Content-Type: multipart/mixed; boundary="-------------0511091450636" This is a multi-part message in MIME format. ---------------0511091450636 Content-Type: text/plain; name="05-383.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-383.keywords" eigenvalues Schroedinger operator PPW spherical rearrangement ---------------0511091450636 Content-Type: application/postscript; name="BL05a.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BL05a.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: BL05a.dvi %%CreationDate: Wed Nov 09 17:48:26 2005 %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: c:\programme\miktex\miktex\bin\DVIPS.EXE BL05a %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.11.09:1748 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict 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b(W)-8 b(e)386 4543 y(call)33 b Fo(\025)624 4558 y Fk(i)652 4543 y Fq(\(\012)p Fo(;)17 b(V)22 b Fq(\))35 b(its)f Fo(i)p Fq(th)i(eigen)m(v)-5 b(alue.)50 b(F)-8 b(urther,)36 b(w)m(e)g(denote)g(b)m(y)g Fo(V)2808 4558 y Fk(?)2882 4543 y Fq(the)f(radially)386 4659 y(increasing)25 b(rearrangemen)m(t)i(of)f Fo(V)21 b Fq(.)41 b(Then)28 b(the)f(follo)m(wing)c(PPW)k(t)m(yp)s(e)g(estimate) 386 4776 y(holds:)386 4972 y Fy(Theorem)49 b(2.1.)e Fh(L)-5 b(et)45 b Fo(S)1332 4987 y Fn(1)1417 4972 y Fm(\032)h Fl(R)1606 4936 y Fk(n)1704 4972 y Fh(b)-5 b(e)44 b(a)g(b)-5 b(al)5 b(l)44 b(c)-5 b(enter)g(e)g(d)44 b(at)h(the)f(origin)g(and)g(of) 386 5091 y(r)-5 b(adius)42 b Fo(R)760 5106 y Fn(1)843 5091 y Fh(and)g(let)1200 5066 y Fq(~)1185 5091 y Fo(V)63 b Fq(:)43 b Fo(S)1435 5106 y Fn(1)1516 5091 y Fm(!)f Fl(R)1724 5055 y Fn(+)1832 5091 y Fh(b)-5 b(e)42 b(some)g(r)-5 b(adial)5 b(ly)42 b(symmetric)g(p)-5 b(ositive)386 5210 y(p)g(otential)40 b(such)g(that)1237 5185 y Fq(~)1222 5210 y Fo(V)22 b Fq(\()p Fo(r)s 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Fq(\(\012)p Fo(;)17 b(V)22 b Fq(\))28 b Fm(\024)g Fo(\025)1976 631 y Fn(2)2015 616 y Fq(\()p Fo(S)2113 631 y Fn(1)2152 616 y Fo(;)2211 590 y Fq(~)2196 616 y Fo(V)22 b Fq(\))p Fo(:)486 781 y Fq(If)36 b Fo(V)59 b Fq(is)36 b(suc)m(h)j(that)d Fo(V)1302 796 y Fk(?)1379 781 y Fq(satis\014es)h(the)h(con)m(v)m(exit)m(y)h (conditions)d(stated)h(in)f(the)386 900 y(theorem,)d(the)h(b)s(est)g(b) s(ound)f(is)g(obtained)g(b)m(y)h(c)m(ho)s(osing)2521 875 y(~)2507 900 y Fo(V)50 b Fq(=)29 b Fo(V)2776 915 y Fk(?)2815 900 y Fq(.)45 b(In)34 b(this)f(case)386 1016 y(the)45 b(theorem)g(is)f(a)h(t)m(ypical)f(PPW)h(result)g(and)g (optimal)d(in)i(the)h(sense)i(that)386 1133 y(equalit)m(y)32 b(holds)f(in)g(\(2\))h(if)f(\012)h(is)g(a)g(ball)e(and)i Fo(V)49 b Fq(=)27 b Fo(V)2295 1148 y Fk(?)2335 1133 y Fq(.)43 b(F)-8 b(or)31 b(a)h(general)f(p)s(oten)m(tial)386 1249 y Fo(V)58 b Fq(w)m(e)38 b(still)c(get)j(a)f(non-trivial)e(b)s (ound)j(on)f Fo(\025)2080 1264 y Fn(2)2120 1249 y Fq(\(\012)p Fo(;)17 b(V)k Fq(\))37 b(though)g(it)e(is)i(not)f(sharp)386 1365 y(an)m(ymore.)43 b(T)-8 b(o)30 b(sho)m(w)h(that)e(our)h(Theorem)g (2.1)g(con)m(tains)g(Haile's)f(result)h([11)o(])g(as)386 1481 y(a)i(sp)s(ecial)g(case,)h(w)m(e)h(state)f(the)g(follo)m(wing)d (corollary:)386 1650 y Fy(Corollary)50 b(2.1.)e Fh(L)-5 b(et)1313 1625 y Fq(~)1298 1650 y Fo(V)69 b Fq(:)47 b Fl(R)1564 1614 y Fk(n)1663 1650 y Fm(!)g Fl(R)1876 1614 y Fn(+)1986 1650 y Fh(b)-5 b(e)45 b(a)g(r)-5 b(adial)5 b(ly)45 b(symmetric)f(p)-5 b(ositive)386 1766 y(p)g(otential)46 b(that)h(satis\014es)e(the)h(c)-5 b(onditions)46 b(a\))g(and)f(b\))h (of)g(The)-5 b(or)g(em)45 b(2.1)h(and)386 1891 y(let)41 b Fo(S)589 1906 y Fn(1)666 1891 y Fm(\032)e Fl(R)848 1854 y Fk(n)941 1891 y Fh(b)-5 b(e)41 b(the)f(b)-5 b(al)5 b(l)40 b(\(c)-5 b(enter)g(e)g(d)40 b(at)h(the)g(origin\))e(such)i(that) g Fo(\025)2953 1906 y Fn(1)2992 1891 y Fq(\(\012)p Fo(;)3159 1865 y Fq(~)3144 1891 y Fo(V)22 b Fq(\))38 b(=)386 2015 y Fo(\025)443 2030 y Fn(1)482 2015 y Fq(\()p Fo(S)580 2030 y Fn(1)620 2015 y Fo(;)678 1990 y Fq(~)664 2015 y Fo(V)21 b Fq(\))p Fh(.)45 b(Then)1421 2140 y Fo(\025)1478 2155 y Fn(2)1517 2140 y Fq(\(\012)p Fo(;)1684 2114 y Fq(~)1669 2140 y Fo(V)22 b Fq(\))28 b Fm(\024)g Fo(\025)1976 2155 y Fn(2)2015 2140 y Fq(\()p Fo(S)2113 2155 y Fn(1)2152 2140 y Fo(;)2211 2114 y Fq(~)2196 2140 y Fo(V)22 b Fq(\))p Fo(:)486 2305 y Fq(The)41 b(pro)s(of)f(of)g(Theorem)g(2.1)g(follo)m(ws) g(the)g(lines)g(of)g(the)h(pro)s(of)e(in)h([3])h(and)386 2421 y(will)j(b)s(e)i(presen)m(ted)j(in)c(Section)i(5.)84 b(Let)46 b(us)h(mak)m(e)g(a)f(few)h(remarks)f(on)g(the)386 2540 y(conditions)c(that)1099 2515 y(~)1084 2540 y Fo(V)64 b Fq(has)44 b(to)e(satisfy)-8 b(.)74 b(Condition)41 b(a\))i(is)f(not)h (a)f(v)m(ery)i(serious)386 2657 y(restriction,)j(b)s(ecause)f(an)m(y)f (b)s(ounded)g(p)s(oten)m(tial)e(can)i(b)s(e)g(shifted)g(suc)m(h)h(that) 386 2773 y Fo(V)443 2788 y Fk(?)482 2773 y Fq(\(0\))d(=)g(0.)71 b(Also)41 b Fo(V)1223 2737 y Fj(0)1202 2797 y Fk(?)1247 2773 y Fq(\(0\))h(=)h(0)f(holds)f(if)g Fo(V)63 b Fq(is)41 b(somewhat)h(regular)f(where)i(it)386 2889 y(tak)m(es)37 b(the)g(v)-5 b(alue)35 b(zero.)54 b(Moreo)m(v)m(er,)38 b(our)e(metho)s(d)g(relies)f(hea)m(vily)h(on)f(the)i(fact)386 3005 y(that)386 3199 y(\(3\))658 b Fo(\025)1226 3214 y Fn(2)1265 3199 y Fq(\()p Fo(B)1377 3214 y Fk(R)1435 3199 y Fo(;)1494 3174 y Fq(~)1479 3199 y Fo(V)21 b Fq(\))28 b Fm(\025)1728 3058 y Fg(\022)1801 3199 y Fq(1)22 b(+)1985 3131 y(2)p 1980 3176 59 4 v 1980 3267 a Fo(n)2048 3058 y Fg(\023)2138 3199 y Fo(\025)2195 3214 y Fn(1)2235 3199 y Fq(\()p Fo(B)2347 3214 y Fk(R)2404 3199 y Fo(;)2463 3174 y Fq(~)2448 3199 y Fo(V)g Fq(\))p Fo(;)386 3412 y Fq(whic)m(h)36 b(is)g(a)g(b)m(ypro)s(duct)h(of)f(our)g(pro)s(of)f (and)h(holds)g(for)f(an)m(y)i(ball)d Fo(B)2936 3427 y Fk(R)3030 3412 y Fq(and)i(an)m(y)386 3531 y(p)s(oten)m(tial)820 3505 y(~)805 3531 y Fo(V)61 b Fq(that)39 b(satis\014es)h(the)g (conditions)e(of)h(Theorem)h(2.1.)63 b(The)41 b(condi-)386 3647 y(tions)36 b(a\))g(and)h(b\))f(will)f(b)s(e)h(needed)i(to)f(sho)m (w)g(the)g(ab)s(o)m(v)m(e)h(inequalit)m(y)-8 b(,)36 b(whic)m(h)h(is)386 3763 y(equiv)-5 b(alen)m(t)29 b(to)h Fo(q)1010 3727 y Fj(00)1052 3763 y Fq(\(0\))d Fm(\024)h Fq(0)i(for)f(a)g(function)g Fo(q)34 b Fq(to)29 b(b)s(e)h(de\014ned)h(in)e(the)h(pro)s(of.)42 b(Nu-)386 3879 y(merical)36 b(studies)i(indicate)f(that)h(b\))g(is)f (somewhat)h(sharp)h(in)e(the)h(sense)i(that,)386 3995 y(for)i(example,)j(a)d(p)s(oten)m(tial)f Fo(r)1527 3959 y Fn(2)p Fj(\000)p Fk(\017)1692 3995 y Fq(\(whic)m(h)i(violates)e(b\))i (only)f(`sligh)m(tly'\))f(do)s(es)386 4112 y(not)33 b(satisfy)h(\(3\))f (for)g(ev)m(ery)i Fo(R)q Fq(.)47 b(In)34 b(this)f(case)h(the)g (statemen)m(t)g(of)f(Theorem)h(2.1)386 4228 y(ma)m(y)k(still)e(b)s(e)j (true,)h(but)f(the)g(t)m(ypical)e(sc)m(heme)j(of)e(the)h(PPW)g(pro)s (of)f(will)e(fail.)386 4344 y(F)-8 b(urthermore,)40 b(condition)d(a\))h (and)h(b\))g(will)d(allo)m(w)h(us)i(to)g(emplo)m(y)f(the)h(crucial)386 4460 y(Baumgartner-Grosse-Martin)32 b(\(BGM\))i(inequalit)m(y)g([7)o(,) h(4]:)47 b(F)-8 b(rom)32 b(a\))i(and)h(b\))386 4577 y(w)m(e)30 b(see)h(that)e Fo(V)22 b Fq(\()p Fo(r)s Fq(\))16 b(+)g Fo(r)s(V)1323 4540 y Fj(0)1347 4577 y Fq(\()p Fo(r)s Fq(\))29 b(is)g(increasing.)41 b(Consequen)m(tly)32 b Fo(r)s(V)21 b Fq(\()p Fo(r)s Fq(\))29 b(is)g(con)m(v)m(ex,)386 4693 y(whic)m(h)k(is)f(just)h(the)g(condition)e(needed)j(to)f(apply)f (the)h(BGM)f(inequalit)m(y)-8 b(.)486 4809 y(As)33 b(men)m(tioned)g(ab) s(o)m(v)m(e,)g(one)h(has)f(to)f(c)m(hose)j(carefully)d(the)h(size)g(of) f(the)i(com-)386 4925 y(parison)f(ball)g(in)g(a)h(PPW)h(estimate)f(if)f Fo(\025)1927 4940 y Fn(2)1966 4925 y Fo(=\025)2072 4940 y Fn(1)2145 4925 y Fq(is)h(a)g(non-constan)m(t)g(function)g(of)386 5042 y(the)d(ball's)e(radius.)42 b(In)31 b(the)f(case)i(of)e(the)g (Laplacian)f(on)h Fl(S)2537 5005 y Fk(n)2578 5042 y Fq(,)h(one)g (compares)f(the)386 5158 y(second)i(eigen)m(v)-5 b(alues)31 b(on)f(\012)h(and)g Fo(S)1684 5173 y Fn(1)1724 5158 y Fq(,)g(the)g(ball)e(that)h(has)h(the)h(same)e(\014rst)h(eigen-)386 5274 y(v)-5 b(alue)36 b(as)i(\012.)57 b(By)37 b(the)h(Ra)m(yleigh-F)-8 b(ab)s(er-Krahn)34 b(\(RFK\))i(inequalit)m(y)g(for)g Fl(S)3232 5238 y Fk(n)3310 5274 y Fq(it)386 5390 y(is)28 b(clear)g(that)g Fo(S)976 5405 y Fn(1)1043 5390 y Fm(\032)h Fq(\012)1219 5354 y Fk(?)1258 5390 y Fq(,)h(where)g(\012)1663 5354 y Fk(?)1731 5390 y Fq(is)e(the)h(spherically)e(symmetric)h (rearrange-)386 5506 y(men)m(t)38 b(of)g(\012.)60 b(It)39 b(has)f(also)g(b)s(e)g(sho)m(wn)h(in)f([2])g(that)g Fo(\025)2379 5521 y Fn(2)2418 5506 y Fo(=\025)2524 5521 y Fn(1)2601 5506 y Fq(on)g(a)g(geo)s(desic)h(ball)386 5623 y(in)c Fl(S)565 5587 y Fk(n)642 5623 y Fq(is)g(an)h(increasing)g(function)f (of)h(the)h(ball's)d(radius.)54 b(One)37 b(can)f(conclude)p eop %%Page: 4 4 4 3 bop 386 259 a Fi(4)604 b(RAF)-9 b(AEL)25 b(D.)h(BENGURIA)e(AND)h (HELMUT)g(LINDE)386 459 y Fq(from)32 b(these)i(t)m(w)m(o)g(facts)f (that)g(in)f Fl(S)1670 423 y Fk(n)1744 459 y Fq(an)h(estimate)f(of)h (the)g(t)m(yp)s(e)h(\(2\))f(is)f(stronger)386 575 y(than)h(the)g (inequalit)m(y)386 747 y(\(4\))716 b Fo(\025)1284 762 y Fn(2)1323 747 y Fq(\(\012\))p Fo(=\025)1575 762 y Fn(1)1615 747 y Fq(\(\012\))28 b Fm(\024)g Fo(\025)1951 762 y Fn(2)1990 747 y Fq(\(\012)2098 705 y Fk(?)2138 747 y Fq(\))p Fo(=\025)2282 762 y Fn(1)2321 747 y Fq(\(\012)2429 705 y Fk(?)2469 747 y Fq(\))p Fo(:)386 918 y Fq(It)48 b(has)g(also)g(b)s(een)g(argued)h (in)e([4])h(wh)m(y)h(the)g(situation)d(is)i(di\013eren)m(t)g(in)f(the) 386 1034 y(h)m(yp)s(erb)s(olic)33 b(space)h Fl(H)1197 998 y Fk(n)1250 1034 y Fq(.)45 b(Here)34 b(an)g(estimate)e(of)h(the)h (t)m(yp)s(e)g(\(4\))f(is)g(not)g(p)s(ossible,)386 1151 y(for)d(the)h(follo)m(wing)d(reason:)43 b(One)31 b(can)g(sho)m(w)h (that)f Fo(\025)2338 1166 y Fn(2)2377 1151 y Fo(=\025)2483 1166 y Fn(1)2553 1151 y Fq(on)f(geo)s(desic)h(balls)e(in)386 1267 y Fl(H)457 1231 y Fk(n)546 1267 y Fq(is)36 b(a)g(decreasing)g (function)g(of)f(the)i(radius.)54 b(No)m(w)36 b(supp)s(ose,)j(for)c (example,)386 1383 y(that)d(\012)h(is)f(the)h(ball)d Fo(B)1229 1398 y Fk(R)1319 1383 y Fq(with)i(v)m(ery)i(long)d(and)i (thin)e(ten)m(tacles)i(attac)m(hed)h(to)e(it.)386 1499 y(Then)d(the)g(\014rst)f(and)g(the)h(second)g(eigen)m(v)-5 b(alue)28 b(of)f(the)i(Laplacian)d(on)i(\012)h(and)f Fo(B)3317 1514 y Fk(R)386 1616 y Fq(are)34 b(almost)f(the)i(same,)g (while)f(the)h(ratio)e Fo(\025)2030 1631 y Fn(2)2069 1616 y Fo(=\025)2175 1631 y Fn(1)2249 1616 y Fq(on)h(\012)2456 1579 y Fk(?)2530 1616 y Fq(can)h(b)s(e)g(considerably)386 1732 y(less)42 b(than)h(on)f Fo(B)1032 1747 y Fk(R)1131 1732 y Fq(\(and)h(th)m(us)g(on)f(\012\).)72 b(W)-8 b(e)43 b(will)c(pro)m(v)m(e)44 b(a)e(PPW)h(inequalit)m(y)386 1848 y(of)37 b(the)g(t)m(yp)s(e)i Fo(\025)956 1863 y Fn(2)995 1848 y Fq(\(\012\))d Fm(\024)g Fo(\025)1347 1863 y Fn(2)1387 1848 y Fq(\()p Fo(S)1485 1863 y Fn(1)1524 1848 y Fq(\))h(for)g Fl(H)1824 1812 y Fk(n)1915 1848 y Fq(and)g(the)h(monotonicit)m(y)d(of)i Fo(\025)3050 1863 y Fn(2)3089 1848 y Fo(=\025)3195 1863 y Fn(1)3272 1848 y Fq(on)386 1964 y(geo)s(desic)c(balls)e(in)g(a)i(future)g (publication.)486 2081 y(T)-8 b(o)41 b(shed)i(ligh)m(t)c(on)j(the)g (question)g(whic)m(h)g(is)f(the)h(righ)m(t)e(t)m(yp)s(e)j(of)e(PPW)h (in-)386 2197 y(equalit)m(y)32 b(for)g(the)h(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erator)f(on)g(\012,)h(w)m(e)h(state)386 2379 y Fy(Theorem)j(2.2.)42 b Fh(L)-5 b(et)36 b Fo(V)50 b Fq(:)28 b Fl(R)1474 2343 y Fk(n)1555 2379 y Fm(!)g Fl(R)1749 2343 y Fn(+)1849 2379 y Fh(b)-5 b(e)35 b(a)g(spheric)-5 b(al)5 b(ly)34 b(symmetric)h(p)-5 b(otential)386 2496 y(that)35 b(satis\014es)f(the)h(c)-5 b(onditions)34 b(of)h(The)-5 b(or)g(em)33 b(2.1,)i(i.e.)572 2640 y Fq(a\))41 b Fo(V)22 b Fq(\(0\))27 b(=)h Fo(V)1113 2603 y Fj(0)1137 2640 y Fq(\(0\))f(=)h(0)34 b Fh(and)567 2756 y Fq(b\))41 b Fo(V)779 2720 y Fj(0)802 2756 y Fq(\()p Fo(r)s Fq(\))35 b Fh(exists)f(and)h(is)f (incr)-5 b(e)g(asing)34 b(and)g(c)-5 b(onvex.)386 2900 y(Then)34 b(the)h(r)-5 b(atio)1667 2998 y Fo(\025)1724 3013 y Fn(2)1764 2998 y Fq(\()p Fo(B)1876 3013 y Fk(R)1933 2998 y Fo(;)17 b(V)22 b Fq(\))p 1667 3042 427 4 v 1667 3133 a Fo(\025)1724 3148 y Fn(1)1764 3133 y Fq(\()p Fo(B)1876 3148 y Fk(R)1933 3133 y Fo(;)17 b(V)22 b Fq(\))386 3271 y Fh(is)35 b(a)f(de)-5 b(cr)g(e)g(asing)34 b(function)g(of)h Fo(R)q Fh(.)486 3453 y Fq(This)44 b(theorem)f(sho)m(ws)j(that)d(one)h (can)g(not)g(replace)g(equation)f(\(2\))h(in)f(our)386 3570 y(Theorem)35 b(2.1)f(b)m(y)i(an)f(inequalit)m(y)e(of)i(the)g(t)m (yp)s(e)g(\(4\),)g(follo)m(wing)d(the)k(same)e(rea-)386 3686 y(soning)43 b(as)h(in)f(the)h(case)h(of)e(the)h(Laplacian)e(on)i Fl(H)2331 3650 y Fk(n)2384 3686 y Fq(.)77 b(Theorem)44 b(2.2)f(will)e(b)s(e)386 3802 y(pro)m(v)m(en)34 b(in)e(Section)g(6.)502 4042 y(3.)49 b Fp(Connection)38 b(to)g(the)f(Lapla)n(cian)h(opera)-7 b(tor)37 b(in)h(Ga)n(ussian)1809 4158 y(sp)-7 b(a)n(ce)486 4332 y Fq(Recen)m(tly)f(,)39 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Fo(r)m(;)17 b(y)48 b Fq(with)c Fo(T)14 b Fq(\()p Fo(r)m(;)j(y)t Fq(\))46 b(=)i(0.)79 b(The)386 2266 y(b)s(eha)m(vior)36 b(of)g Fo(Z)970 2281 y Fk(y)1011 2266 y Fq(\()p Fo(r)s Fq(\))g(at)g Fo(r)h Fq(=)d(0)i(is)g(determined)h(b)m(y)g Fo(M)2418 2281 y Fk(y)2460 2266 y Fq(.)55 b(F)-8 b(rom)35 b(the)i(de\014nition) 386 2382 y(of)32 b Fo(M)591 2397 y Fk(y)665 2382 y Fq(w)m(e)i(get)386 2596 y(\(16\))183 b Fo(y)t(M)889 2611 y Fk(y)958 2596 y Fq(=)1071 2528 y(1)p 1071 2573 49 4 v 1071 2664 a(2)1130 2596 y(\()p Fo(y)1220 2554 y Fn(2)1281 2596 y Fm(\000)22 b Fq(1\))g Fm(\001)g Fq([\()p Fo(y)j Fm(\000)e Fq(1\))f Fm(\000)g Fq(\()p Fo(n)h Fm(\000)f Fq(2\)])g Fm(\001)g Fq([\()p Fo(y)j Fq(+)d(1\))g(+)g(\()p Fo(n)g Fm(\000)h Fq(2\)])p Fo(:)386 2787 y Fq(This)33 b(implies)d(that)1316 2949 y Fo(M)1410 2964 y Fk(y)1535 2949 y Fo(>)83 b Fq(0)97 b(for)32 b(0)c Fo(<)f(y)k(<)d Fq(1)p Fo(;)1318 3094 y(M)1412 3109 y Fn(1)1535 3094 y Fq(=)83 b(0)p Fo(:)386 3256 y Fq(and)33 b(therefore)1176 3418 y(lim)1174 3478 y Fk(r)r Fj(!)p Fn(0)1330 3418 y Fo(Z)1397 3433 y Fk(y)1438 3418 y Fq(\()p Fo(r)s Fq(\))83 b(=)g Fm(1)97 b Fq(for)32 b(0)27 b Fo(<)h(y)j(<)c Fq(1)p Fo(:)386 3632 y Fy(F)-9 b(act)37 b(4.2.)42 b Fh(Ther)-5 b(e)34 b(is)h(some)f Fo(r)1526 3647 y Fn(0)1593 3632 y Fo(>)28 b Fq(0)35 b Fh(such)g(that)g Fo(q)t Fq(\()p Fo(r)s Fq(\))27 b Fm(\024)h Fq(1)35 b Fh(for)g Fq(0)27 b Fo(<)h(r)i(<)e(r)3146 3647 y Fn(0)3220 3632 y Fh(and)386 3748 y Fo(q)t Fq(\()p Fo(r)515 3763 y Fn(0)554 3748 y Fq(\))g Fo(<)f Fq(1)p Fh(.)386 3925 y(Pr)-5 b(o)g(of.)41 b Fq(Supp)s(ose)34 b(the)g(con)m(trary)-8 b(,)34 b(i.e.,)f Fo(q)t Fq(\()p Fo(r)s Fq(\))f(\014rst)i(increases)g(a) m(w)m(a)m(y)h(from)d Fo(r)f Fq(=)e(0.)386 4041 y(Then,)k(b)s(ecause)g Fo(q)t Fq(\(0\))27 b(=)g(1)k(and)h Fo(q)t Fq(\()p Fo(R)q Fq(\))27 b(=)h(0)j(and)h(b)s(ecause)g Fo(q)j Fq(is)c(con)m(tin)m(uous)i (and)386 4158 y(di\013eren)m(tiable,)39 b(w)m(e)g(can)g(\014nd)g(t)m(w) m(o)g(p)s(oin)m(ts)f Fo(r)2074 4173 y Fn(1)2151 4158 y Fo(<)g(r)2309 4173 y Fn(2)2387 4158 y Fq(suc)m(h)i(that)45 b(^)-56 b Fo(q)41 b Fq(:=)d Fo(q)t Fq(\()p Fo(r)3184 4173 y Fn(1)3223 4158 y Fq(\))g(=)386 4274 y Fo(q)t Fq(\()p Fo(r)515 4289 y Fn(2)554 4274 y Fq(\))32 b Fo(>)f Fq(1)k(and)g Fo(q)1054 4238 y Fj(0)1077 4274 y Fq(\()p Fo(r)1159 4289 y Fn(1)1199 4274 y Fq(\))c Fo(>)h Fq(0)g Fo(>)f(q)1611 4238 y Fj(0)1635 4274 y Fq(\()p Fo(r)1717 4289 y Fn(2)1756 4274 y Fq(\).)51 b(Ev)m(en)36 b(more,)f(w)m(e)h(can)f(c)m(hose)i Fo(r)3025 4289 y Fn(1)3099 4274 y Fq(and)e Fo(r)3335 4289 y Fn(2)386 4390 y Fq(suc)m(h)c(that)36 b(^)-56 b Fo(q)33 b Fq(is)c(arbitrarily)e(close)i(to)h(one.)42 b(W)-8 b(riting)35 b(^)-56 b Fo(q)31 b Fq(=)d(1)16 b(+)g Fo(\017)29 b Fq(with)g Fo(\017)f(>)g Fq(0,)i(w)m(e)386 4506 y(can)j(calculate)e(from)h(the)h(de\014nition)e(of)h Fo(Q)1991 4521 y Fk(y)2065 4506 y Fq(that)943 4672 y Fo(Q)1020 4687 y Fn(1+)p Fk(\017)1170 4672 y Fq(=)c Fo(Q)1351 4687 y Fn(1)1413 4672 y Fq(+)22 b Fo(\017n)17 b Fq(\()p Fo(\025)1720 4687 y Fn(2)1781 4672 y Fm(\000)23 b Fq(\(1)f Fm(\000)g Fq(2)p Fo(=n)p Fq(\))17 b Fo(\025)2357 4687 y Fn(1)2396 4672 y Fq(\))22 b(+)g Fm(O)s Fq(\()p Fo(\017)2713 4631 y Fn(2)2753 4672 y Fq(\))p Fo(:)386 4834 y Fq(The)33 b(term)f(in)g(brac)m(k)m(ets)j(can)e(b)s(e)g(estimated)e(b)m(y)1196 4996 y Fo(\025)1253 5011 y Fn(2)1314 4996 y Fm(\000)23 b Fq(\(1)f Fm(\000)g Fq(2)p Fo(=n)p Fq(\))p Fo(\025)1873 5011 y Fn(1)1940 4996 y Fo(>)27 b(\025)2100 5011 y Fn(2)2162 4996 y Fm(\000)c Fo(\025)2319 5011 y Fn(1)2386 4996 y Fo(>)k Fq(0)p Fo(:)386 5158 y Fq(W)-8 b(e)33 b(can)g(also)f(assume)i (that)e Fo(Q)1555 5173 y Fn(1)1623 5158 y Fm(\025)d Fq(0,)k(b)s(ecause) h(otherwise)f Fo(q)2680 5122 y Fj(00)2723 5158 y Fq(\(0\))28 b(=)3011 5119 y Fn(2)p 2990 5135 78 4 v 2990 5192 a Fk(n)3033 5173 y Ff(2)3077 5158 y Fo(Q)3154 5173 y Fn(1)3222 5158 y Fo(<)g Fq(0)386 5274 y(and)22 b(F)-8 b(act)22 b(4.2)g(is)g (immediately)d(true.)41 b(Th)m(us,)26 b(c)m(ho)s(osing)c Fo(r)2490 5289 y Fn(1)2551 5274 y Fq(and)h Fo(r)2775 5289 y Fn(2)2837 5274 y Fq(suc)m(h)g(that)f Fo(\017)h Fq(is)386 5390 y(su\016cien)m(tly)28 b(small,)f(w)m(e)i(can)f(mak)m(e)g (sure)h(that)e Fo(Q)2201 5405 y Fn(^)-40 b Fk(q)2262 5390 y Fo(>)28 b Fq(0)p Fo(:)f Fq(W)-8 b(e)29 b(further)f(note,)h(that) 386 5506 y(in)f(view)h(of)g(\(16\))f(the)h(constan)m(t)h Fo(M)1680 5521 y Fn(^)-40 b Fk(q)1742 5506 y Fq(can)29 b(b)s(e)h(p)s(ositiv)m(e)e(or)g(negativ)m(e)i(\(dep)s(ending)386 5623 y(on)i Fo(n)p Fq(\),)h(but)g(not)f(zero)h(b)s(ecause)h(1)28 b Fo(<)34 b Fq(^)-56 b Fo(q)32 b(<)27 b Fq(2.)p eop %%Page: 9 9 9 8 bop 1237 259 a Fi(A)33 b(SECOND)g(EIGENV)-9 b(ALUE)33 b(BOUND)811 b(9)486 459 y Fq(No)m(w)35 b(consider)h(the)g(function)f Fo(T)14 b Fq(\()p Fo(r)m(;)23 b Fq(^)-55 b Fo(q)s Fq(\).)52 b(W)-8 b(e)35 b(ha)m(v)m(e)i Fo(T)14 b Fq(\()p Fo(r)2557 474 y Fn(1)2596 459 y Fo(;)24 b Fq(^)-56 b Fo(q)s Fq(\))33 b Fo(>)f Fq(0)g Fo(>)g(T)14 b Fq(\()p Fo(r)3207 474 y Fn(2)3246 459 y Fo(;)24 b Fq(^)-56 b Fo(q)t Fq(\))386 575 y(and)35 b(the)g(b)s(oundary)g(b)s(eha)m(vior)g Fo(T)14 b Fq(\(0)p Fo(;)23 b Fq(^)-55 b Fo(q)s Fq(\))31 b(=)g Fm(\0001)k Fq(and)g Fo(T)14 b Fq(\()p Fo(R)2601 590 y Fn(1)2640 575 y Fo(;)24 b Fq(^)-56 b Fo(q)t Fq(\))31 b(=)h(+)p Fm(1)p Fq(.)49 b(Th)m(us)386 691 y Fo(T)14 b Fq(\()p Fo(r)m(;)24 b Fq(^)-56 b Fo(q)s Fq(\))30 b(c)m(hanges)i(its)e (sign)f(at)h(least)g(thrice)g(on)g([0)p Fo(;)17 b(R)2329 706 y Fn(1)2368 691 y Fq(].)43 b(Consequen)m(tly)-8 b(,)33 b(w)m(e)e(can)386 807 y(\014nd)i(three)g(p)s(oin)m(ts)g(0)27 b Fo(<)32 b Fq(^)-54 b Fo(r)1347 822 y Fn(1)1415 807 y Fo(<)31 b Fq(^)-53 b Fo(r)1562 822 y Fn(2)1629 807 y Fo(<)32 b Fq(^)-53 b Fo(r)1777 822 y Fn(3)1844 807 y Fo(<)27 b(R)2021 822 y Fn(1)2094 807 y Fq(suc)m(h)34 b(that)386 989 y(\(17\))497 b Fo(Z)1129 1004 y Fn(^)-40 b Fk(q)1162 989 y Fq(\()t(^)-53 b Fo(r)1244 1004 y Fn(1)1283 989 y Fq(\))28 b Fm(\025)g Fq(0)p Fo(;)114 b(Z)1716 1004 y Fn(^)-40 b Fk(q)1749 989 y Fq(\()t(^)-53 b Fo(r)1831 1004 y Fn(2)1870 989 y Fq(\))28 b Fm(\024)g Fq(0)p Fo(;)114 b(Z)2303 1004 y Fn(^)-40 b Fk(q)2336 989 y Fq(\()t(^)-53 b Fo(r)2418 1004 y Fn(3)2457 989 y Fq(\))28 b Fm(\025)g Fq(0)p Fo(:)386 1170 y Fq(Let)33 b(us)g(de\014ne)1391 1350 y Fo(h)p Fq(\()p Fo(r)s Fq(\))27 b(=)1711 1282 y Fo(E)1789 1246 y Fn(2)1828 1282 y Fo(r)1875 1246 y Fn(2)p 1711 1327 204 4 v 1765 1418 a Fq(2)7 b(^)-56 b Fo(q)1946 1350 y Fm(\000)23 b Fq(2)7 b(^)-56 b Fo(q)2157 1325 y Fq(~)2142 1350 y Fo(V)21 b Fq(\()p Fo(r)s Fq(\))p Fo(:)386 1560 y Fq(Then)386 1769 y(\(18\))784 b Fo(Z)1416 1784 y Fn(^)-40 b Fk(q)1449 1769 y Fq(\()p Fo(r)s Fq(\))27 b(=)1713 1702 y Fo(M)1812 1717 y Fn(^)-40 b Fk(q)p 1713 1746 133 4 v 1736 1837 a Fo(r)1783 1809 y Fn(2)1877 1769 y Fq(+)22 b Fo(Q)2057 1784 y Fn(^)-40 b Fk(q)2113 1769 y Fq(+)22 b Fo(h)p Fq(\()p Fo(r)s Fq(\))p Fo(:)386 2002 y Fq(By)48 b(condition)e(b\))i(on)1302 1977 y(~)1287 2002 y Fo(V)22 b Fq(,)51 b(the)d(function)f Fo(h)2080 1966 y Fj(0)2104 2002 y Fq(\()p Fo(r)s Fq(\))g(is)g(conca)m(v)m(e.)90 b(Also)47 b Fo(h)p Fq(\(0\))53 b(=)386 2118 y Fo(h)442 2082 y Fj(0)465 2118 y Fq(\(0\))28 b(=)f(0.)43 b(W)-8 b(e)31 b(conclude)g(that)f(if)g Fo(h)1758 2082 y Fj(0)1781 2118 y Fq(\()p Fo(r)1863 2133 y Fn(0)1903 2118 y Fq(\))d Fo(<)h Fq(0)i(or)h Fo(h)p Fq(\()p Fo(r)2407 2133 y Fn(0)2446 2118 y Fq(\))d Fo(<)f Fq(0)k(for)f(some)g Fo(r)3128 2133 y Fn(0)3195 2118 y Fo(>)e Fq(0,)386 2235 y(then)33 b Fo(h)664 2199 y Fj(0)688 2235 y Fq(\()p Fo(r)s Fq(\))f(is)h(negativ)m (e)g(and)g(decreasing)g(for)g(all)e Fo(r)g(>)d(r)2498 2250 y Fn(0)2537 2235 y Fq(.)45 b(W)-8 b(e)33 b(will)e(no)m(w)j(sho)m (w)386 2351 y(that)40 b Fo(Z)677 2366 y Fn(^)-40 b Fk(q)750 2351 y Fq(cannot)41 b(ha)m(v)m(e)h(the)f(prop)s(erties)f(\(17\),)i(a)e (con)m(tradiction)g(that)g(pro)m(v)m(es)386 2467 y(F)-8 b(act)32 b(4.2:)486 2583 y(Case)h(1:)43 b(Assume)34 b Fo(M)1299 2598 y Fn(^)-40 b Fk(q)1360 2583 y Fo(>)27 b Fq(0.)44 b(Then)33 b(from)f Fo(Z)2140 2598 y Fn(^)-40 b Fk(q)2173 2583 y Fq(\()t(^)-53 b Fo(r)2255 2598 y Fn(2)2294 2583 y Fq(\))27 b Fm(\024)i Fq(0)j(w)m(e)i(see)f(that)1370 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Fn(^)-40 b Fk(q)2254 3412 y Fo(>)2367 3344 y(M)2466 3359 y Fn(^)g Fk(q)p 2367 3389 V 2395 3480 a Fq(^)-54 b Fo(r)2437 3446 y Fn(2)2434 3504 y(3)2532 3412 y Fq(+)22 b Fo(Q)2712 3427 y Fn(^)-40 b Fk(q)2745 3412 y Fo(;)386 3653 y Fq(suc)m(h)34 b(that)e Fo(Z)889 3668 y Fn(^)-40 b Fk(q)922 3653 y Fq(\()t(^)-53 b Fo(r)1004 3668 y Fn(3)1043 3653 y Fq(\))28 b Fo(<)g Fq(0,)k(con)m(tradicting)f(\(17\).)486 3769 y(Case)23 b(2:)39 b(Assume)23 b Fo(M)1274 3784 y Fn(^)-40 b Fk(q)1335 3769 y Fo(<)28 b Fq(0.)40 b(Then)24 b(from)d Fo(Z)2092 3784 y Fn(^)-40 b Fk(q)2125 3769 y Fq(\()t(^)-53 b Fo(r)2207 3784 y Fn(1)2246 3769 y Fq(\))28 b Fm(\025)g Fq(0)g Fm(\025)g Fo(Z)2671 3784 y Fn(^)-40 b Fk(q)2703 3769 y Fq(\()t(^)-53 b Fo(r)2785 3784 y Fn(2)2825 3769 y Fq(\))22 b(follo)m(ws)g(that)386 3885 y Fo(Z)460 3849 y Fj(0)458 3911 y Fn(^)-40 b Fk(q)491 3885 y Fq(\()t(^)-53 b Fo(r)r Fq(\))28 b Fm(\024)h Fq(0)j(for)g(some)37 b(^)-53 b Fo(r)31 b Fm(2)d Fq([)t(^)-53 b Fo(r)1462 3900 y Fn(1)1501 3885 y Fo(;)22 b Fq(^)-54 b Fo(r)1589 3900 y Fn(2)1629 3885 y Fq(].)44 b(In)33 b(view)g(of)f(\(18\))g(w)m(e)i(ha)m (v)m(e)g Fo(h)2816 3849 y Fj(0)2839 3885 y Fq(\()t(^)-53 b Fo(r)s Fq(\))28 b Fo(<)g Fq(0.)44 b(But)386 4007 y(this)38 b(means)h(b)m(y)h(our)e(ab)s(o)m(v)m(e)i(conca)m(vit)m(y)f(argumen)m(t) g(that)f Fo(h)2646 3971 y Fj(0)2670 4007 y Fq(\()p Fo(r)s Fq(\))g(is)g(decreasing)386 4123 y(and)k(th)m(us)g Fo(h)864 4087 y Fj(0)888 4123 y Fq(\()p Fo(r)s Fq(\))g Fo(<)h Fq(0)e(for)g(all)f Fo(r)45 b(>)j Fq(^)-54 b Fo(r)s Fq(.)70 b(Then)43 b Fo(Z)2255 4087 y Fj(0)2253 4149 y Fn(^)-40 b Fk(q)2327 4123 y Fq(is)41 b(strictly)g(decreasing)h(for)386 4245 y Fo(r)51 b Fm(\025)j Fq(^)-53 b Fo(r)r Fq(.)81 b(T)-8 b(ogether)46 b(with)e Fo(Z)1490 4260 y Fn(^)-40 b Fk(q)1523 4245 y Fq(\()t(^)-53 b Fo(r)1605 4260 y Fn(2)1645 4245 y Fq(\))48 b Fm(\024)i Fq(0)44 b(and)h Fo(Z)2227 4209 y Fj(0)2225 4271 y Fn(^)-40 b Fk(q)2258 4245 y Fq(\()t(^)-53 b Fo(r)s Fq(\))49 b Fm(\024)g Fq(0)c(this)f(implies)f(that)386 4367 y Fo(Z)458 4382 y Fn(^)-40 b Fk(q)491 4367 y Fq(\()t(^)-53 b Fo(r)573 4382 y Fn(3)612 4367 y Fq(\))28 b Fo(<)f Fq(0,)33 b(a)f(con)m(tradiction)f(to)i(\(17\).)1412 b Fe(\003)386 4555 y Fy(F)-9 b(act)37 b(4.3.)42 b Fh(F)-7 b(or)34 b(al)5 b(l)35 b Fq(0)27 b Fm(\024)h Fo(r)i Fm(\024)f Fo(R)1605 4570 y Fn(1)1679 4555 y Fh(the)35 b(ine)-5 b(quality)35 b Fo(q)2329 4519 y Fj(0)2352 4555 y Fq(\()p Fo(r)s Fq(\))27 b Fm(\024)h Fq(0)35 b Fh(holds.)386 4744 y(Pr)-5 b(o)g(of.)41 b Fq(Assume)35 b(the)f(con)m(trary)-8 b(.)47 b(Then)35 b(there)g(are)e(three)i(p)s(oin)m(ts)e Fo(r)2898 4759 y Fn(1)2967 4744 y Fo(<)d(r)3117 4759 y Fn(2)3186 4744 y Fo(<)f(r)3335 4759 y Fn(3)386 4860 y Fq(in)41 b(\(0)p Fo(;)17 b(R)714 4875 y Fn(1)753 4860 y Fq(\))42 b(with)f(0)i Fo(<)50 b Fq(^)-56 b Fo(q)47 b Fq(:=)d Fo(q)t Fq(\()p Fo(r)1641 4875 y Fn(1)1680 4860 y Fq(\))f(=)g Fo(q)t Fq(\()p Fo(r)2009 4875 y Fn(2)2049 4860 y Fq(\))g(=)g Fo(q)t Fq(\()p Fo(r)2378 4875 y Fn(3)2417 4860 y Fq(\))g Fo(<)h Fq(1)d(and)h Fo(q)2954 4824 y Fj(0)2977 4860 y Fq(\()p Fo(r)3059 4875 y Fn(1)3098 4860 y Fq(\))i Fo(<)f Fq(0,)386 4976 y Fo(q)433 4940 y Fj(0)456 4976 y Fq(\()p Fo(r)538 4991 y Fn(2)577 4976 y Fq(\))30 b Fo(>)g Fq(0,)k Fo(q)908 4940 y Fj(0)931 4976 y Fq(\()p Fo(r)1013 4991 y Fn(3)1052 4976 y Fq(\))c Fo(<)g Fq(0.)47 b(Consider)34 b(the)h(function)e Fo(T)14 b Fq(\()p Fo(r)m(;)24 b Fq(^)-56 b Fo(q)s Fq(\),)35 b(whic)m(h)f(is)f(equal)h(to)386 5092 y Fo(q)433 5056 y Fj(0)456 5092 y Fq(\()p Fo(r)s Fq(\))g(at)h Fo(r)779 5107 y Fn(1)818 5092 y Fo(;)17 b(r)906 5107 y Fn(2)945 5092 y Fo(;)g(r)1033 5107 y Fn(3)1072 5092 y Fq(.)50 b(T)-8 b(aking)35 b(in)m(to)f(accoun)m(t)h(its)f(b)s(oundary) i(b)s(eha)m(vior)e(at)h Fo(r)f Fq(=)d(0)386 5209 y(and)g Fo(r)g Fq(=)c Fo(R)826 5224 y Fn(1)866 5209 y Fq(,)32 b(it)e(is)h(clear)g(that)g Fo(T)14 b Fq(\()p Fo(r)m(;)24 b Fq(^)-56 b Fo(q)t Fq(\))31 b(m)m(ust)h(ha)m(v)m(e)g(at)f(least)g(the) h(sign)f(c)m(hanges)386 5325 y(p)s(ositiv)m(e-negativ)m(e-p)s(ositiv)m (e-negativ)m(e-p)s(ositiv)m(e.)62 b(Th)m(us)42 b Fo(T)14 b Fq(\()p Fo(r)m(;)24 b Fq(^)-56 b Fo(q)s Fq(\))40 b(has)g(at)g(least) 386 5441 y(four)32 b(zeros)38 b(^)-53 b Fo(r)878 5456 y Fn(1)945 5441 y Fo(<)32 b Fq(^)-54 b Fo(r)1092 5456 y Fn(2)1159 5441 y Fo(<)32 b Fq(^)-53 b Fo(r)1307 5456 y Fn(3)1374 5441 y Fo(<)32 b Fq(^)-54 b Fo(r)1521 5456 y Fn(4)1593 5441 y Fq(with)33 b(the)g(prop)s(erties)764 5623 y Fo(Z)836 5638 y Fn(^)-40 b Fk(q)868 5623 y Fq(\()t(^)-53 b Fo(r)950 5638 y Fn(1)990 5623 y Fq(\))27 b Fm(\024)h Fq(0)p Fo(;)114 b(Z)1422 5638 y Fn(^)-40 b Fk(q)1455 5623 y Fq(\()t(^)-53 b Fo(r)1537 5638 y Fn(2)1577 5623 y Fq(\))27 b Fm(\025)h Fq(0)p Fo(;)114 b(Z)2009 5638 y Fn(^)-40 b Fk(q)2042 5623 y Fq(\()t(^)-53 b Fo(r)2124 5638 y Fn(3)2164 5623 y Fq(\))27 b Fm(\024)h Fq(0)p Fo(;)114 b(Z)2596 5638 y Fn(^)-40 b Fk(q)2629 5623 y Fq(\()t(^)-53 b Fo(r)2711 5638 y Fn(4)2750 5623 y Fq(\))28 b Fm(\025)g Fq(0)p Fo(:)p eop %%Page: 10 10 10 9 bop 386 259 a Fi(10)566 b(RAF)-9 b(AEL)25 b(D.)h(BENGURIA)e(AND)h (HELMUT)g(LINDE)386 459 y Fq(W)-8 b(e)33 b(also)f(kno)m(w)j(that)d Fo(Z)1289 474 y Fn(^)-40 b Fk(q)1322 459 y Fq(\(0\))28 b(=)g(+)p Fm(1)p Fq(.)45 b(T)-8 b(o)33 b(satisfy)g(all)e(these)j (requiremen)m(ts,)g Fo(Z)3342 474 y Fn(^)-40 b Fk(q)386 575 y Fq(m)m(ust)36 b(either)g(ha)m(v)m(e)h(at)f(least)g(three)g (extremal)g(p)s(oin)m(ts)f(where)j Fo(Z)2810 539 y Fj(0)2808 601 y Fn(^)-40 b Fk(q)2876 575 y Fq(crosses)38 b(zero)386 691 y(or)32 b Fo(Z)577 706 y Fn(^)-40 b Fk(q)642 691 y Fq(m)m(ust)33 b(v)-5 b(anish)33 b(on)f(a)g(\014nite)h(in)m(terv)-5 b(al.)42 b(But)32 b(w)m(e)i(ha)m(v)m(e)1175 957 y Fo(Z)1249 915 y Fj(0)1247 981 y Fn(^)-40 b Fk(q)1280 957 y Fq(\()p Fo(r)s Fq(\))27 b(=)h Fm(\000)1621 889 y Fq(2)p Fo(M)1769 904 y Fn(^)-40 b Fk(q)p 1621 934 182 4 v 1669 1025 a Fo(r)1716 996 y Fn(3)1835 957 y Fq(+)1942 889 y Fo(E)2020 853 y Fn(2)2060 889 y Fo(r)p 1942 934 165 4 v 2008 1025 a Fq(^)-56 b Fo(q)2139 957 y Fm(\000)22 b Fq(2)7 b(^)-56 b Fo(q)2349 931 y Fq(~)2334 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b(e)33 b(start)f(from)g(the)h(basic)f(gap)g(inequalit)m(y)386 2532 y(\(19\))488 b Fo(\025)1105 2547 y Fn(2)1145 2532 y Fq(\(\012)p Fo(;)17 b(V)k Fq(\))h Fm(\000)h Fo(\025)1592 2547 y Fn(1)1631 2532 y Fq(\(\012)p Fo(;)17 b(V)22 b Fq(\))28 b Fm(\024)2043 2379 y Fg(R)2090 2494 y Fn(\012)2162 2459 y Fm(jr)p Fo(P)14 b Fm(j)2378 2423 y Fn(2)2416 2459 y Fo(u)2472 2423 y Fn(2)2472 2484 y(1)2527 2459 y Fq(d)2581 2423 y Fk(n)2629 2459 y Fo(r)p 2043 2510 633 4 v 2112 2522 a Fg(R)2159 2636 y Fn(\012)2231 2602 y Fo(P)2308 2573 y Fn(2)2347 2602 y Fo(u)2403 2567 y Fn(2)2403 2626 y(1)2458 2602 y Fq(d)2512 2573 y Fk(n)2559 2602 y Fo(r)2685 2532 y(;)386 2797 y Fq(where)30 b Fo(u)720 2812 y Fn(1)787 2797 y Fq(is)e(the)h(\014rst)g(Diric)m(hlet)d(eigenfunction)i(of)g Fm(\000)p Fq(\001)14 b(+)g Fo(V)50 b Fq(on)29 b(\012)f(and)h Fo(P)42 b Fq(is)28 b(a)386 2913 y(suitable)35 b(test)i(function)e(that) h(satis\014es)h(the)f(condition)2512 2833 y Fg(R)2560 2947 y Fn(\012)2631 2913 y Fo(P)14 b(u)2764 2877 y Fn(2)2764 2937 y(1)2819 2913 y Fq(d)2873 2877 y Fk(n)2920 2913 y Fo(r)37 b Fq(=)c(0.)53 b(W)-8 b(e)386 3029 y(set)386 3230 y(\(20\))612 b Fo(P)1235 3245 y Fk(i)1263 3230 y Fq(\()m Fo(~)-46 b(r)s Fq(\))28 b(=)f Fo(g)t Fq(\()p Fo(r)s Fq(\))1701 3163 y Fo(r)1745 3178 y Fk(i)p 1701 3208 73 4 v 1713 3299 a Fo(r)1880 3230 y Fq(for)32 b Fo(i)c Fq(=)f(1)p Fo(;)17 b Fq(2)p Fo(;)g(:::;)g(n;)386 3459 y Fq(where)386 3733 y(\(21\))642 b Fo(g)t Fq(\()p Fo(r)s Fq(\))27 b(=)1506 3562 y Fg(\()1638 3600 y Fk(z)1671 3609 y Ff(2)1705 3600 y Fn(\()p Fk(r)r Fn(\))p 1638 3624 157 4 v 1638 3682 a Fk(z)1671 3691 y Ff(1)1705 3682 y Fn(\()p Fk(r)r Fn(\))2038 3647 y Fq(for)32 b Fo(r)e(<)e(R)2439 3662 y Fn(1)1635 3780 y Fq(lim)1628 3842 y Fk(t)p Fj(")p Fk(R)1741 3851 y Ff(1)1793 3780 y Fo(g)t Fq(\()p Fo(t)p Fq(\))83 b(for)32 b Fo(r)e Fm(\025)e Fo(R)2440 3795 y Fn(1)2480 3780 y Fo(:)386 4031 y Fq(Here)37 b Fo(z)665 4046 y Fn(1)741 4031 y Fq(and)g Fo(z)980 4046 y Fn(2)1056 4031 y Fq(are)f(the)h(radial)e(parts)h(\(b)s(oth)g(c)m(hosen)j(p)s (ositiv)m(e\))c(of)h(the)h(\014rst)386 4156 y(t)m(w)m(o)43 b(eigenfunctions)f(of)g Fm(\000)p Fq(\001)29 b(+)1654 4130 y(~)1640 4156 y Fo(V)63 b Fq(on)42 b Fo(S)1965 4171 y Fn(1)2005 4156 y Fq(.)72 b(More)43 b(precisely)-8 b(,)45 b Fo(z)2841 4171 y Fn(2)2880 4156 y Fq(\()p Fo(r)s Fq(\))p Fo(r)3047 4171 y Fk(i)3075 4156 y Fo(r)3122 4119 y Fj(\000)p Fn(1)3258 4156 y Fq(for)386 4272 y Fo(i)29 b Fq(=)g(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)33 b Fq(is)g(a)g(basis)g(of)g(the)h(space)g (of)f(second)i(eigenfunctions.)46 b(It)33 b(follo)m(ws)386 4391 y(from)24 b(the)i(con)m(v)m(exit)m(y)h(of)e Fo(r)1361 4366 y Fq(~)1347 4391 y Fo(V)46 b Fq(and)25 b(the)h(BGM)f(inequalit)m (y)f([1,)i(7)o(])g(that)f(the)g(second)386 4507 y(eigenfunctions)32 b(can)h(b)s(e)g(written)f(in)g(that)g(w)m(a)m(y)-8 b(.)486 4623 y(According)35 b(to)g(an)g(argumen)m(t)g(in)g([3])h(one)f(can)h (alw)m(a)m(ys)g(c)m(hose)h(the)f(origin)d(of)386 4739 y(the)39 b(co)s(ordinate)f(system)i(suc)m(h)h(that)1821 4659 y Fg(R)1868 4774 y Fn(\012)1940 4739 y Fo(P)2003 4754 y Fk(i)2031 4739 y Fo(u)2087 4703 y Fn(2)2087 4764 y(1)2142 4739 y Fq(d)2196 4703 y Fk(n)2243 4739 y Fo(r)h Fq(=)c(0)h(is)f(satis\014ed)i(for)e(all)f Fo(i)p Fq(.)386 4857 y(Putting)32 b(the)h(functions)f Fo(P)1396 4872 y Fk(i)1457 4857 y Fq(in)m(to)g(\(19\))g(and)g(summing)f(o)m(v)m(er)j (all)c Fo(i)j Fq(yields)386 5127 y(\(22\))523 b Fo(\025)1140 5142 y Fn(2)1179 5127 y Fq(\(\012)p Fo(;)17 b(V)22 b Fq(\))g Fm(\000)h Fo(\025)1627 5142 y Fn(1)1666 5127 y Fq(\(\012)p Fo(;)17 b(V)22 b Fq(\))27 b Fm(\024)2083 4973 y Fg(R)2130 5088 y Fn(\012)2201 5053 y Fo(B)5 b Fq(\()p Fo(r)s Fq(\))p Fo(u)2459 5017 y Fn(2)2459 5078 y(1)2515 5053 y Fq(d)2569 5017 y Fk(n)2616 5053 y Fo(r)p 2077 5104 591 4 v 2077 5116 a Fg(R)2124 5231 y Fn(\012)2196 5196 y Fo(g)t Fq(\()p Fo(r)s Fq(\))2370 5167 y Fn(2)2408 5196 y Fo(u)2464 5162 y Fn(2)2464 5220 y(1)2520 5196 y Fq(d)2574 5167 y Fk(n)2621 5196 y Fo(r)386 5391 y Fq(with)1254 5588 y Fo(B)g Fq(\()p Fo(r)s Fq(\))27 b(=)h Fo(g)1638 5546 y Fj(0)1661 5588 y Fq(\()p Fo(r)s Fq(\))1784 5546 y Fn(2)1845 5588 y Fq(+)22 b(\()p Fo(n)g Fm(\000)h Fq(1\))2258 5520 y Fo(g)t Fq(\()p Fo(r)s Fq(\))2432 5484 y Fn(2)p 2258 5565 213 4 v 2320 5656 a Fo(r)2367 5627 y Fn(2)2479 5588 y Fo(:)p eop %%Page: 11 11 11 10 bop 1237 259 a Fi(A)33 b(SECOND)g(EIGENV)-9 b(ALUE)33 b(BOUND)773 b(11)386 459 y Fq(By)24 b(Lemma)f(4.1)h(w)m(e)h(kno)m(w)g (that)f Fo(B)29 b Fq(is)23 b(a)h(decreasing)g(and)g Fo(g)k Fq(an)23 b(increasing)h(func-)386 575 y(tion)37 b(of)g Fo(r)s Fq(.)58 b(Th)m(us,)41 b(denoting)c(b)m(y)i Fo(u)1724 539 y Fk(?)1724 600 y Fn(1)1800 575 y Fq(the)f(spherically)f (decreasing)h(rearrange-)386 691 y(men)m(t)33 b(of)f Fo(u)800 706 y Fn(1)871 691 y Fq(with)g(resp)s(ect)i(to)e(the)h (origin,)e(w)m(e)i(ha)m(v)m(e)745 784 y Fg(Z)801 1010 y Fn(\012)872 920 y Fo(B)5 b Fq(\()p Fo(r)s Fq(\))p Fo(u)1130 879 y Fn(2)1130 944 y(1)1186 920 y Fq(d)1240 879 y Fk(n)1287 920 y Fo(r)85 b Fm(\024)1577 784 y Fg(Z)1632 1010 y Fn(\012)1683 991 y Fd(?)1740 920 y Fo(B)1819 879 y Fk(?)1859 920 y Fq(\()p Fo(r)s Fq(\))17 b Fo(u)2055 879 y Fk(?)2055 944 y Fn(1)2093 874 y(2)2149 920 y Fq(d)2203 879 y Fk(n)2250 920 y Fo(r)-1908 b Fq(\(23\))1416 1183 y Fm(\024)1577 1047 y Fg(Z)1632 1273 y Fn(\012)1683 1254 y Fd(?)1740 1183 y Fo(B)5 b Fq(\()p Fo(r)s Fq(\))17 b Fo(u)2015 1142 y Fk(?)2015 1207 y Fn(1)2053 1137 y(2)2109 1183 y Fq(d)2163 1142 y Fk(n)2210 1183 y Fo(r)31 b Fm(\024)2390 1047 y Fg(Z)2445 1273 y Fk(S)2488 1282 y Ff(1)2544 1183 y Fo(B)5 b Fq(\()p Fo(r)s Fq(\))17 b Fo(z)2812 1142 y Fn(2)2808 1207 y(1)2868 1183 y Fq(d)2922 1142 y Fk(n)2969 1183 y Fo(r)386 1425 y Fq(and)729 1499 y Fg(Z)785 1724 y Fn(\012)856 1634 y Fo(g)t Fq(\()p Fo(r)s Fq(\))1030 1593 y Fn(2)1069 1634 y Fo(u)1125 1593 y Fn(2)1125 1659 y(1)1180 1634 y Fq(d)1234 1593 y Fk(n)1281 1634 y Fo(r)86 b Fm(\025)1572 1499 y Fg(Z)1627 1724 y Fn(\012)1678 1705 y Fd(?)1735 1634 y Fo(g)1782 1649 y Fk(?)1821 1634 y Fq(\()p Fo(r)s Fq(\))1944 1593 y Fn(2)1999 1634 y Fo(u)2055 1593 y Fk(?)2055 1659 y Fn(1)2094 1588 y(2)2150 1634 y Fq(d)2204 1593 y Fk(n)2252 1634 y Fo(r)-1910 b Fq(\(24\))1411 1897 y Fm(\025)1572 1761 y Fg(Z)1627 1987 y Fn(\012)1678 1968 y Fd(?)1735 1897 y Fo(g)t Fq(\()p Fo(r)s Fq(\))1909 1856 y Fn(2)1964 1897 y Fo(u)2020 1856 y Fk(?)2020 1922 y Fn(1)2058 1851 y(2)2115 1897 y Fq(d)2169 1856 y Fk(n)2216 1897 y Fo(r)30 b Fm(\025)2395 1761 y Fg(Z)2451 1987 y Fk(S)2494 1996 y Ff(1)2549 1897 y Fo(g)t Fq(\()p Fo(r)s Fq(\))2723 1856 y Fn(2)2778 1897 y Fo(z)2827 1856 y Fn(2)2823 1922 y(1)2883 1897 y Fq(d)2937 1856 y Fk(n)2985 1897 y Fo(r)386 2139 y Fq(In)39 b(eac)m(h)h(of)f(the)h(ab)s(o)m(v)m(e)g(c)m (hains)f(of)g(inequalities)e(the)i(\014rst)h(step)g(follo)m(ws)e(from) 386 2255 y(general)h(prop)s(erties)g(of)g(rearrangemen)m(ts)h(and)f (the)h(second)g(from)e(the)i(mono-)386 2372 y(tonicit)m(y)28 b(prop)s(erties)h(of)f Fo(g)k Fq(and)d Fo(B)5 b Fq(.)43 b(The)30 b(third)e(step)i(is)e(justi\014ed)h(b)m(y)h(a)f(compar-)386 2488 y(ison)f(result)g(that)h(w)m(e)g(state)g(b)s(elo)m(w)g(and)f(the)h (monotonicit)m(y)e(of)h Fo(g)k Fq(and)c Fo(B)34 b Fq(again.)386 2604 y(Putting)e(\(23\))g(and)h(\(24\))e(in)m(to)h(\(22\))g(w)m(e)i (get)530 2866 y Fo(\025)587 2881 y Fn(2)627 2866 y Fq(\(\012)p Fo(;)17 b(V)k Fq(\))h Fm(\000)h Fo(\025)1074 2881 y Fn(1)1113 2866 y Fq(\(\012)p Fo(;)17 b(V)22 b Fq(\))28 b Fm(\024)1530 2703 y Fg(R)1577 2817 y Fk(S)1620 2826 y Ff(1)1675 2783 y Fo(B)5 b Fq(\()p Fo(r)s Fq(\))17 b Fo(z)1943 2747 y Fn(2)1999 2783 y Fq(d)2053 2747 y Fk(n)2101 2783 y Fo(r)p 1525 2843 628 4 v 1525 2855 a Fg(R)1572 2969 y Fk(S)1615 2978 y Ff(1)1670 2935 y Fo(g)t Fq(\()p Fo(r)s Fq(\))1844 2906 y Fn(2)1899 2935 y Fo(z)1948 2906 y Fn(2)2005 2935 y Fq(d)2059 2906 y Fk(n)2106 2935 y Fo(r)2190 2866 y Fq(=)28 b Fo(\025)2351 2881 y Fn(2)2390 2866 y Fq(\()p Fo(S)2488 2881 y Fn(1)2527 2866 y Fo(;)2586 2840 y Fq(~)2571 2866 y Fo(V)22 b Fq(\))g Fm(\000)g 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3684 y Fo(V)57 b Fh(on)34 b Fo(S)2062 3699 y Fn(1)2102 3684 y Fh(.)46 b(Assume)35 b(b)-5 b(oth)35 b(functions)g(to)g(b)-5 b(e)386 3800 y(p)g(ositive)34 b(and)g(normalize)-5 b(d)34 b(in)g Fo(L)1606 3764 y Fn(2)1646 3800 y Fq(\(\012)1754 3764 y Fk(?)1794 3800 y Fq(\))p Fh(.)45 b(Then)34 b(ther)-5 b(e)35 b(exists)f(an)h Fo(r)2854 3815 y Fn(0)2928 3800 y Fh(such)f(that)1151 3981 y Fo(u)1207 3940 y Fk(?)1207 4006 y Fn(1)1246 3981 y Fq(\()p Fo(r)s Fq(\))83 b Fm(\024)g Fo(z)1657 3996 y Fn(1)1697 3981 y Fq(\()p Fo(r)s Fq(\))99 b Fh(for)35 b Fo(r)30 b Fm(\024)e Fo(r)2298 3996 y Fn(0)2372 3981 y Fh(and)1151 4127 y Fo(u)1207 4086 y Fk(?)1207 4151 y Fn(1)1246 4127 y Fq(\()p Fo(r)s Fq(\))83 b Fm(\025)g Fo(z)1657 4142 y Fn(1)1697 4127 y Fq(\()p Fo(r)s Fq(\))99 b Fh(for)35 b Fo(r)2119 4142 y Fn(0)2186 4127 y Fo(<)27 b(r)k Fm(\024)d Fo(R)2543 4142 y Fn(1)2582 4127 y Fo(:)386 4315 y Fh(Pr)-5 b(o)g(of.)41 b Fq(By)g(a)f(v)m(ersion)g(of)g(the)g(RFK) g(inequalit)m(y)f(for)g(Sc)m(hr\177)-49 b(odinger)41 b(op)s(erators)386 4431 y([14])31 b(and)h(b)m(y)g(domain)e(monotonicit) m(y)g(of)h(the)h(\014rst)g(eigen)m(v)-5 b(alue)31 b(it)g(is)g(clear)g (that)386 4548 y Fo(S)446 4563 y Fn(1)513 4548 y Fm(\032)d Fq(\012)688 4512 y Fk(?)728 4548 y Fq(.)42 b(This)26 b(is)g(wh)m(y)i(w)m(e)g(can)e(view)h Fo(z)1879 4563 y Fn(1)1919 4548 y Fq(\()p Fo(r)s Fq(\))f(as)h(a)f(function)g(in)g Fo(L)2807 4512 y Fn(2)2846 4548 y Fq(\(\012)2954 4512 y Fk(?)2994 4548 y Fq(\),)i(setting)386 4664 y Fo(z)431 4679 y Fn(1)471 4664 y Fq(\()p Fo(r)s Fq(\))f(=)g(0)33 b(for)f Fo(r)e(>)e(R)1207 4679 y Fn(1)1246 4664 y Fq(.)486 4780 y(Both)39 b Fo(u)791 4744 y Fk(?)791 4805 y Fn(1)870 4780 y Fq(and)h Fo(z)1112 4795 y Fn(1)1191 4780 y Fq(are)g(p)s(ositiv)m (e)f(and)g(spherically)g(symmetric.)64 b(Moreo)m(v)m(er,)386 4896 y Fo(u)442 4860 y Fk(?)442 4921 y Fn(1)481 4896 y Fq(\()p Fo(r)s Fq(\))39 b(and)g Fo(z)884 4911 y Fn(1)924 4896 y Fq(\()p Fo(r)s Fq(\))f(are)i(decreasing)f(functions)g(of)g Fo(r)s Fq(.)63 b(F)-8 b(or)38 b Fo(u)2652 4860 y Fk(?)2652 4921 y Fn(1)2731 4896 y Fq(this)g(is)h(clear)g(b)m(y)386 5013 y(de\014nition)d(of)h(the)g(rearrangemen)m(t.)57 b(F)-8 b(or)37 b Fo(z)2036 5028 y Fn(1)2113 5013 y Fq(it)f(follo)m(ws)f (from)h(a)h(simple)f(com-)386 5129 y(parison)e(argumen)m(t)g(using)h Fo(z)1475 5093 y Fk(?)1471 5153 y Fn(1)1550 5129 y Fq(as)g(a)f(test)i (function)e(in)g(the)h(Ra)m(yleigh)e(quotien)m(t)386 5245 y(for)i Fo(\025)595 5260 y Fn(1)635 5245 y Fq(.)54 b(\(Here)37 b(and)f(in)f(the)i(sequel)g(w)m(e)g(write)f(short-hand)g Fo(\025)2718 5260 y Fn(1)2791 5245 y Fq(=)d Fo(\025)2957 5260 y Fn(1)2997 5245 y Fq(\(\012)p Fo(;)17 b(V)k Fq(\))34 b(=)386 5370 y Fo(\025)443 5385 y Fn(1)482 5370 y Fq(\()p Fo(S)580 5385 y Fn(1)620 5370 y Fo(;)678 5344 y Fq(~)664 5370 y Fo(V)21 b Fq(\).\))486 5496 y(W)-8 b(e)33 b(in)m(tro)s(duce)g(a) f(c)m(hange)i(of)e(v)-5 b(ariables)32 b(via)g Fo(s)c Fq(=)g Fo(C)2416 5511 y Fk(n)2463 5496 y Fo(r)2510 5460 y Fk(n)2589 5496 y Fq(and)33 b(write)g Fo(u)3085 5449 y Fn(#)3085 5521 y(1)3147 5496 y Fq(\()p Fo(s)p Fq(\))28 b Fm(\021)386 5623 y Fo(u)442 5587 y Fk(?)442 5647 y Fn(1)481 5623 y Fq(\()p Fo(r)s Fq(\),)k Fo(z)712 5575 y Fn(#)708 5647 y(1)776 5623 y Fq(\()p Fo(s)p Fq(\))27 b Fm(\021)h Fo(z)1075 5638 y Fn(1)1115 5623 y Fq(\()p Fo(r)s Fq(\))k(and)1475 5598 y(~)1460 5623 y Fo(V)1517 5638 y Fn(#)1580 5623 y Fq(\()p Fo(s)p Fq(\))27 b Fm(\021)1849 5598 y Fq(~)1834 5623 y Fo(V)22 b Fq(\()p Fo(r)s Fq(\).)p eop %%Page: 12 12 12 11 bop 386 259 a Fi(12)566 b(RAF)-9 b(AEL)25 b(D.)h(BENGURIA)e(AND)h (HELMUT)g(LINDE)386 469 y Fy(F)-9 b(act)37 b(5.1.)42 b Fh(F)-7 b(or)34 b(the)h(functions)f Fo(u)1671 422 y Fn(#)1671 493 y(1)1734 469 y Fq(\()p Fo(s)p Fq(\))h Fh(and)f Fo(z)2129 422 y Fn(#)2125 493 y(1)2193 469 y Fq(\()p Fo(s)p Fq(\))g Fh(we)h(have)695 707 y Fm(\000)799 639 y Fq(d)p Fo(u)909 592 y Fn(#)909 664 y(1)p 782 684 190 4 v 835 775 a Fq(d)p Fo(s)1064 707 y Fm(\024)84 b Fo(n)1283 666 y Fj(\000)p Fn(2)1377 707 y Fo(C)1454 666 y Fj(\000)p Fn(2)p 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b(e)38 b(in)m(tegrate)g(b)s(oth)g(sides)g(of)g Fm(\000)p Fq(\001)p Fo(u)2078 1237 y Fn(1)2143 1222 y Fq(+)26 b Fo(V)c(u)2380 1237 y Fn(1)2456 1222 y Fq(=)36 b Fo(\025)2625 1237 y Fn(1)2665 1222 y Fo(u)2721 1237 y Fn(1)2798 1222 y Fq(o)m(v)m(er)j(the)f(lev)m(el)386 1338 y(set)f(\012)612 1353 y Fk(t)675 1338 y Fq(:=)d Fm(f)m Fo(~)-46 b(r)36 b Fm(2)e Fq(\012)g(:)f Fo(u)1262 1353 y Fn(1)1301 1338 y Fq(\()m Fo(~)-46 b(r)s Fq(\))33 b Fo(>)h(t)p Fm(g)h Fq(and)i(use)g(Gauss')f(Div)m(ergence)h(Theorem)f(to)386 1454 y(obtain)386 1642 y(\(27\))868 1506 y Fg(Z)923 1732 y Fk(@)t Fn(\012)1015 1740 y Fd(t)1064 1642 y Fm(jr)p Fo(u)1231 1657 y Fn(1)1269 1642 y Fm(j)p Fo(H)1378 1657 y Fk(n)p Fj(\000)p Fn(1)1515 1642 y Fq(\()17 b(d)p Fo(r)s Fq(\))27 b(=)1839 1506 y Fg(Z)1895 1732 y Fn(\012)1946 1740 y Fd(t)1978 1642 y Fq(\()p Fo(\025)2073 1657 y Fn(1)2134 1642 y Fm(\000)c Fo(V)e Fq(\()m Fo(~)-46 b(r)s Fq(\)\))17 b Fo(u)2546 1657 y Fn(1)2584 1642 y Fq(\()m Fo(~)-46 b(r)s Fq(\))17 b(d)2778 1601 y Fk(n)2825 1642 y Fo(r)m(;)386 1867 y Fq(where)44 b Fo(@)5 b Fq(\012)804 1882 y Fk(t)881 1867 y Fq(=)46 b Fm(f)m Fo(~)-46 b(r)48 b Fm(2)e Fq(\012)g(:)g Fo(u)1502 1882 y Fn(1)1541 1867 y Fq(\()m Fo(~)-46 b(r)r Fq(\))46 b(=)f Fo(t)p Fm(g)p Fq(.)76 b(No)m(w)43 b(w)m(e)i(de\014ne)f (the)g(distribution)386 1983 y(function)24 b Fo(\026)p Fq(\()p Fo(t)p Fq(\))j(=)h Fm(j)p Fq(\012)1159 1998 y Fk(t)1189 1983 y Fm(j)p Fq(.)40 b(Using)24 b(the)h(coarea)f(form)m (ula,)g(the)h(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)j(in-)386 2099 y(equalit)m(y)f(and)g(the)g(classical)f(isop)s(erimetric)f (inequalit)m(y)-8 b(,)27 b(T)-8 b(alen)m(ti)26 b(deriv)m(es)i(\([18],) 386 2216 y(p.709,)k(eq.)44 b(\(32\)\))386 2446 y(\(28\))996 2310 y Fg(Z)1052 2536 y Fk(@)t Fn(\012)1144 2544 y Fd(t)1193 2446 y Fm(jr)p Fo(u)1360 2461 y Fn(1)1398 2446 y Fm(j)p Fo(H)1507 2461 y Fk(n)p Fj(\000)p Fn(1)1644 2446 y Fq(\()17 b(d)p Fo(r)s Fq(\))27 b Fm(\025)h(\000)p Fo(n)2105 2405 y Fn(2)2145 2446 y Fo(C)2222 2405 y Fn(2)p Fk(=n)2215 2470 y(n)2350 2378 y Fo(\026)p Fq(\()p Fo(t)p Fq(\))2520 2342 y Fn(2)p Fj(\000)p Fn(2)p Fk(=n)p 2350 2423 378 4 v 2442 2514 a Fo(\026)2501 2485 y Fj(0)2524 2514 y Fq(\()p Fo(t)p Fq(\))2737 2446 y Fo(:)386 2671 y Fq(The)33 b(left)f(sides)h(of)f(\(27\))g(and)h(\(28\))f(are)h(the)g(same,)f(th)m (us)504 2901 y Fm(\000)p Fo(n)639 2860 y Fn(2)679 2901 y Fo(C)756 2860 y Fn(2)p Fk(=n)749 2926 y(n)884 2834 y Fo(\026)p Fq(\()p Fo(t)p Fq(\))1054 2797 y Fn(2)p Fj(\000)p Fn(2)p Fk(=n)p 884 2878 V 976 2969 a Fo(\026)1035 2941 y Fj(0)1058 2969 y Fq(\()p Fo(t)p Fq(\))1354 2901 y Fm(\024)1515 2765 y Fg(Z)1570 2991 y Fn(\012)1621 2999 y Fd(t)1653 2901 y Fq(\()p Fo(\025)1748 2916 y Fn(1)1810 2901 y Fm(\000)22 b Fo(V)g Fq(\()m Fo(~)-46 b(r)r Fq(\)\))17 b Fo(u)2221 2916 y Fn(1)2260 2901 y Fq(\()m Fo(~)-46 b(r)r Fq(\))17 b(d)2453 2860 y Fk(n)2500 2901 y Fo(r)1354 3172 y Fm(\024)1515 3037 y Fg(Z)1570 3262 y Fn(\012)1621 3239 y Fd(?)1621 3282 y(t)1661 3172 y Fq(\()p Fo(\025)1756 3187 y Fn(1)1818 3172 y Fm(\000)22 b Fo(V)1974 3187 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Fq(The)26 b(\014rst)f(eigenfunction)f(of)h Fm(\000)p Fq(\001)6 b(+)g Fo(V)47 b Fq(on)25 b Fo(B)3001 648 y Fk(R)3084 633 y Fq(is)f(radi-)386 749 y(ally)i(symmetric)h(and)h (will)e(b)s(e)i(called)f Fo(z)1848 764 y Fn(1)1888 749 y Fq(\()p Fo(r)s Fq(\).)41 b(F)-8 b(urther,)29 b(a)f(standard)g (separation)386 866 y(of)37 b(v)-5 b(ariables)36 b(and)i(the)g (Baumgartner-Grosse-Martin)e([7)o(,)i(4])f(inequalit)m(y)g(im-)386 982 y(ply)e(that)g(w)m(e)h(can)g(write)f(a)g(basis)g(of)g(the)h(space)g (of)f(second)h(eigenfunctions)f(in)386 1098 y(the)43 b(form)e Fo(z)849 1113 y Fn(2)889 1098 y Fq(\()p Fo(r)s Fq(\))29 b Fm(\001)f Fo(r)1141 1113 y Fk(i)1198 1098 y Fm(\001)h Fo(r)1302 1062 y Fj(\000)p Fn(1)1396 1098 y Fq(.)73 b(The)44 b(radial)c(parts)j Fo(z)2298 1113 y Fn(1)2380 1098 y Fq(and)g Fo(z)2625 1113 y Fn(2)2707 1098 y Fq(of)f(the)h(\014rst)h(and)386 1214 y(the)f(second)h (eigenfunction,)g(whic)m(h)g(w)m(e)f(assume)g(to)g(b)s(e)g(p)s(ositiv)m (e,)h(solv)m(e)f(the)386 1330 y(di\013eren)m(tial)31 b(equations)1078 1589 y Fm(\000)p Fo(z)1204 1548 y Fj(00)1200 1613 y Fn(1)1248 1589 y Fq(\()p Fo(r)s Fq(\))22 b Fm(\000)1502 1521 y Fo(n)h Fm(\000)f Fq(1)p 1502 1566 229 4 v 1593 1657 a Fo(r)1741 1589 y(z)1790 1548 y Fj(0)1786 1613 y Fn(1)1826 1589 y Fq(\()p Fo(r)s Fq(\))g(+)g(\()o Fo(V)g Fq(\()p Fo(r)s Fq(\))g Fm(\000)g Fo(\025)2486 1604 y Fn(1)2526 1589 y Fq(\))16 b Fo(z)2625 1604 y Fn(1)2665 1589 y Fq(\()p Fo(r)s Fq(\))83 b(=)f(0)p Fo(;)-2719 b Fq(\(30\))639 1839 y Fm(\000)p Fo(z)765 1798 y Fj(00)761 1864 y Fn(2)808 1839 y Fq(\()p Fo(r)s Fq(\))22 b Fm(\000)1063 1771 y Fo(n)g Fm(\000)h Fq(1)p 1063 1816 V 1154 1907 a Fo(r)1301 1839 y(z)1350 1798 y Fj(0)1346 1864 y Fn(2)1386 1839 y Fq(\()p Fo(r)s Fq(\))f(+)1629 1698 y Fg(\022)1712 1771 y Fo(n)h Fm(\000)f Fq(1)p 1712 1816 V 1784 1907 a Fo(r)1831 1878 y Fn(2)1973 1839 y Fq(+)g Fo(V)g Fq(\()p Fo(r)s Fq(\))f Fm(\000)i Fo(\025)2451 1854 y Fn(2)2490 1698 y Fg(\023)2580 1839 y Fo(z)2625 1854 y Fn(2)2665 1839 y Fq(\()p Fo(r)s Fq(\))83 b(=)f(0)386 2110 y(with)32 b(the)h(b)s(oundary)g(conditions)386 2325 y(\(31\))290 b Fo(z)899 2284 y Fj(0)895 2350 y Fn(1)935 2325 y Fq(\(0\))27 b(=)g(0)p Fo(;)114 b(z)1425 2340 y Fn(1)1465 2325 y Fq(\()p Fo(R)q Fq(\))28 b(=)f(0)p Fo(;)114 b(z)1982 2340 y Fn(2)2022 2325 y Fq(\(0\))27 b(=)h(0)p Fo(;)114 b(z)2513 2340 y Fn(2)2553 2325 y Fq(\()p Fo(R)q Fq(\))27 b(=)h(0)p Fo(:)386 2540 y Fq(W)-8 b(e)36 b(de\014ne)g(the)g(rescaled)g(functions)41 b(~)-55 b Fo(z)1852 2556 y Fn(1)p Fk(=)p Fn(2)1962 2540 y Fq(\()p Fo(r)s Fq(\))32 b(=)h Fo(z)2271 2556 y Fn(1)p Fk(=)p Fn(2)2381 2540 y Fq(\()p Fo(\014)6 b(r)s Fq(\).)51 b(Putting)35 b Fo(\014)6 b(r)37 b Fq(\(with)386 2660 y Fo(\014)c(>)28 b Fq(0\))e(instead)g(of)f Fo(r)k Fq(in)m(to)c(the)i (equations)f(\(30\))g(and)g(m)m(ultiplying)d(b)m(y)k Fo(\014)3068 2624 y Fn(2)3133 2660 y Fq(yields)386 2776 y(the)33 b(rescaled)g(equations)933 3034 y Fm(\000)6 b Fq(~)-55 b Fo(z)1059 2993 y Fj(00)1055 3059 y Fn(1)1102 3034 y Fq(\()p Fo(r)s 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Fn(2)2524 3284 y Fo(\025)2581 3299 y Fn(2)2621 3144 y Fg(\023)2716 3284 y Fq(~)-54 b Fo(z)2756 3299 y Fn(2)2795 3284 y Fq(\()p Fo(r)s Fq(\))83 b(=)g(0)p Fo(:)386 3556 y Fq(W)-8 b(e)34 b(conclude)g(that)39 b(~)-55 b Fo(z)1214 3571 y Fn(1)1287 3556 y Fq(and)40 b(~)-55 b Fo(z)1523 3571 y Fn(2)1596 3556 y Fq(are)34 b(the)g(radial)d(parts)j(of)f(the)h (\014rst)g(t)m(w)m(o)g(eigen-)386 3672 y(functions)c(of)h Fm(\000)p Fq(\001)18 b(+)g Fo(\014)1245 3636 y Fn(2)1284 3672 y Fo(V)k Fq(\()p Fo(\014)6 b(r)s Fq(\))30 b(on)g Fo(B)1784 3687 y Fk(R=\014)1951 3672 y Fq(to)g(the)h(eigen)m(v)-5 b(alues)31 b Fo(\014)2798 3636 y Fn(2)2837 3672 y Fo(\025)2894 3687 y Fn(1)2964 3672 y Fq(and)f Fo(\014)3212 3636 y Fn(2)3251 3672 y Fo(\025)3308 3687 y Fn(2)3348 3672 y Fq(.)386 3791 y(Consequen)m(tly)-8 b(,)35 b(if)d(w)m(e)i(replace)f Fo(R)g Fq(b)m(y)h Fo(R)q(=\014)k Fq(and)33 b Fo(V)22 b Fq(\()p Fo(r)s Fq(\))32 b(b)m(y)i Fo(\014)2656 3755 y Fn(2)2695 3791 y Fo(V)22 b Fq(\()p Fo(\014)6 b(r)s Fq(\),)32 b(then)h(the)386 3907 y(ratio)e Fo(\025)676 3922 y Fn(2)715 3907 y Fo(=\025)821 3922 y Fn(1)893 3907 y Fq(do)s(esn't)i(c)m(hange.)486 4024 y(F)-8 b(or)51 b(the)i(rest)h(of)e(this)g(section)h(w)m(e)g(shall)f(write)g Fo(\025)2504 4039 y Fn(1)p Fk(=)p Fn(2)2614 4024 y Fq(\()p Fo(R)q(;)17 b(V)k Fq(\))53 b(instead)f(of)386 4140 y Fo(\025)443 4155 y Fn(1)p Fk(=)p Fn(2)553 4140 y Fq(\()p Fo(B)665 4155 y Fk(R)723 4140 y Fo(;)17 b(V)k Fq(\).)86 b(W)-8 b(e)48 b(also)e(\014x)h(t)m(w)m(o)h(radii)d(0)52 b Fo(<)g(R)2285 4155 y Fn(1)2377 4140 y Fo(<)g(R)2579 4155 y Fn(2)2665 4140 y Fq(and)47 b(let)g Fo(\032)p Fq(\()p Fo(\014)6 b Fq(\))46 b(for)386 4256 y Fo(\014)33 b(>)28 b Fq(1)k(b)s(e)h(the)g(function)f(de\014ned)i(implicitly)28 b(b)m(y)386 4471 y(\(32\))539 b Fo(\025)1156 4486 y Fn(1)1195 4471 y Fq(\()p Fo(\032)p Fq(\()p Fo(\014)6 b Fq(\))p Fo(;)17 b(V)k Fq(\()p Fo(r)s Fq(\)\))27 b(=)h Fo(\025)1891 4486 y Fn(1)1930 4471 y Fq(\()p Fo(R)2042 4486 y Fn(2)2082 4471 y Fo(=\014)6 b(;)17 b(\014)2297 4430 y Fn(2)2335 4471 y Fo(V)22 b Fq(\()p Fo(\014)6 b(r)s Fq(\)\))p Fo(:)386 4687 y Fq(Then)35 b(w)m(e)h(ha)m(v)m(e)f Fo(\032)p Fq(\(1\))c(=)f Fo(R)1400 4702 y Fn(2)1440 4687 y Fq(.)49 b(By)34 b(domain)f (monotonicit)m(y)f(of)i Fo(\025)2781 4702 y Fn(1)2855 4687 y Fq(and)g(b)s(ecause)386 4803 y Fo(V)21 b Fq(\()p Fo(r)s Fq(\))26 b(is)g(increasing)g(and)g(p)s(ositiv)m(e)g(w)m(e)h(see) g(that)f(the)h(righ)m(t)f(hand)g(side)h(of)e(\(32\))h(is)386 4919 y(increasing)21 b(in)g Fo(\014)6 b Fq(.)40 b(Therefore,)25 b(again)c(b)m(y)i(domain)d(monotonicit)m(y)-8 b(,)22 b Fo(\032)p Fq(\()p Fo(\014)6 b Fq(\))22 b(m)m(ust)g(b)s(e)386 5035 y(decreasing)31 b(in)f Fo(\014)6 b Fq(.)43 b(One)31 b(can)g(also)f(c)m(hec)m(k)j(that)e Fo(\032)p Fq(\()p Fo(\014)6 b Fq(\))31 b(is)f(a)h(con)m(tin)m(uous)g(function)386 5151 y(and)j(that)g Fo(\032)p Fq(\()p Fo(\014)6 b Fq(\))34 b(go)s(es)g(to)g(zero)h(for)e Fo(\014)j Fm(!)30 b(1)p Fq(.)48 b(Th)m(us)36 b(w)m(e)f(can)f(\014nd)h Fo(\014)2928 5166 y Fn(0)2998 5151 y Fo(>)30 b Fq(1)k(suc)m(h)386 5268 y(that)h Fo(\032)p Fq(\()p Fo(\014)743 5283 y Fn(0)783 5268 y Fq(\))e(=)f Fo(R)1036 5283 y Fn(1)1076 5268 y Fq(.)53 b(Then)36 b(w)m(e)h(can)f(apply)f(Theorem)h(2.1,)g(with)f Fo(B)2915 5283 y Fk(R)2968 5292 y Ff(2)3003 5283 y Fk(=\014)3078 5292 y Ff(0)3152 5268 y Fq(for)g(\012)386 5399 y(and)e Fo(B)650 5415 y Fk(\032)p Fn(\()p Fk(\014)753 5424 y Ff(0)787 5415 y Fn(\))852 5399 y Fq(for)f Fo(S)1061 5414 y Fn(1)1100 5399 y Fq(,)h(as)f(w)m(ell)g(as)h Fo(\014)1658 5363 y Fn(2)1652 5424 y(0)1697 5399 y Fo(V)22 b Fq(\()p Fo(\014)1869 5414 y Fn(0)1908 5399 y Fo(r)s Fq(\))32 b(for)g Fo(V)54 b Fq(and)33 b Fo(V)21 b Fq(\()p Fo(r)s Fq(\))33 b(for)2872 5374 y(~)2858 5399 y Fo(V)21 b Fq(,)33 b(to)f(get)386 5623 y(\(33\))175 b Fo(\025)792 5638 y Fn(2)832 5623 y Fq(\()p Fo(R)944 5638 y Fn(2)983 5623 y Fo(=\014)1087 5638 y Fn(0)1127 5623 y Fo(;)17 b(\014)1232 5582 y Fn(2)1226 5647 y(0)1270 5623 y Fo(V)22 b Fq(\()p Fo(\014)1442 5638 y Fn(0)1481 5623 y Fo(r)s Fq(\))28 b Fm(\024)g Fo(\025)1756 5638 y Fn(2)1795 5623 y Fq(\()p Fo(\032)p Fq(\()p Fo(\014)1976 5638 y Fn(0)2016 5623 y Fq(\))p Fo(;)17 b(V)k Fq(\()p Fo(r)s Fq(\)\))27 b(=)h Fo(\025)2525 5638 y Fn(2)2564 5623 y Fq(\()p Fo(R)2676 5638 y Fn(1)2716 5623 y Fo(;)17 b(V)k Fq(\()p Fo(r)s Fq(\)\))p Fo(:)p eop %%Page: 15 15 15 14 bop 1237 259 a Fi(A)33 b(SECOND)g(EIGENV)-9 b(ALUE)33 b(BOUND)773 b(15)386 459 y Fq(But)38 b(b)m(y)g(what)g(has)g(b)s(een)h (said)e(ab)s(o)m(v)m(e)i(ab)s(out)e(the)h(scaling)e(prop)s(erties)i(of) f(the)386 575 y(problem,)31 b(w)m(e)j(ha)m(v)m(e)386 803 y(\(34\))1092 736 y Fo(\025)1149 751 y Fn(2)1188 736 y Fq(\()p Fo(R)1300 751 y Fn(2)1340 736 y Fo(=\014)1444 751 y Fn(0)1483 736 y Fo(;)17 b(\014)1588 699 y Fn(2)1582 760 y(0)1627 736 y Fo(V)k Fq(\()p Fo(\014)1798 751 y Fn(0)1838 736 y Fo(r)s Fq(\)\))p 1092 780 869 4 v 1092 871 a Fo(\025)1149 886 y Fn(1)1188 871 y Fq(\()p Fo(R)1300 886 y Fn(2)1340 871 y Fo(=\014)1444 886 y Fn(0)1483 871 y Fo(;)c(\014)1588 837 y Fn(2)1582 896 y(0)1627 871 y Fo(V)k Fq(\()p Fo(\014)1798 886 y Fn(0)1838 871 y Fo(r)s Fq(\)\))1998 803 y(=)2111 736 y Fo(\025)2168 751 y Fn(2)2208 736 y Fq(\()p Fo(R)2320 751 y Fn(2)2359 736 y Fo(;)c(V)k Fq(\()p Fo(r)s Fq(\)\))p 2111 780 531 4 v 2111 871 a Fo(\025)2168 886 y Fn(1)2208 871 y Fq(\()p Fo(R)2320 886 y Fn(2)2359 871 y Fo(;)c(V)k Fq(\()p Fo(r)s Fq(\)\))2652 803 y Fo(:)386 1033 y Fq(Com)m(bining)31 b(\(32\))h(for)g Fo(\014)h Fq(=)28 b Fo(\014)1484 1048 y Fn(0)1523 1033 y Fq(,)33 b(\(33\))f(and\(34\),)g(w)m(e)h(get)386 1263 y(\(35\))1260 1195 y Fo(\025)1317 1210 y Fn(2)1356 1195 y Fq(\()p Fo(R)1468 1210 y Fn(1)1508 1195 y Fo(;)17 b(V)k Fq(\()p Fo(r)s Fq(\)\))p 1260 1240 V 1260 1331 a Fo(\025)1317 1346 y Fn(1)1356 1331 y Fq(\()p Fo(R)1468 1346 y Fn(1)1508 1331 y Fo(;)c(V)k Fq(\()p Fo(r)s Fq(\)\))1828 1263 y Fm(\025)1943 1195 y Fo(\025)2000 1210 y Fn(2)2039 1195 y Fq(\()p Fo(R)2151 1210 y Fn(2)2191 1195 y Fo(;)c(V)k Fq(\()p Fo(r)s Fq(\)\))p 1943 1240 V 1943 1331 a Fo(\025)2000 1346 y Fn(1)2039 1331 y Fq(\()p Fo(R)2151 1346 y Fn(2)2191 1331 y Fo(;)c(V)k Fq(\()p Fo(r)s Fq(\)\))2484 1263 y Fo(:)386 1487 y Fq(Because)39 b Fo(R)838 1502 y Fn(1)916 1487 y Fq(and)f Fo(R)1185 1502 y Fn(2)1262 1487 y Fq(w)m(ere)h(c)m (hosen)h(arbitrarily)-8 b(,)36 b(this)h(pro)m(v)m(es)j(Theorem)e(2.2.) 3297 1604 y Fe(\003)1252 1839 y Fq(7.)49 b Fp(Pr)n(oof)38 b(of)h(Theorem)e(3.1)486 2014 y Fq(Before)25 b(w)m(e)i(pro)m(v)m(e)g (Theorem)f(3.1)f(w)m(e)h(need)h(to)e(state)h(the)g(follo)m(wing)d(tec)m (hnical)386 2130 y(Lemma:)386 2312 y Fy(Lemma)37 b(7.1.)42 b Fh(Be)34 b Fo(a;)17 b(b;)g(c;)g(d)27 b(>)h Fq(0)35 b Fh(with)f Fo(a)28 b Fm(\025)g Fo(b)p Fh(,)35 b Fo(d)28 b Fm(\025)g Fo(b)35 b Fh(and)2640 2273 y Fk(a)p 2640 2289 38 4 v 2644 2346 a(b)2715 2312 y Fo(<)2831 2273 y Fk(c)p 2828 2289 37 4 v 2828 2346 a(d)2875 2312 y Fh(.)44 b(Then)386 2533 y Fq(\(36\))1578 2466 y Fo(a)23 b Fq(+)f Fo(x)p 1578 2510 227 4 v 1583 2602 a(b)h Fq(+)f Fo(x)1843 2533 y(<)1960 2466 y(c)h Fq(+)f Fo(x)p 1956 2510 V 1956 2602 a(d)g Fq(+)g Fo(x)386 2741 y Fh(holds)34 b(for)h(any)f Fo(x)29 b(>)e Fq(0)p Fh(.)386 2923 y(Pr)-5 b(o)g(of.)41 b Fq(De\014ne)33 b(the)g(function)1385 3129 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))28 b(:=)1748 3062 y Fo(c)22 b Fq(+)g Fo(x)p 1744 3106 V 1744 3198 a(d)f Fq(+)h Fo(x)2002 3129 y Fm(\000)2112 3062 y Fo(a)g Fq(+)g Fo(x)p 2112 3106 V 2117 3198 a(b)g Fq(+)g Fo(x)2348 3129 y(:)386 3342 y Fq(then)32 b Fo(f)11 b Fq(\(0\))27 b Fo(>)g Fq(0.)43 b(A)31 b(straigh)m(tforw)m(ard)g(calculation)d(sho)m(ws)33 b(that)e Fo(f)42 b Fq(has)31 b(exactly)386 3458 y(one)i(zero)g(at)1430 3617 y Fo(x)1485 3632 y Fn(0)1553 3617 y Fq(=)28 b Fm(\000)1865 3549 y Fo(bc)23 b Fm(\000)f Fo(ad)p 1744 3594 550 4 v 1744 3685 a(b)h Fq(+)f Fo(c)g Fm(\000)g Fo(a)h Fm(\000)f Fo(d)2303 3617 y(:)386 3804 y Fq(The)37 b(n)m(umerator)f Fo(bc)26 b Fm(\000)f Fo(ad)36 b Fq(in)g(the)h(expression)g(for)f Fo(x)2388 3819 y Fn(0)2465 3804 y Fq(is)g(p)s(ositiv)m(e)f(b)s(ecause)j (of)386 3921 y(the)33 b(condition)992 3881 y Fk(a)p 992 3898 38 4 v 996 3955 a(b)1067 3921 y Fo(<)1184 3881 y Fk(c)p 1181 3898 37 4 v 1181 3955 a(d)1227 3921 y Fq(.)43 b(F)-8 b(or)32 b(the)h(denominator)e(w)m(e)j(get)770 4159 y Fo(b)22 b Fq(+)g Fo(c)g Fm(\000)h Fo(a)f Fm(\000)h Fo(d)k(>)h(c)22 b Fq(+)g Fo(b)h Fm(\000)1785 4092 y Fo(bc)p 1785 4137 84 4 v 1801 4228 a(d)1901 4159 y Fm(\000)f Fo(d)28 b Fq(=)2192 4092 y(\()p Fo(d)22 b Fm(\000)h Fo(b)p Fq(\)\()p Fo(c)f Fm(\000)h Fo(d)p Fq(\))p 2192 4137 581 4 v 2457 4228 a Fo(d)2810 4159 y Fm(\025)28 b Fq(0)p Fo(:)386 4365 y Fq(This)33 b(means)f(that)h Fo(x)1174 4380 y Fn(0)1241 4365 y Fo(<)28 b Fq(0,)k(suc)m(h)i(that)e Fo(f)11 b Fq(\()p Fo(x)p Fq(\))28 b Fo(>)g Fq(0)k(for)g(all)f Fo(x)d(>)f Fq(0.)463 b Fe(\003)386 4547 y Fh(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(3.1.)41 b Fq(Cho)s(ose)49 b(some)f Fo(x)55 b(>)g Fq(0.)91 b(F)-8 b(rom)47 b(Theorem)i(2.2)f(w)m (e)386 4663 y(kno)m(w)34 b(that)1069 4764 y Fo(\025)1126 4779 y Fn(2)1165 4764 y Fq(\()p Fo(B)1277 4779 y Fk(R)p Fn(+)p Fk(x)1430 4764 y Fo(;)17 b(r)1521 4728 y Fn(2)1560 4764 y Fq(\))p 1069 4809 529 4 v 1069 4900 a Fo(\025)1126 4915 y Fn(1)1165 4900 y Fq(\()p Fo(B)1277 4915 y Fk(R)p Fn(+)p Fk(x)1430 4900 y Fo(;)g(r)1521 4871 y Fn(2)1560 4900 y Fq(\))1635 4831 y Fo(<)1749 4764 y(\025)1806 4779 y Fn(2)1845 4764 y Fq(\()p Fo(B)1957 4779 y Fk(R)2015 4764 y Fo(;)g(r)2106 4728 y Fn(2)2145 4764 y Fq(\))p 1749 4809 434 4 v 1749 4900 a Fo(\025)1806 4915 y Fn(1)1845 4900 y Fq(\()p Fo(B)1957 4915 y Fk(R)2015 4900 y Fo(;)g(r)2106 4871 y Fn(2)2145 4900 y Fq(\))2290 4831 y(for)32 b Fo(x)c(>)g Fq(0)p Fo(:)386 5046 y Fq(Moreo)m(v)m(er,)34 b Fo(\025)898 5061 y Fn(1)938 5046 y Fq(\()p Fo(B)1050 5061 y Fk(R)1107 5046 y Fo(;)17 b(r)1198 5009 y Fn(2)1237 5046 y Fq(\))28 b Fm(\025)g Fo(\025)1465 5061 y Fn(1)1505 5046 y Fq(\()p Fo(B)1617 5061 y Fk(R)p Fn(+)p Fk(x)1769 5046 y Fo(;)17 b(r)1860 5009 y Fn(2)1899 5046 y Fq(\))32 b(and)h Fo(\025)2216 5061 y Fn(2)2255 5046 y Fq(\()p Fo(B)2367 5061 y Fk(R)p Fn(+)p Fk(x)2520 5046 y Fo(;)17 b(r)2611 5009 y Fn(2)2650 5046 y Fq(\))27 b Fo(>)h(\025)2876 5061 y Fn(1)2916 5046 y Fq(\()p Fo(B)3028 5061 y Fk(R)p Fn(+)p Fk(x)3180 5046 y Fo(;)17 b(r)3271 5009 y Fn(2)3310 5046 y Fq(\).)386 5162 y(Th)m(us)29 b(w)m(e)g(can)f(apply)f(\014rst)h(\(7\),)g(then)h (Lemma)d(7.1)h(and)h(then)g(\(7\))f(again,)g(to)h(get)610 5330 y Fo(\025)667 5289 y Fn(+)667 5355 y(2)726 5330 y Fq(\()p Fo(B)838 5345 y Fk(R)p Fn(+)p Fk(x)990 5330 y Fq(\))p 610 5375 419 4 v 610 5468 a Fo(\025)667 5427 y Fn(+)667 5493 y(1)726 5468 y Fq(\()p Fo(B)838 5483 y Fk(R)p Fn(+)p Fk(x)990 5468 y Fq(\))1065 5398 y(=)1179 5330 y Fo(\025)1236 5345 y Fn(2)1275 5330 y Fq(\()p Fo(B)1387 5345 y Fk(R)p Fn(+)p Fk(x)1540 5330 y Fo(;)17 b(r)1631 5294 y Fn(2)1669 5330 y Fq(\))23 b(+)f Fo(n)p 1179 5375 707 4 v 1179 5466 a(\025)1236 5481 y Fn(1)1275 5466 y Fq(\()p Fo(B)1387 5481 y Fk(R)p Fn(+)p Fk(x)1540 5466 y Fo(;)17 b(r)1631 5437 y Fn(2)1669 5466 y Fq(\))23 b(+)f Fo(n)1923 5398 y(<)2037 5330 y(\025)2094 5345 y Fn(2)2133 5330 y Fq(\()p Fo(B)2245 5345 y Fk(R)2303 5330 y Fo(;)17 b(r)2394 5294 y Fn(2)2433 5330 y Fq(\))22 b(+)g Fo(n)p 2037 5375 613 4 v 2037 5466 a(\025)2094 5481 y Fn(1)2133 5466 y Fq(\()p Fo(B)2245 5481 y Fk(R)2303 5466 y Fo(;)17 b(r)2394 5437 y Fn(2)2433 5466 y Fq(\))22 b(+)g Fo(n)2687 5398 y Fq(=)2800 5330 y Fo(\025)2857 5289 y Fn(+)2857 5355 y(2)2916 5330 y Fq(\()p Fo(B)3028 5345 y Fk(R)3086 5330 y Fq(\))p 2800 5375 324 4 v 2800 5468 a Fo(\025)2857 5427 y Fn(+)2857 5493 y(1)2916 5468 y Fq(\()p Fo(B)3028 5483 y Fk(R)3086 5468 y Fq(\))3134 5398 y Fo(:)3297 5623 y Fe(\003)p eop %%Page: 16 16 16 15 bop 386 259 a Fi(16)566 b(RAF)-9 b(AEL)25 b(D.)h(BENGURIA)e(AND)h (HELMUT)g(LINDE)1590 459 y Fp(References)427 616 y Fx([1])42 b(M.S.)19 b(Ash)n(baugh,)h(R.D.)g(Benguria:)32 b Fc(A)21 b(se)l(c)l(ond)i(pr)l(o)l(of)g(of)g(the)f(Payne-P\023)-42 b(olya-Weinb)l(er)l(ger)557 716 y(c)l(onjetur)l(e)p Fx(,)27 b(Comm)n(un.)h(Math.)f(Ph)n(ys.)g Fb(147)g Fx(\(1992\))f(181{190)427 816 y([2])42 b(M.S.)33 b(Ash)n(baugh,)g(R.D.)g(Benguria:)45 b Fc(A)34 b(sharp)i(b)l(ound)e(for)h(the)g(r)l(atio)g(of)g(the)f (\014rst)g(two)557 915 y(Dirichlet)f(eigenvalues)g(of)g(a)g(domain)g (in)f(a)h(hemispher)l(e)h(of)f Fs(S)2579 885 y Fu(n)2618 915 y Fx(,)e(T)-7 b(ransactions)28 b(of)i(the)557 1015 y(AMS)e Fb(353)f Fx(No.)g(3)g(\(2000\),)g(1055{1087)427 1115 y([3])42 b(M.S.)33 b(Ash)n(baugh,)g(R.D.)g(Benguria:)45 b Fc(A)34 b(sharp)i(b)l(ound)e(for)h(the)g(r)l(atio)g(of)g(the)f (\014rst)g(two)557 1214 y(eigenvalues)23 b(of)g(Dirichlet)g(L)l (aplacians)h(and)f(extensions)p Fx(,)d(Annals)f(of)h(Mathematics)e Fb(135)557 1314 y Fx(\(1992\),)26 b(601{628)427 1413 y([4])42 b(M.S.)62 b(Ash)n(baugh,)69 b(R.D.)62 b(Benguria:)103 b Fc(L)l(o)l(g-c)l(onc)l(avity)62 b(of)g(the)f(gr)l(ound)g(state)g(of) 557 1513 y(Schr\177)-42 b(odinger)42 b(op)l(er)l(ators:)60 b(A)40 b(new)h(pr)l(o)l(of)g(of)g(the)g(Baumgartner-Gr)l(osse-Martin)g (in-)557 1613 y(e)l(quality)p Fx(,)28 b(Ph)n(ysical)e(Letters)h(A)h Fb(131)f Fx(No.)h(4,5)e(\(1988\))h(273{276)427 1712 y([5])42 b(M.)25 b(S.)g(Ash)n(baugh,)g(and)f(R.)i(D.)f(Benguria,)f Fc(Isop)l(erimetric)29 b(ine)l(qualities)f(for)g(eigenvalue)557 1812 y(r)l(atios)p Fx(,)d Fb(P)m(artial)30 b(Di\013eren)m(tial)d (Equations)h(of)g(Elliptic)f(T)m(yp)s(e,)i(Cortona,)g(1992)p Fx(,)557 1912 y(A.)e(Alvino,)g(E.)g(F)-7 b(ab)r(es,)27 b(and)g(G.)g(T)-7 b(alen)n(ti,)27 b(editors,)g(Symp)r(osia)f (Mathematica,)h(v)n(ol.)f Fb(35)p Fx(,)557 2011 y(Cam)n(bridge)g(Univ)n (ersit)n(y)g(Press,)h(Cam)n(bridge,)f(United)i(Kingdom,)f(1994,)f(pp.)i (1{36.)427 2111 y([6])42 b(M.F.)f(Betta,)k(F.)d(Chiacc)n(hio,)h(A.)f(F) -7 b(erone:)63 b Fc(Isop)l(erimetric)44 b(estimates)f(for)g(the)f (\014rst)557 2210 y(eigenfunction)30 b(of)h(a)f(class)g(of)h(line)l(ar) f(el)t(liptic)i(pr)l(oblems)p Fx(,)d(to)e(b)r(e)h(published)g(in)g (ZAMP)427 2310 y([7])42 b(B.)23 b(Baumgartner,)f(H.)i(Grosse,)f(A.)h (Martin:)35 b Fc(The)27 b(L)l(aplacian)h(of)e(the)g(p)l(otential)h(and) g(the)557 2410 y(or)l(der)j(of)h(ener)l(gy)f(levels)p Fx(,)e(Ph)n(ysics)f(Letters)g Fb(146B)g Fx(No.)g(5)g(\(1984\),)g (363{366)427 2509 y([8])42 b(B.)23 b(Baumgartner,)g(H.)h(Grosse)e(and)i (A.)g(Martin,)g Fc(Or)l(der)i(of)h(levels)g(in)f(p)l(otential)h(mo)l (dels)p Fx(,)557 2609 y(Nucl.)h(Ph)n(ys.)e(B)i Fb(254)f Fx(\(1985\),)f(528{542.)427 2709 y([9])42 b(C.)28 b(Borell:)37 b Fc(The)31 b(Brunn-Minkowski)g(ine)l(quality)g(in)g(Gauss)f(sp)l(ac)l (e)p Fx(,)f(In)n(v)n(en)n(t.)f(Math.)g Fb(30)557 2808 y Fx(\(1975\),)e(207-211)386 2908 y([10])41 b(G.)22 b(F)-7 b(ab)r(er,)23 b Fc(Beweis,)k(dass)f(unter)d(al)t(len)j(homo)l(genen)g (Membr)l(anen)f(von)g(gleicher)h(Fl\177)-42 b(ache)557 3008 y(und)35 b(gleicher)i(Sp)l(annung)e(die)h(kr)l(eisf\177)-42 b(ormige)38 b(den)e(tiefsten)g(Grundton)f(gibt)p Fx(,)h(Sitzung-)557 3107 y(b)r(eric)n(h)n(te)j(der)g(mathematisc)n(h-ph)n(ysik)-5 b(alisc)n(hen)37 b(Klasse)h(der)i(Ba)n(y)n(erisc)n(hen)d(Ak)-5 b(ademie)557 3207 y(der)27 b(Wissensc)n(haften)g(zu)h(M)r(\177)-44 b(unc)n(hen)27 b(Jahrgang,)f(1923,)f(pp.)j(169{172.)386 3306 y([11])41 b(C.)36 b(Haile:)53 b Fc(A)37 b(se)l(c)l(ond)h (eigenvalue)h(b)l(ound)f(for)g(the)g(Dirichlet)g(Schr\177)-42 b(odinger)39 b(e)l(quation)557 3406 y(with)d(a)h(r)l(adial)t(ly)i (symmetric)e(p)l(otential)p Fx(,)g(Electronic)c(Journal)h(of)g (Di\013eren)n(tial)h(Equa-)557 3506 y(tions)27 b Fb(2000)g Fx(No.)g(10)g(\(2000\),)f(1{19.)386 3611 y([12])41 b(E.)23 b(Krahn,)953 3590 y Fc(\177)939 3611 y(Ub)l(er)j(eine)g(von)h(R)l (ayleigh)h(formulierte)f(Minimaleigenschaft)j(des)c(Kr)l(eises)p Fx(,)557 3711 y(Math.)h(Ann.)i Fb(94)e Fx(\(1925\),)f(97{100.)386 3816 y([13])41 b(E.)29 b(Krahn,)964 3795 y Fc(\177)949 3816 y(Ub)l(er)j(Minimaleigenschaften)i(der)e(Kugel)g(in)f(dr)l(ei)h (und)f(mehr)h(Dimensio-)557 3916 y(nen)p Fx(,)23 b(Acta)g(Comm.)g (Univ.)g(T)-7 b(artu)22 b(\(Dorpat\))h Fb(A9)g Fx(\(1926\),)g(1{44.)e ([English)h(translation:)557 4015 y Fc(Minimal)j(pr)l(op)l(erties)g(of) g(the)f(spher)l(e)h(in)f(thr)l(e)l(e)g(and)g(mor)l(e)g(dimensions)p Fx(,)g Fb(Edgar)h(Krahn)557 4116 y(1894{1961:)38 b(A)30 b(Cen)m(tenary)g(V)-8 b(olume)p Fx(,)1992 4095 y(\177)1982 4116 y(U.)26 b(Lumiste)g(and)f(J.)g(P)n(eetre,)g(editors,)g(IOS)557 4216 y(Press,)h(Amsterdam,)h(The)h(Netherlands,)f(1994,)f(pp.)i (139{174.])386 4316 y([14])41 b(J.M.)32 b(Luttinger:)47 b Fc(Gener)l(alize)l(d)36 b(isop)l(erimetric)h(ine)l(qualities)p Fx(,)e(Pro)r(c.)d(Nat.)h(Acad.)g(Sci.)557 4415 y(USA)28 b Fb(70)f Fx(\(1973\),)f(1005-1006)386 4515 y([15])41 b(A.)25 b(Mercaldo,)g(M.R.)g(P)n(osteraro,)e(F.)i(Bro)r(c)n(k:)35 b Fc(On)27 b(Schwarz)h(and)g(Steiner)f(symmetriza-)557 4615 y(tion)i(with)i(r)l(esp)l(e)l(ct)e(to)h(a)g(me)l(asur)l(e)k Fx(\(preprin)n(t\))386 4714 y([16])41 b(L.)18 b(E.)h(P)n(a)n(yne,)f(G.) h(P\023)-42 b(oly)n(a,)19 b(and)g(H.)g(F.)g(W)-7 b(ein)n(b)r(erger,)19 b Fc(Sur)i(le)h(quotient)f(de)h(deux)f(fr)n(\023)-40 b(equenc)l(es)557 4814 y(pr)l(opr)l(es)30 b(c)l(ons)n(\023)-40 b(ecutives)p Fx(,)28 b(Comptes)g(Rendus)f(Acad.)h(Sci.)g(P)n(aris)e Fb(241)g Fx(\(1955\),)h(917{919.)386 4913 y([17])41 b(L.)d(E.)h(P)n(a)n (yne,)g(G.)g(P\023)-42 b(oly)n(a,)40 b(and)e(H.)h(F.)g(W)-7 b(ein)n(b)r(erger,)41 b Fc(On)e(the)h(r)l(atio)h(of)g(c)l(onse)l (cutive)557 5013 y(eigenvalues)p Fx(,)29 b(J.)e(Math.)h(and)f(Ph)n(ys.) g Fb(35)g Fx(\(1956\),)f(289{298.)386 5113 y([18])41 b(G.)36 b(T)-7 b(alen)n(ti:)52 b Fc(El)t(liptic)40 b(e)l(quations)d (and)h(r)l(e)l(arr)l(angements)p Fx(,)g(Ann.)e(Scuola)f(Norm.)g(Sup.) 557 5212 y(Pisa)26 b(\(4\))i Fb(3)f Fx(\(1976\),)g(697{718)486 5329 y Fc(E-mail)j(addr)l(ess)7 b Fx(:)38 b Fa(RBenguri@fis.puc.)o(cl)o (,)g(Helmut.Linde@gm)o(x.)o(de)486 5499 y Fw(Dep)-6 b(ar)g(tment)43 b(of)f(Physics,)h(Pontific)1907 5492 y(\023)1917 5499 y(\020a)f(Universid)n(ad)f(Ca)-6 b(t)2745 5492 y(\023)2742 5499 y(olica)42 b(de)g(Chile)386 5598 y(Casilla)31 b(306,)f(Correo)i (22)f(Santia)n(go,)g(Chile.)p eop %%Trailer end userdict /end-hook known{end-hook}if 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