Content-Type: multipart/mixed; boundary="-------------0409031906226" This is a multi-part message in MIME format. ---------------0409031906226 Content-Type: text/plain; name="04-274.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-274.comments" 7 pages and 2 figures ---------------0409031906226 Content-Type: text/plain; name="04-274.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-274.keywords" Entropy, entanglement, quantum spin chains, Riemann-Hilbert problem, integrable Fredholm integral operators ---------------0409031906226 Content-Type: application/postscript; name="qp.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="qp.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: qp.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips qp %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.09.03:1814 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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Fp(G)5 b Fu(\()p Fq(\022)r Fu(\))p Fq(;)99 b Fp(G)5 b Fu(\()p Fq(\022)r Fu(\))24 b(=)3393 2954 y Fm(\022)3610 3020 y Fu(0)179 b Fq(g)s Fu(\()p Fq(\022)r Fu(\))3479 3120 y Fp(\000)p Fq(g)3587 3090 y Fn(\000)p Fl(1)3676 3120 y Fu(\()p Fq(\022)r Fu(\))104 b(0)4005 2954 y Fm(\023)2320 3405 y Fu(and)165 b Fq(g)s Fu(\()p Fq(\022)r Fu(\))24 b(=)2919 3349 y(cos)13 b Fq(\022)20 b Fp(\000)e Fu(i)p Fq(\015)h Fu(sin)14 b Fq(\022)20 b Fp(\000)e Fq(h=)p Fu(2)p 2889 3386 802 4 v 2889 3462 a Fp(j)c Fu(cos)e Fq(\022)21 b Fp(\000)d Fu(i)p Fq(\015)h Fu(sin)13 b Fq(\022)21 b Fp(\000)d Fq(h=)p Fu(2)p Fp(j)3783 3405 y Fq(:)172 b Fu(\(3\))2125 3614 y(One)26 b(can)h(use)f(an)h (orthogonal)d(matrix)j Fq(V)45 b Fu(to)27 b(transform)f Fk(B)4035 3626 y Fo(L)2042 3713 y Fu(to)h(a)g(canonical)g(form:)2438 3906 y Fq(V)19 b Fk(B)2573 3918 y Fo(L)2622 3906 y Fq(V)2689 3872 y Fo(T)2765 3906 y Fu(=)j Fp(\010)2917 3872 y Fo(L)2917 3927 y(m)p Fl(=1)3064 3906 y Fq(\027)3105 3918 y Fo(m)3182 3789 y Fm(\022)3300 3856 y 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5035 a Fu(2)3456 4978 y(ln)3549 4922 y(1)g Fp(\000)g Fq(\027)p 3549 4959 V 3623 5035 a Fu(2)3749 4978 y Fq(:)206 b Fu(\(6\))2042 5166 y(In)27 b(order)f(to)h(calculate)f(the)i(asymptotic)e(form)h(of)g(the)g(en)n (trop)n(y)2042 5266 y(let)h(us)f(in)n(tro)r(duce:)2254 5408 y Fm(e)2243 5430 y Fk(B)2311 5442 y Fo(L)2361 5430 y Fu(\()p Fq(\025)p Fu(\))d(=)f(i)p Fq(\025I)2692 5442 y Fo(L)2761 5430 y Fp(\000)18 b Fk(B)2912 5442 y Fo(L)2961 5430 y Fq(;)97 b(D)3150 5442 y Fo(L)3200 5430 y Fu(\()p Fq(\025)p Fu(\))24 b(=)e(det)3563 5408 y Fm(e)3552 5430 y Fk(B)3620 5442 y Fo(L)3670 5430 y Fu(\()p Fq(\025)p Fu(\))196 b(\(7\))p eop %%Page: 2 2 2 1 bop 4043 -299 a Fu(2)-150 -83 y(and)27 119 y Fq(e)p Fu(\()p Fq(x;)14 b(\027)5 b Fu(\))24 b(=)f Fp(\000)447 62 y Fq(x)18 b Fu(+)g Fq(\027)p 447 99 196 4 v 523 176 a Fu(2)665 119 y(ln)758 62 y Fq(x)h Fu(+)f Fq(\027)p 758 99 V 835 176 a Fu(2)982 119 y Fp(\000)1075 62 y Fq(x)h Fp(\000)f Fq(\027)p 1075 99 V 1152 176 a Fu(2)1294 119 y(ln)1387 62 y Fq(x)h Fp(\000)f Fq(\027)p 1387 99 V 1464 176 a Fu(2)1592 119 y Fq(:)171 b Fu(\(8\))-150 330 y(Here)27 b Fq(I)82 342 y Fo(L)159 330 y Fu(is)h(the)f(iden)n(tit)n(y)h(matrix)e (of)i(dimension)f(2)p Fq(L)p Fu(.)36 b(By)27 b(de\014-)-150 430 y(nition,)h(w)n(e)f(ha)n(v)n(e)g Fq(H)7 b Fu(\()p Fq(\027)e Fu(\))23 b(=)g Fq(e)p Fu(\(1)p Fq(;)14 b(\027)5 b Fu(\))27 b(and)272 699 y Fq(D)341 711 y Fo(L)391 699 y Fu(\()p Fq(\025)p Fu(\))d(=)e(\()p Fp(\000)p Fu(1\))785 665 y Fo(L)897 596 y(L)867 620 y Fm(Y)849 796 y Fo(m)p Fl(=1)992 699 y Fu(\()p Fq(\025)1072 665 y Fl(2)1128 699 y Fp(\000)c Fq(\027)1257 665 y Fl(2)1252 720 y Fo(m)1315 699 y Fu(\))p Fq(:)416 b Fu(\(9\))-150 964 y(In)34 b([28)o(])g(w)n(e)g (used)g(Cauc)n(h)n(y)f(residue)g(theorem)g(to)h(rewrite)f(for-)-150 1063 y(m)n(ula)27 b(\(5\))h(in)g(the)g(follo)n(wing)e(form:)-56 1295 y Fq(S)5 b Fu(\()p Fq(\032)75 1307 y Fo(A)129 1295 y Fu(\))24 b(=)51 b(lim)272 1353 y Fo(\017)p Fn(!)p Fl(0)399 1337 y Fj(+)506 1239 y Fu(1)p 470 1276 115 4 v 470 1352 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b(functions)g Fr(\036)p Fs(\()p Fr(z)s Fs(\))p Fr(;)13 b(w)r Fs(\()p Fr(z)s Fs(\))31 b Ft(ar)l(e)i(lab)l(ele)l(d)f(by)g(b)l (old)g(on)g(line)f Fs(\006)p Ft(.)2042 1737 y Fd(Case)37 b Fs(1)p Ft(a:)58 b Fr(h)41 b(<)g Fs(2)d Ft(and)h Fr(h)2887 1705 y Fe(2)2963 1737 y Fr(>)i Fs(4\(1)25 b Fg(\000)g Fr(\015)3324 1705 y Fe(2)3359 1737 y Fs(\))38 b Ft(then)h Fr(\025)3653 1745 y Ff(A)3744 1737 y Fs(=)i Fr(\025)3890 1745 y Fe(1)3963 1737 y Ft(and)2042 1824 y Fr(\025)2087 1832 y Ff(B)2161 1824 y Fs(=)21 b Fr(\025)2287 1791 y Fh(\000)p Fe(1)2287 1843 y(2)2369 1824 y Ft(,)j Fr(\025)2462 1832 y Ff(C)2535 1824 y Fs(=)d Fr(\025)2661 1832 y Fe(2)2719 1824 y Ft(and)j Fr(\025)2909 1832 y Ff(D)2985 1824 y Fs(=)d Fr(\025)3111 1791 y Fh(\000)p Fe(1)3111 1843 y(1)3194 1824 y Ft(.)33 b Fd(Case)23 b Fs(1)p Ft(b:)34 b Fr(h)3632 1792 y Fe(2)3688 1824 y Fr(<)21 b Fs(4\(1)9 b Fg(\000)g Fr(\015)3997 1792 y Fe(2)4031 1824 y Fs(\))p Ft(,)2042 1911 y(then)29 b Fr(\025)2258 1919 y Ff(A)2330 1911 y Fs(=)23 b 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n(utativit)n(y)2042 5413 y(of)22 b(the)h(asso)r(ciated)d(W)-7 b(einer-Hopf)22 b(factorization.)34 b(This)22 b(creates)p eop %%Page: 3 3 3 2 bop 4043 -299 a Fu(3)-150 -83 y(serious)33 b(tec)n(hnical)h (problems.)55 b(In)35 b(our)e(w)n(ork)g(w)n(e)g(circum)n(v)n(en)n(t) -150 17 y(this)21 b(obstacle)f(b)n(y)g(using)g(an)h(alternativ)n(e)e (approac)n(h)g(to)h(T)-7 b(o)r(eplitz)-150 116 y(determinan)n(ts)33 b(suggested)f(b)n(y)i(Deift)g(in)g([20)o(].)55 b(It)33 b(is)h(based)f(on)-150 216 y(the)41 b(Riemann-Hilb)r(ert)f(tec)n (hnique)h(of)f(the)h(theory)f(of)g(\\in)n(te-)-150 315 y(grable)30 b(in)n(tegral)g(op)r(erators",)g(whic)n(h)h(w)n(as)g(dev)n (elop)r(ed)g(in)g([21)o(],)-150 415 y([43)o(])38 b(for)g(ev)-5 b(aluation)38 b(of)g(correlation)e(functions)i(of)g(quan)n(tum)-150 515 y(completely)29 b(in)n(tegrable)f([exactly)h(solv)-5 b(able])29 b(mo)r(dels)g(\(see)h(also)-150 614 y(commen)n(t)25 b([46]\).)36 b(It)26 b(turns)g(out)f(that,)i(using)e(the)h(blo)r(c)n(k) f(matrix)-150 714 y(v)n(ersion)18 b(of)i([21])g(suggested)f(in)h([25)o (],)i(one)d(can)h(generalize)e(Deift's)-150 814 y(sc)n(heme)33 b(to)g(the)h(blo)r(c)n(k)e(T)-7 b(o)r(eplitz)34 b(matrices.)53 b(In)33 b(addition,)i(w)n(e)-150 913 y(w)n(ere)c(able)h(to)h(\014nd)f (the)h(explicit)g(W)-7 b(einer-Hopf)32 b(factorization)-150 1013 y(of)e(the)g(generator)e(\010\()p Fq(z)t Fu(\))h(whic)n(h)h(ev)n (en)n(tually)f(made)h(it)g(p)r(ossible)-150 1112 y(to)38 b(p)r(erform)h(an)f(explicit)h(ev)-5 b(aluation)38 b(of)g(the)h (asymptotic)f(of)-150 1212 y(the)31 b(en)n(trop)n(y)f Fq(S)5 b Fu(\()p Fq(\032)433 1224 y Fo(A)487 1212 y Fu(\).)46 b(The)31 b(\014nal)g(result)f(is)h(giv)n(en)f(in)h(terms)f(of)-150 1312 y(elliptic)h(functions)f(and)g(is)g(presen)n(ted)g(in)g(Eq.)g (\(61\))g(b)r(elo)n(w.)44 b(In)-150 1411 y(what)36 b(follo)n(ws)f(w)n (e)h(shall)g(outline)g(our)g(calculation)f(pro)n(viding)-150 1511 y(the)h(necessary)e(facts)i(concerning)e(in)n(tegrable)h(F)-7 b(redholm)35 b(op-)-150 1611 y(erators.)54 b(More)33 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b(means)h(the)g(trace)f(taking)h(in) 2042 752 y(the)26 b(space)f Fq(L)2460 764 y Fl(2)2496 752 y Fu(\(\004)p Fq(;)14 b Fc(C)2681 716 y Fl(2)2718 752 y Fu(\),)27 b(while)e(\\tr")g(is)g(the)i(2)14 b Fp(\002)g Fu(2)25 b(matrix)g(trace.)2042 852 y(An)39 b(imp)r(ortan)n(t)g(general) f(fact)h(is)g(that)g(the)h(resolv)n(en)n(t)d(k)n(ernel)2042 952 y(satis\014es)27 b(the)h(equation)f(\(see)g(e.g.)37 b([25)o(]\),)2636 1159 y Fq(R)q Fu(\()p Fq(z)t(;)14 b(z)t Fu(\))22 b(=)3007 1103 y Fq(dF)3115 1073 y Fo(T)3167 1103 y Fu(\()p Fq(z)t Fu(\))p 3007 1140 268 4 v 3098 1216 a Fq(dz)3284 1159 y(H)7 b Fu(\()p Fq(z)t Fu(\))p Fq(:)447 b Fu(\(21\))2042 1340 y(In)38 b(this)g(equation,)i(the)e(4)25 b Fp(\002)g Fu(2)37 b(matrix)h(functions)g Fq(F)12 b Fu(\()p Fq(z)t Fu(\))37 b(and)2042 1439 y Fq(H)7 b Fu(\()p Fq(z)t Fu(\))27 b(are)g(determined)g(b)n(y)h(the)g(relations,)2567 1592 y Fq(F)12 b Fu(\()p Fq(z)t Fu(\))23 b(=)g Fq(Y)2898 1604 y Fl(+)2953 1592 y Fu(\()p Fq(z)t Fu(\))p Fq(f)9 b Fu(\()p Fq(z)t Fu(\))p Fq(;)96 b(z)27 b Fp(2)c Fu(\004)p Fq(;)379 b Fu(\(22\))2478 1797 y Fq(H)7 b Fu(\()p Fq(z)t Fu(\))23 b(=)f(\()p Fq(Y)2870 1762 y Fo(T)2851 1817 y Fl(+)2923 1797 y Fu(\))2955 1762 y Fn(\000)p Fl(1)3044 1797 y Fu(\()p Fq(z)t Fu(\))p Fq(h)p Fu(\()p Fq(z)t Fu(\))p Fq(;)97 b(z)26 b Fp(2)d Fu(\004)p Fq(;)290 b Fu(\(23\))2042 1949 y(where)23 b(the)h(4)10 b Fp(\002)g Fu(4)23 b(matrix)g(function)h Fq(Y)3244 1961 y Fl(+)3299 1949 y Fu(\()p Fq(z)t Fu(\))g(can)f(b)r(e)h (found)g(from)2042 2048 y(the)c(\(unique\))h(solution)e(of)h(the)g (follo)n(wing)f Fk(Riemann-Hilb)s(ert)2042 2148 y(problem)p Fu(:)2152 2290 y(1.)32 b Fq(Y)19 b Fu(\()p Fq(z)t Fu(\))27 b(is)h(analytic)f(for)g Fq(z)36 b(=)-52 b Fp(2)24 b Fu(\004.)2152 2444 y(2.)32 b Fq(Y)19 b Fu(\()p Fp(1)p Fu(\))40 b(=)g Fq(I)2644 2456 y Fl(4)2681 2444 y Fu(,)h(where)c Fq(I)3031 2456 y Fl(4)3106 2444 y Fu(denotes)h(the)g(4)24 b Fp(\002)h Fu(4)37 b(iden)n(tit)n(y)2249 2543 y(matrix.)2152 2697 y(3.)32 b Fq(Y)2297 2709 y Fn(\000)2354 2697 y Fu(\()p 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Fq(z)t Fu(\))133 b Fp(\000)p Fq(z)3141 3422 y Fo(L)3190 3453 y Fu(\()p Fq(I)3258 3465 y Fl(2)3314 3453 y Fp(\000)18 b Fu(\010)3457 3422 y Fo(T)3509 3453 y Fu(\()p Fq(z)t Fu(\)\))2381 3552 y Fq(z)2424 3522 y Fn(\000)p Fo(L)2524 3552 y Fu(\()p Fq(I)2592 3564 y Fl(2)2649 3552 y Fp(\000)g Fu(\010)2792 3522 y Fo(T)2844 3552 y Fu(\()p Fq(z)t Fu(\)\))248 b(\010)3291 3522 y Fo(T)3344 3552 y Fu(\()p Fq(z)t Fu(\))3673 3386 y Fm(\023)3748 3503 y Fq(:)166 b Fu(\(24\))2125 3733 y(Eqs.)28 b(\(20\))g(and)h(\(21\))f(reduce)g(the)i(original)d (question)h(to)h(the)2042 3832 y(asymptotic)22 b(analysis)f(of)i(the)g (solution)f Fq(Y)d Fu(\()p Fq(z)t Fu(\))k(of)f(the)h(Riemann-)2042 3932 y(Hilb)r(ert)d(problem)f(\(1-3\).)33 b(Our)19 b(observ)-5 b(ation)18 b(is)h(that)h(once)e(again)2042 4031 y(w)n(e)j(can)g (generalize)f(the)i(argumen)n(ts)e(of)i([20)o(])f(to)h(the)g(case)e(of) i(ma-)2042 4131 y(trix)c(generator)e(\010\()p Fq(z)t Fu(\))j(and)f(arriv)n(e)f(to)h(the)h(follo)n(wing)f(asymptotic)2042 4231 y(solution)27 b(of)g(the)h(problem)g(\(1-3\))e(\()i Fq(L)23 b Fp(!)g(1)p Fu(\):)2337 4433 y Fq(Y)2385 4445 y Fl(+)2441 4433 y Fu(\()p Fq(z)t Fu(\))f(=)2658 4316 y Fm(\022)2744 4382 y Fq(U)2810 4352 y Fo(T)2801 4403 y Fl(+)2862 4382 y Fu(\()p Fq(z)t Fu(\))124 b Fp(\000)18 b Fq(z)3219 4352 y Fo(L)3268 4382 y Fq(U)3334 4352 y Fo(T)3325 4403 y Fl(+)3386 4382 y Fu(\()p Fq(z)t Fu(\))c Fq(M)9 b Fu(\()p Fq(z)t Fu(\))2817 4482 y(0)2859 4494 y Fl(2)3171 4482 y Fu(\()p Fq(V)3270 4452 y Fo(T)3251 4503 y Fl(+)3322 4482 y Fu(\))3354 4452 y Fn(\000)p Fl(1)3444 4482 y Fu(\()p Fq(z)t Fu(\))3728 4316 y Fm(\023)3937 4433 y Fu(\(25\))2042 4631 y(and)2141 4812 y(\()p Fq(Y)2221 4824 y Fl(+)2277 4812 y Fu(\))2309 4778 y Fn(\000)p Fl(1)2398 4812 y Fu(\()p Fq(z)t Fu(\))23 b(=)2616 4695 y Fm(\022)2702 4761 y Fu(\()p Fq(U)2800 4731 y Fo(T)2791 4782 y Fl(+)2852 4761 y Fu(\))2884 4731 y Fn(\000)p Fl(1)2974 4761 y Fu(\()p Fq(z)t Fu(\))105 b Fq(z)3229 4731 y Fo(L)3278 4761 y Fq(M)9 b Fu(\()p Fq(z)t Fu(\))14 b Fq(V)3555 4731 y Fo(T)3537 4782 y Fl(+)3607 4761 y Fu(\()p Fq(z)t Fu(\))2852 4861 y(0)2894 4873 y Fl(2)3309 4861 y Fq(V)3376 4831 y Fo(T)3357 4882 y Fl(+)3428 4861 y Fu(\()p Fq(z)t Fu(\))3739 4695 y Fm(\023)3814 4812 y Fq(:)100 b Fu(\(26\))2042 5009 y(Here)2635 5161 y Fq(M)9 b Fu(\()p Fq(z)t Fu(\))23 b(=)g Fq(I)2979 5173 y Fl(2)3035 5161 y Fp(\000)18 b Fu(\(\010)3210 5127 y Fo(T)3262 5161 y Fu(\))3294 5127 y Fn(\000)p Fl(1)3384 5161 y Fu(\()p Fq(z)t Fu(\))2042 5313 y(and)44 b Fq(U)2277 5325 y Fn(\006)2333 5313 y Fu(\()p Fq(z)t Fu(\))g(and)g Fq(V)2710 5325 y Fn(\006)2766 5313 y Fu(\()p Fq(z)t Fu(\))g(are)f(2)29 b Fp(\002)h Fu(2)43 b(matrices)h(solving)f(the)2042 5413 y(W)-7 b(einer-Hopf)27 b(factorization)g(problem)g(:)p eop %%Page: 4 4 4 3 bop 4043 -299 a Fu(4)-117 -83 y(\(i\))84 b(\010\()p Fq(z)t Fu(\))23 b(=)f Fq(U)388 -71 y Fl(+)443 -83 y Fu(\()p Fq(z)t Fu(\))p Fq(U)607 -71 y Fn(\000)663 -83 y Fu(\()p Fq(z)t Fu(\))h(=)f Fq(V)928 -71 y Fn(\000)985 -83 y Fu(\()p Fq(z)t Fu(\))p Fq(V)1140 -71 y Fl(+)1195 -83 y Fu(\()p Fq(z)t Fu(\))p Fq(;)97 b(z)26 b Fp(2)e Fu(\004)-117 81 y(\(ii\))112 b Fq(U)162 93 y Fn(\000)218 81 y Fu(\()p Fq(z)t Fu(\))28 b(and)f Fq(V)562 93 y Fn(\000)619 81 y Fu(\()p Fq(z)t Fu(\))h(\()p Fq(U)843 93 y Fl(+)898 81 y Fu(\()p Fq(z)t Fu(\))g(and)f Fq(V)1242 93 y Fl(+)1298 81 y Fu(\()p Fq(z)t Fu(\)\))h(are)f(analytic)58 181 y(outside)g (\(inside\))h(the)g(unit)g(circle)g(\004.)-117 345 y(\(iii\))112 b Fq(U)185 357 y Fn(\000)240 345 y Fu(\()p Fp(1)p Fu(\))24 b(=)f Fq(V)547 357 y Fn(\000)603 345 y Fu(\()p Fp(1)p Fu(\))h(=)e Fq(I)7 b Fu(.)-67 507 y(W)-7 b(e)28 b(can)g(use)g(Eqs.)f (\(25\))h(and)f(\(26\))h(in)g(Eqs.)g(\(20\))f(-)h(\(23\))g(and)-150 607 y(obtain)f(the)h(follo)n(wing)f(asymptotic)g(form)n(ula:)5 768 y Fq(d)p -19 805 92 4 v -19 881 a(d\025)96 824 y Fu(ln)14 b Fq(D)248 836 y Fo(L)298 824 y Fu(\()p Fq(\025)p Fu(\))24 b(=)e Fp(\000)665 768 y Fu(2)p Fq(\025)p 596 805 229 4 v 596 881 a Fu(1)c Fp(\000)g Fq(\025)787 857 y Fl(2)834 824 y Fq(L)g Fu(+)1027 768 y(1)p 1002 805 92 4 v 1002 881 a(2)p Fq(\031)1118 711 y Fm(Z)1164 899 y Fl(\004)1226 824 y Fu(tr)27 b([\011\()p Fq(z)t Fu(\)])h Fq(dz)t(;)72 b Fu(\(27\))-52 1138 y(\011\()p Fq(z)t Fu(\))22 b(=)230 1070 y Fm(\002)264 1138 y Fq(U)330 1103 y Fn(0)321 1158 y Fl(+)376 1138 y Fu(\()p Fq(z)t Fu(\))p Fq(U)549 1102 y Fn(\000)p Fl(1)540 1158 y(+)638 1138 y Fu(\()p Fq(z)t Fu(\))c(+)g Fq(V)913 1102 y Fn(\000)p Fl(1)894 1158 y(+)1002 1138 y Fu(\()p Fq(z)t Fu(\))p Fq(V)1176 1103 y Fn(0)1157 1158 y Fl(+)1213 1138 y Fu(\()p Fq(z)t Fu(\))1320 1070 y Fm(\003)1368 1138 y Fu(\010)1428 1103 y Fn(\000)p Fl(1)1517 1138 y Fu(\()p Fq(z)t Fu(\))p Fq(;)98 b Fu(\(28\))-150 1316 y(as)40 b Fq(L)45 b Fp(!)g(1)c Fu(\(see)f(also)g(commen)n(t)h([47)o(]\).)77 b(Here)1519 1286 y Fn(0)1583 1316 y Fu(means)41 b(a)-150 1415 y(deriv)-5 b(ativ)n(e)27 b(in)h Fq(z)j Fu(v)-5 b(ariable.)-67 1515 y(By)27 b(explicit)h(calculation,)f(one)g(can)g(\014nd)h(that)-37 1738 y(\(1)18 b Fp(\000)g Fq(\025)186 1704 y Fl(2)223 1738 y Fu(\))p Fq(\033)302 1750 y Fl(3)340 1738 y Fu(\010)400 1704 y Fn(\000)p Fl(1)490 1738 y Fu(\()p Fq(z)t Fu(\))p Fq(\033)644 1750 y Fl(3)704 1738 y Fu(=)23 b(\010\()p Fq(z)t Fu(\))p Fq(;)96 b(\033)1125 1750 y Fl(3)1186 1738 y Fu(=)1274 1621 y Fm(\022)1360 1688 y Fu(1)81 b(0)1360 1787 y(0)49 b Fp(\000)p Fu(1)1582 1621 y Fm(\023)1657 1738 y Fq(:)65 b Fu(\(29\))-150 1961 y(Hence,)117 2139 y Fq(V)165 2151 y Fn(\000)222 2139 y Fu(\()p Fq(z)t Fu(\))49 b(=)h Fq(\033)540 2151 y Fl(3)577 2139 y Fq(U)643 2103 y Fn(\000)p Fl(1)634 2160 y Fn(\000)732 2139 y Fu(\()p Fq(z)t Fu(\))p Fq(\033)886 2151 y Fl(3)1745 2139 y Fu(\(30\))118 2273 y Fq(V)166 2285 y Fl(+)222 2273 y Fu(\()p Fq(z)t Fu(\))f(=)h Fq(\033)540 2285 y Fl(3)577 2273 y Fq(U)643 2237 y Fn(\000)p Fl(1)634 2293 y(+)732 2273 y Fu(\()p Fq(z)t Fu(\))p Fq(\033)886 2285 y Fl(3)924 2273 y Fu(\(1)18 b Fp(\000)g Fq(\025)1147 2238 y Fl(2)1185 2273 y Fu(\))p Fq(;)97 b(\025)23 b Fp(6)p Fu(=)g Fp(\006)p Fu(1)p Fq(;)119 b Fu(\(31\))-150 2451 y(and)27 b(one)h(only)f(needs)g(the)h(explicit)g (expressions)e(for)h Fq(U)1625 2463 y Fn(\006)1681 2451 y Fu(\()p Fq(z)t Fu(\).)-67 2550 y(Our)37 b(last)g(principal)g(observ) -5 b(ation)36 b(is)h(that,)k(for)c(all)g Fq(\025)h Fu(out-)-150 2650 y(side)f(of)g(a)f(certain)h(discrete)f(subset)h(of)g(the)g(in)n (terv)-5 b(al)37 b([)p Fp(\000)p Fu(1)p Fq(;)14 b Fu(1],)-150 2750 y(the)38 b(solution)g(to)f(the)h(auxiliary)f(Riemann-Hilb)r(ert)h (problem)-150 2849 y(\(i-iii\))f(exists;)j(moreo)n(v)n(er,)c(the)g (functions)h Fq(U)1303 2861 y Fn(\006)1358 2849 y Fu(\()p Fq(z)t Fu(\))g(can)e(b)r(e)i(ex-)-150 2949 y(pressed)c(in)g(terms)h(of) f(the)h(Jacobi)e(theta-functions.)55 b(Indeed,)-150 3049 y(the)25 b(auxiliary)d(Riemann-Hilb)r(ert)j(problem)e(\(i-iii\))i(can)f (b)r(e)h(eas-)-150 3148 y(ily)d(reduced)g(to)h(a)f(t)n(yp)r(e)g(of)h (the)f(\\\014nite-gap")f(Riemann-Hilb)r(ert)-150 3248 y(problems)39 b(whic)n(h)g(ha)n(v)n(e)f(already)g(app)r(eared)g(in)i (the)g(analysis)-150 3347 y(of)30 b(the)g(in)n(tegrable)e(statistical)i (mec)n(hanics)f(mo)r(dels)g(\(see)h([26)o(]\).)-150 3447 y(Before)25 b(w)n(e)h(giv)n(e)f(detail)g(expressions,)g(let)h(us)g (\014rst)g(de\014ne)g(some)-150 3547 y(basic)h(ob)5 b(jects:)5 3761 y Fq(w)r Fu(\()p Fq(z)t Fu(\))50 b(=)338 3656 y Fm(q)p 421 3656 1280 4 v 105 x Fu(\()p Fq(z)22 b Fp(\000)c Fq(\025)645 3773 y Fl(1)682 3761 y Fu(\)\()p Fq(z)23 b Fp(\000)18 b Fq(\025)939 3773 y Fl(2)976 3761 y Fu(\)\()p Fq(z)23 b Fp(\000)18 b Fq(\025)1233 3725 y Fn(\000)p Fl(1)1233 3783 y(2)1322 3761 y Fu(\)\()p Fq(z)23 b Fp(\000)18 b Fq(\025)1579 3725 y Fn(\000)p Fl(1)1579 3783 y(1)1668 3761 y Fu(\))p Fq(;)k Fu(\(32\))9 3949 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))51 b(=)387 3893 y(1)p 348 3930 121 4 v 348 4006 a(2)p Fq(\031)s(i)492 3949 y Fu(ln)585 3893 y Fq(\025)19 b Fu(+)f(1)p 585 3930 192 4 v 585 4006 a Fq(\025)h Fp(\000)f Fu(1)786 3949 y Fq(;)936 b Fu(\(33\))-150 4162 y(where)22 b Fq(w)r Fu(\()p Fq(z)t Fu(\))i(is)f(analytic)f(on)h (the)g(domain)g Fc(C)e Fp(n)14 b(f)p Fq(J)1411 4174 y Fl(1)1456 4162 y Fp([)9 b Fq(J)1566 4174 y Fl(2)1604 4162 y Fp(g)22 b Fu(sho)n(wn)-150 4262 y(in)28 b(Fig.)f(2)h(and)f (\014xed)h(b)n(y)f(the)h(condition:)37 b Fq(w)r Fu(\()p Fq(z)t Fu(\))23 b Fp(!)g Fq(z)1543 4232 y Fl(2)1607 4262 y Fu(as)-67 4362 y Fq(z)j Fp(!)d(1)p Fu(.)37 b(Next)28 b(w)n(e)g(de\014ne)156 4581 y Fq(\034)k Fu(=)322 4525 y(2)p 322 4562 42 4 v 325 4638 a Fq(c)387 4468 y Fm(Z)470 4489 y Fo(\025)509 4497 y Fb(C)433 4657 y Fo(\025)472 4665 y Fb(B)626 4525 y Fu(d)p Fq(z)p 586 4562 169 4 v 586 4638 a(w)r Fu(\()p Fq(z)t Fu(\))764 4581 y Fq(;)97 b(c)23 b Fu(=)g(2)1087 4468 y Fm(Z)1169 4489 y Fo(\025)1208 4497 y Fb(B)1132 4657 y Fo(\025)1171 4665 y Fb(A)1326 4525 y Fu(d)p Fq(z)p 1286 4562 V 1286 4638 a(w)r Fu(\()p Fq(z)t Fu(\))1464 4581 y Fq(;)258 b Fu(\(34\))-91 4976 y Fq(\016)27 b Fu(=)70 4920 y(2)p 70 4957 42 4 v 73 5033 a Fq(c)136 4834 y Fm( )201 4976 y Fp(\000)p Fq(\031)s(i)18 b Fp(\000)446 4863 y Fm(Z)529 4883 y Fo(\025)568 4891 y Fb(B)492 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Fq(\022)2564 1699 y Fl(3)2601 1687 y Fu(\()p Fq(s)p Fu(\))24 b(=)2887 1583 y Fn(1)2861 1608 y Fm(X)2815 1782 y Fo(n)p Fl(=)p Fn(\0001)3039 1687 y Fq(e)3078 1652 y Fo(\031)r(i\034)7 b(n)3221 1627 y Fj(2)3253 1652 y Fl(+2)p Fo(\031)r(isn)3478 1687 y Fq(:)436 b Fu(\(38\))2042 1938 y(W)-7 b(e)22 b(remind)h(the)f(follo)n(wing)g (prop)r(erties)f(of)h(this)h(theta-function)2042 2037 y(\(see)k(e.g.)37 b([27)o(]\):)2446 2211 y Fq(\022)2485 2223 y Fl(3)2522 2211 y Fu(\()p Fp(\000)p Fq(s)p Fu(\))23 b(=)g Fq(\022)2840 2223 y Fl(3)2877 2211 y Fu(\()p Fq(s)p Fu(\))p Fq(;)97 b(\022)3139 2223 y Fl(3)3177 2211 y Fu(\()p Fq(s)18 b Fu(+)g(1\))23 b(=)g Fq(\022)3573 2223 y Fl(3)3610 2211 y Fu(\()p Fq(s)p Fu(\))224 b(\(39\))2719 2336 y Fq(\022)2758 2348 y Fl(3)2795 2336 y Fu(\()p Fq(s)19 b Fu(+)f Fq(\034)9 b Fu(\))24 b(=)f Fq(e)3196 2301 y Fn(\000)p Fo(\031)r(i\034)7 b Fn(\000)p Fl(2)p Fo(\031)r(is)3534 2336 y Fq(\022)3573 2348 y Fl(3)3610 2336 y Fu(\()p Fq(s)p Fu(\))224 b(\(40\))2313 2510 y Fq(\022)2352 2522 y Fl(3)2403 2393 y Fm(\022)2464 2510 y Fq(n)18 b Fu(+)g Fq(m\034)28 b Fu(+)2845 2454 y(1)p 2845 2491 V 2845 2567 a(2)2915 2510 y(+)3008 2454 y Fq(\034)p 3008 2491 46 4 v 3010 2567 a Fu(2)3064 2393 y Fm(\023)3148 2510 y Fu(=)23 b(0)p Fq(;)96 b(n;)14 b(m)23 b Fp(2)g Fc(Z)217 b Fu(\(41\))2042 2734 y(W)-7 b(e)26 b(also)f(in)n(tro)r(duce)h(the)g(2)15 b Fp(\002)g Fu(2)26 b(matrix)f(v)-5 b(alued)26 b(function)h(\002\()p Fq(z)t Fu(\))2042 2833 y(with)h(the)g(en)n(tries,)2463 3007 y(\002)2528 3019 y Fl(11)2598 3007 y Fu(\()p Fq(z)t Fu(\))23 b(=)g(\()p Fq(z)f Fp(\000)c Fq(\025)3040 3019 y Fl(1)3078 3007 y Fu(\))3110 2973 y Fn(\000)3172 2951 y Fj(1)p 3172 2960 29 4 v 3172 2993 a(2)3214 3007 y Fq(e)3253 2973 y Fl(\001\()p Fo(z)r Fl(\))2712 3193 y Fp(\002)2787 3132 y Fq(\022)2826 3144 y Fl(3)2877 3064 y Fm(\000)2915 3132 y Fq(!)s Fu(\()p Fq(z)t Fu(\))g(+)g Fq(\014)t Fu(\()p Fq(\025)p Fu(\))i Fp(\000)e Fq(\024)g Fu(+)3603 3099 y Fo(\033)r(\034)p 3603 3113 78 4 v 3626 3160 a Fl(2)3691 3064 y Fm(\001)p 2787 3174 943 4 v 2995 3255 a Fq(\022)3034 3267 y Fl(3)3085 3187 y Fm(\000)3123 3255 y Fq(!)s Fu(\()p Fq(z)t Fu(\))g(+)3396 3222 y Fo(\033)r(\034)p 3396 3236 78 4 v 3418 3283 a Fl(2)3483 3187 y Fm(\001)2463 3400 y Fu(\002)2528 3412 y Fl(12)2598 3400 y Fu(\()p Fq(z)t Fu(\))23 b(=)g Fp(\000)p Fu(\()p Fq(z)e Fp(\000)d Fq(\025)3104 3412 y Fl(1)3142 3400 y Fu(\))3174 3366 y Fn(\000)3236 3344 y Fj(1)p 3236 3353 29 4 v 3236 3386 a(2)3279 3400 y Fq(e)3318 3366 y Fn(\000)p Fl(\001\()p Fo(z)r Fl(\))2712 3586 y Fp(\002)2787 3525 y Fq(\022)2826 3537 y Fl(3)2877 3457 y Fm(\000)2915 3525 y Fq(!)s Fu(\()p Fq(z)t Fu(\))g Fp(\000)g Fq(\014)t Fu(\()p Fq(\025)p Fu(\))i(+)e Fq(\024)g Fp(\000)3603 3492 y Fo(\033)r(\034)p 3603 3506 78 4 v 3626 3553 a Fl(2)3691 3457 y Fm(\001)p 2787 3567 943 4 v 2995 3648 a Fq(\022)3034 3660 y Fl(3)3085 3580 y Fm(\000)3123 3648 y Fq(!)s Fu(\()p Fq(z)t Fu(\))g Fp(\000)3396 3615 y Fo(\033)r(\034)p 3396 3629 78 4 v 3418 3676 a Fl(2)3483 3580 y Fm(\001)2463 3793 y 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Fl(\001\()p Fo(z)r Fl(\))2712 4372 y Fp(\002)2787 4311 y Fq(\022)2826 4323 y Fl(3)2877 4243 y Fm(\000)2915 4311 y Fq(!)s Fu(\()p Fq(z)t Fu(\))g Fp(\000)g Fq(\014)t Fu(\()p Fq(\025)p Fu(\))i Fp(\000)e Fq(\024)g Fu(+)3603 4278 y Fo(\033)r(\034)p 3603 4292 78 4 v 3626 4339 a Fl(2)3691 4243 y Fm(\001)p 2787 4353 943 4 v 2995 4434 a Fq(\022)3034 4446 y Fl(3)3085 4366 y Fm(\000)3123 4434 y Fq(!)s Fu(\()p Fq(z)t Fu(\))g(+)3396 4401 y Fo(\033)r(\034)p 3396 4415 78 4 v 3418 4462 a Fl(2)3483 4366 y Fm(\001)3739 4372 y Fq(;)175 b Fu(\(42\))2042 4607 y(where)31 b Fq(\033)j Fu(=)c(1)i(in)g(Case)f(1)h(and)f Fq(\033)j Fu(=)c(0)i(in)g(Case)f(2,)i(and)f Fq(\014)t Fu(\()p Fq(\025)p Fu(\),)2042 4707 y Fq(!)s Fu(\()p Fq(z)t Fu(\))i(and)g Fq(\024)g Fu(are)f(de\014ned)i(in)g(Eqs.)e(\(32-37\).)55 b(The)35 b(branc)n(h)e(of)2042 4815 y(\()p Fq(z)24 b Fp(\000)d Fq(\025)2271 4827 y Fl(1)2308 4815 y Fu(\))2340 4785 y Fn(\000)2402 4763 y Fj(1)p 2402 4772 29 4 v 2402 4805 a(2)2476 4815 y Fu(is)31 b(de\014ned)h(on)f(the)g Fq(z)t Fu(-plane)f(cut)i(along)e(the)i(part)2042 4915 y(of)e(the)h(line)g(\006)g(whic)n(h)f(is)h(to)f(the)h(righ)n(t)f(of)h Fq(\025)3463 4927 y Fl(1)3528 4915 y Fp(\021)d Fq(\025)3669 4927 y Fo(A)3723 4915 y Fu(,)k(and)e(it)h(is)2042 5015 y(\014xed)c(b)n(y)f(the)i(condition)e(arg\()p Fq(z)20 b Fp(\000)d Fq(\025)3200 5027 y Fl(1)3238 5015 y Fu(\))23 b(=)g Fq(\031)s(;)97 b Fu(if)83 b Fq(z)21 b Fp(\000)16 b Fq(\025)3871 5027 y Fl(1)3932 5015 y Fq(<)23 b Fu(0.)2125 5114 y(The)d(matrix)f(function)i(\002\()p Fq(z)t Fu(\))e(is)h (de\014ned)h(on)e Fc(C)i Fp(n)14 b Fu(\006.)34 b(Ho)n(w)n(ev)n(er,)2042 5214 y(analyzing)26 b(the)j(jumps)g(of)f(the)g(in)n(tegrals)e Fq(!)s Fu(\()p Fq(z)t Fu(\))i(and)g(\001\()p Fq(z)t Fu(\))g(o)n(v)n(er) 2042 5313 y(the)37 b(line)g(\006)f(and)h(taking)f(in)n(to)g(accoun)n(t) g(the)h(prop)r(erties)f(\(39\))2042 5413 y(and)i(\(40\))h(of)f(the)h (theta)g(function,)j(one)c(can)h(see)f(that)h(\002\()p Fq(z)t Fu(\))p eop %%Page: 5 5 5 4 bop 4043 -299 a Fu(5)-150 -83 y(is)31 b(actually)g(extended)h(to)f (the)h(analytic)f(function)h(de\014ned)g(on)-150 17 y Fc(C)21 b Fp(n)14 b(f)p Fq(J)54 29 y Fl(1)109 17 y Fp([)k Fq(J)228 29 y Fl(2)266 17 y Fp(g)p Fu(.)36 b(Moreo)n(v)n(er,)25 b(it)j(satis\014es)f(the)h(jump)g(relations)504 181 y(\002)569 193 y Fl(+)623 181 y Fu(\()p Fq(z)t Fu(\))23 b(=)g(\002)906 193 y Fn(\000)962 181 y Fu(\()p Fq(z)t Fu(\))p Fq(\033)1116 193 y Fl(1)1319 181 y Fq(z)j Fp(2)e Fq(J)1509 193 y Fl(1)1745 181 y Fu(\(43\))359 305 y(\002)424 317 y Fl(+)479 305 y Fu(\()p Fq(z)t Fu(\))f(=)f(\002)761 317 y Fn(\000)817 305 y Fu(\()p Fq(z)t Fu(\)\003)p Fq(\033)1029 317 y Fl(1)1066 305 y Fu(\003)1124 271 y Fn(\000)p Fl(1)1296 305 y Fq(z)k Fp(2)e Fq(J)1486 317 y Fl(2)1523 305 y Fq(:)199 b Fu(\(44\))97 480 y(\003)22 b(=)h Fq(i)308 363 y Fm(\022)393 429 y Fq(\025)c Fu(+)f(1)125 b(0)468 529 y(0)g Fq(\025)19 b Fp(\000)f Fu(1)851 363 y Fm(\023)926 480 y Fq(;)97 b(\033)1093 492 y Fl(1)1154 480 y Fu(=)1241 363 y Fm(\022)1327 429 y Fu(0)50 b(1)1327 529 y(1)g(0)1485 363 y Fm(\023)1745 480 y Fu(\(45\))-150 689 y(Also)27 b(note:)308 894 y(\002)373 906 y Fl(11)443 894 y Fu(\()p Fp(1)p Fu(\))50 b(=)g Fq(e)794 860 y Fl(\001)849 868 y Fj(0)895 832 y Fq(\022)934 844 y Fl(3)985 765 y Fm(\000)1023 832 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))19 b(+)1298 800 y Fo(\033)r(\034)p 1298 814 78 4 v 1321 861 a Fl(2)1386 765 y Fm(\001)p 895 875 530 4 v 953 955 a Fq(\022)992 967 y Fl(3)1043 888 y Fm(\000)1081 955 y Fq(\024)f Fu(+)1240 923 y Fo(\033)r(\034)p 1240 937 78 4 v 1263 984 a Fl(2)1328 888 y Fm(\001)1745 894 y Fu(\(46\))308 1150 y(\002)373 1162 y Fl(22)443 1150 y Fu(\()p Fp(1)p Fu(\))50 b(=)g Fq(e)794 1115 y Fl(\001)849 1123 y Fj(0)895 1088 y Fq(\022)934 1100 y Fl(3)985 1021 y Fm(\000)1023 1088 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))19 b Fp(\000)1298 1055 y Fo(\033)r(\034)p 1298 1069 V 1321 1117 a Fl(2)1386 1021 y Fm(\001)p 895 1130 530 4 v 953 1211 a Fq(\022)992 1223 y Fl(3)1043 1144 y Fm(\000)1081 1211 y Fq(\024)f Fu(+)1240 1178 y Fo(\033)r(\034)p 1240 1192 78 4 v 1263 1240 a Fl(2)1328 1144 y Fm(\001)1745 1150 y Fu(\(47\))308 1335 y(\002)373 1347 y Fl(12)443 1335 y Fu(\()p Fp(1)p Fu(\))50 b(=)g(\002)820 1347 y Fl(21)890 1335 y Fu(\()p Fp(1)p Fu(\))23 b(=)g(0)k Fq(;)505 b Fu(\(48\))-150 1500 y(and)304 1705 y(det)14 b(\002\()p Fq(z)t Fu(\))23 b Fp(\021)f Fq(\036)p Fu(\()p Fq(z)t Fu(\))14 b(det)h(\002\()p Fp(1)p Fu(\))1227 1579 y Fm(r)p 1310 1579 106 4 v 1320 1649 a Fq(\025)1368 1661 y Fl(2)p 1320 1686 86 4 v 1320 1762 a Fq(\025)1368 1774 y Fl(1)1415 1705 y Fq(:)307 b Fu(\(49\))-150 1905 y(The)33 b(latter)g(equation)g (follo)n(ws)f(from)h(the)h(comparison)e(of)h(the)-150 2005 y(jumps)24 b(and)e(singularities)g(of)h(its)g(sides.)35 b(Finally)-7 b(,)24 b(w)n(e)f(in)n(tro)r(duce)-150 2105 y(the)28 b(matrix)380 2298 y Fq(Q)p Fu(\()p Fq(z)t Fu(\))23 b(=)663 2181 y Fm(\022)749 2247 y Fq(\036)p Fu(\()p Fq(z)t Fu(\))51 b Fp(\000)p Fq(\036)p Fu(\()p Fq(z)t Fu(\))813 2347 y Fq(i)210 b(i)1201 2181 y Fm(\023)1745 2298 y Fu(\(50\))-150 2512 y(Note)28 b(that)g Fq(Q)p Fu(\()p Fq(z)t Fu(\))f(diagonalizes)e (original)h(jump)j(matrix)e(\010\()p Fq(z)t Fu(\):)487 2676 y(\010\()p Fq(z)t Fu(\))22 b(=)h Fq(Q)p Fu(\()p Fq(z)t Fu(\)\003)p Fq(Q)1061 2642 y Fn(\000)p Fl(1)1149 2676 y Fu(\()p Fq(z)t Fu(\))489 b(\(51\))-150 2840 y(and)27 b Fq(Q)p Fu(\()p Fq(z)t Fu(\))h(is)f(analytic)g(on)g Fc(C)21 b Fp(n)14 b(f)p Fq(J)930 2852 y Fl(1)985 2840 y Fp([)19 b Fq(J)1105 2852 y Fl(2)1142 2840 y Fp(g)27 b Fu(and)273 3004 y Fq(Q)339 3016 y Fl(+)393 3004 y Fu(\()p Fq(z)t Fu(\))c(=)g Fq(Q)677 3016 y Fn(\000)733 3004 y Fu(\()p Fq(z)t Fu(\))p Fq(\033)887 3016 y Fl(1)924 3004 y Fq(;)97 b(z)26 b Fp(2)e Fq(J)1234 3016 y Fl(1)1290 3004 y Fp([)18 b Fq(J)1409 3016 y Fl(2)1447 3004 y Fq(:)275 b Fu(\(52\))-150 3168 y(W)-7 b(e)33 b(are)g(no)n(w)f(ready)g(to)h (presen)n(t)f(the)i(solution)e Fq(U)1481 3180 y Fn(\006)1537 3168 y Fu(\()p Fq(z)t Fu(\))h(of)g(the)-150 3268 y(Riemann-Hilb)r(ert) 28 b(problem)f(\(i-iii\).)37 b(Put)443 3432 y Fq(A)23 b Fu(=)f Fq(Q)p Fu(\()p Fp(1)p Fu(\)\003)886 3398 y Fn(\000)p Fl(1)975 3432 y Fu(\002)1040 3398 y Fn(\000)p Fl(1)1129 3432 y Fu(\()p Fp(1)p Fu(\))p Fq(:)446 b Fu(\(53\))-150 3596 y(Then,)226 3761 y Fq(U)283 3773 y Fn(\000)339 3761 y Fu(\()p Fq(z)t Fu(\))23 b(=)g Fq(A)p Fu(\002\()p Fq(z)t Fu(\)\003)p Fq(Q)915 3726 y Fn(\000)p Fl(1)1003 3761 y Fu(\()p Fq(z)t Fu(\))p Fq(;)96 b Fp(j)p Fq(z)t Fp(j)23 b(\025)g Fu(1)274 b(\(54\))173 3885 y Fq(U)230 3897 y Fl(+)285 3885 y Fu(\()p Fq(z)t Fu(\))22 b(=)h Fq(Q)p Fu(\()p Fq(z)t Fu(\)\002)740 3851 y Fn(\000)p Fl(1)828 3885 y Fu(\()p Fq(z)t Fu(\))p Fq(A)997 3851 y Fn(\000)p Fl(1)1086 3885 y Fq(;)97 b Fp(j)p Fq(z)t Fp(j)23 b(\024)f Fu(1)p Fq(:)275 b Fu(\(55\))-150 4049 y(Indeed,)37 b(b)n(y)e(virtue)g (of)g(Eq.)g(\(51\),)i(w)n(e)d(only)h(need)g(to)g(b)r(e)h(sure)-150 4149 y(that)23 b Fq(U)82 4161 y Fn(\000)138 4149 y Fu(\()p Fq(z)t Fu(\))f(and)h Fq(U)481 4161 y Fl(+)536 4149 y Fu(\()p Fq(z)t Fu(\))f(are)g(analytic)g(for)g Fp(j)p Fq(z)t Fp(j)g Fq(>)h Fu(1)f(and)h Fp(j)p Fq(z)t Fp(j)f Fq(<)h Fu(1)-150 4248 y(resp)r(ectiv)n(ely)-7 b(.)35 b(F)-7 b(rom)23 b(the)h(jump)g(prop)r(erties)f(of)g(\002\()p Fq(z)t Fu(\))g(and)h Fq(Q)p Fu(\()p Fq(z)t Fu(\))-150 4348 y(it)30 b(follo)n(ws)e(that)h Fq(U)447 4360 y Fn(\006)532 4348 y Fu(ha)n(v)n(e)f(no)h(jumps)h(across)d Fq(J)1384 4360 y Fl(1)p Fo(;)p Fl(2)1474 4348 y Fu(,)i(and)g(hence)-150 4448 y(they)41 b(migh)n(t)f(ha)n(v)n(e)g(only)g(p)r(ossible)g(isolated) g(singularities)g(at)-150 4547 y Fq(\025)-102 4559 y Fl(1)p Fo(;)p Fl(2)-12 4547 y Fq(;)14 b(\025)73 4512 y Fn(\000)p Fl(1)73 4569 y(1)p Fo(;)p Fl(2)164 4547 y Fu(.)46 b(The)31 b(analyticit)n(y)f(at)h(these)g(p)r(oin)n(ts)g(can)f (b)r(e)h(sho)n(wn)-150 4647 y(b)n(y)h(observing)f(that)i(the)g (singularities,)g(whic)n(h)f(the)h(functions)-150 4747 y(\002\()p Fq(z)t Fu(\))23 b(and)h Fq(Q)p Fu(\()p Fq(z)t Fu(\))g(do)g(ha)n(v)n(e)e(at)i(the)h(end)f(p)r(oin)n(ts)g(of)g(the)g (segmen)n(ts)-150 4846 y Fq(J)-104 4858 y Fl(1)p Fo(;)p Fl(2)-14 4846 y Fu(,)k(are)e(canceled)i(out)f(in)h(the)g(pro)r(ducts)f (\(54\)-\(55\).)-67 4946 y(The)22 b(excluded)h(v)-5 b(alues)21 b(of)i Fq(\025)f Fu(for)g(whic)n(h)g(the)h(ab)r(o)n(v)n(e)e(construc-) -150 5045 y(tion)29 b(fails)h(are)e Fq(\025)f Fu(=)e Fp(\006)p Fu(1)k(and,)g(in)h(view)f(of)h(Eq.)e(\(49\),)i(the)g(zeros) -150 5145 y(of)e Fq(\022)-16 5157 y Fl(3)35 5078 y Fm(\000)73 5145 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))19 b(+)348 5112 y Fo(\033)r(\034)p 348 5126 78 4 v 371 5174 a Fl(2)436 5078 y Fm(\001)474 5145 y Fu(,)28 b(i.e.)37 b(\(see)28 b(\(41\)\),)-59 5367 y Fp(\006)p Fq(\025)54 5379 y Fo(m)117 5367 y Fq(;)96 b(\025)284 5379 y Fo(m)371 5367 y Fu(=)23 b(tanh)639 5250 y Fm(\022)700 5367 y Fq(m)18 b Fu(+)884 5311 y(1)g Fp(\000)g Fq(\033)p 884 5348 194 4 v 960 5424 a Fu(2)1087 5250 y Fm(\023)1162 5367 y Fq(\031)s(\034)1248 5379 y Fl(0)1286 5367 y Fq(;)97 b(m)23 b Fp(\025)g Fu(0)p Fq(;)90 b Fu(\(56\))2042 -83 y(where,)2543 183 y Fq(\034)2579 195 y Fl(0)2640 183 y Fu(=)22 b Fp(\000)p Fq(i\034)32 b Fu(=)23 b Fp(\000)p Fq(i)3081 41 y Fm(R)3136 61 y Fo(\025)3175 69 y Fb(C)3120 137 y Fo(\025)3159 145 y Fb(B)3285 75 y Fo(dz)p 3252 89 136 4 v 3252 136 a(w)r Fl(\()p Fo(z)r Fl(\))p 3080 164 318 4 v 3080 194 a Fm(R)3135 214 y Fo(\025)3174 222 y Fb(B)3119 290 y Fo(\025)3158 298 y Fb(A)3285 228 y Fo(dz)p 3252 242 136 4 v 3252 289 a(w)r Fl(\()p Fo(z)r Fl(\))3431 183 y Fq(>)f Fu(0)p Fq(:)2125 454 y Fu(Using)41 b(explicit)h(form)n(ulae)e(\(54\)-\(55\))g(one)h(can)g(transform)2042 554 y(our)27 b(basic)g(Eq.)g(\(27\))g(in)n(to)g(the)h(form)2474 720 y Fq(d)p 2450 757 92 4 v 2450 833 a(d\025)2566 776 y Fu(ln)14 b Fq(D)2718 788 y Fo(L)2767 776 y Fu(\()p Fq(\025)p Fu(\))19 b(+)3061 720 y(2)p Fq(\025)p 2991 757 229 4 v 2991 833 a Fu(1)f Fp(\000)g Fq(\025)3182 809 y Fl(2)3230 776 y Fq(L)23 b Fu(=)2326 993 y(=)2608 937 y Fq(i)p 2450 974 344 4 v 2450 1050 a(\031)s Fu(\(1)c Fp(\000)f Fq(\025)2724 1026 y Fl(2)2761 1050 y Fu(\))2817 880 y Fm(Z)2864 1069 y Fl(\004)2925 993 y Fu(tr)3018 876 y Fm(\024)3062 993 y Fu(\002)3127 959 y Fn(\000)p Fl(1)3215 993 y Fu(\()p Fq(z)t Fu(\))3353 937 y Fq(d)p 3332 974 86 4 v 3332 1050 a(dz)3428 993 y Fu(\002\()p Fq(z)t Fu(\))p Fq(\033)3647 1005 y Fl(3)3684 876 y Fm(\025)3741 993 y Fq(dz)t(;)87 b Fu(\(57\))2042 1218 y(here)36 b Fq(\025)h Fp(6)p Fu(=)g Fp(\006)p Fu(1)p Fq(;)k Fp(\006)p Fq(\025)2702 1230 y Fo(m)2765 1218 y Fu(.)63 b(Using)36 b(the)h(same)e(argumen)n(ts)g(as)h(for)2042 1317 y(Eq.)27 b(\(49\),)g(one)g(can)h(see)f(that)2618 1546 y(tr)2711 1429 y Fm(\024)2754 1546 y Fu(\002)2819 1512 y Fn(\000)p Fl(1)2908 1546 y Fu(\()p Fq(z)t Fu(\))3046 1490 y Fq(d)p 3025 1527 V 3025 1603 a(dz)3120 1546 y Fu(\002\()p Fq(z)t Fu(\))p Fq(\033)3339 1558 y Fl(3)3376 1429 y Fm(\025)3443 1546 y Fu(=)2092 1906 y(=)2271 1850 y(1)p 2189 1887 205 4 v 2189 1963 a Fq(cw)r Fu(\()p Fq(z)t Fu(\))2439 1850 y Fq(d)p 2414 1887 95 4 v 2414 1963 a(d\014)2532 1906 y Fu(ln)2615 1814 y Fm(h)2654 1906 y Fq(\022)2693 1918 y Fl(3)2744 1814 y Fm(\020)2794 1906 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))20 b(+)3070 1850 y Fq(\033)s(\034)p 3070 1887 97 4 v 3097 1963 a Fu(2)3176 1814 y Fm(\021)3239 1906 y Fq(\022)3278 1918 y Fl(3)3329 1814 y Fm(\020)3379 1906 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))g Fp(\000)3655 1850 y Fq(\033)s(\034)p 3655 1887 V 3682 1963 a Fu(2)3761 1814 y Fm(\021)o(i)3863 1906 y Fq(:)51 b Fu(\(58\))2042 2134 y(This)41 b(relation)e(allo)n(ws)h(further)h(simpli\014cation)f (of)h(Eq.)f(\(27\).)2042 2234 y(Indeed,)28 b(w)n(e)f(ha)n(v)n(e,)2439 2396 y Fq(d)p 2415 2433 92 4 v 2415 2509 a(d\025)2531 2452 y Fu(ln)14 b Fq(D)2683 2464 y Fo(L)2732 2452 y Fu(\()p Fq(\025)p Fu(\))20 b(+)3026 2396 y(2)p Fq(\025)p 2956 2433 229 4 v 2956 2509 a Fu(1)e Fp(\000)g Fq(\025)3147 2485 y Fl(2)3195 2452 y Fq(L)23 b Fu(=)2291 2663 y(=)2439 2606 y Fq(d)p 2415 2643 92 4 v 2415 2720 a(d\025)2531 2663 y Fu(ln)2614 2570 y Fm(h)2653 2663 y Fq(\022)2692 2675 y Fl(3)2743 2570 y Fm(\020)2792 2663 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))d(+)3068 2606 y Fq(\033)s(\034)p 3068 2643 97 4 v 3096 2720 a Fu(2)3174 2570 y Fm(\021)3238 2663 y Fq(\022)3277 2675 y Fl(3)3328 2570 y Fm(\020)3377 2663 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))g Fp(\000)3653 2606 y Fq(\033)s(\034)p 3653 2643 V 3680 2720 a Fu(2)3759 2570 y Fm(\021i)3862 2663 y Fq(;)52 b Fu(\(59\))2042 2865 y(here)34 b Fq(\025)h Fp(6)p Fu(=)f Fp(\006)p Fu(1)p Fq(;)41 b Fp(\006)p Fq(\025)2695 2877 y Fo(m)2758 2865 y Fu(.)58 b(T)-7 b(aking)33 b(in)n(to)i(accoun)n(t)e(the)i(fact)g(that) 2042 2965 y(as)25 b Fq(\025)f Fp(!)f(1)p Fu(,)k Fq(D)2522 2977 y Fo(L)2571 2965 y Fu(\()p Fq(\025)p Fu(\))d Fp(!)f Fu(\()p Fp(\000)p Fu(1\))2984 2934 y Fo(L)3033 2965 y Fq(\025)3081 2934 y Fl(2)p Fo(L)3165 2965 y Fu(,)j(w)n(e)g(obtain)g (from)g(Eq.)g(\(59\))2042 3064 y(the)33 b(follo)n(wing)e(asymptotic)h (represen)n(tation)e(for)i(the)h Fa(T)-6 b(o)l(eplitz)2042 3164 y(determinant)28 b Fq(D)2577 3176 y Fo(L)2626 3164 y Fu(\()p Fq(\025)p Fu(\):)2042 3398 y Fq(D)2111 3410 y Fo(L)2160 3398 y Fu(\()p Fq(\025)p Fu(\))c(=)2417 3342 y(\()p Fp(\000)p Fu(1\))2588 3312 y Fo(L)p 2394 3379 267 4 v 2394 3460 a Fq(\022)2435 3431 y Fl(2)2433 3482 y(3)2486 3393 y Fm(\000)2534 3427 y Fo(\033)r(\034)p 2534 3441 78 4 v 2557 3489 a Fl(2)2622 3393 y Fm(\001)2670 3398 y Fu(\()p Fq(\025)2750 3364 y Fl(2)2788 3398 y Fp(\000)p Fu(1\))2927 3364 y Fo(L)2976 3398 y Fq(\022)3015 3410 y Fl(3)3066 3306 y Fm(\020)3116 3398 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))c(+)3392 3342 y Fq(\033)s(\034)p 3392 3379 97 4 v 3419 3455 a Fu(2)3498 3306 y Fm(\021)3561 3398 y Fq(\022)3600 3410 y Fl(3)3651 3306 y Fm(\020)3701 3398 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))f Fp(\000)3976 3342 y Fq(\033)s(\034)p 3976 3379 V 4004 3455 a Fu(2)4083 3306 y Fm(\021)2042 3634 y Fu(here)30 b Fq(\025)i Fu(lies)f(outside)g (of)g(\014xed)h(but)f(arbitrary)f(neigh)n(b)r(orho)r(o)r(ds)2042 3734 y(of)d(the)h(p)r(oin)n(ts)g Fp(\006)p Fu(1)f(and)g Fp(\006)p Fq(\025)2937 3746 y Fo(m)3000 3734 y Fu(,)h Fq(m)23 b Fp(\025)f Fu(0.)2125 3834 y(It)h(is)f(w)n(orth)g(noticing)g (that)h(the)g(asymptotic)f(represen)n(tation)2042 3933 y(for)g(the)h(T)-7 b(o)r(eplitz)24 b(determinan)n(t)e(ab)r(o)n(v)n(e)g (sho)n(ws)g(that)h(the)g(p)r(oin)n(ts)2042 4033 y Fq(\025)2090 4045 y Fo(m)2172 4033 y Fu(\(56\))c(are)f(double)g(zeros)g(of)h(the)g Fq(D)3216 4045 y Fo(L)3265 4033 y Fu(\()p Fq(\025)p Fu(\).)36 b(This)18 b(suggests)g(that)2042 4133 y(in)28 b(the)g(large)e Fq(L)h Fu(limit)i(the)f(eigen)n(v)-5 b(alues)26 b Fq(\027)3381 4145 y Fl(2)p Fo(m)3505 4133 y Fu(and)i Fq(\027)3708 4145 y Fl(2)p Fo(m)p Fl(+1)3916 4133 y Fu(from)2042 4232 y(\(5\),)38 b(\(4\))e(merge:)53 b Fq(\027)2686 4244 y Fl(2)p Fo(m)2782 4232 y Fq(;)14 b(\027)2860 4244 y Fl(2)p Fo(m)p Fl(+1)3078 4232 y Fp(!)37 b Fq(\025)3246 4244 y Fo(m)3309 4232 y Fq(:)g Fu(In)f(turn)g(it)g(indicates)2042 4332 y(the)25 b(degeneracy)e(of)h(the)h(sp)r(ectrum)g(of)g(the)g (matrix)f Fk(B)3764 4344 y Fo(L)3838 4332 y Fu(and)g(an)2042 4431 y(app)r(earance)i(of)h(an)h(extra)e(symmetry)i(in)f(the)h(large)f Fq(L)g Fu(limit.)2125 4531 y(Substituting)67 b(Eq.)g(\(59\))f(in)n(to)h (the)g(original)e(equation)2042 4631 y(Eq.)30 b(\(10\),)g(and)g (deforming)g(the)h(original)e(con)n(tour)g(of)h(in)n(tegra-)2042 4730 y(tion)f(to)g(the)g(con)n(tour)f(\000)h(as)f(indicated)h(in)g (Fig.)g(1)g(w)n(e)f(arriv)n(e)f(at)2042 4830 y(the)h(follo)n(wing)e (expression)g(for)h(the)h Fk(en)m(trop)m(y)p Fu(:)2910 5009 y Fq(S)5 b Fu(\()p Fq(\032)3041 5021 y Fo(A)3095 5009 y Fu(\))24 b(=)721 b(\(60\))2184 5342 y(=)2282 5286 y(1)p 2282 5323 42 4 v 2282 5399 a(2)2347 5229 y Fm(Z)2430 5250 y Fn(1)2393 5418 y Fl(1)2514 5342 y Fu(ln)2597 5200 y Fm( )2673 5281 y Fq(\022)2712 5293 y Fl(3)2763 5214 y Fm(\000)2801 5281 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))20 b(+)3077 5248 y Fo(\033)r(\034)p 3077 5262 78 4 v 3099 5310 a Fl(2)3165 5214 y Fm(\001)3217 5281 y Fq(\022)3256 5293 y Fl(3)3307 5214 y Fm(\000)3345 5281 y Fq(\014)t Fu(\()p Fq(\025)p Fu(\))g Fp(\000)3621 5248 y Fo(\033)r(\034)p 3621 5262 V 3643 5310 a Fl(2)3709 5214 y Fm(\001)p 2673 5323 1074 4 v 3077 5404 a Fq(\022)3118 5375 y Fl(2)3116 5426 y(3)3169 5337 y Fm(\000)3217 5371 y Fo(\033)r(\034)p 3217 5385 78 4 v 3239 5433 a Fl(2)3305 5337 y Fm(\001)3757 5200 y(!)3850 5342 y Fq(d\025)p eop %%Page: 6 6 6 5 bop 4043 -299 a Fu(6)-150 -83 y(W)-7 b(e)31 b(remind)g(that)h Fq(\033)g Fu(=)c(0)j(for)f(Case)h(2)f(and)h Fq(\033)h Fu(=)d(1)h(in)i(Case)e(1.)-150 17 y(W)-7 b(e)32 b(can)g(c)n(hange)e(v) -5 b(ariables)31 b(and)g(represen)n(t)g(the)h(\014nal)g(answ)n(er)-150 116 y(in)c(the)g(form:)719 285 y Fq(S)5 b Fu(\()p Fq(\032)850 297 y Fo(A)904 285 y Fu(\))23 b(=)721 b(\(61\))-49 598 y(=)49 542 y Fq(\031)p 49 579 51 4 v 53 655 a Fu(2)123 485 y Fm(Z)206 505 y Fn(1)169 674 y Fl(0)290 598 y Fu(ln)373 456 y Fm( )449 536 y Fq(\022)488 548 y Fl(3)539 469 y Fm(\000)577 536 y Fu(i)p Fq(x)19 b Fu(+)759 504 y Fo(\033)r(\034)p 759 518 78 4 v 781 565 a Fl(2)846 469 y Fm(\001)898 536 y Fq(\022)937 548 y Fl(3)988 469 y Fm(\000)1026 536 y Fu(i)p Fq(x)g Fp(\000)1208 504 y Fo(\033)r(\034)p 1208 518 V 1231 565 a Fl(2)1296 469 y Fm(\001)p 449 579 886 4 v 758 659 a Fq(\022)799 631 y Fl(2)797 681 y(3)851 592 y Fm(\000)899 627 y Fo(\033)r(\034)p 899 641 78 4 v 921 688 a Fl(2)987 592 y Fm(\001)1344 456 y(!)1562 542 y Fq(dx)p 1434 579 348 4 v 1434 663 a Fu(sinh)1582 626 y Fl(2)1619 663 y Fu(\()p Fq(\031)s(x)p Fu(\))-150 837 y(This)32 b(is)f(a)h(limiting)g(expression)e(as)h Fq(L)e Fp(!)h(1)p Fu(.)50 b(Indeed,)33 b(it)f(can)-150 937 y(b)r(e)h(sho)n(wn)f(that)h(the)g(corrections)e(in)i(Eq.)f(\(60\))g (and)h(Eq.)f(\(61\))-150 1061 y(are)27 b(of)g(order)f(of)i Fq(O)474 969 y Fm(\020)524 1061 y Fq(\025)572 1026 y Fn(\000)p Fo(L)572 1085 y(C)674 1061 y Fq(=)716 991 y Fp(p)p 785 991 57 4 v 70 x Fq(L)841 969 y Fm(\021)905 1061 y Fq(:)-67 1186 y Fk(Note)p Fu(:)33 b Fq(S)5 b Fu(\()p Fq(\032)324 1198 y Fo(A)378 1186 y Fu(\))20 b(implicitly)h(dep)r(ends)f (on)g Fq(\015)25 b Fu(and)20 b Fq(h)g Fu(in)n(tro)r(duced)-150 1285 y(in)g(\(1\))h(b)n(y)e(means)h(of)g Fq(\034)30 b Fu(de\014ned)20 b(in)h(Eq.)e(\(34\),)j(this)e Fq(\034)30 b Fu(is)20 b(a)f(ratio)g(of)-150 1385 y(p)r(erio)r(ds)27 b(for)h(the)g(theta)f(function)i Fq(\022)993 1397 y Fl(3)1030 1385 y Fu(\()p Fq(z)t Fu(\).)37 b(The)28 b Fq(\015)k Fu(and)c Fq(h)f Fu(de\014ne)-150 1484 y Fq(\025)-102 1496 y Fo(A)-47 1484 y Fu(,)f Fq(\025)50 1496 y Fo(B)107 1484 y Fu(,)h Fq(\025)205 1496 y Fo(C)287 1484 y Fu(and)e Fq(\025)494 1496 y Fo(D)581 1484 y Fu(b)n(y)g(means)g(of)h(Eqs.)f (\(13,14\))g(and)g(Fig.)h(2.)-150 1584 y(Eq.)h(\(34\))g(and)h(Eqs.)f (\(60\))g(or)g(\(61\))g(de\014ne)h Fq(S)5 b Fu(\()p Fq(\032)1345 1596 y Fo(A)1399 1584 y Fu(\))28 b Fp({)p Fu(.)-67 1684 y(Note)19 b(also,)g(that,)i(b)n(y)e(in)n(tegrating)e(o)n(v)n(er)g(the)i (original)f(con)n(tour)-150 1783 y(\000)-98 1753 y Fn(0)-75 1783 y Fu(,)26 b(one)f(can)g(also)f(suggest)g(an)h(alternativ)n(e)e (represen)n(tation)h(for)-150 1883 y Fq(S)5 b Fu(\()p Fq(\032)-19 1895 y Fo(A)35 1883 y Fu(\))28 b(in)g(terms)f(of)h(the)g (in\014nite)g(series)f(\(cf.)37 b(Eq.)27 b(\(5\)\):)-108 2118 y Fq(S)5 b Fu(\()p Fq(\032)23 2130 y Fo(A)76 2118 y Fu(\))24 b(=)e(2)313 2015 y Fn(1)286 2039 y Fm(X)275 2215 y Fo(m)p Fl(=0)432 2118 y Fq(H)7 b Fu(\()p Fq(\025)588 2130 y Fo(m)651 2118 y Fu(\))24 b(=)875 2015 y Fn(1)848 2039 y Fm(X)794 2214 y Fo(m)p Fl(=)p Fn(\0001)1022 2118 y Fu(\(1)r(+)r Fq(\025)1213 2130 y Fo(m)1277 2118 y Fu(\))14 b(ln)1522 2062 y(2)p 1416 2099 255 4 v 1416 2175 a(1)k(+)g Fq(\025)1607 2187 y Fo(m)1680 2118 y Fq(;)42 b Fu(\(62\))-150 2369 y(here)27 b(the)h(n)n(um)n(b)r(ers)f Fq(\025)556 2381 y Fo(m)647 2369 y Fu(are)g(de\014ned)h(in)g(Eq.)f(\(56\).)-67 2468 y(No)n(w)50 b(let)g(us)h(discuss)f(some)f(in)n(teresting)h (limiting)h(cases)-150 2568 y(whic)n(h)38 b(corresp)r(ond)f Fq(\034)51 b Fp(!)41 b Fu(0.)69 b(The)39 b Fa(L)l(anden)h(tr)l(ansform) f Fu(\(see)-150 2668 y([27)o(]\))34 b(yields)f(the)h(follo)n(wing)f (estimate)h(of)f(the)h(theta-function)-150 2767 y(for)27 b(small)g Fq(\034)38 b Fu(and)27 b(pure)g(imaginary)f Fq(s)-150 2993 y 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y([2])34 b(G.)20 b(Vidal,)h(J.I.)f(Latorre,)h(E.)f(Rico,)h(and)d(A.)h (Kitaev,)i(Ph)n(ys.)e(Rev.)2194 2963 y(Lett.)26 b Fd(90)p Fs(,)h(227902,)h(\(2003\);)2080 3050 y([3])34 b(A.)e(Osterloh,)j(L.)d (Amico,)h(G.)g(F)-6 b(alci)33 b(and)f(R.)g(F)-6 b(azio)32 b(,)i(Nature)2194 3137 y Fd(416)27 b Fs(608,)g(\(2002\))2080 3224 y([4])34 b(T.J.Osb)r(orne,M.A.Nielsen,)28 b(Ph)n(ys.Rev.A)22 b Fd(66)p Fs(,032110\(2002\).)2080 3311 y([5])34 b(V.V)-6 b(edral,)26 b(New)g(J.Ph)n(ys.)p Fd(6)p Fs(,10)i(\(2004\).)2080 3399 y([6])34 b(J.I.Latorre,)28 b(E.Rico,)f(and)e(G.Vidal,)i(QIC.)p Fd(4)p Fs(,048)h(\(2004\).)2080 3486 y([7])34 b(V.E.Korepin,)27 b(Ph)n(ys.Rev.)e(Lett.)p Fd(92)p Fs(,)i(096402)g(\(2004\).)2080 3573 y([8])34 b(M.C.Arnesen,)i(S.Bose,)f(and)d(V.V)-6 b(edral,)34 b(Ph)n(ys.Rev.Lett.)p Fd(87)p Fs(,)2194 3660 y(017901)28 b(\(2001\).)2080 3757 y([9])34 b(F.V)-6 b(erstraete,)23 b(M.A.Mart)2980 3739 y(\023)2989 3757 y(in-Delgado,)g(J.I.Cirac,)h(Ph)n (ys.)d(Rev.)2194 3844 y(Lett.)26 b Fd(92)p Fs(,)h(087201)h(\(2004\).) 2042 3931 y([10])34 b(R.Orus,)26 b(J.I.Latorre,)i(Ph)n(ys.)d(Rev.)g(A)g Fd(69)p Fs(,)i(052308)h(\(2004\).)2042 4018 y([11])34 b(J.K.P)n(ac)n(hos,M.B.Plenio,Ph)n(ys.Rev.Lett.)p Fd(93)p Fs(,05)q(640)q(2\(20)q(04)q(\).)2042 4105 y([12])g(H.F)-6 b(an)26 b(and)f(S.Llo)n(yd,)h(quan)n(t-ph/0405130.)2042 4193 y([13])34 b(V.V)-6 b(edral,)26 b(quan)n(t-ph/0405102.)2042 4280 y([14])34 b(C.H.Bennett,)27 b(D.P)-6 b(.DiVincenzo,Nature)26 b Fd(404)p Fs(,247)h(\(2000\).)2042 4367 y([15])34 b(S.)26 b(Llo)n(yd,Science)g Fd(261)p Fs(,1569\(1993\);)k Ft(ibid)25 b Fd(263)p Fs(,695\(1994\).)2042 4454 y([16])34 b(S.Ghosh,)k(T.F.Rosen) n(baum,)g(G.Aeppli,)g(S.N.Copp)r(ersmith,)2194 4541 y(Nature)22 b Fd(425)p Fs(,)h(48)f(\(2003\);)j(V.V)-6 b(edral,)22 b(Nature)f Fd(425)p Fs(,)i(28)f(\(2003\).)2042 4628 y([17])34 b(J.)43 b(W)-6 b(ang,)45 b(S,)c(Kais,)47 b(F.)41 b(Remacle,)46 b(R.)41 b(D.)g(Levine,)46 b(quan)n(t-)2194 4716 y(ph/0405088;)53 b(J.)43 b(W)-6 b(ang,)46 b(S,)41 b(Kais,)47 b(quan)n(t-ph/0405087;)k (J.)2194 4803 y(W)-6 b(ang,)26 b(S,)g(Kais,)h(quan)n(t-ph/0405085)2042 4890 y([18])34 b(Y.)d(Chen,)g(P)-6 b(.)31 b(Zanardi,)h(Z.)f(D.)f(W)-6 b(ang,)32 b(F.)f(C.)g(Zhang,)h(quan)n(t-)2194 4977 y(ph/0407228;)63 b(Y.)48 b(Zhao,)54 b(P)-6 b(.)48 b(Zanardi,)55 b(G.)49 b(Chen,)54 b(quan)n(t-)2194 5064 y(ph/0407080;)45 b(A.Hamma,)38 b(R.Ionicioiu,)i(P)-6 b(.Zanardi,)41 b(quan)n(t-)2194 5152 y(ph/0406202;)29 b(P)-6 b(.)26 b(Giorda,)h(P)-6 b(.)25 b(Zanardi,)i(quan)n(t-ph/0311058)2042 5239 y([19])34 b(P)-6 b(.)22 b(Zanardi,)h(M.)f(Rasetti,)h(Ph)n(ys.Lett.)e(A264)h (\(1999\))h(94-99;)h(A.)2194 5326 y(Marzuoli,)30 b(M.)e(Rasetti,)g(Ph)n (ys.Lett.)g(A306)f(\(2002\))i(79-87;)g(M.)2194 5413 y(Rasetti,)e (cond-mat/0211081)p eop %%Page: 7 7 7 6 bop 4043 -299 a Fu(7)-150 -83 y Fs([20])35 b(P)-6 b(.)17 b(A.)h(Deift,)i Ft(Inte)l(gr)l(able)h(op)l(er)l(ators,)h Fs(in)17 b(Di\013eren)n(tial)i(Op)r(erators)3 4 y(and)30 b(Sp)r(ectral)g(Theory)-6 b(,)32 b(Amer.)d(Math.)i(So)r(c.)f(T)-6 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b(think)f(it)i(w)n (ould)f(b)r(e)h(of)g(in)n(terest)g(to)f(un-)2194 3591 y(derstand)g(its)g(meaning)f(in)h(con)n(text)f(of)h(the)g(general)g (result)g(of)2194 3678 y(Widom)25 b([40)q(].)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0409031906226--