Content-Type: multipart/mixed; boundary="-------------0009050826179" This is a multi-part message in MIME format. ---------------0009050826179 Content-Type: text/plain; name="00-339.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-339.comments" published in Journees EDP Nantes 5 au 9 juin 2000 pp. XX-1 - XX-14 ---------------0009050826179 Content-Type: text/plain; name="00-339.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-339.keywords" Lieb-Thirring inequalities, eigenvalue estimates ---------------0009050826179 Content-Type: application/postscript; name="nantes.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nantes.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: nantes.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips nantes -o 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Fz(V)2377 337 y Fv(\033)r Fw(+)2484 310 y Fr(d)p 2484 322 33 3 v 2485 363 a Fq(2)2355 419 y Fw(+)2530 397 y Fz(dx)71 743 y FA(ev)-5 b(aluate)32 b(as)h Fz(L)639 758 y Fv(\033)n(;d)766 743 y FA(=)27 b Fz(R)q FA(\()p Fz(\033)n(;)17 b(d)p FA(\))p Fz(L)1234 706 y Fu(cl)1234 768 y Fv(\033)n(;d)1333 743 y FA(.)217 916 y FB(2.)61 b FA(The)39 b(in)m(trinsic)e(link)g(b)s(et)m(w)m(een)k (sp)s(ectral)d(quan)m(tities)g(and)h(their)e(coun)m(terparts)j(in)e (\(1\))71 1037 y(distinguishes)22 b(these)i(b)s(ounds)f(from)f(other)g (v)-5 b(arian)m(ts)23 b(of)f(sp)s(ectral)g(estimates.)40 b(In)23 b(particlar,)g(the)71 1157 y(r.h.s.)44 b(of)31 b(\(1\))h(captures)g(the)h(correct)f(order)g(of)f(the)i(semi-classical) 28 b(W)-8 b(eyl)32 b(t)m(yp)s(e)h(asymptotics)3575 1121 y Fw(2)874 1408 y Fz(S)934 1423 y Fv(\033)n(;d)1033 1408 y FA(\()p Fz(V)22 b FA(;)17 b Fy(~)p FA(\))27 b(=)g(\(1)22 b(+)g Fz(o)p FA(\(1\)\))p Fz(S)1899 1367 y Fu(cl)1893 1433 y Fv(\033)n(;d)1992 1408 y FA(\()p Fz(V)f FA(;)c Fy(~)p FA(\))97 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5348 y Fu(\()p Fk(R)1665 5318 y Fj(d)1710 5348 y Fu(\).)36 b(If)25 b(on)g(the)g(other)f(hand)g(\(1\))h(fails,)g(form)n(ula)f (\(4\))h(do)r(es)f(not)71 5459 y(apply)j(to)h(all)f Fo(V)42 b Fm(2)23 b Fo(L)741 5429 y Fj(\033)r Fp(+)842 5406 y Fi(d)p 842 5415 V 843 5449 a Fh(2)887 5459 y Fu(\()p Fk(R)973 5429 y Fj(d)1018 5459 y Fu(\))28 b(in)g(general)e([5].)183 5528 y Fp(3)220 5558 y Fu(The)i(p)r(oten)n(tial)g Fo(V)46 b Fu(should)27 b(ha)n(v)n(e)g(a)g(p)r(ositiv)n(e)g(mean)h(v)-5 b(alue)27 b(and)g(\(1)19 b(+)f Fm(j)p Fo(x)p Fm(j)p Fu(\))p Fo(V)i Fu(\()p Fo(x)p Fu(\))28 b(should)g(b)r(e)g(in)n(tegrable.)1818 5807 y FA(2)p eop %%Page: 3 3 3 2 bop 236 107 a FA(Estimates)34 b(for)f(Riesz)h(means)g(of)f(eigen)m (v)-5 b(alues)34 b(ha)m(v)m(e)h(\014rst)g(b)s(een)f(studied)h(in)e([22) o(].)48 b(There)35 b(the)236 227 y(cases)i Fz(\033)e(>)d FA(1)p Fz(=)p FA(2)i(for)h Fz(d)c FA(=)h(1)j(and)g Fz(\033)h(>)31 b FA(0)k(for)g Fz(d)c Fx(\025)h FA(2)j(w)m(ere)h(settled.)52 b(The)35 b(metho)s(ds)g(of)g([22])g(do)236 348 y(not)30 b(co)m(v)m(er)i(the)e(minimal)c(admissible)j(v)-5 b(alues)30 b(of)f Fz(\033)t FA(.)43 b(In)30 b(particular,)f(for)h Fz(\033)i FA(=)27 b(0)j(and)g Fz(d)e Fx(\025)g FA(3)i(\(1\))236 468 y(turns)i(in)m(to)f(the)g(celebrated)h(Cwik)m(el-Lieb-Rosen)m(blum) e(estimate)g(on)h(the)h(n)m(um)m(b)s(er)g(of)e(b)s(ound)236 588 y(states)924 759 y Fz(S)984 774 y Fw(0)p Fv(;d)1079 759 y FA(\()p Fz(V)21 b FA(\))28 b Fx(\024)g Fz(L)1432 774 y Fw(0)p Fv(;d)1544 624 y Fs(Z)1661 759 y Fz(V)1739 708 y Fv(d=)p Fw(2)1718 781 y(+)1850 759 y Fz(dx)g FA(=)g Fz(R)q FA(\(0)p Fz(;)17 b(d)p FA(\))p Fz(S)2449 718 y Fu(cl)2443 784 y Fw(0)p Fv(;d)2537 759 y FA(\()p Fz(V)22 b FA(\))p Fz(;)114 b(d)27 b Fx(\025)h FA(3)p Fz(;)563 b FA(\(6\))236 1013 y(whic)m(h)37 b(has)f(b)s(een)h(sho)m(wn)g(in)f ([8,)g(20)o(,)g(24].)54 b(This)36 b(b)s(ound)g(on)g(its)g(turn)g (implies)e(\(1\))h(for)h Fz(\033)h(>)d FA(0)236 1134 y(and)f Fz(d)27 b Fx(\025)h FA(3.)44 b(Indeed,)620 1434 y Fz(S)680 1449 y Fv(\033)n(;d)779 1434 y FA(\()p Fz(V)21 b FA(\))84 b(=)1177 1299 y Fs(Z)1276 1325 y Ft(1)1232 1524 y Fw(0)1368 1434 y FA(#)p Fx(f)p Fz(\025)1556 1449 y Fv(n)1603 1434 y FA(\()p Fz(V)21 b FA(\))28 b Fz(<)f Fx(\000)p Fz(t)p Fx(g)2060 1367 y Fz(t)2095 1331 y Fv(\033)r Ft(\000)p Fw(1)2233 1367 y Fz(dt)p 2060 1411 259 4 v 2160 1503 a(\033)2357 1434 y FA(=)2460 1299 y Fs(Z)2560 1325 y Ft(1)2515 1524 y Fw(0)2651 1434 y Fz(S)2711 1449 y Fw(0)p Fv(;d)2806 1434 y FA(\()p Fz(V)44 b Fx(\000)23 b Fz(t)p FA(\))3128 1367 y Fz(t)3163 1331 y Fv(\033)r Ft(\000)p Fw(1)3300 1367 y Fz(dt)p 3128 1411 V 3227 1503 a(\033)1016 1704 y Fx(\024)84 b Fz(R)q FA(\(0)p Fz(;)17 b(d)p FA(\))1489 1568 y Fs(Z)1587 1594 y Ft(1)1543 1794 y Fw(0)1678 1704 y Fz(S)1744 1663 y Fu(cl)1738 1728 y Fw(0)p Fv(;d)1834 1704 y FA(\()p Fz(V)43 b Fx(\000)23 b Fz(t)p FA(\))2155 1636 y Fz(t)2190 1600 y Fv(\033)r Ft(\000)p Fw(1)2327 1636 y Fz(dt)p 2155 1681 V 2255 1772 a(\033)1017 1973 y FA(=)84 b Fz(R)q FA(\(0)p Fz(;)17 b(d)p FA(\))1489 1838 y Fs(Z)1587 1864 y Ft(1)1543 2063 y Fw(0)1678 1833 y Fs(\032)1753 1838 y(Z)61 b(Z)1925 2063 y Fv(h)p Fw(\()p Fv(\030)s(;x)p Fw(\))p Fv(<)p Ft(\000)p Fv(t)2289 1906 y Fz(dxd\030)p 2279 1950 225 4 v 2279 2042 a FA(\(2)p Fz(\031)t FA(\))2463 2013 y Fv(d)2513 1833 y Fs(\033)2614 1906 y Fz(t)2649 1870 y Fv(\033)r Ft(\000)p Fw(1)2787 1906 y Fz(dt)p 2614 1950 259 4 v 2714 2042 a(\033)1017 2212 y FA(=)84 b Fz(R)q FA(\(0)p Fz(;)17 b(d)p FA(\))p Fz(S)1538 2171 y Fu(cl)1532 2236 y Fv(\033)n(;d)1630 2212 y FA(\()p Fz(V)k FA(\))p Fz(:)236 2459 y FA(This)33 b(is)f(a)g(sp)s(ecial)g(case)h(of)g(an)f(argumen)m(t) g(b)m(y)i(Aizenman)e(and)g(Lieb)g([1],)h(who)g(sho)m(w)h(that)236 2759 y FB(F)-9 b(act)38 b(3)48 b Fl(If)35 b Fz(R)q FA(\()p Fz(\033)n(;)17 b(d)p FA(\))34 b Fl(is)h(\014nite)g(for)f(some)g Fz(d)h Fl(and)f(some)g Fz(\033)e Fx(\025)c FA(0)35 b Fl(then)1242 3006 y Fz(R)q FA(\()p Fz(\033)1414 2965 y Ft(0)1437 3006 y Fz(;)17 b(d)p FA(\))27 b Fx(\024)i Fz(R)q FA(\()p Fz(\033)n(;)17 b(d)p FA(\))99 b Fl(for)51 b(al)5 b(l)100 b Fz(\033)2537 2965 y Ft(0)2588 3006 y Fx(\025)28 b Fz(\033)n(:)882 b FA(\(7\))236 3297 y(The)29 b(other)f(remaining)e(case)j Fz(\033)j FA(=)27 b(1)p Fz(=)p FA(2)g(and)h Fz(d)g FA(=)f(1)h(has)h(b)s(een)f(settled)h(in)e ([28)o(].)42 b(Here)29 b(one)f(\014nds)236 3417 y(in)k(fact)g(a)h Fl(two-side)-5 b(d)42 b FA(estimate)894 3664 y Fz(S)960 3623 y Fu(cl)954 3698 y Fw(1)p Fv(;)1019 3671 y Fq(1)p 1018 3683 31 3 v 1018 3724 a(2)1063 3664 y FA(\()p Fz(V)21 b FA(\))28 b Fx(\024)g Fz(S)1410 3689 y Fw(1)p Fv(;)1475 3662 y Fq(1)p 1475 3674 V 1475 3715 a(2)1520 3664 y FA(\()p Fz(V)21 b FA(\))28 b Fx(\024)g FA(2)p Fz(S)1922 3623 y Fu(cl)1916 3698 y Fw(1)p Fv(;)1981 3671 y Fq(1)p 1980 3683 V 1980 3724 a(2)2025 3664 y FA(\()p Fz(V)22 b FA(\))p Fz(;)114 b(V)49 b Fx(\025)28 b FA(0)p Fz(;)33 b(V)49 b Fx(2)28 b Fz(L)2907 3623 y Fw(1)2947 3664 y FA(\()p Fy(R)5 b FA(\))p Fz(:)539 b FA(\(8\))236 3934 y(The)45 b(sharp)g(lo)m(w)m(er)f(b)s(ound)g(in)g(\(8\))g(w)m(as)h(probably)e (\014rst)i(observ)m(ed)h(in)d([12])h(and)g(the)h(sharp)236 4054 y(constan)m(t)38 b(in)f(the)h(upp)s(er)f(b)s(ound)h(has)g(b)s(een) g(found)f(in)f([15].)58 b(Comparing)35 b(w)m(eak)k(and)e(strong)236 4174 y(coupling)30 b(b)s(eha)m(viours)i(it)e(is)h(easy)h(to)f(see,)h (that)f Fz(\033)h FA(=)c(1)p Fz(=)p FA(2)i(and)i Fz(d)27 b FA(=)g(1)k(is)g(the)h(only)e(case)j(in)d(the)236 4295 y(Lieb-Thirring)38 b(scale,)43 b(where)e(suc)m(h)h(a)e(t)m(w)m(o-sided) h(estimate)f(b)m(y)h(the)g(classical)e(phase)i(space)236 4415 y(a)m(v)m(erage)34 b(is)e(p)s(ossible.)383 4592 y FB(3.)60 b FA(Let)38 b(us)h(men)m(tion)f(that)g(the)g(b)s(ound)h (from)e(b)s(elo)m(w)h(in)f(\(8\))h(induces)h(a)f(lo)m(w)m(er)g (estimate)236 4712 y([12])1633 4871 y Fz(S)1693 4886 y Fw(0)p Fv(;)p Fw(2)1787 4871 y FA(\()p Fz(V)22 b FA(\))27 b Fx(\025)h Fz(S)2140 4830 y Fu(cl)2134 4896 y Fw(0)p Fv(;)p Fw(2)2229 4871 y FA(\()p Fz(V)21 b FA(\))1272 b(\(9\))236 5077 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Fz(\036)1730 2772 y Fv(k)1772 2757 y FA(\()p Fz(\030)5 b FA(\))p Fx(j)1924 2720 y Fw(2)1963 2757 y Fz(d\030)36 b FA(on)d Fy(R)2295 2720 y FA(d)2392 2757 y(w)m(e)g(\014nd)931 2925 y Fs(X)984 3137 y Fv(k)1075 3019 y FA(\()p Fz(\026)1172 3034 y Fv(k)1237 3019 y Fx(\000)22 b FA(\003\))1442 2978 y Fv(\033)1442 3044 y Ft(\000)1585 3019 y FA(=)1745 2925 y Fs(X)1797 3137 y Fv(k)1905 2879 y Fs(\022)1978 2884 y(Z)2034 3109 y Ff(R)2082 3090 y Fr(d)2122 3019 y FA(\()p Fx(j)p Fz(\030)5 b Fx(j)2264 2978 y Fw(2)2324 3019 y Fx(\000)23 b Fz(\025)p FA(\))p Fx(j)2560 2993 y FA(~)2547 3019 y Fz(\036)2605 3034 y Fv(k)2647 3019 y FA(\()p Fz(\030)5 b FA(\))p Fx(j)2799 2978 y Fw(2)2837 3019 y Fz(d\030)2936 2879 y Fs(\023)3009 2901 y Fv(\033)3009 3118 y Ft(\000)3606 3019 y FA(\(15\))1584 3316 y Fx(\024)1745 3180 y Fs(Z)1800 3405 y Ff(R)1848 3387 y Fr(d)1888 3316 y FA(\()p Fx(j)p Fz(\030)g Fx(j)2030 3274 y Fw(2)2090 3316 y Fx(\000)23 b FA(\003\))2296 3274 y Fv(\033)2296 3340 y Ft(\000)2371 3221 y Fs(X)2424 3433 y Fv(k)2532 3316 y Fx(j)2573 3289 y FA(~)2560 3316 y Fz(\036)2618 3331 y Fv(k)2660 3316 y FA(\()p Fz(\030)5 b FA(\))p Fx(j)2812 3274 y Fw(2)2850 3316 y Fz(d\030)g(:)630 b FA(\(16\))236 3626 y(On)39 b(the)h(other)g(hand,)1133 3600 y(~)1121 3626 y Fz(\036)1179 3641 y Fv(k)1221 3626 y FA(\()p Fz(\030)5 b FA(\)are)39 b(the)g(complex)g(conjugates)h(of)f(the)h(F)-8 b(ourier)38 b(co)s(e\016cien)m(ts)i(of)236 3747 y(\(2)p Fz(\031)t FA(\))420 3711 y Ft(\000)p Fv(d=)p Fw(2)586 3747 y Fz(e)631 3711 y Fv(ix\030)765 3747 y FA(on)32 b(\012)h(with)g(resp)s(ect)g(to)g Fx(f)p Fz(\036)1787 3762 y Fv(k)1829 3747 y Fx(g)f FA(in)g Fz(L)2091 3711 y Fw(2)2131 3747 y FA(\(\012\).)44 b(Hence,)974 3904 y Fs(X)1027 4116 y Fv(k)1135 3998 y Fx(j)1176 3972 y FA(~)1163 3998 y Fz(\036)1221 4013 y Fv(k)1263 3998 y FA(\()p Fz(\030)5 b FA(\))p Fx(j)1415 3957 y Fw(2)1481 3998 y FA(=)27 b(\(2)p Fz(\031)t FA(\))1768 3957 y Ft(\000)p Fv(d)1880 3863 y Fs(Z)1935 4088 y Fw(\012)2007 3998 y Fx(j)p Fz(e)2080 3957 y Fv(ix\030)2181 3998 y Fx(j)2209 3957 y Fw(2)2248 3998 y Fz(dx)h FA(=)g(\(2)p Fz(\031)t FA(\))2670 3957 y Ft(\000)p Fv(d)2781 3863 y Fs(Z)2837 4088 y Fw(\012)2908 3998 y Fz(dx:)565 b FA(\(17\))236 4291 y(If)33 b(w)m(e)g(insert)g(\(17\))f(in)m(to)g(\(16\))g(w)m(e)h (obtain)f(\(13\))g(.)43 b Fd(\003)383 4412 y FA(Let)35 b(us)h(consider)f(the)h(Legendre)g(transformation)2330 4376 y Fw(4)2404 4412 y FA(of)f(the)h(inequalit)m(y)e(\(13\))h(for)f Fz(\033)j FA(=)32 b(1.)236 4532 y(It)h(is)f(easy)i(to)e(see)h(that)1058 4835 y(\()1096 4740 y Fs(X)1149 4952 y Fv(k)1240 4835 y FA(\()p Fz(\026)1337 4850 y Fv(k)1401 4835 y Fx(\000)23 b Fz(x)p FA(\))1594 4850 y Ft(\000)1653 4835 y FA(\))1691 4793 y Ft(^)1742 4835 y FA(\()p Fz(p)p FA(\))28 b(=)f(\()p Fz(p)22 b Fx(\000)h FA([)p Fz(p)p FA(]\))p Fz(\026)2407 4850 y Fw([)p Fv(p)p Fw(]+1)2598 4835 y FA(+)2731 4702 y Fw([)p Fv(p)p Fw(])2696 4740 y Fs(X)2704 4952 y Fv(k)r Fw(=1)2857 4835 y Fz(\026)2916 4850 y Fv(k)236 5122 y FA(while)890 5167 y Fs(\020)950 5278 y Fz(L)1016 5237 y Fu(cl)1016 5303 y Fw(1)p Fv(;d)1112 5278 y FA(v)m(ol)o(\(\012\))p Fz(x)1437 5237 y Fw(1+)1537 5210 y Fr(d)p 1538 5222 V 1539 5263 a Fq(2)1584 5167 y Fs(\021)1644 5190 y Ft(^)1712 5278 y FA(\()p Fz(p)p FA(\))k(=)h Fz(p)2017 5237 y Fw(1+)2118 5210 y Fq(2)p 2117 5222 V 2117 5263 a Fr(d)2179 5167 y Fs(\020)2239 5278 y Fz(L)2305 5237 y Fu(cl)2305 5303 y Fw(0)p Fv(;d)2401 5278 y FA(v)m(ol)o(\(\012\))2671 5167 y Fs(\021)2731 5190 y Ft(\000)2797 5163 y Fq(2)p 2796 5175 V 2796 5216 a Fr(d)2953 5211 y Fz(d)p 2869 5255 220 4 v 2869 5346 a FA(2)22 b(+)g Fz(d)3098 5278 y(:)p 236 5367 1418 4 v 348 5429 a Fp(4)386 5459 y Fu(W)-7 b(e)28 b(recall)e(that)i(the)h(Legendre)d(transformation)g Fo(f)2046 5429 y Fn(^)2095 5459 y Fu(\()p Fo(p)p Fu(\))i(of)g(a)f(con)n (v)n(ex,)f(non-negativ)n(e)g(function)i Fo(f)9 b Fu(\()p Fo(x)p Fu(\))29 b(on)236 5558 y Fk(R)290 5570 y Fp(+)379 5558 y Fu(is)e(giv)n(en)g(b)n(y)h Fo(f)845 5528 y Fn(^)893 5558 y Fu(\()p Fo(p)p Fu(\))23 b(=)g(sup)1235 5579 y Fj(x)p Fn(\025)p Fp(0)1376 5558 y Fu(\()p Fo(px)c Fm(\000)f Fo(f)9 b Fu(\()p Fo(x)p Fu(\)\))q(,)27 b Fo(p)c Fm(\025)g Fu(0.)1983 5807 y FA(5)p eop %%Page: 6 6 6 5 bop 71 107 a FA(Since)34 b Fz(f)11 b FA(\()p Fz(x)p FA(\))30 b Fx(\024)h Fz(g)t FA(\()p Fz(x)p FA(\))j(for)f(all)f Fz(x)f Fx(\025)g FA(0)j(implies)d Fz(g)1818 71 y Ft(^)1869 107 y FA(\()p Fz(p)p FA(\))f Fx(\024)h Fz(f)2191 71 y Ft(^)2241 107 y FA(\()p Fz(p)p FA(\))j(for)g(all)e Fz(p)e Fx(\025)h FA(0,)j(from)f(\(13\))g(with)71 227 y Fz(x)28 b FA(=)g(\003)k(for)g Fz(p)27 b FA(=)h Fz(n)g Fx(2)g Fy(N)47 b FA(one)33 b(reco)m(v)m(ers)i(a)d(result)h(b)m(y)g(Li)f(and)h (Y)-8 b(au)32 b([19])1112 388 y Fv(n)1062 418 y Fs(X)1069 631 y Fv(k)r Fw(=1)1222 513 y Fz(\026)1281 528 y Fv(k)1351 513 y Fx(\025)c Fz(n)1514 472 y Fw(1+)1615 445 y Fq(2)p 1615 457 33 3 v 1615 498 a Fr(d)1677 402 y Fs(\020)1737 513 y Fz(L)1803 472 y Fu(cl)1803 538 y Fw(0)p Fv(;d)1899 513 y FA(v)m(ol)o(\(\012\))2169 402 y Fs(\021)2229 425 y Ft(\000)2295 398 y Fq(2)p 2294 410 V 2294 451 a Fr(d)2451 446 y Fz(d)p 2367 490 220 4 v 2367 581 a FA(2)22 b(+)g Fz(d)2596 513 y(:)217 808 y FB(4.)57 b FA(The)39 b(harmonic)c (oscillator)g(is)h(another)i(example)e(whic)m(h)i(has)f(b)s(een)h (studied)g(in)e(con-)71 928 y(nection)c(with)g(Lieb-Thirring)f (inequalities.)41 b(Put)33 b Fz(m)28 b FA(=)g(\()p Fz(m)2324 943 y Fw(1)2364 928 y Fz(;)17 b(:)g(:)g(:)e(;)i(m)2667 943 y Fv(d)2708 928 y FA(\))32 b(and)929 1231 y Fz(V)986 1246 y Fv(m)1052 1231 y FA(\()p Fz(x)p FA(\))c(=)g(\003)22 b Fx(\000)1558 1107 y Fv(d)1504 1137 y Fs(X)1512 1349 y Fv(k)r Fw(=1)1665 1231 y Fz(m)1750 1190 y Fw(2)1750 1256 y Fv(k)1793 1231 y Fz(x)1848 1190 y Fw(2)1848 1256 y Fv(k)1891 1231 y Fz(;)114 b FA(\003)28 b Fz(>)f FA(0)p Fz(;)114 b(m)2506 1246 y Fv(k)2577 1231 y Fz(>)27 b FA(0)p Fz(:)685 b FA(\(18\))71 1531 y(Then)34 b(the)f(op)s(erator)e Fz(H)8 b FA(\()p Fz(V)1070 1546 y Fv(m)1136 1531 y Fz(;)17 b FA(\))33 b(has)g(the)g(eigen)m(v)-5 b(alues)755 1835 y Fz(\025)812 1850 y Fv(\034)855 1835 y FA(\()p Fz(V)950 1850 y Fv(m)1016 1835 y FA(\))28 b(=)g Fx(\000)p FA(\003)22 b(+)1505 1710 y Fv(d)1451 1740 y Fs(X)1459 1952 y Fv(k)r Fw(=1)1611 1835 y Fz(m)1696 1850 y Fv(k)1739 1835 y FA(\(1)g(+)g(2)p Fz(\034)2037 1850 y Fv(k)2080 1835 y FA(\))p Fz(;)114 b(\034)39 b FA(=)28 b(\()p Fz(\034)2524 1850 y Fw(1)2564 1835 y Fz(;)17 b(:)g(:)g(:)f(;)h(\034)2825 1850 y Fv(d)2865 1835 y FA(\))p Fz(;)71 2129 y FA(with)42 b Fz(\034)345 2144 y Fv(k)432 2129 y FA(=)h(0)p Fz(;)17 b FA(1)p Fz(;)g FA(2)p Fz(;)g(:)g(:)g(:)40 b FA(In)j(the)f(dimension)f Fz(d)j FA(=)g(1)e(for)f Fz(\033)48 b FA(=)c(1)e(the)g(classical)f (phase)i(space)71 2250 y(a)m(v)m(erage)33 b(equals)g(to)1384 2370 y Fz(S)1450 2329 y Fu(cl)1444 2395 y Fw(1)p Fv(;)p Fw(1)1538 2370 y FA(\()p Fz(V)1633 2385 y Fv(m)1700 2370 y FA(\))27 b(=)h(\003)1937 2329 y Fw(2)1976 2370 y Fz(=)p FA(\(4)p Fz(m)2197 2385 y Fw(1)2236 2370 y FA(\))p Fz(:)1140 b FA(\(19\))71 2540 y(On)32 b(the)h(other)g(hand,)848 2762 y Fz(S)908 2777 y Fw(1)p Fv(;)p 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FA(.)50 b(With)35 b(the)g(Lieb-Aizenman)f(argumen)m(t) h(w)m(e)h(conclude)71 3425 y(that)1053 3545 y Fz(S)1113 3560 y Fv(\033)n(;)p Fw(1)1211 3545 y FA(\()p Fz(V)1306 3560 y Fv(m)1373 3545 y FA(\))27 b Fx(\024)h Fz(S)1609 3504 y Fu(cl)1603 3570 y Fv(\033)n(;)p Fw(1)1701 3545 y FA(\()p Fz(V)1796 3560 y Fv(m)1863 3545 y FA(\))97 b(for)48 b(all)96 b Fz(\033)32 b Fx(\025)c FA(1)p Fz(:)809 b FA(\(20\))217 3715 y(A)42 b(similar)c(ev)-5 b(aluation)40 b(in)h(the)h Fz(d)p FA(-dimensional)c(case)43 b(is)e(more)g(in)m(v)m (olv)m(ed)h(and)f(has)h(b)s(een)71 3835 y(carried)27 b(out)h(b)m(y)h(De)e(la)g(Bret)m(\022)-46 b(ec)m(he)29 b([9].)42 b(W)-8 b(e)28 b(presen)m(t)i(an)e(alternativ)m(e)f (generalisation)e(to)j(higher)71 3956 y(dimensions,)g(see)g([17].)42 b(Put)28 b Fz(x)1208 3920 y Ft(0)1259 3956 y FA(=)g(\()p Fz(x)1456 3971 y Fw(1)1495 3956 y Fz(;)17 b(:)g(:)g(:)f(;)h(x)1769 3971 y Fv(d)p Ft(\000)p Fw(1)1900 3956 y FA(\))27 b(and)h Fz(V)21 b FA(\()p Fz(x)p FA(\))28 b(=)g Fz(W)14 b FA(\()p Fz(x)2690 3920 y Ft(0)2713 3956 y FA(\))e Fx(\000)g Fz(m)2937 3920 y Fw(2)2937 3982 y Fv(d)2978 3956 y Fz(x)3033 3920 y Fw(2)3033 3982 y Fv(d)3074 3956 y FA(.)42 b(In)m(tegration)71 4076 y(in)32 b(the)h Fz(d)p FA(th)f(co)s(ordinates)g(of)g Fz(x)h FA(and)g Fz(\030)k FA(giv)m(es)835 4330 y Fz(S)901 4289 y Fu(cl)895 4355 y Fv(\033)n(;d)993 4330 y FA(\()p Fz(V)22 b FA(\))83 b(=)1390 4194 y Fs(Z)61 b(Z)1606 4330 y FA(\()p Fx(j)p Fz(\030)5 b Fx(j)1748 4289 y Fw(2)1808 4330 y FA(+)22 b Fz(m)1991 4289 y Fw(2)1991 4355 y Fv(d)2032 4330 y Fz(x)2087 4289 y Fw(2)2087 4355 y Fv(d)2150 4330 y Fx(\000)g Fz(W)14 b FA(\()p Fz(x)2448 4289 y Ft(0)2472 4330 y FA(\)\))2548 4289 y Fv(\033)2548 4355 y Ft(\000)2626 4263 y Fz(dxd\030)p 2617 4307 225 4 v 2617 4398 a FA(\(2)p Fz(\031)t FA(\))2801 4370 y Fv(d)1231 4588 y FA(=)1614 4521 y(1)p 1400 4565 479 4 v 1400 4656 a(2\()p Fz(\033)26 b FA(+)c(1\))p Fz(m)1838 4671 y Fv(d)1888 4588 y Fz(S)1954 4547 y Fu(cl)1948 4613 y Fv(\033)r Fw(+1)p Fv(;d)p Ft(\000)p Fw(1)2231 4588 y FA(\()p Fz(W)14 b FA(\))p Fz(:)71 4854 y FA(Separation)38 b(of)g(v)-5 b(ariables)37 b(implies)g(that)h Fz(\025)1704 4869 y Fv(\034)1743 4850 y Fc(0)1765 4869 y Fv(;\034)1816 4881 y Fr(d)1856 4854 y FA(\()p Fz(V)22 b FA(\))38 b(=)g Fz(\025)2220 4869 y Fv(\034)2259 4850 y Fc(0)2285 4854 y FA(\()p Fz(W)14 b FA(\))26 b(+)g Fz(m)2680 4869 y Fv(d)2721 4854 y FA(\(1)g(+)g(2)p Fz(m)3070 4869 y Fv(d)3111 4854 y FA(\).)62 b(Carrying)71 4974 y(out)32 b(the)h(sum)g(o)m(v)m(er)g Fz(\034)869 4989 y Fv(d)938 4974 y Fx(\025)28 b FA(0)k(\014rst,)i(from)d(\(20\))h(for)g Fz(\033)g Fx(\025)c FA(1)k(it)g(follo)m(ws)f(that)438 5202 y Fz(S)498 5217 y Fv(\033)n(;d)597 5202 y FA(\()p Fz(V)21 b FA(\))84 b(=)997 5107 y Fs(X)994 5320 y Fv(\034)1033 5301 y Fc(0)1056 5320 y Fv(;\034)1107 5332 y Fr(d)1143 5202 y FA(\()p Fz(\025)1238 5217 y Fv(\034)1277 5198 y Fc(0)1303 5202 y FA(\()p Fz(W)14 b FA(\))22 b(+)g Fz(m)1690 5217 y Fv(d)1731 5202 y FA(\(1)g(+)g(2)p Fz(\034)2029 5217 y Fv(d)2069 5202 y FA(\)\))2145 5160 y Fv(\033)2145 5226 y Ft(\000)835 5480 y FA(=)994 5386 y Fs(X)1036 5598 y Fv(\034)1075 5579 y Fc(0)1155 5480 y Fz(S)1215 5495 y Fv(\033)n(;)p Fw(1)1313 5480 y FA(\()p Fz(\025)1408 5495 y Fv(\034)1447 5476 y Fc(0)1473 5480 y FA(\()p Fz(W)14 b FA(\))22 b Fx(\000)g Fz(m)1861 5439 y Fw(2)1861 5505 y Fv(d)1902 5480 y Fz(x)1957 5439 y Fw(2)1957 5505 y Fv(d)1998 5480 y FA(\))28 b Fx(\024)2169 5386 y Fs(X)2210 5598 y Fv(\034)2249 5579 y Fc(0)2329 5480 y Fz(S)2395 5439 y Fu(cl)2389 5505 y Fv(\033)n(;)p Fw(1)2487 5480 y FA(\()p Fz(\025)2582 5495 y Fv(\034)2621 5476 y Fc(0)2648 5480 y FA(\()p Fz(W)14 b FA(\))21 b Fx(\000)i Fz(m)3036 5439 y Fw(2)3036 5505 y Fv(d)3077 5480 y Fz(x)3132 5439 y Fw(2)3132 5505 y Fv(d)3173 5480 y FA(\))1818 5807 y(6)p eop %%Page: 7 7 7 6 bop 999 144 a Fx(\024)1160 49 y Fs(X)1201 262 y Fv(\034)1240 243 y Fc(0)1320 8 y Fs(Z)62 b(Z)1536 144 y FA(\()p Fx(j)p Fz(\030)5 b Fx(j)1678 103 y Fw(2)1738 144 y FA(+)22 b Fz(m)1921 103 y Fw(2)1921 168 y Fv(d)1962 144 y Fz(x)2017 103 y Fw(2)2017 168 y Fv(d)2080 144 y FA(+)g Fz(\025)2235 159 y Fv(\034)2274 140 y Fc(0)2301 144 y FA(\()p Fz(W)14 b FA(\)\))2521 103 y Fv(\033)2521 168 y Ft(\000)2589 76 y Fz(dx)2695 91 y Fv(d)2736 76 y Fz(d\030)2830 91 y Fv(d)p 2589 121 281 4 v 2638 212 a FA(\(2)p Fz(\031)t FA(\))999 437 y Fx(\024)1385 369 y FA(1)p 1170 414 479 4 v 1170 505 a(2\()p Fz(\033)26 b FA(+)c(1\))p Fz(m)1608 520 y Fv(d)1675 342 y Fs(X)1716 555 y Fv(\034)1755 536 y Fc(0)1819 437 y FA(\()p Fx(\000)p Fz(\025)1991 452 y Fv(\034)2030 433 y Fc(0)2056 437 y FA(\()p Fz(W)14 b FA(\)\))2276 395 y Fv(\033)r Fw(+1)2441 437 y FA(=)2769 369 y(1)p 2554 414 V 2554 505 a(2\()p Fz(\033)26 b FA(+)c(1\))p Fz(m)2992 520 y Fv(d)3042 437 y Fz(S)3102 452 y Fv(\033)r Fw(+1)p Fv(;d)p Ft(\000)p Fw(1)3386 437 y Fz(:)236 747 y FA(Hence,)34 b(w)m(e)g(ha)m(v)m(e)1499 855 y Fz(S)1559 870 y Fv(\033)n(;d)1658 855 y FA(\()p Fz(V)21 b FA(\))p 1499 900 314 4 v 1499 1003 a Fz(S)1565 969 y Fu(cl)1559 1031 y Fv(\033)n(;d)1658 1003 y FA(\()p Fz(V)g FA(\))1850 922 y Fx(\024)1965 855 y Fz(S)2025 870 y Fv(\033)r Fw(+1)p Fv(;d)p Ft(\000)p Fw(1)2308 855 y FA(\()p Fz(W)14 b FA(\))p 1965 900 525 4 v 1965 1003 a Fz(S)2031 969 y Fu(cl)2025 1031 y Fv(\033)r Fw(+1)p Fv(;d)p Ft(\000)p Fw(1)2308 1003 y FA(\()p Fz(W)g FA(\))2500 922 y Fz(:)1079 b FA(\(21\))236 1182 y(This)33 b(induction)f(pro)s(cedure)h(giv)m(es)236 1425 y FB(F)-9 b(act)38 b(5)48 b Fl(F)-7 b(or)34 b(the)h(harmonic)f (oscil)5 b(lator)34 b Fz(V)49 b FA(=)28 b(\003)22 b Fx(\000)2192 1350 y Fs(P)2297 1377 y Fv(d)2297 1454 y(k)r Fw(=1)2447 1425 y Fz(m)2532 1389 y Fw(2)2532 1451 y Fv(k)2575 1425 y Fz(x)2630 1389 y Fw(2)2630 1451 y Fv(k)2708 1425 y Fl(it)35 b(holds)879 1666 y Fz(S)939 1681 y Fv(\033)n(;d)1038 1666 y FA(\()p Fz(V)22 b FA(\))27 b Fx(\024)h Fz(S)1391 1625 y Fe(cl)1385 1691 y Fv(\033)n(;d)1484 1666 y FA(\()p Fz(V)22 b FA(\))99 b Fl(for)51 b(al)5 b(l)100 b Fz(\033)31 b Fx(\025)e FA(1)99 b Fl(and)51 b(al)5 b(l)99 b Fz(d)27 b Fx(2)i Fy(N)8 b Fz(:)476 b FA(\(22\))236 1908 y FB(Remark)37 b(2)49 b FA(F)-8 b(rom)33 b(Theorem)i(1)f(in)g(section)h(4)f(it)f(will) g(follo)m(w)f(that)j(in)f(fact)g(\(22\))g(holds)g(true)236 2028 y(for)26 b(all)e Fz(\033)32 b Fx(\025)c FA(1,)f(all)e Fz(d)i Fx(2)h Fy(N)41 b FA(and)27 b(all)d(p)s(oten)m(tials)h(of)h(the)g (t)m(yp)s(e)i Fz(V)21 b FA(\()p Fz(x)p FA(\))28 b(=)g Fz(W)14 b FA(\()p Fz(x)2980 2043 y Fw(1)3019 2028 y Fz(;)j(:)g(:)g(:)f (;)h(x)3293 2043 y Fv(d)p Ft(\000)p Fw(1)3424 2028 y FA(\))9 b Fx(\000)g Fz(m)3642 1992 y Fw(2)3642 2054 y Fv(d)3684 2028 y Fz(x)3739 1992 y Fw(2)3739 2054 y Fv(d)236 2148 y FA(for)32 b(whic)m(h)h Fz(S)730 2112 y Fu(cl)724 2174 y Fv(\033)n(;d)823 2148 y FA(\()p Fz(V)22 b FA(\))32 b(is)g(\014nite.)236 2390 y FB(5.)71 b FA(The)43 b(harmonic)d (oscillator)f(pro)m(vides)j(also)f(certain)g(coun)m(terexamples.)72 b(In)42 b(particular,)236 2510 y(one)30 b(can)g(sho)m(w)h(that)f(if)e Fz(\033)k(<)c FA(1)h(then)i(for)e(certain)g(parameter)g(\003)h(and)g Fz(m)g FA(the)g(strict)g(inequalit)m(y)236 2630 y Fz(S)296 2645 y Fv(\033)n(;d)395 2630 y FA(\()p Fz(V)490 2645 y Fv(m)556 2630 y FA(\))e Fz(>)g(S)792 2594 y Fu(cl)786 2656 y Fv(\033)n(;d)884 2630 y FA(\()p Fz(V)979 2645 y Fv(m)1046 2630 y FA(\))k(holds)h([13)o(],)g(and)g(th)m(us)1210 2856 y Fz(R)q FA(\()p Fz(\033)n(;)17 b(d)p FA(\))28 b Fz(>)f FA(1)98 b(for)48 b(all)95 b Fz(\033)32 b Fx(\024)c FA(1)p Fz(;)114 b(d)28 b Fx(2)g Fy(N)9 b Fz(:)806 b FA(\(23\))236 3081 y(An)31 b(analysis)f(of)h(the)g(lo)m(w)m(est)g(eigen)m(v)-5 b(alue)30 b(via)h(the)g(resp)s(ectiv)m(e)h(Sob)s(olev)e(em)m(b)s (edding)h(constan)m(t)236 3202 y(sho)m(ws,)k(that)d(for)g Fz(d)27 b FA(=)h(2)k(the)h(b)s(ound)g(\(23\))f(extends)i(to)f(all)d Fz(\033)i(<)27 b(\033)2697 3217 y Fw(0)2770 3202 y FA(with)32 b Fz(\033)3047 3217 y Fw(0)3114 3202 y Fx(\031)d FA(1)p Fz(:)p FA(16)j([22)o(].)383 3324 y(The)43 b(metho)s(ds)f(of)g([22)o(,)h (20)o(])f(giv)m(e)h(certain)e(explicite)h(upp)s(er)g(b)s(ounds)h(on)f (the)h(constan)m(ts)236 3444 y Fz(R)q FA(\()p Fz(\033)n(;)17 b(d)p FA(\).)67 b(In)41 b(particular,)g(the)f(b)s(est)i(kno)m(wn)f (estimates)f(on)h Fz(R)q FA(\(0)p Fz(;)17 b(d)p FA(\))39 b(can)i(b)s(e)f(found)h(in)f([20)o(].)236 3565 y(There)c(ha)m(v)m(e)g (b)s(een)g(minor)d(impro)m(v)m(emen)m(ts)i(for)f(certain)g(cases)i(of)e (higher)g(momen)m(ts,)h(see)h(e.g.)236 3685 y([6].)383 3807 y(In)c([22])h(Lieb)f(and)h(Thirring)e(p)s(osed)i(the)236 4050 y FB(Conjecture)38 b(1)49 b Fl(In)39 b(any)h(dimension)f Fz(d)h Fl(ther)-5 b(e)40 b(exists)g(a)g(\014nite)g(critic)-5 b(al)40 b(value)g Fz(\033)3317 4065 y Fe(cr)3395 4050 y FA(\()p Fz(d)p FA(\))p Fl(,)h(such)236 4171 y(that)35 b Fz(R)q FA(\()p Fz(\033)n(;)17 b(d)p FA(\))28 b(=)g(1)34 b Fl(for)h(al)5 b(l)34 b Fz(\033)e Fx(\025)c Fz(\033)1491 4186 y Fe(cr)1569 4171 y FA(\()p Fz(d)p FA(\))p Fl(.)44 b(It)35 b(is)g(exp)-5 b(e)g(cte)g(d)34 b(that)h Fz(\033)2610 4186 y Fe(cr)2688 4171 y FA(\()p Fz(d)p FA(\))27 b(=)h(1)34 b Fl(for)h Fz(d)27 b Fx(\025)i FA(3)p Fl(.)236 4412 y FA(In)k(the)g(sequel)h(w)m(e)f(state)g(our)g(results)g(to)m(w)m(ards)g (the)g(solution)e(of)h(these)i(conjectures.)236 4723 y FC(4.)68 b(Lieb-Thirring)43 b(Inequalities)h(for)f(Op)t(erator)g(V) -11 b(alued)43 b(P)l(oten)l(tials)383 4851 y FB(1.)74 b FA(Our)43 b(results)g(are)g(based)g(on)g(the)g(follo)m(wing)e (generalisation)f(of)i(the)i(Lieb-Thirring)236 4972 y(inequalities)33 b(\(1\).)49 b(Namely)-8 b(,)34 b(let)g Fz(G)g FA(b)s(e)h(a)f(separable) h(Hilb)s(ert)e(space,)j(let)e(1)3054 4987 y Fv(G)3147 4972 y FA(b)s(e)h(the)g(iden)m(tit)m(y)236 5092 y(op)s(erator)22 b(on)g Fz(G)g FA(and)g(let)g Fz(V)44 b FA(b)s(e)22 b(a)g(function)g(on) g Fy(R)2010 5056 y Fv(d)2078 5092 y FA(whic)m(h)h(tak)m(es)h(a.e.)40 b(compact)22 b(self-adjoin)m(t)e(op-)236 5212 y(erators)30 b Fz(V)22 b FA(\()p Fz(x)p FA(\))30 b(on)g Fz(G)g FA(as)g(its)g(v)-5 b(alues.)42 b(W)-8 b(e)31 b(shall)d(study)j(the)g(negativ)m(e)f(eigen)m (v)-5 b(alues)30 b Fx(f)p Fz(\025)3431 5227 y Fv(n)3478 5212 y FA(\()p Fz(V)21 b FA(;)c Fy(~)p FA(\))p Fx(g)236 5333 y FA(of)32 b(the)h(op)s(erator)956 5558 y Fz(H)8 b FA(\()p Fz(V)21 b FA(;)c Fy(~)p FA(\))28 b(=)f Fx(\000)p Fy(~)1559 5517 y Fw(2)1599 5558 y FA(\001)22 b Fx(\012)h FA(1)1851 5573 y Fv(G)1932 5558 y Fx(\000)g Fz(V)e FA(\()p Fz(x)p FA(\))98 b(on)f Fz(L)2605 5517 y Fw(2)2645 5558 y FA(\()p Fy(R)2749 5517 y Fv(d)2796 5558 y FA(\))22 b Fx(\012)g Fz(G:)1983 5807 y FA(7)p eop %%Page: 8 8 8 7 bop 71 107 a FA(W)-8 b(e)33 b(shall)e(\014nd)i(b)s(ounds)1230 227 y Fz(S)1290 242 y Fv(\033)n(;d)1389 227 y FA(\()p Fz(V)21 b FA(;)c Fy(~)p FA(\))28 b Fx(\024)g Fz(r)s FA(\()p Fz(\033)n(;)17 b(d)p FA(\))p Fz(S)2111 186 y Fu(cl)2105 252 y Fv(\033)n(;d)2203 227 y FA(\()p Fz(V)22 b FA(;)17 b Fy(~)p FA(\))985 b(\(24\))71 397 y(of)32 b(the)h(eigen)m(v)-5 b(alue)32 b(momen)m(ts)754 617 y Fz(S)814 632 y Fv(\033)n(;d)913 617 y FA(\()p Fz(V)22 b FA(;)17 b Fy(~)p FA(\))27 b(=)g(tr)1372 636 y Fv(L)1420 617 y Fq(2)1455 636 y Fw(\()p Ff(R)1530 617 y Fr(d)1566 636 y Fw(\))p Ft(\012)p Fv(G)1708 617 y Fz(H)1797 576 y Fv(\033)1789 642 y Ft(\000)1848 617 y FA(\()p Fz(V)21 b FA(;)c Fy(~)p FA(\))27 b(=)2231 522 y Fs(X)2281 732 y Fv(n)2375 617 y FA(\()p Fx(\000)p Fz(\025)2547 632 y Fv(n)2594 617 y FA(\()p Fz(V)22 b FA(;)17 b Fy(~)p FA(\)\))2885 576 y Fv(\033)71 907 y FA(in)32 b(terms)g(of)g(the)h (classical)e(coun)m(terparts)465 1160 y Fz(S)531 1118 y Fu(cl)525 1184 y Fv(\033)n(;d)624 1160 y FA(\()p Fz(V)21 b FA(;)c Fy(~)p FA(\))27 b(=)1007 1024 y Fs(Z)61 b(Z)1239 1160 y FA(tr)1315 1175 y Fv(G)1374 1160 y Fz(h)1430 1118 y Fv(\033)1430 1184 y Ft(\000)1490 1160 y FA(\()p Fz(\030)5 b(;)17 b(x)p FA(\))1759 1092 y Fz(dxd\030)p 1723 1137 278 4 v 1723 1228 a FA(\(2)p Fz(\031)t Fy(~)p FA(\))1961 1199 y Fv(d)2038 1160 y FA(=)27 b Fz(L)2207 1118 y Fu(cl)2207 1184 y Fv(\033)n(;d)2306 1160 y Fy(~)2360 1118 y Ft(\000)p Fv(d)2472 1024 y Fs(Z)2588 1160 y FA(tr)2664 1175 y Fv(G)2723 1160 y Fz(V)2802 1099 y Fv(\033)r Fw(+)2909 1072 y Fr(d)p 2909 1084 33 3 v 2910 1125 a Fq(2)2780 1181 y Ft(\000)2956 1160 y FA(\()p Fz(x)p FA(\))p Fz(dx;)71 1428 y FA(where)40 b Fz(h)p FA(\()p Fz(\030)5 b(;)17 b(x)p FA(\))38 b(=)g Fx(j)p Fz(\030)5 b Fx(j)894 1392 y Fw(2)959 1428 y Fx(\012)27 b FA(1)1112 1443 y Fv(G)1197 1428 y Fx(\000)g Fz(V)22 b FA(\()p Fz(x)p FA(\).)62 b(The)40 b(constan)m(ts)h Fz(r)s FA(\()p Fz(\033)n(;)17 b(d)p FA(\))38 b(should)h(not)f(dep)s (end)i(on)f Fz(G)71 1549 y FA(and)32 b(\(24\))g(should)g(hold)f(whenev) m(er)k(the)e(r.h.s.)44 b(is)32 b(\014nite.)43 b(It)32 b(is)g(ob)m(vious)h(that)f(\(1\))f(is)h(a)g(sp)s(ecial)71 1669 y(case)h(of)f(\(24\))g(and)1387 1789 y(1)27 b Fx(\024)i Fz(R)q FA(\()p Fz(\033)n(;)17 b(d)p FA(\))27 b Fx(\024)h Fz(r)s FA(\()p Fz(\033)n(;)17 b(d)p FA(\))p Fz(:)71 1959 y FA(If)32 b(not)h(needed)h(w)m(e)g(put)e Fy(~)c FA(=)f(1)33 b(and)f(omit)f(it)h(from)f(the)i(notation.)217 2126 y FB(2.)77 b FA(In)44 b([18)o(])g(w)m(e)g(pro)m(v)m(e)h(the)f(follo)m (wing)d(result,)46 b(whic)m(h)f(con\014rms)e(the)h(\014rst)g(part)g(of) f(the)71 2247 y(conjecture)34 b(b)m(y)f(Lieb)f(and)h(Thirring)e(with)h Fz(\033)1750 2262 y Fu(cr)1852 2247 y Fx(\024)c FA(3)p Fz(=)p FA(2.)71 2462 y FB(Theorem)37 b(1)49 b FA([A.)33 b(Laptev,)g(T.)g(W)-8 b(eidl])34 b Fl(The)g(identity)1402 2670 y Fz(R)q FA(\()p Fz(\033)n(;)17 b(d)p FA(\))28 b(=)f Fz(r)s FA(\()p Fz(\033)n(;)17 b(d)p FA(\))28 b(=)f(1)71 2879 y Fl(holds)34 b(true)h(for)g(al)5 b(l)35 b Fz(\033)c Fx(\025)d FA(3)p Fz(=)p FA(2)35 b Fl(and)f(al)5 b(l)34 b Fz(d)28 b Fx(2)g Fy(N)9 b Fl(.)71 3094 y FA(One)33 b(of)f(the)i(most)e(in)m(teresting)g(case)i(for)e(applications)f(is)h Fz(\033)h FA(=)27 b(1)33 b(and)g Fz(d)28 b FA(=)g(3.)44 b(Here)33 b(the)g(b)s(est)71 3214 y(kno)m(w)h(estimate)d(w)m(as)j Fz(R)q FA(\(1)p Fz(;)17 b FA(3\))27 b Fx(\024)h FA(5)p Fz(:)p FA(24)k([6].)43 b(In)33 b([14])f(w)m(e)i(sho)m(w)71 3430 y FB(Theorem)j(2)49 b FA([D.)32 b(Hundertmark,)h(A.)g(Laptev,)g (T.)g(W)-8 b(eidl])34 b Fl(The)g(b)-5 b(ounds)960 3696 y Fz(R)q FA(\()p Fz(\033)n(;)17 b(d)p FA(\))27 b Fx(\024)i Fz(r)s FA(\()p Fz(\033)n(;)17 b(d)p FA(\))27 b Fx(\024)1795 3556 y Fs(\032)1911 3635 y FA(4)83 b Fl(for)2257 3596 y Fw(1)p 2257 3612 36 4 v 2257 3669 a(2)2330 3635 y Fx(\024)28 b Fz(\033)j(<)d FA(1)1911 3755 y(2)83 b Fl(for)g FA(1)27 b Fx(\024)h Fz(\033)k(<)2629 3716 y Fw(3)p 2629 3733 V 2629 3790 a(2)71 3957 y Fl(hold)i(true)i(in)e(al)5 b(l)35 b(dimensions)e Fz(d)27 b Fx(2)h Fy(N)9 b Fl(.)51 b(In)34 b(p)-5 b(articular,)35 b(if)f Fz(d)28 b FA(=)f(1)35 b Fl(then)831 4166 y Fz(R)17 b FA(\(1)p Fz(=)p FA(2)p Fz(;)g FA(1\))27 b(=)g Fz(r)20 b FA(\(1)p Fz(=)p FA(2)p Fz(;)d FA(1\))26 b(=)h(2)83 b Fl(for)g Fz(\033)32 b FA(=)27 b(1)p Fz(=)p FA(2)p Fz(;)864 b FA(\(25\))1046 4311 y Fz(R)q FA(\()p Fz(\033)n(;)17 b FA(1\))28 b Fx(\024)g Fz(r)s FA(\()p Fz(\033)n(;)17 b FA(1\))27 b Fx(\024)h FA(2)83 b Fl(for)g FA(1)p Fz(=)p FA(2)27 b Fz(<)h(\033)j(<)d FA(3)p Fz(=)p FA(2)p Fz(:)586 b FA(\(26\))71 4526 y FB(Remark)37 b(3)49 b FA(The)26 b(metho)s(d)f(of)g([22])g(extends)j(to)d(systems)i 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Fw(2)2626 5493 y Fz(dx)q(:)1794 5807 y FA(10)p eop %%Page: 11 11 11 10 bop 383 107 a FA(The)49 b(term)g(on)g(the)g(r.h.s.)93 b(of)49 b(\(33\))f(do)s(es)i(not)e(dep)s(end)i(on)f Fy(~)p FA(,)k(while)48 b(the)i(quan)m(tities)236 227 y Fz(S)302 191 y Fu(cl)306 239 y Fq(3)p 306 251 31 3 v 306 292 a(2)347 266 y Fv(;)p Fw(1)406 227 y FA(\()p Fz(V)21 b FA(;)c Fy(~)p FA(\))41 b(and)h Fz(S)968 225 y Fq(3)p 968 237 V 968 278 a(2)1008 252 y Fv(;)p Fw(1)1067 227 y FA(\()p Fz(V)22 b FA(;)17 b Fy(~)p FA(\))40 b(sho)m(w)j(the)f(asymptotical)e (order)h Fz(O)s FA(\()p Fy(~)2808 191 y Ft(\000)p Fw(1)2902 227 y FA(\))g(as)h Fy(~)g Fx(!)h FA(0.)70 b(Hence,)236 375 y(the)29 b(inequalit)m(y)e(\(33\))g(pro)m(vides)i(an)f(uniform)e (estimate)h(on)h(the)h(remainder)e(term)g(to)h(the)g(W)-8 b(eyl)236 496 y(asymptotics)35 b(for)g Fz(S)1001 494 y Fq(3)p 1001 506 V 1001 547 a(2)1042 521 y Fv(;)p Fw(1)1101 496 y FA(\()p Fz(V)21 b FA(;)c Fy(~)p FA(\))35 b(for)g(sign-de\014ned)h (p)s(erturbations.)52 b(By)36 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1586 y Fv(n)1433 1571 y FA(b)s(e)47 b(the)g(negativ)m(e)g(eigen)m(v)-5 b(alues)47 b(of)f Fz(H)2886 1586 y Fv(l)2912 1571 y FA(\()p Fz(V)21 b FA(\).)86 b(W)-8 b(e)47 b(study)h(the)236 1691 y(inequalities)1386 1812 y Fz(S)1446 1827 y Fv(\033)n(;d;l)1587 1812 y FA(\()p Fz(V)21 b FA(\))28 b Fx(\024)g Fz(R)q FA(\()p Fz(\033)n(;)17 b(d;)g(l)r FA(\))p Fz(S)2314 1770 y Fu(cl)2308 1836 y Fv(\033)n(;d;l)2448 1812 y FA(\()p Fz(V)k FA(\))p Fz(;)977 b FA(\(34\))236 1982 y(where)34 b Fz(S)578 1997 y Fv(\033)n(;d;l)718 1982 y FA(\()p Fz(V)22 b FA(\))27 b(=)1004 1908 y Fs(P)1109 2011 y Fv(n)1156 1982 y FA(\()p Fx(\000)p Fz(\025)1328 1997 y Fv(n)1375 1982 y FA(\()p Fz(V)22 b FA(\)\))1568 1946 y Fv(\033)1647 1982 y FA(and)920 2247 y Fz(S)986 2206 y Fu(cl)980 2272 y Fv(\033)n(;d;l)1120 2247 y FA(\()p Fz(V)g FA(\))27 b(=)1406 2112 y Fs(Z)61 b(Z)1622 2247 y FA(\()p Fz(h)1716 2262 y Fv(l)1742 2247 y FA(\()p Fz(\030)5 b(;)17 b(x)p FA(\)\))2003 2206 y Fv(\033)2003 2272 y Ft(\000)2081 2180 y Fz(dxd\030)p 2071 2224 225 4 v 2071 2316 a FA(\(2)p Fz(\031)t FA(\))2255 2287 y Fv(d)2333 2247 y FA(=)28 b Fz(L)2503 2206 y Fu(cl)2503 2272 y Fv(\033)n(;d;l)2660 2112 y Fs(Z)2776 2247 y Fz(V)2855 2206 y Fw(1+)p Fv(\024)2833 2272 y Fw(+)2990 2247 y Fz(dx)236 2509 y FA(with)1216 2659 y Fz(L)1282 2618 y Fu(cl)1282 2684 y Fv(\033)n(;d;l)1450 2659 y FA(=)1813 2592 y(\000\()p Fz(\033)f FA(+)22 b(1\)\000\()p Fz(\024)f FA(+)h(1\))p 1564 2637 1227 4 v 1564 2728 a(2)1613 2699 y Fv(d)1653 2728 y Fz(\031)1712 2699 y Fv(d=)p Fw(2)1823 2728 y FA(\000)17 b(\()o Fz(l)r(\024)23 b FA(+)f(1\))16 b(\000\()p Fz(\024)22 b FA(+)g Fz(\033)k FA(+)c(1\))236 2886 y(and)33 b Fz(\024)28 b FA(=)f Fz(d=)p FA(2)p Fz(l)r FA(.)383 3054 y FB(2.)43 b FA(The)34 b(v)-5 b(alidit)m(y)30 b(of)i(\(34\))g(is)g(settled)h(b)m(y)236 3272 y FB(F)-9 b(act)38 b(6)48 b Fl(The)35 b(ine)-5 b(quality)34 b(\(34\))h(holds)f (true)h(if)g(and)f(only)h(if)1524 3481 y Fz(\033)c Fx(\025)d FA(0)261 b Fl(for)83 b Fz(\024)28 b(>)f FA(1)p Fz(;)1524 3601 y(\033)k(>)d 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b(3)49 b Fl(We)35 b(have)521 736 y FA(~)510 762 y Fz(L)576 777 y Fw(1)p Ft(\000)p Fv(\024;d;l)836 762 y FA(=)28 b Fz(L)1006 720 y Fe(cl)1006 786 y Fw(1)p Ft(\000)p Fv(\024;d;l)1338 762 y Fl(and)99 b Fz(L)1658 777 y Fw(1)p Ft(\000)p Fv(\024;d;l)1918 762 y FA(=)2100 694 y Fz(\031)t(\024)p 2032 739 252 4 v 2032 830 a FA(sin)16 b Fz(\031)t(\024)2293 762 y(L)2359 720 y Fe(cl)2359 786 y Fw(1)p Ft(\000)p Fv(\024;d;l)2692 762 y Fl(for)99 b Fz(\024)28 b(<)f FA(1)p Fz(:)217 1035 y FB(3.)56 b FA(Constan)m(ts)38 b(in)e(Lieb-Thirring)e(inequalities)h(for)h(higher)h(order)f(op)s (erators)h(are)g(m)m(uc)m(h)71 1155 y(less)c(studied)h(than)f(their)g (coun)m(terparts)h(for)f Fz(l)e FA(=)d(1.)45 b(No)33 b(sharp)h(v)-5 b(alues)33 b(of)g(the)h(constan)m(ts)g(are)71 1276 y(kno)m(wn,)42 b(not)c(ev)m(en)j(in)d(the)h(dimension)f Fz(d)g FA(=)g(1.)62 b(It)39 b(is)f(also)g(not)h(clear,)g(whether)i(the) e(b)s(ound)71 1396 y(\(35\))d(extends)k(to)c(op)s(erator-v)-5 b(alued)36 b(p)s(oten)m(tials.)56 b(A)38 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b(simple)d(pro)s(of)h(of)h(a)g(theorem)g(b)m(y)h(Laptev)g(and)f (W)-8 b(eidl.)174 2979 y(Preprin)m(t)33 b(\(1999\).)71 3176 y([3])49 b(Berezin)31 b(F.A.:)42 b(Co)m(v)-5 b(arian)m(t)31 b(and)g(con)m(tra)m(v)-5 b(arian)m(t)30 b(sym)m(b)s(ols)h(of)f(op)s (erators.[English)g(T)-8 b(rans-)174 3296 y(lation])30 b(Math.)j(USSR,)g FB(6)p FA(,)g(1117-1151)d(\(1972\))71 3492 y([4])49 b(Birman)27 b(M.S.:)43 b(The)30 b(sp)s(ectrum)g(of)f (singular)f(b)s(oundary)h(problems.)g(\(Russian\))g(Mat.)h(Sb.)174 3613 y(\(N.S.\))44 b FB(55)f FA(\()p FB(97)p FA(\),)j(125-174)41 b(\(1961\).)i(\(English\))f(Amer.)h(Math.)h(So)s(c.)f(T)-8 b(ransl.)43 b FB(53)p FA(,)k(23-80)174 3733 y(\(1966\))71 3930 y([5])i(Birman)57 b(M.)i(S.)g(and)g(Laptev)h(A.:)96 b(The)60 b(negativ)m(e)f(discrete)h(sp)s(ectrum)f(of)f(a)h(t)m(w)m(o-) 174 4050 y(dimensional)23 b(Sc)m(hr\177)-49 b(odinger)25 b(op)s(erator.)g(Comm.)f(Pure)i(and)f(Appl.)g(Math.,)i FB(XLIX)p FA(,)e(967-997)174 4171 y(\(1996\))71 4367 y([6])49 b(Blanc)m(hard)22 b(Ph.)i(and)e(Stubb)s(e)i(J.:)39 b(Bound)23 b(states)g(for)g(Sc)m(hr\177)-49 b(odinger)23 b(Hamiltonians:)35 b(Phase)174 4487 y(Space)e(Metho)s(ds)h(and)e (Applications.)f(Rev.)j(Math.)f(Ph)m(ys.,)h FB(35)p FA(,)f(504-547)d (\(1996\))71 4684 y([7])49 b(Buslaev)23 b(V.S.)h(and)g(F)-8 b(addeev)24 b(L.D:)f(F)-8 b(orm)m(ulas)22 b(for)h(traces)h(for)f(a)g (singular)f(Sturm-Liouville)174 4804 y(di\013eren)m(tial)31 b(op)s(erator.)h([English)f(translation],)g(Dokl.)h(AN)h(SSSR,)g FB(132)p FA(,)f(451-454)f(\(1960\))71 5001 y([8])49 b(Cwik)m(el)38 b(M.:)54 b(W)-8 b(eak)38 b(t)m(yp)s(e)h(estimates)e(for)h(singular)e(v) -5 b(alues)37 b(and)h(the)g(n)m(um)m(b)s(er)h(of)e(b)s(ound)174 5121 y(states)c(of)f(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erators.)g(T) -8 b(rans.)33 b(AMS,)g FB(224)p FA(,)g(93-100)e(\(1977\))71 5318 y([9])49 b(De)43 b(la)g(Bret)m(\022)-46 b(ec)m(he)46 b(R.:)66 b(Preuv)m(e)46 b(de)e(la)f(conjecture)i(de)f(Lieb-Thirring)e (dans)i(le)g(cas)g(des)174 5438 y(p)s(oten)m(tiels)27 b(quadratiques)i(strictemen)m(t)e(con)m(v)m(exes.)k(to)d(app)s(ear)f (in)g(Ann.)i(Inst.)f(H.)g(P)m(oincar)m(\023)-46 b(e)174 5558 y(Ph)m(ys.)34 b(th)m(\023)-46 b(eor.,)33 b FB(70)p FA(,)g(369-380)e(\(1999\))1794 5807 y(12)p eop %%Page: 13 13 13 12 bop 236 107 a FA([10])49 b(Egoro)m(v)37 b(Y)-8 b(u.)37 b(V.)h(and)f(Kon)m(trat'ev)g(V.A.:)53 b(On)37 b(sp)s(ectral)g(theory)h(of)e(elliptic)f(op)s(erators.)339 227 y(Op)s(erator)22 b(Theory:)39 b(Adv)-5 b(ances)24 b(and)e(Applications,)h FB(89)p FA(,)h(Birkh\177)-49 b(auser)23 b(V)-8 b(erlag,)23 b(Basel,)h(1996.)339 348 y(x+328)32 b(pp.)236 565 y([11])49 b(F)-8 b(addeev)37 b(L.D.)f(and)g(Zakharo)m(v)h(V.E.:)52 b(Kortew)m(eg-de)37 b(V)-8 b(ries)36 b(equation:)51 b(A)37 b(completely)339 685 y(in)m(tegrable)32 b(hamiltonian)d(system.)k(F)-8 b(unc.)33 b(Anal.)f(Appl.,)g FB(5)p FA(,)h(18-27)e(\(1971\))236 903 y([12])49 b(Glaser)34 b(V.,)i(Grosse)g(H.)f(and)g(Martin)g(A.:)49 b(Bounds)36 b(on)f(the)h(n)m(um)m(b)s(er)f(of)g(eigen)m(v)-5 b(alues)35 b(of)339 1023 y(the)e(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erator.)f(Comm)m(un.)g(Math.)h(Ph)m(ys.,)i FB(59)p FA(,)d(197-212)f(\(1978\))236 1241 y([13])49 b(Hel\013er)41 b(B.)h(and)g(Rob)s(ert)g(D.:)62 b(Riesz)41 b(means)h(of)g(b)s(ounded)g (states)h(and)f(semi-classical)339 1361 y(limit)c(connected)k(with)f(a) g(Lieb-Thirring)d(conjecture)k(I,I)s(I.)g(I)f(-Jour.)f(Asymp.)i(Anal.,) g FB(3)p FA(,)339 1481 y(91-103)31 b(\(1990\),)h(I)s(I)g(-)g(Ann.)h(de) h(l'Inst.)e(H.)h(P)m(oincare,)g FB(53)f FA(\(2\),)g(139-147)f(\(1990\)) 236 1699 y([14])49 b(Hundertmark)36 b(D.,)g(Laptev)h(A.)f(and)g(W)-8 b(eidl)35 b(T.:)50 b(New)37 b(b)s(ounds)g(on)e(the)i(Lieb-Thirring)339 1819 y(constan)m(ts.)d(to)e(app)s(ear)h(in)f(In)m(v)m(en)m(tiones)i (mathematicae.)236 2037 y([15])49 b(Hundertmark)26 b(D.,)h(Lieb)e(E.H.) i(and)e(Thomas)h(L.E.:)41 b(A)26 b(sharp)g(b)s(ound)g(for)f(an)h(eigen) m(v)-5 b(alue)339 2157 y(momen)m(t)36 b(of)g(the)h(one-dimensional)d (Sc)m(hr\177)-49 b(odinger)37 b(op)s(erator.)g(Adv.)g(Theor.)h(Math.)f (Ph)m(ys.)339 2277 y FB(2)p FA(,)c(719-731)e(\(1998\))236 2495 y([16])49 b(Laptev)40 b(A.:)59 b(Diric)m(hlet)38 b(and)i(Neumann)g(Eigen)m(v)-5 b(alue)40 b(Problems)f(on)h(Domains)e (in)h(Eu-)339 2615 y(clidean)32 b(Spaces.)i(J.)e(F)-8 b(unc.)33 b(Anal.,)f FB(151)p FA(,)h(531-545)e(\(1997\))236 2833 y([17])49 b(Laptev)33 b(A.:)44 b(On)32 b(the)h(Lieb-Thirring)e (conjecture)i(for)f(a)h(class)f(of)h(p)s(oten)m(tials.)e(Op)s(erator) 339 2953 y(Theory:)45 b(Adv.)33 b(and)g(Appl.,)f FB(110)p FA(,)h(227-234)e(\(1999\))236 3170 y([18])49 b(Laptev)27 b(A.,)h(W)-8 b(eidl)26 b(T.:)41 b(Sharp)27 b(Lieb-Thirring)e (inequalities)f(in)i(high)g(dimensions.)g(Acta)339 3291 y(Mathematica,)32 b FB(184)p FA(,)g(87-111)f(\(2000\))236 3508 y([19])49 b(Li)29 b(P)-8 b(.)30 b(and)h(Y)-8 b(au)30 b(S.-T.:)42 b(On)30 b(the)h(Sc)m(hr\177)-49 b(odinger)30 b(equation)g(and)g(the)h(eigen)m(v)-5 b(alue)30 b(problem.)339 3629 y(Comm.)i(Math.)h(Ph)m(ys.,)h FB(88)p FA(,)f(309-318)d(\(1983\)) 236 3846 y([20])49 b(Lieb,)27 b(E.H.:)41 b(The)27 b(n)m(um)m(b)s(er)g (of)f(b)s(ound)g(states)i(of)e(one)g(b)s(o)s(dy)h(Sc)m(hr\177)-49 b(odinger)26 b(op)s(erators)g(and)339 3966 y(the)33 b(W)-8 b(eyl)33 b(problem.)e(Bull.)g(Amer.)i(Math.)g(So)s(c.,)f FB(82)p FA(,)h(751-753)e(\(1976\))236 4184 y([21])49 b(Lieb,)32 b(E.H.:)44 b(Lieb-Thirring)30 b(Inequalities.)i(Preprin)m(t) h(mp)p 2633 4184 30 4 v 35 w(arc)f(00-132)f(\(2000\))236 4401 y([22])49 b(Lieb)30 b(E.H.)h(and)g(Thirring,)f(W.:)43 b(Inequalities)30 b(for)g(the)h(momen)m(ts)f(of)g(the)h(eigen)m(v)-5 b(alues)31 b(of)339 4522 y(the)d(Sc)m(hr\177)-49 b(odinger)28 b(Hamiltonian)c(and)k(their)g(relation)d(to)j(Sob)s(olev)f (inequalities.)f(Studies)i(in)339 4642 y(Math.)h(Ph)m(ys.,)i(Essa)m(ys) g(in)d(Honor)g(of)g(V)-8 b(alen)m(tine)28 b(Bargmann.,)g(Princeton,)i (269-303)d(\(1976\))236 4859 y([23])49 b(Netruso)m(v)30 b(Y.)e(and)h(W)-8 b(eidl)27 b(T.:)42 b(On)28 b(Lieb-Thirring)e (inequalities)h(for)h(higher)g(order)g(op)s(er-)339 4980 y(ators)k(with)g(critical)e(and)j(sub)s(critical)e(p)s(o)m(w)m(ers.)i (Comm.)f(Math.)g(Ph)m(ys.,)j FB(182)d FA(\(1\),)g(355-370)339 5100 y(\(1996\))236 5318 y([24])49 b(Rozen)m(blum,)30 b(G.V.:)42 b(Distribution)28 b(of)i(the)h(discrete)g(sp)s(ectrum)g(of)f (singular)f(di\013eren)m(tial)339 5438 y(op)s(erators.)35 b(Dokl.)f(AN)i(SSSR,)g FB(202)p FA(,)g(1012-1015)c(\(1972\),)j(Izv.)i (VUZo)m(v,)f(Matematik)-5 b(a,)34 b FB(1)p FA(,)339 5558 y(75-86)d(\(1976\))1959 5807 y(13)p eop %%Page: 14 14 14 13 bop 71 107 a FA([25])48 b(Ruelle)29 b(D.:)41 b(Large)30 b(v)m(olume)f(limit)d(of)j(the)h(distribution)e(of)h(c)m(haracteristic) g(exp)s(onen)m(ts)j(in)174 227 y(turbulence.)h(Comm.)f(Math.)h(Ph)m (ys.,)h FB(87)p FA(,)f(287-302)d(\(1982\))71 431 y([26])48 b(Y.Sc)m(h)m(winger:)e(On)32 b(the)h(b)s(ound)g(states)h(for)e(a)h(giv) m(en)g(p)s(oten)m(tial.)e(Pro)s(c.)i(Nat.)f(Acad.)h(Sci.)174 551 y(U.S.A.,)g FB(47)p FA(,)g(122-129)e(\(1961\))71 754 y([27])48 b(B.)41 b(Simon:)59 b(The)42 b(b)s(ound)f(state)g(of)g(w) m(eakly)g(coupled)g(Sc)m(hr\177)-49 b(odinger)41 b(op)s(erators)g(on)g (one)174 875 y(and)33 b(t)m(w)m(o)g(dimensions.)f(Ann.)h(Ph)m(ysics,)h FB(97)f FA(\(2\),)f(279-288,)f(\(1976\))71 1078 y([28])48 b(W)-8 b(eidl,)45 b(T.:)66 b(On)44 b(the)g(Lieb-Thirring)d(constan)m (ts)k Fz(L)2259 1093 y Fv(\015)t(;)p Fw(1)2402 1078 y FA(for)e Fz(\015)51 b Fx(\025)c FA(1)p Fz(=)p FA(2.)42 b(Comm.)h(Math.)174 1199 y(Ph)m(ys.,)34 b FB(178)p FA(,)f(135-146)e (\(1996\))71 1402 y([29])48 b(W)-8 b(eidl,)61 b(T.:)91 b(Remarks)56 b(on)f(virtual)g(b)s(ound)h(states)g(for)g(semi-b)s (ounded)f(op)s(erators.)174 1522 y(Comm.)31 b(P)m(art.)i(Di\013.)e (Equ.,)j FB(24)f FA(\(1&2\),)f(25-60,)f(\(1999\))71 1800 y Fb(A.)g(Laptev:)46 b(Dep)-7 b(ar)g(tment)31 b(of)h(Ma)-7 b(thema)g(tics,)33 b(R)m(o)n(y)-7 b(al)32 b(Institute)g(of)g(Technol-) 71 1921 y(ogy,)38 b(S)f(-)g(100)h(44)g(Stockholm,)h(Sweden)71 2041 y Fa(laptev@math.kth.se)71 2332 y Fb(T.)g(Weidl:)51 b(Dep)-7 b(ar)g(tment)39 b(of)g(Ma)-7 b(thema)g(tics,)39 b(R)m(o)n(y)-7 b(al)39 b(Institute)g(of)g(Technol-)71 2452 y(ogy,)f(S)f(-)g(100)h(44)g(Stockholm,)h(Sweden)71 2573 y Fa(weidl@math.kth.se)71 2693 y FA(h)m(ttp://www.math.kth.se/~w)m (eidl)1794 5807 y(14)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0009050826179--