This is a multi-part message in MIME format. ---------------0010021031603 Content-Type: text/plain; name="00-383.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-383.keywords" Magnetic Schr\"{o}dinger operator, Aharonov-Bohm magnetic field, continuous embedding, compact perturbation. ---------------0010021031603 Content-Type: application/postscript; name="A-B_compact20Sept.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="A-B_compact20Sept.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: A-B_compact20Sept.dvi %%Pages: 5 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o A-B_compact20Sept.ps A-B_compact20Sept.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.09.20:1805 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S 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4190 y Fj(n)17 b(f)p Fr(0)p Fj(g)p Fr(\))231 b(\(1\))538 4476 y(do)r(es)28 b(not)f(hold)h(for)f Fm(n)c Fr(=)f(2;)363 4647 y(\(iii\))42 b(there)28 b(is)f(no)g(Cwik)n(el-Lieb-Rosen)n(blum)f(\(CLR\))j (inequalit)n(y)e(of)g(the)h(form)1020 4875 y Fm(N)9 b Fr(\(\001)18 b(+)g Fm(V)h Fr(\))51 b Fj(\024)g Fr(const)o Fj(\001)1790 4762 y Fh(Z)1776 5004 y Fg(R)1823 4987 y Fd(2)1887 4875 y Fm(V)1935 4887 y Fk(\000)1991 4875 y Fr(\()p Fe(x)p Fr(\))29 b Fm(d)p Fe(x)p Fm(;)98 b(V)2396 4887 y Fk(\000)2475 4875 y Fr(=)23 b(min\()p Fm(V)5 b(;)14 b Fr(0\))p Fm(;)375 b Fr(\(2\))538 5169 y(where)27 b Fm(N)9 b Fr(\(\001)19 b(+)f Fm(V)h Fr(\))28 b(denotes)f(the)h(n)n(um)n (b)r(er)f(of)h(negativ)n(e)e(eigen)n(v)-5 b(alues)27 b(of)g(\001)19 b(+)f Fm(V)h Fr(.)p eop %%Page: 2 2 2 1 bop 331 187 a Fr(2)470 b Fn(A.)30 b(A.)h(Balinsky,)f(W.)g(D.)h(Ev) -7 b(ans)31 b(and)g(R)n(oger)g(T.)g(Lewis)497 527 y Fr(In)38 b([1)o(])g(Laptev)f(and)h(W)-7 b(eidl)38 b(obtained)f(the)h(elegan)n(t) f(result)g(that)h(\(ii\))h(do)r(es)e(hold)g(when)331 637 y Fm(n)23 b Fr(=)f(2)k(if)g(the)h(gradien)n(t)e Fj(r)h Fr(is)g(replaced)f(b)n(y)h(the)g(\\magnetic")e(gradien)n(t)2617 636 y Fm(~)2617 637 y Fe(p)16 b Fr(+)2781 616 y Fm(~)2765 637 y(A)q Fr(,)26 b(where)3115 636 y Fm(~)3116 637 y Fe(p)d Fr(=)3289 604 y Fo(1)p 3289 618 34 4 v 3294 665 a Fl(i)3332 637 y Fj(r)331 754 y Fr(and)j Fm(cur)r(l)670 733 y(~)654 754 y(A)h Fr(is,)g(in)f(particular,)g(a)f(magnetic)h (\014eld)h(of)f(Aharono)n(v-Bohm)e(t)n(yp)r(e.)37 b(T)-7 b(o)26 b(b)r(e)g(sp)r(eci\014c,)331 853 y(they)i(pro)n(v)n(e,)d(in)n (ter)j(alia,)f(the)h(follo)n(wing)e([1,)h(Theorem)g(3]:)37 b(In)28 b(p)r(olar)e(co)r(ordinates)g(\()p Fm(r)n(;)14 b(\022)r Fr(\),)29 b(let)1301 1054 y Fm(~)1285 1075 y(A)p Fr(\()p Fm(r)n(;)14 b(\022)r Fr(\))52 b(=)1701 1019 y(\011\()p Fm(\022)r Fr(\))p 1701 1056 171 4 v 1767 1132 a Fm(r)1909 1008 y Fh(\000)1947 1075 y Fr(sin)14 b Fm(\022)r(;)g Fj(\000)g Fr(cos)f Fm(\022)2386 1008 y Fh(\001)2424 1075 y Fm(;)848 b Fr(\(3\))331 1295 y(where)31 b(\011)f Fj(2)g Fm(L)812 1265 y Fk(1)882 1295 y Fr(\()p Fp(S)967 1265 y Fo(1)998 1295 y Fr(\),)j(so)e(that)h Fm(B)j Fr(=)29 b Fm(cur)r(l)1747 1274 y(~)1732 1295 y(A)h Fr(=)g(0)h(for)h Fm(r)h Fj(6)p Fr(=)c(0,)k(and)2567 1274 y Fm(~)2552 1295 y(A)p Fj(\001)13 b Fm(~)-41 b(x)30 b Fr(=)g(0.)49 b(Then,)34 b(for)d(all)331 1395 y Fm(u)22 b Fj(2)i Fm(C)545 1365 y Fk(1)539 1416 y Fo(0)615 1395 y Fr(\()p Fp(R)702 1365 y Fo(2)763 1395 y Fj(n)18 b(f)p Fr(0)p Fj(g)p Fr(\),)1104 1509 y Fh(Z)1089 1751 y Fg(R)1136 1734 y Fd(2)1210 1566 y Fj(j)p Fm(u)p Fr(\()p Fe(x)p Fr(\))p Fj(j)1418 1536 y Fo(2)p 1210 1603 246 4 v 1266 1679 a Fj(j)p Fe(x)p Fj(j)1362 1655 y Fo(2)1494 1622 y Fm(d)p Fe(x)51 b Fj(\024)g Fm(C)1860 1509 y Fh(Z)1846 1751 y Fg(R)1893 1734 y Fd(2)1957 1622 y Fj(j)p Fr(\()2011 1621 y Fm(~)2012 1622 y Fe(p)19 b Fr(+)2183 1601 y Fm(~)2167 1622 y(A)p Fr(\))p Fm(u)p Fr(\()p Fe(x)p Fr(\))p Fj(j)2446 1588 y Fo(2)2512 1622 y Fm(d)p Fe(x)p Fm(;)667 b Fr(\(4\))331 1971 y(with)28 b(the)g(sharp)e(constan)n(t)h Fm(C)j Fr(=)1400 1904 y Fh(\000)1438 1971 y Fr(min)1447 2026 y Fl(k)q Fk(2)p Fg(Z)1591 1971 y Fj(j)p Fm(k)21 b Fj(\000)1772 1950 y Fr(~)1761 1971 y(\011)p Fj(j)1849 1904 y Fh(\001)1887 1921 y Fk(\000)p Fo(1)1976 1971 y Fr(,)2149 1950 y(~)2137 1971 y(\011)i(:=)2366 1939 y Fo(1)p 2345 1953 75 4 v 2345 2000 a(2)p Fl(\031)2451 1888 y Fo(2)p Fl(\031)2453 1905 y Fh(R)2456 2048 y Fo(0)2531 1971 y Fr(\011\()p Fm(\022)r Fr(\))28 b Fm(d\022)r Fr(.)497 2145 y(An)23 b(imp)r(ortan)n(t)g(consequence)g(is)g(that)g(the)h(Sc)n(hr\177)-42 b(odinger)21 b(op)r(erator)h Fm(S)2706 2157 y Fl(A)2806 2145 y Fr(=)46 b(\()2948 2144 y Fm(~)2949 2145 y Fe(p)10 b Fr(+)3102 2124 y Fm(~)3087 2145 y(A)p Fr(\))3181 2115 y Fo(2)3242 2145 y Fr(do)r(es)331 2245 y(not)27 b(ha)n(v)n(e)g(zero)f (mo)r(des)i(in)g Fm(L)1256 2215 y Fo(2)1292 2245 y Fr(\()p Fp(R)1379 2215 y Fo(2)1422 2245 y Fr(\),)g(i.e.)37 b(zero)26 b(is)i(not)f(an)h(eigen)n(v)-5 b(alue.)497 2354 y(In)29 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Fl(A)1782 3416 y Fr(is)h(de\014ned)h (to)f(b)r(e)h(the)g(non-negativ)n(e)d(self-adjoin)n(t)331 3515 y(op)r(erator)g(in)h Fm(L)819 3485 y Fo(2)856 3515 y Fr(\()p Fp(R)942 3485 y Fo(2)986 3515 y Fr(\))h(asso)r(ciated)e(with) i(the)g(closable,)f(non-negativ)n(e)e(sesquilinear)i(form)969 3732 y Fc(s)1006 3744 y Fl(A)1060 3732 y Fr([)p Fm(')p Fr(])51 b(=)1327 3619 y Fh(Z)1313 3861 y Fg(R)1360 3844 y Fd(2)1424 3732 y Fj(j)p Fr(\()1478 3731 y Fm(~)1479 3732 y Fe(p)18 b Fr(+)1649 3711 y Fm(~)1633 3732 y(A)q Fr(\))p Fm(')p Fj(j)1805 3698 y Fo(2)1870 3732 y Fm(d)p Fe(x)p Fm(;)98 b(')23 b Fj(2)h Fm(C)2305 3698 y Fk(1)2299 3752 y Fo(0)2375 3732 y Fr(\()p Fp(R)2461 3698 y Fo(2)2523 3732 y Fj(n)18 b(f)p Fr(0)p Fj(g)p Fr(\))p Fm(:)531 b Fr(\(5\))331 4019 y(It)28 b(has)g(form)f(domain)h Fj(Q)p Fr(\()p Fm(S)1213 4031 y Fl(A)1267 4019 y Fr(\))h(giv)n(en)e(b)n(y)h (the)g(completion)g(of)g Fm(C)2389 3989 y Fk(1)2383 4040 y Fo(0)2460 4019 y Fr(\()p Fp(R)2546 3989 y Fo(2)2608 4019 y Fj(n)18 b(f)p Fr(0)p Fj(g)p Fr(\))27 b(with)i(resp)r(ect)f(to) 331 4075 y Fh(\000)369 4142 y Fc(s)406 4154 y Fl(A)460 4142 y Fr([)p Fj(\001)14 b Fr(])i(+)g Fj(k\001)e(k)761 4112 y Fo(2)798 4075 y Fh(\001)836 4092 y Fo(1)p Fl(=)p Fo(2)940 4142 y Fr(,)27 b(where)f Fj(k\001)14 b(k)26 b Fr(is)g(the)h Fm(L)1657 4112 y Fo(2)1694 4142 y Fr(\()p Fp(R)1780 4112 y Fo(2)1824 4142 y Fr(\))g(norm.)36 b(W)-7 b(e)27 b(shall)f(use)g(the)h(standard)f(notation)331 4266 y(H)393 4236 y Fo(1)393 4289 y Fl(A)475 4266 y Fr(for)h Fj(Q)p Fr(\()p Fm(S)753 4278 y Fl(A)807 4266 y Fr(\);)h(note)f(that)h (H)1316 4236 y Fo(1)1316 4289 y Fl(A)1394 4266 y Fr(=)22 b Fj(D)r Fr(\()p Fm(S)1635 4223 y Fo(1)p Fl(=)p Fo(2)1630 4290 y Fl(A)1740 4266 y Fr(\))28 b(with)g(the)g(graph)f(norm)g(of)g Fm(S)2735 4223 y Fo(1)p Fl(=)p Fo(2)2730 4290 y Fl(A)2839 4266 y Fr(.)497 4366 y(De\014ne)436 4624 y Fm(X)57 b Fr(=)677 4507 y Fh(\032)740 4624 y Fm(u)50 b Fr(:)h Fm(u)p Fr(\()p Fm(r)n(;)14 b Fj(\001)g Fr(\))23 b Fj(2)g Fm(L)1291 4590 y Fo(2)1328 4624 y Fr(\()p Fp(S)1412 4590 y Fo(1)1444 4624 y Fr(\))55 b(for)g Fm(r)26 b Fj(2)d Fp(R)1881 4590 y Fo(+)1998 4624 y Fr(and)2223 4494 y Fo(2)p Fl(\031)2200 4511 y Fh(Z)2207 4747 y Fo(0)2297 4624 y Fj(j)p Fm(u)p Fr(\()p Fj(\001)14 b Fm(;)g(\022)r Fr(\))p Fj(j)2570 4590 y Fo(2)2636 4624 y Fm(d\022)25 b Fj(2)f Fm(L)2879 4590 y Fk(1)2948 4624 y Fr(\()p Fp(R)3035 4590 y Fo(+)3096 4624 y Fr(\))3128 4507 y Fh(\033)3295 4624 y Fr(\(6\))331 4895 y(with)k(norm)1132 5079 y Fj(k)p Fm(u)p Fj(k)1264 5091 y Fl(X)1376 5079 y Fr(=)51 b(ess)40 b(sup)1639 5146 y Fl(r)r(>)p Fo(0)1760 4962 y Fh(\032)q(\022)1907 4949 y Fo(2)p Fl(\031)1884 4966 y Fh(Z)1890 5202 y Fo(0)1981 5079 y Fj(j)p Fm(u)p Fr(\()p Fm(r)n(;)14 b(\022)r Fr(\))p Fj(j)2252 5045 y Fo(2)2317 5079 y Fm(d\022)2401 4962 y Fh(\023)q(\033)2535 4957 y Fd(1)p 2535 4966 29 3 v 2535 4999 a(2)2577 5079 y Fm(:)695 b Fr(\(7\))p eop %%Page: 3 3 3 2 bop 538 187 a Fn(Schr)742 180 y(\177)739 187 y(odinger)31 b(opera)-6 b(tor)32 b(with)g(an)f(Ahar)n(ono)n(v-Bohm)g(ma)n(gnetic)h (field)131 b(3)331 527 y Fr(In)27 b(other)g(w)n(ords,)g Fm(X)i Fr(=)23 b Fm(L)1156 497 y Fk(1)1226 527 y Fr(\()p Fp(R)1312 497 y Fo(+)1373 527 y Fr(;)42 b Fm(L)1495 497 y Fo(2)1531 527 y Fr(\()p Fp(S)1616 497 y Fo(1)1647 527 y Fr(\);)g Fm(r)r(dr)r Fr(\))30 b(=)d Fm(L)2076 497 y Fk(1)2146 527 y Fr(\()p Fp(R)2232 497 y Fo(+)2293 527 y Fr(\))19 b Fj(\012)f Fm(L)2484 497 y Fo(2)2520 527 y Fr(\()p Fp(S)2605 497 y Fo(1)2636 527 y Fr(\).)331 707 y Fn(Theorem)32 b(1.)64 b Fi(F)-6 b(or)30 b(al)t(l)g Fm(u)23 b Fj(2)g Fr(H)1357 677 y Fo(1)1357 730 y Fl(A)1441 707 y Fi(and)1618 686 y Fm(~)1603 707 y(A)30 b Fi(de\014ne)l(d)g(by)g (\(3\),)1416 865 y Fj(k)p Fm(u)p Fj(k)1548 830 y Fo(2)1548 885 y Fl(X)1662 865 y Fj(\024)52 b Fm(C)6 b Fj(k)p Fr(\()1917 864 y Fm(~)1918 865 y Fe(p)19 b Fr(+)2088 844 y Fm(~)2073 865 y(A)p Fr(\))p Fm(u)p Fj(k)2257 830 y Fo(2)2294 865 y Fm(;)978 b Fr(\(8\))331 1028 y Fi(wher)l(e)30 b Fm(C)f Fr(=)741 960 y Fh(\000)779 1028 y Fr(min)788 1082 y Fl(k)q Fk(2)p Fg(Z)931 1028 y Fj(j)p Fm(k)22 b Fj(\000)1113 1007 y Fr(~)1102 1028 y(\011)o Fj(j)1189 960 y Fh(\001)1227 978 y Fk(\000)p Fo(1)1346 1028 y Fi(is)30 b(sharp.)331 1235 y(Pr)l(o)l(of.)66 b Fr(F)-7 b(ollo)n(wing)26 b([1,)i Fj(x)p Fr(1],)f(w)n(e)g(observ)n(e)f(that)i Fm(S)1903 1247 y Fl(A)1985 1235 y Fr(is)f(the)h(op)r(erator)1124 1435 y Fm(S)1175 1447 y Fl(A)1280 1435 y Fr(=)50 b Fj(\000)1490 1378 y Fm(@)1539 1348 y Fo(2)p 1470 1415 126 4 v 1470 1491 a Fm(@)5 b(r)1558 1467 y Fo(2)1624 1435 y Fj(\000)1717 1378 y Fr(1)p 1717 1415 42 4 v 1718 1491 a Fm(r)1798 1378 y(@)p 1778 1415 89 4 v 1778 1491 a(@)g(r)1895 1435 y Fr(+)2006 1378 y(1)p 1988 1415 78 4 v 1988 1491 a Fm(r)2027 1467 y Fo(2)2075 1317 y Fh(\022)2136 1435 y Fm(i)2196 1378 y(@)p 2175 1415 90 4 v 2175 1491 a(@)g(\022)2293 1435 y Fr(+)18 b(\011\()p Fm(\022)r Fr(\))2546 1317 y Fh(\023)331 1641 y Fr(where)29 b(the)i(op)r(erator)d Fm(K)1127 1653 y Fl(\022)1192 1641 y Fr(=)f Fm(i)1339 1608 y Fl(@)p 1322 1622 73 4 v 1322 1670 a(@)t(\022)1425 1641 y Fr(+)20 b(\011\()p Fm(\022)r Fr(\))31 b(with)f(domain)g(H)2262 1611 y Fo(1)2300 1641 y Fr(\()p Fp(S)2384 1611 y Fo(1)2415 1641 y Fr(\))h(in)f Fm(L)2634 1611 y Fo(2)2671 1641 y Fr(\()p Fp(S)2755 1611 y Fo(1)2787 1641 y Fr(\))g(has)g(eigen)n(v)-5 b(alues)331 1754 y Fm(\025)379 1766 y Fl(k)443 1754 y Fr(=)23 b Fm(k)e Fr(+)690 1733 y(~)678 1754 y(\011,)27 b Fm(k)f Fj(2)e Fp(Z)d Fr(and)27 b(eigenfunctions)1185 1935 y Fm(')1239 1947 y Fl(k)1280 1935 y Fr(\()p Fm(\022)r Fr(\))52 b(=)1622 1879 y(1)p 1562 1916 161 4 v 1562 1932 a Fj(p)p 1632 1932 92 4 v 1632 2001 a Fr(2)p Fm(\031)1733 1935 y(e)1772 1897 y Fk(\000)p Fl(i)1847 1830 y Fh(\002)1882 1897 y Fl(\022)r Fo(\()p Fl(k)q Fo(+)2038 1882 y(~)2029 1897 y(\011\))p Fk(\000)2158 1850 y Fb(R)2202 1871 y Ff(\022)2190 1919 y Fd(0)2248 1897 y Fo(\011\()p Fl(\021)r Fo(\))27 b Fl(d\021)2485 1830 y Fh(\003)2524 1935 y Fm(:)331 2134 y Fr(The)19 b(sequence)f Fj(f)p Fm(')924 2146 y Fl(k)965 2134 y Fr(\()p Fm(\022)r Fr(\))p Fj(g)h Fr(is)g(an)f (orthonormal)f(basis)h(of)h Fm(L)2117 2104 y Fo(2)2154 2134 y Fr(\()p Fp(S)2238 2104 y Fo(1)2269 2134 y Fr(\).)35 b(An)n(y)18 b Fm(u)23 b Fj(2)g Fm(L)2733 2104 y Fo(2)2770 2134 y Fr(\()p Fp(S)2855 2104 y Fo(1)2886 2134 y Fr(\))c(can)g (therefore)331 2234 y(b)r(e)28 b(written)1395 2334 y Fm(u)p Fr(\()p Fm(r)n(;)14 b(\022)r Fr(\))51 b(=)1787 2255 y Fh(X)1786 2434 y Fl(k)q Fk(2)p Fg(Z)1920 2334 y Fm(u)1968 2346 y Fl(k)2009 2334 y Fr(\()p Fm(r)r Fr(\))p Fm(')2166 2346 y Fl(k)2208 2334 y Fr(\()p Fm(\022)r Fr(\))p Fm(;)331 2566 y Fr(where)27 b Fm(u)619 2578 y Fl(k)659 2566 y Fr(\()p Fm(r)r Fr(\))e(=)874 2499 y Fh(R)930 2520 y Fo(2)p Fl(\031)913 2596 y Fo(0)1022 2566 y Fm(u)p Fr(\()p Fm(r)n(;)14 b(\022)r Fr(\))p Fm(')1301 2578 y Fl(k)1342 2566 y Fr(\()p Fm(\022)r Fr(\))29 b Fm(d\022)r Fr(,)f(and,)f(for)g(an)n (y)g Fm(u)c Fj(2)g Fm(H)2304 2536 y Fo(1)2297 2589 y Fl(A)2351 2566 y Fr(,)1021 2813 y Fc(s)1058 2825 y Fl(A)1112 2813 y Fr([)p Fm(u)p Fr(])50 b(=)1371 2734 y Fh(X)1371 2913 y Fl(k)q Fk(2)p Fg(Z)1492 2696 y Fh(\032)1581 2684 y Fk(1)1554 2700 y Fh(Z)1560 2936 y Fo(0)1637 2746 y Fh(\000)1675 2813 y Fj(j)p Fm(u)1746 2779 y Fk(0)1746 2834 y Fl(k)1786 2813 y Fr(\()p Fm(r)r Fr(\))p Fj(j)1912 2779 y Fo(2)1969 2813 y Fr(+)2062 2757 y Fm(\025)2110 2727 y Fo(2)2110 2781 y Fl(k)p 2062 2794 90 4 v 2068 2870 a Fm(r)2107 2846 y Fo(2)2162 2813 y Fj(j)p Fm(u)2233 2825 y Fl(k)2273 2813 y Fr(\()p Fm(r)r Fr(\))p Fj(j)2399 2779 y Fo(2)2438 2746 y Fh(\001)2503 2813 y Fm(r)r(dr)2624 2696 y Fh(\033)2688 2813 y Fm(:)584 b Fr(\(9\))331 3061 y(F)-7 b(or)27 b(an)n(y)g Fm(t)c Fj(2)g Fr(\(0)p Fm(;)14 b Fj(1)p Fr(\))773 3310 y Fj(j)p Fm(u)844 3322 y Fl(k)884 3310 y Fr(\()p Fm(t)p Fr(\))p Fj(j)1001 3276 y Fo(2)1150 3310 y Fr(=)110 b(2)27 b(Re)1554 3181 y Fl(t)1506 3197 y Fh(Z)1513 3433 y Fo(0)1608 3310 y Fr(\026)-47 b Fm(u)1651 3322 y Fl(k)1691 3310 y Fr(\()p Fm(r)r Fr(\))p Fm(u)1842 3276 y Fk(0)1842 3331 y Fl(k)1884 3310 y Fr(\()p Fm(r)r Fr(\))29 b Fm(dr)1150 3648 y Fj(\024)110 b Fr(2)1367 3531 y Fh(\022)1475 3519 y Fl(t)1428 3535 y Fh(Z)1434 3771 y Fo(0)1525 3648 y Fj(j)p Fm(u)1596 3614 y Fk(0)1596 3669 y Fl(k)1636 3648 y Fr(\()p Fm(r)r Fr(\))p Fj(j)1762 3614 y Fo(2)1828 3648 y Fm(r)r(dr)1949 3531 y Fh(\023)2022 3526 y Fd(1)p 2022 3535 29 3 v 2022 3568 a(2)2064 3531 y Fh(\022)2173 3519 y Fl(t)2126 3535 y Fh(Z)2132 3771 y Fo(0)2222 3648 y Fj(j)p Fm(u)2293 3660 y Fl(k)2334 3648 y Fr(\()p Fm(r)r Fr(\))p Fj(j)2460 3614 y Fo(2)2536 3592 y Fm(dr)p 2536 3629 83 4 v 2558 3705 a(r)2629 3531 y Fh(\023)2700 3526 y Fd(1)p 2700 3535 29 3 v 2700 3568 a(2)1150 3986 y Fj(\024)1382 3930 y Fr(2)p 1335 3967 136 4 v 1335 4043 a Fj(j)p Fm(\025)1406 4055 y Fl(k)1447 4043 y Fj(j)1480 3869 y Fh(\022)1589 3857 y Fl(t)1542 3873 y Fh(Z)1548 4109 y Fo(0)1638 3986 y Fj(j)p Fm(u)1709 3952 y Fk(0)1709 4007 y Fl(k)1750 3986 y Fr(\()p Fm(r)r Fr(\))p Fj(j)1876 3952 y Fo(2)1942 3986 y Fm(r)r(dr)2063 3869 y Fh(\023)2136 3864 y Fd(1)p 2136 3873 29 3 v 2136 3906 a(2)2178 3869 y Fh(\022)2239 3986 y Fm(\025)2287 3952 y Fo(2)2287 4007 y Fl(k)2390 3857 y(t)2343 3873 y Fh(Z)2349 4109 y Fo(0)2439 3986 y Fj(j)p Fm(u)2510 3998 y Fl(k)2551 3986 y Fr(\()p Fm(r)r Fr(\))p Fj(j)2677 3952 y Fo(2)2753 3930 y Fm(dr)p 2753 3967 83 4 v 2775 4043 a(r)2846 3869 y Fh(\023)2917 3864 y Fd(1)p 2917 3873 29 3 v 2917 3906 a(2)1150 4313 y Fj(\024)1382 4257 y Fr(1)p 1335 4294 136 4 v 1335 4370 a Fj(j)p Fm(\025)1406 4382 y Fl(k)1447 4370 y Fj(j)1480 4196 y Fh(\032)1570 4184 y Fk(1)1543 4200 y Fh(Z)1549 4436 y Fo(0)1626 4246 y Fh(\000)1664 4313 y Fj(j)p Fm(u)1735 4279 y Fk(0)1735 4334 y Fl(k)1775 4313 y Fr(\()p Fm(r)r Fr(\))p Fj(j)1901 4279 y Fo(2)1958 4313 y Fr(+)2051 4257 y Fm(\025)2099 4227 y Fo(2)2099 4281 y Fl(k)p 2051 4294 90 4 v 2057 4370 a Fm(r)2096 4346 y Fo(2)2150 4313 y Fj(j)p Fm(u)2221 4325 y Fl(k)2262 4313 y Fr(\()p Fm(r)r Fr(\))p Fj(j)2388 4279 y Fo(2)2427 4246 y Fh(\001)2492 4313 y Fm(r)r(dr)2613 4196 y Fh(\033)2677 4313 y Fm(:)331 4562 y Fr(Hence,)28 b(on)f(using)g(\(9\),)1100 4683 y Fo(2)p Fl(\031)1077 4699 y Fh(Z)1083 4935 y Fo(0)1174 4812 y Fj(j)p Fm(u)p Fr(\()p Fm(t;)14 b(\022)r Fr(\))p Fj(j)1440 4778 y Fo(2)1505 4812 y Fm(d\022)113 b Fr(=)1876 4733 y Fh(X)1876 4912 y Fl(k)q Fk(2)p Fg(Z)2010 4812 y Fj(j)p Fm(u)2081 4824 y Fl(k)2121 4812 y Fr(\()p Fm(t)p Fr(\))p Fj(j)2238 4778 y Fo(2)1700 5086 y Fj(\024)2009 5030 y Fr(1)p 1886 5067 288 4 v 1886 5143 a(min)1895 5197 y Fl(k)q Fk(2)p Fg(Z)2038 5143 y Fj(j)p Fm(\025)2109 5155 y Fl(k)2150 5143 y Fj(j)2183 5086 y(k)p Fr(\()2256 5085 y Fm(~)2257 5086 y Fe(p)18 b Fr(+)2427 5065 y Fm(~)2411 5086 y(A)q Fr(\))p Fm(u)p Fj(k)2596 5052 y Fo(2)2632 5086 y Fm(;)p eop %%Page: 4 4 4 3 bop 331 187 a Fr(4)470 b Fn(A.)30 b(A.)h(Balinsky,)f(W.)g(D.)h(Ev) -7 b(ans)31 b(and)g(R)n(oger)g(T.)g(Lewis)331 527 y Fr(whence)c(\(8\).) 331 807 y Fn(3)91 b(Spectral)32 b(consequence)331 990 y Fr(It)h(follo)n(ws)e(from)h(\(8\))h(that)g Fm(S)1279 1002 y Fl(A)1366 990 y Fr(has)f(no)g(zero)g(mo)r(de,)i(and)e(hence)h Fm(S)2531 947 y Fo(1)p Fl(=)p Fo(2)2526 1015 y Fl(A)2667 990 y Fr(is)g(injectiv)n(e,)h(and)e(has)331 1110 y(dense)27 b(domain)h Fj(D)r Fr(\()p Fm(S)1007 1067 y Fo(1)p Fl(=)p Fo(2)1002 1134 y Fl(A)1111 1110 y Fr(\))g(and)g(range)e Fj(R)p Fr(\()p Fm(S)1717 1067 y Fo(1)p Fl(=)p Fo(2)1712 1134 y Fl(A)1822 1110 y Fr(\))i(in)g Fm(L)2036 1080 y Fo(2)2073 1110 y Fr(\()p Fp(R)2159 1080 y Fo(2)2202 1110 y Fr(\).)38 b(Let)28 b Fm(D)2515 1080 y Fo(1)2513 1133 y Fl(A)2594 1110 y Fr(denote)g(the)g(completion)331 1229 y(of)f Fj(D)r Fr(\()p Fm(S)579 1186 y Fo(1)p Fl(=)p Fo(2)574 1254 y Fl(A)684 1229 y Fr(\))h(with)g(resp)r(ect)f(to)1198 1433 y Fj(k)p Fm(')p Fj(k)1336 1451 y Fl(D)1392 1431 y Fd(1)1390 1471 y Ff(A)1491 1433 y Fr(:=)51 b Fj(k)p Fm(S)1728 1390 y Fo(1)p Fl(=)p Fo(2)1723 1458 y Fl(A)1831 1433 y Fm(')p Fj(k)g Fr(=)f Fj(k)p Fr(\()2166 1432 y Fm(~)2167 1433 y Fe(p)18 b Fr(+)2337 1412 y Fm(~)2321 1433 y(A)q Fr(\))p Fm(')p Fj(k)p Fm(:)719 b Fr(\(10\))331 1631 y(This)26 b(is)h(not)f(a)g(subspace)g(of)g Fm(L)1312 1601 y Fo(2)1349 1631 y Fr(\()p Fp(R)1435 1601 y Fo(2)1478 1631 y Fr(\),)i(but)f(in)f(view)h(of)f(the)h(Laptev-W)-7 b(eidl)26 b(inequalit)n(y)g(\(4\))h(it)g(is)331 1731 y(a)g(functional)h(space.)497 1831 y(In)g(the)g(next)f(theorem)g(w)n(e) h(need)f(the)h(space)377 2016 y Fm(Y)42 b Fr(=)555 1948 y Fh(\010)603 2016 y Fm(u)23 b Fr(:)51 b Fm(u)p Fr(\()p Fm(r)n(;)14 b Fj(\001)g Fr(\))23 b Fj(2)g Fm(L)1127 1981 y Fk(1)1197 2016 y Fr(\()p Fp(S)1281 1981 y Fo(1)1313 2016 y Fr(\))k(for)h Fm(r)d Fj(2)f Fp(R)1695 1981 y Fo(+)1811 2016 y Fr(and)69 b Fm(ess)27 b(sup)2022 2089 y Fl(\022)r Fk(2)p Fo(\(0)p Fl(;)p Fo(2)p Fl(\031)r Fo(\))2300 2016 y Fj(j)p Fm(u)p Fr(\()p Fj(\001)14 b Fm(;)g(\022)r Fr(\))p Fj(j)24 b(2)f Fm(L)2732 1981 y Fo(1)2769 2016 y Fr(\()p Fp(R)2855 1981 y Fo(+)2916 2016 y Fm(;)42 b(r)r(dr)r Fr(\))3134 1948 y Fh(\011)3184 2016 y Fm(;)47 b Fr(\(11\))331 2262 y(with)28 b(norm)1277 2436 y Fj(k)p Fm(u)p Fj(k)1409 2448 y Fl(Y)1516 2436 y Fr(=)1658 2306 y Fk(1)1631 2323 y Fh(Z)1638 2559 y Fo(0)1728 2436 y Fm(ess)f(sup)1736 2509 y Fl(\022)r Fk(2)p Fo(\(0)p Fl(;)p Fo(2)p Fl(\031)r Fo(\))2015 2436 y Fj(j)p Fm(u)p Fr(\()p Fm(r)n(;)14 b(\022)r Fr(\))p Fj(j)28 b Fm(r)r(dr)n(;)800 b Fr(\(12\))331 2685 y(that)28 b(is)837 2787 y Fm(Y)69 b Fr(=)50 b Fm(L)1126 2753 y Fo(1)1163 2787 y Fr(\()p Fp(R)1249 2753 y Fo(+)1311 2787 y Fr(;)41 b Fm(L)1432 2753 y Fk(1)1502 2787 y Fr(\()p Fp(S)1586 2753 y Fo(1)1618 2787 y Fr(\);)g Fm(r)r(dr)r Fr(\))53 b(=)e Fm(L)2093 2753 y Fo(1)2129 2787 y Fr(\()p Fp(R)2216 2753 y Fo(+)2277 2787 y Fm(;)41 b(r)r(dr)r Fr(\))21 b Fj(\012)d Fm(L)2655 2753 y Fk(1)2725 2787 y Fr(\()p Fp(S)2809 2753 y Fo(1)2840 2787 y Fr(\))p Fm(:)331 3042 y Fn(Theorem)38 b(2.)87 b Fi(L)l(et)34 b Fm(V)52 b Fj(2)33 b Fm(Y)18 b Fi(.)55 b(Then)36 b Fm(V)54 b Fi(is)35 b(form)h(b)l(ounde)l(d)f(r)l(elative)h(to)f Fm(S)2782 3054 y Fl(A)2871 3042 y Fi(with)h Fm(S)3108 3054 y Fl(A)3162 3042 y Fi(-b)l(ound)331 3141 y(zer)l(o,)g(and)f(henc)l(e)g Fm(S)988 3153 y Fl(A)1064 3141 y Fr(+)21 b Fm(V)53 b Fi(is)35 b(de\014ne)l(d)g(as)f(a)h(form)g(sum)f(with)h(form)g(domain)h Fj(Q)p Fr(\()p Fm(S)3046 3153 y Fl(A)3100 3141 y Fr(\))c(=)f(H)3322 3111 y Fo(1)3322 3164 y Fl(A)3376 3141 y Fi(.)331 3241 y(Mor)l(e)l(over,)g(the)f(essential)h(sp)l(e)l(ctr)l(a)e(of)i Fm(S)1620 3253 y Fl(A)1692 3241 y Fr(+)18 b Fm(V)49 b Fi(and)30 b Fm(S)2084 3253 y Fl(A)2168 3241 y Fi(c)l(oincide.)331 3456 y(Pr)l(o)l(of.)114 b Fr(W)-7 b(e)40 b(shall)g(pro)n(v)n(e)e(that)i Fm(S)1508 3412 y Fk(\000)p Fo(1)p Fl(=)p Fo(2)1503 3480 y Fl(A)1664 3456 y Fm(V)19 b(S)1787 3412 y Fk(\000)p Fo(1)p Fl(=)p Fo(2)1782 3480 y Fl(A)1982 3456 y Fr(is)40 b(compact)f(in)h Fm(L)2588 3425 y Fo(2)2625 3456 y Fr(\()p Fp(R)2711 3425 y Fo(2)2754 3456 y Fr(\).)74 b(It)40 b(is)g(su\016cien)n (t)331 3575 y(to)c(pro)n(v)n(e)g(that)h Fm(P)50 b Fj(\021)38 b Fm(S)1126 3532 y Fk(\000)p Fo(1)p Fl(=)p Fo(2)1121 3599 y Fl(A)1282 3575 y Fj(j)p Fm(V)19 b Fj(j)p Fm(S)1451 3532 y Fk(\000)p Fo(1)p Fl(=)p Fo(2)1446 3599 y Fl(A)1644 3575 y Fr(is)37 b(compact)f(and)h(hence)g(that)g Fm(T)49 b Fr(=)39 b Fj(j)p Fm(V)18 b Fj(j)2992 3545 y Fo(1)p Fl(=)p Fo(2)3097 3575 y Fm(S)3153 3532 y Fk(\000)p Fo(1)p Fl(=)p Fo(2)3148 3599 y Fl(A)3346 3575 y Fr(is)331 3675 y(compact,)27 b(since)g Fm(P)35 b Fr(=)23 b Fm(T)1126 3645 y Fk(\003)1163 3675 y Fm(T)12 b Fr(.)497 3775 y(Giv)n(en)34 b Fm(")g(>)g Fr(0,)i(c)n(ho)r(ose)d Fm(W)46 b Fj(2)35 b Fm(C)1566 3745 y Fk(1)1560 3796 y Fo(0)1637 3775 y Fr(\()p Fp(R)1723 3745 y Fo(+)1784 3775 y Fr(;)14 b Fm(L)1878 3745 y Fk(1)1948 3775 y Fr(\()p Fp(S)2032 3745 y Fo(1)2063 3775 y Fr(\)\))36 b(with)e(supp)r(ort)h(in)f(\012)2836 3787 y Fl(")2906 3775 y Fr(=)g Fm(B)t Fr(\(0)p Fm(;)14 b Fr(1)p Fm(=")p Fr(\))22 b Fj(n)331 3875 y Fm(B)t Fr(\(0)p Fm(;)14 b(")p Fr(\))26 b(and)f(suc)n(h)h(that)g Fj(k)p Fm(W)12 b Fj(k)1303 3890 y Fl(L)1349 3873 y Fa(1)1409 3890 y Fo(\()p Fg(R)1482 3873 y Fd(2)1509 3890 y Fo(\))1562 3875 y Fj(\024)23 b Fm(k)1693 3887 y Fl(")1755 3875 y Fr(for)i(some)g(constan)n(t)h Fm(k)2463 3887 y Fl(")2522 3875 y Fm(>)c Fr(0)k(and)g Fj(k)p Fm(V)33 b Fj(\000)15 b Fm(W)d Fj(k)3172 3887 y Fl(Y)3252 3875 y Fm(<)23 b(")p Fr(.)331 3999 y(Let)k Fm(')533 4011 y Fl(n)602 3999 y Fm(*)c Fr(0)k(in)h Fm(L)931 3969 y Fo(2)968 3999 y Fr(\()p Fp(R)1054 3969 y Fo(2)1097 3999 y Fr(\).)38 b(Then)27 b(with)i Fm( )1650 4011 y Fl(n)1718 3999 y Fr(=)22 b Fm(S)1861 3956 y Fk(\000)p Fo(1)p Fl(=)p Fo(2)1856 4024 y Fl(A)2017 3999 y Fm(')2071 4011 y Fl(n)2117 3999 y Fr(,)55 b Fm( )2249 4011 y Fl(n)2318 3999 y Fm(*)23 b Fr(0)k(in)h Fm(D)2661 3969 y Fo(1)2659 4022 y Fl(A)2740 3999 y Fr(and)765 4200 y Fj(k)p Fm(T)12 b(')922 4212 y Fl(n)966 4200 y Fj(k)1008 4165 y Fo(2)1155 4200 y Fr(=)111 b Fj(kj)p Fm(V)18 b Fj(j)1495 4143 y Fd(1)p 1495 4152 29 3 v 1495 4185 a(2)1538 4200 y Fm( )1592 4212 y Fl(n)1637 4200 y Fj(k)1679 4165 y Fo(2)1766 4200 y Fj(\024)51 b(kj)p Fm(W)12 b Fj(j)2070 4143 y Fd(1)p 2069 4152 V 2069 4185 a(2)2112 4200 y Fm( )2166 4212 y Fl(n)2211 4200 y Fj(k)2253 4165 y Fo(2)2308 4200 y Fr(+)18 b Fj(kj)p Fm(V)37 b Fj(\000)18 b Fm(W)12 b Fj(j)2747 4143 y Fd(1)p 2747 4152 V 2747 4185 a(2)2789 4200 y Fm( )2843 4212 y Fl(n)2889 4200 y Fj(k)2931 4165 y Fo(2)1155 4367 y Fj(\024)111 b Fm(k)1374 4379 y Fl(")1423 4254 y Fh(Z)1407 4492 y Fo(\012)1454 4500 y Ff(")1520 4367 y Fj(j)p Fm( )1597 4379 y Fl(n)1643 4367 y Fj(j)1666 4332 y Fo(2)1730 4367 y Fm(d)p Fe(x)47 b Fr(+)f Fj(k)p Fm(V)37 b Fj(\000)18 b Fm(W)12 b Fj(k)2323 4379 y Fl(Y)2380 4367 y Fj(k)p Fm( )2476 4379 y Fl(n)2520 4367 y Fj(k)2562 4332 y Fo(2)2562 4387 y Fl(X)1155 4655 y Fj(\024)111 b Fm(k)1374 4667 y Fl(")1423 4542 y Fh(Z)1407 4780 y Fo(\012)1454 4788 y Ff(")1520 4655 y Fj(j)p Fm( )1597 4667 y Fl(n)1643 4655 y Fj(j)1666 4620 y Fo(2)1730 4655 y Fm(d)p Fe(x)47 b Fr(+)f Fm(C)6 b(")p Fj(k)p Fm(S)2192 4584 y Fd(1)p 2192 4593 V 2192 4626 a(2)2178 4679 y Fl(A)2234 4655 y Fm( )2288 4667 y Fl(n)2334 4655 y Fj(k)2376 4620 y Fo(2)2463 4655 y Fr(=)50 b Fm(I)2614 4667 y Fo(1)2671 4655 y Fr(+)18 b Fm(I)2790 4667 y Fo(2)3254 4655 y Fr(\(13\))331 4969 y(sa)n(y)-7 b(,)27 b(b)n(y)h(\(8\).)40 b(On)28 b(allo)n(wing)f Fm(n)d Fj(!)h(1)p Fr(,)57 b Fm(I)1622 4981 y Fo(1)1684 4969 y Fj(!)24 b Fr(0)k(since)g Fj(j)2104 4948 y Fm(~)2088 4969 y(A)q Fj(j)g Fr(is)h(b)r(ounded)f(ab)r(o)n(v)n(e)f(and)i(a)n(w)n (a)n(y)d(from)331 5069 y(0)f(on)h(\012)572 5081 y Fl(")634 5069 y Fr(and)f(H)855 5039 y Fo(1)855 5092 y Fl(A)910 5069 y Fr(\(\012)1002 5081 y Fl(")1037 5069 y Fr(\))i(is)f(compactly)f (em)n(b)r(edded)h(in)h Fm(L)2118 5039 y Fo(2)2154 5069 y Fr(\(\012)2246 5081 y Fl(")2282 5069 y Fr(\).)37 b(It)26 b(follo)n(ws)f(that)i Fm(T)12 b(')3027 5081 y Fl(n)3094 5069 y Fj(!)23 b Fr(0)j(and)331 5169 y(the)i(theorem)f(is)g(pro)n(v)n (ed.)p eop %%Page: 5 5 5 4 bop 538 187 a Fn(Schr)742 180 y(\177)739 187 y(odinger)31 b(opera)-6 b(tor)32 b(with)g(an)f(Ahar)n(ono)n(v-Bohm)g(ma)n(gnetic)h (field)131 b(5)331 527 y(Cor)n(ollar)-6 b(y)55 b Fi(L)l(et)29 b Fm(V)42 b Fj(2)23 b Fm(Y)c Fi(.)38 b(Then)31 b(the)f(essential)g(sp)l (e)l(ctrum)f(of)i Fm(S)2468 539 y Fl(A)2551 527 y Fi(and)g Fm(S)2764 539 y Fl(A)2836 527 y Fr(+)18 b Fm(V)49 b Fi(is)30 b Fr([0)p Fm(;)14 b Fj(1)p Fr(\))p Fi(.)331 876 y(A)l(cknow)t(le)l (dgment.)114 b Fr(The)39 b(\014rst)h(t)n(w)n(o)f(authors)f(are)h (grateful)g(to)h(the)g(Europ)r(ean)e(Union)i(for)331 976 y(supp)r(ort)27 b(under)h(the)g(TMR)f(gran)n(t)g(FMRX-CT96-0001.) 331 1300 y Fn(References)372 1482 y Fr([1])41 b(A.)f(Laptev)f(and)h(T.) f(W)-7 b(eidl.)40 b(Hardy)f(inequalities)g(for)g(magnetic)g(Diric)n (hlet)h(forms.)f Fi(In)501 1581 y(Mathematic)l(al)26 b(r)l(esults)c(in)h(quantum)f(me)l(chanics)i(\(Pr)l(ague,)i(1998\))p Fr(,)d(299{305,)c(Op)r(er.)h(The-)501 1681 y(ory)27 b(Adv.)h(Appl.,)g (108,)e(Birkhuser,)h(Basel,)g(1999.)575 2795 y(A.)h(A.)g(Balinsky)575 2894 y(Sc)n(ho)r(ol)f(of)h(Mathematics)575 2994 y(Cardi\013)g(Univ)n (ersit)n(y)575 3094 y(23)f(Senghenn)n(ydd)h(Road)575 3193 y(Cardi\013)g(CF24)f(4YH)575 3293 y(W)-7 b(ales,)28 b(UK)575 3392 y(email:)37 b(BalinskyA@cardi\013.ac.uk)2072 2795 y(W.)28 b(D.)g(Ev)-5 b(ans)2072 2894 y(Sc)n(ho)r(ol)26 b(of)i(Mathematics)2072 2994 y(Cardi\013)f(Univ)n(ersit)n(y)2072 3094 y(23)f(Senghenn)n(ydd)i(Road)2072 3193 y(Cardi\013)f(CF24)g(4YH) 2072 3293 y(W)-7 b(ales,)27 b(UK)2072 3392 y(email:)36 b(Ev)-5 b(ansWD@cardi\013.ac.uk)1324 3640 y(Roger)26 b(T.)h(Lewis)1324 3739 y(Departmen)n(t)g(of)h(Mathematics)1324 3839 y(Univ)n(ersit)n(y)e(of)i(Alabama)1324 3939 y(at)f(Birmingham)1324 4038 y(Birmingham,)f(AL)i(35294-1170)1324 4138 y(USA)1324 4237 y(email:)36 b(lewis@math.uab.edu)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0010021031603--