Content-Type: multipart/mixed; boundary="-------------0010021041449" This is a multi-part message in MIME format. ---------------0010021041449 Content-Type: text/plain; name="00-385.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-385.keywords" Zero modes, Pauli operator, Weyl-Dirac operator ---------------0010021041449 Content-Type: application/postscript; name="ZM-Dirac-Weyl.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ZM-Dirac-Weyl.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ZM-Dirac-Weyl.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o ZM-Dirac-Weyl.ps ZM-Dirac-Weyl %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.08.31:1759 %%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 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a(2)p Fm(\031)1897 682 y Fl(Z)1837 949 y Fj(j)p Fh(x)p Fj(j)p Fn()g Fr(0,)36 b(set)e Fk(j)1121 416 y Fm(~)1104 439 y(A)p Fk(j)e Fr(=)f Fm(a)1379 453 y Fo(1)1441 439 y Fr(+)22 b Fm(a)1582 453 y Fo(2)1622 439 y Fr(,)35 b(where)e Fm(a)1996 453 y Fo(1)2067 439 y Fk(2)d Fm(C)2230 406 y Fj(1)2223 463 y Fo(0)2305 439 y Fr(\(\012)2406 453 y Fn(")2442 439 y Fr(\))35 b(for)e(some)i (ball)d(\012)3132 453 y Fn(")3168 439 y Fr(,)j(and)386 566 y Fk(j)p Fm(a)459 580 y Fo(1)499 566 y Fr(\()p Fu(x)p Fr(\))p Fk(j)26 b(\024)f Fm(k)818 580 y Fn(")885 566 y Fr(sa)m(y)-8 b(,)32 b(and)e Fk(k)p Fm(a)1330 580 y Fo(2)1370 566 y Fk(k)1415 585 y Fn(L)1463 566 y Fd(n)1506 585 y Fo(\()p Fg(R)1581 566 y Fd(n)1624 585 y Fo(\))1681 566 y Fm(<)24 b(")p Fr(.)42 b(Then,)29 b(for)h Fm( )f Fk(2)c Fm(D)2547 503 y Ff(1)p 2548 515 31 3 v 2548 557 a(2)2592 566 y Fr(\()p Fp(R)2687 533 y Fn(n)2740 566 y Fr(\))827 776 y Fk(j)852 702 y Fl(\000)894 776 y Fr([)n Fm(~)-43 b(\033)t Fk(\001)1032 753 y Fm(~)1015 776 y(A)p Fr(])p Fm(')1167 790 y Fn(m)1235 776 y Fm(;)15 b( )1337 702 y Fl(\001)1379 776 y Fk(j)56 b(\024)1586 652 y Fl(Z)1566 915 y Fg(R)1614 896 y Fd(n)1677 776 y Fr(\()p Fm(a)1760 790 y Fo(1)1820 776 y Fr(+)20 b Fm(a)1959 790 y Fo(2)1999 776 y Fr(\))p Fk(j)p Fm(')2118 790 y Fn(m)2185 776 y Fk(jj)p Fm( )s Fk(j)32 b Fm(d)p Fu(x)56 b Fr(:=)f Fm(I)2703 790 y Fo(1)2763 776 y Fr(+)20 b Fm(I)2894 790 y Fo(2)386 1043 y Fr(where,)30 b(b)m(y)g(Kato's)i(inequalit)m(y)c(\(12\),)850 1248 y Fm(I)890 1262 y Fo(1)1043 1248 y Fk(\024)113 b Fm(k)1274 1262 y Fn(")1326 1124 y Fl(Z)1309 1387 y Fo(\012)1360 1395 y Fd(")1417 1120 y Fl(\024)1465 1166 y(p)p 1556 1166 106 4 v 82 x Fk(j)p Fu(x)p Fk(jj)p Fm(')1745 1262 y Fn(m)1813 1248 y Fr(\()p Fu(x)p Fr(\))p Fk(j)1963 1120 y Fl(\025)q(\024)2145 1186 y Fr(1)p 2070 1227 197 4 v 2070 1245 a Fl(p)p 2161 1245 106 4 v 78 x Fk(j)p Fu(x)p Fk(j)2276 1248 y(j)p Fm( )s Fr(\()p Fu(x)p Fr(\))p Fk(j)2513 1120 y Fl(\025)2593 1248 y Fm(d)p Fu(x)1043 1604 y Fk(\024)g Fm(k)1277 1566 y Fj(0)1274 1626 y Fn(")1311 1476 y Fl(\022)1378 1480 y(Z)1361 1743 y Fo(\012)1412 1751 y Fd(")1484 1604 y Fk(j)p Fm(')1568 1618 y Fn(m)1635 1604 y Fr(\()p Fu(x)p Fr(\))p Fk(j)1785 1566 y Fo(2)1856 1604 y Fm(d)p Fu(x)1958 1476 y Fl(\023)2035 1471 y Ff(1)p 2035 1483 31 3 v 2035 1525 a(2)2080 1476 y Fl(\022)2147 1480 y(Z)2127 1743 y Fg(R)2175 1724 y Fd(n)2293 1542 y Fr(1)p 2263 1583 106 4 v 2263 1666 a Fk(j)p Fu(x)p Fk(j)2378 1604 y(j)p Fm( )s Fr(\()p Fu(x)p Fr(\))p Fk(j)2615 1566 y Fo(2)2687 1604 y Fm(d)p Fu(x)2789 1476 y Fl(\023)2866 1471 y Ff(1)p 2866 1483 31 3 v 2866 1525 a(2)1043 1960 y Fk(\024)g Fm(k)1277 1923 y Fj(00)1274 1983 y Fn(")1319 1832 y Fl(\022)1386 1837 y(Z)1369 2099 y Fo(\012)1420 2107 y Fd(")1492 1960 y Fk(j)p Fm(')1576 1974 y Fn(m)1643 1960 y Fr(\()p Fu(x)p Fr(\))p Fk(j)1793 1923 y Fo(2)1864 1960 y Fm(d)p Fu(x)1966 1832 y Fl(\023)2044 1828 y Ff(1)p 2044 1840 V 2044 1881 a(2)2088 1960 y Fk(k)p Fm( )s Fk(k)2240 2011 y Fn(D)2311 1961 y Ff(1)p 2311 1969 V 2311 2004 a(2)2356 2011 y Fo(\()p Fg(R)2431 1992 y Fd(n)2474 2011 y Fo(\))2506 1960 y Fm(;)386 2230 y Fr(and)772 2380 y Fm(I)812 2394 y Fo(2)965 2380 y Fk(\024)g(k)p Fm(a)1242 2394 y Fo(2)1282 2380 y Fk(k)1327 2399 y Fn(L)1375 2380 y Fd(n)1418 2399 y Fo(\()p Fg(R)1493 2380 y Fd(n)1536 2399 y Fo(\))1567 2380 y Fk(k)p Fm(')1671 2394 y Fn(m)1739 2380 y Fk(k)1784 2438 y Fn(L)1866 2381 y Ff(2)p Fd(n)p 1842 2389 117 3 v 1842 2424 a(n)p Fc(\000)p Ff(1)1973 2438 y Fo(\()p Fg(R)2048 2419 y Fd(n)2091 2438 y Fo(\))2123 2380 y Fk(k)p Fm( )s Fk(k)2275 2438 y Fn(L)2358 2381 y Ff(2)p Fd(n)p 2334 2389 V 2334 2424 a(n)p Fc(\000)p Ff(1)2465 2438 y Fo(\()p Fg(R)2540 2419 y Fd(n)2583 2438 y Fo(\))965 2570 y Fk(\024)g Fm("\015)1243 2533 y Fo(2)1238 2593 y Fn(n)1285 2570 y Fk(k)p Fm(')1389 2584 y Fn(m)1457 2570 y Fk(k)1502 2621 y Fn(D)1572 2571 y Ff(1)p 1572 2579 31 3 v 1572 2614 a(2)1617 2621 y Fo(\()p Fg(R)1692 2602 y Fd(n)1735 2621 y Fo(\))1766 2570 y Fk(k)p Fm(')p Fk(k)1915 2621 y Fn(D)1986 2571 y Ff(1)p 1986 2579 V 1986 2614 a(2)2031 2621 y Fo(\()p Fg(R)2106 2602 y Fd(n)2149 2621 y Fo(\))2236 2570 y Fk(\024)55 b Fm("\015)2456 2533 y Fo(2)2451 2593 y Fn(n)2499 2570 y Fm(k)s Fk(k)p Fm(')p Fk(k)2698 2621 y Fn(D)2769 2571 y Ff(1)p 2769 2579 V 2769 2614 a(2)2814 2621 y Fo(\()p Fg(R)2889 2602 y Fd(n)2932 2621 y Fo(\))2963 2570 y Fm(;)386 2756 y Fr(where)30 b Fm(\015)696 2770 y Fn(n)773 2756 y Fr(is)g(the)g(norm)g(of)g(the)h (em)m(b)s(edding)d(\(9\).)42 b(Hence)989 2917 y Fk(k)p Fr([)n Fm(~)-43 b(\033)t Fk(\001)1172 2894 y Fm(~)1155 2917 y(A)p Fr(])p Fm(')1307 2931 y Fn(m)1374 2917 y Fk(k)1419 2968 y Fn(D)1479 2938 y Fc(\000)1538 2918 y Ff(1)p 1538 2926 V 1538 2961 a(2)1582 2968 y Fo(\()p Fg(R)1657 2949 y Fd(n)1700 2968 y Fo(\))1787 2917 y Fk(\024)56 b Fm(k)1964 2880 y Fj(00)1961 2940 y Fn(")2006 2917 y Fk(k)p Fm(')2110 2931 y Fn(m)2178 2917 y Fk(k)2223 2937 y Fn(L)2271 2918 y Ff(2)2306 2937 y Fo(\(\012)2384 2945 y Fd(")2417 2937 y Fo(\))2469 2917 y Fr(+)20 b Fm("\015)2654 2880 y Fo(2)2649 2940 y Fn(n)2697 2917 y Fm(k)s(:)386 3130 y Fr(Since)26 b Fm(D)707 3067 y Ff(1)p 708 3079 V 708 3120 a(2)752 3130 y Fr(\()p Fp(R)847 3097 y Fn(n)900 3130 y Fr(\))g Fm(,)-15 b Fk(!)25 b Fm(L)1149 3097 y Fo(2)1188 3130 y Fr(\(\012)1289 3144 y Fn(")1326 3130 y Fr(\))i(is)f(compact,)j(the)e (compactness)h(of)f(\(13\))h(is)e(established.)386 3238 y(The)k(pro)s(of)f(of)i(\(14\))h(follo)m(ws)d(from)h(that)h(for)f Fm(I)1995 3252 y Fo(2)2065 3238 y Fr(ab)s(o)m(v)m(e.)486 3388 y Fq(Lemma)j(2.)81 b Fb(Supp)-5 b(ose)34 b(that)g(the)f(fol)5 b(lowing)34 b(c)-5 b(onditions)34 b(ar)-5 b(e)34 b(satis\014e)-5 b(d:)460 3523 y Fr(\(i\))42 b Fb(when)33 b Fm(n)25 b Fr(=)g(3)p Fb(,)33 b Fk(j)1154 3500 y Fm(~)1137 3523 y(A)p Fk(j)26 b(2)f Fm(L)1404 3490 y Fo(3)1443 3523 y Fr(\()p Fp(R)1538 3490 y Fo(3)1583 3523 y Fr(\))p Fb(,)435 3642 y Fr(\(ii\))41 b Fb(when)33 b Fm(n)25 b Fr(=)g(2)p Fb(,)33 b Fk(j)1154 3619 y Fm(~)1137 3642 y(A)p Fk(j)26 b(2)f Fm(L)1404 3609 y Fn(r)1441 3642 y Fr(\()p Fp(R)1536 3609 y Fo(2)1582 3642 y Fr(\))33 b Fb(for)g(some)g Fm(r)28 b(>)d Fr(2)p Fb(.)386 3771 y(Then,)30 b(given)f(any)g Fm(")d(>)f Fr(0)30 b Fb(ther)-5 b(e)30 b(exists)f Fm(k)1806 3785 y Fn(")1869 3771 y Fm(>)c Fr(0)k Fb(such)h(that)g(for)g(al)5 b(l)30 b Fm(')25 b Fk(2)g Fm(D)2942 3738 y Fo(1)2981 3771 y Fr(\()p Fp(R)3076 3738 y Fn(n)3129 3771 y Fr(\))13 b Fk(\\)g(H)3328 3785 y Fn(n)1132 3932 y Fk(k)p Fr(\()n Fm(~)-43 b(\033)t Fk(\001)1325 3909 y Fm(~)1308 3932 y(A)p Fr(\))p Fm(')p Fk(k)1515 3895 y Fo(2)1614 3932 y Fk(\024)57 b Fm(")1784 3895 y Fo(2)1824 3932 y Fk(k)p Fm(')p Fk(k)1973 3895 y Fo(2)1973 3960 y Fn(D)2033 3941 y Ff(1)2069 3960 y Fo(\()p Fg(R)2144 3941 y Fd(n)2187 3960 y Fo(\))2239 3932 y Fr(+)20 b Fm(k)2377 3946 y Fn(")2414 3932 y Fk(k)p Fm(')p Fk(k)2563 3895 y Fo(2)2604 3932 y Fm(:)584 b Fr(\(15\))486 4232 y Fb(Pr)-5 b(o)g(of.)41 b Fr(Cho)s(ose)27 b Fk(j)1121 4209 y Fm(~)1103 4232 y(A)q Fk(j)e Fr(=)g Fm(a)1366 4246 y Fo(1)1421 4232 y Fr(+)16 b Fm(a)1556 4246 y Fo(2)1595 4232 y Fr(,)29 b(where)e Fk(j)p Fm(a)1982 4246 y Fo(1)2022 4232 y Fr(\()p Fu(x)p Fr(\))p Fk(j)f(\024)f Fm(k)2341 4246 y Fn(")2406 4232 y Fr(and)j Fk(k)p Fm(a)2674 4246 y Fo(2)2714 4232 y Fk(k)2759 4252 y Fn(L)2807 4233 y Ff(3)2842 4252 y Fo(\()p Fg(R)2917 4233 y Ff(3)2952 4252 y Fo(\))3009 4232 y Fm(<)d(")j Fr(in)f(\(i\))386 4345 y(and)j Fk(k)p Fm(a)656 4359 y Fo(2)696 4345 y Fk(k)741 4365 y Fn(L)789 4346 y Fd(r)824 4365 y Fo(\()p Fg(R)899 4346 y Ff(2)934 4365 y Fo(\))991 4345 y Fm(<)25 b(")30 b Fr(in)f(\(ii\).)40 b(Then)747 4520 y Fk(k)p Fr(\()n Fm(~)-43 b(\033)t Fk(\001)941 4497 y Fm(~)923 4520 y(A)q Fr(\))p Fm(')p Fk(k)1131 4482 y Fo(2)1227 4520 y Fk(\024)55 b(k)p Fm(a)1446 4534 y Fo(1)1486 4520 y Fm(')p Fk(k)1590 4482 y Fo(2)1651 4520 y Fr(+)20 b Fk(k)p Fm(a)1835 4534 y Fo(2)1875 4520 y Fm(')p Fk(k)1979 4482 y Fo(2)2044 4520 y Fk(\024)56 b Fm(k)2218 4534 y Fn(")2255 4520 y Fk(k)p Fm(')p Fk(k)2404 4482 y Fo(2)2465 4520 y Fr(+)20 b Fm(")2598 4482 y Fo(2)2638 4520 y Fk(k)p Fm(')p Fk(k)2787 4482 y Fo(2)2787 4547 y Fn(L)2835 4528 y Fd(t)2864 4547 y Fo(\()p Fg(R)2939 4528 y Fd(n)2982 4547 y Fo(\))386 4692 y Fr(where)36 b Fm(t)f Fr(=)g(6)i(when)f Fm(n)f Fr(=)g(3)i(and)f Fm(t)f Fr(=)1828 4657 y Fo(2)p Fn(r)p 1800 4672 124 4 v 1800 4724 a(r)r Fj(\000)p Fo(2)1970 4692 y Fk(2)g Fr(\(2)p Fm(;)15 b Fk(1)p Fr(\))38 b(when)d Fm(n)g Fr(=)h(2.)59 b(The)36 b(lemma)386 4807 y(follo)m(ws)29 b(from)h(the)h(em)m(b)s(eddings)d(\(10\))k(and)e(\(11\).)486 4957 y(Lemma)38 b(2)h(enables)e(us)h(to)h(de\014ne)e(the)i(op)s(erator) g Fp(D)2341 4971 y Fn(A)2442 4957 y Fr(as)g(the)g(op)s(erator)f(sum)f (of)384 5071 y Fm(~)-43 b(\033)s Fk(\001)480 5070 y Fm(~)481 5071 y Fu(p)29 b Fr(and)d Fm(~)-43 b(\033)s Fk(\001)856 5048 y Fm(~)838 5071 y(A)30 b Fr(in)d(the)i(t)m(w)m(o-)h(and)e (three-dimensional)f(cases.)41 b(Its)28 b(domain)g Fk(D)s Fr(\()p Fp(D)3187 5085 y Fn(A)3250 5071 y Fr(\))h(is)386 5179 y(giv)m(en)h(b)m(y)1011 5335 y Fk(D)s Fr(\()p Fp(D)1179 5349 y Fn(A)1242 5335 y Fr(\))113 b(=)g Fk(f)p Fm(')57 b Fr(:)f Fm(';)44 b(~)-43 b(\033)s Fk(\001)15 b Fr(\()2075 5334 y Fm(~)2076 5335 y Fu(p)21 b Fr(+)2263 5312 y Fm(~)2246 5335 y(A)p Fr(\))p Fm(')56 b Fk(2)g(H)2658 5349 y Fn(n)2704 5335 y Fk(g)1390 5473 y Fr(=)113 b Fk(D)s Fr(\()p Fp(D)1742 5487 y Fo(0)1787 5473 y Fr(\))56 b(=)g Fm(D)2083 5436 y Fo(1)2122 5473 y Fr(\()p Fp(R)2217 5436 y Fn(n)2270 5473 y Fr(\))20 b Fk(\\)g(H)2483 5487 y Fn(n)2530 5473 y Fm(:)658 b Fr(\(16\))p eop %%Page: 5 5 5 4 bop 920 239 a Fo(ON)30 b(THE)g(ZER)n(O)h(MODES)f(OF)h(WEYL-DIRA)n (C)f(OPERA)-6 b(TORS)498 b(5)1424 434 y Fr(3.)51 b Fq(The)34 b(main)g(resul)-6 b(ts)486 638 y Fr(W)e(e)31 b(shall)e(assume)h (hereafter)h(that)624 820 y Fk(j)667 797 y Fm(~)649 820 y(A)q Fk(j)56 b(2)915 692 y Fl(\032)1024 765 y Fm(L)1086 732 y Fo(3)1126 765 y Fr(\()p Fp(R)1221 732 y Fo(3)1266 765 y Fr(\),)491 b(when)29 b Fm(n)c Fr(=)g(3,)1024 873 y Fm(L)1086 840 y Fo(2)1126 873 y Fr(\()p Fp(R)1221 840 y Fo(2)1266 873 y Fr(\))c Fk(\\)e Fm(L)1464 840 y Fn(r)1502 873 y Fr(\()p Fp(R)1597 840 y Fo(2)1643 873 y Fr(\))30 b(,)84 b(for)30 b(some)h Fm(r)c(>)e Fr(2,)31 b(when)f Fm(n)25 b Fr(=)g(2.)3213 820 y(\(17\))386 1020 y(Lemmas)30 b(1)h(and)f(2)h(therefore)f(hold)f(and)h Fp(D)1895 1034 y Fn(A)1988 1020 y Fr(is)g(the)g(op)s(erator)h(sum)1514 1176 y Fp(D)1573 1190 y Fn(A)1692 1176 y Fr(=)53 b Fm(~)-43 b(\033)s Fk(\001)1912 1175 y Fm(~)1913 1176 y Fu(p)21 b Fr(+)d Fm(~)-43 b(\033)s Fk(\001)2196 1153 y Fm(~)2178 1176 y(A)386 1326 y Fr(on)30 b(\(16\).)486 1434 y(Supp)s(ose)e(that) 1086 1584 y Fp(D)1146 1598 y Fn(A)1209 1584 y Fm(')d Fr(=)g(0)p Fm(;)107 b(')26 b Fk(2)f(D)s Fr(\()p Fp(D)1904 1598 y Fn(A)1967 1584 y Fr(\))h(=)f Fm(D)2202 1547 y Fo(1)2241 1584 y Fr(\()p Fp(R)2336 1547 y Fn(n)2389 1584 y Fr(\))c Fk(\\)f(H)2603 1598 y Fn(n)2649 1584 y Fm(:)539 b Fr(\(18\))386 1767 y(Then)29 b Fm( )g Fr(=)807 1689 y Fl(p)p 898 1689 109 4 v 78 x Fk(j)922 1766 y Fm(~)923 1767 y Fu(p)p Fk(j)p Fm(')d Fk(2)f Fm(D)1264 1704 y Ff(1)p 1265 1716 31 3 v 1265 1757 a(2)1309 1767 y Fr(\()p Fp(R)1404 1734 y Fn(n)1457 1767 y Fr(\))c Fk(\\)f(H)1671 1781 y Fn(n)1717 1767 y Fr(,)31 b(and)933 1968 y Fk(k)p Fm( )s Fk(k)1085 1931 y Fo(2)1181 1968 y Fr(=)1307 1845 y Fl(Z)1287 2108 y Fg(R)1335 2089 y Fd(n)1413 1968 y Fk(j)1437 1967 y Fm(~)1438 1968 y Fu(p)q Fk(jj)15 b Fr(^)-60 b Fm(')q Fr(\()1641 1967 y Fm(~)1642 1968 y Fu(p)p Fr(\))p Fk(j)1760 1931 y Fo(2)1831 1968 y Fm(d)p Fu(p)55 b Fk(\024)2128 1907 y Fr(1)p 2128 1948 46 4 v 2128 2031 a(2)2183 1968 y(\()p Fk(k)2262 1967 y Fm(~)2263 1968 y Fu(p)16 b Fr(^)-60 b Fm(')p Fk(k)2426 1931 y Fo(2)2487 1968 y Fr(+)19 b Fk(k)c Fr(^)-60 b Fm(')q Fk(k)2727 1931 y Fo(2)2767 1968 y Fr(\))p Fm(:)386 2229 y Fr(Hence)1567 2379 y(\()p Fm(E)25 b Fk(\000)20 b Fm(K)7 b Fr(\))p Fm( )59 b Fr(=)d(0)1019 b(\(19\))386 2530 y(where)1101 2700 y Fm(E)61 b Fr(=)1442 2639 y(1)p 1365 2679 200 4 v 1365 2697 a Fl(p)p 1456 2697 109 4 v 78 x Fk(j)1480 2774 y Fm(~)1481 2775 y Fu(p)p Fk(j)1605 2700 y Fr(\()n Fm(~)-43 b(\033)s Fk(\001)1734 2699 y Fm(~)1735 2700 y Fu(p)q Fr(\))1946 2639 y(1)p 1869 2679 200 4 v 1869 2697 a Fl(p)p 1960 2697 109 4 v 78 x Fk(j)1984 2774 y Fm(~)1985 2775 y Fu(p)q Fk(j)2134 2700 y Fr(:)56 b Fk(H)2292 2714 y Fn(n)2364 2700 y Fk(\000)-15 b(!)25 b(H)2613 2714 y Fn(n)3213 2700 y Fr(\(20\))386 2917 y(and)1042 3087 y Fm(K)62 b Fr(=)56 b Fk(\000)1465 3026 y Fr(1)p 1389 3066 200 4 v 1389 3084 a Fl(p)p 1479 3084 109 4 v 1479 3162 a Fk(j)1503 3161 y Fm(~)1504 3162 y Fu(p)q Fk(j)1628 3087 y Fr(\()n Fm(~)-43 b(\033)t Fk(\001)1776 3064 y Fm(~)1759 3087 y(A)p Fr(\))1980 3026 y(1)p 1903 3066 200 4 v 1903 3084 a Fl(p)p 1994 3084 109 4 v 78 x Fk(j)2018 3161 y Fm(~)2019 3162 y Fu(p)p Fk(j)2168 3087 y Fr(:)56 b Fk(H)2326 3101 y Fn(n)2398 3087 y Fk(\000)-16 b(!)25 b(H)2646 3101 y Fn(n)2693 3087 y Fm(:)495 b Fr(\(21\))386 3316 y(F)-8 b(urthermore,)33 b Fm(E)1013 3283 y Fo(2)1081 3316 y Fr(=)28 b Fm(I)7 b Fr(,)33 b(the)g(iden)m(tit)m(y)e(on)i Fk(H)1987 3330 y Fn(n)2033 3316 y Fr(,)g Fm(E)2163 3283 y Fj(\003)2232 3316 y Fr(=)28 b Fm(E)37 b Fr(and,)c(b)m(y)f(Lemma)g(1,) i Fm(K)39 b Fr(is)386 3424 y(compact.)j(Th)m(us,)29 b(from)h(\(19\)) 1530 3574 y(\()p Fm(I)e Fk(\000)20 b Fm(E)5 b(K)i Fr(\))p Fm( )59 b Fr(=)d(0)p Fm(:)983 b Fr(\(22\))386 3756 y(Since)29 b Fm( )g Fk(2)c Fm(D)884 3693 y Ff(1)p 885 3705 31 3 v 885 3746 a(2)929 3756 y Fr(\()p Fp(R)1024 3723 y Fn(n)1077 3756 y Fr(\))c Fk(\\)f(H)1291 3770 y Fn(n)1337 3756 y Fr(,)31 b(it)f(follo)m(ws)f(that)1335 3906 y Fm(nul)16 b Fp(D)1545 3920 y Fn(A)1721 3906 y Fk(\024)113 b Fm(nul)r Fr(\()p Fm(I)28 b Fk(\000)19 b Fm(E)5 b(K)i Fr(\))788 b(\(23\))1721 4041 y(=)113 b(0)386 4191 y(unless)29 b(the)h(compact)i (op)s(erator)e Fm(E)5 b(K)38 b Fr(on)30 b Fk(H)1931 4205 y Fn(n)2008 4191 y Fr(has)g(eigen)m(v)-5 b(alue)31 b(1.)486 4341 y Fq(Theorem)40 b(1.)118 b Fb(L)-5 b(et)39 b(\(17\))h(b)-5 b(e)38 b(satis\014e)-5 b(d.)61 b(Then)39 b(for)g(any)g Fm(c)d(>)f Fr(0)p Fb(,)40 b Fm(nul)17 b Fp(D)3099 4355 y Fn(tA)3223 4341 y Fr(=)35 b(0)386 4449 y Fb(exc)-5 b(ept)33 b(at)g(a)g(\014nite)g(numb)-5 b(er)33 b(of)g(values)g(of)g Fm(t)f Fb(in)g Fr([0)p Fm(;)15 b(c)p Fr(])p Fb(.)486 4600 y(Pr)-5 b(o)g(of.)70 b Fr(This)37 b(is)i(a)h(consequence)h(of)f(F) -8 b(redholm)39 b(theory)h(applied)d(to)k Fm(F)3048 4614 y Fn(t)3119 4600 y Fr(=)f Fm(I)34 b Fk(\000)386 4713 y Fm(E)5 b(K)535 4727 y Fn(t)565 4713 y Fr(,)34 b(where)f Fm(K)967 4727 y Fn(t)1027 4713 y Fr(=)d Fk(\000)1271 4678 y Fo(1)p 1209 4693 161 4 v 1209 4703 a Fk(p)p 1284 4703 85 4 v 1284 4766 a Fj(j)1304 4765 y Fn(~)1304 4766 y Fh(p)p Fj(j)1412 4713 y Fr(\()n Fm(~)-43 b(\033)s Fk(\001)1564 4690 y Fm(~)1542 4713 y(tA)q Fr(\))1785 4678 y Fo(1)p 1722 4693 161 4 v 1722 4703 a Fk(p)p 1798 4703 85 4 v 63 x Fj(j)1818 4765 y Fn(~)1818 4766 y Fh(p)p Fj(j)1892 4713 y Fr(.)50 b(Since)32 b Fm(E)5 b(K)2356 4727 y Fn(t)2419 4713 y Fr(is)33 b(compact)h(on)f Fk(H)3086 4727 y Fn(n)3133 4713 y Fr(,)h Fm(F)3250 4727 y Fn(t)3314 4713 y Fr(is)386 4867 y(F)-8 b(redholm)34 b(\([3)q(,)i(Theorem)e(I.3.21]\).)56 b(By)36 b([3,)g(Theorem)f(I.3.22])h(there)f(exists)g Fm(\016)h(>)c Fr(0)386 4975 y(suc)m(h)39 b(that)h(for)f Fk(j)p Fm(t)27 b Fk(\000)f Fm(t)1169 4942 y Fj(0)1192 4975 y Fk(j)41 b Fm(<)f(\016)s Fr(,)j Fm(nul)16 b(F)1688 4990 y Fn(t)1713 4971 y Fc(0)1781 4975 y Fk(\024)40 b Fm(nul)17 b(F)2101 4989 y Fn(t)2131 4975 y Fr(,)42 b(and)d(consequen)m (tly)g Fm(t)i Fk(7!)f Fm(nul)17 b(F)3345 4989 y Fn(t)386 5083 y Fr(is)38 b(upp)s(er)f(semi-con)m(tin)m(uous.)66 b(F)-8 b(urthermore,)42 b(b)m(y)d([3)q(,)i(Theorem)e(I.3.25],)k(for)c (small)386 5191 y(enough)45 b Fm(\016)s Fr(,)50 b Fm(nul)17 b(F)1041 5206 y Fn(t)1066 5187 y Fc(0)1138 5191 y Fr(is)44 b(constan)m(t)j(in)d Fk(f)p Fm(t)1825 5158 y Fj(0)1944 5191 y Fr(:)95 b(0)51 b Fm(<)e Fk(j)p Fm(t)30 b Fk(\000)g Fm(t)2502 5158 y Fj(0)2525 5191 y Fk(j)51 b Fm(<)e(\016)s Fk(g)p Fr(.)87 b(F)-8 b(or)46 b Fm(t)k(<)f(t)3335 5205 y Fo(0)386 5299 y Fr(sa)m(y)-8 b(,)38 b Fm(nul)16 b(F)775 5313 y Fn(t)839 5299 y Fr(=)33 b(0.)55 b(It)36 b(therefore)f(su\016ces) g(to)h(pro)m(v)m(e)g(the)g(theorem)f(for)g([)p Fm(a;)15 b(c)p Fr(],)38 b(where)386 5407 y(0)26 b Fm(<)f(a)g(<)g(t)755 5421 y Fo(0)794 5407 y Fr(.)39 b(But)25 b(this)f(follo)m(ws)g(from)g (the)h(previous)f(remarks)g(and)h(the)g(compactness)386 5515 y(of)40 b([)p Fm(a;)15 b(c)p Fr(].)69 b(An)39 b(alternativ)m(e)h (pro)s(of)f(can)h(b)s(e)f(giv)m(en)h(b)m(y)f(amending)f(the)i(pro)s(of) f(of)h([1,)386 5623 y(Theorem)30 b(4.1])i(to)f(deal)f(with)f(the)h (non-self-adjoin)m(t)g(part)g Fm(E)5 b(K)2599 5637 y Fn(t)2660 5623 y Fr(of)30 b Fm(F)2821 5637 y Fn(t)2851 5623 y Fr(.)p eop %%Page: 6 6 6 5 bop 386 239 a Fo(6)828 b(A.)30 b(A.)g(BALINSKY)g(AND)g(W.)h(D.)f (EV)-8 b(ANS)486 439 y Fq(Theorem)37 b(2.)102 b Fb(L)-5 b(et)36 b Fk(j)1316 416 y Fm(~)1299 439 y(A)p Fk(j)c(2)e Fm(L)1577 406 y Fn(r)1615 439 y Fr(\()p Fp(R)1710 406 y Fo(2)1756 439 y Fr(\))36 b Fb(for)g(some)h Fm(r)c(>)e Fr(2)36 b Fb(when)h Fm(n)30 b Fr(=)h(2)p Fb(.)52 b(Then)36 b(the)386 558 y(set)531 484 y Fl(\010)601 535 y Fm(~)584 558 y(A)67 b Fr(:)f Fm(nul)17 b Fp(D)1021 572 y Fn(A)1115 558 y Fr(=)30 b(0)72 b Fb(and)g Fk(j)1591 535 y Fm(~)1573 558 y(A)q Fk(j)31 b(2)g Fm(L)1852 525 y Fn(n)1898 558 y Fr(\()p Fp(R)1993 525 y Fn(n)2046 558 y Fr(\))2081 484 y Fl(\011)2170 558 y Fb(c)-5 b(ontains)38 b(an)e(op)-5 b(en)37 b(dense)f(subset)386 677 y(of)493 604 y Fl(\002)531 677 y Fm(L)593 644 y Fn(n)640 677 y Fr(\()p Fp(R)735 644 y Fn(n)787 677 y Fr(\))822 604 y Fl(\003)861 626 y Fn(n)486 845 y Fb(Pr)-5 b(o)g(of.)40 b Fr(W)-8 b(e)28 b(shall)c(pro)m(v)m(e)k(that)f(with)e Fm(F)1821 859 y Fn(A)1904 845 y Fr(=)g Fm(I)19 b Fk(\000)12 b Fm(E)5 b(K)2291 859 y Fn(A)2349 845 y Fr(,)27 b Fm(K)2478 859 y Fn(A)2561 845 y Fr(=)e Fk(\000)2800 809 y Fo(1)p 2738 824 161 4 v 2738 835 a Fk(p)p 2813 835 85 4 v 2813 897 a Fj(j)2833 896 y Fn(~)2833 897 y Fh(p)p Fj(j)2934 845 y Fr(\()n Fm(~)-43 b(\033)s Fk(\001)3082 822 y Fm(~)3064 845 y(A)q Fr(\))3267 809 y Fo(1)p 3205 824 161 4 v 3205 835 a Fk(p)p 3280 835 85 4 v 3280 897 a Fj(j)3300 896 y Fn(~)3300 897 y Fh(p)p Fj(j)1114 984 y Fl(\010)1184 1035 y Fm(~)1167 1058 y(A)56 b Fr(:)f Fm(nul)17 b(F)1580 1072 y Fn(A)1663 1058 y Fr(=)25 b(0)61 b(and)f Fk(j)2114 1035 y Fm(~)2097 1058 y(A)p Fk(j)26 b(2)f Fm(L)2364 1020 y Fn(n)2410 1058 y Fr(\()p Fp(R)2505 1020 y Fn(n)2558 1058 y Fr(\))2593 984 y Fl(\011)3213 1058 y Fr(\(24\))386 1222 y(is)33 b(op)s(en)g(in)813 1148 y Fl(\002)851 1222 y Fm(L)913 1189 y Fn(n)960 1222 y Fr(\()p Fp(R)1055 1189 y Fn(n)1108 1222 y Fr(\))1143 1148 y Fl(\003)1181 1171 y Fn(n)1228 1222 y Fr(.)51 b(The)33 b(theorem)h(will)d(follo)m(w)i (from)g(\(23\))i(since)e(the)h(densit)m(y)386 1330 y(of)c(\(24\))i(is)e (implied)d(b)m(y)j(Theorem)g(1.)486 1449 y(Giv)m(en)g Fm(")25 b(>)g Fr(0,)31 b(supp)s(ose)e Fk(k)1418 1426 y Fm(~)1400 1449 y(A)21 b Fk(\000)1597 1426 y Fm(~)1580 1449 y(A)1648 1463 y Fo(0)1688 1449 y Fk(k)1733 1468 y Fn(L)1781 1449 y Fd(n)1824 1468 y Fo(\()p Fg(R)1899 1449 y Fd(n)1942 1468 y Fo(\))1998 1449 y Fm(<)k(")p Fr(.)41 b(Then)1469 1612 y Fk(k)p Fm(K)1591 1626 y Fn(A)1669 1612 y Fk(\000)19 b Fm(K)1836 1626 y Fn(A)1889 1635 y Ff(0)1928 1612 y Fk(k)56 b(\024)g Fm(\015)2208 1575 y Fo(2)2203 1635 y Fn(n)2250 1612 y Fm(")386 1760 y Fr(b)m(y)32 b(Lemma)g(1.)47 b(Let)33 b Fm(nul)17 b Fp(D)1332 1774 y Fn(A)1385 1783 y Ff(0)1458 1760 y Fr(=)28 b(0.)47 b(Since)31 b Fm(E)5 b(K)2062 1774 y Fn(A)2115 1783 y Ff(0)2187 1760 y Fr(is)31 b(compact,)j(this)d(means)h(that)h(1)386 1868 y(is)c(not)i(an)f(eigen)m(v)-5 b(alue)31 b(and)e(there)i(exists)f Fm(\016)f(>)c Fr(0)31 b(suc)m(h)f(that)1372 2015 y(inf)1326 2080 y Fj(k)p Fn(f)7 b Fj(k)p Fo(=1)1543 2015 y Fk(k)31 b Fr(\()p Fm(I)d Fk(\000)19 b Fm(E)5 b(K)1961 2029 y Fn(A)2014 2038 y Ff(0)2054 2015 y Fr(\))p Fm(f)40 b Fk(k)56 b(\025)25 b Fm(\016)n(:)386 2236 y Fr(Hence,)31 b(if)f Fm(")25 b(<)978 2200 y Fn(\016)p 938 2215 115 4 v 938 2269 a Fo(2)p Fn(\015)1013 2250 y Ff(2)1009 2286 y Fd(n)1388 2421 y Fr(inf)1343 2486 y Fj(k)p Fn(f)7 b Fj(k)p Fo(=1)1560 2421 y Fk(k)31 b Fr(\()p Fm(I)c Fk(\000)20 b Fm(E)5 b(K)1978 2435 y Fn(A)2036 2421 y Fr(\))p Fm(f)40 b Fk(k)56 b(\025)2363 2359 y Fm(\016)p 2363 2400 46 4 v 2363 2483 a Fr(2)386 2603 y(and)30 b(hence)g Fm(nul)17 b(F)1024 2617 y Fn(A)1107 2603 y Fr(=)25 b(0.)41 b(The)30 b(theorem)g(therefore)h(follo)m(ws.)486 2751 y Fq(Theorem)i(3.)81 b Fb(L)-5 b(et)33 b(\(17\))h(b)-5 b(e)32 b(satis\014e)-5 b(d.)44 b(Then,)32 b(for)i Fm(n)24 b Fr(=)h(2)p Fm(;)15 b Fr(3)p Fb(,)1400 2945 y Fm(nul)h Fp(D)1610 2959 y Fn(A)1731 2945 y Fk(\024)57 b Fm(c)1913 2821 y Fl(Z)1893 3084 y Fg(R)1941 3065 y Fd(n)2020 2945 y Fk(j)2062 2922 y Fm(~)2045 2945 y(A)p Fk(j)2138 2907 y Fn(n)2218 2945 y Fm(d)p Fu(x)p Fm(;)868 b Fr(\(25\))386 3209 y Fb(wher)-5 b(e)34 b Fm(c)f Fb(is)f(the)h(b)-5 b(est)33 b(c)-5 b(onstant)34 b(in)f(the)g(Daub)-5 b(e)g(chies)33 b(ine)-5 b(quality)33 b Fr([2)q(])p Fb(.)486 3357 y(Pr)-5 b(o)g(of.)486 3465 y Fr(W)d(e)37 b(kno)m(w)g(from)f(\(23\))i(that)g Fm(nul)16 b Fp(D)1725 3479 y Fn(A)1824 3465 y Fk(\024)35 b Fm(nul)q Fr(\()p Fm(I)d Fk(\000)24 b Fm(E)5 b(K)i Fr(\).)60 b(If)36 b Fm( )j Fk(2)d Fr(k)m(er\()p Fm(I)c Fk(\000)24 b Fm(E)5 b(K)i Fr(\),)386 3573 y(then,)30 b(with)f Fk(S)j Fr(=)25 b Fm(E)5 b(K)i Fr(,)31 b Fk(kS)7 b Fm( )s Fk(k)27 b Fr(=)d Fk(k)p Fm( )s Fk(k)p Fr(.)43 b(Hence,)31 b(b)m(y)f(the)h (min-max)e(theorem)1069 3720 y Fm(nul)r Fr(\()p Fm(I)e Fk(\000)20 b Fm(E)5 b(K)i Fr(\))56 b Fk(\024)g Fm(])p Fk(f)p Fm(\025)1905 3734 y Fn(n)1952 3720 y Fr(\()p Fk(S)7 b Fr(\))57 b(:)e Fm(\025)2274 3734 y Fn(n)2321 3720 y Fr(\()p Fk(S)7 b Fr(\))26 b Fk(\025)f Fr(1)p Fk(g)p Fm(;)386 3867 y Fr(where)31 b Fm(\025)703 3881 y Fn(n)750 3867 y Fr(\()p Fk(S)7 b Fr(\))32 b(denotes)g(the)f Fm(n)p Fr(th)g(singular)e(v)-5 b(alue)31 b(of)g Fk(S)7 b Fr(,)32 b(i.e.)44 b(the)31 b Fm(n)p Fr(th)g(eigen)m(v)-5 b(alue)31 b(of)386 3981 y Fk(jS)7 b(j)26 b Fr(=)f(\()p Fk(S)717 3948 y Fj(\003)756 3981 y Fk(S)7 b Fr(\))853 3948 y Fo(1)p Fn(=)p Fo(2)963 3981 y Fr(.)41 b(F)-8 b(urthermore)1103 4200 y Fk(S)1165 4163 y Fj(\003)1204 4200 y Fk(S)120 b Fr(=)1563 4072 y Fl(\022)1717 4139 y Fr(1)p 1640 4179 200 4 v 1640 4197 a Fl(p)p 1731 4197 109 4 v 78 x Fk(j)1755 4274 y Fm(~)1756 4275 y Fu(p)p Fk(j)1880 4200 y Fr(\()n Fm(~)-43 b(\033)s Fk(\001)2028 4177 y Fm(~)2010 4200 y(A)q Fr(\))2232 4139 y(1)p 2155 4179 200 4 v 2155 4197 a Fl(p)p 2246 4197 109 4 v 78 x Fk(j)2270 4274 y Fm(~)2271 4275 y Fu(p)p Fk(j)2364 4072 y Fl(\023)2431 4095 y Fo(2)1379 4486 y Fk(\024)1563 4358 y Fl(\022)1717 4425 y Fr(1)p 1640 4466 200 4 v 1640 4484 a Fl(p)p 1731 4484 109 4 v 77 x Fk(j)1755 4560 y Fm(~)1756 4561 y Fu(p)p Fk(j)1880 4486 y(j)1922 4463 y Fm(~)1905 4486 y(A)p Fk(j)2116 4425 y Fr(1)p 2039 4466 200 4 v 2039 4484 a Fl(p)p 2130 4484 109 4 v 77 x Fk(j)2154 4560 y Fm(~)2155 4561 y Fu(p)p Fk(j)2248 4358 y Fl(\023)2315 4381 y Fo(2)2410 4486 y Fr(=:)56 b(L)2619 4449 y Fo(2)386 4700 y Fr(and)30 b(this)f(giv)m(es) 773 4848 y Fm(nul)q Fr(\()p Fm(I)f Fk(\000)20 b Fm(E)5 b(K)i Fr(\))114 b Fk(\024)f Fm(])p Fk(f)p Fm(\025)1724 4862 y Fn(n)1772 4848 y Fr(\(L\))56 b(:)f Fm(\025)2088 4862 y Fn(n)2135 4848 y Fr(\(L\))26 b Fk(\025)f Fr(1)p Fk(g)1407 4993 y Fr(=)113 b Fm(])p Fk(f)p Fr(negativ)m(e)32 b(eigen)m(v)-5 b(alues)30 b(of)h Fk(j)2628 4992 y Fm(~)2629 4993 y Fu(p)p Fk(j)21 b(\000)f(j)2866 4970 y Fm(~)2849 4993 y(A)p Fk(jg)386 5141 y Fr(b)m(y)30 b(the)h(Birman-Sc)m(h)m(winger) e(principle.)37 b(The)30 b(theorem)h(no)m(w)f(follo)m(ws)f(from)h([2)q (].)486 5407 y(The)36 b(authors)h(are)h(grateful)e(to)i(the)g(Europ)s (ean)e(Union)g(for)g(supp)s(ort)g(under)f(the)386 5515 y(TMR)30 b(gran)m(t)h(FMRX-CT96-0001.)p eop %%Page: 7 7 7 6 bop 920 239 a Fo(ON)30 b(THE)g(ZER)n(O)h(MODES)f(OF)h(WEYL-DIRA)n (C)f(OPERA)-6 b(TORS)498 b(7)1615 434 y Fq(References)424 579 y Ft([1])40 b Fs(A.)23 b(A.)g(Balinsky)g(and)g(W.)f(D.)i(Ev)-7 b(ans)p Ft(,)22 b(`On)f(the)f(zero)i(mo)r(des)e(of)i(P)n(auli)g(op)r (erators',)h(preprin)n(t)544 670 y(mp)p 657 670 24 4 v 26 w(arc/00-130,)54 b Fi(J.)27 b(F)-6 b(unct.)29 b(A)n(nal.)c Ft(to)h(app)r(ear.)424 762 y([2])40 b Fs(I.)34 b(D)n(a)n(ubechies)p Ft(,)e(`An)e(uncertain)n(t)n(y)h(principle)g(for)h(fermions)f(with)h (generalized)g(kinetic)f(en-)544 853 y(ergy',)26 b Fi(Comm.)h(Math.)g (Phys.)f Ft(90)h(\(1983\))g(511-520.)424 944 y([3])40 b Fs(D.)30 b(E.)g(Edmunds)f(and)h(W.)f(D.)h(Ev)-7 b(ans)p Ft(,)28 b Fi(Sp)l(e)l(ctr)l(al)j(the)l(ory)g(and)f(di\013er)l(ential)f (op)l(er)l(ators)i Ft(\(The)544 1036 y(Clarendon)26 b(Press,)h(Oxford)f (Univ)n(ersit)n(y)e(Press,)j(New)f(Y)-6 b(ork,)25 b(1987\).)424 1127 y([4])40 b Fs(E.)35 b(H.)g(Lieb)g(and)g(M.)g(Loss)p Ft(,)h Fi(A)n(nalysis)d Ft(\(American)e(Mathematical)j(So)r(ciet)n(y)-6 b(,)34 b(Pro)n(vidence,)544 1218 y(RI,)25 b(1997\).)424 1310 y([5])40 b Fs(M.)21 b(Loss)i(and)e(H.-T.)g(Y)-7 b(a)n(u)p Ft(,)21 b(`Stabilt)n(y)e(of)h(Coulom)n(b)f(systems)f(with)i (magnetic)f(\014elds.)h(III.)e(Zero)544 1401 y(energy)24 b(b)r(ound)f(states)i(of)g(the)f(Pauli)h(op)r(erator',)i Fi(Comm.)e(Math.)h(Phys.)f Ft(104)g(\(1986\))h(283{290.)486 1664 y Fs(School)e(of)f(Ma)-5 b(thema)g(tics,)24 b(Cardiff)e (University,)g(23)i(Senghennydd)d(R)n(o)n(ad,)j(Cardiff)386 1755 y(CF24)29 b(4YH,)f(UK)486 1846 y Fi(E-mail)f(addr)l(ess)6 b Ft(:)36 b Fa(BalinskyA@cardiff.ac.uk)486 1938 y Fi(E-mail)27 b(addr)l(ess)6 b Ft(:)36 b Fa(EvansWD@cardiff.ac.uk)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0010021041449--