This is a multi-part message in MIME format. ---------------0211222011648 Content-Type: text/plain; name="02-485.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-485.comments" MSC: 82C10;82C24 ---------------0211222011648 Content-Type: text/plain; name="02-485.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-485.keywords" XXZ chain, interfaces, dynamics, kink states, Heisenberg ferromagnet ---------------0211222011648 Content-Type: application/postscript; name="nss_final.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nss_final.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: nss_final.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips nss_final -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.11.21:1516 %%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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(alls)f(and)h(their)g(motion)g(when)h(a)e(magnetic)h(\014eld)h(is)456 2144 y(applied.)f(Although)19 b(the)g(Landau-Lifshitz)f(equation)h(w)n (as)f(rigorously)e(deriv)n(ed)i(in)i(the)f(mean-)456 2243 y(\014eld)25 b(limit)h(b)n(y)f(Moser,)f(Prets,)h(and)g(Spitzer)g ([)p Fz(17)o Fy(],)h(its)g(predictions)e(do)h(not)g(univ)n(ersally)f (hold)456 2343 y(true)g(for)g(all)h(magnetic)f(materials.)35 b(Th)n(us)24 b(w)n(e)h(w)n(an)n(t)f(to)h(understand)f(the)h(dynamical)f (b)r(eha)n(v-)456 2443 y(ior)k(of)h(domain)g(w)n(alls)f(in)i(a)f(quan)n (tum)g(mo)r(del,)h(in)f(whic)n(h)h(they)f(o)r(ccur)g(naturally)f(as)g (ground)456 2542 y(states,)f(namely)g(in)h(the)g(XXZ)g(mo)r(del.)605 2642 y(The)20 b(dynamics)f(w)n(e)g(consider)f(is)h(generated)g(b)n(y)g (the)h(XXZ)g(kink)f(Hamiltonian)g(p)r(erturb)r(ed)456 2742 y(b)n(y)33 b(a)g(magnetic)g(\014eld.)56 b(As)34 b(in)f(man)n(y)h(similar)e(cases,)i(w)n(e)g(in)n(tro)r(duce)f(a)g (scaling)g(where)g(the)456 2841 y(strength)22 b(of)g(the)h(p)r (erturbation,)g(sa)n(y)e(the)i(coupling)f(constan)n(t)g Fw(\025)g Fy(for)g(a)g(magnetic)g(\014eld,)i(tends)456 2941 y(to)31 b(0)g(and)h(the)g(microscopic)e(time)i(scale)f(to)h (in\014nit)n(y)g(suc)n(h)f(that)h(their)g(pro)r(duct,)g(whic)n(h)g(w)n (e)456 3040 y(denote)j(b)n(y)g Fw(\034)9 b Fy(,)38 b(is)d(k)n(ept)g (\014xed.)60 b(W)-7 b(e)36 b(obtain)f(the)g(follo)n(wing)f(results)h (ab)r(out)g(this)h(limit:)53 b(\(i\))456 3140 y(F)-7 b(or)22 b(external)h(magnetic)g(\014elds)g(of)h(b)r(ounded)g(spatial)e (supp)r(ort)i(that)f(are)g(su\016cien)n(tly)g(regular)456 3240 y(functions)36 b(of)h(time,)i(the)e(leading)f(con)n(tribution)g (to)g(the)h(time)g(ev)n(olution)f(is)g(iden)n(ti\014ed)h(as)456 3339 y(the)f(reduced)g(dynamics)g(determined)g(b)n(y)g(the)h(time)g(a)n (v)n(eraged)c(\014eld)k(pro)5 b(jected)36 b(on)n(to)f(the)456 3439 y(ground)24 b(state)i(space)e(of)i(the)g(unp)r(erturb)r(ed)g(mo)r (del)g(with)g(an)f(error)f(term)h(of)h(order)e Fw(\025)3167 3409 y Fx(1)p Fs(\000)p Fv(\016)3289 3439 y Fw(;)14 b(\016)26 b Fr(2)456 3539 y Fy(\(0)p Fw(;)14 b Fy(1\))39 b(\(Theorem)g(3.3\);)46 b(\(ii\))41 b(If)f(the)g(sp)r(ectrum)h(of)e(the)i(unp)r(erturb)r(ed)f (mo)r(del)g(has)g(a)f(gap)456 3638 y(b)r(et)n(w)n(een)19 b(the)h(ground)e(states)h(and)g(the)h(rest)f(of)g(the)h(sp)r(ectrum,)h (and)f(the)f(rest)g(of)h(the)f(sp)r(ectrum)456 3738 y(is)37 b(absolutely)g(con)n(tin)n(uous,)i(the)f(next-to-leading)e(order)g (term)i(is)f(of)h(order)e Fw(\025)i Fy(\(Theorem)456 3837 y(3.5\);)23 b(\(iii\))f(Analysis)f(of)g(the)h(reduced)g(dynamics)f (for)g(uniform,)h(time-indep)r(enden)n(t)h(magnetic)456 3937 y(\014elds,)28 b(rev)n(eals)e(a)i(mark)n(edly)e(di\013eren)n(t)i (b)r(eha)n(vior)f(dep)r(ending)h(on)g(whether)g(or)f(not)h(the)g (\014eld)456 4037 y(has)33 b(a)h(non-v)-5 b(anishing)32 b(comp)r(onen)n(t)i(in)g(the)h Fw(z)i Fy(direction.)55 b(If)35 b(the)f Fw(z)j Fy(comp)r(onen)n(t)d(v)-5 b(anishes,)456 4136 y(the)26 b(reduced)f(dynamics)g(for)h(the)g(magnetization)f (pro\014le)g(is)g(\\ballistic",)g(while)h(if)h(there)e(is)h(a)456 4236 y(non-v)-5 b(anishing)29 b(comp)r(onen)n(t)h(of)g(the)h(\014eld)g (in)f(the)h Fw(z)i Fy(direction,)e(the)g(magnetization)e(pro\014le)456 4336 y(ev)n(olv)n(es)c(p)r(erio)r(dically)-7 b(.)605 4435 y(In)32 b(Section)g(2)g(w)n(e)g(de\014ne)g(the)h(mo)r(del)f(and)g (state)g(the)g(assumptions)g(on)f(the)i(magnetic)456 4535 y(\014eld.)71 b(W)-7 b(e)39 b(deriv)n(e)f(the)h(leading)f (dynamics)h(and)f(its)h(\014rst)g(correction)e(in)i(Section)g(3.)70 b(In)456 4634 y(Section)29 b(4)g(w)n(e)h(analyze)e(the)i(leading)f (dynamical)g(b)r(eha)n(vior)g(in)h(more)e(detail)i(for)f(a)h(uniform) 456 4734 y(\014eld)h(and)g(commen)n(t)g(on)f(small)h(p)r(erturbations)f (thereof.)47 b(Tw)n(o)30 b(app)r(endices)h(pro)n(vide)f(some)456 4834 y(auxiliary)24 b(results)g(on)i(the)f(magnetization)g(pro\014le)g (in)g(the)h(kink)g(ground)e(states)h(of)g(the)h(XXZ)456 4933 y(c)n(hain)h(and)g(on)g(the)h(sp)r(ectrum)g(of)g(the)g (Stark-Jacobi)d(op)r(erator.)p eop %%Page: 3 3 3 2 bop 820 251 a Fx(D)n(YNAMICS)29 b(OF)g(INTERF)-7 b(A)n(CES)29 b(IN)g(THE)g(FERR)n(OMA)n(GNETIC)h(XXZ)f(CHAIN)332 b(3)1665 450 y Fz(2.)47 b(The)32 b(mo)s(del)605 600 y(2.1.)46 b(Finite)d(c)m(hain)g(Hamiltonian.)c Fy(In)f(this)f(pap)r(er)h(w)n(e)f (shall)g(only)g(consider)f(the)456 699 y(spin-)642 666 y Fx(1)p 641 680 34 4 v 641 728 a(2)718 699 y Fy(c)n(hain.)54 b(Unless)33 b(stated)g(otherwise,)h(our)f(results)g(extend)h(to)f (higher)g(v)-5 b(alues)33 b(of)g(the)456 799 y(spin.)75 b(As)41 b(usual,)i(let)e(us)g(denote)f(b)n(y)g(\001)45 b Fw(>)f Fy(1)c(the)h(anisotrop)n(y)e(parameter.)74 b(The)41 b(kink)456 898 y(Hamiltonian)27 b(on)g(the)h(c)n(hain)f([)p Fw(a;)14 b(b)p Fy(])k Fr(\\)h Fq(Z)i Fy(is)28 b(de\014ned)g(as)456 998 y(\(1\))467 1180 y Fw(H)7 b Fy([)p Fw(a;)14 b(b)p Fy(])22 b(=)h Fr(\000)900 1076 y Fv(b)p Fs(\000)p Fx(1)897 1101 y Fp(X)895 1275 y Fv(x)p Fx(=)p Fv(a)1033 1062 y Fp(\024)1101 1123 y Fy(1)p 1087 1160 70 4 v 1087 1236 a(\001)1166 1180 y(\()p Fw(S)1254 1145 y Fx(1)1249 1200 y Fv(x)1291 1180 y Fw(S)1347 1145 y Fx(1)1342 1200 y Fv(x)p Fx(+1)1486 1180 y Fy(+)18 b Fw(S)1625 1145 y Fx(2)1620 1200 y Fv(x)1662 1180 y Fw(S)1718 1145 y Fx(2)1713 1200 y Fv(x)p Fx(+1)1839 1180 y Fy(\))h(+)f Fw(S)2029 1145 y Fx(3)2024 1200 y Fv(x)2066 1180 y Fw(S)2122 1145 y Fx(3)2117 1200 y Fv(x)p Fx(+1)2261 1180 y Fr(\000)2354 1123 y Fy(1)p 2354 1160 42 4 v 2354 1236 a(4)2405 1180 y Fz(1)2453 1062 y Fp(\025)2515 1180 y Fr(\000)2608 1123 y Fy(1)p 2608 1160 V 2608 1236 a(2)2660 1098 y Fp(p)p 2743 1098 302 4 v 82 x Fy(1)g Fr(\000)g Fy(\001)2955 1156 y Fs(\000)p Fx(2)3044 1180 y Fy(\()p Fw(S)3132 1145 y Fx(3)3127 1200 y Fv(a)3188 1180 y Fr(\000)g Fw(S)3327 1145 y Fx(3)3322 1200 y Fv(b)3364 1180 y Fy(\))c Fw(;)456 1416 y Fy(where)33 b(the)h(spin)g(op)r(erators)e(at)h(p)r(osition)h Fw(x)p Fy(,)i Fw(S)1999 1386 y Fx(1)1994 1436 y Fv(x)2036 1416 y Fw(;)14 b(S)2129 1386 y Fx(2)2124 1436 y Fv(x)2165 1416 y Fw(;)g(S)2258 1386 y Fx(3)2253 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b(explicitly)-7 b(,)36 b(let)f(us)f(denote)g(the)g(eigen)n(v)n(ectors)e (of)i(the)g(total)g Fw(S)2855 3339 y Fx(3)2926 3369 y Fy(comp)r(onen)n(t)g(b)n(y)456 3469 y Fr(j)p Fy(\()p Fw(m)584 3481 y Fv(x)626 3469 y Fy(\))p Fr(i)p Fy(,)d(where)e(\()p Fw(m)1091 3481 y Fv(x)1133 3469 y Fy(\))d Fr(2)h(f\006)1390 3436 y Fx(1)p 1389 3450 34 4 v 1389 3498 a(2)1432 3469 y Fr(g)1474 3439 y Fv(b)p Fs(\000)p Fv(a)p Fx(+1)1709 3469 y Fy(and)i Fw(S)1928 3439 y Fx(3)1923 3489 y Fv(y)1965 3469 y Fr(j)p Fy(\()p Fw(m)2093 3481 y Fv(x)2135 3469 y Fy(\))q Fr(i)d Fy(=)h Fw(m)2391 3481 y Fv(y)2430 3469 y Fr(j)p Fy(\()p Fw(m)2558 3481 y Fv(x)2600 3469 y Fy(\))q Fr(i)p Fy(.)43 b(Then,)31 b(as)d(w)n(as)h(found)456 3571 y(b)n(y)h(Alcaraz,)h(Salinas)f(and)g(W)-7 b(reszinski)31 b([)p Fz(2)o Fy(],)h(these)f(ground)f(states)g(are)g(\(up)i(to)e (normaliza-)456 3671 y(tion\))456 3909 y(\(2\))771 b Fw( )1387 3921 y Fv(m)1474 3909 y Fy(=)1575 3830 y Fp(X)1561 4012 y Fx(\()p Fv(m)1646 4020 y Fn(x)1683 4012 y Fx(\))1771 3805 y Fv(b)1732 3830 y Fp(Y)1723 4005 y Fv(x)p Fx(=)p Fv(a)1861 3909 y Fw(q)1901 3875 y Fs(\000)p Fv(x)p Fx(\(1)p Fs(\000)p Fv(m)2161 3883 y Fn(x)2198 3875 y Fx(\))p Fv(=)p Fx(2)2295 3909 y Fr(j)p Fy(\()p Fw(m)2423 3921 y Fv(x)2465 3909 y Fy(\))p Fr(i)14 b Fw(;)456 4162 y Fy(where)32 b(the)h(sum)f(runs)h(o)n(v)n(er)d(all)j(sets)f(\()p Fw(m)1795 4174 y Fv(x)1837 4162 y Fy(\))h(suc)n(h)f(that)2279 4100 y Fp(P)2367 4187 y Fv(x)2423 4162 y Fw(m)2496 4174 y Fv(x)2569 4162 y Fy(=)f Fw(m)p Fy(,)j(whic)n(h)e(is)h(the)g(total)456 4263 y Fw(S)512 4233 y Fx(3)579 4263 y Fy(eigen)n(v)-5 b(alue.)44 b(As)30 b(the)h Fw( )1341 4275 y Fv(m)1435 4263 y Fy(are)e(not)h(normalized,)g(w)n(e)g(also)g(de\014ne)g Fr(j)p Fw(m)p Fr(i)e Fy(=)f Fw( )3016 4275 y Fv(m)3079 4263 y Fw(=)p Fr(k)p Fw( )3217 4275 y Fv(m)3279 4263 y Fr(k)p Fy(.)45 b(If)456 4363 y Fw(m)28 b Fy(=)h Fr(\006)p Fy(\()p Fw(b)20 b Fr(\000)h Fw(a)f Fy(+)h(1\))31 b(is)g (maximal/minimal,)g(then)g(w)n(e)g(ha)n(v)n(e)f(the)i(all)f(spin-up/do) n(wn)f(state,)456 4462 y(i.e.,)d(the)h(magnetization)f(pro\014le)g(in)h (the)g Fw(z)j Fy(direction)c(equals)g Fr(h)p Fw(m)p Fr(j)p Fw(S)2641 4432 y Fx(3)2636 4483 y Fv(x)2678 4462 y Fr(j)p Fw(m)p Fr(i)c Fy(=)g Fr(\006)2992 4430 y Fx(1)p 2991 4444 V 2991 4491 a(2)3062 4462 y Fy(for)k(all)g Fw(x)p Fy(.)605 4656 y Fz(2.2.)46 b(In\014nite)27 b(c)m(hain)i(kink)e (Hamiltonian.)39 b Fy(The)24 b(in\014nite)g(v)n(olume)f(kink)h (Hamilton-)456 4755 y(ian)g(is)h(de\014ned)g(in)h(the)f(standard)f(w)n (a)n(y)g(via)g(the)h(generator)e(of)i(the)g(Heisen)n(b)r(erg)f (dynamics)h(on)456 4855 y(the)31 b(algebra)e(of)i(quasi-lo)r(cal)f (observ)-5 b(ables,)30 b Fr(A)p Fy(.)47 b(This)31 b(algebra)f(is)g(the) i(norm)e(closure)g(of)h(the)456 4954 y(algebra)25 b(of)j(lo)r(cal)f (observ)-5 b(ables,)1160 5123 y Fr(A)1226 5135 y Fx(lo)r(c)1337 5123 y Fy(=)1558 5044 y Fp([)1424 5226 y Fx(\003)p Fs(\032)p Fm(Z)-10 b Fv(;)p Fs(j)p Fx(\003)p Fs(j)p Fv(<)p Fs(1)1797 5123 y Fr(A)1863 5135 y Fx(\003)1913 5123 y Fw(;)97 b Fr(A)2099 5135 y Fx(\003)2172 5123 y Fy(=)2260 5044 y Fp(O)2259 5222 y Fv(x)p Fs(2)p Fx(\003)2401 5123 y Fy(Mat)2551 5135 y Fm(C)2597 5123 y Fy(\(2\))14 b Fw(:)p eop %%Page: 4 4 4 3 bop 456 251 a Fx(4)247 b(BR)n(UNO)23 b(NA)n(CHTER)n(GAELE,)g(W)n (OLF)n(GANG)g(L)g(SPITZER,)f(AND)g(SHANNON)g(ST)-5 b(ARR)456 450 y Fy(W)e(e)31 b(prefer)g(to)g(w)n(ork)f(in)h(the)h Fl(GNS)f Fy(Hilb)r(ert)h(space)f(represen)n(tation)e(of)i(the)h (in\014nite)g(v)n(olume)456 550 y(kink)27 b(states.)36 b(T)-7 b(o)28 b(this)f(end)h(w)n(e)f(in)n(tro)r(duce)g(the)h (incomplete)g(tensor)f(pro)r(duct)g(Hilb)r(ert)h(space)456 843 y(\(3\))705 b Fr(H)24 b Fy(=)p 1448 663 1149 4 v 1536 764 a Fp([)1448 946 y Fx(\003)p Fv(;)p Fs(j)p Fx(\003)p Fs(j)p Fv(<)p Fs(1)1730 676 y Fp(0)1730 825 y(@)1804 764 y(O)1803 942 y Fv(x)p Fs(2)p Fx(\003)1944 843 y Fq(C)1998 819 y Fx(2)2060 843 y Fr(\012)2180 764 y Fp(O)2143 946 y Fv(y)r Fs(2)p Fm(Z)-11 b Fs(n)-6 b Fx(\003)2356 843 y Fy(\012\()p Fw(y)s Fy(\))2524 676 y Fp(1)2524 825 y(A)2610 843 y Fw(;)456 1119 y Fy(where)1468 1295 y(\012\()p Fw(y)s Fy(\))23 b(=)1747 1178 y Fp(\032)1850 1244 y Fr(j"i)111 b Fy(if)28 b Fw(y)e Fr(\024)d Fy(0)1850 1344 y Fr(j#i)111 b Fy(if)28 b Fw(y)e(>)d Fy(0)2409 1295 y Fw(:)456 1504 y Fy(Let)k(us)h(de\014ne)g(the)g(v)n(ector)456 1685 y(\(4\))1130 b(\012)23 b(=)1863 1606 y Fp(O)1866 1785 y Fv(y)r Fs(2)p Fm(Z)2002 1685 y Fy(\012\()p Fw(y)s Fy(\))14 b Fw(;)456 1957 y Fy(whic)n(h)27 b(is)h(a)f(v)n(ector)f(in)i Fr(H)q Fy(.)37 b(W)-7 b(e)28 b(also)e(de\014ne)i(the)g(\(unnormalized\))g Fl(GNS)f Fy(v)n(ector)889 2220 y(\012)949 2186 y Fk(GNS)1095 2220 y Fy(=)1184 2141 y Fp(X)1183 2320 y Fv(k)q Fs(\025)p Fx(0)1630 2141 y Fp(X)1318 2319 y Fv(x)1356 2327 y Fo(1)1388 2319 y Fv(<)p Fs(\001\001\001)p Fv()g(k)s Fy(,)i(although)f(man)n(y)f(b)r(onds)h(can)f(originate)g(from)h(a)f (certain)g(site.)54 b(W)-7 b(e)33 b(call)456 4717 y(the)f(b)r(ond)h Fw(b)p Fy(\()p Fw(k)s Fy(\))e(=)f(\()p Fw(p)p Fy(\()p Fw(k)s Fy(\))p Fw(;)14 b(k)s Fy(\),)35 b(and)d(de\014ne)g Fw(`)p Fy(\()p Fw(b)p Fy(\()p Fw(k)s Fy(\)\))f(=)g Fw(p)p Fy(\()p Fw(k)s Fy(\))22 b Fr(\000)f Fw(k)35 b Fy(the)e(length)f(of)h (this)f(b)r(ond,)456 4817 y(and)i Fw(`)p Fy(\()p Fw(G)p Fy(\))h(=)921 4755 y Fp(P)1023 4817 y Fw(`)p Fy(\()p Fw(b)p Fy(\()p Fw(k)s Fy(\)\))g(the)f(length)h(of)f(the)h(graph)e Fw(G)p Fy(.)58 b(3\))34 b(If)h(there)f(is)g(a)g(b)r(ond)h(b)r(et)n(w)n (een)456 4917 y Fw(i)k Fy(and)g Fw(j)5 b Fy(,)43 b(then)d(there)f(is)g (no)g(b)r(ond)h(p)r(ossible)f(b)r(et)n(w)n(een)h Fw(k)i Fy(and)d Fw(l)i Fy(if)f(\()p Fw(k)46 b(>)d(b)p Fy(\()p Fw(i)p Fy(\))c(and)h Fw(l)k(<)456 5016 y(b)p Fy(\()p Fw(i)p Fy(\)\))38 b(or)g(\()p Fw(k)45 b(>)c(i)d Fy(and)g Fw(l)43 b(<)e(i)p Fy(\))e(or)e(\()p Fw(k)45 b(<)c(b)p Fy(\()p Fw(i)p Fy(\),)g Fw(l)i(>)e(i)p Fy(\).)70 b(In)39 b(other)f(w)n(ords,)i(there)e(are)g(no)456 5116 y(crossing)28 b(b)r(onds,)j(and)f(no)g(b)r(onds)g(prop)r(erly)f(nested.)45 b(\(With)32 b(these)e(rules,)g(the)h(digraphs)e Fw(G)456 5216 y Fy(are)35 b(equiv)-5 b(alen)n(t)37 b(to)g(the)g(set)f(of)h(all)g (comp)r(ositions,)h(i.e.,)h(all)e(\014nite)g(ordered)f(sequences)g(of)p eop %%Page: 6 6 6 5 bop 456 251 a Fx(6)247 b(BR)n(UNO)23 b(NA)n(CHTER)n(GAELE,)g(W)n (OLF)n(GANG)g(L)g(SPITZER,)f(AND)g(SHANNON)g(ST)-5 b(ARR)456 450 y Fy(p)r(ositiv)n(e)27 b(in)n(tegers.)38 b(T)-7 b(o)28 b(a)g(digraph)f Fw(G)d Fy(=)g Fr(f)p Fw(b)p Fy(\(1\))p Fw(;)14 b(:)g(:)g(:)f(;)h(b)p Fy(\()p Fw(n)p Fy(\))p Fr(g)p Fy(,)28 b(w)n(e)g(asso)r(ciate)e(the)j(comp)r(osition)456 550 y([)p Fw(p)p Fy(\(1\))17 b Fr(\000)h Fy(1)p Fw(;)c(p)p Fy(\()p Fw(p)p Fy(\(1\)\))j Fr(\000)g Fw(p)p Fy(\(1\))p Fw(;)d(p)p Fy(\()p Fw(p)p Fy(\()p Fw(p)p Fy(\(1\)\)\))k Fr(\000)g Fw(p)p Fy(\()p Fw(p)p Fy(\(1\)\))p Fw(;)c(:)g(:)g(:)f Fy(].)37 b Fr(G)2338 562 y Fv(n)2411 550 y Fy(is)27 b(the)h(set)f(of)g (all)g(comp)r(ositions)456 649 y(whic)n(h)g(sum)h(to)f Fw(n)p Fy(.\))605 749 y(W)-7 b(e)32 b(can)g(sho)n(w)f(that)h Fr(jG)1375 761 y Fv(n)1421 749 y Fr(j)e Fy(=)f(2)1610 719 y Fv(n)p Fs(\000)p Fx(1)1740 749 y Fy(.)50 b(It)32 b(is)g(clear)f(that)h Fr(jG)2454 761 y Fx(1)2492 749 y Fr(j)e Fy(=)f(1.)50 b(If)32 b(w)n(e)f(ha)n(v)n(e)g(a)h(graph)456 849 y Fw(G)23 b Fr(2)g(G)671 861 y Fv(n)717 849 y Fy(,)i(then)f(w)n(e)g (ma)n(y)f(construct)h(t)n(w)n(o)f(new)h(graphs)f(in)h Fr(G)2334 861 y Fv(n)p Fx(+1)2464 849 y Fy(.)35 b(Let)24 b(us)g(denote)g(the)h Fw(n)11 b Fy(+)g(2nd)456 948 y(v)n(ertex)23 b(b)n(y)h(0,)h(then)f(0)g(ma)n(y)g(b)r(e)h(connected)f(to)g(1)g(whose)g (graph)f(w)n(e)h(denote)g(b)n(y)g Fw(G)3026 960 y Fx(1)3064 948 y Fy(.)35 b(The)25 b(only)456 1048 y(other)d(p)r(ossibilit)n(y)h (is)h(to)f(join)g(0)g(with)h Fw(p)p Fy(\(1\))g(and)f(w)n(e)g(call)g (the)h(graph)e Fw(G)2675 1060 y Fx(2)2712 1048 y Fy(.)36 b(W)-7 b(e)24 b(de\014ne)f(signs)g(for)456 1147 y(the)g(graphs,)f (inductiv)n(ely)-7 b(.)36 b(Let)23 b Fq(y)g Fy(b)r(e)g(the)g(unique)g (graph)f(with)h(v)n(ertex)f(set)h Fr(f)p Fy(2)p Fw(;)14 b Fy(1)p Fr(g)21 b Fy(and)h(b)r(ond)456 1247 y(from)27 b(2)g(\(left\))h(to)g(1)f(\(righ)n(t\),)g(then)h(w)n(e)f(set)h Fw(\033)s Fy(\()p Fq(y)p Fy(\))c(=)f(1.)36 b(Using)27 b(the)h(ab)r(o)n(v)n(e)e(construction,)h(w)n(e)456 1347 y(de\014ne)h(the)h(sign)f(b)n(y)g Fw(\033)s Fy(\()p Fw(G)1275 1359 y Fx(1)1313 1347 y Fy(\))d(=)f Fr(\000)p Fw(\033)s Fy(\()p Fw(G)p Fy(\),)30 b(and)e Fw(\033)s Fy(\()p Fw(G)2065 1359 y Fx(2)2103 1347 y Fy(\))d(=)f Fw(\033)s Fy(\()p Fw(G)p Fy(\).)41 b(Some)28 b(graphs)f(are)h(sho)n(wn)f(in)456 1446 y(Figure)g(1.)605 1546 y(Giv)n(en)f(a)g(sequence)g(of)g(n)n(um)n (b)r(ers)f Fw(k)1724 1558 y Fv(j)1796 1546 y Fy(:)37 b(1)23 b Fr(\024)g Fw(j)28 b Fr(\024)22 b Fw(n)p Fy(,)27 b(for)e(an)n(y)h(graph)f Fw(G)e Fr(2)h(G)2989 1558 y Fv(n)3060 1546 y Fy(and)i(v)n(ertex)456 1646 y Fw(i)h Fy(w)n(e)g(de\014ne)h Fw(k)s Fy(\()p Fw(i)p Fy(;)14 b Fw(G)p Fy(\))23 b(=)1226 1583 y Fp(P)1313 1671 y Fv(i)p Fs(\024)p Fv(j)s Fs(\024)p Fv(i)p Fx(+)p Fv(`)p Fx(\()p Fv(b)p Fx(\()p Fv(i)p Fx(\)\))p Fs(\000)p Fx(1)1833 1646 y Fw(k)1876 1658 y Fv(j)1911 1646 y Fy(.)605 1781 y Fi(Lemma)36 b Fy(3.2)p Fi(.)43 b Fh(L)l(et)34 b Fw(E)1305 1793 y Fv(j)1371 1781 y Fy(:)d(1)f Fr(\024)g Fw(j)36 b Fr(\024)31 b Fw(n)21 b Fy(+)g(1)34 b Fh(and)g Fw(k)2199 1793 y Fv(j)2265 1781 y Fy(:)d(1)f Fr(\024)h Fw(j)k Fr(\024)c Fw(n)j Fh(b)l(e)g(two)g (se)l(quenc)l(es)g(of)456 1881 y(numb)l(ers,)29 b(then)580 1989 y Fp(Z)627 2178 y Fx(0)p Fs(\024)p Fv(t)737 2186 y Fn(n)777 2178 y Fs(\024)p Fv(:::)o Fs(\024)p Fv(t)965 2186 y Fo(1)998 2178 y Fs(\024)p Fv(t)1093 2102 y Fw(d)p Fz(t)1224 1998 y Fv(n)1191 2023 y Fp(Y)1187 2200 y Fv(j)s Fx(=1)1316 2102 y Fw(e)1355 2068 y Fs(\000)p Fv(it)1455 2076 y Fn(j)1486 2068 y Fx(\()p Fv(E)1561 2076 y Fn(j)r Fo(+1)1662 2068 y Fs(\000)p Fv(E)1763 2076 y Fn(j)1794 2068 y Fx(+)p Fv(i\025k)1942 2076 y Fn(j)1974 2068 y Fx(\))2027 2102 y Fy(=)775 2344 y Fp(X)746 2522 y Fv(G)p Fs(2G)883 2530 y Fn(n)937 2423 y Fw(\033)s Fy(\()p Fw(G)p Fy(\))1130 2256 y Fp(0)1130 2405 y(@)1241 2319 y Fv(n)1209 2344 y Fp(Y)1204 2521 y Fv(j)s Fx(=1)1734 2367 y Fw(i)p 1343 2404 811 4 v 1343 2480 a(E)1404 2495 y Fv(p)p Fx(\()p Fv(j)s Fx(\))1544 2480 y Fr(\000)18 b Fw(E)1688 2492 y Fv(j)1741 2480 y Fy(+)h Fw(i\025k)s Fy(\()p Fw(j)5 b Fy(;)14 b Fw(G)p Fy(\))2163 2256 y Fp(1)2163 2405 y(A)2250 2331 y(\020)2299 2423 y Fw(e)2338 2389 y Fs(\000)p Fv(it)p Fx(\()p Fv(E)2513 2400 y Fn(p)p Fo(\(1\))2621 2389 y Fs(\000)p Fv(E)2722 2397 y Fo(1)2754 2389 y Fx(+)p Fv(i\025k)q Fx(\(1;)p Fv(G)p Fx(\)\))3109 2423 y Fr(\000)k Fy(1)3234 2331 y Fp(\021)3296 2423 y Fw(:)-2863 b Fy(\(8\))605 2703 y Fi(Pr)n(oof.)41 b Fy(W)-7 b(e)19 b(can)g(easily)f(c)n(hec)n(k)g (this)h(b)n(y)f(induction.)35 b(The)19 b(form)n(ula)f(is)g(ob)n (viously)g(correct)456 2802 y(for)31 b Fw(n)f Fy(=)f(1.)49 b(Assuming)31 b(that)h(the)g(form)n(ula)f(is)h(true)f(for)h Fw(n)21 b Fr(\000)f Fy(1,)33 b(w)n(e)e(ma)n(y)g(write)h(the)g Fw(n)f Fy(fold)456 2902 y(in)n(tegral)26 b(as)835 3078 y Fp(X)770 3256 y Fv(G)p Fs(2G)907 3264 y Fn(n)p Fg(\000)p Fo(1)1034 3157 y Fw(\033)s Fy(\()p Fw(G)p Fy(\))1227 2990 y Fp(0)1227 3139 y(@)1338 3053 y Fv(n)1305 3078 y Fp(Y)1301 3255 y Fv(j)s Fx(=2)1830 3100 y Fw(i)p 1439 3137 V 1439 3213 a(E)1500 3228 y Fv(p)p Fx(\()p Fv(j)s Fx(\))1640 3213 y Fr(\000)18 b Fw(E)1784 3225 y Fv(j)1838 3213 y Fy(+)g Fw(i\025k)s Fy(\()p Fw(j)5 b Fy(;)14 b Fw(G)p Fy(\))2259 2990 y Fp(1)2259 3139 y(A)936 3447 y Fr(\002)1015 3334 y Fp(Z)1097 3354 y Fv(t)1060 3522 y Fx(0)1140 3447 y Fw(dt)1213 3459 y Fx(1)1265 3355 y Fp(\020)1314 3447 y Fw(e)1353 3412 y Fs(\000)p Fv(it)1453 3420 y Fo(1)1486 3412 y Fx(\()p Fv(E)1561 3423 y Fn(p)p Fo(\(2\))1668 3412 y Fs(\000)p Fv(E)1769 3420 y Fo(1)1801 3412 y Fx(+)p Fv(i\025k)q Fx(\(2;)p Fv(G)p Fx(\)+)p Fv(i\025k)2254 3420 y Fo(1)2289 3412 y Fx(\))2338 3447 y Fr(\000)k Fw(e)2460 3412 y Fs(\000)p Fv(it)2560 3420 y Fo(1)2592 3412 y Fx(\()p Fv(E)2667 3420 y Fo(2)2699 3412 y Fs(\000)p Fv(E)2800 3420 y Fo(1)2832 3412 y Fx(+)p Fv(i\025k)2980 3420 y Fo(1)3014 3412 y Fx(\))3044 3355 y Fp(\021)3107 3447 y Fw(:)456 3653 y Fy(The)h(v)n(ertex)f(set)g(of)h(the)h(graphs)d Fw(G)24 b Fr(2)f(G)1677 3665 y Fv(n)p Fs(\000)p Fx(1)1827 3653 y Fy(is)18 b(here)h Fr(f)p Fw(n)q Fy(+)q(1)p Fw(;)14 b(:)g(:)g(:)e(;)i Fy(2)p Fr(g)p Fy(.)33 b(No)n(w,)20 b(in)g(the)f(\014rst)g(term)g(of)456 3752 y(the)24 b(in)n(tegrand)f(w)n (e)h(ha)n(v)n(e)f(that)i Fw(p)p Fy(\(2;)14 b Fw(G)p Fy(\))23 b(=)f Fw(p)p Fy(\(1;)14 b Fw(H)7 b Fy(\))p Fw(;)14 b(k)s Fy(\(1;)g Fw(H)7 b Fy(\))23 b(=)f Fw(k)s Fy(\(2;)14 b Fw(G)p Fy(\))e(+)g Fw(k)2866 3764 y Fx(1)2903 3752 y Fw(;)i(\033)s Fy(\()p Fw(H)7 b Fy(\))23 b(=)g Fw(\033)s Fy(\()p Fw(G)p Fy(\),)456 3852 y(where)30 b Fw(H)39 b Fy(is)31 b(the)h(graph)e(on)h Fr(f)p Fw(n)20 b Fy(+)h(1)p Fw(;)14 b(:)g(:)g(:)f(;)h Fy(1)p Fr(g)30 b Fy(whic)n(h)i(agrees)d(on)i Fw(n)21 b Fy(+)g(1)p Fw(;)14 b(:)g(:)g(:)f(;)h Fy(2)31 b(with)h Fw(G)f Fy(plus)456 3952 y(a)e(b)r(ond)i(connecting)f(1)g(with) g(the)h(same)f(site)g(as)g(2.)44 b(In)31 b(the)f(second)g(term)g(w)n(e) g(ha)n(v)n(e)f(a)h(b)r(ond)456 4051 y(b)r(et)n(w)n(een)24 b(1)g(and)h(2,)g(and)f(b)n(y)h(our)e(rules)h(hence)h(a)f(c)n(hange)g (of)g(the)h(sign.)36 b(W)-7 b(e)25 b(can)f(com)n(bine)g(this)456 4151 y(in)n(to)j(a)g(sum)h(o)n(v)n(er)e(all)h(graphs)f Fw(H)k Fr(2)23 b(G)1656 4163 y Fv(n)1702 4151 y Fy(,)28 b(whic)n(h)f(v)n(eri\014es)g(the)h(form)n(ula.)668 b Ff(\003)605 4336 y Fy(In)32 b(the)g(follo)n(wing)e(theorem)h(w)n(e)g (iden)n(tify)h(the)g(leading)f(term)h(in)f(the)h(limit)h(when)e Fw(\025)f Fr(!)456 4435 y Fy(0,)39 b(and)f Fw(t)h Fr(!)h(1)p Fy(,)g(suc)n(h)e(that)f Fw(\034)50 b Fy(=)39 b Fw(\025t)f Fy(is)g(constan)n(t.)66 b(Here,)39 b Fw(\034)48 b Fy(can)37 b(b)r(e)h(in)n(terpreted)f(as)g(a)456 4535 y(macroscopic)19 b(time)i(scale.)34 b(W)-7 b(e)21 b(shall)g(use)g(the)g(sp)r(ectral)g (decomp)r(osition)f(of)h(the)h(Hamiltonian)456 4737 y(\(9\))1027 b Fw(H)29 b Fy(=)1775 4624 y Fp(Z)1858 4644 y Fs(1)1821 4812 y Fx(0)1942 4737 y Fw(E)19 b(dP)12 b Fy(\()p Fw(E)5 b Fy(\))28 b Fw(;)456 4938 y Fy(suc)n(h)f(that)h(for)f(an)n(y)g Fw(\036)c Fr(2)h(H)q Fy(,)j(w)n(e)g(ha)n(v)n(e)1610 5140 y Fw(\036)c Fy(=)1770 5027 y Fp(Z)1853 5047 y Fs(1)1816 5216 y Fx(0)1951 5140 y Fw(dP)12 b Fy(\()p Fw(E)5 b Fy(\))p Fw(\036)29 b(:)p eop %%Page: 7 7 7 6 bop 820 251 a Fx(D)n(YNAMICS)29 b(OF)g(INTERF)-7 b(A)n(CES)29 b(IN)g(THE)g(FERR)n(OMA)n(GNETIC)h(XXZ)f(CHAIN)332 b(7)605 450 y Fi(Theorem)34 b Fy(3.3)f(\(Leading)d(dynamics\))p Fi(.)42 b Fh(L)l(et)31 b Fw(\036)d Fr(2)f(H)q Fh(,)33 b(and)f(let)g Fw(V)51 b Fh(satisfy)33 b(Assumption)456 550 y(\(2.1\).)40 b(Then,)31 b(for)g(al)t(l)g Fw(\016)s Fh(,)f Fy(1)23 b Fw(>)g(\016)j(>)d Fy(0)p Fh(,)30 b(ther)l(e)g(exists)g (a)g Fw(\025)2243 562 y Fx(0)2311 550 y Fh(dep)l(ending)h(on)f Fr(k)p Fw(V)18 b Fr(k)p Fw(;)c(\034)5 b(;)14 b Fr(k)p Fw(\036)p Fr(k)29 b Fh(and)h Fw(\016)456 649 y Fh(such)f(that)h(for)h Fy(0)22 b Fw(<)h(\025)g(<)g(\025)1306 661 y Fx(0)456 834 y Fy(\(10\))732 714 y Fp(\015)732 764 y(\015)732 813 y(\015)732 863 y(\015)779 834 y Fw(e)818 800 y Fv(i\025)880 775 y Fg(\000)p Fo(1)957 800 y Fv(\034)7 b(H)1058 834 y Fw(U)1124 800 y Fv(\025)1167 834 y Fy(\()p Fw(\025)1247 800 y Fs(\000)p Fx(1)1337 834 y Fw(\034)i Fy(\))p Fw(\036)20 b Fr(\000)1566 721 y Fp(Z)1649 742 y Fs(1)1612 910 y Fx(0)1733 834 y Fq(T)1803 742 y Fp(\020)1852 834 y Fw(e)1891 800 y Fs(\000)p Fv(iP)9 b Fx(\()p Fv(E)s Fx(\))2132 753 y Fj(R)2177 774 y Fn(\034)2164 822 y Fo(0)2226 800 y Fv(dt)i(V)j Fx(\()p Fv(t)p Fx(\))p Fv(P)9 b Fx(\()p Fv(E)s Fx(\))2586 742 y Fp(\021)2663 834 y Fw(dP)j Fy(\()p Fw(E)5 b Fy(\))p Fw(\036)2950 714 y Fp(\015)2950 764 y(\015)2950 813 y(\015)2950 863 y(\015)3021 834 y Fw(<)22 b(\025)3156 800 y Fx(1)p Fs(\000)p Fv(\016)3292 834 y Fw(:)456 1024 y Fq(T)28 b Fh(me)l(ans)i(time)f(or)l(dering,)i(i.e.,)h Fq(T)p Fy(\()p Fw(V)18 b Fy(\()p Fw(s)p Fy(\))p Fw(V)h Fy(\()p Fw(t)p Fy(\)\))24 b(=)f Fw(V)c Fy(\()p Fw(s)p Fy(\))p Fw(V)g Fy(\()p Fw(t)p Fy(\))30 b Fh(if)g Fw(s)23 b(>)g(t)p Fh(,)30 b(and)g(zer)l(o)g(otherwise.)605 1181 y Fi(Pr)n(oof.)41 b Fy(W)-7 b(e)29 b(ma)n(y)g(assume)f(that)i Fr(k)p Fw(\036)p Fr(k)25 b Fy(=)g(1.)41 b(Let)29 b Fw(N)38 b Fy(b)r(e)30 b(an)f(in)n(teger)f(whic)n(h)h(w)n(e)g(c)n(ho)r(ose)456 1280 y(later.)36 b(W)-7 b(e)28 b(ma)n(y)f(write)g(\()p Fw(t)c Fy(=)g Fw(\025)1438 1250 y Fs(\000)p Fx(1)1528 1280 y Fw(\034)9 b Fy(\))951 1349 y Fp(\015)951 1399 y(\015)951 1449 y(\015)951 1499 y(\015)997 1470 y Fw(e)1036 1435 y Fv(itH)1147 1470 y Fw(U)1213 1435 y Fv(\025)1257 1470 y Fy(\()p Fw(\025)1337 1435 y Fs(\000)p Fx(1)1427 1470 y Fw(\034)g Fy(\))p Fw(\036)19 b Fr(\000)1655 1357 y Fp(Z)1738 1377 y Fs(1)1702 1545 y Fx(0)1823 1470 y Fq(T)1893 1378 y Fp(\020)1941 1470 y Fw(e)1980 1435 y Fs(\000)p Fv(iP)9 b Fx(\()p Fv(E)s Fx(\))2222 1389 y Fj(R)2267 1409 y Fn(\034)2254 1457 y Fo(0)2315 1435 y Fv(dt)i(V)k Fx(\()p Fv(t)p Fx(\))p Fv(P)9 b Fx(\()p Fv(E)s Fx(\))2676 1378 y Fp(\021)2753 1470 y Fw(dP)j Fy(\()p Fw(E)5 b Fy(\))p Fw(\036)3054 1349 y Fp(\015)3054 1399 y(\015)3054 1449 y(\015)3054 1499 y(\015)1117 1648 y Fr(\024)23 b(j)p Fz(E)1291 1660 y Fv(N)1354 1648 y Fr(j)18 b Fy(+)g Fr(j)p Fy(\006)1561 1660 y Fv(N)1642 1648 y Fr(\000)h Fy(\006)1786 1614 y Fs(0)1786 1669 y Fv(N)1848 1648 y Fr(j)g Fy(+)f Fr(j)p Fz(E)2059 1614 y Fs(0)2059 1669 y Fv(N)2122 1648 y Fr(j)c Fw(:)456 1788 y Fy(Here,)30 b Fz(E)741 1800 y Fv(N)834 1788 y Fy(is)f(the)i(error)d(term)i(from)f (equation)h(\(7\),)g(and)g Fz(E)2398 1758 y Fs(0)2398 1811 y Fv(N)2491 1788 y Fy(the)g(remainder)f(term)h(of)g(the)456 1915 y(p)r(o)n(w)n(er)20 b(series)i(for)f(the)i(exp)r(onen)n(tial)e (function)i(in)f Fq(T)2089 1823 y Fp(\020)2138 1915 y Fw(e)2177 1884 y Fs(\000)p Fv(iP)9 b Fx(\()p Fv(E)s Fx(\))2418 1837 y Fj(R)2463 1858 y Fn(\034)2451 1906 y Fo(0)2512 1884 y Fv(dt)i(V)j Fx(\()p Fv(t)p Fx(\))p Fv(P)9 b Fx(\()p Fv(E)s Fx(\))2872 1823 y Fp(\021)2950 1915 y Fw(\036)22 b Fy(to)g(the)h(order)456 2039 y Fw(N)k Fr(\000)18 b Fy(1,)27 b(whereas)f(\006)1099 2051 y Fv(N)1190 2039 y Fy(and)h(\006)1411 2009 y Fs(0)1411 2062 y Fv(N)1502 2039 y Fy(refers)g(to)g(their)h(\014nite)g(sums.)605 2139 y(First)21 b(w)n(e)h(tak)n(e)f(care)f(of)i(the)f(error)f(terms)h Fz(E)1972 2151 y Fv(N)2057 2139 y Fy(and)g Fz(E)2275 2109 y Fs(0)2275 2162 y Fv(N)2359 2139 y Fy(b)n(y)h(c)n(ho)r(osing)e Fw(N)32 b Fy(=)22 b Fw(N)9 b Fy(\()p Fw(\025)p Fy(;)14 b Fr(k)p Fw(V)19 b Fr(k)p Fw(;)14 b(\034)9 b Fy(\))456 2239 y(suc)n(h)27 b(that)1572 2322 y(1)p 1518 2359 150 4 v 1518 2435 a Fr(k)p Fw(V)19 b Fr(k)1688 2322 y Fy(\()p Fw(\034)24 b Fr(k)p Fw(V)18 b Fr(k)p Fy(\))1962 2292 y Fv(N)p 1688 2359 337 4 v 1807 2435 a Fw(N)9 b Fy(!)2058 2378 y Fw(<)2156 2345 y Fx(1)p 2156 2359 34 4 v 2156 2407 a(3)2199 2378 y Fw(\025)2247 2344 y Fx(1)p Fs(\000)p Fv(\016)2369 2378 y Fw(:)456 2549 y Fy(This)23 b(can)g(b)r(e)h (accomplished)f(with)h Fw(N)32 b Fy(=)23 b Fr(\000)p Fw(C)c Fy(ln)14 b Fw(\025)24 b Fy(and)g(the)g(constan)n(t)e Fw(C)30 b Fy(dep)r(ending)24 b(on)f Fr(k)p Fw(V)c Fr(k)456 2649 y Fy(and)27 b Fw(\034)9 b Fy(.)605 2748 y(Second,)37 b(w)n(e)f(in)n(v)n(estigate)e(the)i(limit)g(\()p Fw(\025)h Fr(!)f Fy(0)p Fw(;)14 b(t)36 b Fr(!)g(1)p Fw(;)14 b(\025t)37 b Fr(!)f Fw(\034)9 b Fy(\))37 b(for)e(the)h Fw(n)p Fy(-th)g(term)456 2848 y(in)29 b(the)h(\014nite)g(Dyson)f(series.)41 b(By)30 b(inserting)e(the)i(sp)r(ectral)f(resolution)f(for)h Fw(H)37 b Fy(\(9\),)30 b(the)f Fw(n)p Fy(-th)456 2948 y(term)e(in)h Fw(e)790 2917 y Fv(itH)901 2948 y Fw(U)967 2917 y Fv(\025)1010 2948 y Fy(\()p Fw(t)p Fy(\))p Fw(\036)h Fy(equals)456 3187 y(\(11\))46 b(\()p Fr(\000)p Fw(i\025)p Fy(\))856 3153 y Fv(n)915 3074 y Fp(Z)961 3262 y Fx([0)p Fv(;t)p Fx(])1077 3246 y Fn(n)1136 3187 y Fw(d)p Fz(t)1230 3074 y Fp(Z)1276 3262 y Fm(R)1323 3236 y Fn(n)p Fo(+1)1323 3281 y(+)1448 3187 y Fw(d)p Fz(E)1554 3199 y Fv(n)1649 3083 y(n)1617 3108 y Fp(Y)1613 3285 y Fv(j)s Fx(=1)1741 3187 y Fw(e)1780 3153 y Fs(\000)p Fv(it)1880 3161 y Fn(j)1911 3153 y Fx(\()p Fv(E)1986 3161 y Fn(j)r Fo(+1)2088 3153 y Fs(\000)p Fv(E)2189 3161 y Fn(j)2220 3153 y Fx(\))2250 3187 y Fq(T)2320 3020 y Fp(0)2320 3169 y(@)2429 3083 y Fv(n)2396 3108 y Fp(Y)2392 3285 y Fv(j)s Fx(=1)2520 3187 y Fw(P)12 b Fy(\()p Fw(E)2678 3199 y Fv(j)2714 3187 y Fy(\))p Fw(V)19 b Fy(\()p Fw(\025t)2923 3199 y Fv(j)2959 3187 y Fy(\))2991 3020 y Fp(1)2991 3169 y(A)3077 3187 y Fw(dE)3181 3199 y Fv(n)p Fx(+1)3311 3187 y Fw(\036)14 b(:)456 3422 y Fy(In)29 b(order)f(to)h(p)r(erform)g(the)h Fw(t)g Fy(in)n(tegration)e(w)n(e)h(write)g Fw(V)48 b Fy(as)29 b(its)g(in)n(v)n(erse)f(Laplace)h(transform.)456 3521 y(Let)e Fw(c)640 3533 y Fx(1)701 3521 y Fw(<)22 b(c)824 3533 y Fx(2)884 3521 y Fw(<)h Fr(\001)14 b(\001)g(\001)23 b Fw(<)g(c)1216 3533 y Fv(n)1284 3521 y Fw(<)f Fy(0,)28 b(then)g(w)n(e)f(obtain)627 3717 y(\()p Fr(\000)p Fw(i\025)p Fy(\))833 3682 y Fv(n)983 3660 y Fy(1)p 888 3697 231 4 v 888 3774 a(\(2)p Fw(\031)s(i)p Fy(\))1073 3750 y Fv(n)1142 3604 y Fp(Z)1225 3624 y Fv(c)1255 3632 y Fo(1)1287 3624 y Fx(+)p Fv(i)p Fs(1)1189 3792 y Fv(c)1219 3800 y Fo(1)1251 3792 y Fs(\000)p Fv(i)p Fs(1)1446 3717 y Fr(\001)14 b(\001)g(\001)1557 3604 y Fp(Z)1640 3624 y Fv(c)1670 3632 y Fn(n)1710 3624 y Fx(+)p Fv(i)p Fs(1)1603 3792 y Fv(c)1633 3800 y Fn(n)1673 3792 y Fs(\000)p Fv(i)p Fs(1)1869 3717 y Fw(d)p Fz(k)1976 3604 y Fp(Z)2022 3792 y Fm(R)2069 3765 y Fn(n)p Fo(+1)2069 3811 y(+)2194 3717 y Fw(d)p Fz(E)2300 3729 y Fv(n)793 4022 y Fr(\002)885 3909 y Fp(Z)931 4097 y Fx([0)p Fv(;t)p Fx(])1047 4081 y Fn(n)1106 4022 y Fw(d)p Fz(t)1237 3918 y Fv(n)1204 3943 y Fp(Y)1200 4119 y Fv(j)s Fx(=1)1329 4022 y Fw(e)1368 3987 y Fs(\000)p Fv(it)1468 3995 y Fn(j)1499 3987 y Fx(\()p Fv(E)1574 3995 y Fn(j)r Fo(+1)1675 3987 y Fs(\000)p Fv(E)1776 3995 y Fn(j)1807 3987 y Fx(+)p Fv(i\025k)1955 3995 y Fn(j)1987 3987 y Fx(\))2031 4022 y Fq(T)2101 3855 y Fp(0)2101 4004 y(@)2210 3918 y Fv(n)2177 3943 y Fp(Y)2173 4119 y Fv(j)s Fx(=1)2302 4022 y Fw(P)e Fy(\()p Fw(E)2460 4034 y Fv(j)2495 4022 y Fy(\))2540 4001 y(^)2527 4022 y Fw(V)19 b Fy(\()p Fw(k)2669 4034 y Fv(j)2705 4022 y Fy(\))2737 3855 y Fp(1)2737 4004 y(A)2823 4022 y Fw(dP)12 b Fy(\()p Fw(E)3024 4034 y Fv(n)p Fx(+1)3154 4022 y Fy(\))p Fw(\036)i(:)456 4261 y Fy(By)27 b(using)g(form)n(ula)g(\(8\))h(w)n(e)f(can)g(do)g(the)h Fw(t)g Fy(in)n(tegration,)e(and)i(obtain)696 4411 y(1)p 602 4448 V 602 4524 a(\(2)p Fw(\031)s(i)p Fy(\))787 4500 y Fv(n)856 4354 y Fp(Z)939 4375 y Fv(c)969 4383 y Fo(1)1001 4375 y Fx(+)p Fv(i)p Fs(1)902 4543 y Fv(c)932 4551 y Fo(1)964 4543 y Fs(\000)p Fv(i)p Fs(1)1160 4467 y Fr(\001)14 b(\001)g(\001)1270 4354 y Fp(Z)1353 4375 y Fv(c)1383 4383 y Fn(n)1424 4375 y Fx(+)p Fv(i)p Fs(1)1316 4543 y Fv(c)1346 4551 y Fn(n)1387 4543 y Fs(\000)p Fv(i)p Fs(1)1582 4467 y Fw(d)p Fz(k)1689 4354 y Fp(Z)1736 4543 y Fm(R)1783 4516 y Fn(n)p Fo(+1)1783 4562 y(+)1908 4467 y Fw(d)p Fz(E)2028 4354 y Fp(Z)2073 4543 y Fx([0)p Fv(;t)p Fx(])2189 4526 y Fn(n)2248 4467 y Fw(d)p Fz(t)2379 4363 y Fv(n)2346 4388 y Fp(Y)2342 4565 y Fv(j)s Fx(=1)2471 4467 y Fw(e)2510 4433 y Fs(\000)p Fv(it)2610 4441 y Fn(j)2641 4433 y Fx(\()p Fv(E)2716 4441 y Fn(j)r Fo(+1)2817 4433 y Fs(\000)p Fv(E)2918 4441 y Fn(j)2949 4433 y Fx(+)p Fv(i\025k)3097 4441 y Fn(j)3129 4433 y Fx(\))787 4709 y Fp(X)758 4887 y Fv(G)p Fs(2G)895 4895 y Fn(n)949 4788 y Fw(\033)s Fy(\()p Fw(G)p Fy(\))1142 4621 y Fp(0)1142 4771 y(@)1253 4684 y Fv(n)1220 4709 y Fp(Y)1216 4886 y Fv(j)s Fx(=1)1735 4732 y Fw(\025)p 1354 4769 811 4 v 1354 4845 a(E)1415 4860 y Fv(p)p Fx(\()p Fv(j)s Fx(\))1555 4845 y Fr(\000)k Fw(E)1699 4857 y Fv(j)1753 4845 y Fy(+)g Fw(i\025k)s Fy(\()p Fw(j)5 b Fy(;)14 b Fw(G)p Fy(\))2175 4621 y Fp(1)2175 4771 y(A)2261 4696 y(\020)2311 4788 y Fw(e)2350 4754 y Fs(\000)p Fv(it)p Fx(\()p Fv(E)2525 4765 y Fn(p)p Fo(\(1\))2633 4754 y Fs(\000)p Fv(E)2734 4762 y Fo(1)2766 4754 y Fx(+)p Fv(i\025k)q Fx(\(1;)p Fv(G)p Fx(\)\))3120 4788 y Fr(\000)k Fy(1)3245 4696 y Fp(\021)758 5120 y Fr(\002)c Fq(T)907 4953 y Fp(0)907 5103 y(@)1015 5016 y Fv(n)983 5041 y Fp(Y)978 5218 y Fv(j)s Fx(=1)1107 5120 y Fw(P)e Fy(\()p Fw(E)1265 5132 y Fv(j)1300 5120 y Fy(\))1345 5099 y(^)1332 5120 y Fw(V)20 b Fy(\()p Fw(k)1475 5132 y Fv(j)1510 5120 y Fy(\))1542 4953 y Fp(1)1542 5103 y(A)1629 5120 y Fw(dP)12 b Fy(\()p Fw(E)1830 5132 y Fv(n)p Fx(+1)1960 5120 y Fy(\))p Fw(\036)i(:)p eop %%Page: 8 8 8 7 bop 456 251 a Fx(8)247 b(BR)n(UNO)23 b(NA)n(CHTER)n(GAELE,)g(W)n (OLF)n(GANG)g(L)g(SPITZER,)f(AND)g(SHANNON)g(ST)-5 b(ARR)456 450 y Fy(Here,)26 b(it)i(is)f(con)n(v)n(enien)n(t)f(to)h(ha)n(v)n(e)e Fw(c)1577 462 y Fv(i)1605 450 y Fy('s)i(suc)n(h)f(that)i(there)e(are)g (no)h(zeros)f(in)h(the)g(denominator.)456 550 y(W)-7 b(e)28 b(observ)n(e)e(here)i(the)g(crucial)f(fact)i(that)f(in)g(order)f (to)h(\014nd)g(a)g(term)g(of)g(the)g(order)f(1,)g(all)h Fw(E)3409 562 y Fv(j)456 649 y Fy(ha)n(v)n(e)h(to)h(b)r(e)g(equal.)44 b(W)-7 b(e)30 b(ma)n(y)g(no)n(w)f(go)h(bac)n(k)f(to)h(\(11\),)g(c)n (hange)f(co)r(ordinates)g(and)h(conclude)456 749 y(that)d(in)h(this)g (limit)h(w)n(e)e(get)1064 1002 y(\()p Fr(\000)p Fw(i\025)p Fy(\))1270 968 y Fv(n)1329 889 y Fp(Z)1412 910 y Fs(1)1375 1078 y Fx(0)1496 889 y Fp(Z)1542 1078 y Fx([0)p Fv(;t)p Fx(])1658 1061 y Fn(n)1717 1002 y Fw(d)p Fz(t)14 b Fq(T)1881 836 y Fp(0)1881 985 y(@)1989 899 y Fv(n)1957 924 y Fp(Y)1953 1100 y Fv(j)s Fx(=1)2081 1002 y Fw(P)e Fy(\()p Fw(E)5 b Fy(\))p Fw(V)20 b Fy(\()p Fw(\025t)2454 1014 y Fv(j)2489 1002 y Fy(\))2521 836 y Fp(1)2521 985 y(A)2608 1002 y Fw(dP)12 b Fy(\()p Fw(E)5 b Fy(\))p Fw(\036)1147 1289 y Fy(=)82 b(\()p Fr(\000)p Fw(i)p Fy(\))1452 1255 y Fv(n)1511 1176 y Fp(Z)1594 1197 y Fs(1)1557 1365 y Fx(0)1692 1289 y Fq(T)1762 1172 y Fp(\022)1823 1176 y(Z)1906 1197 y Fv(\034)1869 1365 y Fx(0)1961 1289 y Fw(dtP)12 b Fy(\()p Fw(E)5 b Fy(\))p Fw(V)19 b Fy(\()p Fw(t)p Fy(\))2390 1172 y Fp(\023)2452 1189 y Fv(n)2511 1289 y Fw(dP)12 b Fy(\()p Fw(E)5 b Fy(\))p Fw(\036)14 b(:)605 1493 y Fy(Finally)-7 b(,)27 b(w)n(e)g(ha)n(v)n(e)f(to)g(b)r(ound)i(the)f(sum)g (of)g(all)g(other)f(terms)h(in)g(the)g(\014nite)h(Dyson)f(series)456 1592 y(whic)n(h)i(individually)g(tend)h(to)f(0)f(as)h Fw(\025)d Fr(!)g Fy(0,)j(and)g(are)f(con)n(tained)h(in)g Fr(j)p Fy(\006)2760 1604 y Fv(N)2842 1592 y Fr(\000)18 b Fy(\006)2985 1562 y Fs(0)2985 1615 y Fv(N)3048 1592 y Fr(j)p Fy(.)41 b(As)30 b(there)456 1692 y(are)c(2)636 1662 y Fv(n)p Fs(\000)p Fx(1)794 1692 y Fy(terms)h(in)h(form)n(ula)e (\(8\),)i(w)n(e)f(immediately)h(obtain)f(the)h(b)r(ound)1376 1932 y Fr(j)p Fy(\006)1459 1944 y Fv(N)1541 1932 y Fr(\000)18 b Fy(\006)1684 1897 y Fs(0)1684 1952 y Fv(N)1747 1932 y Fr(j)23 b(\024)1890 1876 y Fw(\025)p 1890 1913 49 4 v 1893 1989 a Fy(2)1963 1828 y Fv(N)6 b Fs(\000)p Fx(1)1974 1853 y Fp(X)1972 2029 y Fv(n)p Fx(=1)2130 1876 y Fy(\(2)p Fw(\034)j Fr(k)p Fw(V)19 b Fr(k)p Fy(\))2432 1845 y Fv(n)p 2130 1913 347 4 v 2267 1989 a Fw(n)p Fy(!)2501 1932 y Fw(;)456 2178 y Fy(whic)n(h)24 b(in)h(turn)g(is)f(b)r(ounded)h(b)n(y) 1505 2145 y Fx(1)p 1505 2159 34 4 v 1505 2206 a(3)1548 2178 y Fw(\025)1596 2148 y Fx(1)p Fs(\000)p Fv(\016)1742 2178 y Fy(if)g(w)n(e)f(c)n(ho)r(ose)g Fw(\025)f(<)g(\025)2402 2190 y Fx(0)2464 2178 y Fy(with)i Fw(\025)2698 2190 y Fx(0)2761 2178 y Fy(suc)n(h)f(that)h Fw(e)3161 2148 y Fx(2)p Fv(\034)7 b Fs(k)p Fv(V)14 b Fs(k)3380 2178 y Fw(<)466 2258 y Fx(2)p 466 2272 V 466 2319 a(3)509 2291 y Fw(\025)557 2255 y Fs(\000)p Fv(\016)557 2313 y Fx(0)645 2291 y Fy(.)2712 b Ff(\003)605 2474 y Fy(If)28 b Fw(H)7 b(\036)23 b Fy(=)g(0)k(is)h(a)f(kink)g(ground)g(state)g(of)h Fw(H)7 b Fy(,)27 b(then)h(the)g(leading)f(term)h(equals)456 2653 y(\(12\))899 b Fq(T)1573 2561 y Fp(\020)1622 2653 y Fw(e)1661 2619 y Fs(\000)p Fv(i)1747 2572 y Fj(R)1792 2592 y Fn(\034)1779 2641 y Fo(0)1840 2619 y Fv(P)9 b Fx(\(0\))p Fv(V)15 b Fx(\()p Fv(t)p Fx(\))p Fv(P)9 b Fx(\(0\))2247 2561 y Fp(\021)2311 2653 y Fw(\036)14 b(:)456 2832 y Fy(This)27 b(expression)f(will)i(b)r(e)g(further)g(analyzed)e (for)h(a)g(sp)r(eci\014c)h(c)n(hoice)f(of)h Fw(V)46 b Fy(in)28 b(Section)g(4.)605 2931 y(Note)j(that)f(so)g(far)g(w)n(e)g (did)h(not)g(use)f(an)n(y)g(sp)r(eci\014c)h(sp)r(ectral)f(prop)r (erties)f(of)i(the)g(Hamil-)456 3031 y(tonian,)c(and)g(th)n(us)h (Theorem)f(3.3)g(is)g(v)-5 b(alid,)28 b(for)f(instance,)g(for)g(an)n(y) g(v)-5 b(alue)28 b(of)f(the)h(spin.)605 3131 y(W)-7 b(e)25 b(kno)n(w)e(that)i Fw(\033)s Fy(\()p Fw(H)7 b Fy(\))24 b Fr(\032)e(f)p Fy(0)p Fr(g)12 b([)g Fy([1)g Fr(\000)g Fy(\001)1864 3100 y Fs(\000)p Fx(1)1952 3131 y Fw(;)i Fr(1)p Fy(\),)25 b(using)f(the)h(theorem)f(of)g([)p Fz(12)p Fy(])g(that)h(there)456 3230 y(is)37 b(a)g(gap)g(of)h(1)25 b Fr(\000)f Fy(\001)1125 3200 y Fs(\000)p Fx(1)1253 3230 y Fy(ab)r(o)n(v)n(e)36 b(the)i(in\014nite)g(v)n(olume)f(ground)g (states.)67 b(If)38 b(w)n(e)f(decomp)r(ose)456 3330 y(the)31 b(kink)g(Hamiltonian)g(on)g Fq(Z)25 b Fy(as)31 b(the)g(sum)h(of)f(a)g (kink)g(plus)g(a)g(droplet)g(Hamiltonian,)h(i.e.,)456 3429 y Fw(H)532 3394 y Fx(+)p Fs(\000)525 3454 y Fm(Z)672 3429 y Fy(=)i Fw(H)847 3394 y Fx(+)p Fs(\000)840 3458 y Fx(\()p Fs(\0001)p Fv(;y)r Fx(])1085 3429 y Fy(+)22 b Fw(H)1248 3394 y Fs(\000\000)1241 3458 y Fx(\()p Fv(y)r(;)p Fs(1)p Fx(\))1453 3429 y Fy(with)35 b Fw(y)i Fr(2)d Fq(Z)o Fy(,)c(then)35 b(w)n(e)f(get)g(appro)n(ximate)e(eigen)n(v)n(ectors)g(b) n(y)456 3543 y(tensoring)f(a)h(kink)g(ground)f(on)h(\()p Fr(\0001)p Fw(;)14 b(y)s Fy(])32 b(\(and)g(with)h(in)n(terface)f(far)g (to)g(the)g(left)h(of)g Fw(y)s Fy(\))f(with)456 3643 y(eigen)n(v)n(ectors)d(of)k Fw(H)1100 3608 y Fs(\000\000)1093 3672 y Fx(\()p Fv(y)r(;)p Fs(1)p Fx(\))1270 3643 y Fy(.)51 b(In)33 b(the)f(case)g Fw(y)h Fy(=)e Fr(\0001)h Fy(the)g(sp)r(ectrum)h (of)f(the)g(latter)g(has)g(b)r(een)456 3762 y(sho)n(wn)26 b(b)n(y)h(Babbitt)h(and)f(Gutkin)h([)p Fz(6)p Fy(])f(to)h(b)r(e)f Fr(f)p Fy(0)p Fr(g)17 b([)h Fy([1)g Fr(\000)f Fy(\001)2360 3732 y Fs(\000)p Fx(1)2449 3762 y Fw(;)d Fr(1)p Fy(\).)37 b(Their)27 b(analysis)f(follo)n(ws)456 3862 y(the)32 b(earlier)f(w)n(ork)g(b)n(y)h(Babbitt)h(and)f(Thomas)g([)p Fz(7)o Fy(])h(for)f(\001)f(=)g(1)g(pro)n(ving)g(the)i(completeness)456 3961 y(of)j(the)h(Bethe)f(ansatz)g(eigenfunctions)g(in)h(the)f (in\014nite)i(v)n(olume)d(limit)i(and)g(thereb)n(y)f(also)456 4061 y(sho)n(wing)26 b(that)i(the)g(sp)r(ectrum)g(is)f(absolutely)g (con)n(tin)n(uous.)605 4161 y(This)h(w)n(a)n(y)-7 b(,)27 b(one)g(obtains)h Fw(\033)s Fy(\()p Fw(H)7 b Fy(\))24 b(=)f Fr(f)p Fy(0)p Fr(g)17 b([)i Fy([1)g Fr(\000)f Fy(\001)2179 4130 y Fs(\000)p Fx(1)2268 4161 y Fw(;)c Fr(1)p Fy(\).)38 b(But)29 b(this)f(construction)f(do)r(es)456 4260 y(not)32 b(giv)n(e)g(the)h(nature)g(of)f(the)i(sp)r(ectrum)e(in)h(the)h(in)n (terv)-5 b(al)32 b(in)h([1)21 b Fr(\000)h Fy(\001)2716 4230 y Fs(\000)p Fx(1)2805 4260 y Fw(;)14 b Fr(1)p Fy(\).)53 b(W)-7 b(e)33 b(plan)g(to)456 4360 y(address)26 b(this)i(issue)f(in)h (the)g(future;)g(for)f(no)n(w)g(w)n(e)g(state)h(the)g(follo)n(wing)e(h) n(yp)r(othesis.)605 4489 y Fi(Hypothesis)k Fy(3.4)p Fi(.)39 b Fh(The)30 b(sp)l(e)l(ctrum)e(of)i(the)e(spin-)2211 4456 y Fx(1)p 2212 4470 V 2212 4517 a(2)2283 4489 y Fh(kink)i (Hamiltonian)f Fw(H)36 b Fh(with)29 b Fy(\001)23 b Fw(>)g Fy(1)456 4588 y Fh(has)31 b(no)f(other)h(p)l(oint)g(sp)l(e)l(ctrum)e (than)i(0,)g(and)g(the)g(r)l(est)e(is)i(absolutely)h(c)l(ontinuous,)e (e)l(qual)g(to)456 4688 y Fy([1)18 b Fr(\000)g Fy(\001)691 4658 y Fs(\000)p Fx(1)780 4688 y Fw(;)c Fr(1)p Fy(\))p Fh(.)605 4817 y Fy(By)24 b(assuming)f(this)h(information)f(w)n(e)g(can) h(iden)n(tify)g(the)g(\014rst)g(correction)e(to)i(the)g(leading)456 4917 y(term)34 b(giv)n(en)h(in)g(Theorem)f(3.3)g(for)g(ground)g (states,)i(and)f(w)n(e)g(sho)n(w)f(that)h(it)g(is)g Fr(O)r Fy(\()p Fw(\025)p Fy(\).)60 b(F)-7 b(or)456 5016 y(simplicit)n(y)g(,)24 b(w)n(e)f(state)g(the)h(result)e(only)h(for)g(time)h(indep)r(enden)n(t) g(\014elds)f(and)g(lea)n(v)n(e)f(the)i(general)456 5116 y(case)i(to)h(the)h(reader.)35 b(The)27 b(main)g(argumen)n(t)g(in)g (the)h(pro)r(of)e(b)r(elo)n(w,)h(ho)n(w)n(ev)n(er,)f(is)h(carried)e (out)456 5216 y(for)i(a)g(time)h(dep)r(enden)n(t)g(\014eld.)p eop %%Page: 9 9 9 8 bop 820 251 a Fx(D)n(YNAMICS)29 b(OF)g(INTERF)-7 b(A)n(CES)29 b(IN)g(THE)g(FERR)n(OMA)n(GNETIC)h(XXZ)f(CHAIN)332 b(9)605 450 y Fi(Theorem)38 b Fy(3.5)e(\(First)d(order)e(correction\))p Fi(.)43 b Fh(L)l(et)34 b Fw(\036)h Fh(b)l(e)g(a)g(\(normalize)l(d\))h (gr)l(ound)f(state)456 550 y(of)g(the)g(spin-)889 517 y Fx(1)p 889 531 34 4 v 889 578 a(2)967 550 y Fh(kink)g(Hamiltonian,)i Fw(V)53 b Fh(a)35 b(p)l(erturb)l(ation)g(as)g(in)g(Assumption)f (\(2.1\))i(and)f Fw(H)456 649 y Fh(satisfying)c(Hyp)l(othesis)g (\(3.4\).)40 b(Then,)749 777 y Fp(\015)749 827 y(\015)749 877 y(\015)749 926 y(\015)795 897 y Fw(e)834 863 y Fv(i\025)896 838 y Fg(\000)p Fo(1)974 863 y Fv(\034)7 b(H)1074 897 y Fw(e)1113 863 y Fs(\000)p Fv(i\025)1227 838 y Fg(\000)p Fo(1)1305 863 y Fv(\034)g Fx(\()p Fv(H)t Fx(+)p Fv(\025V)15 b Fx(\))1601 897 y Fw(\036)k Fr(\000)f Fw(e)1791 863 y Fs(\000)p Fv(i\034)7 b(P)i Fx(\(0\))p Fv(V)2097 897 y Fw(\036)19 b Fr(\000)f Fw(\025)2310 784 y Fp(Z)2393 805 y Fs(1)2356 973 y Fx(1)p Fs(\000)p Fx(\001)2496 956 y Fg(\000)p Fo(1)2601 841 y Fw(dE)p 2601 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y(2)1202 5216 y Fw(a)28 b Fy(and)f Fw(\015)h Fy(=)22 b Fw(B)1656 5228 y Fx(3)1694 5216 y Fy(.)p eop %%Page: 12 12 12 11 bop 456 251 a Fx(12)214 b(BR)n(UNO)23 b(NA)n(CHTER)n(GAELE,)g(W)n (OLF)n(GANG)g(L)g(SPITZER,)f(AND)g(SHANNON)g(ST)-5 b(ARR)605 450 y Fy(The)41 b(op)r(erator)f Fw(K)1209 462 y Fx(0)1287 450 y Fy(is)i(selfadjoin)n(t)f(with)h(dense)f(domain)g Fr(D)r Fy(\()p Fw(K)2719 462 y Fx(0)2756 450 y Fy(\))47 b(=)e Fr(f)p Fw(f)54 b Fr(2)47 b Fw(`)3219 420 y Fx(2)3255 450 y Fy(\()p Fq(Z)p Fy(\))40 b(:)456 550 y(\()p Fw(nf)9 b Fy(\()p Fw(n)p Fy(\)\))734 562 y Fv(n)802 550 y Fr(2)24 b Fw(`)916 520 y Fx(2)952 550 y Fy(\()p Fq(Z)p Fy(\))p Fr(g)p Fy(.)30 b(This)d(follo)n(ws)e(from)h(the)h(fact)g(that)f Fw(W)39 b Fy(is)26 b(selfadjoin)n(t)g(on)g Fr(D)r Fy(\()p Fw(K)3191 562 y Fx(0)3229 550 y Fy(\),)h(and)456 649 y(that)g(\001)h(is)g(b)r(ounded)g(and)f(selfadjoin)n(t.)605 749 y(As)33 b(for)g(the)g(con)n(tin)n(uous)f(case,)h(one)g(can)g 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y(Ho)n(w)n(ev)n(er,)f(as)h Fw(\015)g Fr(!)c Fy(0)921 4904 y Fw(m)994 4870 y Fx(3)994 4924 y Fv(\015)t Fx(=0)1121 4904 y Fy(\()p Fw(x;)14 b(t)p Fy(\))84 b(=)1542 4825 y Fp(X)1531 5003 y Fv(m)p Fs(2)p Fm(Z)1687 4904 y Fw(J)1741 4870 y Fx(2)1733 4924 y Fv(m)1796 4904 y Fy(\(2)p Fw(\013t)p Fy(\))14 b Fr(h)p Fw(m)p Fr(j)p Fw(S)2183 4870 y Fx(3)2178 4924 y Fv(x)2220 4904 y Fr(j)p Fw(m)p Fr(i)p Fw(;)-1915 b Fy(\(17\))921 5137 y Fw(m)994 5102 y Fx(1)994 5157 y Fv(\015)t Fx(=0)1121 5137 y Fy(\()p Fw(x;)14 b(t)p Fy(\))84 b(=)f Fw(b)1591 5058 y Fp(X)1581 5236 y Fv(m)p Fs(2)p Fm(Z)1736 5137 y Fw(J)1782 5149 y Fv(m)1845 5137 y Fy(\(2)p Fw(\013t)p Fy(\))p Fw(J)2080 5149 y Fv(m)p Fx(+1)2228 5137 y Fy(\(2)p Fw(\013t)p Fy(\))14 b Fr(h)p Fw(m)p Fr(j)p Fw(S)2615 5102 y Fx(+)2610 5157 y Fv(x)2670 5137 y Fr(j)p Fw(m)19 b Fr(\000)f Fy(1)p Fr(i)c Fw(:)-2523 b Fy(\(18\))p eop %%Page: 14 14 14 13 bop 456 251 a Fx(14)214 b(BR)n(UNO)23 b(NA)n(CHTER)n(GAELE,)g(W)n (OLF)n(GANG)g(L)g(SPITZER,)f(AND)g(SHANNON)g(ST)-5 b(ARR)456 450 y Fy(Next,)42 b(w)n(e)c(analyze)g Fw(m)1211 420 y Fx(3)1211 471 y Fv(\015)t Fx(=0)1338 450 y Fy(\()p Fw(x;)14 b(t)p Fy(\))40 b(in)f(more)f(detail.)71 b(A)39 b(v)n(ery)f(similar)g (situation)h(has)f(b)r(een)456 553 y(studied)33 b(b)n(y)g(An)n(tal)g (et)g(al)g([)p Fz(3)p Fy(])g(in)h(the)f(con)n(text)g(of)g(the)g(time)h (ev)n(olution)e(of)h(the)h(XX)f(mo)r(del.)456 652 y(W)-7 b(e)32 b(follo)n(w)e(their)i(analysis.)47 b(Although)32 b(w)n(e)f(ha)n(v)n(e)f(just)i(deriv)n(ed)f(explicit)h(form)n(ulas)e (for)h(the)456 752 y(pro\014les,)26 b(whic)n(h)h(can)g(b)r(e)h (analyzed)e(n)n(umerically)-7 b(,)27 b(w)n(e)f(prefer)h(to)g(ha)n(v)n (e)f(simpler)h(expressions.)456 851 y(As)22 b(w)n(e)g(shall)g(sho)n(w,) h(for)f(large)f Fw(x)h Fy(and)h Fw(t)p Fy(,)g Fw(m)1799 821 y Fx(3)1836 851 y Fy(\()p Fw(x;)14 b(t)p Fy(\))24 b(will)e(b)r(e)h(a)f(function)h(of)g(the)f(v)n(elo)r(cit)n(y)g Fw(v)k Fy(=)3374 819 y Fv(x)p 3374 833 38 4 v 3380 880 a(t)3421 851 y Fy(,)456 956 y(only)-7 b(.)36 b(T)-7 b(o)27 b(this)h(end)g(w)n(e)f(de\014ne)h(the)g(discrete)f(deriv)-5 b(ativ)n(e)27 b(of)h Fw(m)2463 926 y Fx(3)2500 956 y Fy(\()p Fw(x;)14 b(t)p Fy(\))28 b(as)1054 1145 y Fw(\036)1103 1110 y Fs(0)1103 1165 y Fv(x)1145 1145 y Fy(\()p Fw(v)s Fy(\))84 b(:=)e Fw(t)1536 1053 y Fp(h)1576 1145 y Fw(m)1649 1110 y Fx(3)1649 1165 y Fv(\015)t Fx(=0)1775 1145 y Fy(\()p Fw(x)19 b Fy(+)f(1)p Fw(;)c(t)p Fy(\))k Fr(\000)g Fw(m)2271 1110 y Fx(3)2271 1165 y Fv(\015)t Fx(=0)2398 1145 y Fy(\()p Fw(x;)c(t)p Fy(\))2576 1053 y Fp(i)2626 1181 y Fn(x)p 2626 1190 33 3 v 2631 1223 a(t)2669 1203 y Fx(=)p Fv(v)1347 1343 y Fy(=)1516 1311 y Fv(x)p 1516 1325 38 4 v 1517 1372 a(v)1589 1264 y Fp(X)1578 1443 y Fv(m)p Fs(2)p Fm(Z)1734 1343 y Fw(J)1788 1309 y Fx(2)1780 1364 y Fv(m)1843 1343 y Fy(\()1885 1311 y Fv(x\013)p 1885 1325 81 4 v 1908 1372 a(v)1976 1343 y Fy(\))g Fr(h)p Fy(0)p Fr(j)p Fw(S)2175 1309 y Fx(3)2170 1364 y Fv(x)p Fs(\000)p Fv(m)p Fs(\000)p Fx(1)2426 1343 y Fr(\000)k Fw(S)2565 1309 y Fx(3)2560 1364 y Fv(x)p Fs(\000)p Fv(m)2713 1343 y Fr(j)p Fy(0)p Fr(i)c Fw(;)456 1596 y Fy(and)36 b(study)g(the)g(limit)h(lim)1336 1608 y Fv(x)p Fs(!1)1524 1596 y Fw(\036)1573 1565 y Fs(0)1573 1616 y Fv(x)1616 1596 y Fy(\()p Fw(v)s Fy(\).)63 b(Note)36 b(that)g Fr(h)p Fy(0)p Fr(j)p Fw(S)2359 1565 y Fx(3)2354 1616 y Fv(x)p Fs(\000)p Fv(m)p Fs(\000)p Fx(1)2616 1596 y Fr(\000)23 b Fw(S)2760 1565 y Fx(3)2755 1616 y Fv(x)p Fs(\000)p Fv(m)2908 1596 y Fr(j)p Fy(0)p Fr(i)37 b(\025)g Fy(0)e(for)h(all)456 1695 y Fw(x)h Fy(and)g Fw(m)h Fy(b)r(ecause)e(the) i(pro\014le)e(of)i(the)f(kinks)g(is)g(a)g(decreasing)e(function.)67 b(If)37 b(w)n(e)g(de\014ne)456 1795 y Fw(p)p Fy(\()p Fw(m)p Fy(\))23 b(=)g Fr(h)p Fy(0)p Fr(j)p Fw(S)899 1765 y Fx(3)894 1815 y Fv(m)p Fs(\000)p Fx(1)1060 1795 y Fr(\000)18 b Fw(S)1199 1765 y Fx(3)1194 1815 y Fv(m)1257 1795 y Fr(j)p Fy(0)p Fr(i)p Fy(,)27 b(then)h(since)g(it)g(is)f(a)h (telescoping)e(sum)1095 1928 y Fs(1)1068 1953 y Fp(X)1014 2127 y Fv(m)p Fx(=)p Fs(\0001)1256 2032 y Fw(p)p Fy(\()p Fw(m)p Fy(\))d(=)61 b(lim)1546 2082 y Fv(m)p Fs(!1)1737 2032 y Fr(h)p Fy(0)p Fr(j)p Fw(S)1890 1998 y Fx(3)1885 2052 y Fv(m)1948 2032 y Fr(j)p Fy(0)p Fr(i)18 b(\000)82 b Fy(lim)2146 2082 y Fv(m)p Fs(!\0001)2389 2032 y Fr(h)p Fy(0)p Fr(j)p Fw(S)2542 1998 y Fx(3)2537 2052 y Fv(m)2600 2032 y Fr(j)p Fy(0)p Fr(i)23 b Fy(=)g(1)14 b Fw(;)456 2277 y Fy(i.e.,)36 b Fw(p)p Fy(\()p Fw(m)p Fy(\))f(is)f(a)g(probabilit) n(y)f(distribution.)58 b(All)35 b(momen)n(ts)f(of)g Fw(p)g Fy(are)g(\014nite)h(and)f(the)h(\014rst)456 2380 y(momen)n(t)30 b(of)g Fw(p)g Fy(is)1044 2347 y Fx(1)p 1044 2361 34 4 v 1044 2409 a(2)1087 2380 y Fy(.)45 b(In)30 b(fact)g Fw(p)p Fy(\()p Fw(m)p Fy(\))e Fr(\030)f Fw(e)1765 2350 y Fs(\000)p Fv(c)p Fs(j)p Fv(m)p Fs(j)1979 2380 y Fy(for)i(some)h(p)r (ositiv)n(e)g Fw(c)g Fy(dep)r(ending)h(on)e(\001)f Fw(>)f Fy(1)456 2480 y(\(see)g(Lemma)g(A.1.1\).)37 b(Th)n(us,)1359 2660 y Fw(\036)1408 2626 y Fs(0)1408 2681 y Fv(x)1451 2660 y Fy(\()p Fw(v)s Fy(\))24 b(=)e Fr(\000)1759 2581 y Fp(X)1748 2760 y Fv(m)p Fs(2)p Fm(Z)1914 2628 y Fv(x)p 1914 2642 38 4 v 1915 2689 a(v)1961 2660 y Fw(J)2015 2626 y Fx(2)2007 2681 y Fv(m)p Fx(+)p Fv(x)2159 2660 y Fy(\()2201 2628 y Fv(x\013)p 2201 2642 81 4 v 2224 2689 a(v)2292 2660 y Fy(\))p Fw(p)p Fy(\()p Fw(m)p Fy(\))14 b Fw(:)456 2913 y Fy(W)-7 b(e)41 b(used)g(the)h(fact)f(that)g Fw(J)1388 2925 y Fs(\000)p Fv(n)1485 2913 y Fy(\()p Fw(x)p Fy(\))1596 2882 y Fx(2)1680 2913 y Fy(=)k Fw(J)1836 2925 y Fv(n)1881 2913 y Fy(\()p Fr(\000)p Fw(x)p Fy(\))2057 2882 y Fx(2)2141 2913 y Fy(=)g Fw(J)2297 2925 y Fv(n)2342 2913 y Fy(\()p Fw(x)p Fy(\))2453 2882 y Fx(2)2491 2913 y Fy(.)78 b(One)40 b(ma)n(y)h(easily)f(b)r(ound)466 2980 y Fv(x)p 466 2994 38 4 v 467 3041 a(v)513 3012 y Fw(J)567 2982 y Fx(2)559 3033 y Fv(x)p Fs(\000)p Fv(m)712 3012 y Fy(\()754 2980 y Fv(x\013)p 754 2994 81 4 v 777 3041 a(v)845 3012 y Fy(\))33 b(uniformly)g(in)g Fw(x)h Fy(and)f Fw(m)p 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Fr(\000)1468 4567 y Fx(1)p 1467 4581 34 4 v 1467 4628 a(2)1511 4599 y Fy(.)51 b(This)33 b(b)r(eha)n(vior)e(is)i(not)f(surprising)g(for)g(it)h(just)g (means)f(a)456 4699 y(\014nite)c(sp)r(eed)g(of)f(propagation,)f(whic)n (h)h(has)g(to)h(b)r(e)g(less)f(than)h(2)p Fw(\013)p Fy(.)605 4798 y(Ho)n(w)n(ev)n(er)18 b(for)i(0)i Fr(\024)h Fw(v)j(<)d Fy(2)p Fw(\013)p Fy(,)f(the)e(b)r(eha)n(vior)f(of)h(the)h(Bessel)e (functions)h(c)n(hanges,)h(cf)f(\(9.3.3\),)456 4898 y([)p Fz(1)o Fy(]:)713 5072 y Fx(2)p Fv(x\013)p 713 5086 114 4 v 753 5134 a(v)837 5105 y Fw(J)891 5071 y Fx(2)883 5126 y Fv(m)p Fx(+)p Fv(x)1035 5105 y Fy(\()1077 5072 y Fx(2)p Fv(x\013)p 1077 5086 V 1116 5134 a(v)1201 5105 y Fy(\))j Fr(\030)1536 5049 y Fy(2)p 1354 5086 405 4 v 1354 5206 a Fw(\031)1404 5103 y Fp(q)p 1487 5103 272 4 v 103 x Fy(1)18 b Fr(\000)1661 5173 y Fv(v)1696 5148 y Fo(2)p 1640 5187 109 4 v 1640 5234 a Fx(4)p Fv(\013)1716 5218 y Fo(2)1783 5105 y Fy(cos)1894 5071 y Fx(2)1945 5013 y Fp(\020)1994 5105 y Fw(x)2041 4998 y Fp(q)p 2125 4998 264 4 v 2147 5072 a Fv(\013)2190 5047 y Fo(2)p 2135 5086 101 4 v 2135 5134 a Fx(4)p Fv(v)2203 5117 y Fo(2)2264 5105 y Fr(\000)g Fy(1)g Fr(\000)g Fw(x)c Fy(arccos)e(\()2850 5072 y Fv(v)p 2829 5086 77 4 v 2829 5134 a Fx(2)p Fv(\013)2916 5105 y Fy(\))18 b Fr(\000)3059 5072 y Fv(\031)p 3059 5086 41 4 v 3063 5134 a Fx(4)3110 5013 y Fp(\021)3174 5105 y Fw(:)p eop %%Page: 15 15 15 14 bop 820 251 a Fx(D)n(YNAMICS)29 b(OF)g(INTERF)-7 b(A)n(CES)29 b(IN)g(THE)g(FERR)n(OMA)n(GNETIC)h(XXZ)f(CHAIN)299 b(15)456 450 y Fy(This)27 b(implies)h(that)g(for)f(large)f Fw(x)778 644 y(\036)827 610 y Fs(0)827 665 y Fv(x)869 644 y Fy(\()p Fw(v)s Fy(\))84 b Fr(\030)1426 588 y Fy(1)p 1218 625 459 4 v 1218 745 a Fw(\013\031)1321 642 y Fp(q)p 1404 642 272 4 v 103 x Fy(1)18 b Fr(\000)1578 712 y Fv(v)1613 687 y Fo(2)p 1557 726 109 4 v 1557 773 a Fx(4)p Fv(\013)1633 757 y Fo(2)1700 644 y Fy(cos)1811 610 y Fx(2)1862 552 y Fp(\020)1912 644 y Fw(x)1959 537 y Fp(q)p 2042 537 272 4 v 2052 611 a Fx(4)p Fv(\013)2128 586 y Fo(2)p 2052 625 109 4 v 2073 673 a Fv(v)2108 656 y Fo(2)2189 644 y Fr(\000)g Fy(1)g Fr(\000)g Fw(x)c Fy(arccos)e(\()2775 611 y Fv(v)p 2755 625 77 4 v 2755 673 a Fx(2)p Fv(\013)2841 644 y Fy(\))19 b Fr(\000)2985 611 y Fv(\031)p 2985 625 41 4 v 2989 673 a Fx(4)3035 552 y Fp(\021)3099 644 y Fw(:)456 914 y Fy(F)-7 b(or)30 b Fw(v)i(<)c Fy(2)p Fw(\013)j Fy(w)n(e)g(reco)n(v)n(er)e(the)i(function)h Fw(\036)p Fy(\()p Fw(v)s Fy(\))g(b)n(y)f(in)n(tegration,)g(\(w)n(e)g(switc)n(h)f (the)i(limit)g(and)456 1014 y(in)n(tegral)g(b)n(y)h(dominated)g(con)n (v)n(ergence,)g(since)g Fw(\036)2049 983 y Fs(0)2049 1034 y Fv(x)2091 1014 y Fy(\()p Fw(v)s Fy(\))h(is)f(uniformly)h(b)r (ounded)f(in)h Fw(x)g Fy(and)f Fw(v)456 1113 y Fy(for)27 b Fw(v)f Fr(2)d Fy([0)p Fw(;)844 1080 y Fx(2)p 839 1094 44 4 v 839 1142 a Fv(\013)910 1113 y Fr(\000)18 b Fw(\017)p Fy(]\),)28 b(i.e.,)839 1323 y Fw(\036)p Fy(\()p Fw(v)s Fy(\))84 b(:=)110 b(lim)1250 1373 y Fv(x)p Fs(!1)1434 1323 y Fw(\036)1483 1335 y Fv(x)1525 1323 y Fy(\()p Fw(v)s Fy(\))24 b(=)1743 1210 y Fp(Z)1826 1231 y Fv(v)1789 1399 y Fx(0)1880 1323 y Fw(dy)58 b Fy(lim)1994 1373 y Fv(x)p Fs(!1)2178 1323 y Fw(\036)2227 1289 y Fs(0)2227 1344 y Fv(x)2269 1323 y Fy(\()p Fw(y)s Fy(\))24 b(=)e Fr(\000)2567 1267 y Fy(2)p 2563 1304 51 4 v 2563 1380 a Fw(\031)2637 1323 y Fy(arcsin)13 b(\()2926 1291 y Fv(v)p 2906 1305 77 4 v 2906 1352 a Fx(2)p Fv(\013)2992 1323 y Fy(\))h Fw(:)456 1533 y Fy(The)30 b(oscillating)e(cosine)h(con)n(tributes)h(an) f(a)n(v)n(erage)e(factor)2372 1500 y Fx(1)p 2372 1514 34 4 v 2372 1562 a(2)2445 1533 y Fy(to)j(the)g(in)n(tegral.)42 b(Using)30 b(sym-)456 1633 y(metry)-7 b(,)27 b(and)h(collecting)f (terms,)g(w)n(e)g(ha)n(v)n(e)g(sho)n(wn)f(that)456 1884 y(\(19\))212 b(lim)711 1933 y Fv(t)p Fs(!1)p Fx(:)p Fv(x)p Fx(=)p Fv(v)r(t)1050 1884 y Fw(m)1123 1849 y Fx(3)1123 1904 y Fv(\015)t Fx(=0)1250 1884 y Fy(\()p Fw(x;)14 b(t)p Fy(\))24 b(=)f Fw(\036)p Fy(\()p Fw(v)s Fy(\))h(=)1808 1713 y Fp(8)1808 1788 y(<)1808 1938 y(:)2127 1783 y Fy(+)2202 1750 y Fx(1)p 2202 1764 V 2202 1812 a(2)2533 1783 y Fy(for)185 b Fw(v)26 b(<)d Fr(\000)p Fy(2)p Fw(\013)1923 1883 y Fr(\000)2001 1850 y Fx(2)p 1997 1864 41 4 v 1997 1912 a Fv(\031)2062 1883 y Fy(arcsin)13 b(\()2352 1850 y Fv(v)p 2331 1864 77 4 v 2331 1912 a Fx(2)p Fv(\013)2417 1883 y Fy(\))84 b(for)e Fr(\000)p Fy(2)p Fw(\013)23 b(<)f(v)k(<)d Fy(2)p Fw(\013)2127 1983 y Fr(\000)2202 1950 y Fx(1)p 2202 1964 34 4 v 2202 2012 a(2)2533 1983 y Fy(for)217 b Fw(v)26 b(>)d Fy(2)p Fw(\013)3313 1884 y(:)605 2135 y Fy(The)33 b(imp)r(ortance)g(b)r(ehind)h(a)f(further)g(analysis)f(of)i (the)f(pro\014le)g(p)r(erp)r(endicular)g(to)g(the)456 2234 y Fw(z)i Fy(direction)d(is)g(that)h(it)f(allo)n(ws)f(us)h(to)h (decide)f(whether)g(the)h(state)f(is)g(rotated)f(b)n(y)h(the)h(p)r(er-) 456 2334 y(turbation)d(in)n(to)g(the)h Fw(xy)j Fy(plane)c(or)f(whether) i(there)f(is)h(ballistic)f(di\013usion)h(of)f(the)h(in)n(terface.)456 2433 y(Comparing)f(equations)i(\(17\))f(and)h(\(18\),)h(w)n(e)f(see)g (that)g(the)h(main)f(di\013erence)g(is)g(a)g(replace-)456 2533 y(men)n(t)g(of)g(the)g(probabilit)n(y)f(measure)g Fw(p)p Fy(\()p Fw(m)p Fy(\))f(=)g Fr(h)p Fy(0)p Fr(j)p Fw(S)2130 2503 y Fx(3)2125 2554 y Fv(m)p Fs(\000)p Fx(1)2294 2533 y Fr(\000)21 b Fw(S)2436 2503 y Fx(3)2431 2554 y Fv(m)2494 2533 y Fr(j)p Fy(0)p Fr(i)32 b Fy(b)n(y)g(the)g(signed)g (measure)463 2643 y(~)-49 b Fw(p)o Fy(\()p Fw(m)p Fy(\))32 b(=)f Fr(h)p Fy(1)p Fr(j)p Fw(S)915 2612 y Fx(+)910 2663 y Fv(m)995 2643 y Fr(\000)21 b Fw(S)1137 2607 y Fx(+)1132 2665 y Fv(m)p Fs(\000)p Fx(1)1280 2643 y Fr(j)p Fy(0)p Fr(i)p Fy(.)52 b(One)32 b(can)g(easily)g(determine)h(that)g Fr(j)7 b Fy(~)-49 b Fw(p)p Fy(\()p Fw(m)p Fy(\))p Fr(j)32 b Fw(<)f(C)3013 2655 y Fx(1)3050 2643 y Fw(e)3089 2612 y Fs(\000)p Fv(c)p Fs(j)p Fv(m)p Fs(j)3306 2643 y Fy(\(see)456 2742 y(Lemma)f(A.1.2\).)47 b(Ho)n(w)n(ev)n(er,)30 b(no)n(w)38 b(~)-50 b Fw(p)p Fy(\()p Fw(m)p Fy(\))32 b(oscillates,)f(and)f(again)g (b)r(ecause)h(of)g(a)f(telescoping)456 2842 y(sum,)947 2920 y Fs(1)920 2945 y Fp(X)866 3120 y Fv(m)p Fx(=)p Fs(\0001)1115 3024 y Fy(~)-49 b Fw(p)p Fy(\()p Fw(m)p Fy(\))23 b(=)61 b(lim)1398 3074 y Fv(m)p Fs(!1)1589 3024 y Fr(h)p Fy(1)p Fr(j)p Fw(S)1742 2990 y Fx(+)1737 3045 y Fv(m)1800 3024 y Fr(j)p Fy(0)p Fr(i)18 b(\000)82 b Fy(lim)1999 3074 y Fv(m)p Fs(!\0001)2242 3024 y Fr(h)p Fy(1)p Fr(j)p Fw(S)2395 2990 y Fx(+)2390 3045 y Fv(m)2453 3024 y Fr(j)p Fy(0)p Fr(i)22 b Fy(=)h(0)18 b Fr(\000)g Fy(0)23 b(=)f(0)14 b Fw(:)456 3252 y Fy(Since)37 b(w)n(e)f(ha)n(v)n(e)g (determined)h(that)h(for)e(an)n(y)g(\014xed)h(\014nite)h Fw(m)p Fy(,)h(lim)2631 3264 y Fv(x)p Fs(!1)2829 3219 y Fv(x)p 2829 3233 38 4 v 2830 3280 a(v)2877 3252 y Fw(J)2931 3221 y Fx(2)2923 3272 y Fv(m)p Fx(+)p Fv(x)3075 3252 y Fy(\()3117 3219 y Fx(2)p Fv(x\013)p 3117 3233 114 4 v 3156 3280 a(v)3241 3252 y Fy(\))e(is)g(a)456 3351 y(\014xed)32 b(function)h(of)f Fw(v)s(=)p Fy(2)p Fw(\013)h Fy(indep)r(enden)n(t)g (of)f Fw(m)p Fy(,)i(and)e(in)h(particular)e(this)i(implies)f(the)h (same)456 3451 y(limit)28 b(for)f(lim)896 3463 y Fv(x)p Fs(!1)1094 3418 y Fv(x)p 1094 3432 38 4 v 1095 3479 a(v)1142 3451 y Fw(J)1188 3463 y Fv(m)p Fx(+)p Fv(x)1340 3451 y Fy(\()1382 3418 y Fx(2)p Fv(x\013)p 1382 3432 114 4 v 1421 3479 a(v)1506 3451 y Fy(\))p Fw(J)1584 3463 y Fv(m)p Fx(+)p Fv(x)p Fx(+1)1820 3451 y Fy(\()1862 3418 y Fx(2)p Fv(x\013)p 1862 3432 V 1901 3479 a(v)1986 3451 y Fy(\),)h(the)g(oscillation)e(of)35 b(~)-49 b Fw(p)27 b Fy(implies)h(that)807 3639 y(lim)695 3689 y Fv(x)p Fs(!1)p Fv(;x)p Fx(=)p Fv(v)r(t)1047 3639 y Fw( )1104 3605 y Fs(0)1101 3659 y Fv(x)1143 3639 y Fy(\()p Fw(v)s Fy(\))84 b(:=)f Fw(t)1535 3547 y Fp(h)1574 3639 y Fw(m)1647 3605 y Fx(1)1647 3659 y Fv(\015)t Fx(=0)1774 3639 y Fy(\()p Fw(x)19 b Fy(+)f(1)p Fw(;)c(t)p Fy(\))k Fr(\000)g Fw(m)2270 3605 y Fx(1)2270 3659 y Fv(\015)t Fx(=0)2396 3639 y Fy(\()p Fw(x;)c(t)p Fy(\))2574 3547 y Fp(i)2624 3675 y Fn(x)p 2624 3684 33 3 v 2629 3717 a(t)2667 3697 y Fx(=)p Fv(v)1346 3837 y Fy(=)121 b(lim)1505 3887 y Fv(x)p Fs(!1)1689 3837 y Fw(b)1749 3759 y Fp(X)1739 3937 y Fv(m)p Fs(2)p Fm(Z)1901 3837 y Fy(~)-49 b Fw(p)p Fy(\()p Fw(m)p Fy(\))42 b(lim)2087 3887 y Fv(x)p Fs(!1)2281 3805 y Fv(x)p 2281 3819 38 4 v 2282 3866 a(v)2329 3837 y Fw(J)2375 3849 y Fv(m)p Fx(+)p Fv(x)2527 3837 y Fy(\()2569 3805 y Fx(2)p Fv(x\013)p 2569 3819 114 4 v 2608 3866 a(v)2693 3837 y Fy(\))p Fw(J)2771 3849 y Fv(m)p Fx(+)p Fv(x)p Fx(+1)3007 3837 y Fy(\()3049 3805 y Fx(2)p Fv(x\013)p 3049 3819 V 3088 3866 a(v)3173 3837 y Fy(\))1346 4036 y(=)94 b(0)14 b Fw(:)456 4192 y Fy(This)19 b(do)r(es)g(not)g(sa)n(y)f(that)h(the)h(pro\014le)e(in)i (the)g Fw(xy)i Fy(plane)d(is)g(v)-5 b(anishing.)34 b(Quite)19 b(on)g(the)g(con)n(trary)-7 b(,)456 4291 y(it)32 b(means)f(that)h(it)g (sta)n(ys)e(exp)r(onen)n(tially)h(concen)n(trated)g(around)f(the)i (kink)g(at)f(0,)h(and)g(do)r(es)456 4391 y(not)27 b(follo)n(w)g(the)h 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Fw(m)p Fr(j)p Fw(S)2501 4848 y Fx(3)2496 4903 y Fv(x)2539 4882 y Fr(j)p Fw(m)p Fr(i)g Fw(:)456 5116 y Fy(W)-7 b(e)32 b(ha)n(v)n(e)f(not)h(b)r(een)g(able)g (to)f(deriv)n(e)h(explicit)g(expressions)e(in)i(the)h(general)d(case,)i (but)h(one)456 5216 y(can)27 b(sho)n(w,)f(for)h(instance,)g(b)n(y)g (rep)r(eated)g(use)g(of)h(Graf)6 b('s)27 b(addition)g(theorem,)g(that)h (there)f(is)g(a)p eop %%Page: 16 16 16 15 bop 456 251 a Fx(16)214 b(BR)n(UNO)23 b(NA)n(CHTER)n(GAELE,)g(W)n (OLF)n(GANG)g(L)g(SPITZER,)f(AND)g(SHANNON)g(ST)-5 b(ARR)456 450 y Fy(function)28 b Fw(F)39 b Fy(suc)n(h)28 b(that)1173 592 y Fw(m)1246 557 y Fx(3)1283 592 y Fy(\()p Fw(x;)14 b(t)p Fy(\))24 b(=)1584 513 y Fp(X)1573 691 y Fv(m)p Fs(2)p Fm(Z)1729 592 y Fw(J)1783 557 y Fx(2)1775 612 y Fv(m)1838 592 y Fy(\()p Fw(F)12 b Fy(\()p Fw(\013)p Fy(\()p Fw(t)p Fy(\))p Fw(;)i(\015)5 b Fy(\()p Fw(t)p Fy(\)\))14 b Fr(h)p Fw(m)p Fr(j)p Fw(S)2523 557 y Fx(3)2518 612 y Fv(x)2562 592 y Fr(j)p Fw(m)p Fr(i)g Fw(:)456 812 y Fy(In)27 b(these)h(time)g(dep)r(enden)n(t)g(cases,)f(a)g(scaling)g (limit)h(as)f(b)r(efore)g(do)r(es)h(not)f(exist.)605 975 y Fz(4.2.)46 b(General)24 b(in\014nitely)e(extended)h(\014elds.)40 b Fy(Since)21 b(a)e(thorough)g(discussion)h(w)n(ould)456 1075 y(b)r(e)35 b(out)f(of)h(the)g(scop)r(e)g(of)f(this)h(pap)r(er,)h (w)n(e)f(only)f(w)n(an)n(t)g(to)h(con)n(vince)f(the)h(reader)e(that)i (the)456 1174 y(leading)i(dynamics)g(of)h(the)g(\\man)n(y-b)r(o)r(dy")e (Hamiltonian)h(leads)h(to)f(a)h(w)n(ell)f(studied)h(one-)456 1274 y(b)r(o)r(dy)27 b(problem)h(whic)n(h)f(has)g(seen)g(great)g (progress)e(in)j(recen)n(t)f(y)n(ears.)605 1379 y(Let)f(us)g(start)f (with)h(the)g(simplest)g(case)f(when)h(the)g(magnetic)f(\014eld)2779 1358 y Fw(~)2765 1379 y(B)t Fy(\()p Fw(n)p Fy(\))i(is)e(asymptoti-)456 1478 y(cally)e(uniform)g(with)h(non-zero)e(third)i(comp)r(onen)n(t.)35 b(I.e.,)25 b(w)n(e)e(assume)g(that)h(there)f(is)h(a)f(v)n(ector)469 1562 y Fw(~)456 1583 y(B)31 b Fy(and)c(some)g Fw(p)c(>)f Fy(1)27 b(suc)n(h)g(that)h(\()p Fw(B)1602 1595 y Fx(3)1639 1583 y Fy(\()p Fw(n)p Fy(\))18 b Fr(\000)g Fw(B)1917 1595 y Fx(3)1954 1583 y Fy(\))p Fw(n)23 b Fr(2)g Fw(`)2172 1553 y Fv(p)2210 1583 y Fy(\()p Fq(Z)p Fy(\),)f(and)27 b(\()p Fw(B)2637 1595 y Fx(1)p Fv(;)p Fx(2)2727 1583 y Fy(\()p Fw(n)p Fy(\))18 b Fr(\000)g Fw(B)3005 1595 y Fx(1)p Fv(;)p Fx(2)3095 1583 y Fy(\))23 b Fr(2)g Fw(`)3263 1553 y Fv(p)3301 1583 y Fy(\()p Fq(Z)p Fy(\).)456 1683 y(Let)29 b Fw(K)35 b Fy(denote)29 b(the)h(Jacobi)e(op)r(erator)f (corresp)r(onding)h(to)h(this)g(v)n(ector)f(\014eld.)42 b(Then)30 b Fw(K)35 b Fy(is)29 b(a)456 1782 y(compact)20 b(p)r(erturbation)h(of)g(the)g(Stark-Jacobi)e(op)r(erator)g Fw(K)2362 1794 y Fx(0)2420 1782 y Fy(in)i(\(15\).)34 b(By)21 b(W)-7 b(eyl's)21 b(Theorem,)456 1882 y Fw(K)28 b Fy(has)23 b(pure)g(p)r(oin)n(t)g(sp)r(ectrum)h(and)f(the)g(motion)g (is)g(quasi-p)r(erio)r(dic)f(and)h(the)h(in)n(terface)f(sta)n(ys)456 1982 y(lo)r(calized)k(at)g(0.)605 2081 y(The)i(case)g(of)g(a)g(p)r (erturbation)g(of)g(a)g(uniform)g(\014eld)h(with)g Fw(B)2527 2093 y Fx(3)2590 2081 y Fy(=)25 b(0)k(is)h(m)n(uc)n(h)f(more)f(com-)456 2181 y(plicated.)36 b(If)27 b(w)n(e)f(p)r(erturb)g(b)n(y)h(a)f(\014eld) g(in)h(the)g Fw(z)i Fy(direction,)e(whic)n(h)f(corresp)r(onds)f(to)h (adding)g(a)456 2280 y(p)r(oten)n(tial)f(to)g(the)h(\(discrete\))f (Laplace)g(op)r(erator,)f(then)i(w)n(e)f(are)f(in)h(the)h(widely)g (studied)f(case)456 2380 y(of)i(a)g(\(discrete\))h(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erator.)605 2489 y(The)g(e\013ect)h(of)f(a)g(p)r (erturbation)f(of)1744 2468 y Fw(~)1731 2489 y(B)30 b Fy(in)d(the)f Fw(xy)k Fy(plane,)c(ho)n(w)n(ev)n(er,)e(is)i(non-lo)r (cal.)35 b(I.e.,)27 b(if)456 2589 y(w)n(e)d(c)n(hange)f Fw(B)909 2601 y Fx(1)970 2589 y Fy(at)i(0)e(in)n(to)i Fw(B)1363 2601 y Fx(1)1400 2589 y Fy(\(0\),)g(then)g(the)g (o\013-diagonal)d(matrix)i(elemen)n(t)g(computed)h(from)606 2747 y Fr(h)p Fw(n)p Fr(j)726 2668 y Fp(X)725 2847 y Fv(x)p Fs(2)p Fm(Z)860 2747 y Fw(B)923 2759 y Fx(1)961 2747 y Fy(\()p Fw(x)p Fy(\))p Fw(S)1128 2713 y Fx(+)1123 2768 y Fv(x)1183 2747 y Fr(j)p Fw(n)19 b Fr(\000)f Fy(1)o Fr(i)24 b Fy(=)e Fw(B)1605 2759 y Fx(1)1642 2747 y Fr(h)p Fw(n)p Fr(j)1762 2668 y Fp(X)1761 2847 y Fv(x)p Fs(2)p Fm(Z)1896 2747 y Fw(S)1952 2713 y Fx(+)1947 2768 y Fv(x)2007 2747 y Fr(j)p Fw(n)d Fr(\000)f Fy(1)o Fr(i)h Fy(+)f(\()p Fw(B)2452 2759 y Fx(1)2489 2747 y Fy(\(0\))h Fr(\000)f Fw(B)2760 2759 y Fx(1)2797 2747 y Fy(\))p Fr(h)p Fw(n)p Fr(j)p Fw(S)2990 2712 y Fx(+)2985 2769 y(0)3045 2747 y Fr(j)p Fw(n)h Fr(\000)f Fy(1)o Fr(i)456 2967 y Fy(is)33 b(e\013ected)i(for)e(all)h Fw(n)p Fy(,)h(although)f(exp)r(onen)n (tially)f(lo)r(calized)g(at)h(the)g(impurit)n(y)g(at)g(0.)55 b(By)34 b(a)456 3067 y(rotation,)26 b(one)i(can)f(map)h(this)g(Jacobi)f (op)r(erator)f(in)n(to)i(a)f(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erator.)36 b(An)n(y)28 b(non-)456 3166 y(uniform)j(p)r (erturbation)h(of)f(the)i(uniform)f(case)e(b)n(y)i(a)f(b)r(ounded)i(p)r (erturbation)e(will)h(mo)r(dify)456 3266 y(the)27 b(sp)r(ectrum)g(of)g (the)h(reduced)f(Jacobi)f(op)r(erator,)f(as)i(w)n(as)f(recen)n(tly)g (pro)n(v)n(en)g(b)n(y)h(Killip)g(and)456 3366 y(Simon)g([)p Fz(11)p Fy(].)605 3465 y(There)h(are)g(man)n(y)h(more)f(in)n(teresting) g(\014elds)h(for)g(whic)n(h)f(w)n(e)h(w)n(ould)f(lik)n(e)h(to)g (understand)456 3565 y(the)j(time)h(ev)n(olution)f(of)h(in)n(terfaces.) 50 b(A)33 b(particularly)e(simple)i(case)e(is)i(a)f(sharply)f(lo)r (calized)456 3670 y(\014eld)g(at)g(a)f(single)h(site)g Fw(y)s Fy(,)g(i.e.,)h Fw(V)48 b Fy(=)1668 3649 y Fw(~)1655 3670 y(B)25 b Fr(\001)1794 3649 y Fw(~)1786 3670 y(S)1837 3682 y Fv(y)1877 3670 y Fy(.)47 b(Here,)32 b(one)e(should)h(rather)f (appro)n(ximate)f(the)456 3782 y(time)e(ev)n(olution)e(b)n(y)i(the)g (pro)5 b(jection)25 b(of)i Fw(H)c Fy(+)1932 3761 y Fw(~)1919 3782 y(B)d Fr(\001)2049 3761 y Fw(~)2042 3782 y(S)2093 3794 y Fv(y)2159 3782 y Fy(on)n(to)26 b(its)h(lo)n(w)e(energy)h(sp)r (ectrum)h(whic)n(h)456 3882 y(seems)e(to)g(b)r(e)i(separated)d(b)n(y)h (a)h(gap)f(from)g(the)h(rest)f(of)h(the)g(sp)r(ectrum,)g(and)g(whic)n (h)g(is)f(similar)456 3981 y(to)38 b(the)g(sp)r(ectrum)h(of)f(the)g (unp)r(erturb)r(ed)h(kink)f(Hamiltonian.)69 b(So)37 b(far)h(only)g(the) g(ground)456 4081 y(state)24 b(and)h(the)g(gap)f(ab)r(o)n(v)n(e)g(the)h (ground)f(state)g(of)h(this)g(Hamiltonian)g(ha)n(v)n(e)e(b)r(een)j (studied)f(b)n(y)456 4181 y(Con)n(tucci,)i(Nac)n(h)n(tergaele)e(and)j (Spitzer)f([)p Fz(9)p Fy(].)37 b(W)-7 b(e)28 b(plan)g(to)f(pursue)g (this)h(in)g(the)g(future.)1191 4370 y Fz(App)s(endix)k(A.)47 b(Magnetization)32 b(pro\014les)605 4519 y Fy(The)k(exact)f(form)n(ula) g(for)g(the)h(spin-)1796 4487 y Fx(1)p 1796 4501 34 4 v 1796 4548 a(2)1875 4519 y Fy(magnetization)e(pro\014le)i(of)f(kink)h (states)f(in)h(the)456 4619 y Fw(z)44 b Fy(direction)c(w)n(as)g(deriv)n (ed)g(in)i([)p Fz(20)o Fy(],)j(pp)c(55{58.)75 b(Note)41 b(that)g(b)n(y)f(rotational)g(symmetry)-7 b(,)456 4719 y Fr(h)p Fw(n)p Fr(j)p Fw(S)617 4688 y Fx(1)612 4739 y Fv(x)654 4719 y Fr(j)p Fw(n)p Fr(i)23 b Fy(=)g Fr(h)p Fw(n)p Fr(j)p Fw(S)1031 4688 y Fx(2)1026 4739 y Fv(x)1068 4719 y Fr(j)p Fw(n)p Fr(i)g Fy(=)g(0.)36 b(W)-7 b(e)28 b(in)n(tro)r(duce)f(the)h(function)456 4877 y(\(20\))866 b Fw(f)9 b Fy(\()p Fw(z)t Fy(\))22 b(=)1738 4798 y Fp(X)1737 4977 y Fv(k)q Fs(\025)p Fx(0)1858 4877 y Fy(\()p Fr(\000)p Fy(1\))2029 4842 y Fv(k)2070 4877 y Fw(z)2113 4842 y Fv(k)2153 4877 y Fw(q)2193 4842 y Fv(k)q Fx(\()p Fv(k)q Fs(\000)p Fx(1\))2407 4877 y Fw(:)605 5116 y Fi(Lemma)35 b Fy(A.1)p Fi(.)44 b Fh(F)-6 b(or)33 b(the)g(spin-)1603 5083 y Fx(1)p 1604 5097 V 1604 5145 a(2)1680 5116 y Fh(in\014nite)g (chain)h(kink)g(Hamiltonian)g(we)g(have)g(the)g(fol-)456 5216 y(lowing)c(pr)l(o\014les)h(of)f(the)g(normalize)l(d)h(kink)g (state,)f Fr(j)p Fy(0)p Fr(i)f Fh(\(which)i(has)g(total)f Fw(z)j Fh(c)l(omp)l(onent)d(0\):)p eop %%Page: 17 17 17 16 bop 820 251 a Fx(D)n(YNAMICS)29 b(OF)g(INTERF)-7 b(A)n(CES)29 b(IN)g(THE)g(FERR)n(OMA)n(GNETIC)h(XXZ)f(CHAIN)299 b(17)647 450 y Fy(\(1\))41 b Fh(If)30 b Fw(x)24 b(>)e Fy(0)p Fh(,)30 b(then)456 659 y Fy(\(21\))597 b Fr(h)p Fy(0)p Fr(j)p Fw(S)1354 624 y Fx(3)1349 679 y Fv(x)1391 659 y Fr(j)p Fy(0)p Fr(i)23 b Fy(=)f Fr(\000)1673 602 y Fy(1)p 1673 639 42 4 v 1673 715 a(2)1743 659 y(+)c Fw(q)1866 624 y Fx(2)p Fv(x)1982 555 y Fs(1)1955 580 y Fp(X)1955 758 y Fv(k)q Fx(=0)2075 659 y Fy(\()p Fr(\000)p Fy(1\))2246 624 y Fv(k)2287 659 y Fw(q)2327 624 y Fv(k)q Fx(\()p Fv(k)q Fx(+2)p Fv(x)p Fx(+1\))2676 659 y Fw(;)794 885 y Fh(while)46 b(for)g Fw(x)k Fr(\024)g Fy(0)44 b Fh(we)h(have)h Fr(h)p Fy(0)p Fr(j)p Fw(S)1964 855 y Fx(3)1959 905 y Fv(x)2000 885 y Fr(j)p Fy(0)p Fr(i)k Fy(=)g Fr(\000h)p Fy(0)p Fr(j)p Fw(S)2480 855 y Fx(3)2475 905 y(1)p Fs(\000)p Fv(x)2601 885 y Fr(j)p Fy(0)p Fr(i)p Fh(.)83 b(F)-6 b(urther,)49 b Fw(p)p Fy(\()p Fw(m)p Fy(\))h(=)794 994 y Fr(h)p Fy(0)p Fr(j)p Fw(S)947 964 y Fx(3)942 1015 y Fv(m)p Fs(\000)p Fx(1)1108 994 y Fr(\000)18 b Fw(S)1247 964 y Fx(3)1242 1015 y Fv(m)1305 994 y Fr(j)p Fy(0)p Fr(i)23 b(\024)g Fw(e)1552 964 y Fs(\000)p Fv(c)p Fs(j)p Fv(m)p Fs(j)1765 994 y Fh(with)31 b Fw(c)e Fh(dep)l(ending)i(on)f Fw(q)s Fh(.)647 1104 y Fy(\(2\))41 b Fr(h)p Fw(n)p Fr(j)p Fw(S)955 1068 y Fs(\000)950 1126 y Fx(0)1011 1104 y Fr(j)p Fw(n)27 b Fr(\000)h Fy(1)p Fr(i)45 b Fy(=)g Fw(q)1473 1074 y Fs(j)p Fv(n)p Fs(j)1558 1104 y Fw(f)9 b Fy(\()p Fw(q)1680 1074 y Fx(2)p Fs(j)p Fv(n)p Fs(j)p Fx(+2)1882 1104 y Fy(\))p Fh(,)46 b(and)k Fy(~)-49 b Fw(p)o Fy(\()p Fw(m)p Fy(\))46 b(=)g Fr(h)p Fy(1)p Fr(j)p Fw(S)2647 1074 y Fx(+)2642 1124 y Fv(m)2732 1104 y Fr(\000)27 b Fw(S)2880 1068 y Fx(+)2875 1126 y Fv(m)p Fs(\000)p Fx(1)3023 1104 y Fr(j)p Fy(0)p Fr(i)42 b Fh(satis\014es)794 1215 y Fr(j)7 b Fy(~)-49 b Fw(p)p Fy(\()p Fw(m)p Fy(\))p Fr(j)24 b Fw(<)e(C)1189 1227 y Fx(1)1227 1215 y Fw(e)1266 1185 y Fs(\000)p Fv(c)p Fs(j)p Fv(m)p Fs(j)1479 1215 y Fh(with)31 b Fw(c)e Fh(dep)l(ending)i(on)f Fw(q)s Fh(.)647 1315 y Fy(\(3\))41 b Fh(The)31 b(numb)l(er)e Fw(a)g Fh(in)h(the)g(Stark-Jac) l(obi)h(op)l(er)l(ator)f(\(15\))h(is)f(e)l(qual)g(to)456 1513 y Fy(\(22\))386 b Fw(a)23 b Fy(=)f Fr(h)p Fw(n)p Fr(j)1264 1434 y Fp(X)1263 1612 y Fv(x)p Fs(2)p Fm(Z)1398 1513 y Fw(S)1454 1478 y Fx(1)1449 1533 y Fv(x)1491 1513 y Fr(j)p Fw(n)d Fr(\000)f Fy(1)p Fr(i)23 b Fy(=)1860 1456 y(1)p 1860 1493 V 1860 1570 a(2)1926 1434 y Fp(X)1926 1612 y Fv(k)q Fs(\025)p Fx(0)2047 1513 y Fy(\()p Fr(\000)p Fy(1\))2218 1478 y Fv(k)2259 1513 y Fw(q)2299 1478 y Fv(k)q Fx(\()p Fv(k)q Fx(+1\))2522 1456 y Fy(1)18 b(+)g Fw(q)2705 1426 y Fx(1+2)p Fv(k)p 2522 1493 342 4 v 2522 1570 a Fy(1)g Fr(\000)g Fw(q)2705 1546 y Fx(1+2)p Fv(k)2887 1513 y Fw(:)605 1758 y Fi(Pr)n(oof.)41 b Fy(Here,)27 b(w)n(e)g(only)g(sho)n(w)g(the)h(k)n(ey)f(steps)g(for)g(P)n(art)f(2)h (and)h(3;)f(the)h(details)f(can)g(b)r(e)456 1857 y(found)g(in)h([)p Fz(20)o Fy(].)37 b(The)27 b(exp)r(onen)n(tial)g(lo)r(calization)f(of)h (the)g(measures)f Fw(p)p Fy(\()p Fw(m)p Fy(\))i(and)34 b(~)-49 b Fw(p)p Fy(\()p Fw(m)p Fy(\))27 b(can)g(b)r(e)456 1957 y(deduced)33 b(from)f(the)h(explicit)h(form)n(ulas)d(but)j(it)f (can)f(also)g(b)r(e)h(done)g(m)n(uc)n(h)g(easier)e(b)n(y)i(using)456 2057 y(the)28 b(follo)n(wing)e(estimate)456 2201 y(\(23\))788 b Fr(h)p Fw( )1478 2213 y Fx(0)1515 2201 y Fr(j)p Fy([)1571 2169 y Fx(1)p 1571 2183 34 4 v 1571 2230 a(2)1633 2201 y Fy(+)18 b(sgn)o(\()p Fw(x)p Fy(\))p Fw(S)2003 2167 y Fx(3)1998 2222 y Fv(x)2041 2201 y Fy(])p Fr(j)p Fw( )2141 2213 y Fx(0)2179 2201 y Fr(i)23 b(\024)g Fw(C)6 b(q)2427 2167 y Fs(j)p Fv(x)p Fs(j)456 2355 y Fy(for)25 b(some)g Fw(C)30 b Fy(=)22 b Fw(C)6 b Fy(\()p Fw(q)s Fy(\))27 b(indep)r(enden)n(t)g(of)f Fw(x)p Fy(.)37 b(Let)26 b(\012)2035 2325 y Fx(+)p Fs(\000)2165 2355 y Fy(=)2253 2293 y Fp(N)2345 2380 y Fv(x)p Fs(\024)p Fx(0)2485 2288 y Fp(\000)2523 2318 y Fx(1)2523 2383 y(0)2557 2288 y Fp(\001)2610 2355 y Fr(\012)2689 2293 y Fp(N)2782 2380 y Fv(x>)p Fx(0)2922 2288 y Fp(\000)2960 2318 y Fx(0)2960 2383 y(1)2993 2288 y Fp(\001)3031 2355 y Fy(.)37 b(W)-7 b(e)26 b(de\014ne)456 2456 y(the)i(so-called)e(grand-canonical)f(states)456 2673 y(\(24\))429 b Fw( )s Fy(\()p Fw(z)t Fy(\))23 b(=)1395 2569 y Fx(0)1358 2594 y Fp(Y)1308 2768 y Fv(x)p Fx(=)p Fs(\0001)1514 2673 y Fy(\(1)c(+)f Fw(z)1733 2638 y Fs(\000)p Fx(1)1821 2673 y Fw(q)1861 2638 y Fs(\000)p Fv(x)1955 2673 y Fw(S)2011 2638 y Fs(\000)2006 2693 y Fv(x)2067 2673 y Fy(\))2140 2569 y Fs(1)2120 2594 y Fp(Y)2113 2770 y Fv(x)p Fx(=1)2234 2673 y Fy(\(1)h(+)f Fw(z)t(q)2493 2638 y Fv(x)2534 2673 y Fw(S)2590 2638 y Fx(+)2585 2693 y Fv(x)2645 2673 y Fy(\)\012)2737 2638 y Fx(+)p Fs(\000)2844 2673 y Fw(:)456 2892 y Fy(The)27 b(reason)f(for)h(this)h(de\014nition)g (is)g(that)456 3033 y(\(25\))1035 b Fw( )s Fy(\()p Fw(z)t Fy(\))23 b(=)1916 2955 y Fp(X)1913 3133 y Fv(n)p Fs(2)p Fm(Z)2052 3033 y Fw( )2106 3045 y Fv(n)2151 3033 y Fw(z)2194 2999 y Fv(n)2238 3033 y Fw(:)456 3262 y Fy(It)28 b(is)f(clear)g(that)g (in)h(the)g(grand)f(canonical)f(ground)h(state,)g(for)g(real)g Fw(z)f Fy(=)d Fw(q)2837 3232 y Fs(\000)p Fv(\026)2933 3262 y Fy(,)1196 3403 y Fr(h)p Fw( )s Fy(\()p Fw(z)t Fy(\))p Fr(j)p Fw(S)1471 3373 y Fx(3)1466 3423 y Fv(x)1508 3403 y Fr(j)p Fw( )s Fy(\()p Fw(z)t Fy(\))p Fr(i)p 1196 3440 532 4 v 1254 3516 a(h)p Fw( )s Fy(\()p Fw(z)t Fy(\))p Fr(j)p Fw( )s Fy(\()p Fw(z)t Fy(\))p Fr(i)1760 3459 y Fy(=)1858 3403 y(1)p 1858 3440 42 4 v 1858 3516 a(2)1928 3459 y Fr(\001)1979 3403 y Fw(q)2019 3373 y Fx(\()p Fv(x)p Fs(\000)p Fv(\026)p Fx(\))p Fv(=)p Fx(2)2291 3403 y Fr(\000)18 b Fw(q)2414 3373 y Fx(\()p Fv(\026)p Fs(\000)p Fv(x)p Fx(\))p Fv(=)p Fx(2)p 1979 3440 688 4 v 2057 3456 a Fp(p)p 2140 3456 450 4 v 73 x Fw(q)2180 3505 y Fv(x)p Fs(\000)p Fv(\026)2332 3529 y Fy(+)h Fw(q)2456 3505 y Fv(\026)p Fs(\000)p Fv(x)2691 3459 y Fw(;)456 3661 y Fy(whic)n(h)j(pro)n(v)n(es)e (exp)r(onen)n(tial)i(lo)r(calization)f(of)h(the)h(in)n(terface)e(in)i (the)f(grand)f(canonical)g(ground)456 3761 y(state.)36 b(E.g.,)27 b(setting)h Fw(\026)23 b Fy(=)f(0,)28 b(one)f(has)g(for)g Fw(x)d(>)e Fy(0,)962 3963 y Fr(\000)1037 3907 y Fy(1)p 1037 3944 42 4 v 1037 4020 a(2)1111 3963 y Fr(\024)1209 3907 y Fy(1)p 1209 3944 V 1209 4020 a(2)1279 3963 y Fr(\001)1330 3907 y Fw(q)1370 3877 y Fv(x=)p Fx(2)1497 3907 y Fr(\000)c Fw(q)1620 3877 y Fs(\000)p Fv(x=)p Fx(2)p 1330 3944 452 4 v 1356 3961 a Fp(p)p 1439 3961 318 4 v 72 x Fw(q)1479 4009 y Fv(x)1539 4033 y Fy(+)g Fw(q)1662 4009 y Fs(\000)p Fv(x)1814 3963 y Fy(=)23 b Fr(\000)1977 3907 y Fy(1)p 1977 3944 42 4 v 1977 4020 a(2)2046 3963 y(+)2139 3907 y(1)p 2139 3944 V 2139 4020 a(2)2209 3963 y Fr(\001)2261 3834 y Fp(p)p 2344 3834 258 4 v 73 x Fy(1)18 b(+)g Fw(q)2527 3883 y Fx(2)p Fv(x)2620 3907 y Fr(\000)g Fy(1)g(+)g Fw(q)2886 3877 y Fv(x)p 2261 3944 668 4 v 2424 3961 a Fp(p)p 2507 3961 258 4 v 72 x Fy(1)g(+)g Fw(q)2690 4009 y Fx(2)p Fv(x)1814 4200 y Fr(\024)23 b(\000)1977 4144 y Fy(1)p 1977 4181 42 4 v 1977 4257 a(2)2046 4200 y(+)2139 4144 y(1)p 2139 4181 V 2139 4257 a(2)2191 4200 y([1)18 b(+)2367 4144 y(1)p 2367 4181 V 2367 4257 a(2)2418 4200 y Fw(q)2458 4165 y Fx(2)p Fv(x)2552 4200 y Fr(\000)g Fy(1)g(+)g Fw(q)2818 4165 y Fv(x)2860 4200 y Fy(])1814 4400 y Fr(\024)23 b(\000)1977 4343 y Fy(1)p 1977 4380 V 1977 4456 a(2)2046 4400 y(+)2139 4343 y(3)p 2139 4380 V 2139 4456 a(4)2191 4400 y Fw(q)2231 4365 y Fv(x)2287 4400 y Fw(;)456 4163 y Fy(\(26\))456 4569 y(and)k(a)g(similar)g(inequalit)n(y)g(holds)g(for)g Fw(x)d(<)f Fy(0.)36 b(Setting)28 b Fw(\026)23 b Fy(=)g(0,)k(\(i.e.)37 b Fw(z)27 b Fy(=)22 b(1\))28 b(w)n(e)f(see)g(that)771 4728 y Fr(h)p Fw( )s Fy(\(1\))p Fr(j)p Fy(\()1031 4695 y Fx(1)p 1032 4709 34 4 v 1032 4756 a(2)1093 4728 y Fr(\006)18 b Fw(S)1232 4694 y Fx(3)1227 4748 y Fv(x)1269 4728 y Fy(\))p Fr(j)p Fw( )s Fy(\(1\))q Fr(i)23 b Fy(=)1633 4649 y Fp(X)1630 4827 y Fv(n)p Fs(2)p Fm(Z)1755 4728 y Fr(h)p Fw( )1841 4740 y Fv(n)1887 4728 y Fr(j)p Fy(\()1952 4695 y Fx(1)p 1952 4709 V 1952 4756 a(2)2013 4728 y Fr(\006)18 b Fw(S)2152 4694 y Fx(3)2147 4748 y Fv(x)2189 4728 y Fy(\))p Fr(j)p Fw( )2298 4740 y Fv(n)2344 4728 y Fr(i)23 b(\025)g(h)p Fw( )2573 4740 y Fx(0)2611 4728 y Fr(j)p Fy(\()2676 4695 y Fx(1)p 2676 4709 V 2676 4756 a(2)2737 4728 y Fr(\006)18 b Fw(S)2876 4694 y Fx(3)2871 4748 y Fv(x)2913 4728 y Fy(\))p Fr(j)p Fw( )3022 4740 y Fx(0)3060 4728 y Fr(i)c Fw(;)456 4961 y Fy(using)27 b(the)h(fact)g(that)f(\()1201 4928 y Fx(1)p 1202 4942 V 1202 4989 a(2)1263 4961 y Fr(\006)18 b Fw(S)1402 4931 y Fx(3)1397 4981 y Fv(x)1439 4961 y Fy(\))28 b(is)g(a)f(nonnegativ)n(e)f(op)r(erator.)35 b(Hence,)1026 5110 y Fr(h)p Fw( )1112 5122 y Fx(0)1149 5110 y Fr(j)p Fy([)1205 5077 y Fx(1)p 1205 5091 V 1205 5139 a(2)1267 5110 y Fr(\006)18 b Fw(S)1406 5080 y Fx(3)1401 5131 y Fv(x)1443 5110 y Fy(])p Fr(j)p Fw( )1543 5122 y Fx(0)1580 5110 y Fr(i)p 1026 5152 587 4 v 1184 5228 a(h)p Fw( )1270 5240 y Fx(0)1307 5228 y Fr(j)p Fw( )1384 5240 y Fx(0)1422 5228 y Fr(i)1645 5171 y(\024)1743 5115 y(k)p Fw( )s Fy(\()p Fw(z)t Fy(\))p Fr(k)1991 5085 y Fx(2)p 1743 5152 285 4 v 1779 5228 a Fr(k)p Fw( )1875 5240 y Fx(0)1912 5228 y Fr(k)1954 5204 y Fx(2)2056 5171 y Fr(\001)2107 5110 y(h)p Fw( )s Fy(\(1\))p Fr(j)p Fy([)2358 5077 y Fx(1)p 2359 5091 34 4 v 2359 5139 a(2)2420 5110 y Fr(\006)g Fw(S)2559 5080 y Fx(3)2554 5131 y Fv(x)2596 5110 y Fy(])p Fr(j)p Fw( )s Fy(\(1\))p Fr(i)p 2107 5152 731 4 v 2265 5228 a(h)p Fw( )s Fy(\(1\))p Fr(j)p Fw( )s Fy(\(1\))q Fr(i)2861 5171 y Fw(:)p eop %%Page: 18 18 18 17 bop 456 251 a Fx(18)214 b(BR)n(UNO)23 b(NA)n(CHTER)n(GAELE,)g(W)n (OLF)n(GANG)g(L)g(SPITZER,)f(AND)g(SHANNON)g(ST)-5 b(ARR)456 450 y Fy(Since)23 b(\()p Fr(k)p Fw( )s Fy(\()p Fw(z)t Fy(\))p Fr(k)948 420 y Fx(2)984 450 y Fw(=)p Fr(k)p Fw( )1122 462 y Fx(0)1159 450 y Fr(k)1201 420 y Fx(2)1237 450 y Fy(\))h Fw(<)e Fr(1)p Fy(,)i(and)f(using)g(equation)f(\(26\))h(w)n(e)f (deriv)n(e)g(equation)h(\(23\).)35 b(This)456 550 y(pro)n(v)n(es)25 b(P)n(art)h(1.)605 649 y(Next,)i(using)f(the)h Fw(q)s Fy(-binomial)f(pro)r(duct)h(theorem)f(w)n(e)g(get)h(that)689 874 y Fr(h)p Fw( )s Fy(\()p Fw(z)t Fy(\))p Fr(j)p Fw(S)964 839 y Fx(+)959 896 y(0)1019 874 y Fr(j)p Fw( )s Fy(\()p Fw(z)t Fy(\))p Fr(i)84 b Fy(=)e Fw(z)1512 840 y Fs(\000)p Fx(1)1675 770 y Fs(\000)p Fx(1)1665 795 y Fp(Y)1615 970 y Fv(x)p Fx(=)p Fs(\0001)1821 874 y Fy(\(1)19 b(+)f Fw(z)2040 840 y Fs(\000)p Fx(2)2128 874 y Fw(q)2168 840 y Fs(\000)p Fx(2)p Fv(x)2295 874 y Fy(\))2369 770 y Fs(1)2349 795 y Fp(Y)2341 971 y Fv(x)p Fx(=1)2463 874 y Fy(\(1)g(+)g Fw(z)2681 840 y Fx(2)2717 874 y Fw(q)2757 840 y Fx(2)p Fv(x)2832 874 y Fy(\))1322 1163 y(=)82 b Fw(z)1512 1129 y Fs(\000)p Fx(1)1615 1021 y Fp( )1710 1059 y Fs(1)1683 1084 y Fp(X)1680 1260 y Fv(n)p Fx(=0)1819 1163 y Fr(j)p Fw(z)t Fr(j)1908 1129 y Fs(\000)p Fx(2)p Fv(n)2067 1107 y Fw(q)2107 1077 y Fv(n)p Fx(\()p Fv(n)p Fx(+1\))p 2048 1144 302 4 v 2048 1220 a Fy(\()p Fw(q)2120 1196 y Fx(2)2157 1220 y Fy(;)14 b Fw(q)2234 1196 y Fx(2)2272 1220 y Fy(\))2304 1232 y Fv(n)2359 1021 y Fp(!)g( )2534 1059 y Fs(1)2507 1084 y Fp(X)2504 1260 y Fv(n)p Fx(=0)2643 1163 y Fr(j)p Fw(z)t Fr(j)2732 1129 y Fx(2)p Fv(n)2839 1107 y Fw(q)2879 1077 y Fv(n)p Fx(\()p Fv(n)p Fx(+1\))p 2820 1144 V 2820 1220 a Fy(\()p Fw(q)2892 1196 y Fx(2)2929 1220 y Fy(;)g Fw(q)3006 1196 y Fx(2)3044 1220 y Fy(\))3076 1232 y Fv(n)3131 1021 y Fp(!)1322 1439 y Fy(=)82 b Fw(z)1512 1405 y Fs(\000)p Fx(1)1687 1335 y Fs(1)1660 1360 y Fp(X)1615 1534 y Fv(n)p Fx(=)p Fs(\0001)1838 1439 y Fr(j)p Fw(z)t Fr(j)1927 1405 y Fx(2)p Fv(n)2015 1383 y Fw(q)2055 1353 y Fs(j)p Fv(n)p Fs(j)p Fx(\()p Fs(j)p Fv(n)p Fs(j)p Fx(+1\))2356 1383 y Fw(f)9 b Fy(\()p Fw(q)2478 1353 y Fx(2\()p Fs(j)p Fv(n)p Fs(j)p Fx(+1\))2732 1383 y Fy(\))p 2015 1420 749 4 v 2226 1496 a(\()p Fw(q)2298 1472 y Fx(2)2336 1496 y Fy(;)14 b Fw(q)2413 1472 y Fx(2)2450 1496 y Fy(\))2482 1508 y Fs(1)2788 1439 y Fw(:)456 1671 y Fy(Comparing)26 b(p)r(o)n(w)n(ers)g (of)h Fw(z)k Fy(in)d(\(25\))g(w)n(e)f(ha)n(v)n(e)1340 1878 y Fr(h)p Fw(n)p Fr(j)p Fw(S)1501 1843 y Fx(+)1496 1900 y(0)1556 1878 y Fr(j)p Fw(n)18 b Fr(\000)g Fy(1)p Fr(i)83 b Fy(=)2045 1822 y Fr(h)p Fw( )2131 1834 y Fv(n)2176 1822 y Fr(j)p Fw(S)2255 1787 y Fx(+)2250 1844 y(0)2310 1822 y Fr(j)p Fw( )2387 1834 y Fv(n)p Fs(\000)p Fx(1)2518 1822 y Fr(i)p 2045 1859 506 4 v 2073 1935 a(k)p Fw( )2169 1947 y Fv(n)p Fs(\000)p Fx(1)2298 1935 y Fr(kk)p Fw( )2436 1947 y Fv(n)2481 1935 y Fr(k)1887 2067 y Fy(=)g Fw(q)2075 2033 y Fs(j)p Fv(n)p Fs(j)2160 2067 y Fw(f)9 b Fy(\()p Fw(q)2282 2033 y Fx(2)p Fs(j)p Fv(n)p Fs(j)p Fx(+2)2483 2067 y Fy(\))14 b Fw(:)456 2220 y Fy(W)-7 b(e)28 b(ha)n(v)n(e)e(used)i (that)f Fr(k)p Fw( )1255 2232 y Fv(n)1300 2220 y Fr(k)1342 2190 y Fx(2)1402 2220 y Fy(=)c Fw(q)1530 2190 y Fv(n)p Fx(\()p Fv(n)p Fx(+1\))1752 2220 y Fw(=)p Fy(\()p Fw(q)1866 2190 y Fx(2)1903 2220 y Fy(;)14 b Fw(q)1980 2190 y Fx(2)2017 2220 y Fy(\))2049 2232 y Fs(1)2120 2220 y Fy(.)37 b(Finally)-7 b(,)947 2309 y Fp(X)946 2487 y Fv(x)p Fs(2)p Fm(Z)1067 2387 y Fr(h)p Fw(n)p Fr(j)p Fw(S)1228 2353 y Fx(+)1223 2408 y Fv(x)1283 2387 y Fr(j)p Fw(n)19 b Fr(\000)f Fy(1)p Fr(i)83 b Fy(=)1763 2309 y Fp(X)1762 2487 y Fv(x)p Fs(2)p Fm(Z)1883 2387 y Fr(h)p Fw(x)p Fr(j)p Fw(S)2041 2352 y Fx(+)2036 2410 y(0)2097 2387 y Fr(j)p Fw(x)19 b Fr(\000)f Fy(1)p Fr(i)1615 2620 y Fy(=)1763 2541 y Fp(X)1762 2720 y Fv(x)p Fs(2)p Fm(Z)1897 2620 y Fw(q)1937 2586 y Fs(j)p Fv(x)p Fs(j)2033 2541 y Fp(X)2032 2720 y Fv(k)q Fs(\025)p Fx(0)2154 2620 y Fy(\()p Fr(\000)p Fy(1\))2325 2586 y Fv(k)2365 2620 y Fw(q)2405 2586 y Fx(2\()p Fs(j)p Fv(x)p Fs(j)p Fx(+1\))p Fv(k)q Fx(+)p Fv(k)q Fx(\()p Fv(k)q Fs(\000)p Fx(1\))1615 2899 y Fy(=)1763 2820 y Fp(X)1762 2999 y Fv(k)q Fs(\025)p Fx(0)1884 2899 y Fy(\()p Fr(\000)p Fy(1\))2055 2865 y Fv(k)2095 2899 y Fw(q)2135 2865 y Fv(k)q Fx(\()p Fv(k)q Fx(+1\))2359 2843 y Fy(1)g(+)g Fw(q)2542 2813 y Fx(1+2)p Fv(k)p 2359 2880 342 4 v 2359 2956 a Fy(1)g Fr(\000)g Fw(q)2542 2932 y Fx(1+2)p Fv(k)2724 2899 y Fw(:)3380 3133 y Ff(\003)1062 3347 y Fz(App)s(endix)31 b(B.)47 b(Stark-Jacobi)34 b(Op)s(erator)e(on)f Fq(Z)2806 3317 y Fv(d)605 3497 y Fi(Lemma)d Fy(B.1)p Fi(.)37 b Fh(L)l(et)26 b Fw(K)6 b(f)j Fy(\()n Fw(~)-40 b(n)o Fy(\))24 b(=)e(\()p Fw(\013)p Fy(\001)11 b(+)d Fw(~)-39 b(\015)18 b Fr(\001)9 b Fw(~)-40 b(n)p Fy(\))p Fw(f)9 b Fy(\()n Fw(~)-40 b(n)p Fy(\))27 b Fh(b)l(e)g(selfadjointly)i(de\014ne)l(d)e(on) f Fr(D)r Fy(\()p Fw(K)6 b Fy(\))24 b(=)456 3596 y Fr(f)p Fw(f)52 b Fr(2)45 b Fw(`)p Fy(\()p Fq(Z)819 3566 y Fv(d)852 3596 y Fy(\))g(:)f(\()m Fw(~)-39 b(\015)32 b Fr(\001)25 b Fw(~)-40 b(n)p Fy(\))p Fw(f)9 b Fy(\()n Fw(~)-40 b(n)p Fy(\))45 b Fr(2)g Fw(`)1579 3566 y Fx(2)1616 3596 y Fy(\()p Fq(Z)1709 3566 y Fv(d)1742 3596 y Fy(\))p Fr(g)p Fh(.)74 b(If)42 b(al)t(l)g Fw(\015)2187 3608 y Fv(j)2264 3596 y Fh(ar)l(e)g(non-zer)l(o)f(and)h(ther)l(e)g(is)g(some)456 3696 y Fw(\015)499 3708 y Fx(0)574 3696 y Fh(such)c(that)g Fw(\015)992 3708 y Fv(j)1066 3696 y Fy(=)g Fw(a)1213 3708 y Fv(j)1247 3696 y Fw(\015)1290 3708 y Fx(0)1366 3696 y Fh(with)h Fw(a)1599 3708 y Fv(j)1672 3696 y Fr(2)f Fq(Z)32 b Fh(for)39 b(al)t(l)g Fw(j)k Fy(=)38 b(1)p Fw(;)14 b(:)g(:)g(:)f(;)h(d)p Fh(,)41 b(i.e.)f(if)f Fy(\()p Fw(\015)2959 3708 y Fx(1)2996 3696 y Fw(;)14 b(:)g(:)g(:)g(;)g(\015)3224 3708 y Fv(d)3263 3696 y Fy(\))38 b Fh(ar)l(e)456 3796 y(c)l(ommensur)l(able,)30 b(then)1385 3960 y(sp)l(e)l(c)p Fy(\()p Fw(K)6 b Fy(\))23 b(=)2018 3904 y Fr(k)p Fw(\015)5 b Fr(k)2150 3873 y Fx(2)p 1791 3941 623 4 v 1791 4017 a Fy(lcm\()p Fw(a)1996 4029 y Fx(1)2033 4017 y Fw(;)14 b(:)g(:)g(:)g(;)g(a)2262 4029 y Fv(d)2300 4017 y Fy(\))p Fw(\015)2375 4029 y Fx(0)2423 3960 y Fq(Z)f Fw(;)456 4141 y Fh(wher)l(e)31 b Fy(lcm\()p Fw(a)896 4153 y Fx(1)933 4141 y Fw(;)14 b(:)g(:)g(:)g(;)g(a)1162 4153 y Fv(d)1200 4141 y Fy(\))31 b Fh(is)f(the)h(le)l(ast)f(c)l(ommon)g(multiple.)41 b(If)31 b(al)t(l)g Fw(g)2612 4153 y Fv(j)2677 4141 y Fh(ar)l(e)f(nonzer)l(o,)h(but)f(they)456 4241 y(ar)l(e)d(not)f(c)l (ommensur)l(able,)i(i.e.,)i(ther)l(e)d(exists)f(a)h(p)l(air)h Fw(\015)2212 4253 y Fv(j)2247 4241 y Fh(,)g Fw(\015)2343 4253 y Fv(k)2411 4241 y Fh(such)e(that)h Fw(\015)2806 4253 y Fv(j)2841 4241 y Fw(=\015)2926 4253 y Fv(k)2993 4241 y Fh(is)g(irr)l(ational,)456 4341 y(then)e(sp)l(e)l(c)p Fy(\()p Fw(K)6 b Fy(\))24 b(=)e Fq(R)32 b Fh(and)26 b(the)g(sp)l(e)l (ctrum)f(is)h(dense)g(pur)l(e)g(p)l(oint)g(sp)l(e)l(ctrum.)37 b(In)25 b(this)h(c)l(ase)g(ther)l(e)456 4440 y(is)k(an)g(eigenvalue)h (at)e(every)i(p)l(oint)f(of)g(the)g(lattic)l(e)1152 4564 y Fv(d)1109 4589 y Fp(X)1112 4766 y Fv(j)s Fx(=1)1253 4612 y Fr(k)p Fw(\015)5 b Fr(k)1385 4581 y Fx(2)p 1253 4649 168 4 v 1298 4725 a Fw(\015)1341 4737 y Fv(j)1431 4668 y Fq(Z)16 b Fy(=)23 b Fr(f)1681 4564 y Fv(d)1639 4589 y Fp(X)1642 4766 y Fv(j)s Fx(=1)1772 4668 y Fw(a)1816 4680 y Fv(j)1851 4668 y Fr(k)p Fw(\015)5 b Fr(k)1983 4633 y Fx(2)2019 4668 y Fw(=\015)2104 4680 y Fv(j)2161 4668 y Fy(:)23 b Fw(a)2251 4680 y Fx(1)2289 4668 y Fw(;)14 b(:)g(:)g(:)f(;)h(a)2517 4680 y Fv(d)2579 4668 y Fr(2)23 b Fq(Z)o Fr(g)14 b Fw(:)456 4904 y Fh(However,)31 b(if)f(ther)l(e)g(ar) l(e)g Fy(1)23 b Fr(\024)f Fw(l)j(<)e(d)29 b Fh(non-zer)l(o)h(c)l(omp)l (onents)g(of)d Fw(~)-39 b(\015)5 b Fh(,)30 b(say)g(the)g(\014rst)f Fw(l)r Fh(,)h(then)985 5131 y(sp)l(e)l(c)p Fy(\()p Fw(K)6 b Fy(\))23 b(=)g(cl)1441 5039 y Fp(\020)1491 5131 y Fy([)p Fr(\000)p Fy(2)p Fw(\013)p Fy(\()p Fw(d)18 b Fr(\000)g Fw(l)r Fy(\))p Fw(;)c Fy(2)p Fw(\013)p Fy(\()p Fw(d)19 b Fr(\000)f Fw(l)r Fy(\)])g(+)2450 5027 y Fv(l)2401 5052 y Fp(X)2403 5229 y Fv(j)s Fx(=1)2521 5131 y Fy(\()2563 5075 y Fr(k)p Fw(\015)5 b Fr(k)2695 5045 y Fx(2)p 2563 5112 V 2608 5188 a Fw(\015)2651 5200 y Fv(j)2741 5131 y Fq(Z)o Fy(\))2834 5039 y Fp(\021)2892 5131 y Fw(;)p eop %%Page: 19 19 19 18 bop 820 251 a Fx(D)n(YNAMICS)29 b(OF)g(INTERF)-7 b(A)n(CES)29 b(IN)g(THE)g(FERR)n(OMA)n(GNETIC)h(XXZ)f(CHAIN)299 b(19)456 450 y Fh(wher)l(e)32 b Fy(cl\()p Fw(:)p Fy(\))g Fh(is)g(norm)f(closur)l(e)h(in)g Fq(R)p Fh(,)38 b(and)32 b(ther)l(e)f(is)h(essential)g(sp)l(e)l(ctrum)f(even)h(when)g(al)t(l)g Fw(\015)3409 462 y Fv(j)456 550 y Fh(ar)l(e)e(c)l(ommensur)l(able.)605 706 y Fi(Pr)n(oof.)41 b Fy(Let)23 b(us)g(assume)f(\014rst)h(that)g(all) g(comp)r(onen)n(ts)f(are)g(non-zero.)34 b(In)23 b(F)-7 b(ourier)22 b(space)456 806 y(w)n(e)27 b(ha)n(v)n(e)f(the)i(eigen)n(v) -5 b(alue)27 b(equation)1013 873 y Fp(0)1013 1022 y(@)1086 1040 y Fy(2)p Fw(\013)1237 936 y Fv(d)1194 961 y Fp(X)1197 1138 y Fv(j)s Fx(=1)1328 1040 y Fy(cos)13 b(\()p Fw(k)1528 1052 y Fv(j)1564 1040 y Fy(\))18 b(+)g Fw(i)p Fr(k)p Fw(\015)5 b Fr(k)1913 936 y Fv(d)1870 961 y Fp(X)1873 1138 y Fv(j)s Fx(=1)2041 984 y Fw(\015)2084 996 y Fv(j)p 2014 1021 131 4 v 2014 1097 a Fr(k)p Fw(\015)g Fr(k)2204 984 y Fw(@)p 2165 1021 127 4 v 2165 1097 a(@)g(k)2257 1109 y Fv(j)2320 1040 y Fr(\000)18 b Fw(\025)2451 873 y Fp(1)2451 1022 y(A)2538 1040 y Fw(\036)p Fy(\()p Fw(k)s Fy(\))24 b(=)f(0)14 b Fw(:)456 1299 y Fy(Let)40 b Fw(R)45 b Fy(=)e Fw(R)q Fy(\()p Fw(\015)5 b Fy(\))40 b(b)r(e)h(an)f(orthogonal) e(matrix)h(suc)n(h)h(that)2406 1236 y Fp(P)2493 1257 y Fv(d)2493 1323 y(j)s Fx(=1)2626 1299 y Fw(R)2689 1311 y Fv(ij)2778 1256 y(\015)2813 1264 y Fn(j)p 2758 1279 107 4 v 2758 1327 a Fs(k)p Fv(\015)t Fs(k)2918 1299 y Fy(=)k Fw(\016)s Fy(\()p Fw(i;)14 b Fy(1\).)74 b(\(In)456 1427 y(particular)26 b(this)i(means)f Fw(R)1318 1439 y Fx(1)p Fv(j)1409 1427 y Fy(=)1527 1385 y Fv(\015)1562 1393 y Fn(j)p 1507 1408 V 1507 1456 a Fs(k)p Fv(\015)t Fs(k)1623 1427 y Fy(.\))37 b(Setting)2004 1405 y(~)2001 1427 y Fw(k)26 b Fy(=)d Fw(R)q(k)s Fy(,)k(w)n(e)g(get)973 1633 y Fw(@)p 932 1670 130 4 v 932 1757 a(@)983 1735 y Fy(~)981 1757 y Fw(k)1024 1769 y Fx(1)1072 1689 y Fw(\036)p Fy(\()1155 1667 y(~)1153 1689 y Fw(k)s Fy(\))d(=)e Fr(\000)1468 1633 y Fw(i)p 1417 1670 131 4 v 1417 1746 a Fr(k)p Fw(\015)5 b Fr(k)1571 1522 y Fp(0)1571 1671 y(@)1644 1689 y Fw(\025)19 b Fr(\000)f Fy(2)p Fw(\013)1945 1585 y Fv(d)1902 1610 y Fp(X)1905 1787 y Fv(j)s Fx(=1)2036 1689 y Fy(cos)2161 1597 y Fp(\020)2279 1585 y Fv(d)2236 1610 y Fp(X)2225 1786 y Fv(m)p Fx(=1)2381 1689 y Fw(R)2444 1701 y Fv(mj)2541 1667 y Fy(~)2538 1689 y Fw(k)2581 1701 y Fv(m)2645 1597 y Fp(\021)2694 1522 y(1)2694 1671 y(A)2781 1689 y Fw(\036)p Fy(\()2864 1667 y(~)2862 1689 y Fw(k)s Fy(\))c Fw(:)456 1923 y Fy(The)27 b(solution)g(is)h(th)n(us)1005 2067 y Fw(\036)p Fy(\()1088 2045 y(~)1086 2067 y Fw(k)t Fy(\))23 b(=)g Fw(C)6 b(e)1380 2027 y Fs(\000)1478 2005 y Fn(i)p 1442 2014 96 3 v 1442 2048 a Fg(k)p Fn(\015)s Fg(k)1546 2033 y Fy(\()1579 2027 y Fv(\025)1620 2012 y Fx(~)1618 2027 y Fv(k)1653 2035 y Fo(1)1686 2027 y Fs(\000)p Fx(2)p Fv(\013)1825 1983 y Fj(P)1895 2002 y Fn(d)1895 2046 y(j)r Fo(=1)1997 2027 y Fx(\()p Fv(R)2073 2035 y Fo(1)p Fn(j)2132 2027 y Fx(\))2158 2002 y Fg(\000)p Fo(1)2247 2027 y Fx(sin)11 b(\()2366 1983 y Fj(P)2436 2002 y Fn(d)2436 2046 y(m)p Fo(=1)2573 2027 y Fv(R)2623 2035 y Fn(mj)2707 2012 y Fx(~)2705 2027 y Fv(k)2740 2035 y Fn(m)2796 2027 y Fx(\))2821 2033 y Fy(\))2872 2067 y Fw(:)456 2206 y Fy(Using)27 b(that)h Fw(R)932 2218 y Fx(1)p Fv(j)1023 2206 y Fy(=)1141 2164 y Fv(\015)1176 2172 y Fn(j)p 1121 2187 107 4 v 1121 2235 a Fs(k)p Fv(\015)t Fs(k)1265 2206 y Fy(w)n(e)f(ha)n(v)n(e)456 2415 y(\(27\))477 b Fw(\036)p Fy(\()p Fw(k)s Fy(\))24 b(=)f Fw(C)6 b(e)1456 2364 y Fs(\000)1554 2342 y Fn(i)p 1517 2351 96 3 v 1517 2385 a Fg(k)p Fn(\015)s Fg(k)1622 2300 y Fj(\020)1663 2364 y Fv(\025)1713 2321 y Fj(P)1783 2339 y Fn(d)1783 2383 y(j)r Fo(=1)1923 2324 y Fn(\015)1954 2337 y(j)p 1906 2351 V 1906 2385 a Fg(k)p Fn(\015)s Fg(k)2011 2364 y Fv(k)2046 2372 y Fn(j)2078 2364 y Fs(\000)p Fx(2)p Fv(\013)2217 2321 y Fj(P)2287 2339 y Fn(d)2287 2383 y(j)r Fo(=1)2410 2335 y Fg(k)p Fn(\015)s Fg(k)p 2410 2351 V 2427 2384 a Fn(\015)2458 2397 y(j)2526 2364 y Fx(sin)11 b(\()p Fv(k)2680 2372 y Fn(j)2712 2364 y Fx(\))2738 2300 y Fj(\021)2796 2415 y Fw(:)456 2554 y Fy(Finally)-7 b(,)27 b(the)h(p)r(erio)r(dicit)n(y)g(condition,)f Fw(\036)p Fy(\()p Fw(k)1824 2566 y Fv(j)1878 2554 y Fy(+)18 b(2)p Fw(\031)s Fy(\))24 b(=)e Fw(\036)p Fy(\()p Fw(k)2320 2566 y Fv(j)2356 2554 y Fy(\))p Fw(;)14 b(j)28 b Fy(=)23 b(1)p Fw(;)14 b(:)g(:)g(:)f(;)h(d)28 b Fy(requires)e(that)1735 2749 y Fw(\025)e Fr(2)1895 2693 y(k)p Fw(\015)5 b Fr(k)2027 2663 y Fx(2)p 1895 2730 168 4 v 1940 2806 a Fw(\015)1983 2818 y Fv(j)2073 2749 y Fq(Z)13 b Fw(:)605 2937 y Fy(No)n(w)21 b(supp)r(ose)h(that)g(the)g(\014rst)f Fw(l)i Fy(comp)r(onen)n(ts)e(of)e Fw(~)-39 b(\015)26 b Fy(are)21 b(non-zero)f(and)i(the)g(rest)f(are)g (zero.)456 3037 y(According)f(to)h(the)h(decomp)r(osition)e Fw(`)1651 3007 y Fx(2)1688 3037 y Fy(\()p Fq(Z)1782 3007 y Fv(d)1815 3037 y Fy(\))j(=)g Fw(`)1993 3007 y Fx(2)2029 3037 y Fy(\()p Fq(Z)2123 3007 y Fv(l)2143 3037 y Fy(\))2189 2975 y Fp(L)2295 3037 y Fw(`)2330 3007 y Fx(2)2367 3037 y Fy(\()p Fq(Z)2460 3007 y Fv(d)p Fs(\000)p Fv(l)2566 3037 y Fy(\),)g(w)n(e)e(write)g Fw(K)27 b Fy(as)21 b(the)g(sum)456 3140 y(of)i(Stark-Jacobi)e(op)r(erator)g Fw(K)1432 3110 y Fx(\()p Fv(l)p Fx(\))1532 3140 y Fy(acting)h(on)h Fw(`)1922 3110 y Fx(2)1959 3140 y Fy(\()p Fq(Z)2053 3110 y Fv(l)2072 3140 y Fy(\))h(and)f Fw(\013)p Fy(\001)2407 3110 y Fx(\()p Fv(l)p Fx(\))2508 3140 y Fy(acting)f(on)h Fw(`)2898 3110 y Fx(2)2935 3140 y Fy(\()p Fq(Z)3029 3110 y Fv(d)o Fs(\000)p Fv(l)3135 3140 y Fy(\).)35 b(Hence)456 3239 y(the)28 b(result)f(follo)n(ws.)2281 b Ff(\003)1726 3422 y Fz(References)491 3555 y FA([1])35 b(M.)17 b(Abramo)n(witz)h(and)i(I.A.)e(Stegun,)j Fu(Handb)l(o)l(ok)i(of)e(mathematic)l(al)h(functions)p FA(,)e(Do)n(v)n(er)f(Bo)r(oks)g(in)f(Math-)601 3638 y(ematics,)k(1970.) 491 3721 y([2])35 b(F.)e(C.)h(Alcaraz,)j(S.)c(R.)h(Salinas,)i(and)f(W.) f(F.)g(W)-6 b(reszinski,)36 b Fu(A)n(nisotr)l(opic)h(ferr)l(omagnetic)f (quantum)601 3804 y(domains)p FA(,)24 b(Ph)n(ys.)g(Rev.)g(Lett)g Fc(75)f FA(\(1995\),)i(930{933.)491 3887 y([3])35 b(T.)23 b(An)n(tal,)g(Z.)h(R\023)-35 b(acz,)24 b(A.)f(R\023)-35 b(ak)n(os,)23 b(and)i(G.M.)d(Sc)n(h)r(\177)-37 b(utz,)25 b Fu(T)-5 b(r)l(ansp)l(ort)27 b(in)f(the)f(XX)h(chain)g(at)f(zer)l(o)i (temp)l(er)l(a-)601 3970 y(tur)l(e:)32 b(Emer)l(genc)l(e)26 b(of)g(\015at)g(magnetization)g(pr)l(o\014les)p FA(,)f(Ph)n(ys.)e(Rev.) h(E,)f Fc(59)g FA(\(1999\),)i(4912{4919.)491 4053 y([4])35 b(H.)21 b(Araki,)f Fu(On)k(the)f(XY)g(mo)l(del)i(on)g(two-side)l(d)f (in\014nite)f(chain)p FA(,)f(Publ.)f(RIMS)h(Ky)n(oto)g(Univ.)f Fc(20)g FA(\(1984\),)601 4136 y(277{296.)491 4219 y([5])35 b(W.H.)20 b(Asc)n(h)n(bac)n(her)j(and)f(C.-A.)e(Pillet,)h Fu(Non-e)l(quilibrium)j(ste)l(ady)f(states)h(of)g(the)f(XY)g(mo)l(del)p FA(,)g(Preprin)n(t,)601 4302 y(2002,)h(mp-02-277)o(.)491 4385 y([6])35 b(D.)c(Babbitt)h(and)g(E.)f(Gutkin,)j Fu(The)f(Plancher)l (el)h(formula)h(for)e(the)g(in\014nite)f(XXZ)g(Heisenb)l(er)l(g)h(spin) 601 4468 y(chain)p FA(,)23 b(Lett.)i(Math.)f(Ph)n(ys.)f Fc(20)g FA(\(1990\),)i(no.)f(2,)f(91{99.)491 4551 y([7])35 b(D.)f(Babbitt)i(and)f(L.)g(Thomas,)h Fu(Gr)l(ound)i(state)d(r)l(epr)l (esentation)i(of)g(the)f(in\014nite)f(one-dimensional)601 4634 y(Heisenb)l(er)l(g)18 b(ferr)l(omagnet.)h(II:)h(An)e(explicit)h (Plancher)l(el)h(formula)p FA(,)f(Comm)n(un.)14 b(Math.)i(Ph)n(ys.)g Fc(54)g FA(\(1977\),)601 4717 y(no.)23 b(3,)h(255{278.)491 4800 y([8])35 b(P)-6 b(.)25 b(Caputo)i(and)f(F.)f(Martinelli,)f Fu(R)l(elaxation)k(time)f(of)h(anisotr)l(opic)h(simple)f(exclusion)f (pr)l(o)l(c)l(esses)i(and)601 4883 y(quantum)d(Heisenb)l(er)l(g)f(mo)l (dels)p FA(,)g(preprin)n(t,)e(2002,)i(math.)d(PR/0202025)s(.)491 4967 y([9])35 b(P)-6 b(.)26 b(Con)n(tucci,)k(B.)c(Nac)n(h)n(tergaele,)k (and)e(W.L.)e(Spitzer,)j Fu(The)g(ferr)l(omagnetic)g(Heisenb)l(er)l(g)f (XXZ)h(chain)601 5050 y(in)c(a)h(pinning)g(\014eld)p FA(,)d(Ph)n(ys.)h(Rev.)g(B)f Fc(66)g FA(\(2002\),)i(Art.#)e(064429.)456 5133 y([10])35 b(C.-T.)23 b(Gottstein)k(and)e(R.)f(F.)g(W)-6 b(erner,)25 b Fu(Gr)l(ound)k(states)d(of)h(the)g(in\014nite)f (q-deforme)l(d)h(Heisenb)l(er)l(g)f(fer-)601 5216 y(r)l(omagnet)p FA(,)e(Preprin)n(t,)f(cond-mat/9501123)r(.)p eop %%Page: 20 20 20 19 bop 456 251 a Fx(20)214 b(BR)n(UNO)23 b(NA)n(CHTER)n(GAELE,)g(W)n (OLF)n(GANG)g(L)g(SPITZER,)f(AND)g(SHANNON)g(ST)-5 b(ARR)456 450 y FA([11])35 b(R.)c(Killip)g(and)h(B.)g(Simon,)h Fu(Sum)h(rules)g(for)g(Jac)l(obi)g(matric)l(es)g(and)g(their)g(applic)l (ations)h(to)f(sp)l(e)l(ctr)l(al)601 533 y(the)l(ory)p FA(,)23 b(mp-arc/01-453,)g(to)h(app)r(ear)h(in)e(Ann.)h(of)f(Math.)456 616 y([12])35 b(T.)18 b(Koma)h(and)h(B.)f(Nac)n(h)n(tergaele,)i Fu(The)h(sp)l(e)l(ctr)l(al)h(gap)g(of)e(the)h(ferr)l(omagnetic)g(XXZ)f (chain)p FA(,)f(Lett.)g(Math.)601 699 y(Ph)n(ys.)j Fc(40)g FA(\(1997\),)i(1{16.)456 782 y([13])p 601 782 212 4 v 258 w(,)k Fu(The)h(c)l(omplete)h(set)f(of)g(gr)l(ound)i(states)d(of)i (the)f(ferr)l(omagnetic)g(XXZ)f(chains)p FA(,)h(Adv.)e(Theor.)601 865 y(Math.)23 b(Ph)n(ys.)h Fc(2)f FA(\(1998\),)i(533{558,)g (cond-mat/9709208)s(.)456 948 y([14])35 b(T.)23 b(Koma,)g(B.)h(Nac)n(h) n(tergaele,)i(and)e(S.)g(Starr,)g Fu(The)i(sp)l(e)l(ctr)l(al)i(gap)e (for)h(the)f(ferr)l(omagnetic)g(spin-j)g(XXZ)601 1031 y(chain)p FA(,)d(Adv.)h(Theor.)f(Math.)h(Ph)n(ys.)f Fc(5)h FA(\(2001\),)h(1047{1090.)456 1114 y([15])35 b(E.)22 b(Lieb,)h(T.)f(Sc)n(h)n(ultz,)i(and)f(D.)f(Mattis,)h Fu(Two)i(soluble)h(mo)l(dels)g(of)f(an)g(antiferr)l(omagnetic)g(chain)p FA(,)f(Ann.)601 1197 y(Ph)n(ys.)f Fc(16)g FA(\(1961\),)i(407{466.)456 1280 y([16])35 b(T.)20 b(Matsui,)g Fu(On)i(gr)l(ound)i(states)f(of)f (the)h(one-dimensional)h(ferr)l(omagnetic)f Fb(X)5 b(X)g(Z)27 b Fu(mo)l(del)p FA(,)22 b(Lett.)f(Math.)601 1363 y(Ph)n(ys.)i Fc(37)g FA(\(1996\),)i(397.)456 1446 y([17])35 b(M.)c(Moser,)h(A.)f (Prets,)j(and)e(W.)g(Spitzer,)i Fu(Time)f(evolution)g(of)h(spin)f (waves)p FA(,)h(Ph)n(ys.)e(Rev.)f(Lett.)i Fc(83)601 1529 y FA(\(1999\),)25 b(3542{3545.)456 1612 y([18])35 b(Y.)i(Ogata,)43 b Fu(The)d(di\013usion)g(of)f(the)g(magnetization)h(pr)l(o\014le)h(in)e (the)g(XX-mo)l(del)p FA(,)j(Preprin)n(t,)g(2002,)601 1695 y(cond-mat/0210011)r(.)456 1778 y([19])35 b(R.)25 b(H.)g(Sc)n(honmann)i(and)f(S.)g(Shlosman,)f Fu(Wul\013)j(dr)l(oplets)h (and)g(the)e(metastable)h(r)l(elaxation)h(of)f(kinetic)601 1861 y(Ising)d(mo)l(dels)p FA(,)g(Comm)n(un.)d(Math.)h(Ph)n(ys.)h Fc(194)e FA(\(1998\),)k(389{462.)456 1944 y([20])35 b(S.)22 b(Starr,)g Fu(Some)j(pr)l(op)l(erties)h(of)e(the)h(low)g(lying)f(sp)l (e)l(ctrum)h(of)g(the)f(quantum)i(XXZ)d(spin)i(system)p FA(,)d(Ph.D.)601 2028 y(thesis,)h(U.C.)g(Da)n(vis,)g(Da)n(vis,)g(CA)g (95616,)i(June)f(2001,)g(math-ph/0106024)r(.)605 2183 y FB(Dep)l(ar)l(tment)g(of)i(Ma)l(thema)l(tics,)f(University)g(of)g (Calif)o(ornia,)g(D)n(a)-6 b(vis,)26 b(One)g(Shields)g(A)-8 b(venue,)456 2266 y(D)n(a)i(vis)25 b(95616-8366,)e(USA)605 2349 y Fu(E-mail)j(addr)l(ess)5 b FA(:)33 b Fa(bxn@math.ucdavis.edu)605 2490 y FB(Dep)l(ar)l(tment)24 b(of)i(Ma)l(thema)l(tics,)f(University)g (of)g(Calif)o(ornia,)g(D)n(a)-6 b(vis,)26 b(One)g(Shields)g(A)-8 b(venue,)456 2573 y(D)n(a)i(vis)25 b(95616-8366,)e(USA)605 2656 y Fu(E-mail)j(addr)l(ess)5 b FA(:)33 b Fa(spitzer@math.ucdavis.e)q (du)605 2798 y FB(Dep)l(ar)l(tment)24 b(of)h(Physics,)h(Princeton)g (University,)f(Princeton,)h(NJ)f(08544,)f(USA)605 2881 y Fu(E-mail)i(addr)l(ess)5 b FA(:)33 b Fa(sstarr@math.princeton.)q(edu) p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0211222011648--