This is a multi-part message in MIME format. ---------------0205100907496 Content-Type: text/plain; name="02-220.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-220.comments" PACS-Code: 75.20.-g, 03.65.Ge ---------------0205100907496 Content-Type: text/plain; name="02-220.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-220.keywords" Schroedinger operators with magnetic fields, diamagnetic inequality ---------------0205100907496 Content-Type: application/postscript; name="leschke.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="leschke.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ruder.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ruder %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.05.10:1550 %%BeginProcSet: texc.pro %! 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b(c)n(haracterized)d(b)n(y)j(a)f Fl(Schr\177)-42 b(odinger)29 b(op)l(er)l(ator)p Fn(,)d(or)e(standard)g (Hamiltonian,)h(informally)f(giv)n(en)199 3470 y(b)n(y)k(the)g (expression)797 3624 y Fi(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\))23 b(:=)1197 3591 y Fj(1)p 1197 3605 34 4 v 1197 3652 a(2)1240 3557 y Fh(\000)1278 3624 y Fg(P)32 b Ff(\000)18 b Fg(a)p Fn(\()p Fg(Q)p Fn(\))1642 3557 y Fh(\001)1680 3574 y Fj(2)1736 3624 y Fn(+)g Fi(v)s Fn(\()p Fg(Q)p Fn(\))1184 b(\(1\))199 3765 y(where)44 b Fg(Q)51 b Fn(=)f(\()p Fi(Q)792 3777 y Fj(1)830 3765 y Fi(;)14 b(Q)933 3777 y Fj(2)969 3765 y Fn(\))45 b(and)f Fg(P)64 b Fn(=)51 b(\()p Fi(P)1549 3777 y Fj(1)1586 3765 y Fi(;)14 b(P)1676 3777 y Fj(2)1714 3765 y Fn(\))44 b(denote)h(the)f(usual)g(t)n(w)n (o-comp)r(onen)n(t)f(v)n(ector)199 3865 y(op)r(erators)27 b(of)i(p)r(osition)g(and)f(canonical)g(momen)n(tum,)i(resp)r(ectiv)n (ely)-7 b(.)40 b(Since)29 b(w)n(e)f(are)g(in)n(terested)199 3964 y(in)22 b(gauge-indep)r(enden)n(t)e(quan)n(tities)g(only)-7 b(,)23 b(w)n(e)d(tak)n(e)h(the)g(lib)r(ert)n(y)g(to)g(sligh)n(tly)f (abuse)h(the)g(notation)199 4064 y(in)34 b(that)f(the)g(op)r(erator)f (\(1\))h(actually)g(dep)r(ends)g(on)g(the)g(v)n(ector)f(p)r(oten)n (tial)h Fg(a)f Fn(=)g(\()p Fi(a)2901 4076 y Fj(1)2938 4064 y Fi(;)14 b(a)3019 4076 y Fj(2)3056 4064 y Fn(\))34 b(\(and)199 4163 y(the)42 b(scalar)d(p)r(oten)n(tial)i Fi(v)s Fn(\))h(and)f(not)g(just)h(on)f(the)g(magnetic)g(\014eld)g Fi(b)k Fn(:)h Fk(R)2651 4133 y Fj(2)2740 4163 y Ff(!)g Fk(R)h Fn(giv)n(en)40 b(b)n(y)199 4263 y Fi(b)p Fn(\()p Fg(x)p Fn(\))23 b(=)g Fi(@)5 b(a)558 4275 y Fj(2)595 4263 y Fn(\()p Fg(x)p Fn(\))p Fi(=@)g(x)852 4275 y Fj(1)904 4263 y Ff(\000)14 b Fi(@)5 b(a)1076 4275 y Fj(1)1113 4263 y Fn(\()p Fg(x)p Fn(\))p Fi(=@)g(x)1370 4275 y Fj(2)1407 4263 y Fn(,)27 b(where)e(\()p Fi(x)1774 4275 y Fj(1)1812 4263 y Fi(;)14 b(x)1896 4275 y Fj(2)1933 4263 y Fn(\))27 b(is)e(the)h(pair)g(of)f(Cartesian)g(co-ordinates)199 4363 y(of)g(the)g(p)r(oin)n(t)g Fg(x)d Ff(2)i Fk(R)855 4333 y Fj(2)898 4363 y Fn(.)36 b(W)-7 b(e)25 b(write)f Fg(x)12 b Ff(\001)g Fg(y)27 b Fn(:=)c Fi(x)1642 4375 y Fj(1)1680 4363 y Fi(y)1721 4375 y Fj(1)1770 4363 y Fn(+)12 b Fi(x)1894 4375 y Fj(2)1932 4363 y Fi(y)1973 4375 y Fj(2)2034 4363 y Fn(for)24 b(the)h(Euclidean)f(scalar)f(pro)r (duct)i(of)199 4462 y Fg(x)p Fn(,)j Fg(y)f Ff(2)d Fk(R)514 4432 y Fj(2)585 4462 y Fn(and)k(also)e Fg(x)968 4432 y Fj(2)1029 4462 y Fn(:=)d Fg(x)c Ff(\001)f Fg(x)28 b Fn(and)g Ff(j)p Fg(x)p Ff(j)23 b Fn(:=)1735 4390 y Ff(p)p 1805 4390 92 4 v 1805 4462 a Fg(x)1860 4438 y Fj(2)1897 4462 y Fn(.)38 b(T)-7 b(o)27 b(a)n(v)n(oid)f(tec)n(hnical)i (complications,)f(w)n(e)199 4562 y(will)21 b(assume)e(throughout)h (that)g Fg(a)j Fn(:)g Fk(R)1403 4532 y Fj(2)1469 4562 y Ff(!)g Fk(R)1629 4532 y Fj(2)1693 4562 y Fn(is)d(con)n(tin)n(uously)f (di\013eren)n(tiable)h(and)g Fi(v)26 b Fn(:)d Fk(R)3055 4532 y Fj(2)3122 4562 y Ff(!)g Fk(R)199 4662 y Fn(is)34 b(con)n(tin)n(uous)f(and)g(b)r(ounded)h(from)g(b)r(elo)n(w.)54 b(This)34 b(in)g(particular)e(implies)i(that)g Fi(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\))34 b(can)199 4761 y(b)r(e)g(de\014ned)g (straigh)n(tforw)n(ardly)d(and)i(uniquely)h(as)f(a)g(self-adjoin)n(t)g (b)r(ounded)h(b)r(elo)n(w)f(op)r(erator)199 4861 y(acting)h(on)g(a)g (dense)h(domain)f(in)g(the)h(separable)e(Hilb)r(ert)i(space)f(L)2388 4831 y Fj(2)2425 4861 y Fn(\()p Fk(R)2511 4831 y Fj(2)2555 4861 y Fn(\))g(of)h(all)f(\(equiv)-5 b(alence)199 4960 y(classes)22 b(of)6 b(\))24 b(Leb)r(esgue)f(square-in)n(tegrable)d (complex-v)-5 b(alued)23 b(functions)h(on)f(the)g(plane)h(with)f(the) 199 5060 y(usual)k(scalar)f(pro)r(duct)i Ff(h)p Fi(';)g( )s Ff(i)c Fn(:=)1324 4993 y Fh(R)1364 5090 y Fe(R)1411 5073 y Fd(2)1456 5060 y Fi(d)1499 5030 y Fj(2)1536 5060 y Fi(x)g(')p Fn(\()p Fg(x)p Fn(\))1780 5030 y Fc(\003)1833 5060 y Fi( )s Fn(\()p Fg(x)p Fn(\),)k(for)f Fi(')p Fn(,)h Fi( )e Ff(2)d Fn(L)2502 5030 y Fj(2)2539 5060 y Fn(\()p Fk(R)2626 5030 y Fj(2)2669 5060 y Fn(\).)p eop %%Page: 3 3 3 2 bop 199 100 a Fl(Simple)31 b(diamagnetic)g(monotonicities)1782 b Fn(3)365 282 y(The)20 b(simplest)g(quan)n(tit)n(y)f(w)n(e)g(are)g (going)f(to)i(lo)r(ok)f(at)g(is)h(the)g Fl(gr)l(ound-state)i(ener)l(gy) 27 b Fn(of)20 b Fi(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\),)199 382 y(that)28 b(is,)g(the)g(b)r(ottom)g(of)f(its)h(sp)r(ectrum,)g(in)g (sym)n(b)r(ols)797 523 y Fi(e)836 535 y Fj(0)873 523 y Fn(\()p Fi(b;)14 b(v)s Fn(\))23 b(:=)g(inf)30 b(sp)r(ec)14 b Fi(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\))23 b(=)103 b(inf)1847 580 y Fc(h)p Fb( )r(;)11 b( )r Fc(i)p Fj(=1)2121 523 y Ff(h)p Fi( )s(;)28 b(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\))g Fi( )s Ff(i)p Fi(:)539 b Fn(\(2\))199 711 y(Here)50 b(the)h(second)e (in\014m)n(um)i(is)f(tak)n(en)g(o)n(v)n(er)e(all)i(normalized)f(w)n(a)n (v)n(e-functions)g Fi( )k Fn(in)d(the)199 810 y(domain)29 b(of)f Fi(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\).)41 b(The)28 b(other)g(quan)n(tit)n(y)h(w)n(e)f(will)h(deal)f(with)i(is)e(the)h (canonical-equilibrium)199 910 y Fl(density)53 b(matrix)62 b Fn(of)52 b Fi(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\),)59 b(that)52 b(is,)58 b Ff(h)p Fg(x)p Ff(j)14 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\)])p Ff(j)p Fg(y)s Ff(i)p Fn(.)111 b(It)52 b(is)g(de\014ned)h(as)e(the)199 1010 y(in)h Fg(x)p Fn(,)58 b Fg(y)67 b Ff(2)d Fk(R)745 980 y Fj(2)840 1010 y Fn(join)n(tly)52 b(con)n(tin)n(uous)f(Hermitian)g (in)n(tegral)g(k)n(ernel)g(of)h(the)g(self-adjoin)n(t,)199 1109 y(non-negativ)n(e)k(and)h(b)r(ounded)h(\\Boltzmann)f(op)r(erator") e(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\)])58 b(corresp)r(onding)199 1209 y(to)53 b(in)n(v)n(erse)f(temp)r (erature)g Fi(\014)70 b(>)65 b Fn(0.)113 b(In)53 b(particular,)58 b(the)c(image)e(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\)])g Fi( )68 b Ff(2)199 1309 y Fn(L)251 1278 y Fj(2)288 1309 y Fn(\()p Fk(R)375 1278 y Fj(2)418 1309 y Fn(\))41 b(of)h(an)f(arbitrary)e Fi( )49 b Ff(2)d Fn(L)1352 1278 y Fj(2)1389 1309 y Fn(\()p Fk(R)1475 1278 y Fj(2)1518 1309 y Fn(\))c(has)f(a)g(con)n(tin)n(uous)f(represen)n (tativ)n(e)f(giv)n(en)h(b)n(y)h(the)199 1408 y(function)47 b Fg(x)53 b Ff(7!)788 1341 y Fh(R)827 1438 y Fe(R)874 1421 y Fd(2)919 1408 y Fi(d)962 1378 y Fj(2)1000 1408 y Fi(y)16 b Ff(h)p Fg(x)p Ff(j)e Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\)])p Ff(j)p Fg(y)s Ff(i)g Fi( )s Fn(\()p Fg(y)s Fn(\))47 b(and)f(the)g(densit)n (y-matrix)f(diagonal)199 1508 y Ff(h)p Fg(x)p Ff(j)14 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\)])p Ff(j)p Fg(x)p Ff(i)51 b Fn(is)g(a)g(w)n(ell-de\014ned,) 57 b(non-negativ)n(e)49 b(and)i(con)n(tin)n(uous)f(function)i(of)199 1607 y Fg(x)23 b Ff(2)g Fk(R)409 1577 y Fj(2)453 1607 y Fn(.)35 b(F)-7 b(or)23 b(the)h(precise)f(de\014nition)h(of)g Fi(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\))24 b(and)f(the)h(existence)g(of)f (a)g(con)n(tin)n(uous)g(in)n(tegral)199 1707 y(k)n(ernel)30 b(of)g(the)g(corresp)r(onding)f(\\Boltzmann)g(op)r(erator")f(\(under)i (m)n(uc)n(h)g(w)n(eak)n(er)f(assumptions)199 1807 y(on)f Fi(b)f Fn(and)g Fi(v)s Fn(\),)i(see)e([5)o(,)h(23)o(,)g(3)o(,)g(4].)199 2031 y Fr(2.)42 b(Monotonicit)m(y)31 b(of)h(ground-state)g(energies)199 2213 y Fn(The)23 b(follo)n(wing)g(t)n(w)n(o)f(facts,)i(whic)n(h)f(w)n (e)g(recall)f(from)h([6)o(],)i(sho)n(w)d(that)h(diamagnetic)g (monotonicit)n(y)199 2313 y(fails)28 b(in)g(general)e(already)g(for)h (ground-state)f(energies.)241 2454 y Fr(F)-8 b(act)54 b(1)f(\([6)o(,)58 b(I)s(I.10]\):)72 b Fn(There)45 b(exists)h(a)g(cen)n (trally)f(symmetric)g(p)r(oten)n(tial)50 b(^)-46 b Fi(v)50 b Fn(with)c(the)365 2554 y(function)25 b Ff(j)p Fg(x)p Ff(j)d(7!)27 b Fn(^)-45 b Fi(v)s Fn(\()p Ff(j)p Fg(x)p Ff(j)p Fn(\))24 b(b)r(eing)g(decreasing)e(in)i(some)f(neigh)n(b)r (ourho)r(o)r(d)g(of)g(some)g Ff(j)p Fg(x)3000 2523 y Fj(\(0\))3089 2554 y Ff(j)g Fi(>)g Fn(0.)365 2662 y(Moreo)n(v)n(er,)30 b(there)i(exist)f(t)n(w)n(o)g(constan)n(t)g(magnetic)g(\014elds)h Fi(b)f Fn(and)2481 2640 y(^)2484 2662 y Fi(b)g Fn(suc)n(h)h(that)f(0)f Fi(<)f(b)h(<)3249 2640 y Fn(^)3252 2662 y Fi(b)365 2762 y Fn(but)f Fi(e)557 2774 y Fj(0)593 2762 y Fn(\()p Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))24 b Fi(>)e(e)923 2774 y Fj(0)960 2762 y Fn(\()989 2740 y(^)992 2762 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\).)241 2886 y Fr(F)-8 b(act)35 b(2)e(\([6,)g(I)s(I.11]\):) 39 b Fn(There)29 b(exist)g(t)n(w)n(o)f(inhomogeneous)g(cen)n(trally)g (symmetric)h(magnetic)365 2995 y(\014elds)f Fi(b)f Fn(and)800 2973 y(^)803 2995 y Fi(b)g Fn(suc)n(h)g(that)h(0)23 b Ff(\024)f Fi(b)p Fn(\()p Ff(j)p Fg(x)p Ff(j)p Fn(\))h Ff(\024)1694 2973 y Fn(^)1697 2995 y Fi(b)p Fn(\()p Ff(j)p Fg(x)p Ff(j)p Fn(\))28 b(for)f(all)g Fg(x)c Ff(2)g Fk(R)2378 2965 y Fj(2)2449 2995 y Fn(but)28 b Fi(e)2640 3007 y Fj(0)2677 2995 y Fn(\()p Fi(b;)14 b Fn(0\))23 b Fi(>)g(e)3006 3007 y Fj(0)3043 2995 y Fn(\()3072 2973 y(^)3075 2995 y Fi(b)o(;)14 b Fn(0\).)199 3145 y(In)23 b(b)r(oth)f(situations)g(the)g (ground-state)f(w)n(a)n(v)n(e-function)2018 3123 y(^)2001 3145 y Fi( 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3728 y(assumption)27 b(in)199 3869 y Fr(Theorem)k(1.)40 b Fl(L)l(et)29 b(the)h(Schr\177)-42 b(odinger)31 b(op)l(er)l(ator)g Fi(H)7 b Fn(\()1893 3847 y(^)1896 3869 y Fi(b)o(;)17 b Fn(^)-45 b Fi(v)t Fn(\))29 b Fl(in)h(the)f(Poinc)l(ar)n(\023)-40 b(e)31 b(gauge)f(\(3\))g(p)l (ossess)199 3978 y(a)h(r)l(e)l(al-value)l(d)h(gr)l(ound-state)e (wave-function,)i(that)f(is,)h Fi(e)2042 3990 y Fj(0)2079 3978 y Fn(\()2108 3956 y(^)2111 3978 y Fi(b)o(;)18 b Fn(^)-46 b Fi(v)t Fn(\))25 b(=)f Ff(h)2422 3956 y Fn(^)2405 3978 y Fi( )2459 3990 y Fj(0)2497 3978 y Fi(;)j(H)7 b Fn(\()2652 3956 y(^)2655 3978 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))2834 3956 y(^)2817 3978 y Fi( )2871 3990 y Fj(0)2909 3978 y Ff(i)31 b Fl(for)g(some)199 4086 y(r)l(e)l(al-value) l(d)642 4064 y Fn(^)625 4086 y Fi( )679 4098 y Fj(0)762 4086 y Fl(in)45 b(the)g(domain)h(of)g Fi(H)7 b Fn(\()1562 4064 y(^)1565 4086 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))45 b Fl(with)h Ff(h)2003 4064 y Fn(^)1986 4086 y Fi( )2040 4098 y Fj(0)2078 4086 y Fi(;)2145 4064 y Fn(^)2128 4086 y Fi( )2182 4098 y Fj(0)2220 4086 y Ff(i)51 b Fn(=)f(1)p Fl(.)85 b(Then)45 b(the)h(p)l(ointwise)199 4195 y(ine)l(qualities)31 b Ff(j)p Fi(b)p Fn(\()p Fg(x)p Fn(\))p Ff(j)23 b(\024)941 4173 y Fn(^)944 4195 y Fi(b)o Fn(\()p Fg(x)p Fn(\))30 b Fl(and)h Fi(v)s Fn(\()p Fg(x)p Fn(\))23 b Ff(\024)j Fn(^)-45 b Fi(v)s Fn(\()p Fg(x)p Fn(\))30 b Fl(for)h(al)t(l)g Fg(x)22 b Ff(2)i Fk(R)2217 4165 y Fj(2)2290 4195 y Fl(imply)31 b(the)f(ine)l(quality)797 4345 y Fi(e)836 4357 y Fj(0)873 4345 y Fn(\()p Fi(b;)14 b(v)s Fn(\))23 b Ff(\024)g Fi(e)1203 4357 y Fj(0)1240 4345 y Fn(\()1269 4323 y(^)1272 4345 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))1762 b(\(4\))199 4486 y Fl(for)31 b(the)f(c)l(orr)l(esp)l(onding)h(gr)l(ound-state)e (ener)l(gies.)199 4627 y Fn(Before)j(giving)g(a)g(simple)g(pro)r(of)g (of)h(the)g(theorem)f(at)g(the)h(end)g(of)f(the)h(section,)g(four)f (remarks)199 4727 y(are)27 b(in)h(order:)282 4868 y Ff(\017)41 b Fn(An)32 b(unpleasan)n(t)f(feature)g(of)g(Theorem)g(1)g(is)g(that)h (the)g(additional)e(assumption)h(uses)g(the)365 4968 y(sp)r(ecial)c(gauge)f(\(3\))h(and)g(therefore)f(is)h(not)g(gauge)e (indep)r(enden)n(t.)38 b(Ho)n(w)n(ev)n(er,)25 b(this)i(gauge)f(is)365 5076 y(often)g(con)n(v)n(enien)n(t,)f(in)g(particular,)g(if)1575 5055 y(^)1578 5076 y Fi(b)g Fn(and)j(^)-45 b Fi(v)28 b Fn(are)d(cen)n(trally)f(symmetric,)i(see)f(b)r(elo)n(w.)35 b(F)-7 b(or)p eop %%Page: 4 4 4 3 bop 199 100 a Fl(Simple)31 b(diamagnetic)g(monotonicities)1782 b Fn(4)365 282 y(other)34 b(gauges)e(the)i(existence)g(of)g(a)g(real-v) -5 b(alued)32 b(ground-state)h(w)n(a)n(v)n(e-function)3020 260 y(^)3003 282 y Fi( )3057 294 y Fj(0)3129 282 y Fn(is)g(in)365 382 y(general)19 b(not)i(su\016cien)n(t)f(to)h(imply)g(the)f (monotonicit)n(y)g(\(4\).)35 b(Con)n(v)n(ersely)-7 b(,)20 b(ev)n(en)g(in)g(the)h(gauge)365 491 y(\(3\))30 b(the)h(real-v)-5 b(aluedness)28 b(of)1334 469 y(^)1317 491 y Fi( )1371 503 y Fj(0)1438 491 y Fn(is)i(not)g(necessary)e(for)h(\(4\))h(to)g (hold)g(as)f(the)h(diamagnetic)365 599 y(inequalit)n(y)i Fi(e)794 611 y Fj(0)830 599 y Fn(\(0)p Fi(;)17 b Fn(^)-45 b Fi(v)s Fn(\))31 b Ff(\024)e Fi(e)1180 611 y Fj(0)1217 599 y Fn(\()1246 577 y(^)1249 599 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))32 b(illustrates)f([21)o(].)50 b(\(See)32 b(also)e(the)i(In)n(tro)r(duction)g(and)f(the)365 699 y(\014rst)d(remark)e(follo)n(wing)g(Theorem)h(2)g(b)r(elo)n(w.\))282 832 y Ff(\017)41 b Fn(If)53 b Fi(e)512 844 y Fj(0)549 832 y Fn(\()578 810 y(^)581 832 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))53 b(do)r(es)f(not)h(b)r(elong)f(to)g(the)h(p)r(oin)n(t) g(sp)r(ectrum)g(of)f Fi(H)7 b Fn(\()2605 810 y(^)2608 832 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\),)59 b(equiv)-5 b(alen)n(tly)e(,)365 941 y(if)48 b(there)f(is)g(no)f(square-in)n (tegrable)f(ground-state)g(w)n(a)n(v)n(e-function)2666 919 y(^)2649 941 y Fi( )2703 953 y Fj(0)2740 941 y Fn(,)52 b(assertion)46 b(\(4\))365 1041 y(remains)36 b(true,)i(pro)n(vided)d (there)h(is)g(a)f(generalized)g(real-v)-5 b(alued)35 b(ground-state,)h(that)h(is,)365 1140 y(a)26 b(sequence)g Ff(f)835 1118 y Fn(^)818 1140 y Fi( )872 1152 y Fj(0)p Fb(;n)970 1140 y Ff(g)1012 1152 y Fb(n)p Fc(2)p Fe(N)1170 1140 y Fn(of)g(real-v)-5 b(alued)26 b(functions)g(in)h(the)g(domain)f (of)h Fi(H)7 b Fn(\()2772 1118 y(^)2775 1140 y Fi(b)o(;)17 b Fn(^)-45 b Fi(v)s Fn(\))27 b(suc)n(h)f(that)365 1252 y Ff(h)415 1230 y Fn(^)397 1252 y Fi( )451 1264 y Fj(0)p Fb(;n)550 1252 y Fi(;)617 1230 y Fn(^)600 1252 y Fi( )654 1264 y Fj(0)p Fb(;n)752 1252 y Ff(i)e Fn(=)e(1)28 b(for)f(all)g Fi(n)c Ff(2)g Fk(N)38 b Fn(and)27 b Fi(e)1646 1264 y Fj(0)1683 1252 y Fn(\()1712 1230 y(^)1715 1252 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))24 b(=)e(inf)2075 1264 y Fb(n)p Fc(2)p Fe(N)2221 1252 y Ff(h)2270 1230 y Fn(^)2253 1252 y Fi( )2307 1264 y Fj(0)p Fb(;n)2405 1252 y Fi(;)28 b(H)7 b Fn(\()2561 1230 y(^)2564 1252 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))2743 1230 y(^)2726 1252 y Fi( )2780 1264 y Fj(0)p Fb(;n)2878 1252 y Ff(i)p Fn(.)282 1388 y Ff(\017)41 b Fn(If)445 1366 y(^)448 1388 y Fi(b)28 b Fn(is)f(constan)n(t,)g(one)g (ma)n(y)g(use)h(Lieb's)f(inequalit)n(y)g([1,)g(App.])963 1538 y Fi(e)1002 1550 y Fj(0)1039 1538 y Fn(\()1068 1517 y(^)1071 1538 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))23 b Ff(\024)1340 1506 y Fj(1)p 1340 1520 34 4 v 1340 1567 a(2)1397 1538 y Ff(j)1417 1517 y Fn(^)1420 1538 y Fi(b)p Ff(j)18 b Fn(+)g Fi(e)1619 1550 y Fj(0)1656 1538 y Fn(\(0)p Fi(;)f Fn(^)-45 b Fi(v)s Fn(\))1340 b(\(5\))365 1680 y(to)28 b(further)f(estimates)h(the)g(r.h.s.)f(of)g(\(4\).)282 1804 y Ff(\017)41 b Fn(The)35 b(subsequen)n(t)f(pro)r(of)g(sho)n(ws)g (that)h(analogous)d(statemen)n(ts)i(hold)h(in)g(more)f(than)h(t)n(w)n (o)365 1904 y(dimensions)45 b(and)h(also)e(for)h(spinless)g(\(in)n (teracting\))g(man)n(y-particle)f(systems)h(ob)r(eying)365 2003 y(either)28 b(Boltzmann)f(or)g(Bose)f(statistics.)199 2145 y(If)42 b(b)r(oth)504 2123 y(^)507 2145 y Fi(b)f Fn(and)k(^)-45 b Fi(v)45 b Fn(are)40 b(cen)n(trally)h(symmetric)g(\(or) g(axially)g(symmetric,)k(when)d(taking)f(in)n(to)199 2244 y(accoun)n(t)25 b(the)h(third)g(dimension)f(along)g(the)h (magnetic-\014eld)f(direction\),)h(a)f(ground-state)f(w)n(a)n(v)n(e-) 199 2353 y(function)549 2331 y(^)532 2353 y Fi( )586 2365 y Fj(0)659 2353 y Fn(of)35 b Fi(H)7 b Fn(\()866 2331 y(^)869 2353 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\),)38 b(in)e(the)g(gauge)e(\(3\),)j(is)e(real-v)-5 b(alued)35 b(if)g(and)h(only)f(if)h(it)f(is)h(cen)n(trally)199 2461 y(symmetric)e(or,)i(equiv)-5 b(alen)n(tly)e(,)36 b(an)e(eigenfunction)g (of)h Fi(L)2027 2473 y Fj(3)2098 2461 y Fn(with)g(eigen)n(v)-5 b(alue)33 b(zero,)j Fi(L)2962 2473 y Fj(3)3015 2439 y Fn(^)2998 2461 y Fi( )3052 2473 y Fj(0)3124 2461 y Fn(=)e(0.)199 2561 y(F)-7 b(or)31 b(v)-5 b(anishing)31 b(magnetic)f(\014eld)i(it)g (is)f(standard)f(textb)r(o)r(ok)h(wisdom)g([12)o(,)h(8)o(])g(that)f (the)h(ground)199 2661 y(state)27 b(in)g(a)f(cen)n(trally)f(symmetric)i (p)r(oten)n(tial)f(is)h(alw)n(a)n(ys)e(giv)n(en)g(b)n(y)i(a)f(zero)f (angular-momen)n(tum)199 2760 y(eigenfunction.)35 b(F)-7 b(or)19 b(non-v)-5 b(anishing)19 b(magnetic)h(\014elds)g(this)g(is)g (wrong)f(in)i(general,)f(cf.)g([13,)g(1)o(,)h(23)o(].)199 2869 y(Ho)n(w)n(ev)n(er,)38 b(in)f(case)g(the)g(magnetic)g(\014eld)1565 2847 y(^)1567 2869 y Fi(b)i(>)g Fn(0)e(is)g(constan)n(t,)i(the)e(p)r (oten)n(tial)j(^)-45 b Fi(v)41 b Fn(is)c(cen)n(trally)199 2968 y(symmetric)32 b(and)g(the)g(function)g Ff(j)p Fg(x)p Ff(j)e(7!)k Fn(^)-45 b Fi(v)s Fn(\()p Ff(j)p Fg(x)p Ff(j)p Fn(\))32 b(is)g(non-decreasing,)f(then)h(there)g(exists,)h(cf.)f([2,) 199 3077 y(Thm.)g(4.6],)g(at)f(least)g(one)1090 3055 y(^)1073 3077 y Fi( )1127 3089 y Fj(0)1194 3077 y Ff(6)p Fn(=)e(0)i(with)h Fi(L)1611 3089 y Fj(3)1665 3055 y Fn(^)1648 3077 y Fi( )1702 3089 y Fj(0)1769 3077 y Fn(=)d(0)i(in)h(the)g(\(in)g (general)e(m)n(ulti-dimensional\))199 3186 y(ground-state)37 b(eigenspace)f(of)i Fi(H)7 b Fn(\()1321 3164 y(^)1324 3186 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))38 b(and)g(the)g(additional)f (assumption)h(of)f(Theorem)g(1)h(is)199 3285 y(ful\014lled.)64 b(This)36 b(applies)f(for)h(instance)g(to)g(the)g(\(cen)n(trally)g (symmetric\))g(harmonic-oscillator)199 3385 y(p)r(oten)n(tial)797 3575 y Fi(v)837 3587 y Fj(osc)930 3575 y Fn(\()p Fg(x)p Fn(\))24 b(:=)1193 3519 y Fi(!)1248 3488 y Fj(2)p 1193 3556 92 4 v 1219 3632 a Fn(2)1295 3575 y Fg(x)1350 3540 y Fj(2)1387 3575 y Fi(;)222 b(!)25 b Ff(\025)e Fn(0)p Fi(;)1320 b Fn(\(6\))199 3739 y(and)28 b(the)g(attractiv)n(e)e(Coulom)n (b-t)n(yp)r(e)h(p)r(oten)n(tial,)797 3897 y Fi(v)837 3909 y Fj(cou)941 3897 y Fn(\()p Fg(x)p Fn(\))c(:=)g Ff(\000)1421 3841 y Fi(g)p 1269 3878 349 4 v 1269 3894 a Ff(p)p 1337 3894 280 4 v 1337 3964 a Fg(x)1392 3940 y Fj(2)1448 3964 y Fn(+)18 b Fi(\025)1579 3940 y Fj(2)1627 3897 y Fi(;)221 b(g)s(;)27 b(\025)d(>)e Fn(0)p Fi(:)994 b Fn(\(7\))199 4082 y(Accordingly)-7 b(,)20 b(Theorem)e(1)h(implies)g (the)h(follo)n(wing)e(upp)r(er)h(b)r(ounds)g(on)g(the)g(ground-state)f (energy)199 4181 y(of)25 b(a)g(spinless)g(c)n(harged)f(particle)g(in)i (the)g(plane)f Fk(R)1768 4151 y Fj(2)1836 4181 y Fn(sub)5 b(ject)25 b(to)h(one)e(of)i(these)f(p)r(oten)n(tials)g(and)g(to)199 4281 y(a)32 b(p)r(erp)r(endicular,)h(p)r(ossibly)f(inhomogeneous)f (magnetic)g(\014eld)i Fi(b)f Fn(the)h(strength)e(of)i(whic)n(h)f(not) 199 4381 y(exceeding)27 b(a)g(certain)g(v)-5 b(alue,)28 b(that)g(is,)f Ff(j)p Fi(b)p Fn(\()p Fg(x)p Fn(\))p Ff(j)c(\024)1754 4359 y Fn(^)1757 4381 y Fi(b)k Fn(for)g(all)h Fg(x)22 b Ff(2)i Fk(R)2273 4350 y Fj(2)2344 4381 y Fn(with)k(some)f(constan)n (t)3073 4359 y(^)3076 4381 y Fi(b)22 b(>)h Fn(0,)797 4566 y Fi(e)836 4578 y Fj(0)873 4566 y Fn(\()p Fi(b;)14 b(v)1018 4578 y Fj(osc)1111 4566 y Fn(\))23 b Ff(\024)g Fi(e)1293 4578 y Fj(0)1330 4566 y Fn(\()1359 4544 y(^)1362 4566 y Fi(b;)14 b(v)1475 4578 y Fj(osc)1568 4566 y Fn(\))23 b(=)1711 4458 y Fh(q)p 1794 4458 426 4 v 41 x(\000)1829 4544 y Fn(^)1832 4566 y Fi(b=)p Fn(2)1952 4499 y Fh(\001)1989 4516 y Fj(2)2044 4566 y Fn(+)c Fi(!)2183 4542 y Fj(2)2219 4566 y Fi(;)940 b Fn(\(8\))797 4717 y Fi(e)836 4729 y Fj(0)873 4717 y Fn(\()p Fi(b;)14 b(v)1018 4729 y Fj(cou)1121 4717 y Fn(\))24 b Ff(\024)e Fi(e)1303 4729 y Fj(0)1340 4717 y Fn(\()1369 4695 y(^)1372 4717 y Fi(b;)14 b(v)1485 4729 y Fj(cou)1589 4717 y Fn(\))p Fi(:)1538 b Fn(\(9\))199 4858 y(The)37 b(equalit)n(y)f(in)h(\(8\))g(follo)n(ws)e(from)i(the)g (explicitly)g(kno)n(wn)f([7)o(,)h(17)o(])g(sp)r(ectral)f(prop)r(erties) g(of)199 4966 y Fi(H)7 b Fn(\()304 4944 y(^)307 4966 y Fi(b;)14 b(v)420 4978 y Fj(osc)513 4966 y Fn(\))28 b(with)g(a)f(constan)n(t)1163 4944 y(^)1166 4966 y Fi(b)p Fn(,)h(see)f(also)f(\(13\))i(b)r(elo)n(w.)p eop %%Page: 5 5 5 4 bop 199 100 a Fl(Simple)31 b(diamagnetic)g(monotonicities)1782 b Fn(5)365 282 y(T)-7 b(aking)19 b Fi(!)26 b Fn(=)d(0)c(and)1012 260 y(^)1015 282 y Fi(b)j Fn(=)h(sup)1286 303 y Fa(x)p Fc(2)p Fe(R)1421 286 y Fd(2)1466 282 y Ff(j)p Fi(b)p Fn(\()p Fg(x)p Fn(\))p Ff(j)d Fn(in)g(\(8\))g(yields)f(the)h(upp)r(er)g (b)r(ound)h(in)f(the)g(follo)n(wing)199 382 y(sandwic)n(hing)33 b(estimate)g(on)g(the)g(ground-state)f(energy)g(of)i(a)e(spinless)h(c)n (harged)f(particle)h(in)g(a)199 482 y(globally)38 b(b)r(ounded)i(and)f (con)n(tin)n(uously)f(di\013eren)n(tiable)h(but)h(otherwise)e (arbitrary)f(magnetic)199 581 y(\014eld)28 b Fi(b)23 b Fn(:)g Fk(R)538 551 y Fj(2)604 581 y Ff(!)g Fk(R)p Fn(,)807 690 y Fj(1)p 807 704 34 4 v 807 751 a(2)908 722 y Fn(inf)878 779 y Fa(x)p Fc(2)p Fe(R)1013 763 y Fd(2)1054 652 y Fh(\014)1054 702 y(\014)1081 722 y Fi(b)p Fn(\()p Fg(x)p Fn(\))1236 652 y Fh(\014)1236 702 y(\014)1287 722 y Ff(\024)f Fi(e)1413 734 y Fj(0)1450 722 y Fn(\()p Fi(b;)14 b Fn(0\))23 b Ff(\024)1750 690 y Fj(1)p 1750 704 V 1750 751 a(2)1839 722 y Fn(sup)1820 796 y Fa(x)p Fc(2)p Fe(R)1955 779 y Fd(2)1996 652 y Fh(\014)1996 702 y(\014)2024 722 y Fi(b)p Fn(\()p Fg(x)p Fn(\))2179 652 y Fh(\014)2179 702 y(\014)2206 722 y Fi(:)911 b Fn(\(10\))199 920 y(The)37 b(lo)n(w)n(er)d(b)r(ound)j(in)g(\(10\))f(is)g(a)g (consequence)g(of)g(the)h(con)n(tin)n(uit)n(y)f(of)g Fi(b)g Fn(together)f(with)i(the)199 1020 y(non-negativit)n(y)f(of)h Fi(H)7 b Fn(\()p Fi(b;)14 b Fn(0\))36 b(and)h(that)h(of)f(its)g(asso)r (ciated)f(P)n(auli)g(op)r(erator)f([5)o(],)40 b(in)d(sym)n(b)r(ols,)199 1120 y Fi(H)7 b Fn(\()p Fi(b;)14 b Fn(0\))23 b Ff(\025)f Fn(0)28 b(and)f Fi(H)7 b Fn(\()p Fi(b;)14 b Fn(0\))k Ff(\006)g Fi(b)p Fn(\()p Fg(Q)p Fn(\))p Fi(=)p Fn(2)k Ff(\025)h Fn(0.)365 1319 y(W)-7 b(e)28 b(close)f(this)h(section)f(with) h(a)f(simple)199 1485 y Fr(Pro)s(of)32 b(of)g(Theorem)e(1.)41 b Fn(The)j(v)n(ector)e(p)r(oten)n(tials)h Fg(a)g Fn(and)2223 1484 y(^)2218 1485 y Fg(a)g Fn(de\014ned)h(b)n(y)f(\(3\))h(satisfy)f (the)199 1585 y(p)r(oin)n(t)n(wise)27 b(inequalit)n(y)199 1789 y Ff(j)p Fg(a)p Fn(\()p Fg(x)p Fn(\))p Ff(j)c(\024)g(j)p Fg(x)p Ff(j)643 1676 y Fh(Z)726 1696 y Fj(1)689 1864 y(0)777 1789 y Fi(d\030)18 b(\030)27 b Ff(j)p Fi(b)p Fn(\()p Fi(\030)t Fg(x)p Fn(\))p Ff(j)c(\024)g(j)p Fg(x)p Ff(j)1404 1676 y Fh(Z)1487 1696 y Fj(1)1450 1864 y(0)1538 1789 y Fi(d\030)32 b(\030)1709 1767 y Fn(^)1712 1789 y Fi(b)p Fn(\()p Fi(\030)t Fg(x)p Fn(\))23 b(=)g Ff(j)2047 1788 y Fn(^)2041 1789 y Fg(a)o Fn(\()p Fg(x)p Fn(\))p Ff(j)905 b Fn(\(11\))199 2002 y(b)n(y)38 b(the)g(triangle)f(inequalit)n (y)h(and)g(the)g(assumption)f Ff(j)p Fi(b)p Fn(\()p Fg(x)p Fn(\))p Ff(j)k(\024)2307 1980 y Fn(^)2310 2002 y Fi(b)o Fn(\()p Fg(x)p Fn(\))e(for)e(all)h Fg(x)i Ff(2)g Fk(R)3010 1972 y Fj(2)3054 2002 y Fn(.)68 b(The)199 2102 y(non-comm)n(utativ)n(e) 27 b(binomial)g(form)n(ula)f(giv)n(es)199 2251 y Ff(h)p Fi( )s(;)i(H)7 b Fn(\()444 2229 y(^)447 2251 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))14 b Fi( )s Ff(i)24 b Fn(=)e Ff(h)p Fi( )s(;)28 b(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\))g Fi( )s Ff(i)19 b Fn(+)f(Re)c Ff(h)p Fi( )s(;)g Fn(\()p Fg(a)p Fn(\()p Fg(Q)p Fn(\))19 b Ff(\000)1977 2250 y Fn(^)1971 2251 y Fg(a)p Fn(\()p Fg(Q)p Fn(\)\))g Ff(\001)g Fg(P)27 b Fi( )s Ff(i)828 2376 y Fn(+)921 2343 y Fj(1)p 921 2357 V 921 2404 a(2)978 2376 y Ff(h)p Fi( )s(;)14 b Fn(\()1142 2375 y(^)1136 2376 y Fg(a)p Fn(\()p Fg(Q)p Fn(\))1325 2341 y Fj(2)1381 2376 y Ff(\000)k Fg(a)p Fn(\()p Fg(Q)p Fn(\))1653 2341 y Fj(2)1691 2376 y Fn(\))c Fi( )s Ff(i)k Fn(+)h Ff(h)p Fi( )s(;)14 b Fn(\()s(^)-45 b Fi(v)s Fn(\()p Fg(Q)p Fn(\))19 b Ff(\000)f Fi(v)s Fn(\()p Fg(Q)p Fn(\)\))c Fi( )s Ff(i)459 b Fn(\(12\))199 2542 y(for)38 b(an)n(y)g Fi( )k Fn(in)d(the)g(domain)g(of)f Fi(H)7 b Fn(\(0)p Fi(;)14 b Fn(0\),)41 b(hence)e(in)g(those)f(of)h Fi(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\))39 b(and)f Fi(H)7 b Fn(\()2856 2520 y(^)2859 2542 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))39 b(b)n(y)f(the)199 2642 y(Kato-Rellic)n(h)f(theorem)g ([5].)69 b(The)38 b(real)f(part)g(of)h Ff(h)p Fi( )s(;)14 b Fn(\()p Fg(a)p Fn(\()p Fg(Q)p Fn(\))27 b Ff(\000)2323 2641 y Fn(^)2318 2642 y Fg(a)o Fn(\()p Fg(Q)p Fn(\)\))g Ff(\001)e Fg(P)i Fi( )s Ff(i)39 b Fn(on)f(the)g(r.h.s.)199 2741 y(v)-5 b(anishes)28 b(if)h Fi( )i Fn(is)d(real-v)-5 b(alued,)28 b(b)r(ecause)f Fg(P)42 b Fn(acts)28 b(as)f(the)i(gradien)n (t)e(divided)i(b)n(y)f(the)g(imaginary)199 2841 y(unit)36 b Fi(i)g Fn(=)547 2776 y Ff(p)p 616 2776 107 4 v 65 x(\000)p Fn(1)o(.)60 b(Emplo)n(ying)34 b(\(11\))h(and)g(the)h(assumption)e(on)h (the)h(p)r(oten)n(tials)f(the)h(last)f(t)n(w)n(o)199 2941 y(terms)e(on)g(the)h(r.h.s.)e(of)i(\(12\))e(are)h(b)r(oth)g(seen)g (to)g(b)r(e)h(non-negativ)n(e.)52 b(Altogether)32 b(this)i(yields)199 3049 y Fi(e)238 3061 y Fj(0)275 3049 y Fn(\()p Fi(b;)14 b(v)s Fn(\))25 b Ff(\024)f(h)619 3027 y Fn(^)601 3049 y Fi( )655 3061 y Fj(0)707 3049 y Fi(;)14 b(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\))1031 3027 y(^)1014 3049 y Fi( )1068 3061 y Fj(0)1105 3049 y Ff(i)25 b(\024)f(h)1301 3027 y Fn(^)1283 3049 y Fi( )1337 3061 y Fj(0)1389 3049 y Fi(;)14 b(H)7 b Fn(\()1531 3027 y(^)1534 3049 y Fi(b)o(;)17 b Fn(^)-45 b Fi(v)t Fn(\))1713 3027 y(^)1696 3049 y Fi( )1750 3061 y Fj(0)1787 3049 y Ff(i)25 b Fn(=)f Fi(e)1972 3061 y Fj(0)2009 3049 y Fn(\()2038 3027 y(^)2041 3049 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))29 b(where)f(the)h(\014rst)g(inequalit)n(y)f (is)g(a)199 3149 y(consequence)f(of)g(the)h(Ra)n(yleigh-Ritz)f(v)-5 b(ariational)26 b(principle.)p 3227 3149 4 57 v 3231 3096 50 4 v 3231 3149 V 3280 3149 4 57 v 199 3373 a Fr(3.)42 b(Monotonicit)m(y)31 b(of)h(densit)m(y)g(matrices)199 3555 y Fn(Diamagnetic)40 b(monotonicit)n(y)g(of)g(densit)n(y)g (matrices)g(\(or)g(only)g(of)g(their)h(diagonals\))e(implies)199 3655 y(monotonicit)n(y)20 b(of)g(the)g(corresp)r(onding)e(ground-state) h(energies)g(in)h(the)h(zero-temp)r(erature)d(limit.)199 3755 y(Therefore,)28 b(F)-7 b(acts)29 b(1)g(and)g(2)f(sho)n(w)g(that)i (the)f(former)f(monotonicit)n(y)g(can)h(also)f(neither)h(hold)f(in)199 3854 y(the)g(presence)f(of)h(general)e(p)r(oten)n(tials)i(for)f(t)n(w)n (o)g(homogeneous)f(magnetic)h(\014elds)h(nor)f(in)h(case)f(of)199 3954 y(v)-5 b(anishing)25 b(p)r(oten)n(tial)h(for)f(t)n(w)n(o)g (general)f(magnetic)h(\014elds.)37 b(Accordingly)-7 b(,)25 b(one)g(can)g(hop)r(e)h(to)f(\014nd)199 4053 y(a)f(diamagnetic)g (monotonicit)n(y)f(of)i(densit)n(y)f(matrices)f(only)h(under)h (additional)e(assumptions)h(on)199 4153 y(the)k(magnetic)f(\014elds)h (and/or)e(the)i(p)r(oten)n(tials.)199 4369 y Fl(3.1.)40 b(Glob)l(al)t(ly)32 b(c)l(onstant)d(magnetic)h(\014elds)h(and)f (harmonic-oscil)t(lator)j(p)l(otential)199 4522 y Fn(In)k(case)e(of)h (a)g(homogeneous,)h(that)f(is,)i(globally)d(constan)n(t)h(magnetic)g (\014eld)g Fi(b)h Ff(2)h Fk(R)k Fn(and)36 b(the)199 4622 y(harmonic-oscillator)16 b(p)r(oten)n(tial)i(\(6\),)j(the)e(densit)n(y) g(matrix)f(is)h(exactly)f(kno)n(wn)g(and,)i(in)f(the)g(gauge)p eop %%Page: 6 6 6 5 bop 199 100 a Fl(Simple)31 b(diamagnetic)g(monotonicities)1782 b Fn(6)199 282 y(\(3\),)28 b(explicitly)g(giv)n(en)f([19)o(,)h(17)o(])g (in)g(terms)f(of)h(the)g(ground-state)e(energy)g(\012)2600 294 y Fb(b)2657 282 y Fn(:=)2768 211 y Fh(p)p 2851 211 414 4 v 71 x Fn(\()p Fi(b=)p Fn(2\))3035 258 y Fj(2)3090 282 y Fn(+)18 b Fi(!)3228 258 y Fj(2)3265 282 y Fn(,)199 382 y(cf.)37 b(\(8\),)28 b(and)g(h)n(yp)r(erb)r(olic)f(functions)h(b)n (y)f(the)h(Mehler-t)n(yp)r(e)f(of)h(form)n(ula)199 561 y Ff(h)p Fg(x)p Ff(j)p Fn(e)346 527 y Fc(\000)p Fb(\014)s(H)t Fj(\()p Fb(b;v)605 535 y Fd(osc)686 527 y Fj(\))716 561 y Ff(j)p Fg(y)s Ff(i)c Fn(=)1136 505 y(\012)1196 517 y Fb(b)p 944 542 477 4 v 944 618 a Fn(2)p Fi(\031)17 b Fn(sinh)d(\()p Fi(\014)t Fn(\012)1355 630 y Fb(b)1389 618 y Fn(\))1445 561 y(exp)1572 444 y Fh(\032)1634 561 y Ff(\000)1884 505 y Fn(\012)1944 517 y Fb(b)p 1709 542 445 4 v 1709 618 a Fn(2)g(tanh)f(\()p Fi(\014)t Fn(\012)2087 630 y Fb(b)2121 618 y Fn(\))2177 444 y Fh(\024)2221 561 y Fg(x)2276 527 y Fj(2)2331 561 y Fn(+)18 b Fg(y)2466 527 y Fj(2)2508 561 y Ff(\000)g Fn(2)p Fg(x)g Ff(\001)g Fg(y)2809 505 y Fn(cosh\()p Fi(\014)t(b=)p Fn(2\))p 2809 542 392 4 v 2815 618 a(cosh)13 b(\()p Fi(\014)t Fn(\012)3129 630 y Fb(b)3163 618 y Fn(\))3211 444 y Fh(\025\033)953 819 y Ff(\002)18 b Fn(exp)1177 677 y Fh(\()1243 819 y Fi(i)c Fn(\012)1346 831 y Fb(b)1403 763 y Fn(sinh\()p Fi(\014)t(b=)p Fn(2\))p 1403 800 383 4 v 1416 876 a(sinh\()p Fi(\014)t Fn(\012)1707 888 y Fb(b)1741 876 y Fn(\))1796 751 y Fh(\000)1834 819 y Fi(x)1881 831 y Fj(2)1918 819 y Fi(y)1959 831 y Fj(1)2015 819 y Ff(\000)k Fi(x)2145 831 y Fj(1)2182 819 y Fi(y)2223 831 y Fj(2)2260 751 y Fh(\001)2298 677 y(\))2365 819 y Fi(:)795 b Fn(\(13\))365 1023 y(By)24 b(discriminating)f(the)h(cases)f Fg(x)11 b Ff(\001)g Fg(y)26 b Ff(\025)c Fn(0)i(and)g Fg(x)11 b Ff(\001)g Fg(y)25 b Fi(<)e Fn(0,)h(an)g(elemen)n(tary)f(\(but)h (somewhat)199 1123 y(tedious\))34 b(calculation)f(sho)n(ws)f(that)i (the)g(function)g Ff(j)p Fi(b)p Ff(j)f(7!)g(jh)p Fg(x)p Ff(j)14 b Fn(exp)o([)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)2724 1135 y Fj(osc)2817 1123 y Fn(\)])p Ff(j)p Fg(y)s Ff(ij)35 b Fn(is)e(non-)199 1222 y(increasing)24 b(for)g(all)g Fi(!)i Ff(\025)d Fn(0,)i(all)f Fg(x)p Fn(,)h Fg(y)h Ff(2)e Fk(R)1500 1192 y Fj(2)1568 1222 y Fn(and)g(all)h Fi(\014)i(>)c Fn(0.)35 b(This)25 b(monotonicit)n(y)f(extends)h(\(8\))f (in)199 1322 y(case)i(of)h(a)f(globally)g(constan)n(t)g Fi(b)h Fn(and,)f(in)h(view)g(of)g(F)-7 b(act)27 b(1,)f(is)h(a)f (particularit)n(y)g(of)h(the)g(harmonic)199 1421 y(oscillator.)365 1521 y(As)i(for)f(the)h(monotonicit)n(y)e(in)i(the)g(p)r(oten)n(tial,)g (w)n(e)f(remark)f(the)i(follo)n(wing.)39 b(The)28 b(function)199 1621 y Fi(!)37 b Ff(7!)d(h)p Fg(x)p Ff(j)28 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)1030 1633 y Fj(osc)1123 1621 y Fn(\)])g Ff(j)p Fg(x)p Ff(i)34 b Fn(is)g(ob)n(viously)f(non-increasing)f(for)i(all)g Fi(b)g Ff(2)g Fk(R)p Fn(,)42 b(all)34 b Fg(x)g Ff(2)g Fk(R)3244 1591 y Fj(2)199 1720 y Fn(and)42 b(all)f Fi(\014)51 b(>)46 b Fn(0.)78 b(This)41 b(prop)r(ert)n(y)g(of)g(the)h(harmonic-oscillator) d(densit)n(y-matrix)h(diagonal)199 1820 y(is)52 b(a)f(stronger)f(one)i (than)g(the)g(univ)n(ersally)e(v)-5 b(alid)52 b(\(rev)n(erse\))e (monotonicit)n(y)h(of)h(the)g(free)199 1920 y(energy)34 b Ff(\000)p Fi(\014)588 1890 y Fc(\000)p Fj(1)691 1920 y Fn(ln)774 1853 y Fh(R)813 1949 y Fe(R)860 1933 y Fd(2)906 1920 y Fi(d)949 1890 y Fj(2)986 1920 y Fi(x)14 b Ff(h)p Fg(x)p Ff(j)28 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\)])g Ff(j)p Fg(x)p Ff(i)36 b Fn(when)f Fi(v)k Fn(is)c(p)r(oin)n(t)n(wise)g(increased,)h(cf.)f(the) 199 2019 y(In)n(tro)r(duction.)58 b(Ho)n(w)n(ev)n(er,)35 b(the)g(harmonic)f(oscillator)f(already)g(illustrates)h(that,)j(in)e (con)n(trast)199 2119 y(to)g(the)h(situation)f(with)h Fi(b)f Fn(=)g(0,)i(monotonicit)n(y)e(of)g(the)g(densit)n(y-matrix)g (o\013-diagonal)e(in)i(the)199 2219 y(p)r(oten)n(tial)20 b(in)g(general)f(ceases)f(to)i(hold)g(for)f Fi(b)j Ff(6)p Fn(=)h(0.)34 b(More)19 b(precisely)-7 b(,)21 b(b)n(y)e(elemen)n(tary)g (calculations)199 2318 y(one)27 b(\014nds)241 2448 y Fr(F)-8 b(act)37 b(3:)45 b Fn(There)31 b(exist)h(t)n(w)n(o)f(constan)n (ts)g Fi(b)p Fn(,)h Fi(\014)j(>)29 b Fn(0)j(and)f(a)h(non-zero)e Fg(x)f Ff(2)i Fk(R)2698 2418 y Fj(2)2741 2448 y Fn(,)i(suc)n(h)e(that)h (the)365 2547 y(function)f Fi(!)e Ff(7!)e(jh)p Fg(x)p Ff(j)h Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)1532 2559 y Fj(osc)1625 2547 y Fn(\)])g Ff(j)20 b(\000)g Fg(x)p Ff(ij)30 b Fn(is)g(increasing)e(in)i(some)g(neigh)n(b)r(ourho)r (o)r(d)365 2647 y(of)e(some)f Fi(!)720 2659 y Fj(0)780 2647 y Fi(>)22 b Fn(0.)199 2859 y Fl(3.2.)40 b(Non-c)l(onstant)29 b(magnetic)h(\014elds)g(and)g(vanishing)i(p)l(otential)199 3013 y Fn(Loss)g(and)g(Thaller)g([15)o(])g(studied)h(the)g(densit)n(y)f (matrix)g(asso)r(ciated)f(with)i(an)f(inhomogeneous)199 3121 y(magnetic)22 b(\014eld)721 3099 y(^)724 3121 y Fi(b)g Fn(whic)n(h)g(is)g(globally)f(b)r(ounded)i(from)f(b)r(elo)n(w)g (b)n(y)g(a)g(non-negativ)n(e)e(constan)n(t)i(one.)199 3221 y(In)28 b(particular,)e(they)i(ha)n(v)n(e)f(sho)n(wn)f([15,)h (Thm.)h(1.3])f(that)h(the)g(inequalit)n(y)797 3308 y Fh(\014)797 3358 y(\014)797 3408 y(\014)825 3404 y Ff(h)p Fg(x)p Ff(j)14 b Fn(e)986 3369 y Fc(\000)p Fb(\014)s(H)t Fj(\()1161 3354 y(^)1163 3369 y Fb(b;)p Fj(0\))1289 3404 y Ff(j)p Fg(y)s Ff(i)1396 3308 y Fh(\014)1396 3358 y(\014)1396 3408 y(\014)1447 3404 y Ff(\024)1771 3348 y Fi(b)p 1544 3385 489 4 v 1544 3461 a Fn(4)p Fi(\031)j Fn(sinh\()p Fi(\014)t(b=)p Fn(2\))2070 3404 y(exp)2211 3312 y Fh(h)2269 3404 y Ff(\000)2362 3348 y Fn(\()p Fg(x)h Ff(\000)g Fg(y)s Fn(\))2634 3317 y Fj(2)p 2362 3385 311 4 v 2470 3461 a Fn(2)p Fi(\014)2682 3312 y Fh(i)3140 3404 y Fn(\(14\))199 3599 y(holds)32 b(for)f(all)h Fg(x)p Fn(,)h Fg(y)h Ff(2)d Fk(R)1006 3569 y Fj(2)1081 3599 y Fn(and)h(all)g Fi(\014)i(>)c Fn(0)i(as)f(long)h(as)f Fi(b)f Ff(\024)2172 3577 y Fn(^)2175 3599 y Fi(b)p Fn(\()p Fg(x)p Fn(\))i(for)g(all)f Fg(x)g Ff(2)f Fk(R)2838 3569 y Fj(2)2914 3599 y Fn(with)i(some)199 3699 y(constan)n(t)e Fi(b)f Ff(\025)f Fn(0.)46 b(The)31 b(Gaussian)f(in)h(\(14\))g(coincides)f(with)i(that)f(of)g(the)g(free)g (densit)n(y)f(matrix)199 3799 y Ff(h)p Fg(x)p Ff(j)14 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\(0)p Fi(;)14 b Fn(0\)])g Ff(j)p Fg(y)s Ff(i)42 b Fn(and)g(the)h(pre-factor)d(with)j (the)g(diagonal)e Ff(h)p Fg(x)p Ff(j)14 b Fn(exp)o([)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b Fn(0\)])g Ff(j)p Fg(x)p Ff(i)p Fn(,)199 3898 y(whic)n(h)28 b(is)f(actually)g (indep)r(enden)n(t)i(of)e Fg(x)c Ff(2)g Fk(R)1611 3868 y Fj(2)1655 3898 y Fn(,)k(see)h(\(13\))f(with)h Fi(!)e Fn(=)c(0.)365 3998 y(Erd})-42 b(os)34 b([6])h(has)g(sho)n(wn)g(that)h (\(14\))f(pro)n(vides)f(basically)g(the)i(b)r(est)g(upp)r(er)f(b)r (ound)h(on)f(the)199 4106 y(densit)n(y-matrix)22 b(o\013-diagonal)e (one)i(can)g(hop)r(e)h(for,)g(unless)2068 4084 y(^)2071 4106 y Fi(b)f Fn(:)i Fk(R)2230 4076 y Fj(2)2296 4106 y Ff(!)f Fk(R)28 b Fn(has)22 b(further)h(prop)r(erties.)199 4206 y(More)32 b(precisely)-7 b(,)32 b(he)h(pro)n(v)n(ed)e(that)h (\(14\))g(cannot)g(b)r(e)h(impro)n(v)n(ed)e(univ)n(ersally)f(b)n(y)i (replacing)g(its)199 4306 y(r.h.s.)21 b(b)n(y)f Ff(jh)p Fg(x)p Ff(j)14 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b Fn(0\)])p Ff(j)p Fg(y)s Ff(ij)p Fn(,)22 b(whic)n(h)e(has)h(more)e(rapid)h(Gaussian)g(deca)n(y)g(as)g Ff(j)p Fg(x)5 b Ff(\000)g Fg(y)s Ff(j)22 b(!)h(1)p Fn(,)199 4405 y(cf.)37 b(\(13\).)241 4544 y Fr(F)-8 b(act)30 b(4)g(\([6)o(,)f(I) s(I.16]\):)34 b Fn(De\014ne)1269 4522 y(^)1272 4544 y Fi(b)o Fn(\()p Fg(x)p Fn(\))24 b(:=)e(\(1)14 b(+)g Fi(x)1774 4514 y Fj(2)1774 4564 y(1)1812 4544 y Fi(=\025)1902 4514 y Fj(2)1939 4544 y Fn(\))g Fi(b)26 b Fn(with)g(t)n(w)n(o)e(constan)n (ts)h Fi(b)p Fn(,)h Fi(\025)d(>)g Fn(0.)36 b(Then)365 4652 y Fi(b)25 b Ff(\024)512 4631 y Fn(^)515 4652 y Fi(b)o Fn(\()p Fg(x)p Fn(\))k(for)f(all)h Fg(x)24 b Ff(2)h Fk(R)1156 4622 y Fj(2)1199 4652 y Fn(,)k(but)g Ff(jh)p Fg(x)p Ff(j)14 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()1922 4631 y(^)1925 4652 y Fi(b;)14 b Fn(0\)])p Ff(j)p Fg(y)s Ff(ij)25 b Fi(>)f Ff(jh)p Fg(x)p Ff(j)14 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b Fn(0\)])p Ff(j)p Fg(y)s Ff(ij)28 b Fn(for)365 4752 y(some)f Fi(\025)d(>)e Fn(0,)28 b(some)f Fg(x)p Fn(,)g Fg(y)f Ff(2)e Fk(R)1346 4722 y Fj(2)1417 4752 y Fn(and)j(some)g Fi(\014)g(>)c Fn(0.)199 4882 y(Nev)n(ertheless,)i(in)h(the)g(next)f(subsection)g(it)h(will)g (turn)f(out)h(that)f(\(14\))g(can)g(still)h(b)r(e)g(impro)n(v)n(ed)e (at)199 4981 y(the)f(cost)f(of)g(allo)n(wing)e(only)i(a)g(restricted)f (class)h(of)g(inhomogeneous)e(magnetic)i(\014elds)g(\(including)199 5081 y(the)28 b(one)f(of)h(F)-7 b(act)28 b(4\).)p eop %%Page: 7 7 7 6 bop 199 100 a Fl(Simple)31 b(diamagnetic)g(monotonicities)1782 b Fn(7)199 282 y Fl(3.3.)40 b(Magnetic)31 b(\014elds)f(and)h(p)l (otentials)f(which)h(ar)l(e)f(c)l(onstant)f(along)i(one)f(dir)l(e)l (ction)199 436 y Fn(In)35 b(this)g(subsection)f(w)n(e)g(will)g (restrict)g(ourselv)n(es)f(to)h(the)h(sp)r(ecial)f(class)f(of)i (magnetic)f(\014elds)g Fi(b)199 535 y Fn(whic)n(h)39 b(do)f(not)h(dep)r(end)g(on)g(the)g(second)f(co-ordinate)f Fi(x)2084 547 y Fj(2)2122 535 y Fn(.)70 b(This)39 b(class)e(co)n(v)n (ers)g(the)i(case)f(of)199 635 y(globally)24 b(constan)n(t)h(\014elds,) g(for)g(whic)n(h)g(the)h(sp)r(ectrum)f(of)g(the)h(Sc)n(hr\177)-42 b(odinger)23 b(op)r(erator)g Fi(H)7 b Fn(\()p Fi(b;)14 b Fn(0\))25 b(is)199 734 y(w)n(ell)f(kno)n(wn)f([7,)h(11)o(,)g(9)o(,)g (17)o(])g(to)g(consist)f(only)h(of)g(isolated)f(harmonic-oscillator)d (lik)n(e)k(eigen)n(v)-5 b(alues)199 834 y(\()p Fi(n)22 b Ff(\000)e Fn(1)p Fi(=)p Fn(2\))p Ff(j)p Fi(b)p Ff(j)p Fn(,)32 b Fi(n)d Ff(2)i Fk(N)t Fn(,)39 b(of)31 b(in\014nite)i (degeneracy)-7 b(,)31 b(the)h(so-called)e(Landau)h(lev)n(els.)49 b(Ho)n(w)n(ev)n(er,)30 b(in)199 934 y(case)35 b(suc)n(h)f(a)h(\014eld)h (is)f(not)g(globally)f(constan)n(t,)i Fi(H)7 b Fn(\()p Fi(b;)14 b Fn(0\))35 b(is)g(conjectured)g([5])g(to)g(p)r(ossess)f(only) 199 1033 y(\(absolutely\))k(con)n(tin)n(uous)f(sp)r(ectrum.)69 b(The)38 b(\014rst)g(rigorous)e(pro)r(of)i(of)g(this)g(conjecture,)j(w) n(as)199 1133 y(giv)n(en)g(b)n(y)f(Iw)n(atsuk)-5 b(a)41 b([10)o(])g(under)g(certain)g(additional)f(assumptions,)k(see)d(also)f ([5,)h(16)o(].)78 b(In)199 1233 y(ac)n(kno)n(wledgemen)n(t)37 b(of)i(this)g(ac)n(hiev)n(emen)n(t,)i(for)d(suc)n(h)g(\014elds)h Fi(H)7 b Fn(\()p Fi(b;)14 b Fn(0\))39 b(often)g(go)r(es)f(under)g(the) 199 1332 y(name)28 b Fl(Iwatsuka)i(mo)l(del)p Fn(.)365 1432 y(In)c(what)f(follo)n(ws,)g(it)h(is)f(most)h(con)n(v)n(enien)n(t)e (to)h(c)n(ho)r(ose)g(the)g(asymmetric)g(gauge)f(de\014ned)i(b)n(y)797 1615 y Fi(a)841 1627 y Fj(1)878 1615 y Fn(\()p Fg(x)p Fn(\))d(:=)g(0)p Fi(;)221 b(a)1461 1627 y Fj(2)1498 1615 y Fn(\()p Fi(x)1577 1627 y Fj(1)1615 1615 y Fn(\))24 b(:=)1781 1502 y Fh(Z)1864 1523 y Fb(x)1902 1531 y Fd(1)1827 1691 y Fj(0)1939 1615 y Fi(dx)2029 1581 y Fc(0)2029 1636 y Fj(1)2080 1615 y Fi(b)p Fn(\()p Fi(x)2195 1581 y Fc(0)2195 1636 y Fj(1)2233 1615 y Fn(\))p Fi(:)852 b Fn(\(15\))199 1798 y(Then)28 b(the)g(resulting)f(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erator)797 1952 y Fi(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\))23 b(=)1174 1919 y Fj(1)p 1174 1933 34 4 v 1174 1980 a(2)1217 1952 y Fi(P)1282 1918 y Fj(2)1270 1972 y(1)1337 1952 y Fn(+)1430 1919 y Fj(1)p 1430 1933 V 1430 1980 a(2)1473 1885 y Fh(\000)1511 1952 y Fi(P)1564 1964 y Fj(2)1620 1952 y Ff(\000)18 b Fi(a)1747 1964 y Fj(2)1785 1952 y Fn(\()p Fi(Q)1883 1964 y Fj(1)1920 1952 y Fn(\))1952 1885 y Fh(\001)1990 1902 y Fj(2)2046 1952 y Fn(+)g Fi(v)s Fn(\()p Fi(Q)2270 1964 y Fj(1)2307 1952 y Fn(\))p Fi(;)778 b Fn(\(16\))199 2093 y(is)43 b(translation)g(in)n(v)-5 b(arian)n(t)42 b(along)g(the)i Fi(x)1542 2105 y Fj(2)1579 2093 y Fn(-direction,)j(pro)n(vided)42 b(the)i(p)r(oten)n(tial)f Fi(v)k Fn(do)r(es)c(not)199 2193 y(dep)r(end)26 b(on)f Fi(x)645 2205 y Fj(2)683 2193 y Fn(,)g(to)r(o.)36 b(The)25 b(op)r(erator)f(\(16\))h(can)f(therefore)h(b)r(e)g(decomp)r(osed)g(b)n (y)f(partial)h(F)-7 b(ourier)199 2292 y(transformation)18 b(in)n(to)h(the)g(one-parameter)e(family)i Fi(H)1906 2304 y Fj(1)1943 2292 y Fn(\()p Fi(k)s Fn(\))24 b(:=)e Fi(P)2252 2262 y Fj(2)2240 2313 y(1)2289 2292 y Fi(=)p Fn(2)q(+)q(\()p Fi(k)t Ff(\000)q Fi(a)2629 2304 y Fj(2)2667 2292 y Fn(\()p Fi(Q)2765 2304 y Fj(1)2802 2292 y Fn(\)\))2866 2262 y Fj(2)2904 2292 y Fi(=)p Fn(2)q(+)q Fi(v)s Fn(\()p Fi(Q)3196 2304 y Fj(1)3232 2292 y Fn(\),)199 2392 y Fi(k)31 b Ff(2)e Fk(R)p Fn(,)38 b(of)31 b(Sc)n(hr\177)-42 b(odinger)29 b(op)r(erators)f(for)j(the)g Fi(x)1722 2404 y Fj(1)1759 2392 y Fn(-direction.)46 b(As)31 b(a)f(consequence,)h(one)f(obtains)199 2491 y(for)d(the)h(densit)n(y)g(matrix)f(of)g(\(16\))199 2676 y Ff(h)p Fg(x)p Ff(j)14 b Fn(e)360 2642 y Fc(\000)p Fb(\014)s(H)t Fj(\()p Fb(b;v)r Fj(\))665 2676 y Ff(j)p Fg(y)s Ff(i)48 b Fn(=)908 2563 y Fh(Z)954 2751 y Fe(R)1025 2620 y Fi(dk)p 1024 2657 92 4 v 1024 2733 a Fn(2)p Fi(\031)1149 2676 y Ff(h)p Fi(x)1228 2688 y Fj(1)1266 2676 y Ff(j)14 b Fn(e)1340 2642 y Fc(\000)p Fb(\014)s(H)1487 2650 y Fd(1)1519 2642 y Fj(\()p Fb(k)q Fj(\))1625 2676 y Ff(j)p Fi(y)1689 2688 y Fj(1)1726 2676 y Ff(i)24 b Fn(e)1819 2642 y Fb(i)p 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y(of)f(con)n(tin)n(uous)g(paths)g Fi(w)25 b Fn(:)e([0)p Fi(;)14 b(\014)t Fn(])23 b Ff(!)g Fk(R)p Fn(,)k Fi(\034)33 b Ff(7!)23 b Fi(w)r Fn(\()p Fi(\034)9 b Fn(\))21 b(with)f(mean)f (function)h Fi(\034)33 b Ff(7!)23 b Fi(x)2749 3722 y Fj(1)2788 3710 y Fn(+)r(\()p Fi(y)2928 3722 y Fj(1)2966 3710 y Ff(\000)r Fi(x)3080 3722 y Fj(1)3117 3710 y Fn(\))p Fi(\034)9 b(=\014)199 3809 y Fn(and)32 b(co)n(v)-5 b(ariance)31 b(function)i(\()p Fi(\034)5 b(;)14 b(\034)1257 3779 y Fc(0)1281 3809 y Fn(\))31 b Ff(7!)g Fn(min)q Ff(f)p Fi(\034)5 b(;)14 b(\034)1762 3779 y Fc(0)1785 3809 y Ff(g)21 b(\000)g Fi(\034)9 b(\034)2024 3779 y Fc(0)2049 3809 y Fi(=\014)t Fn(.)51 b(In)33 b(\(18\))f(w)n(e)g(are)f(making)h(use)g(of)199 3909 y(the)c(notations)797 4092 y Fi(\026)847 4104 y Fb(\014)892 4092 y Fn(\()p Fi(f)f Ff(\016)18 b Fi(w)r Fn(\))24 b(:=)f Fi(\014)1331 4057 y Fc(\000)p Fj(1)1434 3979 y Fh(Z)1517 3999 y Fb(\014)1480 4167 y Fj(0)1576 4092 y Fi(d\034)h(f)1729 4024 y Fh(\000)1766 4092 y Fi(w)r Fn(\()p 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b(T)-7 b(o)28 b(obtain)g(\(18\))199 4640 y(w)n(e)j(also)f(in)n(terc)n(hanged)f (the)i(\(Leb)r(esgue\))g(in)n(tegration)f(with)h(resp)r(ect)g(to)g Fi(k)j Fn(and)d(the)g(Bro)n(wian-)199 4740 y(bridge)f(in)n(tegration)f (b)n(y)i(referring)e(to)h(the)h(F)-7 b(ubini-T)g(onelli)31 b(theorem.)46 b(Thanks)29 b(to)i(translation)199 4839 y(in)n(v)-5 b(ariance)35 b(along)h(the)h Fi(x)1030 4851 y Fj(2)1067 4839 y Fn(-direction)f(the)h(densit)n(y)f(matrix)g(of)h (\(16\))f(dep)r(ends)h(on)f Fi(x)2965 4851 y Fj(2)3039 4839 y Fn(and)h Fi(y)3251 4851 y Fj(2)199 4939 y Fn(only)c(through)g (their)g(di\013erence)h Fi(x)1336 4951 y Fj(2)1396 4939 y Ff(\000)22 b Fi(y)1524 4951 y Fj(2)1561 4939 y Fn(.)54 b(Moreo)n(v)n(er,)32 b(a)h(nice)h(feature)f(of)g(the)h(path-in)n (tegral)199 5038 y(represen)n(tation)22 b(\(18\))g(is)h(that)h(its)f (in)n(tegrand)f(con)n(tains)g(the)h(magnetic)g(\014eld)g(but)h(is)f (nev)n(ertheless)p eop %%Page: 8 8 8 7 bop 199 100 a Fl(Simple)31 b(diamagnetic)g(monotonicities)1782 b Fn(8)199 282 y(non-negativ)n(e)30 b(if)i Fi(x)819 294 y Fj(2)885 282 y Fn(=)d Fi(y)1020 294 y Fj(2)1057 282 y Fn(.)48 b(This)31 b(enables)g(one)f(to)h(estimate)h(the)f(densit)n(y) g(matrix)g(of)g(\(16\))g(b)n(y)199 382 y(using)26 b(standard)g (inequalities)g(of)g(general)f(in)n(tegration)g(theory)g(\(cf.)i ([14]\))f(without)h(lo)r(osing)e(the)199 482 y(dep)r(endence)f(on)e (the)i(magnetic)e(\014eld.)36 b(F)-7 b(or)22 b(example,)i(the)f (triangle)f(inequalit)n(y)g(applied)h(to)g(\(18\))199 581 y(giv)n(es)k(the)h(\(gauge-indep)r(enden)n(t\))f(estimate)199 656 y Fh(\014)199 706 y(\014)199 756 y(\014)227 684 y(\012)266 751 y Fg(x)p Ff(j)14 b Fn(e)395 717 y Fc(\000)p Fb(\014)s(H)t Fj(\()p Fb(b;v)r Fj(\))700 751 y Ff(j)p Fg(y)775 684 y Fh(\013)814 656 y(\014)814 706 y(\014)814 756 y(\014)865 751 y Ff(\024)953 656 y Fh(\014)953 706 y(\014)953 756 y(\014)980 684 y(\012)1020 751 y Fn(\()p Fi(x)1099 763 y Fj(1)1137 751 y Fi(;)g Fn(0\))p Ff(j)g Fn(e)1322 717 y Fc(\000)p Fb(\014)s(H)t Fj(\()p 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Fi(v)s Fl(,)44 b Fn(^)-45 b Fi(v)42 b Fl(b)l(e)d(two)g(magnetic)g(\014elds)g(and)g(two)g(p)l(otentials,)j(al) t(l)d(four)199 1337 y(of)33 b(them)f(not)g(dep)l(ending)h(on)g(the)f (se)l(c)l(ond)g(c)l(o-or)l(dinate)h Fi(x)2052 1349 y Fj(2)2090 1337 y Fl(.)46 b(Then)32 b(the)g(p)l(ointwise)i(ine)l (qualities)199 1437 y Ff(j)p Fi(b)p Fn(\()p Fi(x)337 1449 y Fj(1)375 1437 y Fn(\))p Ff(j)23 b(\024)538 1415 y Fn(^)541 1437 y Fi(b)p Fn(\()p Fi(x)656 1449 y Fj(1)694 1437 y Fn(\))30 b Fl(and)g Fi(v)s Fn(\()p Fi(x)1039 1449 y Fj(1)1077 1437 y Fn(\))23 b Ff(\024)j Fn(^)-45 b Fi(v)s Fn(\()p Fi(x)1342 1449 y Fj(1)1380 1437 y Fn(\))30 b Fl(for)h(al)t(l)g Fi(x)1741 1449 y Fj(1)1801 1437 y Ff(2)24 b Fk(R)35 b Fl(imply)c(the)f(ine)l(quality)199 1512 y Fh(\014)199 1562 y(\014)199 1612 y(\014)227 1540 y(\012)266 1607 y Fg(x)p Ff(j)14 b Fn(e)395 1573 y Fc(\000)p Fb(\014)s(H)t Fj(\()570 1558 y(^)572 1573 y Fb(b;)s Fj(^)-36 b Fb(v)r Fj(\))700 1607 y Ff(j)p Fg(y)775 1540 y Fh(\013)814 1512 y(\014)814 1562 y(\014)814 1612 y(\014)865 1607 y Ff(\024)953 1512 y Fh(\014)953 1562 y(\014)953 1612 y(\014)980 1540 y(\012)1020 1607 y Fn(\()p Fi(x)1099 1619 y Fj(1)1137 1607 y Fi(;)14 b Fn(0\))p Ff(j)g Fn(e)1322 1573 y Fc(\000)p Fb(\014)s(H)t Fj(\()p Fb(b;v)r Fj(\))1627 1607 y Ff(j)p Fn(\()p Fi(y)1723 1619 y Fj(1)1760 1607 y Fi(;)g Fn(0\))1871 1540 y Fh(\013)1910 1512 y(\014)1910 1562 y(\014)1910 1612 y(\014)1965 1607 y Fn(exp)2092 1515 y Fh(h)2150 1607 y Ff(\000)2243 1551 y Fn(\()p Fi(x)2322 1563 y Fj(2)2378 1551 y Ff(\000)k Fi(y)2502 1563 y Fj(2)2539 1551 y Fn(\))2571 1521 y Fj(2)p 2243 1588 V 2379 1664 a Fn(2)p Fi(\014)2618 1515 y Fh(i)3140 1607 y Fn(\(22\))199 1774 y Fl(for)31 b(al)t(l)g Fg(x)22 b Fn(=)h(\()p Fi(x)695 1786 y Fj(1)733 1774 y Fi(;)14 b(x)817 1786 y Fj(2)855 1774 y Fn(\))23 b Ff(2)g Fk(R)1042 1744 y Fj(2)1086 1774 y Fl(,)30 b(al)t(l)h Fg(y)26 b Fn(=)c(\()p Fi(y)1495 1786 y Fj(1)1533 1774 y Fi(;)14 b(y)1611 1786 y Fj(2)1647 1774 y Fn(\))24 b Ff(2)f Fk(R)1835 1744 y Fj(2)1908 1774 y Fl(and)30 b(al)t(l)h Fi(\014)d(>)22 b Fn(0)p Fl(.)199 1891 y Fn(Three)27 b(sp)r(ecial)h (cases)e(of)i(\(22\))f(are)g(w)n(orth)f(to)i(b)r(e)g(men)n(tioned)f (separately:)282 2008 y Ff(\017)41 b Fn(Theorem)21 b(2)h(and)g (translation)e(in)n(v)-5 b(ariance)21 b(along)f(the)j Fi(x)2131 2020 y Fj(2)2168 2008 y Fn(-direction)f(implies)g (monotonicit)n(y)365 2108 y(of)41 b(the)g(densit)n(y-matrix)e(diagonal) g(in)i(the)f(magnetic)g(\014eld)h(and)f(the)h(p)r(oten)n(tial)g(in)f (the)365 2207 y(sense)35 b(that)774 2140 y Fh(\012)814 2207 y Fg(x)p Ff(j)27 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()1290 2185 y(^)1293 2207 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\)])14 b Ff(j)p Fg(x)1556 2140 y Fh(\013)1631 2207 y Ff(\024)1732 2140 y Fh(\012)1772 2207 y Fg(x)p Ff(j)27 b Fn(exp[)p Ff(\000)p Fi(\014)t(H)7 b Fn(\()p Fi(b;)14 b(v)s Fn(\)])g Ff(j)p Fg(x)2514 2140 y Fh(\013)2589 2207 y Fn(for)35 b(all)g Fg(x)h Ff(2)h Fk(R)3084 2177 y Fj(2)3127 2207 y Fn(.)62 b(In)365 2307 y(particular,)41 b(in)e(the)g(zero-temp)r(erature)e(limit)i Fi(\014)46 b Ff(!)c(1)d Fn(the)g(monotonicit)n(y)f Fi(e)2964 2319 y Fj(0)3001 2307 y Fn(\()p Fi(b;)14 b(v)s Fn(\))42 b Ff(\024)365 2415 y Fi(e)404 2427 y Fj(0)441 2415 y Fn(\()470 2394 y(^)473 2415 y Fi(b;)17 b Fn(^)-45 b Fi(v)s Fn(\))44 b(of)f(the)g(corresp)r(onding)e(ground-state)h(energies)g(emerges.)82 b(In)43 b(general,)i(this)365 2515 y(monotonicit)n(y)27 b(is)h(not)f(co)n(v)n(ered)f(b)n(y)h(Theorem)g(1)g(as)g(the)h(case)f Fi(b)22 b Fn(=)h(0)k(already)f(illustrates.)282 2632 y Ff(\017)41 b Fn(If)g Fi(b)i(>)g Fn(0)c(is)h(globally)e(constan)n(t)h (and)h Fi(v)47 b Fn(=)f(^)-45 b Fi(v)46 b Fn(=)d(0,)g(\(22\))c (together)g(with)h(\(13\))g(yields)365 2740 y(the)30 b(follo)n(wing)f(impro)n(v)n(emen)n(t)f(of)i(\(14\))f(for)g(the)h (presen)n(t)g(situation,)g(in)f(whic)n(h)2910 2718 y(^)2913 2740 y Fi(b)g Fn(do)r(es)h(not)365 2840 y(dep)r(end)f(on)e Fi(x)816 2852 y Fj(2)854 2840 y Fn(,)365 2910 y Fh(\014)365 2960 y(\014)365 3010 y(\014)393 2938 y(\012)432 3006 y Fg(x)p Ff(j)14 b Fn(e)561 2971 y Fc(\000)p Fb(\014)s(H)t Fj(\()736 2956 y(^)738 2971 y Fb(b;)p Fj(0\))864 3006 y Ff(j)p Fg(y)939 2938 y Fh(\013)978 2910 y(\014)978 2960 y(\014)978 3010 y(\014)1029 3006 y Ff(\024)1353 2949 y Fi(b)p 1127 2986 489 4 v 1127 3062 a Fn(4)p Fi(\031)i Fn(sinh\()p Fi(\014)t(b=)p Fn(2\))1652 3006 y(exp)1793 2913 y Fh(h)1851 3006 y Ff(\000)1947 2949 y Fi(b)p 1944 2986 42 4 v 1944 3062 a Fn(4)2023 2949 y(\()p Fi(x)2102 2961 y Fj(1)2158 2949 y Ff(\000)i Fi(y)2282 2961 y Fj(1)2319 2949 y Fn(\))2351 2919 y Fj(2)p 2005 2986 401 4 v 2005 3062 a Fn(tanh\()p Fi(\014)t(b=)p Fn(2\))2434 3006 y Ff(\000)2527 2949 y Fn(\()p Fi(x)2606 2961 y Fj(2)2663 2949 y Ff(\000)g Fi(y)2787 2961 y Fj(2)2824 2949 y Fn(\))2856 2919 y Fj(2)p 2527 2986 366 4 v 2664 3062 a Fn(2)p Fi(\014)2903 2913 y Fh(i)2942 3006 y Fi(:)175 b Fn(\(23\))365 3171 y(The)28 b(Gaussian)f(deca)n(y)f(on)i(the)g(r.h.s.)f(of)h(\(14\))f(as)g Ff(j)p Fi(x)2032 3183 y Fj(1)2088 3171 y Ff(\000)18 b 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y Ff(\017)41 b Fn(If)451 3682 y(^)454 3704 y Fi(b)31 b(>)g Fn(0)i(is)g(globally)e(constan)n(t)h (and)h Fi(v)i Fn(=)f(^)-45 b Fi(v)35 b Fn(=)d(0,)h(\(22\))g(together)f (with)h(\(13\))g(yields)f(the)365 3804 y(follo)n(wing)27 b(lo)n(w)n(er)f(estimate)h(on)h(certain)f(densit)n(y-matrix)f (o\013-diagonals)611 3913 y(^)614 3935 y Fi(b)p 375 3972 514 4 v 375 4061 a Fn(4)p Fi(\031)17 b Fn(sinh)643 3994 y Fh(\000)681 4061 y Fi(\014)729 4040 y Fn(^)732 4061 y Fi(b=)p Fn(2)852 3994 y Fh(\001)927 3991 y Fn(exp)1067 3849 y Fh(")1134 3991 y Ff(\000)1227 3913 y Fn(^)1230 3935 y Fi(b)p 1227 3972 42 4 v 1227 4060 a Fn(4)1319 3935 y(\()p Fi(x)1398 3947 y Fj(1)1454 3935 y Ff(\000)h Fi(y)1578 3947 y Fj(1)1615 3935 y Fn(\))1647 3905 y Fj(2)p 1289 3972 426 4 v 1289 4061 a Fn(tanh)1468 3994 y Fh(\000)1506 4061 y Fi(\014)1554 4040 y Fn(^)1557 4061 y Fi(b=)p Fn(2)1677 3994 y Fh(\001)1724 3849 y(#)1796 3991 y Ff(\024)1884 3896 y Fh(\014)1884 3945 y(\014)1884 3995 y(\014)1911 3924 y(\012)1950 3991 y Fn(\()p 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y(SPP)g(1033\).)199 3658 y Fr(References)235 3819 y Ft([1])41 b(Avron)31 b(J,)f(Herbst)h(I)h(and)f(Simon)f(B)h(1978)h(Sc)n(hr\177)-35 b(odinger)32 b(op)r(erators)f(with)g(magnetic)g(\014elds.)g(I.)g (General)434 3898 y(in)n(teractions)24 b Fu(Duke)h(Math.)h(J.)e Fp(45)f Ft(847{883)235 3977 y([2])41 b(Avron)18 b(J)h(E,)f(Herbst)h(I)g (W)g(and)g(Simon)f(B)g(1981)i(Sc)n(hr\177)-35 b(odinger)19 b(op)r(erators)g(with)g(magnetic)g(\014elds.)f(I)r(I)r(I.)h(A)n(toms) 434 4056 y(in)k(homogeneous)i(magnetic)e(\014eld)h Fu(Commun.)j(Math.)f (Phys.)e Fp(79)f Ft(529{572)235 4134 y([3])41 b(Bro)r(derix)46 b(K,)h(Hundertmark)g(D)g(and)h(Lesc)n(hk)n(e)h(H)e(2000)i(Con)n(tin)n (uit)n(y)f(prop)r(erties)g(of)f(Sc)n(hr\177)-35 b(odinger)434 4213 y(semigroups)22 b(with)i(magnetic)g(\014elds)f Fu(R)l(ev.)j(Math.) f(Phys.)f Fp(12)f Ft(181{225)235 4292 y([4])41 b(Bro)r(derix)29 b(K,)g(Lesc)n(hk)n(e)i(H)e(and)i(M)r(\177)-37 b(uller)28 b(P)i(Con)n(tin)n(uous)h(in)n(tegral)f(k)n(ernels)f(for)g(un)n(b)r 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b(Iw)n(atsuk)l(a)31 b(A)f(1985)h (Examples)e(of)h(absolutely)h(con)n(tin)n(uous)g(Sc)n(hr\177)-35 b(odinger)31 b(op)r(erators)g(in)e(magnetic)h(\014elds)434 5081 y Fu(Publ.)c(R)l(es.)f(Inst.)h(Math.)g(Sci.)d Fp(21)g Ft(385{401)p eop %%Page: 10 10 10 9 bop 199 100 a Fl(Simple)31 b(diamagnetic)g(monotonicities)1741 b Fn(10)199 282 y Ft([11])42 b(Landau)24 b(L)g(1930)h(Diamagnetism)n (us)d(der)h(Metalle)h(\(in)g(German\))f Fu(Z.)i(Physik)f Fp(64)f Ft(629{637)199 361 y([12])42 b(Landau)25 b(L)f(D)g(and)g (Lifshitz)g(E)g(M)f(1959)i Fu(Quantum)i(me)l(chanics:)34 b(The)26 b(non-r)l(elativistic)g(the)l(ory)e Ft(\(London:)434 440 y(P)n(ergamon\))199 519 y([13])42 b(La)n(vine)25 b(R)g(and)g(O'Carroll)f(M)g(1977)i(Ground)g(state)g(prop)r(erties)f (and)h(lo)n(w)n(er)e(b)r(ounds)i(for)e(energy)i(lev)n(els)f(of)434 598 y(a)e(particle)h(in)g(a)f(uniform)f(magnetic)i(\014eld)g(and)g (external)h(p)r(oten)n(tial)g Fu(J.)g(Math.)h(Phys.)48 b Fp(18)22 b Ft(1908{1912)199 677 y([14])42 b(Lieb)23 b(E)h(H)f(and)h(Loss)g(M)f(2001)i Fu(A)n(nalysis)f Ft(\(2)1551 653 y Fo(nd)1647 677 y Ft(edition\))g(\(Pro)n(vidence:)33 b(Amer.)22 b(Math.)h(So)r(c.\))199 756 y([15])42 b(Loss)36 b(M)h(and)g(Thaller)f(B)h(1997)h(Optimal)d(heat)j(k)n(ernel)f (estimates)g(for)f(Sc)n(hr\177)-35 b(odinger)37 b(op)r(erators)g(with) 434 834 y(magnetic)23 b(\014elds)h(in)g(t)n(w)n(o)g(dimensions)e Fu(Commun.)27 b(Math.)f(Phys.)e Fp(186)e Ft(95{107)199 913 y([16])42 b(M\024)-35 b(an)n(toiu)27 b(M)g(and)g(Purice)g(R)g(1997) h(Some)f(propagation)h(prop)r(erties)g(of)f(the)g(Iw)n(atsuk)l(a)i(mo)r (del)d Fu(Commun.)434 992 y(Math.)f(Phys.)f Fp(188)f Ft(691{708)199 1071 y([17])42 b(Matsumoto)29 b(H)f(1996)i(Quadratic)g (Hamiltonians)e(and)i(asso)r(ciated)g(orthogonal)g(p)r(olynomials)e Fu(J.)j(F)-5 b(unct.)434 1150 y(A)n(nal.)24 b Fp(136)e Ft(214{225)199 1229 y([18])42 b(Mattis)23 b(D)h(C)f(1988)i Fu(The)h(the)l(ory)f(of)h(magnetism)g Ft(I)e(\(corr.)f(2)1973 1205 y Fo(nd)2069 1229 y Ft(prin)n(ting\))h(\(Berlin:)30 b(Springer\))199 1308 y([19])42 b(P)n(apadop)r(oulos)26 b(G)e(J)h(1971)g(Magnetization)h(of)e(harmonically)g(b)r(ound)h(c)n (harges)g Fu(J.)i(Phys.)f(A:)g(Math.)h(Gen.)434 1386 y Fp(4)c Ft(773{781)199 1465 y([20])42 b(Ro)r(epstor\013)24 b(G)g(1996)g Fu(Path)i(inte)l(gr)l(al)g(appr)l(o)l(ach)j(to)c(quantum)i (physics)c Ft(\(2)2344 1442 y Fo(nd)2440 1465 y Ft(prin)n(ting\))h (\(Berlin:)30 b(Springer\))199 1544 y([21])42 b(Simon)21 b(B)h(1976)i(Univ)n(ersal)e(diamagnetism)f(of)h(spinless)g(Bose)h (systems)f Fu(Phys.)i(R)l(ev.)h(L)l(ett.)43 b Fp(36)22 b Ft(1083{1084)199 1623 y([22])42 b(Simon)24 b(B)g(1977)i(An)f (abstract)i(Kato's)e(inequalit)n(y)g(for)g(generators)h(of)e(p)r (ositivit)n(y)i(preserving)f(semigroups)434 1702 y Fu(Ind.)h(Math.)g (J.)d Fp(26)g Ft(1067{1073)199 1781 y([23])42 b(Simon)22 b(B)i(1979)g Fu(F)-5 b(unctional)27 b(inte)l(gr)l(ation)f(in)f(quantum) h(physics)e Ft(\(New)h(Y)-6 b(ork:)31 b(Academic\))199 1860 y([24])42 b(Ueki)23 b(N)g(2000)h(Simple)e(examples)h(of)g(Lifsc)n (hitz)g(tails)g(in)g(Gaussian)h(random)f(magnetic)g(\014elds)h Fu(A)n(nn.)h(Henri)434 1939 y(Poinc)l(ar)n(\023)-35 b(e)25 b Fp(1)e Ft(473{498)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0205100907496--