%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: x.dvi %%Pages: 34 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Times-Italic Times-Bold RMTMI MTSY %%+ Helvetica MTEX %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 x.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.10.23:1409 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry", % version = "0.6", % date = "1 July 1998", % filename = "8r.enc", % email = "tex-fonts@@tug.org", % docstring = "Encoding for TrueType or Type 1 fonts % to be used with TeX." % @} % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MTSY %!PS-AdobeFont-1.1: MTSY 1.1 %%CreationDate: 1993 May 30 16:26:28 % Copyright (c) 1992, 1993 The TeXplorators Corporation % Hinting Copyright (c) 1992, 1993 Y&Y, Inc. 11 dict begin /FontInfo 9 dict dup begin /version (1.1) readonly def /Notice (Copyright (C) 1992, 1993 The TeXplorators Corporation) readonly def /FullName (MTSY) readonly def /FamilyName (MathTime) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /MTSY def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 10 /circlemultiply put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 41 /arrowdblright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 67 /plus put dup 68 /equal put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{0 -954 1043 796}readonly def /UniqueXX 5018947 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d70c1e2528f01045f8caec9829f31d6 48c8d0a29ea851af41c327a0d569abaafe5afe94dad818c312d3cee72f1acea0 b701b6a5608521a2866790bdd5776d6cd0c7d971b9a48b96aa970dcbb8b76edc b90da356dc2529b665eb4bb80ac4f5b0f4c0ed76861e399638ad3db1be4759c7 8d4f2e81a2ff688d366b91d729d63ab5fc9556fe10a07b81904d879a7446da82 dc107ff41b0e3b7c2245d57b2ee9bab31eccfe9b79e3ec32cb1f10c622f4bce1 8df0e5c4b98ec714593d6f127c5cf6d719a79b83c627433d3aa39cb88ef85d27 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(limit)f(e)o(xists,)i(is)f(independent)f(of)i(almost)f(all)120 4696 y(realizations)24 b(of)g(the)f(random)h(potential)e(and)i(of)g (the)g(chosen)g(boundary)f(condition.)29 b(These)24 b(are)g(the)120 4816 y(questions)g(of)h(e)o(xistence,)f(non-randomness,)f(and)i (uniqueness.)270 4938 y(F)o(or)30 b(v)n(anishing)d(magnetic)i(\002eld)h (these)f(questions)g(were)h(settled)f(se)n(v)o(eral)g(years)h(ago)f ([44,)h(43)o(,)120 5059 y(32,)g(31)o(,)g(13,)f(46],)i(see)f(also)g([35) o(])g(for)g(a)g(more)g(recent)g(approach.)46 b(F)o(or)29 b(non-zero)h(magnetic)f(\002elds)120 5179 y(the)j(e)o(xistence)f(and)h (non-randomness)e(of)39 b FD(N)k FE(are)32 b(kno)n(wn)f(since)g([41,)h (55,)f(9].)51 b(Uniqueness,)33 b(that)120 5299 y(is,)h(the)e (independence)g(of)g(the)g(boundary)f(condition)g(follo)n(ws)g(from)g (recent)i(results)e(in)h([19])g(and)120 5420 y([42])24 b(for)g(bounded)f(belo)n(w)g(or)g(bounded)g(random)g(potentials,)g (respecti)n(v)o(ely)-6 b(.)28 b(Ho)n(we)n(v)o(er)l(,)23 b(a)h(proof)f(of)120 5540 y(uniqueness)h(is)g(lacking)h(for)g(random)f (potentials)f(which)i(are)g(unbounded)f(from)h(belo)n(w)-6 b(.)270 5662 y(The)29 b(main)f(goal)g(of)h(the)g(present)f(paper)h(is)f (to)h(gi)n(v)o(e)e(a)i(detailed)g(proof)f(of)h(the)g(e)o(xistence,)g (non-)120 5783 y(randomness,)35 b(and)f(uniqueness)f(of)41 b FD(N)46 b FE(for)34 b(the)f(case)i(of)f(constant)f(magnetic)g (\002elds)h(and)f(a)h(wide)p eop %%Page: 3 3 3 2 bop 238 -183 a FH(Inte)l(gr)o(ated)26 b(Density)e(of)g(States)h (for)f(Random)f(Sc)o(hr\366ding)o(er)k(Oper)o(ator)o(s)e(with)f(Ma)o (gnetic)h(F)l(ields)237 b FE(3)120 125 y(class)41 b(of)h(er)n(godic)f (random)f(potentials)g(which)h(may)g(be)g(unbounded)f(from)h(abo)o(v)o (e)f(as)h(well)g(as)120 245 y(from)30 b(belo)n(w)e(and)h(which)g (satisfy)g(a)g(simple)f(moment)g(condition.)43 b(In)29 b(particular)l(,)38 b FD(N)j FE(is)29 b(sho)n(wn)f(to)120 365 y(coincide)d(with)f(the)h(e)o(xpectation)f(of)h(the)g(trace)h(of)f 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b(the)g(in\002nite-v)n(olume)120 1333 y(operator)f(may)f(be) g(unbounded)f(from)h(belo)n(w)-6 b(,)28 b(we)g(ha)n(v)o(e)g(to)g(mak)o (e)g(sure)h(that)e(the)i(sequence)f(of)g(the)120 1453 y(underlying)j(\002nite-v)n(olume)f(density-of-states)g(measures)i(is)f (\223tight)f(near)i(minus)e(in\002nity\224.)50 b(Our)120 1573 y(proof)34 b(of)g(the)g(independence)f(of)h(the)g(boundary)f (condition)f(uses)i(an)g(approximation)e(ar)n(gument)120 1694 y(which)g(reduces)f(the)h(problem)e(to)h(that)g(of)h(bounded)e (random)h(potentials)f(and)i(therefore)g(hea)n(vily)120 1814 y(relies)25 b(on)g(results)f(of)h(Doi,)f(Iw)o(atsuka)g(and)h(Mine) f([19])h(or)g(Nakamura)g([42].)120 2181 y FA(2)144 b(Random)45 b(Schr\366dinger)f(Operators)h(with)h(Constant)f(Mag-)336 2363 y(netic)34 b(Fields)120 2625 y FG(2.1)119 b(Basic)30 b(Notation)120 2821 y FE(As)i(usual,)g(let)f Fy(N)d FE(:)p Fv(D)g(f)p FE(1)p Fz(;)14 b FE(2)p Fz(;)g FE(3)p Fz(;)g(:)g(:)g(:)f Fv(g)32 b FE(denote)f(the)g(set)g(of)h(natural)f(numbers.)49 b(Let)32 b Fy(R)p FE(,)h(respecti)n(v)o(ely)120 2941 y Fy(C)p FE(,)25 b(denote)g(the)g(algebraic)g(\002eld)g(of)g(real,)g (respecti)n(v)o(ely)e(comple)o(x,)h(numbers.)30 b(An)25 b(open)f(cube)h Fz(3)h FE(in)120 3062 y FD(d)7 b FE(-dimensional)25 b(Euclidean)f(space)i Fy(R)1475 3026 y Fw(d)1523 3062 y FE(,)f FD(d)33 b Fv(2)25 b Fy(N)p FE(,)g(is)g(a)h(translate)f(of)g (the)g FD(d)7 b FE(-fold)26 b(Cartesian)g(product)127 3182 y FD(I)31 b Fv(\002)17 b(\001)d(\001)g(\001)j(\002)24 b FD(I)36 b FE(of)22 b(an)h(open)f(interv)n(al)28 b FD(I)36 b Fv(\022)22 b Fy(R)p FE(.)31 b(The)22 b(open)g(unit)f(cube)i(in)f Fy(R)2623 3146 y Fw(d)2693 3182 y FE(which)g(is)g(centered)g(at)h(site) 126 3303 y FD(y)34 b Fv(D)28 b Fz(.)6 b FD(y)397 3318 y FC(1)439 3303 y Fz(;)14 b(:)g(:)g(:)g(;)20 b FD(y)706 3318 y Fw(d)753 3303 y Fz(/)28 b Fv(2)g Fy(R)990 3266 y Fw(d)1068 3303 y FE(and)i(whose)g(edges)h(are)g(oriented)f(parallel)g (to)g(the)g(co-ordinate)h(ax)o(es)f(is)120 3423 y(the)22 b(product)f Fz(3.)6 b 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y Fz(.3/)p FE(,)i(respecti)n(v)o(e-)120 855 y(ly)c Fm(C)302 819 y Ft(1)285 882 y FC(0)383 855 y Fz(.3/)p FE(.)30 b(Finally)-6 b(,)23 b FD(W)1011 819 y FC(1)p Fx(;)p FC(2)1115 855 y Fz(.3/)g FE(:)p Fv(D)1417 784 y Fs(\010)1466 855 y Fz(\036)k Fv(2)21 b FE(L)1695 819 y FC(2)1736 855 y Fz(.3/)h FE(:)f Fv(r)g Fz(\036)27 b Fv(2)21 b Fz(.)p FE(L)2317 819 y FC(2)2359 855 y Fz(.3//)2549 819 y Fw(d)2596 784 y Fs(\011)2666 855 y FE(is)g(the)g(\002rst-order)g (Sobole)n(v)120 976 y(space)i(of)g(L)531 939 y FC(2)572 976 y FE(-type)g(where)g Fv(r)29 b FE(stands)22 b(for)g(the)h(gradient) f(in)g(the)g(sense)g(of)h(distrib)n(utions)d(on)i Fm(C)3381 939 y Ft(1)3364 1003 y FC(0)3462 976 y Fz(.3/)p FE(.)270 1098 y(The)33 b(absolute)f(v)n(alue)g(of)h(a)g(closed)g(operator)39 b FD(F)g FE(:)29 b Fm(D)9 b Fz(.)e FD(F)i Fz(/)30 b Fv(!)g FE(L)2545 1062 y FC(2)2586 1098 y Fz(.3/)p FE(,)36 b(densely)c (de\002ned)h(with)120 1218 y(domain)e Fm(D)9 b Fz(.)e FD(F)i Fz(/)30 b Fv(\022)e FE(L)885 1182 y FC(2)927 1218 y Fz(.3/)33 b FE(and)e(adjoint)38 b FD(F)1674 1182 y Fl(y)1712 1218 y FE(,)c(is)d(the)h(positi)n(v)o(e)e(operator)2731 1217 y Fv(j)2766 1218 y FD(F)2836 1217 y Fv(j)2893 1218 y FE(:)p Fv(D)f Fz(.)7 b FD(F)3142 1182 y Fl(y)3187 1218 y FD(F)i Fz(/)3294 1182 y FC(1)p Fx(=)p FC(2)3408 1218 y FE(.)52 b(The)120 1339 y(\(uniform\))c(norm)f(of)h(a)g(bounded)g (operator)55 b FD(F)46 b FE(:)38 b(L)2050 1303 y FC(2)2091 1339 y Fz(.3/)g Fv(!)h FE(L)2491 1303 y FC(2)2532 1339 y Fz(.3/)49 b FE(is)e(de\002ned)h(as)g Fo(k)7 b FD(F)i Fo(k)38 b FE(:)p Fv(D)120 1459 y FE(sup)273 1388 y Fs(\010)335 1458 y Fv(j)370 1459 y FD(F)19 b(f)499 1458 y Fv(j)527 1483 y FC(2)607 1459 y FE(:)61 b FD(f)45 b Fv(2)25 b FE(L)918 1423 y FC(2)959 1459 y Fz(.3/)14 b(;)1173 1458 y Fv(j)1224 1459 y FD(f)1272 1458 y Fv(j)1301 1483 y FC(2)1367 1459 y Fv(D)25 b FE(1)1520 1388 y Fs(\011)1568 1459 y FE(.)31 b(Finally)-6 b(,)23 b(for)37 b FD(p)28 b Fv(2)c FE([1)p Fz(;)14 b Fv(1)p FE([)25 b(we)g(will)f(use)h(the)f (notation)1417 1724 y Fo(k)7 b FD(F)i Fo(k)1603 1753 y Fw(p)1671 1724 y FE(:)p Fv(D)1801 1628 y Fs(\020)1865 1724 y FE(T)m(r)1969 1723 y Fv(j)2004 1724 y FD(F)2074 1723 y Fv(j)2112 1683 y Fw(p)2169 1628 y Fs(\021)2219 1651 y FC(1)p Fx(=)g Fw(p)3449 1724 y FE(\(2.3\))120 1978 y(for)30 b(the)e(\(v)n(on)h(Neumann-\))f(Schatten)h(norm)f(of)h (an)g(operator)36 b FD(F)i FE(on)28 b(L)2610 1942 y FC(2)2652 1978 y Fz(.3/)h FE(in)g(the)f(Banach)i(space)120 2098 y Fm(J)234 2113 y Fw(p)278 2027 y Fs(\000)316 2098 y FE(L)377 2062 y FC(2)418 2098 y Fz(.3/)571 2027 y Fs(\001)610 2098 y FE(.)42 b(F)o(or)28 b(these)h Fm(J)1190 2113 y Fw(p)1233 2098 y FE(-spaces)g(of)f(compact)h(operators,)g(see)g([53)o (,)g(7].)42 b(In)29 b(particular)l(,)g Fm(J)3599 2113 y FC(1)120 2219 y FE(is)c(the)f(space)i(of)f(trace-class)g(and)g Fm(J)1432 2234 y FC(2)1498 2219 y FE(the)f(space)i(of)f (Hilbert-Schmidt)e(operators.)120 2528 y FG(2.2)119 b(Basic)30 b(Assumptions)f(and)i(De\002nitions)g(of)f(the)g(Operators)120 2720 y FE(Let)d Fz(.\177)5 b(;)14 b Fm(A)22 b Fz(;)14 b Fy(P)p Fz(/)26 b FE(be)h(a)f(complete)g(probability)e(space)j(and)f Fy(E)p Fv(f\001g)f FE(:)p Fv(D)2533 2639 y Fn(R)2578 2754 y Fx(\177)2640 2720 y Fy(P)p Fz(.)p FE(d)p Fz(!)r(/.)p Fv(\001)p Fz(/)i FE(be)f(the)g(e)o(xpecta-)120 2840 y(tion)31 b(induced)g(by)g(the)g(probability)f(measure)h Fy(P)p FE(.)51 b(By)31 b(a)h FD(r)o(andom)e(potential)g FE(we)i(mean)f(a)g (\(scalar\))120 2960 y(random)25 b(\002eld)k FD(V)40 b FE(:)e Fz(\177)20 b Fv(\002)f Fy(R)1099 2924 y Fw(d)1172 2960 y Fv(!)26 b Fy(R)14 b FE(,)25 b Fz(.!)5 b(;)17 b FD(x)9 b Fz(/)26 b Fv(7!)i FD(V)1938 2924 y Fx(.!)q(/)2049 2960 y Fz(.)s FD(x)9 b Fz(/)25 b FE(which)g(is)g(assumed)f(to)h(be)g (jointly)f(mea-)120 3081 y(surable)33 b(with)g(respect)g(to)g(the)g (product)g(of)g(the)g(sigma-algebra)g Fm(A)56 b FE(of)33 b(e)n(v)o(ent)f(sets)h(in)g Fz(\177)f FE(and)h(the)120 3201 y(sigma-algebra)26 b Fm(B)t Fz(.)p Fy(R)923 3165 y Fw(d)971 3201 y Fz(/)f FE(of)h(Borel)g(sets)f(in)h Fy(R)1749 3165 y Fw(d)1796 3201 y FE(.)33 b(W)-8 b(e)26 b(will)f(al)o(w)o(ays)h(assume)f FD(d)32 b Fv(\025)26 b FE(2,)f(because)h(mag-)120 3322 y(netic)f(\002elds)g(in)f(one)h (space)h(dimension)d(may)h(be)h(\223gauged)g(a)o(w)o(ay\224)g(and)g (are)g(therefore)h(of)f(no)f(phys-)120 3442 y(ical)35 b(rele)n(v)n(ance.)59 b(Furthermore,)36 b(for)f FD(d)j Fv(D)29 b FE(1)35 b(f)o(ar)g(more)f(is)g(kno)n(wn)f([13,)h(46)o(])h (thanks)f(to)g(methods)120 3562 y(which)25 b(only)f(w)o(ork)h(for)g (one)f(dimension.)120 3805 y(W)-8 b(e)26 b(list)d(three)i(properties)g (which)i FD(V)40 b FE(may)24 b(ha)n(v)o(e)h(or)g(not:)247 4018 y(\()p Fu(S)p FE(\))100 b(There)29 b(e)o(xists)e(some)h(pair)g(of) g(reals)40 b FD(p)1795 4033 y FC(1)1864 4018 y Fz(>)d FD(p)s Fz(.)p FD(d)7 b Fz(/)30 b FE(and)39 b FD(p)2426 4033 y FC(2)2495 4018 y Fz(>)e FD(p)2660 4033 y FC(1)2702 4018 y FD(d)7 b Fz(=)2820 3946 y Fs(\002)2855 4018 y FE(2)p Fz(.)k FD(p)3003 4033 y FC(1)3064 4018 y Fv(\000)30 b FD(p)s Fz(.)p FD(d)7 b Fz(//)3393 3946 y Fs(\003)3458 4018 y FE(such)469 4138 y(that)1254 4389 y(sup)1240 4485 y Fw(y)t Ft(2)p Fk(Z)1380 4463 y Fp(d)1439 4389 y Fy(E)1515 4293 y Fs(n)1570 4318 y(\002)1619 4253 y Fn(Z)1671 4479 y Fx(3.)e Fw(y)t Fx(/)1805 4389 y FE(d)1855 4348 y Fw(d)1905 4389 y FD(x)23 b Fo(j)s FD(V)14 b Fz(.)s FD(x)9 b Fz(/)p Fo(j)2255 4348 y Fw(p)2292 4360 y Fj(1)2330 4318 y Fs(\003)2373 4341 y Fw(p)2410 4353 y Fj(2)2443 4341 y Fx(=)g Fw(p)2524 4353 y Fj(1)2562 4293 y Fs(o)2642 4389 y Fz(<)25 b Fv(1)p Fz(:)575 b FE(\(2.4\))469 4691 y(Here)39 b FD(p)s Fz(.)p FD(d)7 b Fz(/)29 b FE(is)e(de\002ned)h(as)g(follo)n(ws:)45 b FD(p)s Fz(.)p FD(d)7 b Fz(/)28 b FE(:)p Fv(D)e FE(2)h(if)h FD(d)33 b Fv(\024)27 b FE(3,)39 b FD(p)s Fz(.)p FD(d)7 b Fz(/)27 b FE(:)p Fv(D)f FD(d)7 b Fz(=)p FE(2)28 b(if)g FD(d)33 b Fv(\025)27 b FE(5)g(and)480 4812 y FD(p)s Fz(.)p FE(4)p Fz(/)f(>)e FE(2,)h(otherwise)f(arbitrary)-6 b(.)247 5001 y(\()p Fu(E)p FE(\))103 b FD(V)40 b FE(is)24 b Fy(Z)733 4965 y Fw(d)780 5001 y FE(-er)n(godic)h(or)g Fy(R)1324 4965 y Fw(d)1372 5001 y FE(-er)n(godic.)280 5191 y(\()p Fu(I)p FE(\))102 b FD(V)40 b FE(satis\002es)24 b(the)h(\002niteness)f (condition)1352 5459 y(sup)1338 5555 y Fw(y)t Ft(2)p Fk(Z)1477 5533 y Fp(d)1537 5459 y Fy(E)1613 5388 y Fs(\002)1661 5323 y Fn(Z)1714 5549 y Fx(3.)5 b Fw(y)t Fx(/)1847 5459 y FE(d)1897 5418 y Fw(d)1947 5459 y FD(x)23 b Fo(j)s FD(V)14 b Fz(.)s FD(x)9 b Fz(/)p Fo(j)2278 5418 y FC(2)p Fx(#)e Ft(C)p FC(1)2466 5388 y Fs(\003)2525 5459 y Fz(<)25 b Fv(1)p Fz(;)686 b FE(\(2.5\))469 5761 y(where)25 b Fz(#)34 b Fv(2)25 b Fy(N)f FE(is)h(the)f(smallest)g(inte)o(ger)g(with)g Fz(#)33 b(>)25 b FD(d)7 b Fz(=)p FE(4.)p eop %%Page: 5 5 5 4 bop 238 -183 a FH(Inte)l(gr)o(ated)26 b(Density)e(of)g(States)h (for)f(Random)f(Sc)o(hr\366ding)o(er)k(Oper)o(ator)o(s)e(with)f(Ma)o (gnetic)h(F)l(ields)237 b FE(5)120 125 y FB(Remarks)26 b(2.1.)204 b FE(\(i\))100 b(Property)37 b(\()p Fu(E)p FE(\))i(requires)e(the)g(e)o(xistence)g(of)h(a)g(group)e Fm(T)3111 140 y Fw(x)3156 125 y FE(,)43 b FD(x)e Fv(2)31 b Fy(Z)3472 88 y Fw(d)3557 125 y FE(or)120 245 y Fy(R)201 209 y Fw(d)249 245 y FE(,)36 b(of)d(probability-preserving)f(and)h(er)n (godic)h(transformations)e(on)h Fz(\177)g FE(such)h(that)i FD(V)48 b FE(is)33 b Fy(Z)3443 209 y Fw(d)3490 245 y FE(-)h(or)120 365 y Fy(R)201 329 y Fw(d)249 365 y FE(-homogeneous)29 b(in)i(the)g(sense)f(that)k FD(V)1634 329 y Fx(.)p Fi(T)1732 340 y Fp(x)1767 329 y Fx(!)q(/)1849 365 y Fz(.)6 b FD(y)g Fz(/)28 b Fv(D)j FD(V)2192 329 y Fx(.!)q(/)2302 365 y Fz(.)6 b FD(y)28 b Fv(\000)d FD(x)9 b Fz(/)31 b FE(for)g(all)j FD(x)j Fv(2)28 b Fy(Z)3163 329 y Fw(d)3241 365 y FE(or)j Fy(R)3436 329 y Fw(d)3483 365 y FE(,)i(all)126 486 y FD(y)e Fv(2)25 b Fy(R)370 450 y Fw(d)417 486 y FE(,)g(and)g(all)f Fz(!)j Fv(2)e Fz(\177)p FE(;)f(see)h([31].)248 682 y(\(ii\))99 b(Property)36 b(\()p Fu(S)p FE(\))g(assures)f(that)f(the)h(realization) j FD(V)2205 646 y Fx(.!)q(/)2346 682 y FE(:)33 b FD(x)40 b Fv(7!)34 b FD(V)2709 646 y Fx(.!)q(/)2820 682 y Fz(.)s FD(x)9 b Fz(/)35 b FE(of)j FD(V)50 b FE(belongs)35 b(to)120 815 y(L)190 763 y Fw(p)r Fx(.)p Fw(d)6 b Fx(/)181 842 y FC(loc)332 815 y Fz(.)p Fy(R)450 779 y Fw(d)498 815 y Fz(/)35 b FE(for)f(each)h Fz(!)i FE(in)d(some)g(subset)g Fz(\177)1758 830 y Fh(S)1834 815 y Fv(2)c Fm(A)57 b FE(of)35 b Fz(\177)f FE(with)g(full)f(probability)-6 b(,)35 b(in)f(symbols,)120 935 y Fg(P)p Fz(.\177)294 950 y Fh(S)343 935 y Fz(/)i Fv(D)g FE(1.)92 b(If)45 b FD(d)f Fv(6D)36 b FE(4,)50 b(property)45 b(\()p Fu(I)p FE(\))g(in)g(general)g(does)g(not)g(imply)f (property)h(\()p Fu(S)p FE(\))h(e)n(v)o(en)e(if)120 1056 y(property)35 b(\()p Fu(E)p FE(\))h(is)f(supposed.)61 b(Gi)n(v)o(en)33 b(\()p Fu(E)p FE(\),)j(a)g(suf)n(\002cient)e (criterion)h(for)h(both)e(\()p Fu(S)p FE(\))i(and)f(\()p Fu(I)p FE(\))g(to)g(hold)120 1176 y(is)25 b(the)f(\002niteness)1347 1447 y Fy(E)1423 1376 y Fs(\002)1471 1311 y Fn(Z)1524 1536 y Fx(3.)p FC(0)p Fx(/)1653 1447 y FE(d)1703 1406 y Fw(d)1753 1447 y FD(x)e Fo(j)s FD(V)15 b Fz(.)s FD(x)9 b Fz(/)p Fo(j)2093 1406 y Fw(p)2150 1376 y Fs(\003)2209 1447 y Fz(<)25 b Fv(1)1035 b FE(\(2.6\))120 1751 y(for)25 b(some)e(real)36 b FD(p)27 b Fz(>)d FD(d)i Fv(C)19 b FE(1.)30 b(T)-8 b(o)24 b(pro)o(v)o(e)f(this)g(claim)h(for)g(property)g (\()p Fu(S)p FE(\))h(we)f(choose)35 b FD(p)3077 1766 y FC(1)3143 1751 y Fv(D)g FD(p)3306 1766 y FC(2)3372 1751 y Fv(D)g FD(p)28 b FE(in)120 1872 y(\(2.4\).)i(F)o(or)20 b(\()p Fu(I)p FE(\))g(the)g(claim)f(follo)n(ws)g(from)h(2)p Fz(#)29 b Fv(\024)20 b FD(d)7 b FE(.)30 b(If)20 b(the)g(random)g (potential)f(is)g Fy(R)2981 1835 y Fw(d)3029 1872 y FE(-homogeneous,) 120 1992 y(Fubini')-5 b(s)24 b(theorem)g(gi)n(v)o(es)g Fy(E)1131 1921 y Fs(\002)1165 1992 y Fo(j)s FD(V)15 b Fz(.)p FE(0)p Fz(/)p Fo(j)1443 1956 y Fw(p)1486 1921 y Fs(\003)1546 1992 y FE(for)25 b(the)f(l.h.s.)g(of)h(\(2.6\).)120 2239 y(In)37 b(the)f(present)h(paper)f(we)h(mainly)e(consider)h(the)h (case)g(of)f(a)h FD(constant)e(ma)o(gnetic)h(\002eld)j FE(in)d Fy(R)3568 2203 y Fw(d)3615 2239 y FE(.)120 2360 y(This)f(is)g(characterized)h(by)f(a)g(sk)o(e)n(w-symmetric)f(tensor)g (with)h(real)g(constant)g(components)41 b FD(B)3546 2375 y Fw(j)8 b(k)3615 2360 y FE(,)136 2480 y FD(j)d Fz(;)14 b FD(k)38 b Fv(2)32 b(f)p FE(1)p Fz(;)14 b(:)g(:)g(:)g(;)g FD(d)7 b Fv(g)p FE(.)72 b(On)38 b(account)g(of)g(gauge)g(equi)n(v)n (alence,)j(there)d(is)g(no)g(loss)f(of)i(generality)f(in)120 2600 y(assuming)32 b(that)g(the)h FD(vector)g(potential)41 b(A)31 b FE(:)e Fy(R)1776 2564 y Fw(d)1853 2600 y Fv(!)h Fy(R)2073 2564 y Fw(d)2120 2600 y FE(,)38 b FD(x)g Fv(7!)i FD(A)r Fz(.)s FD(x)9 b Fz(/)p FE(,)35 b(generating)d(the)h(magnetic)120 2721 y(\002eld)26 b(according)e(to)33 b FD(B)924 2736 y Fw(j)8 b(k)1017 2721 y Fv(D)25 b Fz(@)1178 2736 y Fw(j)1220 2721 y FD(A)1283 2736 y Fw(k)1344 2721 y Fv(\000)20 b Fz(@)1488 2736 y Fw(k)1539 2721 y FD(A)1614 2736 y Fw(j)1646 2721 y FE(,)25 b(satis\002es)g(property)242 2939 y(\()p Fu(C)p FE(\))110 b FD(A)35 b FE(is)e(the)f(v)o(ector)h(potential)f(of)h (a)g(constant)f(magnetic)h(\002eld)g(in)f(the)h(symmetric)f(gauge,)469 3060 y(that)24 b(is,)h(its)f(components)f(are)j(gi)n(v)o(en)d(by)34 b FD(A)1971 3075 y Fw(k)2013 3060 y Fz(.)s FD(x)9 b Fz(/)25 b Fv(D)2281 3020 y FC(1)p 2281 3037 38 4 v 2281 3095 a(2)2342 2993 y Fs(P)2429 3022 y Fw(d)2441 3087 y(j)8 b Ft(D)p FC(1)2586 3060 y FD(x)2645 3075 y Fw(j)2686 3060 y FD(B)2759 3075 y Fw(j)g(k)2852 3060 y FE(with)24 b FD(k)32 b Fv(2)24 b(f)p FE(1)p Fz(;)14 b(:)g(:)g(:)g(;)g FD(d)7 b Fv(g)p FE(.)120 3279 y(W)-8 b(e)34 b(are)g(no)n(w)e(prepared)i (to)e(precisely)h(de\002ne)h(magnetic)e(Schr\366dinger)i(operators)e (with)h(random)120 3399 y(potentials)24 b(on)g(the)h(Hilbert)f(spaces)h (L)1457 3363 y FC(2)1498 3399 y Fz(.3/)h FE(and)f(L)1907 3363 y FC(2)1948 3399 y Fz(.)p Fy(R)2066 3363 y Fw(d)2114 3399 y Fz(/)p FE(.)31 b(The)24 b(\002nite-v)n(olume)g(case)i(is)e (treated)h(in)120 3646 y FB(Pr)n(oposition)g(2.2.)49 b FD(Let)26 b Fz(3)f Fv(\032)g Fy(R)1276 3610 y Fw(d)1348 3646 y FD(be)g(a)g(bounded)f(open)h(cube)o(.)30 b(Let)36 b(A)26 b(be)f(a)g(vector)g(potential)f(with)120 3767 y(pr)l(operty)f FE(\()p Fu(C)p FE(\))h FD(and)i(V)39 b(be)24 b(a)f(r)o(andom)g(potential)f(with)h(pr)l(operty)g FE(\()p Fu(S)p FE(\))p FD(.)h FE([)p FD(Recall)g(fr)l(om)e(Remark)i (2.1\(ii\))120 3887 y(the)h(de\002nition)e(of)i(the)f(set)j Fz(\177)1133 3902 y Fh(S)1179 3887 y FD(.)p FE(])k FD(Then)275 4104 y(\(i\))100 b(the)19 b(sesquilinear)e(form)h Fm(Q)f Fv(\002)d Fm(Q)23 b Fv(3)c Fz(.')5 b(;)14 b( )9 b(/)20 b Fv(7!)2132 4064 y FC(1)p 2132 4081 V 2132 4139 a(2)2194 4038 y Fs(P)2281 4066 y Fw(d)2293 4131 y(j)8 b Ft(D)p FC(1)2435 4104 y Fo(h)p Fz(.)p FE(i)p Fv(r)2621 4119 y Fw(j)2673 4104 y Fv(C)29 b FD(A)2855 4119 y Fw(j)2887 4104 y Fz(/)14 b(')19 b(;)28 b(.)p FE(i)p Fv(r)3221 4119 y Fw(j)3273 4104 y Fv(C)h FD(A)3455 4119 y Fw(j)3487 4104 y Fz(/)14 b( )9 b Fo(i)469 4241 y FD(with)36 b(form)g(domain)f Fm(Q)f Fv(D)f FD(W)1565 4205 y FC(1)p Fx(;)p FC(2)1670 4241 y Fz(.3/)k FD(or)f Fm(Q)e Fv(D)d Fm(C)2299 4205 y Ft(1)2282 4269 y FC(0)2380 4241 y Fz(.3/)36 b FD(is)g(positive)o(,)i (symmetric)e(and)469 4362 y(closed,)48 b(r)l(espectively)c(closable)o (.)87 b(Accor)l(dingly)-5 b(,)47 b(both)c(forms)g(uniquely)g(de\002ne)h (pos-)469 4482 y(itive)k(self-adjoint)e(oper)o(ator)o(s)g(on)i FE(L)1833 4446 y FC(2)1875 4482 y Fz(.3/)g FD(whic)o(h)g(we)h(denote)f (by)56 b(H)3053 4497 y Fx(3;)p FC(N)3196 4482 y Fz(.)10 b FD(A)5 b Fz(;)14 b FE(0)p Fz(/)48 b FD(and)477 4603 y(H)546 4618 y Fx(3;)p FC(D)689 4603 y Fz(.)10 b FD(A)5 b Fz(;)14 b FE(0)p Fz(/)p FD(,)24 b(r)l(espectively)-5 b(.)248 4798 y(\(ii\))99 b(the)25 b(two)g(oper)o(ator)o(s)692 5044 y(H)761 5059 y Fx(3;)p FC(X)903 5044 y Fz(.)10 b FD(A)5 b Fz(;)17 b FD(V)1142 5003 y Fx(.!)q(/)1252 5044 y Fz(/)25 b FE(:)p Fv(D)32 b FD(H)1521 5059 y Fx(3;)p FC(X)1664 5044 y Fz(.)10 b FD(A)5 b Fz(;)14 b FE(0)p Fz(/)19 b Fv(C)j FD(V)2106 5003 y Fx(.!)q(/)2216 5044 y Fz(;)214 b FE(X)24 b Fv(D)h FE(D)50 b(or)g(X)24 b Fv(D)h FE(N)p Fz(;)228 b FE(\(2.7\))469 5301 y FD(ar)l(e)36 b(well)g(de\002ned)g(on)g FE(L)1357 5265 y FC(2)1398 5301 y Fz(.3/)h FD(as)e(form)g(sums)h(for)f(all)g Fz(!)e Fv(2)e Fz(\177)2735 5316 y Fh(S)2781 5301 y FD(,)39 b(hence)d(for)f Fy(P)p FD(-almost)469 5421 y(all)e Fz(!)e Fv(2)f Fz(\177)p FD(.)55 b(The)m(y)34 b(ar)l(e)f(self-adjoint)f(and)g(bounded)h(below)-7 b(.)55 b(Mor)l(eo)o(ver)-11 b(,)35 b(the)e(mapping)477 5542 y(H)546 5557 y Fx(3;)6 b Fw(X)694 5542 y Fz(.)k FD(A)5 b Fz(;)17 b FD(V)d Fz(/)30 b FE(:)g Fz(\177)1135 5557 y Fh(S)1210 5542 y Fv(3)g Fz(!)i Fv(7!)38 b FD(H)1616 5557 y Fx(3;)p FC(X)1759 5542 y Fz(.)10 b FD(A)5 b Fz(;)17 b FD(V)1997 5506 y Fx(.!)q(/)2107 5542 y Fz(/)34 b FD(is)g(measur)o (able)o(.)55 b(W)-9 b(e)33 b(call)h(it)f(the)g FE(\002nite-)469 5662 y(v)n(olume)c(magnetic)h(Schr\366dinger)g(operator)h(with)e (random)h(potential)i FD(V)45 b(and)30 b(Diric)o(hlet)469 5783 y(or)25 b(Neumann)f(boundary)g(condition)f(if)i FE(X)g Fv(D)f FE(D)h FD(or)g FE(X)f Fv(D)h FE(N)p FD(,)g(r)l (espectively)-5 b(.)p eop %%Page: 6 6 6 5 bop 120 -183 a FE(6)261 b FH(T)-7 b(.)23 b(Hupfer)-10 b(,)24 b(H.)e(Lesc)o(hk)o(e)o(,)i(P)-12 b(.)21 b(M\374ller)k(&)d(S.)h (W)-8 b(arzel)220 125 y FD(\(iii\))99 b(the)32 b(spectrum)f(of)39 b(H)1206 140 y Fx(3;)p FC(X)1349 125 y Fz(.)10 b FD(A)5 b Fz(;)17 b FD(V)1587 88 y Fx(.!)q(/)1697 125 y Fz(/)32 b FD(is)f(pur)l(ely)h(discr)l(ete)g(for)f(all)g Fz(!)g Fv(2)d Fz(\177)3041 140 y Fh(S)3119 125 y FD(suc)o(h)j(that)f(the)469 245 y(\(r)o(andom\))24 b FE(\002nite-v)n(olume)g(density-of-states)f (measure)p FD(,)i(de\002ned)f(by)1307 479 y Fz(\027)1360 429 y Fx(.!)q(/)1354 505 y(3;)p FC(X)1496 479 y Fz(.)7 b FD(I)12 b Fz(/)27 b FE(:)p Fv(D)d FE(T)m(r)1883 383 y Fs(h)1922 461 y Fz(\037)1996 494 y Fw(I)2035 407 y Fs(\000)2081 479 y FD(H)2150 494 y Fx(3;)p FC(X)2293 479 y Fz(.)10 b FD(A)5 b Fz(;)17 b FD(V)2531 438 y Fx(.!)q(/)2641 479 y Fz(/)2678 407 y Fs(\001)2717 383 y(i)2756 479 y Fz(;)660 b FE(\(2.8\))469 701 y FD(is)53 b(a)g(positive)f(Bor)l(el)h (measur)l(e)g(on)g(the)g(r)l(eal)g(line)g Fy(R)h FD(for)e(all)h Fz(!)k Fv(2)d Fz(\177)3234 716 y Fh(S)3280 701 y FD(.)116 b(Her)l(e)o(,)469 804 y Fz(\037)543 837 y Fw(I)582 750 y Fs(\000)628 822 y FD(H)697 837 y Fx(3;)p FC(X)840 822 y Fz(.)10 b FD(A)5 b Fz(;)17 b FD(V)1078 785 y Fx(.!)q(/)1188 822 y Fz(/)1225 750 y Fs(\001)1302 822 y FD(is)39 b(the)g(spectr)o(al)e (pr)l(ojection)g(oper)o(ator)g(of)47 b(H)2951 837 y Fx(3;)p FC(X)3094 822 y Fz(.)10 b FD(A)5 b Fz(;)17 b FD(V)3332 785 y Fx(.!)q(/)3442 822 y Fz(/)39 b FD(as-)469 942 y(sociated)g(with)g (the)g(ener)l(gy)g(r)l(e)l(gime)47 b(I)f Fv(2)32 b Fm(B)t Fz(.)p Fy(R)p Fz(/)p FD(.)75 b(Mor)l(eo)o(ver)-11 b(,)43 b(the)c(\(unbounded)g(left-)469 1062 y(continuous\))24 b(distrib)n(ution)d(function)626 1295 y(N)704 1245 y Fx(.!)q(/)694 1321 y(3;)p FC(X)836 1295 y Fz(.)7 b FD(E)i Fz(/)25 b FE(:)p Fv(D)f Fz(\027)1195 1245 y Fx(.!)q(/)1189 1321 y(3;)p FC(X)1332 1224 y Fs(\000)1384 1295 y FE(])p Fv(\0001)p Fz(;)d FD(E)9 b FE([)1767 1224 y Fs(\001)1830 1295 y Fv(D)24 b FE(T)m(r)2037 1199 y Fs(h)2076 1295 y Fz(2)2156 1224 y Fs(\000)2201 1295 y FD(E)k Fv(\000)g FD(H)2465 1310 y Fx(3;)p FC(X)2607 1295 y Fz(.)10 b FD(A)5 b Fz(;)17 b FD(V)2845 1254 y Fx(.!)q(/)2956 1295 y Fz(/)2993 1224 y Fs(\001)3031 1199 y(i)3095 1295 y Fz(<)25 b Fv(1)149 b FE(\(2.9\))469 1559 y FD(of)39 b Fz(\027)639 1509 y Fx(.!)q(/)633 1585 y(3;)p FC(X)775 1559 y FD(,)k(called)c(the)g FE(\002nite-v)n(olume)f(inte)o(grated)g(density)g(of)h(states)p FD(,)j(is)d(\002nite)f(for)h(all)469 1679 y(ener)l(gies)32 b(E)i Fv(2)24 b Fy(R)p FD(.)120 1888 y FB(Pr)n(oof)o(.)50 b FE(The)25 b(assumptions)d(of)j(Proposition)e(2.2)i(imply)e(those)i (of)f([28,)h(Prop.)g(2.1].)p 3568 1888 4 68 v 3572 1824 60 4 v 3572 1888 V 3631 1888 4 68 v 120 2106 a FB(Remark)h(2.3.)49 b FE(Counting)38 b(multiplicity)-6 b(,)38 b Fz(\027)1681 2056 y Fx(.!)q(/)1675 2132 y(3;)p FC(X)1817 2106 y Fz(.)7 b FD(I)12 b Fz(/)40 b FE(is)e(just)f(the)i(number)e(of)i(eigen)l(v)n (alues)e(of)h(the)120 2238 y(operator)32 b FD(H)553 2253 y Fx(3;)p FC(X)696 2238 y Fz(.)10 b FD(A)5 b Fz(;)17 b FD(V)934 2202 y Fx(.!)q(/)1044 2238 y Fz(/)24 b FE(in)g(the)f(Borel)h (set)31 b FD(I)37 b Fv(\022)24 b Fy(R)p FE(.)30 b(Since)24 b(this)f(number)g(is)h(almost)e(surely)h(\002nite)120 2373 y(for)28 b(e)n(v)o(ery)e(bounded)33 b FD(I)12 b FE(,)28 b(the)f(mapping)e Fz(\027)1554 2388 y Fx(3;)p FC(X)1723 2373 y FE(:)g Fz(\177)1854 2388 y Fh(S)1926 2373 y Fv(3)h Fz(!)i Fv(7!)f Fz(\027)2297 2323 y Fx(.!)q(/)2291 2399 y(3;)p FC(X)2460 2373 y FE(is)f(a)h(random)g(Borel)g(measure)g(in) 120 2519 y(the)e(sense)g(that)f Fz(\027)735 2469 y Fx(.!)q(/)729 2545 y(3;)p FC(X)896 2519 y FE(assigns)g(a)h(\002nite)g(length)f(to)g (each)i(bounded)e(Borel)h(set.)120 2726 y(The)g(in\002nite-v)n(olume)f (case)h(is)f(treated)h(in)120 2934 y FB(Pr)n(oposition)g(2.4.)49 b FD(Let)j(A)43 b(be)f(a)f(vector)h(potential)d(with)j(pr)l(operty)e FE(\()p Fu(C)p FE(\))i FD(and)i(V)56 b(be)41 b(a)h(r)o(andom)120 3054 y(potential)24 b(with)g(pr)l(operty)g FE(\()p Fu(S)p FE(\))p FD(.)h(Then)275 3263 y(\(i\))100 b(the)45 b(oper)o(ator)e Fm(C)1111 3226 y Ft(1)1094 3290 y FC(0)1191 3263 y Fz(.)p Fy(R)1309 3227 y Fw(d)1357 3263 y Fz(/)36 b Fv(3)g Fz( )45 b Fv(7!)1799 3223 y FC(1)p 1799 3240 38 4 v 1799 3298 a(2)1860 3197 y Fs(P)1948 3225 y Fw(d)1960 3290 y(j)8 b Ft(D)p FC(1)2088 3263 y Fz(.)p FE(i)p Fz(@)2211 3278 y Fw(j)2270 3263 y Fv(C)37 b FD(A)2460 3278 y Fw(j)2492 3263 y Fz(/)2529 3227 y FC(2)2584 3263 y Fz( )g Fv(C)29 b FD(V)2874 3227 y Fx(.!)q(/)2984 3263 y Fz( )54 b FD(is)45 b(essentially)469 3400 y(self-adjoint)35 b(for)g(all)h Fz(!)e Fv(2)d Fz(\177)1525 3415 y Fh(S)1571 3400 y FD(.)65 b(Its)36 b(self-adjoint)f(closur)l(e)h(on)g FE(L)2811 3364 y FC(2)2852 3400 y Fz(.)p Fy(R)2970 3364 y Fw(d)3018 3400 y Fz(/)g FD(is)g(denoted)g(by)477 3521 y(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)798 3485 y Fx(.!)q(/)908 3521 y Fz(/)p FD(.)248 3708 y(\(ii\))99 b(the)33 b(mapping)41 b(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)d Fz(/)30 b FE(:)g Fz(\177)1538 3723 y Fh(S)1613 3708 y Fv(3)g Fz(!)i Fv(7!)38 b FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)2270 3672 y Fx(.!)q(/)2381 3708 y Fz(/)33 b FD(is)g(measur)o(able)o(.)55 b(W)-9 b(e)34 b(call)e(it)h(the)469 3828 y FE(in\002nite-v)n(olume)23 b(magnetic)i(Schr\366dinger)f(operator)h(with)f(random)h(potential)h FD(V)15 b(.)120 4037 y FB(Pr)n(oof)o(.)50 b FE(See)26 b(for)f(e)o(xample)f([28)o(,)h(Prop.)g(2.2].)p 3568 4037 4 68 v 3572 3973 60 4 v 3572 4037 V 3631 4037 4 68 v 120 4245 a FB(Remarks)h(2.5.)204 b FE(\(i\))100 b(F)o(or)25 b(alternati)n(v)o(e)f(or)i(weak)o(er)g(criteria)g(instead)e(of)i(\()p Fu(S)p FE(\))g(guaranteeing)f(the)120 4365 y(almost-sure)f (self-adjointness)f(of)33 b FD(H)10 b Fz(.)p FE(0)p Fz(;)17 b FD(V)e Fz(/)p FE(,)25 b(see)g([46,)g(Thm.)f(5.8])h(or)g([31)o(,)g (Thm.)f(1)h(on)f(p.)h(299].)248 4553 y(\(ii\))99 b(The)44 b(in\002nite-v)n(olume)f(magnetic)g(Schr\366dinger)h(operator)g (without)f(scalar)i(potential,)128 4674 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)14 b FE(0)p Fz(/)p FE(,)27 b(is)g(unitarily)e(in)l(v)n (ariant)h(under)h(so-called)f FD(ma)o(gnetic)g(tr)o(anslations)e FE([60)o(,)j(37].)37 b(The)26 b(latter)120 4794 y(form)f(a)g(f)o(amily) f(of)h(unitary)f(operators)h Fv(f)r FD(T)1596 4809 y Fw(x)1640 4794 y Fv(g)1679 4818 y Fw(x)6 b Ft(2)p Fr(R)1829 4796 y Fp(d)1896 4794 y FE(on)25 b(L)2082 4758 y FC(2)2123 4794 y Fz(.)p Fy(R)2241 4758 y Fw(d)2288 4794 y Fz(/)g FE(de\002ned)h(by)218 5097 y Fz(.)257 5096 y FD(T)311 5111 y Fw(x)355 5096 y Fz( )434 5097 y(/)485 5096 y(.)6 b FD(y)g Fz(/)25 b FE(:)p Fv(D)g FE(e)o(xp)927 4950 y Fs(")996 5027 y FE(i)p 985 5073 50 4 v 985 5167 a(2)1134 4989 y Fw(d)1096 5013 y Fs(X)1071 5202 y Fw(j)t Fx(;)p Fw(k)5 b Ft(D)p FC(1)1253 5096 y Fz(.)s FD(x)1352 5111 y Fw(j)1404 5096 y Fv(\000)25 b FD(y)1563 5111 y Fw(j)1596 5096 y Fz(/)8 b FD(B)1714 5111 y Fw(j)g(k)1800 5096 y FD(x)1847 5111 y Fw(k)1888 4950 y Fs(#)1951 5096 y Fz( )h(.)d FD(y)25 b Fv(\000)d FD(x)9 b Fz(/;)413 b( )34 b Fv(2)24 b FE(L)3030 5055 y FC(2)3072 5096 y Fz(.)p Fy(R)3190 5055 y Fw(d)3237 5096 y Fz(/:)98 b FE(\(2.10\))120 5404 y(In)35 b(the)e(situation)g(of)h(Proposition)f(2.4)g(and)h(if)g(the)g (random)g(potential)i FD(V)48 b FE(has)34 b(property)g(\()p Fu(E)p FE(\),)h(we)120 5524 y(ha)n(v)o(e)1171 5746 y FD(T)1225 5761 y Fw(x)1290 5746 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)1611 5705 y Fx(.!)q(/)1721 5746 y Fz(/)f FD(T)1845 5705 y Fl(y)1828 5773 y Fw(x)1908 5746 y Fv(D)33 b FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)2339 5705 y Fx(.)p Fi(T)2437 5716 y Fp(x)2472 5705 y Fx(!)q(/)2555 5746 y Fz(/)807 b FE(\(2.11\))p eop %%Page: 7 7 7 6 bop 238 -183 a FH(Inte)l(gr)o(ated)26 b(Density)e(of)g(States)h (for)f(Random)f(Sc)o(hr\366ding)o(er)k(Oper)o(ator)o(s)e(with)f(Ma)o (gnetic)h(F)l(ields)237 b FE(7)120 125 y(for)25 b(all)g Fz(!)i Fv(2)d Fz(\177)643 140 y Fh(S)714 125 y FE(and)h(all)i FD(x)34 b Fv(2)24 b Fy(Z)1244 88 y Fw(d)1316 125 y FE(or)k FD(x)34 b Fv(2)24 b Fy(R)1673 88 y Fw(d)1720 125 y FE(,)h(depending)f (on)g(whether)k FD(V)40 b FE(is)24 b Fy(Z)2944 88 y Fw(d)2991 125 y FE(-)h(or)g Fy(R)3238 88 y Fw(d)3285 125 y FE(-er)n(godic.)120 245 y(Hence,)35 b(follo)n(wing)30 b(standard)h(ar)n(guments,)41 b FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)e Fz(/)32 b FE(is)f(an)h FD(er)l(godic)g(oper)o(ator)g FE(and)g(its)f(spectral) 120 365 y(components)41 b(are)i(non-random,)i(see)e([55,)j(Thm.)82 b(2.1].)h(Moreo)o(v)o(er)l(,)45 b(the)d(discrete)g(spectrum)120 486 y(of)i FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)567 450 y Fx(.!)q(/)678 486 y Fz(/)35 b FE(is)g(empty)f(for)h Fy(P)p FE(-almost)f(all)h Fz(!)e Fv(2)d Fz(\177)p FE(,)37 b(see)f([31)o(,)f(13,)g(55],)j(because)d(the)g(f)o(amily)120 606 y Fv(f)r FD(T)213 621 y Fw(x)257 606 y Fv(g)296 630 y Fw(x)6 b Ft(2)p Fr(Z)437 608 y Fp(d)512 606 y FE(and)32 b(hence)h Fv(f)r FD(T)1046 621 y Fw(x)1090 606 y Fv(g)1129 630 y Fw(x)6 b Ft(2)p Fr(R)1279 608 y Fp(d)1354 606 y FE(is)32 b FD(total)p FE(.)52 b(The)33 b(latter)f(is)g(true)h(by)f (de\002nition,)h(since)g(the)f(subset)120 726 y Fv(f)r FD(T)213 741 y Fw(x)257 726 y Fz( )9 b Fv(g)375 750 y Fw(x)d Ft(2)p Fr(Z)516 729 y Fp(d)586 726 y Fv(\032)28 b FE(L)753 690 y FC(2)794 726 y Fz(.)p Fy(R)912 690 y Fw(d)960 726 y Fz(/)i FE(contains)f(an)i(in\002nite)e(set)h(of)g (pairwise)g(orthogonal)f(functions)g(for)i(each)120 847 y Fz( )j Fv(2)25 b Fm(C)378 862 y FC(0)420 847 y Fz(.)p Fy(R)538 811 y Fw(d)585 847 y Fz(/)h FE(which)e(is)g(dense)h(in)f(L) 1422 811 y FC(2)1464 847 y Fz(.)p Fy(R)1582 811 y Fw(d)1629 847 y Fz(/)p FE(.)120 1200 y FA(3)144 b(The)35 b(Integrated)e(Density)h (of)h(States)120 1458 y FG(3.1)119 b(Existence)30 b(and)h(Uniqueness) 120 1650 y FE(The)e(quantity)f(of)h(main)f(interest)g(in)g(the)h (present)g(paper)g(is)f(the)h(inte)o(grated)e(density)h(of)h(states)f (and)120 1770 y(its)e(corresponding)e(measure,)i(called)g(the)g (density-of-states)e(measure.)34 b(The)26 b(ne)o(xt)f(theorem)g(deals) 120 1891 y(with)f(its)g(de\002nition)f(and)h(its)g(representation)f(as) i(an)f(in\002nite-v)n(olume)f(limit)g(of)h(the)g(suitably)f(scaled)120 2011 y(\002nite-v)n(olume)h(counterparts)h(\(2.9\).)30 b(It)25 b(is)f(the)h(main)f(result)h(of)f(the)h(present)g(paper)-5 b(.)120 2225 y FB(Theor)n(em)27 b(3.1.)49 b FD(Let)43 b Fz(0)c Fv(\032)34 b Fy(R)1196 2189 y Fw(d)1286 2225 y FD(be)43 b(a)f(bounded)g(open)h(cube)f(compatible)g(with)g(the)g (lattice)g Fy(Z)3593 2189 y Fw(d)120 2346 y FE([)p FD(r)l(ecall)24 b(\(2.1\))p FE(])g FD(and)g(let)952 2328 y Fz(\037)1020 2361 y Fx(0)1099 2346 y FD(denote)g(the)g(multiplication)d(oper)o(ator) h(associated)g(with)i(the)g(indicator)120 2466 y(function)j(of)h Fz(0)t FD(.)42 b(Assume)28 b(that)f(the)h(potentials)37 b(A)30 b(and)h(V)43 b(have)29 b(the)f(pr)l(operties)e FE(\()p Fu(C)p FE(\))p FD(,)j FE(\()p Fu(S)p FE(\))p FD(,)g FE(\()p Fu(I)p FE(\))p FD(,)g(and)120 2586 y FE(\()p Fu(E)p FE(\))p FD(.)d(Then)f(the)g FE(\(in\002nite-v)n(olume\))f(inte)o 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FD(and)e(independent)h (of)f Fz(0)t FD(.)120 3492 y(Mor)l(eo)o(ver)-11 b(,)29 b(let)f Fz(3)f Fv(\032)f Fy(R)970 3456 y Fw(d)1045 3492 y FD(stand)h(for)g(bounded)g(open)h(cubes)g(center)l(ed)g(at)g(the)f (origin.)39 b(Then)28 b(ther)l(e)120 3612 y(is)d(a)g(set)f Fz(\177)500 3627 y FC(0)566 3612 y Fv(2)h Fm(A)47 b FD(of)25 b(full)f(pr)l(obability)-5 b(,)21 b Fy(P)p Fz(.\177)1715 3627 y FC(0)1756 3612 y Fz(/)k Fv(D)g FE(1)p FD(,)g(suc)o(h)f(that)1399 3921 y(N)12 b Fz(.)7 b FD(E)i Fz(/)25 b Fv(D)57 b FE(lim)1756 3996 y Fx(3)p Ft(")p Fk(R)1912 3974 y Fp(d)1997 3848 y FD(N)2075 3799 y Fx(.!)q(/)2065 3874 y(3;)p FC(X)2207 3848 y Fz(.)7 b FD(E)i Fz(/)p 1990 3898 369 4 v 2106 3991 a Fv(j)2135 3992 y Fz(3)2214 3991 y Fv(j)3449 3921 y FE(\(3.2\))120 4201 y FD(holds)23 b(for)f(both)g(boundary)g (conditions)g FE(X)h Fv(D)g FE(D)g FD(and)g FE(X)g Fv(D)g FE(N)p FD(,)h(all)e Fz(!)k Fv(2)d Fz(\177)2707 4216 y FC(0)2748 4201 y FD(,)g(and)g(all)29 b(E)j Fv(2)23 b Fy(R)h FD(e)n(xcept)120 4321 y(the)h(\(at)g(most)f(countably)g(many\))h (discontinuity)d(points)i(of)31 b(N)12 b(.)120 4536 y FB(Pr)n(oof)o(.)50 b FE(See)26 b(Section)e(4.)p 3568 4536 4 68 v 3572 4472 60 4 v 3572 4536 V 3631 4536 4 68 v 120 4750 a FB(Remarks)i(3.2.)204 b FE(\(i\))100 b(As)28 b(to)g(the)h(limit)e Fz(3)g Fv(")h Fg(R)1955 4714 y Fw(d)2008 4750 y FE(,)h(we)g(here)h(and)e(in)g(the)h(follo)n (wing)e(think)g(of)i(a)120 4870 y(sequence)j(of)f(open)f(cubes)h (centered)h(at)e(the)h(origin)f(whose)h(edge)g(lengths)f(tend)g(to)h (in\002nity)-6 b(.)47 b(But)120 4991 y(there)34 b(e)o(xist)d(more)i (general)h(sequences)e(of)i(e)o(xpanding)d(re)o(gions)h(in)h Fy(R)2653 4955 y Fw(d)2733 4991 y FE(for)g(which)g(the)g(theorem)120 5111 y(remains)25 b(true,)g(see)g(for)g(e)o(xample)f([46)o(,)h(Rem.)g (1)g(on)g(p.)f(105])h(and)f([13,)h(p.)g(304].)248 5301 y(\(ii\))99 b(The)34 b(homogeneity)e(of)h(the)h(random)f(potential)f (and)i(the)f(magnetic)g(\002eld)h(with)f(respect)120 5421 y(to)h Fy(Z)301 5385 y Fw(d)382 5421 y FE(renders)h(the)f(r)-5 b(.h.s.)33 b(of)h(\(3.1\))g(independent)f(of)h Fz(0)t FE(.)58 b(In)34 b(case)k FD(V)49 b FE(is)33 b(e)n(v)o(en)g Fy(R)3057 5385 y Fw(d)3105 5421 y FE(-er)n(godic,)j(one)120 5542 y(may)f(pick)g(an)h(arbitrarily)f(shaped)g(bounded)g(subset)f Fz(0)h Fv(2)30 b Fm(B)t Fz(.)p Fy(R)2477 5506 y Fw(d)2525 5542 y Fz(/)36 b FE(with)2810 5541 y Fv(j)2839 5542 y Fz(0)2907 5541 y Fv(j)2966 5542 y Fz(>)31 b FE(0)k(or)g(e)n(v)o(en)g (an)o(y)120 5662 y(non-zero)29 b(square-inte)o(grable)f(function)g (instead)g(of)g(the)h(indicator)f(function;)h(for)g(details)f(see)g (the)120 5783 y(ne)o(xt)c(corollary)-6 b(.)p eop %%Page: 8 8 8 7 bop 120 -183 a FE(8)261 b FH(T)-7 b(.)23 b(Hupfer)-10 b(,)24 b(H.)e(Lesc)o(hk)o(e)o(,)i(P)-12 b(.)21 b(M\374ller)k(&)d(S.)h (W)-8 b(arzel)220 125 y FE(\(iii\))99 b(A)40 b(proof)f(of)h(the)g FD(e)n(xistence)f FE(of)h(the)g(in\002nite-v)n(olume)e(limits)g(in)h (\(3.2\))h(under)f(slightly)120 245 y(dif)n(ferent)h(hypotheses)e(w)o 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y(\(i)n(v\))99 b(Property)24 b(\()p Fu(S)p FE(\))g(is)g(only)e(assumed)h(to)g (guarantee)h(the)g(almost-sure)e(essential)h(self-adjoint-)120 1998 y(ness)46 b(of)f(the)g(in\002nite-v)n(olume)f(operator)i(on)f Fm(C)1897 1961 y Ft(1)1880 2025 y FC(0)1977 1998 y Fz(.)p Fy(R)2095 1962 y Fw(d)2143 1998 y Fz(/)p FE(.)92 b(Property)46 b(\()p Fu(I)p FE(\))f(is)g(mainly)g(technical.)120 2118 y(It)h(ensures)f(the)g(e)o(xistence)g(of)g(a)h(suf)n(\002ciently)e (high)g(inte)o(ger)h(moment)f(of)k FD(V)60 b FE(needed)46 b(for)f(the)120 2239 y(applicability)c(of)g(standard)h(resolv)o(ent)f (techniques.)80 b(In)42 b(particular)l(,)k(\()p Fu(I)p FE(\))c(does)g(not)f(distinguish)120 2359 y(between)d(the)g(positi)n(v) o(e)e(part)41 b FD(V)1254 2374 y Ft(C)1348 2359 y FE(:)p Fv(D)32 b FE(max)o Fv(f)p FE(0)p Fz(;)17 b FD(V)e Fv(g)38 b FE(and)f(the)h(ne)o(gati)n(v)o(e)e(part)41 b FD(V)2910 2374 y Ft(\000)3004 2359 y FE(:)p Fv(D)32 b FE(max)o Fv(f)p FE(0)p Fz(;)14 b Fv(\000)s FD(V)g Fv(g)120 2480 y FE(of)44 b 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(gnetic)h(F)l(ields)237 b FE(9)198 125 y(\(vii\))99 b(As)50 b(a)g(by-product,)55 b(our)49 b(proof)h(of)f(Theorem)h(3.1)f(yields)g (\(see)h(\(4.20\))f(belo)n(w\))g(the)120 245 y(follo)n(wing)23 b(rough)i(upper)f(bound)g(on)h(the)g(lo)n(w-ener)n(gy)f(f)o(all-of)n(f) g(of)32 b FD(N)12 b FE(,)1449 471 y FD(N)g Fz(.)7 b FD(E)i Fz(/)25 b Fv(\024)g FD(C)1905 470 y Fv(j)1941 471 y FD(E)2011 470 y Fv(j)2039 430 y Fw(d)6 b Fx(=)p FC(2)p Ft(\000)p FC(2)p Fx(#)3449 471 y FE(\(3.5\))120 697 y(for)29 b(all)35 b FD(E)h Fv(2)p FE(])21 b Fv(\000)g(1)p Fz(;)14 b Fv(\000)p FE(1])27 b(with)h(some)g(constant)g FD(C)35 b Fv(\025)27 b FE(0,)j(see)e(also)h([46)o(,)h(Thm.)d(5.29])i(for)f(the)h(case)130 818 y FD(A)h Fv(D)d FE(0.)44 b(The)29 b(true)h(leading)e(beha)n(vior)h (of)37 b FD(N)12 b Fz(.)7 b FD(E)i Fz(/)29 b FE(for)36 b FD(E)h Fv(!)27 b(\0001)i FE(is,)h(of)g(course,)g(consistent)e(with) 120 938 y(\(3.5\),)k(b)n(ut)e(typically)g(much)g(f)o(aster)-5 b(.)47 b(F)o(or)31 b(e)o(xample,)g(in)f(the)g(case)h(of)g(a)g(Gaussian) f(random)g(poten-)120 1059 y(tial,)36 b(in)d(the)g(sense)h(of)f (Subsection)g(3.3)g(belo)n(w)-6 b(,)35 b(it)e(is)g(kno)n(wn)f(that)h (lim)2678 1074 y Fw(E)7 b Ft(!\0001)2972 1059 y FD(E)3042 1022 y Ft(\000)p FC(2)3156 1059 y FE(log)20 b FD(N)12 b Fz(.)7 b FD(E)i Fz(/)29 b Fv(D)120 1179 y(\000)p Fz(.)p FE(2)p FD(C)9 b Fz(.)p FE(0)p Fz(//)521 1143 y Ft(\000)p FC(1)621 1179 y FE(,)39 b(also)d(in)g(the)g(presence)h(of)f(a)g (constant)g(magnetic)f(\002eld)i([41,)f(9,)g(55].)65 b(The)36 b(lead-)120 1299 y(ing)31 b(lo)n(w-ener)n(gy)g(beha)n(vior)g (is)g(less)f(uni)n(v)o(ersal)g(in)h(case)g(of)h(a)f(positi)n(v)o(e)e (Poissonian)h(potential)g(and)120 1420 y(a)37 b(constant)e(magnetic)g (\002eld)h([10,)g(21,)g(26)o(,)g(27,)g(22,)g(57)o(],)j(where)44 b FD(N)j FE(v)n(anishes)35 b(for)h(ne)o(gati)n(v)o(e)e(en-)120 1540 y(er)n(gies)42 b(an)o(yw)o(ay)-6 b(.)78 b(In)41 b(this)f(conte)o(xt)g(we)i(recall)f(from)g([41,)g(55)o(])h(that)e(the)h (high-ener)n(gy)g(asymp-)120 1660 y(totics)f(is)g(neither)h(af)n (fected)g(by)f(the)g(magnetic)g(\002eld)h(nor)f(by)h(the)f(random)g (potential)g(and)g(gi)n(v-)120 1781 y(en)27 b(by)f(lim)505 1796 y Fw(E)7 b Ft(!1)741 1781 y FD(E)811 1745 y Ft(\000)p Fw(d)f Fx(=)p FC(2)1010 1781 y FD(N)12 b Fz(.)7 b FD(E)i Fz(/)25 b Fv(D)h FE([)p Fz(.)p FD(d)7 b Fz(=)p FE(2)p Fz(/)p FE(!)14 b Fz(.)p FE(2)p Fz(\031)c(/)1870 1745 y Fw(d)c Fx(=)p FC(2)1990 1781 y FE(])2023 1745 y Ft(\000)p FC(1)2149 1781 y FE(in)26 b(accordance)i(with)d(W)-8 b(e)o(yl')j(s)26 b(celebrated)120 1901 y(asymptotics)d(for)i(the)g (free)h(particle)f([59].)170 2091 y(\(viii\))99 b(In)25 b(case)34 b FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)d Fz(/)25 b FE(is)g(unbounded)e(from)i(belo)n(w)f(almost)g(surely)g(and)h (serv)o(es)f(as)h(the)g(one-)120 2211 y(particle)f(Hamiltonian)f(of)h (a)g(macroscopic)f(system)g(of)h(non-interacting)f(\(spinless\))g (fermions,)g(the)120 2332 y(corresponding)31 b(free)i(ener)n(gy)g(and)f 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y(In)g(analogy)g(to)f([46,)h(Prob)l(.)g(II.4])g(Theorem)f(3.1)h (implies)120 3361 y FB(Cor)n(ollary)g(3.3.)49 b FD(Assume)24 b(that)h(the)g(potentials)33 b(A)27 b(and)h(V)40 b(have)25 b(the)g(pr)l(operties)e FE(\()p Fu(C)p FE(\))p FD(,)j FE(\()p Fu(S)p FE(\))p FD(,)g(and)e FE(\()p Fu(I)p FE(\))p FD(.)120 3482 y(Mor)l(eo)o(ver)-11 b(,)25 b(let)j(V)40 b(be)25 b Fy(R)976 3445 y Fw(d)1023 3482 y FD(-er)l(godic)g(\(and)f (not)g(only)h Fy(Z)2007 3445 y Fw(d)2054 3482 y FD(-er)l(godic\).)31 b(Then)689 3754 y(N)12 b Fz(.)7 b FD(E)i Fz(/)25 b Fv(D)1115 3684 y FE(1)p 1056 3730 169 4 v 1056 3829 a Fv(j)1106 3830 y FD(f)1154 3829 y Fv(j)1183 3791 y FC(2)1183 3856 y(2)1248 3754 y Fy(E)1324 3657 y Fs(n)1393 3754 y FE(T)m(r)1498 3657 y Fs(h)p 1537 3677 71 4 v 1559 3754 a FD(f)34 b Fz(2)1701 3682 y Fs(\000)1746 3754 y FD(E)28 b Fv(\000)g FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)d Fz(/)2298 3682 y Fs(\001)2372 3754 y FD(f)2420 3657 y Fs(i)q(o)2515 3754 y Fz(;)220 b FD(E)34 b Fv(2)24 b Fy(R)p Fz(;)385 b FE(\(3.6\))120 4057 y FD(for)37 b(any)f(non-zer)l(o)58 b(f)52 b Fv(2)31 b FE(L)1098 4021 y FC(2)1139 4057 y Fz(.)p Fy(R)1257 4021 y Fw(d)1305 4057 y Fz(/)37 b FD(whic)o(h)f(is)h (to)f(be)h(under)o(stood)e(as)h(a)h(multiplication)d(oper)o(ator)120 4178 y(inside)24 b(the)h(tr)o(ace)o(.)120 4392 y FB(Remark)h(3.4.)49 b FE(Assume)43 b(the)i(situation)d(of)i(Corollary)g(3.3)g(and)g(that)g (the)g(spectral)g(projection)120 4512 y Fz(2)s(.)7 b FD(E)31 b Fv(\000)e FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)e Fz(//)31 b FE(possesses)f Fy(P)p FE(-almost)f(surely)i(a)g (jointly)e(continuous)g(inte)o(gral)h(k)o(ernel)h Fy(R)3493 4476 y Fw(d)3562 4512 y Fv(\002)120 4632 y Fy(R)201 4596 y Fw(d)274 4632 y Fv(3)25 b Fz(.)s FD(x)9 b Fz(;)20 b FD(y)6 b Fz(/)25 b Fv(7!)h Fz(2)s(.)7 b FD(E)28 b Fv(\000)g FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)e Fz(//.)s FD(x)9 b Fz(;)20 b FD(y)6 b Fz(/)25 b Fv(2)g Fy(C)p FE(.)33 b(Then)25 b(\(3.6\))h(with)46 b FD(f)g Fv(2)25 b Fm(C)2855 4647 y FC(0)2897 4632 y Fz(.)p 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Fm(B)t Fz(.)p Fy(R)524 5422 y Fw(d)572 5458 y Fz(/)h Fy(P)p FE(-almost)f(surely)g(to)h(the)f(Kato)h (class)358 5696 y Fm(K)32 b Fz(.)p Fy(R)599 5654 y Fw(d)646 5696 y Fz(/)25 b FE(:)p Fv(D)838 5599 y Fs(n)894 5696 y Fz(v)j FE(:)d Fy(R)1105 5654 y Fw(d)1177 5696 y Fv(!)g Fy(R)39 b FE(:)64 b Fz(v)42 b FE(Borel)25 b(measurable)g(and)52 b(lim)2546 5760 y Fw(t)7 b Ft(#)p FC(0)2682 5696 y Fg({)2748 5711 y Fw(t)2780 5696 y Fz(.v)t(/)25 b Fv(D)f FE(0)3109 5599 y Fs(o)3165 5696 y Fz(;)251 b FE(\(3.8\))p eop %%Page: 10 10 10 9 bop 120 -183 a FE(10)261 b FH(T)-7 b(.)22 b(Hupfer)-10 b(,)24 b(H.)e(Lesc)o(hk)o(e)o(,)j(P)-12 b(.)21 b(M\374ller)j(&)f(S.)g (W)-8 b(arzel)120 128 y FE(where)37 b Fg({)466 143 y Fw(t)498 128 y Fz(.v)t(/)32 b FE(:)p Fv(D)e FE(sup)934 153 y Fw(x)6 b Ft(2)p Fr(R)1084 131 y Fp(d)1140 47 y Fn(R)1205 76 y Fw(t)1186 162 y FC(0)1237 128 y FE(d)p FD(s)1359 47 y Fn(R)1405 162 y Fr(R)1469 140 y Fp(d)1510 128 y FE(d)1560 91 y Fw(d)1607 128 y Fz(\030)26 b FE(e)1719 91 y Ft(\000)p 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1275 y(are)21 b(assumed)e(to)h(be)g(of)g (the)f(form)42 b FD(f)1348 1290 y Fw(n)1393 1275 y Fz(.)s FD(x)9 b Fz(/)21 b Fv(D)1642 1209 y Fs(P)1729 1237 y Fw(n)1729 1303 y(k)5 b Ft(D)p FC(1)1902 1275 y FD(f)1933 1290 y Fw(n)t Fx(;)p Fw(k)2041 1257 y Fz(\037)2110 1290 y Fx(0)2158 1301 y Fp(n)s Fq(;)p Fp(k)2246 1275 y Fz(.)s FD(x)k Fz(/)20 b FE(with)f(suitable)g(constants)41 b FD(f)3367 1290 y Fw(n)t Fx(;)p Fw(k)3495 1275 y Fv(\025)20 b FE(0)120 1404 y(and)h(bounded)e(Borel)i(sets)f Fz(0)1124 1419 y Fw(n)t Fx(;)p Fw(k)1252 1404 y Fv(2)g Fm(B)t Fz(.)p Fy(R)1547 1368 y Fw(d)1595 1404 y Fz(/)h FE(which)f(are)h(pairwise)f (disjoint)e(for)j(each)f(\002x)o(ed)h FD(n)t FE(.)29 b(Using)120 1525 y(\(3.1\))f(and)g(the)f Fy(R)741 1489 y Fw(d)788 1525 y FE(-homogeneity)g(of)g(the)h(random)f(potential)f (\(see)i(Remark)h(3.2\(ii\))o(\))f(one)f(v)o(eri\002es)120 1645 y(that)c(\(3.6\))g(is)g(v)n(alid)e(for)j(all)e(simple)g (functions.)29 b(Thanks)23 b(to)f(the)h(con)l(v)o(er)n(gence)45 b FD(f)2927 1660 y Fw(n)2996 1645 y Fv(!)g FD(f)f FE(as)23 b FD(n)28 b Fv(!)23 b(1)120 1766 y FE(in)i(L)284 1729 y FC(2)325 1766 y Fz(.)p Fy(R)443 1729 y Fw(d)491 1766 y Fz(/)g FE(this)f(implies)454 1998 y(lim)379 2048 y Fw(n)t Fx(;)p Fw(m)t Ft(!1)689 1863 y Fn(Z)742 2088 y Fx(\177)804 1998 y Fy(P)p Fz(.)p FE(d)p Fz(!)r(/)k Fo(k)p Fz(2)1229 1957 y Fx(.!)q(/)1339 1927 y Fs(\000)1399 1998 y FD(f)1430 2013 y Fw(n)1495 1998 y Fv(\000)41 b FD(f)1645 2013 y Fw(m)1708 1927 y Fs(\001)1746 1998 y Fo(k)1796 1951 y FC(2)1796 2028 y(2)1862 1998 y Fv(D)32 b FD(N)12 b Fz(.)7 b FD(E)i Fz(/)102 b FE(lim)2228 2048 y Fw(n)t Fx(;)p Fw(m)t Ft(!1)2538 1923 y Fs(\014)2538 1973 y(\014)2588 1998 y FD(f)2619 2013 y Fw(n)2684 1998 y Fv(\000)41 b FD(f)2834 2013 y Fw(m)2897 1923 y Fs(\014)2897 1973 y(\014)2925 1951 y FC(2)2925 2028 y(2)2991 1998 y Fv(D)25 b FE(0)p Fz(;)272 b FE(\(3.9\))120 2251 y(where)34 b(we)e(are)i(using)d(the)i (abbre)n(viation)e Fz(2)1711 2214 y Fx(.!)q(/)1850 2251 y FE(:)p Fv(D)e Fz(2)2065 2179 y Fs(\000)2110 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FE(\(3.10\))120 2807 y(for)30 b(all)45 b FD(j)37 b Fv(2)27 b Fy(N)i FE(by)g(Jensen')-5 b(s)28 b(inequality)g(and)i(\(3.9\).)44 b(Thanks)28 b(to)h (monotonicity)e(the)i(r)-5 b(.h.s.)43 b(of)29 b(the)120 2927 y(estimate)673 3154 y Fo(k)p Fz(2)803 3113 y Fx(.!)q(/)935 3154 y FD(f)966 3169 y Fw(n)1007 3180 y Fp(i)1057 3154 y Fv(\000)19 b Fz(2)1234 3113 y Fx(.!)q(/)1366 3154 y FD(f)1397 3169 y Fw(n)1448 3180 y Fp(j)1478 3154 y Fo(k)1528 3169 y FC(2)1594 3154 y Fv(\024)1834 3047 y Ft(1)1812 3071 y Fs(X)1694 3260 y Fw(k)5 b Ft(D)p FC(min)q Ft(f)o Fw(i)s Fx(;)12 b Fw(j)c Ft(g)2064 3154 y Fo(k)p Fz(2)2194 3113 y Fx(.!)q(/)2326 3154 y FD(f)2357 3169 y Fw(n)2398 3181 y Fp(k)r Ff(C)p Fj(1)2530 3154 y Fv(\000)19 b Fz(2)2707 3113 y Fx(.!)q(/)2839 3154 y FD(f)2870 3169 y Fw(n)2911 3180 y Fp(k)2949 3154 y Fo(k)2999 3169 y FC(2)3040 3154 y Fz(;)326 b FE(\(3.11\))120 3436 y(con)l(v)o(er)n(ges)22 b(pointwise)f(for)h(all)g Fz(!)i Fv(2)e Fz(\177)1461 3451 y Fh(S)1530 3436 y FE(as)f FD(i)t Fz(;)30 b FD(j)i Fv(!)23 b(1)p FE(.)29 b(Since)23 b(lim)2453 3451 y Fw(i)s Fx(;)12 b Fw(j)c Ft(!1)2719 3369 y Fs(P)2806 3398 y Ft(1)2806 3462 y Fw(k)d Ft(D)p FC(min)p Ft(f)o Fw(i)s Fx(;)12 b Fw(j)c Ft(g)3180 3436 y Fy(E)3256 3364 y Fs(\002)3291 3436 y Fo(k)p Fz(2)35 b FD(f)3488 3451 y Fw(n)3529 3463 y Fp(k)r Ff(C)p Fj(1)120 3556 y Fv(\000)p Fz(2)h FD(f)345 3571 y Fw(n)386 3582 y Fp(k)424 3556 y Fo(k)474 3571 y FC(2)515 3485 y Fs(\003)581 3556 y Fv(D)30 b FE(0)36 b(by)g(\(3.10\),)i(the)e(monotone-)f(and)g(dominated-con)l(v)o(er)n (gence)g(theorems)h(imply)120 3676 y(that)j(the)f(r)-5 b(.h.s.)38 b(\(and)h(hence)g(the)g(l.h.s.\))72 b(of)38 b(\(3.11\))h(con)l(v)o(er)n(ges)f(in)h(f)o(act)g(to)f(zero)h(for)g Fy(P)p FE(-almost)120 3797 y(all)k Fz(!)37 b Fv(2)e Fz(\177)p FE(.)84 b(In)43 b(other)g(w)o(ords,)k(the)c(subsequence)2050 3725 y Fs(\000)2088 3797 y Fz(2)2168 3761 y Fx(.!)q(/)2300 3797 y FD(f)2331 3812 y Fw(n)2382 3823 y Fp(j)2412 3725 y Fs(\001)2462 3827 y Fw(j)2538 3797 y FE(is)f(Cauchy)h(in)f Fm(J)3219 3812 y FC(2)3261 3797 y Fz(.)p FE(L)3359 3761 y FC(2)3400 3797 y Fz(.)p Fy(R)3518 3761 y Fw(d)3566 3797 y Fz(//)120 3937 y FE(for)47 b Fy(P)p FE(-almost)f(all)g Fz(!)39 b Fv(2)e Fz(\177)p FE(.)95 b(Since)47 b(the)f(space)h Fm(J)2062 3952 y FC(2)2103 3937 y Fz(.)p FE(L)2201 3901 y FC(2)2243 3937 y Fz(.)p Fy(R)2361 3901 y Fw(d)2408 3937 y Fz(//)g FE(is)f(complete,)51 b(this)46 b(sequence)120 4058 y(con)l(v)o(er)n(ges)30 b(with)e(respect)i(to)f(the)g (Hilbert-Schmidt)f(norm)h Fo(k)p Fv(\001)p Fo(k)2367 4086 y FC(2)2438 4058 y FE(to)g(some)36 b FD(F)2862 4021 y Fx(.!)q(/)2999 4058 y Fv(2)28 b Fm(J)3195 4073 y FC(2)3236 4058 y Fz(.)p FE(L)3334 4021 y FC(2)3375 4058 y Fz(.)p Fy(R)3493 4021 y Fw(d)3541 4058 y Fz(//)p FE(.)120 4178 y(Let)58 b FD(f)51 b FE(:)30 b Fm(C)532 4141 y Ft(1)514 4205 y FC(0)612 4178 y Fz(.)p Fy(R)730 4142 y Fw(d)778 4178 y Fz(/)h Fv(!)g FE(L)1047 4142 y FC(2)1088 4178 y Fz(.)p Fy(R)1206 4142 y Fw(d)1254 4178 y Fz(/)36 b FE(denote)f(a)g(multiplication)e(operator)j(associated)f(with)56 b FD(f)21 b FE(.)62 b(The)120 4298 y(abo)o(v)o(e)24 b(con)l(v)o(er)n (gence)h(and)g(lim)1203 4313 y Fw(n)t Ft(!1)1434 4297 y Fv(j)1462 4298 y Fz(.)d FD(f)40 b Fv(\000)h FD(f)1740 4313 y Fw(n)1785 4298 y Fz(/ )1901 4297 y Fv(j)1930 4322 y FC(2)1996 4298 y Fv(D)25 b FE(0)f(for)h(all)g Fz( )34 b Fv(2)24 b Fm(C)2713 4262 y Ft(1)2696 4326 y FC(0)2794 4298 y Fz(.)p Fy(R)2912 4262 y Fw(d)2959 4298 y Fz(/)h FE(imply)f(that)31 b FD(F)3530 4262 y Fx(.!)q(/)120 4419 y FE(is)f(the)g(unique)g(continuous)e(e)o(xtension)h(of)h Fz(2)1742 4383 y Fx(.!)q(/)1874 4419 y FD(f)51 b FE(from)29 b Fm(C)2260 4382 y Ft(1)2242 4446 y FC(0)2340 4419 y Fz(.)p Fy(R)2458 4383 y Fw(d)2506 4419 y Fz(/)h FE(to)g(the)g(whole)f (Hilbert)h(space)120 4539 y(L)181 4503 y FC(2)223 4539 y Fz(.)p Fy(R)341 4503 y Fw(d)388 4539 y Fz(/)p FE(.)h(Denoting)24 b(this)g(e)o(xtension)f(also)i(by)f Fz(2)1841 4503 y Fx(.!)q(/)1973 4539 y FD(f)d FE(,)j(we)i(thus)e(ha)n(v)o(e)1290 4727 y(lim)1269 4791 y Fw(j)8 b Ft(!1)1484 4727 y Fo(k)o Fz(2)1613 4685 y Fx(.!)q(/)1745 4727 y FD(f)1777 4742 y Fw(n)1828 4753 y Fp(j)1877 4727 y Fv(\000)19 b Fz(2)2054 4685 y Fx(.!)q(/)2186 4727 y FD(f)i Fo(k)2285 4757 y FC(2)2351 4727 y Fv(D)k FE(0)895 b(\(3.12\))120 4957 y(for)26 b Fy(P)p FE(-almost)d(all)i Fz(!)i Fv(2)d Fz(\177)p FE(.)31 b(W)-8 b(e)25 b(therefore)g(get)203 5181 y Fy(E)279 5085 y Fs(h)319 5181 y Fo(k)o Fz(2)448 5110 y Fs(\000)493 5181 y FD(E)k Fv(\000)e FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)d Fz(/)1045 5110 y Fs(\001)1105 5181 y FD(f)21 b Fo(k)1204 5134 y FC(2)1204 5212 y(2)1245 5085 y Fs(i)1309 5181 y Fv(D)k Fy(E)1488 5085 y Fs(h)1574 5181 y FE(lim)1553 5245 y Fw(j)8 b Ft(!1)1767 5181 y Fo(k)p Fz(2)1897 5110 y Fs(\000)1942 5181 y FD(E)28 b Fv(\000)g FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)d Fz(/)2494 5110 y Fs(\001)2554 5181 y FD(f)2585 5196 y Fw(n)2636 5207 y Fp(j)2666 5181 y Fo(k)2716 5134 y FC(2)2716 5212 y(2)2758 5085 y Fs(i)394 5404 y Fv(D)57 b FE(lim)508 5468 y Fw(j)8 b Ft(!1)723 5404 y Fy(E)799 5308 y Fs(h)838 5404 y Fo(k)p Fz(2)968 5333 y Fs(\000)1013 5404 y FD(E)28 b Fv(\000)f FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)e Fz(/)1565 5333 y Fs(\001)1625 5404 y FD(f)1656 5419 y Fw(n)1707 5430 y Fp(j)1737 5404 y Fo(k)1787 5356 y FC(2)1787 5434 y(2)1828 5308 y Fs(i)1892 5404 y Fv(D)32 b FD(N)12 b Fz(.)7 b FD(E)i Fz(/)60 b FE(lim)2270 5468 y Fw(j)8 b Ft(!1)2485 5329 y Fs(\014)2485 5379 y(\014)2534 5404 y FD(f)2565 5419 y Fw(n)2616 5430 y Fp(j)2646 5329 y Fs(\014)2646 5379 y(\014)2674 5356 y FC(2)2674 5434 y(2)2740 5404 y Fv(D)32 b FD(N)12 b Fz(.)7 b FD(E)i Fz(/)3106 5403 y Fv(j)3157 5404 y FD(f)3205 5403 y Fv(j)3234 5363 y FC(2)3234 5431 y(2)3289 5404 y Fz(:)83 b FE(\(3.13\))120 5634 y(F)o(or)56 b(the)f(second)g(equality) f(we)i(used)f(the)g(monotone-con)l(v)o(er)n(gence)f(theorem.)122 b(Note)55 b(that)120 5755 y Fz(.)p Fo(k)p Fz(2)287 5718 y Fx(.!)q(/)419 5755 y FD(f)451 5770 y Fw(n)496 5755 y Fo(k)546 5718 y FC(2)546 5782 y(2)587 5755 y Fz(/)624 5770 y Fw(n)694 5755 y FE(is)25 b(monotone)e(increasing)h(since)h Fo(k)p Fz(2)2003 5718 y Fx(.!)q(/)2135 5755 y FD(f)2166 5770 y Fw(n)2211 5755 y Fo(k)2261 5718 y FC(2)2261 5782 y(2)2328 5755 y Fv(D)f Fo(k)p Fz(2)2560 5718 y Fx(.!)q(/)2692 5755 y FD(f)2741 5718 y FC(2)2723 5781 y Fw(n)2782 5755 y Fz(2)2862 5718 y Fx(.!)q(/)2972 5755 y Fo(k)3022 5770 y FC(1)3064 5755 y FE(.)p 3568 5755 4 68 v 3572 5691 60 4 v 3572 5755 V 3631 5755 4 68 v eop %%Page: 11 11 11 10 bop 188 -183 a FH(Inte)l(gr)o(ated)26 b(Density)f(of)e(States)i (for)f(Random)g(Sc)o(hr\366ding)o(er)i(Oper)o(ator)o(s)g(with)d(Ma)o (gnetic)i(F)l(ields)238 b FE(11)120 125 y FG(3.2)119 b(Some)30 b(Pr)n(operties)g(of)g(the)g(Density-of-States)f(Measur)n(e) 120 315 y FE(The)40 b(proof)f(of)h(Theorem)f(3.1)g(will)g(be)g(based)g (on)h(the)f(\(almost-sure\))g(v)n(ague)g(con)l(v)o(er)n(gence)g([6,)120 446 y(Def.)k(30.1])e(of)i(the)e(tw)o(o)h FD(spatial)e(eig)o(en)l(value) 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FE(:)p Fv(D)1061 1330 y FE(1)p 1024 1377 125 4 v 1024 1470 a Fv(j)1052 1471 y Fz(0)1120 1470 y Fv(j)1172 1400 y Fy(E)1248 1304 y Fs(n)1317 1400 y FE(T)m(r)1422 1329 y Fs(\002)1456 1382 y Fz(\037)1525 1415 y Fx(0)1594 1382 y Fz(\037)1667 1415 y Fw(I)1707 1400 y Fz(.)8 b FD(H)i Fz(.)g FD(A)5 b Fz(;)17 b FD(V)d Fz(//)2160 1382 y(\037)2229 1415 y Fx(0)2284 1329 y Fs(\003)2319 1304 y(o)2374 1400 y Fz(;)220 b FD(I)38 b Fv(2)25 b Fm(B)t Fz(.)p Fy(R)p Fz(/;)331 b FE(\(3.14\))120 1669 y FD(is)38 b(a)g(positive)f(Bor)l(el)h(measur)l(e)g(on)f(the)h(r)l(eal)g(line)g Fy(R)p FD(,)j(well)e(de\002ned)e(in)h(terms)f(of)h(the)g(spatially)120 1789 y(localized)25 b(pr)l(ojection-valued)f(spectr)o(al)g(measur)l(e)i (of)g(the)f(in\002nite-volume)g(r)o(andom)f(Sc)o(hr\366ding)o(er)120 1910 y(oper)o(ator)-11 b(,)23 b(and)i(independent)f(of)g Fz(0)t FD(.)31 b(Mor)l(eo)o(ver)-11 b(,)25 b(in)f(the)h(sense)f(of)h (va)o(gue)g(con)l(ver)l(g)o(ence)1573 2217 y Fz(\027)31 b Fv(D)57 b FE(lim)1754 2291 y Fx(3)p Ft(")p Fk(R)1910 2269 y Fp(d)1988 2143 y Fz(\027)2041 2094 y Fx(.!)q(/)2035 2169 y(3;)p FC(X)p 1988 2193 190 4 v 2015 2286 a Fv(j)2043 2287 y Fz(3)2122 2286 y Fv(j)3399 2217 y FE(\(3.15\))120 2494 y FD(for)25 b(both)f FE(X)h Fv(D)f FE(D)h FD(and)f FE(X)h Fv(D)g FE(N)g FD(and)f Fy(P)p FD(-almost)g(all)g Fz(!)j Fv(2)d Fz(\177)p FD(.)120 2705 y FB(Pr)n(oof)o(.)50 b FE(See)26 b(Section)e(4.)p 3568 2705 4 68 v 3572 2641 60 4 v 3572 2705 V 3631 2705 4 68 v 120 2916 a FB(Remarks)i(3.6.)204 b FE(\(i\))100 b(Lemma)38 b(3.5)h(generalizes)g([46,)k(Thm.)38 b(5.20])h(which)g(deals)g(with)f(the)120 3036 y(case)43 b FD(A)31 b Fv(D)e FE(0.)54 b(In)32 b(f)o(act,)j(our)d(proof)h(in)f (Section)g(4)g(closely)g(follo)n(ws)f(the)h(ar)n(guments)g(gi)n(v)o(en) f(there.)120 3156 y(Concerning)f(the)g(independence)f(of)h(X)g(of)g (the)f(in\002nite-v)n(olume)g(limit)f(in)h(\(3.15\),)i(we)f(b)n(uild)e (on)i(a)120 3277 y(result)25 b(in)f([42])h(for)g(bounded)i FD(V)15 b FE(.)31 b(\(Alternati)n(v)o(ely)-6 b(,)22 b(one)j(may)f(use)h (a)g(result)f(in)h([19].\))248 3466 y(\(ii\))99 b(Lemma)23 b(3.5)f(alone)h(does)f(not)h(imply)e(the)i(e)o(xistence)f(of)h(the)g (inte)o(grated)f(density)g(of)h(states)127 3586 y FD(N)12 b FE(.)67 b(Moreo)o(v)o(er)l(,)38 b(e)n(v)o(en)e(if)h(the)g (\002niteness)f(of)h Fz(\027)6 b(.)p FE(])24 b Fv(\000)g(1)p Fz(;)d FD(E)9 b FE([)p Fz(/)36 b FE(for)h(all)44 b FD(E)c Fv(2)31 b Fy(R)37 b FE(were)h(kno)n(wn,)g(see)120 3707 y(\(3.5\),)f(the)d(v)n(ague)g(con)l(v)o(er)n(gence)h(\(3.15\))f(alone)g (w)o(ould)f(not)h(imply)f(the)h(pointwise)f(con)l(v)o(er)n(gence)120 3827 y(\(3.2\))45 b(of)g(the)f(distrib)n(ution)e(functions)i(in)g(case) h(their)f(supports)g(are)h(not)f(uniformly)f(bounded)120 3947 y(from)30 b(belo)n(w)-6 b(.)43 b(The)30 b(latter)f(occurs)h(for)f (random)g(potentials)f(with)h(realizations)j FD(V)3028 3911 y Fx(.!)q(/)3168 3947 y FE(which)d(yield)120 4068 y(operators)39 b FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)851 4032 y Fx(.!)q(/)961 4068 y Fz(/)31 b FE(unbounded)f(from)h (belo)n(w)-6 b(.)47 b(On)31 b(the)g(other)f(hand,)i(\(3.2\))f(implies)f (\(3.15\),)120 4188 y(see)c(Proposition)d(4.3)h(belo)n(w)-6 b(.)270 4399 y(Using)29 b(\(3.14\))h(one)h(may)f(relate)g(properties)g (of)h(the)f(density-of-states)f(measure)h Fz(\027)37 b FE(to)29 b(simple)120 4519 y(spectral)g(properties)g(of)g(the)g (in\002nite-v)n(olume)e(magnetic)i(Schr\366dinger)g(operator)-5 b(.)43 b(Examples)28 b(are)120 4640 y(the)41 b(support)g(of)g Fz(\027)47 b FE(and)41 b(the)g(location)g(of)g(the)g(almost-sure)f (spectrum)h(of)49 b FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)3205 4603 y Fx(.!)q(/)3316 4640 y Fz(/)41 b FE(or)g(the)120 4760 y(absence)34 b(of)f(a)g(point)f(component)g(in)g(the)h(Lebesgue)g (decomposition)d(of)j Fz(\027)39 b FE(and)33 b(the)g(absence)g(of)120 4880 y(\223immobile)24 b(eigen)l(v)n(alues\224)g(of)32 b FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)1541 4844 y Fx(.!)q(/)1651 4880 y Fz(/)p FE(.)31 b(This)24 b(is)h(the)f(content)h (of)120 5091 y FB(Cor)n(ollary)g(3.7.)49 b FD(Under)23 b(the)h(assumptions)d(of)j(Theor)l(em)g(3.1)g(and)f(letting)29 b(I)38 b Fv(2)23 b Fm(B)t Fz(.)p Fy(R)p Fz(/)i FD(the)f(follow-)120 5212 y(ing)29 b(equivalence)g(holds:)67 b Fz(\027)6 b(.)h FD(I)12 b Fz(/)29 b Fv(D)e FE(0)57 b FD(if)29 b(and)f(only)h(if)2096 5194 y Fz(\037)2170 5227 y Fw(I)2209 5140 y Fs(\000)2255 5212 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)2575 5175 y Fx(.!)q(/)2686 5212 y Fz(/)2723 5140 y Fs(\001)2788 5212 y Fv(D)27 b FE(0)58 b FD(for)28 b Fg(P)p FD(-almost)i(all)120 5332 y Fz(!)e Fv(2)c Fz(\177)p FD(.)30 b(This)25 b(immediately)f (implies:)275 5542 y(\(i\))141 b FE(supp)13 b Fz(\027)40 b Fv(D)34 b FE(spec)22 b FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)1430 5506 y Fx(.!)q(/)1541 5542 y Fz(/)41 b FD(for)f Fg(P)p FD(-almost)i(all)f Fz(!)36 b Fv(2)d Fz(\177)p FD(.)79 b FE([)p FD(Her)l(e)42 b FE(spec)22 b FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)3493 5506 y Fx(.!)q(/)3603 5542 y Fz(/)469 5662 y FD(denotes)35 b(the)h(spectrum)f(of)43 b(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)1810 5626 y Fx(.!)q(/)1920 5662 y Fz(/)36 b FD(and)71 b FE(supp)13 b Fz(\027)37 b FE(:)p Fv(D)30 b(f)7 b FD(E)40 b Fv(2)30 b Fy(R)h FE(:)g Fz(\027)6 b(.)p FE(])h FD(E)33 b Fv(\000)23 b Fz(";)e FD(E)32 b Fv(C)469 5783 y Fz(")s FE([)p Fz(/)25 b(>)g FE(0)50 b(for)25 b(all)f Fz(")k(>)d FE(0)p Fv(g)f FD(is)h(the)f(topolo) o(gical)f(support)h(of)g Fz(\027)6 b FD(.)p FE(])p eop %%Page: 12 12 12 11 bop 120 -183 a FE(12)261 b FH(T)-7 b(.)22 b(Hupfer)-10 b(,)24 b(H.)e(Lesc)o(hk)o(e)o(,)j(P)-12 b(.)21 b(M\374ller)j(&)f(S.)g (W)-8 b(arzel)248 125 y FD(\(ii\))134 b FE(0)30 b Fv(D)h Fz(\027)6 b(.)p Fv(f)h FD(E)i Fv(g)p Fz(/)1001 53 y Fs(\000)1069 125 y Fv(D)30 b FE(lim)1310 140 y Fx(")r Ft(#)p FC(0)1448 125 y FE([)7 b FD(N)12 b Fz(.)7 b FD(E)28 b Fv(C)19 b Fz(")s(/)h Fv(\000)26 b FD(N)12 b Fz(.)7 b FD(E)i Fz(/)p FE(])2281 53 y Fs(\001)2389 125 y FD(if)34 b(and)h(only)f(if)42 b(E)d Fv(2)30 b Fy(R)35 b FD(is)f(not)h(an)469 245 y(eig)o(en)l(value) 25 b(of)32 b(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)1352 209 y Fx(.!)q(/)1462 245 y Fz(/)25 b FD(for)f Fg(P)p FD(-almost)i(all)e Fz(!)j Fv(2)e Fz(\177)p FD(.)120 460 y FB(Pr)n(oof)o(.)50 b FE(If)538 442 y Fz(\037)611 475 y Fw(I)650 389 y Fs(\000)696 460 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)1017 424 y Fx(.!)q(/)1127 460 y Fz(/)1164 389 y Fs(\001)1237 460 y Fv(D)33 b FE(0)42 b(for)f Fg(P)p FE(-almost)i(all)e Fz(!)36 b Fv(2)e Fz(\177)p FE(,)45 b(then)c Fz(\027)6 b(.)h FD(I)12 b Fz(/)36 b Fv(D)e FE(0)41 b(using)g(\(3.14\).)120 580 y(Con)l(v)o(ersely)-6 b(,)47 b(for)d(e)n(v)o(ery)e Fz( )j Fv(2)34 b Fm(C)1333 595 y FC(0)1375 580 y Fz(.)p Fy(R)1493 544 y Fw(d)1541 580 y Fz(/)h Fv(\032)g FE(L)1787 544 y FC(2)1828 580 y Fz(.)p Fy(R)1946 544 y Fw(d)1994 580 y Fz(/)p FE(,)48 b(normalized)42 b(in)h(the)g(sense)g Fo(h)p Fz( )s(;)14 b( )9 b Fo(i)36 b Fv(D)e FE(1,)120 701 y(there)j(e)o(xists)d(a)i(bounded)g(open)f(cube) i Fz(0)e Fv(\032)30 b Fy(R)1819 665 y Fw(d)1903 701 y FE(compatible)35 b(with)g Fy(Z)2664 665 y Fw(d)2747 701 y FE(such)h(that)f(supp)13 b Fz( )40 b Fv(\022)31 b Fz(0)120 821 y FE(and)25 b(therefore)730 965 y Fs(D)780 1061 y Fz( )e(;)934 1043 y(\037)1007 1076 y Fw(I)1047 990 y Fs(\000)1093 1061 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)1413 1020 y Fx(.!)q(/)1523 1061 y Fz(/)1560 990 y Fs(\001)1613 1061 y Fz( )1692 965 y Fs(E)1767 1061 y Fv(\024)25 b FE(T)m(r)1972 965 y Fs(h)2011 1043 y Fz(\037)2079 1076 y Fx(0)2148 1043 y Fz(\037)2222 1076 y Fw(I)2261 990 y Fs(\000)2307 1061 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)2628 1020 y Fx(.!)q(/)2738 1061 y Fz(/)2775 990 y Fs(\001)2827 1043 y Fz(\037)2896 1076 y Fx(0)2951 965 y Fs(i)3004 1061 y Fz(:)368 b FE(\(3.16\))120 1302 y(T)-8 b(aking)42 b(the)h(probabilistic)d(e)o(xpectation)i(on)g(both)f(sides)h (and)h(using)e(\(3.14\))h(we)h(arri)n(v)o(e)f(at)g(the)120 1422 y(sandwiching)30 b(estimate)h(0)d Fv(\024)h Fy(E)1291 1350 y Fs(\002)1325 1422 y Fo(h)p Fz( )23 b(;)1517 1404 y(\037)1591 1437 y Fw(I)1630 1422 y Fz(.)8 b FD(H)i Fz(.)g FD(A)5 b Fz(;)17 b FD(V)e Fz(//)f( )9 b Fo(i)2202 1350 y Fs(\003)2265 1422 y Fv(\024)28 b Fo(j)p Fz(0)t Fo(j)14 b Fz(\027)6 b(.)h FD(I)12 b Fz(/)30 b Fv(D)e FE(0)j(by)g(the)g (assumption)120 1542 y Fz(\027)6 b(.)h FD(I)12 b Fz(/)28 b Fv(D)d FE(0.)35 b(Since)26 b(the)g(magnetic)f(translations)g Fv(f)r FD(T)1905 1557 y Fw(x)1949 1542 y Fv(g)h FE(with)i FD(x)34 b Fv(2)26 b Fy(R)2466 1506 y Fw(d)2539 1542 y FE(or)g Fy(Z)2717 1506 y Fw(d)2791 1542 y FE(are)h(total,)e(the)h (proof)g(of)120 1663 y([13,)f(Lemma)f(V)-13 b(.2.1])25 b(sho)n(ws)e(that)1345 1645 y Fz(\037)1418 1678 y Fw(I)1457 1663 y Fz(.)8 b FD(H)i Fz(.)g FD(A)5 b Fz(;)17 b FD(V)1823 1626 y Fx(.!)q(/)1933 1663 y Fz(//)26 b Fv(D)e FE(0)h(for)g Fy(P)p FE(-almost)f(all)g Fz(!)j Fv(2)e Fz(\177)p FE(.)p 3568 1663 4 68 v 3572 1599 60 4 v 3572 1663 V 3631 1663 4 68 v 120 1900 a FB(Remark)h(3.8.)49 b FE(The)22 b(equi)n(v)n(alence)f (\(ii\))g(of)h(the)g(abo)o(v)o(e)f(corollary)g(is)h(a)g(continuum)e (analogue)i(of)f([16,)120 2020 y(Prop.)38 b(1.1],)j(see)d(also)f([46,)k (Thm.)c(3.3].)69 b(In)38 b(the)f(one-dimensional)f(case)j([45)o(])f (and)g(the)g(multi-)120 2140 y(dimensional)26 b(lattice)h(case)h([18)o (],)h(the)e(equi)n(v)n(alence)f(has)h(been)h(e)o(xploited)d(to)i(sho)n (w)g(in)f(case)38 b FD(A)28 b Fv(D)e FE(0)120 2261 y(the)k(\(global\))e (continuity)g(of)h(the)g(inte)o(grated)g(density)f(of)h(states)36 b FD(N)41 b FE(under)29 b(practically)g(no)g(further)120 2381 y(assumptions)35 b(on)h(the)g(random)g(potential)g(be)o(yond)f (those)h(ensuring)g(the)g(e)o(xistence)g(of)44 b FD(N)12 b FE(.)65 b(The)120 2502 y(proof)41 b(of)g(such)g(a)g(statement)f(in)h (the)g(multi-dimensional)c(continuum)i(case)j(is)e(considered)h(an)120 2622 y(important)25 b(open)h(problem)g([54].)35 b(In)26 b(case)37 b FD(A)28 b Fv(6D)d FE(0)h(one)g(certainly)g(needs)h (additional)e(assumptions)120 2742 y(as)30 b([20])f(illustrates.)42 b(Under)29 b(certain)h(additional)d(assumptions)g(the)i(inte)o(grated)f (density)h(of)g(states)120 2863 y(is)36 b(not)g(only)g(continuous)f(b)n (ut)g(e)n(v)o(en)h(\(locally\))g(H\366lder)g(continuous)f(of)h (arbitrary)h(order)f(strictly)120 2983 y(smaller)23 b(than)g(one)h([15) o(,)g(25)o(])g(or)g(e)n(v)o(en)e(equal)h(to)g(one)h([14)o(,)g(4,)f(5,)g (28].)30 b(The)24 b(latter)f(is)g(equi)n(v)n(alent)e(to)30 b FD(N)120 3104 y FE(being)25 b(absolutely)e(continuous)g(with)h (locally)h(bounded)f(deri)n(v)n(ati)n(v)o(e)e([50,)i(Chap.)i(7,)e(Exc.) h(10].)120 3442 y FG(3.3)119 b(Examples)120 3643 y FE(In)31 b(this)f(subsection)f(we)h(list)g(three)h(e)o(xamples)e(of)h (\(possibly)f(unbounded\))h(random)g(potentials)f(to)120 3764 y(which)23 b(the)h(results)e(of)i(the)f(preceding)g(subsections)f (can)h(be)h(applied.)30 b(While)22 b(the)i(\002rst)f(one)g(models)120 3884 y(\(crystalline\))29 b(disordered)h(allo)o(ys,)f(the)h(other)f(tw) o(o)g(model)g(\(non-crystalline\))f(amorphous)h(solids.)120 4005 y(These)c(are)f(typical)g(e)o(xamples)f(considered)g(in)h(the)g (literature.)30 b(Each)24 b(of)g(them)g(is)f(characterized)j(by)120 4125 y(one)k(of)g(the)g(follo)n(wing)e(properties.)46 b(W)-8 b(e)30 b(recall)h(from)e(properties)h(\()p Fu(S)p FE(\))h(and)f(\()p Fu(I)p FE(\))g(the)f(de\002nitions)g(of)120 4245 y(the)c(constants)35 b FD(p)s Fz(.)p FD(d)7 b Fz(/)26 b FE(and)e Fz(#)9 b FE(.)247 4462 y(\()p Fu(A)p FE(\))103 b FD(V)46 b FE(is)31 b(an)g FD(alloy-type)f(r)o(andom)g(\002eld)p FE(,)i(that)f(is,)h(a)f(random)g(\002eld)g(with)f(realizations)h(gi)n (v)o(en)469 4583 y(by)1464 4833 y FD(V)1539 4792 y Fx(.!)q(/)1650 4833 y Fz(.)s FD(x)9 b Fz(/)25 b Fv(D)1937 4750 y Fs(X)1919 4949 y Fw(j)8 b Ft(2)p Fr(Z)2050 4927 y Fp(d)2101 4833 y Fz(\025)2155 4784 y Fx(.!)q(/)2167 4860 y Fw(j)2265 4833 y FD(u)d Fz(.)s FD(x)29 b Fv(\000)35 b FD(j)11 b Fz(/:)750 b FE(\(3.17\))469 5169 y(The)39 b(random)g(v)n(ariables)g Fv(f)p Fz(\025)1507 5184 y Fw(j)1539 5169 y Fv(g)h FE(are)g Fy(P)p FE(-independent)e(and)h(identically)f(distrib)n(uted)g(ac-)469 5289 y(cording)33 b(to)g(the)h(common)e(probability)g(measure)i Fm(B)t Fz(.)p Fy(R)p Fz(/)d Fv(3)36 b FD(I)43 b Fv(7!)31 b Fy(P)p Fv(f)p Fz(\025)3055 5304 y FC(0)3126 5289 y Fv(2)37 b FD(I)12 b Fv(g)p FE(.)58 b(More-)469 5409 y(o)o(v)o(er)l(,)31 b(we)g(suppose)e(that)h(the)h(Borel-measurable)f(function)g FD(u)j FE(:)28 b Fy(R)2848 5373 y Fw(d)2923 5409 y Fv(!)h Fy(R)h FE(satis\002es)g(the)469 5530 y(Birman-Solomyak)18 b(condition)1639 5463 y Fs(P)1739 5554 y Fw(j)8 b Ft(2)p Fr(Z)1870 5532 y Fp(d)1924 5458 y Fs(\000)1976 5450 y Fn(R)2022 5564 y Fx(3.)k Fw(j)c Fx(/)2182 5530 y FE(d)2232 5494 y Fw(d)2281 5530 y FD(x)23 b Fo(j)p FD(u)5 b Fz(.)s FD(x)k Fz(/)p Fo(j)2598 5494 y Fw(p)2635 5506 y Fj(1)2672 5458 y Fs(\001)2710 5482 y FC(1)p Fx(=)g Fw(p)2828 5494 y Fj(1)2886 5530 y Fz(<)19 b Fv(1)g FE(with)f(some)h(re-)469 5662 y(al)33 b FD(p)624 5677 y FC(1)687 5662 y Fv(\025)22 b FE(2)p Fz(#)k Fv(C)17 b FE(1)j(and)i(that)f Fy(E)1511 5591 y Fs(\000)1548 5662 y Fo(j)p Fz(\025)1630 5677 y FC(0)1672 5662 y Fo(j)1709 5626 y Fw(p)1746 5638 y Fj(2)1783 5591 y Fs(\001)1843 5662 y Fz(<)h Fv(1)f FE(for)h(some)f(real)33 b FD(p)2668 5677 y FC(2)2731 5662 y FE(satisfying)e FD(p)3201 5677 y FC(2)3265 5662 y Fv(\025)22 b FE(2)p Fz(#)j Fv(C)17 b FE(1)469 5783 y(and)36 b FD(p)699 5798 y FC(2)766 5783 y Fz(>)f FD(p)929 5798 y FC(1)971 5783 y FD(d)7 b Fz(=)p FE([2)p Fz(.)k FD(p)1256 5798 y FC(1)1318 5783 y Fv(\000)30 b FD(p)s Fz(.)p FD(d)7 b Fz(//)p FE(].)p eop %%Page: 13 13 13 12 bop 188 -183 a FH(Inte)l(gr)o(ated)26 b(Density)f(of)e(States)i (for)f(Random)g(Sc)o(hr\366ding)o(er)i(Oper)o(ator)o(s)g(with)d(Ma)o (gnetic)i(F)l(ields)238 b FE(13)247 125 y(\()p Fu(P)p FE(\))103 b FD(V)40 b FE(is)24 b(a)h FD(P)-8 b(oissonian)23 b(\002eld)p FE(,)i(that)f(is,)g(a)h(random)g(\002eld)g(with)f (realizations)g(gi)n(v)o(en)f(by)1343 390 y FD(V)1419 349 y Fx(.!)q(/)1529 390 y Fz(.)s FD(x)9 b Fz(/)25 b Fv(D)1787 254 y Fn(Z)1839 479 y Fr(R)1903 458 y Fp(d)1945 390 y Fz(\026)2013 349 y Fx(.!)q(/)2013 416 y(\045)2124 390 y Fz(.)p FE(d)2211 349 y Fw(d)2264 390 y FD(y)6 b Fz(/)14 b FD(u)5 b Fz(.)s FD(x)28 b Fv(\000)e FD(y)6 b Fz(/;)643 b FE(\(3.18\))469 673 y(where)27 b Fz(\026)807 688 y Fx(\045)881 673 y FE(denotes)e(the)i(\(random\))f(Poissonian)f (measure)h(on)g Fy(R)2780 637 y Fw(d)2854 673 y FE(with)g(parameter)g Fz(\045)j Fv(\025)469 794 y FE(0.)44 b(Moreo)o(v)o(er)l(,)30 b(we)f(suppose)g(that)g(the)g(Borel-measurable)h(function)e FD(u)33 b FE(:)27 b Fy(R)3174 758 y Fw(d)3249 794 y Fv(!)h Fy(R)h FE(sat-)469 922 y(is\002es)k(the)g(Birman-Solomyak)g(condition) 2041 856 y Fs(P)2141 946 y Fw(j)8 b Ft(2)p Fr(Z)2272 925 y Fp(d)2326 851 y Fs(\000)2378 842 y Fn(R)2424 957 y Fx(3.)k Fw(j)c Fx(/)2584 922 y FE(d)2634 886 y Fw(d)2684 922 y FD(x)22 b Fo(j)p FD(u)5 b Fz(.)s FD(x)k Fz(/)p Fo(j)2991 886 y FC(2)p Fx(#)e Ft(C)p FC(1)3179 851 y Fs(\001)3217 875 y FC(1)p Fx(=.)p FC(2)p Fx(#)g Ft(C)p FC(1)p Fx(/)3562 922 y Fz(<)469 1043 y 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FE(-er)n(godic)h(and)g(hence)h(has)f(property)g(\()p Fu(E)p FE(\).)h(Moreo)o(v)o(er)l(,)i FD(V)43 b FE(is)28 b(a)g(random)g(\002eld)g(of)g(the)g(form)120 2784 y(\(3.21\))34 b(belo)n(w)-6 b(,)34 b(since)f(one)h(may)f(choose)g Fz(\026)h FE(there)g(as)g(the)f(random)g(signed)g(pure-point)f(measure)120 2915 y(gi)n(v)o(en)27 b(by)h Fz(\026)562 2879 y Fx(.!)q(/)699 2915 y Fv(D)803 2849 y Fs(P)903 2939 y Fw(j)8 b Ft(2)p Fr(Z)1034 2918 y Fp(d)1089 2915 y Fz(\025)1143 2866 y Fx(.!)q(/)1155 2942 y Fw(j)1267 2915 y Fz(\016)1322 2930 y Fw(j)1383 2915 y FE(where)28 b Fz(\016)1702 2930 y Fw(y)1771 2915 y FE(denotes)g(the)f(Dirac)i(measure)f(on)g Fy(R)3072 2879 y Fw(d)3147 2915 y FE(supported)f(at)126 3052 y FD(y)36 b Fv(2)30 b Fy(R)380 3015 y Fw(d)427 3052 y FE(.)59 b(Lemma)34 b(3.10)f(belo)n(w)g(with)h FD(q)j Fv(D)j FD(p)1800 3067 y FC(1)1877 3052 y FE(and)32 b FD(r)39 b Fv(D)i FD(p)2300 3067 y FC(2)2376 3052 y FE(sho)n(ws)33 b(that)k FD(V)49 b FE(has)34 b(property)f(\()p Fu(S)p FE(\).)120 3172 y(Choosing)24 b FD(q)32 b Fv(D)23 b FD(r)34 b Fv(D)25 b FE(2)p Fz(#)j Fv(C)19 b FE(1)25 b(it)f(is)h(seen)g(to)f (obe)o(y)g(property)h(\()p Fu(I)p FE(\).)248 3360 y(\(ii\))99 b(Consider)32 b(a)g FD(P)-8 b(oissonian)29 b(potential)p FE(,)j(that)f(is,)i(a)f(random)f(potential)f(with)h(property)g(\()p Fu(P)p FE(\).)120 3480 y(Then)39 b FD(V)51 b FE(is)36 b Fy(R)660 3444 y Fw(d)707 3480 y FE(-er)n(godic)h(and)f(hence)g(has)g (property)g(\()p Fu(E)p FE(\).)g(Using)g(the)f(f)o(act)i(that)f(the)f (Poissonian)120 3600 y(measure)28 b Fz(\026)548 3615 y Fx(\045)623 3600 y FE(is)f(a)h(random)f(Borel)h(measure)g(which)f(is) h(pure)f(point)g(and)h(positi)n(v)o(e-inte)o(ger)c(v)n(alued,)120 3741 y(each)47 b(realization)f(of)j FD(V)60 b FE(is)46 b(informally)f(gi)n(v)o(en)f(by)49 b FD(V)2137 3704 y Fx(.!)q(/)2247 3741 y Fz(.)s FD(x)9 b Fz(/)37 b Fv(D)2528 3674 y Fs(P)2628 3761 y Fw(j)2674 3741 y FD(u)2729 3669 y Fs(\000)2770 3741 y FD(x)f Fv(\000)30 b FD(x)3011 3691 y Fx(.!)q(/)3017 3767 y Fw(j)3122 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4463 y Fx(3)2153 4462 y Ft(j)2179 4504 y Fz(;)213 b FD(n)29 b Fv(2)c Fy(N)19 b Fv([)g(f)p FE(0)p Fv(g)p Fz(;)462 b FE(\(3.19\))120 4752 y(so)28 b(that)g(the)g(parameter)h Fz(\045)i FE(is)d(identi\002ed)g(as)g(the)h(mean)f(spatial)f (concentration)h(of)g(impurities.)40 b(By)120 4873 y(choosing)30 b Fz(\026)e Fv(D)g Fz(\026)781 4888 y Fx(\045)828 4873 y FE(,)k FD(q)j Fv(D)j FD(p)1136 4888 y FC(1)1206 4873 y Fv(D)28 b FE(2)p Fz(#)i Fv(C)22 b FE(1,)32 b(and)c FD(r)38 b Fv(D)g FD(p)2073 4888 y FC(2)2143 4873 y Fz(>)h FD(p)2310 4888 y FC(1)2352 4873 y FD(d)7 b Fz(=)p FE([2)k FD(p)2600 4888 y FC(1)2664 4873 y Fv(\000)32 b FD(p)s Fz(.)p FD(d)7 b Fz(/)p FE(])32 b(in)e(Lemma)g(3.10)120 4993 y(belo)n(w)-6 b(,)22 b(one)h(v)o(eri\002es)f(that)h(a)g (Poissonian)e(potential)h(statis\002es)g(property)g(\()p Fu(S)p FE(\).)h(Moreo)o(v)o(er)l(,)f(choosing)120 5113 y FD(q)42 b Fv(D)32 b FD(r)44 b Fv(D)34 b FE(2)p Fz(#)g Fv(C)26 b FE(1)42 b(there,)k(it)c(is)f(seen)h(to)g(obe)o(y)f(property)g (\()p Fu(I)p 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FB(Pr)n(oof)h(of)g(Pr)n(oposition)g(4.3.)49 b FD(\(i\))f Fv(\))g FD(\(ii\):)177 b FE(Equation)47 b(\(4.9\))h(follo)n (ws)e(from)i(\(4.8\))g(and)f(the)120 2353 y(\002niteness)25 b Fz(\026)590 2282 y Fs(\000)642 2353 y FE(])p Fv(\0001)p Fz(;)c FD(E)9 b FE([)1025 2282 y Fs(\001)1088 2353 y Fz(<)25 b Fv(1)p FE(.)30 b(Moreo)o(v)o(er)l(,)24 b(for)h(e)n(v)o(ery)46 b FD(f)f Fv(2)25 b Fm(C)2433 2316 y FC(1)2416 2380 y(0)2475 2353 y Fz(.)p Fy(R)p Fz(/)g FE(one)g(has)877 2458 y Fn(Z)930 2683 y Fr(R)998 2594 y Fz(\026)1066 2609 y Fw(n)1111 2594 y Fz(.)p FE(d)7 b FD(E)i Fz(/)36 b FD(f)20 b Fz(.)7 b FD(E)i Fz(/)25 b Fv(D)g(\000)1767 2458 y Fn(Z)1819 2683 y Fr(R)1887 2594 y FE(d)7 b FD(E)23 b Fz(\026)2096 2609 y Fw(n)2141 2594 y Fz(.)p FE(])d Fv(\000)f(1)p Fz(;)i FD(E)9 b FE([)p Fz(/)35 b FD(f)2708 2553 y Ft(0)2732 2594 y Fz(.)7 b FD(E)i Fz(/)516 b FE(\(4.10\))120 2829 y(by)38 b(partial)f(inte)o(gration.)66 b(V)-11 b(ague)38 b(con)l(v)o(er)n(gence)f(of)h Fz(.\026)2119 2844 y Fw(n)2164 2829 y Fz(/)g 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b(sho)n(w)e(that)i Fz(\027)31 b FE(is)24 b(a)h(positi)n(v)o(e)d(Borel)k (measure)f(on)f Fy(R)p FE(,)h(it)f(suf)n(\002ces)h(that)351 1810 y Fz(\027)6 b(.)h FD(I)12 b Fz(/)559 1809 y Fv(j)588 1810 y Fz(0)656 1809 y Fv(j)709 1810 y(D)25 b Fy(E)888 1714 y Fs(n)957 1810 y FE(T)m(r)1061 1714 y Fs(h)1100 1792 y Fz(\037)1169 1825 y Fx(0)1238 1792 y Fz(\037)1311 1825 y Fw(I)1351 1739 y Fs(\000)1397 1810 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)d Fz(/)1754 1739 y Fs(\001)1806 1792 y Fz(\037)1875 1825 y Fx(0)1930 1739 y Fs(\003)1964 1714 y(o)709 2017 y Fv(\024)809 1921 y Fs(\020)859 1929 y Fv(p)p 943 1929 50 5 v 88 x FE(2)p Fz(")1041 1921 y Fs(\021)1091 1946 y FC(2)p Fx(#)1197 2017 y Fy(E)1273 1921 y Fs(n)1342 2017 y FE(T)m(r)1446 1946 y Fs(\002)1481 1999 y Fz(\037)1549 2032 y Fx(0)1618 2017 y Fo(j)8 b FD(H)i Fz(.)g FD(A)5 b Fz(;)17 b FD(V)d Fz(/)20 b Fv(\000)26 b FD(E)2200 2032 y FC(0)2261 2017 y Fv(\000)20 b FE(i)p Fz(")s Fo(j)2463 1976 y Ft(\000)p FC(2)p Fx(#)2626 1999 y Fz(\037)2695 2032 y 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y(states)d(measure)g(associated)f(with)j FD(V)1421 3324 y Fw(n)1466 3309 y FE(,)e(see)g(\(2.8\).)30 b(Moreo)o(v)o(er)l(,)709 3526 y Fz(\027)756 3541 y Fw(n)800 3526 y Fz(.)7 b FD(I)12 b Fz(/)27 b FE(:)p Fv(D)1130 3456 y FE(1)p 1093 3503 124 4 v 1093 3597 a Fo(j)p Fz(0)t Fo(j)1240 3526 y Fy(E)1316 3430 y Fs(n)1385 3526 y FE(T)m(r)1490 3455 y Fs(\002)1524 3508 y Fz(\037)1593 3541 y Fx(0)1648 3508 y Fz(\037)1722 3541 y Fw(I)1761 3455 y Fs(\000)1807 3526 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)2110 3541 y Fw(n)2154 3526 y Fz(/)2191 3455 y Fs(\001)2229 3508 y Fz(\037)2298 3541 y Fx(0)2353 3455 y Fs(\003)2388 3430 y(o)2443 3526 y Fz(;)120 b FD(I)39 b Fv(2)24 b Fm(B)t Fz(.)p Fy(R)p Fz(/;)362 b FE(\(4.16\))120 3751 y(de\002nes)22 b(the)f(approximate)f (\(in\002nite-v)n(olume\))f(density-of-states)h(measure.)29 b(It)21 b(is)g(a)g(positi)n(v)o(e)e(Borel)120 3871 y(measure)k(on)f Fy(R)p FE(,)h(see)g(\(4.14\),)g(and)f(independent)f(of)i(the)f(bounded) g(open)g(cube)h Fz(0)j Fv(\032)d Fy(R)3155 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Fz(\027)1134 4328 y Fx(.!)q(/)1128 4404 y(3;)p FC(D)p Fx(;)p Fw(n)1336 4378 y Fz(.)i FD(f)e Fz(/)g Fv(\000)1601 4370 y Fs(b)1597 4378 y Fz(\027)1650 4328 y Fx(.!)q(/)1644 4404 y(3;)p FC(D)1787 4378 y Fz(.)i FD(f)e Fz(/)1945 4303 y Fs(\014)1945 4353 y(\014)1992 4378 y Fv(C)2090 4303 y Fs(\014)2090 4353 y(\014)2145 4377 y Fv(j)2174 4378 y Fz(3)2253 4377 y Fv(j)2281 4337 y Ft(\000)p FC(1)2398 4370 y Fs(b)2395 4378 y Fz(\027)2448 4328 y Fx(.!)q(/)2442 4404 y(3;)p FC(D)p Fx(;)p Fw(n)2650 4378 y Fz(.)i FD(f)e Fz(/)g Fv(\000)2915 4370 y Fs(b)2911 4378 y Fz(\027)2958 4393 y Fw(n)3003 4378 y Fz(.)i FD(f)f Fz(/)3162 4303 y Fs(\014)3162 4353 y(\014)3189 4378 y Fz(:)183 b FE(\(4.17\))120 4563 y(W)-8 b(e)23 b(\002rst)f(consider)f(the)h(limit)f Fz(3)h Fv(")h Fy(R)1430 4526 y Fw(d)1477 4563 y FE(.)30 b(In)22 b(this)f(limit,)g(the)g(third)h(dif)n(ference)g(on)g(the)f(r)-5 b(.h.s.)21 b(of)h(\(4.17\))120 4683 y(v)n(anishes)e(for)h(all)g Fz(!)j Fv(2)929 4653 y Fs(b)913 4683 y Fz(\177)d FE(:)p Fv(D)1139 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y Fs(b)3187 5349 y Fz(\027)3240 5299 y Fx(.!)q(/)3234 5375 y(3;)p FC(D)3376 5349 y Fz(.)i FD(f)f Fz(/)3535 5274 y Fs(\014)3535 5324 y(\014)3563 5349 y Fz(:)3399 5478 y FE(\(4.18\))120 5662 y(As)28 b Fz(3)f Fv(")h Fy(R)535 5626 y Fw(d)610 5662 y FE(the)g(\002rst)h(term)e(on)h(the)g(r)-5 b(.h.s.)27 b(con)l(v)o(er)n(ges)h(to)g(zero)h(for)f Fy(P)p FE(-almost)f(all)h Fz(!)h Fv(2)e Fz(\177)g FE(and)h(the)120 5783 y(same)d(is)f(true)h(for)g(the)g(second)g(term)f(thanks)g(to)h (Proposition)e(4.8)h(belo)n(w)-6 b(.)p 3568 5783 4 68 v 3572 5719 60 4 v 3572 5783 V 3631 5783 4 68 v eop %%Page: 19 19 19 18 bop 188 -183 a FH(Inte)l(gr)o(ated)26 b(Density)f(of)e(States)i (for)f(Random)g(Sc)o(hr\366ding)o(er)i(Oper)o(ator)o(s)g(with)d(Ma)o (gnetic)i(F)l(ields)238 b FE(19)120 125 y(W)-8 b(e)26 b(no)n(w)e(pro)o(v)o(e)f(our)i(main)f(result.)145 315 y FB(Pr)n(oof)h(of)g(Theor)n(em)i(3.1.)49 b FE(Since)41 b(we)f(ha)n(v)o(e)g(already)g(established)f(the)h(v)n(ague)g(con)l(v)o (er)n(gence)g(of)120 436 y(the)48 b(density-of-states)e(measures)h(in)g (Lemma)f(3.5,)53 b(it)47 b(remains)f(to)h(v)o(erify)g(relation)g (\(4.9\))g(of)120 567 y(Proposition)39 b(4.3)g(for)h(the)g (corresponding)e(random)i(distrib)n(ution)d(functions)2963 566 y Fv(j)2992 567 y Fz(3)3071 566 y Fv(j)3100 528 y Ft(\000)p FC(1)3234 567 y FD(N)3312 517 y Fx(.!)q(/)3302 593 y(3;)p FC(X)3524 567 y FE(for)120 687 y Fy(P)p FE(-almost)24 b(all)h Fz(!)i Fv(2)d Fz(\177)p FE(.)270 807 y(T)-8 b(o)30 b(this)f(end,)i(we)f(emplo)o(y)f(the)h(elementary)g(inequality)e Fz(2)s(.)7 b FD(E)i Fz(/)28 b Fv(\024)g FE(2)2746 727 y Fn(R)2816 756 y Fw(E)2792 842 y Ft(\0001)2931 807 y FE(d)7 b FD(E)3058 771 y Ft(0)3095 807 y Fz(\016)3142 758 y Fx(.")r(/)3138 835 y FC(0)3238 807 y Fz(.)g FD(E)3352 771 y Ft(0)3376 807 y Fz(/)30 b FE(v)n(alid)120 957 y(for)g(all)36 b FD(E)h Fv(2)27 b Fy(R)p FE(,)k Fz(")f(>)d FE(0)j(with)f Fz(\016)1241 907 y Fx(.")r(/)1237 984 y FC(0)1366 957 y FE(de\002ned)h(in)f(\(4.3\).)45 b(Choosing)28 b Fz(")j Fv(D)c 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1513 5458 137 4 v 1513 5551 a Fv(j)1541 5552 y Fz(3)1620 5551 y Fv(j)1673 5407 y Fs(\014)1673 5456 y(\014)1718 5473 y(b)1714 5481 y Fz(\027)1767 5432 y Fx(.!)q(/)1761 5507 y(3;)p FC(X)p Fx(;)p Fw(n)1969 5481 y Fz(.)22 b FD(f)f Fz(/)e Fv(\000)2234 5473 y Fs(b)2231 5481 y Fz(\027)2284 5432 y Fx(.!)q(/)2278 5507 y(3;)p FC(X)2420 5481 y Fz(.)j FD(f)f Fz(/)2579 5407 y Fs(\014)2579 5456 y(\014)2631 5481 y Fv(D)k FE(0)615 b(\(4.22\))120 5783 y FD(for)25 b(all)46 b(f)f Fv(2)25 b Fm(C)658 5746 y FC(1)641 5810 y(0)699 5783 y Fz(.)p Fy(R)p Fz(/)p FD(,)h(all)e Fz(!)j Fv(2)1230 5753 y Fs(e)1214 5783 y Fz(\177)e FD(and)f(both)g(boundary)g (conditions)f FE(X)i Fv(D)g FE(D)f FD(and)h FE(X)g Fv(D)f FE(N)p FD(.)p eop %%Page: 20 20 20 19 bop 120 -183 a FE(20)261 b FH(T)-7 b(.)22 b(Hupfer)-10 b(,)24 b(H.)e(Lesc)o(hk)o(e)o(,)j(P)-12 b(.)21 b(M\374ller)j(&)f(S.)g (W)-8 b(arzel)120 125 y FB(Pr)n(oof)o(.)50 b FE(Thanks)24 b(to)g(\(4.31\),)h(Proposition)e(4.10\(i\))i(belo)n(w)f(and)g(property) h(\()p Fu(I)p FE(\),)f(the)h(inte)o(grals)483 394 y Fs(e)480 402 y Fz(\027)533 352 y Fx(.!)q(/)527 428 y(3;)p FC(X)p Fx(;)p Fw(n)734 402 y Fz(.)r FD(z)5 b Fz(;)14 b FE(2)p Fz(#)9 b(/)24 b Fv(D)1146 266 y Fn(Z)1199 491 y Fr(R)1291 329 y Fz(\027)1344 279 y Fx(.!)q(/)1338 354 y(3;)p FC(X)p Fx(;)p Fw(n)1546 329 y Fz(.)p FE(d)7 b FD(E)i Fz(/)p 1291 378 456 4 v 1325 472 a Fo(j)e FD(E)28 b Fv(\000)22 b FD(z)5 b Fo(j)1621 444 y FC(2)p Fx(#)1781 402 y Fv(D)25 b FE(T)m(r)1988 305 y Fs(h)2027 327 y(\014)2027 377 y(\014)2063 402 y FD(H)2132 417 y Fx(3;)p FC(X)2275 402 y Fz(.)10 b FD(A)5 b Fz(;)17 b FD(V)2513 360 y Fx(.!)q(/)2496 428 y Fw(n)2623 402 y Fz(/)j Fv(\000)h FD(z)2823 327 y Fs(\014)2823 377 y(\014)2850 354 y Ft(\000)p FC(2)p Fx(#)3001 305 y Fs(i)3399 402 y FE(\(4.23\))120 680 y(and)31 b(\(analogously\))878 672 y Fs(e)874 680 y Fz(\027)927 630 y Fx(.!)q(/)921 706 y(3;)p FC(X)1064 680 y Fz(.)r FD(z)5 b Fz(;)14 b FE(2)p Fz(#)9 b(/)30 b FE(are)i(\002nite)f(for)g(all)i FD(z)f Fv(2)c Fy(C)p Fv(n)p Fy(R)k 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1138 y Fs(\014)1662 1188 y(\014)1693 1205 y(e)1690 1213 y Fz(\027)1743 1163 y Fx(.!)q(/)1737 1239 y(3;)p FC(X)p Fx(;)p Fw(n)1944 1213 y Fz(.)7 b FD(E)29 b Fv(C)19 b FE(i)p Fz(";)14 b FE(2)p Fz(#)9 b(/)19 b Fv(\000)2570 1205 y Fs(e)2567 1213 y Fz(\027)2620 1163 y Fx(.!)q(/)2614 1239 y(3;)p FC(X)2756 1213 y Fz(.)7 b FD(E)29 b Fv(C)19 b FE(i)p Fz(";)14 b FE(2)p Fz(#)9 b(/)3262 1138 y Fs(\014)3262 1188 y(\014)3289 1213 y Fz(:)83 b FE(\(4.24\))120 1473 y(Here)25 b(the)g(quantity)e FD(C)9 b Fz(.")s(/)p FE(,)24 b(which)g(depends)g(on)g Fz(")j FE(and)46 b FD(f)21 b FE(,)j(w)o(as)h(introduced)e(in)h(\(4.5\))g(and)h(v)n(anishes)120 1593 y(for)c Fz(")j Fv(#)e FE(0.)29 b(W)-8 b(e)21 b(further)g(estimate) e(the)i(\002rst)f(term)h(with)e(the)i(help)f(of)h(\(4.31\))f(and)h (Proposition)e(4.10\(i\))120 1714 y(choosing)34 b FD(E)580 1729 y FC(1)647 1714 y Fv(D)26 b(\000)p FE(1)i(there.)38 b(The)28 b(upper)f(limit)f Fz(3)h Fv(")g Fy(R)2097 1677 y Fw(d)2171 1714 y FE(of)h(the)f(\002rst)h(term)f(after)h(di)n(viding)d (by)i(the)120 1834 y(v)n(olume)454 1833 y Fv(j)482 1834 y Fz(3)561 1833 y Fv(j)627 1834 y FE(is)36 b(then)g(seen)h(to)f(be)h (\002nite)f(by)g(the)h(Birkhof)n(f-Khintchine)e(er)n(godic)h(theorem)g ([46,)120 1954 y(Prop.)25 b(1.13],)199 2195 y(lim)14 b(sup)238 2291 y Fx(3)p Ft(")p Fr(R)407 2269 y Fp(d)565 2125 y FE(1)p 522 2171 137 4 v 522 2264 a Fv(j)550 2265 y Fz(3)629 2264 y Fv(j)696 2098 y Fs(h)738 2187 y(e)735 2195 y Fz(\027)788 2145 y Fx(.!)q(/)782 2221 y(3;)p FC(X)p Fx(;)p Fw(n)990 2195 y Fz(.)g FE(i)p Fz(;)g FE(2)p Fz(#)9 b(/)19 b Fv(C)1390 2187 y Fs(e)1387 2195 y Fz(\027)1440 2145 y Fx(.!)q(/)1434 2221 y(3;)p FC(X)1576 2195 y Fz(.)14 b FE(i)p Fz(;)g FE(2)p Fz(#)9 b(/)1857 2098 y Fs(i)1921 2195 y Fv(\024)25 b FE(2)14 b FD(C)2154 2210 y FC(1)2194 2195 y Fz(.)p FE(1)p Fz(/)g Fy(E)2408 2098 y Fs(h)2461 2059 y Fn(Z)2514 2284 y Fx(3.)p FC(0)p Fx(/)2657 2195 y FE(d)2707 2154 y Fw(d)2757 2195 y FD(x)2823 2123 y Fs(\000)2861 2195 y 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FD(V)15 b Fz(.)s FD(x)9 b Fz(/)20 b Fv(\000)i FD(V)2639 3788 y Fw(n)2684 3773 y Fz(.)s FD(x)9 b Fz(/)2814 3698 y Fs(\014)2814 3748 y(\014)2842 3725 y FC(2)p Fx(#)e Ft(C)p FC(1)3029 3676 y Fs(i)3069 3626 y(\))3202 3622 y Fj(1)p 3146 3635 V 3146 3678 a(2)p Fq(#)e Ff(C)p Fj(1)3399 3773 y FE(\(4.26\))120 4025 y(for)36 b Fy(P)p FE(-almost)f(all)g Fz(!)e Fv(2)e Fz(\177)p FE(.)62 b(In)36 b(the)f(limit)f FD(n)h Fv(!)c(1)p FE(,)38 b(the)e(r)-5 b(.h.s.)34 b(and)i(hence)g(the)f(l.h.s.)g(of)g (\(4.26\))120 4145 y(v)n(anishes)c(for)g Fy(P)p FE(-almost)g(all)g Fz(!)g Fv(2)d Fz(\177)j FE(thanks)g(to)g(property)g(\()p Fu(I)p FE(\).)g(This)g(completes)g(the)g(proof)g(since)120 4265 y(the)25 b(\002rst)g(term)g(on)f(the)h(l.h.s.)f(of)h(\(4.24\))f (may)h(be)g(made)f(arbitrarily)h(small)e(as)i Fz(")j Fv(#)d FE(0.)p 3568 4265 4 68 v 3572 4201 60 4 v 3572 4265 V 3631 4265 4 68 v 120 4453 a(The)d(last)f(lemma)f(sho)n(ws)g(in)h (which)g(sense)h(the)f(approximate)f(\(in\002nite-v)n(olume\))h 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b(v)n(ague)g(con)l(v)o(er)n(gence)h(by)g(Proposition) e(4.1.)p 3568 5783 4 68 v 3572 5719 60 4 v 3572 5783 V 3631 5783 4 68 v eop %%Page: 21 21 21 20 bop 188 -183 a FH(Inte)l(gr)o(ated)26 b(Density)f(of)e(States)i (for)f(Random)g(Sc)o(hr\366ding)o(er)i(Oper)o(ator)o(s)g(with)d(Ma)o (gnetic)i(F)l(ields)238 b FE(21)120 125 y(In)24 b(the)f(follo)n(wing)e (proposition)g(we)i(e)o(xploit)e(recent)j(results)e(of)h(Nakamura)g ([42])h(or)f(Doi,)g(Iw)o(atsuka)120 245 y(and)46 b(Mine)g([19])g(on)g (the)g(independence)g(of)g(the)g(density-of-states)e(measure)i(of)h (the)e(chosen)120 365 y(boundary)38 b(condition)f(for)i(the)f(present)g (setting,)i(thereby)e(hea)n(vily)g(relying)g(on)g(either)g(of)g(these) 120 486 y(results.)120 666 y FB(Pr)n(oposition)25 b(4.8.)49 b FD(Let)40 b Fz(3)33 b Fv(\032)f Fy(R)1305 630 y Fw(d)1392 666 y FD(stand)38 b(for)g(bounded)g(open)h(cubes.)73 b(Assume)48 b(A)41 b(is)d(a)h(vector)120 786 y(potential)23 b(with)g(pr)l(operty)f FE(\()p Fu(C)p FE(\))j FD(and)h(V)39 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Fn(Z)1942 4612 y Ft(1)1896 4808 y FC(0)2009 4719 y FE(d)p Fz(\030)25 b FE(e)2170 4678 y Ft(\000)p Fx(\030)2286 4719 y Fz(\030)2339 4678 y Fx(\013)s Ft(\000)p FC(1)2499 4623 y Fs(\020)2548 4719 y FE(1)20 b Fv(C)f Fz(.)p FE(2)p Fz(\031)10 b(\030)h(/)2962 4678 y Ft(\000)p FC(1)p Fx(=)p FC(2)3135 4623 y Fs(\021)3184 4648 y Fw(d)3256 4719 y Fz(<)25 b Fv(1)469 4941 y FD(for)f(all)56 b(E)837 4956 y FC(1)903 4941 y Fv(2)p FE(])20 b Fv(\000)f(1)p Fz(;)14 b Fv(\000)p FE(1])p FD(.)120 5150 y FB(Remark)26 b(4.13.)49 b FE(The)35 b(v)n(alidity)e(of)i(\(4.33\))f(for)h(all)42 b FD(Q)36 b Fv(2)30 b FE(L)2218 5114 y Ft(1)2298 5150 y Fz(.3/)36 b FE(may)e(be)h(e)o(xtended)g(to)f(all)42 b FD(Q)35 b Fv(2)120 5270 y FE(L)190 5234 y Fw(p)234 5270 y Fz(.3/)29 b FE(by)f(an)h(approximation)d(ar)n(gument.)41 b(In)29 b(this)e(re)o(gard,)i(Lemma)f(4.12\(i\))g(is)g(a)g(\002nite-v)n (olume)120 5390 y(analogue)23 b(of)f([53,)h(Thm.)f(4.1].)29 b(Ho)n(we)n(v)o(er)l(,)22 b(the)g(bound)g(in)g([53,)h(Thm.)f(4.1])g(is) 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FD(H)526 715 y Fx(3;)p FC(X)669 700 y Fz(.)p FE(0)p Fz(;)14 b FE(0)p Fz(//)21 b FD(Q)5 b Fo(k)1084 652 y Fw(p)1084 730 y(p)1153 700 y Fv(\024)25 b Fo(k)1316 625 y Fs(\014)1316 675 y(\014)1366 700 y FD(f)20 b Fz(.)8 b FD(H)1528 715 y Fx(3;)p FC(X)1671 700 y Fz(.)p FE(0)p Fz(;)14 b FE(0)p Fz(//)1929 625 y Fs(\014)1929 675 y(\014)1984 699 y Fv(j)2020 700 y FD(Q)2097 699 y Fv(j)2140 700 y Fo(k)2198 652 y Fw(p)2198 730 y(p)2267 700 y Fv(\024)25 b Fo(k)2430 625 y Fs(\014)2430 675 y(\014)2480 700 y FD(f)20 b Fz(.)8 b FD(H)2642 715 y Fx(3;)p FC(X)2785 700 y Fz(.)p FE(0)p Fz(;)14 b FE(0)p Fz(//)3043 625 y Fs(\014)3043 675 y(\014)3088 617 y Fp(p)p 3081 635 38 4 v 3085 679 a Fj(2)3160 699 y Fv(j)3196 700 y FD(Q)3273 699 y Fv(j)3319 623 y Fp(p)p 3312 642 V 3316 685 a Fj(2)3377 700 y Fo(k)3427 652 y FC(2)3427 730 y(2)1153 881 y Fv(D)24 b FE(T)m(r)1360 810 y Fs(\002)1408 880 y Fv(j)1443 881 y FD(Q)1520 880 y Fv(j)1566 805 y Fp(p)p 1559 823 V 1563 867 a Fj(2)1625 807 y Fs(\014)1625 856 y(\014)1674 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FE(\(4.72\))120 3853 y(It)20 b(is)g Fy(R)369 3816 y Fw(d)416 3853 y FE(-er)n(godic)g(by)g(construction)e ([29])i(and)g(enjo)o(ys)f(properties)g(\()p Fu(S)p FE(\))i(and)f(\()p Fu(I)p FE(\).)f(The)h(latter)g(assertion)120 3973 y(is)25 b(pro)o(v)o(en)e(by)i(tracing)g(the)f(claimed)g(properties)h(of)g Fm(V)52 b FE(back)25 b(to)f(the)h(respecti)n(v)o(e)f(properties)g(of)k FD(V)15 b FE(.)270 4094 y(It)35 b(remains)f(to)g(pro)o(v)o(e)g(that)g (the)h(v)n(alidity)d(of)j(Proposition)e(4.15)h(for)h Fm(V)63 b FE(implies)33 b(the)h(one)h(for)123 4215 y FD(V)15 b FE(.)49 b(F)o(or)31 b(this)f(purpose,)i(we)f(note)f(that)g (the)h(inte)o(gral)f(transform)g(\(4.27\))h(of)g(the)f(\(in\002nite-v)n (olume\))120 4335 y(density-of-states)24 b(measure)h(corresponding)f (to)g Fm(V)52 b FE(obe)o(ys)611 4535 y(1)p 574 4582 125 4 v 574 4675 a Fv(j)602 4676 y Fz(0)670 4675 y Fv(j)736 4469 y Fn(Z)789 4694 y Fx(\177)p Ft(\002)p Fx(3.)p FC(0)p Fx(/)966 4605 y Fy(P)p Fz(.)p FE(d)p Fz(!)r(/)19 b Fv(\012)h FE(d)1402 4564 y Fw(d)1455 4605 y FD(y)47 b FE(T)m(r)1650 4509 y Fs(h)1689 4587 y Fz(\037)1758 4620 y Fx(0)1827 4530 y Fs(\014)1827 4580 y(\014)1863 4605 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)14 b Fm(V)2194 4564 y Fx(.!)t(;)5 b Fw(y)t Fx(/)2373 4605 y Fz(/)20 b Fv(\000)h FD(z)2573 4530 y Fs(\014)2573 4580 y(\014)2600 4557 y Ft(\000)p FC(2)p Fx(#)2765 4587 y Fz(\037)2833 4620 y Fx(0)2888 4509 y Fs(i)645 4878 y Fv(D)784 4808 y FE(1)p 746 4855 V 746 4948 a Fv(j)775 4949 y Fz(0)843 4948 y Fv(j)909 4742 y Fn(Z)961 4968 y Fx(3.)p FC(0)p Fx(/)1104 4878 y FE(d)1154 4837 y Fw(d)1207 4878 y FD(y)1285 4742 y Fn(Z)1337 4968 y Fx(\177)1386 4878 y Fy(P)p Fz(.)p FE(d)p Fz(!)r(/)28 b FE(T)m(r)1785 4782 y Fs(h)1824 4860 y Fz(\037)1893 4893 y Fx(0)s Ft(\000)5 b Fw(y)2062 4803 y Fs(\014)2062 4853 y(\014)2098 4878 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)2418 4837 y Fx(.!)q(/)2528 4878 y Fz(/)j Fv(\000)i FD(z)2729 4803 y Fs(\014)2729 4853 y(\014)2756 4831 y Ft(\000)p FC(2)p Fx(#)2920 4860 y Fz(\037)2989 4893 y Fx(0)s Ft(\000)5 b Fw(y)3144 4782 y Fs(i)645 5151 y Fv(D)784 5081 y FE(1)p 746 5128 V 746 5221 a Fv(j)775 5222 y Fz(0)843 5221 y Fv(j)909 5016 y Fn(Z)961 5241 y Fx(\177)1010 5151 y Fy(P)p Fz(.)p FE(d)p Fz(!)r(/)28 b FE(T)m(r)1409 5055 y Fs(h)1448 5133 y Fz(\037)1517 5166 y Fx(0)1586 5076 y Fs(\014)1586 5126 y(\014)1622 5151 y FD(H)10 b Fz(.)g FD(A)5 b Fz(;)17 b FD(V)1942 5110 y Fx(.!)q(/)2052 5151 y Fz(/)j Fv(\000)i FD(z)2253 5076 y Fs(\014)2253 5126 y(\014)2280 5104 y Ft(\000)p FC(2)p Fx(#)2444 5133 y Fz(\037)2513 5166 y Fx(0)2568 5055 y Fs(i)3399 5151 y FE(\(4.73\))120 5421 y(Here)j(the)e(\002rst)h(equality)e(results)h (from)g(\(2.11\))h(and)f(the)g(de\002nitions)g(of)g Fm(V)51 b FE(and)24 b(the)f(cube)h Fz(0)e Fv(\000)i FD(y)30 b FE(:)p Fv(D)120 5542 y(f)s FD(x)i Fv(\000)d FD(y)35 b Fv(2)30 b Fy(R)596 5506 y Fw(d)703 5542 y FE(:)47 b FD(x)38 b Fv(2)30 b Fz(0)t Fv(g)k FE(together)f(with)g(Fubini')-5 b(s)33 b(theorem.)57 b(T)-8 b(o)34 b(obtain)f(the)h(second)f(equality) 120 5662 y(we)f(ha)n(v)o(e)e(used)h(the)f(f)o(act)i(that)e(the)h(trace) g(does)g(not)f(depend)h(on)37 b FD(y)f FE(after)c(performing)e(the)h Fy(P)p Fz(.)p FE(d)p Fz(!)r(/)p FE(-)120 5783 y(inte)o(gration.)56 b(This)33 b(follo)n(ws)f(from)h Fy(Z)1470 5746 y Fw(d)1517 5783 y FE(-homogeneity)f(as)i(well)f(as)h(from)f(the)h(f)o(act)g(that)f (one)h(may)p eop %%Page: 30 30 30 29 bop 120 -183 a FE(30)261 b FH(T)-7 b(.)22 b(Hupfer)-10 b(,)24 b(H.)e(Lesc)o(hk)o(e)o(,)j(P)-12 b(.)21 b(M\374ller)j(&)f(S.)g (W)-8 b(arzel)120 125 y FE(\223re-arrange\224)34 b Fz(0)26 b Fv(\000)i FD(y)37 b FE(in)31 b(the)h(form)f(of)h Fz(0)j FE(by)d Fy(Z)1830 88 y Fw(d)1877 125 y FE(-translations)e(since)h Fz(0)36 b FE(is)31 b(compatible)g(with)f(the)120 245 y(lattice.)h(Moreo)o(v)o(er)l(,)23 b(one)i(computes)217 361 y Fn(Z)270 586 y Fx(\177)p Ft(\002)p Fx(3.)p FC(0)p Fx(/)447 496 y Fy(P)p Fz(.)p FE(d)p Fz(!)r(/)19 b Fv(\012)h FE(d)883 455 y Fw(d)936 496 y FD(y)1027 400 y Fs(\020)1076 496 y FE(1)g Fv(C)f Fo(j)r FD(z)24 b Fv(\000)i FD(E)1505 511 y FC(2)1546 496 y Fo(j)20 b Fv(C)f Fo(j)p Fm(V)1808 455 y Fx(.!)t(;)5 b Fw(y)t Fx(/)1987 496 y Fz(.)p FE(0)p Fz(/)p Fo(j)2139 400 y Fs(\021)2189 425 y FC(2)p Fx(#)1248 768 y Fv(D)1350 632 y Fn(Z)1403 857 y Fx(\177)1451 768 y Fy(P)p Fz(.)p FE(d)p Fz(!)r(/)1746 632 y Fn(Z)1799 857 y Fx(3.)p FC(0)p Fx(/)1928 768 y FE(d)1978 726 y Fw(d)2028 768 y FD(x)2122 671 y Fs(\020)2172 768 y FE(1)19 b Fv(C)g Fo(j)r FD(z)24 b Fv(\000)i FD(E)2600 783 y FC(2)2642 768 y Fo(j)19 b Fv(C)g Fo(j)s FD(V)2893 726 y Fx(.!)q(/)3003 768 y Fz(.)s FD(x)9 b Fz(/)p Fo(j)3161 671 y Fs(\021)3210 697 y FC(2)p Fx(#)3399 768 y FE(\(4.74\))120 1007 y(using)26 b(\(4.72\))h(and)g(Fubini')-5 b(s)26 b(theorem.)36 b(This)26 b(completes)g(the)h(proof)g(of)g(Proposition)f(4.15\(i\))g(with)123 1127 y FD(V)181 1142 y Fw(n)252 1127 y FE(replaced)g(by)i FD(V)15 b FE(.)33 b(The)25 b(other)g(parts)h(of)f(Proposition)f(4.15)h (in)g(the)g Fy(Z)2619 1091 y Fw(d)2667 1127 y FE(-er)n(godic)g(case)h (are)g(pro)o(v)o(en)120 1248 y(similarly)-6 b(.)p 3568 1248 4 68 v 3572 1184 60 4 v 3572 1248 V 3631 1248 4 68 v 120 1581 a FA(Ackno)o(wledgment)120 1805 y FE(It')h(s)24 b(a)g(pleasure)g(to)g(thank)f(K)o(urt)g(Broderix)h(\(1962)19 b Fv(\000)f FE(2000\),)24 b(Eckhard)g(Giere,)g(J\366rn)f(Lembck)o(e,)h (and)120 1925 y(Geor)n(gi)e(D.)g(Raik)o(o)o(v)g(for)g(helpful)f (remarks)h(and)g(stimulating)d(discussions.)28 b(The)22 b(present)f(w)o(ork)h(w)o(as)120 2045 y(supported)k(by)g(the)g (Deutsche)g(F)o(orschungsgemeinschaft)e(under)i(grant)g(no.)35 b(Le)26 b(330/12)f(which)h(is)120 2166 y(a)32 b(project)f(within)f(the) h(Schwerpunktprogramm)g(\223Interagierende)h(stochastische)e(Systeme)h (v)n(on)120 2286 y(hoher)25 b(K)m(omple)o(xit\344t\224)e(\(DFG)i (Priority)f(Programme)h(SPP)h(1033\).)120 2620 y FA(Refer)m(ences)258 2836 y FI([1])51 b(R.)22 b(J.)h(Adler)l(,)g FH(The)h(g)o(eometry)h(of)e (r)o(andom)h(\002elds)p FI(,)h(Chichester)l(,)g(W)l(ile)o(y)-6 b(,)24 b(1981.)258 3021 y([2])51 b(T)-7 b(.)22 b(Ando,)i(A.)f(B.)f(F)o (o)n(wler)i(and)g(F)-7 b(.)22 b(Stern,)i(\223Electronic)i(properties)h (of)d(tw)o(o-dimensional)j(systems\224,)414 3134 y FH(Re)o(v)-7 b(.)22 b(Mod.)h(Phys.)h Fe(54)f FI(\(1982\))i(437\226672.)258 3318 y([3])51 b(J.)20 b(A)-7 b(vron)21 b(and)g(B.)e(Simon,)h(\223)-7 b(Almost)21 b(periodic)h(Schr\366dinger)h(operators)g(II.)d(The)h(inte) o(grated)i(density)414 3431 y(of)g(states\224,)i FH(Duk)o(e)f(Math.)f (J)n(.)g Fe(50)g FI(\(1983\))i(369\226391.)258 3616 y([4])51 b(J.-M.)37 b(Barbaroux,)i(J.)e(M.)f(Combes)i(and)g(P)-10 b(.)36 b(D.)h(Hislop,)h(\223Localization)i(near)f(band)f(edges)h(for) 414 3729 y(random)24 b(Schr\366dinger)i(operators\224,)g FH(Helv)-7 b(.)24 b(Phys.)f(Acta)g Fe(70)h FI(\(1997\))h(16\22643.)258 3914 y([5])51 b(J.-M.)21 b(Barbaroux,)i(J.)e(M.)g(Combes)h(and)g(P)-10 b(.)20 b(D.)g(Hislop,)j(\223Landau)g(Hamiltonians)h(with)d(unbounded) 414 4027 y(random)j(potentials\224,)j FH(Lett.)22 b(Math.)i(Phys.)f Fe(40)h FI(\(1997\))h(335\226369.)258 4212 y([6])51 b(H.)28 b(Bauer)l(,)i FH(Measur)m(e)g(and)g(Inte)l(gr)o(ation)j(Theory)p FI(,)d(Berlin,)g(de)f(Gruyter)l(,)i(2001)f([German)g(original:)414 4325 y(Berlin,)23 b(de)h(Gruyter)l(,)h(1992].)258 4510 y([7])51 b(M.)29 b(S.)g(Birman)i(and)g(M.)f(Z.)f(Solomjak,)i FH(Spectr)o(al)i(theory)f(of)f(self-adjoint)j(oper)o(ator)o(s)f(in)e (Hilbert)414 4622 y(space)p FI(,)41 b(Dordrecht,)h(Reidel,)f(1987)g ([Russian)h(original:)g(Leningrad,)g(Leningrad)h(Uni)n(v)-6 b(.)40 b(Press,)414 4735 y(1980].)258 4920 y([8])51 b(V)-12 b(.)42 b(L.)h(Bonch-Brue)n(vich,)j(R.)d(Enderlein,)i(B.)d(Esser)l(,)i (R.)f(K)n(eiper)l(,)h(A.)e(G.)h(Mirono)o(v)i(and)f(I.)f(P)-10 b(.)414 5033 y(Zvyagin,)28 b FH(Elektr)l(onentheorie)k(ung)o(eor)m (dneter)f(Halbleiter)p FI(,)e(Berlin,)f(VEB)d(Deutscher)k(V)-10 b(erlag)28 b(der)414 5146 y(W)l(issenschaften,)f(1984)e([in)e(German.)h (Russian)g(original:)i(Mosco)n(w)-6 b(,)23 b(Nauka,)h(1981].)258 5331 y([9])51 b(K.)45 b(Broderix,)j(D.)d(Hundertmark)k(and)e(H.)e (Leschk)o(e,)j(\223Self-a)n(v)o(eraging,)i(decomposition)g(and)414 5444 y(asymptotic)40 b(properties)h(of)c(the)i(density)h(of)d(states)j (for)e(random)h(Schr\366dinger)h(operators)h(with)414 5557 y(constant)35 b(magnetic)f(\002eld\224,)f(in)f FH(P)-7 b(ath)33 b(inte)l(gr)o(als)i(fr)l(om)e(meV)e(to)i(MeV)-6 b(:)32 b(T)-5 b(utzing)34 b('92)p FI(,)e(H.)g(Grabert,)414 5670 y(A.)25 b(Inomata,)i(L.)e(S.)g(Schulman,)j(U.)d(W)-7 b(eiss)26 b(\(eds.\))i(Singapore,)f(W)-7 b(orld)27 b(Scienti\002c,)g (1993,)g(pp.)g(98\226)414 5783 y(107.)p eop %%Page: 31 31 31 30 bop 188 -183 a FH(Inte)l(gr)o(ated)26 b(Density)f(of)e(States)i (for)f(Random)g(Sc)o(hr\366ding)o(er)i(Oper)o(ator)o(s)g(with)d(Ma)o (gnetic)i(F)l(ields)238 b FE(31)213 125 y FI([10])51 b(K.)29 b(Broderix,)j(D.)d(Hundertmark,)j(W)-8 b(.)29 b(Kirsch)i(and)g(H.)e(Leschk)o(e,)j(\223The)e(f)o(ate)h(of)g(Lifshits)g (tails)g(in)414 237 y(magnetic)25 b(\002elds\224,)f FH(J)n(.)e(Stat.)i (Phys.)f Fe(80)h FI(\(1995\))h(1\22622.)213 450 y([11])51 b(K.)35 b(Broderix,)j(D.)e(Hundertmark)j(and)e(H.)f(Leschk)o(e,)i (\223Continuity)h(properties)h(of)d(Schr\366dinger)414 563 y(semigroups)26 b(with)d(magnetic)i(\002elds\224,)f FH(Re)o(v)-7 b(.)22 b(Math.)i(Phys.)f Fe(12)h FI(\(2000\))h (181\226225.)213 775 y([12])51 b(K.)39 b(Broderix,)i(H.)e(Leschk)o(e)i (and)g(P)-10 b(.)38 b(M\374ller)l(,)j(\223Continuous)i(inte)o(gral)f(k) o(ernels)g(for)e(unbounded)414 888 y(Schr\366dinger)26 b(semigroups)g(and)e(their)g(spectral)i(projections\224,)g(in)e (preparation.)213 1101 y([13])51 b(R.)29 b(Carmona)i(and)g(J.)f (Lacroix,)h FH(Spectr)o(al)h(theory)g(of)f(r)o(andom)g(Sc)o(hr\366ding) o(er)j(oper)o(ator)o(s)p FI(,)f(Boston,)414 1214 y(Birkh\344user)l(,)25 b(1990.)213 1426 y([14])51 b(J.)21 b(M.)g(Combes)h(and)g(P)-10 b(.)21 b(D.)f(Hislop,)j(\223Landau)g(Hamiltonians)h(with)e(random)h (potentials:)i(Localiza-)414 1539 y(tion)f(and)g(the)g(density)h(of)f (states\224,)g FH(Commun.)f(Math.)g(Phys.)h Fe(177)g FI(\(1996\))h(603\226629.)213 1752 y([15])51 b(J.)40 b(M.)f(Combes,)i(P)-10 b(.)39 b(D.)g(Hislop)j(and)f(S.)e(Nakamura,)i (\223The)46 b FH(L)2548 1719 y Fb(p)2587 1752 y FI(-theory)c(of)f(the)g (spectral)i(shift)414 1864 y(function,)e(the)f(W)-7 b(e)o(gner)40 b(estimate,)g(and)g(the)g(inte)o(grated)i(density)f(of)e(states)i(for)f (some)f(random)414 1977 y(operators\224,)26 b FH(Commun.)d(Math.)g (Phys.)g Fe(218)h FI(\(2001\))h(113\226130.)213 2190 y([16])51 b(W)-8 b(.)31 b(Craig)i(and)g(B.)e(Simon,)h(\223Log)h (H\366lder)g(continuity)j(of)c(the)h(inte)o(grated)i(density)g(of)d (states)i(for)414 2303 y(stochastic)26 b(Jacobi)f(matrices\224,)g FH(Commun.)e(Math.)g(Phys.)g Fe(90)h FI(\(1983\))h(207\226218.)213 2515 y([17])51 b(H.)36 b(L.)f(Cycon,)j(R.)e(G.)g(Froese,)h(W)-8 b(.)36 b(Kirsch)i(and)f(B.)f(Simon,)h FH(Sc)o(hr\366ding)o(er)k(oper)o (ator)o(s)p FI(,)e(Berlin,)414 2628 y(Springer)l(,)25 b(1987.)213 2841 y([18])51 b(F)-7 b(.)22 b(Delyon)j(and)g(B.)d (Souillard,)k(\223Remark)e(on)g(the)h(continuity)i(of)d(the)g(density)i (of)e(states)h(of)f(er)n(godic)414 2954 y(\002nite)f(dif)n(ference)j (operators\224,)g FH(Commun.)d(Math.)g(Phys.)h Fe(94)f FI(\(1984\))i(289\226291.)213 3166 y([19])51 b(S.)24 b(Doi,)i(A.)e(Iw)o(atsuka)k(and)f(T)-7 b(.)25 b(Mine,)g(\223The)i (uniqueness)i(of)d(the)g(inte)o(grated)j(density)f(of)e(states)h(for) 414 3279 y(the)d(Schr\366dinger)i(operators)g(with)d(magnetic)i (\002elds\224,)f FH(Math.)f(Z.)f Fe(237)i FI(\(2001\))h(335\226371.)213 3491 y([20])51 b(T)-7 b(.)26 b(C.)f(Dorlas,)j(N.)d(Macris)j(and)g(J.)e (V)-12 b(.)26 b(Pul\351,)h(\223Characterization)k(of)c(the)h(spectrum)g (of)f(the)h(Landau)414 3604 y(Hamiltonian)d(with)e(delta)i (impurities\224,)g FH(Commun.)e(Math.)g(Phys.)h Fe(204)g FI(\(1999\))h(367\226396.)213 3817 y([21])51 b(L.)29 b(Erd)661 3816 y(\005)653 3817 y(os,)h(\223Lifschitz)j(tail)e(in)f(a)g (magnetic)i(\002eld:)f(the)g(non-classical)j(re)o(gime\224,)d FH(Pr)l(obab)l(.)h(Theory)414 3930 y(Relat.)23 b(F)l(ields)h Fe(112)g FI(\(1998\))h(321\226371.)213 4142 y([22])51 b(L.)20 b(Erd)652 4141 y(\005)644 4142 y(os,)h(\223Lifshitz)i(tail)f (in)g(a)f(magnetic)i(\002eld:)f(coe)o(xistence)i(of)e(classical)i(and)e (quantum)h(beha)n(vior)414 4255 y(in)g(the)h(borderline)i(case\224,)f FH(Pr)l(obab)l(.)f(Theory)g(Relat.)g(F)l(ields)g Fe(121)g FI(\(2001\))h(219\226236.)213 4468 y([23])51 b(X.)34 b(M.)h(Fernique,)h(\223Re)o(gularit\351)i(des)e(trajectoires)j(des)d (fonctions)j(al\351atoires)f(Gaussiennes\224,)g(in)414 4581 y FH(Ecole)e(d'Et\351)g(de)g(Pr)l(obabilit\351s)i(de)e (Saint-Flour)i(IV)d(-)g(1974)p FI(,)i(P)-10 b(.-L.)34 b(Hennequin)k(\(ed.\),)e(Lecture)414 4693 y(Notes)24 b(in)f(Mathematics)i Fe(480)p FI(,)f(Berlin,)g(Springer)l(,)h(1975,)f (pp.)f(1\22696)i([in)f(French].)213 4906 y([24])51 b(I.)23 b(S.)f(Gradshte)o(yn)k(and)e(I.)f(M.)f(Ryzhik,)j FH(T)-8 b(able)23 b(of)h(inte)l(gr)o(als,)i(series,)f(and)f(pr)l(oducts)p FI(,)i(corrected)g(and)414 5019 y(enlar)n(ged)g(edition)f(San)e(Die)o (go,)g(Academic,)h(1980.)213 5231 y([25])51 b(P)-10 b(.)27 b(D.)g(Hislop)i(and)g(F)-7 b(.)26 b(Klopp,)j(\223The)f(inte)o(grated)j (density)f(of)f(states)g(for)g(some)g(random)g(operators)414 5344 y(with)23 b(nonsign)j(de\002nite)e(potentials\224,)j(e-print)e (mp_arc)f(01-139)h(\(2001\).)213 5557 y([26])51 b(T)-7 b(.)20 b(Hupfer)l(,)i(H.)e(Leschk)o(e)j(and)f(S.)e(W)-7 b(arzel,)22 b(\223Poissonian)i(obstacles)g(with)d(Gaussian)i(w)o(alls)f (discrim-)414 5670 y(inate)j(between)g(classical)h(and)e(quantum)h (Lifshits)g(tailing)g(in)f(magnetic)h(\002elds\224,)g FH(J)n(.)e(Stat.)h(Phys.)f Fe(97)414 5783 y FI(\(1999\))i(725\226750.)p eop %%Page: 32 32 32 31 bop 120 -183 a FE(32)261 b FH(T)-7 b(.)22 b(Hupfer)-10 b(,)24 b(H.)e(Lesc)o(hk)o(e)o(,)j(P)-12 b(.)21 b(M\374ller)j(&)f(S.)g (W)-8 b(arzel)213 125 y FI([27])51 b(T)-7 b(.)39 b(Hupfer)l(,)i(H.)e (Leschk)o(e)j(and)f(S.)e(W)-7 b(arzel,)40 b(\223The)h(multiformity)h (of)f(Lifshits)g(tails)g(caused)h(by)414 237 y(random)c(Landau)g (Hamiltonians)h(with)e(repulsi)n(v)o(e)i(impurity)g(potentials)h(of)d (dif)n(ferent)i(decay)g(at)414 350 y(in\002nity\224,)24 b FH(AMS/IP)f(Studies)i(in)f(Advanced)h(Mathematics)g Fe(16)f FI(\(2000\))h(233-247.)213 544 y([28])51 b(T)-7 b(.)19 b(Hupfer)l(,)h(H.)f(Leschk)o(e,)i(P)-10 b(.)18 b(M\374ller)j(and)g(S.)d(W)-7 b(arzel,)20 b(\223The)g(absolute)j (continuity)g(of)d(the)g(inte)o(grated)414 657 y(density)40 b(of)e(states)h(for)f(magnetic)i(Schr\366dinger)h(operators)f(with)e (certain)i(unbounded)h(random)414 770 y(potentials\224,)26 b FH(Commun.)d(Math.)g(Phys.)g Fe(221)h FI(\(2001\))i(229-254.)213 963 y([29])51 b(W)-8 b(.)33 b(Kirsch,)i(\223On)f(a)g(class)h(of)g (random)g(Schr\366dinger)j(operators\224,)f FH(Adv)-7 b(.)34 b(Appl.)g(Math.)h Fe(6)f FI(\(1985\))414 1076 y(177\226187.)213 1269 y([30])51 b(W)-8 b(.)39 b(Kirsch,)i(\223Random)h (Schr\366dinger)h(operators)h(and)d(the)g(density)i(of)e(states\224,)h (in)f FH(Stoc)o(hastic)414 1382 y(aspects)31 b(of)e(classical)i(and)f (quantum)g(systems)p FI(,)g(S.)d(Albe)n(v)o(erio,)j(Ph.)e(Combe,)h(M.)f (Sirugue-Collin)414 1495 y(\(eds.\),)c(Lecture)g(Notes)g(in)f (Mathematics)j Fe(1109)p FI(,)e(Berlin,)g(Springer)l(,)h(1985,)f(pp.)f (68\226102.)213 1688 y([31])51 b(W)-8 b(.)37 b(Kirsch,)j(\223Random)g (Schr\366dinger)h(operators:)h(a)d(course\224,)i(in)e FH(Sc)o(hr\366ding)o(er)j(oper)o(ator)o(s)p FI(,)g(H.)414 1801 y(Holden,)24 b(A.)f(Jensen)i(\(eds.\),)f(Lecture)h(Notes)f(in)g (Physics)g Fe(345)p FI(,)g(Berlin,)g(Springer)l(,)h(1989,)g(pp.)e (264\226)414 1914 y(370.)213 2107 y([32])51 b(W)-8 b(.)30 b(Kirsch)j(and)f(F)-7 b(.)30 b(Martinelli,)k(\223On)e(the)g(density)i (of)e(states)h(of)f(Schr\366dinger)i(operators)h(with)d(a)414 2220 y(random)24 b(potential\224,)i FH(J)n(.)d(Phys.)g(A)f Fe(15)i FI(\(1982\))h(2139\2262156.)213 2414 y([33])51 b(W)-8 b(.)27 b(Kirsch)i(and)g(F)-7 b(.)27 b(Martinelli,)j(\223On)f (the)g(essential)h(self)o(adjointness)k(of)28 b(stochastic)k (Schr\366dinger)414 2527 y(operators\224,)26 b FH(Duk)o(e)d(Math.)h(J)n (.)e Fe(50)i FI(\(1983\))h(1255\2261260.)213 2720 y([34])51 b(C.)22 b(Kittel,)h FH(Intr)l(oduction)28 b(to)23 b(solid-state)k (physics)p FI(,)e(7)2099 2687 y Fa(th)2178 2720 y FI(edition,)g(Ne)n(w) d(Y)-10 b(ork,)23 b(W)l(ile)o(y)-6 b(,)24 b(1996.)213 2913 y([35])51 b(V)-12 b(.)34 b(K)m(ostrykin)j(and)f(R.)e(Schrader)l(,) j(\223The)e(density)j(of)d(states)i(and)f(the)f(spectral)j(shift)e (density)h(of)414 3026 y(random)24 b(Schr\366dinger)i(operators\224,)g FH(Re)o(v)-7 b(.)23 b(Math.)g(Phys.)h Fe(12)f FI(\(2000\))i (807\226847.)213 3220 y([36])51 b(I.)25 b(V)-12 b(.)25 b(K)o(ukushkin,)j(S.)d(V)-12 b(.)25 b(Meshk)o(o)o(v)i(and)g(V)-12 b(.)25 b(B.)g(T)m(imofee)n(v)-6 b(,)26 b(\223T)-7 b(w)o(o-dimensional) 29 b(electron)g(density)414 3332 y(of)k(states)i(in)f(a)f(transv)o (erse)j(magnetic)f(\002eld\224,)f FH(So)o(v)-7 b(.)34 b(Phys.)g(Usp.)f Fe(31)h FI(\(1988\))h(511\226534)h([Russian)414 3445 y(original:)26 b FH(Usp.)c(F)l(iz.)h(Nauk)i Fe(155)f FI(\(1988\))h(219\226264].)213 3639 y([37])51 b(I.)19 b(B.)f(Le)n(vinson,)j(\223T)m(ranslational)h(in)l(v)n(ariance)h(in)d (uniform)h(\002elds)e(and)i(the)f(equation)i(for)e(the)g(density)414 3752 y(matrix)41 b(in)g(the)h(W)l(igner)g(representation\224,)j FH(So)o(v)-7 b(.)42 b(Phys.)f(JETP)e Fe(30)i FI(\(1970\))i(362\226367)g ([Russian)414 3865 y(original:)26 b FH(Zh.)c(Eksp.)h(T)-8 b(er)e(.)22 b(F)l(iz.)h Fe(57)h FI(\(1969\))h(660\226672].)213 4058 y([38])51 b(E.)22 b(H.)g(Lieb)h(and)h(M.)e(Loss,)i FH(Analysis)p FI(,)g(2)1711 4025 y Fa(nd)1805 4058 y FI(edition,)h(Pro)o(vidence,)g(RI,)d(Amer)-5 b(.)23 b(Math.)g(Soc.,)g (2001.)213 4251 y([39])51 b(I.)24 b(M.)g(Lifshits,)j(S.)c(A.)h (Gredeskul)k(and)d(L.)f(A.)g(P)o(astur)l(,)i FH(Intr)l(oduction)j(to)c (the)h(theory)h(of)e(disor)m(der)m(ed)414 4364 y(systems)p FI(,)f(Ne)n(w)e(Y)-10 b(ork,)23 b(W)l(ile)o(y)-6 b(,)24 b(1988)h([Russian)f(original:)i(Mosco)n(w)-6 b(,)24 b(Nauka,)f(1982].) 213 4557 y([40])51 b(M.)22 b(A.)g(Lifshits,)i FH(Gaussian)i(r)o(andom)e (functions)p FI(,)i(Dordrecht,)f(Kluwer)l(,)e(1995.)213 4751 y([41])51 b(H.)30 b(Matsumoto,)j(\223On)f(the)g(inte)o(grated)j (density)f(of)e(states)h(for)f(the)g(Schr\366dinger)j(operators)g(with) 414 4864 y(certain)25 b(random)f(electromagnetic)k(potentials\224,)e FH(J)n(.)d(Math.)g(Soc.)h(J)m(apan)g Fe(45)g FI(\(1993\))h(197\226214.) 213 5057 y([42])51 b(S.)19 b(Nakamura,)i(\223)-7 b(A)19 b(remark)i(on)f(the)h(Dirichlet-Neumann)j(decoupling)f(and)e(the)g (inte)o(grated)i(density)414 5170 y(of)g(states\224,)i FH(J)n(.)e(Funct.)g(Anal.)g Fe(173)h FI(\(2001\))h(136\226152.)213 5363 y([43])51 b(S.)26 b(Nakao,)i(\223On)g(the)g(spectral)i(distrib)n (ution)h(of)d(the)g(Schr\366dinger)i(operator)g(with)e(random)h(poten-) 414 5476 y(tial\224,)24 b FH(J)m(apan.)g(J)n(.)f(Math.)g Fe(3)g FI(\(1977\))i(111\226139.)213 5670 y([44])51 b(L.)25 b(P)o(astur)l(,)h(\223On)g(the)h(Schr\366dinger)i(equation)g(with)d(a)g (random)h(potential\224,)i FH(Theor)-10 b(.)26 b(Math.)h(Phys.)f Fe(6)414 5783 y FI(\(1971\))f(299-306)h([Russian)e(original:)i FH(T)-8 b(eor)e(.)23 b(Mat.)g(F)l(iz.)f Fe(6)i FI(\(1971\))h (415\226424].)p eop %%Page: 33 33 33 32 bop 188 -183 a FH(Inte)l(gr)o(ated)26 b(Density)f(of)e(States)i (for)f(Random)g(Sc)o(hr\366ding)o(er)i(Oper)o(ator)o(s)g(with)d(Ma)o (gnetic)i(F)l(ields)238 b FE(33)213 125 y FI([45])51 b(L.)33 b(P)o(astur)l(,)h(\223Spectral)i(properties)h(of)d(disordered)k (systems)d(in)f(the)h(one-body)i(approximation\224,)414 237 y FH(Commun.)22 b(Math.)i(Phys.)f Fe(75)h FI(\(1980\))h (179\226196.)213 433 y([46])51 b(L.)40 b(P)o(astur)i(and)h(A.)d (Figotin,)j FH(Spectr)o(a)g(of)f(r)o(andom)h(and)f(almost-periodic)k (oper)o(ator)o(s)p FI(,)e(Berlin,)414 546 y(Springer)l(,)25 b(1992.)213 742 y([47])51 b(M.)21 b(Reed)h(and)h(B.)d(Simon,)i FH(Methods)h(of)g(modern)g(mathematical)h(physics)g(I:)e(Functional)i (analysis,)414 855 y FI(re)n(vised)h(and)f(enlar)n(ged)i(edition,)f (San)e(Die)o(go,)g(Academic,)h(1980.)213 1051 y([48])51 b(M.)23 b(Reed)i(and)g(B.)f(Simon,)g FH(Methods)i(of)e(modern)i (mathematical)h(physics)f(III:)f(Scattering)j(theory)p FI(,)414 1164 y(Ne)n(w)22 b(Y)-10 b(ork,)23 b(Academic,)h(1979.)213 1360 y([49])51 b(M.)45 b(Reed)h(and)h(B.)e(Simon,)h FH(Methods)i(of)f (modern)g(mathematical)i(physics)f(IV)-6 b(:)46 b(Analysis)h(of)414 1473 y(oper)o(ator)o(s)p FI(,)25 b(Ne)n(w)e(Y)-10 b(ork,)23 b(Academic,)h(1978.)213 1669 y([50])51 b(W)-8 b(.)22 b(Rudin,)h FH(Real)h(and)g(comple)n(x)h(analysis)p FI(,)g(3)1865 1636 y Fa(rd)1948 1669 y FI(edition,)g(Ne)n(w)d(Y)-10 b(ork,)23 b(McGra)o(w-Hill,)g(1987.)213 1865 y([51])51 b(B.)25 b(I.)g(Shklo)o(vskii)j(and)f(A.)e(L.)f(Efros,)i FH(Electr)l(onic)i(pr)l(operties)h(of)e(doped)g(semiconductor)o(s)p FI(,)j(Berlin,)414 1978 y(Springer)l(,)25 b(1984)f([Russian)h (original:)h(Mosco)n(w)-6 b(,)23 b(Nauka,)h(1979].)213 2174 y([52])51 b(B.)22 b(Simon,)h FH(Functional)i(inte)l(gr)o(ation)i (and)d(quantum)h(physics)p FI(,)g(Ne)n(w)d(Y)-10 b(ork,)24 b(Academic,)g(1979.)213 2369 y([53])51 b(B.)20 b(Simon,)i FH(T)-5 b(r)o(ace)22 b(ideals)h(and)g(their)g(applications)p FI(,)i(Cambridge,)e(Cambridge)g(Uni)n(v)-6 b(.)22 b(Press,)g(1979.)213 2565 y([54])51 b(B.)26 b(Simon,)h(\223Schr\366dinger)k(operators)f(in)e (the)g(twenty-\002rst)h(century\224,)h(in)d FH(Mathematical)j(Physics) 414 2678 y(2000)p FI(,)37 b(A.)d(F)o(okas,)i(A.)f(Grigoryan,)i(T)-7 b(.)35 b(Kibble,)i(and)f(B.)f(Ze)o(garlinski)j(\(eds.\))e(London,)h (Imperial)414 2791 y(Colle)o(ge)24 b(Press,)f(2000,)i(pp.)e (283\226288.)213 2987 y([55])51 b(N.)28 b(Ueki,)h(\223On)g(spectra)j (of)d(random)i(Schr\366dinger)h(operators)g(with)d(magnetic)i (\002elds\224,)f FH(Osaka)g(J)n(.)414 3100 y(Math.)23 b Fe(31)h FI(\(1994\))h(177\226187.)213 3296 y([56])51 b(W)-7 b(ei-Min)52 b(W)-7 b(ang,)52 b(\223)-7 b(Asymptotic)54 b(e)o(xpansion)h(for)d(the)h(density)h(of)e(states)h(of)f(the)h (magnetic)414 3409 y(Schr\366dinger)30 b(operator)f(with)e(a)g(random)i (potential\224,)h FH(Commun.)c(Math.)h(Phys.)g Fe(172)h FI(\(1995\))h(401\226)414 3522 y(425.)213 3718 y([57])51 b(S.)22 b(W)-7 b(arzel,)25 b FH(On)e(Lifshits)i(tails)g(in)g(ma)o (gnetic)g(\002elds)p FI(,)g(Berlin,)g(Logos,)f(2001)h([PhD-Thesis,)g (Uni)n(v)o(er)n(-)414 3831 y(sit\344t)f(Erlangen-N\374rnber)n(g].)213 4027 y([58])51 b(J.)23 b(W)-7 b(eidmann,)24 b FH(Linear)g(oper)o(ator)o (s)i(in)d(Hilbert)i(space)p FI(,)f(Berlin,)g(Springer)l(,)h(1980.)213 4223 y([59])51 b(H.)21 b(W)-7 b(e)o(yl,)23 b(\223Das)f(asymptotische)27 b(V)-10 b(erteilungsgesetz)27 b(der)d(Eigenwerte)g(linearer)h (partieller)g(Dif)n(fer)n(-)414 4335 y(entialgleichungen)f(\(mit)19 b(einer)g(Anwendung)h(auf)g(die)f(Theorie)g(der)g (Hohlraumstrahlung\)\224,)24 b FH(Math.)414 4448 y(Ann.)f Fe(71)g FI(\(1912\))i(441\226479)h([in)e(German].)213 4644 y([60])51 b(J.)23 b(Zak,)f(\223Magnetic)k(translation)g (group\224,)f FH(Phys.)e(Re)o(v)-7 b(.)23 b Fe(134)h FI(\(1964\))h(A1602\226A1606.)p eop %%Page: 34 34 34 33 bop 120 -183 a FE(34)261 b FH(T)-7 b(.)22 b(Hupfer)-10 b(,)24 b(H.)e(Lesc)o(hk)o(e)o(,)j(P)-12 b(.)21 b(M\374ller)j(&)f(S.)g (W)-8 b(arzel)120 125 y FA(Citation)34 b(Index)120 329 y FI([1])646 b(13)120 442 y([2])691 b(2)120 555 y([3])g(8)120 668 y([4])646 b(12)120 781 y([5])g(12)120 894 y([6])237 b(11,)23 b(15,)g(17,)h(18)120 1006 y([7])691 b(4)120 1119 y([8])g(2)120 1232 y([9])146 b(2,)23 b(3,)g(8,)f(9,)h(15,)h(19)120 1345 y([10])646 b(9)120 1458 y([11])555 b(8,)23 b(9)120 1571 y([12])601 b(10)120 1684 y([13])214 b(2,)23 b(4,)g(7,)g(8,)g(12,) 871 1797 y(14)120 1910 y([14])601 b(12)120 2023 y([15])g(12)120 2136 y([16])g(12)120 2248 y([17])646 b(8)120 2361 y([18])601 b(12)120 2474 y([19])169 b(2,)23 b(3,)g(8,)g(11,)g(15,)871 2587 y(21)120 2700 y([20])601 b(12)1460 329 y([21])645 b(9)1460 442 y([22])g(9)1460 555 y([23])600 b(13)1460 668 y([24])645 b(3)1460 781 y([25])600 b(12)1460 894 y([26])645 b(9)1460 1006 y([27])g(9)1460 1119 y([28])214 b(6,)23 b(12,)g(21,)g(22,)2074 1232 y(24,)g(27)1460 1345 y([29])464 b(26,)23 b(29)1460 1458 y([30])373 b(8,)23 b(26,)g(29)1460 1571 y([31])464 b(2,)23 b(5\2268)1460 1684 y([32])555 b(2,)22 b(8)1460 1797 y([33])600 b(14)1460 1910 y([34])g(13)1460 2023 y([35])645 b(2)1460 2136 y([36])g(2)1460 2248 y([37])g(6)1460 2361 y([38])600 b(14)1460 2474 y([39])645 b(2)1460 2587 y([40])600 b(13)1460 2700 y([41])373 b(2,)23 b(3,)g(8,)f(9)2799 329 y([42])168 b(2,)23 b(3,)g(8,)g(11,)h(15,)3549 442 y(21)2799 555 y([43])555 b(2,)23 b(8)2799 668 y([44])555 b(2,)23 b(8)2799 781 y([45])600 b(12)2799 894 y([46])214 b(2\2264,)24 b(6\2269,)g(11,)2867 1006 y(12,)g(15,)f(19\22621,)i (25\22628)2799 1119 y([47])328 b(14,)23 b(25,)g(28)2799 1232 y([48])646 b(9)2799 1345 y([49])373 b(8,)23 b(23,)g(26)2799 1458 y([50])600 b(12)2799 1571 y([51])646 b(2)2799 1684 y([52])509 b(8,)23 b(27)2799 1797 y([53])214 b(4,)23 b(22\22624,)i(26,)3549 1910 y(27)2799 2023 y([54])600 b(12)2799 2136 y([55])214 b(2,)23 b(3,)g(7\2269,)h(15,)3549 2248 y(19)2799 2361 y([56])646 b(8)2799 2474 y([57])g(9)2799 2587 y([58])464 b(24,)23 b(27)2799 2700 y([60])646 b(6)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF