This is a multi-part message in MIME format. ---------------9810130346406 Content-Type: text/plain; name="98-639.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-639.keywords" SCHRODINGER OPERATORS, MAGNETIC FIELDS, TIGHT-BINDING LIMIT ---------------9810130346406 Content-Type: application/postscript; name="ahi.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ahi.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: ahi.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips ahi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1998.10.08:1343 %%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 length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B 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b(width)h(of)e(the)i("minibands")d(lo)q(cated)j(b)q(elo)o (w)f(or)g(in)g(b)q(et)o(w)o(een)i(the)f(Landau)f(lev)o(els)262 2449 y(shrinks)g(gaussian)g(lik)o(e)f(in)h(the)h(limit)c(when)k(the)f (in)o(terw)o(ell)g(distance)h(go)q(es)f(to)g(in\014nit)o(y)m(.)967 2574 y(2)p eop %%Page: 3 3 3 2 bop 262 307 a Fp(Notice)15 b(that)g(the)g(limit)d(considered)17 b(in)d([H-H;)g(Na)h(1])f(is)h(the)g(strong)h(\014eld)f(case)g(i.e.)21 b(the)262 357 y(magnetic)12 b(\014eld)i(outside)g(the)h(w)o(ells)e(go)q (es)i(to)e(in\014nit)o(y)m(.)324 407 y(The)h(con)o(ten)o(ts)h(of)e(the) i(pap)q(er)f(is)g(as)g(follo)o(ws:)324 457 y(Section)k(2)f(\014xes)h (some)f(notations)g(and)h(giv)o(es)f(a)g(few)h(results)h(needed)g(in)e (the)h(next)262 506 y(section.)g(Lemma)11 b(2.1)i(outlines)g(the)h (gaussian)g(deca)o(y)g(of)f(the)h(k)o(ernel)g(of)f(the)h("free)g (resol-)262 556 y(v)o(en)o(t")e(\(the)i(magnetic)d(\014eld)i(is)g (constan)o(t)g(here)h(and)f(the)g(scalar)g(p)q(oten)o(tial)f(is)h (absen)o(t\);)g(in)262 606 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Fl(')p Fp(\()p Fk(x)p Fl(;)7 b Fk(x)485 1701 y Fh(0)497 1718 y Fp(\))k(=)h Fj(\000)607 1660 y Ff(\022)638 1662 y(Z)680 1672 y Fi(1)661 1756 y(0)705 1718 y Fl(dt)749 1662 y Ff(Z)790 1672 y Fi(1)772 1756 y(0)816 1718 y Fl(ds)d(s)h Fk(B)p Fp(\()p Fl(s)f(t)h Fp(\()p Fk(x)f Fj(\000)h Fk(x)1115 1701 y Fh(0)1127 1718 y Fp(\))f(+)g Fl(s)h Fk(x)1247 1701 y Fh(0)1259 1718 y Fp(\))1275 1660 y Ff(\023)1315 1718 y Fj(\001)f Fp(\()p Fk(x)g Fj(^)g Fk(x)1448 1701 y Fh(0)1460 1718 y Fp(\))112 b(\(2.10\))324 1816 y(If)13 b Fl(B)r Fp(\()p Fk(x)p Fp(\))g(=)f Fl(B)543 1822 y Fi(0)576 1816 y Fp(is)h(constan)o(t,)h(then)633 1908 y Fl(')660 1914 y Fi(0)678 1908 y Fp(\()p Fk(x)p Fl(;)7 b Fk(x)763 1890 y Fh(0)775 1908 y Fp(\))12 b(=)f Fj(\000)883 1879 y Fp(1)p 883 1898 21 2 v 883 1936 a(2)909 1908 y Fl(B)940 1914 y Fi(0)969 1908 y Fp(\()p Fl(x)1009 1914 y Fi(1)1036 1908 y Fl(x)1060 1890 y Fh(0)1060 1918 y Fi(2)1088 1908 y Fj(\000)f Fl(x)1154 1890 y Fh(0)1154 1918 y Fi(1)1181 1908 y Fl(x)1205 1914 y Fi(2)1223 1908 y Fp(\))755 2007 y Fk(a)778 2013 y Fi(0)797 2007 y Fp(\()p Fk(x)p Fl(;)d Fk(x)882 1990 y Fh(0)894 2007 y Fp(\))k(=)970 1979 y(1)p 970 1998 V 970 2036 a(2)996 2007 y Fk(B)1030 2013 y Fi(0)1058 2007 y Fj(^)e Fp(\()p Fk(x)g Fj(\000)h Fk(x)1212 1990 y Fh(0)1223 2007 y Fp(\))349 b(\(2.11\))324 2092 y(The)18 b(hamiltonian)c(of)j(a)g(particle)h(in)f(the)h(presence)i(of)d(the)i (magnetic)d(\014eld)h(and)h(a)262 2141 y(scalar)13 b(p)q(oten)o(tial)h Fl(V)23 b Fp(is)14 b(\(in)f(the)i(transv)o(ersal)f(gauge\):)551 2218 y Fl(H)45 b Fp(=)d(\()p Fk(p)9 b Fj(\000)g Fk(a)p Fp(\()p Fk(x)p Fp(\)\))895 2197 y Fi(2)923 2218 y Fp(+)g Fl(V)h Fp(\()p Fk(x)p Fp(\))563 2305 y Fk(p)41 b Fp(=)705 2247 y Ff(\022)735 2305 y Fj(\000)p Fl(i)808 2277 y(@)p 786 2296 67 2 v 786 2334 a(@)r(x)834 2340 y Fi(1)858 2305 y Fl(;)7 b Fj(\000)p Fl(i)950 2277 y(@)p 928 2296 V 928 2334 a(@)r(x)976 2340 y Fi(2)1000 2247 y Ff(\023)1588 2305 y Fp(\(2.12\))509 2426 y Fk(a)p Fp(\()p Fk(x)p Fp(\))42 b(=)705 2368 y Ff(\022)735 2426 y Fj(\000)p Fl(x)791 2432 y Fi(2)817 2370 y Ff(Z)858 2380 y Fi(1)840 2464 y(0)884 2426 y Fl(ds)9 b(s)h(B)r Fp(\()p Fl(s)g Fk(x)p Fp(\))p Fl(;)d(x)1125 2432 y Fi(1)1150 2370 y Ff(Z)1192 2380 y Fi(1)1173 2464 y(0)1217 2426 y Fl(ds)j(s)f(B)r Fp(\()p Fl(s)i Fk(x)p Fp(\))1416 2368 y Ff(\023)967 2574 y Fp(4)p eop %%Page: 5 5 5 4 bop 324 307 a Fp(In)14 b(the)g(case)h(of)e(a)h(constan)o(t)g (magnetic)f(\014eld,)g(one)h(has)h(the)f(hamiltonian)501 395 y Fl(H)536 401 y Fi(0)596 395 y Fp(=)42 b(\()p Fk(p)9 b Fj(\000)g Fk(a)786 401 y Fi(0)805 395 y Fp(\()p Fk(x)p Fp(\)\))878 378 y Fi(2)906 395 y Fp(where)577 b(\(2.13\))528 482 y Fk(p)41 b Fp(=)h Fj(\000)p Fl(i)p Fj(r)751 488 y Fe(x)814 482 y Fp(and)g Fk(a)946 488 y Fi(0)964 482 y Fp(\()p Fk(x)p Fp(\))12 b(=)1077 424 y Ff(\022)1108 482 y Fj(\000)1145 454 y Fp(1)p 1145 473 21 2 v 1145 511 a(2)1171 482 y Fl(B)1202 488 y Fi(0)1230 482 y Fl(x)1254 488 y Fi(2)1273 482 y Fl(;)1296 454 y Fp(1)p 1296 473 V 1296 511 a(2)1322 482 y Fl(B)1353 488 y Fi(0)1381 482 y Fl(x)1405 488 y Fi(1)1423 424 y Ff(\023)1588 482 y Fp(\(2.14\))262 599 y(whic)o(h)g(is)h(essen)o(tially)f(self-adjoin)o(t) g(on)g Fl(C)919 584 y Fh(1)916 610 y Fi(0)961 566 y Ff(\000)980 599 y Fk(R)1016 584 y Fi(2)1034 566 y Ff(\001)1066 599 y Fp(and)g(its)h(sp)q(ectrum)g(is)g(the)g(w)o(ell)f(kno)o(wn)262 649 y(Landau)h(sp)q(ectrum)383 737 y Fl(\033)q Fp(\()p Fl(H)459 743 y Fi(0)477 737 y Fp(\))f(=)g Fl(\033)573 743 y Fg(ess)622 737 y Fp(\()p Fl(H)673 743 y Fi(0)691 737 y Fp(\))g Fj(\021)g Fl(\033)787 743 y Fg(L)811 737 y Fp(\()p Fl(B)858 743 y Fi(0)877 737 y Fp(\))g(=)g Fj(f)p Fp(\(2)p Fl(n)d Fp(+)g(1\))p Fl(B)1150 743 y Fi(0)1178 737 y Fj(j)g Fl(n)j Fp(=)f(0)p Fl(;)c Fp(1)p Fl(;)g Fp(2)p Fl(;)g(:)g(:)g(:)l Fj(g)122 b Fp(\(2.15\))262 825 y(F)m(or)13 b Fl(z)h Fj(62)d Fl(\033)q Fp(\()p Fl(H)484 831 y Fi(0)502 825 y Fp(\))j(and)g Fl(g)f Fj(2)e Fl(L)713 810 y Fi(2)739 791 y Ff(\000)758 825 y Fk(R)794 810 y Fi(2)812 791 y Ff(\001)831 825 y Fp(,)j(w)o(e)g(write)624 904 y Ff(\002)641 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Fi(0)991 1703 y Fp(\()p Fk(x)p Fl(;)g Fk(x)1076 1686 y Fh(0)1087 1703 y Fp(;)g Fl(z)r Fp(\))k Fj(\021)691 1790 y(\021)770 1761 y Fp(\000\()p Fl(\013)p Fp(\))p 770 1780 85 2 v 785 1818 a(4)d Fl(\031)860 1790 y(e)879 1772 y Fg(i)f(')920 1776 y Fd(0)936 1772 y Fi(\()p Fe(x)p Fg(;)p Fe(x)999 1760 y Fc(0)1010 1772 y Fi(\))1034 1790 y Fl(e)1053 1772 y Fh(\000)p Fg( )q Fi(\()p Fe(x)p Fg(;)p Fe(x)1165 1760 y Fc(0)1176 1772 y Fi(\))1191 1790 y Fl(U)e Fp(\()p Fl(\013;)i Fp(1;)g(2)h Fl( )q Fp(\()p Fk(x)p Fl(;)f Fk(x)1468 1772 y Fh(0)1480 1790 y Fp(\)\))76 b(\(2.17\))262 1891 y Fb(wher)n(e)19 b Fp(\000)h Fb(is)f(the)h(Euler)g(function)g(and)h Fl(U)5 b Fp(\()p Fl(\013;)i(\015)r Fp(;)g Fl(\030)r Fp(\))19 b Fb(is)h(the)g(c)n(on\015uent)h(hyp)n(er)n(ge)n(ometric)262 1941 y(function)15 b([A-S].)324 2029 y Fp(F)m(rom)d(Lemma)f(2.1)i(one)h (sees)i(that)d Fl(K)932 2035 y Fi(0)951 2029 y Fp(\()p Fk(x)p Fl(;)7 b Fk(x)1036 2014 y Fh(0)1048 2029 y Fp(;)g Fl(z)r Fp(\))14 b(has)g(a)f(gaussian)h(deca)o(y)g(as)262 2079 y Fj(j)p Fk(x)8 b Fj(\000)i Fk(x)374 2064 y Fh(0)386 2079 y Fj(j)h(!)g(1)p Fp(.)17 b(W)m(e)d(shall)f(use)i(this)f(in)f(the)i (follo)o(wing)c(form:)262 2167 y Fk(Corollary)j(2.1)21 b Fb(L)n(et)14 b Fl(\037)653 2173 y Fi(1)672 2167 y Fl(;)7 b(\037)717 2173 y Fi(2)746 2167 y Fj(2)12 b Fk(L)815 2152 y Fh(1)850 2167 y Fp(\()p Fk(R)902 2152 y Fi(2)920 2167 y Fp(\))j Fb(such)h(that)491 2255 y Fj(j)p Fl(\037)529 2261 y Fi(1)547 2255 y Fj(j)p Fl(;)7 b Fj(j)p Fl(\037)616 2261 y Fi(2)633 2255 y Fj(j)k(\024)h Fl(M)47 b Fb(and)c Fp(dist)p Fj(f)p Fl(supp)9 b(\037)1103 2261 y Fi(2)1122 2255 y Fl(;)e(supp)i(\037)1261 2261 y Fi(1)1279 2255 y Fj(g)i Fp(=)h Fl(d)f(>)h Fp(0)p Fl(:)262 2343 y Fb(Then)j(for)f(al)r (l)g Fp(0)e Fl(<)g(\016)h(<)652 2326 y Fg(B)677 2330 y Fd(0)p 652 2333 42 2 v 664 2357 a Fi(4)713 2343 y Fb(and)j Fl(z)d Fj(2)e Fl(\032)p Fp(\()p Fl(H)937 2349 y Fi(0)956 2343 y Fp(\))p Fb(,)k(one)g(has)h(that)551 2439 y Fj(jj)p Fl(\037)601 2445 y Fi(1)626 2439 y Fp(\()p Fl(H)677 2445 y Fi(0)704 2439 y Fj(\000)10 b Fl(z)r Fp(\))783 2418 y Fh(\000)p Fi(1)835 2439 y Fl(\037)861 2445 y Fi(2)879 2439 y Fj(jj)h(\024)21 b Fl(M)1012 2422 y Fi(2)1039 2439 y Fl(const)p Fp(\()p Fl(z)r Fp(\))c(exp)q(\()p Fj(\000)p Fl(\016)12 b(d)1370 2422 y Fi(2)1388 2439 y Fp(\))184 b(\(2.18\))967 2574 y(5)p eop %%Page: 6 6 6 5 bop 262 307 a Fb(Pr)n(o)n(of:)324 357 y Fp(Use)18 b(the)h(explicit)e(form)f(of)h Fl(K)821 363 y Fi(0)858 357 y Fp(and)g(Y)m(oung)g(inequalities)g(\(see)j([C-N])c(for)i(further) 262 407 y(discussions\).)324 457 y Fb(R)n(emark:)324 506 y Fp(Since)c(under)h(a)e(gauge)h(transformation)616 593 y(\()p Fl(U)660 599 y Fg(\037)683 593 y Fl(f)t Fp(\))8 b(\()p Fk(x)p Fp(\))42 b(=)g Fl(e)923 576 y Fg(i)7 b(\037)p Fi(\()p Fe(x)p Fi(\))1011 593 y Fl(f)t Fp(\()p Fk(x)p Fp(\))43 b(and)386 b(\(2.19\))356 644 y Ff(\000)375 678 y Fl(U)408 661 y Fh(\003)403 688 y Fg(\037)427 678 y Fp(\()p Fl(H)478 684 y Fi(0)506 678 y Fj(\000)9 b Fl(z)r Fp(\))584 661 y Fh(\000)p Fi(1)629 678 y Fl(U)657 684 y Fg(\037)680 678 y Fl(f)704 644 y Ff(\001)731 678 y Fp(\()p Fk(x)p Fp(\))42 b(=)904 621 y Ff(Z)927 715 y Fe(R)955 707 y Fd(2)980 678 y Fl(d)p Fk(x)1027 661 y Fh(0)1047 678 y Fl(K)1082 684 y Fg(\037)1105 678 y Fp(\()p Fk(x)p Fl(;)7 b Fk(x)1190 661 y Fh(0)1202 678 y Fp(;)g Fl(z)r Fp(\))k(=)830 789 y(=)904 732 y Ff(Z)927 826 y Fe(R)955 818 y Fd(2)980 789 y Fl(d)p Fk(x)1027 771 y Fh(0)1047 789 y Fl(e)1066 771 y Fh(\000)p Fg(i)d(\037)p Fi(\()p Fe(x)p Fi(\))1181 789 y Fl(K)1216 795 y Fi(0)1234 789 y Fp(\()p Fk(x)p Fl(;)f Fk(x)1319 771 y Fh(0)1331 789 y Fp(;)g Fl(z)r Fp(\))p Fl(e)1406 771 y Fg(i)g(\037)p Fi(\()p Fe(x)1479 759 y Fc(0)1490 771 y Fi(\))1505 789 y Fl(f)t Fp(\()p Fk(x)1570 771 y Fh(0)1583 789 y Fp(\))262 896 y(one)14 b(has)719 946 y Fj(j)p Fl(K)766 952 y Fg(\037)788 946 y Fp(\()p Fk(x)p Fl(;)7 b Fk(x)873 928 y Fh(0)885 946 y Fp(;)g Fl(z)r Fp(\))p Fj(j)k Fp(=)g Fj(j)p Fl(K)1054 952 y Fi(0)1073 946 y Fp(\()p Fk(x)p Fl(;)c Fk(x)1158 928 y Fh(0)1169 946 y Fp(;)g Fl(z)r Fp(\))p Fj(j)351 b Fp(\(2.20\))262 1017 y(i.e.)17 b(the)e(gaussian)e(deca)o(y)h(is)g(v)n (alid)e(for)i(an)g(arbitrary)f(gauge.)324 1067 y(Supp)q(ose)g(no)o(w)e (that)h(the)h(scalar)f(p)q(oten)o(tial)f Fl(V)21 b Fp(and)12 b(the)g(magnetic)f(\014eld)h(whic)o(h)g(corre-)262 1117 y(sp)q(onds)h(to)g Fk(a)g Fp(describ)q(e)i(the)e(one)g(w)o(ell)f(case)i (studied)g(in)e([C-N])g(i.e.)17 b(satisfy)c(the)g(follo)o(wing)262 1167 y(conditions)g(:)618 1253 y Fl(B)h Fp(=)e Fl(B)738 1259 y Fi(0)766 1253 y Fp(+)e Fl(B)841 1236 y Fh(0)853 1253 y Fl(;)48 b(B)944 1259 y Fi(0)974 1253 y Fl(>)12 b Fp(0)618 1315 y Fl(B)651 1298 y Fh(0)674 1315 y Fj(2)g Fl(C)747 1298 y Fi(1)772 1282 y Ff(\000)791 1315 y Fk(R)827 1298 y Fi(2)845 1282 y Ff(\001)880 1315 y Fp(;)31 b(lim)908 1340 y Fg(n)p Fh(!1)1002 1315 y Fj(jj)p Fl(B)1059 1298 y Fh(0)1070 1315 y Fj(jj)1094 1322 y Fg(C)1120 1314 y Fd(1)1135 1322 y Fi(\()p Fe(R)1176 1314 y Fd(2)1192 1322 y Fh(nfj)p Fe(x)p Fh(j\024)p Fg(n)p Fh(g)p Fi(\))1356 1315 y Fp(=)12 b(0)618 1397 y Fl(V)21 b Fp(=)12 b Fl(V)731 1403 y Fi(1)759 1397 y Fp(+)d Fl(V)824 1403 y Fi(2)852 1397 y Fp(;)16 b Fl(V)904 1403 y Fi(1)934 1397 y Fj(2)11 b Fl(L)1001 1380 y Fi(2)1027 1363 y Ff(\000)1046 1397 y Fk(R)1082 1380 y Fi(2)1101 1363 y Ff(\001)1127 1397 y Fl(;)k(V)1178 1403 y Fi(2)1209 1397 y Fj(2)c Fl(L)1276 1380 y Fh(1)1318 1363 y Ff(\000)1337 1397 y Fk(R)1373 1380 y Fi(2)1392 1363 y Ff(\001)632 1459 y Fp(lim)618 1484 y Fg(n)p Fh(!1)723 1459 y Fp(sup)711 1496 y Fh(j)p Fe(x)p Fh(j\025)p Fg(n)804 1459 y Fj(j)p Fl(V)840 1465 y Fi(2)859 1459 y Fp(\()p Fk(x)p Fp(\))p Fj(j)g Fp(=)h(0)584 b(\(2.21\))324 1583 y(In)10 b(particular,)g(under)h(these)g(conditions) f Fl(H)i Fp(is)e(essen)o(tially)g(self-adjoin)o(t)f(on)g Fl(C)1559 1568 y Fh(1)1556 1593 y Fi(0)1601 1549 y Ff(\000)1620 1583 y Fk(R)1656 1568 y Fi(2)1675 1549 y Ff(\001)262 1633 y Fp(\(see)k(e.g.)k([C-F-K-S])10 b(\).)i(Moreo)o(v)o(er,)g Fl(V)21 b Fp(is)11 b(relativ)o(ely)g(compact)g(with)h(resp)q(ect)i(to)d (\()p Fk(p)5 b Fj(\000)g Fk(a)p Fp(\))1675 1618 y Fi(2)262 1683 y Fp([C-F-K-S])12 b(whic)o(h)i(together)h(with)f(the)g(results)h (in)e([I,)g(H])h(it)f(implies)f(that)537 1769 y Fl(\033)561 1775 y Fg(ess)610 1769 y Fp(\()p Fl(H)s Fp(\))f(=)h Fl(\033)q Fp(\()p Fl(H)811 1775 y Fi(0)830 1769 y Fp(\))f(=)h Fj(f)p Fp(\(2)p Fl(n)d Fp(+)g(1\))p Fl(B)1102 1775 y Fi(0)1131 1769 y Fj(j)f Fl(n)k Fp(=)g(0)p Fl(;)7 b Fp(1)p Fl(;)g Fp(2)p Fl(;)g(:)g(:)f(:)m Fj(g)169 b Fp(\(2.22\))324 1855 y(In)14 b(the)g(rest)h(of)f(this)f(section,)620 1941 y Fl(g)f Fj(2)g Fl(C)725 1924 y Fh(1)766 1908 y Ff(\000)785 1941 y Fk(R)821 1924 y Fi(2)840 1941 y Fp(;)7 b Fk(R)895 1908 y Ff(\001)929 1941 y Fp(;)16 b Fj(jj)p Fl(g)q Fj(jj)1026 1948 y Fg(C)1052 1941 y Fd(2)1066 1948 y Fi(\()p Fe(R)1107 1941 y Fd(2)1123 1948 y Fi(\))1150 1941 y Fp(=)c Fl(M)k(<)c Fj(1)252 b Fp(\(2.23\))262 2028 y(Let)20 b Fl(E)k Fj(2)e Fl(\033)471 2034 y Fg(disc)533 2028 y Fp(\()p Fl(H)s Fp(\))e(\(the)h(discrete)h(sp)q(ectrum)f(of)e Fl(H)s Fp(\))h(and)g(let)h Fl( )g Fp(b)q(e)g(a)f(normalised)262 2077 y(eigenfunction)12 b(corresp)q(onding)h(to)g Fl(E)r Fp(.)k(W)m(e)12 b(are)h(in)o(terested)h(no)o(w)e(in)g(con)o(trolling)f (as)i(go)q(o)q(d)262 2127 y(as)g(p)q(ossible)i(the)f(term)f Fj(jj)p Fp([)p Fl(H)q(;)7 b(g)q Fp(])p Fl( )q Fj(jj)p Fp(.)324 2177 y(Under)12 b(the)f(conditions)g(\(2.21\),)f(one)i(has)f Fl(D)q Fp(\()p Fl(H)s Fp(\))h(=)g Fl(D)q Fp(\(\()p Fk(p)e Fj(\000)f Fk(a)p Fp(\))1330 2162 y Fi(2)1349 2177 y Fp(\))i(and)g(\()p Fl(p)1491 2183 y Fg(j)1512 2177 y Fj(\000)t Fl(a)1570 2183 y Fg(j)1588 2177 y Fp(\)\()p Fl(H)6 b Fp(+)262 2227 y Fl(i)p Fp(\))292 2212 y Fh(\000)p Fi(1)350 2227 y Fp(is)14 b(b)q(ounded,)g Fl(j)g Fj(2)d(f)p Fp(1)p Fl(;)16 b Fp(2)p Fj(g)p Fp(.)h(Moreo)o(v)o(er,)d(b)q(ecause)613 2313 y([)p Fl(H)q(;)7 b(g)q Fp(])j(=)h Fj(\000)p Fl(i)p Fj(f)p Fp(\()p Fk(p)f Fj(\000)f Fk(a)p Fp(\))h Fj(\001)e(r)p Fl(g)j Fp(+)e Fj(r)p Fl(g)h Fj(\001)f Fp(\()p Fk(p)g Fj(\000)g Fk(a)q Fp(\))p Fj(g)245 b Fp(\(2.24\))262 2399 y(it)13 b(follo)o(ws)f(that)701 2449 y Fj(jj)p Fp([)p Fl(H)q(;)7 b(g)q Fp(]\()p Fl(H)i Fp(+)h Fl(i)p Fp(\))957 2432 y Fh(\000)p Fi(1)1002 2449 y Fj(jj)g(\024)i Fl(const)p Fp(\()p Fl(M)5 b Fp(\))334 b(\(2.25\))967 2574 y(6)p eop %%Page: 7 7 7 6 bop 262 307 a Fp(whic)o(h)13 b(giv)o(es)h(the)g(follo)o(wing)e (rough)h(result:)668 407 y Fj(jj)p Fp([)p Fl(H)q(;)7 b(g)q Fp(])p Fl( )q Fj(jj)h(\024)k Fl(const)p Fp(\()p Fl(M)5 b Fp(\))10 b(\()p Fl(E)1129 390 y Fi(2)1157 407 y Fp(+)f(1\))1235 390 y Fi(1)p Fg(=)p Fi(2)1588 407 y Fp(\(2.26\))262 481 y(In)j(order)h(to)g(obtain)f(a)g(sharp)q(er)i (estimate)e(on)g(this)h(term,)f(w)o(e)h(use)g(the)g(follo)o(wing)d (form)h(of)262 531 y(the)j(I.B.M.)f(lo)q(calisation)f(lemma)f ([C-F-K-S]:)262 613 y Fk(Lemma)k(2.2)21 b Fb(L)n(et)14 b Fl(')e Fj(2)f Fp(D\()p Fl(H)s Fp(\))p Fb(.)19 b(Then:)534 703 y Fl(<)12 b(';)7 b(g)q(H)s(g)q(')12 b(>)p Fp(=)g Fj(<)p Fp(\()p Fl(<)g(';)7 b(g)976 686 y Fi(2)994 703 y Fl(H)s(')12 b(>)p Fp(\)+)g Fl(<)g(';)7 b Fj(jr)p Fl(g)q Fj(j)1333 686 y Fi(2)1350 703 y Fl(')12 b(>)167 b Fp(\(2.27\))324 793 y Fb(and)651 843 y Fl(<)12 b(g)q( )q(;)7 b Fp(\()p Fl(H)13 b Fj(\000)c Fl(E)r Fp(\))p Fl(g)q( )14 b(>)p Fp(=)p Fl(<)e( )q(;)7 b Fj(jr)p Fl(g)q Fj(j)1214 825 y Fi(2)1232 843 y Fl( )13 b(>)284 b Fp(\(2.28\))324 925 y(Under)15 b(the)f(conditions)g(\(2.21\),)e(there)k(exists)e(a)g (constan)o(t)g Fl(c)e(>)g Fp(0)i(indep)q(enden)o(t)h(of)e Fl( )262 974 y Fp(suc)o(h)h(that:)519 1074 y Fj(j)d Fl(<)h(g)q( )q(;)7 b(V)j(g)q( )j(>)f Fj(j)f(\024)865 1046 y Fp(1)p 865 1064 21 2 v 865 1102 a(2)902 1074 y Fl(<)h(g)q( )q(;)7 b Fp(\()p Fk(p)i Fj(\000)h Fk(a)p Fp(\))1147 1056 y Fi(2)1166 1074 y Fl(g)q( )j(>)f Fp(+)p Fl(c)p Fj(jj)p Fl(g)q( )q Fj(jj)1418 1056 y Fi(2)1588 1074 y Fp(\(2.29\))324 1178 y(F)m(rom)g(\(2.28\))h (and)h(\(2.29\))e(it)i(follo)o(ws)e(that:)673 1268 y Fj(jj)p Fp(\()p Fk(p)c Fj(\000)i Fk(a)p Fp(\))p Fl(g)q( )q Fj(jj)902 1251 y Fi(2)932 1268 y Fj(\021)p Fl(<)i(g)q( )q(;)7 b Fp(\()p Fk(p)i Fj(\000)h Fk(a)p Fp(\))1209 1251 y Fi(2)1228 1268 y Fl(g)q( )j(>)p Fj(\024)673 1351 y(\024)f Fp(2)p Fl(M)783 1334 y Fi(2)801 1351 y Fp([)p Fj(j)p Fl(E)r Fj(j)c Fp(+)h Fl(c)h Fp(+)f(1])1028 1295 y Ff(Z)1050 1389 y Fg(supp)e(g)1154 1351 y Fl(d)p Fk(x)p Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)1310 1334 y Fi(2)1588 1351 y Fp(\(2.30\))324 1470 y(After)14 b(a)g(little)f(calculation,)f(from)h (\(2.24\))f(and)i(\(2.30\))f(one)h(obtains:)526 1598 y Fj(jj)p Fp([)p Fl(H)q(;)7 b(g)q Fp(])p Fl( )q Fj(jj)702 1581 y Fi(2)729 1598 y Fj(\024)21 b Fl(const)p Fp(\()p Fl(E)r(;)7 b(M)e Fp(\))1015 1527 y Ff( )1048 1541 y(Z)1071 1635 y Fg(supp)i Fh(jr)p Fg(g)q Fh(j)1222 1598 y Fl(d)p Fk(x)p Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)1378 1581 y Fi(2)1396 1527 y Ff(!)1588 1598 y Fp(\(2.31\))262 1723 y(whic)o(h)13 b(is)h(the)h(needed)g(estimate.)324 1773 y(W)m(e'll)10 b(sho)o(w)j(no)o(w)f(that)g(there)i(are)f(man)o(y)d (examples)h(of)h(one)h(w)o(ell)e(hamiltonians)f(with)262 1822 y(discrete)19 b(sp)q(ectrum)g(in)e(b)q(et)o(w)o(een)i(the)g (Landau)e(lev)o(els.)30 b(W)m(e)18 b(put)g(this)g(in)g(the)g(form)e(of) 262 1872 y(t)o(w)o(o)e(prop)q(ositions:)21 b(the)15 b(\014rst)h(one)g (constructs)h(a)e(purely)g(electric)h(w)o(ell)f(whic)o(h)g(giv)o(es)g (an)262 1922 y(eigen)o(v)n(alue)d(lo)q(cated)h(an)o(ywhere)g(w)o(e)g(w) o(an)o(t)g(outside)g Fl(\033)1112 1928 y Fg(L)1136 1922 y Fp(\()p Fl(B)1183 1928 y Fi(0)1203 1922 y Fp(\))g(and)f(the)i(second) f(one)g(states)262 1972 y(that)20 b(an)o(y)g(su\016cien)o(tly)g (\\small")d(purely)k(magnetic)e(w)o(ell)g(with)h(de\014nite)h(sign)f (creates)262 2022 y(eigen)o(v)n(alues)13 b Fb(ne)n(ar)h Fp(an)o(y)g(Landau)f(lev)o(el)h(w)o(e)g(c)o(ho)q(ose.)262 2112 y Fk(Prop)q(ositi)o(on)f(2.1)20 b Fb(T)m(ake)h Fl(\025)i Fj(2)f Fk(R)p Fb(,)f Fl(\025)28 b(=)-26 b Fj(2)22 b Fl(\033)996 2118 y Fg(L)1020 2112 y Fp(\()p Fl(B)1067 2118 y Fi(0)1087 2112 y Fp(\))p Fb(.)37 b(Then)21 b(ther)n(e)f(exists)h(a)g(b)n(ounde)n (d,)262 2162 y(c)n(omp)n(actly)13 b(supp)n(orte)n(d)h(p)n(otential)f Fl(V)21 b Fj(2)12 b Fk(L)917 2146 y Fh(1)958 2128 y Ff(\000)977 2162 y Fk(R)1013 2146 y Fi(2)1032 2128 y Ff(\001)1064 2162 y Fb(such)j(that)e Fl(\025)h Fb(is)f(a)h(discr)n(ete)f(eigenvalue) 262 2211 y(for)h(the)h(op)n(er)n(ator)f(sum)h Fl(H)g Fp(=)c Fl(H)776 2217 y Fi(0)804 2211 y Fp(+)e Fl(V)h Fb(.)324 2301 y(Pr)n(o)n(of:)324 2351 y Fp(Fix)j Fl(\025)h Fp(as)g(men)o(tioned)f(ab)q(o)o(v)o(e.)18 b(F)m(rom)12 b(Lemma)f(2.1,)h(one)i(can)h(easily)e(see)i(that)632 2449 y Fl(K)667 2455 y Fi(0)686 2449 y Fp(\()p Fk(x)p Fl(;)7 b Fp(0;)g Fl(\025)p Fp(\))k(=)p 881 2407 194 2 v 12 w Fl(K)916 2455 y Fi(0)935 2449 y Fp(\(0)p Fl(;)c Fk(x)p Fp(;)p 1035 2415 25 2 v 7 w Fl(\025)o Fp(\))12 b(=)p 1130 2413 194 2 v 12 w Fl(K)1165 2455 y Fi(0)1183 2449 y Fp(\()p Fk(x)p Fl(;)7 b Fp(0;)g Fl(\025)p Fp(\))265 b(\(2.32\))967 2574 y(7)p eop %%Page: 8 8 8 7 bop 262 307 a Fp(where)14 b(the)f(o)o(v)o(erline)g(means)f(complex) f(conjugation.)17 b(Because)e(the)e(con\015uen)o(t)h(h)o(yp)q(erge-)262 357 y(ometric)h(function)h Fl(U)5 b Fp(\()p Fl(\013;)i Fp(1)p Fl(;)g(\030)r Fp(\))15 b(is)i(analytic)f(in)g Fj(f)p Fl(\030)h Fj(2)f Fk(C)p Fl(;)23 b Fj(<)p Fl(\030)18 b(>)e Fp(0)p Fj(g)g Fp(and)g(together)i(with)262 407 y(the)g(realt)o(y)f(of)g Fl(K)544 413 y Fi(0)563 407 y Fp(\()p Fk(x)p Fl(;)7 b Fp(0;)g Fl(\025)p Fp(\))17 b(one)h(obtains)f(the)h(existence)i(of)c Fl(A)i(>)g Fp(0,)g(0)g Fl(<)g(\017)f(<)h(A)g Fp(and)262 457 y(0)11 b Fl(<)h(\016)h(<)f Fp(1)g(suc)o(h)i(that)f(if)f Fl(A)7 b Fj(\000)g Fl(\017)k Fj(\024)h(j)p Fk(x)p Fj(j)f(\024)h Fl(A)7 b Fp(+)g Fl(\017)p Fp(,)12 b(then)i Fl(K)1165 463 y Fi(0)1184 457 y Fp(\()p Fk(x)p Fl(;)7 b Fp(0;)g Fl(\025)p Fp(\))12 b(is)g(not)h(c)o(hanging)f (sign)262 506 y(and)h(moreo)o(v)o(er,)g(one)h(can)g(supp)q(ose)h (without)f(loss)f(that)843 598 y Fl(K)878 604 y Fi(0)897 598 y Fp(\()p Fk(x)p Fl(;)7 b Fp(0;)g Fl(\025)p Fp(\))k Fj(\025)h Fl(\016)478 b Fp(\(2.33\))262 689 y(De\014ne)14 b(no)o(w:)733 830 y Fl(\021)754 836 y Fi(1)784 830 y Fj(2)d Fl(C)856 813 y Fh(1)853 840 y Fi(0)891 830 y Fp(\()p Fk(R)943 813 y Fi(2)961 830 y Fp(\))p Fl(;)49 b Fp(0)11 b Fj(\024)h Fl(\021)1135 836 y Fi(1)1164 830 y Fj(\024)g Fp(1)d(and)733 917 y Fl(\021)754 923 y Fi(1)772 917 y Fp(\()p Fk(x)p Fp(\))j(=)885 859 y Ff(\032)937 892 y Fp(1)41 b(if)g Fj(j)p Fk(x)p Fj(j)11 b(\024)h Fl(A)937 942 y Fp(0)41 b(if)g Fj(j)p Fk(x)p Fj(j)11 b(\025)h Fl(A)d Fp(+)h Fl(\017)1588 917 y Fp(\(2.34\))733 1100 y Fl(\021)754 1106 y Fi(2)784 1100 y Fj(2)h Fl(C)856 1083 y Fh(1)891 1100 y Fp(\()p Fk(R)943 1083 y Fi(2)961 1100 y Fp(\))p Fl(;)49 b Fp(0)11 b Fj(\024)h Fl(\021)1135 1106 y Fi(2)1164 1100 y Fj(\024)g Fp(1)d(and)733 1187 y Fl(\021)754 1193 y Fi(2)772 1187 y Fp(\()p Fk(x)p Fp(\))j(=)885 1129 y Ff(\032)937 1162 y Fp(0)41 b(if)g Fj(j)p Fk(x)p Fj(j)11 b(\024)h Fl(A)d Fj(\000)h Fl(\017)937 1212 y Fp(1)41 b(if)g Fj(j)p Fk(x)p Fj(j)11 b(\025)h Fl(A)1588 1187 y Fp(\(2.35\))688 1328 y Fl( )q Fp(\()p Fk(x)p Fp(\))h(=)f Fl(\021)851 1334 y Fi(1)869 1328 y Fp(\()p Fk(x)p Fp(\))e(+)f Fl(\021)998 1334 y Fi(2)1016 1328 y Fp(\()p Fk(x)p Fp(\))p Fl(K)1108 1334 y Fi(0)1128 1328 y Fp(\()p Fk(x)p Fl(;)e Fp(0;)g Fl(\025)p Fp(\))320 b(\(2.36\))324 1403 y(W)m(e)13 b(also)h(require)g Fl(\021)639 1409 y Fi(1)671 1403 y Fp(and)g Fl(\021)773 1409 y Fi(2)805 1403 y Fp(to)g(b)q(e)g(radially)f (symmetric.)324 1453 y(Using)j(\(2.33\))f(and)h(the)h(de\014nitions)f (of)g(the)h(cutt-o\013)g(functions,)f(one)h(obtains)f(that)262 1503 y Fl( )d Fj(2)e Fk(L)370 1488 y Fi(2)395 1469 y Ff(\000)414 1503 y Fk(R)450 1488 y Fi(2)468 1469 y Ff(\001)501 1503 y Fp(and)j Fl( )q Fp(\()p Fk(x)p Fp(\))f Fj(\025)e Fl(\016)16 b Fp(if)d Fj(j)p Fk(x)p Fj(j)e(\024)h Fl(A)d Fp(+)h Fl(\017)p Fp(.)324 1557 y(T)m(ak)o(e)j(no)o(w)h(\010)d Fj(2)g Fk(L)620 1542 y Fh(1)662 1523 y Ff(\000)681 1557 y Fk(R)717 1542 y Fi(2)736 1523 y Ff(\001)768 1557 y Fp(suc)o(h)k(that:)632 1716 y(\010\()p Fk(x)p Fp(\))d(=)775 1657 y Ff(\032)858 1671 y Fi(1)p 831 1678 69 2 v 831 1702 a Fg( )q Fi(\()p Fe(x)p Fi(\))947 1687 y Fp(if)40 b Fj(j)p Fk(x)p Fj(j)11 b(\024)h Fl(A)d Fp(+)h Fl(\017)884 1744 y Fp(1)42 b(if)e Fj(j)p Fk(x)p Fj(j)11 b Fl(>)h(A)d Fp(+)h Fl(\017)1588 1716 y Fp(\(2.37\))324 1833 y(Finally)m(,)h(the)k (p)q(oten)o(tial)e(w)o(e)h(are)g(lo)q(oking)f(for)g(will)g(b)q(e:)592 1933 y Fl(V)21 b Fp(=)12 b Fj(\000)p Fp(\010)e Fj(\001)e(f)p Fp(\()p Fl(H)845 1939 y Fi(0)873 1933 y Fj(\000)h Fl(\025)p Fp(\))p Fl(\021)975 1939 y Fi(1)1003 1933 y Fp(+)g([)p Fl(H)1091 1939 y Fi(0)1109 1933 y Fl(;)e(\021)1149 1939 y Fi(2)1167 1933 y Fp(])p Fl(K)s Fp(\()p Fj(\001)p Fl(;)g Fp(0;)g Fl(\025)p Fp(\))p Fj(g)223 b Fp(\(2.38\))324 2007 y(Due)17 b(to)f(the)h(fact)g(that)g Fk(a)742 2013 y Fi(0)777 2007 y Fp(is)g(written)g(in)f(the)h(transv)o(ersal)h(gauge)e (\(whic)o(h)h(implies)262 2057 y Fk(a)285 2063 y Fi(0)303 2057 y Fp(\()p Fk(x)p Fp(\))12 b Fj(\001)e Fk(x)15 b Fp(=)h(0\),)g(it)g(follo)o(ws)e(that)i Fl(H)856 2063 y Fi(0)890 2057 y Fp(maps)f(radially)g(symmetric)f(functions)i(in)o(to) g(real)262 2107 y(functions,)i(and)f(that)h Fl(V)27 b Fp(is)18 b(real,)g(b)q(ounded)h(and)e(compactly)g(supp)q(orted.)31 b(Moreo)o(v)o(er,)262 2157 y Fl(H)14 b( )f Fp(=)f Fl(\025)g( )q Fp(.)262 2248 y Fk(Prop)q(ositi)o(on)h(2.2)20 b Fb(L)n(et)d Fl(B)708 2233 y Fh(0)734 2248 y Fj(2)d Fl(C)809 2233 y Fi(1)806 2259 y(0)835 2215 y Ff(\000)854 2248 y Fk(R)890 2233 y Fi(2)908 2248 y Fp(;)7 b Fk(R)963 2215 y Ff(\001)998 2248 y Fb(b)n(e)16 b(a)h(nonne)n(gative,)h(c)n(omp)n(actly)e(supp)n (orte)n(d)262 2298 y(function)f(and)h(let)e Fk(a)587 2283 y Fh(0)599 2298 y Fp(\()p Fk(x)p Fp(\))h Fb(b)n(e)g(the)g(tr)n (ansversal)f(gauge)i(which)e(gives)h Fl(B)1368 2283 y Fh(0)1380 2298 y Fb(.)324 2348 y(F)m(or)h Fl(b)f(>)g Fp(0)p Fb(,)i(de\014ne)h Fl(H)692 2354 y Fg(b)723 2348 y Fp(=)d(\()p Fk(p)c Fj(\000)g Fk(a)890 2354 y Fi(0)919 2348 y Fj(\000)g Fl(b)p Fk(a)1003 2333 y Fh(0)1015 2348 y Fp(\))1031 2333 y Fi(2)1049 2348 y Fb(.)25 b(L)n(et)17 b Fl(E)1192 2354 y Fg(n)1229 2348 y Fp(=)e(\(2)p Fl(n)10 b Fp(+)h(1\))p Fl(B)1459 2354 y Fi(0)1495 2348 y Fb(b)n(e)17 b(the)g Fl(n)p Fb(-th)262 2398 y(L)n(andau)d(level.)k(Then)c(for)f Fl(b)h Fb(su\016ciently)f(smal)r(l,)g Fl(H)1087 2404 y Fg(b)1117 2398 y Fb(wil)r(l)g(have)h(at)f(le)n(ast)h(one)g (eigenvalue)262 2447 y(ne)n(ar)g Fl(E)386 2453 y Fg(n)409 2447 y Fb(.)967 2574 y Fp(8)p eop %%Page: 9 9 9 8 bop 262 307 a Fb(Pr)n(o)n(of)p Fp(:)324 357 y(Because)16 b Fl(B)516 342 y Fh(0)543 357 y Fp(has)f(compact)f(supp)q(ort,)i(one)f (has)g Fj(j)p Fk(a)1139 342 y Fh(0)1150 357 y Fp(\()p Fk(x)p Fp(\))p Fj(j)e(\024)26 b Fl(const)p Fj(\001)13 b Fl(<)h Fk(x)f Fl(>)1528 342 y Fh(\000)p Fi(1)1587 357 y Fp(where)262 421 y Fl(<)e Fk(x)h Fl(>)p Fj(\021)418 387 y Ff(\000)437 421 y Fp(1)d(+)h Fk(x)534 406 y Fi(2)552 387 y Ff(\001)577 385 y Fd(1)p 577 390 15 2 v 577 406 a(2)598 421 y Fp(.)324 471 y(Denote)16 b(with)f Fl(W)6 b Fp(\()p Fl(b)p Fp(\))14 b(=)g Fj(\000)p Fl(b)p Fp(\()p Fk(p)c Fj(\000)g Fk(a)887 477 y Fi(0)906 471 y Fp(\))h Fj(\001)e Fk(a)977 456 y Fh(0)999 471 y Fj(\000)i Fl(b)p Fk(a)1083 456 y Fh(0)1105 471 y Fj(\001)f Fp(\()p Fk(p)f Fj(\000)i Fk(a)1245 477 y Fi(0)1264 471 y Fp(\))f(+)h Fl(b)1351 456 y Fi(2)1369 471 y Fk(a)1392 456 y Fh(0)p Fi(2)1436 471 y Fp(and)k(with)g Fl(V)23 b Fp(=)262 521 y Fj(\000)p Fp(\()p Fk(p)7 b Fj(\000)g Fk(a)406 527 y Fi(0)424 521 y Fp(\))g Fj(\001)g Fk(a)489 506 y Fh(0)507 521 y Fj(\000)g Fk(a)569 506 y Fh(0)588 521 y Fj(\001)g Fp(\()p Fk(p)g Fj(\000)g Fk(a)719 527 y Fi(0)736 521 y Fp(\).)18 b(It)13 b(is)f(easy)h(to)g(see)h(that)e Fl(W)6 b Fp(\()p Fl(b)p Fp(\))13 b(is)g(relativ)o(ely)e(b)q(ounded)j(to)262 571 y Fl(H)297 577 y Fi(0)315 571 y Fp(;)h(one)g(can)g(then)g(aply)f (the)i(analytic)e(p)q(erturbation)h(theory)h(around)f Fl(E)1464 577 y Fg(n)1501 571 y Fp(if)f Fl(b)g Fp(is)h(k)o(ept)262 620 y(small)c(enough.)18 b(The)d(reduced)g(op)q(erator)g(de\014ned)g (in)e(Ran)p Fl(P)1230 626 y Fg(n)1266 620 y Fp(\()p Fl(P)1309 626 y Fg(n)1345 620 y Fp(b)q(eing)h(the)h(pro)r(jector)262 670 y(asso)q(ciated)f(with)g Fl(E)585 676 y Fg(n)607 670 y Fp(\))g(will)e(ha)o(v)o(e)i(the)h(form:)755 754 y Fl(H)790 760 y Fg(ef)s(f)846 754 y Fp(\()p Fl(b)p Fp(\))d(=)g Fl(E)983 760 y Fg(n)1005 754 y Fl(P)1032 760 y Fg(n)1063 754 y Fp(+)e Fl(P)1132 760 y Fg(n)1154 754 y Fl(T)c Fp(\()p Fl(b)p Fp(\))p Fl(P)1261 760 y Fg(n)755 816 y Fl(T)g Fp(\()p Fl(b)p Fp(\))11 b Fj(\021)h Fl(bP)935 822 y Fg(n)957 816 y Fl(V)e(P)1018 822 y Fg(n)1049 816 y Fp(+)g Fj(O)q Fp(\()p Fl(b)1159 799 y Fi(2)1178 816 y Fp(\))p Fl(:)382 b Fp(\(2.39\))262 899 y(The)14 b(only)f(thing)g(w)o(e)h(should)g(c)o (hec)o(k)h(is)e(that)h Fl(P)1009 905 y Fg(n)1031 899 y Fl(T)6 b Fp(\()p Fl(b)p Fp(\))p Fl(P)1138 905 y Fg(n)1175 899 y Fp(is)13 b(not)h(zero;)g(this)g(w)o(ould)f(imply)262 949 y(that)e Fl(H)384 955 y Fg(ef)s(f)441 949 y Fp(\()p Fl(b)p Fp(\))5 b Fj(\000)g Fl(E)564 955 y Fg(n)598 949 y Fp(has)12 b(nonzero)h(sp)q(ectrum,)f(therefore)i Fl(H)1222 955 y Fg(b)1249 949 y Fp(will)d(ha)o(v)o(e)h(\(discrete\))h(sp)q(ec-) 262 999 y(trum)f(near)i Fl(E)488 1005 y Fg(n)510 999 y Fp(.)k(T)m(o)12 b(ac)o(hiev)o(e)i(that,)f(one)g(can)h(compute)f Fl(<)e(f)1227 1005 y Fg(n)1250 999 y Fl(;)c(V)i(f)1322 1005 y Fg(n)1357 999 y Fl(>)k Fp(where)i Fl(f)1542 1005 y Fg(n)1578 999 y Fp(stands)262 1049 y(for)e(the)i(sferically)e (symetric,)g(real)h(eigenfunction)g(of)g Fl(H)1168 1055 y Fi(0)1186 1049 y Fp(,)f(corresp)q(onding)i(to)f Fl(E)1560 1055 y Fg(n)1582 1049 y Fp(.)19 b(This)262 1098 y(computation)12 b(giv)o(es:)315 1212 y Fl(<)g(f)379 1218 y Fg(n)401 1212 y Fl(;)7 b(V)i(f)473 1218 y Fg(n)508 1212 y Fl(>)p Fp(=)j(2)f Fl(<)h(f)680 1218 y Fg(n)703 1212 y Fl(;)7 b Fk(a)745 1218 y Fi(0)772 1212 y Fj(\001)i Fk(a)816 1195 y Fh(0)828 1212 y Fl(f)848 1218 y Fg(n)882 1212 y Fl(>)p Fp(=)k Fl(B)990 1218 y Fi(0)1016 1156 y Ff(Z)1064 1212 y Fl(d)p Fk(x)e Fl(f)1146 1195 y Fi(2)1142 1222 y Fg(n)1166 1212 y Fp(\()p Fk(x)p Fp(\))1230 1156 y Ff(Z)1272 1166 y Fh(j)p Fe(x)p Fh(j)1253 1250 y Fi(0)1320 1212 y Fl(d\032)h(\032B)1429 1195 y Fh(0)1442 1212 y Fp(\()p Fl(\032;)7 b(\022)q Fp(\))54 b(\(2.40\))262 1319 y(This)12 b(quan)o(tit)o(y)f(is)h(not)g(zero)i(b)q (ecause)f Fl(B)902 1304 y Fh(0)927 1319 y Fp(is)f(not)g(c)o(hanging)g (sign;)g(therefore,)h(if)f Fl(b)g Fp(is)g(small)262 1369 y(enough,)h Fl(P)443 1375 y Fg(n)465 1369 y Fl(T)6 b Fp(\()p Fl(b)p Fp(\))p Fl(P)572 1375 y Fg(n)606 1369 y Fj(6)p Fp(=)12 b(0.)324 1419 y Fb(R)n(emark)p Fp(:)324 1468 y(This)k(t)o(yp)q(e)g(of)f(argumen)o(t)g(also)g(w)o(orks)h(in)f (the)i(case)g(of)e(a)h(purely)g(electric)h(w)o(ell;)e(one)262 1518 y(only)g(has)i(to)f(c)o(hec)o(k)h(that)f(the)h(term)f(in)g (\(2.40\))f(\(where)j Fl(V)26 b Fp(stands)17 b(no)o(w)f(for)g(the)h (scalar)262 1568 y(p)q(oten)o(tial\))10 b(is)g(di\013eren)o(t)h(from)e (zero.)18 b(In)10 b(conclusion,)h(it)f(is)g(not)h(di\016cult)e(to)i (giv)o(e)f(examples)262 1618 y(of)j(one)i(w)o(ell)e(hamiltonians)e (with)j(discrete)i(sp)q(ectrum)f(outside)f(the)h(Landau)f(lev)o(els;)g (the)262 1668 y(really)f(hard)h(problem)e(is)i(to)g(study)g(their)g(b)q (eha)o(viour)g(near)g(the)h(essen)o(tial)f(sp)q(ectrum.)262 1804 y Fo(3)66 b(Gaps)22 b(in)i(the)e(essen)n(tial)h(sp)r(ectrum)262 1894 y Fp(Consider)684 1944 y(\000)710 1950 y Fg(N)753 1944 y Fp(=)12 b Fj(f)p Fk(x)p Fp(\()p Fl(i)p Fp(\))p Fj(g)910 1923 y Fg(N)910 1957 y(i)p Fi(=1)978 1944 y Fj(\032)g Fk(R)1058 1927 y Fi(2)1076 1944 y Fl(;)48 b(N)16 b Fj(\024)c(1)337 b Fp(\(3.1\))324 2014 y(Without)11 b(loss)h(of)f(generalit)o(y)m(,)g(one)i(can)f(alw)o(a)o(ys)f(tak)o(e)h Fk(x)p Fp(\(1\))f(=)h(0.)17 b(The)c(main)d(assump-)262 2064 y(tion)j(ab)q(out)h(\000)492 2070 y Fg(N)537 2064 y Fp(is)g(that:)758 2114 y(inf)753 2141 y Fg(j)r Fh(6)p Fi(=)p Fg(k)819 2114 y Fj(j)p Fk(x)p Fp(\()p Fl(j)r Fp(\))c Fj(\000)g Fk(x)p Fp(\()p Fl(k)q Fp(\))p Fj(j)h Fp(=)h Fl(r)g(>)g Fp(0)406 b(\(3.2\))262 2204 y(and)13 b(since)i(the)g(limit)c (to)j(b)q(e)g(considered)i(is)d Fl(r)g Fj(!)e(1)p Fp(,)i(w)o(e)h (assume)g(for)f(tec)o(hnical)i(reasons)262 2254 y(that)e Fl(r)i Fp(is)f(su\016cien)o(tly)g(large,)f(sa)o(y)h Fl(r)e Fj(\025)g Fp(1000.)324 2304 y(Concerning)i(the)g(magnetic)f(\014eld)h (and)g(the)g(p)q(oten)o(tial,)f(w)o(e)h(assume:)598 2387 y Fl(B)629 2393 y Fi(0)659 2387 y Fl(>)e Fp(0)d Fl(;)16 b(B)794 2370 y Fh(0)817 2387 y Fj(2)c Fl(C)890 2370 y Fi(1)908 2387 y Fp(\()p Fk(R)960 2370 y Fi(2)978 2387 y Fp(\))e(and)f Fl(supp)g(B)1207 2370 y Fh(0)1231 2387 y Fj(\032)j(fj)p Fk(x)p Fj(j)e(\024)i Fp(1)p Fj(g)598 2449 y Fl(V)21 b Fj(2)11 b Fl(L)710 2432 y Fi(2)729 2449 y Fp(\()p Fk(R)781 2432 y Fi(2)799 2449 y Fp(\))p Fl(;)16 b(supp)9 b(V)21 b Fj(\032)12 b(fj)p Fk(x)p Fj(j)f(\024)g Fp(1)p Fj(g)416 b Fp(\(3.3\))967 2574 y(9)p eop %%Page: 10 10 10 9 bop 324 307 a Fp(Let)411 424 y Fl(B)442 430 y Fg(N)474 424 y Fp(\()p Fk(x)p Fp(\))42 b(=)f Fl(B)677 430 y Fi(0)706 424 y Fp(+)762 372 y Fg(N)747 384 y Ff(X)748 472 y Fg(j)r Fi(=1)814 424 y Fl(B)847 406 y Fh(0)859 424 y Fp(\()p Fk(x)10 b Fj(\000)f Fk(x)p Fp(\()p Fl(j)r Fp(\)\))p Fl(;)17 b(V)1096 430 y Fg(N)1128 424 y Fp(\()p Fk(x)p Fp(\))12 b(=)1256 372 y Fg(N)1241 384 y Ff(X)1242 472 y Fg(j)r Fi(=1)1308 424 y Fl(V)d Fp(\()p Fk(x)h Fj(\000)f Fk(x)p Fp(\()p Fl(j)r Fp(\)\))p Fl(;)419 564 y Fk(a)442 570 y Fg(N)474 564 y Fp(\()p Fk(x)p Fp(\))42 b(=)646 507 y Ff(Z)688 518 y Fi(1)669 602 y(0)713 564 y Fl(ds)10 b(s)f Fk(B)826 570 y Fg(N)858 564 y Fp(\()p Fl(s)h Fk(x)p Fp(\))f Fj(^)g Fk(x)593 b Fp(\(3.4\))324 671 y(Consider)14 b(no)o(w)g(for)f Fl(N)k Fp(=)11 b(1)p Fl(;)c Fp(2)p Fl(;)g(:)g(:)g(:)t (;)g Fj(1)13 b Fp(the)h(follo)o(wing)e(family)e(of)k(hamiltonia)o(ns:) 767 757 y Fl(H)802 763 y Fg(N)844 757 y Fp(=)e(\()p Fk(p)d Fj(\000)g Fk(a)1004 763 y Fg(N)1036 757 y Fp(\))1052 740 y Fi(2)1080 757 y Fp(+)g Fl(V)1145 763 y Fg(N)1177 757 y Fl(:)419 b Fp(\(3.5\))324 843 y(These)15 b(op)q(erators)g(are)f (essen)o(tialy)g(self-adjoin)o(t)f(on)h Fl(C)1189 828 y Fh(1)1186 853 y Fi(0)1223 843 y Fp(\()p Fk(R)1275 828 y Fi(2)1294 843 y Fp(\))g(and)g(for)f Fl(N)k(<)11 b Fj(1)799 929 y Fl(\033)823 935 y Fg(ess)872 929 y Fp(\()p Fl(H)923 935 y Fg(N)954 929 y Fp(\))h(=)f Fl(\033)1049 935 y Fg(L)1074 929 y Fp(\()p Fl(B)1121 935 y Fi(0)1140 929 y Fp(\))452 b(\(3.6\))324 1015 y(In)14 b(particular,)792 1065 y Fl(H)827 1071 y Fi(1)856 1065 y Fp(=)e(\()p Fk(p)d Fj(\000)g Fk(a)1016 1071 y Fi(1)1035 1065 y Fp(\))1051 1048 y Fi(2)1079 1065 y Fp(+)h Fl(V)1145 1071 y Fi(1)1608 1065 y Fp(\(3.7\))262 1137 y(is)j(the)i("one)f(w)o(ell")f(hamiltonia)o(n.)324 1187 y(The)h(main)e(result)j(of)e(this)h(section)g(is)g(con)o(tained)g (in)262 1265 y Fk(Theorem)g(3.1)21 b Fb(L)n(et)14 b Fp(1)d Fj(\024)h Fl(N)k(<)c Fj(1)p Fb(,)h Fl(c)f(<)f(d)p Fb(,)j Fl(K)g Fj(\021)e Fp([)p Fl(c;)7 b(d)p Fp(])p Fb(,)12 b Fl(K)e Fj(\\)d Fl(\033)1281 1271 y Fg(L)1305 1265 y Fp(\()p Fl(B)1352 1271 y Fi(0)1371 1265 y Fp(\))12 b(=)g Fj(;)h Fb(and)i(supp)n(ose)262 1315 y(that)499 1401 y Fl(\033)q Fp(\()p Fl(H)575 1407 y Fi(1)593 1401 y Fp(\))9 b Fj(\\)g Fl(K)45 b Fp(=)d Fj(f)p Fl(E)861 1407 y Fi(1)890 1401 y Fl(<)12 b(:)7 b(:)g(:)j(<)i(E)1069 1407 y Fg(s)1086 1401 y Fj(g)g(\032)g Fl(\033)1187 1407 y Fg(disc)1248 1401 y Fp(\()p Fl(H)1299 1407 y Fi(1)1317 1401 y Fp(\))p Fl(;)k(s)c Fj(\025)g Fp(1)529 1464 y(m)o(ult)n(\()p Fl(E)660 1470 y Fg(j)677 1464 y Fp(\))42 b(=)g Fl(m)845 1470 y Fg(j)863 1464 y Fl(;)16 b(j)e Fj(2)d(f)p Fp(1)p Fl(;)c(:)g(:)g(:)t(s)p Fj(g)324 1550 y Fb(If)13 b Fl(c;)7 b(d)13 b Fb(ar)n(e)g(not)h (eigenvalues)g(for)f Fl(H)895 1556 y Fi(1)913 1550 y Fb(,)h(then)g(ther)n(e)f(exist)h Fl(r)1248 1556 y Fi(0)1266 1550 y Fp(\()p Fl(K)q(;)7 b(m)1373 1556 y Fi(1)1391 1550 y Fl(;)g(:)g(:)g(:)e(;)i(m)1520 1556 y Fg(s)1538 1550 y Fp(\))p Fl(;)g Fj(C)13 b Fl(<)f Fj(1)262 1600 y Fb(and)j Fl(u)c(>)h Fp(0)j(indep)q(enden)o(t)g(of)e(N)i Fb(such)h(that)331 1717 y Fl(i)p Fp(\))10 b Fl(\033)q Fp(\()p Fl(H)447 1723 y Fg(N)478 1717 y Fp(\))f Fj(\\)g Fl(K)15 b Fj(\032)655 1665 y Fg(s)639 1677 y Ff([)634 1766 y Fg(j)r Fi(=1)691 1717 y Fp([)p Fl(E)734 1723 y Fg(j)760 1717 y Fj(\000)10 b Fl(\016)o(;)d(E)869 1723 y Fg(j)895 1717 y Fp(+)j Fl(\016)r Fp(])p Fl(;)15 b Fp(0)c Fj(\024)h Fl(\016)h Fj(\024)f(C)r Fl(e)1190 1699 y Fh(\000)p Fg(u)c(r)1260 1687 y Fd(2)1279 1717 y Fl(;)15 b Fb(for)g(al)r(l)8 b Fl(r)13 b Fj(\025)f Fl(r)1521 1723 y Fi(0)1608 1717 y Fp(\(3.8\))639 1921 y Fl(ii)p Fp(\))e(dim)n Fj(f)p Fp(Ran)e Fj(P)895 1927 y Fg(N)927 1921 y Fp([)p Fl(\033)q Fp(\()p Fl(H)1015 1927 y Fg(N)1046 1921 y Fp(\))h Fj(\\)g Fl(K)s Fp(])p Fj(g)i Fp(=)h Fl(N)1301 1869 y Fg(s)1279 1881 y Ff(X)1280 1970 y Fg(j)r Fi(=1)1345 1921 y Fl(m)1381 1927 y Fg(j)1608 1921 y Fp(\(3.9\))262 2049 y Fb(wher)n(e)i Fj(P)408 2055 y Fg(N)454 2049 y Fb(is)h(the)f(sp)n(e)n(ctr)n(al)g(me)n(asur)n(e)h (asso)n(ciate)n(d)g(with)f Fl(H)1195 2055 y Fg(N)1226 2049 y Fb(.)324 2128 y(R)n(emark:)k Fp(If)c(one)g(drops)h(the)g (compactness)f(condition)g(in)f(\(3.3\))h(but)g(imp)q(oses)f(addi-)262 2177 y(tional)e(conditions)i(to)g(\(2.21\))f(in)h(order)g(to)g(ensure)i (the)f(\014niteness)g(of)f(the)g("total)f(p)q(ertur-)262 2227 y(bations")h(in)g Fl(H)511 2233 y Fh(1)546 2227 y Fp(,)g(suc)o(h)i(as:)539 2313 y(max)n Fj(fj)p Fl(B)682 2296 y Fh(0)693 2313 y Fp(\()p Fk(x)p Fp(\))p Fj(j)p Fl(;)7 b Fj(j)p Fl(V)i Fp(\()p Fk(x)p Fp(\))p Fj(jg)i(\024)21 b Fl(const)10 b Fp(\(1)f(+)g Fj(j)p Fk(x)p Fj(j)p Fp(\))1239 2296 y Fh(\000)p Fg(\014)1287 2313 y Fl(;)16 b(\014)e(>)e Fp(2)171 b(\(3.10\))262 2399 y(then)13 b(the)g(pro)q(of)g(of)f(Theorem) g(3.1)g(can)h(b)q(e)g(adapted)g(suc)o(h)h(that)f Fb(ii\))f Fp(remains)g(true)h(and)g Fb(i\))262 2449 y Fp(is)h(c)o(hanging)g(in)g (the)i(sense)g(that)f(instead)g(of)f(a)g(gaussian)g(deca)o(y)i(in)e Fl(r)q Fp(,)g(w)o(e)h(can)g(only)f(sa)o(y)957 2574 y(10)p eop %%Page: 11 11 11 10 bop 262 307 a Fp(that)13 b Fl(\016)i Fp(go)q(es)f(to)f(zero)i (when)e Fl(r)i 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(helpful)g(tec)o(hnical)g(lemmas)d(and)j(w)o(e)h(start)g(with)e(some)h (de\014ni-)262 825 y(tions.)j(F)m(or)d(\()p Fl(p)491 831 y Fi(1)510 825 y Fl(;)7 b(p)550 831 y Fi(2)567 825 y Fp(\))12 b Fj(2)f Fk(Z)663 810 y Fi(2)696 825 y Fp(and)j Fl(\016)f(>)f Fp(0,)h(de\014ne:)353 933 y Fl(K)s Fp(\()p Fl(p)428 939 y Fi(1)447 933 y Fl(;)7 b(p)487 939 y Fi(2)505 933 y Fp(;)g Fl(\016)r Fp(\))k(=)615 874 y Ff(\032)646 933 y Fk(x)h Fj(2)f Fk(R)758 916 y Fi(2)777 933 y Fj(j)d(j)p Fl(x)833 939 y Fg(j)859 933 y Fj(\000)927 905 y Fl(r)p 906 923 63 2 v 906 961 a Fp(100)980 874 y Ff(\022)1010 933 y Fl(p)1031 939 y Fg(j)1058 933 y Fp(+)1105 905 y(1)p 1105 923 21 2 v 1105 961 a(2)1130 874 y Ff(\023)1168 933 y Fj(j)j(\024)1240 905 y Fl(\016)p 1240 923 V 1240 961 a Fp(2)1265 933 y Fl(;)16 b(j)e Fj(2)d(f)p Fp(1)p Fl(;)c Fp(2)p Fj(g)1466 874 y Ff(\033)1588 933 y Fp(\(3.12\))324 1038 y(It)14 b(is)f(easily)h(seen)h(that)f(for)g(an)o(y)f Fl(\016)g Fj(\025)944 1022 y Fg(r)p 927 1029 50 2 v 927 1053 a Fi(100)982 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Fp(suc)o(h)262 1367 y(that:)393 1450 y Fk(x)p Fp(\()p Fl(j)r Fp(\))g Fj(2)g Fl(K)s Fp(\()p Fl(p)596 1430 y Fg(j)596 1461 y Fi(1)615 1450 y Fl(;)7 b(p)655 1430 y Fg(j)655 1461 y Fi(2)672 1450 y Fp(;)g Fl(r)q(=)p Fp(100\))h(and)h Fk(x)p Fp(\()p Fl(k)q Fp(\))j Fj(62)f Fl(K)s Fp(\()p Fl(p)1101 1430 y Fg(j)1101 1461 y Fi(1)1120 1450 y Fl(;)c(p)1160 1430 y Fg(j)1160 1461 y Fi(2)1178 1450 y Fp(;)g Fl(r)q(=)p Fp(100\))h(if)g Fl(j)14 b Fj(6)p Fp(=)e Fl(k)133 b Fp(\(3.14\))324 1533 y(If)13 b Fl(\014)r(;)7 b(\015)15 b Fj(2)c(f\000)p Fp(1)p Fl(;)c Fp(0)p Fl(;)g Fp(1)p Fj(g)12 b Fp(then)i(de\014ne)614 1624 y Fj(K)646 1630 y Fg(j)663 1624 y Fp(\()p Fl(\016)r Fp(\))e(=)773 1585 y Ff([)771 1674 y Fg(\014)q(;\015)827 1624 y Fl(K)s Fp(\()p Fl(p)902 1604 y Fg(j)902 1635 y Fi(1)930 1624 y Fp(+)e Fl(\014)r(;)d(p)1037 1604 y Fg(j)1037 1635 y Fi(2)1065 1624 y Fp(+)i Fl(\015)r Fp(;)e Fl(\016)r Fp(\))p Fl(;)49 b(\016)13 b(>)f Fp(0)247 b(\(3.15\))324 1749 y(By)14 b(construction,)686 1832 y(dist)p Fj(f)p Fk(x)p Fp(\()p Fl(j)r Fp(\))p Fl(;)7 b(@)r Fj(K)925 1838 y Fg(j)943 1832 y Fp(\()p Fl(r)q(=)p Fp(100\))p Fj(g)k(\025)g Fl(r)q(=)p Fp(100)p Fl(:)318 b Fp(\(3.16\))324 1915 y(Denote)14 b(with)795 1995 y Fj(F)825 2001 y Fg(N)868 1995 y Fp(=)926 1943 y Fg(N)917 1956 y Ff([)912 2044 y Fg(j)r Fi(=1)976 1995 y Fj(K)1008 2001 y Fg(j)1025 1995 y Fp(\()p Fl(r)q(=)p Fp(100\))427 b(\(3.17\))262 2120 y Fk(Lemma)15 b(3.1)21 b Fb(T)m(ake)670 2203 y Fl(K)s Fp(\()p Fl(p;)7 b(q)q Fp(;)g Fl(r)q(=)p Fp(100\))j Fj(6\032)i(F)1007 2209 y Fg(N)1047 2203 y Fb(and)e Fl(\021)j Fj(2)e Fl(C)1228 2186 y Fh(1)1225 2213 y Fi(0)1263 2203 y Fp(\()p Fk(R)1315 2186 y Fi(2)1333 2203 y Fp(\))p Fl(;)670 2265 y(supp)e(\021)k Fj(\032)f Fl(K)s Fp(\()p Fl(p;)7 b(q)q Fp(;)g Fl(r)q(=)p Fp(98\))p Fl(:)262 2348 y Fb(Denote)15 b(with)f Fk(m)e Fp(=)590 2314 y Ff(\000)630 2332 y Fg(r)p 614 2339 V 614 2362 a Fi(100)675 2314 y Ff(\000)694 2348 y Fl(p)d Fp(+)771 2332 y Fi(1)p 771 2339 17 2 v 771 2362 a(2)792 2314 y Ff(\001)818 2348 y Fl(;)858 2332 y Fg(r)p 842 2339 50 2 v 842 2362 a Fi(100)903 2314 y Ff(\000)922 2348 y Fl(q)h Fp(+)998 2332 y Fi(1)p 998 2339 17 2 v 998 2362 a(2)1019 2314 y Ff(\001\001)1064 2348 y Fl(:)15 b Fb(Then:)619 2439 y Fl(H)654 2445 y Fg(N)685 2439 y Fl(\021)d Fp(=)g Fl(e)781 2422 y Fg(i)c(')823 2426 y Fa(N)849 2422 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))928 2439 y Fl(t)943 2445 y Fe(m)986 2439 y Fl(H)1021 2445 y Fi(0)1048 2439 y Fl(t)1063 2445 y Fh(\000)p Fe(m)1122 2439 y Fl(e)1141 2422 y Fh(\000)p Fg(i)g(')1209 2426 y Fa(N)1235 2422 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))1314 2439 y Fl(\021)253 b Fp(\(3.18\))957 2574 y(11)p eop %%Page: 12 12 12 11 bop 324 307 a Fb(Pr)n(o)n(of:)324 357 y Fp(If)13 b Fk(x)f Fj(2)f Fl(supp)e(\021)q Fp(,)14 b(then:)545 448 y Fk(a)568 454 y Fg(N)600 448 y Fp(\()p Fk(x)p Fp(\))42 b(=)g Fk(a)796 454 y Fg(N)827 448 y Fp(\()p Fk(x)p Fl(;)7 b Fk(m)p Fp(\))i(+)h Fj(r)p Fl(')1056 454 y Fg(N)1087 448 y Fp(\()p Fk(x)p Fl(;)d Fk(m)p Fp(\))385 b(\(3.19\))544 511 y Fl(V)568 517 y Fg(N)600 511 y Fp(\()p Fk(x)p Fp(\))42 b(=)g(0)487 603 y Fk(a)510 609 y Fg(N)541 603 y Fp(\()p Fk(x)p Fl(;)7 b Fk(m)p Fp(\))42 b(=)773 546 y Ff(Z)814 556 y Fi(1)796 640 y(0)840 603 y Fl(ds)9 b(s)h Fk(B)953 609 y Fg(N)984 603 y Fp(\()p Fk(m)f Fp(+)h Fl(s)p Fp(\()p Fk(x)g Fj(\000)f Fk(m)p Fp(\)\))g Fj(^)g Fp(\()p Fk(x)h Fj(\000)f Fk(m)p Fp(\))324 717 y(Because)16 b(for)d(all)g Fl(y)g Fj(2)e(f)p Fk(m)e Fp(+)h Fl(s)p Fp(\()p Fk(x)g Fj(\000)f Fk(m)p Fp(\))p Fl(;)16 b Fp(0)11 b Fj(\024)h Fl(s)g Fj(\024)g Fp(1)p Fj(g)h Fp(one)h(has)864 808 y Fk(B)898 814 y Fg(N)929 808 y Fp(\()p Fl(y)q Fp(\))f(=)f Fk(B)1073 814 y Fi(0)1588 808 y Fp(\(3.20\))262 900 y(then)480 991 y Fk(a)503 997 y Fg(N)535 991 y Fp(\()p Fk(x)p Fl(;)7 b Fk(m)p Fp(\))41 b(=)h Fk(a)789 997 y Fi(0)808 991 y Fp(\()p Fk(x)9 b Fj(\000)h Fk(m)p Fp(\))41 b(and)563 1053 y Fl(H)598 1059 y Fg(N)629 1053 y Fl(\021)h Fp(=)g Fl(e)785 1036 y Fg(i)8 b(')827 1040 y Fa(N)853 1036 y Fi(\()p Fe(x)p Fg(;)p Fe(m)p Fi(\))942 1053 y Fp([)p Fk(p)g Fj(\000)i Fk(a)1054 1059 y Fi(0)1073 1053 y Fp(\()p Fk(x)f Fj(\000)h Fk(m)p Fp(\)])1233 1036 y Fi(2)1251 1053 y Fl(e)1270 1036 y Fh(\000)p Fg(i)d(')1337 1040 y Fa(N)1364 1036 y Fi(\()p Fe(x)p Fg(;)p Fe(m)p Fi(\))1453 1053 y Fl(\021)692 1116 y Fp(=)42 b Fl(e)785 1099 y Fg(i)8 b(')827 1103 y Fa(N)853 1099 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))932 1116 y Fl(t)947 1122 y Fe(m)990 1116 y Fl(H)1025 1122 y Fi(0)1052 1116 y Fl(t)1067 1122 y Fh(\000)p Fe(m)1126 1116 y Fl(e)1145 1099 y Fh(\000)p Fg(i)g(')1213 1103 y Fa(N)1239 1099 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))1318 1116 y Fl(\021)262 1207 y Fk(Lemma)15 b(3.2)21 b Fb(Fix)15 b Fl(j)g Fj(2)d(f)p Fp(1)p Fl(;)7 b(:)g(:)g(:)t(;)g(N)e Fj(g)p Fb(.)20 b(T)m(ake)15 b Fl(\021)1007 1213 y Fg(j)1036 1207 y Fj(2)e Fl(C)1110 1192 y Fh(1)1107 1217 y Fi(0)1144 1207 y Fp(\()p Fk(R)1196 1192 y Fi(2)1215 1207 y Fp(\))i Fb(and)i Fl(supp)9 b(\021)1443 1213 y Fg(j)1473 1207 y Fj(\032)j(K)1549 1213 y Fg(j)1566 1207 y Fp(\()p Fl(r)q(=)p Fp(98\))p Fb(.)262 1257 y(Then:)543 1307 y Fl(H)578 1313 y Fg(N)609 1307 y Fl(\021)630 1313 y Fg(j)658 1307 y Fp(=)g Fl(e)721 1289 y Fg(i)c(')763 1293 y Fa(N)789 1289 y Fi(\()p Fg(:;)p Fe(x)p Fi(\()p Fg(j)r Fi(\)\))898 1307 y Fl(t)913 1314 y Fe(x)p Fi(\()p Fg(j)r Fi(\))986 1307 y Fl(H)1021 1313 y Fi(1)1048 1307 y Fl(t)1063 1314 y Fh(\000)p Fe(x)p Fi(\()p Fg(j)r Fi(\))1152 1307 y Fl(e)1171 1289 y Fh(\000)p Fg(i)g(')1239 1293 y Fa(N)1265 1289 y Fi(\()p Fg(:;)p Fe(x)p Fi(\()p Fg(j)r Fi(\)\))1374 1307 y Fl(\021)1395 1313 y Fg(j)1588 1307 y Fp(\(3.21\))324 1398 y Fb(Pr)n(o)n(of:)324 1448 y Fp(As)14 b(b)q(efore,)642 1498 y Fk(a)665 1504 y Fg(N)696 1498 y Fp(\()p Fk(x)p Fp(\))e(=)g Fk(a)832 1504 y Fg(N)864 1498 y Fp(\()p Fk(x)p Fl(;)7 b Fk(x)p Fp(\()p Fl(j)r Fp(\)\))j(+)f Fj(r)p Fl(')1129 1504 y Fg(N)1161 1498 y Fp(\()p Fk(x)p Fl(;)e Fk(x)p Fp(\()p Fl(j)r Fp(\)\))275 b(\(3.22\))324 1580 y(Since)14 b Fl(r)c Fj(\000)508 1564 y Fi(3)525 1540 y Fh(p)p 552 1540 17 2 v 24 x Fi(2)p 508 1571 61 2 v 521 1595 a(98)573 1580 y Fl(r)j(>)e Fp(1)j(and)g Fk(x)d Fj(2)h Fl(supp)d(\021)955 1586 y Fg(j)972 1580 y Fp(,)k(then)308 1672 y Fj(j)p Fk(x)p Fp(\()p Fl(j)r Fp(\))d Fj(\000)f Fk(x)p Fp(\()p Fl(k)q Fp(\))h(+)f Fl(s)p Fp(\()p Fk(x)h Fj(\000)g Fk(x)p Fp(\()p Fl(j)r Fp(\)\))p Fj(j)i(\025)g(j)p Fk(x)p Fp(\()p Fl(j)r Fp(\))d Fj(\000)h Fk(x)p Fp(\()p Fl(k)q Fp(\))p Fj(j)f(\000)h(j)p Fk(x)e Fj(\000)i Fk(x)p Fp(\()p Fl(j)r Fp(\))p Fj(j)i Fl(>)g Fp(1)p Fl(;)48 b(j)14 b Fj(6)p Fp(=)d Fl(k)48 b Fp(\(3.23\))324 1763 y(therefore:)509 1854 y Fk(a)532 1860 y Fg(N)564 1854 y Fp(\()p Fk(x)p Fl(;)7 b Fk(x)p Fp(\()p Fl(j)r Fp(\)\))13 b(=)e Fk(a)795 1860 y Fi(0)814 1854 y Fp(\()p Fk(x)f Fj(\000)f Fk(x)p Fp(\()p Fl(j)r Fp(\)\))i(+)436 1957 y(+)509 1901 y Ff(Z)551 1911 y Fi(1)532 1995 y(0)576 1957 y Fl(ds)f(s)678 1905 y Fg(N)663 1918 y Ff(X)662 2007 y Fg(k)q Fi(=1)730 1957 y Fk(B)764 1940 y Fh(0)775 1957 y Fp(\()p Fk(x)p Fp(\()p Fl(j)r Fp(\))h Fj(\000)e Fk(x)p Fp(\()p Fl(k)q Fp(\))h(+)f Fl(s)p Fp(\()p Fk(x)h Fj(\000)g Fk(x)p Fp(\()p Fl(j)r Fp(\)\)\))g Fj(^)f Fp(\()p Fk(x)h Fj(\000)f Fk(x)p Fp(\()p Fl(k)q Fp(\)\))j(=)436 2059 y(=)41 b Fk(a)532 2065 y Fi(1)551 2059 y Fp(\()p Fk(x)10 b Fj(\000)f Fk(x)p Fp(\()p Fl(j)r Fp(\)\))i(+)436 2151 y(+)509 2094 y Ff(Z)551 2105 y Fi(1)532 2189 y(0)576 2151 y Fl(ds)f(s)662 2112 y Ff(X)663 2201 y Fg(k)q Fh(6)p Fi(=)p Fg(j)729 2151 y Fk(B)763 2134 y Fh(0)775 2151 y Fp(\()p Fk(x)p Fp(\()p Fl(j)r Fp(\))g Fj(\000)g Fk(x)p Fp(\()p Fl(k)q Fp(\))f(+)h Fl(s)p Fp(\()p Fk(x)g Fj(\000)f Fk(x)p Fp(\()p Fl(j)r Fp(\)\)\))i Fj(^)e Fp(\()p Fk(x)g Fj(\000)h Fk(x)p Fp(\()p Fl(k)q Fp(\)\))i(=)436 2258 y(=)41 b Fk(a)532 2264 y Fi(1)551 2258 y Fp(\()p Fk(x)10 b Fj(\000)f Fk(x)p Fp(\()p Fl(j)r Fp(\)\))853 b(\(3.24\))324 2350 y(If)13 b Fk(x)f Fj(2)f Fl(supp)d(\021)555 2356 y Fg(j)572 2350 y Fp(,)13 b(then)i Fl(V)716 2356 y Fg(N)747 2350 y Fp(\()p Fk(x)p Fp(\))d(=)g Fl(V)d Fp(\()p Fk(x)g Fj(\000)g Fk(x)p Fp(\()p Fl(j)r Fp(\)\);)14 b(putting)f(all)g(these)i(together,)f(\(3.21\))262 2400 y(follo)o(ws.)957 2574 y(12)p eop %%Page: 13 13 13 12 bop 262 307 a Fk(Lemma)15 b(3.3)21 b Fb(Under)d(the)g(same)g (assumptions)g(made)h(in)f(L)n(emma)g(3.1,)h(supp)n(ose)g(now)262 357 y(that)695 448 y Fp(0)11 b Fj(\024)h Fl(\021)h Fj(\024)f Fp(1)42 b Fb(and)695 533 y Fl(\021)q Fp(\()p Fk(x)p Fp(\))12 b(=)830 475 y Ff(\032)882 508 y Fp(1)41 b Fb(if)g Fk(x)12 b Fj(2)f Fl(K)s Fp(\()p Fl(p;)c(q)q Fp(;)g Fl(r)q(=)p Fp(99\))882 558 y(0)41 b Fb(if)g Fk(x)12 b Fj(62)f Fl(K)s Fp(\()p Fl(p;)c(q)q Fp(;)g Fl(r)q(=)p Fp(98\))1588 533 y(\(3.25\))324 649 y Fb(Then)15 b(ther)n(e)g(exist)g Fj(C)655 655 y Fi(1)686 649 y Fl(>)e Fp(0)i Fb(and)h Fl(u)c(>)g Fp(0)j Fb(\(which)g(ar)n(e)g Fp(indep)q(enden)o(t)i Fb(of)e Fl(N)5 b Fb(,)15 b Fp(\()p Fl(p;)7 b(q)q Fp(\))15 b Fb(and)262 699 y Fl(E)e Fj(2)e Fp(\006)375 705 y Fg(N)407 699 y Fb(\))k(such)g(that:)340 755 y Ff(Z)363 850 y Fg(K)r Fi(\()p Fg(p;q)q Fi(;)p Fg(r)q(=)p Fi(100\))564 812 y Fl(d)p Fk(x)9 b Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)729 795 y Fi(2)759 812 y Fj(\024)j Fl(e)822 795 y Fh(\000)p Fg(ur)884 782 y Fd(2)902 812 y Fj(C)924 818 y Fi(1)950 755 y Ff(Z)p 973 823 369 2 v 95 x Fg(K)r Fi(\()p Fg(p;q)q Fi(;)p Fg(r)q(=)p Fi(98\))p Fh(n)p Fg(K)r Fi(\()p Fg(p;q)q Fi(;)p Fg(r)q(=)p Fi(99\))1348 812 y Fl(d)p Fk(x)d Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)1513 795 y Fi(2)1588 812 y Fp(\(3.26\))324 931 y Fb(Pr)n(o)n(of:)324 981 y Fp(\(3.18\))k(implies)f(that)405 1072 y([)p Fl(H)452 1078 y Fg(N)483 1072 y Fl(;)7 b(\021)q Fp(])p Fl( )42 b Fp(=)g Fl(e)698 1055 y Fg(i)8 b(')740 1059 y Fa(N)766 1055 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))845 1072 y Fl(t)860 1078 y Fe(m)902 1072 y Fp(\()p Fl(H)953 1078 y Fi(0)981 1072 y Fj(\000)h Fl(E)r Fp(\))h Fl(t)1096 1078 y Fh(\000)p Fe(m)1155 1072 y Fl(e)1174 1055 y Fh(\000)p Fg(i)e(')1242 1059 y Fa(N)1268 1055 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))1347 1072 y Fl(\021)q( )192 b Fp(\(3.27\))527 1135 y(or)513 1197 y Fl(\021)q( )43 b Fp(=)f Fl(e)698 1180 y Fg(i)8 b(')740 1184 y Fa(N)766 1180 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))845 1197 y Fl(t)860 1203 y Fe(m)902 1197 y Fp(\()p Fl(H)953 1203 y Fi(0)981 1197 y Fj(\000)h Fl(E)r Fp(\))1071 1180 y Fh(\000)p Fi(1)1125 1197 y Fl(t)1140 1203 y Fh(\000)p Fe(m)1200 1197 y Fl(e)1219 1180 y Fh(\000)p Fg(i)e(')1286 1184 y Fa(N)1313 1180 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))1392 1197 y Fp([)p Fl(H)1439 1203 y Fg(N)1469 1197 y Fl(;)g(\021)q Fp(])p Fl( )39 b Fp(\(3.28\))324 1288 y(If)477 1338 y Fk(x)12 b Fj(2)f Fl(K)s Fp(\()p Fl(p;)c(q)q Fp(;)g Fl(r)q(=)p Fp(100\))h(and)h Fk(x)915 1321 y Fh(0)938 1338 y Fj(2)p 978 1302 501 2 v 12 w Fl(K)s Fp(\()p Fl(p;)e(q)q Fp(;)g Fl(r)q(=)p Fp(98\))g Fj(n)i Fl(K)s Fp(\()p Fl(p;)e(q)q Fp(;)g Fl(r)q(=)p Fp(99\))109 b(\(3.29\))262 1413 y(then)718 1486 y Fj(j)p Fk(x)9 b Fj(\000)g Fk(x)830 1469 y Fh(0)842 1486 y Fj(j)854 1469 y Fi(2)884 1486 y Fj(\025)933 1458 y Fl(r)953 1443 y Fi(2)p 933 1476 39 2 v 942 1514 a Fp(4)983 1427 y Ff(\022)1029 1458 y Fp(1)p 1019 1476 42 2 v 1019 1514 a(99)1074 1486 y Fj(\000)1142 1458 y Fp(1)p 1121 1476 63 2 v 1121 1514 a(100)1188 1427 y Ff(\023)1219 1436 y Fi(2)1588 1486 y Fp(\(3.30\))262 1598 y(and)k(using)h(Corollary) e(2.1,)h(one)h(has)g(\(0)d Fl(<)h(u)g(<)1039 1581 y Fg(B)1064 1585 y Fd(0)p 1039 1588 42 2 v 1043 1612 a Fi(64)1092 1564 y Ff(\000)1125 1582 y Fi(1)p 1116 1589 34 2 v 1116 1612 a(99)1164 1598 y Fj(\000)1227 1582 y Fi(1)p 1210 1589 50 2 v 1210 1612 a(100)1265 1564 y Ff(\001)1284 1573 y Fi(2)1303 1598 y Fp(\):)560 1658 y Ff(Z)583 1752 y Fg(K)r Fi(\()p Fg(p;q)q Fi(;)p Fg(r)q(=)p Fi(100\))784 1715 y Fl(d)p Fk(x)d Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)949 1697 y Fi(2)979 1715 y Fj(\024)j Fl(e)1042 1697 y Fh(\000)p Fg(ur)1104 1685 y Fd(2)1122 1715 y Fj(C)1144 1721 y Fi(1)1172 1715 y Fj(jj)p Fp([)p Fl(H)1243 1721 y Fg(N)1273 1715 y Fl(;)7 b(\021)q Fp(])p Fl( )q Fj(jj)1378 1697 y Fi(2)1588 1715 y Fp(\(3.31\))324 1836 y(Using)13 b(\(2.31\))g(in)h(\(3.31\),)e(one)i(obtains)g(\(3.26\).)262 1920 y Fk(Lemma)h(3.4)21 b Fb(Ther)n(e)13 b(exist)h Fl(u)d(>)h Fp(0)p Fb(,)h Fj(C)h Fl(<)e Fj(1)h Fb(with)g(the)h(pr)n(op)n(erties)f (given)i(in)f(L)n(emma)f(3.3)262 1969 y(such)i(that:)725 1973 y Ff(Z)748 2068 y Fi(\()p Fh(F)785 2072 y Fa(N)811 2068 y Fi(\))824 2056 y Fa(c)858 2030 y Fl(d)p Fk(x)9 b Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)1023 2013 y Fi(2)1053 2030 y Fj(\024)j(C)g Fl(e)1150 2013 y Fh(\000)p Fg(ur)1212 2000 y Fd(2)1588 2030 y Fp(\(3.32\))324 2141 y Fb(Pr)n(o)n(of:)324 2191 y Fp(Adding)18 b(the)h(con)o(tributions)g (giv)o(en)f(b)o(y)g(all)g Fl(K)s Fp(\()p Fl(p;)7 b(q)q Fp(;)g Fl(r)q(=)p Fp(100\))17 b Fj(6\032)j(F)1399 2197 y Fg(N)1449 2191 y Fp(in)e(\(3.26\))g(and)262 2241 y(b)q(ecause)615 2262 y Ff(X)610 2353 y Fi(\()p Fg(p;q)q Fi(\))686 2245 y Ff(Z)p 709 2313 369 2 v 94 x Fg(K)r Fi(\()p Fg(p;q)q Fi(;)p Fg(r)q(=)p Fi(98\))p Fh(n)p Fg(K)r Fi(\()p Fg(p;q)q Fi(;)p Fg(r)q(=)p Fi(99\))1085 2302 y Fl(d)p Fk(x)9 b Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)1250 2284 y Fi(2)1280 2302 y Fj(\024)j Fp(4)243 b(\(3.33\))262 2422 y(the)14 b(result)h(follo)o(ws.)957 2574 y(13)p eop %%Page: 14 14 14 13 bop 324 307 a Fp(W)m(e)18 b(are)i(no)o(w)e(able)h(to)g(pro)o(v)o (e)g(the)g(\014rst)h(a\016rmation)c(of)i(Theorem)h(3.1.)32 b(F)m(or)19 b Fl(j)j Fj(2)262 357 y(f)p Fp(1)p Fl(;)7 b(:)g(:)g(:)t(;)g(N)e Fj(g)13 b Fp(tak)o(e)721 442 y Fl(\021)742 448 y Fg(j)771 442 y Fj(2)e Fl(C)843 424 y Fh(1)840 452 y Fi(0)878 442 y Fp(\()p Fk(R)930 424 y Fi(2)948 442 y Fp(\))p Fl(;)48 b Fp(0)12 b Fj(\024)f Fl(\021)1121 448 y Fg(j)1150 442 y Fj(\024)h Fp(1)d(and)721 530 y Fl(\021)742 536 y Fg(j)759 530 y Fp(\()p Fk(x)p Fp(\))j(=)872 472 y Ff(\032)924 505 y Fp(1)41 b(if)g Fk(x)12 b Fj(2)f(K)1160 511 y Fg(j)1177 505 y Fp(\()p Fl(r)q(=)p Fp(99\))924 555 y(0)41 b(if)g Fk(x)12 b Fj(62)f(K)1160 561 y Fg(j)1177 555 y Fp(\()p Fl(r)q(=)p Fp(98\))1588 530 y(\(3.34\))324 637 y(Let)877 706 y(~)869 717 y Fl( )i Fp(=)968 665 y Fg(N)953 678 y Ff(X)954 766 y Fg(j)r Fi(=1)1020 717 y Fl(\021)1041 723 y Fg(j)1058 717 y Fl( )503 b Fp(\(3.35\))262 830 y(then)14 b(from)e(Lemma)f(3.2)i(one)h(has:)331 945 y(\()p Fl(H)382 951 y Fg(N)422 945 y Fj(\000)c Fl(E)r Fp(\))521 934 y(~)513 945 y Fl( )j Fp(=)612 893 y Fg(N)597 905 y Ff(X)598 994 y Fg(j)r Fi(=1)664 945 y Fl(e)683 927 y Fg(i)7 b(')724 931 y Fa(N)751 927 y Fi(\()p Fg(:;)p Fe(x)p Fi(\()p Fg(j)r Fi(\)\))860 945 y Fl(t)875 952 y Fe(x)p Fi(\()p Fg(j)r Fi(\))947 945 y Fp(\()p Fl(H)998 951 y Fi(1)1026 945 y Fj(\000)i Fl(E)r Fp(\))h Fl(t)1141 952 y Fh(\000)p Fe(x)p Fi(\()p Fg(j)r Fi(\))1230 945 y Fl(e)1249 927 y Fh(\000)p Fg(i)d(')1316 931 y Fa(N)1343 927 y Fi(\()p Fg(:;)p Fe(x)p Fi(\()p Fg(j)r Fi(\)\))1452 945 y Fl(\021)1473 951 y Fg(j)1490 945 y Fl( )71 b Fp(\(3.36\))262 1060 y(or)412 1088 y Fg(N)397 1101 y Ff(X)398 1189 y Fg(j)r Fi(=1)464 1140 y Fj(jj)p Fp(\()p Fl(H)539 1146 y Fi(1)566 1140 y Fj(\000)9 b Fl(E)r Fp(\))p Fl(t)671 1147 y Fh(\000)p Fe(x)p Fi(\()p Fg(j)r Fi(\))760 1140 y Fl(e)779 1123 y Fh(\000)p Fg(i)f(')847 1127 y Fa(N)874 1123 y Fi(\()p Fg(:;)p Fe(x)p Fi(\()p Fg(j)r Fi(\)\))983 1140 y Fl(\021)1004 1146 y Fg(j)1021 1140 y Fl( )q Fj(jj)1073 1123 y Fi(2)1102 1140 y Fp(=)1161 1088 y Fg(N)1146 1101 y Ff(X)1147 1189 y Fg(j)r Fi(=1)1213 1140 y Fj(jj)p Fp([)p Fl(H)1284 1146 y Fg(N)1314 1140 y Fl(;)f(\021)1354 1146 y Fg(j)1370 1140 y Fp(])p Fl( )q Fj(jj)1434 1123 y Fi(2)1588 1140 y Fp(\(3.37\))324 1252 y(But)385 1337 y Fj(jj)p Fp(\()p Fl(H)460 1343 y Fi(1)486 1337 y Fj(\000)j Fl(E)r Fp(\))p Fl(t)592 1344 y Fh(\000)p Fe(x)p Fi(\()p Fg(j)r Fi(\))681 1337 y Fl(e)700 1320 y Fh(\000)p Fg(i)e(')768 1324 y Fa(N)794 1320 y Fi(\()p Fg(:;)p Fe(x)p Fi(\()p Fg(j)r Fi(\)\))903 1337 y Fl(\021)924 1343 y Fg(j)941 1337 y Fl( )q Fj(jj)993 1320 y Fi(2)1023 1337 y Fj(\025)k Fp(dist)1134 1318 y Fi(2)1153 1337 y Fj(f)p Fl(E)r(;)7 b(\033)q Fp(\()p Fl(H)1302 1343 y Fi(1)1319 1337 y Fp(\))p Fj(gjj)p Fl(\021)1401 1343 y Fg(j)1417 1337 y Fl( )q Fj(jj)1469 1320 y Fi(2)1588 1337 y Fp(\(3.38\))262 1422 y(therefore,)14 b(together)h(with)f(\(3.37\))f(and)g(\(2.31\))g(one)h (obtains:)529 1547 y(dist)596 1528 y Fi(2)614 1547 y Fj(f)p Fl(E)r(;)7 b(\033)q Fp(\()p Fl(H)763 1553 y Fi(1)781 1547 y Fp(\))p Fj(g)840 1495 y Fg(N)825 1507 y Ff(X)826 1596 y Fg(j)r Fi(=1)892 1547 y Fj(jj)p Fl(\021)937 1553 y Fg(j)953 1547 y Fl( )q Fj(jj)1005 1530 y Fi(2)1035 1547 y Fj(\024)k(C)1110 1490 y Ff(Z)1133 1585 y Fi(\()p Fh(F)1170 1589 y Fa(N)1196 1585 y Fi(\))1209 1573 y Fa(c)1243 1547 y Fl(d)p Fk(x)e Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)1408 1530 y Fi(2)1588 1547 y Fp(\(3.39\))324 1662 y(or)366 1774 y(dist)433 1755 y Fi(2)452 1774 y Fj(f)p Fl(E)r(;)e(\033)q Fp(\()p Fl(H)601 1780 y Fi(1)618 1774 y Fp(\))p Fj(g)662 1703 y Ff( )695 1774 y Fp(1)i Fj(\000)767 1717 y Ff(Z)790 1812 y Fi(\()p Fh(F)827 1816 y Fa(N)853 1812 y Fi(\))866 1800 y Fa(c)891 1774 y Fl(d)p Fk(x)p Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)1047 1757 y Fi(2)1065 1703 y Ff(!)1110 1774 y Fj(\024)i(C)1185 1717 y Ff(Z)1208 1812 y Fi(\()p Fh(F)1245 1816 y Fa(N)1271 1812 y Fi(\))1284 1800 y Fa(c)1309 1774 y Fl(d)p Fk(x)p Fj(j)p Fl( )q Fp(\()p Fk(x)p Fp(\))p Fj(j)1465 1757 y Fi(2)1588 1774 y Fp(\(3.40\))262 1896 y(and)i(together)i(with)f (\(3.32\),)e(the)j(a\016rmation)c(stated)k(in)e(\(3.8\))g(follo)o(ws.) 324 1946 y Fb(Pr)n(o)n(of)h(of)h(ii\):)324 1996 y Fp(Let's)f(sho)o(w)g (\014rst)h(that)746 2080 y(dim)n Fj(f)p Fb(R)n(an)q Fp([)p Fj(P)950 2086 y Fg(N)981 2080 y Fp(\(\006)1027 2086 y Fg(N)1058 2080 y Fp(\)])p Fj(g)c(\025)h Fl(N)r(:)379 b Fp(\(3.41\))324 2165 y(Denote)14 b(with)g Fl( )588 2171 y Fi(1)620 2165 y Fp(the)h(normalized)d(eigen)o(v)o(ector)j(of)e Fl(H)1202 2171 y Fi(1)1234 2165 y Fp(corresp)q(onding)i(to)f Fl(E)1583 2171 y Fi(1)1601 2165 y Fp(:)739 2249 y Fl(H)774 2255 y Fi(1)792 2249 y Fl( )819 2255 y Fi(1)849 2249 y Fp(=)e Fl(E)924 2255 y Fi(1)943 2249 y Fl( )970 2255 y Fi(1)988 2249 y Fl(;)48 b Fj(jj)p Fl( )1099 2255 y Fi(1)1117 2249 y Fj(jj)10 b Fp(=)i(1)372 b(\(3.42\))324 2334 y(With)13 b(the)i(notations)e(in)o(tro)q(duced)i(in)e(\(3.34\),)f (let)615 2431 y Fj(V)640 2437 y Fg(N)684 2431 y Fp(=)728 2385 y Ff(n)764 2420 y Fp(~)755 2431 y Fl( )782 2437 y Fg(j)811 2431 y Fp(=)g 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y Fp(\()p Fl(K)s Fp(\))p Fl( )14 b(>)d Fj(j)g Fp(=)h(1)134 b(\(3.59\))262 1705 y(whic)o(h)13 b(con)o(tradicts)i(\(3.58\).)324 1755 y(Let's)e (construct)h(no)o(w)e Fl(P)724 1739 y Fh(0)718 1766 y Fg(N)749 1755 y Fp(.)18 b(Using)12 b(\(3.8\),)g(one)h(obtains)f(the)h (existence)i(of)d Fl(r)1534 1761 y Fi(0)1552 1755 y Fp(\()p Fl(\017)p Fp(\))h(suc)o(h)262 1804 y(that)726 1854 y Fj(fj)p Fl(z)d Fj(\000)g Fl(E)861 1860 y Fi(1)879 1854 y Fj(j)h Fp(=)h Fl(\017)p Fj(g)d(\\)g Fl(\033)q Fp(\()p Fl(H)1106 1860 y Fg(N)1137 1854 y Fp(\))j(=)g Fj(;)358 b Fp(\(3.60\))262 1921 y(as)13 b(so)q(on)h(as)g Fl(r)f Fj(\025)f Fl(r)554 1927 y Fi(0)572 1921 y Fp(\()p Fl(\017)p Fp(\))i(\()p Fl(\017)g Fp(b)q(eing)g(c)o(hosen)g(su\016cien)o(tly)g (small)e(then)i(k)o(ept)h(\014xed\).)324 1971 y(The)f(idea)g(consists)h (\(see)g(for)f(similar)e(reasoning)i([B-C-D])e(and)i([Na)g(1]\))f(in)h (appro)o(xi-)262 2021 y(mating)d(the)j(resolv)o(en)o(t)g(\()p Fl(H)698 2027 y Fg(N)737 2021 y Fj(\000)8 b Fl(z)r Fp(\))814 2006 y Fh(\000)p Fi(1)872 2021 y Fp(for)13 b Fj(j)p Fl(z)d Fj(\000)f Fl(E)1048 2027 y Fi(1)1066 2021 y Fj(j)i Fp(=)h Fl(\017)h Fp(and)g(then)h(in)o(tegrating)f(o)o(v)o(er)g(the)262 2070 y(con)o(tour.)324 2120 y(Let)328 2198 y Fk(m)p Fp(\()p Fl(p;)7 b(q)q Fp(\))k(=)h(\()p Fl(r)q(=)p Fp(100\()p Fl(p)c Fp(+)i(1)p Fl(=)p Fp(2\))p Fl(;)d(r)q(=)p Fp(100\()p Fl(q)h Fp(+)i(1)p Fl(=)p Fp(2\)\))15 b(if)9 b Fl(K)s Fp(\()p Fl(p;)e(q)q Fp(;)g Fl(r)q(=)p Fp(100\))j Fj(6\032)h(F)1489 2204 y Fg(N)1588 2198 y Fp(\(3.61\))262 2277 y(and)742 2326 y(\000)768 2332 y Fh(1)815 2326 y Fp(=)g(\000)884 2332 y Fg(N)923 2287 y Ff([)969 2326 y Fj(f)p Fk(m)p Fp(\()p Fl(p;)c(q)q Fp(\))p Fj(g)1143 2333 y Fi(\()p Fg(p;q)q Fi(\))1588 2326 y Fp(\(3.62\))324 2399 y(It)15 b(is)h(p)q(ossible)g(to)f(construct)i(a)e(quadratic)h(partition)f(of)g (unit)o(y)g(whic)o(h)g(has)h(the)g(fol-)262 2449 y(lo)o(wing)c(prop)q (erties)j(\(see)g([C-F-K-S]\):)957 2574 y(16)p eop %%Page: 17 17 17 16 bop 324 307 a Fj(\017)638 318 y Ff(X)616 407 y Fe(m)p Fh(2)p Fi(\000)690 411 y Fc(1)728 357 y Fl(\021)750 340 y Fi(2)749 367 y Fe(m)793 357 y Fp(=)12 b(1)p Fl(;)48 b(\021)939 363 y Fe(m)983 357 y Fj(2)11 b Fl(C)1055 340 y Fh(1)1052 367 y Fi(0)1090 357 y Fp(\()p Fk(R)1142 340 y Fi(2)1161 357 y Fp(\))p Fl(;)48 b Fp(0)11 b Fj(\024)h Fl(\021)1334 363 y Fe(m)1378 357 y Fj(\024)g Fp(1)145 b(\(3.63\))324 472 y Fj(\017)606 522 y Fl(\021)627 528 y Fe(m)659 522 y Fp(\()p Fk(x)p Fp(\))12 b(=)g(0)42 b(if)e Fk(x)12 b Fj(62)f(K)1008 528 y Fe(m)1041 522 y Fp(\()p Fl(r)q(=)p Fp(99\))41 b(and)g Fk(m)12 b Fj(2)f Fp(\000)1422 528 y Fg(N)1588 522 y Fp(\(3.64\))324 606 y Fj(\017)434 688 y Fl(\021)455 694 y Fe(m)487 688 y Fp(\()p Fk(x)p Fp(\))h(=)g(0)42 b(if)e Fk(x)12 b Fj(62)f Fl(K)s Fp(\()p Fl(p;)c(q)q Fp(;)g Fl(r)q(=)p Fp(99\))40 b(and)i Fk(m)11 b Fj(2)g(f)p Fk(m)p Fp(\()p Fl(p;)c(q)q Fp(\))p Fj(g)1449 695 y Fi(\()p Fg(p;q)q Fi(\))1588 688 y Fp(\(3.65\))324 786 y Fj(\017)606 843 y(jj)p Fp(\()p Fl(@)r(\021)691 849 y Fe(m)723 843 y Fl(=@)r(x)792 849 y 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Fi(2)621 1548 y Fp(\).)21 b(Then,)15 b(from)e(\(3.60\))h(and)h(\(3.63\),)e Fl(A)1256 1554 y Fg(N)1288 1548 y Fp(\()p Fl(z)r Fp(\))p Fl(f)20 b Fp(consists)c(in)e (a)h(sum)262 1598 y(in)i(whic)o(h)g(eac)o(h)h(term)f(is)g(b)q(ounded)h (and)g(from)e(\(3.64\))g(and)h(\(3.65\))g(there)i(results)g(that)262 1647 y(eac)o(h)14 b(term)f(is)h(orthogonal)f(to)g(all)g(others)i (except)g(at)f(most)f(16)g("neigh)o(b)q(ours".)324 1697 y Fl(A)355 1703 y Fg(N)386 1697 y Fp(\()p Fl(z)r Fp(\))i(is)e(our)h (appro)o(ximation)d(of)i(the)h(resolv)o(en)o(t.)19 b(F)m(rom)12 b(Lemma)f(3.1)i(and)g(Lemma)262 1747 y(3.2,)f(one)i(obtains:)385 1829 y(\()p Fl(H)436 1835 y Fg(N)476 1829 y Fj(\000)c Fl(z)r Fp(\))p Fl(A)586 1835 y Fg(N)618 1829 y Fp(\()p Fl(z)r Fp(\))i(=)311 1900 y(=)405 1861 y Ff(X)385 1950 y Fe(m)p Fh(2)p Fi(\000)459 1954 y Fa(N)493 1900 y Fl(e)512 1883 y Fg(i)7 b(')553 1887 y Fa(N)580 1883 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))659 1900 y Fl(t)674 1906 y Fe(m)707 1900 y Fp(\()p Fl(H)758 1906 y Fi(1)785 1900 y Fj(\000)j Fl(z)r Fp(\))p Fl(t)879 1906 y Fh(\000)p Fe(m)938 1900 y Fl(\021)959 1906 y Fe(m)992 1900 y Fl(t)1007 1906 y Fe(m)1040 1900 y Fp(\()p Fl(H)1091 1906 y Fi(1)1119 1900 y Fj(\000)f Fl(z)r Fp(\))1197 1883 y Fh(\000)p Fi(1)1242 1900 y Fl(t)1257 1906 y Fh(\000)p Fe(m)1317 1900 y Fl(\021)1338 1906 y Fe(m)1370 1900 y Fl(e)1389 1883 y Fh(\000)p Fg(i)f(')1457 1887 y Fa(N)1483 1883 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))1572 1900 y Fp(+)311 2018 y(+)439 1979 y Ff(X)385 2070 y Fe(m)p Fh(2)p Fi(\000)459 2074 y Fc(1)489 2070 y Fh(n)p Fi(\000)527 2074 y Fa(N)560 2018 y Fl(e)579 2001 y Fg(i)g(')621 2005 y Fa(N)648 2001 y Fi(\()p Fg(:;)p Fe(m)p Fi(\))726 2018 y Fl(t)741 2024 y Fe(m)775 2018 y Fp(\()p Fl(H)826 2024 y Fi(0)853 2018 y Fj(\000)i Fl(z)r Fp(\))p Fl(t)947 2024 y Fh(\000)p Fe(m)1006 2018 y Fl(\021)1027 2024 y Fe(m)1060 2018 y Fl(t)1075 2024 y Fe(m)1108 2018 y Fp(\()p Fl(H)1159 2024 y Fi(0)1187 2018 y Fj(\000)f Fl(z)r Fp(\))1265 2001 y Fh(\000)p Fi(1)1310 2018 y Fl(t)1325 2024 y Fh(\000)p 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