Content-Type: multipart/mixed; boundary="-------------0012120553192" This is a multi-part message in MIME format. ---------------0012120553192 Content-Type: text/plain; name="00-498.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-498.keywords" Strong magnetic fields, integrated density of states, random Schroedinger operators ---------------0012120553192 Content-Type: application/postscript; name="Zeller.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Zeller.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: zeller.dvi %%CreationDate: Tue Dec 12 11:15:14 2000 %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: C:\EMTEX\BIN\dvips32.EXE zeller -o %+ -pj:c:\temp\dv3.mfj %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2000.12.12:1114 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup 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y Fi(!)1062 1969 y Fr(\()p Fn(x)p Fr(\),)g Fq(!)g Fp(2)e Fq(\012)r Fr(,)i Fn(x)e Fp(2)h Fm(R)1390 1954 y Fl(3)1405 1969 y Fr(,)h(b)q(e)h(a)e(real)h(random)262 2019 y(\014eld.)d(W)m(e)h(assume)g(that)g Fq(V)681 2025 y Fi(!)716 2019 y Fr(is)g Fm(G)784 2004 y Fl(3)806 2019 y Fr(-ergo)q(dic)g(with)g Fm(G)19 b Fr(=)12 b Fm(Z)-6 b Fr(or)12 b Fm(G)18 b Fr(=)12 b Fm(R)5 b Fr(\(see)13 b([K,)d(Section)i(3.1]\).)262 2069 y(In)e(other)g(w)o(ords,)g(there)i (exists)f(an)e(ergo)q(dic)i(group)f(of)f(measure)h(preserving)i (automorphisms)262 2119 y Fp(T)285 2125 y Fh(k)319 2119 y Fr(:)h Fq(\012)i Fp(!)e Fq(\012)r Fr(,)i Fn(k)e Fp(2)g Fm(G)615 2104 y Fl(3)637 2119 y Fr(,)h(suc)o(h)i(that)f Fq(V)873 2125 y Fi(!)897 2119 y Fr(\()p Fn(x)c Fr(+)f Fn(k)p Fr(\))j(=)h Fq(V)1115 2125 y Fj(T)1134 2129 y Fg(k)1153 2125 y Fi(!)1177 2119 y Fr(\()p Fn(x)p Fr(\))h(for)g Fn(x)e Fp(2)g Fm(R)1426 2104 y Fl(3)1456 2119 y Fr(and)i Fq(!)g Fp(2)e Fq(\012)r Fr(.)h(W)m(e)262 2168 y(recall)k(that)h(ergo)q (dicit)o(y)g(of)f(a)g(group)h Fq(G)f Fr(of)g(automorphisms)e(of)i Fq(\012)j Fr(means)d(that)h(the)g Fq(G)p Fr(-)262 2218 y(in)o(v)n(ariance)13 b(of)g(a)h(giv)o(en)f(set)i Fp(A)c(2)h(F)17 b Fr(implies)12 b(either)j Fm(P)-6 b Fr(\()p Fp(A)p Fr(\))9 b(=)j(1)i(or)g Fm(P)-6 b Fr(\()p Fp(A)p Fr(\))9 b(=)j(0.)262 2268 y(F)m(or)f Fn(x)h Fp(2)f Fm(R)443 2253 y Fl(3)470 2268 y Fr(w)o(e)i(write)f Fn(x)g Fr(=)f(\()p Fq(X)q(;)c(z)r Fr(\))13 b(with)e Fq(X)k Fp(2)d Fm(R)1048 2253 y Fl(2)1063 2268 y Fr(,)g Fq(z)h Fp(2)e Fm(R)m Fr(.)e(Hence,)k Fq(z)h Fr(is)e(the)g(v)n(ariable)f(along)262 2318 y(the)j(magnetic)f(\014eld)h Fn(b)d Fr(=)g(\(0)p Fq(;)c Fr(0)p Fq(;)g(b)p Fr(\),)12 b(while)i Fq(X)j Fr(runs)e(o)o(v)o(er)f(the)g(plane)g(p)q(erp)q (endicular)h(to)f Fn(b)p Fr(.)262 2368 y(W)m(e)i(supp)q(ose)i(that)f Fq(V)612 2374 y Fi(!)653 2368 y Fr(is)g Fm(G)t Fr(-ergo)q(dic)j(with)c Fm(G)24 b Fr(=)17 b Fm(Z)p Fr(or)g Fm(G)24 b Fr(=)16 b Fm(R)11 b Fr(in)17 b(the)g(direction)g(of)f(the)p eop %%Page: 2 2 2 1 bop 262 116 a Fu(2)1341 b(Raik)o(o)o(v)262 224 y Fr(magnetic)10 b(\014eld,)h(i.e.)g(that)h(the)h(subgroup)f Fp(f)o(T)980 230 y Fh(k)1002 224 y Fp(j)p Fn(k)f Fr(=)h(\(0)p Fq(;)7 b Fr(0)p Fq(;)g(k)q Fr(\))p Fq(;)17 b(k)12 b Fp(2)f Fm(G)5 b Fp(g)14 b Fr(is)e(ergo)q(dic.)f(F)m(urther,)262 274 y(w)o(e)j(assume)f(that)h(the)h(realizations)e(of)g Fq(V)918 280 y Fi(!)956 274 y Fr(are)i(almost)d(surely)i(uniformly)d(b) q(ounded,)j(i.e.)690 365 y Fq(c)708 371 y Fl(0)739 365 y Fr(:=)d(ess)f Fp(\000)k Fr(sup)896 399 y Fi(!)q Fj(2)p Fi(\012)994 365 y Fr(sup)986 401 y Fh(x)p Fj(2)p Ff(R)1052 393 y Fe(3)1072 365 y Fp(j)p Fq(V)1108 371 y Fi(!)1132 365 y Fr(\()p Fn(x)p Fr(\))p Fp(j)d Fq(<)h Fp(1)p Fq(:)343 b Fr(\(1.3\))262 481 y(Finally)m(,)11 b(w)o(e)j(supp)q(ose)h(that)f (the)h(realizations)e(of)g Fq(V)1080 487 y Fi(!)1118 481 y Fr(are)i(almost)d(surely)i(con)o(tin)o(uous.)262 558 y(Let)i Fq(T)21 b Fr(b)q(e)c(a)e(selfadjoin)o(t)g(op)q(erator)h(in) f(a)h(Hilb)q(ert)g(space.)g(Denote)g(b)o(y)g Fq(P)1436 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Fq(h)355 1709 y Fl(0)p Fi(;R)420 1703 y Fr(:=)j Fp(\000)522 1687 y Fi(d)539 1674 y Fe(2)p 513 1694 51 2 v 513 1717 a Fi(dz)547 1709 y Fe(2)578 1703 y Fr(b)q(e)e(the)g(self-adjoin)o(t)d(op)q(erator)j(de\014ned)h(on)e(H) 1291 1688 y Fl(2)1316 1669 y Fk(\000\000)1354 1703 y Fp(\000)1391 1687 y Fi(R)p 1392 1694 26 2 v 1396 1717 a Fl(2)1422 1703 y Fq(;)1445 1687 y Fi(R)p 1445 1694 V 1449 1717 a Fl(2)1475 1669 y Fk(\001\001)1523 1703 y Fr(with)g(Diric)o(h-)262 1753 y(let)k(b)q(oundary)h(conditions.)262 1862 y Fn(Prop)q(ositi)o(on)f(2.1)20 b Fr(\([K,)15 b(Chapter7],)g([P)m (.Fi)o(,)g(Chapter)h(I)q(I)q(I]\))g Fo(L)n(et)h Fm(G)k Fr(=)15 b Fm(Z)p Fo(or)h Fm(G)21 b Fr(=)15 b Fm(R)m Fo(.)e(L)n(et)262 1912 y Fq(f)282 1918 y Fi(!)306 1912 y Fr(\()p Fq(z)r Fr(\))p Fo(,)f Fq(!)g Fp(2)g Fq(\012)r Fo(,)f Fq(z)j Fp(2)d Fm(R)m Fo(,)d(b)n(e)k(a)g(r)n(e)n(al)e Fm(G)5 b Fo(-er)n(go)n(dic)14 b(r)n(andom)e(\014eld)g(whose)f(r)n(e)n (alizations)g(ar)n(e)g(almost)262 1961 y(sur)n(ely)j(uniformly)g(b)n 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Fr(,)i(satis\014es)h(the)g(h) o(yp)q(otheses)g(of)f(Prop)q(osition)f(2.1.)p eop %%Page: 3 3 3 2 bop 262 116 a Fu(In)o(tegrated)13 b(Densit)o(y)i(of)d(States)i(in)g (Strong)f(Magnetic)i(Fields)538 b(3)262 224 y Fr(Moreo)o(v)o(er,)18 b(if)f Fm(G)25 b Fr(=)19 b Fm(Z)-13 b Fr(,)14 b(then)19 b(the)g(function)e Fq(\045)p Fr(\()p Fq(\025)p Fr(;)7 b Fq(V)j Fr(\()p Fq(X)q(;)d(:)p Fr(\)\))18 b(is)g(p)q(erio)q(dic)g (with)g(resp)q(ect)i(to)262 274 y Fq(X)i Fp(2)c Fm(R)397 259 y Fl(2)413 274 y Fr(,)g(while)g(in)f(the)i(case)g Fm(G)27 b Fr(=)19 b Fm(R)12 b Fr(the)19 b(quan)o(tit)o(y)e Fq(\045)p Fr(\()p Fq(\025)p Fr(;)7 b Fq(V)j Fr(\()p Fq(X)q(;)d(:)p Fr(\)\))18 b(is)g(indep)q(enden)o(t)h(of)262 324 y Fq(X)c Fp(2)c Fm(R)383 309 y Fl(2)412 324 y Fr(\(see)16 b([K.R)n(]\).)e(F)m (or)f Fq(\025)f Fp(2)f Fm(R)d Fr(set)459 443 y Fq(k)q Fr(\()p Fq(\025)p Fr(\))k(=)g Fq(k)q Fr(\()p Fq(\025)p Fr(;)7 b Fq(V)i Fr(\))j(:=)792 372 y Fk(\()832 377 y(R)852 430 y Fr(\()868 427 y Fj(\000)899 416 y Fe(1)p 899 421 15 2 v 899 437 a(2)918 427 y Fi(;)933 416 y Fe(1)p 933 421 V 933 437 a(2)952 430 y Fr(\))968 409 y Fe(2)993 410 y Fq(\045)p Fr(\()p Fq(\025;)7 b(V)j Fr(\()p Fq(X)q(;)d(:)p Fr(\)\))k Fq(dX)45 b Fr(if)f Fm(G)19 b Fr(=)12 b Fm(Z)-14 b Fq(;)832 476 y(\045)p Fr(\()p Fq(\025;)7 b(V)j Fr(\(0)p Fq(;)d(:)p Fr(\)\))40 b(if)k Fm(G)19 b Fr(=)12 b Fm(R)m Fq(:)262 563 y Fr(Ob)o(viously)m(,)g Fq(k)q Fr(\()p Fq(\025)p Fr(\))i(is)g(non-decreasing)g(and)g(con)o(tin)o(uous)g(with)f(resp)q (ect)k(to)c Fq(\025)p Fr(.)262 664 y Fn(Theorem)h(2.1.)21 b Fo(L)n(et)c Fm(G)24 b Fr(=)16 b Fm(Z)q Fo(or)h Fm(G)24 b Fr(=)16 b Fm(R)m Fo(.)e(L)n(et)j Fq(V)1086 670 y Fi(!)1128 664 y Fo(b)n(e)g(a)h(r)n(e)n(al)e Fm(G)1330 649 y Fl(3)1352 664 y Fo(-er)n(go)n(dic)h(r)n(andom)h(\014eld)262 714 y(whose)f(r)n(e)n(alizations)g(ar)n(e)g(almost)h(sur)n(ely)f(uniformly) g(b)n(ounde)n(d)i(and)f(c)n(ontinuous.)h(Assume)262 764 y(in)e(addition)h(that)f Fq(V)591 770 y Fi(!)633 764 y Fo(is)g Fm(G)t Fo(-er)n(go)n(dic)j(in)e(the)f(dir)n(e)n(ction)g(of)g (the)h(magnetic)f(\014eld.)h(Then)f(for)262 814 y(e)n(ach)e Fq(q)d Fp(2)g Fm(N)458 820 y Fj(\003)475 814 y Fo(,)i(and)i Fq(\025)607 820 y Fl(1)626 814 y Fq(;)7 b(\025)669 820 y Fl(2)699 814 y Fp(2)k Fm(R)m Fo(,)g Fq(\025)819 820 y Fl(1)850 814 y Fq(<)h(\025)918 820 y Fl(2)936 814 y Fo(,)j(we)f(have)304 913 y Fr(lim)292 940 y Fi(b)p Fj(!1)380 913 y Fq(b)398 896 y Fj(\000)p Fl(1)449 913 y Fr(\()p Fp(D)497 919 y Fi(b)514 913 y Fr(\()p Fq(\025)554 919 y Fl(2)582 913 y Fr(+)c(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))h Fp(\000)f(D)885 919 y Fi(b)902 913 y Fr(\()p Fq(\025)942 919 y Fl(1)970 913 y Fr(+)g(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)p Fr(\)\))j(=)1279 885 y(1)p 1266 904 46 2 v 1266 942 a(2)p Fq(\031)1324 913 y Fr(\()p Fq(k)q Fr(\()p Fq(\025)1403 919 y Fl(2)1422 913 y Fr(\))e Fp(\000)f Fq(k)q Fr(\()p Fq(\025)1552 919 y Fl(1)1571 913 y Fr(\)\))e Fq(:)31 b Fr(\(2.2\))262 1033 y(Theorem)17 b(2.1)f(con)o(tains)i(the)g(asymptotics)f(as)g Fq(b)h Fp(!)f(1)g Fr(of)g(the)h(IDOS)g Fp(D)1473 1039 y Fi(b)1507 1033 y Fr(near)g(the)h Fq(q)q Fr(th)262 1083 y(Landau)12 b(lev)o(el,)g Fq(q)g Fp(\025)g Fr(1.)g(Since)h(\(2.2\))f(has)h(b)q(een) g(pro)o(v)o(ed)g(in)f([K.R)o(])g(for)h Fq(q)f Fr(=)g(1,)g(w)o(e)h (shall)f(pro)o(v)o(e)262 1132 y(it)f(here)i(for)e Fq(q)i(>)f Fr(1.)f(The)h(metho)q(ds)f(applied)g(in)h(this)f(pap)q(er)i(are)f (similar)d(to)j(the)g(ones)h(used)f(in)262 1182 y([K.R)n(],)j(and)g (are)h(based)g(on)f(the)h(Birman-Sc)o(h)o(winger)e(principle)h(\(see)i ([B,)e(Lemma)e(1.1]\),)g(a)262 1232 y(suitable)f(v)o(ersion)g(of)f(the) i(Kac-Murdo)q(c)o(k-Szeg\177)-21 b(o)13 b(theorem)f(\(see)i([R)o(,)d (Lemma)f(3.2]\),)g(and)i(the)262 1282 y(Birkho\013-Khin)o(tc)o(hine)17 b(ergo)q(dic)g(theorem.)f(Ho)o(w)o(ev)o(er,)h(the)g(analysis)f(near)h (higher)g(Landau)262 1332 y(lev)o(els)c(is)g(more)f(complicated)g (since)i(the)g(\014rst)g(Landau)f(lev)o(el)g(coincides)h(with)f(lo)o(w) o(er)g(b)q(ound)262 1382 y(of)k(the)i(sp)q(ectrum)g(of)f Fq(H)662 1388 y Fl(0)680 1382 y Fr(\()p Fq(b)p Fr(\))g(\(see)i (\(1.2\)\),)d(while)h(the)h(higher)g(Landau)f(lev)o(els)g(\(2)p Fq(q)13 b Fp(\000)g Fr(1\))p Fq(b)p Fr(,)262 1431 y Fq(q)g(>)f Fr(1,)i(are)g(in)o(ternal)g(p)q(oin)o(ts)g(of)g Fq(\033)q Fr(\()p Fq(H)857 1437 y Fl(0)876 1431 y Fr(\()p Fq(b)p Fr(\)\).)g(The)g(pro)q(of)g(of)g(Theorem)g(2.1)f(can)i(b)q(e)f(found)g (in)262 1481 y(Section)g(4,)f(while)g(Section)i(3)e(con)o(tains)h (preliminary)e(estimates.)262 1630 y Fs(3)56 b(Preliminary)15 b(Estimates)262 1757 y Fn(3.1.)f Fr(Let)h Fq(H)463 1742 y Fi(N)460 1768 y Fl(0)p Fi(;R)513 1757 y Fr(\()p Fq(b)p Fr(\))f(b)q(e)h(the)g(self-adjoin)o(t)e(op)q(erator)i(generated)h(in)e Fq(L)1355 1742 y Fl(2)1381 1711 y Fk(\020)1405 1723 y(\000)1424 1757 y Fp(\000)1461 1740 y Fi(R)p 1462 1747 26 2 v 1466 1771 a Fl(2)1492 1757 y Fq(;)1515 1740 y Fi(R)p 1515 1747 V 1519 1771 a Fl(2)1545 1723 y Fk(\001)1564 1731 y Fl(3)1583 1711 y Fk(\021)1622 1757 y Fr(b)o(y)g(the)262 1836 y(closed)g(quadratic)g(form)668 1802 y Fk(R)687 1856 y Fr(\()704 1853 y Fj(\000)735 1841 y Fd(R)p 735 1846 22 2 v 738 1863 a Fe(2)761 1853 y Fi(;)776 1841 y Fd(R)p 776 1846 V 779 1863 a Fe(2)802 1856 y Fr(\))818 1835 y Fe(3)844 1800 y Fk(\014)844 1825 y(\014)857 1836 y Fq(i)p Fp(r)p Fq(u)9 b Fr(+)986 1819 y Fh(b)p Fj(^)p Fh(x)p 986 1826 64 2 v 1009 1850 a Fl(2)1054 1836 y Fq(u)1078 1800 y Fk(\014)1078 1825 y(\014)1091 1809 y Fl(2)1117 1836 y Fq(d)p Fn(x)p Fr(,)k Fq(u)e Fp(2)h Fr(H)1295 1821 y Fl(1)1320 1790 y Fk(\020)1345 1802 y(\000)1364 1836 y Fp(\000)1401 1819 y Fi(R)p 1401 1826 26 2 v 1405 1850 a Fl(2)1431 1836 y Fq(;)1455 1819 y Fi(R)p 1455 1826 V 1459 1850 a Fl(2)1485 1802 y Fk(\001)1504 1810 y Fl(3)1523 1790 y Fk(\021)1547 1836 y Fr(.)262 1946 y Fn(Lemma)j(3.1.)21 b Fr(\([N,)15 b(Theorem)g(1],)g([H.L.M.W)n(,)g(Theorem)g(3.1]\))g Fo(L)n(et)i Fm(G)22 b Fr(=)15 b Fm(Z)p Fo(or)h Fm(G)22 b Fr(=)15 b Fm(R)m Fo(.)262 1996 y(Assume)j(that)g Fq(V)532 2002 y Fi(!)574 1996 y Fo(is)g(a)h(r)n(e)n(al)e Fm(G)773 1981 y Fl(3)795 1996 y Fo(-er)n(go)n(dic)h(r)n(andom)g(\014eld)h(whose) f(r)n(e)n(alizations)f(ar)n(e)h(almost)262 2046 y(sur)n(ely)d (uniformly)g(b)n(ounde)n(d)j(and)e(c)n(ontinuous.)h(L)n(et)f Fq(\026)e Fp(2)f Fm(R)d Fo(b)n(e)16 b(a)g(c)n(ontinuity)g(p)n(oint)h (of)e Fp(D)1708 2052 y Fi(b)1725 2046 y Fo(.)262 2095 y(Then)g(almost)f(sur)n(ely)644 2178 y Fp(D)676 2184 y Fi(b)692 2178 y Fr(\()p Fq(\026)p Fr(\))e(=)29 b(lim)805 2205 y Fi(R)p Fj(!1)910 2178 y Fq(R)942 2161 y Fj(\000)p Fl(3)993 2178 y Fq(N)5 b Fr(\()p Fq(\026)p Fr(;)i Fq(H)1129 2161 y Fi(N)1126 2188 y Fl(0)p Fi(;R)1179 2178 y Fr(\()p Fq(b)p Fr(\))i(+)h Fq(V)1304 2184 y Fi(!)1328 2178 y Fr(\))p Fq(:)297 b Fr(\(3.1\))262 2302 y(Set)14 b Fq(\037)359 2308 y Fi(R)386 2302 y Fr(\()p Fn(x)p Fr(\))e(:=)g Fn(1)535 2322 y Fr(\()551 2319 y Fj(\000)581 2308 y Fd(R)p 581 2313 22 2 v 585 2329 a Fe(2)608 2319 y Fi(;)623 2308 y Fd(R)p 623 2313 V 626 2329 a Fe(2)649 2322 y Fr(\))665 2301 y Fe(3)683 2302 y Fr(\()p Fn(x)p Fr(\),)i Fn(x)e Fp(2)f Fm(R)875 2287 y Fl(3)891 2302 y Fr(.)i(On)h Fq(D)q Fr(\()p Fq(H)1071 2308 y Fl(0)1090 2302 y Fr(\()p Fq(b)p Fr(\)\))g(in)o(tro)q(duce)h(the)f(op)q(erator)g Fq(H)1628 2308 y Fl(0)1646 2302 y Fr(\()p Fq(b)p Fr(\))c(+)262 2368 y(\()p Fq(V)302 2374 y Fi(!)335 2368 y Fp(\000)g Fq(\026)p Fr(\))p Fq(\037)444 2374 y Fi(R)485 2368 y Fr(with)j Fq(b)f(>)g Fr(0,)h Fq(!)g Fp(2)e Fq(\012)r Fr(,)i Fq(\026)f Fp(2)f Fm(R)m Fr(,)f Fq(R)i(>)g Fr(0.)p eop %%Page: 4 4 4 3 bop 262 116 a Fu(4)1341 b(Raik)o(o)o(v)262 224 y Fn(Prop)q(ositi)o(on)13 b(3.1)20 b Fo(Under)15 b(the)g(hyp)n(otheses)h (of)f(L)n(emma)g(3.1)g(almost)f(sur)n(ely)583 310 y Fp(D)615 316 y Fi(b)631 310 y Fr(\()p Fq(\026)p Fr(\))e(=)29 b(lim)744 337 y Fi(R)p Fj(!1)849 310 y Fq(R)881 293 y Fj(\000)p Fl(3)932 310 y Fq(N)5 b Fr(\(0;)i Fq(H)1061 316 y Fl(0)1079 310 y Fr(\()p Fq(b)p Fr(\))i(+)h(\()p Fq(V)1220 316 y Fi(!)1253 310 y Fp(\000)g Fq(\026)p Fr(\))p Fq(\037)1362 316 y Fi(R)1389 310 y Fr(\))p Fq(:)236 b Fr(\(3.2\))262 435 y Fo(Pr)n(o)n(of)p Fr(:)12 b(By)j(the)f(minim)o(ax)d(principle,)323 521 y Fq(N)5 b Fr(\()p Fq(\026)p Fr(;)i Fq(H)459 504 y Fi(D)456 532 y Fl(0)p Fi(;R)518 521 y Fr(+)i Fq(V)583 527 y Fi(!)608 521 y Fr(\))i(=)h Fq(N)5 b Fr(\(0;)i Fq(H)811 504 y Fi(D)808 532 y Fl(0)p Fi(;R)870 521 y Fr(+)i Fq(V)935 527 y Fi(!)969 521 y Fp(\000)g Fq(\026)p Fr(\))j Fp(\024)g Fq(N)5 b Fr(\(0;)i Fq(H)1236 527 y Fl(0)1253 521 y Fr(\()p Fq(b)p Fr(\))i(+)h(\()p Fq(V)1394 527 y Fi(!)1428 521 y Fp(\000)f Fq(\026)p Fr(\))p Fq(\037)1536 527 y Fi(R)1563 521 y Fr(\))p Fq(:)62 b Fr(\(3.3\))262 622 y(Set)14 b Fp(O)366 628 y Fi(R)405 622 y Fr(:=)e Fm(R)494 607 y Fl(3)519 622 y Fp(n)549 589 y Fk(\002)566 622 y Fp(\000)603 606 y Fi(R)p 604 613 26 2 v 608 637 a Fl(2)634 622 y Fq(;)657 606 y Fi(R)p 657 613 V 661 637 a Fl(2)687 589 y Fk(\003)705 596 y Fl(3)723 622 y Fr(.)i(Denote)h(b)o(y)960 612 y(~)949 622 y Fq(H)987 607 y Fi(N)984 634 y Fl(0)p Fi(;R)1052 622 y 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Fi(!)1195 832 y Fp(\000)g Fq(\026)p Fr(\))h(+)f Fq(N)c Fr(\(0;)1432 821 y(~)1422 832 y Fq(H)1460 815 y Fi(N)1457 842 y Fl(0)p Fi(;R)1509 832 y Fr(\))p Fq(:)116 b Fr(\(3.4\))262 926 y(Since)381 916 y(~)370 926 y Fq(H)408 911 y Fi(N)405 938 y Fl(0)p Fi(;R)470 926 y Fp(\025)11 b Fr(0,)j(w)o(e)g(ha)o(v)o(e)f Fq(N)5 b Fr(\(0;)820 916 y(~)810 926 y Fq(H)848 911 y Fi(N)845 938 y Fl(0)p Fi(;R)898 926 y Fr(\))11 b(=)h(0.)h(Hence,)i(\(3.4\))e(can)i(b)q(e)f(re-written)h (as)323 1021 y Fq(N)5 b Fr(\(0;)i Fq(H)452 1027 y Fl(0)469 1021 y Fr(\()p Fq(b)p Fr(\))j(+)f(\()p Fq(V)610 1027 y Fi(!)644 1021 y Fp(\000)g Fq(\026)p Fr(\))p Fq(\037)752 1027 y Fi(R)780 1021 y Fr(\))i Fp(\024)h Fq(N)5 b Fr(\(0;)i Fq(H)983 1004 y Fi(N)980 1031 y Fl(0)p Fi(;R)1042 1021 y Fr(+)i Fq(V)1107 1027 y Fi(!)1141 1021 y Fp(\000)g Fq(\026)p Fr(\))j(=)g Fq(N)5 b Fr(\()p Fq(\026)p Fr(;)i Fq(H)1415 1004 y Fi(N)1412 1031 y Fl(0)p Fi(;R)1474 1021 y Fr(+)i Fq(V)1539 1027 y Fi(!)1563 1021 y Fr(\))p Fq(:)62 b Fr(\(3.5\))262 1107 y(Com)o(bining)11 b(\(3.3\))i(and)g(\(3.5\))h (with)f(\(1.4\))g(and)h(\(3.1\),)f(w)o(e)h(get)g(\(3.2\).)f Fp(})262 1181 y Fo(R)n(emark)p Fr(.)k(Prop)q(osition)g(3.1)f(is)i(v)o (ery)g(similar)d(to)i([K.R)o(,)g(Prop)q(osition)g(4.1].)f(Ho)o(w)o(ev)o (er,)h(the)262 1231 y(pro)q(of)c(presen)o(ted)j(here)f(is)f(m)o(uc)o(h) f(simpler)f(b)q(ecause)k(no)o(w)d(w)o(e)h(disp)q(ose)h(of)e(Lemma)e (3.1.)262 1306 y(In)o(tro)q(duce)k(the)f(compact)f(Birman-Sc)o(h)o (winger-t)o(yp)q(e)g(op)q(erators)453 1392 y Fq(T)6 b Fr(\()p Fq(\026)p Fr(\))12 b(=)f Fq(T)619 1398 y Fi(b;!)q(;R)703 1392 y Fr(\()p Fq(\026)p Fr(\))h(:=)f Fq(H)862 1398 y Fl(0)880 1392 y Fr(\()p Fq(b)p Fr(\))930 1375 y Fj(\000)p Fl(1)p Fi(=)p Fl(2)1009 1392 y Fr(\()p Fq(V)1049 1398 y Fi(!)1082 1392 y Fp(\000)f Fq(\026)p Fr(\))p Fq(\037)1191 1398 y Fi(R)1218 1392 y Fq(H)1253 1398 y Fl(0)1271 1392 y Fr(\()p Fq(b)p Fr(\))1321 1375 y Fj(\000)p Fl(1)p Fi(=)p Fl(2)1399 1392 y Fq(;)18 b(\026)12 b Fp(2)f Fm(R)m Fq(;)103 b Fr(\(3.6\))680 1473 y(~)672 1483 y Fq(T)17 b Fr(=)765 1473 y(~)757 1483 y Fq(T)781 1489 y Fi(b;R)845 1483 y Fr(:=)11 b Fq(H)935 1489 y Fl(0)953 1483 y Fr(\()p Fq(b)p Fr(\))1003 1466 y Fj(\000)p Fl(1)p Fi(=)p Fl(2)1081 1483 y Fq(\037)1107 1489 y Fi(R)1135 1483 y Fq(H)1170 1489 y Fl(0)1188 1483 y Fr(\()p Fq(b)p Fr(\))1238 1466 y Fj(\000)p Fl(1)p Fi(=)p Fl(2)1316 1483 y Fq(;)325 b Fr(\(3.7\))262 1559 y(so)13 b(that)h(w)o(e)h(ha)o(v)o(e)e Fq(T)6 b Fr(\()p Fq(\026)p Fr(\))12 b(=)g Fq(T)6 b Fr(\(0\))j Fp(\000)h Fq(\026)869 1549 y Fr(~)861 1559 y Fq(T)c Fr(.)262 1663 y Fn(Corollary)14 b(3.1)21 b Fo(Under)15 b(the)f(assumptions)i(of)f(L)n (emma)g(3.1)g(almost)f(sur)n(ely)674 1749 y Fp(D)706 1755 y Fi(b)723 1749 y Fr(\()p Fq(\026)p Fr(\))e(=)28 b(lim)836 1776 y Fi(R)p Fj(!1)941 1749 y Fq(R)973 1732 y Fj(\000)p Fl(3)1024 1749 y Fq(n)1049 1755 y Fj(\000)1077 1749 y Fr(\(1;)7 b Fq(T)1157 1755 y Fi(b;!)q(;R)1240 1749 y Fr(\()p Fq(\026)p Fr(\)\))p Fq(:)328 b Fr(\(3.8\))262 1876 y Fo(Pr)n(o)n(of)p Fr(:)12 b(It)i(su\016ces)i(to)d(recall)h (\(3.2\),)f(and)h(to)f(apply)g(the)i(Birman-Sc)o(h)o(winger)d (principle.)i Fp(})262 1971 y Fn(3.2.)g Fr(Let)h Fp(H)460 1977 y Fl(0)479 1971 y Fr(\()p Fq(b)p Fr(\))d(:=)598 1925 y Fk(\020)622 1971 y Fq(i)651 1955 y Fi(@)p 642 1962 39 2 v 642 1986 a(@)r(x)694 1971 y Fp(\000)741 1953 y Fi(by)p 741 1962 33 2 v 749 1986 a Fl(2)778 1925 y Fk(\021)803 1934 y Fl(2)831 1971 y Fr(+)873 1925 y Fk(\020)898 1971 y Fq(i)926 1955 y Fi(@)p 917 1962 38 2 v 917 1986 a(@)r(y)969 1971 y Fr(+)1015 1955 y Fi(bx)p 1015 1962 34 2 v 1024 1986 a Fl(2)1054 1925 y Fk(\021)1078 1934 y Fl(2)1111 1971 y Fr(b)q(e)j(the)h(selfadjoin)o(t)d(op)q(erator)i (de\014ned)262 2036 y(originally)8 b(on)j Fq(C)530 2021 y Fj(1)527 2047 y Fl(0)565 2036 y Fr(\()p Fm(R)614 2021 y Fl(2)630 2036 y Fr(\),)g(and)g(then)h(closed)g(in)f Fq(L)1033 2021 y Fl(2)1051 2036 y Fr(\()p Fm(R)1101 2021 y Fl(2)1116 2036 y Fr(\).)g(The)h(sp)q(ectrum)g(of)e Fp(H)1496 2042 y Fl(0)1515 2036 y Fr(\()p Fq(b)p Fr(\))h(coincides)262 2086 y(with)20 b(the)h(set)g(of)f(the)h(Landau)f(lev)o(els,)g(i.e.)g Fq(\033)q Fr(\()p Fp(H)1080 2092 y Fl(0)1099 2086 y Fr(\()p Fq(b)p Fr(\)\))j(=)1242 2055 y Fk(S)1277 2065 y Fj(1)1277 2099 y Fi(q)q Fl(=1)1337 2086 y Fp(f)p Fr(\(2)p Fq(q)15 b Fp(\000)f Fr(1\))p Fq(b)p Fp(g)p Fr(.)19 b(Fix)h Fq(q)k Fp(\025)262 2142 y Fr(1.)17 b(Denote)h(b)o(y)g Fq(p)541 2148 y Fi(q)578 2142 y Fr(=)h Fq(p)650 2148 y Fi(q)q(;b)711 2142 y Fr(:)f Fq(L)769 2127 y Fl(2)788 2142 y Fr(\()p Fm(R)837 2127 y Fl(2)853 2142 y Fr(\))h Fp(!)f Fq(L)976 2127 y Fl(2)995 2142 y Fr(\()p Fm(R)1044 2127 y Fl(2)1059 2142 y Fr(\))h(the)f(orthogonal)f(pro)r(jection)i(on)o(to)e(the)262 2192 y(eigenspace)12 b(of)f Fp(H)542 2198 y Fl(0)561 2192 y Fr(\()p Fq(b)p Fr(\))g(asso)q(ciated)i(with)e(the)h Fq(q)q Fr(th)g(Landau)f(lev)o(el)g(\(2)p Fq(q)6 b Fp(\000)t Fr(1\))p Fq(b)p Fr(.)k(In)i(other)g(w)o(ords,)262 2242 y Fq(p)283 2248 y Fi(q)301 2242 y Fq(w)g Fr(=)g Fq(w)j Fr(implies)c Fq(w)i Fp(2)e Fq(D)q Fr(\()p Fp(H)740 2248 y Fl(0)759 2242 y Fr(\()p Fq(b)p Fr(\)\))j(and)g Fp(H)955 2248 y Fl(0)974 2242 y Fr(\()p Fq(b)p Fr(\))p Fq(w)e Fr(=)g(\(2)p Fq(q)e Fp(\000)g Fr(1\))p Fq(bw)q Fr(.)j(It)h(is)f(w)o (ell-kno)o(wn)g(that)438 2351 y(\()p Fq(p)475 2357 y Fi(q)494 2351 y Fq(w)q Fr(\)\()p Fq(x;)7 b(y)q Fr(\))12 b(=)692 2294 y Fk(Z)715 2388 y Ff(R)739 2380 y Fe(2)762 2351 y Fp(P)791 2357 y Fi(q)809 2351 y Fr(\()p Fq(x;)7 b(y)q Fr(;)g Fq(x)932 2333 y Fj(0)943 2351 y Fq(;)g(y)983 2333 y Fj(0)995 2351 y Fr(\))p Fq(w)q Fr(\()p Fq(x)1082 2333 y Fj(0)1093 2351 y Fq(;)g(y)1133 2333 y Fj(0)1145 2351 y Fr(\))g Fq(dx)1214 2333 y Fj(0)1225 2351 y Fq(dy)1268 2333 y Fj(0)1280 2351 y Fq(;)48 b(w)13 b Fp(2)e Fq(L)1450 2333 y Fl(2)1469 2351 y Fr(\()p Fm(R)1518 2333 y Fl(2)1533 2351 y Fr(\))p Fq(;)p eop %%Page: 5 5 5 4 bop 262 116 a Fu(In)o(tegrated)13 b(Densit)o(y)i(of)d(States)i(in)g (Strong)f(Magnetic)i(Fields)538 b(5)262 224 y Fr(with)847 274 y Fp(P)876 280 y Fi(q)895 274 y Fr(\()p Fq(x;)7 b(y)q Fr(;)g Fq(x)1018 257 y Fj(0)1029 274 y Fq(;)g(y)1069 257 y Fj(0)1081 274 y Fr(\))k(:=)404 343 y Fq(b)p 390 362 46 2 v 390 400 a Fr(2)p Fq(\031)440 371 y(e)459 353 y Fj(\000)491 342 y Fd(b)p 491 347 15 2 v 491 363 a Fe(4)510 356 y Fr([)521 353 y Fl(\()p Fi(x)p Fj(\000)p Fi(x)598 341 y Fc(0)609 353 y Fl(\))622 341 y Fe(2)638 353 y Fl(+\()p Fi(y)q Fj(\000)p Fi(y)738 341 y Fc(0)750 353 y Fl(\))763 341 y Fe(2)779 353 y Fl(+2)p Fi(i)p Fl(\()p Fi(xy)883 341 y Fc(0)894 353 y Fj(\000)p Fi(y)q(x)957 341 y Fc(0)967 353 y Fl(\))980 356 y Fr(])994 371 y Fq(L)1022 377 y Fi(q)q Fj(\000)p Fl(1)1090 313 y Fk(\022)1126 343 y Fr(\()p Fq(x)e Fp(\000)g Fq(x)1240 328 y Fj(0)1251 343 y Fr(\))1267 328 y Fl(2)1295 343 y Fr(+)h(\()p Fq(y)h Fp(\000)f Fq(y)1447 328 y Fj(0)1459 343 y Fr(\))1475 328 y Fl(2)p 1126 362 369 2 v 1299 400 a Fr(2)1499 313 y Fk(\023)1653 371 y Fr(\(3.9\))262 472 y(where)k Fq(L)409 478 y Fi(s)427 472 y Fr(\()p Fq(\030)r Fr(\))e(:=)556 456 y Fl(1)p 551 463 26 2 v 551 487 a Fi(s)p Fl(!)581 472 y Fq(e)600 457 y Fi(\030)632 456 y(d)649 444 y Fd(s)p 624 463 50 2 v 624 487 a Fi(d\030)657 478 y Fd(s)685 439 y Fk(\000)704 472 y Fq(\030)724 457 y Fi(s)742 472 y Fq(e)761 457 y Fj(\000)p Fi(\030)806 439 y Fk(\001)825 472 y Fr(,)h Fq(s)f Fp(\025)f Fr(0,)i Fq(\030)h Fp(2)d Fm(R)m Fr(,)f(is)j(the)i (Laguerre)f(p)q(olynomial)c(of)j(order)262 522 y Fq(s)p Fr(.)g(Note)i(that)f(w)o(e)g(ha)o(v)o(e)626 616 y Fp(P)655 622 y Fi(q)673 616 y Fr(\()p Fq(x;)7 b(y)q Fr(;)g Fq(x;)g(y)q Fr(\))12 b(=)926 588 y Fq(b)p 912 606 46 2 v 912 644 a Fr(2)p Fq(\031)963 616 y(;)18 b Fr(\()p Fq(x;)7 b(y)q Fr(\))12 b Fp(2)f Fm(R)1173 599 y Fl(2)1189 616 y Fq(;)18 b Fp(8)p Fq(q)13 b Fp(2)e Fm(N)1345 622 y Fj(\003)1362 616 y Fq(:)258 b Fr(\(3.10\))262 722 y(De\014ne)16 b(the)g(orthogonal)e (pro)r(jection)h Fq(P)899 728 y Fi(q)931 722 y Fr(=)g Fq(P)1005 728 y Fi(q)q(;b)1061 722 y Fr(:)f Fq(L)1115 707 y Fl(2)1134 722 y Fr(\()p Fm(R)1183 707 y Fl(3)1198 722 y Fr(\))g Fp(!)g Fq(L)1312 707 y Fl(2)1331 722 y Fr(\()p Fm(R)1380 707 y Fl(3)1395 722 y Fr(\))i(b)o(y)f Fq(P)1513 728 y Fi(q)1531 722 y Fr(:=)1575 689 y Fk(R)1603 699 y Fj(\010)1594 737 y Ff(R)1637 722 y Fq(p)1658 728 y Fi(q)1684 722 y Fq(dz)r Fr(,)262 772 y(i.e.)406 833 y(\()p Fq(P)449 839 y Fi(q)468 833 y Fq(u)p Fr(\)\()p Fq(x;)7 b(y)q(;)g(z)r Fr(\))k(=)699 776 y Fk(Z)722 871 y Ff(R)746 862 y Fe(2)768 833 y Fp(P)797 839 y Fi(q)815 833 y Fr(\()p Fq(x;)c(y)q Fr(;)g Fq(x)938 816 y Fj(0)949 833 y Fq(;)g(y)989 816 y Fj(0)1001 833 y Fr(\))p Fq(u)p Fr(\()p Fq(x)1081 816 y Fj(0)1093 833 y Fq(;)g(y)1133 816 y Fj(0)1144 833 y Fq(;)g(z)r Fr(\))g Fq(dx)1253 816 y Fj(0)1264 833 y Fq(dy)1307 816 y Fj(0)1319 833 y Fq(;)48 b(u)11 b Fp(2)h Fq(L)1482 816 y Fl(2)1501 833 y Fr(\()p Fm(R)1550 816 y Fl(3)1565 833 y Fr(\))p Fq(:)262 931 y Fr(Then)i Fq(P)397 937 y Fi(q)429 931 y Fr(comm)o(utes)e(with)h Fq(H)753 937 y Fl(0)772 931 y Fr(\()p Fq(b)p Fr(\))g(and)930 914 y Fi(@)p 921 921 37 2 v 921 945 a(@)r(z)963 931 y Fr(,)g(and)h(w)o(e)g(ha)o(v)o(e)499 1044 y Fq(H)534 1050 y Fl(0)553 1044 y Fr(\()p Fq(b)p Fr(\))p Fq(P)630 1050 y Fi(q)648 1044 y Fq(u)d Fr(=)727 985 y Fk(\022)757 1044 y Fp(\000)805 1016 y Fq(@)829 1001 y Fl(2)p 795 1034 65 2 v 795 1072 a Fq(@)r(z)840 1060 y Fl(2)873 1044 y Fr(+)e(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)1077 985 y Fk(\023)1114 1044 y Fq(P)1141 1050 y Fi(q)1159 1044 y Fq(u;)48 b(u)11 b Fp(2)g Fq(D)q Fr(\()p Fq(H)1403 1050 y Fl(0)1422 1044 y Fr(\()p Fq(b)p Fr(\)\))p Fq(;)132 b Fr(\(3.11\))262 1151 y(\(see)15 b(\(1.1\)\).)e(F)m(or)g Fq(\015)h(>)e Fr(0)i(de\014ne)h(the)f(op)q(erator)759 1272 y Fq(r)q Fr(\()p Fq(\015)r Fr(\))f(:=)902 1213 y Fk(\022)932 1272 y Fp(\000)980 1244 y Fq(@)1004 1228 y Fl(2)p 970 1262 V 970 1300 a Fq(@)r(z)1015 1288 y Fl(2)1048 1272 y Fr(+)c Fq(\015)1112 1213 y Fk(\023)1144 1220 y Fj(\000)p Fl(1)p Fi(=)p Fl(2)1229 1272 y Fq(;)391 b Fr(\(3.12\))262 1382 y(b)q(ounded)14 b(and)g(selfadjoin)o(t)e(in)i Fq(L)788 1367 y Fl(2)807 1382 y Fr(\()p Fm(R)856 1367 y Fl(3)872 1382 y Fr(\).)f(Eviden)o(tly)m(,)356 1485 y(\()p Fq(r)q Fr(\()p Fq(\015)r Fr(\))447 1468 y Fl(2)466 1485 y Fq(u)p Fr(\)\()p Fq(x;)7 b(y)q(;)g(z)r Fr(\))12 b(=)732 1457 y(1)p 702 1476 80 2 v 702 1514 a(2)723 1488 y Fp(p)p 758 1488 24 2 v 26 x Fq(\015)794 1429 y Fk(Z)817 1523 y Ff(R)847 1485 y Fq(e)866 1468 y Fj(\000)892 1450 y(p)p 919 1450 20 2 v 18 x Fi(\015)r Fj(j)p Fi(z)q Fj(\000)p Fi(z)1008 1456 y Fc(0)1019 1468 y Fj(j)1031 1485 y Fq(u)p Fr(\()p Fq(x;)7 b(y)q(;)g(z)1175 1468 y Fj(0)1187 1485 y Fr(\))g Fq(dz)1253 1468 y Fj(0)1264 1485 y Fq(;)48 b(u)11 b Fp(2)g Fq(L)1426 1468 y Fl(2)1445 1485 y Fr(\()p Fm(R)1494 1468 y Fl(3)1510 1485 y Fr(\))p Fq(:)94 b Fr(\(3.13\))262 1596 y(Moreo)o(v)o(er,)13 b(the)i(op)q(erators)g Fq(P)738 1602 y Fi(q)770 1596 y Fr(and)e Fq(r)q Fr(\()p Fq(\015)r Fr(\))i(comm)o(ute.)262 1668 y Fn(3.3.)g Fr(In)g(this)g(subsection)i(w) o(e)e(estimate)g(a)g(quan)o(tit)o(y)f(whic)o(h)h(yields)g(the)h(main)d (asymptotic)262 1718 y(term)g(as)h Fq(b)d Fp(!)g(1)j Fr(of)f Fp(D)629 1724 y Fi(b)645 1718 y Fr(\()p Fq(\025)685 1724 y Fl(2)714 1718 y Fr(+)c(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)p Fr(\))f Fp(\000)g(D)1016 1724 y Fi(b)1033 1718 y Fr(\()p Fq(\025)1073 1724 y Fl(1)1101 1718 y Fr(+)h(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)p Fr(\).)262 1817 y Fn(Prop)q(ositi)o (on)j(3.2)20 b Fo(L)n(et)15 b(the)f(hyp)n(otheses)i(of)e(The)n(or)n(em) g(2.1)h(hold.)g(Then)g(for)f(every)g Fq(\025)e Fp(2)f Fm(R)m Fo(,)262 1867 y Fq(q)h Fp(2)f Fm(N)364 1873 y Fj(\003)381 1867 y Fo(,)k Fq(s)d(>)g Fr(0)p Fo(,)i(and)i Fq(\015)e(>)e Fr(0)i Fo(almost)h(sur)n(ely)312 1972 y Fr(lim)300 1999 y Fi(b)p Fj(!1)405 1972 y Fr(lim)388 1999 y Fi(R)p Fj(!1)515 1944 y Fr(1)p 491 1962 69 2 v 491 2000 a Fq(bR)541 1988 y Fl(3)564 1972 y Fq(n)589 1978 y Fj(\000)617 1972 y Fr(\()p Fq(s)p Fr(;)7 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))p Fq(P)773 1978 y Fi(q)792 1972 y Fr(\()p Fq(V)832 1978 y Fi(!)866 1972 y Fp(\000)j Fq(\025)f Fp(\000)g Fq(\015)r Fr(\))p Fq(\037)1047 1978 y Fi(R)1076 1972 y Fq(P)1103 1978 y Fi(q)1121 1972 y Fq(r)q Fr(\()p Fq(\015)r Fr(\)\))j(=)1286 1944 y(1)p 1273 1962 46 2 v 1273 2000 a(2)p Fq(\031)1324 1972 y(k)1354 1913 y Fk(\022)1389 1944 y Fq(\025)e Fr(+)f Fq(\015)p 1389 1962 99 2 v 1429 2000 a(s)1502 1972 y Fp(\000)h Fq(\015)r Fr(;)1591 1944 y(1)p 1591 1962 21 2 v 1592 2000 a Fq(s)1617 1972 y(V)1650 1913 y Fk(\023)1688 1972 y Fq(:)1632 2047 y Fr(\(3.14\))262 2145 y Fo(Ide)n(a)15 b(of)g(the)g(pr)n(o)n(of)p Fr(:)e(The)i(pro)q(of)e (is)h(analogous)f(to)h(the)h(one)f(of)f([K.R)o(,)h(Corollary)e(4.3])h (whic)o(h)262 2195 y(corresp)q(onds)19 b(to)f Fq(q)g Fr(=)h(1)e(and)g Fq(\015)k Fr(=)d(1.)f(By)h(\(3.9\),)e(\(3.10\),)h(and) g(\(3.13\),)f(the)j(extension)f(to)262 2245 y(general)c Fq(q)g Fr(and)g Fq(\015)j Fr(can)d(b)q(e)g(carried)h(out)f(in)f(a)h (quite)g(straigh)o(tforw)o(ard)f(manner.)g Fp(})262 2318 y Fr(Let)f Fq(\036)f Fr(:)h([)p Fq(b)424 2324 y Fl(0)442 2318 y Fq(;)7 b Fr(+)p Fp(1)p Fr(\))j Fp(!)h Fm(R)647 2324 y Fl(+)672 2318 y Fr(,)h Fq(b)714 2324 y Fl(0)744 2318 y Fq(>)f Fr(0,)h(b)q(e)h(a)f(strictly)g(decreasing)h(function)f (suc)o(h)h(that)g Fq(\036)p Fr(\()p Fq(b)1660 2324 y Fl(0)1678 2318 y Fr(\))f Fq(<)262 2368 y Fr(1)h(and)h(lim)434 2374 y Fi(b)p Fj(!1)524 2368 y Fq(b\036)p Fr(\()p Fq(b)p Fr(\))d(=)h(0.)p eop %%Page: 6 6 6 5 bop 262 116 a Fu(6)1341 b(Raik)o(o)o(v)262 224 y Fn(Corollary)14 b(3.2)21 b Fo(L)n(et)c(the)i(hyp)n(otheses)f(of)g(The)n (or)n(em)g(2.1)g(hold.)g(Then)g(for)g(every)g Fq(\025)f Fp(2)g Fm(R)m Fo(,)262 274 y Fq(q)12 b Fp(2)f Fm(N)364 280 y Fj(\003)381 274 y Fo(,)k(almost)g(sur)n(ely)349 385 y Fr(lim)337 412 y Fi(b)p Fj(!1)442 385 y Fr(lim)425 412 y Fi(R)p Fj(!1)523 385 y Fq(b)541 368 y Fj(\000)p Fl(1)586 385 y Fq(R)618 368 y Fj(\000)p Fl(3)662 385 y Fq(n)687 391 y Fj(\000)715 385 y Fr(\(1)9 b Fp(\006)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)924 391 y Fi(q)941 385 y Fq(T)f Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)p Fr(\))p Fq(P)1268 391 y Fi(q)1286 385 y Fr(\))h(=)1375 357 y(1)p 1362 375 46 2 v 1362 413 a(2)p Fq(\031)1413 385 y(k)q Fr(\()p Fq(\025)p Fr(;)c Fq(V)i Fr(\))p Fq(;)76 b Fr(\(3.15\))262 496 y Fo(the)14 b(op)n(er)n(ator)h Fq(T)6 b Fr(\()p Fq(\026)p Fr(\))15 b Fo(b)n(eing)g(de\014ne)n(d)h(in)f (\(3.6\).)262 610 y(Pr)n(o)n(of)p Fr(:)d(First)j(of)e(all)g(note)h (that)307 704 y Fq(P)334 710 y Fi(q)352 704 y Fq(T)6 b Fr(\()p Fq(\025)j Fr(+)h(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)p Fr(\))p Fq(P)679 710 y Fi(q)708 704 y Fr(=)i Fq(r)q Fr(\(\(2)p Fq(q)e Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fq(P)993 710 y Fi(q)1011 704 y Fr(\()p Fq(V)1051 710 y Fi(!)1085 704 y Fp(\000)g Fq(\025)h Fp(\000)f Fr(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1406 710 y Fi(R)1433 704 y Fq(P)1460 710 y Fi(q)1478 704 y Fq(r)q Fr(\(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)p Fr(\))262 798 y(\(see)15 b(\(3.6\),)d(\(3.11\))h(and)h(\(3.12\)\).)f(Then)h(the)h(Birman-Sc)o(h) o(winger)d(principle)i(en)o(tails)650 892 y Fq(n)675 898 y Fj(\000)703 892 y Fr(\(1)9 b Fp(\006)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)912 898 y Fi(q)929 892 y Fq(T)f Fr(\()p Fq(\025)j Fr(+)h(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)p Fr(\))p Fq(P)1256 898 y Fi(q)1273 892 y Fr(\)\))i(=)371 1017 y Fq(N)416 959 y Fk(\022)447 1017 y Fr(0;)7 b Fp(\000)534 989 y Fq(@)558 974 y Fl(2)p 523 1008 65 2 v 523 1046 a Fq(@)r(z)568 1034 y Fl(2)601 1017 y Fr(+)j(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)f Fr(+)g Fq(')883 1023 y Fj(\006)911 1017 y Fr(\()p Fq(b)p Fr(\))p Fq(P)988 1023 y Fi(q)1006 1017 y Fr(\()p Fq(V)1046 1023 y Fi(!)1080 1017 y Fp(\000)g Fq(\025)h Fp(\000)f Fr(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1401 1023 y Fi(R)1428 1017 y Fq(P)1455 1023 y Fi(q)1473 959 y Fk(\023)1511 1017 y Fq(;)109 b Fr(\(3.16\))262 1121 y(where)13 b Fq(')407 1127 y Fj(\006)435 1121 y Fr(\()p Fq(b)p Fr(\))f(:=)f(1)p Fq(=)p Fr(\(1)6 b Fp(\006)g Fq(\036)p Fr(\()p Fq(b)p Fr(\)\).)k(Note)j(that)f(lim)1032 1127 y Fi(b)p Fj(!1)1122 1121 y Fq(')1149 1127 y Fj(\006)1177 1121 y Fr(\()p Fq(b)p Fr(\))g(=)g(1,)f(lim)1385 1127 y Fi(b)p Fj(!1)1475 1121 y Fq(b)p Fr(\(1)6 b Fp(\000)g Fq(')1601 1127 y Fj(\006)1628 1121 y Fr(\()p Fq(b)p Fr(\)\))12 b(=)262 1170 y(0.)f(Let)i(us)h(re-arrange)f(the)g(terms)g(app)q(earing) f(at)g(the)i(righ)o(t-hand)e(side)h(of)f(\(3.16\).)f(W)m(e)h(ha)o(v)o (e)282 1291 y Fp(\000)330 1263 y Fq(@)354 1248 y Fl(2)p 320 1281 V 320 1319 a Fq(@)r(z)365 1307 y Fl(2)389 1291 y Fr(+\(2)p Fq(q)r Fp(\000)p Fr(1\))p Fq(b)p Fr(+)p Fq(')625 1297 y Fj(\006)653 1291 y Fr(\()p Fq(b)p Fr(\))p Fq(P)730 1297 y Fi(q)748 1291 y Fr(\()p Fq(V)788 1297 y Fi(!)813 1291 y Fp(\000)p Fq(\025)p Fp(\000)p Fr(\(2)p Fq(q)r Fp(\000)p Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1088 1297 y Fi(R)1116 1291 y Fq(P)1143 1297 y Fi(q)1172 1291 y Fr(=)g Fp(\000)1264 1263 y Fq(@)1288 1248 y Fl(2)p 1253 1281 V 1253 1319 a Fq(@)r(z)1298 1307 y Fl(2)1323 1291 y Fr(+)p Fq(\015)r Fr(+)p Fq(E)1441 1297 y Fi(q)q(;b)1485 1291 y Fr(+)p Fq(W)1562 1273 y Fj(\006)1556 1303 y Fi(q)q(;b)1600 1291 y Fq(;)20 b Fr(\(3.17\))262 1403 y(where)15 b Fq(\015)f(>)e Fr(0)i(is)f(a)h(\014xed)g(n)o(um)o(b)q(er,)f Fq(E)866 1409 y Fi(q)q(;b)920 1403 y Fr(:=)f(\(\(2)p Fq(q)e Fp(\000)g Fr(1\))p Fq(b)e Fp(\000)i Fq(\015)r Fr(\)\(1)g Fp(\000)g Fq(P)1360 1409 y Fi(q)1378 1403 y Fq(\037)1404 1409 y Fi(R)1431 1403 y Fq(P)1458 1409 y Fi(q)1476 1403 y Fr(\),)j(and)h Fq(W)1643 1385 y Fj(\006)1637 1416 y Fi(q)q(;b)1680 1403 y Fr(:)g(=)262 1453 y Fq(P)289 1459 y Fi(q)313 1453 y Fp(f)p Fq(V)358 1459 y Fi(!)392 1453 y Fp(\000)9 b Fq(\025)h Fp(\000)f Fq(\015)j Fp(\000)e Fr(\(1)f Fp(\000)g Fq(')697 1459 y Fj(\006)725 1453 y Fr(\()p Fq(b)p Fr(\)\)\()p Fq(V)831 1459 y Fi(!)865 1453 y Fp(\000)h Fq(\025)f Fp(\000)h Fr(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fp(g)e Fq(\037)1214 1459 y Fi(R)1241 1453 y Fq(P)1268 1459 y Fi(q)1286 1453 y Fr(.)16 b(Fix)g Fq(\016)j(>)d Fr(0,)g(and)g(assume)262 1503 y(that)d Fq(b)h Fr(is)g(so)g(large)f(that)h(w)o(e)g(ha)o(v)o(e)401 1597 y Fq(P)428 1603 y Fi(q)446 1597 y Fr(\()p Fq(V)486 1603 y Fi(!)519 1597 y Fp(\000)c Fq(\025)f Fp(\000)h Fq(\015)i Fp(\000)d Fq(\016)r Fr(\))p Fq(\037)772 1603 y Fi(R)800 1597 y Fq(P)827 1603 y Fi(q)856 1597 y Fp(\024)j Fq(W)945 1579 y Fj(\006)939 1609 y Fi(q)q(;b)993 1597 y Fp(\024)g Fq(P)1064 1603 y Fi(q)1082 1597 y Fr(\()p Fq(V)1122 1603 y Fi(!)1156 1597 y Fp(\000)d Fq(\025)h Fp(\000)f Fq(\015)j Fr(+)e Fq(\016)r Fr(\))p Fq(\037)1409 1603 y Fi(R)1436 1597 y Fq(P)1463 1603 y Fi(q)1481 1597 y Fq(:)139 b Fr(\(3.18\))262 1690 y(Assume)14 b(as)g(w)o(ell)f(that)i Fq(b)d(>)g(\015)r(=)p Fr(\(2)p Fq(q)f Fp(\000)f Fr(1\))k(so)g(that)g (the)h(op)q(erator)g Fq(E)1332 1696 y Fi(q)q(;b)1388 1690 y Fr(is)f(non-negativ)o(e.)g(By)262 1740 y(the)g(minim)o(ax)d (principle,)i(\(3.16\))g({)h(\(3.18\))f(imply)505 1861 y Fq(N)550 1802 y Fk(\022)581 1861 y Fr(0;)7 b Fp(\000)668 1832 y Fq(@)692 1817 y Fl(2)p 657 1851 V 657 1889 a Fq(@)r(z)702 1877 y Fl(2)736 1861 y Fr(+)i Fq(\015)j Fr(+)e Fq(E)883 1867 y Fi(q)q(;b)934 1861 y Fr(+)g Fq(P)1003 1867 y Fi(q)1021 1861 y Fr(\()p Fq(V)1061 1867 y Fi(!)1094 1861 y Fp(\000)g Fq(\025)f Fp(\000)h Fq(\015)i Fr(+)d Fq(\016)r Fr(\))p Fq(\037)1347 1867 y Fi(R)1375 1861 y Fq(P)1402 1867 y Fi(q)1420 1802 y Fk(\023)1462 1861 y Fp(\024)262 2014 y Fq(n)287 2020 y Fj(\000)314 2014 y Fr(\(1)g Fp(\006)f Fq(\036)p Fr(\()p Fq(b)p Fr(\);)f Fq(P)521 2020 y Fi(q)538 2014 y Fq(T)f Fr(\()p Fq(\025)j Fr(+)g(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fq(P)861 2020 y Fi(q)879 2014 y Fr(\))k Fp(\024)g Fq(N)5 b Fr(\(0;)i Fp(\000)1092 1986 y Fq(@)1116 1971 y Fl(2)p 1081 2004 V 1081 2042 a Fq(@)r(z)1126 2030 y Fl(2)1159 2014 y Fr(+)h Fq(\015)j Fr(+)e Fq(P)1299 2020 y Fi(q)1317 2014 y Fr(\()p Fq(V)1357 2020 y Fi(!)1390 2014 y Fp(\000)f Fq(\025)h Fp(\000)f Fq(\015)j Fp(\000)e Fq(\016)r Fr(\))p Fq(\037)1638 2020 y Fi(R)1665 2014 y Fq(P)1692 2020 y Fi(q)1710 2014 y Fr(\))p Fq(:)1632 2077 y Fr(\(3.19\))262 2127 y(By)14 b(the)g(Birman-Sc)o(h)o(winger)f (principle,)567 2245 y Fq(N)612 2187 y Fk(\022)643 2245 y Fr(0;)7 b Fp(\000)730 2217 y Fq(@)754 2202 y Fl(2)p 719 2236 V 719 2274 a Fq(@)r(z)764 2262 y Fl(2)798 2245 y Fr(+)i Fq(\015)j Fr(+)e Fq(P)941 2251 y Fi(q)959 2245 y Fr(\()p Fq(V)999 2251 y Fi(!)1032 2245 y Fp(\000)g Fq(\025)f Fp(\000)h Fq(\015)i Fp(\000)d Fq(\016)r Fr(\))p Fq(\037)1285 2251 y Fi(R)1313 2245 y Fq(P)1340 2251 y Fi(q)1358 2187 y Fk(\023)1400 2245 y Fr(=)634 2368 y Fq(n)659 2374 y Fj(\000)687 2368 y Fr(\(1;)e Fq(r)q Fr(\()p Fq(\015)r Fr(\))p Fq(P)845 2374 y Fi(q)863 2368 y Fr(\()p Fq(V)903 2374 y Fi(!)937 2368 y Fp(\000)i Fq(\025)h Fp(\000)f Fq(\015)j Fp(\000)e Fq(\016)r Fr(\))p Fq(\037)1190 2374 y Fi(R)1217 2368 y Fq(P)1244 2374 y Fi(q)1262 2368 y Fq(r)q Fr(\()p Fq(\015)r Fr(\)\))p Fq(:)267 b Fr(\(3.20\))p eop %%Page: 7 7 7 6 bop 262 116 a Fu(In)o(tegrated)13 b(Densit)o(y)i(of)d(States)i(in)g (Strong)f(Magnetic)i(Fields)538 b(7)262 224 y Fr(It)13 b(follo)o(ws)g(from)f(\(3.14\))h(with)g Fq(s)f Fr(=)g(1)i(that)304 330 y(lim)292 357 y Fi(b)p Fj(!1)397 330 y Fr(lim)380 357 y Fi(R)p Fj(!1)507 302 y Fr(1)p 483 320 69 2 v 483 359 a Fq(bR)533 347 y Fl(3)556 330 y Fq(n)581 336 y Fj(\000)609 330 y Fr(\(1;)7 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))p Fq(P)767 336 y Fi(q)785 330 y Fr(\()p Fq(V)825 336 y Fi(!)859 330 y Fp(\000)j Fq(\025)f Fp(\000)h Fq(\015)i Fp(\000)d Fq(\016)r Fr(\))p Fq(\037)1112 336 y Fi(R)1139 330 y Fq(P)1166 336 y Fi(q)1184 330 y Fq(r)q Fr(\()p Fq(\015)r Fr(\)\))k(=)1349 302 y(1)p 1337 320 46 2 v 1337 359 a(2)p Fq(\031)1388 330 y(k)q Fr(\()p Fq(\025)c Fr(+)h Fq(\016)r Fr(;)d Fq(V)i Fr(\))p Fq(:)30 b Fr(\(3.21\))262 433 y(The)14 b(second)h(inequalit)o(y)d(in)i(\(3.19\),)e(and)i(\(3.20\))f(-)h (\(3.21\))f(imply)e(that)j(for)f(an)o(y)h Fq(\016)f(>)f Fr(0)291 539 y(lim)7 b(sup)314 574 y Fi(b)p Fj(!1)425 539 y Fr(lim)g(sup)443 574 y Fi(R)p Fj(!1)588 511 y Fr(1)p 564 529 69 2 v 564 567 a Fq(bR)614 555 y Fl(3)638 539 y Fq(n)663 545 y Fj(\000)691 539 y Fr(\(1)i Fp(\006)g Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)899 545 y Fi(q)916 539 y Fq(T)f Fr(\()p Fq(\025)k Fr(+)g(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fq(P)1243 545 y Fi(q)1261 539 y Fr(\))j Fp(\024)1350 511 y Fr(1)p 1337 529 46 2 v 1337 567 a(2)p Fq(\031)1388 539 y(k)q Fr(\()p Fq(\025)e Fr(+)f Fq(\016)r Fr(;)e Fq(V)i Fr(\))p Fq(:)30 b Fr(\(3.22\))262 650 y(No)o(w)10 b(assume)g(that)g Fq(\015)15 b(>)d Fr(0)e(is)g(so)h (large)f(that)h(almost)d(surely)j Fq(V)1233 656 y Fi(!)1258 650 y Fr(\()p Fn(x)p Fr(\))s Fp(\000)s Fq(\025)s Fp(\000)s Fq(\015)5 b Fr(+)s Fq(\016)13 b Fp(\024)f Fr(0)e(for)g(ev)o(ery)262 700 y Fn(x)k Fp(2)f Fm(R)375 685 y Fl(3)391 700 y Fr(.)h(Then)i(w)o(e)g (ha)o(v)o(e)f Fq(P)714 706 y Fi(q)732 700 y Fr(\()p Fq(V)772 706 y Fi(!)806 700 y Fp(\000)10 b Fq(\025)h Fp(\000)f Fq(\015)j Fr(+)e Fq(\016)r Fr(\))p Fq(\037)1064 706 y Fi(R)1091 700 y Fq(P)1118 706 y Fi(q)1150 700 y Fr(=)j Fp(\000)p Fq(S)1255 685 y Fj(\003)1275 700 y Fq(S)k Fr(with)d Fq(S)i Fr(:=)c Fq(U)5 b(\037)1572 706 y Fi(R)1599 700 y Fq(P)1626 706 y Fi(q)1644 700 y Fr(,)15 b(and)262 775 y Fq(U)23 b Fr(:=)375 744 y Fp(p)p 410 744 269 2 v 31 x Fq(\015)12 b Fr(+)e Fq(\025)f Fp(\000)h Fq(V)584 781 y Fi(!)617 775 y Fp(\000)g Fq(\016)r Fr(.)17 b(In)o(tro)q(duce)i(the)g (op)q(erator)i(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))20 b(:=)1305 729 y Fk(\020)1330 775 y Fp(\000)1376 759 y Fi(@)1396 746 y Fe(2)p 1367 766 53 2 v 1367 790 a Fi(@)r(z)1404 781 y Fe(2)1434 775 y Fr(+)9 b Fq(\015)k Fr(+)c Fq(E)1581 781 y Fi(q)q(;b)1623 729 y Fk(\021)1648 738 y Fj(\000)p Fl(1)p Fi(=)p Fl(2)1726 775 y Fr(,)262 840 y(b)q(ounded)14 b(and)g(self-adjoin)o(t)e(in)i Fq(L)802 825 y Fl(2)821 840 y Fr(\()p Fm(R)870 825 y Fl(3)885 840 y Fr(\).)g(By)g(the)g (Birman-Sc)o(h)o(winger)f(principle,)302 953 y Fq(N)347 895 y Fk(\022)378 953 y Fr(0;)7 b Fp(\000)465 925 y Fq(@)489 910 y Fl(2)p 454 944 65 2 v 454 982 a Fq(@)r(z)499 970 y Fl(2)532 953 y Fr(+)j Fq(\015)i Fr(+)e Fq(E)680 959 y Fi(q)q(;b)731 953 y Fr(+)g Fq(P)800 959 y Fi(q)818 953 y Fr(\()p Fq(V)858 959 y Fi(!)891 953 y Fp(\000)g Fq(\025)f Fp(\000)h Fq(\015)i Fr(+)d Fq(\016)r Fr(\))p Fq(\037)1144 959 y Fi(R)1172 953 y Fq(P)1199 959 y Fi(q)1217 895 y Fk(\023)1259 953 y Fr(=)j Fq(n)1328 959 y Fl(+)1355 953 y Fr(\(1;)c(~)-22 b Fq(r)p Fr(\()p Fq(\015)r Fr(\))p Fq(S)1512 936 y Fj(\003)1533 953 y Fq(S)t Fr(~)f Fq(r)r Fr(\()p Fq(\015)r Fr(\)\))13 b(=)563 1069 y Fq(n)588 1075 y Fj(\003)607 1069 y Fr(\(1;)7 b Fq(S)t Fr(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\)\))13 b(=)f Fq(n)863 1075 y Fj(\003)882 1069 y Fr(\(1;)c(~)-22 b Fq(r)p Fr(\()p Fq(\015)r Fr(\))p Fq(S)1039 1052 y Fj(\003)1060 1069 y Fr(\))12 b(=)f Fq(n)1156 1075 y Fl(+)1184 1069 y Fr(\(1;)c Fq(S)t Fr(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))1342 1052 y Fl(2)1362 1069 y Fq(S)1389 1052 y Fj(\003)1409 1069 y Fr(\))p Fq(:)195 b Fr(\(3.23\))262 1141 y(Applying)12 b(the)j(resolv)o(en)o(t)f(iden)o(tit)o(y)i(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))916 1126 y Fl(2)947 1141 y Fr(=)12 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))1066 1126 y Fl(2)1094 1141 y Fp(\000)e Fq(r)q Fr(\()p Fq(\015)r Fr(\))1211 1126 y Fl(2)1231 1141 y Fq(E)1262 1147 y Fi(q)q(;b)1306 1141 y Fr(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))1379 1126 y Fl(2)1399 1141 y Fr(,)13 b(w)o(e)h(get)308 1227 y Fq(n)333 1233 y Fl(+)361 1227 y Fr(\(1;)7 b Fq(S)t Fr(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))519 1210 y Fl(2)539 1227 y Fq(S)566 1210 y Fj(\003)585 1227 y Fr(\))12 b Fp(\025)g Fq(n)682 1233 y Fl(+)709 1227 y Fr(\(1)e(+)f Fq(")p Fr(;)e Fq(S)r(r)q Fr(\()p Fq(\015)r Fr(\))937 1210 y Fl(2)957 1227 y Fq(S)984 1210 y Fj(\003)1004 1227 y Fr(\))j Fp(\000)f Fq(n)1096 1233 y Fl(+)1124 1227 y Fr(\()p Fq(")p Fr(;)e Fq(S)r(r)q Fr(\()p Fq(\015)r Fr(\))1280 1210 y Fl(2)1300 1227 y Fq(E)1331 1233 y Fi(q)q(;b)1375 1227 y Fr(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))1448 1210 y Fl(2)1468 1227 y Fq(S)1495 1210 y Fj(\003)1515 1227 y Fr(\))p Fq(;)18 b Fp(8)p Fq(")12 b(>)g Fr(0)p Fq(:)1632 1277 y Fr(\(3.24\))262 1327 y(Let)i(us)g(estimate)f(the)i (\014rst)g(term)e(at)h(the)g(righ)o(t-hand)f(side)i(of)e(\(3.24\).)f(W) m(e)i(ha)o(v)o(e)558 1413 y Fq(n)583 1419 y Fl(+)611 1413 y Fr(\(1)9 b(+)g Fq(")p Fr(;)e Fq(S)r(r)q Fr(\()p Fq(\015)r Fr(\))838 1396 y Fl(2)858 1413 y Fq(S)885 1396 y Fj(\003)905 1413 y Fr(\))12 b(=)g Fq(n)1002 1419 y Fl(+)1029 1413 y Fr(\(1)d(+)h Fq(")p Fr(;)d Fq(r)q Fr(\()p Fq(\015)r Fr(\))p Fq(S)1257 1396 y Fj(\003)1278 1413 y Fq(S)r(r)q Fr(\()p Fq(\015)r Fr(\)\))13 b(=)524 1499 y Fq(n)549 1505 y Fj(\000)577 1499 y Fr(\(1)d(+)f Fq(")p Fr(;)e Fq(r)q Fr(\()p Fq(\015)r Fr(\))p Fq(P)805 1505 y Fi(q)824 1499 y Fr(\()p Fq(V)864 1505 y Fi(!)897 1499 y Fp(\000)j Fq(\025)f Fp(\000)h Fq(\015)i Fr(+)d Fq(\016)r Fr(\))p Fq(\037)1150 1505 y Fi(R)1178 1499 y Fq(P)1205 1505 y Fi(q)1223 1499 y Fq(r)q Fr(\()p Fq(\015)r Fr(\)\))p Fq(;)19 b Fp(8)p Fq(")12 b(>)g Fr(0)p Fq(:)156 b Fr(\(3.25\))262 1571 y(By)14 b(\(3.14\))f(with)g Fq(s)f Fr(=)g(1)d(+)g Fq(")p Fr(,)454 1657 y(lim)442 1685 y Fi(b)p Fj(!1)547 1657 y Fr(lim)530 1684 y Fi(R)p Fj(!1)628 1657 y Fq(b)646 1640 y Fj(\000)p Fl(1)690 1657 y Fq(R)722 1640 y Fj(\000)p Fl(3)767 1657 y Fq(n)792 1663 y Fj(\000)820 1657 y Fr(\(1)g(+)g Fq(")p Fr(;)e Fq(r)q Fr(\()p Fq(\015)r Fr(\))p Fq(P)1047 1663 y Fi(q)1066 1657 y Fr(\()p Fq(V)1106 1663 y Fi(!)1140 1657 y Fp(\000)i Fq(\025)h Fp(\000)f Fq(\015)j Fr(+)e Fq(\016)r Fr(\))p Fq(\037)1393 1663 y Fi(R)1420 1657 y Fq(P)1447 1663 y Fi(q)1465 1657 y Fq(r)q Fr(\()p Fq(\015)r Fr(\)\))709 1790 y(=)770 1761 y(1)p 757 1780 46 2 v 757 1818 a(2)p Fq(\031)808 1790 y(k)838 1731 y Fk(\022)874 1761 y Fq(\025)f Fr(+)h Fq(\015)i Fp(\000)d Fq(\016)p 874 1780 170 2 v 913 1818 a Fr(1)g(+)g Fq(")1057 1790 y Fp(\000)h Fq(\015)r Fr(;)1175 1761 y Fq(V)p 1146 1780 91 2 v 1146 1818 a Fr(1)f(+)h Fq(")1242 1731 y Fk(\023)1279 1790 y Fq(:)341 b Fr(\(3.26\))262 1886 y(Let)14 b(us)g(no)o(w)g (estimate)f(the)h(second)h(term)f(at)f(the)i(righ)o(t-hand)e(side)h(of) g(\(3.24\).)e(W)m(e)i(ha)o(v)o(e)356 1973 y Fq(S)r(r)q Fr(\()p Fq(\015)r Fr(\))458 1955 y Fl(2)478 1973 y Fq(E)509 1979 y Fi(q)q(;b)554 1973 y Fr(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))627 1955 y Fl(2)646 1973 y Fq(S)673 1955 y Fj(\003)705 1973 y Fr(=)12 b(\(\(2)p Fq(q)e Fp(\000)f Fr(1\))p Fq(b)g Fp(\000)h Fq(\015)r Fr(\))p Fq(U)5 b(\037)1076 1979 y Fi(R)1104 1973 y Fq(P)1131 1979 y Fi(q)1149 1973 y Fq(r)q Fr(\()p Fq(\015)r Fr(\))1224 1955 y Fl(2)1243 1973 y Fr(\(1)10 b Fp(\000)f Fq(P)1358 1979 y Fi(q)1376 1973 y Fq(\037)1402 1979 y Fi(R)1429 1973 y Fq(P)1456 1979 y Fi(q)1474 1973 y Fr(\))r(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))1565 1955 y Fl(2)1585 1973 y Fq(S)1612 1955 y Fj(\003)1632 1973 y Fq(:)262 2059 y Fr(The)14 b(op)q(erators)h(\(\(2)p Fq(q)c Fp(\000)f Fr(1\))p Fq(b)f Fp(\000)h Fq(\015)r Fr(\))p Fq(U)20 b Fr(and)c(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))1005 2044 y Fl(2)1024 2059 y Fq(S)1051 2044 y Fj(\003)1085 2059 y Fr(are)15 b(uniformly)d(b)q(ounded)j(with)f(resp)q(ect)262 2109 y(to)f Fq(R)p Fr(.)g(Therefore,)411 2195 y Fq(n)436 2201 y Fl(+)463 2195 y Fr(\()p Fq(")p Fr(;)7 b Fq(S)r(r)q Fr(\()p Fq(\015)r Fr(\))619 2178 y Fl(2)640 2195 y Fq(E)671 2201 y Fi(q)q(;b)715 2195 y Fr(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))788 2178 y Fl(2)808 2195 y Fq(S)835 2178 y Fj(\003)855 2195 y Fr(\))11 b Fp(\024)h Fq(n)951 2201 y Fj(\003)970 2195 y Fr(\()p Fq(\021)q Fr(;)7 b Fq(\037)1053 2201 y Fi(R)1080 2195 y Fq(P)1107 2201 y Fi(q)1125 2195 y Fq(r)q Fr(\()p Fq(\015)r Fr(\))1200 2178 y Fl(2)1220 2195 y Fr(\(1)i Fp(\000)g Fq(P)1334 2201 y Fi(q)1352 2195 y Fq(\037)1378 2201 y Fi(R)1405 2195 y Fq(P)1432 2201 y Fi(q)1450 2195 y Fr(\)\))150 b(\(3.27\))262 2281 y(where)15 b Fq(\021)d(>)g Fr(0)i(is)f(indep)q(enden)o(t)j(of)d Fq(R)p Fr(.)g(F)m(urther,)334 2368 y Fq(n)359 2374 y Fj(\003)378 2368 y Fr(\()p Fq(\021)q Fr(;)7 b Fq(\037)461 2374 y Fi(R)488 2368 y Fq(P)515 2374 y Fi(q)533 2368 y Fq(r)q Fr(\()p Fq(\015)r Fr(\))608 2351 y Fl(2)628 2368 y Fr(\(1)i Fp(\000)g Fq(P)742 2374 y Fi(q)760 2368 y Fq(\037)786 2374 y Fi(R)814 2368 y Fq(P)841 2374 y Fi(q)859 2368 y Fr(\)\))i Fp(\024)h Fq(\021)968 2351 y Fj(\000)p Fl(2)1013 2368 y Fp(k)p Fq(\037)1060 2374 y Fi(R)1087 2368 y Fq(P)1114 2374 y Fi(q)1132 2368 y Fq(r)q Fr(\()p Fq(\015)r Fr(\))1207 2351 y Fl(2)1226 2368 y Fr(\(1)d Fp(\000)h Fq(P)1341 2374 y Fi(q)1359 2368 y Fq(\037)1385 2374 y Fi(R)1412 2368 y Fq(P)1439 2374 y Fi(q)1457 2368 y Fr(\))p Fp(k)1494 2351 y Fl(2)1494 2378 y Fi(H)r(S)1548 2368 y Fq(;)72 b Fr(\(3.28\))p eop %%Page: 8 8 8 7 bop 262 116 a Fu(8)1341 b(Raik)o(o)o(v)262 224 y Fp(k)p Fq(:)p Fp(k)316 230 y Fi(H)r(S)382 224 y Fr(b)q(eing)14 b(the)g(Hilb)q(ert-Sc)o(hmidt)f(norm.)f(A)i(straigh)o(tforw)o(ard)f (calculation)g(yields)618 310 y(lim)601 336 y Fi(R)p Fj(!1)699 310 y Fq(R)731 293 y Fj(\000)p Fl(3)776 310 y Fp(k)p Fq(\037)823 316 y Fi(R)850 310 y Fq(P)877 316 y Fi(q)895 310 y Fq(r)q Fr(\()p Fq(\015)r Fr(\))970 293 y Fl(2)989 310 y Fr(\(1)d Fp(\000)f Fq(P)1104 316 y Fi(q)1122 310 y Fq(\037)1148 316 y Fi(R)1175 310 y Fq(P)1202 316 y Fi(q)1220 310 y Fr(\))p Fp(k)1257 293 y Fl(2)1257 320 y Fi(H)r(S)1322 310 y Fr(=)j(0)p Fq(:)233 b Fr(\(3.29\))262 412 y(Putting)13 b(together)i(\(3.27\){\(3.29\),)c(w)o(e)k(obtain)581 497 y(lim)564 524 y Fi(R)p Fj(!1)663 497 y Fq(R)695 480 y Fj(\000)p Fl(3)739 497 y Fq(n)764 503 y Fl(+)791 497 y Fr(\()p Fq(")p Fr(;)7 b Fq(S)r(r)q Fr(\()p Fq(\015)r Fr(\))947 480 y Fl(2)968 497 y Fq(E)999 503 y Fi(q)q(;b)1043 497 y Fr(~)-23 b Fq(r)q Fr(\()p Fq(\015)r Fr(\))1116 480 y Fl(2)1136 497 y Fq(S)1163 480 y Fj(\003)1183 497 y Fr(\))11 b(=)h(0)p Fq(;)18 b Fp(8)p Fq(")12 b(>)g Fr(0)p Fq(:)196 b Fr(\(3.30\))262 599 y(Com)o(bining)11 b(the)j(\014rst)h (inequalit)o(y)d(in)i(\(3.19\))f(with)g(\(3.23\){\(3.26\))f(and)i (\(3.30\),)e(w)o(e)i(get)467 684 y(lim)7 b(inf)484 711 y Fi(b)p Fj(!1)589 684 y Fr(lim)g(inf)600 711 y Fi(R)p Fj(!1)711 684 y Fq(b)729 667 y Fj(\000)p Fl(1)773 684 y Fq(R)805 667 y Fj(\000)p Fl(3)849 684 y Fq(n)874 690 y Fj(\000)902 684 y Fr(\(1)i Fp(\006)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)1111 690 y Fi(q)1128 684 y Fq(T)f Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)p Fr(\))p Fq(P)1455 690 y Fi(q)1473 684 y Fr(\))i Fp(\025)748 788 y Fr(1)p 735 806 46 2 v 735 844 a(2)p Fq(\031)786 816 y(k)816 757 y Fk(\022)852 788 y Fq(\025)d Fr(+)h Fq(\015)i Fp(\000)d Fq(\016)p 852 806 170 2 v 891 844 a Fr(1)g(+)h Fq(")1035 816 y Fp(\000)g Fq(\015)r Fr(;)1153 788 y Fq(V)p 1124 806 91 2 v 1124 844 a Fr(1)f(+)h Fq(")1220 757 y Fk(\023)1257 816 y Fq(:)363 b Fr(\(3.31\))262 912 y(Letting)11 b Fq(")h Fp(#)f Fr(0)g(in)g(\(3.31\),)f(and)i(then)g Fq(\016)h Fp(#)f Fr(0)f(in)g(\(3.22\))f(and)i(\(3.31\),)e(and)h(taking) g(in)o(to)f(accoun)o(t)262 962 y(the)k(b)q(oundedness)i(of)d Fq(V)648 968 y Fi(!)686 962 y Fr(and)h(the)g(con)o(tin)o(uit)o(y)f(of)h Fq(k)q Fr(,)f(w)o(e)h(arriv)o(e)g(at)g(\(3.15\).)e Fp(})262 1036 y Fn(3.4.)i Fr(In)h(this)g(subsection)h(w)o(e)f(estimate)f (certain)h(quan)o(tities)g(whic)o(h)f(do)h(not)g(con)o(tribute)g(to)262 1085 y(the)f(main)e(asymptotic)g(term)i(as)g Fq(b)d Fp(!)g(1)i Fr(of)h Fp(D)1018 1091 y Fi(b)1034 1085 y Fr(\()p Fq(\025)1074 1091 y Fl(2)1102 1085 y Fr(+)c(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)p Fr(\))f Fp(\000)g(D)1405 1091 y Fi(b)1422 1085 y Fr(\()p Fq(\025)1462 1091 y Fl(1)1490 1085 y Fr(+)h(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\).)262 1140 y(In)o(tro)q(duce)19 b(the)g(orthogonal)e(pro)r(jections)i Fq(P)993 1125 y Fj(\000)987 1150 y Fi(q)1040 1140 y Fr(:=)1103 1108 y Fk(P)1146 1119 y Fi(q)q Fj(\000)p Fl(1)1146 1152 y Fi(s)p Fl(=1)1214 1140 y Fq(P)1241 1146 y Fi(s)1259 1140 y Fr(,)e Fq(q)j(>)g Fr(1,)1439 1129 y(~)1429 1140 y Fq(P)1462 1125 y Fj(\000)1456 1150 y Fi(q)1509 1140 y Fr(:=)1571 1108 y Fk(P)1615 1119 y Fi(q)1615 1152 y(s)p Fl(=1)1682 1140 y Fq(P)1709 1146 y Fi(s)1726 1140 y Fr(,)262 1189 y Fq(q)12 b Fp(\025)g Fr(1,)h(and)h Fq(P)497 1174 y Fl(+)491 1200 y Fi(q)535 1189 y Fr(:=)591 1158 y Fk(P)634 1169 y Fj(1)634 1202 y Fi(s)p Fl(=)p Fi(q)q Fl(+1)743 1189 y Fq(P)770 1195 y Fi(s)787 1189 y Fr(,)g Fq(q)e Fp(\025)g Fr(1.)262 1292 y Fn(Prop)q(ositi)o(on)h(3.3)20 b Fo(Assume)f(that)g (\(1.3\))g(holds.)g(L)n(et)g Fq(\026)1181 1298 y Fj(\000)1209 1292 y Fq(;)7 b(\026)1253 1298 y Fl(+)1299 1292 y Fp(2)18 b Fm(R)m Fo(,)e Fq(\026)1432 1298 y Fj(\000)1479 1292 y Fq(<)j(\026)1555 1298 y Fl(+)1583 1292 y Fo(,)f Fq(q)i(>)f Fr(1)p Fo(.)262 1342 y(Then)c(almost)f(sur)n(ely)447 1427 y Fr(lim)435 1454 y Fi(b)p Fj(!1)523 1427 y Fr(lim)7 b(sup)541 1462 y Fi(R)p Fj(!1)657 1427 y Fq(b)675 1410 y Fj(\000)p Fl(1)719 1427 y Fq(R)751 1410 y Fj(\000)p Fl(3)796 1427 y Fr(\()p Fq(n)837 1433 y Fj(\000)865 1427 y Fr(\(1)i Fp(\000)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)1080 1410 y Fj(\000)1074 1438 y Fi(q)1106 1427 y Fq(T)f Fr(\()p Fq(\026)1177 1433 y Fl(+)1214 1427 y Fr(+)k(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1460 1433 y Fi(R)1487 1427 y Fq(P)1520 1410 y Fj(\000)1514 1438 y Fi(q)1548 1427 y Fr(\))563 1541 y Fp(\000)p Fq(n)620 1547 y Fj(\000)649 1541 y Fr(\(1)g(+)g Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)863 1524 y Fj(\000)857 1552 y Fi(q)890 1541 y Fq(T)f Fr(\()p Fq(\026)961 1547 y Fj(\000)998 1541 y Fr(+)k(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1245 1547 y Fi(R)1272 1541 y Fq(P)1305 1524 y Fj(\000)1299 1552 y Fi(q)1332 1541 y Fr(\))i(=)g(0)p Fq(:)195 b Fr(\(3.32\))262 1644 y Fo(Pr)n(o)n(of)p Fr(.)12 b(Assume)i(that)g Fq(b)g Fr(is)g(so)g(large)f(that)h(\(2)p Fq(q)c Fp(\000)g Fr(1\))p Fq(b)f Fr(+)g Fq(\026)1181 1650 y Fj(\006)1218 1644 y Fp(\006)h Fq(c)1278 1650 y Fl(0)1308 1644 y Fq(>)i Fr(0,)h Fq(c)1416 1650 y Fl(0)1448 1644 y Fr(b)q(eing)h(de\014ned)h(in)262 1694 y(\(1.3\).)d(Set)j Fq( )471 1700 y Fj(\006)499 1694 y Fr(\()p Fq(b)p Fr(\))c(:=)h(\(1)d Fp(\007)g Fq(\036)p Fr(\()p Fq(b)p Fr(\)\))p Fq(=)p Fr(\(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)f Fr(+)g Fq(\026)1069 1700 y Fj(\006)1107 1694 y Fp(\006)g Fq(c)1166 1700 y Fl(0)1185 1694 y Fr(\).)k(Eviden)o(tly)m(,)358 1779 y Fq(n)383 1785 y Fj(\000)411 1779 y Fr(\(1)c Fp(\000)g Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)625 1762 y Fj(\000)619 1789 y Fi(q)652 1779 y Fq(T)f Fr(\()p Fq(\026)723 1785 y Fl(+)760 1779 y Fr(+)k(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1006 1785 y Fi(R)1033 1779 y Fq(P)1066 1762 y Fj(\000)1060 1789 y Fi(q)1094 1779 y Fr(\))i Fp(\024)h Fq(n)1190 1785 y Fl(+)1218 1779 y Fr(\()p Fq( )1261 1785 y Fl(+)1288 1779 y Fr(\()p Fq(b)p Fr(\);)7 b Fq(P)1390 1762 y Fj(\000)1384 1789 y Fi(q)1425 1769 y Fr(~)1417 1779 y Fq(T)f(P)1480 1762 y Fj(\000)1474 1789 y Fi(q)1508 1779 y Fr(\))p Fq(;)96 b Fr(\(3.33\))357 1873 y Fq(n)382 1879 y Fj(\000)410 1873 y Fr(\(1)9 b(+)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)625 1856 y Fj(\000)619 1883 y Fi(q)652 1873 y Fq(T)f Fr(\()p Fq(\026)723 1879 y Fj(\000)760 1873 y Fr(+)k(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1006 1879 y Fi(R)1033 1873 y Fq(P)1066 1856 y Fj(\000)1060 1883 y Fi(q)1094 1873 y Fr(\))i Fp(\025)h Fq(n)1190 1879 y Fl(+)1218 1873 y Fr(\()p Fq( )1261 1879 y Fj(\000)1289 1873 y Fr(\()p Fq(b)p Fr(\);)7 b Fq(P)1391 1856 y Fj(\000)1385 1883 y Fi(q)1426 1863 y Fr(~)1418 1873 y Fq(T)f(P)1481 1856 y Fj(\000)1475 1883 y Fi(q)1508 1873 y Fr(\))p Fq(;)96 b Fr(\(3.34\))270 1942 y(~)262 1953 y Fq(T)19 b Fr(b)q(eing)14 b(de\014ned)h(in)f(\(3.7\).)e(Ob)o (viously)m(,)h Fq(P)956 1938 y Fj(\000)950 1963 y Fi(q)991 1942 y Fr(~)983 1953 y Fq(T)6 b(P)1046 1938 y Fj(\000)1040 1963 y Fi(q)1087 1953 y Fr(is)14 b(a)g(trace-class)h(op)q(erator,)f (and)429 2085 y(lim)412 2112 y Fi(R)p Fj(!1)510 2085 y Fq(R)542 2068 y Fj(\000)p Fl(3)586 2085 y Fr(T)m(r)e(\()p Fq(P)690 2068 y Fj(\000)684 2096 y Fi(q)725 2075 y Fr(~)717 2085 y Fq(T)6 b(P)780 2068 y Fj(\000)774 2096 y Fi(q)808 2085 y Fr(\))824 2068 y Fi(l)848 2085 y Fr(=)937 2057 y Fq(b)p 897 2076 97 2 v 897 2114 a Fr(\(2)p Fq(\031)q Fr(\))975 2102 y Fl(2)1006 2031 y Fi(q)q Fj(\000)p Fl(1)1006 2046 y Fk(X)1007 2134 y Fi(s)p Fl(=1)1073 2029 y Fk(Z)1096 2123 y Ff(R)1258 2057 y Fq(d\020)p 1131 2076 298 2 v 1131 2114 a Fr(\()p Fq(\020)1168 2102 y Fl(2)1196 2114 y Fr(+)j(\(2)p Fq(s)h Fp(\000)f Fr(1\))p Fq(b)p Fr(\))1415 2102 y Fi(l)1433 2085 y Fq(;)18 b(l)13 b Fp(2)e Fm(N)1559 2091 y Fj(\003)1576 2085 y Fq(:)262 2209 y Fr(Applying)h(the)j (Kac-Murdo)q(c)o(k-Szeg\177)-21 b(o)15 b(theorem)e(\(see)j([R)o(,)d (Lemma)e(3.2]\),)h(w)o(e)j(get)322 2336 y(lim)305 2363 y Fi(R)p Fj(!1)404 2336 y Fq(R)436 2319 y Fj(\000)p Fl(3)480 2336 y Fq(n)505 2342 y Fl(+)533 2336 y Fr(\()p Fq(")p Fr(;)7 b Fq(P)620 2319 y Fj(\000)614 2346 y Fi(q)655 2325 y Fr(~)647 2336 y Fq(T)f(P)710 2319 y Fj(\000)704 2346 y Fi(q)737 2336 y Fr(\))12 b(=)853 2308 y Fq(b)p 814 2326 97 2 v 814 2364 a Fr(\(2)p Fq(\031)q Fr(\))892 2352 y Fl(2)923 2282 y Fi(q)q Fj(\000)p Fl(1)923 2296 y Fk(X)924 2384 y Fi(s)p Fl(=1)990 2336 y Fr(meas)e Fp(f)p Fq(\020)15 b Fp(2)c Fm(R)m Fp(j)o Fr(\()p Fq(\020)1265 2319 y Fl(2)1291 2336 y Fr(+)e(\(2)p Fq(s)h Fp(\000)f Fr(1\))p Fq(b)p Fr(\))1510 2319 y Fj(\000)p Fl(1)1566 2336 y Fq(>)j(")p Fp(g)g Fr(=)p eop %%Page: 9 9 9 8 bop 262 116 a Fu(In)o(tegrated)13 b(Densit)o(y)i(of)d(States)i(in)g (Strong)f(Magnetic)i(Fields)538 b(9)619 231 y Fq(b)p 596 250 65 2 v 596 288 a Fr(2)p Fq(\031)642 276 y Fl(2)673 206 y Fi(q)q Fj(\000)p Fl(1)672 220 y Fk(X)674 308 y Fi(s)p Fl(=1)739 226 y Fk(\000)758 259 y Fq(")777 242 y Fj(\000)p Fl(1)831 259 y Fp(\000)10 b Fr(\(2)p Fq(s)g Fp(\000)f Fr(1\))p Fq(b)1035 226 y Fk(\001)1054 234 y Fl(1)p Fi(=)p Fl(2)1054 276 y(+)1113 259 y Fq(;)18 b Fp(8)p Fq(")12 b(>)g Fr(0)p Fq(;)7 b Fp(8)p Fq(b)j(>)i Fr(0)p Fq(:)223 b Fr(\(3.35\))262 368 y(Com)o(bining)11 b(\(3.33\){\(3.35\),)g(w)o(e)j(get)510 453 y(lim)7 b(sup)528 488 y Fi(R)p Fj(!1)644 453 y Fq(R)676 436 y Fj(\000)p Fl(3)721 453 y Fr(\()p Fq(n)762 459 y Fj(\000)790 453 y Fr(\(1)i Fp(\000)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)1005 436 y Fj(\000)999 463 y Fi(q)1031 453 y Fq(T)f Fr(\()p Fq(\026)1102 459 y Fl(+)1139 453 y Fr(+)k(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1385 459 y Fi(R)1412 453 y Fq(P)1445 436 y Fj(\000)1439 463 y Fi(q)1473 453 y Fr(\))577 567 y Fp(\000)p Fq(n)634 573 y Fj(\000)663 567 y Fr(\(1)g(+)g Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)877 549 y Fj(\000)871 577 y Fi(q)904 567 y Fq(T)f Fr(\()p Fq(\026)975 573 y Fj(\000)1012 567 y Fr(+)k(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1258 573 y Fi(R)1286 567 y Fq(P)1319 549 y Fj(\000)1313 577 y Fi(q)1346 567 y Fr(\)\))j Fp(\024)392 656 y Fq(b)p 369 675 V 369 713 a Fr(2)p Fq(\031)415 701 y Fl(2)446 631 y Fi(q)q Fj(\000)p Fl(1)445 645 y Fk(X)446 733 y Fi(s)p Fl(=1)512 638 y Fk(n)540 684 y Fr(\()p Fq( )583 690 y Fl(+)610 684 y Fr(\()p Fq(b)p Fr(\))660 667 y Fj(\000)p Fl(1)714 684 y Fp(\000)e Fr(\(2)p Fq(s)f Fp(\000)h Fr(1\))p Fq(b)p Fr(\))934 663 y Fl(1)p Fi(=)p Fl(2)934 695 y(+)995 684 y Fp(\000)g Fr(\()p Fq( )1080 690 y Fj(\000)1108 684 y Fr(\()p Fq(b)p Fr(\))1158 667 y Fj(\000)p Fl(1)1212 684 y Fp(\000)f Fr(\(2)p Fq(s)h Fp(\000)f Fr(1\))p Fq(b)p Fr(\))1431 663 y Fl(1)p Fi(=)p Fl(2)1431 695 y(+)1483 638 y Fk(o)1518 684 y Fq(:)102 b Fr(\(3.36\))262 791 y(Rationalizing,)10 b(w)o(e)k(\014nd)g(that)g(for)g(eac)o(h)g Fq(s)e Fr(=)g(1)p Fq(;)7 b(:)g(:)g(:)e(;)i(q)i Fp(\000)h Fr(1,)j(w)o(e)h(ha)o(v)o(e)371 886 y(lim)359 913 y Fi(b)p Fj(!1)447 840 y Fk(n)475 886 y Fr(\()p Fq( )518 892 y Fl(+)546 886 y Fr(\()p Fq(b)p Fr(\))596 869 y Fj(\000)p Fl(1)649 886 y Fp(\000)c Fr(\(2)p Fq(s)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\))869 865 y Fl(1)p Fi(=)p Fl(2)869 896 y(+)930 886 y Fp(\000)h Fr(\()p Fq( )1015 892 y Fj(\000)1043 886 y Fr(\()p Fq(b)p Fr(\))1093 869 y Fj(\000)p Fl(1)1147 886 y Fp(\000)f Fr(\(2)p Fq(s)h Fp(\000)g Fr(1\))p Fq(b)p Fr(\))1367 865 y Fl(1)p Fi(=)p Fl(2)1367 896 y(+)1419 840 y Fk(o)1458 886 y Fr(=)i(0)p Fq(:)97 b Fr(\(3.37\))262 988 y(No)o(w,)13 b(\(3.32\))g(follo)o(ws)f(from)g(\(3.36\))h(and)h (\(3.37\).)e Fp(})262 1097 y Fn(Prop)q(ositi)o(on)h(3.4)20 b Fo(Assume)e(that)f(\(1.3\))h(holds.)g(Fix)f Fq(\025)g Fp(2)f Fm(R)m Fo(.)e(Then)k(her)n(e)f(exists)g Fq(b)1633 1103 y Fj(\003)1669 1097 y Fq(>)f Fr(0)262 1146 y Fo(indep)n(endent)g (of)f Fq(R)p Fo(,)f(such)h(that)g Fq(b)d(>)f(b)870 1152 y Fj(\003)904 1146 y Fo(implies)506 1231 y Fq(n)531 1237 y Fj(\000)559 1231 y Fr(\(1)e Fp(\006)h Fq(\036)p Fr(\()p Fq(b)p Fr(\)\);)d Fq(P)790 1214 y Fl(+)784 1242 y Fi(q)816 1231 y Fq(T)f Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)p Fr(\))p Fq(\037)1142 1237 y Fi(R)1169 1231 y Fq(P)1202 1214 y Fl(+)1196 1242 y Fi(q)1229 1231 y Fr(\))h(=)h(0)p Fq(;)18 b Fp(8)p Fq(R)11 b(>)h Fr(0)p Fq(:)138 b Fr(\(3.38\))262 1340 y Fo(Pr)n(o)n(of)p Fr(:)14 b(It)i(su\016ces)h(to)f(note)g(that)g(lim)871 1346 y Fi(b)p Fj(!1)954 1340 y Fr(\(1)10 b Fp(\006)h Fq(\036)p Fr(\()p Fq(b)p Fr(\)\))k(=)g(1,)g(and)g(lim)7 b(sup)1454 1350 y Fi(b)p Fj(!1)1544 1340 y Fp(k)p Fq(P)1598 1325 y Fl(+)1592 1351 y Fi(q)1625 1340 y Fq(T)f Fr(\()p Fq(\025)11 b Fr(+)262 1402 y(\(2)p Fq(q)f Fp(\000)f Fr(1\))p Fq(b)p Fr(\))p Fq(\037)466 1408 y Fi(R)493 1402 y Fq(P)526 1387 y Fl(+)520 1413 y Fi(q)553 1402 y Fp(k)14 b(\024)g Fr(lim)692 1408 y Fi(b)p Fj(!1)786 1382 y Fi(c)801 1386 y Fe(0)817 1382 y Fl(+)p Fj(j)p Fi(\025)p Fj(j)p Fl(+\(2)p Fi(q)q Fj(\000)p Fl(1\))p Fi(b)p 786 1393 238 2 v 847 1417 a Fl(\(2)p Fi(q)q Fl(+1\))p Fi(b)1042 1402 y Fr(=)1094 1384 y Fl(2)p Fi(q)q Fj(\000)p Fl(1)p 1094 1393 76 2 v 1094 1417 a(2)p Fi(q)q Fl(+1)1185 1402 y Fq(<)e Fr(1.)h Fp(})262 1516 y Fn(Prop)q(ositi)o(on)g(3.5)20 b Fo(F)m(or)15 b(e)n(ach)g Fq(")d(>)g Fr(0)j Fo(and)g Fq(q)e(<)f Fr(1)i Fo(we)h(have)599 1601 y Fr(lim)582 1628 y Fi(R)p Fj(!1)680 1601 y Fq(R)712 1584 y Fj(\000)p Fl(3)757 1601 y Fq(n)782 1607 y Fj(\006)810 1601 y Fr(\()p Fq(")p Fr(;)7 b(2Re)k(\()p Fq(P)994 1584 y Fj(\000)988 1611 y Fi(q)1029 1590 y Fr(~)1021 1601 y Fq(T)6 b(P)1078 1607 y Fi(q)1106 1601 y Fr(+)1156 1590 y(~)1147 1601 y Fq(P)1180 1584 y Fj(\000)1174 1611 y Fi(q)1216 1590 y Fr(~)1208 1601 y Fq(T)g(P)1271 1584 y Fl(+)1265 1611 y Fi(q)1298 1601 y Fr(\)\))11 b(=)h(0)p Fq(:)214 b Fr(\(3.39\))262 1733 y Fo(Pr)n(o)n(of)p Fr(:)13 b(W)m(rite)h(the)h(estimates)g Fq(n)781 1739 y Fj(\006)809 1733 y Fr(\()p Fq(")p Fr(;)7 b(2Re)o(\()p Fq(P)981 1718 y Fj(\000)975 1744 y Fi(q)1017 1723 y Fr(~)1009 1733 y Fq(T)f(P)1066 1739 y Fi(q)1093 1733 y Fr(+)1145 1723 y(~)1135 1733 y Fq(P)1168 1718 y Fj(\000)1162 1744 y Fi(q)1204 1723 y Fr(~)1196 1733 y Fq(T)g(P)1259 1718 y Fl(+)1253 1744 y Fi(q)1286 1733 y Fr(\)\))15 b Fp(\024)f Fr(2)p Fq(n)1425 1739 y Fj(\003)1444 1733 y Fr(\()p Fq("=)p Fr(2;)7 b Fq(P)1573 1718 y Fj(\000)1567 1744 y Fi(q)1608 1723 y Fr(~)1600 1733 y Fq(T)f(P)1657 1739 y Fi(q)1675 1733 y Fr(\))15 b(+)262 1801 y(2)p Fq(n)308 1807 y Fj(\003)326 1801 y Fr(\()p Fq("=)p Fr(2;)431 1791 y(~)422 1801 y Fq(P)455 1786 y Fj(\000)449 1811 y Fi(q)490 1791 y Fr(~)482 1801 y Fq(T)6 b(P)545 1786 y Fl(+)539 1811 y Fi(q)572 1801 y Fr(\))21 b Fp(\024)f Fr(8)p Fq(")701 1786 y Fj(\000)p Fl(2)753 1755 y Fk(\020)778 1801 y Fp(k)p Fq(P)832 1786 y Fj(\000)826 1811 y Fi(q)867 1791 y Fr(~)859 1801 y Fq(T)6 b(P)916 1807 y Fi(q)934 1801 y Fp(k)955 1786 y Fl(2)955 1813 y Fi(H)r(S)1017 1801 y Fr(+)k Fp(k)1089 1791 y Fr(~)1080 1801 y Fq(P)1113 1786 y Fj(\000)1107 1811 y Fi(q)1148 1791 y Fr(~)1140 1801 y Fq(T)c(P)1203 1786 y Fl(+)1197 1811 y Fi(q)1230 1801 y Fp(k)1251 1786 y Fl(2)1251 1813 y Fi(H)r(S)1304 1755 y Fk(\021)1329 1801 y Fr(,)20 b(and)g(v)o(erify)g(b)o(y)g(direct)262 1870 y(calculation)12 b(that)i(lim)619 1876 y Fi(R)p Fj(!1)719 1870 y Fq(R)751 1855 y Fj(\000)p Fl(3)796 1870 y Fp(k)p Fq(P)850 1855 y Fj(\000)844 1881 y Fi(q)885 1860 y Fr(~)877 1870 y Fq(T)6 b(P)934 1876 y Fi(q)952 1870 y Fp(k)973 1855 y Fl(2)973 1882 y Fi(H)r(S)1040 1870 y Fr(=)14 b(lim)1144 1876 y Fi(R)p Fj(!1)1244 1870 y Fq(R)1276 1855 y Fj(\000)p Fl(3)1320 1870 y Fp(k)1351 1860 y Fr(~)1341 1870 y Fq(P)1374 1855 y Fj(\000)1368 1881 y Fi(q)1410 1860 y Fr(~)1402 1870 y Fq(T)6 b(P)1465 1855 y Fl(+)1459 1881 y Fi(q)1492 1870 y Fp(k)1513 1855 y Fl(2)1513 1882 y Fi(H)r(S)1580 1870 y Fr(=)14 b(0.)f Fp(})262 2022 y Fs(4)56 b(Pro)r(of)18 b(of)h(Theorem)e(2.1)262 2141 y Fr(Fix)12 b Fq(q)h(>)f Fr(1,)h Fq(\025)481 2147 y Fl(1)499 2141 y Fq(;)7 b(\025)542 2147 y Fl(2)572 2141 y Fp(2)k Fm(R)m Fr(,)f Fq(\025)690 2147 y Fl(1)721 2141 y Fq(<)i(\025)789 2147 y Fl(2)807 2141 y Fr(.)h(In)h(order)g(to)f(pro)o (v)o(e)h(\(2.2\),)e(it)h(su\016ces)i(to)e(sho)o(w)h(that)f(for)262 2191 y(eac)o(h)h(sequence)i Fp(f)p Fq(b)566 2197 y Fi(j)583 2191 y Fp(g)604 2197 y Fi(j)r Fj(\025)p Fl(1)678 2191 y Fr(suc)o(h)e(that)g Fq(b)879 2197 y Fi(j)908 2191 y Fp(!)d(1)i Fr(as)h Fq(j)g Fp(!)d(1)p Fr(,)i(w)o(e)h(ha)o(v)o(e)311 2296 y(lim)299 2323 y Fi(j)r Fj(!1)387 2296 y Fq(b)405 2279 y Fj(\000)p Fl(1)405 2308 y Fi(j)456 2263 y Fk(\000)475 2296 y Fp(D)507 2302 y Fi(b)522 2306 y Fd(j)540 2296 y Fr(\()p Fq(\025)580 2302 y Fl(2)608 2296 y Fr(+)9 b(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)812 2302 y Fi(j)829 2296 y Fr(\))h Fp(\000)f(D)928 2302 y Fi(b)943 2306 y Fd(j)960 2296 y Fr(\()p Fq(\025)1000 2302 y Fl(1)1029 2296 y Fr(+)g(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)1233 2302 y Fi(j)1250 2296 y Fr(\))1266 2263 y Fk(\001)1297 2296 y Fr(=)1358 2268 y(1)p 1345 2287 46 2 v 1345 2325 a(2)p Fq(\031)1403 2296 y Fr(\()p Fq(k)q Fr(\()p Fq(\025)1482 2302 y Fl(2)1501 2296 y Fr(\))g Fp(\000)f Fq(k)q Fr(\()p Fq(\025)1631 2302 y Fl(1)1650 2296 y Fr(\)\))e Fq(:)1653 2368 y Fr(\(4.1\))p eop %%Page: 10 10 10 9 bop 262 116 a Fu(10)1322 b(Raik)o(o)o(v)262 224 y Fr(Fix)19 b(four)g(sequences)j Fp(f)p Fq(\025)673 206 y Fj(\006)673 237 y Fi(l;m)725 224 y Fp(g)746 230 y Fi(m)p Fj(\025)p Fl(1)820 224 y Fr(,)d Fq(l)j Fr(=)g(1)p Fq(;)7 b Fr(2,)18 b(suc)o(h)i(that)g Fq(\025)1249 206 y Fj(\000)1249 237 y Fi(l;m)1322 224 y Fq(<)i(\025)1400 230 y Fi(l)1434 224 y Fq(<)f(\025)1511 206 y Fl(+)1511 237 y Fi(l;m)1564 224 y Fr(,)e Fq(m)i Fp(\025)h Fr(1,)262 282 y(lim)319 288 y Fi(m)p Fj(!1)424 282 y Fq(\025)448 264 y Fj(\006)448 295 y Fi(l;m)512 282 y Fr(=)11 b Fq(\025)579 288 y Fi(l)592 282 y Fr(,)h(and)g Fq(\025)719 264 y Fj(\006)719 295 y Fi(l;m)777 282 y Fr(+)6 b(\(2)p Fq(q)h Fp(\000)f Fr(1\))p Fq(b)971 288 y Fi(j)1001 282 y Fr(are)12 b(con)o(tin)o(uit)o(y)g(p)q (oin)o(ts)g(of)f Fp(D)1463 288 y Fi(b)1478 292 y Fd(j)1508 282 y Fr(for)g(all)g Fq(m)h Fp(\025)g Fr(1)262 332 y(and)h Fq(j)h Fp(\025)e Fr(1,)h Fq(l)g Fr(=)f(1)p Fq(;)7 b Fr(2.)12 b(Then)i(b)o(y)g(Corollary)f(3.1)g(w)o(e)h(ha)o(v)o(e)462 419 y(lim)7 b(sup)485 454 y Fi(j)r Fj(!1)596 419 y Fq(b)614 402 y Fj(\000)p Fl(1)614 431 y Fi(j)665 386 y Fk(\000)684 419 y Fp(D)716 425 y Fi(b)731 429 y Fd(j)748 419 y Fr(\()p Fq(\025)788 425 y Fl(2)817 419 y Fr(+)i(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)1021 425 y Fi(j)1038 419 y Fr(\))f Fp(\000)h(D)1137 425 y Fi(b)1152 429 y Fd(j)1169 419 y Fr(\()p Fq(\025)1209 425 y Fl(1)1237 419 y Fr(+)g(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)1441 425 y Fi(j)1459 419 y Fr(\))1475 386 y Fk(\001)1505 419 y Fp(\024)267 542 y Fr(lim)e(sup)290 576 y Fi(j)r Fj(!1)418 542 y Fr(lim)401 568 y Fi(R)p Fj(!1)500 542 y Fq(b)518 524 y Fj(\000)p Fl(1)518 553 y Fi(j)562 542 y Fq(R)594 524 y Fj(\000)p Fl(3)638 542 y Fr(\()p Fq(n)679 548 y Fj(\000)708 542 y Fr(\(1;)g Fq(T)f Fr(\()p Fq(\025)834 524 y Fl(+)834 553 y(2)p Fi(;m)900 542 y Fr(+)k(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)1105 548 y Fi(j)1122 542 y Fr(\)\))f Fp(\000)h Fq(n)1230 548 y Fj(\000)1258 542 y Fr(\(1;)d Fq(T)f Fr(\()p Fq(\025)1384 524 y Fj(\000)1384 553 y Fl(1)p Fi(;m)1451 542 y Fr(+)j(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)1655 548 y Fi(j)1672 542 y Fr(\)\)\))p Fq(;)1653 621 y Fr(\(4.2\))468 671 y(lim)d(inf)485 697 y Fi(j)r Fj(!1)590 671 y Fq(b)608 653 y Fj(\000)p Fl(1)608 682 y Fi(j)659 637 y Fk(\000)678 671 y Fp(D)710 677 y Fi(b)725 681 y Fd(j)742 671 y Fr(\()p Fq(\025)782 677 y Fl(2)811 671 y Fr(+)i(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)1015 677 y Fi(j)1032 671 y Fr(\))f Fp(\000)h(D)1131 677 y Fi(b)1146 681 y Fd(j)1163 671 y Fr(\()p Fq(\025)1203 677 y Fl(1)1231 671 y Fr(+)g(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)1435 677 y Fi(j)1453 671 y Fr(\))1469 637 y Fk(\001)1499 671 y Fp(\025)273 770 y Fr(lim)e(inf)290 796 y Fi(j)r Fj(!1)412 770 y Fr(lim)395 797 y Fi(R)p Fj(!1)494 770 y Fq(b)512 752 y Fj(\000)p Fl(1)512 781 y Fi(j)556 770 y Fq(R)588 753 y Fj(\000)p Fl(3)632 770 y Fr(\()p Fq(n)673 776 y Fj(\000)702 770 y Fr(\(1;)g Fq(T)f Fr(\()p Fq(\025)828 752 y Fj(\000)828 781 y Fl(2)p Fi(;m)894 770 y Fr(+)k(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)1099 776 y Fi(j)1116 770 y Fr(\)\))f Fp(\000)h Fq(n)1224 776 y Fj(\000)1252 770 y Fr(\(1;)d Fq(T)f Fr(\()p Fq(\025)1378 752 y Fl(+)1378 781 y(1)p Fi(;m)1445 770 y Fr(+)j(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)1649 776 y Fi(j)1666 770 y Fr(\)\)\))p Fq(:)1653 841 y Fr(\(4.3\))262 891 y(F)m(urther,)k(note)g(that)g(the)g(elemen)o(tary)g(op)q(erator)g (inequalities)547 978 y Fq(T)6 b Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)p Fr(\))h Fp(\025)h Fq(P)929 984 y Fi(q)947 978 y Fq(T)6 b Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)f Fr(+)g(2)p Fq(c)1320 984 y Fl(0)1339 978 y Fq(\016)r Fr(\))p Fq(P)1402 984 y Fi(q)1420 978 y Fr(+)282 1066 y Fq(P)315 1049 y Fj(\000)309 1076 y Fi(q)343 1066 y Fq(T)d Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)f Fr(+)g Fq(c)695 1072 y Fl(0)714 1066 y Fr(\(1)g(+)g Fq(\016)821 1049 y Fj(\000)p Fl(1)866 1066 y Fr(\)\))p Fq(P)931 1049 y Fj(\000)925 1076 y Fi(q)968 1066 y Fr(+)g Fq(P)1042 1049 y Fl(+)1036 1076 y Fi(q)1069 1066 y Fq(T)d Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)g Fr(+)h Fq(c)1422 1072 y Fl(0)1440 1066 y Fr(\(1)f(+)h Fq(\016)1548 1049 y Fj(\000)p Fl(1)1592 1066 y Fr(\)\))p Fq(P)1657 1049 y Fl(+)1651 1076 y Fi(q)1685 1066 y Fp(\000)640 1146 y Fr(2\()p Fq(\025)f Fr(+)h(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\)Re\()p Fq(P)1028 1129 y Fl(+)1022 1157 y Fi(q)1063 1136 y Fr(~)1055 1146 y Fq(T)1095 1136 y Fr(~)1085 1146 y Fq(P)1118 1129 y Fj(\000)1112 1157 y Fi(q)1155 1146 y Fr(+)g Fq(P)1229 1129 y Fj(\000)1223 1157 y Fi(q)1265 1136 y Fr(~)1257 1146 y Fq(T)d(P)1314 1152 y Fi(q)1332 1146 y Fr(\))p Fq(;)547 1219 y(T)g Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)p Fr(\))h Fp(\024)h Fq(P)929 1225 y Fi(q)947 1219 y Fq(T)6 b Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)f Fp(\000)g Fr(2)p Fq(c)1320 1225 y Fl(0)1339 1219 y Fq(\016)r Fr(\))p Fq(P)1402 1225 y Fi(q)1420 1219 y Fr(+)282 1291 y Fq(P)315 1274 y Fj(\000)309 1301 y Fi(q)343 1291 y Fq(T)d Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)f Fp(\000)g Fq(c)695 1297 y Fl(0)714 1291 y Fr(\(1)g(+)g Fq(\016)821 1274 y Fj(\000)p Fl(1)866 1291 y Fr(\)\))p Fq(P)931 1274 y Fj(\000)925 1301 y Fi(q)968 1291 y Fr(+)g Fq(P)1042 1274 y Fl(+)1036 1301 y Fi(q)1069 1291 y Fq(T)d Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)g Fp(\000)h Fq(c)1422 1297 y Fl(0)1440 1291 y Fr(\(1)f(+)h Fq(\016)1548 1274 y Fj(\000)p Fl(1)1592 1291 y Fr(\)\))p Fq(P)1657 1274 y Fl(+)1651 1301 y Fi(q)1685 1291 y Fp(\000)640 1372 y Fr(2\()p Fq(\025)f Fr(+)h(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\)Re\()p Fq(P)1028 1355 y Fl(+)1022 1382 y Fi(q)1063 1361 y Fr(~)1055 1372 y Fq(T)1095 1361 y Fr(~)1085 1372 y Fq(P)1118 1355 y Fj(\000)1112 1382 y Fi(q)1155 1372 y Fr(+)g Fq(P)1229 1355 y Fj(\000)1223 1382 y Fi(q)1265 1361 y Fr(~)1257 1372 y Fq(T)d(P)1314 1378 y Fi(q)1332 1372 y Fr(\))p Fq(;)262 1444 y Fr(are)14 b(v)n(alid)e(for)i(eac)o(h)g Fq(\025)e Fp(2)f Fm(R)m Fr(,)f Fq(b)i(>)g Fr(0,)h(and)g Fq(\016)h(>)e Fr(0.)h(Therefore,)360 1532 y Fq(n)385 1538 y Fj(\000)413 1532 y Fr(\(1;)7 b Fq(T)f Fr(\()p Fq(\025)j Fr(+)g(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)p Fr(\)\))j Fp(\024)f Fq(n)864 1538 y Fj(\000)892 1532 y Fr(\(1)e Fp(\000)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)1101 1538 y Fi(q)1118 1532 y Fq(T)f Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)g Fr(+)g(2)p Fq(c)1491 1538 y Fl(0)1510 1532 y Fq(\016)r Fr(\))p Fq(P)1573 1538 y Fi(q)1591 1532 y Fr(\)+)520 1619 y Fq(n)545 1625 y Fj(\000)573 1619 y Fr(\(1)h Fp(\000)f Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)788 1602 y Fj(\000)782 1629 y Fi(q)815 1619 y Fq(T)f Fr(\()p Fq(\025)j Fr(+)h(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)e Fr(+)i Fq(c)1167 1625 y Fl(0)1186 1619 y Fr(\(1)f(+)g Fq(\016)1293 1602 y Fj(\000)p Fl(1)1338 1619 y Fr(\)\))p Fq(P)1403 1602 y Fj(\000)1397 1629 y Fi(q)1431 1619 y Fr(\)+)521 1698 y Fq(n)546 1704 y Fj(\000)574 1698 y Fr(\(1)g Fp(\000)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)789 1680 y Fl(+)783 1708 y Fi(q)815 1698 y Fq(T)f Fr(\()p Fq(\025)j Fr(+)h(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)e Fr(+)i Fq(c)1167 1704 y Fl(0)1186 1698 y Fr(\(1)f(+)g Fq(\016)1293 1680 y Fj(\000)p Fl(1)1338 1698 y Fr(\)\))p Fq(P)1403 1680 y Fl(+)1397 1708 y Fi(q)1430 1698 y Fr(\)+)551 1778 y Fq(n)576 1784 y Fl(+)603 1778 y Fr(\()p Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e(2\()p Fq(\025)i Fr(+)h(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\)Re\()p Fq(P)1101 1761 y Fl(+)1095 1789 y Fi(q)1136 1768 y Fr(~)1128 1778 y Fq(T)1167 1768 y Fr(~)1158 1778 y Fq(P)1191 1761 y Fj(\000)1185 1789 y Fi(q)1228 1778 y Fr(+)g Fq(P)1302 1761 y Fj(\000)1296 1789 y Fi(q)1338 1768 y Fr(~)1330 1778 y Fq(T)d(P)1387 1784 y Fi(q)1405 1778 y Fr(\)\))p Fq(;)262 1851 y Fr(and)360 1938 y Fq(n)385 1944 y Fj(\000)413 1938 y Fr(\(1;)h Fq(T)f Fr(\()p Fq(\025)j Fr(+)g(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)p Fr(\)\))j Fp(\025)f Fq(n)864 1944 y Fj(\000)892 1938 y Fr(\(1)e(+)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)1101 1944 y Fi(q)1118 1938 y Fq(T)f Fr(\()p Fq(\025)k Fr(+)f(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)g Fp(\000)g Fr(2)p Fq(c)1491 1944 y Fl(0)1510 1938 y Fq(\016)r Fr(\))p Fq(P)1573 1944 y Fi(q)1591 1938 y Fr(\)+)520 2025 y Fq(n)545 2031 y Fj(\000)573 2025 y Fr(\(1)h(+)f Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)788 2008 y Fj(\000)782 2036 y Fi(q)815 2025 y Fq(T)f Fr(\()p Fq(\025)j Fr(+)h(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)e Fp(\000)i Fq(c)1167 2031 y Fl(0)1186 2025 y Fr(\(1)f(+)g Fq(\016)1293 2008 y Fj(\000)p Fl(1)1338 2025 y Fr(\)\))p Fq(P)1403 2008 y Fj(\000)1397 2036 y Fi(q)1431 2025 y Fr(\)+)521 2104 y Fq(n)546 2110 y Fj(\000)574 2104 y Fr(\(1)g(+)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)789 2087 y Fl(+)783 2114 y Fi(q)815 2104 y Fq(T)f Fr(\()p Fq(\025)j Fr(+)h(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)e Fp(\000)i Fq(c)1167 2110 y Fl(0)1186 2104 y Fr(\(1)f(+)g Fq(\016)1293 2087 y Fj(\000)p Fl(1)1338 2104 y Fr(\)\))p Fq(P)1403 2087 y Fl(+)1397 2114 y Fi(q)1430 2104 y Fr(\))p Fp(\000)551 2185 y Fq(n)576 2191 y Fj(\000)603 2185 y Fr(\()p Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e(2\()p Fq(\025)i Fr(+)h(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\)Re\()p Fq(P)1101 2168 y Fl(+)1095 2195 y Fi(q)1136 2174 y Fr(~)1128 2185 y Fq(T)1168 2174 y Fr(~)1158 2185 y Fq(P)1191 2168 y Fj(\000)1185 2195 y Fi(q)1228 2185 y Fr(+)h Fq(P)1303 2168 y Fj(\000)1297 2195 y Fi(q)1338 2174 y Fr(~)1330 2185 y Fq(T)c(P)1387 2191 y Fi(q)1405 2185 y Fr(\)\))p Fq(;)262 2257 y Fr(the)14 b(n)o(umerical)e(function)i Fq(\036)g Fr(b)q(eing)f(in)o(tro)q(duced)i(b)q(efore)g(Corollary)d (3.2.)h(Hence,)i(w)o(e)f(get)262 2344 y(lim)7 b(sup)284 2379 y Fi(j)r Fj(!1)412 2344 y Fr(lim)396 2371 y Fi(R)p Fj(!1)494 2344 y Fq(b)512 2327 y Fj(\000)p Fl(1)512 2356 y Fi(j)556 2344 y Fq(R)588 2327 y Fj(\000)p Fl(3)633 2344 y Fr(\()p Fq(n)674 2350 y Fj(\000)702 2344 y Fr(\(1;)g Fq(T)f Fr(\()p Fq(\025)828 2327 y Fl(+)828 2355 y(2)p Fi(;m)892 2344 y Fr(+)i(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)1091 2350 y Fi(j)1108 2344 y Fr(\)\))f Fp(\000)h Fq(n)1212 2350 y Fj(\000)1240 2344 y Fr(\(1;)f Fq(T)f Fr(\()p Fq(\025)1366 2327 y Fj(\000)1366 2355 y Fl(1)p Fi(;m)1430 2344 y Fr(+)i(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)1628 2350 y Fi(j)1646 2344 y Fr(\)\)\))12 b Fp(\024)p eop %%Page: 11 11 11 10 bop 262 116 a Fu(In)o(tegrated)13 b(Densit)o(y)i(of)d(States)i (in)g(Strong)f(Magnetic)i(Fields)518 b(11)367 224 y Fr(lim)7 b(sup)389 258 y Fi(j)r Fj(!1)501 224 y Fr(lim)g(sup)519 259 y Fi(R)p Fj(!1)635 224 y Fq(b)653 206 y Fj(\000)p Fl(1)653 236 y Fi(j)697 224 y Fq(R)729 207 y Fj(\000)p Fl(3)773 224 y Fr(\()p Fq(n)814 230 y Fj(\000)843 224 y Fr(\(1)i Fp(\000)g Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)1051 230 y Fi(q)1068 224 y Fq(T)f Fr(\()p Fq(\025)1138 206 y Fl(+)1138 235 y(2)p Fi(;m)1206 224 y Fr(+)j(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)1410 230 y Fi(j)1437 224 y Fr(+)g(2)p Fq(c)1517 230 y Fl(0)1535 224 y Fq(\016)r Fr(\))p Fq(P)1598 230 y Fi(q)1617 224 y Fr(\))538 321 y Fp(\000)p Fq(n)595 327 y Fj(\000)623 321 y Fr(\(1)g(+)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)832 327 y Fi(q)849 321 y Fq(T)f Fr(\()p Fq(\025)919 303 y Fj(\000)919 332 y Fl(1)p Fi(;m)986 321 y Fr(+)k(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)1191 327 y Fi(j)1217 321 y Fp(\000)g Fr(2)p Fq(c)1298 327 y Fl(0)1316 321 y Fq(\016)r Fr(\))p Fq(P)1379 327 y Fi(q)1397 321 y Fr(\)\)+)287 391 y(lim)d(sup)310 425 y Fi(j)r Fj(!1)421 391 y Fr(lim)g(sup)439 426 y Fi(R)p Fj(!1)555 391 y Fq(b)573 373 y Fj(\000)p Fl(1)573 403 y Fi(j)618 391 y Fq(R)650 374 y Fj(\000)p Fl(3)694 391 y Fr(\()p Fq(n)735 397 y Fj(\000)763 391 y Fr(\(1)i Fp(\000)h Fq(\036)p Fr(\()p Fq(b)p Fr(\);)d Fq(P)978 374 y Fj(\000)972 401 y Fi(q)1005 391 y Fq(T)f Fr(\()p Fq(\025)1075 373 y Fl(+)1075 402 y(2)p Fi(;m)1142 391 y Fr(+)k(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)1346 397 y Fi(j)1373 391 y Fr(+)g Fq(c)1432 397 y Fl(0)1451 391 y Fr(\(1)g(+)h Fq(\016)1559 374 y Fj(\000)p Fl(1)1603 391 y Fr(\)\))p Fq(P)1668 374 y Fj(\000)1662 401 y Fi(q)1696 391 y Fr(\))459 488 y Fp(\000)p Fq(n)516 494 y Fj(\000)544 488 y Fr(\(1)f(+)g Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)758 471 y Fj(\000)752 498 y Fi(q)785 488 y Fq(T)f Fr(\()p Fq(\025)855 470 y Fj(\000)855 499 y Fl(1)p Fi(;m)923 488 y Fr(+)j(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)1127 494 y Fi(j)1153 488 y Fp(\000)g Fq(c)1213 494 y Fl(0)1231 488 y Fr(\(1)f(+)h Fq(\016)1339 471 y Fj(\000)p Fl(1)1384 488 y Fr(\)\))p Fq(P)1449 471 y Fj(\000)1443 498 y Fi(q)1476 488 y Fr(\)\)+)288 558 y(lim)d(sup)311 592 y Fi(j)r Fj(!1)422 558 y Fr(lim)g(sup)440 593 y Fi(R)p Fj(!1)556 558 y Fq(b)574 540 y Fj(\000)p Fl(1)574 570 y Fi(j)618 558 y Fq(R)650 541 y Fj(\000)p Fl(3)695 558 y Fq(n)720 564 y Fj(\000)748 558 y Fr(\(1)i Fp(\000)g Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)962 541 y Fl(+)956 568 y Fi(q)989 558 y Fq(T)f Fr(\()p Fq(\025)1059 540 y Fl(+)1059 569 y(2)p Fi(;m)1126 558 y Fr(+)k(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)1330 564 y Fi(j)1357 558 y Fr(+)g Fq(c)1416 564 y Fl(0)1435 558 y Fr(\(1)g(+)g Fq(\016)1542 541 y Fj(\000)p Fl(1)1587 558 y Fr(\))p Fq(P)1636 541 y Fl(+)1630 568 y Fi(q)1663 558 y Fr(\)+)307 657 y(lim)e(sup)330 691 y Fi(j)r Fj(!1)441 657 y Fr(lim)g(sup)459 692 y Fi(R)p Fj(!1)575 657 y Fq(b)593 639 y Fj(\000)p Fl(1)593 669 y Fi(j)638 657 y Fq(R)670 640 y Fj(\000)p Fl(3)714 657 y Fr(\()p Fq(n)755 663 y Fl(+)783 657 y Fr(\()p Fq(\036)p Fr(\()p Fq(b)p Fr(\);)g(2\()p Fq(\025)954 639 y Fl(+)954 668 y(2)p Fi(;m)1020 657 y Fr(+)j(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)1225 663 y Fi(j)1242 657 y Fr(\)Re\()p Fq(P)1356 640 y Fl(+)1350 667 y Fi(q)1391 647 y Fr(~)1383 657 y Fq(T)1422 647 y Fr(~)1413 657 y Fq(P)1446 640 y Fj(\000)1440 667 y Fi(q)1483 657 y Fr(+)f Fq(P)1557 640 y Fj(\000)1551 667 y Fi(q)1593 647 y Fr(~)1585 657 y Fq(T)d(P)1642 663 y Fi(q)1660 657 y Fr(\)\))497 756 y(+)p Fq(n)554 762 y Fj(\000)583 756 y Fr(\()p Fq(\036)p Fr(\()p Fq(b)p Fr(\);)h(2\()p Fq(\025)754 739 y Fj(\000)754 767 y Fl(1)p Fi(;m)820 756 y Fr(+)j(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\)Re\()p Fq(P)1138 739 y Fl(+)1132 767 y Fi(q)1174 746 y Fr(~)1165 756 y Fq(T)1205 746 y Fr(~)1195 756 y Fq(P)1228 739 y Fj(\000)1222 767 y Fi(q)1265 756 y Fr(+)h Fq(P)1340 739 y Fj(\000)1334 767 y Fi(q)1375 746 y Fr(~)1367 756 y Fq(T)c(P)1424 762 y Fi(q)1442 756 y Fr(\)\)\))p Fq(;)151 b Fr(\(4.4\))262 819 y(and)262 890 y(lim)7 b(inf)278 916 y Fi(j)r Fj(!1)400 890 y Fr(lim)384 917 y Fi(R)p Fj(!1)482 890 y Fq(b)500 872 y Fj(\000)p Fl(1)500 902 y Fi(j)544 890 y Fq(R)576 873 y Fj(\000)p Fl(3)621 890 y Fr(\()p Fq(n)662 896 y Fj(\000)690 890 y Fr(\(1;)g Fq(T)f Fr(\()p Fq(\025)816 872 y Fj(\000)816 901 y Fl(2)p Fi(;m)882 890 y Fr(+)i(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)1083 896 y Fi(j)1101 890 y Fr(\)\))g Fp(\000)h Fq(n)1207 896 y Fj(\000)1235 890 y Fr(\(1;)e Fq(T)f Fr(\()p Fq(\025)1361 872 y Fl(+)1361 901 y(1)p Fi(;m)1427 890 y Fr(+)i(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)1628 896 y Fi(j)1646 890 y Fr(\)\)\))k Fp(\025)379 988 y Fr(lim)7 b(inf)395 1014 y Fi(j)r Fj(!1)501 988 y 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Fr(\)\))p Fq(P)1449 1218 y Fj(\000)1443 1246 y Fi(q)1476 1236 y Fr(\)\))p Fp(\000)292 1306 y Fr(lim)d(inf)309 1332 y Fi(j)r Fj(!1)414 1306 y Fr(lim)g(inf)426 1332 y Fi(R)p Fj(!1)536 1306 y Fq(b)554 1288 y Fj(\000)p Fl(1)554 1317 y Fi(j)598 1306 y Fq(R)630 1289 y Fj(\000)p Fl(3)675 1306 y Fq(n)700 1312 y Fj(\000)728 1306 y Fr(\(1)i Fp(\000)g Fq(\036)p Fr(\()p Fq(b)p Fr(\);)e Fq(P)942 1289 y Fl(+)936 1316 y Fi(q)968 1306 y Fq(T)f Fr(\()p Fq(\025)1038 1288 y Fl(+)1038 1317 y(1)p Fi(;m)1106 1306 y Fr(+)j(\(2)p Fq(q)i Fp(\000)e Fr(1\))p Fq(b)1310 1312 y Fi(j)1337 1306 y Fr(+)g Fq(c)1396 1312 y Fl(0)1415 1306 y Fr(\(1)g(+)g Fq(\016)1522 1289 y Fj(\000)p Fl(1)1567 1306 y Fr(\)\))p Fq(P)1632 1289 y Fl(+)1626 1316 y Fi(q)1659 1306 y Fr(\))p Fp(\000)319 1397 y Fr(lim)e(inf)336 1423 y Fi(j)r Fj(!1)441 1397 y Fr(lim)g(inf)453 1424 y Fi(R)p Fj(!1)563 1397 y Fq(b)581 1379 y Fj(\000)p Fl(1)581 1408 y Fi(j)626 1397 y Fq(R)658 1380 y Fj(\000)p Fl(3)702 1397 y Fr(\()p Fq(n)743 1403 y Fl(+)771 1397 y Fr(\()p Fq(\036)p Fr(\()p Fq(b)p Fr(\);)g(2\()p Fq(\025)942 1379 y Fl(+)942 1408 y(1)p Fi(;m)1008 1397 y Fr(+)j(\(2)p Fq(q)g Fp(\000)g Fr(1\))p Fq(b)1213 1403 y Fi(j)1230 1397 y Fr(\)Re\()p Fq(P)1344 1380 y Fl(+)1338 1407 y Fi(q)1379 1386 y Fr(~)1371 1397 y Fq(T)1410 1386 y Fr(~)1401 1397 y Fq(P)1434 1380 y Fj(\000)1428 1407 y Fi(q)1471 1397 y Fr(+)f Fq(P)1545 1380 y Fj(\000)1539 1407 y Fi(q)1581 1386 y Fr(~)1573 1397 y Fq(T)d(P)1630 1403 y Fi(q)1648 1397 y Fr(\)\))497 1488 y(+)p Fq(n)554 1494 y Fj(\000)583 1488 y Fr(\()p Fq(\036)p Fr(\()p Fq(b)p Fr(\);)h(2\()p Fq(\025)754 1470 y Fj(\000)754 1499 y Fl(2)p Fi(;m)820 1488 y Fr(+)j(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)p Fr(\)Re\()p Fq(P)1138 1471 y Fl(+)1132 1498 y Fi(q)1174 1477 y Fr(~)1165 1488 y Fq(T)1205 1477 y Fr(~)1195 1488 y Fq(P)1228 1471 y Fj(\000)1222 1498 y Fi(q)1265 1488 y Fr(+)h Fq(P)1340 1471 y Fj(\000)1334 1498 y Fi(q)1375 1477 y Fr(~)1367 1488 y Fq(T)c(P)1424 1494 y Fi(q)1442 1488 y Fr(\)\)\))p Fq(:)151 b Fr(\(4.5\))262 1551 y(Emplo)o(ying)16 b(Corollary)i(3.2,)g (w)o(e)i(\014nd)g(that)f(the)h(\014rst)h(term)d(at)i(the)g(righ)o (t-hand)f(side)g(of)262 1600 y(\(4.4\))g(is)h(equal)g(to)604 1584 y Fl(1)p 594 1591 38 2 v 594 1615 a(2)p Fi(\031)643 1567 y Fk(\000)662 1600 y Fq(k)q Fr(\()p Fq(\025)725 1583 y Fl(+)725 1611 y(2)p Fi(;m)792 1600 y Fr(+)10 b(2)p Fq(c)873 1606 y Fl(0)891 1600 y Fq(\016)r Fr(\)\))f Fp(\000)h Fq(k)q Fr(\()p Fq(\025)1057 1583 y Fj(\000)1057 1611 y Fl(1)p Fi(;m)1124 1600 y Fp(\000)g Fr(2)p Fq(c)1205 1606 y Fl(0)1223 1600 y Fq(\016)r Fr(\))1259 1567 y Fk(\001)1278 1600 y Fr(,)20 b(while)g(the)h(\014rst)g(term)f(at)262 1657 y(the)d(side)h(of)e(\(4.5\))g(is)h(equal)g(to)802 1641 y Fl(1)p 792 1648 V 792 1671 a(2)p Fi(\031)841 1623 y Fk(\000)860 1657 y Fq(k)q Fr(\()p Fq(\025)923 1639 y Fj(\000)923 1668 y Fl(2)p Fi(;m)990 1657 y Fp(\000)9 b Fr(2)p Fq(c)1070 1663 y Fl(0)1089 1657 y Fq(\016)r Fr(\)\))g Fp(\000)h Fq(k)q Fr(\()p Fq(\025)1255 1639 y Fl(+)1255 1668 y(1)p Fi(;m)1322 1657 y Fr(+)g(2)p Fq(c)1403 1663 y Fl(0)1421 1657 y Fq(\016)r Fr(\))1457 1623 y Fk(\001)1476 1657 y Fr(.)17 b(Assume)g(that)262 1707 y Fq(\016)c(>)f Fr(0)g(is)h(small)d(enough,)i(and)h(apply)e(Prop)q(osition)i(3.3)e(to)i (the)g(second)g(terms)g(at)f(the)h(righ)o(t-)262 1757 y(hand)h(sides)h(of)e(\(4.4\))h(and)g(\(4.5\))f(in)h(order)h(to)f(c)o (hec)o(k)h(that)f(these)i(terms)e(v)n(anish.)f(Similarly)l(,)262 1806 y(utilize)j(Prop)q(osition)h(3.4)f(\(resp)q(ectiv)o(ely)m(,)h (Prop)q(osition)g(3.5\))f(in)g(order)i(to)f(v)o(erify)f(that)h(the)262 1856 y(third)12 b(\(resp)q(ectiv)o(ely)m(,)i(fourth\))e(terms)h(at)f (the)i(righ)o(t-hand)e(sides)h(of)f(\(4.4\))g(and)h(\(4.5\))f(v)n (anish.)262 1906 y(Putting)h(together)i(\(4.2\))e({)h(\(4.5\),)f(w)o(e) h(get)462 1977 y(lim)7 b(sup)485 2011 y Fi(j)r Fj(!1)596 1977 y Fq(b)614 1959 y Fj(\000)p Fl(1)614 1989 y Fi(j)665 1944 y Fk(\000)684 1977 y Fp(D)716 1983 y Fi(b)731 1987 y Fd(j)748 1977 y Fr(\()p Fq(\025)788 1983 y Fl(2)817 1977 y Fr(+)i(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)1021 1983 y Fi(j)1038 1977 y Fr(\))f Fp(\000)h(D)1137 1983 y Fi(b)1152 1987 y Fd(j)1169 1977 y Fr(\()p Fq(\025)1209 1983 y Fl(1)1237 1977 y Fr(+)g(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)1441 1983 y Fi(j)1459 1977 y Fr(\))1475 1944 y Fk(\001)1505 1977 y Fp(\024)659 2073 y Fr(1)p 646 2092 46 2 v 646 2130 a(2)p Fq(\031)704 2068 y Fk(\000)723 2102 y Fq(k)q Fr(\()p Fq(\025)786 2084 y Fl(+)786 2113 y(2)p Fi(;m)853 2102 y Fr(+)h(2)p Fq(c)934 2108 y Fl(0)952 2102 y Fq(\016)r Fr(\)\))g Fp(\000)f Fq(k)q Fr(\()p Fq(\025)1118 2084 y Fj(\000)1118 2113 y Fl(1)p Fi(;m)1186 2102 y Fp(\000)g Fr(2)p Fq(c)1266 2108 y Fl(0)1285 2102 y Fq(\016)r Fr(\))1321 2068 y Fk(\001)1347 2102 y Fq(;)294 b Fr(\(4.6\))468 2183 y(lim)7 b(inf)485 2210 y Fi(j)r Fj(!1)590 2183 y Fq(b)608 2166 y Fj(\000)p Fl(1)608 2195 y Fi(j)659 2150 y Fk(\000)678 2183 y Fp(D)710 2189 y Fi(b)725 2193 y Fd(j)742 2183 y Fr(\()p Fq(\025)782 2189 y Fl(2)811 2183 y Fr(+)i(\(2)p Fq(q)h Fp(\000)g Fr(1\))p Fq(b)1015 2189 y Fi(j)1032 2183 y Fr(\))f Fp(\000)h(D)1131 2189 y Fi(b)1146 2193 y Fd(j)1163 2183 y Fr(\()p Fq(\025)1203 2189 y Fl(1)1231 2183 y Fr(+)g(\(2)p Fq(q)g Fp(\000)f Fr(1\))p Fq(b)1435 2189 y Fi(j)1453 2183 y Fr(\))1469 2150 y Fk(\001)1499 2183 y Fp(\025)659 2263 y Fr(1)p 646 2282 V 646 2320 a(2)p Fq(\031)704 2258 y Fk(\000)723 2291 y Fq(k)q Fr(\()p Fq(\025)786 2273 y Fj(\000)786 2302 y Fl(2)p Fi(;m)853 2291 y Fp(\000)h Fr(2)p Fq(c)934 2297 y Fl(0)952 2291 y Fq(\016)r Fr(\)\))g Fp(\000)f Fq(k)q Fr(\()p Fq(\025)1118 2273 y Fl(+)1118 2302 y(1)p Fi(;m)1186 2291 y Fr(+)g(2)p Fq(c)1266 2297 y Fl(0)1285 2291 y Fq(\016)r Fr(\))1321 2258 y Fk(\001)1347 2291 y Fq(:)294 b Fr(\(4.7\))262 2368 y(Letting)13 b Fq(m)f Fp(!)f(1)p Fr(,)i(and)h Fq(\016)g Fp(#)d Fr(0)i(in)h(\(4.6\)-\(4.7\),)e(w)o(e)i(obtain)f(\(4.1\).)p eop %%Page: 12 12 12 11 bop 262 116 a Fu(12)1322 b(Raik)o(o)o(v)262 224 y Fs(Ac)n(kno)n(wledgemen)n(ts)262 350 y Fr(The)15 b(\014nancial)f (supp)q(ort)i(of)e(the)i(Organizing)e(Committee)f(of)h(the)i (Conference)g(on)f(P)o(artial)262 400 y(Di\013eren)o(tial)e(Equations,) h(Clausthal,)f(July)i(24)f(-28,)f(2000,)g(and)h(of)g(Gran)o(t)g(MM)h (706/97)d(of)262 450 y(the)h(Bulgarian)g(Science)i(F)m(oundation,)c(is) i(gratefully)g(ac)o(kno)o(wledged.)g(The)g(author)h(thanks)262 499 y(the)f(referee)i(whose)e(v)n(aluable)f(remarks)g(con)o(tributed)i (to)f(the)g(impro)o(v)o(emen)o(t)d(of)i(the)i(exp)q(osi-)262 549 y(tion.)262 708 y Fs(References)262 803 y Fu([A.H.S])89 b Fb(J.A)l(vr)o(on,)15 b(I.Herbst,)g(B.Simon)p Fu(,)e Fa(Schr\177)-20 b(odinger)11 b(op)n(er)n(ators)i(with)h(magnetic)e (\014elds.)474 849 y(I.)h(Gener)n(al)f(inter)n(actions)o Fu(,)e(Duk)o(e.)j(Math.)g(J.)g Fv(45)g Fu(\(1978\),)g(847-883.)262 894 y([B])163 b Fb(M.)524 885 y(\025)523 894 y(S.Birman)p Fu(,)15 b Fa(On)i(the)e(sp)n(e)n(ctrum)h(of)f(singular)g(b)n(oundary)f (value)h(pr)n(oblems)p Fu(,)f(Mat.)474 940 y(Sb)q(ornik)19 b Fv(55)e Fu(\(1961\),)h(125-174)g(\(Russian\);)h(Engl.)f(transl.)g(in) g(Amer.)f(Math.)h(So)q(c.)474 986 y(T)m(ransl.,)13 b(\(2\))g Fv(53)g Fu(\(1966\),)g(23-80.)262 1031 y([H.L.M.W])30 b Fb(T.Hupfer,)25 b(H.Leschke,)f(P.M)999 1028 y(\177)998 1031 y(uller,)h(S.W)l(arzel)p Fu(,)d Fa(The)h(inte)n(gr)n(ate)n(d)d (density)474 1077 y(of)g(states)f(and)h(its)g(absolute)e(c)n(ontinuity) g(for)i(magnetic)f(Schr\177)-20 b(odinger)18 b(op)n(er)n(ators)474 1122 y(with)e(unb)n(ounde)n(d)d(r)n(andom)i(p)n(otentials)p Fu(,)d(math-ph)17 b(Preprin)o(t)g(0010013)g(a)o(v)n(ailable)i(at)474 1168 y(h)o(ttp://xxx.uni-augsburg)q(.de/abs)q(/math-ph)262 1214 y([K])160 b Fb(W.Kirsch)p Fu(,)14 b Fa(R)n(andom)e(Schr\177)-20 b(odinger)11 b(op)n(er)n(ators:)g(a)j(c)n(ourse)p Fu(.)c(In:)j(Sc)o (hr\177)-19 b(odinger)15 b(op)q(er-)474 1259 y(ators,)f(Pro)q(c.)g (Nord.)f(Summer)i(Sc)o(h.)f(Math.,)g(Sandb)r(jerg)h(Slot,)g(So)q (enderb)q(org/Den-)474 1305 y(mark)e(1988,)g(Lect.)g(Notes)g(Ph)o(ys.)g Fv(345)p Fu(,)g(\(1989\))g(264-370.)262 1351 y([K.R])121 b Fb(W.Kirsch,)11 b(G.D.Raik)o(o)o(v)p Fu(,)d Fa(Str)n(ong)g(magnetic)g (\014eld)h(asymptotics)e(of)i(the)g(inte)n(gr)n(ate)n(d)474 1396 y(density)g(of)i(states)e(for)i(a)g(r)n(andom)g(3D)g(Schr\177)-20 b(odinge)o(r)9 b(op)n(er)n(ator)p Fu(,)f(mp)p 1462 1396 12 2 v 14 w(arc)j(Preprin)o(t)h(99-)474 1442 y(345)j(a)o(v)n(ailable)k (at)c(h)o(ttp://www.ma.utexas.edu/mp)p 1284 1442 V 15 w(arc)h(\(to)f(app)q(ear)h(in)g(Annales)474 1488 y(Henri)d(P)o(oincar)o (\023)-18 b(e\))262 1533 y([N])161 b Fb(S.Nakamura)p Fu(,)10 b Fa(A)i(r)n(emark)f(on)g(the)g(Dirichlet-Neuma)o(nn)e(de)n(c)n (ouplin)o(g)g(and)i(the)f(inte-)474 1579 y(gr)n(ate)n(d)f(density)f(of) i(states)p Fu(,)d(mp)p 923 1579 V 14 w(arc)j(Preprin)o(t)h(00-198)f(a)o (v)n(ailable)i(at)e(h)o(ttp://www.ma.)474 1624 y(utexas.edu/mp)p 725 1624 V 15 w(arc)k(\(to)e(app)q(ear)i(in)g(J.F)m(unct.Anal.\))262 1670 y([P)m(.Fi])121 b Fb(L.P)m(astur,)13 b(A.Figotin)p Fu(,)g Fa(Sp)n(e)n(ctr)n(a)f(of)h(R)n(andom)f(and)h(A)o(lmost-Perio)n (dic)d(Op)n(er)n(ators.)474 1716 y Fu(Grundlehren)26 b(der)f(Mathematisc)o(hen)h(Wissensc)o(haften)h Fv(297)p Fu(.)d(Springer-V)m(erlag,)474 1761 y(Berlin)14 b(etc.)f(\(1992\).)262 1807 y([R])162 b Fb(G.D.Raik)o(o)o(v)p Fu(,)12 b Fa(Eigenvalue)f (asymptotics)g(for)j(the)f(Pauli)f(op)n(er)n(ator)h(in)g(str)n(ong)f (non-)474 1852 y(c)n(onstant)f(magnetic)g(\014elds)p Fu(,)g(Ann.Inst.F)m(ourier,)j Fv(49)p Fu(,)e(\(1999\),)i(1603-1636.)262 2011 y Fs(Address)321 2111 y Fw(Geor)o(gi)h(D.)h(Raik)o(o)o(v)p Fr(,)e(Institute)g(of)f(Mathematics,)f(Bulgarian)h(Academ)o(y)f(of)h (Sci-)380 2160 y(ences,)i(Acad.G.Bonc)o(hev)f(Str.,)f(bl.8,)f(1113)h (So\014a,)g(Bulgaria)380 2228 y Fw(E-mail:)h Fr(gdraik)o(o)o(v@y)o(aho) q(o.com)262 2360 y Fu(1991)f(Mathematics)i(Sub)r(ject)e (Classi\014cation)q(.)i(Primary)f(82B44;)g(Secondary)g(81Q10,)f(47B80)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0012120553192--