 1504 y Fv(2)2292 1490 y Fu(;)g(\016)s Fx(\))p Fu(:)244 1675 y Fy(Let)1378 1788 y Fu(D)28 b Fx(:=)d Fo(1)c Fw(\012)f Fx(e)1806 1750 y Ft(\000)p Fq(\024)p Ft(h)p Fq(P)10 b Ft(i)2036 1788 y Fw(\012)20 b Fx(e)2167 1750 y Ft(\000)p Fq(\024)p Ft(h)p Fq(P)10 b Ft(i)2376 1788 y Fu(:)244 1945 y Fy(Ob)o(viously)-6 b(,)26 b Fx(Ran)o Fu(D)h Fy(is)c(dense)j(in)e Fw(H)g Fy(and)g(the)h(v)o(ectors)g(of)f(the)h(form)f Fu( )g Fw(\012)c Fx(\012)2682 1897 y Fv(\(1\))2682 1975 y(f)2797 1945 y Fw(\012)g Fx(\012)2954 1897 y Fv(\(2\))2954 1975 y(f)3048 1945 y Fy(,)j Fu( )30 b Fw(2)c(H)3347 1959 y Fv(s)3378 1945 y Fy(,)e(are)244 2058 y(in)l(v)n(ariant)i(under)f Fu(D)s Fy(.)i(Recall)d(that)g Fi(P)p Fx(\()p Fu(\026)p Fx(\))i Fw(\021)f(f)p Fu(z)20 b Fw(j)15 b Fx(Im)p Fu(z)30 b(>)25 b(\026)p Fw(g)p Fy(.)380 2171 y(W)-7 b(e)23 b(deal)h(\002rst)f (with)h(the)f(standard)j(Liouvillean)g(and)e(Theorem)g(1.3.)244 2374 y Fz(Theor)n(em)46 b(4.1)k Fj(F)-10 b(or)34 b(any)i Fu(\026)46 b(>)g Fw(\000)p Fu(\024)34 b Fj(ther)m(e)i(is)e(a)h (constant)i Fx(\003)47 b Fu(>)f Fx(0)34 b Fj(suc)o(h)i(that)f(for)g Fw(j)p Fu(\025)p Fw(j)47 b Fu(<)f Fx(\003)35 b Fj(the)244 2487 y(oper)o(ator)n(-valued)28 b(function)1490 2600 y Fu(z)i Fw(7!)25 b Fu(D)s Fx(\()p Fu(z)g Fw(\000)20 b(L)2012 2615 y Fq(\025)2057 2600 y Fx(\))2092 2562 y Ft(\000)p Fv(1)2186 2600 y Fu(D)s(;)1027 b Fy(\(4.33\))244 2757 y Fj(originally)31 b(de\002ned)f(for)f Fx(Im)p Fu(z)39 b(>)34 b Fx(0)p Fj(,)29 b(has)g(a)f(mer)l(omorphic)i(continuation)j(to) 28 b(the)h(half-plane)i Fi(P)p Fx(\()p Fu(\026)p Fx(\))p Fj(.)244 2870 y(The)19 b(function)i(\(4.33\))f(has)f(at)g(most)g(four)h (poles)g(in)f Fi(P)p Fx(\()p Fw(\000)p Fu(\026)p Fx(\))p Fj(.)27 b(If)19 b(in)g(addition)i Fy(\(A3\))e Fj(holds)h(and)g Fu(\014)3261 2884 y Fv(1)3326 2870 y Fw(6)p Fx(=)25 b Fu(\014)3473 2884 y Fv(2)3513 2870 y Fj(,)244 2983 y(then)k(ther)m(e)h (is)f(a)f(constant)j Fu(`)k(>)g Fx(0)28 b Fj(suc)o(h)h(that)h(for)f Fx(0)36 b Fu(<)e Fw(j)p Fu(\025)p Fw(j)i Fu(<)f(`)27 b Fj(none)j(of)f(the)g(poles)h(is)f(on)f(the)i(r)m(eal)244 3096 y(axis.)g(In)24 b(particular)-10 b(,)27 b(for)d Fx(0)i Fu(<)g Fw(j)p Fu(\025)p Fw(j)g Fu(<)g(`)c Fj(the)j(spectrum)g (of)f Fw(L)2177 3111 y Fq(\025)2245 3096 y Fj(is)f(pur)m(ely)i (absolutely)i(continuous)g(and)244 3209 y(ther)m(e)d(ar)m(e)g(no)f Fu(\034)744 3224 y Fq(\025)789 3209 y Fj(-in)l(variant)k(states)e(in)e (the)h(set)g Fw(N)36 b Fj(of)23 b(normal)h(states.)244 3384 y Fy(The)29 b(last)g(part)h(of)f(Theorem)g(4.1,)i(the)e(absence)i (of)e Fu(\034)1964 3399 y Fq(\025)2009 3384 y Fy(-in)l(v)n(ariant)j (states)e(in)f Fw(N)13 b Fy(,)30 b(is)f(the)g(statement)i(of)244 3496 y(Theorem)24 b(1.3.)380 3609 y(The)f(proof)h(of)e(Theorem)i(4.1)e (follo)n(ws)i(the)f(ar)n(gument)i(in)e([JP1,)f(JP2].)28 b(Although)d(in)e(these)g(w)o(orks)244 3722 y(the)35 b(Bose)f(reserv)n(oirs)j(are)e(studied,)k(the)c(same)f(\(in)h(f)o(act,) i(slightly)g(simpler\))e(ar)n(gument)i(applies)f(to)244 3835 y(Fermi)22 b(reserv)n(oirs.)32 b(F)o(or)22 b(the)h(reader)i(con)l (v)o(enience)h(and)e(for)f(latter)h(applications,)j(we)22 b(recall)i(the)g(main)244 3948 y(steps)g(of)g(the)g(ar)n(gument)h(in)f ([JP1,)f(JP2].)244 4148 y Fz(Sk)o(etch)f(of)i(the)f(pr)n(oof)h(of)f (Theor)n(em)h(4.1.)k Fy(Let)1409 4333 y Fu(u)p Fx(\()p Fu(\022)s Fx(\))d Fw(\021)g Fx(e)1738 4296 y Ft(\000)p Fv(i)p Fq(\022)r(P)1932 4333 y Fx(=)g(\000\(e)2160 4296 y Ft(\000)p Fv(i)p Fq(\022)r(p)2310 4333 y Fx(\))p Fu(;)244 4519 y Fy(be)h(the)g(second)i(quantization)i(of)c(the)h(group)g(of)f (translations)k(and)c Fu(U)10 b Fx(\()p Fu(\022)s Fx(\))30 b(=)g Fo(1)22 b Fw(\012)g Fu(u)p Fx(\()p Fu(\022)s Fx(\))g Fw(\012)g Fu(u)p Fx(\()p Fu(\022)s Fx(\))p Fy(.)36 b(W)-7 b(e)244 4631 y(set)1427 4744 y Fw(L)1490 4759 y Fq(\025)1535 4744 y Fx(\()p Fu(\022)s Fx(\))25 b Fw(\021)g Fu(U)10 b Fx(\()p Fu(\022)s Fx(\))p Fw(L)2023 4759 y Fq(\025)2068 4744 y Fu(U)g Fx(\()p Fw(\000)p Fu(\022)s Fx(\))p Fu(:)244 4902 y Fy(Let)23 b Fu(N)35 b Fx(=)25 b Fu(N)665 4916 y Fv(1)725 4902 y Fx(+)20 b Fu(N)889 4916 y Fv(2)928 4902 y Fy(.)28 b(Note)23 b(that)1424 5075 y Fu(U)10 b Fx(\()p Fu(\022)s Fx(\))p Fw(L)p Fu(U)g Fx(\()p Fw(\000)p Fu(\022)s Fx(\))24 b(=)h Fw(L)20 b Fx(+)g Fu(\022)s(N)5 b(;)1403 5284 y(U)10 b Fx(\()p Fu(\022)s Fx(\))p Fu(N)g(U)g Fx(\()p Fw(\000)p Fu(\022)s Fx(\))25 b(=)g Fu(N)5 b(;)p eop %%Page: 25 25 25 24 bop 3680 100 a FA(25)244 407 y Fy(and)892 594 y Fu(U)10 b Fx(\()p Fu(\022)s Fx(\))p Fu(V)20 b(U)10 b Fx(\()p Fw(\000)p Fu(\022)s Fx(\))25 b(=)1533 508 y Fm(X)1586 703 y Fq(i)1664 594 y Fx(\()p Fu(\033)1751 608 y Fq(x)1815 594 y Fw(\012)20 b Fo(1)p Fx(\))h Fw(\012)f Fu(')2164 556 y Fv(\()p Fq(i)p Fv(\))2248 594 y Fx(\(e)2323 556 y Ft(\000)p Fv(i)p Fq(\022)r(p)2472 594 y Fu(\013)2530 609 y Fq(i\014)2594 619 y Fn(i)2625 594 y Fx(\))p Fu(;)774 912 y(U)10 b Fx(\()p Fu(\022)s Fx(\))p Fu(J)f(V)20 b(J)9 b(U)h Fx(\()p Fw(\000)p Fu(\022)s Fx(\))25 b(=)g(i)1574 825 y Fm(X)1627 1021 y Fq(i)1705 912 y Fx(\()p Fo(1)20 b Fw(\012)g Fu(\033)1955 926 y Fq(x)1999 912 y Fx(\))h Fw(\012)f Fx(\()p Fw(\000)p Fo(1)p Fx(\))2339 874 y Fq(N)2395 884 y Fn(i)2426 912 y Fu(')2485 874 y Fv(\()p Fq(i)p Fv(\))2568 912 y Fx(\(e)2643 874 y Ft(\000)p Fv(i)p Fq(\022)r(p)2793 912 y Fu(\013)2851 867 y Fv(#)2851 941 y Fq(i\014)2915 951 y Fn(i)2945 912 y Fx(\))p Fu(:)244 1183 y Fy(If)1238 1295 y Fu(V)1291 1309 y Fv(tot)1386 1295 y Fx(\()p Fu(\022)s Fx(\))25 b(=)g Fu(U)10 b Fx(\()p Fu(\022)s Fx(\)\()p Fu(V)41 b Fw(\000)19 b Fu(J)9 b(V)21 b(J)9 b Fx(\))p Fu(U)h Fx(\()p Fw(\000)p Fu(\022)s Fx(\))p Fu(;)244 1458 y Fy(then)1340 1571 y Fw(L)1403 1586 y Fq(\025)1448 1571 y Fx(\()p Fu(\022)s Fx(\))25 b(=)g Fw(L)20 b Fx(+)g Fu(\022)s(N)29 b Fx(+)20 b Fu(\025V)2204 1585 y Fv(tot)2298 1571 y Fx(\()p Fu(\022)s Fx(\))p Fu(:)244 1733 y Fy(By)29 b(\(A2\))i(and)f(\(3.31\))h (the)g(operator)h Fu(V)1516 1747 y Fv(tot)1611 1733 y Fx(\()p Fu(\022)s Fx(\))d Fy(is)h(de\002ned)h(for)g(all)f Fu(\022)40 b Fw(2)d Fu(I)7 b Fx(\()p Fu(\024)p Fx(\))31 b Fy(and)g(the)f(map)g Fu(I)7 b Fx(\()p Fu(\024)p Fx(\))39 b Fw(3)244 1846 y Fu(\022)27 b Fw(7!)e Fu(V)483 1860 y Fv(tot)578 1846 y Fx(\()p Fu(\022)s Fx(\))d Fy(is)i(an)f(analytic)j (operator)n(-v)n(alued)i(function)e(satisfying)908 2055 y Fu(C)31 b Fx(:=)64 b(sup)1125 2138 y Fq(\022)r Ft(2)p Fq(I)5 b Fv(\()p Fq(\024)p Fv(\))1354 2055 y Fw(k)p Fu(V)1452 2069 y Fv(tot)1547 2055 y Fx(\()p Fu(\022)s Fx(\))p Fw(k)25 b(\024)g Fx(2)1874 1975 y Fw(p)p 1950 1975 46 4 v 80 x Fx(2)2011 1969 y Fm(X)2065 2164 y Fq(i)2158 2055 y Fu(C)7 b Fx(\()p Fu(\024;)15 b(\014)2408 2069 y Fq(i)2437 2055 y Fx(\))p Fw(k)9 b Fx(~)-54 b Fu(\013)2575 2069 y Fq(i)2604 2055 y Fw(k)2649 2075 y Fq(H)2712 2056 y Fl(2)2747 2075 y Fv(\()p Fq(\024)p Fv(\))2847 2055 y Fu(:)244 2332 y Fy(Ob)o(viously)-6 b(,)24 b(the)e(operator)i Fw(L)1179 2347 y Fq(\025)1224 2332 y Fx(\()p Fu(\022)s Fx(\))d Fy(is)h(also)h(de\002ned)g(for)f Fu(\022)27 b Fw(2)e Fu(I)7 b Fx(\()p Fu(\024)p Fx(\))p Fy(.)29 b(F)o(or)21 b Fx(Im)p Fu(\022)27 b Fw(6)p Fx(=)e(0)p Fy(,)d Fw(L)2978 2347 y Fq(\025)3023 2332 y Fx(\()p Fu(\022)s Fx(\))f Fy(is)h(a)f(closed)244 2445 y(operator)29 b(with)e(domain)h Fw(D)s Fx(\()p Fw(L)p Fx(\))23 b Fw(\\)g(D)s Fx(\()p Fu(N)10 b Fx(\))p Fy(.)39 b(Let)26 b Fu(I)1855 2412 y 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12001 5286 l 12001 5429 l cp % Polyline n 12858 9001 m 12072 9001 l 12072 5429 l 12858 5429 l cp gs col7 1.00 shd ef gr % Polyline n 657 9286 m 14 9286 l 14 5429 l 657 5429 l cp gs col7 1.00 shd ef gr % Polyline 15.000 slw n 10658 1600 m 10658 1857 l gs col-1 s gr % Polyline n 2085 1600 m 2085 1857 l gs col-1 s gr % Polyline n 6372 1600 m 6372 1857 l gs col-1 s gr % Polyline [90] 0 sd n 642 6586 m 12072 6586 l gs col-1 s gr [] 0 sd % Polyline 7.500 slw gs clippath 8629 6887 m 8686 6887 l 8686 6740 l 8658 6855 l 8629 6740 l cp 8686 1713 m 8629 1713 l 8629 1859 l 8658 1745 l 8686 1859 l cp eoclip n 8658 1728 m 8658 6872 l gs col-1 s gr gr % arrowhead n 8686 1859 m 8658 1745 l 8629 1859 l col-1 s % arrowhead n 8629 6740 m 8658 6855 l 8686 6740 l col-1 s % Polyline gs clippath 3914 6601 m 3971 6601 l 3971 6454 l 3943 6569 l 3914 6454 l cp 3971 1713 m 3914 1713 l 3914 1859 l 3943 1745 l 3971 1859 l cp eoclip n 3943 1728 m 3943 6586 l gs col-1 s gr gr % arrowhead n 3971 1859 m 3943 1745 l 3914 1859 l col-1 s % arrowhead n 3914 6454 m 3943 6569 l 3971 6454 l col-1 s % Polyline gs clippath 9986 6601 m 10043 6601 l 10043 6454 l 10015 6569 l 9986 6454 l cp 10043 5714 m 9986 5714 l 9986 5860 l 10015 5746 l 10043 5860 l cp eoclip n 10015 5729 m 10015 6586 l gs col-1 s gr gr % arrowhead n 10043 5860 m 10015 5746 l 9986 5860 l col-1 s % arrowhead n 9986 6454 m 10015 6569 l 10043 6454 l col-1 s $F2psEnd rs %%EndDocument @endspecial 690 1390 a Fd(\000)p FD(2)2170 b(2)-1163 b(0)900 2117 y Fd(\000)p FD(Im\()p FC(\022)r FD(\))2851 2684 y FC(O)r FD(\()p FC(\025)p FD(\))2551 2117 y(min)p FC(\031)s(=\014)2828 2129 y Fc(i)838 3690 y FA(Figure)25 b(1:)30 b(Resonances)25 b(of)g(the)g(standard)f(Liouvillean)g Fb(L)2897 3705 y Fq(\025)380 4014 y Fy(If)19 b(\(A3\))g(holds,)i(then)e Fx(Im)p Fu(a)1219 3975 y Ft(\006)1219 4040 y Fv(2)1303 4014 y Fu(<)25 b Fx(0)19 b Fy(and)g(for)g Fu(\025)g Fy(non-zero)i(and)e (suf)n(\002ciently)i(small,)f Fx(Im)15 b Fu(e)3128 4028 y Ft(\006)p Fv(2)3223 4014 y Fx(\()p Fu(\025)p Fx(\))26 b Fu(<)e Fx(0)p Fy(.)244 4127 y(Notice)h(also)g(that)g(the)g(matrices)h Fu(T)1360 4141 y Fq(i)1412 4127 y Fy(are)e(self-adjoint)k(and)d(non-ne) o(gati)n(v)o(e)i(with)d(a)h(simple)g(eigen)l(v)n(alue)244 4240 y Fx(0)e Fy(and)h(corresponding)k(eigen)l(v)o(ector)1559 4498 y Fu( )1618 4513 y Fq(\014)1658 4523 y Fn(i)1714 4498 y Fx(=)1810 4370 y Fm(\022)1877 4442 y Fx(e)1917 4409 y Ft(\000)p Fq(\014)2012 4419 y Fn(i)2038 4409 y Fq(=)p Fv(2)1904 4558 y Fx(e)1944 4525 y Fq(\014)1984 4535 y Fn(i)2011 4525 y Fq(=)p Fv(2)2113 4370 y Fm(\023)2195 4498 y Fu(:)244 4779 y Fy(Thus,)23 b(unless)i Fu(\014)772 4793 y Fv(1)837 4779 y Fx(=)g Fu(\014)984 4793 y Fv(2)1024 4779 y Fy(,)e Fx(i\006)1162 4793 y Fv(2)1225 4779 y Fu(>)i Fx(0)p Fy(,)e(and)h(for)g Fu(\025)e Fy(non-zero)k(and)e(suf)n (\002ciently)i(small,)d Fx(Im)15 b Fu(e)3108 4735 y Fv(1)p Fq(;)p Fv(2)3108 4805 y(0)3203 4779 y Fx(\()p Fu(\025)p Fx(\))26 b Fu(<)f Fx(0)p Fy(.)380 4892 y(T)-7 b(o)22 b(\002nish)h(the)g(proof,)h(we)e(ha)n(v)o(e)h(to)g(relate)g Fu(D)s Fx(\()p Fu(z)f Fw(\000)17 b(L)2057 4907 y Fq(\025)2102 4892 y Fx(\))2137 4859 y Ft(\000)p Fv(1)2232 4892 y Fu(D)24 b Fy(and)g Fx(\()p Fu(z)d Fw(\000)c(L)2734 4907 y Fq(\025)2779 4892 y Fx(\()p Fu(\022)s Fx(\)\))2930 4859 y Ft(\000)p Fv(1)3025 4892 y Fy(.)28 b(T)-7 b(o)21 b(do)i(so,)g(we)244 5005 y(\002x)g Fu(z)k Fy(with)c Fx(Im)p Fu(z)k Fy(lar)n(ge)d(enough.)31 b(Then,)23 b(one)h(sho)n(ws)g(that)1048 5209 y Fx(s)c Fw(\000)52 b Fx(lim)1195 5270 y Fv(Im)o Fq(\022)r Ft(")p Fv(0)1385 5209 y Fx(\()p Fu(z)25 b Fw(\000)20 b(L)1641 5224 y Fq(\025)1686 5209 y Fx(\()p Fu(\022)s Fx(\)\))1837 5171 y Ft(\000)p Fv(1)1956 5209 y Fx(=)25 b(\()p Fu(z)g Fw(\000)20 b(L)2308 5224 y Fq(\025)2353 5209 y Fx(\(Re)q Fu(\022)s Fx(\)\))2612 5171 y Ft(\000)p Fv(1)2706 5209 y Fu(:)585 b Fy(\(4.34\))p eop %%Page: 27 27 27 26 bop 3680 100 a FA(27)244 407 y Fy(Let)1189 520 y Fu(D)s Fx(\()p Fu(\022)s Fx(\))25 b(=)f Fo(1)d Fw(\012)f Fx(e)1707 482 y Ft(\000)p Fq(\024)p Ft(h)p Fq(P)10 b Ft(i)p Fv(+)p Fq(\022)r(P)2081 520 y Fw(\012)20 b Fx(e)2212 482 y Ft(\000)p Fq(\024)p Ft(h)p Fq(P)10 b Ft(i)p Fv(+)p Fq(\022)r(P)2566 520 y Fu(;)244 687 y Fy(and)24 b(consider)i(the)d (function)1111 891 y Fu(I)1158 853 y Ft(\000)1218 891 y Fx(\()p Fu(\024)p Fx(\))j Fw(3)f Fu(\022)i Fw(7!)e Fu(D)s Fx(\()p Fu(\022)s Fx(\)\()p Fu(z)g Fw(\000)20 b(L)2088 906 y Fq(\025)2133 891 y Fx(\()p Fu(\022)s Fx(\)\))2284 853 y Ft(\000)p Fv(1)2378 891 y Fu(D)s Fx(\()p Fw(\000)p Fu(\022)s Fx(\))p Fu(:)244 1095 y Fy(By)j(analyticity)-6 b(,)26 b(this)e(function)i(is)d(constant)j(in)d Fu(\022)s Fy(.)28 b(By)22 b(\(4.34\))j(and)f(continuity)-6 b(,)26 b(the)e(relation)1021 1299 y Fu(D)s Fx(\()p Fu(z)h Fw(\000)20 b(L)1355 1314 y Fq(\025)1400 1299 y Fx(\))1435 1262 y Ft(\000)p Fv(1)1530 1299 y Fu(D)28 b Fx(=)d Fu(D)s Fx(\()p Fu(\022)s Fx(\)\()p Fu(z)f Fw(\000)c(L)2178 1314 y Fq(\025)2223 1299 y Fx(\()p Fu(\022)s Fx(\)\))2374 1262 y Ft(\000)p Fv(1)2469 1299 y Fu(D)s Fx(\()p Fw(\000)p Fu(\022)s Fx(\))p Fu(;)557 b Fy(\(4.35\))244 1504 y(holds)31 b(for)f Fw(\000)p Fu(\024)36 b(<)g Fx(Im)p Fu(\022)j Fw(\024)d Fx(0)p Fy(.)47 b(If)29 b Fu(\022)i Fy(in)e(\(4.35\))i(satis\002es)g Fw(\000)p Fu(\024)36 b(<)g Fx(Im)p Fu(\022)j(<)d Fw(\000)p Fx(\()p Fu(\024)25 b Fx(+)g Fw(j)p Fu(\026)p Fw(j)p Fx(\))p Fu(=)p Fx(2)p Fy(,)31 b(then)g(the)244 1616 y(right-hand)36 b(side)e(in)f(\(4.35\))h(pro)o(vides)h(the)f(desired)h(meromorphic)g (continuation)i(of)c(the)h(function)244 1729 y Fu(D)s Fx(\()p Fu(z)24 b Fw(\000)c(L)577 1744 y Fq(\025)622 1729 y Fx(\))657 1696 y Ft(\000)p Fv(1)752 1729 y Fu(D)s Fy(.)380 1842 y(Since)37 b Fx(Ran)p Fu(D)h Fy(is)e(dense)i(in)e Fw(H)h Fy(and)g Fu(D)s Fx(\()p Fu(z)d Fw(\000)29 b(L)1972 1857 y Fq(\025)2017 1842 y Fx(\))2052 1809 y Ft(\000)p Fv(1)2147 1842 y Fu(D)38 b Fy(has)f(no)g(poles)g(on)g(the)g(real)g (axis,)j(the)244 1955 y(spectrum)25 b(of)f Fw(L)762 1970 y Fq(\025)830 1955 y Fy(is)g(purely)h(absolutely)i(continuous)g(for)d Fx(0)i Fu(<)g Fw(j)p Fu(\025)p Fw(j)g Fu(<)g(`)p Fy(.)j(In)24 b(particular)l(,)i Fx(Ker)p Fw(L)3300 1970 y Fq(\025)3371 1955 y Fx(=)f(0)p Fy(,)244 2068 y(and,)f(by)f(Proposition)j(2.1,)d (there)i(are)e(no)h Fu(\034)1627 2083 y Fq(\025)1672 2068 y Fy(-in)l(v)n(ariant)i(states)f(in)f(the)f(set)h Fw(N)36 b Fy(of)23 b(normal)i(states.)30 b Fe(2)380 2231 y Fy(In)c(the)f(proof)i(of)e(Theorem)h(4.1)f(we)g(ha)n(v)o(e)h(not)g (used)g(the)g(full)g(strength)h(of)f(the)f(assumption)j(\(A2\))244 2344 y(and)37 b(for)f(this)h(theorem)g(it)f(suf)n(\002ces)h(that)45 b Fx(~)-54 b Fu(\013)1678 2358 y Fq(i)1755 2344 y Fw(2)49 b Fu(H)1948 2311 y Fv(2)1987 2344 y Fx(\()p Fu(\024)p Fx(\))p Fy(.)67 b(In)36 b(f)o(act,)k(if)c(the)g(comple)o(x)h (deformation)244 2457 y(technique)28 b(is)d(replaced)j(with)d(Mourre)i (theory)-6 b(,)27 b(then)f(the)g(main)g(conclusion)i(of)e(the)f (theorem)i(can)f(be)244 2570 y(deri)n(v)o(ed)f(under)f(much)g(weak)o (er)g(re)o(gularity)i(condition)g(on)33 b Fx(~)-54 b Fu(\013)2187 2584 y Fq(i)2215 2570 y Fy(,)23 b(see)h([DJ1,)f(DJ2].)380 2682 y(W)-7 b(e)21 b(no)n(w)g(deal)h(with)f(the)g Fu(C)7 b Fy(-Liouvillean)24 b(and)e(Theorem)g(1.2.)27 b(As)21 b(we)g(ha)n(v)o(e)h(remark)o(ed)h(at)e(the)g(end)244 2795 y(of)f(the)h(last)g(section,)h(it)f(is)f(con)l(v)o(enient)j(to)e (tak)o(e)g(for)f(the)h(initial)h(state)f(of)g(the)f(small)h(system)g (the)g(state)g Fu(!)3504 2809 y Fv(s)244 2908 y Fy(de\002ned)e(by)f (\(3.32\).)28 b(In)18 b(what)g(follo)n(ws)h Fu(L)1543 2923 y Fq(\025)1606 2908 y Fy(is)f(the)g Fu(C)7 b Fy(-Liouvillean)21 b(associated)g(to)d Fu(!)28 b Fx(=)d Fu(!)3068 2922 y Fv(s)3100 2908 y Fw(\012)q Fu(!)3229 2923 y Fq(\014)3269 2932 y Fl(1)3307 2908 y Fw(\012)q Fu(!)3436 2923 y Fq(\014)3476 2932 y Fl(2)3513 2908 y Fy(.)244 3021 y(Let)1395 3134 y Fw(R)1472 3148 y Fq(D)1536 3134 y Fx(\()p Fu(z)t Fx(\))h(=)f Fu(D)s Fx(\()p Fu(z)g Fw(\000)20 b Fu(L)2107 3097 y Ft(\003)2107 3157 y Fq(\025)2152 3134 y Fx(\))2187 3097 y Ft(\000)p Fv(1)2282 3134 y Fu(D)s(:)244 3347 y Fz(Theor)n(em)46 b(4.2)k Fj(F)-10 b(or)34 b(any)i Fu(\026)46 b(>)g Fw(\000)p Fu(\024)34 b Fj(ther)m(e)i(is)e(a)h(constant)i Fx(\003)47 b Fu(>)f Fx(0)34 b Fj(suc)o(h)i(that)f(for)g Fw(j)p Fu(\025)p Fw(j)47 b Fu(<)f Fx(\003)35 b Fj(the)244 3460 y(oper)o(ator)n(-valued) 41 b(function)c Fw(R)1276 3474 y Fq(D)1341 3460 y Fx(\()p Fu(z)t Fx(\))p Fj(,)h(originally)h(de\002ned)e(for)g Fu(z)52 b Fw(2)47 b Fi(P)p Fx(\()p Fu(a)p Fx(\))p Fj(,)39 b(has)d(a)g(mer)l(omorphic)244 3572 y(continuation)29 b(to)d(the)f(half-plane)k Fi(P)p Fx(\()p Fu(\026)p Fx(\))p Fj(.)34 b(The)25 b(function)j Fw(R)2176 3586 y Fq(D)2240 3572 y Fx(\()p Fu(z)t Fx(\))d Fj(has)h(at)g(most)f(four)h(poles)h(in)e Fi(P)p Fx(\()p Fu(\026)p Fx(\))p Fj(,)244 3685 y(and)e(zer)l(o)h(is)f (one)g(of)g(its)g(poles.)30 b(Let)22 b Fu(P)1466 3647 y Fv(+)1453 3715 y Fq(\025)1547 3685 y Fj(be)h(the)g(r)m(esidue)h(of)f Fw(R)2244 3699 y Fq(D)2308 3685 y Fx(\()p Fu(z)t Fx(\))g Fj(at)g Fx(0)p Fj(.)28 b(If)23 b(in)g(addition)i Fx(\(A3\))f Fj(holds)244 3798 y(and)e Fu(\025)k Fw(6)p Fx(=)f(0)p Fj(,)c(then)i Fx(dim)14 b(Ran)p Fu(P)1244 3760 y Fv(+)1231 3828 y Fq(\025)1328 3798 y Fx(=)25 b(1)d Fj(and)g(all)h(singularities)i (of)d Fw(R)2410 3812 y Fq(D)2475 3798 y Fx(\()p Fu(z)t Fx(\))g Fj(e)n(xcept)h(zer)l(o)g(ar)m(e)f(contained)244 3911 y(in)h(the)h(half-plane)j Fx(Im)o Fu(z)j(<)25 b Fx(0)p Fj(.)j(Mor)m(eo)o(ver)-10 b(,)25 b Fu(P)1701 3873 y Fv(+)1688 3941 y Fq(\025)1782 3911 y Fj(is)f(analytic)h(function)h (of)d Fu(\025)g Fj(for)h Fw(j)p Fu(\025)p Fw(j)i Fu(<)f Fx(\003)p Fj(.)244 4090 y Fy(The)i(proof)i(of)e(this)h(theorem)h(is)e (a)g(slight)i(elaboration)i(of)c(the)h(ar)n(guments)i(in)d([JP1)q(,)f (JP2)q(])h(which)h(we)244 4203 y(ha)n(v)o(e)c(already)h(sk)o(etched)h (abo)o(v)o(e.)k(W)-7 b(e)22 b(gi)n(v)o(e)i(belo)n(w)f(an)h(outline)h (of)f(the)f(proof.)244 4416 y Fz(Sk)o(etch)35 b(of)i(the)f(pr)n(oof)h (of)f(Theor)n(em)h(4.2.)67 b Fy(W)-7 b(e)35 b(use)i(the)g(notation)i (introduced)g(in)e(the)f(proof)i(of)244 4529 y(Theorem)24 b(4.1.)k(F)o(or)23 b(real)h Fu(\022)h Fy(let)1054 4709 y Fu(L)1116 4671 y Ft(\003)1116 4732 y Fq(\025)1161 4709 y Fx(\()p Fu(\022)s Fx(\))g Fw(\021)g Fu(U)10 b Fx(\()p Fu(\022)s Fx(\))p Fu(L)1648 4671 y Ft(\003)1648 4732 y Fq(\025)1693 4709 y Fu(U)g Fx(\()p Fw(\000)p Fu(\022)s Fx(\))p Fu(;)1027 4907 y Fx(~)1013 4930 y Fu(V)1066 4944 y Fv(tot)1161 4930 y Fx(\()p Fu(\022)s Fx(\))25 b Fw(\021)g Fu(U)10 b Fx(\()p Fu(\022)s Fx(\))p Fu(V)20 b(U)10 b Fx(\()p Fw(\000)p Fu(\022)s Fx(\))20 b Fw(\000)g Fu(U)10 b Fx(\()p Fu(\022)s Fx(\))p Fu(J)f(V)2329 4949 y Fv(i)p Fq(=)p Fv(2)2423 4930 y Fu(J)g(U)h Fx(\()p Fw(\000)p Fu(\022)s Fx(\))p Fu(:)244 5127 y Fy(Clearly)-6 b(,)1340 5240 y Fu(L)1402 5202 y Ft(\003)1402 5263 y Fq(\025)1447 5240 y Fx(\()p Fu(\022)s Fx(\))26 b(=)f Fw(L)20 b Fx(+)f Fu(\022)s(N)30 b Fx(+)20 b Fu(\025)2165 5217 y Fx(~)2151 5240 y Fu(V)2204 5254 y Fv(tot)2298 5240 y Fx(\()p Fu(\022)s Fx(\))p Fu(:)p eop %%Page: 28 28 28 27 bop 3680 100 a FA(28)244 407 y Fy(Assumption)27 b(\(A2\))e(implies)i(that)f Fu(I)7 b Fx(\()p Fu(\024)p Fx(\))30 b Fw(3)e Fu(\022)j Fw(7!)1854 384 y Fx(~)1840 407 y Fu(V)1893 421 y Fv(tot)1988 407 y Fx(\()p Fu(\022)s Fx(\))24 b Fy(is)i(an)f(analytic)j(operator)n(-v)n(alued)i(function)244 520 y(satisfying)362 736 y Fx(~)341 759 y Fu(C)i Fx(:=)63 b(sup)559 842 y Fq(\022)r Ft(2)p Fq(I)5 b Fv(\()p Fq(\024)p Fv(\))788 759 y Fw(k)847 736 y Fx(~)833 759 y Fu(V)886 773 y Fv(tot)980 759 y Fx(\()p Fu(\022)s Fx(\))p Fw(k)26 b(\024)1311 698 y Fx(1)p 1273 738 122 4 v 1273 756 a Fw(p)p 1349 756 46 4 v 76 x Fx(2)1419 673 y Fm(X)1473 868 y Fq(i)1566 759 y Fu(C)7 b Fx(\()p Fu(\024;)15 b(\014)1816 773 y Fq(i)1845 759 y Fx(\))1895 658 y Fm(\020)1950 759 y Fx(3)p Fw(k)9 b Fx(~)-54 b Fu(\013)2098 773 y Fq(i)2127 759 y Fw(k)2172 779 y Fq(H)2235 760 y Fl(2)2270 779 y Fv(\()p Fq(\024)p Fv(\))2390 759 y Fx(+)20 b Fw(k)p Fx(e)2566 722 y Ft(\000)p Fq(\014)2661 732 y Fn(i)2688 722 y Fq(s=)p Fv(2)2804 759 y Fx(~)-54 b Fu(\013)2853 773 y 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9750 l 15 5700 l 690 5700 l cp gs col7 1.00 shd ef gr % Polyline 15.000 slw [90] 0 sd n 690 7215 m 12690 7215 l gs col-1 s gr [] 0 sd $F2psEnd rs %%EndDocument @endspecial 691 1378 a Fd(\000)p FD(2)1044 b(0)1082 b(2)1336 3677 y FA(Figure)25 b(2:)30 b(Resonances)25 b(of)g Fa(L)2398 3641 y Ft(\003)2398 3703 y Fq(\025)380 4012 y Fy(One)k(no)n(w)g(repeats)i(the)e(analysis)j(outlined)f(in)e (the)h(proof)g(of)f(Theorem)h(4.1.)45 b(F)o(or)29 b Fx(\003)3160 3989 y(~)3139 4012 y Fu(C)42 b(<)36 b Fx(\()p Fu(\024)25 b Fw(\000)244 4125 y(j)p Fu(\026)p Fw(j)p Fx(\))p Fu(=)p Fx(4)19 b Fy(and)f Fw(j)p Fu(\025)p Fw(j)26 b Fu(<)f Fx(\003)17 b Fy(the)i(essential)h(spectrum)g(of)e Fu(L)1911 4092 y Ft(\003)1911 4153 y Fq(\025)1956 4125 y Fx(\()p Fu(\022)s Fx(\))f Fy(is)h(contained)j(in)d(the)g(half-plane)j Fw(f)p Fu(z)f Fw(j)15 b Fx(Im)p Fu(z)29 b(<)244 4238 y(\026)p Fw(g)p Fy(.)34 b(Here,)26 b(again,)g(the)g(location)i(of)e (the)g(discrete)h(spectrum)g(inside)g Fi(P)p Fx(\()p Fu(\026)p Fx(\))f 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b(Moreo)o(v)o(er)l(,)24 b(the)g(functions)i Fu(\025)g Fw(7!)i Fx(~)-48 b Fu(e)2116 4895 y Ft(\006)p Fv(2)2210 4881 y Fx(\()p Fu(\025)p Fx(\))24 b Fy(are)g(analytic)h(for)f Fw(j)p Fu(\025)p Fw(j)i Fu(<)f Fx(\003)p Fy(,)1381 5156 y Fx(~)-49 b Fu(e)1419 5170 y Ft(\006)p Fv(2)1514 5156 y Fx(\()p Fu(\025)p Fx(\))26 b(=)f Fw(\006)p Fx(2)20 b(+)2016 5042 y Ft(1)1986 5069 y Fm(X)1990 5265 y Fq(j)t Fv(=1)2132 5156 y Fu(\025)2185 5118 y Fv(2)p Fq(j)2259 5156 y Fx(~)-47 b Fu(a)2305 5118 y Ft(\006)2305 5183 y Fv(2)p Fq(j)2377 5156 y Fu(;)p eop %%Page: 29 29 29 28 bop 3680 100 a FA(29)244 407 y Fy(and)24 b(one)g(\002nds)f(that) 1711 520 y Fx(~)-47 b Fu(a)1757 481 y Ft(\006)1757 546 y Fv(2)1842 520 y Fx(=)25 b Fu(a)1986 481 y Ft(\006)1986 546 y Fv(2)2045 520 y Fu(:)244 695 y Fy(The)30 b(resonances)37 b Fx(~)-49 b Fu(e)885 651 y Fv(1)p Fq(;)p Fv(2)885 721 y(0)980 695 y Fx(\()p Fu(\025)p Fx(\))30 b Fy(are)h(the)f(eigen)l(v)n (alues)k(of)d(a)f Fx(2)25 b Fw(\002)g Fx(2)30 b Fy(matrix)2561 672 y Fx(~)2551 695 y(\006\()p Fu(\025)p Fx(\))g Fy(which)h(is)f (analytic)j(for)244 808 y Fw(j)p Fu(\025)p Fw(j)26 b Fu(<)f Fx(\003)p Fy(,)1528 967 y Fx(~)1517 990 y(\006\()p Fu(\025)p Fx(\))h(=)1858 877 y Ft(1)1828 904 y Fm(X)1832 1099 y Fq(j)t Fv(=1)1975 990 y Fu(\025)2028 953 y Fv(2)p Fq(j)2109 967 y Fx(~)2099 990 y(\006)2165 1004 y Fv(2)p Fq(j)2237 990 y Fu(;)244 1246 y Fy(and)1386 1336 y Fx(~)1376 1359 y(\006)1442 1373 y Fv(2)1506 1359 y Fx(=)f Fw(\000)p Fx(i)p Fu(\031)1768 1272 y Fm(X)1822 1468 y Fq(i)1915 1359 y Fw(k)9 b Fx(~)-54 b Fu(\013)2018 1373 y Fq(i)2047 1359 y Fx(\(2\))p Fw(k)2207 1321 y Fv(2)2207 1382 y Fg(G)2315 1336 y Fx(~)2297 1359 y Fu(T)2350 1373 y Fq(i)2378 1359 y Fu(;)244 1601 y Fy(where)913 1723 y Fx(~)895 1746 y Fu(T)948 1760 y Fq(i)1002 1746 y Fx(=)25 b(e)1138 1709 y Ft(\000)p Fq(\014)1233 1719 y Fn(i)1259 1709 y Fq(\033)1299 1717 y Fn(z)1335 1709 y Fq(=)p Fv(2)1410 1746 y Fu(T)1463 1760 y Fq(i)1491 1746 y Fx(e)1531 1709 y Fq(\014)s(\033)1614 1717 y Fn(z)1651 1709 y Fq(=)p Fv(2)1751 1746 y Fx(=)1998 1685 y(1)p 1857 1725 328 4 v 1857 1809 a(2)15 b(cosh)h Fu(\014)2156 1823 y Fq(i)2209 1618 y Fm(\022)2312 1690 y Fx(e)2352 1657 y Fq(\014)2392 1667 y Fn(i)2541 1690 y Fw(\000)p Fx(e)2652 1657 y Ft(\000)p Fq(\014)2747 1667 y Fn(i)2276 1803 y Fw(\000)p Fx(e)2387 1770 y Fq(\014)2427 1780 y Fn(i)2576 1803 y Fx(e)2616 1770 y Ft(\000)p Fq(\014)2711 1780 y Fn(i)2777 1618 y Fm(\023)2859 1746 y Fu(:)244 1961 y Fy(Notice)24 b(that)1662 2083 y Fx(~)1644 2106 y Fu(T)1710 2068 y Ft(\003)1697 2128 y Fq(i)1764 1978 y Fm(\022)1831 2049 y Fx(1)1831 2162 y(1)1877 1978 y Fm(\023)1969 2106 y Fx(=)h(0)p Fu(;)244 2342 y Fy(and)f(so)f(zero)i(is) e(al)o(w)o(ays)h(an)g(eigen)l(v)n(alue)i(of)1665 2319 y Fx(~)1655 2342 y(\006)1721 2356 y Fv(2)1760 2342 y Fy(.)i(The)23 b(second)i(eigen)l(v)n(alue)i(of)2769 2319 y Fx(~)2759 2342 y(\006)2825 2356 y Fv(2)2887 2342 y Fy(is)c(equal)i(to)1350 2557 y Fx(T)-8 b(r)o(\()1488 2534 y(~)1478 2557 y(\006)1544 2571 y Fv(2)1584 2557 y Fx(\))25 b(=)g Fw(\000)p Fx(i)p Fu(\031)1906 2470 y Fm(X)1960 2666 y Fq(i)2053 2557 y Fw(k)9 b Fx(~)-54 b Fu(\013)2156 2571 y Fq(i)2185 2557 y Fx(\(2\))p Fw(k)2345 2519 y Fv(2)2345 2580 y Fg(G)2405 2557 y Fu(:)244 2838 y Fy(If)27 b(\(A3\))h(holds,)h(then)f(this)g(eigen)l(v)n(alue)j(has)d (ne)o(gati)n(v)o(e)g(imaginary)i(part.)41 b(Thus,)28 b(for)g Fu(\025)f Fy(non-zero)i(and)244 2950 y(suf)n(\002ciently)c (small,)f Fx(Im)s(~)-49 b Fu(e)1066 2918 y Fv(2)1066 2975 y(0)1106 2950 y Fx(\()p Fu(\025)p Fx(\))26 b Fu(<)f Fx(0)p Fy(.)380 3063 y(F)o(ollo)n(wing)f(the)g(ar)n(gument)h(in)f(the)g (proof)g(of)g(Theorem)g(4.1,)f(we)f(see)i(that)1186 3268 y Fw(R)1263 3282 y Fq(D)1327 3268 y Fx(\()p Fu(z)t Fx(\))i(=)f Fu(D)s Fx(\()p Fu(\022)s Fx(\)\()p Fu(z)f Fw(\000)c Fu(L)2013 3230 y Ft(\003)2013 3291 y Fq(\025)2059 3268 y Fx(\()p Fu(\022)s Fx(\)\))2210 3230 y Ft(\000)p Fv(1)2304 3268 y Fu(D)s Fx(\()p Fw(\000)p Fu(\022)s Fx(\))p Fu(;)244 3472 y Fy(pro)o(vides)29 b(the)e(required)j(meromorphic)f(continuation) i(of)c Fw(R)2225 3486 y Fq(D)2289 3472 y Fy(.)39 b(By)27 b(this)g(formula,)i(the)f(residue)h Fu(P)3477 3434 y Fv(+)3464 3501 y Fq(\025)244 3607 y Fy(is)c(related)j(to)d(the)h (spectral)i(projection)g Fu(Q)1606 3559 y Fv(\(1\))1606 3637 y Fq(\025)1701 3607 y Fx(\()p Fu(\022)s Fx(\))c Fy(corresponding)30 b(to)c(the)g(zero)g(eigen)l(v)n(alue)j(of)d Fu(L)3375 3574 y Ft(\003)3375 3635 y Fq(\025)3420 3607 y Fx(\()p Fu(\022)s Fx(\))244 3720 y Fy(by)1381 3846 y Fu(P)1452 3808 y Fv(+)1439 3875 y Fq(\025)1537 3846 y Fx(=)f Fu(D)s Fx(\()p Fu(\022)s Fx(\))p Fu(Q)1899 3798 y Fv(\(1\))1899 3875 y Fq(\025)1993 3846 y Fx(\()p Fu(\022)s Fx(\))p Fu(D)s Fx(\()p Fw(\000)p Fu(\022)s Fx(\))p Fu(:)917 b Fy(\(4.37\))244 4013 y(This)23 b(implies)i(that)f Fx(dim)13 b(Ran)p Fu(P)1278 3975 y Fv(+)1265 4042 y Fq(\025)1363 4013 y Fx(=)25 b(1)p Fy(.)380 4148 y(T)-7 b(o)29 b(pro)o(v)o(e)i(the)g (last)f(statement)i(of)e(the)h(theorem)g(we)f(must)g(sho)n(w)g(that)h Fu(Q)2783 4100 y Fv(\(1\))2783 4177 y Fq(\025)2877 4148 y Fx(\()p Fu(\022)s Fx(\))e Fy(is)h(analytic)j(for)244 4269 y Fw(j)p Fu(\025)p Fw(j)26 b Fu(<)f Fx(\003)p Fy(.)i(W)-7 b(e)19 b(pro)o(v)o(e)i(this)h(by)e(relating)j(this)e(operator)i(to)d (the)h(spectral)h(projection)2901 4247 y Fx(~)2890 4269 y(\005\()p Fu(\025)p Fx(\))e Fy(correspond-)244 4382 y(ing)k(to)g(the)g(zero)g(eigen)l(v)n(alue)j(of)d(the)g(analytic)i (matrix)f Fu(A)p Fx(\()p Fu(\025)p Fx(\))h(=)g Fu(\025)2373 4349 y Ft(\000)p Fv(2)2477 4359 y Fx(~)2467 4382 y(\006\()p Fu(\025)p Fx(\))p Fy(.)j(Notice)c(that)f(since)h Fx(0)f Fy(is)f(a)244 4495 y(simple)h(eigen)l(v)n(alue)j(of)c Fu(A)p Fx(\(0\))p Fy(,)1259 4472 y Fx(~)1248 4495 y(\005\()p Fu(\025)p Fx(\))h Fy(is)f(analytic)i(for)f Fu(\025)f Fy(small)h(enough.)380 4608 y(Let)32 b(us)g(recall)h(the)f (construction)k(of)31 b(the)i(operator)2094 4585 y Fx(~)2084 4608 y(\006)o(\()p Fu(\025)p Fx(\))f Fy([JP1)q(,)f(HP)o(].)54 b(By)31 b(taking)i Fx(\003)f Fy(possibly)244 4721 y(smaller)l(,)23 b(one)f(can)h(\002nd)e(a)h(contour)i Fu(\015)i Fy(around)e Fx(0)d Fy(such)i(that)g(for)f Fu(\022)h Fy(with)f Fx(Im)o Fu(\022)i Fy(suf)n(\002ciently)g(close)f(to)f Fw(\000)p Fu(\024)244 4834 y Fy(and)h(for)g Fw(j)p Fu(\025)p Fw(j)j Fu(<)f Fx(\003)p Fy(,)d(the)i(spectral)h(projection)g(corresponding)j (to)22 b(the)i(group)g Fw(f)s Fx(~)-48 b Fu(e)2777 4801 y Fv(1)2777 4858 y(0)2817 4834 y Fx(\()p Fu(\025)p Fx(\))p Fu(;)20 b Fx(~)-50 b Fu(e)3022 4801 y Fv(2)3022 4858 y(0)3063 4834 y Fx(\()p Fu(\025)p Fx(\))p Fw(g)23 b Fy(is)g(gi)n(v)o (en)244 4947 y(by)1251 5099 y Fu(Q)1323 5114 y Fq(\025)1368 5099 y Fx(\()p Fu(\022)s Fx(\))i(=)1655 5037 y(1)p 1615 5078 126 4 v 1615 5161 a(2)p Fu(\031)s Fx(i)1766 4975 y Fm(I)1817 5181 y Fq(\015)1861 5099 y Fx(\()p Fu(z)g Fw(\000)20 b Fu(L)2116 5061 y Ft(\003)2116 5122 y Fq(\025)2161 5099 y Fx(\()p Fu(\022)s Fx(\)\))2312 5061 y Ft(\000)p Fv(1)2407 5099 y Fx(d)p Fu(z)t(:)787 b Fy(\(4.38\))p eop %%Page: 30 30 30 29 bop 3680 100 a FA(30)244 407 y Fu(Q)316 422 y Fq(\025)361 407 y Fx(\()p Fu(\022)s Fx(\))23 b Fy(is)g(an)h(analytic)h(function)h (of)d Fu(\025)g Fy(and)1463 599 y Fw(k)p Fu(Q)1580 614 y Fq(\025)1625 599 y Fx(\()p Fu(\022)s Fx(\))d Fw(\000)g Fu(Q)1924 613 y Fv(0)1964 599 y Fx(\()p Fu(\022)s Fx(\))p Fw(k)25 b Fu(<)g Fx(1)p Fu(:)244 791 y Fy(Notice)30 b(that)h Fu(Q)753 805 y Fv(0)792 791 y Fx(\()p Fu(\022)s Fx(\))37 b(=)g Fu(Q)1125 805 y Fv(0)1194 791 y Fy(does)31 b(not)f(depend)i(on)e Fu(\022)h Fy(and)g(is)f(the)g(spectral)i(projection)h(of)d Fw(L)f Fy(corre-)244 904 y(sponding)d(to)d(double)j(de)o(generate)g (eigen)l(v)n(alue)g Fx(0)p Fy(.)j(It)23 b(follo)n(ws)h(that)g(the)g (maps)1378 1096 y Fu(Q)1450 1110 y Fv(0)1514 1096 y Fx(:)i(Ran)p Fu(Q)1800 1111 y Fq(\025)1845 1096 y Fx(\()p Fu(\022)s Fx(\))f Fw(!)g Fx(Ran)p Fu(Q)2337 1110 y Fv(0)2376 1096 y Fu(;)1317 1288 y(Q)1389 1303 y Fq(\025)1434 1288 y Fx(\()p Fu(\022)s Fx(\))g(:)h(Ran)p Fu(Q)1861 1302 y Fv(0)1925 1288 y Fw(!)f Fx(Ran)p Fu(Q)2276 1303 y Fq(\025)2321 1288 y Fx(\()p Fu(\022)s Fx(\))p Fu(;)244 1449 y Fy(are)f (isomorphisms.)31 b(Setting)1494 1561 y Fu(T)13 b Fx(\()p Fu(\025)p Fx(\))26 b Fw(\021)f Fu(Q)1877 1575 y Fv(0)1916 1561 y Fu(Q)1988 1576 y Fq(\025)2033 1561 y Fx(\()p Fu(\022)s Fx(\))p Fu(Q)2221 1575 y Fv(0)2260 1561 y Fu(;)244 1722 y Fy(one)f(easily)h(checks)g(that)f(the)g(operator)1092 1914 y Fu(S)1148 1929 y Fq(\025)1193 1914 y Fx(\()p Fu(\022)s Fx(\))h(=)g Fu(Q)1502 1928 y Fv(0)1542 1914 y Fu(Q)1614 1929 y Fq(\025)1659 1914 y Fx(\()p Fu(\022)s Fx(\))g(:)h(Ran)o Fu(Q)2085 1929 y Fq(\025)2131 1914 y Fx(\()p Fu(\022)s Fx(\))f Fw(!)g Fx(Ran)p Fu(Q)2623 1928 y Fv(0)2662 1914 y Fu(;)244 2106 y Fy(has)f(in)l(v)o(erse)1347 2219 y Fu(S)1403 2234 y Fq(\025)1448 2219 y Fx(\()p Fu(\022)s Fx(\))1564 2182 y Ft(\000)p Fv(1)1683 2219 y Fx(=)h Fu(Q)1851 2234 y Fq(\025)1896 2219 y Fx(\()p Fu(\022)s Fx(\))p Fu(Q)2084 2233 y Fv(0)2124 2219 y Fu(T)13 b Fx(\()p Fu(\025)p Fx(\))2313 2182 y Ft(\000)p Fv(1)2407 2219 y Fu(:)244 2380 y Fy(Using)37 b(the)f(isomorphism)i Fu(S)1209 2395 y Fq(\025)1254 2380 y Fx(\()p Fu(\022)s Fx(\))p Fy(,)h(we)c(transport)k (the)e(reduced)h(operator)g Fu(Q)2812 2395 y Fq(\025)2857 2380 y Fx(\()p Fu(\022)s Fx(\))p Fu(L)3035 2347 y Ft(\003)3035 2408 y Fq(\025)3080 2380 y Fx(\()p Fu(\022)s Fx(\))p Fu(Q)3268 2395 y Fq(\025)3314 2380 y Fx(\()p Fu(\022)s Fx(\))d Fy(to)244 2493 y Fx(Ran)p Fu(Q)479 2507 y Fv(0)543 2493 y Fx(=)25 b Fp(C)699 2460 y Fv(2)744 2493 y Fy(.)j(A)23 b(simple)h(calculation)i(yields:)810 2662 y Fx(~)800 2685 y(\006)o(\()p Fu(\025)p Fx(\))g Fw(\021)f Fu(S)1166 2700 y Fq(\025)1211 2685 y Fx(\()p Fu(\022)s Fx(\))p Fu(Q)1399 2700 y Fq(\025)1444 2685 y Fx(\()p Fu(\022)s Fx(\))p Fu(L)1622 2647 y Ft(\003)1622 2708 y Fq(\025)1667 2685 y Fx(\()p Fu(\022)s Fx(\))p Fu(Q)1855 2700 y Fq(\025)1901 2685 y Fx(\()p Fu(\022)s Fx(\))p Fu(S)2073 2700 y Fq(\025)2118 2685 y Fx(\()p Fu(\022)s Fx(\))2234 2647 y Ft(\000)p Fv(1)2353 2685 y Fx(=)g Fu(M)10 b Fx(\()p Fu(\025)p Fx(\))p Fu(T)j Fx(\()p Fu(\025)p Fx(\))2859 2647 y Ft(\000)p Fv(1)2955 2685 y Fu(;)336 b Fy(\(4.39\))244 2877 y(where)1250 2990 y Fu(M)10 b Fx(\()p Fu(\025)p Fx(\))26 b Fw(\021)f Fu(Q)1665 3004 y Fv(0)1704 2990 y Fu(Q)1776 3005 y Fq(\025)1821 2990 y Fx(\()p Fu(\022)s Fx(\))p Fu(L)1999 2952 y Ft(\003)1999 3013 y Fq(\025)2044 2990 y Fx(\()p Fu(\022)s Fx(\))p Fu(Q)2232 3005 y Fq(\025)2277 2990 y Fx(\()p Fu(\022)s Fx(\))p Fu(Q)2465 3004 y Fv(0)2505 2990 y Fu(:)244 3150 y Fy(The)j(operators)i Fu(T)13 b Fx(\()p Fu(\025)p Fx(\))28 b Fy(and)h Fu(M)10 b Fx(\()p Fu(\025)p Fx(\))28 b Fy(are)h(independent) j(of)c Fu(\022)i Fy(as)e(long)h(as)f Fw(j)p Fu(\025)p Fw(j)35 b Fu(<)e Fx(\003)28 b Fy(and)h Fx(Im)o Fu(\022)h Fy(is)e(suf)n(\002-)244 3263 y(ciently)d(close)g(to)e Fw(\000)p Fu(\024)p Fy(.)28 b(Moreo)o(v)o(er)l(,)d(the)o(y)e(are)h (analytic)i(functions)g(of)d Fu(\025)p Fy(.)380 3376 y(F)o(ormula)h(\(4.39\))g(yields)h(that)1327 3558 y Fx(~)1315 3581 y(\005)q(\()p Fu(\025)p Fx(\))h(=)f Fu(S)1685 3596 y Fq(\025)1730 3581 y Fx(\()p Fu(\022)s Fx(\))p Fu(Q)1918 3533 y Fv(\(1\))1918 3610 y Fq(\025)2012 3581 y Fx(\()p Fu(\022)s Fx(\))p Fu(S)2184 3596 y Fq(\025)2229 3581 y Fx(\()p Fu(\022)s Fx(\))2345 3544 y Ft(\000)p Fv(1)2439 3581 y Fu(:)852 b Fy(\(4.40\))244 3796 y(In)l(v)o(erting)28 b(this)d(formula)i(we)d(deri)n(v)o(e)i(that)g Fu(Q)1676 3748 y Fv(\(1\))1676 3825 y Fq(\025)1770 3796 y Fx(\()p Fu(\022)s Fx(\))e Fy(\(and)i(hence)h Fu(P)2404 3757 y Fv(+)2391 3825 y Fq(\025)2463 3796 y Fy(\))d(is)h(an)h(analytic)h (function)g(for)f Fu(\025)244 3908 y Fy(small)e(enough.)p Fe(2)244 4161 y Fz(Theor)n(em)46 b(4.3)k Fj(Assume)23 b(that)i Fx(\(A3\))g Fj(holds.)31 b(Then)24 b(ther)m(e)h(is)e Fx(\003)k Fu(>)e Fx(0)f Fj(suc)o(h)h(that)f(for)g Fx(0)j Fu(<)f Fw(j)p Fu(\025)p Fw(j)h Fu(<)e Fx(\003)f Fj(all)244 4274 y(the)g(assumptions)i(of)d(Theor)m(em)h(2.6)f(hold.)244 4450 y Fz(Pr)n(oof)o(.)65 b Fy(Choose)35 b Fx(0)43 b Fu(>)g(\026)g(>)g Fw(\000)p Fu(\024)32 b Fy(and)i Fx(\003)f Fy(so)g(that)h(Theorem)f(4.2)g(holds.)59 b(This)33 b(theorem)h(v)o (eri\002es)244 4563 y(assumptions)e(\(DL6\),)d(\(b\))g(and)h(\(c\))f (of)g(Theorem)g(2.6.)46 b(T)-7 b(o)28 b(v)o(erify)h(\(d\))h(it)f(suf)n (\002ces)g(to)g(sho)n(w)g(that)h(for)244 4676 y(some)24 b Fu(r)j(>)e Fx(0)e Fy(lar)n(ge)i(enough,)g(all)f Fx(\011)h Fw(2)g(H)e Fy(and)h Fu(j)31 b Fx(=)25 b(0)p Fu(;)15 b Fx(1)1040 4906 y(sup)1041 4979 y Fq(y)r(>\026)1192 4783 y Fm(Z)1243 4989 y Ft(j)p Fq(x)p Ft(j)p Fq(>r)1430 4906 y Fw(j)p Fu(@)1508 4869 y Fq(j)1503 4929 y(x)1547 4906 y Fx(\(\011)p Fu(;)g Fw(R)1770 4920 y Fq(D)1835 4906 y Fx(\()p Fu(x)20 b Fx(+)g(i)p Fu(y)s Fx(\)\011\))p Fw(j)2273 4869 y Fv(2)p Ft(\000)p Fq(j)2399 4906 y Fx(d)p Fu(x)25 b(<)g Fw(1)p Fu(:)244 5155 y Fy(Since)1186 5268 y Fw(R)1263 5282 y Fq(D)1327 5268 y Fx(\()p Fu(z)t Fx(\))h(=)f Fu(D)s Fx(\()p Fu(\022)s Fx(\)\()p Fu(z)f Fw(\000)c Fu(L)2013 5230 y Ft(\003)2013 5291 y Fq(\025)2059 5268 y Fx(\()p Fu(\022)s Fx(\)\))2210 5230 y Ft(\000)p Fv(1)2304 5268 y Fu(D)s Fx(\()p Fw(\000)p Fu(\022)s Fx(\))p Fu(;)p eop %%Page: 31 31 31 30 bop 3680 100 a FA(31)244 407 y Fy(it)36 b(suf)n(\002ces)g(to)g (sho)n(w)g(that)h(for)f Fx(Im)p Fu(\022)h Fy(close)g(enough)h(to)e Fw(\000)p Fu(\024)p Fy(,)i Fu(\025)d Fy(small)i(enough,)j(all)c Fx(\011)48 b Fw(2)g(H)36 b Fy(and)244 520 y Fu(j)31 b Fx(=)25 b(0)p Fu(;)15 b Fx(1)p Fy(,)899 669 y Fx(sup)900 742 y Fq(y)r(>\026)1051 546 y Fm(Z)1101 752 y Ft(j)p Fq(x)p Ft(j)p Fq(>r)1288 669 y Fw(j)p Fx(\(\011)p Fu(;)g Fx(\()p Fu(x)22 b Fx(+)e(i)p Fu(y)i Fw(\000)e Fu(L)1905 632 y Ft(\003)1905 693 y Fq(\025)1950 669 y Fx(\()p Fu(\022)s Fx(\)\))2101 632 y Ft(\000)p Fv(1)p Ft(\000)p Fq(j)2283 669 y Fx(\011\))p Fw(j)2414 632 y Fv(2)p Ft(\000)p Fq(j)2541 669 y Fx(d)p Fu(x)k(<)h Fw(1)p Fu(:)436 b Fy(\(4.41\))244 892 y(Note)23 b(that)h Fu(L)664 859 y Ft(\003)664 917 y Fv(0)704 892 y Fx(\()p Fu(\022)s Fx(\))h(=)g Fw(L)20 b Fx(+)g Fu(\022)s(N)32 b Fy(is)23 b(a)g(normal)h(operator)l(,)i(and)e (that)g(the)g(bounds)1060 1127 y Fx(sup)1061 1199 y Fq(y)r(>\026)1212 1003 y Fm(Z)1262 1209 y Ft(j)p Fq(x)p Ft(j)p Fq(>r)1449 1127 y Fw(k)p Fx(\()p Fu(x)d Fx(+)f(i)p Fu(y)i Fw(\000)e Fu(L)1939 1089 y Ft(\003)1939 1149 y Fv(0)1979 1127 y Fx(\()p Fu(\022)s Fx(\)\))2130 1089 y Ft(\000)p Fv(1)2224 1127 y Fx(\011)p Fw(k)2340 1089 y Fv(2)2380 1127 y Fx(d)p Fu(x)25 b(<)g Fw(1)p Fu(;)1417 1410 y Fx(sup)1325 1493 y Fq(y)r(>\026;)p Ft(j)p Fq(x)p Ft(j)p Fq(>r)1662 1410 y Fw(k)p Fx(\()p Fu(x)c Fx(+)f(i)p Fu(y)i Fw(\000)e Fu(L)2152 1372 y Ft(\003)2152 1432 y Fv(0)2191 1410 y Fx(\()p Fu(\022)s Fx(\)\))2342 1372 y Ft(\000)p Fv(1)2437 1410 y Fw(k)26 b Fu(<)f Fw(1)p Fu(;)3316 1284 y Fy(\(4.42\))244 1667 y(follo)n(w)i(from)g(the)g(spectral)i(theorem.)40 b(The)26 b(second)j(relation)g(in)e(\(4.42\))g(and)h(the)f(resolv)o(ent)i (identity)244 1780 y(yield)24 b(that)h(for)e Fu(\025)g Fy(small)h(enough,)1048 1966 y Fx(\()p Fu(x)d Fx(+)f(i)p Fu(y)i Fw(\000)e Fu(L)1493 1928 y Ft(\003)1493 1989 y Fq(\025)1538 1966 y Fx(\()p Fu(\022)s Fx(\)\))1689 1928 y Ft(\000)p Fv(1)1809 1966 y Fx(=)25 b Fu(G)p Fx(\()p Fu(x)c Fx(+)f(i)p 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b(\031)s Fx(\()p Fw(O)1690 4355 y Fv(test)1809 4341 y Fx(\))p Fu(J)42 b Fw(\032)31 b Fi(M)2133 4308 y Ft(0)2133 4368 y Fq(D)2198 4341 y Fy(.)39 b(Since)28 b Fw(O)2564 4308 y Fv(cl)2561 4364 y(test)2712 4341 y Fx(=)k Fw(O)d Fy(and)f Fu(\031)s Fx(\()p Fw(O)3236 4355 y Fv(test)3355 4341 y Fx(\))3390 4308 y Ft(00)3465 4341 y Fx(=)244 4454 y Fi(M)p Fy(,)23 b Fu(\031)s Fx(\()p Fw(O)547 4468 y Fv(test)666 4454 y Fx(\)\012)g Fy(is)g(dense)i(in)e Fw(H)q Fy(.)28 b(Thus,)23 b Fi(M)1647 4421 y Ft(0)1647 4481 y Fq(D)1711 4454 y Fx(\012)g Fy(is)g(also)h(dense)h(in)e Fw(H)q Fy(.)380 4567 y(F)o(ollo)n(wing)40 b(the)f(abo)o(v)o(e)g(ar)n (gument)i(one)f(can)f(also)h(easily)g(v)o(erify)g(Hypothesis)h(\(DL3\)) e(in)f(our)244 4680 y(model.)29 b(W)-7 b(e)23 b(will)g(not)h(mak)o(e)g (use)f(of)h(this)g(hypothesis)i(belo)n(w)-6 b(.)29 b Fe(2)380 4838 y Fy(W)-7 b(e)23 b(are)h(no)n(w)f(ready)h(to)g(\002nish:) 244 5042 y Fz(Pr)n(oof)f(of)g(Theor)n(em)g(1.2.)28 b Fy(P)o(arts)22 b(\(i\))h(and)g(\(ii\))g(follo)n(w)g(from)g(Theorems)g (2.6)g(and)g(4.3)f(with)h Fw(N)3237 5056 y Fv(0)3301 5042 y Fx(=)i Fw(N)3472 5056 y Fq(D)244 5155 y Fy(and)i Fw(O)473 5169 y Fv(0)544 5155 y Fx(=)k Fw(O)718 5169 y Fq(D)783 5155 y Fy(.)37 b(From)26 b(the)h(construction)k(of)26 b Fw(N)1855 5169 y Fq(D)1945 5155 y Fy(and)h Fw(O)2174 5169 y Fq(D)2265 5155 y Fy(it)f(is)h(immediate)h(that)f Fw(N)3075 5169 y Fv(s)3138 5155 y Fw(\032)k(N)3315 5169 y Fq(D)3404 5155 y Fy(and)244 5268 y Fw(O)316 5282 y Fv(s)373 5268 y Fw(\032)25 b(O)541 5282 y Fq(D)606 5268 y Fy(.)p eop %%Page: 32 32 32 31 bop 3680 100 a FA(32)380 407 y Fy(Since)24 b(for)g Fu(A)h Fw(2)g(O)985 421 y Fq(D)1049 407 y Fy(,)1010 611 y Fu(!)1070 573 y Fv(+)1067 640 y Fq(\025)1129 611 y Fx(\()p Fu(A)p Fx(\))h(=)f(\()p Fu(D)1502 574 y Ft(\000)p Fv(1)1597 611 y Fu(A)1665 574 y Ft(\003)1704 611 y Fx(\012)p Fu(;)15 b Fx(\012)1876 573 y Fv(+)1876 640 y Fq(\025)1935 611 y Fx(\))26 b(=)f(\()p Fu(D)2205 574 y Ft(\000)p Fv(1)2299 611 y Fu(A)2367 574 y Ft(\003)2407 611 y Fx(\012)p Fu(;)15 b(P)2584 573 y Fv(+)2571 640 y Fq(\025)2643 611 y Fx(\012\))p Fu(;)244 815 y Fy(P)o(art)23 b(\(iii\))h(follo)n(ws)g(from)f(the)h (last)g(statement)h(of)f(Theorem)g(4.2.)k Fe(2)380 978 y Fy(As)22 b(we)f(ha)n(v)o(e)i(pointed)i(in)d(the)g(Remark)h(3)f(after) h(Theorem)g(1.2,)f(P)o(art)f(\(iii\))i(of)g(Theorem)f(1.2)g(yields)244 1091 y(that)i(for)g Fu(A)h Fw(2)g(O)783 1105 y Fq(D)870 1091 y Fy(we)e(ha)n(v)o(e)h(an)g(e)o(xpansion)1438 1362 y Fu(!)1498 1323 y Fv(+)1495 1391 y Fq(\025)1557 1362 y Fx(\()p Fu(A)p Fx(\))i(=)1847 1248 y Ft(1)1817 1275 y Fm(X)1818 1473 y Fq(k)r Fv(=0)1963 1362 y Fu(\025)2016 1324 y Fq(k)2059 1362 y Fu(!)2119 1323 y Fv(+)2116 1391 y Fq(k)2178 1362 y Fx(\()p Fu(A)p Fx(\))p Fu(:)975 b Fy(\(4.45\))244 1660 y(It)28 b(is)g(an)h(important)h(question)h (whether)f(the)e(functionals)k Fu(!)2197 1622 y Fv(+)2194 1689 y Fq(k)2284 1660 y Fy(can)d(be)f(\(at)h(least)g(in)g(principle\))i (com-)244 1773 y(puted.)f(If)1571 1952 y Fx(\012)1637 1914 y Fv(+)1637 1981 y Fq(\025)1721 1952 y Fx(=)1847 1838 y Ft(1)1817 1866 y Fm(X)1818 2063 y Fq(k)r Fv(=0)1963 1952 y Fu(\025)2016 1915 y Fq(k)2059 1952 y Fx(\012)2125 1914 y Fv(+)2125 1981 y Fq(k)2184 1952 y Fu(;)244 2213 y Fy(then)g Fu(!)489 2175 y Fv(+)486 2242 y Fq(k)548 2213 y Fx(\()p Fu(A)p Fx(\))38 b(=)e(\()p Fu(D)944 2180 y Ft(\000)p Fv(1)1039 2213 y Fu(A)1107 2180 y Ft(\003)1147 2213 y Fu(;)15 b Fx(\012)1253 2175 y Fv(+)1253 2242 y Fq(k)1312 2213 y Fx(\))p Fy(,)30 b(so)g Fu(!)1570 2175 y Fv(+)1567 2242 y Fq(k)1658 2213 y Fy(is)f(determined)j Fx(\012)2251 2175 y Fv(+)2251 2242 y Fq(k)2339 2213 y Fy(\(\(DL3\))e(implies)g(that)h(the)f(opposite)244 2326 y(is)g(also)h(true\).)50 b(T)-7 b(o)29 b(compute)i(the)g(e)o(xpansion)h (of)f Fx(\012)1927 2288 y Fv(+)1927 2355 y Fq(\025)1985 2326 y Fy(,)g(one)g(uses)g(that)g Fu(P)2623 2288 y Fv(+)2610 2355 y Fq(\025)2682 2326 y Fx(\012)37 b(=)g(\012)2959 2288 y Fv(+)2959 2355 y Fq(\025)3048 2326 y Fy(and)30 b(the)h(iden-)244 2439 y(tity)e(\(4.37\).)45 b(First,)29 b(using)h(\(4.38\),)g(one)f(e)o(xpands)i Fu(Q)1949 2454 y Fq(\025)1994 2439 y Fx(\()p Fu(\022)s Fx(\))c Fy(in)i(po)n(wers)g(of) f Fu(\025)p Fy(.)44 b(Using)28 b(this)i(result,)g(one)244 2552 y(e)o(xpands)i Fu(T)13 b Fx(\()p Fu(\025)p Fx(\))p Fy(,)31 b Fu(S)869 2567 y Fq(\025)914 2552 y Fx(\()p Fu(\022)s Fx(\))p Fy(,)f Fu(M)10 b Fx(\()p Fu(\025)p Fx(\))30 b Fy(and)1505 2529 y Fx(~)1495 2552 y(\006\()p Fu(\025)p Fx(\))p Fy(.)48 b(The)29 b(e)o(xpansion)k(of)2437 2529 y Fx(~)2427 2552 y(\006)o(\()p Fu(\025)p Fx(\))d Fy(and)h(re)o(gular)g(perturbation)244 2665 y(theory)d(yield)g(the)g(e) o(xpansion)h(of)1354 2642 y Fx(~)1343 2665 y(\005\()p Fu(\025)p Fx(\))p Fy(.)39 b(The)26 b(formulas)j(\(4.40\))f(and)f (\(4.37\))h(then)f(yield)h(the)g(e)o(xpan-)244 2794 y(sions)i(of)f Fu(Q)635 2746 y Fv(\(1\))635 2823 y Fq(\025)730 2794 y Fx(\()p Fu(\022)s Fx(\))f Fy(and)i Fu(P)1105 2755 y Fv(+)1092 2823 y Fq(\025)1164 2794 y Fx(\()p Fu(\022)s Fx(\))p Fy(.)45 b(Although)31 b(clearly)g(the)e(resulting)j(formulas)f (are)e(complicated,)k(at)244 2906 y(least)28 b(in)f(principle)i(it)e (is)g(possible)i(to)e(compute)i(an)o(y)e(term)g(in)g(the)g(e)o (xpansion)j(\(4.45\).)40 b(In)27 b(particular)l(,)244 3019 y(the)d(\002rst)f(term)g Fu(!)788 3033 y Fv(0)850 3019 y Fy(is)g(determined)j(by)e(the)g(v)o(ector)1148 3236 y Fx(\012)1214 3198 y Fv(+)1214 3262 y(0)1298 3236 y Fx(=)h Fu(P)1465 3198 y Fv(+)1452 3262 y(0)1524 3236 y Fx(\012)g(=)g(\()1757 3213 y(~)1746 3236 y(\005)q(\(0\)\012)1996 3250 y Fv(s)2028 3236 y Fx(\))c Fw(\012)f Fx(\012)2241 3188 y Fv(\(1\))2241 3266 y(f)2355 3236 y Fw(\012)g Fx(\012)2512 3188 y Fv(\(2\))2512 3266 y(f)2606 3236 y Fu(:)244 3562 y Fs(5)143 b(Entr)m(opy)34 b(pr)m(oduction)244 3869 y Fz(Pr)n(oof)i(of)g(Theor)n(em)f(1.1.)65 b Fy(W)-7 b(e)34 b(assume)j(that)f(the)g(reader)h(is)e(f)o(amiliar)i(with)e(basic)i (properties)h(of)244 3982 y(relati)n(v)o(e)29 b(entrop)o(y)h(\(a)e (particularly)k(clear)d(re)n(vie)n(w)f(is)g(gi)n(v)o(en)g(in)g([Don)q (]\).)42 b(Let)28 b Fi(M)34 b Fx(=)g Fu(\031)3021 3996 y Fq(!)3071 3982 y Fx(\()p Fw(O)s Fx(\))3216 3949 y Ft(00)3286 3982 y Fy(and)29 b(let)244 4095 y Fi(M)339 4109 y Ft(\003)401 4095 y Fy(be)24 b(the)g(predual)h(of)f Fi(M)p Fy(.)380 4208 y(Assume)i(that)g(\(a\))g(and)g(\(b\))f(hold,)i(and)f(that)g Fx(Ep)o(\()p Fu(!)2005 4169 y Fv(+)2002 4236 y Fq(V)2064 4208 y Fx(\))k(=)e Fu(!)2288 4169 y Fv(+)2285 4236 y Fq(V)2347 4208 y Fx(\()p Fu(\033)2434 4222 y Fq(V)2495 4208 y Fx(\))i(=)e(0)p Fy(.)35 b(Then,)25 b(by)h(the)g(formula)244 4320 y(\(1.2\))e(and)g(\(b\),)760 4575 y Fx(En)m(t\()p Fu(!)g Fw(\016)c Fu(\034)1136 4537 y Fq(t)1126 4597 y(V)1202 4575 y Fw(j)15 b Fu(!)s Fx(\))26 b(=)f Fw(\000)1545 4451 y Fm(Z)1636 4477 y Fq(t)1595 4657 y Fv(0)1681 4501 y Fm(\000)1722 4575 y Fu(!)s Fx(\()p Fu(\034)1867 4537 y Fq(s)1857 4597 y(V)1918 4575 y Fx(\()p Fu(\033)2005 4589 y Fq(V)2066 4575 y Fx(\)\))c Fw(\000)f Fu(!)2308 4536 y Fv(+)2305 4603 y Fq(V)2367 4575 y Fx(\()p Fu(\033)2454 4589 y Fq(V)2515 4575 y Fx(\)\))2585 4501 y Fm(\001)2643 4575 y Fx(d)p Fu(s)k Fw(\025)h(\000)p Fu(C)q(;)244 4819 y Fy(for)f(all)f Fu(t)i(>)g Fx(0)e Fy(and)h(some)g Fu(C)32 b(>)25 b Fx(0)p Fy(.)j(Set)1465 5065 y Fu(!)1522 5079 y Fq(T)1602 5065 y Fw(\021)1718 5003 y Fx(1)p 1708 5044 66 4 v 1708 5127 a Fu(T)1799 4941 y Fm(Z)1889 4967 y Fq(T)1849 5147 y Fv(0)1960 5065 y Fu(!)23 b Fw(\016)d Fu(\034)2155 5027 y Fq(t)2145 5087 y(V)2206 5065 y Fx(d)p Fu(t:)p eop %%Page: 33 33 33 32 bop 3680 100 a FA(33)244 407 y Fy(The)23 b(con)l(v)o(e)o(xity)j (and)e(the)g(upper)g(semicontinuity)k(of)23 b(the)h(relati)n(v)o(e)h (entrop)o(y)g(yield)g(that)1004 660 y Fx(En)m(t)q(\()p Fu(!)1242 674 y Fq(T)1312 660 y Fw(j)15 b Fu(!)s Fx(\))26 b Fw(\025)1589 598 y Fx(1)p 1579 639 66 4 v 1579 722 a Fu(T)1669 536 y Fm(Z)1760 562 y Fq(T)1720 742 y Fv(0)1830 660 y Fx(En)m(t)q(\()p Fu(!)d Fw(\016)e Fu(\034)2207 622 y Fq(t)2197 682 y(V)2273 660 y Fw(j)15 b Fu(!)s Fx(\)d)p Fu(t)25 b Fw(\025)g(\000)p Fu(C)q(:)244 901 y Fy(Since)i(the)h(set)g (of)f(all)h(states)g Fu(\021)36 b Fw(2)c(N)1435 915 y Fq(!)1512 901 y Fy(such)c(that)g Fx(En)m(t\()p Fu(\021)19 b Fw(j)c Fu(!)s Fx(\))33 b Fw(\025)f(\000)p Fu(C)h Fy(is)27 b Fu(\033)s Fx(\()p Fi(M)2824 915 y Ft(\003)2865 901 y Fu(;)15 b Fi(M)p Fx(\))p Fy(-compact,)31 b(the)244 1014 y(set)c(of)h(weak-*)g(limit)f(points)i(of)f Fw(f)p Fu(!)1420 1028 y Fq(T)1490 1014 y Fw(j)15 b Fu(T)45 b(>)33 b Fx(0)p Fw(g)27 b Fy(is)g(contained)j(in)e Fw(N)2489 1028 y Fq(!)2539 1014 y Fy(.)39 b(It)28 b(follo)n(ws)g(that)g Fu(!)3204 976 y Fv(+)3201 1043 y Fq(V)3295 1014 y Fw(2)k(N)3463 1028 y Fq(!)3513 1014 y Fy(,)244 1127 y(and)24 b(this)g(contradicts)j (\(a\).)h Fe(2)244 1380 y Fz(Pr)n(oof)e(of)f(Theor)n(em)h(1.4.)34 b Fy(Theorem)26 b(1.3)f(yields)i(that)f(the)f(assumption)j(\(a\))e(of)f (Theorem)h(1.1)f(holds.)244 1493 y(Let)f(us)h(v)o(erify)h(\(b\))f(for)g (the)g(initial)h(state)g Fu(!)31 b Fx(=)c Fu(!)1799 1507 y Fv(s)1852 1493 y Fw(\012)21 b Fu(!)2001 1508 y Fq(\014)2041 1517 y Fl(1)2100 1493 y Fw(\012)g Fu(!)2249 1508 y Fq(\014)2289 1517 y Fl(2)2327 1493 y Fy(,)j(where)h Fu(!)2677 1507 y Fv(s)2733 1493 y Fy(is)g(gi)n(v)o(en)g(by)g(\(3.32\).)34 b(By)244 1606 y(T)-7 b(ak)o(esaki')i(s)25 b(theorem)g([BR1],)1510 1719 y Fu(\016)1550 1733 y Fq(!)1626 1719 y Fx(=)g Fw(\000)p Fu(\014)1844 1733 y Fv(1)1883 1719 y Fu(\016)1923 1733 y Fv(1)1983 1719 y Fw(\000)20 b Fu(\014)2125 1733 y Fv(2)2165 1719 y Fu(\016)2205 1733 y Fv(2)2245 1719 y Fu(;)1046 b Fy(\(5.46\))244 1881 y(and)851 1994 y Fu(\016)891 2008 y Fq(!)942 1994 y Fx(\()p Fu(V)21 b Fx(\))k(=)g Fw(\000)p Fu(\014)1329 2008 y Fv(1)1369 1994 y Fu(\033)1421 2008 y Fq(x)1485 1994 y Fw(\012)20 b Fu(')p Fx(\(i)p Fu(s\013)1797 2008 y Fv(1)1836 1994 y Fx(\))h Fw(\012)f Fo(1)g Fw(\000)g Fu(\014)2197 2008 y Fv(2)2237 1994 y Fu(\033)2289 2008 y Fq(x)2353 1994 y Fw(\012)g Fo(1)g Fw(\012)g Fu(')p Fx(\(i)p Fu(s\013)2828 2008 y Fv(2)2867 1994 y Fx(\))p Fu(:)244 2157 y Fy(Since)25 b Fx(i)p Fu(s\013)597 2171 y Fq(i)652 2157 y Fw(2)i Fi(h)787 2171 y Fv(test)906 2157 y Fy(,)d(\()p Fi(h)1030 2171 y Fv(test)1173 2157 y Fy(is)g(gi)n(v)o(en)i(by)f(\(4.44\)\),)h Fu(\016)1932 2171 y Fq(!)1983 2157 y Fx(\()p Fu(V)20 b Fx(\))28 b Fw(2)g(O)2315 2171 y Fv(0)2354 2157 y Fy(.)k(Hence,)25 b(by)g(P)o(art)g(\(ii\))g(of)f(Theorem)244 2269 y(1.2,)f(the)h (assumption)i(\(b\))d(of)h(Theorem)g(1.1)f(holds,)i(and)f Fx(Ep)o(\()p Fu(!)2292 2231 y Fv(+)2289 2299 y Fq(\025)2351 2269 y Fx(\))i Fu(>)f Fx(0)p Fy(.)380 2382 y(It)32 b(remains)g(to)g (sho)n(w)f(that)h(the)g(entrop)o(y)i(production)h(does)d(not)g(depend)h (on)f(the)g(choice)h(of)f(the)244 2495 y(initial)25 b(state)f(in)788 2472 y Fm(b)762 2495 y Fw(N)837 2509 y Fv(s)868 2495 y Fy(.)k(Let)23 b Fu(\021)29 b Fx(=)c Fu(\021)1278 2509 y Fv(s)1330 2495 y Fw(\012)20 b Fu(!)1478 2510 y Fq(\014)1518 2519 y Fl(1)1576 2495 y Fw(\012)g Fu(!)1724 2510 y Fq(\014)1764 2519 y Fl(2)1827 2495 y Fw(2)1939 2472 y Fm(b)1913 2495 y Fw(N)1988 2509 y Fv(s)2019 2495 y Fy(.)28 b(Then,)c(by)f(Theorem)h (1.1)g(in)f([JP3)q(],)906 2752 y Fx(En)m(t)q(\()p Fu(!)g Fw(\016)e Fu(\034)1283 2714 y Fq(t)1273 2774 y(V)1348 2752 y Fw(j)15 b Fu(\021)s Fx(\))27 b(=)e(En)m(t\()p Fu(!)18 b Fw(j)d Fu(\021)s Fx(\))22 b Fw(\000)2085 2628 y Fm(Z)2176 2654 y Fq(t)2135 2834 y Fv(0)2220 2752 y Fu(\021)s Fx(\()p Fu(\034)2353 2714 y Fq(s)2343 2774 y(V)2405 2752 y Fx(\()p Fu(\016)2480 2766 y Fq(\021)2522 2752 y Fx(\()p Fu(\025V)f Fx(\)\)d)p Fu(s:)244 2987 y Fy(By)i(the)h(proof)g(of)g(Proposition)i(1.3)d(in)g([JP3)q(],)1095 3182 y Fx(En)m(t\()p Fu(!)h Fw(\016)c Fu(\034)1471 3145 y Fq(t)1461 3205 y(V)1537 3182 y Fw(j)15 b Fu(!)s Fx(\))26 b(=)f(En)m(t\()p Fu(!)f Fw(\016)c Fu(\034)2170 3145 y Fq(t)2160 3205 y(V)2236 3182 y Fw(j)15 b Fu(\021)s Fx(\))21 b(+)f Fu(O)s Fx(\(1\))p Fu(;)244 3377 y Fy(uniformly)25 b(for)f Fu(t)h(>)g Fx(0)p Fy(.)j(This)c(implies)g(that)1343 3573 y Fu(!)1403 3534 y Fv(+)1400 3602 y Fq(\025)1462 3573 y Fx(\()p Fu(\016)1537 3587 y Fq(!)1588 3573 y Fx(\()p Fu(\025V)d Fx(\)\))26 b(=)f Fu(!)2002 3534 y Fv(+)1999 3602 y Fq(\025)2061 3573 y Fx(\()p Fu(\016)2136 3587 y Fq(\021)2179 3573 y Fx(\()p Fu(\025V)20 b Fx(\)\))p Fu(:)881 b Fy(\(5.47\))244 3768 y Fe(2)380 3927 y Fy(Relation)29 b(\(5.47\))g(has)f(one)g(important)i(consequence.)45 b(Let)27 b Fu(!)j Fy(and)f Fu(\021)h Fy(be)e(as)g(in)f(the)h(abo)o(v)o (e)h(proof)244 4040 y(and)1356 4153 y Fu(\021)1401 4167 y Fv(s)1433 4153 y Fx(\()p Fu(A)p Fx(\))e(=)e(T)-8 b(r)o(\()p Fu(A)p Fx(e)1930 4116 y Fq(H)1988 4124 y Fl(s)2022 4153 y Fx(\))p Fu(=)p Fx(T)g(r\(e)2271 4116 y Fq(H)2329 4124 y Fl(s)2363 4153 y Fx(\))p Fu(:)244 4316 y Fy(Then,)1408 4428 y Fu(\016)1448 4442 y Fq(\021)1491 4428 y Fx(\()15 b Fw(\001)g Fx(\))26 b(=)f(i[)p Fu(H)1865 4442 y Fv(s)1896 4428 y Fu(;)31 b Fw(\001)15 b Fx(])21 b(+)f Fu(\016)2169 4442 y Fq(!)2220 4428 y Fx(\()15 b Fw(\001)g Fx(\))p Fu(;)244 4591 y Fy(and)24 b(\(5.47\))g(yield)h(that)1563 4704 y Fu(!)1623 4666 y Fv(+)1620 4733 y Fq(\025)1682 4704 y Fx(\([)p Fu(H)1818 4718 y Fv(s)1850 4704 y Fu(;)15 b(V)21 b Fx(]\))26 b(=)f(0)p Fu(:)1100 b Fy(\(5.48\))244 4960 y Fz(Pr)n(oof)22 b(of)f(Theor)n(em)g(1.5.)27 b Fy(The)21 b(second)i(relation)g(in)e(\(1.11\))h(follo)n(ws)g(from)f(the)g (de\002nition)i(of)e(entrop)o(y)244 5073 y(production)27 b(and)d(Relation)g(\(5.46\).)30 b(T)-7 b(o)22 b(pro)o(v)o(e)i(the)g (\002rst,)f(note)h(that)1275 5268 y Fu(\016)s Fx(\()15 b Fw(\001)g Fx(\))27 b(=)e(i[)p Fu(H)1693 5282 y Fv(s)1724 5268 y Fu(;)31 b Fw(\001)15 b Fx(])20 b(+)g Fu(\016)1996 5282 y Fv(1)2036 5268 y Fx(\()15 b Fw(\001)g Fx(\))22 b(+)e Fu(\016)2314 5282 y Fv(2)2353 5268 y Fx(\()15 b Fw(\001)g Fx(\))p Fu(;)p eop %%Page: 34 34 34 33 bop 3680 100 a FA(34)244 407 y Fy(and)1429 520 y Fu(\016)1469 535 y Fq(\025)1514 520 y Fx(\()15 b Fw(\001)g Fx(\))27 b(=)e Fu(\016)s Fx(\()15 b Fw(\001)g Fx(\))22 b(+)e(i)p Fu(\025)p Fx([)p Fu(V)5 b(;)30 b Fw(\001)15 b Fx(])p Fu(;)244 685 y Fy(are)32 b(the)g(generators)j(of)c(the)h(free) h(and)f(the)g(perturbed)i(dynamics.)55 b(Since)32 b Fu(!)2759 647 y Fv(+)2756 714 y Fq(\025)2849 685 y Fy(is)g Fu(\034)2981 700 y Fq(\025)3026 685 y Fy(-in)l(v)n(ariant)i(and)244 798 y Fu(V)45 b Fw(2)25 b(D)s Fx(\()p Fu(\016)s Fx(\))i(=)e Fw(D)s Fx(\()p Fu(\016)885 813 y Fq(\025)930 798 y Fx(\))p Fy(,)1131 996 y Fx(0)h(=)f Fu(!)1358 958 y Fv(+)1355 1025 y Fq(\025)1416 996 y Fx(\()p Fu(\016)1491 1011 y Fq(\025)1537 996 y Fx(\()p Fu(\025V)c Fx(\)\))26 b(=)f Fu(!)1951 958 y Fv(+)1948 1025 y Fq(\025)2010 996 y Fx(\()p Fu(\016)s Fx(\()p Fu(\025V)d Fx(\)\))1202 1205 y(=)j(i)p Fu(\025!)1437 1166 y Fv(+)1434 1234 y Fq(\025)1495 1205 y Fx(\([)p Fu(H)1631 1219 y Fv(s)1663 1205 y Fu(;)15 b(V)20 b Fx(]\))h(+)f Fu(!)2008 1166 y Fv(+)2005 1234 y Fq(\025)2067 1205 y Fx(\(\010)2168 1219 y Fv(1)2208 1205 y Fx(\))g(+)g Fu(!)2414 1166 y Fv(+)2411 1234 y Fq(\025)2473 1205 y Fx(\(\010)2574 1219 y Fv(2)2613 1205 y Fx(\))1202 1413 y(=)25 b Fu(!)1358 1375 y Fv(+)1355 1443 y Fq(\025)1416 1413 y Fx(\(\010)1517 1427 y Fv(1)1557 1413 y Fx(\))20 b(+)g Fu(!)1763 1375 y Fv(+)1760 1443 y Fq(\025)1822 1413 y Fx(\(\010)1923 1427 y Fv(2)1963 1413 y Fx(\))p Fu(;)244 1605 y Fy(where)k(we)e(used)j(\(5.48\).)k Fe(2)244 1979 y Fs(Refer)m(ences)244 2186 y Fy([Ar1])110 b(Araki)37 b(H.:)55 b(Relati)n(v)o(e)37 b(entrop)o(y)i(of)e(states)i (of)d(v)n(on)i(Neumann)g(algebras,)k(Pub)l(.)37 b(R.I.M.S.,)555 2299 y(K)n(yoto)24 b(Uni)n(v)-6 b(.)23 b Fz(11)p Fy(,)g(809)h (\(1976\).)244 2452 y([Ar2])110 b(Araki)30 b(H.:)40 b(Relati)n(v)o(e)31 b(entrop)o(y)h(of)d(states)i(of)f(v)n(on)h(Neumann)f(algebras)i(II,)d (Pub)l(.)h(R.I.M.S.,)555 2565 y(K)n(yoto)24 b(Uni)n(v)-6 b(.)23 b Fz(13)p Fy(,)g(173)h(\(1977\).)244 2719 y([A)-8 b(W])107 b(Araki)21 b(H.,)f(W)-7 b(yss)22 b(W)-8 b(.:)27 b(Representations)e(of)c(canonical)j(anticommutation)g(relations,)g (Helv)-6 b(.)555 2832 y(Phys.)23 b(Acta)g Fz(37)p Fy(,)h(136)g (\(1964\).)244 2985 y([BSZ])83 b(Baez)25 b(J.C.,)f(Se)o(gal)g(I.E.,)g (Zhou)h(Z.:)31 b Fj(Intr)l(oduction)d(to)d(Alg)o(ebr)o(aic)i(and)f (Constructive)h(Quan-)555 3098 y(tum)c(F)l(ield)h(Theory)p Fy(,)g(Princeton)h(Uni)n(v)o(ersity)g(Press,)e(Princeton)i(\(1991\).) 244 3251 y([BFS])88 b(Bach)37 b(V)-12 b(.,)40 b(Fr)8 b(\250)-38 b(ohlich)38 b(J.,)i(Sigal)d(I.:)56 b(Return)38 b(to)g(equilibrium,)43 b(J.)36 b(Math.)i(Phys.)e Fz(41)p Fy(,)41 b(3985)555 3364 y(\(2000\).)244 3517 y([BR1])84 b(Brattelli)36 b(O.,)g(Robinson)h(D.)d(W)-8 b(.:)50 b Fj(Oper)o(ator)36 b(Alg)o(ebr)o(as)g(and)g(Quantum)f(Statistical)j(Me-) 555 3630 y(c)o(hanics)25 b(1)p Fy(,)e(Springer)n(-V)-10 b(erlag,)26 b(Berlin,)e(second)h(edition)g(\(1987\).)244 3783 y([BR2])84 b(Brattelli)36 b(O.,)g(Robinson)h(D.)d(W)-8 b(.:)50 b Fj(Oper)o(ator)36 b(Alg)o(ebr)o(as)g(and)g(Quantum)f (Statistical)j(Me-)555 3896 y(c)o(hanics)25 b(2)p Fy(,)e(Springer)n(-V) -10 b(erlag,)26 b(Berlin,)e(second)h(edition)g(\(1996\).)244 4050 y([Da])145 b(Da)n(vies)24 b(E.B.:)j(Mark)o(o)o(vian)f(master)e (equations,)i(Commun.)d(Math.)g(Phys.)g Fz(39)p Fy(,)g(91)h(\(1974\).) 244 4203 y([Don])95 b(Donald)28 b(M.J.:)35 b(Relati)n(v)o(e)27 b(Hamiltonians)j(which)d(are)g(not)g(bounded)j(from)d(abo)o(v)o(e,)h (J.)e(Func.)555 4316 y(Anal.)d Fz(91)p Fy(,)h(143)g(\(1990\).)244 4469 y([DG1])74 b(Derezinski)27 b(J.,)c(Gerard)j(C.:)j(Asymptotic)e (completeness)g(in)e(quantum)h(\002eld)e(theory)-6 b(.)26 b(Mas-)555 4582 y(si)n(v)o(e)e(P)o(auli-Fierz)g(Hamiltonians,)h(Re)n(v) -6 b(.)23 b(Math.)g(Phys.)g Fz(11)p Fy(,)h(383)g(\(2000\).)244 4735 y([DG2])74 b(Derezinski)26 b(J.,)d(Gerard)i(C.:)j(Spectral)d(and)g (scattering)h(theory)g(of)e(spatially)i(cut-of)n(f)f Fu(P)13 b Fx(\()p Fu(')p Fx(\))3495 4749 y Fv(2)555 4848 y Fy(Hamiltonians,)25 b(Commun.)e(Math.)h(Phys.)e Fz(213)p Fy(,)i(39)g(\(2000\).)244 5002 y([DJ1])105 b(Derezinski)33 b(J.,)f(Jak)-5 b Fx(\024)-40 b(s)q Fy(i)5 b(\264)-35 b(c)31 b(V)-12 b(.:)43 b(Spectral)32 b(theory)h(of)e(P)o(auli-Fierz)h (operators,)j(J.)30 b(Func.)h(Anal.)555 5115 y Fz(180)p Fy(,)24 b(243)g(\(2001\).)244 5268 y([DJ2])105 b(Derezinski)26 b(J.,)c(Jak)-5 b Fx(\024)-40 b(s)q Fy(i)5 b(\264)-35 b(c)23 b(V)-12 b(.:)29 b(Return)24 b(to)f(equilibrium)j(for)e(P)o (auli-Fierz)g(systems,)h(preprint.)p eop %%Page: 35 35 35 34 bop 3680 100 a FA(35)244 407 y Fy([DJP])99 b(Derezinski)26 b(J.,)c(Jak)-5 b Fx(\024)-40 b(s)q Fy(i)5 b(\264)-35 b(c)23 b(V)-12 b(.,)23 b(Pillet)g(C.-A.:)28 b(Perturbation)e(theory)f (of)e(KMS-states,)h(preprint.)244 558 y([Fr])170 b(Frigerio)32 b(A.:)43 b(Quantum)31 b(dynamical)i(semigroups)h(and)d(approach)j(to)c (equilibrium,)35 b(Lett.)555 671 y(Math.)23 b(Phys.)g Fz(2)p Fy(,)g(79)h(\(1977\).)244 821 y([FGS])83 b(Fr)8 b(\250)-38 b(ohlich)24 b(J.,)e(Griesmer)h(M.,)f(Schlein)h(B.:)28 b(Asymptotic)c(completeness)i(for)d(Rayleigh)h(scat-)555 934 y(tering,)h(preprint.)244 1085 y([GFV])68 b(Gorini)23 b(V)-12 b(.,)21 b(Frigerio)j(A.,)c(V)-10 b(erri)23 b(M.,)e(K)m(ossak)o (o)n(wski)j(A.,)d(Sudarshan)j(E.C.G.:)i(Properties)e(of)555 1198 y(quantum)h(Mark)o(o)o(vian)g(master)f(equations,)i(Rep.)d(Math.)h (Phys.)e Fz(13)p Fy(,)i(149)g(\(1978\).)244 1349 y([Ha])145 b(Haag)24 b(R.:)j Fj(Local)d(Quantum)g(Physics)p Fy(.)g(Springer)n(-V) -10 b(erlag,)27 b(Berlin)d(\(1993\).)244 1500 y([Hak])100 b(Haak)o(e)29 b(F)-7 b(.:)39 b Fj(Statistical)32 b(tr)m(eatment)e(of)f (open)h(systems)g(by)f(g)o(ener)o(alized)j(master)d(equations)p Fy(,)555 1613 y(Springer)c(tracts)f(in)g(modern)g(physics)h Fz(66)p Fy(,)f(Springer)n(-V)-10 b(erlag,)26 b(Berlin)e(\(1973\).)244 1764 y([HP])134 b(Hunzik)o(er)33 b(W)-8 b(.,)30 b(Pillet)i(C-A.:)42 b(De)o(generate)33 b(asymptotic)g(perturbation)h(theory)-6 b(,)34 b(Commun.)555 1877 y(Math.)23 b(Phys.)g Fz(90)p Fy(,)h(219)g(\(1983\).)244 2027 y([JP1])120 b(Jak)-5 b Fx(\024)-40 b(s)q Fy(i)5 b(\264)-35 b(c)21 b(V)-12 b(.,)20 b(Pillet)h(C.-A.:)26 b(On)20 b(a)g(model)i(for)f(quantum)h (friction)g(II.)f(Fermi')-5 b(s)21 b(golden)h(rule)f(and)555 2140 y(dynamics)k(at)f(positi)n(v)o(e)h(temperature,)g(Commun.)e(Math.) g(Phys.)g Fz(176)p Fy(,)h(619)g(\(1996\).)244 2291 y([JP2])120 b(Jak)-5 b Fx(\024)-40 b(s)q Fy(i)5 b(\264)-35 b(c)25 b(V)-12 b(.,)25 b(Pillet)h(C.-A.:)31 b(On)25 b(a)g(model)h(for)g (quantum)h(friction)g(III.)f(Er)n(godic)h(properties)h(of)555 2404 y(the)c(spin-boson)j(system,)d(Commun.)e(Math.)i(Phys.)f Fz(178)p Fy(,)g(627)i(\(1996\).)244 2555 y([JP3])120 b(Jak)-5 b Fx(\024)-40 b(s)q Fy(i)5 b(\264)-35 b(c)31 b(V)-12 b(.,)32 b(Pillet)f(C.-A.:)43 b(On)30 b(entrop)o(y)j(production) i(in)c(quantum)h(statistical)i(mechanics,)555 2668 y(Commun.)23 b(Math.)g(Phys.)g Fz(217)p Fy(,)h(285)g(\(2001\).)244 2819 y([JP4])120 b(Jak)-5 b Fx(\024)-40 b(s)q Fy(i)5 b(\264)-35 b(c)23 b(V)-12 b(.,)23 b(Pillet)g(C.-A.:)28 b(In)23 b(preparation.)244 2970 y([LS])144 b(Lebo)n(witz)26 b(J.,)f(Spohn)h(S.:)32 b(Irre)n(v)o(ersible)d(thermodynamics)f(for)e (quantum)h(systems)g(weakly)555 3083 y(coupled)i(to)d(thermal)i(reserv) n(oirs,)h(Adv)-6 b(.)27 b(Chem.)e(Phys.,)i Fz(38)p Fy(,)g(109,)g(John)h (W)l(ille)o(y)f(and)g(Sons,)555 3196 y(Ne)n(w-Y)-10 b(ork)23 b(\(1978\).)244 3346 y([O1])140 b(Ojima)21 b(I.:)27 b(Entrop)o(y)c (production)h(and)e(non-equilibrium)j(stationarity)g(in)c(quantum)i (dynam-)555 3459 y(ical)h(systems:)30 b(physical)c(meaning)e(of)g(v)n (an)g(Ho)o(v)o(e)f(limit,)g(J.)g(Stat.)g(Phys.)f Fz(56)p Fy(,)i(203)g(\(1989\).)244 3610 y([O2])140 b(Ojima)21 b(I.:)27 b(Entrop)o(y)c(production)h(and)e(non-equilibrium)j (stationarity)g(in)c(quantum)i(dynam-)555 3723 y(ical)h(systems,)g(in)g Fj(Pr)l(oceedings)i(of)d(international)28 b(workshop)d(on)f(quantum)h (aspects)g(of)e(op-)555 3836 y(tical)i(communications,)j Fy(Lecture)e(Notes)f(in)f(Physics)h Fz(378)p Fy(,)h(164,)f(Springer)n (-V)-10 b(erlag,)27 b(Berlin)555 3949 y(\(1991\).)244 4100 y([OHI])89 b(Ojima)33 b(I.,)i(Hase)o(ga)o(w)o(a)e(H.,)h (Ichiyanagi,M.:)51 b(Entrop)o(y)35 b(production)i(and)c(its)h(positi)n (vity)i(in)555 4213 y(nonlinear)d(response)g(theory)f(of)f(quantum)h (dynamical)h(systems,)g(J.)d(Stat.)g(Phys.)g Fz(50)p Fy(,)i(633)555 4326 y(\(1988\).)244 4476 y([OP])134 b(Ohya)23 b(M.,)f(Petz)h(D.:)28 b Fj(Quantum)c(Entr)l(opy)h(and)f(its)g(Use)p Fy(,)e(Springer)n(-V)-10 b(erlag,)27 b(Berlin)d(\(1993\).)244 4627 y([M])170 b(Merkli)26 b(M.:)32 b(Positi)n(v)o(e)26 b(commutators)h(in)e(non-equilibrium)30 b(quantum)d(statistical)h (mechan-)555 4740 y(ics,)23 b(preprint.)244 4891 y([Si])175 b(Simon)32 b(B.:)46 b Fj(Functional)34 b(Inte)l(gr)o(ation)i(and)d (Quantum)g(Physics)p Fy(,)j(Academic)d(Press,)h(Ne)n(w)555 5004 y(Y)-10 b(ork)24 b(\(1979\).)244 5155 y([Ru1])100 b(Ruelle)24 b(D.:)k(Natural)c(nonequilibrium)k(states)d(in)e(quantum)i (statistical)h(mechanics,)g(J.)c(Stat.)555 5268 y(Phys.)h Fz(98)p Fy(,)g(57)h(\(2000\).)p eop %%Page: 36 36 36 35 bop 3680 100 a FA(36)244 407 y Fy([Ru2])100 b(Ruelle)24 b(D.:)k(Entrop)o(y)d(production)h(in)e(quantum)h(spin)f(systems,)g (preprint.)244 561 y([Rud])100 b(Rudin)24 b(W)-8 b(.:)28 b Fj(Real)23 b(and)h(Comple)n(x)g(Analysis)p Fy(,)h(McGra)o(w)d(Hill,)h (Inc,)h(Ne)n(w)e(Y)-10 b(ork)23 b(\(1974\).)244 716 y([Sp1])110 b(Spohn)24 b(H.:)k(Entrop)o(y)c(production)j(for)d(quantum)h(dynamical) g(semigroups,)h(J.)c(Math.)i(Phys.)555 829 y Fz(19)p Fy(,)f(227)h(\(1978\).)244 983 y([Sp2])110 b(Spohn)28 b(H.:)34 b(An)27 b(algebraic)i(condition)h(for)d(the)h(approach)h(to)e (equilibrium)i(of)e(an)g(open)h Fu(N)10 b Fy(-)555 1096 y(le)n(v)o(el)24 b(system,)g(Lett.)f(Math.)g(Phys.)g Fz(2)p Fy(,)g(33)g(\(1977\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0107090905164--