Content-Type: multipart/mixed; boundary="-------------0310160210408" This is a multi-part message in MIME format. ---------------0310160210408 Content-Type: text/plain; name="03-470.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-470.comments" Mathematics Subject Classification: 81Q10, 35J10, 47A10 ---------------0310160210408 Content-Type: text/plain; name="03-470.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-470.keywords" Integrated density of states, interacting particles ---------------0310160210408 Content-Type: application/postscript; name="ids.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ids.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86f Copyright 2001 Radical Eye Software %%Title: ids6.dvi %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ids6 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.10.15:1048 %%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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/DEST pdfmark end 2449 4648 a 1455 4949 a Fo(H)i Fp(:=)28 b Fk(\000)23 b(4)e Fp(+)p Fo(V)2045 4964 y Fm(i)2095 4949 y Fp(+)h Fo(V)2250 4964 y Fm(n)2297 4949 y Fo(;)1255 b Fp(\(1.5\))0 5130 y(where)282 5010 y SDict begin H.S end 282 5010 a 282 5010 a SDict begin 14.5 H.A end 282 5010 a 282 5010 a SDict begin [ /View [/XYZ H.V] /Dest (equation.1.6) cvn H.B /DEST pdfmark end 282 5010 a 1159 5251 a Fo(V)1216 5266 y Fm(i)1244 5251 y Fp(\()p Fo(x)1337 5266 y Fi(1)1377 5251 y Fo(;)17 b(:::;)g(x)1601 5266 y Fm(n)1648 5251 y Fp(\))28 b(:=)1924 5156 y Fj(X)1845 5368 y Fi(1)p Fg(\024)p Fm(k)r()1801 2740 y Fm(nd)p 1801 2756 80 4 v 1823 2813 a Fi(2)1923 2779 y Fp(and)g(sp)s(ecify)g(the)h(appropriate)e(c)m(hoice)h(later)f(on.)0 2899 y(By)h(assumption)f(\(H.1.a\))g(and)h(\(H.2\),)f(there)i(exists)f Fo(\020)i(>)27 b Fp(0)33 b(suc)m(h)h(that)2689 2779 y SDict begin H.S end 2689 2779 a 2689 2779 a SDict begin 14.5 H.A end 2689 2779 a 2689 2779 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.4) cvn H.B /DEST pdfmark end 2689 2779 a 297 3174 a Fk(\0001)28 b Fo(<)g Fk(\000)p Fo(\020)35 b Fk(\024)28 b Fp(min)1045 3033 y Fj(\022)1129 3174 y Fp(inf)1119 3236 y Fm(L)p Fg(\025)p Fi(1)1257 3174 y Fk(f)p Fp(inf)6 b([)p Fo(\033)t Fp(\()p Fo(H)1630 3189 y Fm(n)p Fg(j)p Fi(\003)1742 3166 y Fd(n)1742 3212 y(L)1792 3174 y Fp(\))22 b Fk([)g Fo(\033)t Fp(\()p Fo(H)2118 3189 y Fg(j)p Fi(\003)2187 3166 y Fd(n)2187 3212 y(L)2237 3174 y Fp(\)])p Fk(g)p Fo(;)17 b Fp(inf)6 b([)p Fo(\033)t Fp(\()p Fo(H)2719 3189 y Fm(n)2765 3174 y Fp(\))22 b Fk([)h Fo(\033)t Fp(\()p Fo(H)8 b Fp(\)])3165 3033 y Fj(\023)3254 3174 y Fo(:)298 b Fp(\(2.4\))0 3448 y(Let)33 b Fo(\015)f Fp(=)c Fo(\015)5 b Fp(\(1)p Fo(=)p Fp(2\))32 b(b)s(e)h(giv)m(en)f(b)m(y)j(\()p 0 0 1 TeXcolorrgb 1235 3448 a SDict begin H.S end 1235 3448 a 0 0 1 TeXcolorrgb Fp(1.3)p 0 0 1 TeXcolorrgb 1360 3384 a SDict begin H.R end 1360 3384 a 1360 3448 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1.3) cvn H.B /ANN pdfmark end 1360 3448 a Black Fp(\))d(for)g Fo(\013)c Fp(=)g(1)p Fo(=)p Fp(2.)43 b(Fix)32 b Fo(\025)2222 3463 y Fi(0)2289 3448 y Fo(>)27 b(\020)i Fp(+)22 b(2)p Fo(\015)27 b Fp(+)22 b(1.)0 3568 y(By)i(\()p 0 0 1 TeXcolorrgb 182 3568 a SDict begin H.S end 182 3568 a 0 0 1 TeXcolorrgb Fp(2.4)p 0 0 1 TeXcolorrgb 307 3504 a SDict begin H.R end 307 3504 a 307 3568 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.4) cvn H.B /ANN pdfmark end 307 3568 a Black Fp(\),)h(w)m(e)e(only)f(need)i(to)e(pro)m(v)m (e)i(\()p 0 0 1 TeXcolorrgb 1352 3568 a SDict begin H.S end 1352 3568 a 0 0 1 TeXcolorrgb Fp(2.3)p 0 0 1 TeXcolorrgb 1477 3504 a SDict begin H.R end 1477 3504 a 1477 3568 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.3) cvn H.B /ANN pdfmark end 1477 3568 a Black Fp(\))e(for)g Fo(')28 b Fk(2)g Fo(C)1939 3532 y Fg(1)1932 3593 y Fi(0)2014 3568 y Fp(\()p Fn(R)t Fp(\))h(supp)s(orted)23 b(in)f(\()p Fk(\000)p Fo(\020)9 b Fk(\000)r Fp(1)p Fo(;)17 b Fp(+)p Fk(1)p Fp(\).)40 b(F)-8 b(or)22 b(suc)m(h)i(a)0 3689 y(function,)h(let)39 b(~)-64 b Fo(')24 b Fp(b)s(e)g(an)g(almost)e (analytic)h(extension)i(of)e(the)i(function)f Fo(x)k Fk(7!)f Fp(\()p Fo(x)5 b Fp(+)g Fo(\025)3048 3704 y Fi(0)3087 3689 y Fp(\))3125 3653 y Fm(q)3163 3689 y Fo(')p Fp(\()p Fo(x)p Fp(\))28 b Fk(2)g Fo(C)3557 3653 y Fg(1)3550 3714 y Fi(0)3632 3689 y Fp(\()p Fn(R)5 b Fp(\))0 3809 y(i.e.)59 b(~)-65 b Fo(')32 b Fp(satis\014es)p Black 145 4010 a Fk(\017)p Black 65 w Fp(~)-65 b Fo(')27 b Fk(2)i(S)7 b Fp(\()p Fk(f)p Fo(z)33 b Fk(2)28 b Fn(C)53 b Fp(:)28 b Fk(j=)p Fo(z)t Fk(j)g Fo(<)f Fp(1)p Fk(g)p Black 145 4213 a(\017)p Black 49 w Fp(for)37 b(an)m(y)h Fo(k)i Fk(2)c Fn(N)9 b Fp(,)45 b(the)38 b(family)d(of)i(functions)h(\()p Fo(x)f Fk(7!)2211 4169 y Fm(@)16 b Fi(~)-47 b Fm(')p 2211 4190 88 4 v 2216 4247 a(@)p 2257 4210 36 3 v 4 w(z)2309 4213 y Fp(\()p Fo(x)25 b Fp(+)h Fo(iy)t Fp(\))p Fk(j)p Fo(y)t Fk(j)2760 4177 y Fg(\000)p Fm(k)2855 4213 y Fp(\))2893 4228 y Fi(0)p Fm(<)p Fg(j)p Fm(y)r Fg(j)p Fm(<)p Fi(1)3192 4213 y Fp(is)37 b(b)s(ounded)h(in)244 4333 y Fk(S)7 b Fp(\()p Fn(R)f Fp(\).)0 4534 y(The)34 b(functional)c(calculus)j(based)g (on)g(the)g(Hel\013er-Sj\177)-49 b(ostrand)32 b(form)m(ula)e(implies) 684 4751 y SDict begin H.S end 684 4751 a 684 4751 a SDict begin 14.5 H.A end 684 4751 a 684 4751 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.5) cvn H.B /DEST pdfmark end 684 4751 a Fo(')p Fp(\()p Fo(H)867 4767 y Fg(j)p Fi(\003)936 4744 y Fd(n)936 4790 y(L)986 4751 y Fp(\))22 b Fk(\000)h Fo(')p Fp(\()p Fo(H)1329 4767 y Fm(n)p Fg(j)p Fi(\003)1441 4744 y Fd(n)1441 4790 y(L)1490 4751 y Fp(\))28 b(=)p Black 1947 w(\(2.5\))p Black 768 4959 a(=)974 4892 y Fo(i)p 936 4937 108 4 v 936 5028 a Fp(2)p Fo(\031)1071 4824 y Fj(Z)1126 5049 y Fe(C)1205 4892 y Fo(@)21 b Fp(~)-65 b Fo(')p 1205 4937 121 4 v 1212 5028 a(@)p 1268 4973 50 4 v 5 w(z)1335 4959 y Fp(\()p Fo(z)t Fp(\)[\()p Fo(H)1606 4975 y Fg(j)p Fi(\003)1675 4952 y Fd(n)1675 4998 y(L)1748 4959 y Fp(+)22 b Fo(\025)1903 4974 y Fi(0)1942 4959 y Fp(\))1980 4918 y Fg(\000)p Fm(q)2073 4959 y Fp(\()p Fo(H)2192 4975 y Fg(j)p Fi(\003)2261 4952 y Fd(n)2261 4998 y(L)2333 4959 y Fk(\000)h Fo(z)t Fp(\))2520 4918 y Fg(\000)p Fi(1)2637 4959 y Fk(\000)1517 5167 y Fp(\()p Fo(H)1636 5182 y Fm(n)p Fg(j)p Fi(\003)1748 5159 y Fd(n)1748 5205 y(L)1820 5167 y Fp(+)f Fo(\025)1975 5182 y Fi(0)2014 5167 y Fp(\))2052 5125 y Fg(\000)p Fm(q)2145 5167 y Fp(\()p Fo(H)2264 5182 y Fm(n)p Fg(j)p Fi(\003)2376 5159 y Fd(n)2376 5205 y(L)2448 5167 y Fk(\000)g Fo(z)t Fp(\))2634 5125 y Fg(\000)p Fi(1)2729 5167 y Fp(])p Fo(dz)27 b Fk(^)22 b Fo(d)p 3018 5112 V(z)5 b(:)0 5384 y Fp(In)37 b(the)g(follo)m(wing,)e (w)m(e)j(apply)e(an)h(idea,)g(whic)m(h)g(has)g(already)f(b)s(een)i (used)g(in)e([)p 0 1 0 TeXcolorrgb 2956 5384 a SDict begin H.S end 2956 5384 a 0 1 0 TeXcolorrgb Fp(Kl)p 0 1 0 TeXcolorrgb 3059 5314 a SDict begin H.R end 3059 5314 a 3059 5384 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Kl) cvn H.B /ANN pdfmark end 3059 5384 a Black Fp(])g(and)h([)p 0 1 0 TeXcolorrgb 3343 5384 a SDict begin H.S end 3343 5384 a 0 1 0 TeXcolorrgb Fp(KlP)m(a)p 0 1 0 TeXcolorrgb 3559 5314 a SDict begin H.R end 3559 5314 a 3559 5384 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.KlPa) cvn H.B /ANN pdfmark end 3559 5384 a Black Fp(])f(and)0 5504 y(whic)m(h)24 b(simpli\014es)f(in)g(this)g(situation.)40 b(Using)23 b(resolv)m(en)m(t)i(equalit)m(y)-8 b(,)25 b(the)g(in)m(tegrand)f(in)f (\()p 0 0 1 TeXcolorrgb 3195 5504 a SDict begin H.S end 3195 5504 a 0 0 1 TeXcolorrgb Fp(2.5)p 0 0 1 TeXcolorrgb 3319 5440 a SDict begin H.R end 3319 5440 a 3319 5504 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.5) cvn H.B /ANN pdfmark end 3319 5504 a Black Fp(\))h(is)g(written)p Black 1865 5753 a(5)p Black eop %%Page: 6 6 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 0 -137 a SDict begin H.S end 0 -137 a 0 TeXcolorgray 0 TeXcolorgray 0 -137 a SDict begin H.R end 0 -137 a 0 -137 a SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end 0 -137 a Black Black 308 x Fp(as)120 50 y SDict begin H.S end 120 50 a 120 50 a SDict begin 14.5 H.A end 120 50 a 120 50 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.6) cvn H.B /DEST pdfmark end 120 50 a 140 316 a Fp(\()p Fo(H)259 331 y Fg(j)p Fi(\003)328 308 y Fd(n)328 354 y(L)400 316 y Fp(+)22 b Fo(\025)555 331 y Fi(0)594 316 y Fp(\))632 274 y Fg(\000)p Fm(q)725 316 y Fp(\()p Fo(H)844 331 y Fg(j)p Fi(\003)913 308 y Fd(n)913 354 y(L)985 316 y Fk(\000)h Fo(z)t Fp(\))1172 274 y Fg(\000)p Fi(1)1289 316 y Fk(\000)g Fp(\()p Fo(H)1508 331 y Fm(n)p Fg(j)p Fi(\003)1620 308 y Fd(n)1620 354 y(L)1691 316 y Fp(+)f Fo(\025)1846 331 y Fi(0)1886 316 y Fp(\))1924 274 y Fg(\000)p Fm(q)2017 316 y Fp(\()p Fo(H)2136 331 y Fm(n)p Fg(j)p Fi(\003)2248 308 y Fd(n)2248 354 y(L)2319 316 y Fk(\000)h Fo(z)t Fp(\))2506 274 y Fg(\000)p Fi(1)2629 316 y Fp(=)753 472 y(=)28 b(\()p Fo(H)976 487 y Fm(n)p Fg(j)p Fi(\003)1088 464 y Fd(n)1088 510 y(L)1159 472 y Fp(+)22 b Fo(\025)1314 487 y Fi(0)1354 472 y Fp(\))1392 431 y Fg(\000)p Fm(q)1484 472 y Fp([\()p Fo(H)1630 487 y Fg(j)p Fi(\003)1699 464 y Fd(n)1699 510 y(L)1771 472 y Fk(\000)h Fo(z)t Fp(\))1958 431 y Fg(\000)p Fi(1)2075 472 y Fk(\000)g Fp(\()p Fo(H)2294 487 y Fm(n)p Fg(j)p Fi(\003)2406 464 y Fd(n)2406 510 y(L)2478 472 y Fk(\000)f Fo(z)t Fp(\))2664 431 y Fg(\000)p Fi(1)2759 472 y Fp(])1456 628 y(+)g([\()p Fo(H)1700 643 y Fg(j)p Fi(\003)1769 620 y Fd(n)1769 666 y(L)1841 628 y Fp(+)g Fo(\025)1996 643 y Fi(0)2036 628 y Fp(\))2074 587 y Fg(\000)p Fm(q)2189 628 y Fk(\000)g Fp(\()p Fo(H)2407 643 y Fm(n)p Fg(j)p Fi(\003)2519 620 y Fd(n)2519 666 y(L)2591 628 y Fp(+)g Fo(\025)2746 643 y Fi(0)2785 628 y Fp(\))2823 587 y Fg(\000)p Fm(q)2916 628 y Fp(]\()p Fo(H)3062 643 y Fg(j)p Fi(\003)3131 620 y Fd(n)3131 666 y(L)3203 628 y Fk(\000)h Fo(z)t Fp(\))3390 587 y Fg(\000)p Fi(1)753 784 y Fp(=)28 b Fk(\000)p Fp(\()p Fo(H)1053 799 y Fm(n)p Fg(j)p Fi(\003)1165 776 y Fd(n)1165 822 y(L)1237 784 y Fp(+)22 b Fo(\025)1392 799 y Fi(0)1431 784 y Fp(\))1469 743 y Fg(\000)p Fm(q)1562 784 y Fp(\()p Fo(H)1681 799 y Fm(n)p Fg(j)p Fi(\003)1793 776 y Fd(n)1793 822 y(L)1865 784 y Fk(\000)g Fo(z)t Fp(\))2051 743 y Fg(\000)p Fi(1)2146 784 y Fo(V)2203 799 y Fm(i)2231 784 y Fp(\()p Fo(H)2350 799 y Fg(j)p Fi(\003)2419 776 y Fd(n)2419 822 y(L)2491 784 y Fk(\000)h Fo(z)t Fp(\))2678 743 y Fg(\000)p Fi(1)1456 1023 y Fk(\000)1611 894 y Fm(q)1556 928 y Fj(X)1572 1140 y Fm(l)q Fi(=1)1700 1023 y Fp(\()p Fo(H)1819 1038 y Fm(n)p Fg(j)p Fi(\003)1931 1015 y Fd(n)1931 1061 y(L)2002 1023 y Fp(+)f Fo(\025)2157 1038 y Fi(0)2197 1023 y Fp(\))2235 982 y Fm(l)q Fg(\000)p Fm(q)r Fg(\000)p Fi(1)2440 1023 y Fo(V)2497 1038 y Fm(i)2525 1023 y Fp(\()p Fo(H)2644 1038 y Fg(j)p Fi(\003)2713 1015 y Fd(n)2713 1061 y(L)2785 1023 y Fp(+)g Fo(\025)2940 1038 y Fi(0)2979 1023 y Fp(\))3017 982 y Fg(\000)p Fm(l)3098 1023 y Fp(\()p Fo(H)3217 1038 y Fg(j)p Fi(\003)3286 1015 y Fd(n)3286 1061 y(L)3358 1023 y Fk(\000)g Fo(z)t Fp(\))3544 982 y Fg(\000)p Fi(1)3579 1243 y Fp(\(2.6\))0 1414 y(Estimating)30 b(the)j(trace)g(of)f(\()p 0 0 1 TeXcolorrgb 1061 1414 a SDict begin H.S end 1061 1414 a 0 0 1 TeXcolorrgb Fp(2.6)p 0 0 1 TeXcolorrgb 1186 1350 a SDict begin H.R end 1186 1350 a 1186 1414 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.6) cvn H.B /ANN pdfmark end 1186 1414 a Black Fp(\),)h(w)m(e)g(c)m(ho)s(ose)h Fo(")27 b(>)h Fp(0)k(and)g(write)2434 1294 y SDict begin H.S end 2434 1294 a 2434 1294 a SDict begin 14.5 H.A end 2434 1294 a 2434 1294 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.7) cvn H.B /DEST pdfmark end 2434 1294 a 1240 1587 a Fo(V)1297 1602 y Fm(i)1352 1587 y Fp(=)c Fo(V)1513 1602 y Fm(i)1563 1587 y Fk(\001)22 b Fh(1)1669 1603 y Fg(fj)p Fm(V)1765 1613 y Fd(i)1791 1603 y Fg(j\024)p Fm(")p Fg(g)1960 1587 y Fp(+)g Fo(V)2115 1602 y Fm(i)2165 1587 y Fk(\001)g Fh(1)2271 1603 y Fg(fj)p Fm(V)2367 1613 y Fd(i)2393 1603 y Fg(j)p Fm(>")p Fg(g)3579 1587 y Fp(\(2.7\))0 1760 y(and)35 b(note,)i(that)e Fo(V)711 1775 y Fm(i)763 1760 y Fk(\001)24 b Fh(1)871 1776 y Fg(fj)p Fm(V)967 1786 y Fd(i)993 1776 y Fg(j\024)p Fm(")p Fg(g)1175 1760 y Fp(is)35 b(b)s(ounded)h(b)m(y)g Fk(k)p Fo(V)1922 1775 y Fm(i)1974 1760 y Fk(\001)23 b Fh(1)2081 1776 y Fg(fj)p Fm(V)2177 1786 y Fd(i)2204 1776 y Fg(j\024)p Fm(")p Fg(g)2350 1760 y Fk(k)32 b(\024)h Fo(")p Fp(.)52 b(As)36 b Fo(V)56 b Fp(is)35 b(non-negativ)m(e,)h(one)0 1880 y(has)174 1760 y SDict begin H.S end 174 1760 a 174 1760 a SDict begin 14.5 H.A end 174 1760 a 174 1760 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.8) cvn H.B /DEST pdfmark end 174 1760 a 195 2104 a Fp(supp)q(\()p Fo(V)491 2119 y Fm(i)541 2104 y Fk(\001)22 b Fh(1)647 2120 y Fg(fj)p Fm(V)743 2130 y Fd(i)769 2120 y Fg(j)p Fm(>")p Fg(g)915 2104 y Fp(\))28 b Fk(\022)1126 1980 y Fm(n)1092 2010 y Fj([)1086 2220 y Fm(j)t Fi(=1)1261 1980 y Fm(n)1227 2010 y Fj([)1225 2220 y Fm(i)p Fi(=1)1227 2276 y Fm(i)p Fg(6)p Fi(=)p Fm(j)1356 1964 y Fj(\032)1431 2104 y Fp(\()p Fo(x)1524 2119 y Fi(1)1564 2104 y Fo(;)17 b(:)g(:)g(:)f(;)h(x)1838 2119 y Fm(n)1885 2104 y Fp(\))27 b Fk(2)h Fn(R)2110 2063 y Fm(nd)2227 2104 y Fp(:)45 b Fo(V)21 b Fp(\()p Fo(x)2470 2119 y Fm(i)2521 2104 y Fk(\000)h Fo(x)2675 2119 y Fm(j)2712 2104 y Fp(\))28 b Fk(\025)3052 2037 y Fo(")p 2893 2081 363 4 v 2893 2173 a(n)p Fp(\()p Fo(n)23 b Fk(\000)f Fp(1\))3266 1964 y Fj(\033)3357 2104 y Fo(:)195 b Fp(\(2.8\))0 2434 y(As,)25 b(b)m(y)e(assumption)f(\(H.2\),)i Fo(V)44 b Fp(tends)23 b(to)f(0)g(at)g(in\014nit)m(y)-8 b(,)24 b(\()p 0 0 1 TeXcolorrgb 2095 2434 a SDict begin H.S end 2095 2434 a 0 0 1 TeXcolorrgb Fp(2.8)p 0 0 1 TeXcolorrgb 2219 2370 a SDict begin H.R end 2219 2370 a 2219 2434 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.8) cvn H.B /ANN pdfmark end 2219 2434 a Black Fp(\))e(implies)e(that)i(there)h(exists)g(0)28 b Fo(<)f(C)7 b Fp(\()p Fo(n)p Fp(;)17 b Fo(")p Fp(\))0 2555 y(\(indep)s(enden)m(t)34 b(of)e Fo(L)p Fp(\))h(suc)m(h)h(that)1270 2434 y SDict begin H.S end 1270 2434 a 1270 2434 a SDict begin 14.5 H.A end 1270 2434 a 1270 2434 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.9) cvn H.B /DEST pdfmark end 1270 2434 a 1121 2728 a Fo(\026)p Fp(\()p Fk(fj)p Fo(V)1353 2743 y Fm(i)1380 2728 y Fk(j)28 b Fo(>)f(")p Fk(g)22 b(\\)g Fp(\003)1813 2686 y Fm(n)1813 2752 y(L)1865 2728 y Fp(\))28 b Fk(\024)g Fo(C)7 b Fp(\()p Fo(n;)17 b(")p Fp(\))p Fo(L)2403 2686 y Fi(\()p Fm(n)p Fg(\000)p Fi(1\))p Fm(d)2631 2728 y Fo(;)921 b Fp(\(2.9\))0 2900 y(where)34 b Fo(\026)p Fp(\()p Fk(\001)p Fp(\))e(denotes)h(the)g(Leb)s(esgue)h (measure.)44 b(Using)32 b(decomp)s(osition)f(\()p 0 0 1 TeXcolorrgb 2792 2900 a SDict begin H.S end 2792 2900 a 0 0 1 TeXcolorrgb Fp(2.7)p 0 0 1 TeXcolorrgb 2917 2836 a SDict begin H.R end 2917 2836 a 2917 2900 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.7) cvn H.B /ANN pdfmark end 2917 2900 a Black Fp(\))h(of)g Fo(V)3155 2915 y Fm(i)3183 2900 y Fp(,)h(w)m(e)g(obtain)3690 2780 y SDict begin H.S end 3690 2780 a 3690 2780 a SDict begin 14.5 H.A end 3690 2780 a 3690 2780 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.10) cvn H.B /DEST pdfmark end 3690 2780 a 153 3070 a Fk(j)p Fp(T)-8 b(race\()p Fo(H)535 3086 y Fm(n)p Fg(j)p Fi(\003)647 3063 y Fd(n)647 3109 y(L)719 3070 y Fp(+)22 b Fo(\025)874 3085 y Fi(0)913 3070 y Fp(\))951 3029 y Fg(\000)p Fm(q)1044 3070 y Fp(\()p Fo(H)1163 3086 y Fm(n)p Fg(j)p Fi(\003)1275 3063 y Fd(n)1275 3109 y(L)1347 3070 y Fk(\000)h Fo(z)t Fp(\))1534 3029 y Fg(\000)p Fi(1)1629 3070 y Fo(V)1686 3085 y Fm(i)1714 3070 y Fp(\()p Fo(H)1833 3086 y Fg(j)p Fi(\003)1902 3063 y Fd(n)1902 3109 y(L)1974 3070 y Fk(\000)f Fo(z)t Fp(\))2160 3029 y Fg(\000)p Fi(1)2255 3070 y Fk(j)444 3252 y(\024)645 3185 y Fo(")p 559 3229 217 4 v 559 3321 a Fk(j=)p Fo(z)t Fk(j)736 3292 y Fi(2)786 3252 y Fp(T)-8 b(race)p Fk(j)p Fp(\()p Fo(H)1168 3268 y Fm(n)p Fg(j)p Fi(\003)1280 3245 y Fd(n)1280 3291 y(L)1352 3252 y Fp(+)22 b Fo(\025)1507 3267 y Fi(0)1546 3252 y Fp(\))1584 3211 y Fg(\000)p Fm(q)1677 3252 y Fk(j)1147 3510 y Fp(+)1319 3443 y(1)p 1255 3488 178 4 v 1255 3579 a Fk(j=)p Fo(z)t Fk(j)1442 3510 y(k)p Fo(V)1549 3525 y Fm(i)1577 3510 y Fp(\()p Fo(H)1696 3526 y Fg(j)p Fi(\003)1765 3503 y Fd(n)1765 3549 y(L)1837 3510 y Fk(\000)h Fo(z)t Fp(\))2024 3469 y Fg(\000)p Fi(1)2119 3510 y Fk(k)f(\001)f Fp(T)-8 b(race)q Fk(j)p Fp(\()p Fo(H)2623 3526 y Fm(n)p Fg(j)p Fi(\003)2735 3503 y Fd(n)2735 3549 y(L)2806 3510 y Fp(+)22 b Fo(\025)2961 3525 y Fi(0)3001 3510 y Fp(\))3039 3469 y Fg(\000)p Fm(q)3132 3510 y Fh(1)3188 3526 y Fg(fj)p Fm(V)3284 3536 y Fd(i)3310 3526 y Fg(j)p Fm(>")p Fg(g\\)p Fi(\003)3549 3503 y Fd(n)3549 3549 y(L)3599 3510 y Fk(j)444 3768 y(\024)645 3701 y Fo(")p 559 3746 217 4 v 559 3837 a Fk(j=)p Fo(z)t Fk(j)736 3808 y Fi(2)786 3768 y Fk(k)p Fp(\()p Fo(H)955 3784 y Fm(n)p Fg(j)p Fi(\003)1067 3761 y Fd(n)1067 3807 y(L)1138 3768 y Fp(+)g Fo(\025)1293 3783 y Fi(0)1333 3768 y Fp(\))1371 3727 y Fg(\000)p Fi(1)1465 3768 y Fk(k)1515 3721 y Fm(q)1515 3795 y Fg(T)1554 3803 y Fd(q)1614 3768 y Fp(+)1806 3701 y(1)p 1722 3746 V 1722 3837 a Fk(j=)p Fo(z)t Fk(j)1899 3808 y Fi(2)1949 3768 y Fk(k)p Fp(\()p Fo(H)2118 3784 y Fm(n)p Fg(j)p Fi(\003)2230 3761 y Fd(n)2230 3807 y(L)2301 3768 y Fp(+)g Fo(\025)2456 3783 y Fi(0)2496 3768 y Fp(\))2534 3727 y Fg(\000)p Fi(1)2628 3768 y Fk(k)2678 3721 y Fm(q)r Fg(\000)p Fi(1)2678 3795 y Fg(T)2717 3803 y Fd(q)1383 3979 y Fk(\001)g(k)p Fp(\()p Fo(H)1602 3995 y Fm(n)p Fg(j)p Fi(\003)1714 3972 y Fd(n)1714 4018 y(L)1786 3979 y Fp(+)g Fo(\025)1941 3994 y Fi(0)1980 3979 y Fp(\))2018 3938 y Fg(\000)p Fi(1)2112 3979 y Fh(1)2168 3995 y Fg(fj)p Fm(V)2264 4005 y Fd(i)2291 3995 y Fg(j)p Fm(>")p Fg(g\\)p Fi(\003)2530 3972 y Fd(n)2530 4018 y(L)2580 3979 y Fk(k)2630 3994 y Fg(T)2669 4002 y Fd(q)2729 3979 y Fk(\001)g(k)p Fo(V)2886 3994 y Fm(i)2914 3979 y Fp(\()p Fo(H)3033 3995 y Fm(n)p Fg(j)p Fi(\003)3145 3972 y Fd(n)3145 4018 y(L)3216 3979 y Fp(+)g Fo(\025)3371 3994 y Fi(0)3411 3979 y Fp(\))3449 3938 y Fg(\000)p Fi(1)3543 3979 y Fk(k)3530 4110 y Fp(\(2.10\))0 4282 y(where)i Fk(k)r(\001)r(k)404 4297 y Fg(T)443 4305 y Fd(q)504 4282 y Fp(denotes)h(the)e Fo(q)t Fp(-th)f(Sc)m(hatten)i(class)f(norm)f(\(see)i([)p 0 1 0 TeXcolorrgb 2271 4282 a SDict begin H.S end 2271 4282 a 0 1 0 TeXcolorrgb Fp(Si)p 0 1 0 TeXcolorrgb 2353 4214 a SDict begin H.R end 2353 4214 a 2353 4282 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Si1) cvn H.B /ANN pdfmark end 2353 4282 a Black Fp(]\))f(and)g(w)m(e)h(used)g(H\177)-49 b(older's)23 b(inequalit)m(y)-8 b(.)0 4402 y(In)33 b(the)g(same)f(w)m (a)m(y)-8 b(,)34 b(the)f(cyclicit)m(y)f(of)g(the)h(trace)g(yields)2098 4282 y SDict begin H.S end 2098 4282 a 2098 4282 a SDict begin 14.5 H.A end 2098 4282 a 2098 4282 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.11) cvn H.B /DEST pdfmark end 2098 4282 a 49 4570 a Fk(j)p Fp(T)-8 b(race\()p Fo(H)431 4585 y Fm(n)p Fg(j)p Fi(\003)543 4562 y Fd(n)543 4608 y(L)615 4570 y Fp(+)22 b Fo(\025)770 4585 y Fi(0)809 4570 y Fp(\))847 4529 y Fm(l)q Fg(\000)p Fm(q)r Fg(\000)p Fi(1)1052 4570 y Fo(V)1109 4585 y Fm(i)1137 4570 y Fp(\()p Fo(H)1256 4585 y Fg(j)p Fi(\003)1325 4562 y Fd(n)1325 4608 y(L)1397 4570 y Fp(+)g Fo(\025)1552 4585 y Fi(0)1591 4570 y Fp(\))1629 4529 y Fg(\000)p Fm(l)1710 4570 y Fp(\()p Fo(H)1829 4585 y Fg(j)p Fi(\003)1898 4562 y Fd(n)1898 4608 y(L)1970 4570 y Fk(\000)h Fo(z)t Fp(\))2157 4529 y Fg(\000)p Fi(1)2252 4570 y Fk(j)340 4729 y(\024)28 b Fp(T)-8 b(race)q Fk(j)p Fp(\()p Fo(H)828 4744 y Fg(j)p Fi(\003)897 4721 y Fd(n)897 4767 y(L)968 4729 y Fp(+)22 b Fo(\025)1123 4744 y Fi(0)1163 4729 y Fp(\))1201 4688 y Fg(\000)p Fm(l)1282 4729 y Fp(\()p Fo(H)1401 4744 y Fm(n)p Fg(j)p Fi(\003)1513 4721 y Fd(n)1513 4767 y(L)1584 4729 y Fp(+)g Fo(\025)1739 4744 y Fi(0)1779 4729 y Fp(\))1817 4688 y Fm(l)q Fg(\000)p Fm(q)r Fg(\000)p Fi(1)2022 4729 y Fo(V)2079 4744 y Fm(i)2107 4729 y Fp(\()p Fo(H)2226 4744 y Fg(j)p Fi(\003)2295 4721 y Fd(n)2295 4767 y(L)2366 4729 y Fk(\000)h Fo(z)t Fp(\))2553 4688 y Fg(\000)p Fi(1)2648 4729 y Fk(j)340 4888 y(\024)28 b(k)p Fp(\()p Fo(H)614 4903 y Fg(j)p Fi(\003)683 4880 y Fd(n)683 4926 y(L)755 4888 y Fp(+)22 b Fo(\025)910 4903 y Fi(0)949 4888 y Fp(\))987 4847 y Fg(\000)p Fm(l)1068 4888 y Fp(\()p Fo(H)1187 4903 y Fm(n)p Fg(j)p Fi(\003)1299 4880 y Fd(n)1299 4926 y(L)1371 4888 y Fp(+)g Fo(\025)1526 4903 y Fi(0)1565 4888 y Fp(\))1603 4847 y Fm(l)1629 4888 y Fk(k)g(\001)g Fp(T)-8 b(race)p Fk(j)p Fp(\()p Fo(H)2133 4903 y Fm(n)p Fg(j)p Fi(\003)2245 4880 y Fd(n)2245 4926 y(L)2317 4888 y Fp(+)22 b Fo(\025)2472 4903 y Fi(0)2511 4888 y Fp(\))2549 4847 y Fg(\000)p Fm(q)r Fg(\000)p Fi(1)2732 4888 y Fo(V)2789 4903 y Fm(i)2817 4888 y Fp(\()p Fo(H)2936 4903 y Fg(j)p Fi(\003)3005 4880 y Fd(n)3005 4926 y(L)3077 4888 y Fk(\000)h Fo(z)t Fp(\))3264 4847 y Fg(\000)p Fi(1)3359 4888 y Fk(j)340 5095 y(\024)505 5028 y Fo(C)p 455 5072 178 4 v 455 5164 a Fk(j=)p Fo(z)t Fk(j)642 5095 y(k)p Fp(\()p Fo(H)811 5111 y Fm(n)p Fg(j)p Fi(\003)923 5088 y Fd(n)923 5134 y(L)995 5095 y Fp(+)f Fo(\025)1150 5110 y Fi(0)1189 5095 y Fp(\))1227 5054 y Fg(\000)p Fi(1)1321 5095 y Fk(k)1371 5048 y Fm(q)r Fg(\000)p Fi(1)1371 5122 y Fg(T)1410 5130 y Fd(q)1521 5095 y Fk(\001)g(k)p Fp(\()p Fo(H)1740 5111 y Fm(n)p Fg(j)p Fi(\003)1852 5088 y Fd(n)1852 5134 y(L)1924 5095 y Fp(+)g Fo(\025)2079 5110 y Fi(0)2118 5095 y Fp(\))2156 5054 y Fg(\000)p Fi(1)2250 5095 y Fh(1)2306 5111 y Fg(fj)p Fm(V)2402 5121 y Fd(i)2429 5111 y Fg(j)p Fm(>")p Fg(g\\)p Fi(\003)2668 5088 y Fd(n)2668 5134 y(L)2717 5095 y Fk(k)2767 5110 y Fg(T)2806 5118 y Fd(q)2867 5095 y Fk(\001)g(k)p Fo(V)3024 5110 y Fm(i)3052 5095 y Fp(\()p Fo(H)3171 5111 y Fm(n)p Fg(j)p Fi(\003)3283 5088 y Fd(n)3283 5134 y(L)3354 5095 y Fp(+)g Fo(\025)3509 5110 y Fi(0)3549 5095 y Fp(\))3587 5054 y Fg(\000)p Fi(1)3681 5095 y Fk(k)2224 5332 y Fp(+)g Fo(C)2475 5265 y(")p 2409 5309 V 2409 5400 a Fk(j=)p Fo(z)t Fk(j)2596 5332 y(k)p Fp(\()p Fo(H)2765 5347 y Fm(n)p Fg(j)p Fi(\003)2877 5325 y Fd(n)2877 5371 y(L)2949 5332 y Fp(+)g Fo(\025)3104 5347 y Fi(0)3143 5332 y Fp(\))3181 5291 y Fg(\000)p Fi(1)3275 5332 y Fk(k)3325 5285 y Fm(q)3325 5359 y Fg(T)3364 5367 y Fd(q)3402 5332 y Fo(:)3530 5518 y Fp(\(2.11\))p Black 1865 5753 a(6)p Black eop %%Page: 7 7 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 0 -137 a SDict begin H.S end 0 -137 a 0 TeXcolorgray 0 TeXcolorgray 0 -137 a SDict begin H.R end 0 -137 a 0 -137 a SDict begin [ /View [/XYZ H.V] /Dest (page.7) cvn H.B /DEST pdfmark end 0 -137 a Black Black 308 x Fp(W)-8 b(e)29 b(are)f(no)m(w)h(left)e (with)h(estimating)e Fk(k)p Fp(\()p Fo(H)1547 186 y Fm(n)p Fg(j)p Fi(\003)1659 163 y Fd(n)1659 209 y(L)1721 171 y Fp(+)13 b Fo(\025)1867 186 y Fi(0)1906 171 y Fp(\))1944 134 y Fg(\000)p Fi(1)2039 171 y Fk(k)2089 186 y Fg(T)2128 194 y Fd(q)2194 171 y Fp(and)28 b Fk(k)p Fp(\()p Fo(H)2548 186 y Fm(n)p Fg(j)p Fi(\003)2660 163 y Fd(n)2660 209 y(L)2723 171 y Fp(+)13 b Fo(\025)2869 186 y Fi(0)2908 171 y Fp(\))2946 134 y Fg(\000)p Fi(1)3040 171 y Fh(1)3096 186 y Fg(fj)p Fm(V)3192 196 y Fd(i)3219 186 y Fg(j)p Fm(>")p Fg(g\\)p Fi(\003)3458 163 y Fd(n)3458 209 y(L)3508 171 y Fk(k)3558 186 y Fg(T)3597 194 y Fd(q)3663 171 y Fp(for)0 291 y Fo(q)36 b Fp(su\016cien)m(tly)e(large,)d(dep)s(ending)i (on)f Fo(nd)p Fp(.)44 b(Therefore,)34 b(w)m(e)f(compute)83 501 y Fk(k)p Fp(\()p Fo(H)252 517 y Fm(n)p Fg(j)p Fi(\003)364 494 y Fd(n)364 540 y(L)436 501 y Fp(+)22 b Fo(\025)591 516 y Fi(0)630 501 y Fp(\))668 460 y Fg(\000)p Fi(1)762 501 y Fh(1)818 517 y Fg(fj)p Fm(V)914 527 y Fd(i)941 517 y Fg(j)p Fm(>")p Fg(g\\)p Fi(\003)1180 494 y Fd(n)1180 540 y(L)1229 501 y Fk(k)1279 516 y Fg(T)1318 524 y Fd(q)1361 501 y Fk(\024)28 b(k)p Fp(\()p Fo(H)1635 517 y Fm(n)p Fg(j)p Fi(\003)1747 494 y Fd(n)1747 540 y(L)1818 501 y Fp(+)22 b Fo(\025)1973 516 y Fi(0)2013 501 y Fp(\))2051 460 y Fg(\000)p Fi(1)2145 501 y Fp(\()p Fk(\000)h(4)2372 517 y Fg(j)p Fi(\003)2441 494 y Fd(n)2441 540 y(L)2512 501 y Fp(+)p Fo(\025)2645 516 y Fi(0)2684 501 y Fp(\))2732 433 y Ff(1)p 2732 445 31 3 v 2732 486 a(2)2777 501 y Fk(k)2827 516 y Fg(T)2866 525 y Ff(2)p Fd(q)2036 674 y Fk(\001)f(k)p Fp(\()p Fk(\000)g(4)2362 689 y Fg(j)p Fi(\003)2431 666 y Fd(n)2431 712 y(L)2503 674 y Fp(+)p Fo(\025)2636 689 y Fi(0)2675 674 y Fp(\))2713 632 y Fg(\000)2778 605 y Ff(1)p 2778 617 V 2778 658 a(2)2822 674 y Fh(1)2878 689 y Fg(fj)p Fm(V)2974 699 y Fd(i)3001 689 y Fg(j)p Fm(>")p Fg(g\\)p 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1688 y(L)1165 1649 y Fp(+)p Fo(V)1319 1608 y Fg(\000)1298 1680 y Fm(n)p Fg(j)p Fi(\003)1410 1658 y Fd(n)1410 1704 y(L)1481 1649 y Fp(+)f Fo(\025)1636 1664 y Fi(0)1703 1649 y Fk(\025)1819 1582 y Fp(1)p 1819 1626 V 1819 1717 a(2)1877 1649 y(\()p Fk(\000)h(4)2104 1665 y Fg(j)p Fi(\003)2173 1642 y Fd(n)2173 1688 y(L)2245 1649 y Fk(\000)p Fp(2)p Fo(\015)k Fp(+)22 b(2)p Fo(\025)2653 1664 y Fi(0)2692 1649 y Fp(\))28 b Fk(\025)2873 1582 y Fp(1)p 2873 1626 V 2873 1717 a(2)2932 1649 y(\()p Fk(\000)22 b(4)3158 1665 y Fg(j)p Fi(\003)3227 1642 y Fd(n)3227 1688 y(L)3299 1649 y Fp(+)p Fo(\025)3432 1664 y Fi(0)3471 1649 y Fp(\))p Fo(:)0 1860 y Fp(Th)m(us,)34 b(the)f(op)s(erator)f Fo(H)916 1875 y Fm(n)p Fg(j)p Fi(\003)1028 1853 y Fd(n)1028 1899 y(L)1100 1860 y Fp(+)22 b Fo(\025)1255 1875 y Fi(0)1327 1860 y Fp(is)32 b(in)m(v)m(ertible)g(and)1135 2056 y(\()p Fo(H)1254 2071 y Fm(n)p Fg(j)p Fi(\003)1366 2048 y Fd(n)1366 2094 y(L)1438 2056 y Fp(+)22 b Fo(\025)1593 2071 y Fi(0)1632 2056 y Fp(\))1670 2015 y Fg(\000)p Fi(1)1792 2056 y Fk(\024)28 b Fp(2\()p Fk(\000)22 b(4)2172 2071 y Fg(j)p Fi(\003)2241 2048 y Fd(n)2241 2094 y(L)2313 2056 y Fp(+)p Fo(\025)2446 2071 y Fi(0)2485 2056 y Fp(\))2523 2015 y Fg(\000)p Fi(1)2618 2056 y Fo(:)0 2241 y Fp(Let)28 b(\()p Fo(\026)267 2256 y Fm(j)p 267 2281 37 3 v 303 2241 a Fp(\))341 2256 y Fm(j)p 341 2281 V 405 2241 a Fp(and)g(\()p Fo(\036)686 2256 y Fm(j)p 686 2281 V 722 2241 a Fp(\))760 2256 y Fm(j)p 760 2281 V 824 2241 a Fp(resp)s(ectiv)m(ely)h(denote)f(the)g (eigen)m(v)-5 b(alues)28 b(and)g(eigenfunctions)f(of)h(the)g(Diric)m (hlet)0 2375 y(Laplacian)43 b Fk(\0004)624 2390 y Fg(j)p Fi(\003)693 2367 y Fd(n)693 2413 y(L)789 2375 y Fp(\(the)i(index)h Fo(j)p 1283 2410 46 4 v 51 w Fp(runs)g(o)m(v)m(er)g(\()p Fn(N)1929 2339 y Fm(nd)2018 2375 y Fp(\))2056 2339 y Fg(\003)2095 2375 y Fp(\).)82 b(F)-8 b(or)44 b Fo(q)53 b Fk(2)d Fn(N)60 b Fp(suc)m(h)46 b(that)f(2)p Fo(q)53 b(>)c(nd)p Fp(,)f(w)m(e)0 2495 y(compute)89 2692 y Fk(k)p Fp(\()p Fo(H)258 2708 y Fm(n)p Fg(j)p Fi(\003)370 2685 y Fd(n)370 2731 y(L)442 2692 y Fp(+)22 b Fo(\025)597 2707 y Fi(0)636 2692 y Fp(\))674 2651 y Fg(\000)p Fi(1)769 2692 y Fp(\()p Fk(\000)g(4)995 2708 y Fg(j)p Fi(\003)1064 2685 y Fd(n)1064 2731 y(L)1136 2692 y Fp(+)p Fo(\025)1269 2707 y Fi(0)1308 2692 y Fp(\))1356 2624 y Ff(1)p 1356 2636 31 3 v 1356 2677 a(2)1400 2692 y Fk(k)1450 2645 y Fi(2)p Fm(q)1450 2719 y Fg(T)1489 2728 y Ff(2)p Fd(q)1586 2692 y Fp(=)1718 2598 y Fj(X)1689 2817 y Fm(j)p 1689 2842 33 3 v 4 w Fg(2)p Fe(N)1813 2798 y Fd(nd)1891 2692 y Fp(\()p Fo(\026)1988 2707 y Fm(j)p 1988 2732 37 3 v 2024 2692 a Fp(\()p Fk(\0004)2228 2708 y Fg(j)p Fi(\003)2297 2685 y Fd(n)2297 2731 y(L)2347 2692 y Fp(\))g(+)g Fo(\025)2562 2707 y Fi(0)2602 2692 y Fp(\))2640 2651 y Fm(q)2678 2692 y Fk(h)p Fo(\036)2775 2707 y Fm(j)p 2775 2732 V 2811 2692 a Fo(;)17 b Fp(\()p Fo(H)2974 2708 y Fm(n)p Fg(j)p Fi(\003)3086 2685 y Fd(n)3086 2731 y(L)3157 2692 y Fp(+)22 b Fo(\025)3312 2707 y Fi(0)3352 2692 y Fp(\))3390 2651 y Fg(\000)p Fi(1)3484 2692 y Fo(\036)3542 2707 y Fm(j)p 3542 2732 V 3578 2692 a Fk(i)3617 2651 y Fi(2)p Fm(q)796 2993 y Fk(\024)28 b Fp(2)950 2952 y Fi(2)p Fm(q)1069 2898 y Fj(X)1040 3118 y Fm(j)p 1040 3143 33 3 v 3 w Fg(2)p Fe(N)1163 3099 y Fd(nd)1242 2993 y Fp(\()p Fo(\026)1339 3008 y Fm(j)p 1339 3033 37 3 v 1375 2993 a Fp(\()p Fk(\0004)1579 3009 y Fg(j)p Fi(\003)1648 2986 y Fd(n)1648 3032 y(L)1698 2993 y Fp(\))22 b(+)g Fo(\025)1913 3008 y Fi(0)1952 2993 y Fp(\))1990 2952 y Fm(q)2028 2993 y Fk(h)p Fo(\036)2125 3008 y Fm(j)p 2125 3033 V 2161 2993 a Fo(;)17 b Fp(\()p Fk(\000)23 b(4)2432 3009 y Fg(j)p Fi(\003)2501 2986 y Fd(n)2501 3032 y(L)2572 2993 y Fp(+)p Fo(\025)2705 3008 y Fi(0)2745 2993 y Fp(\))2783 2952 y Fg(\000)p Fi(1)2877 2993 y Fo(\036)2935 3008 y Fm(j)p 2935 3033 V 2971 2993 a Fk(i)p Fp(\))3048 2952 y Fi(2)p Fm(q)796 3294 y Fp(=)28 b(2)949 3252 y Fi(2)p Fm(q)1067 3199 y Fj(X)1038 3418 y Fm(j)p 1038 3443 33 3 v 4 w Fg(2)p Fe(N)1162 3400 y Fd(nd)1240 3294 y Fp(\()p Fo(\026)1337 3309 y Fm(j)p 1337 3333 37 3 v 1373 3294 a Fp(\()p Fk(\0004)1577 3309 y Fg(j)p Fi(\003)1646 3286 y Fd(n)1646 3332 y(L)1696 3294 y Fp(\))23 b(+)f Fo(\025)1912 3309 y Fi(0)1951 3294 y Fp(\))1989 3252 y Fg(\000)p Fm(q)2109 3294 y Fk(\024)29 b Fo(C)7 b(L)2358 3252 y Fm(nd)2441 3294 y Fo(:)0 3599 y Fp(The)37 b(last)e(estimate)h(is)f(a)h(direct)g(computation)f(using)h (the)g(explicit)f(form)g(of)h(the)g(Diric)m(hlet)e(eigen-)0 3719 y(v)-5 b(alues.)0 3840 y(By)32 b(Lemma)e(2.2)i(in)e([)p 0 1 0 TeXcolorrgb 795 3840 a SDict begin H.S end 795 3840 a 0 1 0 TeXcolorrgb Fp(KlP)m(a)p 0 1 0 TeXcolorrgb 1011 3770 a SDict begin H.R end 1011 3770 a 1011 3840 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.KlPa) cvn H.B /ANN pdfmark end 1011 3840 a Black Fp(],)i(w)m(e)g(kno)m(w)h(that,)f(for)f Fo(q)g Fk(2)d Fn(N)47 b Fp(suc)m(h)33 b(that)e(2)p Fo(q)h(>)27 b(nd)p Fp(,)32 b(there)g(exists)h Fo(C)3562 3855 y Fm(q)3627 3840 y Fo(>)28 b Fp(0)0 3960 y(suc)m(h)34 b(that,)e(for)h(an)m(y)g (measurable)f(subset)i(\003)1669 3924 y Fg(0)1720 3960 y Fk(\022)28 b Fp(\003)1893 3924 y Fm(n)1893 3985 y(L)1945 3960 y Fp(,)k(one)h(has)2357 3840 y SDict begin H.S end 2357 3840 a 2357 3840 a SDict begin 14.5 H.A end 2357 3840 a 2357 3840 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.13) cvn H.B /DEST pdfmark end 2357 3840 a 1130 4159 a Fk(k)p Fp(\()p Fk(\000)23 b(4)1407 4174 y Fg(j)p Fi(\003)1476 4152 y Fd(n)1476 4198 y(L)1547 4159 y Fp(+)p Fo(\025)1680 4174 y Fi(0)1719 4159 y Fp(\))1757 4118 y Fg(\000)1822 4091 y Ff(1)p 1822 4103 31 3 v 1822 4144 a(2)1867 4159 y Fh(1)1923 4174 y Fi(\003)1972 4155 y Fa(0)1998 4159 y Fk(k)2048 4112 y Fi(2)p Fm(q)2048 4186 y Fg(T)2087 4195 y Ff(2)p Fd(q)2184 4159 y Fk(\024)28 b Fo(C)2359 4174 y Fm(q)2397 4159 y Fo(\026)p Fp(\(\003)2562 4118 y Fg(0)2584 4159 y Fp(\))p Fo(:)881 b Fp(\(2.13\))0 4353 y(Cho)s(osing)26 b(\003)488 4317 y Fg(0)539 4353 y Fp(=)h Fk(fj)p Fo(V)777 4368 y Fm(i)805 4353 y Fk(j)g Fo(>)h(")p Fk(g)10 b(\\)g Fp(\003)1214 4317 y Fm(n)1214 4378 y(L)1293 4353 y Fp(and)27 b(taking)f(\()p 0 0 1 TeXcolorrgb 1810 4353 a SDict begin H.S end 1810 4353 a 0 0 1 TeXcolorrgb Fp(2.9)p 0 0 1 TeXcolorrgb 1934 4289 a SDict begin H.R end 1934 4289 a 1934 4353 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.9) cvn H.B /ANN pdfmark end 1934 4353 a Black Fp(\))h(in)m(to)f(accoun)m(t,)j(then)e(b)m(y)h(com)m(bining)d (estimates)0 4473 y(\()p 0 0 1 TeXcolorrgb 38 4473 a SDict begin H.S end 38 4473 a 0 0 1 TeXcolorrgb Fp(2.10)p 0 0 1 TeXcolorrgb 211 4409 a SDict begin H.R end 211 4409 a 211 4473 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.10) cvn H.B /ANN pdfmark end 211 4473 a Black Fp(\){\()p 0 0 1 TeXcolorrgb 336 4473 a SDict begin H.S end 336 4473 a 0 0 1 TeXcolorrgb Fp(2.13)p 0 0 1 TeXcolorrgb 509 4409 a SDict begin H.R end 509 4409 a 509 4473 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.13) cvn H.B /ANN pdfmark end 509 4473 a Black Fp(\),)f(w)m(e)f(get)f(that)g(there)g(exists)h Fo(c)p Fp(,)h(dep)s(ending)e(only)f(on)h Fo(q)k Fp(\(and)c(the)g(b)s (ound)g(in)f(assumption)0 4594 y(\(H.3\)\),)32 b(suc)m(h)i(that)83 4804 y(T)-8 b(race)q Fk(j)p Fp(\()p Fo(H)466 4820 y Fg(j)p Fi(\003)535 4797 y Fd(n)535 4843 y(L)606 4804 y Fp(+)22 b Fo(\025)761 4819 y Fi(0)801 4804 y Fp(\))839 4763 y Fg(\000)p Fm(q)932 4804 y Fp(\()p Fo(H)1051 4820 y Fg(j)p Fi(\003)1120 4797 y Fd(n)1120 4843 y(L)1191 4804 y Fk(\000)h Fo(z)t Fp(\))1378 4763 y Fg(\000)p Fi(1)1495 4804 y Fk(\000)g Fp(\()p Fo(H)1714 4820 y Fm(n)p Fg(j)p Fi(\003)1826 4797 y Fd(n)1826 4843 y(L)1898 4804 y Fp(+)f Fo(\025)2053 4819 y Fi(0)2092 4804 y Fp(\))2130 4763 y Fg(\000)p Fm(q)2223 4804 y Fp(\()p Fo(H)2342 4820 y Fm(n)p Fg(j)p Fi(\003)2454 4797 y Fd(n)2454 4843 y(L)2526 4804 y Fk(\000)g Fo(z)t Fp(\))2712 4763 y Fg(\000)p Fi(1)2807 4804 y Fk(j)1350 5008 y(\024)28 b Fo(c)p Fp(\()1630 4940 y Fo(")p 1545 4985 217 4 v 1545 5076 a Fk(j=)p Fo(z)t Fk(j)1722 5047 y Fi(2)1771 5008 y Fo(L)1837 4966 y Fm(nd)1943 5008 y Fp(+)2135 4940 y(1)p 2051 4985 V 2051 5076 a Fk(j=)p Fo(z)t Fk(j)2228 5047 y Fi(2)2277 5008 y Fo(L)2343 4961 y Fm(nd)p Fg(\000)2502 4934 y Fd(d)p 2488 4946 61 3 v 2488 4987 a Ff(2)p Fd(q)2585 5008 y Fp(+)2758 4940 y Fo(")p 2693 4985 178 4 v 2693 5076 a Fk(j=)p Fo(z)t Fk(j)2880 5008 y Fo(L)2946 4966 y Fm(nd)3051 5008 y Fp(+)3223 4940 y(1)p 3159 4985 V 3159 5076 a Fk(j=)p Fo(z)t Fk(j)3346 5008 y Fo(L)3412 4961 y Fm(nd)p Fg(\000)3571 4934 y Fd(d)p 3557 4946 61 3 v 3557 4987 a Ff(2)p Fd(q)3631 5008 y Fp(\))p Fo(:)0 5263 y Fp(Using)39 b(this)g(inequalit)m(y)f(in)h(\()p 0 0 1 TeXcolorrgb 1096 5263 a SDict begin H.S end 1096 5263 a 0 0 1 TeXcolorrgb Fp(2.5)p 0 0 1 TeXcolorrgb 1221 5199 a SDict begin H.R end 1221 5199 a 1221 5263 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.5) cvn H.B /ANN pdfmark end 1221 5263 a Black Fp(\),)i(w)m(e)f(get)f(\()p 0 0 1 TeXcolorrgb 1684 5263 a SDict begin H.S end 1684 5263 a 0 0 1 TeXcolorrgb Fp(2.3)p 0 0 1 TeXcolorrgb 1809 5199 a SDict begin H.R end 1809 5199 a 1809 5263 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.3) cvn H.B /ANN pdfmark end 1809 5263 a Black Fp(\))g(as,)57 b(~)-64 b Fo(')39 b Fp(b)s(eing)g(almost)e(analytic,)p 3143 5182 57 4 v 40 w Fo(@)22 b Fp(~)-65 b Fo(')p Fp(\()p Fo(z)t Fp(\))40 b(v)-5 b(anishes)0 5384 y(to)48 b(an)m(y)h(order)f(in)f Fk(=)p Fo(z)53 b Fp(as)48 b Fo(z)53 b Fp(approac)m(hes)d(the)e(real)g (line.)89 b(Th)m(us,)53 b(w)m(e)c(completed)f(the)h(pro)s(of)e(of)0 5504 y(Theorem)p 0 0 1 TeXcolorrgb 412 5504 a SDict begin H.S end 412 5504 a 0 0 1 TeXcolorrgb Fp(2.2)p 0 0 1 TeXcolorrgb 537 5440 a SDict begin H.R end 537 5440 a 537 5504 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.2.2) cvn H.B /ANN pdfmark end 537 5504 a Black Fp(.)p 3709 5504 4 66 v 3713 5442 59 4 v 3713 5504 V 3771 5504 4 66 v Black 1865 5753 a(7)p Black eop %%Page: 8 8 8 7 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 0 -137 a SDict begin H.S end 0 -137 a 0 TeXcolorgray 0 TeXcolorgray 0 -137 a SDict begin H.R end 0 -137 a 0 -137 a SDict begin [ /View [/XYZ H.V] /Dest (page.8) cvn H.B /DEST pdfmark end 0 -137 a Black Black 308 x Fq(References)0 269 y SDict begin H.S end 0 269 a 0 269 a SDict begin 14.5 H.A end 0 269 a 0 269 a SDict begin [ /View [/XYZ H.V] /Dest (section*.1) cvn H.B /DEST pdfmark end 0 269 a Black 0 390 a SDict begin H.S end 0 390 a Fp([CHK])273 390 y SDict begin 14.5 H.A end 273 390 a 273 390 a SDict begin [ /View [/XYZ H.V] /Dest (cite.CHK) cvn H.B /DEST pdfmark end 273 390 a Black 123 w Fp(Jean-Mic)m(hel)23 b(Com)m(b)s(es,)j(P)m(eter)f (Hislop,)f(F)-8 b(r)m(\023)-46 b(ed)m(\023)g(eric)23 b(Klopp:)38 b(Lo)s(cal)23 b(and)g(Global)e(Con)m(tin)m(uit)m(y)396 510 y(of)45 b(the)g(In)m(tegrated)i(Densit)m(y)f(of)e(States;)53 b(Con)m(temp)s(orary)46 b(Mathematics;)51 b Fh(327)p Fp(,)e(61-74)396 630 y(\(2003\))p Black 0 834 a SDict begin H.S end 0 834 a Fp([Kl])158 834 y SDict begin 14.5 H.A end 158 834 a 158 834 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Kl) cvn H.B /DEST pdfmark end 158 834 a Black 238 w Fp(F)-8 b(r)m(\023)-46 b(ed)m(\023)g(eric)50 b(Klopp:)79 b(In)m(ternal)51 b(Lifshits)e(tails)g(for)i(random)e(p)s(erturbations)i (of)f(p)s(erio)s(dic)396 954 y(Sc)m(hr\177)-49 b(odinger)32 b(op)s(erators;)h(Duk)m(e)g(Mathematical)e(Journal;)g Fh(98)p Fp(\(2\))i(335-396,)d(\(1999\))p Black 0 1158 a SDict begin H.S end 0 1158 a Fp([KlP)m(a])270 1158 y SDict begin 14.5 H.A end 270 1158 a 270 1158 a SDict begin [ /View [/XYZ H.V] /Dest (cite.KlPa) cvn H.B /DEST pdfmark end 270 1158 a Black 126 w Fp(F)-8 b(r)m(\023)-46 b(ed)m(\023)g(eric)33 b(Klopp,)h(Leonid)f(P)m(astur:)47 b(Lifshitz)33 b(tails)f(for)h(random) h(Sc)m(hr\177)-49 b(odinger)34 b(op)s(erators)396 1278 y(with)23 b(negativ)m(e)h(P)m(oisson)g(p)s(oten)m(tial;)h(Comm)m (unications)d(in)g(Mathematical)g(Ph)m(ysics;)28 b Fh(206)p Fp(,)396 1398 y(57-103)j(\(1999\))p Black 0 1602 a SDict begin H.S end 0 1602 a Fp([P)m(aFi])258 1602 y SDict begin 14.5 H.A end 258 1602 a 258 1602 a SDict begin [ /View [/XYZ H.V] /Dest (cite.PaFi) cvn H.B /DEST pdfmark end 258 1602 a Black 138 w Fp(Leonid)47 b(P)m(astur,)53 b(Alexander)c (Figotin:)72 b(Sp)s(ectra)48 b(of)g(Random)f(and)h(Almost-P)m(erio)s (dic)396 1722 y(Op)s(erators;)32 b(Springer)g(V)-8 b(erlag)32 b(\(1992\))p Black 0 1926 a SDict begin H.S end 0 1926 a Fp([RS2])229 1926 y SDict begin 14.5 H.A end 229 1926 a 229 1926 a SDict begin [ /View [/XYZ H.V] /Dest (cite.RS2) cvn H.B /DEST pdfmark end 229 1926 a Black 167 w Fp(Mic)m(hael)24 b(Reed,)k(Barry)e(Simon:)38 b(Metho)s(ds)26 b(of)f(Mo)s(dern)h (Mathematical)d(Ph)m(ysics;)30 b(V)-8 b(olume)396 2046 y(I)s(I:)32 b(F)-8 b(ourier)32 b(Analysis,)g(Self-Adjoin)m(tness,)h (Academic)f(Press,)i(San)f(Diego)e(1975)p Black 0 2249 a SDict begin H.S end 0 2249 a Fp([RS4])229 2249 y SDict begin 14.5 H.A end 229 2249 a 229 2249 a SDict begin [ /View [/XYZ H.V] /Dest (cite.RS4) cvn H.B /DEST pdfmark end 229 2249 a Black 167 w Fp(Mic)m(hael)24 b(Reed,)k(Barry)e(Simon:)38 b(Metho)s(ds)26 b(of)f(Mo)s(dern)h(Mathematical)d(Ph)m(ysics;)30 b(V)-8 b(olume)396 2370 y(IV:)33 b(Analysis)f(of)g(Op)s(erators,)g (Academic)h(Press,)h(San)e(Diego)f(1978)p Black 0 2573 a SDict begin H.S end 0 2573 a Fp([Si])136 2573 y SDict begin 14.5 H.A end 136 2573 a 136 2573 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Si1) cvn H.B /DEST pdfmark end 136 2573 a Black 260 w Fp(Barry)38 b(Simon:)53 b(T)-8 b(race)39 b(ideals)f(and)g(their)g(application;)g(Cam)m(bridge)g(Univ)m(ersit)m (y)h(Press,)396 2693 y(Cam)m(bridge)31 b(1979)p Black 0 2897 a SDict begin H.S end 0 2897 a Fp([Stl])174 2897 y SDict begin 14.5 H.A end 174 2897 a 174 2897 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Stl) cvn H.B /DEST pdfmark end 174 2897 a Black 222 w Fp(P)m(eter)i(Stollman:)41 b(Caugh)m(t)33 b(b)m(y)h(disorder;)e(Birkh\177)-49 b(auser)33 b(2001)p Black 0 3100 a SDict begin H.S end 0 3100 a Fp([Stz])189 3100 y SDict begin 14.5 H.A end 189 3100 a 189 3100 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Stz) cvn H.B /DEST pdfmark end 189 3100 a Black 207 w Fp(G)s(\177)-51 b(un)m(ter)54 b(Stolz:)88 b(Strategies)55 b(in)f(lo)s(calization)e(pro) s(ofs)i(for)h(one-dimensional)d(random)396 3221 y(Sc)m(hr\177)-49 b(odinger)25 b(op)s(erators;)i(Pro)s(c.)e(Indian)g(Acad.)g(Sci.)g (\(Math.)g(Sci.\);)i Fh(112)p Fp(,)g(229-243)c(\(2002\))p Black 0 3424 a SDict begin H.S end 0 3424 a Fp([Ze])157 3424 y SDict begin 14.5 H.A end 157 3424 a 157 3424 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Ze2) cvn H.B /DEST pdfmark end 157 3424 a Black 239 w Fp(Herib)s(ert)35 b(Zenk:)50 b(An)35 b(in)m(teracting)g(m)m(ultielectron)e(Anderson)k(mo)s(del;)e (Preprin)m(t)h(mp)p 3621 3424 30 4 v 34 w(arc)396 3544 y(03-410)p Black 1865 5753 a(8)p Black eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0310160210408--