Content-Type: multipart/mixed; boundary="-------------0004260356523" This is a multi-part message in MIME format. ---------------0004260356523 Content-Type: text/plain; name="00-199.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-199.keywords" current, magnetic fields ---------------0004260356523 Content-Type: application/postscript; name="current_easy_main.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="current_easy_main.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: current_easy_main.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips current_easy_main %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.04.26:1023 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 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Fs(\).)43 b(Here)30 b Fo(V)51 b Fs(is)29 b(a)h(real)e(p)s(oten)m(tial,)e Fo(~)-46 b(\033)31 b Fs(=)d(\()p Fo(\033)2657 3170 y Fm(1)2697 3155 y Fo(;)17 b(\033)2796 3170 y Fm(2)2835 3155 y Fo(;)g(\033)2934 3170 y Fm(3)2974 3155 y Fs(\))29 b(is)h(the)g(v)m(ector)324 3275 y(of)i(P)m(auli)f(spin)i(matrices:)1508 3548 y Fo(\033)1563 3563 y Fm(1)1686 3548 y Fs(=)1845 3407 y Fk(\022)1960 3487 y Fs(0)82 b(1)1960 3607 y(1)g(0)2182 3407 y Fk(\023)2272 3548 y Fo(;)1508 3822 y(\033)1563 3837 y Fm(2)1686 3822 y Fs(=)1845 3681 y Fk(\022)1960 3761 y Fs(0)g Fn(\000)p Fo(i)1967 3881 y(i)122 b Fs(0)2244 3681 y Fk(\023)2334 3822 y Fo(;)1508 4096 y(\033)1563 4111 y Fm(3)1686 4096 y Fs(=)1845 3955 y Fk(\022)1960 4035 y Fs(1)f(0)1960 4155 y(0)82 b Fn(\000)p Fs(1)2259 3955 y Fk(\023)2349 4096 y Fo(;)324 4395 y Fs(and)544 4370 y Fo(~)529 4395 y(B)58 b Fs(=)c Fn(r)33 b(\002)1035 4370 y Fo(~)1017 4395 y(A)p Fs(.)89 b(The)49 b(op)s(erator)e Ft(P)g Fs(con)m(tains)h(t)m(w)m(o)h(parameters)e Fo(h;)17 b(\026)54 b Fn(2)g Fl(R)3470 4410 y Fm(+)3535 4395 y Fs(,)324 4516 y(where)29 b Fo(h)e Fs(is)h(a)f(semi-classical)e (parameter,)j(whic)m(h)g(w)m(e)h(will)c(let)i(tend)i(to)e(zero,)i(and)f Fo(\026)f Fs(is)324 4636 y(parameter)d(measuring)h(the)g(strength)h(of) f(the)h(magnetic)e(\014eld.)40 b(W)-8 b(e)26 b(will)d(let)h Fo(\026)k Fn(!)f Fs(+)p Fn(1)324 4756 y Fs(as)40 b Fo(h)h Fn(!)g Fs(0)f(in)g(suc)m(h)i(a)e(w)m(a)m(y)h(that)g(the)f(pro)s(duct)h Fo(\026h)f Fs(remains)g(b)s(ounded)h(b)s(elo)m(w,)h(i.e.)324 4877 y Fo(\026h)27 b Fn(\025)h Fo(c)g(>)g Fs(0.)324 4997 y(Let)42 b Fo( )k Fs(b)s(e)d(an)m(y)f(state,)k(then)c(the)h Fp(curr)-5 b(ent)44 b(in)f(the)h(state)g Fo( )i Fs(is)c(the)g (distribution)3460 4975 y Fo(~)3469 4997 y(j)3509 5012 y Fj( )1918 5251 y Fs(3)p eop %%Page: 4 4 4 3 bop 324 548 a Fs(giv)m(en)32 b(b)m(y:)1442 576 y Fk(Z)1549 689 y Fo(~)1558 711 y(j)1598 726 y Fj( )1673 711 y Fn(\001)16 b Fo(~)-43 b(a)28 b Fs(=)g Fn(h)p Fo( )t Fn(j)p Ft(J)p Fs(\()p Fo(\026)-6 b(~)-43 b(a)o Fs(\))p Fn(j)p Fo( )t Fn(i)p Fo(;)324 934 y Fs(where)34 b Ft(J)p Fs(\()p Fo(\026)-6 b(~)-43 b(a)o Fs(\))33 b(is)f(the)h(op)s(erator:)826 1164 y Ft(J)p Fs(\()p Fo(\026)-6 b(~)-43 b(a)p Fs(\))28 b(=)f Fo(\026)-6 b(~)-43 b(a)p Fs(\()p Fn(\000)p Fo(ih)p Fn(r)23 b Fs(+)f Fo(\026)1796 1139 y(~)1778 1164 y(A)p Fs(\))g(+)g(\()p Fn(\000)p Fo(ih)p Fn(r)h Fs(+)f Fo(\026)2494 1139 y(~)2476 1164 y(A)p Fs(\))p Fo(\026)-6 b(~)-43 b(a)22 b Fs(+)g Fo(h\026)m(~)-46 b(\033)2979 1138 y(~)2991 1164 y(b;)324 1394 y Fs(with)537 1368 y Fo(~)548 1394 y(b)31 b Fs(=)g(rot)-6 b Fo(~)-43 b(a)p Fs(.)50 b(W)-8 b(e)35 b(will)d(study)k(the)f Fp(total)i(curr)-5 b(ent)37 b(of)f(a)h (non-inter)-5 b(acting)35 b(ele)-5 b(ctr)g(on)324 1515 y(gas)38 b(at)i(zer)-5 b(o)38 b(temp)-5 b(er)g(atur)g(e)37 b Fs(i.e.)57 b(w)m(e)38 b(sum)f(the)h(curren)m(t)g(of)f(all)e (eigenfunctions)i(b)s(elo)m(w)324 1635 y(zero)30 b(\(w)m(e)h(set)f(the) h(c)m(hemical)d(p)s(oten)m(tial)g(equal)i(to)f(zero\).)43 b(Th)m(us)32 b(the)e(de\014nition)f(of)g(the)324 1755 y(curren)m(t)34 b(\(as)e(a)g(distribution\))f(is:)1256 1888 y Fk(Z)1363 2001 y Fo(~)1373 2024 y(j)d Fn(\001)16 b Fo(~)-44 b(a)17 b(dx)28 b Fs(=)g(tr)o([)p Ft(J)p Fs(\()p Fo(\026)-6 b(~)-43 b(a)p Fs(\)1)2191 2039 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2423 2024 y Fs(\()p Ft(P)p Fs(\)])p Fo(:)324 2314 y Fs(The)39 b(curren)m(t)g(is)f(giv)m(en)h(as)f(the)h(rotation)e (of)g(the)i(magnetisation:)2878 2291 y Fo(~)2887 2314 y(j)k Fs(=)37 b(rot)3236 2289 y Fo(~)3208 2314 y(M)11 b Fs(,)40 b(th)m(us)324 2434 y(results)33 b(for)f(the)h(curren)m(t)h (translate)e(directly)g(in)m(to)f(results)i(for)f(the)h(magnetisation.) 324 2723 y Fh(1.1)135 b(Semiclassics)46 b(for)f(the)h(energy)324 2908 y Fs(The)33 b(energy)h(of)e(the)h(electron)g(gas)f(is)g(giv)m(en)h (b)m(y:)1039 3128 y Fo(E)6 b Fs(\()1173 3103 y Fo(~)1155 3128 y(A;)17 b(V)22 b Fs(\))27 b(=)h(tr[)p Ft(P)p Fs(\()p Fo(\026)1814 3103 y(~)1797 3128 y(A)o(;)17 b(V)k Fs(\)1)2078 3143 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2310 3128 y Fs(\()p Ft(P)p Fs(\()p Fo(\026)2539 3103 y(~)2522 3128 y(A)o(;)c(V)k Fs(\)\)])p Fo(:)324 3348 y Fs(Notice,)26 b(that)f(this)g(is)g(clearly)f (a)i(negativ)m(e)f(quan)m(tit)m(y)-8 b(.)42 b(The)26 b(semiclassical)d(asymptotics)324 3468 y(of)31 b(the)h(energy)g(has)g (b)s(een)g(calculated)f(b)m(y)63 b([LSY94])32 b(\(constan)m(t)g (magnetic)e(\014eld\))h(and)324 3589 y([ES97])e(\(non)g(constan)m(t)h (\014elds\),)g(and)f(it)e(w)m(as)j(found,)g(under)g(v)m(ery)g(general)f (conditions)324 3718 y(on)j Fo(V)5 b(;)583 3693 y(~)565 3718 y(A)p Fs(,)33 b(that)f Fo(E)6 b Fs(\()1043 3693 y Fo(~)1025 3718 y(A;)17 b(V)22 b Fs(\))27 b Fn(\031)1418 3674 y Fj(\026)p 1401 3695 76 4 v 1401 3752 a(h)1442 3733 y Fg(2)1487 3718 y Fo(E)1559 3733 y Fj(scl)1648 3718 y Fs(\()1704 3693 y Fo(~)1686 3718 y(A;)17 b(V)k Fs(\),)33 b(where)795 4023 y Fo(E)867 4038 y Fj(scl)957 4023 y Fs(\()1013 3998 y Fo(~)995 4023 y(A;)17 b(V)k Fs(\))28 b(=)f Fn(\000)1476 3956 y Fs(2)p 1446 4000 108 4 v 1446 4092 a(3)p Fo(\031)1581 3888 y Fk(Z)1734 3899 y Fi(1)1697 3929 y Fk(X)1703 4138 y Fj(n)p Fm(=0)1858 4023 y Fo(d)1909 4038 y Fj(n)1955 4023 y Fn(j)1999 3998 y Fo(~)1983 4023 y(B)5 b Fn(j)p Fs([2)p Fo(n\026h)p Fn(j)2383 3998 y Fo(~)2367 4023 y(B)g Fn(j)21 b Fs(+)h Fo(V)g Fs(\()p Fo(x)p Fs(\)])2830 3972 y Fm(3)p Fj(=)p Fm(2)2830 4045 y Fi(\000)2957 4023 y Fo(dx;)324 4338 y Fs(with)38 b Fo(d)603 4353 y Fm(0)679 4338 y Fs(=)824 4299 y Fm(1)p 803 4315 79 4 v 803 4372 a(2)p Fj(\031)929 4338 y Fs(and)g Fo(d)1175 4353 y Fj(n)1260 4338 y Fs(=)1387 4299 y Fm(1)p 1383 4315 43 4 v 1383 4372 a Fj(\031)1474 4338 y Fs(for)g Fo(n)f Fn(\025)h Fs(1.)61 b(Here)39 b(and)f(in)g(what)g(follo)m(ws)g(w) m(e)h(use)g(the)324 4458 y(notation)1482 4612 y([)p Fo(x)p Fs(])1591 4627 y Fi(\000)1679 4612 y Fs(=)1782 4471 y Fk(\032)1899 4551 y Fn(\000)p Fo(x)83 b(x)29 b Fn(\024)f Fs(0)1941 4671 y(0)124 b Fo(x)29 b(>)e Fs(0)1918 5251 y(4)p eop %%Page: 5 5 5 4 bop 324 548 a Fs(No)m(w,)33 b(formally)928 512 y Fm(1)988 526 y Fo(~)997 548 y(j)h Fs(=)1184 509 y Fj(\016)r(E)p 1184 525 90 4 v 1185 593 a(\016)1233 577 y(~)1218 593 y(A)1284 548 y Fs(,)e(so)h(w)m(e)h(w)m(ould)e(exp)s(ect)i(that)489 750 y Fo(h)545 714 y Fm(2)p 489 794 96 4 v 507 886 a Fo(\026)611 682 y Fk(Z)718 795 y Fo(~)727 817 y(j)28 b Fn(\001)16 b Fo(~)-43 b(a)17 b(dx)83 b Fn(\031)1295 750 y Fo(d)p 1272 794 97 4 v 1272 886 a(ds)1379 817 y Fn(j)1407 832 y Fj(s)p Fm(=0)1534 817 y Fo(E)1606 832 y Fj(scl)1695 817 y Fs(\()1751 792 y Fo(~)1733 817 y(A)22 b Fs(+)g Fo(s)-6 b(~)-43 b(a;)17 b(V)22 b Fs(\))1103 1101 y(=)1272 1034 y Fn(\000)p Fs(2)p 1272 1078 127 4 v 1281 1170 a(3)p Fo(\031)1462 977 y Fi(1)1425 1007 y Fk(X)1431 1216 y Fj(n)p Fm(=0)1586 1101 y Fo(d)1637 1116 y Fj(n)1700 966 y Fk(Z)1800 1101 y Fs(\()p Fo(@)1889 1116 y Fj(x)1929 1125 y Fg(1)1968 1101 y Fo(a)2019 1116 y Fm(2)2080 1101 y Fn(\000)h Fo(@)2231 1116 y Fj(x)2271 1125 y Fg(2)2310 1101 y Fo(a)2361 1116 y Fm(1)2401 1101 y Fs(\))1262 1433 y Fn(\002)1339 1263 y Fk( )1419 1433 y Fs([2)p Fo(nh\026)f Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\)])2024 1382 y Fm(3)p Fj(=)p Fm(2)2024 1455 y Fi(\000)2157 1433 y Fn(\000)i Fs(3)p Fo(nh\026)p Fs([2)p Fo(nh\026)f Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\)])3084 1382 y Fm(1)p Fj(=)p Fm(2)3084 1455 y Fi(\000)3195 1263 y Fk(!)3273 1433 y Fo(dx;)324 1705 y Fs(as)33 b Fo(h)f Fs(tends)i(to)e(zero.)44 b(This)33 b(is)f(indeed)h(the)g(result)f(of)g(the)h(pap)s(er:)324 1882 y Ft(Theorem)44 b(1.1.)i Fp(Supp)-5 b(ose)1421 1857 y Fo(~)1403 1882 y(A)39 b Fs(=)1639 1843 y Fm(1)p 1639 1859 36 4 v 1639 1916 a(2)1684 1882 y Fs(\()p Fn(\000)p Fo(x)1854 1897 y Fm(2)1895 1882 y Fo(;)17 b(x)1994 1897 y Fm(1)2033 1882 y Fo(;)g Fs(0\))p Fp(,)42 b(that)35 b Fo(~)-43 b(a)39 b Fs(=)g(\()p Fo(a)2735 1897 y Fm(1)2775 1882 y Fo(;)17 b(a)2870 1897 y Fm(2)2909 1882 y Fo(;)g Fs(0\))38 b Fn(2)h Fo(C)3260 1846 y Fi(1)3253 1906 y Fm(0)3335 1882 y Fp(,)j(and)324 2002 y(that)g Fo(V)5 b(;)18 b Fs(~)-50 b Fo(a)27 b Fn(\001)g(r)p Fo(V)62 b Fn(2)41 b Fo(L)1144 1966 y Fm(3)p Fj(=)p Fm(2)1282 2002 y Fn(\\)28 b Fo(L)1442 1966 y Fm(5)p Fj(=)p Fm(2)1552 2002 y Fp(,)43 b(wher)-5 b(e)43 b Fs(~)-51 b Fo(a)41 b Fs(=)f(\()p Fn(\000)p Fo(a)2281 2017 y Fm(2)2322 2002 y Fo(;)17 b(a)2417 2017 y Fm(1)2456 2002 y Fo(;)g Fs(0\))p Fp(.)65 b(Supp)-5 b(ose)41 b(furthermor)-5 b(e)324 2123 y(that)35 b Fo(\026h)28 b Fn(!)f Fo(\014)33 b Fn(2)28 b Fs(\(0)p Fo(;)17 b Fs(+)p Fn(1)p Fs(\))34 b Fp(as)g Fo(h)28 b Fn(!)g Fs(0)p Fp(.)44 b(Then)491 2388 y Fs(lim)485 2451 y Fj(h)p Fi(!)p Fm(0)658 2321 y Fo(h)714 2284 y Fm(2)p 658 2365 96 4 v 677 2456 a Fo(\026)780 2252 y Fk(Z)887 2366 y Fo(~)896 2388 y(j)29 b Fn(\001)16 b Fo(~)-44 b(a)17 b(dx)83 b Fs(=)1440 2321 y Fn(\000)p Fs(2)p 1440 2365 127 4 v 1449 2456 a(3)p Fo(\031)1630 2264 y Fi(1)1593 2293 y Fk(X)1600 2503 y Fj(\027)t Fm(=0)1754 2388 y Fo(d)1805 2403 y Fj(\027)1864 2252 y Fk(Z)1964 2388 y Fs(\()p Fo(@)2053 2403 y Fj(x)2093 2412 y Fg(1)2132 2388 y Fo(a)2183 2403 y Fm(2)2244 2388 y Fn(\000)23 b Fo(@)2395 2403 y Fj(x)2435 2412 y Fg(2)2474 2388 y Fo(a)2525 2403 y Fm(1)2565 2388 y Fs(\))1430 2720 y Fn(\002)1507 2550 y Fk( )1587 2720 y Fs([2)p Fo(\027)6 b(\014)28 b Fs(+)22 b Fo(V)f Fs(\()p Fo(x)p Fs(\)])2134 2669 y Fm(3)p Fj(=)p Fm(2)2134 2742 y Fi(\000)2267 2720 y Fn(\000)h Fs(3)p Fo(\027)6 b(\014)g Fs([2)p Fo(\027)g(\014)28 b Fs(+)22 b Fo(V)g Fs(\()p Fo(x)p Fs(\)])3078 2669 y Fm(1)p Fj(=)p Fm(2)3078 2742 y Fi(\000)3188 2550 y Fk(!)3267 2720 y Fo(dx:)470 2991 y Fs(Theorem)68 b(1.1)33 b(only)g(deals)h(with)f(the)h(curren)m(t)h(p)s (erp)s(endicular)e(to)h(the)g(magnetic)324 3112 y(\014eld.)63 b(It)39 b(turns)h(out)f(that)f(the)i(curren)m(t)g(parallel)d(to)h(the)i (\014eld)f(is)f(more)h(di\016cult)f(to)324 3232 y(analyse.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m(wing)d(result:)324 3409 y Ft(Theorem)25 b(1.2.)35 b Fp(L)-5 b(et)26 b(the)g(assumptions)f (b)-5 b(e)25 b(as)g(in)h(The)-5 b(or)g(em)50 b(1.1)26 b(exc)-5 b(ept)25 b(that)20 b Fo(~)-43 b(a)28 b Fn(2)g Fo(C)3487 3373 y Fi(1)3480 3434 y Fm(0)324 3530 y Fp(is)46 b(arbitr)-5 b(ary)47 b(\(i.e.)80 b Fo(a)1165 3545 y Fm(3)1252 3530 y Fp(is)46 b(not)h(ne)-5 b(c)g(essarily)46 b(vanishing\))f(and)h (supp)-5 b(ose)46 b Fo(V)68 b Fp(satis\014es)324 3650 y(the)29 b(fol)5 b(lowing)27 b(additional)h(symmetry)i(c)-5 b(onstr)g(aint:)41 b Fo(V)22 b Fs(\()p Fo(x)2442 3665 y Fm(1)2482 3650 y Fo(;)17 b(x)2581 3665 y Fm(2)2620 3650 y Fo(;)g Fn(\000)p Fo(x)2796 3665 y Fm(3)2836 3650 y Fs(\))28 b(=)f Fo(V)22 b Fs(\()p Fo(x)3177 3665 y Fm(1)3217 3650 y Fo(;)17 b(x)3316 3665 y Fm(2)3355 3650 y Fo(;)g(x)3454 3665 y Fm(3)3494 3650 y Fs(\))p Fp(.)324 3771 y(Then)491 4011 y Fs(lim)485 4074 y Fj(h)p Fi(!)p Fm(0)658 3944 y Fo(h)714 3908 y Fm(2)p 658 3988 96 4 v 677 4079 a Fo(\026)780 3875 y Fk(Z)887 3989 y Fo(~)896 4011 y(j)29 b Fn(\001)16 b Fo(~)-44 b(a)17 b(dx)83 b Fs(=)1440 3944 y Fn(\000)p Fs(2)p 1440 3988 127 4 v 1449 4079 a(3)p Fo(\031)1630 3887 y Fi(1)1593 3916 y Fk(X)1600 4126 y Fj(\027)t Fm(=0)1754 4011 y Fo(d)1805 4026 y Fj(\027)1864 3875 y Fk(Z)1964 4011 y Fs(\()p Fo(@)2053 4026 y Fj(x)2093 4035 y Fg(1)2132 4011 y Fo(a)2183 4026 y Fm(2)2244 4011 y Fn(\000)23 b Fo(@)2395 4026 y Fj(x)2435 4035 y Fg(2)2474 4011 y Fo(a)2525 4026 y Fm(1)2565 4011 y Fs(\))1430 4343 y Fn(\002)1507 4173 y Fk( )1587 4343 y Fs([2)p Fo(\027)6 b(\014)28 b Fs(+)22 b Fo(V)f Fs(\()p Fo(x)p Fs(\)])2134 4292 y Fm(3)p Fj(=)p Fm(2)2134 4365 y Fi(\000)2267 4343 y Fn(\000)h Fs(3)p Fo(\027)6 b(\014)g Fs([2)p Fo(\027)g(\014)28 b Fs(+)22 b Fo(V)g Fs(\()p Fo(x)p Fs(\)])3078 4292 y Fm(1)p Fj(=)p Fm(2)3078 4365 y Fi(\000)3188 4173 y Fk(!)3267 4343 y Fo(dx:)p 324 4527 1296 4 v 436 4588 a Ff(1)473 4618 y Fq(It)28 b(is)g(easy)e(to)i(pro)n(v)n(e)e(that)1170 4818 y(tr[)p Fe(J)p Fq(\()p Fd(\026)-5 b(~)-37 b(a)q Fq(\)1)1508 4833 y Ff(\()p Fr(\0001)p Fc(;)p Ff(0])1727 4818 y Fq(\()p Fe(P)p Fq(\)])24 b(=)2016 4762 y Fd(d)p 2001 4799 74 4 v 2001 4875 a(dt)2084 4818 y Fx(j)2107 4830 y Fc(t)p Ff(=0)2220 4818 y Fd(E)5 b Fq(\()2334 4797 y Fd(~)2318 4818 y(A)19 b Fq(+)f Fd(t)-5 b(~)-37 b(a;)14 b(V)19 b Fq(\))p Fd(;)324 5002 y Fq(if)28 b(the)g(deriv)-5 b(ativ)n(e)27 b(on)g(the)h(righ)n(t)f(hand)g(exists.)1918 5251 y Fs(5)p eop %%Page: 6 6 6 5 bop 324 548 a Fh(1.2)135 b(Di\016culties)324 733 y Fs(Let)33 b(us)g(recall)e(ho)m(w)i(the)g(densit)m(y)h(is)e (calculated)64 b([LS77]:)324 853 y(The)33 b(densit)m(y)h Fo(\032)f Fs(is)f(de\014ned)i(as)1215 969 y Fk(Z)1331 1105 y Fo(\032\036)17 b(dx)27 b Fs(=)h(tr[)p Fo(\036)p Fs(1)1903 1120 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2134 1105 y Fs(\()p Ft(P)p Fs(\()p Fo(\026)2363 1079 y(~)2346 1105 y(A)o(;)17 b(V)22 b Fs(\)\)])p Fo(;)324 1362 y Fs(for)40 b(all)f Fo(\036)i Fn(2)h Fo(C)909 1325 y Fi(1)902 1386 y Fm(0)983 1362 y Fs(\()p Fl(R)5 b Fs(\).)74 b(F)-8 b(ormally)g(,)40 b Fo(\032)h Fs(is)f(the)h(v)-5 b(ariational)37 b(deriv)-5 b(ativ)m(e)40 b(of)g(the)i(energy)324 1511 y(with)h(resp)s(ect)i(to)e Fo(V)65 b Fs(i.e.)76 b Fo(\032)1452 1451 y Fj(f)7 b(or)r(mal)q(l)q(y) 1559 1511 y Fs(=)1799 1472 y Fj(\016)r(E)p 1798 1488 91 4 v 1798 1545 a(\016)r(V)1898 1511 y Fs(.)77 b(T)-8 b(o)43 b(calculate)g(the)h(asymptotics)f(of)g(the)324 1631 y(densit)m(y)33 b(w)m(e)h(use)f(the)g(follo)m(wing)d(v)-5 b(ariational)29 b(principle:)1274 1840 y Fo(E)6 b Fs(\()1408 1815 y Fo(~)1390 1840 y(A;)17 b(V)k Fs(\))28 b(=)79 b(inf)1755 1901 y Fm(0)p Fi(\024)p Fj(\015)t Fi(\024)p Fm(1)1992 1840 y Fs(tr[)p Fo(\015)5 b Ft(P)p Fs(\()p Fo(\026)2342 1815 y(~)2325 1840 y(A)o(;)17 b(V)k Fs(\)])p Fo(:)324 2116 y Fs(Let)31 b Fo(H)8 b Fs(\()p Fo(s)p Fs(\))27 b(=)g Ft(P)p Fs(\()p Fo(\026)1030 2091 y(~)1012 2116 y(A;)17 b(V)39 b Fs(+)19 b Fo(s\036)p Fs(\))30 b(and)h(let)f Fo(E)6 b Fs(\()p Fo(s)p Fs(\))31 b(b)s(e)g(the)g(corresp)s(onding)g (energy)-8 b(.)43 b(Then,)324 2246 y(b)m(y)33 b(using)g(1)764 2261 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])995 2246 y Fs(\()p Ft(P)p Fs(\()p Fo(\026)1224 2220 y(~)1207 2246 y(A)o(;)17 b(V)k Fs(\)\))33 b(in)f(the)h(v)-5 b(ariational)29 b(principle)i(for)h Fo(E)6 b Fs(\()p Fo(s)p Fs(\),)32 b(w)m(e)i(obtain:)1396 2510 y Fo(E)6 b Fs(\()p Fo(s)p Fs(\))22 b Fn(\000)g Fo(E)6 b Fs(\(0\))28 b Fn(\024)g Fo(s)2116 2374 y Fk(Z)2232 2510 y Fo(\032\036)17 b(dx:)324 2771 y Fs(If)32 b(w)m(e)i(no)m(w)g (divide)e(b)m(y)h Fo(s)28 b Fn(6)p Fs(=)g(0)33 b(on)f(b)s(oth)h(sides)g (of)g(the)g(inequalit)m(y)-8 b(,)31 b(m)m(ultiply)g(b)m(y)i Fo(h)3387 2735 y Fm(2)3427 2771 y Fo(=\026)p Fs(,)324 2891 y(and)f(let)g Fo(h)h Fs(and)g Fo(s)f Fs(tend)h(successiv)m(ely)i (to)e(zero,)g(then)g(w)m(e)g(get:)1416 3008 y Fk(Z)1532 3144 y Fo(\032\036)17 b(dx)27 b Fn(!)1918 3008 y Fk(Z)2044 3077 y Fo(\016)t(E)2163 3092 y Fj(scl)p 2044 3121 209 4 v 2085 3212 a Fo(\016)t(V)2262 3144 y(\036)17 b(dx:)324 3395 y Fs(Unfortunately)-8 b(,)32 b(this)g(tec)m(hnique)i(do)s(es)f (not)g(w)m(ork)g(for)f(the)h(curren)m(t:)45 b(If)32 b(w)m(e)i(de\014ne) 1376 3579 y(~)1350 3604 y Fo(H)8 b Fs(\()p Fo(s)p Fs(\))27 b(=)h Ft(P)p Fs(\()p Fo(\026)1883 3579 y(~)1866 3604 y(A)o(;)17 b(V)k Fs(\))h(+)g Fo(s)p Ft(J)p Fs(\()p Fo(\026)-6 b(~)-43 b(a)p Fs(\))p Fo(;)324 3813 y Fs(and)32 b(let)677 3787 y(~)654 3813 y Fo(E)6 b Fs(\()p Fo(s)p Fs(\))33 b(b)s(e)g(the)g(corresp)s(onding)f(energy)-8 b(,)34 b(then)f(w)m(e)g (get)656 4005 y(~)633 4030 y Fo(E)6 b Fs(\()p Fo(s)p Fs(\))83 b Fn(\024)h Fs(tr[1)1229 4046 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])1460 4030 y Fs(\()p Ft(P)p Fs(\()p Fo(\026)p Fs(\()1727 4005 y Fo(~)1710 4030 y(A)21 b Fs(+)h Fo(s)-6 b(~)-43 b(a)p Fs(\))p Fo(;)17 b(V)22 b Fs(\)\))2261 4005 y(~)2236 4030 y Fo(H)7 b Fs(\()p Fo(s)p Fs(\)])917 4192 y(=)84 b Fo(E)6 b Fs(\()1211 4166 y Fo(~)1193 4192 y(A)22 b Fs(+)g Fo(s)-6 b(~)-43 b(a;)17 b(V)k Fs(\))h Fn(\000)h Fo(s)1811 4150 y Fm(2)1850 4192 y Fo(\026)1909 4150 y Fm(2)1949 4192 y Fs(tr)o([)-6 b Fo(~)-43 b(a)2102 4150 y Fm(2)2142 4192 y Fs(1)2191 4207 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2422 4192 y Fs(\()p Ft(P)p Fs(\()p Fo(\026)p Fs(\()2690 4166 y Fo(~)2672 4192 y(A)22 b Fs(+)g Fo(s)-6 b(~)-43 b(a)p Fs(\))p Fo(;)17 b(V)k Fs(\)\)])p Fo(:)324 4400 y Fs(The)41 b(\014rst)g(term)g(on)f(the)h(righ)m(t)f(hand)h(side)f (is)h(kno)m(wn)g(to)g(b)s(e)g(of)f(order)3101 4356 y Fj(\026)p 3084 4377 76 4 v 3084 4435 a(h)3125 4416 y Fg(2)3170 4400 y Fs(,)i(but)f(the)324 4521 y(second)34 b(term)e(is)g(of)h(order)g Fo(\026)1396 4484 y Fm(2)1461 4477 y Fj(\026)p 1445 4498 V 1445 4555 a(h)1486 4536 y Fg(2)1563 4521 y Fs(!)44 b(Th)m(us,)35 b(this)d(term)g(-)h(whic)m(h)g (is)f(quadratic)h(in)f Fo(s)g Fs(and)324 4641 y(therefore)h(without)f (in)m(terest)h(for)f(us)h(-)f(sp)s(oils)g(the)h(asymptotic)f(picture.) 324 4761 y(The)f(morale)c(of)j(this)f(calculation)f(is,)i(that)f(the)h (op)s(erator)f Ft(J)h Fs(is)f(to)s(o)g(big)g(-)g(adding)g(just)324 4882 y(a)c(bit)f(of)h(it,)g(c)m(hanges)i(the)e(energy)i(dramatically)-8 b(.)37 b(The)26 b(w)m(a)m(y)h(out)e(of)g(this)f(problem)g(is)h(to)324 5002 y(realize)k(that)i Fn(h)p Fo(\036)p Fn(j)p Ft(J)p Fn(j)p Fo(\036)p Fn(i)e Fs(is)h('small')f(for)h(all)f Fo(\036)h Fs(whic)m(h)h(are)g(eigenfunctions)g(of)f Ft(P)p Fs(\()p Fo(\026)3319 4977 y(~)3302 5002 y(A)o(;)17 b(V)22 b Fs(\).)1918 5251 y(6)p eop %%Page: 7 7 7 6 bop 324 548 a Fh(1.3)135 b(Organisation)47 b(of)e(the)g(pap)t(er) 324 733 y Fs(In)34 b(Section)67 b(2)34 b(w)m(e)h(pro)m(v)m(e)g(a)e (comm)m(utator)g(form)m(ula)e(for)j Ft(J)p Fs(.)47 b(This)34 b(form)m(ula)e(expresses)324 853 y Ft(J)46 b Fs(as)h(a)g(comm)m(utator) e(with)h Ft(P)g Fs(plus)h(an)f(op)s(erator)g(whic)m(h)i(is)e(a)g (factor)g Fo(\026)h Fs(smaller)324 973 y(than)37 b Ft(J)p Fs(.)56 b(Since)38 b(the)f(comm)m(utator)f(do)s(es)h(not)g(con)m (tribute)g(to)g(the)g(trace,)i(w)m(e)f(hereb)m(y)324 1094 y(reduce)i(the)f(problem)f(of)g(calculating)e(the)j(curren)m(t)h (considerably)-8 b(.)62 b(Unfortunately)-8 b(,)324 1214 y(this)23 b(comm)m(utator)f(form)m(ula)f(only)i(giv)m(es)h(information) c(ab)s(out)j(the)h(curren)m(t)g(orthogonal)324 1335 y(to)41 b(the)h(magnetic)f(\014eld)g(-)g(this)h(is)f(the)h(reason)g(wh)m(y)h (the)f(parallel)d(curren)m(t)k(is)e(more)324 1455 y(di\016cult.)324 1575 y(In)32 b(Section)63 b(4)31 b(w)m(e)i(pro)m(v)m(e)g(that)f(the)g (curren)m(t)h(is)e Fp(gauge)j(invariant)d Fs(i.e.)42 b(that)3174 1553 y Fo(~)3183 1575 y(j)26 b Fn(\001)14 b Fo(~)-43 b(a)32 b Fs(only)324 1696 y(dep)s(ends)i(on)e(the)h (magnetic)f(\014eld)1622 1669 y Fo(~)1633 1696 y(b)c Fs(=)g(rot)-7 b Fo(~)-43 b(a)33 b Fs(generated)h(b)m(y)27 b Fo(~)-43 b(a)p Fs(.)324 1816 y(Then,)52 b(in)47 b(Sections)95 b(5)47 b(and)95 b(6)47 b(w)m(e)i(use)f(the)g('v)-5 b(ariational)44 b(principle')i(-)h(i.e.)87 b(the)324 1936 y(metho)s(d)44 b(used)j(ab)s(o)m(v)m(e)f(to)f(calculate)f(the)i(densit)m(y)g(-)f(to)g (calculate)f(the)h(orthogonal)324 2057 y(curren)m(t.)f(Using)31 b(symmetry)-8 b(,)31 b(gauge)g(in)m(v)-5 b(ariance)31 b(and)g(the)h(result)f(on)g(the)g(orthogonal)324 2177 y(curren)m(t,)j(w)m(e)f(can)g(pro)m(v)m(e)h(Theorem)65 b(1.2.)324 2298 y(Finally)-8 b(,)37 b(in)h(App)s(endix)77 b(A,)41 b(w)m(e)e(giv)m(e)g(some)f(argumen)m(ts)h(to)f(supp)s(ort)h (the)g(necessit)m(y)324 2418 y(of)32 b(using)g(our)h(comm)m(utator)e (form)m(ula:)41 b(W)-8 b(e)33 b(construct)h(a)e(densit)m(y)i(matrix)d (whic)m(h)i(has)324 2538 y(asymptotically)c(\(as)j Fo(h)c Fn(!)f Fs(0\))32 b(the)g(same)g(energy)g(and)g(densit)m(y)h(as)f(the)g (ground)g(state,)324 2659 y(but)h(do)s(es)g(not)f(ha)m(v)m(e)i(the)f (righ)m(t)f(curren)m(t.)324 2948 y Fh(1.4)135 b(Notation)47 b(and)d(preliminaries)324 3132 y Fs(The)26 b(results)f(in)g(Section)49 b(3)25 b(and)g(the)h(calculations)d(in)h(Sections)51 b(5)25 b(and)50 b(6)24 b(are)i(only)e(for)324 3253 y(a)32 b(constan)m(t)i(magnetic)e(\014eld)h(and)g(there)g(w)m(e)h(\014x)g(the) f(c)m(hoice)g(of)g(the)g(v)m(ector)h(p)s(oten)m(tial)324 3373 y(as)475 3348 y Fo(~)456 3373 y(A)q Fs(\()p Fo(x)p Fs(\))50 b(=)846 3334 y Fm(1)p 846 3350 36 4 v 846 3407 a(2)891 3373 y Fs(\()p Fn(\000)p Fo(x)1061 3388 y Fm(2)1102 3373 y Fo(;)17 b(x)1201 3388 y Fm(1)1240 3373 y Fo(;)g Fs(0\).)82 b(The)46 b(comm)m(utator)e(form)m(ula)g(in)g(Section)91 b(2)45 b(is)g(v)-5 b(alid)324 3512 y(for)41 b(general,)i(ev)m(erywhere) i(non)m(v)-5 b(anishing)41 b(magnetic)f(\014elds,)45 b(so)c(in)g(that)h(section)3507 3487 y Fo(~)3489 3512 y(A)324 3632 y Fs(denotes)34 b(an)e(arbitrary)g(v)m(ector)h(p)s(oten)m (tial.)324 3753 y(W)-8 b(e)33 b(will)d(denote)j(the)g(magnetic)f(momen) m(tum)f(op)s(erator)h(as)g Fo(p)2662 3766 y Fj(~)2649 3783 y(A)2734 3753 y Fs(=)c(\()p Fn(\000)p Fo(ih)p Fn(r)23 b Fs(+)f Fo(\026)3322 3728 y(~)3305 3753 y(A)o Fs(\).)324 3873 y(F)-8 b(urthermore,)34 b(w)m(e)h(will)d(denote)j(the)g(closed)g (ball)d(of)i(radius)g Fo(r)j Fs(cen)m(tered)f(around)e(the)324 3993 y(p)s(oin)m(t)d Fo(x)i Fs(b)m(y)h Fo(B)5 b Fs(\()p Fo(x;)17 b(r)s Fs(\).)324 4114 y(All)40 b(through)j(the)f(pap)s(er)h(w) m(e)g(will)d(apply)i(the)h(standard)g(con)m(v)m(en)m(tion)g(that)f Fo(c)h Fs(or)f Fo(C)324 4234 y Fs(denote)33 b(arbitrary)f(constan)m (ts.)324 4355 y(Finally)e(a)i(few)h(w)m(ords)h(on)e(the)h(P)m(auli)f (op)s(erator)g(in)f(a)i(constan)m(t)g(magnetic)e(\014eld:)324 4475 y(With)g(our)h(c)m(hoice)h(of)1166 4450 y Fo(~)1148 4475 y(A)q Fs(,)f(the)g(magnetic)f(\014eld)h(is)g(parallel)d(to)j(the)g (3rd)g(unit)g(v)m(ector)d Fo(~)-45 b(e)3522 4490 y Fm(3)324 4595 y Fs(and)32 b(therefore)989 4857 y Ft(P)27 b Fs(=)1197 4716 y Fk(\022)1311 4788 y Fo(p)1360 4751 y Fm(2)1374 4815 y Fj(~)1360 4832 y(A)1440 4788 y Fs(+)22 b Fo(\026h)g Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\))395 b(0)1623 4916 y(0)e Fo(p)2114 4880 y Fm(2)2128 4943 y Fj(~)2114 4960 y(A)2194 4916 y Fn(\000)22 b Fo(\026h)g Fs(+)g Fo(V)g Fs(\()p Fo(x)p Fs(\))2780 4716 y Fk(\023)2869 4857 y Fo(;)1918 5251 y Fs(7)p eop %%Page: 8 8 8 7 bop 324 548 a Fs(and)32 b(th)m(us)324 768 y(tr[)p Ft(J)p Fs(\()p Fo(\026)-6 b(~)-43 b(a)o Fs(\)1)719 783 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])951 768 y Fs(\()p Ft(P)p Fs(\)])82 b(=)h Fo(\026)p Fs(tr[\()-6 b Fo(~)-43 b(a)22 b Fn(\001)g Fo(p)1757 781 y Fj(~)1744 798 y(A)1823 768 y Fs(+)g Fo(p)1983 781 y Fj(~)1970 798 y(A)2049 768 y Fn(\001)16 b Fo(~)-43 b(a)22 b Fs(+)g Fo(hb)2367 783 y Fm(3)2407 768 y Fs(\)1)2494 783 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2726 768 y Fs(\()p Fo(p)2813 727 y Fm(2)2826 790 y Fj(~)2813 807 y(A)2892 768 y Fs(+)g Fo(\026h)g Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\)\)])1372 924 y(+)p Fo(\026)p Fs(tr[\()-6 b Fo(~)-43 b(a)22 b Fn(\001)g Fo(p)1833 937 y Fj(~)1820 954 y(A)1899 924 y Fs(+)g Fo(p)2059 937 y Fj(~)2046 954 y(A)2125 924 y Fn(\001)16 b Fo(~)-43 b(a)22 b Fn(\000)h Fo(hb)2445 939 y Fm(3)2485 924 y Fs(\)1)2572 940 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2803 924 y Fs(\()p Fo(p)2890 883 y Fm(2)2904 947 y Fj(~)2890 964 y(A)2969 924 y Fn(\000)g Fo(\026h)f Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\)\)])p Fo(:)324 1144 y Fs(W)-8 b(e)27 b(therefore)h(can)f(\(and)g (will\))e(calculate)h(the)h(curren)m(t)h(as)g(the)f(sum)g(of)f(the)i(t) m(w)m(o)g(terms)324 1265 y(on)k(the)h(righ)m(t)f(hand)h(side.)324 1597 y Fu(2)161 b(Comm)l(utator)51 b(iden)l(tit)l(y)324 1829 y Fs(Let)33 b(us)g(assume)g Fn(j)1005 1803 y Fo(~)989 1829 y(B)5 b Fs(\()p Fo(x)p Fs(\))p Fn(j)27 b(6)p Fs(=)h(0)k(for)g(all) f Fo(x)p Fs(,)i(where)2136 1803 y Fo(~)2121 1829 y(B)f Fs(=)c Fn(r)22 b(\002)2554 1803 y Fo(~)2536 1829 y(A)p Fs(.)43 b(De\014ne)1346 2058 y Fo(H)35 b Fs(=)27 b(\()p Fn(\000)p Fo(ih)p Fn(r)c Fs(+)f Fo(\026)2050 2032 y(~)2032 2058 y(A)p Fs(\))2143 2016 y Fm(2)2205 2058 y Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\))p Fo(;)324 2287 y Fs(and)32 b(write)h Fo(J)817 2302 y Fj(p)856 2287 y Fs(\()-6 b Fo(~)-43 b(a)p Fs(\))28 b(=)21 b Fo(~)-42 b(a)22 b Fn(\001)g Fs(\()p Fn(\000)p Fo(ih)p Fn(r)h Fs(+)f Fo(\026)1723 2261 y(~)1705 2287 y(A)p Fs(\))g(+)g(\()p Fn(\000)p Fo(ih)p Fn(r)h Fs(+)f Fo(\026)2421 2261 y(~)2403 2287 y(A)p Fs(\))g Fn(\001)16 b Fo(~)-44 b(a)q Fs(.)43 b(Let)33 b(furthermore)828 2586 y(~)-50 b Fo(a)83 b Fs(=)1146 2493 y Fo(~)1130 2519 y(B)27 b Fn(\002)16 b Fo(~)-42 b(a)p 1130 2563 253 4 v 1169 2671 a Fn(j)1213 2646 y Fo(~)1197 2671 y(B)5 b Fn(j)1304 2642 y Fm(2)1392 2586 y Fo(;)812 2935 y Fl(B)104 b Fs(=)1120 2735 y Fk(0)1120 2914 y(@)1320 2814 y Fs(0)192 b Fo(B)1635 2829 y Fm(3)1797 2814 y Fn(\000)p Fo(B)1948 2829 y Fm(2)1249 2934 y Fn(\000)p Fo(B)1400 2949 y Fm(3)1594 2934 y Fs(0)h Fo(B)1910 2949 y Fm(1)1287 3054 y Fo(B)1361 3069 y Fm(2)1523 3054 y Fn(\000)p Fo(B)1674 3069 y Fm(1)1868 3054 y Fs(0)2029 2735 y Fk(1)2029 2914 y(A)2144 2935 y Fs(=)28 b Fn(f)p Fo(@)2349 2950 y Fj(x)2389 2960 y Fb(j)2425 2935 y Fo(A)2498 2950 y Fj(k)2563 2935 y Fn(\000)23 b Fo(@)2714 2950 y Fj(x)2754 2962 y Fb(k)2796 2935 y Fo(A)2869 2950 y Fj(j)2906 2935 y Fn(g)2956 2950 y Fj(j;k)3047 2935 y Fo(:)324 3295 y Fs(If)32 b(no)m(w)27 b Fo(~)-42 b(a)p Fs(\()p Fo(x)p Fs(\))22 b Fn(\001)895 3269 y Fo(~)879 3295 y(B)5 b Fs(\()p Fo(x)p Fs(\))28 b(=)g(0)k(for)g(all)e Fo(x)p Fs(,)k(then)f Fl(B)16 b Fs(~)-50 b Fo(a)34 b Fs(=)21 b Fo(~)-42 b(a)p Fs(.)324 3507 y Ft(Remark)52 b(2.1.)d Fs(Notice,)g(that)d(if)1681 3482 y Fo(~)1665 3507 y(B)56 b Fs(=)50 b(\(1)p Fo(;)17 b Fs(0)p Fo(;)g Fs(0\))45 b(and)40 b Fo(~)-43 b(a)51 b Fs(=)g(\()p Fo(a)2798 3522 y Fm(1)2837 3507 y Fo(;)17 b(a)2932 3522 y Fm(2)2972 3507 y Fo(;)g Fs(0\))45 b(then)j(~)-50 b Fo(a)51 b Fs(=)324 3627 y(\()p Fn(\000)p Fo(a)490 3642 y Fm(2)530 3627 y Fo(;)17 b(a)625 3642 y Fm(1)664 3627 y Fo(;)g Fs(0\).)470 3831 y(Let)27 b(us)h(denote)g(b)m(y)g(\(;)17 b(\))27 b(the)h(inner)f(pro)s(duct)g(in) g Fl(R)2286 3795 y Fm(3)2359 3831 y Fs(and)g(b)m(y)h Fn(h)p Fs(;)17 b Fn(i)27 b Fs(the)g(inner)g(pro)s(duct)324 3951 y(in)38 b Fo(L)510 3915 y Fm(2)550 3951 y Fs(\()p Fl(R)654 3915 y Fm(3)699 3951 y Fs(\).)63 b(Let)40 b(us)f(\014nally)f (write)h(the)h(magnetic)e(momen)m(tum)f(op)s(erator)i(as)g Fo(p)3403 3964 y Fj(~)3390 3981 y(A)3486 3951 y Fs(=)324 4086 y(\()p Fn(\000)p Fo(ih)p Fn(r)23 b Fs(+)f Fo(\026)809 4061 y(~)791 4086 y(A)p Fs(\).)43 b(Then)34 b(w)m(e)f(get:)324 4298 y Ft(Lemma)k(2.2.)42 b Fp(If)34 b Fn(j)1083 4273 y Fo(~)1067 4298 y(B)5 b Fs(\()p Fo(x)p Fs(\))p Fn(j)28 b(6)p Fs(=)f(0)35 b Fp(and)28 b Fo(~)-43 b(a)p Fs(\()p Fo(x)p Fs(\))23 b Fn(\001)1979 4273 y Fo(~)1963 4298 y(B)6 b Fs(\()p Fo(x)p Fs(\))28 b(=)f(0)35 b Fp(for)f(al)5 b(l)35 b Fo(x)p Fp(,)g(then)752 4518 y Fs([)p Fo(H)r(;)17 b(J)960 4533 y Fj(p)999 4518 y Fs(\()q(~)-50 b Fo(a)p Fs(\)])83 b(=)g(2)p Fo(ih)q Fs(~)-50 b Fo(a)23 b Fn(\001)e(r)p Fo(V)44 b Fn(\000)23 b Fs(2)p Fo(ih\026J)2191 4533 y Fj(p)2230 4518 y Fs(\()-6 b Fo(~)-43 b(a)p Fs(\))1395 4663 y Fn(\000)p Fs(2)p Fo(ih)p Fs(\()p Fo(p)1710 4676 y Fj(~)1697 4694 y(A)1755 4663 y Fs(;)17 b(\()p Fo(D)t Fs(~)-50 b Fo(a)21 b Fs(+)h(\()p Fo(D)t Fs(~)-50 b Fo(a)p Fs(\))2302 4622 y Fj(t)2332 4663 y Fs(\))p Fo(p)2432 4676 y Fj(~)2419 4694 y(A)2476 4663 y Fs(\))22 b Fn(\000)g Fo(ih)2724 4622 y Fm(3)2764 4663 y Fs(\001div)r(\()q(~)-50 b Fo(a)p Fs(\))p Fo(:)470 4884 y Fs(Before)33 b(w)m(e)h(giv)m(e)e(the)h (pro)s(of)f(of)g(Lemma)64 b(2.2)32 b(w)m(e)h(state)g(an)g(easy)g (consequence:)1918 5251 y(8)p eop %%Page: 9 9 9 8 bop 324 548 a Ft(Corollary)37 b(2.3.)42 b Fp(L)-5 b(et)35 b Fo(\036)g Fp(b)-5 b(e)34 b(an)h(eigenfunction)f(for)g Fo(H)8 b Fp(,)35 b(i.e.)44 b Fo(H)8 b(\036)27 b Fs(=)g Fo(\025\036)p Fp(,)35 b(then)490 768 y Fo(\026)p Fn(h)p Fo(\036)p Fs(;)17 b Fo(J)744 783 y Fj(p)783 768 y Fs(\()-6 b Fo(~)-43 b(a)p Fs(\))p Fo(\036)p Fn(i)82 b Fs(=)h Fn(h)p Fo(\036)p Fs(;)17 b(~)-49 b Fo(a)22 b Fn(\001)f(r)p Fo(V)h(\036)p Fn(i)1248 961 y(\000h)p Fo(\036)p Fs(;)17 b(\()p Fo(p)1566 974 y Fj(~)1553 991 y(A)1610 961 y Fs(;)g(\(\()p Fo(D)t Fs(~)-50 b Fo(a)21 b Fs(+)h(\()p Fo(D)t Fs(~)-50 b Fo(a)p Fs(\))2195 919 y Fj(t)2225 961 y Fs(\))p Fo(p)2325 974 y Fj(~)2312 991 y(A)2369 961 y Fs(\))p Fo(\036)p Fn(i)21 b(\000)2635 893 y Fs(1)p 2635 938 49 4 v 2635 1029 a(2)2693 961 y Fo(h)2749 919 y Fm(2)2789 961 y Fn(h)p Fo(\036)p Fs(;)c(\001div)q(\()q(~)-50 b Fo(a)p Fs(\))p Fo(\036)p Fn(i)p Fo(:)324 1212 y Fp(Pr)-5 b(o)g(of.)48 b Fn(h)p Fo(\036)p Fs(;)17 b([)p Fo(H)r(;)g(J)978 1227 y Fj(p)1017 1212 y Fs(\()q(~)-50 b Fo(a)p Fs(\)])p Fo(\036)p Fn(i)27 b Fs(=)h(0)k(for)g(all)f(eigenfunctions)h Fo(\036)p Fs(.)p 3491 1212 4 66 v 3495 1149 59 4 v 3495 1212 V 3553 1212 4 66 v 470 1407 a(No)m(w)h(w)m(e)h(pro)m(v)m(e)g(Lemma)63 b(2.2.)324 1602 y Fp(Pr)-5 b(o)g(of.)48 b Fs(The)34 b(pro)s(of)d(of)i (Lemma)63 b(2.2)32 b(is)g(essen)m(tially)h(just)g(a)f(calculation:)822 1822 y([)p Fo(H)r(;)17 b(p)1038 1835 y Fj(~)1025 1852 y(A)1082 1822 y Fs(])83 b(=)g([)p Fo(p)1427 1781 y Fm(2)1441 1844 y Fj(~)1427 1861 y(A)1484 1822 y Fo(;)17 b(p)1590 1835 y Fj(~)1577 1852 y(A)1634 1822 y Fs(])22 b(+)g([)p Fo(V)5 b(;)17 b(p)1976 1835 y Fj(~)1963 1852 y(A)2020 1822 y Fs(])1192 2100 y(=)83 b Fo(ih)p Fn(r)p Fo(V)44 b Fn(\000)23 b Fo(ih\026)1889 1899 y Fk(0)1889 2079 y(@)2017 1979 y Fs(\()p Fo(p)2117 1992 y Fj(~)2104 2009 y(A)2161 1979 y Fs(;)17 b Fl(B)e Fo(e)2311 1994 y Fm(1)2356 1979 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)2658 1994 y Fm(1)2703 1979 y Fo(;)i(p)2809 1992 y Fj(~)2796 2009 y(A)2853 1979 y Fs(\))2017 2099 y(\()p Fo(p)2117 2112 y Fj(~)2104 2129 y(A)2161 2099 y Fs(;)g Fl(B)e Fo(e)2311 2114 y Fm(2)2356 2099 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)2658 2114 y Fm(2)2703 2099 y Fo(;)i(p)2809 2112 y Fj(~)2796 2129 y(A)2853 2099 y Fs(\))2017 2219 y(\()p Fo(p)2117 2232 y Fj(~)2104 2249 y(A)2161 2219 y Fs(;)g Fl(B)e Fo(e)2311 2234 y Fm(3)2356 2219 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)2658 2234 y Fm(3)2703 2219 y Fo(;)i(p)2809 2232 y Fj(~)2796 2249 y(A)2853 2219 y Fs(\))2932 1899 y Fk(1)2932 2079 y(A)3036 2100 y Fo(;)892 2568 y Fs([)p Fo(H)r(;)g(f)11 b Fs(])83 b(=)g([)p Fo(p)1450 2526 y Fm(2)1463 2590 y Fj(~)1450 2607 y(A)1507 2568 y Fo(;)17 b(f)11 b Fs(])27 b(=)1768 2473 y Fk(X)1823 2683 y Fj(j)1928 2568 y Fo(p)1990 2581 y Fj(~)1977 2598 y(A;j)2086 2568 y Fs([)p Fo(p)2175 2581 y Fj(~)2162 2598 y(A;j)2271 2568 y Fo(;)17 b(f)11 b Fs(])22 b(+)g([)p Fo(p)2610 2581 y Fj(~)2597 2598 y(A;j)2706 2568 y Fo(;)17 b(f)11 b Fs(])p Fo(p)2898 2581 y Fj(~)2885 2598 y(A)o(;j)1215 2844 y Fs(=)83 b Fn(\000)p Fo(ih)1557 2749 y Fk(X)1613 2959 y Fj(j)1701 2844 y Fs(\()p Fo(p)1801 2857 y Fj(~)1788 2874 y(A;j)1897 2844 y Fo(@)1948 2859 y Fj(x)1988 2869 y Fb(j)2025 2844 y Fo(f)33 b Fs(+)22 b Fo(@)2255 2859 y Fj(x)2295 2869 y Fb(j)2332 2844 y Fo(f)11 b(p)2453 2857 y Fj(~)2440 2874 y(A)o(;j)2548 2844 y Fs(\))1215 3090 y(=)83 b Fn(\000)p Fo(ih)p Fs(\()p Fo(p)1640 3103 y Fj(~)1627 3120 y(A)1707 3090 y Fn(\001)22 b(r)p Fo(f)32 b Fs(+)22 b Fn(r)p Fo(f)33 b Fn(\001)22 b Fo(p)2294 3103 y Fj(~)2281 3120 y(A)2338 3090 y Fs(\))p Fo(:)324 3310 y Fs(Based)33 b(on)g(these)h(t)m(w)m(o)f(equalities)e(w)m (e)j(get:)566 3530 y([)p Fo(H)r(;)17 b(J)774 3545 y Fj(p)813 3530 y Fs(\()q(~)-50 b Fo(a)p Fs(\)])407 3675 y(=)84 b(~)-50 b Fo(a)22 b Fn(\001)g Fs([)p Fo(H)r(;)17 b(p)905 3688 y Fj(~)892 3705 y(A)949 3675 y Fs(])22 b(+)g([)p Fo(H)r(;)17 b(p)1312 3688 y Fj(~)1299 3705 y(A)1356 3675 y Fs(])22 b Fn(\001)h Fs(~)-50 b Fo(a)22 b Fs(+)g Fo(p)1688 3688 y Fj(~)1675 3705 y(A)1754 3675 y Fn(\001)g Fs([)p Fo(H)r(;)c Fs(~)-50 b Fo(a)p Fs(])23 b(+)f([)p Fo(H)r(;)c Fs(~)-50 b Fo(a)p Fs(])22 b Fn(\001)g Fo(p)2523 3688 y Fj(~)2510 3705 y(A)407 3820 y Fs(=)83 b(2)p Fo(ih)q Fs(~)-50 b Fo(a)22 b Fn(\001)g(r)p Fo(V)566 4059 y Fn(\000)p Fo(ih\026)808 3855 y Fk(8)808 3944 y(<)808 4124 y(:)898 4059 y Fs(~)-51 b Fo(a)23 b Fn(\001)1020 3859 y Fk(0)1020 4038 y(@)1148 3938 y Fs(\()p Fo(p)1248 3951 y Fj(~)1235 3968 y(A)1292 3938 y Fs(;)17 b Fl(B)e Fo(e)1442 3953 y Fm(1)1487 3938 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)1789 3953 y Fm(1)1834 3938 y Fo(;)i(p)1940 3951 y Fj(~)1927 3968 y(A)1984 3938 y Fs(\))1148 4058 y(\()p Fo(p)1248 4071 y Fj(~)1235 4088 y(A)1292 4058 y Fs(;)g Fl(B)e Fo(e)1442 4073 y Fm(2)1487 4058 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)1789 4073 y Fm(2)1834 4058 y Fo(;)i(p)1940 4071 y Fj(~)1927 4088 y(A)1984 4058 y Fs(\))1148 4178 y(\()p Fo(p)1248 4191 y Fj(~)1235 4209 y(A)1292 4178 y Fs(;)g Fl(B)e Fo(e)1442 4193 y Fm(3)1487 4178 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)1789 4193 y Fm(3)1834 4178 y Fo(;)i(p)1940 4191 y Fj(~)1927 4209 y(A)1984 4178 y Fs(\))2063 3859 y Fk(1)2063 4038 y(A)2173 4059 y Fs(+)2271 3859 y Fk(0)2271 4038 y(@)2399 3938 y Fs(\()p Fo(p)2499 3951 y Fj(~)2486 3968 y(A)2543 3938 y Fs(;)g Fl(B)e Fo(e)2692 3953 y Fm(1)2738 3938 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)3040 3953 y Fm(1)3085 3938 y Fo(;)i(p)3191 3951 y Fj(~)3178 3968 y(A)3235 3938 y Fs(\))2399 4058 y(\()p Fo(p)2499 4071 y Fj(~)2486 4088 y(A)2543 4058 y Fs(;)g Fl(B)e Fo(e)2692 4073 y Fm(2)2738 4058 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)3040 4073 y Fm(2)3085 4058 y Fo(;)i(p)3191 4071 y Fj(~)3178 4088 y(A)3235 4058 y Fs(\))2399 4178 y(\()p Fo(p)2499 4191 y Fj(~)2486 4209 y(A)2543 4178 y Fs(;)g Fl(B)e Fo(e)2692 4193 y Fm(3)2738 4178 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)3040 4193 y Fm(3)3085 4178 y Fo(;)i(p)3191 4191 y Fj(~)3178 4209 y(A)3235 4178 y Fs(\))3314 3859 y Fk(1)3314 4038 y(A)3423 4059 y Fn(\001)23 b Fs(~)-50 b Fo(a)3524 3855 y Fk(9)3524 3944 y(=)3524 4124 y(;)566 4453 y Fn(\000)p Fo(ih)749 4249 y Fk(8)749 4339 y(<)749 4518 y(:)838 4253 y(0)838 4432 y(@)966 4332 y Fo(p)1028 4345 y Fj(~)1015 4362 y(A)1095 4332 y Fn(\001)21 b(r)q Fs(~)-50 b Fo(a)1278 4347 y Fm(1)1340 4332 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)1572 4347 y Fm(1)1634 4332 y Fn(\001)22 b Fo(p)1746 4345 y Fj(~)1733 4362 y(A)966 4452 y Fo(p)1028 4465 y Fj(~)1015 4483 y(A)1095 4452 y Fn(\001)f(r)q Fs(~)-50 b Fo(a)1278 4467 y Fm(2)1340 4452 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)1572 4467 y Fm(2)1634 4452 y Fn(\001)22 b Fo(p)1746 4465 y Fj(~)1733 4483 y(A)966 4573 y Fo(p)1028 4586 y Fj(~)1015 4603 y(A)1095 4573 y Fn(\001)f(r)q Fs(~)-50 b Fo(a)1278 4588 y Fm(3)1340 4573 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)1572 4588 y Fm(3)1634 4573 y Fn(\001)22 b Fo(p)1746 4586 y Fj(~)1733 4603 y(A)1831 4253 y Fk(1)1831 4432 y(A)1941 4453 y Fn(\001)f Fo(p)2052 4466 y Fj(~)2039 4483 y(A)2119 4453 y Fs(+)h Fo(p)2279 4466 y Fj(~)2266 4483 y(A)2345 4453 y Fn(\001)2394 4253 y Fk(0)2394 4432 y(@)2523 4332 y Fo(p)2585 4345 y Fj(~)2572 4362 y(A)2651 4332 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)2835 4347 y Fm(1)2897 4332 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)3129 4347 y Fm(1)3191 4332 y Fn(\001)22 b Fo(p)3303 4345 y Fj(~)3290 4362 y(A)2523 4452 y Fo(p)2585 4465 y Fj(~)2572 4483 y(A)2651 4452 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)2835 4467 y Fm(2)2897 4452 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)3129 4467 y Fm(2)3191 4452 y Fn(\001)22 b Fo(p)3303 4465 y Fj(~)3290 4483 y(A)2523 4573 y Fo(p)2585 4586 y Fj(~)2572 4603 y(A)2651 4573 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)2835 4588 y Fm(3)2897 4573 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)3129 4588 y Fm(3)3191 4573 y Fn(\001)22 b Fo(p)3303 4586 y Fj(~)3290 4603 y(A)3388 4253 y Fk(1)3388 4432 y(A)3475 4249 y(9)3475 4339 y(=)3475 4518 y(;)3580 4453 y Fo(:)1918 5251 y Fs(9)p eop %%Page: 10 10 10 9 bop 324 548 a Fs(No)m(w)625 637 y Fk(X)787 731 y Fs(~)-50 b Fo(a)837 746 y Fj(j)874 731 y Fn(f)p Fs(\()p Fo(p)1024 744 y Fj(~)1011 761 y(A)1067 731 y Fs(;)17 b Fl(B)f Fo(e)1217 746 y Fj(j)1259 731 y Fs(\))22 b(+)g(\()p Fl(B)16 b Fo(e)1561 746 y Fj(j)1603 731 y Fs(;)h Fo(p)1709 744 y Fj(~)1696 761 y(A)1753 731 y Fs(\))p Fn(g)22 b Fs(+)1961 637 y Fk(X)2105 731 y Fn(f)p Fs(\()p Fo(p)2255 744 y Fj(~)2242 761 y(A)2299 731 y Fs(;)17 b Fl(B)e Fo(e)2448 746 y Fj(j)2491 731 y Fs(\))22 b(+)g(\()p Fl(B)15 b Fo(e)2792 746 y Fj(j)2835 731 y Fs(;)i Fo(p)2941 744 y Fj(~)2928 761 y(A)2984 731 y Fs(\))p Fn(g)q Fs(~)-50 b Fo(a)3123 746 y Fj(j)467 934 y Fs(=)625 839 y Fk(X)654 1051 y Fj(j;k)786 823 y Fk(n)854 934 y Fs(~)f Fo(a)903 949 y Fj(j)940 934 y Fs(\()p Fo(p)1040 947 y Fj(~)1027 964 y(A;k)1142 934 y Fl(B)1203 949 y Fj(k)r(;j)1325 934 y Fs(+)22 b Fl(B)1484 949 y Fj(k)r(;j)1584 934 y Fo(p)1646 947 y Fj(~)1633 964 y(A;k)1749 934 y Fs(\))g(+)g(\()p Fo(p)2007 947 y Fj(~)1994 964 y(A)o(;k)2109 934 y Fl(B)2169 949 y Fj(k)r(;j)2292 934 y Fs(+)g Fl(B)2450 949 y Fj(k)r(;j)2551 934 y Fo(p)2613 947 y Fj(~)2600 964 y(A;k)2715 934 y Fs(\))q(~)-50 b Fo(a)2804 949 y Fj(j)2841 823 y Fk(o)467 1222 y Fs(=)82 b(2)691 1127 y Fk(X)720 1339 y Fj(j;k)851 1111 y Fk(n)919 1222 y Fs(~)-50 b Fo(a)969 1237 y Fj(j)1006 1222 y Fl(B)1066 1237 y Fj(k)r(;j)1167 1222 y Fo(p)1229 1235 y Fj(~)1216 1252 y(A)o(;k)1353 1222 y Fs(+)22 b Fo(p)1513 1235 y Fj(~)1500 1252 y(A;k)1615 1222 y Fl(B)1675 1237 y Fj(k)r(;j)1777 1222 y Fs(~)-50 b Fo(a)1827 1237 y Fj(j)1864 1111 y Fk(o)1952 1222 y Fs(+)2050 1127 y Fk(X)2079 1339 y Fj(j;k)2211 1111 y Fk(n)2278 1222 y Fs(~)g Fo(a)2328 1237 y Fj(j)2365 1222 y Fs([)p Fo(p)2454 1235 y Fj(~)2441 1252 y(A;k)2556 1222 y Fo(;)17 b Fl(B)2660 1237 y Fj(k)r(;j)2761 1222 y Fs(])22 b(+)g([)p Fl(B)2996 1237 y Fj(k)r(;j)3096 1222 y Fo(;)17 b(p)3202 1235 y Fj(~)3189 1252 y(A;k)3304 1222 y Fs(])q(~)-50 b Fo(a)3382 1237 y Fj(j)3419 1111 y Fk(o)467 1470 y Fs(=)82 b(2\()p Fl(B)17 b Fs(~)-50 b Fo(a)6 b(;)17 b(p)936 1483 y Fj(~)923 1500 y(A)979 1470 y Fs(\))23 b(+)f(\()p Fo(p)1238 1483 y Fj(~)1225 1500 y(A)1281 1470 y Fo(;)17 b Fl(B)g Fs(~)-50 b Fo(a)6 b Fs(\))467 1616 y(=)82 b(2)p Fo(J)728 1631 y Fj(p)768 1616 y Fs(\()-6 b Fo(~)-43 b(a)p Fs(\))p Fo(:)324 1799 y Fs(Finally)-8 b(,)651 1895 y Fk(0)651 2074 y(@)780 1974 y Fo(p)842 1987 y Fj(~)829 2004 y(A)908 1974 y Fn(\001)22 b(r)q Fs(~)-50 b Fo(a)1092 1989 y Fm(1)1153 1974 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)1385 1989 y Fm(1)1447 1974 y Fn(\001)22 b Fo(p)1559 1987 y Fj(~)1546 2004 y(A)780 2094 y Fo(p)842 2107 y Fj(~)829 2124 y(A)908 2094 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)1092 2109 y Fm(2)1153 2094 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)1385 2109 y Fm(2)1447 2094 y Fn(\001)22 b Fo(p)1559 2107 y Fj(~)1546 2124 y(A)780 2214 y Fo(p)842 2227 y Fj(~)829 2245 y(A)908 2214 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)1092 2229 y Fm(3)1153 2214 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)1385 2229 y Fm(3)1447 2214 y Fn(\001)22 b Fo(p)1559 2227 y Fj(~)1546 2245 y(A)1644 1895 y Fk(1)1644 2074 y(A)1754 2095 y Fn(\001)g Fo(p)1866 2108 y Fj(~)1853 2125 y(A)1932 2095 y Fs(+)g Fo(p)2092 2108 y Fj(~)2079 2125 y(A)2158 2095 y Fn(\001)2208 1895 y Fk(0)2208 2074 y(@)2336 1974 y Fo(p)2398 1987 y Fj(~)2385 2004 y(A)2464 1974 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)2648 1989 y Fm(1)2710 1974 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)2942 1989 y Fm(1)3004 1974 y Fn(\001)22 b Fo(p)3116 1987 y Fj(~)3103 2004 y(A)2336 2094 y Fo(p)2398 2107 y Fj(~)2385 2124 y(A)2464 2094 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)2648 2109 y Fm(2)2710 2094 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)2942 2109 y Fm(2)3004 2094 y Fn(\001)22 b Fo(p)3116 2107 y Fj(~)3103 2124 y(A)2336 2214 y Fo(p)2398 2227 y Fj(~)2385 2245 y(A)2464 2214 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)2648 2229 y Fm(3)2710 2214 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)2942 2229 y Fm(3)3004 2214 y Fn(\001)22 b Fo(p)3116 2227 y Fj(~)3103 2245 y(A)3201 1895 y Fk(1)3201 2074 y(A)492 2398 y Fs(=)651 2304 y Fk(X)707 2514 y Fj(j)811 2288 y Fk(n)878 2398 y Fs(\()p Fo(p)978 2411 y Fj(~)965 2428 y(A)1044 2398 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)1228 2413 y Fj(j)1287 2398 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)1519 2413 y Fj(j)1578 2398 y Fn(\001)21 b Fo(p)1689 2411 y Fj(~)1676 2428 y(A)1733 2398 y Fs(\))p Fo(p)1833 2411 y Fj(~)1820 2428 y(A;j)1951 2398 y Fs(+)h Fo(p)2111 2411 y Fj(~)2098 2428 y(A;j)2207 2398 y Fs(\()p Fo(p)2307 2411 y Fj(~)2294 2428 y(A)2373 2398 y Fn(\001)g(r)q Fs(~)-50 b Fo(a)2557 2413 y Fj(j)2616 2398 y Fs(+)22 b Fn(r)q Fs(~)-50 b Fo(a)2848 2413 y Fj(j)2907 2398 y Fn(\001)22 b Fo(p)3019 2411 y Fj(~)3006 2428 y(A)3063 2398 y Fs(\))3101 2288 y Fk(o)492 2684 y Fs(=)651 2590 y Fk(X)680 2802 y Fj(j;k)811 2574 y Fk(n)878 2684 y Fs(\()p Fo(p)978 2697 y Fj(~)965 2714 y(A;k)1080 2684 y Fs(\()p Fo(@)1169 2699 y Fj(k)1213 2684 y Fs(~)-50 b Fo(a)1263 2699 y Fj(j)1300 2684 y Fs(\))22 b(+)g(\()p Fo(@)1547 2699 y Fj(k)1591 2684 y Fs(~)-50 b Fo(a)1641 2699 y Fj(j)1678 2684 y Fs(\))p Fo(p)1778 2697 y Fj(~)1765 2714 y(A)o(;k)1880 2684 y Fs(\))p Fo(p)1980 2697 y Fj(~)1967 2714 y(A)o(;j)2098 2684 y Fs(+)22 b Fo(p)2258 2697 y Fj(~)2245 2714 y(A;j)2354 2684 y Fs(\()p Fo(p)2454 2697 y Fj(~)2441 2714 y(A)o(;k)2556 2684 y Fs(\()p Fo(@)2645 2699 y Fj(k)2689 2684 y Fs(~)-50 b Fo(a)2739 2699 y Fj(j)2775 2684 y Fs(\))23 b(+)f(\()p Fo(@)3023 2699 y Fj(k)3067 2684 y Fs(~)-50 b Fo(a)3117 2699 y Fj(j)3153 2684 y Fs(\))p Fo(p)3253 2697 y Fj(~)3240 2714 y(A;k)3355 2684 y Fs(\))3393 2574 y Fk(o)492 2972 y Fs(=)83 b(2)717 2878 y Fk(X)745 3090 y Fj(j;k)877 2862 y Fk(n)943 2972 y Fo(p)1005 2985 y Fj(~)992 3003 y(A;k)1107 2972 y Fs(\()p Fo(@)1196 2987 y Fj(k)1241 2972 y Fs(~)-51 b Fo(a)1290 2987 y Fj(j)1327 2972 y Fs(\))p Fo(p)1427 2985 y Fj(~)1414 3003 y(A;j)1545 2972 y Fs(+)22 b Fo(p)1705 2985 y Fj(~)1692 3003 y(A;j)1801 2972 y Fs(\()p Fo(@)1890 2987 y Fj(k)1934 2972 y Fs(~)-50 b Fo(a)1984 2987 y Fj(j)2021 2972 y Fs(\))p Fo(p)2121 2985 y Fj(~)2108 3003 y(A;k)2223 2862 y Fk(o)651 3261 y Fs(+)744 3166 y Fk(X)772 3378 y Fj(j;k)904 3150 y Fk(n)970 3261 y Fs([)p Fo(@)1048 3276 y Fj(k)1093 3261 y Fs(~)g Fo(a)1143 3276 y Fj(j)1179 3261 y Fo(;)17 b(p)1285 3274 y Fj(~)1272 3291 y(A;k)1387 3261 y Fs(])p Fo(p)1476 3274 y Fj(~)1463 3291 y(A;j)1594 3261 y Fn(\000)23 b Fo(p)1756 3274 y Fj(~)1743 3291 y(A;j)1852 3261 y Fs([)p Fo(@)1930 3276 y Fj(k)1974 3261 y Fs(~)-50 b Fo(a)2024 3276 y Fj(j)2061 3261 y Fo(;)17 b(p)2167 3274 y Fj(~)2154 3291 y(A)o(;k)2269 3261 y Fs(])2296 3150 y Fk(o)492 3539 y Fs(=)83 b(2\()p Fo(p)800 3552 y Fj(~)787 3569 y(A)844 3539 y Fs(;)17 b(\()p Fo(D)t Fs(~)-50 b Fo(a)21 b Fs(+)h(\()p Fo(D)t Fs(~)-50 b Fo(a)p Fs(\))1391 3498 y Fj(t)1421 3539 y Fs(\))p Fo(p)1521 3552 y Fj(~)1508 3569 y(A)1565 3539 y Fs(\))22 b Fn(\000)g Fo(ih)1830 3444 y Fk(X)1859 3656 y Fj(j;k)1974 3539 y Fs([)p Fo(p)2063 3552 y Fj(~)2050 3569 y(A;j)2159 3539 y Fo(;)17 b(@)2259 3498 y Fm(2)2254 3564 y Fj(k)2301 3539 y Fs(~)-51 b Fo(a)2350 3554 y Fj(j)2387 3539 y Fs(])492 3817 y(=)83 b(2\()p Fo(p)800 3830 y Fj(~)787 3847 y(A)844 3817 y Fs(;)17 b(\()p Fo(D)t Fs(~)-50 b Fo(a)21 b Fs(+)h(\()p Fo(D)t Fs(~)-50 b Fo(a)p Fs(\))1391 3776 y Fj(t)1421 3817 y Fs(\))p Fo(p)1521 3830 y Fj(~)1508 3847 y(A)1565 3817 y Fs(\))22 b Fn(\000)g Fo(h)1780 3776 y Fm(2)1837 3722 y Fk(X)1865 3935 y Fj(j;k)1997 3817 y Fo(@)2048 3832 y Fj(j)2085 3817 y Fo(@)2141 3776 y Fm(2)2136 3842 y Fj(k)2183 3817 y Fs(~)-51 b Fo(a)2232 3832 y Fj(j)2269 3817 y Fo(:)p 3491 4095 4 66 v 3495 4032 59 4 v 3495 4095 V 3553 4095 4 66 v 324 4422 a Fu(3)161 b(Kno)l(wn)52 b(results)324 4641 y Fs(In)36 b(this)g(section)g(w)m(e)i(will)33 b(recall)i(some)h(results)h(on)f(semiclassics)f(of)h(the)g(energy)h (and)324 4761 y(densit)m(y)c(in)f(a)g(constan)m(t)i(magnetic)d (\014eld.)43 b(These)35 b(are)d(all)f(tak)m(en)i(from)64 b([LSY94].)470 4882 y(First)32 b(w)m(e)h(ha)m(v)m(e)h(a)e(magnetic)f (Lieb-Thirring)f(inequalit)m(y)i(for)g(constan)m(t)h(magnetic)324 5002 y(\014eld:)1894 5251 y(10)p eop %%Page: 11 11 11 10 bop 324 548 a Ft(Theorem)42 b(3.1.)i Fp(L)-5 b(et)39 b Fn(j)p Fo(V)22 b Fn(j)1329 563 y Fi(\000)1423 548 y Fn(2)35 b Fo(L)1590 512 y Fm(3)p Fj(=)p Fm(2)1701 548 y Fs(\()p Fl(R)1805 512 y Fm(3)1850 548 y Fs(\))25 b Fn(\\)h Fo(L)2071 512 y Fm(5)p Fj(=)p Fm(2)2181 548 y Fs(\()p Fl(R)2285 512 y Fm(3)2330 548 y Fs(\))39 b Fp(and)g(let)g Fo(e)2787 563 y Fj(j)2823 548 y Fs(\()p Fo(\026;)17 b(V)k Fs(\))39 b Fp(denote)g(the)324 668 y(ne)-5 b(gative)34 b(eigenvalues)f(of)i(the)g(op)-5 b(er)g(ator)35 b Fo(p)1920 632 y Fm(2)1933 696 y Fj(~)1920 713 y(A)1999 668 y Fn(\000)22 b Fo(\026h)g Fs(+)g Fo(V)g Fs(\()p Fo(x)p Fs(\))p Fp(.)45 b(Then)554 819 y Fk(X)610 1029 y Fj(j)715 914 y Fn(j)p Fo(e)788 929 y Fj(j)824 914 y Fs(\()p Fo(\026;)17 b(V)k Fs(\))p Fn(j)27 b(\024)h Fo(L)1307 929 y Fm(1)1347 914 y Fo(\026h)1462 872 y Fi(\000)p Fm(2)1573 778 y Fk(Z)1689 914 y Fn(j)p Fo(V)21 b Fs(\()p Fo(x)p Fs(\))p Fn(j)1954 863 y Fi(\000)p Fm(3)p Fj(=)p Fm(2)1954 935 y Fi(\000)2136 914 y Fo(dx)h Fs(+)g Fo(L)2428 929 y Fm(2)2468 914 y Fo(h)2524 872 y Fi(\000)p Fm(3)2635 778 y Fk(Z)2751 914 y Fn(j)p Fo(V)g Fs(\()p Fo(x)p Fs(\))p Fn(j)3017 863 y Fi(\000)p Fm(5)p Fj(=)p Fm(2)3017 935 y Fi(\000)3198 914 y Fo(dx;)324 1186 y Fp(wher)-5 b(e)34 b(the)h(c)-5 b(onstants)34 b Fo(L)1255 1201 y Fm(1)1295 1186 y Fo(;)17 b(L)1405 1201 y Fm(2)1479 1186 y Fp(ar)-5 b(e)35 b(indep)-5 b(endent)34 b(of)g Fo(h;)17 b(\026)35 b Fp(and)f Fo(V)21 b Fp(.)470 1353 y Fs(The)35 b(result)e(on)h(the)g(semiclassics)f(of)g (the)h(energy)h(in)e(a)g(constan)m(t)h(magnetic)f(\014eld)324 1474 y(is:)324 1641 y Ft(Theorem)50 b(3.2.)d Fp(Supp)-5 b(ose)45 b Fn(j)p Fo(V)21 b Fn(j)1549 1656 y Fi(\000)1654 1641 y Fn(2)48 b Fo(L)1834 1605 y Fm(3)p Fj(=)p Fm(2)1944 1641 y Fs(\()p Fl(R)2048 1605 y Fm(3)2093 1641 y Fs(\))30 b Fn(\\)g Fo(L)2323 1605 y Fm(5)p Fj(=)p Fm(2)2433 1641 y Fs(\()p Fl(R)2537 1605 y Fm(3)2583 1641 y Fs(\))45 b Fp(and)f(let)i Fo(E)6 b Fs(\()3147 1616 y Fo(~)3129 1641 y(A;)17 b(V)k Fs(\))45 b Fp(and)324 1770 y Fo(E)396 1785 y Fj(scl)485 1770 y Fs(\()541 1745 y Fo(~)523 1770 y(A;)17 b(V)22 b Fs(\))34 b Fp(b)-5 b(e)35 b(as)f(given)g(in)h(Se)-5 b(ction)69 b(1.)45 b(Then)1410 2035 y Fs(lim)1405 2098 y Fj(h)p Fi(!)p Fm(0)1568 1865 y Fk( )1746 1967 y Fo(E)6 b Fs(\()1880 1942 y Fo(~)1862 1967 y(A;)17 b(V)k Fs(\))p 1657 2012 528 4 v 1683 2076 a Fj(\026)p 1667 2097 76 4 v 1667 2154 a(h)1708 2136 y Fg(2)1752 2120 y Fo(E)1824 2135 y Fj(scl)1913 2120 y Fs(\()1969 2095 y Fo(~)1951 2120 y(A;)c(V)22 b Fs(\))2195 1865 y Fk(!)2301 2035 y Fs(=)28 b(1)p Fo(;)324 2294 y Fp(uniformly)34 b(in)h(the)g(magnetic)f (\014eld)g(str)-5 b(ength)35 b Fo(\026)p Fp(.)470 2461 y Fs(By)e(the)g(v)-5 b(ariational)29 b(principle,)i(w)m(e)j(get)f(as)f (in)g(Section)65 b(1.2:)324 2628 y Ft(Corollary)46 b(3.3.)f Fp(L)-5 b(et)43 b(us)e(ke)-5 b(ep)41 b(the)h(assumptions)f(fr)-5 b(om)41 b(The)-5 b(or)g(em)83 b(3.2.)64 b(Supp)-5 b(ose)324 2748 y Fo(\036)27 b Fn(2)h Fo(L)569 2712 y Fm(5)p Fj(=)p Fm(2)680 2748 y Fs(\()p Fl(R)784 2712 y Fm(3)829 2748 y Fs(\))22 b Fn(\\)h Fo(L)1044 2712 y Fm(3)p Fj(=)p Fm(2)1154 2748 y Fs(\()p Fl(R)1258 2712 y Fm(3)1303 2748 y Fs(\))p Fp(,)35 b(then)446 2934 y Fo(h)502 2898 y Fm(2)p 446 2979 96 4 v 464 3070 a Fo(\026)551 3002 y Fs(tr[)p Fo(\036)p Fs(1)761 3017 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])992 3002 y Fs(\()p Ft(P)p Fs(\()p Fo(\026)1222 2977 y(~)1204 3002 y(A;)17 b(V)k Fs(\)\)])28 b(=)1648 2934 y(1)p 1643 2979 59 4 v 1643 3070 a Fo(\031)1729 2866 y Fk(Z)1881 2877 y Fi(1)1845 2907 y Fk(X)1850 3116 y Fj(n)p Fm(=0)2005 3002 y Fo(d)2056 3017 y Fj(n)2103 3002 y Fs([2)p Fo(n\026h)22 b Fs(+)g Fo(V)g Fs(\()p Fo(x)p Fs(\)])2709 2951 y Fm(1)p Fj(=)p Fm(2)2709 3024 y Fi(\000)2819 3002 y Fo(\036)p Fs(\()p Fo(x)p Fs(\))17 b Fo(dx)22 b Fs(+)g Fo(o)p Fs(\(1\))p Fo(;)324 3261 y Fp(as)34 b Fo(h)28 b Fn(!)f Fs(0)p Fp(.)324 3588 y Fu(4)161 b(Gauge)53 b(in)l(v)-9 b(ariance)55 b(of)f(the)e (curren)l(t)324 3816 y Fs(In)46 b(this)f(section)h(w)m(e)h(will)d(pro)m (v)m(e)j(that)e(the)i(curren)m(t)2390 3736 y Fk(R)2464 3793 y Fo(~)2473 3816 y(j)37 b Fn(\001)24 b Fo(~)-42 b(a)p Fs(,)49 b(as)d(a)g(function)f(of)39 b Fo(~)-43 b(a)q Fs(,)324 3952 y(only)38 b(dep)s(ends)j(on)e(the)h(magnetic)e (\014eld)1876 3926 y Fo(~)1887 3952 y(b)h Fs(=)g Fn(r)27 b(\002)21 b Fo(~)-43 b(a)40 b Fs(generated)g(b)m(y)33 b Fo(~)-42 b(a)p Fs(,)41 b(i.e.)63 b(that)39 b(if)317 4072 y Fo(~)-42 b(a)28 b Fs(=)g(~)-50 b Fo(a)23 b Fs(+)f Fn(r)p Fo(\036)32 b Fs(then)1073 3992 y Fk(R)1147 4050 y Fo(~)1156 4072 y(j)c Fn(\001)16 b Fo(~)-43 b(a)28 b Fs(=)1456 3992 y Fk(R)1530 4050 y Fo(~)1539 4072 y(j)h Fn(\001)23 b Fs(~)-51 b Fo(a)q Fs(:)324 4239 y Ft(Lemma)55 b(4.1.)50 b Fp(Supp)-5 b(ose)49 b Fo(V)72 b Fp(is)49 b(r)-5 b(elatively)50 b(b)-5 b(ounde)g(d)48 b(with)i(r)-5 b(esp)g(e)g(ct)49 b(to)h Fn(\000)p Fo(h)3236 4203 y Fm(2)3276 4239 y Fs(\001)g Fp(and)324 4360 y(that)43 b Fs(Sp)s(ec)q(\()p Ft(P)p Fs(\()p Fo(h;)17 b(\026;)g(V)j Fs(\)\))43 b Fp(b)-5 b(elow)43 b(zer)-5 b(o)43 b(is)g(discr)-5 b(ete.)69 b(Then)42 b Fn(8)p Fo(\036)h Fn(2)h Fo(C)2903 4323 y Fi(1)2896 4384 y Fm(0)2977 4360 y Fs(\()p Ft(R)3099 4323 y Fm(3)3138 4360 y Fs(\))g Fp(we)e(have)324 4400 y Fk(R)398 4458 y Fo(~)407 4480 y(j)28 b Fn(\001)22 b(r)p Fo(\036)27 b Fs(=)h(0)p Fp(.)324 4647 y(Pr)-5 b(o)g(of.)48 b Fs(The)34 b(pro)s(of)d(follo)m(ws)h(from)f(the)i(comm)m(utator)e(form)m(ula)g(b)s (elo)m(w:)1565 4825 y([)p Ft(P)p Fo(;)17 b(\036)p Fs(])27 b(=)h Fo(ih)p Ft(J)p Fs(\()p Fn(r)p Fo(\036)p Fs(\))p Fo(:)p 3491 5002 4 66 v 3495 4940 59 4 v 3495 5002 V 3553 5002 4 66 v 1894 5251 a Fs(11)p eop %%Page: 12 12 12 11 bop 324 548 a Fu(5)161 b(Lo)l(w)l(er)51 b(b)t(ound)324 860 y Fs(Let)45 b Fo(M)59 b Fs(=)788 660 y Fk(0)788 839 y(@)917 739 y Fo(M)1011 754 y Fm(11)1169 739 y Fo(M)1263 754 y Fm(12)1421 739 y Fo(M)1515 754 y Fm(13)917 860 y Fo(M)1011 875 y Fm(21)1169 860 y Fo(M)1263 875 y Fm(22)1421 860 y Fo(M)1515 875 y Fm(23)917 980 y Fo(M)1011 995 y Fm(31)1169 980 y Fo(M)1263 995 y Fm(32)1481 980 y Fs(0)1631 660 y Fk(1)1631 839 y(A)1767 860 y Fn(2)49 b Fo(C)1959 824 y Fi(1)1952 885 y Fm(0)2033 860 y Fs(\()p Fl(R)2137 824 y Fm(3)2183 860 y Fs(\))44 b(and)h(let)f Fo(b)2661 875 y Fm(3)2701 860 y Fs(\()p Fo(x)p Fs(\))49 b(=)3015 813 y Fj(M)3083 822 y Fg(11)3148 813 y Fm(\()p Fj(x)p Fm(\)+)p Fj(M)3365 822 y Fg(22)3430 813 y Fm(\()p Fj(x)p Fm(\))p 3015 837 510 4 v 3252 895 a(2)3535 860 y Fs(.)324 1094 y(In)39 b(this)g(section)g(w)m(e)h(pro)m(v)m(e)g(a)f 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y(then)500 2067 y Fs(2)p Fn(h)p Fo(f)11 b(g)t Fn(j)p Fo(p)788 2080 y Fj(~)775 2097 y(A)853 2067 y Fn(\001)22 b Fo(S)6 b Fs(\()p Fo(x)p Fs(\))p Fo(p)1162 2080 y Fj(~)1149 2097 y(A)1206 2067 y Fn(j)p Fo(g)t(f)11 b Fn(i)81 b Fs(=)i Fn(h)p Fo(f)11 b Fn(j)p Fo(g)1800 2026 y Fm(2)1838 2067 y Fo(p)1900 2080 y Fj(~)1887 2097 y(A)1966 2067 y Fn(\001)22 b Fo(S)6 b Fs(\()p Fo(x)p Fs(\))p Fo(p)2275 2080 y Fj(~)2262 2097 y(A)2319 2067 y Fn(j)p Fo(f)11 b Fn(i)21 b Fs(+)h Fn(h)p Fo(f)11 b Fn(j)p Fo(p)2752 2080 y Fj(~)2739 2097 y(A)2817 2067 y Fn(\001)22 b Fo(S)6 b Fs(\()p Fo(x)p Fs(\))p Fo(p)3126 2080 y Fj(~)3113 2097 y(A)3170 2067 y Fo(g)3221 2026 y Fm(2)3260 2067 y Fn(j)p Fo(f)11 b Fn(i)1623 2213 y(\000)p Fs(2)p Fo(h)1805 2172 y Fm(2)1845 2213 y Fn(h)p Fo(f)g Fn(jr)p Fo(g)24 b Fn(\001)e Fo(S)6 b Fs(\()p Fo(x)p Fs(\))p Fn(r)p Fo(g)t Fn(j)p Fo(f)11 b Fn(i)p Fo(;)324 2386 y Fp(for)34 b(al)5 b(l)35 b Fo(f)j Fn(2)28 b Fo(L)865 2350 y Fm(2)905 2386 y Fp(.)470 2550 y Fs(W)-8 b(e)42 b(will)c(also)j(need)h(to)e(diagonalise)f(the)j ('kinetic)e(energy)i(part')f(of)g Fo(H)8 b Fs(\()p Fo(t)p Fs(\))41 b(-)f(for)324 2671 y(constan)m(t)33 b(matrices)f Fo(S)6 b Fs(\()p Fo(t)p Fs(\),)32 b(this)h(is)f(the)h(con)m(ten)m(t)h (of)e(the)h(next)g(lemma:)324 2835 y Ft(Lemma)d(5.2.)37 b Fp(L)-5 b(et)30 b Fo(S)1148 2850 y Fj(t)1206 2835 y Fs(=)d Fo(I)18 b Fs(+)10 b Fo(tM)g Fp(,)31 b(wher)-5 b(e)29 b Fo(M)40 b Fp(is)29 b(a)h(c)-5 b(onstant)29 b(matrix,)h(and)f Fo(t)g Fp(is)h(smal)5 b(l.)324 2955 y(L)-5 b(et)41 b(the)g(matrix)f Fo(N)1062 2970 y Fj(t)1133 2955 y Fp(solve)g(the)g(e)-5 b(quation)41 b Fo(e)1985 2919 y Fm(2)p Fj(tN)2101 2927 y Fb(t)2172 2955 y Fs(=)e Fo(I)34 b Fs(+)26 b Fo(tM)10 b Fp(,)44 b(and)c(de\014ne)f(a)i(unitary)324 3075 y(op)-5 b(er)g(ator)34 b Fo(U)773 3090 y Fj(t)838 3075 y Fp(on)h Fo(L)1044 3039 y Fm(2)1118 3075 y Fp(by:)1419 3249 y Fs(\()p Fo(U)1523 3264 y Fj(t)1553 3249 y Fo(f)11 b Fs(\)\()p Fo(x)p Fs(\))28 b(=)f(\003)1980 3198 y Fm(1)p Fj(=)p Fm(2)1980 3272 y Fj(t)2090 3249 y Fo(f)11 b Fs(\()p Fo(e)2232 3208 y Fj(tN)2313 3216 y Fb(t)2346 3249 y Fo(x)p Fs(\))p Fo(;)324 3423 y Fs(\003)392 3438 y Fj(t)449 3423 y Fs(=)27 b Fn(j)17 b Fs(det)g Fo(e)794 3387 y Fj(tN)875 3395 y Fb(t)907 3423 y Fn(j)p Fp(.Then)1220 3597 y Fo(U)1286 3612 y Fj(t)1316 3597 y Fo(p)1378 3610 y Fj(~)1365 3627 y(A)1444 3597 y Fn(\001)22 b Fo(S)1554 3612 y Fj(t)1583 3597 y Fo(p)1645 3610 y Fj(~)1632 3627 y(A)1689 3597 y Fo(U)1765 3556 y Fi(\000)p Fm(1)1755 3622 y Fj(t)1888 3597 y Fs(=)28 b(\()p Fn(\000)p Fo(ih)p Fn(r)23 b Fs(+)f Fo(\026)2484 3572 y Fs(~)2459 3597 y Fo(A)2532 3612 y Fj(t)2561 3597 y Fs(\))2599 3556 y Fm(2)2638 3597 y Fo(;)324 3785 y Fp(wher)-5 b(e)625 3760 y Fs(~)599 3785 y Fo(A)672 3800 y Fj(t)702 3785 y Fs(\()p Fo(x)p Fs(\))28 b(=)f Fo(e)1009 3749 y Fj(tN)1090 3757 y Fb(t)1141 3760 y Fo(~)1123 3785 y(A)p Fs(\()p Fo(e)1279 3749 y Fj(tN)1360 3757 y Fb(t)1393 3785 y Fo(x)p Fs(\))p Fp(.)324 4067 y Ft(Remark)43 b(5.3.)h Fs(If)38 b Fo(M)48 b Fs(=)1330 3867 y Fk(0)1330 4046 y(@)1459 3946 y Fo(M)1553 3961 y Fm(11)1711 3946 y Fo(M)1805 3961 y Fm(12)1963 3946 y Fo(M)2057 3961 y Fm(13)1459 4066 y Fo(M)1553 4081 y Fm(21)1711 4066 y Fo(M)1805 4081 y Fm(22)1963 4066 y Fo(M)2057 4081 y Fm(23)1459 4187 y Fo(M)1553 4202 y Fm(31)1711 4187 y Fo(M)1805 4202 y Fm(32)2023 4187 y Fs(0)2174 3867 y Fk(1)2174 4046 y(A)2261 4067 y Fs(,)39 b(then)f Fn(j)p Fs(rot)2734 4042 y(~)2707 4067 y Fo(A)2780 4082 y Fj(t)2810 4067 y Fn(j)e Fs(=)g(1)26 b(+)3173 4028 y Fm(1)p 3173 4044 36 4 v 3173 4102 a(2)3218 4067 y Fo(t)p Fs(\()p Fo(M)3385 4082 y Fm(11)3486 4067 y Fs(+)324 4310 y Fo(M)418 4325 y Fm(22)493 4310 y Fs(\))19 b(+)g Fo(t)680 4274 y Fm(2)720 4310 y Fo(c)g Fs(+)g Fo(O)s Fs(\()p Fo(t)1027 4274 y Fm(3)1066 4310 y Fs(\),)31 b(where)i Fo(c)e Fs(is)g(a)f(negativ)m(e)i(constan)m(t)g(\(dep)s(ending)f(on)g Fo(M)10 b Fs(\).)44 b(Th)m(us,)324 4430 y(w)m(e)33 b(see)h(from)d(the)i (ab)s(o)m(v)m(e,)h(that)e(for)g Fo(t)c Fn(6)p Fs(=)g(0:)769 4656 y(inf)22 b(Sp)s(ec)1118 4516 y Fk(\022)1191 4656 y Fo(p)1253 4669 y Fj(~)1240 4686 y(A)1319 4656 y Fn(\001)g Fo(S)1429 4671 y Fj(t)1459 4656 y Fo(p)1521 4669 y Fj(~)1508 4686 y(A)1587 4656 y Fn(\000)g Fo(\026h)p Fs(\(1)g(+)2018 4589 y(1)p 2018 4633 49 4 v 2018 4724 a(2)2077 4656 y Fo(t)p Fs(\()p Fo(M)2244 4671 y Fm(11)2341 4656 y Fs(+)g Fo(M)2533 4671 y Fm(22)2608 4656 y Fs(\)\))2684 4516 y Fk(\023)2785 4656 y Fn(!)27 b(\0001)p Fo(;)324 4882 y Fs(as)j Fo(\026h)e Fn(!)f(1)p Fs(.)42 b(This)30 b(is)g(the)g(reason)h (wh)m(y)g(the)g(lo)m(w)m(er)f(b)s(ound)g(b)s(elo)m(w)g(do)s(es)h(not)e (w)m(ork)i(in)324 5002 y(that)h(case.)1894 5251 y(12)p eop %%Page: 13 13 13 12 bop 324 548 a Ft(Theorem)37 b(5.4.)42 b Fp(Supp)-5 b(ose)34 b(that)h Fs([)p Fo(V)22 b Fs(])1718 563 y Fi(\000)1805 548 y Fn(2)28 b Fo(L)1965 512 y Fm(3)p Fj(=)p Fm(2)2097 548 y Fn(\\)23 b Fo(L)2252 512 y Fm(5)p Fj(=)p Fm(2)2397 548 y Fp(and)34 b(that)1143 884 y Fo(M)10 b Fs(\()p Fo(x)p Fs(\))29 b(=)1511 684 y Fk(0)1511 864 y(@)1639 763 y Fo(M)1733 778 y Fm(11)1892 763 y Fo(M)1986 778 y Fm(12)2144 763 y Fo(M)2238 778 y Fm(13)1639 884 y Fo(M)1733 899 y Fm(21)1892 884 y Fo(M)1986 899 y Fm(22)2144 884 y Fo(M)2238 899 y Fm(23)1639 1004 y Fo(M)1733 1019 y Fm(31)1892 1004 y Fo(M)1986 1019 y Fm(32)2204 1004 y Fs(0)2354 684 y Fk(1)2354 864 y(A)2469 884 y Fn(2)f Fo(C)2640 843 y Fi(1)2633 909 y Fm(0)2715 884 y Fo(:)324 1221 y Fp(L)-5 b(et)32 b Fo(E)6 b Fs(\()p Fo(t)p Fs(\))28 b(=)g(tr)o([)p Fo(H)8 b Fs(\()p Fo(t)p Fs(\)1)1161 1236 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])1393 1221 y Fs(\()p Fo(H)g Fs(\()p Fo(t)p Fs(\)\)])p Fp(,)32 b(and)f(supp)-5 b(ose,)32 b(that)g Fo(\026h)c Fn(!)f Fo(\014)38 b Fp(as)32 b Fo(h)c Fn(!)f Fs(0)p Fp(.)43 b(Then)324 1341 y(we)34 b(have)g(the)h(fol)5 b(lowing)34 b(lower)g(b)-5 b(ound)34 b(on)h Fo(E)6 b Fs(\()p Fo(t)p Fs(\))p Fp(:)761 1619 y Fs(lim)17 b(inf)822 1681 y Fj(h)p Fi(!)p Fm(0)1057 1551 y Fo(h)1113 1515 y Fm(2)p 1057 1596 96 4 v 1076 1687 a Fo(\026)1163 1619 y(E)6 b Fs(\()p Fo(t)p Fs(\))28 b Fn(\025)781 1809 y(\000)p Fs(1)p 770 1853 148 4 v 770 1945 a(3)p Fo(\031)878 1916 y Fm(2)944 1782 y Fk(X)997 1991 y Fj(\027)1105 1741 y Fk(Z)1244 1809 y Fo(b)1285 1824 y Fj(u;t)p 1231 1853 158 4 v 1231 1945 a Fs(\003)1299 1960 y Fj(u;t)1415 1876 y Fs([\(2)p Fo(\027)h Fs(+)22 b(1\))p Fo(\014)6 b(b)1893 1891 y Fj(u;t)2005 1876 y Fn(\000)22 b Fo(\014)6 b Fs(\(1)22 b(+)g Fo(tb)2448 1891 y Fm(3)2488 1876 y Fs(\()p Fo(u)p Fs(\)\))f(+)h Fo(V)g Fs(\()p Fo(u)p Fs(\)])3014 1828 y Fm(3)p Fj(=)p Fm(2)3014 1908 y Fi(\000)3158 1876 y Fo(du;)324 2203 y Fp(wher)-5 b(e)24 b Fo(b)630 2218 y Fj(u;t)748 2203 y Fs(=)k Fn(j)p Fs(rot)1004 2218 y Fj(x)1048 2118 y Fk(p)p 1148 2118 441 4 v 85 x Fs(1)22 b(+)g Fo(tM)10 b Fs(\()p Fo(u)p Fs(\))1606 2178 y Fo(~)1588 2203 y(A)p Fs(\()1699 2118 y Fk(p)p 1799 2118 V 85 x Fs(1)22 b(+)g Fo(tM)10 b Fs(\()p Fo(u)p Fs(\))p Fo(x)p Fs(\))p Fn(j)25 b Fp(and)f Fs(\003)2632 2218 y Fj(u;t)2750 2203 y Fs(=)j Fn(j)17 b Fs(det)3050 2118 y Fk(p)p 3149 2118 V 3149 2203 a Fs(1)22 b(+)g Fo(tM)10 b Fs(\()p Fo(u)p Fs(\))q Fn(j)p Fo(:)324 2405 y Ft(Remark)47 b(5.5.)g Fs(Notice,)c(that)f(due)g (to)g(the)g(2nd)f(order)h(discrepancy)h(b)s(et)m(w)m(een)h Fo(b)3471 2420 y Fj(u;t)324 2525 y Fs(and)29 b Fo(\026)p Fs(\(1)15 b(+)g Fo(tb)838 2540 y Fm(3)878 2525 y Fs(\()p Fo(u)p Fs(\)\))28 b(\(see)i(Remark)58 b(5.3\),)30 b(w)m(e)g(really)e (need)i(the)g(matrix)d Fo(M)10 b Fs(\()p Fo(x)p Fs(\))30 b(to)f(ha)m(v)m(e)324 2646 y(compact)42 b(supp)s(ort,)j(since)d(this)g (assures)i(the)e(con)m(v)m(ergence)j(of)d(the)g(in)m(tegral)f(in)g(the) 324 2766 y(lo)m(w)m(er)32 b(b)s(ound)h(for)f Fo(\027)i Fs(=)28 b(0.)324 2968 y Fp(Pr)-5 b(o)g(of.)48 b Fs(It)29 b(is)e(clear,)i(that)f(w)m(e)h(get)g(a)f(lo)m(w)m(er)g(energy)h(b)m(y)h (replacing)d Fo(V)21 b Fs(\()p Fo(x)p Fs(\))29 b(b)m(y)g Fn(\000)p Fs([)p Fo(V)22 b Fs(\()p Fo(x)p Fs(\)])3475 2983 y Fi(\000)3535 2968 y Fs(,)324 3088 y(so)33 b(w)m(e)g(will)d (assume)k Fo(V)21 b Fs(\()p Fo(x)p Fs(\))28 b(=)g Fn(\000)p Fs([)p Fo(V)22 b Fs(\()p Fo(x)p Fs(\)])1791 3103 y Fi(\000)1883 3088 y Fs(in)31 b(the)i(pro)s(of.)470 3209 y Ft(Coheren)m(t)38 b(states:)p 470 3225 800 4 v 324 3329 a Fs(If)436 3304 y Fo(~)420 3329 y(B)f Fs(is)31 b(a)g(\(constan)m(t\))h(v)m(ector,)h (then)g(the)f(pro)5 b(jection)31 b(on)m(to)g(the)h Fo(\027)6 b Fs(th)32 b(Landau)g(lev)m(el)f(of)324 3458 y(\()p Fn(\000)p Fo(ih)p Fn(r)23 b Fs(+)f Fo(\026)801 3419 y Fm(1)p 801 3435 36 4 v 801 3493 a(2)861 3433 y Fo(~)846 3458 y(B)27 b Fn(\002)c Fo(x)p Fs(\))1140 3422 y Fm(2)1212 3458 y Fs(has)33 b(in)m(tegral)e(k)m(ernel)i([LSY94,)g(p.95]:)364 3762 y(\005)437 3721 y Fm(\(2\))437 3787 y Fj(\027)531 3762 y Fs(\()p Fo(x)624 3777 y Fi(?)684 3762 y Fo(;)17 b(y)776 3777 y Fi(?)834 3762 y Fs(\))28 b(=)1045 3695 y Fo(\026b)p 1013 3740 164 4 v 1013 3831 a Fs(2)p Fo(\031)t(h)1204 3762 y Fs(exp)q Fn(f)p Fo(i)p Fs(\()p Fo(x)1529 3777 y Fi(?)1610 3762 y Fn(\002)23 b Fo(y)1758 3777 y Fi(?)1817 3762 y Fs(\))1865 3695 y Fo(\026)1940 3670 y(~)1924 3695 y(B)p 1865 3740 138 4 v 1881 3831 a Fs(2)p 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Fs(\()p Fo(x;)g(y)t Fs(\))p Fo(;)1894 5251 y Fs(13)p eop %%Page: 14 14 14 13 bop 324 548 a Fs(with)32 b Fo(\017)585 563 y Fj(\027;p)680 548 y Fs(\()p Fo(h;)17 b(b)p Fs(\))28 b(=)f(\(2)p Fo(\027)i Fs(+)22 b(1\))p Fo(h\026b)g Fs(+)g Fo(h)1709 512 y Fm(2)1748 548 y Fo(p)1797 512 y Fm(2)1837 548 y Fs(,)32 b(and)1573 794 y(\005)1646 809 y Fj(\027;p)1740 794 y Fs(\()p Fo(x;)17 b(x)p Fs(\))28 b(=)2144 727 y Fo(\026b)p 2112 772 164 4 v 2112 863 a Fs(2)p Fo(\031)t(h)2286 794 y(:)324 1025 y Fs(Let)42 b(us)g(\014nally)f(in)m(tro)s(duce)g(a)h(lo)s(calisation)c (function)j Fo(g)46 b Fn(2)e Fo(C)2673 989 y Fi(1)2666 1050 y Fm(0)2748 1025 y Fs(\()p Fl(R)2852 989 y Fm(3)2897 1025 y Fs(\))p Fo(;)2979 945 y Fk(R)3062 1025 y Fo(g)3113 989 y Fm(2)3195 1025 y Fs(=)f(1)e(and)324 1145 y(write)32 b Fo(g)620 1160 y Fj(r)658 1145 y Fs(\()p Fo(x)p Fs(\))c(=)f Fo(r)967 1109 y Fi(\000)p Fm(3)p Fj(=)p Fm(2)1132 1145 y Fo(g)t Fs(\()p Fo(x=r)s Fs(\),)k(where)j Fo(r)d Fs(=)c Fo(h)1984 1109 y Fm(1)p Fi(\000)p 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Fs(\()p Fo(h;)17 b(b)2047 1443 y Fj(t;u)2137 1428 y Fs(\))p Fn(h)p Fo(f)11 b Fn(j)p Fo(Q)p Fs(\()p Fo(\027)q(;)17 b(u;)g(p;)g(t)p Fs(\))p Fn(j)p Fo(f)11 b Fn(i)17 b Fo(dp)g(du:)324 1729 y Fs(F)-8 b(or)32 b(the)h(p)s(oten)m(tial)d(w)m(e)k(get:)739 1992 y Fn(h)p Fo(f)11 b Fn(j)p Fo(V)42 b Fn(\003)22 b Fo(g)1087 1951 y Fm(2)1083 2017 y Fj(r)1126 1992 y Fn(j)p Fo(f)11 b Fn(i)82 b Fs(=)1493 1857 y Fk(Z)1609 1992 y Fo(V)22 b Fs(\()p Fo(u)p Fs(\))p Fn(h)p Fo(f)11 b Fn(j)p Fo(g)1997 1951 y Fm(2)1993 2017 y Fj(r)2034 1992 y Fs(\()p Fn(\001)22 b(\000)h Fo(u)p Fs(\))p Fn(j)p Fo(f)11 b Fn(i)17 b Fo(du)795 b Fs(\(5.4\))1334 2247 y(=)1493 2152 y Fk(X)1546 2361 y Fj(\027)1654 2111 y Fk(Z)11 b(Z)1836 2247 y Fo(V)22 b Fs(\()p Fo(u)p Fs(\))2086 2179 y(1)p 2057 2224 108 4 v 2057 2315 a(2)p Fo(\031)2174 2247 y Fn(h)p Fo(f)11 b Fn(j)p Fo(Q)p Fs(\()p Fo(\027)q(;)17 b(u;)g(p;)g(t)p Fs(\))p Fn(j)p Fo(f)11 b Fn(i)17 b Fo(dpdu:)470 2548 y Ft(Lo)m(w)m(er)37 b(b)s(ound:)p 470 2564 672 4 v 324 2669 a Fs(No)m(w)27 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Fm(3)3009 3130 y Fs(\))g(+)g Fo(V)g Fs(\()p Fo(x)p Fs(\))p Fo(:)324 3350 y Fs(Let)33 b(us)g(furthermore)f(tak)m(e)h(an)g Fo(\017)28 b(>)f Fs(0)33 b(and)f(c)m(ho)s(ose)i Fo(R)f Fs(suc)m(h)h(that)795 3460 y Fk(Z)850 3686 y Fi(j)p Fj(x)p Fi(j\025)p Fj(R)1058 3596 y Fn(j)p Fo(V)21 b Fs(\()p Fo(x)p Fs(\))p Fn(j)1323 3555 y Fm(3)p Fj(=)p Fm(2)1450 3596 y Fo(dx)28 b(<)f(\017)116 b Fs(and)2132 3460 y Fk(Z)2187 3686 y Fi(j)p Fj(x)p Fi(j\025)p Fj(R)2395 3596 y Fn(j)p Fo(V)21 b Fs(\()p Fo(x)p Fs(\))p Fn(j)2660 3555 y Fm(5)p Fj(=)p Fm(2)2787 3596 y Fo(dx)28 b(<)f(\017:)324 3885 y Fs(Since)h Fo(M)10 b Fs(\()p Fo(x)p Fs(\))29 b Fn(2)f Fo(C)1009 3849 y Fi(1)1002 3910 y Fm(0)1111 3885 y Fs(w)m(e)h(will)d(assume)i(that)g Fo(M)10 b Fs(\()p Fo(x)p Fs(\))29 b(=)e(0)h(for)g Fn(j)p Fo(x)p Fn(j)f(\025)h Fo(R)q Fs(.)42 b(Cho)s(ose)29 b(\014nally)-8 b(,)324 4005 y(a)32 b(partition)f(of)i(unit)m(y)g Fo(\022)1227 3969 y Fm(2)1224 4030 y(1)1266 4005 y Fo(;)17 b(\022)1358 3969 y Fm(2)1355 4030 y(2)1431 4005 y Fs(of)32 b(p)s(ositiv)m(e)g(real) g(functions,)i(satisfying:)43 b Fo(\022)3060 4020 y Fm(1)3100 4005 y Fs(\()p Fo(x)p Fs(\))28 b(=)g(0)33 b(for)324 4126 y Fn(j)p Fo(x)p Fn(j)27 b(\025)h Fs(2)p Fo(R)34 b Fs(and)e Fo(\022)958 4141 y Fm(2)998 4126 y Fs(\()p Fo(x)p Fs(\))c(=)g(0)k(for)g Fn(j)p Fo(x)p Fn(j)c(\024)g Fo(R)q Fs(.)1894 5251 y(15)p eop %%Page: 16 16 16 15 bop 324 548 a Fs(Then)837 706 y Fj(N)797 735 y Fk(X)808 945 y Fj(j)t Fm(=1)941 830 y Fn(h)p Fo(f)1028 845 y Fj(j)1064 830 y Fn(j)p Fo(H)8 b Fs(\()p Fo(t)p Fs(\))p Fn(j)p Fo(f)1368 845 y Fj(j)1404 830 y Fn(i)638 1181 y Fs(=)837 1057 y Fj(N)797 1087 y Fk(X)808 1297 y Fj(j)t Fm(=1)941 1181 y Fn(h)p Fo(f)1028 1196 y Fj(j)1064 1181 y Fn(j)p Fo(\022)1137 1196 y Fm(1)1177 1181 y Fo(H)g Fs(\()p Fo(t)p Fs(\))p Fo(\022)1422 1196 y Fm(1)1461 1181 y Fn(j)p Fo(f)1537 1196 y Fj(j)1573 1181 y Fn(i)22 b Fs(+)1773 1057 y Fj(N)1732 1087 y Fk(X)1743 1297 y Fj(j)t Fm(=1)1876 1181 y Fn(h)p Fo(f)1963 1196 y Fj(j)2000 1181 y Fn(j)p Fo(\022)2073 1196 y Fm(2)2112 1181 y Fo(H)8 b Fs(\()p Fo(t)p Fs(\))p Fo(\022)2357 1196 y Fm(2)2397 1181 y Fn(j)p Fo(f)2473 1196 y Fj(j)2509 1181 y Fn(i)797 1533 y(\000)p Fo(h)930 1492 y Fm(2)1027 1408 y Fj(N)987 1438 y Fk(X)997 1648 y Fj(j)t Fm(=1)1130 1533 y Fn(h)p Fo(f)1217 1548 y Fj(j)1254 1533 y Fn(j)p Fs(\()p Fn(r)p Fo(\022)1448 1548 y Fm(1)1487 1533 y Fs(\))1525 1492 y Fm(2)1587 1533 y Fs(+)22 b(\()p Fn(r)p Fo(\022)1851 1548 y Fm(2)1891 1533 y Fs(\))1929 1492 y Fm(2)1968 1533 y Fn(j)p Fo(f)2044 1548 y Fj(j)2080 1533 y Fn(i)638 1884 y Fs(=)83 b(\(1)22 b Fn(\000)g Fo(\016)t Fs(\))1147 1759 y Fj(N)1107 1789 y Fk(X)1118 1999 y Fj(j)t Fm(=1)1251 1884 y Fn(h)p Fo(f)1338 1899 y Fj(j)1374 1884 y Fn(j)p Fo(\022)1447 1899 y Fm(1)1487 1773 y Fk(\020)1546 1884 y Fo(p)1608 1897 y Fj(~)1595 1914 y(A)1674 1884 y Fn(\001)g Fs(\()p Fo(I)30 b Fs(+)2034 1817 y Fo(t)p 1943 1861 218 4 v 1943 1952 a Fs(1)22 b Fn(\000)h Fo(\016)2170 1884 y(M)10 b Fs(\()p Fo(x)p Fs(\)\))p Fo(p)2505 1897 y Fj(~)2492 1914 y(A)1395 2197 y Fn(\000)p Fo(\026h)p Fs(\(1)22 b(+)1895 2130 y Fo(t)p 1804 2175 V 1804 2266 a Fs(1)g Fn(\000)g Fo(\016)2031 2197 y(b)2072 2212 y Fm(3)2134 2197 y Fn(\003)g Fo(g)2257 2156 y Fm(2)2253 2222 y Fj(r)2296 2197 y Fs(\))g(+)2468 2130 y Fo(V)f Fs(\()p Fo(x)p Fs(\))p 2464 2175 V 2464 2266 a(1)h Fn(\000)h Fo(\016)2714 2197 y Fn(\003)f Fo(g)2837 2156 y Fm(2)2833 2222 y Fj(r)2875 2087 y Fk(\021)2935 2197 y Fo(\022)2980 2212 y Fm(1)3020 2197 y Fn(j)p Fo(f)3096 2212 y Fj(j)3132 2197 y Fn(i)797 2487 y Fs(+)930 2363 y Fj(N)889 2393 y Fk(X)900 2602 y Fj(j)t Fm(=1)1033 2487 y Fn(h)p Fo(f)1120 2502 y Fj(j)1157 2487 y Fn(j)p Fo(\022)1230 2502 y Fm(1)1269 2317 y Fk( )1348 2487 y Fo(\016)t Fs(\()p Fo(p)1482 2446 y Fm(2)1495 2510 y Fj(~)1482 2526 y(A)1561 2487 y Fn(\000)h Fo(\026h)p Fs(\))f(+)g(\()p Fo(V)43 b Fn(\000)23 b Fo(V)44 b Fn(\003)21 b Fo(g)2395 2446 y Fm(2)2391 2512 y Fj(r)2434 2487 y Fs(\))797 2833 y Fn(\000)p Fo(\026ht)p Fs(\()p Fo(b)1103 2848 y Fm(3)1165 2833 y Fn(\000)i Fo(b)1306 2848 y Fm(3)1368 2833 y Fn(\003)f Fo(g)1491 2792 y Fm(2)1487 2858 y Fj(r)1530 2833 y Fs(\))g Fn(\000)g Fo(h)1745 2792 y Fm(2)1785 2833 y Fs(\()p Fn(r)p Fo(\022)1951 2848 y Fm(1)1991 2833 y Fs(\))2029 2792 y Fm(2)2090 2833 y Fn(\000)h Fo(h)2246 2792 y Fm(2)2286 2833 y Fs(\()p Fn(r)p Fo(\022)2452 2848 y Fm(2)2492 2833 y Fs(\))2530 2792 y Fm(2)2569 2663 y Fk(!)2648 2833 y Fo(\022)2693 2848 y Fm(1)2733 2833 y Fn(j)p Fo(f)2809 2848 y Fj(j)2845 2833 y Fn(i)797 3171 y Fs(+)930 3046 y Fj(N)889 3076 y Fk(X)900 3286 y Fj(j)t Fm(=1)1033 3171 y Fn(h)p Fo(f)1120 3186 y Fj(j)1157 3171 y Fn(j)p Fo(\022)1230 3186 y Fm(2)1286 3090 y Fk(\000)1331 3171 y Fo(p)1380 3129 y Fm(2)1394 3193 y Fj(~)1380 3210 y(A)1460 3171 y Fn(\000)f Fo(\026h)g Fs(+)g Fo(V)g Fs(\()p Fo(x)p Fs(\))g Fn(\000)h Fo(h)2182 3129 y Fm(2)2221 3171 y Fs(\()p Fn(r)p Fo(\022)2387 3186 y Fm(1)2427 3171 y Fs(\))2465 3129 y Fm(2)2527 3171 y Fn(\000)f Fo(h)2682 3129 y Fm(2)2722 3171 y Fs(\()p Fn(r)p Fo(\022)2888 3186 y Fm(2)2928 3171 y Fs(\))2966 3129 y Fm(2)3005 3090 y Fk(\001)3067 3171 y Fo(\022)3112 3186 y Fm(2)3152 3171 y Fn(j)p Fo(f)3228 3186 y Fj(j)3265 3171 y Fn(i)p Fo(:)30 b Fs(\(5.5\))470 3480 y(The)k(\014rst)f(term)f(can)h(b)s(e)f(b)s(ounded)i(b)s(elo)m(w)e (b)m(y)500 3679 y(1)22 b Fn(\000)g Fo(\016)p 500 3723 V 555 3815 a Fs(2)p Fo(\031)744 3652 y Fk(X)796 3861 y Fj(\027)904 3611 y Fk(Z)12 b(Z)1087 3606 y(\032)1162 3746 y Fs(\(1)22 b Fn(\000)g Fo(F)1433 3761 y Fj(r)1471 3746 y Fs(\()p Fo(u)p Fs(\)\))p Fo(\017)1680 3761 y Fj(p;\027)1778 3746 y Fs(\()p Fo(h;)17 b(b)1957 3770 y Fj(u;)2070 3743 y Fb(t)p 2028 3755 109 3 v 2028 3796 a Fg(1)p Fa(\000)p Fb(\016)2150 3746 y Fs(\))23 b Fn(\000)f Fo(\026h)p Fs(\(1)g(+)2733 3679 y Fo(t)p 2642 3723 218 4 v 2642 3815 a Fs(1)g Fn(\000)g Fo(\016)2869 3746 y(b)2910 3761 y Fm(3)2950 3746 y Fs(\()p Fo(u)p Fs(\)\))g(+)3253 3679 y Fo(V)g Fs(\()p Fo(u)p Fs(\))p 3250 3723 V 3250 3815 a(1)g Fn(\000)g Fo(\016)3477 3606 y Fk(\033)490 4084 y Fn(\002)624 3959 y Fj(N)584 3989 y Fk(X)595 4199 y Fj(j)t Fm(=1)728 4084 y Fn(h)p Fo(f)815 4099 y Fj(j)851 4084 y Fn(j)p Fo(\022)924 4099 y Fm(1)964 4084 y Fo(Q)p Fs(\()p Fo(\027)q(;)17 b(u;)g(p;)1465 4016 y(t)p 1374 4061 V 1374 4152 a Fs(1)k Fn(\000)i Fo(\016)1601 4084 y Fs(\))p Fo(\022)1684 4099 y Fm(1)1724 4084 y Fn(j)p Fo(f)1800 4099 y Fj(j)1836 4084 y Fn(i)17 b Fo(dpdu:)324 4392 y Fs(Since)904 4581 y(0)28 b Fn(\024)1126 4456 y Fj(N)1086 4486 y Fk(X)1097 4696 y Fj(j)t Fm(=1)1230 4581 y Fn(h)p Fo(f)1317 4596 y Fj(j)1353 4581 y Fn(j)p Fo(\022)1426 4596 y Fm(1)1466 4581 y Fo(Q)p Fs(\()p Fo(\027)q(;)17 b(u;)g(p;)1967 4513 y(t)p 1876 4558 V 1876 4649 a Fs(1)k Fn(\000)i Fo(\016)2103 4581 y Fs(\))p Fo(\022)2186 4596 y Fm(1)2226 4581 y Fn(j)p Fo(f)2302 4596 y Fj(j)2338 4581 y Fn(i)28 b(\024)2585 4485 y Fo(\026b)2685 4509 y Fj(u;)2798 4482 y Fb(t)p 2756 4494 109 3 v 2756 4535 a Fg(1)p Fa(\000)p Fb(\016)p 2520 4558 425 4 v 2520 4649 a Fs(2)p Fo(\031)t(h)p Fs(\003)2803 4645 y Fb(t)p 2761 4657 109 3 v 2761 4698 a Fg(1)p Fa(\000)p Fb(\016)2879 4672 y Fj(;u)2954 4581 y Fo(;)324 4859 y Fs(and)36 b Fn(h)p Fo(f)604 4874 y Fj(j)640 4859 y Fn(j)p Fo(\022)713 4874 y Fm(1)752 4859 y Fo(Q)p Fs(\()p Fo(\027)q(;)17 b(u;)g(p;)1211 4819 y Fj(t)p 1162 4836 124 4 v 1162 4893 a Fm(1)p Fi(\000)p Fj(\016)1296 4859 y Fs(\))p Fo(\022)1379 4874 y Fm(1)1419 4859 y Fn(j)p Fo(f)1495 4874 y Fj(j)1531 4859 y Fn(i)33 b Fs(=)f(0)k(if)e Fn(j)p Fo(u)p Fn(j)e(\025)h Fs(3)p Fo(R)25 b Fs(+)f Fo(r)s Fs(,)37 b(w)m(e)f(get)g(a)f(lo)m(w)m(er) h(b)s(ound)g(b)m(y)324 5002 y(replacing)739 4927 y Fk(P)844 4954 y Fj(N)844 5031 y(j)t Fm(=1)971 5002 y Fn(h)p Fo(f)1058 5017 y Fj(j)1094 5002 y Fn(j)p Fo(\022)1167 5017 y Fm(1)1207 5002 y Fo(Q)p Fs(\()p Fo(\027)q(;)17 b(u;)g(p;)1666 4963 y Fj(t)p 1617 4979 V 1617 5036 a Fm(1)p Fi(\000)p Fj(\016)1751 5002 y Fs(\))p Fo(\022)1834 5017 y Fm(1)1873 5002 y Fn(j)p Fo(f)1949 5017 y Fj(j)1986 5002 y Fn(i)30 b Fs(b)m(y)h(a)g(function)f Fo(M)10 b Fs(\()p Fo(\027)q(;)17 b(u;)g(p;)3134 4963 y Fj(t)p 3085 4979 V 3085 5036 a Fm(1)p Fi(\000)p Fj(\016)3219 5002 y Fs(\),)31 b(whic)m(h)1894 5251 y(16)p eop %%Page: 17 17 17 16 bop 324 548 a Fs(is)32 b(the)h(c)m(haracteristic)f(function)g(of) g(the)h(set)580 798 y Fn(f)p Fs(\()p Fo(\027)q(;)17 b(u;)g(p)p Fs(\))p Fn(j)p Fs(\(1)j Fn(\000)j Fo(F)1246 813 y Fj(r)1284 798 y Fs(\()p Fo(u)p Fs(\)\))p Fo(\017)1493 813 y Fj(p;\027)1591 798 y Fs(\()p Fo(h;)17 b(b)1770 821 y Fj(u;)1883 794 y Fb(t)p 1841 806 109 3 v 1841 848 a Fg(1)p Fa(\000)p Fb(\016)1963 798 y Fs(\))22 b Fn(\000)h Fo(\026h)p Fs(\(1)f(+)2546 731 y Fo(t)p 2455 775 218 4 v 2455 866 a Fs(1)f Fn(\000)i Fo(\016)2682 798 y(b)2723 813 y Fm(3)2763 798 y Fs(\()p Fo(u)p Fs(\)\))e(+)3066 731 y Fo(V)h Fs(\()p Fo(u)p Fs(\))p 3062 775 V 3062 866 a(1)g Fn(\000)h Fo(\016)3317 798 y Fn(\024)29 b Fs(0)613 983 y(and)j Fn(j)p Fo(u)p Fn(j)27 b(\024)h Fs(3)p Fo(R)23 b Fs(+)f Fo(r)s Fn(g)p Fo(;)324 1252 y Fs(times)31 b Fn(\000)719 1162 y Fj(\026b)791 1189 y Fb(u;)900 1169 y(t)p 858 1177 109 3 v 858 1215 a Fg(1)p Fa(\000)p Fb(\016)p 672 1229 356 4 v 672 1287 a Fm(2)p Fj(\031)r(h)p Fm(\003)892 1295 y Fb(t)p 850 1303 109 3 v 850 1340 a Fg(1)p Fa(\000)p Fb(\016)968 1314 y(;u)1038 1252 y Fs(.)43 b(Th)m(us,)34 b(the)f(lo)m(w)m(er)g(b)s (ound)g(b)s(ecomes)653 1619 y Fn(\000)740 1552 y Fs(1)22 b Fn(\000)h Fo(\016)p 740 1596 218 4 v 795 1687 a Fs(2)p Fo(\031)984 1525 y Fk(X)1037 1734 y Fj(\027)1145 1484 y Fk(Z)11 b(Z)1267 1709 y Fi(fj)p Fj(u)p Fi(j\024)p Fm(3)p Fj(R)p Fm(+)p Fj(r)r Fi(g)1746 1524 y Fo(\026b)1846 1547 y Fj(u;)1959 1520 y Fb(t)p 1917 1532 109 3 v 1917 1573 a Fg(1)p Fa(\000)p Fb(\016)p 1680 1596 425 4 v 1680 1687 a Fs(2)p Fo(\031)t(h)p Fs(\003)1964 1683 y Fb(t)p 1922 1695 109 3 v 1922 1737 a Fg(1)p Fa(\000)p Fb(\016)2040 1711 y Fj(;u)2137 1619 y Fn(\002)653 1781 y Fk(\024)705 1922 y Fs(\(1)22 b Fn(\000)h Fo(F)977 1937 y Fj(r)1015 1922 y Fs(\()p Fo(u)p Fs(\)\))p Fo(\017)1224 1937 y Fj(p;\027)1322 1922 y Fs(\()p Fo(h;)17 b(b)1501 1945 y Fj(u;)1614 1918 y Fb(t)p 1572 1930 V 1572 1971 a Fg(1)p Fa(\000)p Fb(\016)1694 1922 y Fs(\))22 b Fn(\000)h Fo(\026h)p Fs(\(1)f(+)2277 1854 y Fo(t)p 2186 1899 218 4 v 2186 1990 a Fs(1)f Fn(\000)i Fo(\016)2413 1922 y(b)2454 1937 y Fm(3)2494 1922 y Fs(\()p Fo(u)p Fs(\)\))e(+)2797 1854 y Fo(V)h Fs(\()p Fo(u)p Fs(\))p 2793 1899 V 2793 1990 a(1)g Fn(\000)h Fo(\016)3021 1781 y Fk(\025)3073 2020 y Fi(\000)3166 1922 y Fo(dpdu:)324 2204 y Fs(W)-8 b(e)33 b(do)f(the)h Fo(p)g Fs(in)m(tegration)e (explicitly)f(and)j(get:)490 2496 y Fn(\000)p Fs(\(1)22 b Fn(\000)h Fo(\016)t Fs(\))878 2402 y Fk(X)930 2611 y Fj(\027)1038 2361 y Fk(Z)1093 2586 y Fi(fj)p Fj(u)p Fi(j\024)p Fm(3)p Fj(R)p Fm(+)p Fj(r)r Fi(g)1507 2429 y Fs(1)f Fn(\000)g Fo(F)1740 2444 y Fj(r)1779 2429 y Fs(\()p Fo(u)p Fs(\))p 1507 2473 404 4 v 1655 2565 a(2)p Fo(\031)1991 2401 y Fs(4)p Fo(\026b)2140 2424 y Fj(u;)2253 2397 y Fb(t)p 2210 2409 109 3 v 2210 2450 a Fg(1)p Fa(\000)p Fb(\016)p 1930 2473 464 4 v 1930 2565 a Fs(6)p Fo(\031)t(h)2094 2536 y Fm(2)2133 2565 y Fs(\003)2253 2561 y Fb(t)p 2211 2573 109 3 v 2211 2614 a Fg(1)p Fa(\000)p Fb(\016)2329 2588 y Fj(;u)490 2822 y Fn(\002)584 2682 y Fk(\024)637 2822 y Fs(\(2)p Fo(\027)28 b Fs(+)22 b(1\))p Fo(h\026b)1141 2846 y Fj(u;)1254 2819 y Fb(t)p 1211 2831 V 1211 2872 a Fg(1)p Fa(\000)p Fb(\016)1356 2822 y Fs(+)1641 2755 y(1)p 1464 2799 404 4 v 1464 2891 a(1)g Fn(\000)g Fo(F)1697 2906 y Fj(r)1736 2891 y Fs(\()p Fo(u)p Fs(\))1893 2682 y Fk(\022)1967 2822 y Fn(\000)p Fo(\026h)p Fs(\(1)g(+)2467 2755 y Fo(t)p 2376 2799 218 4 v 2376 2891 a Fs(1)g Fn(\000)h Fo(\016)2603 2822 y(b)2644 2837 y Fm(3)2684 2822 y Fs(\()p Fo(u)p Fs(\)\))f(+)2988 2755 y Fo(V)f Fs(\()p Fo(u)p Fs(\))p 2984 2799 V 2984 2891 a(1)h Fn(\000)g Fo(\016)3211 2682 y Fk(\023)q(\025)3337 2704 y Fm(3)p Fj(=)p Fm(2)3337 2921 y Fi(\000)3480 2822 y Fo(du:)324 3111 y Fs(The)29 b(last)f(t)m(w)m(o)i(terms)e(in)g(\(5.5\))g(are)h(error)g(terms,)g(and) g(will)d(b)s(e)j(b)s(ounded)g(b)s(elo)m(w)g(using)324 3231 y(the)j(magnetic)e(Lieb-Thirring)f(inequalit)m(y)-8 b(.)42 b(Since)32 b Fo(r)e Fs(=)e Fo(h)2484 3195 y Fm(1)p Fi(\000)p Fj(\013)2655 3231 y Fs(and)j Fo(\026h)d Fn(\024)g Fo(C)7 b Fs(,)32 b(w)m(e)g(ha)m(v)m(e)324 3351 y(for)g(small)e Fo(h)j Fs(that)324 3459 y Fk(Z)440 3510 y(\014)440 3570 y(\014)473 3595 y Fs(\()p Fo(V)22 b Fs(\()p Fo(x)p Fs(\))g Fn(\000)h Fo(V)43 b Fn(\003)22 b Fo(g)1066 3554 y Fm(2)1062 3619 y Fj(r)1105 3595 y Fs(\()p Fo(x)p Fs(\)\))h Fn(\000)f Fo(\026h)p Fs(\()p Fo(b)1590 3610 y Fm(3)1630 3595 y Fs(\()p Fo(x)p Fs(\))g Fn(\000)h Fo(b)1924 3610 y Fm(3)1986 3595 y Fn(\003)f Fo(g)2109 3554 y Fm(2)2105 3619 y Fj(r)2148 3595 y Fs(\()p Fo(x)p Fs(\)\))g Fn(\000)h Fo(h)2495 3554 y Fm(2)2534 3595 y Fs(\()p Fn(r)p Fo(\022)2700 3610 y Fm(1)2740 3595 y Fs(\))2778 3554 y Fm(2)2840 3595 y Fn(\000)f Fo(h)2995 3554 y Fm(2)3035 3595 y Fs(\()p Fo(\022)3118 3610 y Fm(2)3158 3595 y Fs(\))3196 3554 y Fm(2)3235 3510 y Fk(\014)3235 3570 y(\014)3268 3536 y Fj(q)3339 3595 y Fo(dx)28 b(<)g(\017;)324 3863 y Fs(for)35 b Fo(q)j Fs(=)33 b(3)p Fo(=)p Fs(2)i(and)h Fo(q)i Fs(=)33 b(5)p Fo(=)p Fs(2.)53 b(Therefore,)38 b(w)m(e)f(get)f(b)m(y)h(application)d (of)h(the)i(magnetic)324 3984 y(Lieb-Thirring)i(inequalit)m(y)i(that)g (the)i(\014rst)f(error)g(term)f(in)g(\(5.5\))g(can)h(b)s(e)g(b)s (ounded)324 4104 y(b)s(elo)m(w)32 b(b)m(y)1447 4224 y Fn(\000)p Fo(C)7 b(\017h)1696 4183 y Fi(\000)p Fm(3)1791 4224 y Fs(\()p Fo(\016)1876 4183 y Fi(\000)p Fm(1)p Fj(=)p Fm(2)2063 4224 y Fs(+)22 b Fo(\016)2208 4183 y Fi(\000)p Fm(3)p Fj(=)p Fm(2)2373 4224 y Fs(\))p Fo(:)324 4399 y Fs(W)-8 b(e)35 b(can)h(use)g(the)g(Lieb-Thirring)d(inequalit)m(y)h (directly)h(to)g(b)s(ound)g(the)h(second)h(error)324 4519 y(term)32 b(from)f(b)s(elo)m(w)i(b)m(y)1757 4640 y Fn(\000)p Fo(C)7 b(\017h)2006 4598 y Fi(\000)p Fm(3)2101 4640 y Fo(;)324 4814 y Fs(where)34 b(w)m(e)f(used)h(the)f(de\014nition) f(of)g Fo(\022)1730 4829 y Fm(2)1770 4814 y Fs(.)p 3491 4814 4 66 v 3495 4751 59 4 v 3495 4814 V 3553 4814 4 66 v 1894 5251 a(17)p eop %%Page: 18 18 18 17 bop 324 548 a Fu(6)161 b(Calculation)55 b(of)f(the)f(curren)l(t) 324 767 y Fs(In)33 b(this)f(section)h(w)m(e)g(will)d(\014nally)i (\014nd)h(the)g(asymptotics)f(of)g(the)h(curren)m(t:)470 887 y(By)g(applying)f(the)h(comm)m(utator)e(iden)m(tit)m(y)h(from)f (Section)65 b(2,)33 b(w)m(e)g(get)595 1103 y Fn(\000)p Fs(tr[)p Ft(J)p Fs(\()p Fo(\026)-6 b(~)-43 b(a)p Fs(\)1)1068 1118 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])1299 1103 y Fs(\()p Ft(P)p Fs(\)])83 b(=)g(tr[\()p Fo(p)1924 1116 y Fj(~)1911 1133 y(A)1990 1103 y Fn(\001)22 b Fo(M)10 b Fs(\()p Fo(x)p Fs(\))p Fo(p)2337 1116 y Fj(~)2324 1133 y(A)2404 1103 y Fs(+)22 b Fo(\026h\033)2672 1118 y Fm(3)2712 1103 y Fo(b)2753 1118 y Fm(3)2793 1103 y Fs(\)1)2880 1118 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])3111 1103 y Fs(\()p Ft(P)p Fs(\)])1721 1248 y(+tr[)q(~)-50 b Fo(a)22 b Fn(\001)g(r)p Fo(V)f Fs(1)2233 1264 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2465 1248 y Fs(\()p Ft(P)p Fs(\)])p Fo(;)689 b Fs(\(6.1\))324 1463 y(where)34 b Fo(M)10 b Fs(\()p Fo(x)p Fs(\))28 b(=)g Fn(\000)17 b Fs(\()p Fo(D)t Fs(~)-50 b Fo(a)p Fs(\()p Fo(x)p Fs(\))22 b(+)g(\()p Fo(D)t Fs(~)-50 b Fo(a)p Fs(\()p Fo(x)p Fs(\)\))1833 1427 y Fj(t)1863 1463 y Fs(\).)470 1584 y(The)30 b(asymptotics)e(of)g(the)h(second)h(term)e(in)g(\(6.1\))g (is)g(easy)h(to)f(calculate)g(using)g(the)324 1704 y(results)33 b(on)f(the)h(densit)m(y)h(from)64 b([LSY94].)324 1904 y Ft(Lemma)37 b(6.1.)42 b Fp(When)34 b Fs([)p Fo(V)22 b Fs(])1351 1919 y Fi(\000)1410 1904 y Fo(;)c Fs(~)-50 b Fo(a)22 b Fn(\001)g(r)p Fo(V)49 b Fn(2)28 b Fo(L)1926 1867 y Fm(3)p Fj(=)p Fm(2)2059 1904 y Fn(\\)22 b Fo(L)2213 1867 y Fm(5)p Fj(=)p Fm(2)2324 1904 y Fp(,)35 b(then)490 2032 y Fk(\014)490 2092 y(\014)490 2151 y(\014)490 2211 y(\014)490 2271 y(\014)523 2206 y Fs(tr[\()q(~)-50 b Fo(a)22 b Fn(\001)g(r)p Fo(V)g Fs(\)1)1036 2222 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])1267 2206 y Fs(\()p Ft(P)p Fs(\)])g Fn(\000)1587 2139 y(\000)p Fs(2)p Fo(\026)p 1578 2183 204 4 v 1578 2274 a Fs(3)p Fo(\031)t(h)1742 2246 y Fm(2)1845 2082 y Fi(1)1808 2111 y Fk(X)1815 2321 y Fj(\027)t Fm(=0)1969 2206 y Fo(d)2020 2221 y Fj(\027)2079 2070 y Fk(Z)2179 2206 y Fs(\()p Fo(@)2268 2221 y Fj(x)2308 2230 y Fg(1)2347 2206 y Fo(a)2398 2221 y Fm(2)2459 2206 y Fn(\000)h Fo(@)2610 2221 y Fj(x)2650 2230 y Fg(2)2689 2206 y Fo(a)2740 2221 y Fm(1)2780 2206 y Fs(\)[2)p Fo(\027)6 b(\026h)22 b Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\)])3419 2155 y Fm(3)p Fj(=)p Fm(2)3419 2228 y Fi(\000)3546 2206 y Fo(dx)3652 2032 y Fk(\014)3652 2092 y(\014)3652 2151 y(\014)3652 2211 y(\014)3652 2271 y(\014)490 2448 y Fs(=)27 b Fo(o)p Fs(\()p Fo(h)734 2407 y Fi(\000)p Fm(3)851 2448 y Fs(+)22 b Fo(\026h)1064 2407 y Fi(\000)p Fm(2)1158 2448 y Fs(\))p Fo(:)470 2663 y Fs(F)-8 b(or)26 b(the)i(\014rst)g(term)e(in)h(\(6.1\))f(w)m(e)i(need) h(the)e(result)g(from)f(Section)54 b(5.)42 b(Let)27 b(us)h(write)324 2784 y(the)33 b(term)f(as)940 2999 y(tr[\()p Fo(p)1143 3012 y Fj(~)1130 3029 y(A)1209 2999 y Fn(\001)22 b Fo(M)10 b Fs(\()p Fo(x)p Fs(\))p Fo(p)1556 3012 y Fj(~)1543 3029 y(A)1623 2999 y Fn(\000)22 b Fo(\026hb)1878 3014 y Fm(3)1918 2999 y Fs(\)1)2005 3014 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2236 2999 y Fs(\()p Fo(p)2323 2958 y Fm(2)2337 3021 y Fj(~)2323 3038 y(A)2402 2999 y Fn(\000)h Fo(\026h)f Fs(+)g Fo(V)g Fs(\()p Fo(x)p Fs(\)\)])940 3155 y(+tr[\()p Fo(p)1219 3168 y Fj(~)1206 3185 y(A)1285 3155 y Fn(\001)f Fo(M)10 b Fs(\()p Fo(x)p Fs(\))p Fo(p)1631 3168 y Fj(~)1618 3185 y(A)1699 3155 y Fs(+)22 b Fo(\026hb)1953 3170 y Fm(3)1992 3155 y Fs(\)1)2079 3171 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2311 3155 y Fs(\()p Fo(p)2398 3114 y Fm(2)2411 3178 y Fj(~)2398 3194 y(A)2477 3155 y Fs(+)g Fo(\026h)g Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\)\)])p Fo(;)250 b Fs(\(6.2\))324 3371 y(and)32 b(analyse)h(eac)m(h)h(term)e(sep)s(erately)-8 b(.)324 3570 y Ft(Lemma)45 b(6.2.)g Fp(Supp)-5 b(ose)40 b Fs([)p Fo(V)22 b Fs(])1458 3585 y Fi(\000)1556 3570 y Fn(2)40 b Fo(L)1728 3534 y Fm(3)p Fj(=)p Fm(2)1865 3570 y Fn(\\)27 b Fo(L)2024 3534 y Fm(5)p Fj(=)p Fm(2)2176 3570 y Fp(and)42 b Fs(~)-51 b Fo(a)40 b Fs(=)f(\()p Fn(\000)p Fo(a)2743 3585 y Fm(2)2783 3570 y Fo(;)17 b(a)2878 3585 y Fm(1)2917 3570 y Fo(;)g Fs(0\))39 b Fn(2)h Fo(C)3270 3534 y Fi(1)3263 3595 y Fm(0)3344 3570 y Fs(\()p Fl(R)3448 3534 y Fm(3)3494 3570 y Fs(\))p Fp(.)324 3690 y(Write)d Fo(M)10 b Fs(\()p Fo(x)p Fs(\))31 b(=)g Fn(\000)17 b Fs(\()p Fo(D)t Fs(~)-50 b Fo(a)p Fs(\()p Fo(x)p Fs(\))22 b(+)g(\()p Fo(D)t Fs(~)-50 b Fo(a)p Fs(\()p Fo(x)p Fs(\)\))1839 3654 y Fj(t)1869 3690 y Fs(\))p Fp(.)49 b(Supp)-5 b(ose)36 b(furthermor)-5 b(e,)37 b(that)f Fo(\026h)31 b Fn(!)f Fo(\014)36 b Fn(2)324 3811 y Fs(\(0)p Fo(;)17 b Fs(+)p Fn(1)p Fs(\))34 b Fp(as)g Fo(h)28 b Fn(!)f Fs(0)p Fp(.)45 b(Then)624 4085 y Fs(lim)619 4148 y Fj(h)p Fi(!)p Fm(0)792 4018 y Fo(h)848 3982 y Fm(2)p 792 4062 96 4 v 810 4154 a Fo(\026)897 4085 y Fs(tr[\()p Fo(p)1100 4098 y Fj(~)1087 4115 y(A)1166 4085 y Fn(\001)22 b Fo(M)10 b Fs(\()p Fo(x)p Fs(\))p Fo(p)1513 4098 y Fj(~)1500 4115 y(A)1580 4085 y Fn(\000)23 b Fo(\026hb)1836 4100 y Fm(3)1876 4085 y Fs(\)1)1963 4101 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2194 4085 y Fs(\()p Fo(p)2281 4044 y Fm(2)2294 4108 y Fj(~)2281 4125 y(A)2360 4085 y Fn(\000)g Fo(\026h)f Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\)\)])28 b(=)1176 4245 y Fi(1)1139 4275 y Fk(X)1147 4484 y Fj(\027)t Fm(=0)1310 4302 y Fs(2)p Fo(\027)6 b(\014)p 1310 4346 164 4 v 1338 4438 a Fs(2)p Fo(\031)1500 4234 y Fk(Z)1600 4369 y Fs(\()p Fo(@)1689 4384 y Fj(x)1729 4393 y Fg(1)1768 4369 y Fo(a)1819 4384 y Fm(2)1880 4369 y Fn(\000)23 b Fo(@)2031 4384 y Fj(x)2071 4393 y Fg(2)2110 4369 y Fo(a)2161 4384 y Fm(1)2201 4369 y Fs(\)[2)p Fo(\027)6 b(\026h)22 b Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\)])2840 4318 y Fm(1)p Fj(=)p Fm(2)2840 4391 y Fi(\000)2968 4369 y Fo(dx:)324 4666 y Fp(Pr)-5 b(o)g(of.)48 b Fs(The)38 b(pro)s(of)e(is)h(easy)-8 b(,)39 b(using)e(the)g(v)-5 b(ariational)34 b(principle)h(for)i(the)g(energy)h(and)324 4787 y(the)33 b(lo)m(w)m(er)g(b)s(ound)f(from)g(Section)65 b(5:)43 b(W)-8 b(e)33 b(write)893 5002 y Fo(H)8 b Fs(\()p Fo(t)p Fs(\))27 b(=)h Fo(p)1273 4961 y Fm(2)1286 5024 y Fj(~)1273 5041 y(A)1352 5002 y Fn(\000)23 b Fo(\026h)f Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\))i(+)f Fo(t)p Fs(\()p Fo(p)2152 5015 y Fj(~)2139 5032 y(A)2218 5002 y Fn(\001)g Fo(M)10 b Fs(\()p Fo(x)p Fs(\))p Fo(p)2565 5015 y Fj(~)2552 5032 y(A)2632 5002 y Fn(\000)23 b Fo(\026hb)2888 5017 y Fm(3)2927 5002 y Fs(\))p Fo(;)1894 5251 y Fs(18)p eop %%Page: 19 19 19 18 bop 324 548 a Fs(and)52 b Fo(E)6 b Fs(\()p Fo(t)p Fs(\))61 b(=)g(tr[)p Fo(H)8 b Fs(\()p Fo(t)p Fs(\)1)1272 563 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])1503 548 y Fs(\()p Fo(H)g Fs(\()p Fo(t)p Fs(\)\)])60 b(=)h(inf)2121 563 y Fm(0)p Fi(\024)p Fj(\015)t Fi(\024)p Fm(1)2363 548 y Fs(tr[)p Fo(\015)5 b(H)j Fs(\()p Fo(t)p Fs(\)].)102 b(Then)53 b(the)f(lo)m(w)m(er)324 668 y(b)s(ound)33 b(com)m(bined)f (with)g(the)h(v)-5 b(ariational)29 b(principle)i(giv)m(es:)765 942 y(lim)760 1005 y Fj(h)p Fi(!)p Fm(0)933 875 y Fo(h)989 838 y Fm(2)p 933 919 96 4 v 951 1010 a Fo(\026)1038 942 y Fs(tr[\()p Fo(p)1241 955 y Fj(~)1228 972 y(A)1307 942 y Fn(\001)22 b Fo(M)10 b Fs(\()p Fo(x)p Fs(\))p Fo(p)1654 955 y Fj(~)1641 972 y(A)1721 942 y Fn(\000)23 b Fo(\026hb)1977 957 y Fm(3)2017 942 y Fs(\)1)2104 957 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2335 942 y Fs(\()p Fo(p)2422 901 y Fm(2)2435 964 y Fj(~)2422 981 y(A)2501 942 y Fn(\000)g Fo(\026h)e Fs(+)i Fo(V)e Fs(\()p Fo(x)p Fs(\)\)])28 b(=)505 1132 y Fo(d)p 488 1177 86 4 v 488 1268 a(dt)584 1199 y Fn(j)612 1214 y Fj(t)p Fm(=0)748 1105 y Fk(X)800 1314 y Fj(\027)908 1064 y Fk(Z)1082 1132 y Fn(\000)p Fo(b)1200 1147 y Fj(u;t)p 1034 1177 305 4 v 1034 1268 a Fs(3)p Fo(\031)1142 1239 y Fm(2)1182 1268 y Fs(\003)1250 1283 y Fj(u;t)1366 1199 y Fs([\(2)p Fo(\027)g Fs(+)22 b(1\))p Fo(\014)6 b(b)1843 1214 y Fj(u;t)1955 1199 y Fn(\000)23 b Fo(\014)6 b Fs(\(1)21 b(+)h Fo(tb)2398 1214 y Fm(3)2439 1199 y Fs(\()p Fo(u)p Fs(\)\))f(+)h Fo(V)g Fs(\()p Fo(u)p Fs(\)])2965 1151 y Fm(3)p Fj(=)p Fm(2)2965 1231 y Fi(\000)3108 1199 y Fo(du:)324 1511 y Fs(No)m(w)38 b(w)m(e)g(obtain)e(the)i(result,)g(if)e (w)m(e)j(remem)m(b)s(er)e(that)g Fo(b)2444 1526 y Fj(u;t)2570 1511 y Fs(=)e(\(1)26 b(+)f Fo(tb)2971 1526 y Fm(3)3011 1511 y Fs(\()p Fo(u)p Fs(\))g(+)g Fo(O)s Fs(\()p Fo(t)3420 1475 y Fm(2)3459 1511 y Fs(\)\),)324 1642 y(and)32 b(that)744 1593 y Fj(b)774 1601 y Fb(u;t)p 735 1619 133 4 v 735 1676 a Fm(\003)784 1684 y Fb(u;t)905 1642 y Fs(=)27 b(1)22 b(+)g Fo(O)s Fs(\()p Fo(t)1328 1606 y Fm(2)1367 1642 y Fs(\).)p 3491 1642 4 66 v 3495 1580 59 4 v 3495 1642 V 3553 1642 4 66 v 470 1847 a(No)m(w,)30 b(since)e Fo(\026h)g Fs(is)g(b)s(ounded,)i(it)d(is)h(easy)h(to)f(use)h(the)f(same)g(metho)s (ds)g(to)g(treat)g(the)324 1967 y(spin-up)k(part.)43 b(The)34 b(result)e(is)h(similar:)324 2170 y Ft(Lemma)45 b(6.3.)g Fp(Supp)-5 b(ose)40 b Fs([)p Fo(V)22 b Fs(])1458 2185 y Fi(\000)1556 2170 y Fn(2)40 b Fo(L)1728 2134 y Fm(3)p Fj(=)p Fm(2)1865 2170 y Fn(\\)27 b Fo(L)2024 2134 y Fm(5)p Fj(=)p Fm(2)2176 2170 y Fp(and)42 b Fs(~)-51 b Fo(a)40 b Fs(=)f(\()p Fn(\000)p Fo(a)2743 2185 y Fm(2)2783 2170 y Fo(;)17 b(a)2878 2185 y Fm(1)2917 2170 y Fo(;)g Fs(0\))39 b Fn(2)h Fo(C)3270 2134 y Fi(1)3263 2195 y Fm(0)3344 2170 y Fs(\()p Fl(R)3448 2134 y Fm(3)3494 2170 y Fs(\))p Fp(.)324 2291 y(Supp)-5 b(ose)34 b(furthermor)-5 b(e,)35 b(that)g Fo(\026h)27 b Fn(!)h Fo(\014)33 b Fn(2)28 b Fs(\(0)p Fo(;)17 b Fs(+)p Fn(1)p Fs(\))34 b Fp(as)g Fo(h)28 b Fn(!)f Fs(0)p Fp(.)45 b(Then)626 2570 y Fs(lim)620 2633 y Fj(h)p Fi(!)p Fm(0)793 2503 y Fo(h)849 2466 y Fm(2)p 793 2547 96 4 v 812 2638 a Fo(\026)899 2570 y Fs(tr[\()p Fo(p)1102 2583 y Fj(~)1089 2600 y(A)1168 2570 y Fn(\001)22 b Fo(M)10 b Fs(\()p Fo(x)p Fs(\))p Fo(p)1515 2583 y Fj(~)1502 2600 y(A)1582 2570 y Fs(+)22 b Fo(\026hb)1836 2585 y Fm(3)1876 2570 y Fs(\)1)1963 2585 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2194 2570 y Fs(\()p Fo(p)2281 2529 y Fm(2)2294 2592 y Fj(~)2281 2609 y(A)2360 2570 y Fs(+)g Fo(\026h)g Fs(+)g Fo(V)g Fs(\()p Fo(x)p Fs(\)\)])28 b(=)685 2729 y Fi(1)648 2759 y Fk(X)655 2969 y Fj(\027)t Fm(=0)818 2787 y Fs(2\()p Fo(\027)h Fs(+)22 b(1\))p Fo(\014)p 818 2831 409 4 v 969 2922 a Fs(2)p Fo(\031)1254 2718 y Fk(Z)1353 2854 y Fs(\()p Fo(@)1442 2869 y Fj(x)1482 2878 y Fg(1)1521 2854 y Fo(a)1572 2869 y Fm(2)1634 2854 y Fn(\000)h Fo(@)1785 2869 y Fj(x)1825 2878 y Fg(2)1864 2854 y Fo(a)1915 2869 y Fm(1)1954 2854 y Fs(\)[2\()p Fo(\027)29 b Fs(+)22 b(1\))p Fo(\026h)f Fs(+)h Fo(V)g Fs(\()p Fo(x)p Fs(\)])2839 2803 y Fm(1)p Fj(=)p Fm(2)2839 2876 y Fi(\000)2966 2854 y Fo(dx:)470 3156 y Fs(If)33 b(w)m(e)g(put)g(the)g(three)g(lemmas)e (together)i(w)m(e)h(obtain)d(Theorem)65 b(1.1.)470 3276 y(Finally)-8 b(,)30 b(w)m(e)k(pro)m(v)m(e)f(Theorem)66 b(1.2:)324 3471 y Fp(Pr)-5 b(o)g(of.)48 b Fs(Using)29 b(Theorem)60 b(1.1)29 b(w)m(e)h(ma)m(y)g(assume)23 b Fo(~)-42 b(a)28 b Fs(=)f(\(0)p Fo(;)17 b Fs(0)p Fo(;)g(a)2633 3486 y Fm(3)2672 3471 y Fs(\),)30 b(with)f Fo(a)3037 3486 y Fm(3)3104 3471 y Fn(2)f Fo(C)3275 3435 y Fi(1)3268 3496 y Fm(0)3350 3471 y Fs(.)42 b(Let)324 3591 y Fo(U)h Fs(b)s(e)33 b(the)g(unitary)f(op)s(erator)g(on)g Fo(L)1672 3555 y Fm(2)1712 3591 y Fs(\()p Fl(R)1816 3555 y Fm(3)1861 3591 y Fo(;)17 b Fl(C)1971 3555 y Fm(2)2016 3591 y Fs(\))33 b(de\014ned)h(b)m(y)1280 3811 y Fo(U)10 b(f)h Fs(\()p Fo(x)1508 3826 y Fm(1)1548 3811 y Fo(;)17 b(x)1647 3826 y Fm(2)1687 3811 y Fo(;)g(x)1786 3826 y Fm(3)1825 3811 y Fs(\))28 b(=)f Fo(f)11 b Fs(\()p Fo(x)2146 3826 y Fm(1)2186 3811 y Fo(;)17 b(x)2285 3826 y Fm(2)2324 3811 y Fo(;)g Fn(\000)p Fo(x)2500 3826 y Fm(3)2541 3811 y Fs(\))p Fo(;)324 4032 y Fs(and)31 b(write)g Fo(a)811 4047 y Fm(3)882 4032 y Fs(as)h Fo(a)1052 4047 y Fm(3)1119 4032 y Fs(=)c Fo(a)1274 4047 y Fm(3)p Fj(;ev)r(en)1498 4032 y Fs(+)19 b Fo(a)1644 4047 y Fm(3)p Fj(;odd)1841 4032 y Fs(where)33 b Fo(a)2173 4047 y Fm(3)p Fj(;ev)r(en)2409 4032 y Fs(\()p Fo(a)2498 4047 y Fm(3)p Fj(;odd)2664 4032 y Fs(\))e(is)g(ev)m(en)i(\(o)s(dd\))e (under)324 4152 y(the)k(re\015ection)f Fo(x)973 4167 y Fm(3)1044 4152 y Fn(7!)c(\000)p Fo(x)1306 4167 y Fm(3)1347 4152 y Fs(.)49 b(No)m(w,)35 b(since)g Fo(V)56 b Fs(is)34 b(in)m(v)-5 b(arian)m(t)33 b(under)i(conjugation)f(b)m(y)h Fo(U)324 4272 y Fs(w)m(e)e(get:)324 4492 y(tr[)p Ft(J)p Fs(\()p Fo(\026)-6 b(~)-43 b(a)o Fs(\)1)719 4508 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])951 4492 y Fs(\()p Ft(P)p Fs(\)])27 b(=)g(tr[)p Fo(U)10 b Ft(J)p Fs(\()p Fo(\026)-6 b(~)-43 b(a)q Fs(\))p Fo(U)10 b Fs(1)1810 4508 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])2042 4492 y Fs(\()p Ft(P)p Fs(\)])27 b(=)g(tr[)p Ft(J)p Fs(\()p Fo(\026)p Fs(\(0)p Fo(;)17 b Fs(0)p Fo(;)g(a)2885 4507 y Fm(3)p Fj(;odd)3050 4492 y Fs(\)\)1)3175 4508 y Fm(\()p Fi(\0001)p Fj(;)p Fm(0])3406 4492 y Fs(\()p Ft(P)p Fs(\)])p Fo(:)324 4712 y Fs(W)-8 b(e)22 b(can)g(easily)f(\014nd) h(a)g(function)g(~)-50 b Fo(a)28 b Fs(=)f(\()q(~)-50 b Fo(a)1804 4727 y Fm(1)1844 4712 y Fo(;)18 b Fs(~)-50 b Fo(a)1939 4727 y Fm(2)1978 4712 y Fo(;)17 b Fs(0\))27 b Fn(2)h Fo(C)2307 4676 y Fi(1)2300 4737 y Fm(0)2404 4712 y Fs(suc)m(h)23 b(that)e(rot)q(~)-50 b Fo(a)28 b Fs(=)f(rot\(0)p Fo(;)17 b Fs(0)p Fo(;)g(a)3520 4727 y Fm(3)p Fj(;odd)3685 4712 y Fs(\).)324 4833 y(W)-8 b(e)31 b(th)m(us)g(\014nish)f(the)h(pro)s(of)f(b)m(y)h(app)s(ealing)d(to)i (Theorem)61 b(1.1)30 b(and)h(the)g(gauge)f(in)m(v)-5 b(ari-)324 4953 y(ance)33 b(of)f(the)h(curren)m(t.)p 3491 4953 4 66 v 3495 4891 59 4 v 3495 4953 V 3553 4953 4 66 v 1894 5251 a(19)p eop %%Page: 20 20 20 19 bop 324 548 a Fu(A)161 b(A)54 b(densit)l(y)e(matrix)j(with)f(a)g (strong)e(curren)l(t)324 767 y Fs(W)-8 b(e)32 b(w)m(an)m(t)g(to)f(pro)m (v)m(e)i(that)e(it)g(is)g(necessary)j(to)d(use)h(something)f(lik)m(e)g (our)g(comm)m(utator)324 887 y(argumen)m(t)41 b(in)g(order)h(to)f (calculate)g(the)h(curren)m(t.)72 b(Therefore)43 b(w)m(e)g(will)c(pro)s (duce)k(an)324 1008 y(example)33 b(of)g(a)g(densit)m(y)h(matrix)e Fo(\015)39 b Fs(that)33 b(giv)m(es)h(the)g(righ)m(t)e(energy)j(to)e (highest)h(order)f(-)324 1128 y(but)g(giv)m(es)g(a)f(curren)m(t)i(of)e (to)s(o)f(high)h(order.)470 1249 y(W)-8 b(e)42 b(will)d(w)m(ork)k(with) e Fo(\026h)h Fs(=)h(1)e(i.e)g Fo(\026)i Fs(=)g Fo(h)2101 1212 y Fi(\000)p Fm(1)2237 1249 y Fs(and)e(will)e(only)i(lo)s(ok)g(at)g (one)h(spin)324 1369 y(comp)s(onen)m(t)32 b(i.e.)1227 1489 y Fo(H)k Fs(=)27 b(\()p Fn(\000)p Fo(ih)p Fn(r)c Fs(+)f Fo(\026)1932 1464 y(~)1914 1489 y(A)p Fs(\))2025 1448 y Fm(2)2086 1489 y Fn(\000)h Fo(\026h)f Fs(+)g Fo(V)g Fs(\()p Fo(x)p Fs(\))p Fo(:)324 1693 y Ft(Lemma)42 b(A.1.)i Fp(Ther)-5 b(e)38 b(exists)g(a)h(p)-5 b(otential)38 b Fo(V)22 b Fs(\()p Fo(x)p Fs(\))35 b Fn(2)h Fo(C)2442 1657 y Fi(1)2435 1717 y Fm(0)2517 1693 y Fs(\()p Fl(R)2620 1657 y Fm(3)2666 1693 y Fs(\))j Fp(and)f(a)h(test)g(function)329 1797 y Fo(~)324 1823 y(\036)c Fs(=)g(\()p Fo(\036)624 1838 y Fm(1)663 1823 y Fo(;)17 b(\036)765 1838 y Fm(2)804 1823 y Fo(;)g Fs(0\))35 b Fn(2)g Fo(C)1148 1787 y Fi(1)1141 1848 y Fm(0)1223 1823 y Fs(\()p Fl(R)1327 1787 y Fm(3)1372 1823 y Fs(\))k Fp(to)-5 b(gether)39 b(with)g(a)g(density)g(matrix)g (i.e.)56 b(an)39 b(op)-5 b(er)g(ator)39 b Fo(\015)324 1944 y Fp(satisfying)34 b Fs(0)28 b Fn(\024)g Fo(\015)k Fn(\024)d Fs(1)34 b Fp(such)h(that)1466 2181 y Fs(tr[)p Fo(H)8 b(\015)d Fs(])28 b(=)f Fo(E)1944 2196 y Fj(scl)2056 2181 y Fs(+)22 b Fo(o)p Fs(\()2267 2114 y Fo(\026)p 2249 2158 96 4 v 2249 2250 a(h)2305 2221 y Fm(2)2354 2181 y Fs(\))p Fo(;)324 2427 y Fp(and)1506 2514 y Fo(h)1562 2478 y Fm(2)p 1506 2558 V 1524 2650 a Fo(\026)1611 2581 y Fn(j)p Fs(tr[)p Fo(J)9 b Fs(\()p Fo(\026\036)p Fs(\))p Fo(\015)c Fs(])p Fn(j)27 b(!)g(1)p Fo(;)324 2800 y Fp(as)34 b Fo(h)28 b Fn(!)f Fs(0)p Fp(.)470 3004 y Fs(Th)m(us)41 b(the)e(lemma)d(sa)m(ys)41 b(that)d(a)h(densit)m(y)h(matrix)d(that)i (giv)m(es)g(the)g(righ)m(t)g(energy)324 3124 y(do)s(es)g Fp(not)f Fs(necessarily)h(giv)m(e)f(the)h(righ)m(t)f(curren)m(t.)62 b(This)38 b(is)g(unlik)m(e)g(the)h(situation)e(for)324 3245 y(the)27 b(densit)m(y)-8 b(,)29 b(since)e(it)f(is)g(easy)i(to)f (pro)m(v)m(e)h(that)f(a)f(densit)m(y)i(matrix)d(that)i(giv)m(es)g(the)g (righ)m(t)324 3365 y(energy)33 b(also)f(giv)m(es)h(the)g(righ)m(t)f (densit)m(y)-8 b(.)470 3485 y(The)40 b(trial)c(densit)m(y)k(matrix)e Fo(\015)43 b Fs(will)37 b(b)s(e)i(constructed)h(as)f(a)g(p)s (erturbation)f(of)g(the)324 3606 y(densit)m(y)26 b(matrix)e(used)i(in) 50 b([LSY94].)41 b(The)26 b(k)m(ey)h(to)e(the)g(construction)h(is)f (the)g(follo)m(wing:)324 3726 y(The)31 b(curren)m(t)h(op)s(erator)e(-)h (as)g(opp)s(osed)g(to)f(the)h(energy)h(op)s(erator)e(\(the)h (Hamiltonian\))324 3847 y(-)41 b(mixes)g(the)h(Landau)f(lev)m(els.)70 b(In)42 b(fact,)i(the)e(main)e(part)h(of)g(the)h(curren)m(t)g(op)s (erator)324 3967 y(do)s(es)g(not)g(resp)s(ect)i(the)e(Landau)g(lev)m (els)g(-)g(the)h(part)e(that)h(do)s(es)h(is)e(m)m(uc)m(h)i(smaller)3497 3931 y Fm(2)3535 3967 y Fs(.)324 4087 y(Th)m(us,)33 b(a)e(densit)m(y)i (matrix)d(that)h(giv)m(es)h(the)g(righ)m(t)f(energy)h(but)g(con)m (tains)f(a)h(small)d(part)324 4208 y(whic)m(h)35 b(mixes)f(neigh)m(b)s (oring)f(Landau)i(lev)m(els)g(should)f(ha)m(v)m(e)i(to)s(o)e(large)f(a) h(curren)m(t.)51 b(As)324 4328 y(the)33 b(pro)s(of)f(b)s(elo)m(w)g(sho) m(ws)i(this)e(turns)i(out)e(to)g(b)s(e)h(the)g(case.)324 4523 y Fp(Pr)-5 b(o)g(of.)48 b Fs(Let)37 b(us)g(c)m(ho)s(ose)g Fo(V)56 b Fn(2)34 b Fo(C)1540 4487 y Fi(1)1533 4548 y Fm(0)1615 4523 y Fs(\()p Fl(R)1719 4487 y Fm(3)1764 4523 y Fs(\),)j(whic)m(h)g(satis\014es)g([)p Fo(V)22 b Fs(\()p Fo(x)p Fs(\)])2776 4538 y Fi(\000)2870 4523 y Fs(=)34 b(10)i(for)f(all)g Fo(x)f Fn(2)324 4654 y Fo(B)5 b Fs(\(0)p Fo(;)17 b Fs(2\).)56 b(W)-8 b(e)37 b(will)e(c)m(ho)s(ose)j(a)e(test)i (v)m(ector)1961 4627 y Fo(~)1956 4654 y(\036)d Fs(=)g(\()p Fo(\036)2256 4669 y Fm(1)2295 4654 y Fo(;)17 b(\036)2397 4669 y Fm(2)2436 4654 y Fo(;)g Fs(0\),)37 b(whic)m(h)h(is)e(supp)s (orted)i(in)324 4774 y Fo(B)5 b Fs(\(0)p Fo(;)17 b Fs(1\).)p 324 4842 1296 4 v 436 4903 a Ff(2)473 4933 y Fq(This)28 b(can)f(b)r(e)h(seen)f(from)h(the)g(comm)n(utator)e(form)n(ula)1894 5251 y Fs(20)p eop %%Page: 21 21 21 20 bop 470 548 a Fs(The)34 b(densit)m(y)f(matrix)e Fo(\015)1380 512 y Fi(0)1436 548 y Fs(constructed)j(in)65 b([LSY94])32 b(is)1063 842 y Fo(\015)1119 800 y Fi(0)1170 842 y Fs(=)1310 717 y Fi(1)1274 747 y Fk(X)1281 956 y Fj(\027)t Fm(=1)1474 774 y Fs(1)p 1444 819 108 4 v 1444 910 a(2)p Fo(\031)1578 706 y Fk(Z)12 b(Z)1761 842 y Fo(M)e Fs(\()p Fo(\027)q(;)17 b(u;)g(p)p Fs(\)\005\()p Fo(\027)q(;)g(u;)g(p)p Fs(\))g Fo(dudp;)324 1146 y Fs(where)42 b Fo(M)10 b Fs(\()p Fo(\027)q(;)17 b(u;)g(p)p Fs(\))41 b(is)g(the)h(c)m(haracteristic)f (function)g(of)g(the)h(set)g(\(in)e(\()p Fl(N)3082 1161 y Fm(+)3175 1146 y Fn([)29 b(f)p Fs(0)p Fn(g)p Fs(\))e Fn(\002)324 1267 y Fl(R)390 1231 y Fm(3)390 1291 y Fj(u)463 1267 y Fn(\002)c Fl(R)628 1282 y Fj(p)674 1267 y Fs(\))1190 1387 y Fn(f)p Fs(\()p Fo(\027)q(;)17 b(u;)g(p)p Fs(\))p Fn(j)p Fs(2)p Fo(\027)6 b(\026h)21 b Fs(+)h Fo(h)1979 1346 y Fm(2)2018 1387 y Fo(p)2067 1346 y Fm(2)2129 1387 y Fs(+)g Fo(V)f Fs(\()p Fo(x)p Fs(\))28 b Fn(\024)h Fs(0)p Fn(g)p Fo(;)324 1560 y Fs(and)j(where)i(\005\()p Fo(\027)q(;)17 b(u;)g(p)p Fs(\))32 b(is)g(an)g(op)s(erator)g(with)g(k)m(ernel)729 1778 y(\005\()p Fo(\027)q(;)17 b(u;)g(p)p Fs(\)\()p Fo(x;)g(y)t Fs(\))26 b(=)h Fo(g)1523 1793 y Fj(r)1561 1778 y Fs(\()p Fo(x)c Fn(\000)f Fo(u)p Fs(\)\005)1943 1737 y Fm(\(2\))1943 1803 y Fj(\027)2037 1778 y Fs(\()p Fo(x)2130 1793 y Fi(?)2190 1778 y Fo(;)17 b(y)2282 1793 y Fi(?)2340 1778 y Fs(\))p Fo(e)2423 1737 y Fj(ip)p Fm(\()p Fj(x)2550 1746 y Fg(3)2584 1737 y Fi(\000)p Fj(y)2674 1746 y Fg(3)2709 1737 y Fm(\))2740 1778 y Fo(g)2787 1793 y Fj(r)2825 1778 y Fs(\()p Fo(y)25 b Fn(\000)e Fo(u)p Fs(\))p Fo(:)324 1996 y Fs(In)35 b(this)f(last)g (expression)h Fo(g)1345 2011 y Fj(r)1418 1996 y Fs(is)f(a)g(lo)s (calisation)d(function)j Fo(g)2547 2011 y Fj(r)2584 1996 y Fs(\()p Fo(x)p Fs(\))e(=)f Fo(r)2901 1960 y Fi(\000)p Fm(3)p Fj(=)p Fm(2)3065 1996 y Fo(g)t Fs(\()p Fo(x=r)s Fs(\),)j(0)d Fn(\024)324 2116 y Fo(g)46 b Fn(2)e Fo(C)604 2080 y Fi(1)597 2141 y Fm(0)679 2116 y Fs(\()p Fl(R)782 2080 y Fm(3)828 2116 y Fs(\),)937 2036 y Fk(R)1020 2116 y Fo(g)1071 2080 y Fm(2)1153 2116 y Fs(=)f(1)e(and)h Fo(r)k Fs(=)d Fo(h)1826 2080 y Fm(1)p Fi(\000)p Fj(\013)2008 2116 y Fs(for)e(some)g(0)i Fo(<)g(\013)h(<)f Fs(1.)71 b(F)-8 b(urthermore,)324 2256 y(\005)397 2205 y Fm(\(2\))397 2266 y Fj(\027)491 2256 y Fs(\()p Fo(x)584 2271 y Fi(?)644 2256 y Fo(;)17 b(y)736 2271 y Fi(?)794 2256 y Fs(\))32 b(is)f(the)i(\(t)m(w)m(o-dimensional\))d(in)m(tegral)g(k)m(ernel)j(of)e (the)i(pro)5 b(jection)31 b(to)h(the)324 2376 y Fo(\027)6 b Fs(-th)32 b(Landau)h(lev)m(el:)364 2645 y(\005)437 2604 y Fm(\(2\))437 2670 y Fj(\027)531 2645 y Fs(\()p Fo(x)624 2660 y Fi(?)684 2645 y Fo(;)17 b(y)776 2660 y Fi(?)834 2645 y Fs(\))28 b(=)1066 2578 y Fo(\026)p 1013 2622 164 4 v 1013 2713 a Fs(2)p Fo(\031)t(h)1204 2645 y Fs(exp)q Fn(f)p Fo(i)p Fs(\()p Fo(x)1529 2660 y Fi(?)1610 2645 y Fn(\002)23 b Fo(y)1758 2660 y Fi(?)1817 2645 y Fs(\))1865 2578 y Fo(\026)1940 2552 y(~)1924 2578 y(B)p 1865 2622 138 4 v 1881 2713 a Fs(2)p Fo(h)2034 2645 y Fn(\000)g(j)p Fo(x)2217 2660 y Fi(?)2298 2645 y Fn(\000)g Fo(y)2446 2660 y Fi(?)2505 2645 y Fn(j)2533 2604 y Fm(2)2605 2578 y Fo(\026)p 2582 2622 105 4 v 2582 2713 a Fs(4)p Fo(h)2697 2645 y Fn(g)p Fo(L)2813 2660 y Fj(\027)2856 2645 y Fs(\()p Fn(j)p Fo(x)2977 2660 y Fi(?)3058 2645 y Fn(\000)g Fo(y)3206 2660 y Fi(?)3264 2645 y Fn(j)3292 2604 y Fm(2)3365 2578 y Fo(\026)p 3341 2622 V 3341 2713 a Fs(2)p Fo(h)3456 2645 y Fs(\))p Fo(;)324 2915 y Fs(where)39 b(w)m(e)f(ha)m(v)m(e)h(written)e Fo(x)g Fn(2)f Fl(R)1595 2879 y Fm(3)1678 2915 y Fs(as)i(\()p Fo(x)1896 2930 y Fi(?)1955 2915 y Fo(;)17 b(x)2054 2931 y Fi(k)2094 2915 y Fs(\),)39 b(with)e Fo(x)2480 2930 y Fi(?)2575 2915 y Fn(?)2705 2890 y Fo(~)2689 2915 y(B)43 b Fs(and)37 b Fo(x)3055 2931 y Fi(k)3131 2915 y Fn(k)3233 2890 y Fo(~)3217 2915 y(B)5 b Fs(.)58 b(F)-8 b(ur-)324 3036 y(thermore,)32 b Fo(L)834 3051 y Fj(\027)910 3036 y Fs(are)h(Laguerre)g(p)s(olynomials)c(normalised)i(b)m(y)i Fo(L)2718 3051 y Fj(\027)2762 3036 y Fs(\(0\))27 b(=)h(1.)470 3156 y(Let)k(no)m(w)882 3131 y(~)846 3156 y Fo(M)42 b Fs(b)s(e)32 b(the)g(c)m(haracteristic)g(function)f(of)g Fo(B)5 b Fs(\(0)p Fo(;)17 b Fs(1\))2668 3171 y Fj(u)2732 3156 y Fn(\002)k Fs([)p Fn(\000)p Fo(h)2990 3120 y Fi(\000)p Fm(1)3085 3156 y Fo(;)c(h)3185 3120 y Fi(\000)p Fm(1)3279 3156 y Fs(])3306 3171 y Fj(p)3346 3156 y Fs(,)31 b(and)324 3276 y(write)1296 3420 y(~)-53 b Fo(\015)33 b Fs(=)27 b Fo(\017)1535 3285 y Fk(Z)12 b(Z)1754 3395 y Fs(~)1718 3420 y Fo(M)f Fs(\()p Fo(u;)17 b(p)p Fs(\))2060 3395 y(~)2048 3420 y(\005)o(\()p Fo(u;)g(p)p Fs(\))g Fo(dudp;)324 3636 y Fs(where)34 b Fo(\017)28 b Fn(!)f Fs(0)32 b(as)h Fo(h)28 b Fn(!)f Fs(0)32 b(and)h(where)833 3829 y(~)821 3854 y(\005\()p Fo(u;)17 b(p)p Fs(\)\()p Fo(x;)g(y)t Fs(\))26 b(=)i Fo(g)1523 3869 y Fj(r)1560 3854 y Fs(\()p Fo(x)23 b Fn(\000)f Fo(u)p Fs(\))p Fo(P)14 b Fs(\()p Fo(x)2039 3869 y Fi(?)2098 3854 y Fo(;)j(y)2190 3869 y Fi(?)2248 3854 y Fs(\))p Fo(e)2331 3813 y Fj(ip)p Fm(\()p Fj(x)2458 3822 y Fg(3)2492 3813 y Fi(\000)p Fj(y)2582 3822 y Fg(3)2617 3813 y Fm(\))2648 3854 y Fo(g)2695 3869 y Fj(r)2733 3854 y Fs(\()p Fo(y)25 b Fn(\000)e Fo(u)p Fs(\))p Fo(:)324 4072 y Fs(In)33 b(this)f(\014nal)g(expression)i Fo(P)45 b Fs(is)33 b(the)g(op)s(erator)1359 4289 y Fo(P)42 b Fs(=)27 b(\005)1640 4238 y Fm(\(2\))1640 4314 y(1)1735 4289 y Fo(a)1786 4248 y Fi(\003)1825 4289 y Fs(\005)1898 4238 y Fm(\(2\))1898 4314 y(0)2015 4289 y Fs(+)22 b(\005)2186 4238 y Fm(\(2\))2186 4314 y(0)2280 4289 y Fo(a)p Fs(\005)2404 4238 y Fm(\(2\))2404 4314 y(1)2499 4289 y Fo(;)324 4507 y Fs(with)h Fo(a)28 b Fs(=)f Fo(p)781 4520 y Fj(~)768 4537 y(A;)p Fm(1)884 4507 y Fn(\000)t Fo(ip)1060 4520 y Fj(~)1047 4537 y(A)q(;)p Fm(2)1160 4507 y Fo(;)17 b(a)1255 4471 y Fi(\003)1322 4507 y Fs(=)27 b Fo(p)1487 4520 y Fj(~)1474 4537 y(A;)p Fm(1)1590 4507 y Fs(+)t Fo(ip)1765 4520 y Fj(~)1752 4537 y(A;)p Fm(2)1887 4507 y Fs(b)s(eing)c(the)h (raising)e(and)i(lo)m(w)m(ering)f(op)s(erators)324 4627 y(that)32 b(de\014ne)i(the)f(Landau)g(lev)m(els.)470 4748 y(W)-8 b(e)39 b(\014nally)e(de\014ne)i Fo(\015)j Fs(=)c Fo(\015)1495 4712 y Fi(0)1544 4748 y Fs(+)29 b(~)-52 b Fo(\015)5 b Fs(.)60 b(Since)38 b(the)h(op)s(erator)e Fo(P)52 b Fs(satis\014es)39 b(\(remem)m(b)s(er)324 4868 y Fo(\026h)27 b Fs(=)h(1\))1146 5002 y Fn(\000)p Fo(c)p Fs(\(\005)1376 4951 y Fm(\(2\))1376 5027 y(0)1493 5002 y Fs(+)22 b(\005)1664 4951 y Fm(\(2\))1664 5027 y(1)1759 5002 y Fs(\))28 b Fn(\024)g Fo(P)41 b Fn(\024)28 b Fo(c)p Fs(\(\005)2292 4951 y Fm(\(2\))2292 5027 y(0)2408 5002 y Fs(+)22 b(\005)2579 4951 y Fm(\(2\))2579 5027 y(1)2674 5002 y Fs(\))p Fo(;)1894 5251 y Fs(21)p eop %%Page: 22 22 22 21 bop 324 548 a Fs(it)39 b(is)g(easy)i(to)e(see)i(that)e(0)h Fn(\024)g Fo(\015)45 b Fs(for)39 b(su\016cien)m(tly)i(small)c Fo(\017)p Fs(.)66 b(In)40 b(order)f(to)h(get)g Fo(\015)45 b Fn(\024)40 b Fs(1)324 668 y(w)m(e)c(should)e(m)m(ultiply)f(b)m(y)i(a) g(factor)1731 629 y Fm(1)p 1686 645 124 4 v 1686 703 a(1+)p Fj(\016)1820 668 y Fs(,)h(where)g Fo(\016)f Fn(!)c Fs(0)k(as)g Fo(h)c Fn(!)g Fs(0.)50 b(W)-8 b(e)36 b(will)c(not)j(do)324 789 y(this,)j(since)h(it)d(will)f(not)j(a\013ect)g(order)g(of)f (magnitude)f(estimates)h(and)h(only)f(obscure)324 909 y(notation.)470 1029 y(W)-8 b(e)33 b(need)h(to)e(calculate)1448 1249 y(tr[)p Fo(H)8 b(\015)d Fs(])27 b(=)h Fo(E)1926 1264 y Fj(scl)2038 1249 y Fs(+)22 b(tr)o([)p Fo(H)11 b Fs(~)-52 b Fo(\015)5 b Fs(])p Fo(;)324 1469 y Fs(and)1647 1590 y(tr[)p Fo(h)1806 1549 y Fi(\000)p Fm(1)1901 1590 y Fo(J)1955 1605 y Fj(p)1994 1590 y Fs(\()2037 1564 y Fo(~)2032 1590 y(\036)p Fs(\))p Fo(\015)g Fs(])p Fo(:)324 1764 y Fs(Notice,)33 b(that)f(since)i Fo(\015)k Fs(giv)m(es)33 b(the)h(righ)m(t)e Fp(density)h Fs(to)f(highest)h(order,)h(w)m(e)g(do)f (not)g(need)324 1885 y(to)d(calculate)g(the)h(spin)g(curren)m(t)h(i.e.) 42 b(tr[)p Fo(\026hb)1980 1900 y Fm(3)2020 1885 y Fo(\015)5 b Fs(],)31 b(since)g(w)m(e)h(kno)m(w)g(this)f(to)f(b)s(e)h(of)f(order) 350 1961 y Fj(\026)p 334 1982 76 4 v 334 2039 a(h)375 2020 y Fg(2)452 2005 y Fs(once)j(w)m(e)h(ha)m(v)m(e)g(pro)m(v)m(ed)g (that)f Fo(\015)38 b Fs(giv)m(es)33 b(the)g(righ)m(t)f(energy)-8 b(.)46 b(F)-8 b(urthermore,)32 b(w)m(e)i(ma)m(y)324 2125 y(assume)c(that)g Fo(\015)923 2089 y Fi(0)976 2125 y Fs(do)s(es)g Fp(not)g Fs(satisfy)g(the)g(requiremen)m(ts)g(of)g(the)g (lemma)e(-)h(if)g(it)f(do)s(es)j(w)m(e)324 2246 y(do)h(not)h(ha)m(v)m (e)h(to)e(construct)i(an)m(ything.)470 2366 y Ft(The)k(energy:)p 470 2401 586 4 v 1073 2649 a Fs(tr[)p Fo(H)11 b Fs(~)-52 b Fo(\015)5 b Fs(])28 b(=)f Fo(\017)1535 2513 y Fk(Z)12 b(Z)1754 2624 y Fs(~)1718 2649 y Fo(M)f Fs(\()p Fo(u;)17 b(p)p Fs(\)tr)o([)p Fo(H)2250 2624 y Fs(~)2239 2649 y(\005)o(\()p Fo(u;)g(p)p Fs(\)])g Fo(dudp:)324 2912 y Fs(w)m(e)33 b(use)h(the)f(AMS-lo)s(calisation)c(form)m(ula:)1180 3132 y(2)p Fo(g)t(p)1329 3091 y Fm(2)1342 3154 y Fj(~)1329 3171 y(A)1385 3132 y Fo(g)d Fn(\000)c Fs(\()p Fo(p)1644 3091 y Fm(2)1658 3154 y Fj(~)1644 3171 y(A)1701 3132 y Fo(g)1752 3091 y Fm(2)1813 3132 y Fs(+)g Fo(g)1962 3091 y Fm(2)2001 3132 y Fo(p)2050 3091 y Fm(2)2063 3154 y Fj(~)2050 3171 y(A)2107 3132 y Fs(\))27 b(=)h([[)p Fo(g)t(;)17 b(p)2474 3091 y Fm(2)2486 3154 y Fj(~)2474 3171 y(A)2530 3132 y Fs(])p Fo(;)g(g)t Fs(])p Fo(:)324 3352 y Fs(Let)33 b(us)g(\014rst)g(lo)s(ok)e(at)h(the)h(p)s(oten)m(tial) e(energy:)1285 3572 y(tr[)p Fo(V)1479 3547 y Fs(~)1467 3572 y(\005\()p Fo(u;)17 b(p)p Fs(\)])27 b(=)g(tr[)p Fo(g)2072 3587 y Fj(r)2110 3572 y Fs(\()p Fn(\001)22 b(\000)g Fo(u)p Fs(\))p Fo(V)f(P)14 b Fs(])p Fo(:)324 3805 y Fs(This)39 b(is)f(small)e(\(i.e)i Fo(o)p Fs(\()p Fo(\026=h)p Fs(\)\))g(since)h(\005)1773 3755 y Fm(\(2\))1773 3830 y(1)1868 3805 y Fo(f)11 b Fs(\005)2000 3755 y Fm(\(2\))2000 3830 y(0)2132 3805 y Fs(is)38 b(small)f(for)h Fo(f)49 b Fn(2)38 b Fo(C)2931 3769 y Fi(1)2924 3830 y Fm(0)3044 3805 y Fs(\(see)i(Lemma)324 3926 y(A.2)32 b(b)s(elo)m(w\).)44 b(F)-8 b(or)31 b(the)i(kinetic)f(energy)i(term)e(w)m(e)i(get:)566 4146 y(tr[\()p Fo(p)756 4105 y Fm(2)769 4168 y Fj(~)756 4185 y(A)835 4146 y Fn(\000)22 b Fo(\026h)p Fs(\))1099 4121 y(~)1087 4146 y(\005\()p Fo(u;)17 b(p)p Fs(\)])407 4350 y(=)576 4282 y(1)p 576 4327 49 4 v 576 4418 a(2)634 4350 y(tr)727 4269 y Fk(\000)773 4350 y Fs(\()p Fo(p)860 4309 y Fm(2)873 4372 y Fj(~)860 4389 y(A)916 4350 y Fo(g)967 4309 y Fm(2)963 4374 y Fj(r)1006 4350 y Fs(\()p Fn(\001)22 b(\000)h Fo(u)p Fs(\))e(+)h Fo(g)1458 4309 y Fm(2)1454 4374 y Fj(r)1497 4350 y Fs(\()p Fn(\001)g(\000)g Fo(u)p Fs(\))p Fo(p)1827 4309 y Fm(2)1840 4372 y Fj(~)1827 4389 y(A)1906 4350 y Fn(\000)g Fs(2)p Fo(\026h)p Fs(\))p Fo(P)36 b Fs(+)22 b(2[[)p Fo(g)2554 4365 y Fj(r)2591 4350 y Fs(\()p Fn(\001)g(\000)h Fo(u)p Fs(\))p Fo(;)17 b(p)2966 4309 y Fm(2)2978 4372 y Fj(~)2966 4389 y(A)3022 4350 y Fs(])p Fo(;)g(g)3140 4365 y Fj(r)3177 4350 y Fs(\()p Fn(\001)22 b(\000)h Fo(u)p Fs(\)])p Fo(P)3563 4269 y Fk(\001)407 4583 y Fs(=)576 4516 y(1)p 576 4560 V 576 4651 a(2)634 4583 y(tr)727 4502 y Fk(\000)773 4583 y Fs(4)p Fo(\026hg)988 4542 y Fm(2)984 4608 y Fj(r)1026 4583 y Fs(\()p Fn(\001)f(\000)g Fo(u)p Fs(\))p Fo(P)35 b Fs(+)22 b(2)p Fo(h)1608 4542 y Fm(2)1648 4583 y Fs(\()p Fn(r)p Fo(g)1816 4598 y Fj(r)1853 4583 y Fs(\()p Fn(\001)g(\000)h Fo(u)p Fs(\)\))2173 4542 y Fm(2)2211 4583 y Fo(P)2288 4502 y Fk(\001)2350 4583 y Fo(:)324 4828 y Fs(This)36 b(term)g(is)f(small)f(for)h(the)i(same)f (reason)g(as)g(ab)s(o)m(v)m(e.)55 b(Th)m(us)38 b(w)m(e)f(ma)m(y)f(c)m (ho)s(ose)h Fo(\017)f Fs(to)324 4949 y(go)c(to)g(zero)h(slo)m(wly)f (with)g Fo(h)h Fs(-)f(for)g(de\014niteness)i(let)e(us)i(tak)m(e)f Fo(\017)28 b Fs(=)g Fn(j)17 b Fs(log)e Fo(h)p Fn(j)3037 4913 y Fi(\000)p Fm(1)3132 4949 y Fs(.)1894 5251 y(22)p eop %%Page: 23 23 23 22 bop 470 548 a Ft(The)38 b(curren)m(t:)p 470 564 618 4 v 324 668 a Fs(In)33 b(order)f(to)h(calculate)e(the)i(curren)m(t) h(w)m(e)f(write)1186 879 y(tr[)p Fo(J)1343 894 y Fj(p)1382 879 y Fs(\()1425 853 y Fo(~)1420 879 y(\036)p Fs(\))t(~)-53 b Fo(\015)5 b Fs(])28 b(=)f(2)p Fn(<)p Fs(tr[)1959 853 y Fo(~)1954 879 y(\036)p Fs(\()p Fn(\000)p Fo(ih)p Fn(r)c Fs(+)f Fo(\026)2496 854 y(~)2479 879 y(A)p Fs(\))t(~)-53 b Fo(\015)t Fs(])p Fo(;)324 1090 y Fs(so)33 b(w)m(e)g(only)f(need)i(to) e(consider)844 1300 y(tr[)952 1274 y Fo(~)947 1300 y(\036)p Fs(\()p Fn(\000)p Fo(ih)p Fn(r)23 b Fs(+)f Fo(\026)1490 1275 y(~)1472 1300 y(A)p Fs(\))t(~)-53 b Fo(\015)5 b Fs(])685 1494 y(=)83 b Fo(\017)900 1358 y Fk(Z)12 b(Z)1119 1469 y Fs(~)1083 1494 y Fo(M)f Fs(\()p Fo(u;)17 b(p)p Fs(\)tr)o([)1520 1468 y Fo(~)1515 1494 y(\036)o Fs(\()p Fn(\000)p Fo(ih)p Fn(r)23 b Fs(+)f Fo(\026)2057 1469 y(~)2039 1494 y(A)p Fs(\))2162 1469 y(~)2150 1494 y(\005\()p Fo(u;)17 b(p)p Fs(\)])g Fo(dudp)685 1749 y Fs(=)83 b Fo(\017)900 1613 y Fk(Z)12 b(Z)1119 1723 y Fs(~)1083 1749 y Fo(M)f Fs(\()p Fo(u;)17 b(p)p Fs(\)tr)o([)1520 1722 y Fo(~)1515 1749 y(\036)o Fs(\()p Fn(\000)p Fo(ih)p Fn(r)p Fo(g)1906 1764 y Fj(r)1945 1749 y Fs(\()p Fn(\001)22 b(\000)g Fo(u)p Fs(\)\))p Fo(P)14 b(g)2388 1764 y Fj(r)2425 1749 y Fs(\()p Fn(\001)22 b(\000)g Fo(u)p Fs(\)])17 b Fo(dudp)844 2003 y Fs(+)p Fo(\017)976 1868 y Fk(Z)12 b(Z)1195 1978 y Fs(~)1159 2003 y Fo(M)f Fs(\()p Fo(u;)17 b(p)p Fs(\)tr)n([)1595 1977 y Fo(~)1590 2003 y(\036g)1695 2018 y Fj(r)1733 2003 y Fs(\()p Fn(\001)22 b(\000)g Fo(u)p Fs(\)\()p Fn(\000)p Fo(ih)p Fn(r)h Fs(+)f Fo(\026)2499 1978 y(~)2481 2003 y(A)p Fs(\))p Fo(P)14 b(g)2716 2018 y Fj(r)2753 2003 y Fs(\()p Fn(\001)21 b(\000)i Fo(u)p Fs(\)])17 b Fo(dudp)324 2288 y Fs(Since)34 b(\005)653 2237 y Fm(\(2\))653 2314 y Fj(j)747 2288 y Fo(f)11 b Fs(\005)879 2237 y Fm(\(2\))879 2316 y Fj(k)1007 2288 y Fs(is)33 b(small)f(when)i Fo(j)i Fn(6)p Fs(=)29 b Fo(k)s Fs(,)35 b Fo(f)40 b Fn(2)30 b Fo(C)2176 2252 y Fi(1)2169 2313 y Fm(0)2251 2288 y Fs(,)k(w)m(e)g(get)g(that)g(the)g(highest)f (order)324 2409 y(con)m(tribution)e(comes)i(from)e(a)i(part)f(of)g(the) h(second)h(term,)e(namely:)567 2732 y Fo(\017)623 2597 y Fk(Z)12 b(Z)842 2707 y Fs(~)806 2732 y Fo(M)f Fs(\()p Fo(u;)17 b(p)p Fs(\)tr)o([)p Fo(g)1285 2747 y Fj(r)1322 2732 y Fs(\()p Fn(\001)22 b(\000)h Fo(u)p Fs(\))1609 2706 y Fo(~)1604 2732 y(\036)1678 2532 y Fk(0)1678 2711 y(@)1862 2611 y Fs(\()p Fo(a)f Fs(+)g Fo(a)2122 2575 y Fi(\003)2162 2611 y Fs(\))p Fo(=)p Fs(2)1806 2731 y(\()p Fo(a)1895 2695 y Fi(\003)1957 2731 y Fn(\000)h Fo(a)p Fs(\))p Fo(=)p Fs(\(2)p Fo(i)p Fs(\))2055 2852 y(0)2394 2532 y Fk(1)2394 2711 y(A)2498 2732 y Fo(P)14 b(g)2622 2747 y Fj(r)2659 2732 y Fs(\()p Fn(\001)22 b(\000)g Fo(u)p Fs(\)])17 b Fo(dudp)407 3127 y Fn(\031)83 b Fo(\017)623 2991 y Fk(Z)12 b(Z)842 3101 y Fs(~)806 3127 y Fo(M)f Fs(\()p Fo(u;)17 b(p)p Fs(\)tr)o([)p Fo(g)1289 3085 y Fm(2)1285 3151 y Fj(r)1327 3127 y Fs(\()p Fn(\001)22 b(\000)h Fo(u)p Fs(\))1626 2922 y Fk(8)1626 3012 y(<)1626 3191 y(:)1718 3100 y Fo(~)1713 3127 y(\036)1788 2926 y Fk(0)1788 3106 y(@)1933 3005 y Fo(\026h)1916 3126 y(i\026h)1966 3246 y Fs(0)2106 2926 y Fk(1)2106 3106 y(A)2210 3127 y Fs(\005)2283 3076 y Fm(\(2\))2283 3151 y(0)2400 3127 y Fs(+)2502 3100 y Fo(~)2498 3127 y(\036)2572 2926 y Fk(0)2572 3106 y(@)2756 3005 y Fo(\026h)2701 3126 y Fn(\000)p Fo(i\026h)2789 3246 y Fs(0)2968 2926 y Fk(1)2968 3106 y(A)3071 3127 y Fs(\005)3144 3076 y Fm(\(2\))3144 3151 y(1)3239 2922 y Fk(9)3239 3012 y(=)3239 3191 y(;)3327 3127 y Fs(])17 b Fo(dudp:)324 3450 y Fs(If)37 b(w)m(e)h(remem)m(b)s(er) f(that)g Fo(\026h)e Fs(=)g(1)i(and)g(c)m(ho)s(ose)h Fo(\036)2163 3465 y Fm(2)2238 3450 y Fs(=)d(0)i(w)m(e)h(can)g(calculate)e(the)h (trace)324 3570 y(as:)651 3805 y Fo(\017)707 3669 y Fk(Z)11 b(Z)926 3779 y Fs(~)890 3805 y Fo(M)f Fs(\()p Fo(u;)17 b(p)p Fs(\))p Fo(g)1270 3763 y Fm(2)1266 3829 y Fj(r)1308 3805 y Fs(\()p Fo(x)23 b Fn(\000)f Fo(u)p Fs(\))p Fo(\036)1675 3820 y Fm(1)1714 3805 y Fs(\()p Fo(x)p Fs(\))1862 3694 y Fk(\020)1921 3805 y Fs(\005)1994 3754 y Fm(\(2\))1994 3829 y(0)2089 3805 y Fs(\()p Fo(x)2182 3820 y Fi(?)2241 3805 y Fo(;)17 b(x)2340 3820 y Fi(?)2399 3805 y Fs(\))22 b(+)g(\005)2630 3754 y Fm(\(2\))2630 3829 y(1)2725 3805 y Fs(\()p Fo(x)2818 3820 y Fi(?)2877 3805 y Fo(;)17 b(x)2976 3820 y Fi(?)3035 3805 y Fs(\))3073 3694 y Fk(\021)3166 3805 y Fo(dxdudp)491 4059 y Fs(=)669 3992 y Fo(\017\026)p 661 4036 116 4 v 661 4127 a(\031)t(h)803 3924 y Fk(Z)11 b(Z)g(Z)1088 4034 y Fs(~)1052 4059 y Fo(M)f Fs(\()p Fo(u;)17 b(p)p Fs(\))p Fo(g)1432 4018 y Fm(2)1428 4084 y Fj(r)1470 4059 y Fs(\()p Fo(x)23 b Fn(\000)f Fo(u)p Fs(\))p Fo(\036)1837 4074 y Fm(1)1876 4059 y Fs(\()p Fo(x)p Fs(\))17 b Fo(dxdudp)491 4343 y Fs(=)669 4276 y Fo(\017\026)p 661 4320 V 661 4411 a(\031)t(h)803 4207 y Fk(Z)902 4234 y Fj(h)943 4210 y Fa(\000)p Fg(1)858 4433 y Fi(\000)p Fj(h)954 4414 y Fa(\000)p Fg(1)1057 4343 y Fo(dp)1174 4207 y Fk(Z)1228 4433 y Fi(j)p Fj(u)p Fi(j\024)p Fm(1)1419 4207 y Fk(Z)1536 4343 y Fo(r)1583 4302 y Fi(\000)p Fm(3)1677 4343 y Fo(g)1728 4302 y Fm(2)1766 4343 y Fs(\(\()p Fo(x)23 b Fn(\000)f Fo(u)p Fs(\))p Fo(=r)s Fs(\))p Fo(\036)2305 4358 y Fm(1)2343 4343 y Fs(\()p Fo(x)p Fs(\))17 b Fo(dxdu)490 4618 y Fn(\031)689 4551 y Fo(\017\026)p 661 4595 155 4 v 661 4687 a(\031)t(h)776 4658 y Fm(2)842 4483 y Fk(Z)958 4618 y Fo(\036)1016 4633 y Fm(1)1055 4618 y Fs(\()p Fo(x)p Fs(\))g Fo(dx:)324 4882 y Fs(If)35 b(w)m(e)h(remem)m(b)s(er)f(that)h(this)f(term)f(has)i (to)f(b)s(e)h(m)m(ultiplied)c(b)m(y)k Fo(h)2766 4846 y Fi(\000)p Fm(1)2896 4882 y Fs(it)e(is)h(easy)h(to)f(see)324 5002 y(that)d(w)m(e)i(ha)m(v)m(e)g(reac)m(hed)g(our)e(aim.)p 3491 5002 4 66 v 3495 4940 59 4 v 3495 5002 V 3553 5002 4 66 v 1894 5251 a(23)p eop %%Page: 24 24 24 23 bop 324 548 a Ft(Lemma)37 b(A.2.)k Fp(L)-5 b(et)36 b Fo(\036)27 b Fn(2)h Fo(C)1390 512 y Fi(1)1383 573 y Fm(0)1465 548 y Fs(\()p Fl(R)1569 512 y Fm(3)1614 548 y Fs(\))p Fp(,)35 b(then)1214 758 y Fn(k)p Fs([\005)1364 707 y Fm(\(2\))1364 784 y Fj(j)1459 758 y Fo(;)17 b(\036)p Fs(])p Fn(k)1638 775 y Fi(B)r Fm(\()p Fj(L)1761 756 y Fg(2)1795 775 y Fm(\))1854 758 y Fn(\024)28 b Fo(C)2029 773 y Fj(j)2066 668 y Fk(p)p 2165 668 164 4 v 2165 758 a Fo(h=\026)p Fn(kr)p Fo(\036)p Fn(k)2570 773 y Fi(1)2644 758 y Fo(;)324 955 y Fp(wher)-5 b(e)31 b(the)h(norm)f(of)h(the)g(op)-5 b(er)g(ator)32 b(on)f(the)h(left)g(is)g(the)g(op)-5 b(er)g(ator)32 b(norm)f(as)h(a)f(b)-5 b(ounde)g(d)324 1075 y(op)g(er)g(ator)34 b(in)h Fo(L)893 1039 y Fm(2)933 1075 y Fp(.)324 1258 y(Pr)-5 b(o)g(of.)48 b Fs(W)-8 b(e)40 b(use)g(Sc)m(h)m(ur's)h(Lemma)c (i.e.)63 b(the)39 b(follo)m(wing)d(b)s(ound)k(on)f(the)g(norm)f(of)h (an)324 1379 y(in)m(tegral)31 b(k)m(ernel:)638 1618 y Fn(k)p Fo(K)7 b Fs(\()p Fo(x;)17 b(y)t Fs(\))p Fn(k)1055 1635 y Fi(B)r Fm(\()p Fj(L)1178 1616 y Fg(2)1211 1635 y Fm(\))1271 1618 y Fn(\024)28 b Fs(max)o(\(sup)1649 1697 y Fj(x)1759 1483 y Fk(Z)1875 1618 y Fn(j)p Fo(K)7 b Fs(\()p Fo(x;)17 b(y)t Fs(\))p Fn(j)g Fo(dy)t(;)g Fs(sup)2464 1697 y Fj(y)2572 1483 y Fk(Z)2688 1618 y Fn(j)p Fo(K)7 b Fs(\()p Fo(x;)17 b(y)t Fs(\))p Fn(j)g Fo(dx)p Fs(\))p Fo(:)324 1905 y Fs(The)33 b(in)m(tegral)e(k)m(ernel)j Fo(K)7 b Fs(\()p Fo(x;)17 b(y)t Fs(\))31 b(of)h([\005)1726 1855 y Fm(\(2\))1726 1931 y Fj(j)1821 1905 y Fo(;)17 b(\036)p Fs(])32 b(is)324 2181 y Fo(K)7 b Fs(\()p Fo(x;)17 b(y)t Fs(\))27 b(=)g(\005)844 2130 y Fm(\(2\))844 2206 y Fj(j)939 2181 y Fs(\()p Fo(x;)17 b(y)t Fs(\)\()p Fo(\036)p Fs(\()p Fo(y)t Fs(\))c Fn(\000)g Fo(\036)p Fs(\()p Fo(x)p Fs(\)\))25 b(=)j(\005)1922 2130 y Fm(\(2\))1922 2206 y Fj(j)2016 2181 y Fs(\()p Fo(x;)17 b(y)t Fs(\))2260 2045 y Fk(Z)2359 2071 y Fm(1)2315 2270 y(0)2398 2181 y Fs(\()p Fo(y)f Fn(\000)d Fo(x)p Fs(\))g Fn(\001)g(r)p Fo(\036)p Fs(\()p Fo(x)g Fs(+)g Fo(t)p Fs(\()p Fo(y)j Fn(\000)d Fo(x)p Fs(\)\))k Fo(dt:)324 2423 y Fs(So)32 b(w)m(e)i(estimate:)591 2508 y Fk(Z)708 2643 y Fn(j)p Fo(K)7 b Fs(\()p Fo(x;)17 b(y)t Fs(\))p Fn(j)g Fo(dy)84 b Fn(\024)f(kr)p Fo(\036)p Fn(k)1682 2658 y Fi(1)1773 2508 y Fk(Z)1889 2643 y Fn(j)p Fs(\005)1990 2592 y Fm(\(2\))1990 2669 y Fj(j)2084 2643 y Fs(\()p Fo(x;)17 b(y)t Fs(\))p Fn(jj)p Fo(x)k Fn(\000)i Fo(y)t Fn(j)17 b Fo(dy)1282 2898 y Fs(=)83 b Fn(kr)p Fo(\036)p Fn(k)1682 2913 y Fi(1)1756 2808 y Fk(p)p 1856 2808 V 90 x Fo(h=\026)2036 2762 y Fk(Z)2153 2898 y Fn(j)p Fs(\005)2254 2847 y Fm(\(2\))2254 2923 y Fj(j)2348 2898 y Fs(\()p Fo(x;)17 b(y)t Fs(\))p Fn(j)2603 2808 y Fk(p)p 2701 2808 V 2701 2898 a Fo(\026=h)p Fn(j)p Fo(x)22 b Fn(\000)h Fo(y)t Fn(j)17 b Fo(dy)t(:)324 3169 y Fs(No)m(w)37 b(w)m(e)h(use)g(the)f(fact)g(that)g(\005)1529 3118 y Fm(\(2\))1529 3194 y Fj(j)1623 3169 y Fs(\()p Fo(x;)17 b(y)t Fs(\))35 b(=)g Fo(f)11 b Fs(\()2093 3084 y Fk(p)p 2192 3084 V 2192 3169 a Fo(\026=h)o Fs(\()p Fo(x)26 b Fn(\000)g Fo(y)t Fs(\)\),)37 b(where)h Fo(f)47 b Fs(has)38 b(exp)s(o-)324 3289 y(nen)m(tial)27 b(deca)m(y)-8 b(,)30 b(to)e(b)s(ound)h(the)g(last)e(in)m(tegral)g(uniformly)f(in)h Fo(x)p Fs(.)43 b(It)28 b(is)g(easy)h(to)f(see)i(that)324 3410 y(the)j(ab)s(o)m(v)m(e)g(estimate)f(w)m(orks)i(equally)e(w)m(ell)g (for)g(sup)2266 3433 y Fj(y)2324 3329 y Fk(R)2407 3410 y Fn(j)p Fo(K)7 b Fs(\()p Fo(x;)17 b(y)t Fs(\))p Fn(j)g Fo(dx)p Fs(.)p 3491 3410 4 66 v 3495 3347 59 4 v 3495 3410 V 3553 3410 4 66 v 1362 3611 a(A)m(CKNO)m(WLEDGEMENTS)502 3731 y(The)33 b(author)g(wishes)g(to)f(thank)h(the)g(Sc)m(hr\177)-49 b(odinger)33 b(Institute)g(in)f(Vienna)g(for)508 3852 y(hospitalit)m(y)f(during)h(the)h(fall)e(term)h(1999,)f(esp)s(ecially)h (Thomas)g(and)h(Maria)398 3972 y(Ho\013mann-Ostenhof.)43 b(F)-8 b(urthermore,)32 b(the)h(author)f(ac)m(kno)m(wledges)j(man)m(y)d (useful)446 4093 y(discussions)i(with)e(Thomas)g(\037stergaard)h (S\034rensen)h(and)f(Jan)f(Philip)f(Solo)m(v)m(ej.)1565 4422 y Fu(References)324 4641 y Fs([ES97])117 b(L.)39 b(Erd\177)-49 b(os)41 b(and)f(J.P)-8 b(.)40 b(Solo)m(v)m(ej,)i Fp(Semiclassic)-5 b(al)39 b(Eigenvalue)i(Estimates)g(for)713 4761 y(the)i(Pauli)h(Op)-5 b(er)g(ator)43 b(with)h(Str)-5 b(ong)43 b(non-homo)-5 b(gene)g(ous)42 b(magnetic)h(\014elds.)713 4882 y(II.)30 b(Le)-5 b(ading)30 b(or)-5 b(der)31 b(asymptotic)h (estimates)p Fs(,)d(Comm)m(un.)f(Math.)h(Ph)m(ys.)i Ft(188)713 5002 y Fs(\(1997\),)g(599{656.)1894 5251 y(24)p eop %%Page: 25 25 25 24 bop 324 548 a Fs([F)-8 b(ou98])78 b(S.)26 b(F)-8 b(ournais,)26 b Fp(Semiclassics)h(of)i(the)g(Quantum)g(Curr)-5 b(ent)p Fs(,)27 b(Comm.)e(in)g(P)-8 b(.D.E)713 668 y Ft(23)32 b Fs(\(1998\),)g(no.)g(3-4,)g(601{628.)324 872 y([F)-8 b(ou99])78 b(S.)38 b(F)-8 b(ournais,)40 b Fp(Semiclassics)e(of) i(the)h(quantum)f(curr)-5 b(ent)41 b(in)f(a)g(str)-5 b(ong)40 b(c)-5 b(on-)713 992 y(stant)35 b(magnetic)f(\014eld)p Fs(,)e(Univ)m(ersit)m(y)h(of)f(Aarh)m(us)h(Preprin)m(t)g(\(1999\),)f (no.)g(9.)324 1196 y([Ivr98])113 b(V.)22 b(Ivrii,)h Fp(Micr)-5 b(olo)g(c)g(al)24 b(Analysis)h(and)g(Semiclassic)-5 b(al)23 b(Sp)-5 b(e)g(ctr)g(al)24 b(Asymptotics)p Fs(,)713 1316 y(Springer)32 b(V)-8 b(erlag,)31 b(1998.)324 1519 y([LS77])122 b(E.H.)49 b(Lieb)f(and)h(B.)f(Simon,)k Fp(The)d(Thomas-Fermi)e(The)-5 b(ory)49 b(of)h(Atoms,)713 1640 y(Mole)-5 b(cules)34 b(and)g(Solids)p Fs(,)e(Adv.)h(Math.)g(\(1977\),)f(no.)h(23,)f(22{116.) 324 1843 y([LSY94])49 b(E.)29 b(Lieb,)g(J.P)-8 b(.)30 b(Solo)m(v)m(ej,)g(and)f(J.)g(Yngv)-5 b(ason,)31 b Fp(Asymptotics)g(of) h(he)-5 b(avy)31 b(atoms)713 1963 y(in)j(high)g(magnetic)g(\014elds:)44 b(II.)33 b(Semiclassic)-5 b(al)33 b(r)-5 b(e)g(gions.)p Fs(,)31 b(Comm)m(un.)h(Math.)713 2084 y(Ph)m(ys.)i(\(1994\),)e(no.)g (161,)g(77{124.)324 2287 y([Sob94])80 b(A.V.)38 b(Sob)s(olev,)h Fp(The)g(quasiclassic)-5 b(al)39 b(asymptotics)g(of)h(lo)-5 b(c)g(al)39 b(Riesz)h(me)-5 b(ans)713 2408 y(for)52 b(the)h(Schr\177) -50 b(odinger)52 b(op)-5 b(er)g(ator)52 b(in)h(a)g(str)-5 b(ong)52 b(homo)-5 b(gene)g(ous)52 b(magnetic)713 2528 y(\014eld)p Fs(,)32 b(Duk)m(e)h(Math)g(Journal)e Ft(74)i Fs(\(1994\),)e(no.)i(2,)f(319{429.)1894 5251 y(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0004260356523--