Content-Type: multipart/mixed; boundary="-------------0003281023773" This is a multi-part message in MIME format. ---------------0003281023773 Content-Type: text/plain; name="00-130.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-130.keywords" Pauli operator, zero modes, magnetic field, Birman-Schwinger operator ---------------0003281023773 Content-Type: application/postscript; name="zero_modex.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="zero_modex.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: zero_modex.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o zero_modex.ps zero_modex %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.03.28:1710 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls 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b(signi\014can)n(t)f(mathematical)h(implication)f(of)599 927 y(zero)18 b(mo)r(des)h(is)g(that)g(there)g(can't)g(b)r(e)g(an)g (analogue)e(of)i(the)g(Cwik)n(el-Lieb-Rosen)n(blum)599 1027 y(inequalit)n(y)h(for)g(the)h(n)n(um)n(b)r(er)f(of)g(negativ)n(e)f (eigen)n(v)-5 b(alues)19 b(of)h Ft(P)2496 1039 y Fs(A)2553 1027 y FG(+)t Fo(V)38 b FG(in)21 b(terms)f(of)g(some)599 1126 y Fo(L)656 1096 y Fs(p)726 1126 y FG(norm)31 b(of)g(the)h(scalar)e (p)r(oten)n(tial)i Fo(V)19 b FG(,)32 b(since)g(an)n(y)e(small)i (negativ)n(e)e(p)r(erturbation)599 1226 y Fo(V)46 b FG(w)n(ould)27 b(pro)r(duce)f(negativ)n(e)g(eigen)n(v)-5 b(alues,)26 b(con)n(trary)f(to)h(suc)n(h)h(an)g(inequalit)n(y)f(if)h Fo(V)599 1326 y FG(is)h(su\016cien)n(tly)f(small.)682 1425 y(The)f(\014rst)f(example)f(of)h(a)g(magnetic)g(\014eld)2045 1404 y Fo(~)2031 1425 y(B)30 b FG(whic)n(h)25 b(yields)g(zero)f(mo)r (des)h(w)n(as)f(the)599 1525 y(follo)n(wing)j(constructed)g(in)h([11)o (]:)750 1715 y Fo(~)736 1736 y(B)5 b FG(\()p Fq(x)p 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Fp(j)1355 2242 y Fo(~)1342 2263 y(B)5 b FG(\()p Fq(x)p FG(\))p Fp(j)23 b(2)g Fo(L)1710 2229 y Fs(p)1748 2263 y FG(\()p Ft(R)1835 2229 y Fm(3)1878 2263 y FG(\))111 b(for)27 b(an)n(y)54 b Fo(p)23 b(>)2495 2207 y FG(3)p 2495 2244 42 4 v 2495 2320 a(4)2546 2263 y Fo(;)614 b FG(\(3\))1079 2504 y(1)p 1079 2541 V 1079 2617 a(3)1131 2560 y(\(3)18 b(+)g(2)p Fo(l)r FG(\))1420 2539 y Fo(~)1407 2560 y(B)s FG(\()p Fq(x)p FG(\))p Fo(;)42 b(l)24 b Fp(2)g Ft(N)t Fo(;)130 b FG(also)27 b(yields)g(zero)g(mo)r(des.)363 b(\(4\))599 2713 y(W)-7 b(e)21 b(shall)f(reserv)n(e)e(commen)n(t)i(on)f (these)h(till)h(later.)34 b(Other)19 b(examples)h(of)g(zero)f(mo)r (des,)599 2812 y(based)34 b(on)f(the)h(construction)f(of)h([11)o(],)h (are)e(giv)n(en)g(in)h([6)o(],)i([1)o(].)56 b(In)34 b(an)f(attempt)i (to)599 2912 y(explain)40 b(the)g(origin)f(of)h(zero)e(mo)r(des,)43 b(Erd})-42 b(os)38 b(and)i(Solo)n(v)n(ej)e(in)i([7])g(giv)n(e)f(a)g (more)599 3012 y(geometric)d(viewp)r(oin)n(t.)66 b(Using)36 b(the)i(kno)n(wn)e(b)r(eha)n(viour)g(of)h(the)h(Dirac)e(op)r(erator)599 3111 y(under)25 b(conformal)f(transformations,)g(and)h(that)g Ft(R)2225 3081 y Fm(3)2293 3111 y FG(is)g(conformally)f(equiv)-5 b(alen)n(t)25 b(to)599 3211 y(a)i(punctured)f(sphere)g Ft(S)1373 3181 y Fm(3)1405 3211 y FG(,)g(they)h(establish)f(their)h (zero)e(mo)r(des)i(on)f Ft(S)2775 3181 y Fm(3)2833 3211 y FG(as)g(w)n(ell)g(as)g(on)599 3311 y Ft(R)653 3280 y Fm(3)732 3311 y FG(as)34 b(pull-bac)n(ks)g(of)g(zero)g(mo)r(des)h(on) g Ft(S)1964 3280 y Fm(2)2030 3311 y FG(under)g(the)g(Hopf)g(map)g Ft(S)2877 3280 y Fm(3)2944 3311 y Fp(!)g Ft(S)3114 3280 y Fm(2)3145 3311 y FG(.)59 b(It)599 3410 y(is)29 b(also)e(sho)n(wn)h (in)g([7])g(that)h(arbitrary)e(degeneracy)f(is)j(p)r(ossible;)f (examples)g(of)g(this)599 3510 y(ma)n(y)f(also)g(b)r(e)h(found)g(in)g ([2)o(].)682 3609 y(In)42 b(ev)n(en-dimensional)f(manifolds,)k(the)d(A) n(tiy)n(ah-Singer)e(index)i(theorem)f(is)h(a)599 3709 y(p)r(o)n(w)n(erful)e(to)r(ol)f(for)h(in)n(v)n(estigating)e(the)j(k)n (ernel)e(of)h Ft(P)2340 3721 y Fs(A)2392 3709 y FG(,)k(since)39 b(the)i(de\014ciency)e(of)599 3809 y Ft(P)651 3821 y Fs(A)737 3809 y FG(can)32 b(v)-5 b(anish,)34 b(in)f(whic)n(h)g(case)e (the)i(index)g(is)g(equal)f(to)g(n)n(ul)h Ft(P)2699 3821 y Fs(A)2752 3809 y FG(,)h(the)f(n)n(ullit)n(y)f(of)599 3908 y Ft(P)651 3920 y Fs(A)744 3908 y FG(\(i.e.)72 b(the)39 b(dimension)h(of)f(the)g(k)n(ernel)g(k)n(er)13 b Ft(P)2207 3920 y Fs(A)2259 3908 y FG(\).)73 b(A)39 b(celebrated)g(example)g(is) 599 4008 y(the)33 b(Aharono)n(v-Casher)28 b(Theorem)j(in)h Ft(R)1919 3978 y Fm(2)1995 4008 y FG(and)f(its)h(analogue)f(due)h(to)f (Avron)h(and)599 4108 y(T)-7 b(omaras)36 b(in)h Ft(S)1107 4078 y Fm(2)1176 4108 y FG(\(see)g([5)o(]\).)66 b(In)37 b Ft(R)1727 4078 y Fm(2)1771 4108 y FG(,)i(this)f(assert)e(that)h(for)g (suitable)2909 4087 y Fo(~)2896 4108 y(B)k FG(\(e.g.)3236 4087 y Fo(~)3222 4108 y(B)599 4207 y FG(b)r(ounded)28 b(and)g(of)f(compact)h(supp)r(ort\),)f(the)h(n)n(ullit)n(y)g(of)f Ft(P)2439 4219 y Fs(A)2520 4207 y FG(is)1608 4312 y Fr(\032)1705 4373 y FG(1)p 1680 4410 92 4 v 1680 4486 a(2)p Fo(\031)1782 4309 y Fr(\014)1782 4359 y(\014)1782 4408 y(\014)1782 4458 y(\014)1810 4316 y(Z)1796 4558 y Fk(R)1843 4541 y Fj(2)1907 4429 y Fo(B)t FG(\()p Fq(x)p FG(\))p Fo(d)p Fq(x)2191 4309 y Fr(\014)2191 4359 y(\014)2191 4408 y(\014)2191 4458 y(\014)2219 4312 y(\033)p eop %%Page: 3 3 3 2 bop 1120 75 a FH(ON)24 b(THE)f(ZER)n(O)h(MODES)f(OF)g(P)-6 b(A)n(ULI)24 b(OPERA)-6 b(TORS)479 b FG(3)599 320 y(where)38 b Fp(f)p Fo(y)s Fp(g)e FG(denotes)i(the)g(largest)e(in)n(teger)h (strictly)g(less)h(than)g Fo(y)i FG(and)e Fp(f)p FG(0)p Fp(g)g FG(=)i(0;)599 419 y(note)28 b(that)g(in)g Ft(R)1115 389 y Fm(2)1159 419 y FG(,)g(the)g(magnetic)f(\014eld)i(has)e(only)g (one)h(comp)r(onen)n(t,)f(and)h(is)g(th)n(us)g(a)599 519 y(scalar)22 b(\014eld.)35 b(Th)n(us)23 b(in)g Ft(R)1393 489 y Fm(2)1436 519 y FG(,)h(zero)e(mo)r(des)h(are)e(abundan)n(t;)k (they)e(exist)f(as)h(long)f(as)g(the)599 619 y(magnetic)32 b(\015ux)1159 586 y Fm(1)p 1138 600 75 4 v 1138 647 a(2)p Fs(\031)1245 552 y Fr(R)1228 701 y Fk(R)1275 684 y Fj(2)1324 619 y Fo(B)t FG(\()p Fq(x)p FG(\))p Fo(d)p Fq(x)h FG(tak)n(es)f(v)-5 b(alues)32 b(outside)g([)p Fp(\000)p FG(1)p Fo(:)p FG(1].)50 b(The)33 b(situation)f(in)599 785 y Ft(S)652 755 y Fm(2)711 785 y FG(is)27 b(v)n(ery)g(di\013eren)n(t.)38 b(There)27 b(are)g(no)n(w)g(zero)f(mo)r(des)i(if)g(and)g(only)f(if)i(the)f (magnetic)599 885 y(\015ux)i(is)g(an)f(in)n(teger,)h(a)f(picture)h (whic)n(h)g(is)f(somewhat)g(reminiscen)n(t)g(of)h(that)g(in)g(\(4\).) 599 985 y(In)g(fact,)f(this)g(is)g(t)n(ypical)g(of)g(what)f(happ)r(ens) h(on)g(an)n(y)f(compact)h(manifold)g(\(of)g(ev)n(en)599 1084 y(or)e(o)r(dd)h(dimension\),)g(as)f(sho)n(wn)g(b)n(y)g(Anghel)g (in)h([3].)682 1184 y(Apart)e(from)g(the)h(examples)e(in)i([11)o(,)f (7])g(men)n(tioned)g(ab)r(o)n(v)n(e)f(v)n(ery)g(little)i(is)f(kno)n(wn) 599 1283 y(for)33 b Ft(R)786 1253 y Fm(3)829 1283 y FG(,)h(and)f (indeed)g(for)g(non-compact)f(manifolds)g(of)h(o)r(dd)g(dimension,)h (since)f(it)599 1383 y(is)27 b(not)f(easy)f(to)h(obtain)g(information)g (from)g(the)g(A)n(tiy)n(ah-Singer)f(index)h(theorem)g(in)599 1483 y(this)g(case.)35 b(Ho)n(w)n(ev)n(er,)24 b(w)n(e)g(pro)n(v)n(e)g (that)h(the)h(situation)f(in)g Ft(R)2473 1453 y Fm(3)2542 1483 y FG(is)g(lik)n(e)f(that)i(describ)r(ed)599 1582 y(ab)r(o)n(v)n(e)40 b(for)g(compact)h(manifolds,)j(and)d(is)g(th)n(us)g (dramatically)e(di\013eren)n(t)i(to)g Ft(R)3223 1552 y Fm(2)3266 1582 y FG(.)599 1682 y(Sp)r(eci\014cally)-7 b(,)28 b(w)n(e)f(pro)n(v)n(e)f(in)i(Theorems)f(4.1)f(and)i(4.2)682 1840 y Fp(\017)62 b FG(for)29 b Fp(j)951 1819 y Fo(~)938 1840 y(B)t Fp(j)c(2)h Fo(L)1191 1810 y Fm(3)p Fs(=)p Fm(2)1295 1840 y FG(\()p Ft(R)1381 1810 y Fm(3)1424 1840 y FG(\),)59 b(n)n(ul)29 b Ft(P)1732 1852 y Fs(tA)1835 1840 y FG(=)c(0)k(except)g(for)f(a)h(\014nite)g(n)n(um)n(b)r(er)g(of)g (v)-5 b(alues)599 1939 y(of)28 b Fo(t)g FG(in)g(an)n(y)e(compact)h (subset)h(of)g([0)p Fo(;)14 b Fp(1)p FG(\);)682 2055 y Fp(\017)789 1988 y Fr(\010)882 2034 y Fo(~)868 2055 y(B)34 b FG(:)60 b(n)n(ul)32 b Ft(P)1245 2067 y Fs(A)1327 2055 y FG(=)c(0)p Fo(;)45 b FG(curl)1716 2034 y Fo(~)1700 2055 y(A)30 b FG(=)1899 2034 y Fo(~)1885 2055 y(B)67 b FG(and)c Fp(j)2248 2034 y Fo(~)2235 2055 y(B)t Fp(j)29 b(2)h Fo(L)2496 2025 y Fm(3)p Fs(=)p Fm(2)2599 2055 y FG(\()p Ft(R)2686 2025 y Fm(3)2729 2055 y FG(\))2793 1988 y Fr(\011)2872 2055 y FG(con)n(tains)h(an)599 2166 y(op)r(en)d(dense)g(subset)f(of)h([)p Fo(L)1456 2136 y Fm(3)p Fs(=)p Fm(2)1560 2166 y FG(\()p Ft(R)1646 2136 y Fm(3)1689 2166 y FG(\)])1744 2136 y Fm(3)1782 2166 y FG(.)682 2324 y(This)k(explains)f(wh)n(y)g(zero)g(mo)r(des)g(are)g (so)g(di\016cult)h(to)g(obtain.)48 b(Note)32 b(that)g(the)599 2423 y(Loss-Y)-7 b(au)28 b(example)g(satis\014es)g(our)f(h)n(yp)r (othesis.)40 b(The)28 b(analogous)e(result)j(holds)f(for)599 2523 y Ft(R)653 2493 y Fs(n)732 2523 y FG(with)g Fo(n)23 b(>)g FG(3.)1495 2789 y Fu(2.)33 b(PRELIMINARIES)682 2942 y FG(W)-7 b(e)28 b(can)g(write)f(\(1\))h(as)1383 3143 y Ft(P)1435 3155 y Fs(A)1511 3143 y FG(=)22 b Fo(S)1649 3155 y Fs(A)1722 3143 y FG(+)16 b Fo(~)-40 b(\033)s Fp(\001)1906 3122 y Fo(~)1892 3143 y(B)t(;)2093 3122 y(~)2079 3143 y(B)27 b FG(=)c Fo(cur)r(l)2436 3122 y(~)2421 3143 y(A;)677 b FG(\(5\))599 3334 y(where)28 b Fo(S)891 3346 y Fs(A)972 3334 y FG(is)g(the)g(magnetic)f(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erator)1103 3599 y Fo(S)1154 3611 y Fs(A)1231 3599 y FG(=)1319 3482 y Fr(\022)1390 3543 y FG(1)p 1390 3580 42 4 v 1396 3656 a Fo(i)1442 3599 y Fn(r)18 b FG(+)1638 3578 y Fo(~)1623 3599 y(A)1685 3482 y Fr(\023)1746 3500 y Fm(2)1811 3599 y Ft(I)1849 3611 y Fm(2)1931 3599 y Fp(\021)2090 3496 y Fm(3)2047 3521 y Fr(X)2049 3697 y Fs(j)s Fm(=1)2166 3482 y Fr(\022)2238 3543 y FG(1)p 2238 3580 V 2244 3656 a Fo(i)2289 3599 y(@)2333 3611 y Fs(j)2387 3599 y FG(+)g Fo(A)2532 3611 y Fs(j)2567 3482 y Fr(\023)2628 3500 y Fm(2)2693 3599 y Ft(I)2731 3611 y Fm(2)2762 3599 y Fo(;)398 b FG(\(6\))599 3897 y Ft(I)638 3909 y Fm(2)700 3897 y FG(b)r(eing)32 b(the)g(2)20 b Fp(\002)h FG(2)31 b(iden)n(tit)n(y)g(matrix)g(and)e Fo(~)-40 b(\033)t Fp(\001)2146 3876 y Fo(~)2133 3897 y(B)35 b FG(the)d(Zeeman)f(term.)48 b(Note)31 b(that)599 3997 y(a)g(gauge)e(transformation)1494 3976 y Fo(~)1479 3997 y(A)f Fp(7!)1696 3976 y Fo(~)1680 3997 y(A)21 b FG(+)f Fo(d)-14 b(f)40 b FG(do)r(es)30 b(not)h(alter)e(the)i(n)n(ullit)n(y)-7 b(,)32 b(and)e(hence)599 4096 y(n)n(ul)36 b Ft(P)800 4108 y Fs(A)888 4096 y FG(is)g(indep)r (enden)n(t)g(of)f(the)h(gauge.)59 b(W)-7 b(e)36 b(denote)f([)p Fo(L)2508 4066 y Fm(2)2545 4096 y FG(\()p Ft(R)2631 4066 y Fm(3)2674 4096 y FG(\)])2729 4066 y Fm(2)2802 4096 y FG(b)n(y)h Fp(H)g FG(and)f(its)599 4196 y(standard)27 b(inner-pro)r(duct)g(and)g(norm)g(b)n(y)h(\()p Fp(\001)14 b Fo(;)g Fp(\001)g FG(\))28 b(and)f Fp(k\001)14 b(k)27 b FG(resp)r(ectiv)n(ely:)1518 4429 y Fp(k)p Fo(f)9 b Fp(k)1652 4395 y Fm(2)1739 4429 y FG(=)1854 4316 y Fr(Z)1840 4558 y Fk(R)1887 4541 y Fj(3)1978 4429 y Fp(j)p Fo(f)g FG(\()p Fu(x)p FG(\))p Fp(j)2188 4395 y Fm(2)2254 4429 y Fo(d)p Fu(x)p Fo(;)p eop %%Page: 4 4 4 3 bop 599 75 a FG(4)753 b FH(A.)23 b(BALINSKY)g(AND)g(W.D.)g(EV)-8 b(ANS)599 320 y FG(where)24 b Fp(j\001)14 b(j)25 b FG(is)f(the)h (Euclidean)f(norm)f(on)i Ft(C)1924 290 y Fm(2)1968 320 y FG(.)36 b(It)24 b(will)h(b)r(e)g(assumed)f(throughout)f(that)1433 504 y Fo(A)1495 516 y Fs(j)1581 504 y Fp(2)51 b Fo(L)1744 470 y Fm(2)1744 524 y Fs(loc)1831 504 y FG(\()p Ft(R)1917 470 y Fm(3)1961 504 y FG(\))p Fo(;)69 b(j)28 b FG(=)23 b(1)p Fo(;)14 b FG(2)p Fo(;)g FG(3)p Fo(:)725 b FG(\(7\))599 688 y(W)-7 b(e)25 b(con)n(tin)n(ue)e(to)h(denote)g(b)n(y)g Fo(S)1592 700 y Fs(A)1670 688 y FG(the)h(F)-7 b(riedric)n(hs)23 b(extension)g(of)i(\(6\))f(on)g([)p Fo(C)2974 658 y Fi(1)2968 709 y Fm(0)3044 688 y FG(\()p Ft(R)3130 658 y Fm(3)3174 688 y FG(\)])3229 658 y Fm(2)3266 688 y FG(.)599 788 y(It)k(is)e(a)h(non-negativ)n(e)e(self-adjoin)n(t)h(op)r(erator)f(with) j(no)e(zero)g(mo)r(des,)h(and)g(its)g(form)599 887 y(domain)34 b Fp(Q)p FG(\()p Fo(S)1052 899 y Fs(A)1106 887 y FG(\))g(is)g(the)g (completion)g(of)f([)p Fo(C)2030 857 y Fi(1)2024 908 y Fm(0)2101 887 y FG(\()p Ft(R)2187 857 y Fm(3)2231 887 y FG(\)])2286 857 y Fm(2)2357 887 y FG(with)h(resp)r(ect)g(to)g(the)g (norm)599 987 y(giv)n(en)27 b(b)n(y)1170 1240 y Fp(k)p Fo(')p Fp(k)1308 1252 y Fm(1)p Fs(;A)1465 1240 y FG(=)1580 1123 y Fr(\032)1642 1119 y(\015)1642 1169 y(\015)1642 1219 y(\015)1642 1269 y(\015)1689 1123 y(\022)1760 1183 y FG(1)p 1760 1220 42 4 v 1766 1297 a Fo(i)1811 1240 y Fn(r)18 b FG(+)2008 1219 y Fo(~)1992 1240 y(A)2054 1123 y Fr(\023)2116 1240 y Fo(')2170 1119 y Fr(\015)2170 1169 y(\015)2170 1219 y(\015)2170 1269 y(\015)2216 1140 y Fm(2)2272 1240 y FG(+)g Fp(k)p Fo(')p Fp(k)2493 1205 y Fm(2)2529 1123 y Fr(\033)2591 1140 y Fm(1)p Fs(=)p Fm(2)2696 1240 y Fo(:)464 b FG(\(8\))599 1469 y(The)28 b(op)r(erator)e(realisation)g(of)h Ft(P)1653 1481 y Fs(A)1734 1469 y FG(is)g(giv)n(en)g(in)h(the)g(\014rst)f(lemma.)682 1620 y Fh(Lemma)j(2.1.)69 b Fg(L)l(et)59 b Fp(j)1392 1599 y Fo(~)1379 1620 y(B)t Fp(j)23 b(2)h Fo(L)1628 1590 y Fm(3)p Fs(=)p Fm(2)1732 1620 y FG(\()p Ft(R)1818 1590 y Fm(3)1861 1620 y FG(\))p Fg(.)39 b(Then)30 b(the)g(sesquiline)l(ar)h (form)1192 1804 y Fo(p)1234 1816 y Fs(A)1288 1804 y FG([)p Fo(';)14 b( )s FG(])53 b(=)f(\()p Ft(P)1736 1816 y Fs(A)1789 1804 y Fo(';)14 b( )s FG(\))p Fo(;)74 b(';)14 b( )27 b Fp(2)c FG([)p Fo(C)2404 1770 y Fi(1)2398 1825 y Fm(0)2475 1804 y FG(\()p Ft(R)2561 1770 y Fm(3)2604 1804 y FG(\)])2659 1770 y Fm(2)3183 1804 y FG(\(9\))599 1989 y Fg(is)33 b(symmetric,)g(closable)g(and)g(non-ne)l(gative)f(in)g Fp(H)q Fg(.)44 b(The)33 b(asso)l(ciate)l(d)g(self-adjoint)599 2088 y(op)l(er)l(ator)e Ft(P)978 2100 y Fs(A)1061 2088 y Fg(has)f(form)h(domain)g Fp(Q)p FG(\()p Fo(S)1858 2100 y Fs(A)1912 2088 y FG(\))p Fg(.)682 2368 y(Pr)l(o)l(of.)79 b FG(Giv)n(en)24 b Fo(")f(>)f FG(0,)i(w)n(e)g(ma)n(y)f(write)h Fp(j)1993 2347 y Fo(~)1980 2368 y(B)t Fp(j)f FG(=)g Fo(B)2244 2380 y Fm(1)2292 2368 y FG(+)11 b Fo(B)2431 2380 y Fm(2)2467 2368 y FG(,)25 b(where)e Fp(k)p Fo(B)2856 2380 y Fm(1)2893 2368 y Fp(k)2935 2387 y Fs(L)2981 2370 y Fj(3)p Ff(=)p Fj(2)3071 2387 y Fm(\()p Fk(R)3144 2370 y Fj(3)3172 2387 y Fm(\))3225 2368 y Fo(<)599 2467 y(")28 b FG(and)f Fp(k)p Fo(B)932 2479 y Fm(2)969 2467 y Fp(k)1011 2482 y Fs(L)1057 2466 y Fe(1)1117 2482 y Fm(\()p Fk(R)1190 2466 y Fj(3)1217 2482 y Fm(\))1270 2467 y Fo(<)c(C)1417 2479 y Fs(")1453 2467 y FG(,)k(for)g(some)g(constan)n(t)g Fo(C)2232 2479 y Fs(")2296 2467 y FG(dep)r(ending)h(on)f Fo(")p FG(.)37 b(Then)1240 2670 y(\()p Ft(P)1324 2682 y Fs(A)1404 2670 y Fo(';)14 b(')p FG(\))52 b(=)e(\()p Fo(S)1831 2682 y Fs(A)1913 2670 y Fo(';)14 b(')p FG(\))19 b(+)f(\(\()n Fo(~)-40 b(\033)t Fp(\001)2358 2649 y Fo(~)2344 2670 y(B)t FG(\))28 b Fo(';)14 b(')p FG(\))599 2854 y(and)992 3039 y Fp(j)p FG(\(\()n Fo(~)-40 b(\033)t Fp(\001)1181 3018 y Fo(~)1167 3039 y(B)t FG(\))28 b Fo(';)14 b(')p FG(\))p Fp(j)51 b(\024)e FG(\()p Fo(B)1754 3051 y Fm(1)1820 3039 y Fo(';)14 b(')p FG(\))19 b(+)f(\()p Fo(B)2194 3051 y Fm(2)2259 3039 y Fo(';)c(')p FG(\))1545 3163 y Fp(\024)49 b(k)p Fo(B)1764 3175 y Fm(1)1801 3163 y Fp(k)1843 3182 y Fs(L)1889 3166 y Fj(3)p Ff(=)p Fj(2)1979 3182 y Fm(\()p Fk(R)2052 3166 y Fj(3)2080 3182 y Fm(\))2110 3163 y Fp(k)p Fo(')p Fp(k)2248 3129 y Fm(2)2248 3186 y([)p Fs(L)2313 3169 y Fj(6)2344 3186 y Fm(\()p Fk(R)2417 3169 y Fj(3)2444 3186 y Fm(\)])2489 3169 y Fj(2)2544 3163 y FG(+)18 b Fo(C)2686 3175 y Fs(")2722 3163 y Fp(k)p Fo(')p Fp(k)2860 3129 y Fm(2)1545 3305 y Fp(\024)49 b Fo("\015)1746 3271 y Fm(2)1811 3305 y Fp(k)p Fn(r)o Fp(j)p Fo(')p Fp(jk)2074 3271 y Fm(2)2129 3305 y FG(+)18 b Fo(C)2271 3317 y Fs(")2307 3305 y Fp(k)p Fo(')p Fp(k)2445 3271 y Fm(2)629 3490 y FG(b)n(y)30 b(the)g(Sob)r(olev)g(Em)n(b)r(edding)f(Theorem,)h(with)g Fo(\015)35 b FG(the)30 b(norm)f(of)h(the)g(em)n(b)r(edding)599 3589 y Fo(H)675 3559 y Fm(1)712 3589 y FG(\()p Ft(R)799 3559 y Fm(3)842 3589 y FG(\))23 b Fo(,)-14 b Fp(!)24 b Fo(L)1070 3559 y Fm(6)1106 3589 y FG(\()p Ft(R)1193 3559 y Fm(3)1236 3589 y FG(\),)1376 3840 y Fp(\024)e Fo("\015)1550 3806 y Fm(2)1615 3720 y Fr(\015)1615 3770 y(\015)1615 3819 y(\015)1615 3869 y(\015)1661 3723 y(\022)1732 3784 y FG(1)p 1732 3821 V 1738 3897 a Fo(i)1783 3840 y Fn(r)c FG(+)1980 3819 y Fo(~)1964 3840 y(A)2026 3723 y Fr(\023)2088 3840 y Fo(')2142 3720 y Fr(\015)2142 3770 y(\015)2142 3819 y(\015)2142 3869 y(\015)2188 3740 y Fm(2)2244 3840 y FG(+)g Fo(C)2386 3852 y Fs(")2422 3840 y Fp(k)p Fo(')p Fp(k)2560 3806 y Fm(2)599 4074 y FG(b)n(y)23 b(the)h(diamagnetic)e(inequalit)n(y)g(\(see)h([9,)h(Thm)f(7.21]\).)35 b(The)23 b(lemma)g(follo)n(ws)f(from)599 4199 y(this.)p 840 4216 25 75 v 682 4349 a(Hereafter,)27 b(w)n(e)h(shall)f(alw)n(a)n (ys)e(assume)i(that)1637 4533 y Fp(j)1673 4512 y Fo(~)1660 4533 y(B)t Fp(j)51 b(2)g Fo(L)1964 4499 y Fm(3)p Fs(=)p Fm(2)2067 4533 y FG(\()p Ft(R)2154 4499 y Fm(3)2197 4533 y FG(\))p Fo(:)890 b FG(\(10\))p eop %%Page: 5 5 5 4 bop 1120 75 a FH(ON)24 b(THE)f(ZER)n(O)h(MODES)f(OF)g(P)-6 b(A)n(ULI)24 b(OPERA)-6 b(TORS)479 b FG(5)599 320 y(The)28 b(op)r(erator)1665 499 y Ft(P)49 b FG(:=)i Ft(P)1957 511 y Fs(A)2028 499 y FG(+)18 b Fp(j)2147 478 y Fo(~)2134 499 y(B)t Fp(j)918 b FG(\(11\))599 679 y(ma)n(y)28 b(b)r(e)h(de\014ned) g(as)f(in)h(Lemma)f(2.1,)g(namely)-7 b(,)29 b(the)g(self-adjoin)n(t)f (op)r(erator)f(asso)r(ci-)599 778 y(ated)h(with)g(the)g(form)1449 958 y Fo(p)p FG([)p Fo(')p FG(])51 b Fp(\021)f Fo(p)p FG([)p Fo(';)14 b(')p FG(])51 b(=)f(\()p Ft(P)p Fo(';)14 b(')p FG(\))p Fo(;)702 b FG(\(12\))599 1137 y(with)30 b(form)e(domain)h Fp(Q)p FG(\()p Fo(S)1435 1149 y Fs(A)1489 1137 y FG(\).)41 b(As)29 b(for)g Fo(S)1889 1149 y Fs(A)1942 1137 y FG(,)h Ft(P)e FG(has)g(no)h(zero)e(mo)r(des.)41 b(Th)n(us)29 b Fo(S)3073 1149 y Fs(A)3156 1137 y FG(and)599 1237 y Ft(P)35 b FG(are)g(injectiv)n(e)i(and)f(ha)n(v)n(e)f(dense)h (domains)f(and)h(ranges)f(in)h Fp(H)q FG(.)62 b(F)-7 b(urthermore,)599 1348 y Fp(D)r FG(\()p Ft(P)749 1318 y Fm(1)p Fs(=)p Fm(2)853 1348 y FG(\))24 b(=)e Fp(D)r FG(\()p Fo(S)1150 1305 y Fm(1)p Fs(=)p Fm(2)1145 1372 y Fs(A)1255 1348 y FG(\))h(=)g Fp(Q)p FG(\()p Fo(S)1549 1360 y Fs(A)1603 1348 y FG(\).)682 1461 y(The)k(op)r(erator)e(of)i (prime)f(in)n(terest)h(is)f Ft(P)1946 1473 y Fs(A)1999 1461 y FG(.)37 b(W)-7 b(e)27 b(shall)f(write)h(it)g(as)f Ft(P)2841 1473 y Fs(A)2917 1461 y FG(=)c Ft(P)16 b Fp(\000)g(j)3189 1440 y Fo(~)3176 1461 y(B)t Fp(j)p FG(,)599 1561 y(and)31 b(then,)h(initially)-7 b(,)32 b(pro)r(ceed)e(along)f(lines)i(whic)n(h)g (are)e(reminiscen)n(t)i(of)f(those)h(de-)599 1660 y(scrib)r(ed)37 b(in)g([4])g(for)g(pro)n(ving)e(the)i(Cwik)n(el-Lieb-Rosen)n(bljum)f (inequalit)n(y)h(for)f(the)599 1760 y(Sc)n(hr\177)-42 b(odinger)25 b(op)r(erator.)35 b(The)26 b(problem)g(is)g(essen)n (tially)f(reduced)h(to)g(one)g(for)f(an)h(as-)599 1859 y(so)r(ciated)i(op)r(erator)e(of)h(Birman-Sc)n(h)n(winger)f(t)n(yp)r (e.)37 b(The)28 b(follo)n(wing)f(spaces)g(feature)599 1959 y(prominen)n(tly)h(in)f(the)h(analysis.)682 2117 y Fp(\017)61 b Fo(H)861 2087 y Fm(1)854 2140 y Fs(A)936 2117 y FG(is)27 b(the)h(completion)f(of)h Fp(D)r FG(\()p Fo(S)1835 2074 y Fm(1)p Fs(=)p Fm(2)1830 2141 y Fs(A)1940 2117 y FG(\))g(with)g(resp)r(ect)f(to)h(the)g(norm)1568 2316 y Fp(k)p Fo(')p Fp(k)1706 2334 y Fs(H)1764 2314 y Fj(1)1760 2354 y Ff(A)1862 2316 y FG(:=)50 b Fp(k)p Fo(S)2098 2273 y Fm(1)p Fs(=)p Fm(2)2093 2341 y Fs(A)2202 2316 y Fo(')p Fp(k)p FG(;)821 b(\(13\))599 2510 y Fo(H)675 2480 y Fm(1)668 2530 y(0)768 2510 y FG(has)27 b(norm)g Fp(k)p Fo(')p Fp(k)1271 2528 y Fs(H)1329 2508 y Fj(1)1325 2546 y(0)1416 2510 y FG(:=)51 b Fp(k)p Fn(r)o Fo(')p Fp(k)p FG(.)682 2631 y Fp(\017)61 b Ft(H)843 2601 y Fm(1)843 2654 y Fs(B)934 2631 y FG(is)28 b(the)g(completion)f(of)h Fp(D)r FG(\()p Ft(P)1830 2601 y Fm(1)p Fs(=)p Fm(2)1933 2631 y FG(\))g(with)g(resp)r(ect)g(to)f(the)h(norm)1574 2820 y Fp(k)p Fo(')p Fp(k)1712 2838 y Fk(H)1751 2818 y Fj(1)1751 2858 y Ff(B)1861 2820 y FG(:=)50 b Fp(k)p Ft(P)2093 2786 y Fm(1)p Fs(=)p Fm(2)2196 2820 y Fo(')p Fp(k)p Fo(:)827 b FG(\(14\))682 3028 y Fd(Remarks)p 682 3041 367 4 v 682 3174 a FG(1.)61 b([)p Fo(C)896 3144 y Fi(1)890 3195 y Fm(0)967 3174 y FG(\()p Ft(R)1053 3144 y Fm(3)1096 3174 y FG(\)])1151 3144 y Fm(2)1216 3174 y FG(is)28 b(dense)f(in)h Fo(H)1699 3144 y Fm(1)1692 3197 y Fs(A)1774 3174 y FG(and)f Ft(H)1994 3144 y Fm(1)1994 3197 y Fs(B)2057 3174 y FG(.)682 3277 y(2.)59 b(The)26 b(space)g Fo(H)1272 3247 y Fm(1)1265 3298 y(0)1335 3277 y FG(is)g(not)g(a)f(subspace)h Fp(H)q FG(.)36 b(Ho)n(w)n(ev)n(er,)25 b(for)g Fo(')e Fp(2)h FG([)p Fo(C)2832 3247 y Fi(1)2826 3298 y Fm(0)2902 3277 y FG(\()p Ft(R)2989 3247 y Fm(3)3032 3277 y FG(\)])3087 3247 y Fm(2)3125 3277 y FG(,)i(the)599 3377 y(Hardy)h(inequalit)n(y)1332 3492 y Fr(Z)1318 3733 y Fk(R)1365 3717 y Fj(3)1439 3549 y Fp(j)p Fo(')p FG(\()p Fq(x)p FG(\))p Fp(j)1658 3519 y Fm(2)p 1439 3586 257 4 v 1498 3662 a Fp(j)p Fq(x)p Fp(j)1599 3638 y Fm(2)1706 3605 y Fo(d)p Fq(x)51 b Fp(\024)f FG(4)2026 3492 y Fr(Z)2012 3733 y Fk(R)2059 3717 y Fj(3)2122 3605 y Fp(j)p Fn(r)p Fo(')p FG(\()p Fq(x)p FG(\))p Fp(j)2421 3571 y Fm(2)2459 3605 y Fo(d)p Fq(x)599 3898 y FG(is)24 b(v)-5 b(alid,)25 b(and)e(this)h(implies)g(that)g Fo(H)1745 3867 y Fm(1)1738 3918 y(0)1806 3898 y FG(ma)n(y)f(b)r(e)h(iden)n(ti\014ed)g(with)h(the)f (function)g(space)984 4077 y Fo(H)1060 4043 y Fm(1)1053 4098 y(0)1148 4077 y FG(=)1263 4010 y Fr(\010)1312 4077 y Fo(u)e Fp(2)i FG([)p Fo(H)1560 4043 y Fm(1)1553 4098 y Fs(loc)1641 4077 y FG(\()p Ft(R)1727 4043 y Fm(3)1770 4077 y FG(\)])1825 4043 y Fm(2)1914 4077 y FG(:)50 b Fp(k)p Fo(u)p Fp(k)2119 4043 y Fm(2)2119 4103 y Fs(H)2177 4083 y Fj(1)2173 4121 y(0)2232 4077 y FG(+)18 b Fp(k)27 b Fo(u=)p Fp(j\001)14 b(j)27 b(k)2626 4043 y Fm(2)2685 4077 y Fo(<)c Fp(1)2856 4010 y Fr(\011)3142 4077 y FG(\(15\))599 4271 y(and)28 b Fp(k\001)14 b(k)882 4289 y Fs(H)940 4269 y Fj(1)936 4307 y(0)1004 4271 y FG(is)27 b(equiv)-5 b(alen)n(t)28 b(to)f(the)h(norm)f(de\014ned)h(b)n(y)1477 4422 y Fr(\000)1515 4489 y Fp(k)p Fo(u)p Fp(k)1647 4455 y Fm(2)1647 4515 y Fs(H)1705 4495 y Fj(1)1701 4533 y(0)1788 4489 y FG(+)45 b Fp(k)27 b Fo(u=)p Fp(j\001)14 b(j)27 b(k)2209 4455 y Fm(2)2246 4422 y Fr(\001)2284 4439 y Fm(1)p Fs(=)p Fm(2)2388 4489 y Fo(:)p eop %%Page: 6 6 6 5 bop 599 75 a FG(6)753 b FH(A.)23 b(BALINSKY)g(AND)g(W.D.)g(EV)-8 b(ANS)682 320 y FG(3.)62 b(F)-7 b(or)29 b(the)g(spaces)g Fo(H)1437 290 y Fm(1)1430 343 y Fs(A)1513 320 y FG(and)g Ft(H)1734 290 y Fm(1)1734 343 y Fs(B)1798 320 y FG(,)g(whic)n(h)g(also) f(do)h(not)g(lie)g(in)h Fp(H)q FG(,)f(w)n(e)g(ha)n(v)n(e)f(the)599 419 y(natural)f(em)n(b)r(edding)1741 616 y Ft(H)1800 582 y Fm(1)1800 637 y Fs(B)1886 616 y Fo(,)-14 b Fp(!)23 b Fo(H)2077 582 y Fm(1)2070 637 y Fs(A)2124 616 y Fo(:)995 b FG(\(16\))599 813 y(Also,)33 b(b)n(y)e(the)h(diamagnetic)f(inequalit) n(y)-7 b(,)32 b Fo(')e Fp(7!)g(j)p Fo(')p Fp(j)i FG(maps)f Fo(H)2583 783 y Fm(1)2576 836 y Fs(A)2662 813 y FG(con)n(tin)n(uously)g (in)n(to)599 912 y Fo(H)675 882 y Fm(1)668 933 y(0)712 912 y FG(,)d(whic)n(h,)g(in)g(turn,)f(is)h(con)n(tin)n(uously)e(em)n(b) r(edded)i(in)g([)p Fo(L)2463 882 y Fm(6)2500 912 y FG(\()p Ft(R)2586 882 y Fm(3)2629 912 y FG(\)])2684 882 y Fm(2)2750 912 y FG(b)n(y)f(the)h(Sob)r(olev)599 1012 y(Em)n(b)r(edding)g (Theorem.)38 b(In)28 b(fact)h(the)f(spaces)f(in)i(\(16\))e(are)g (isomorphic)g(when)i(\(10\))599 1112 y(is)f(satis\014ed.)682 1275 y(F)-7 b(or)24 b(a)g(magnetic)g(p)r(oten)n(tial)1609 1254 y Fo(~)1594 1275 y(A)g FG(satisfying)g Fp(j)2086 1254 y Fo(~)2070 1275 y(A)p Fp(j)f(2)h Fo(L)2314 1245 y Fm(3)2351 1275 y FG(\()p Ft(R)2437 1245 y Fm(3)2480 1275 y FG(\),)i Fo(H)2637 1245 y Fm(1)2630 1298 y Fs(A)2708 1275 y FG(can)e(b)r(e)h(sho)n(wn)f(to)599 1387 y(b)r(e)j(con)n(tin)n (uously)e(em)n(b)r(edded)i(in)f Fo(H)1754 1356 y Fm(1)1747 1407 y(0)1791 1387 y FG(.)36 b(Suc)n(h)27 b(a)e(c)n(hoice)h(of)2471 1366 y Fo(~)2456 1387 y(A)g FG(is)g(p)r(ossible)g(in)h(view)e(of)599 1486 y(the)j(next)g(lemma)g(whic)n(h)f(is)h(similar)f(to)g(Theorem)g (A1)g(in)h(App)r(endix)h(A)f(of)f([8].)682 1659 y Fh(Lemma)j(2.2.)69 b Fg(L)l(et)59 b Fp(j)1392 1638 y Fo(~)1379 1659 y(B)t Fp(j)23 b(2)h Fo(L)1628 1629 y Fm(3)p Fs(=)p Fm(2)1732 1659 y FG(\()p Ft(R)1818 1629 y Fm(3)1861 1659 y FG(\))30 b Fg(and)g(de\014ne)1315 1883 y Fo(~)1300 1904 y(A)p FG(\()p Fq(x)p FG(\))53 b(=)1687 1848 y(1)p 1661 1885 92 4 v 1661 1961 a(4)p Fo(\031)1777 1791 y Fr(Z)1763 2033 y Fk(R)1810 2016 y Fj(3)1893 1848 y FG(\()p Fq(x)19 b Fp(\000)f Fq(y)s FG(\))p 1884 1885 292 4 v 1884 1961 a Fp(j)p Fq(x)g Fp(\000)g Fq(y)s Fp(j)2138 1937 y Fm(3)2204 1904 y Fp(\002)2300 1883 y Fo(~)2287 1904 y(B)t FG(\()p Fq(y)s FG(\))p Fo(d)p Fq(y)t Fo(:)553 b FG(\(17\))599 2227 y Fg(Then)31 b Fp(j)854 2206 y Fo(~)839 2227 y(A)p Fp(j)23 b(2)h Fo(L)1083 2197 y Fm(3)1119 2227 y FG(\()p Ft(R)1206 2197 y Fm(3)1249 2227 y FG(\))p Fg(,)30 b Fo(cur)r(l)1515 2206 y(~)1500 2227 y(A)23 b FG(=)1686 2206 y Fo(~)1673 2227 y(B)t Fg(,)60 b Fo(div)1969 2206 y(~)1954 2227 y(A)23 b FG(=)g(0)29 b Fg(in)h Fp(D)2366 2197 y Fi(0)2419 2227 y Fg(and)1456 2433 y Fp(k)1513 2412 y Fo(~)1498 2433 y(A)p Fp(k)1602 2448 y Fs(L)1648 2432 y Fj(3)1679 2448 y Fm(\()p Fk(R)1752 2432 y Fj(3)1780 2448 y Fm(\))1833 2433 y Fp(\024)22 b Fo(C)36 b Fp(k)2070 2412 y Fo(~)2057 2433 y(B)t Fp(k)2166 2452 y Fs(L)2212 2436 y Fj(3)p Ff(=)p Fj(2)2302 2452 y Fm(\()p Fk(R)2375 2436 y Fj(3)2403 2452 y Fm(\))599 2630 y Fg(for)31 b(some)f(c)l(onstant)f Fo(C)6 b Fg(.)682 2891 y(Pr)l(o)l(of.)93 b FG(The)32 b(pro)r(of)g(is)f (similar)h(to)f(that)h(in)h([8)o(].)50 b(The)32 b(follo)n(wing)f (formal)g(argu-)599 2991 y(men)n(t)f(for)f(deriving)g(\(17\))g(is)g (instructiv)n(e,)g(and)h(will)f(b)r(e)h(helpful)g(for)f(obtaining)g (the)599 3091 y(analogous)d(result)h(in)h Ft(R)1365 3061 y Fs(n)1444 3091 y FG(for)f Fo(n)c(>)f FG(3.)682 3190 y(The)32 b(set)f(of)g(Hamiltonian)g(quaternions)f Ft(H)52 b FG(is)31 b(the)g(unitary)g Ft(R)p Fp(\000)p FG(algebra)k(gener-)599 3290 y(ated)28 b(b)n(y)f(the)h(sym)n(b)r(ols)f Fo(i;)14 b(j;)g(k)30 b FG(with)e(the)g(relations)1613 3487 y Fo(i)1642 3452 y Fm(2)1702 3487 y FG(=)22 b Fo(j)1828 3452 y Fm(2)1889 3487 y FG(=)g Fo(k)2022 3452 y Fm(2)2082 3487 y FG(=)h Fp(\000)p FG(1)1137 3733 y Fo(ij)28 b FG(=)23 b Fp(\000)p Fo(j)5 b(i)22 b FG(=)h Fo(k)s(;)69 b(j)5 b(k)25 b FG(=)e Fp(\000)p Fo(k)s(j)k FG(=)c Fo(i:)55 b(k)s(i)23 b FG(=)f Fp(\000)p Fo(ik)j FG(=)e Fo(j:)599 3881 y FG(Multiplication)28 b(is)g(asso)r(ciativ)n(e)e(but)i(ob)n(viously)e(not)i(comm)n(utativ)n (e.)682 3980 y(If)23 b(w)n(e)e(iden)n(tify)h(a)g(magnetic)f(\014eld) 1773 3959 y Fo(~)1759 3980 y(B)28 b FG(=)22 b(\()p Fo(B)2032 3992 y Fm(1)2070 3980 y Fo(;)14 b(B)2170 3992 y Fm(2)2207 3980 y Fo(;)g(B)2307 3992 y Fm(3)2344 3980 y FG(\))22 b(and)g(a)f(magnetic)g(p)r(oten)n(tial)615 4069 y Fo(~)599 4090 y(A)j FG(=)e(\()p Fo(A)866 4102 y Fm(1)904 4090 y Fo(;)14 b(A)1003 4102 y Fm(2)1041 4090 y Fo(;)g(A)1140 4102 y Fm(3)1177 4090 y FG(\))28 b(with)g(purely)f(imaginary)f (quaternionic)h(\014elds)h(on)f Ft(R)2932 4060 y Fm(3)1368 4286 y Fq(b)d FG(=)e Fo(B)1585 4298 y Fm(1)1623 4286 y FG(\()p Fq(x)p FG(\))p Fo(i)c FG(+)g Fo(B)1935 4298 y Fm(2)1972 4286 y FG(\()p Fq(x)p FG(\))p Fo(j)24 b FG(+)18 b Fo(B)2295 4298 y Fm(3)2332 4286 y FG(\()p Fq(x)p FG(\))p Fo(k)s(;)1365 4533 y Fq(a)k FG(=)h Fo(A)1590 4545 y Fm(1)1628 4533 y FG(\()p Fq(x)p FG(\))p Fo(i)18 b FG(+)g Fo(A)1939 4545 y Fm(2)1977 4533 y FG(\()p Fq(x)p FG(\))p Fo(j)23 b FG(+)18 b Fo(A)2298 4545 y Fm(3)2336 4533 y FG(\()p Fq(x)p FG(\))p Fo(k)s(;)p eop %%Page: 7 7 7 6 bop 1120 75 a FH(ON)24 b(THE)f(ZER)n(O)h(MODES)f(OF)g(P)-6 b(A)n(ULI)24 b(OPERA)-6 b(TORS)479 b FG(7)599 320 y(then)29 b(the)f(equation)1754 448 y(~)1734 469 y Fo(D)r FG(\()p Fq(a)p FG(\))51 b(=)f Fq(b)q Fo(;)599 613 y FG(where)859 592 y(~)840 613 y Fo(D)25 b FG(=)d Fo(i)1095 581 y Fs(@)p 1060 595 110 4 v 1060 642 a(@)t(x)1137 650 y Fj(1)1197 613 y FG(+)c Fo(j)1364 581 y Fs(@)p 1329 595 V 1329 642 a(@)t(x)1406 650 y Fj(2)1467 613 y FG(+)g Fo(k)1641 581 y Fs(@)p 1606 595 V 1606 642 a(@)t(x)1683 650 y Fj(3)1725 613 y FG(,)27 b(is)h(equiv)-5 b(alen)n(t)27 b(to)1398 832 y Fo(cur)r(l)1577 811 y(~)1561 832 y(A)d FG(=)1748 811 y Fo(~)1734 832 y(B)5 b(;)96 b FG(and)69 b Fo(div)2269 811 y(~)2253 832 y(A)24 b FG(=)e(0)p Fo(:)599 1024 y FG(W)-7 b(e)35 b(can)e(solv)n(e)g(the)i(equation)1631 1003 y(~)1612 1024 y Fo(D)r FG(\()p Fq(a)p FG(\))68 b(=)f Fq(b)35 b FG(b)n(y)e(the)i(con)n(v)n(olution)d(of)i Fq(b)h FG(with)f(the)599 1124 y(Green's)27 b(function)h(of)1338 1103 y(~)1319 1124 y Fo(D)r FG(.)36 b(Since)1685 1103 y(~)1666 1124 y Fo(D)1737 1094 y Fm(2)1797 1124 y FG(=)23 b Fp(\000)p FG(\001)j(then)2253 1103 y(~)2234 1124 y Fo(D)r FG(\()p Fo(G)p FG(\()p Fq(x)q FG(\)\))h(is)g(the)h(Green's)e (func-)599 1223 y(tion)21 b(for)903 1202 y(~)883 1223 y Fo(D)i FG(if)e Fo(G)p FG(\()p Fq(x)p FG(\))g(is)g(the)g(Green's)f (function)2104 1191 y Fm(1)p 2084 1205 75 4 v 2084 1252 a(4)p Fs(\031)2202 1191 y Fm(1)p 2178 1205 83 4 v 2178 1252 a Fi(j)p Fc(x)o Fi(j)2291 1223 y FG(for)g Fp(\000)p FG(\001.)34 b(The)21 b(iden)n(tit)n(y)f(\(17\))h(is)599 1358 y(exactly)27 b(this)h(con)n(v)n(olution)e(of)i Fq(b)g FG(with)1866 1337 y(~)1847 1358 y Fo(D)r FG(\()p Fo(G)p FG(\()p Fq(x)q FG(\)\).)p 2273 1374 25 75 v 682 1527 a Fh(Lemma)i(2.3.)69 b Fg(L)l(et)59 b Fp(j)1395 1506 y Fo(~)1379 1527 y(A)p Fp(j)24 b(2)f Fo(L)1623 1497 y Fm(3)1660 1527 y FG(\()p Ft(R)1746 1497 y Fm(3)1789 1527 y FG(\))p Fg(,)31 b Fp(j)1913 1506 y Fo(~)1900 1527 y(B)t Fp(j)23 b(2)g Fo(L)2148 1497 y Fm(3)p Fs(=)p Fm(2)2252 1527 y FG(\()p Ft(R)2338 1497 y Fm(3)2382 1527 y FG(\))p Fg(.)39 b(Then)599 1626 y(\(i\))30 b(for)h(al)t(l)g Fo(f)h Fp(2)23 b Ft(H)1184 1596 y Fm(1)1184 1649 y Fs(B)1247 1626 y Fg(,)1060 1830 y Fp(j)29 b FG(\()h([)1210 1809 y Fo(~)1197 1830 y(D)s Fp(\001)1321 1809 y Fo(~)1306 1830 y(A)18 b FG(+)1485 1809 y Fo(~)1469 1830 y(A)p Fp(\001)1581 1809 y Fo(~)1568 1830 y(D)r FG(])p Fo(f)t(;)c(f)9 b FG(\))30 b Fp(j)50 b(\024)79 b FG(2)p Fo(\015)34 b Fp(k)2249 1809 y Fo(~)2234 1830 y(A)o Fp(k)2337 1845 y Fs(L)2383 1829 y Fj(3)2415 1845 y Fm(\()p Fk(R)2488 1829 y Fj(3)2515 1845 y Fm(\))2575 1830 y Fp(k)p Fo(f)9 b Fp(k)2709 1796 y Fm(2)2709 1856 y Fs(H)2767 1836 y Fj(1)2763 1874 y(0)3142 1830 y FG(\(18\))1929 1988 y Fp(\024)79 b FG(2)p Fo(\015)34 b Fp(k)2249 1967 y Fo(~)2234 1988 y(A)o Fp(k)2337 2003 y Fs(L)2383 1986 y Fj(3)2415 2003 y Fm(\()p Fk(R)2488 1986 y Fj(3)2515 2003 y Fm(\))2575 1988 y Fp(k)p Fo(f)9 b Fp(k)2709 1954 y Fm(2)2709 2014 y Fk(H)2747 1994 y Fj(1)2747 2033 y Ff(B)2806 1988 y Fo(;)313 b FG(\(19\))624 2215 y Fg(wher)l(e)866 2194 y Fo(~)853 2215 y(D)25 b FG(=)1044 2182 y Fm(1)p 1044 2196 34 4 v 1049 2243 a Fs(i)1087 2215 y Fn(r)f Fg(and)h Fo(\015)k Fg(is)c(the)f(norm)h(of)g (the)f(Sob)l(olev)i(emb)l(e)l(dding)f Fo(H)2833 2184 y Fm(1)2826 2235 y(0)2893 2215 y Fo(,)-14 b Fp(!)23 b Fo(L)3065 2184 y Fm(6)3102 2215 y FG(\()p Ft(R)3188 2184 y Fm(3)3232 2215 y FG(\))p Fg(;)599 2314 y(\(ii\))1377 2507 y Ft(H)1436 2472 y Fm(1)1436 2527 y Fs(B)1522 2507 y Fo(,)-14 b Fp(!)23 b Fo(H)1713 2472 y Fm(1)1706 2527 y Fs(A)1783 2507 y Fo(,)-14 b Fp(!)24 b Fo(H)1975 2472 y Fm(1)1968 2527 y(0)2035 2507 y Fo(,)-14 b Fp(!)23 b FG([)p Fo(L)2230 2472 y Fm(6)2267 2507 y FG(\()p Ft(R)2353 2472 y Fm(3)2396 2507 y FG(\)])2451 2472 y Fm(2)2489 2507 y Fo(:)630 b FG(\(20\))682 2866 y Fg(Pr)l(o)l(of.)108 b FG(\(i\))28 b(Let)g Fo(')c Fp(2)f FG([)p Fo(C)1515 2836 y Fi(1)1509 2887 y Fm(0)1586 2866 y FG(\()p Ft(R)1672 2836 y Fm(3)1715 2866 y FG(\)])1770 2836 y Fm(2)1808 2866 y FG(.)37 b(Then)875 3017 y Fr(\014)875 3067 y(\014)875 3117 y(\014)875 3167 y(\014)930 3137 y FG(\()28 b([)1026 3116 y Fo(~)1013 3137 y(D)r Fp(\001)1137 3116 y Fo(~)1121 3137 y(A)19 b FG(+)1300 3116 y Fo(~)1285 3137 y(A)p Fp(\001)1397 3116 y Fo(~)1384 3137 y(D)r FG(])p Fo(';)14 b(')p FG(\))1684 3017 y Fr(\014)1684 3067 y(\014)1684 3117 y(\014)1684 3167 y(\014)1789 3137 y FG(=)1931 3017 y Fr(\014)1931 3067 y(\014)1931 3117 y(\014)1931 3167 y(\014)1986 3137 y FG(2)27 b(Re)2211 3034 y Fm(3)2167 3059 y Fr(X)2170 3235 y Fs(j)s Fm(=1)2287 3137 y FG(\()p Fo(A)2381 3149 y Fs(j)2417 3137 y Fo(';)14 b(D)2577 3149 y Fs(j)2612 3137 y Fo(')p FG(\))2726 3017 y Fr(\014)2726 3067 y(\014)2726 3117 y(\014)2726 3167 y(\014)1789 3369 y Fp(\024)77 b FG(2)p Fp(k)2029 3348 y Fo(~)2015 3369 y(A)o Fp(k)2118 3384 y Fs(L)2164 3367 y Fj(3)2196 3384 y Fm(\()p Fk(R)2269 3367 y Fj(3)2296 3384 y Fm(\))2354 3369 y Fp(k)p Fo(')p Fp(k)2492 3384 y Fm([)p Fs(L)2557 3367 y Fj(6)2588 3384 y Fm(\()p Fk(R)2661 3367 y Fj(3)2688 3384 y Fm(\)])2733 3367 y Fj(2)2797 3369 y Fp(k)p Fn(r)o Fo(')p Fp(k)1789 3512 y(\024)g FG(2)p Fo(\015)32 b Fp(k)2105 3491 y Fo(~)2090 3512 y(A)o Fp(k)2193 3527 y Fs(L)2239 3510 y Fj(3)2271 3527 y Fm(\()p Fk(R)2344 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Fp(g)27 b Fo(';)14 b(')3299 4038 y Fr(\001)1208 4248 y FG(+)1300 4181 y Fr(\000)1338 4248 y FG([)p Fo(k)s Fp(j)1445 4227 y Fo(~)1430 4248 y(A)p Fp(j)1515 4214 y Fm(2)1571 4248 y Fp(\000)k Fo(k)1700 4214 y Fm(2)1737 4248 y Fp(j)1776 4227 y Fo(~)1760 4248 y(A)p Fp(j)1845 4214 y Fm(2)1883 4248 y FG(])p Fo(';)c(')2051 4181 y Fr(\001)1066 4390 y FG(=)49 b(\()p Fo(k)21 b Fp(\000)d FG(1\)\()p Fp(\000)p FG(\001)p Fo(';)c(')p FG(\))20 b(+)e(\()p Fo(S)1962 4418 y Fs(k)2011 4403 y(~)1998 4418 y(A)2080 4390 y Fo(';)c(')p FG(\))20 b Fp(\000)e FG(\()p Fo(k)2438 4356 y Fm(2)2493 4390 y Fp(\000)g Fo(k)s FG(\))2654 4323 y Fr(\000)2693 4390 y Fp(j)2731 4369 y Fo(~)2716 4390 y(A)p Fp(j)2801 4356 y Fm(2)2838 4390 y Fo(';)c(')2983 4323 y Fr(\001)1066 4533 y Fp(\025)49 b FG(\()p Fo(k)21 b Fp(\000)d FG(1\)\()p Fp(\000)p FG(\001)p Fo(';)c(')p FG(\))20 b Fp(\000)e FG(\()p Fo(k)1957 4499 y Fm(2)2012 4533 y Fp(\000)g Fo(k)s FG(\))2173 4466 y Fr(\000)2212 4533 y Fp(j)2250 4512 y Fo(~)2235 4533 y(A)p Fp(j)2320 4499 y Fm(2)2357 4533 y Fo(';)c(')2502 4466 y Fr(\001)2541 4533 y Fo(;)p eop %%Page: 8 8 8 7 bop 599 75 a FG(8)753 b FH(A.)23 b(BALINSKY)g(AND)g(W.D.)g(EV)-8 b(ANS)627 320 y FG(whence)839 512 y(\()p Fo(k)22 b Fp(\000)c FG(1\))p Fp(k)p Fn(r)n Fo(')p Fp(k)1309 478 y Fm(2)1396 512 y Fp(\024)50 b Fo(k)s Fp(k)p Fo(')p Fp(k)1695 478 y Fm(2)1695 538 y Fs(H)1753 518 y Fj(1)1749 557 y Ff(A)1818 512 y FG(+)18 b Fo(k)s FG(\()p Fo(k)j Fp(\000)e FG(1\))p Fo(\015)2249 478 y Fm(2)2313 512 y Fp(k)2370 491 y Fo(~)2355 512 y(A)p Fp(k)2459 478 y Fm(2)2459 534 y Fs(L)2505 518 y Fj(3)2536 534 y Fm(\()p Fk(R)2609 518 y Fj(3)2636 534 y Fm(\))2694 512 y Fp(k)27 b Fn(r)p Fp(j)p Fo(')p Fp(j)h(k)3013 478 y Fm(2)1396 671 y Fp(\024)1511 604 y Fr(\010)1559 671 y Fo(k)21 b FG(+)d Fo(k)s FG(\()p Fo(k)k Fp(\000)c FG(1\))p Fo(\015)2054 637 y Fm(2)2118 671 y Fp(k)2175 650 y Fo(~)2160 671 y(A)p Fp(k)2264 637 y Fm(2)2264 693 y Fs(L)2310 677 y Fj(3)2341 693 y Fm(\()p Fk(R)2414 677 y Fj(3)2442 693 y Fm(\))2472 604 y 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FG(,)g Fo(p)e FG(resp)r(ectiv)n(ely)g (ha)n(v)n(e)g(the)h(same)599 1947 y(form)i(domain)f Fp(Q)p FG(,)g(and)h(this)g(is)f Fp(D)r FG(\()p Ft(P)1766 1917 y Fm(1)p Fs(=)p Fm(2)1870 1947 y FG(\))h(with)g(the)g(graph)e(norm)1531 2096 y Fr(\000)1596 2163 y Fp(k)p Ft(P)1690 2129 y Fm(1)p Fs(=)p Fm(2)1793 2163 y Fo(')p Fp(k)18 b FG(+)g Fp(k)p Fo(')p Fp(k)2128 2129 y Fm(2)2193 2096 y Fr(\001)2231 2113 y Fm(1)p Fs(=)p Fm(2)2335 2163 y Fo(:)784 b FG(\(22\))599 2355 y(Also)28 b([)p Fo(C)875 2325 y Fi(1)869 2376 y Fm(0)945 2355 y FG(\()p Ft(R)1032 2325 y Fm(3)1075 2355 y FG(\)])1130 2325 y Fm(2)1195 2355 y FG(is)g(a)f(form)g(core.)36 b(It)28 b(follo)n(ws)e(that)1657 2547 y Fp(Q)51 b FG(=)f Ft(H)1950 2513 y Fm(1)1950 2568 y Fs(B)2059 2547 y Fp(\\)d(H)911 b FG(\(23\))599 2740 y(with)27 b(norm)f(\(22\);)g(the)h(em)n(b)r (edding)f Ft(H)1819 2709 y Fm(1)1819 2762 y Fs(B)1905 2740 y Fo(,)-14 b Fp(!)23 b FG([)p Fo(L)2100 2709 y Fm(6)2137 2740 y FG(\()p Ft(R)2223 2709 y Fm(3)2266 2740 y FG(\)])2321 2709 y Fm(2)2385 2740 y FG(guaran)n(tees)h(the)j(complete-)599 2839 y(ness,)21 b(since)f(con)n(v)n(ergen)n(t)d(sequences)i(in)h Ft(H)1911 2809 y Fm(1)1911 2862 y Fs(B)1994 2839 y FG(therefore)f(con)n (v)n(erge)e(p)r(oin)n(t)n(wise)i(to)g(their)599 2939 y(limits,)29 b(almost)e(ev)n(erywhere.)682 3038 y(F)-7 b(rom)1674 3188 y(0)22 b Fp(\024)h Fo(b)p FG([)p Fo(')p FG(])g Fp(\024)g Fo(p)p FG([)p Fo(')p FG(])599 3332 y(it)31 b(follo)n(ws)f(that)g(there)h(exists)f(a)g(b)r(ounded)g(self-adjoin)n (t)g(op)r(erator)f Fp(B)j FG(on)f Ft(H)3036 3302 y Fm(1)3036 3355 y Fs(B)3130 3332 y FG(suc)n(h)599 3432 y(that)1386 3624 y Fo(b)p FG([)p Fo(')p FG(])51 b(=)g(\()p Fp(B)30 b Fo(';)14 b(')p FG(\))1983 3642 y Fk(H)2023 3622 y Fj(1)2023 3662 y Ff(B)2082 3624 y Fo(;)97 b(')23 b Fp(2)h Ft(H)2416 3590 y Fm(1)2416 3644 y Fs(B)2479 3624 y Fo(:)640 b FG(\(24\))682 3836 y(F)-7 b(or)27 b Fo(')d Fp(2)f(R)p FG(\()p Ft(P)1141 3806 y Fm(1)p Fs(=)p Fm(2)1245 3836 y FG(\),)28 b(the)g(range)e(of)i Ft(P)1844 3806 y Fm(1)p Fs(=)p Fm(2)1947 3836 y FG(,)1557 4029 y Fp(k)p Ft(P)1651 3994 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1834 4029 y Fo(')p Fp(k)1930 4047 y Fk(H)1969 4027 y Fj(1)1969 4066 y Ff(B)2079 4029 y FG(=)50 b Fp(k)p Fo(')p Fp(k)810 b FG(\(25\))599 4241 y(and)34 b(hence,)g(since)f Fp(D)r FG(\()p Ft(P)1386 4211 y Fm(1)p Fs(=)p Fm(2)1490 4241 y FG(\))h(and)f Fp(R)p FG(\()p Ft(P)1877 4211 y Fm(1)p Fs(=)p Fm(2)1981 4241 y FG(\))g(are)g(dense)g(subspaces)f(of)h Ft(H)2967 4211 y Fm(1)2967 4264 y Fs(B)3030 4241 y FG(,)i Fp(H)f FG(re-)599 4341 y(sp)r(ectiv)n(ely)-7 b(,)28 b Ft(P)1051 4311 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1234 4341 y FG(extends)f(to)h(a)f(unitary)g(map)1156 4533 y Fo(U)59 b FG(:)51 b Fp(H)24 b(\000)-14 b(!)23 b Ft(H)1655 4499 y Fm(1)1655 4554 y Fs(B)1718 4533 y Fo(;)97 b(U)32 b FG(=)23 b Ft(P)2067 4499 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)2277 4533 y FG(on)55 b Fp(R)p FG(\()p Ft(P)2574 4499 y Fm(1)p Fs(=)p Fm(2)2678 4533 y FG(\))p Fo(:)409 b FG(\(26\))p eop %%Page: 9 9 9 8 bop 1120 75 a FH(ON)24 b(THE)f(ZER)n(O)h(MODES)f(OF)g(P)-6 b(A)n(ULI)24 b(OPERA)-6 b(TORS)479 b FG(9)599 320 y(De\014ne)1457 509 y Fp(S)57 b FG(:=)50 b Fp(j)1738 488 y Fo(~)1725 509 y(B)5 b Fp(j)1816 475 y Fm(1)p Fs(=)p Fm(2)1920 509 y Fo(U)59 b FG(:)51 b Fp(H)24 b(\000)-14 b(!)23 b(H)711 b FG(\(27\))599 699 y(Note)28 b(that)g(for)f Fo(u)c Fp(2)g Ft(H)1315 669 y Fm(1)1315 722 y Fs(B)1378 699 y FG(,)972 901 y Fp(k)28 b(j)1078 880 y Fo(~)1065 901 y(B)t Fp(j)1155 866 y Fm(1)p Fs(=)p Fm(2)1259 901 y Fo(u)f Fp(k)1376 866 y Fm(2)1436 901 y Fp(\024)c(k)1579 880 y Fo(~)1566 901 y(B)t Fp(k)1675 920 y Fs(L)1721 903 y Fj(3)p Ff(=)p Fj(2)1811 920 y Fm(\()p Fk(R)1884 903 y Fj(3)1911 920 y Fm(\))1941 901 y Fp(k)p Fo(u)p Fp(k)2073 866 y Fm(2)2073 923 y([)p Fs(L)2138 906 y Fj(6)2169 923 y Fm(\()p Fk(R)2242 906 y Fj(3)2269 923 y Fm(\)])2314 906 y Fj(2)2373 901 y Fp(\024)g FG(const)o Fp(\001)14 b(k)p Fo(u)p Fp(k)2819 866 y Fm(2)2819 926 y Fk(H)2857 906 y Fj(1)2857 946 y Ff(B)3142 901 y FG(\(28\))599 1107 y(b)n(y)28 b(\(20\).)682 1263 y Fh(Theorem)j(3.1.)1136 1453 y FG(n)n(ul)f Ft(P)1331 1465 y Fs(A)1434 1453 y FG(=)49 b(dim)1686 1385 y Fr(\010)1735 1453 y Fo(u)22 b FG(:)53 b Fp(B)s Fo(u)22 b FG(=)h Fo(u;)43 b(u)22 b Fp(2)i Ft(H)2418 1418 y Fm(1)2418 1473 y Fs(B)2530 1453 y Fp(\\)48 b(H)2704 1385 y Fr(\011)1434 1577 y Fp(\024)h FG(n)n(ul)30 b Fo(F)r(;)629 1767 y Fg(wher)l(e)h Fo(F)k FG(=)22 b(1)c Fp(\000)g(S)6 b(S)1294 1737 y Fi(\003)1333 1767 y Fg(.)682 2040 y(Pr)l(o)l(of.)80 b FG(Let)28 b Fo(u;)14 b(')23 b Fp(2)g(D)r FG(\()p Ft(P)1518 2010 y Fm(1)p Fs(=)p Fm(2)1622 2040 y FG(\).)37 b(Then)1448 2230 y Fo(p)1490 2242 y Fs(A)1543 2230 y FG([)p Fo(u;)14 b(')p FG(])50 b(=)g Fo(p)p FG([)p Fo(u;)14 b(')p FG(])k Fp(\000)g Fo(b)p FG([)p Fo(u;)c(')p FG(])1778 2354 y(=)50 b(\()p Fo(u)18 b Fp(\000)g(B)s Fo(u;)c(')p FG(\))2303 2372 y Fk(H)2341 2352 y Fj(1)2341 2392 y Ff(B)2400 2354 y Fo(:)624 2564 y FG(Hence,)26 b Fo(u)c Fp(2)i FG(k)n(er)13 b Ft(P)1218 2576 y Fs(A)1293 2564 y Fp(\032)23 b(D)r FG(\()p Ft(P)1531 2534 y Fm(1)p Fs(=)p Fm(2)1635 2564 y FG(\))i(if)g(and)g(only)f(if)h Fp(B)s Fo(u)d FG(=)h Fo(u)h FG(with)h Fo(u)e Fp(2)g(H)q FG(.)36 b(Moreo)n(v)n(er,)599 2664 y(for)28 b(an)n(y)e Fo(f)t(;)14 b(g)26 b Fp(2)d(H)1380 2746 y Fr(\000)1446 2813 y Fp(S)6 b Fo(f)t(;)14 b Fp(S)6 b Fo(g)1711 2746 y Fr(\001)1800 2813 y FG(=)1915 2746 y Fr(\000)1981 2813 y Fp(B)29 b Fo(U)9 b(f)t(;)14 b(U)9 b(g)2349 2746 y Fr(\001)2387 2846 y Fk(H)2426 2826 y Fj(1)2426 2866 y Ff(B)2486 2813 y Fo(;)599 2974 y FG(whence)1183 3057 y Fr(\000)1249 3124 y FG([)p Fp(S)1328 3090 y Fi(\003)1367 3124 y Fp(S)25 b(\000)18 b FG(1])p Fo(f)t(;)c(g)1742 3057 y Fr(\001)1830 3124 y FG(=)1946 3057 y Fr(\000)2011 3124 y FG([)p Fp(B)21 b(\000)d FG(1])27 b Fo(U)9 b(f)t(;)14 b(U)9 b(g)2569 3057 y Fr(\001)2607 3157 y Fk(H)2646 3137 y Fj(1)2646 3177 y Ff(B)599 3311 y FG(The)28 b(result)f(follo)n(ws)g (since)g(n)n(ul)h([)p Fp(S)1697 3281 y Fi(\003)1736 3311 y Fp(S)d(\000)18 b FG(1])k(=)h(n)n(ul)28 b([)p Fp(S)6 b(S)2345 3281 y Fi(\003)2402 3311 y Fp(\000)18 b FG(1].)p 2656 3327 25 75 v 682 3460 a(The)34 b(op)r(erator)e Fp(S)6 b(S)1312 3430 y Fi(\003)1385 3460 y FG(is)34 b(of)g(Birman-Sc)n(h)n (winger)d(t)n(yp)r(e.)56 b(W)-7 b(e)34 b(ha)n(v)n(e,)g(in)g(terms)f(of) 599 3560 y(\(27\))1261 3759 y Fp(S)6 b(S)1373 3725 y Fi(\003)1463 3759 y FG(=)50 b Fp(j)1614 3738 y Fo(~)1601 3759 y(B)t Fp(j)1691 3725 y Fm(1)p Fs(=)p Fm(2)1796 3759 y Fo(U)1862 3725 y Fm(2)1898 3759 y Fp(j)1934 3738 y Fo(~)1921 3759 y(B)5 b Fp(j)2012 3725 y Fm(1)p Fs(=)p Fm(2)2199 3759 y FG(on)55 b Fp(D)r FG(\()p Ft(P)2492 3725 y Fm(1)p Fs(=)p Fm(2)2596 3759 y FG(\))514 b(\(29\))599 3949 y(and)30 b(this)h(extends)f(b)n(y)g(con)n(tin)n(uit)n(y)g(to)g(a)f (b)r(ounded)i(op)r(erator)e(on)g Fp(H)q FG(.)45 b(T)-7 b(o)30 b(see)g(\(29\),)599 4048 y(\014rst)e(observ)n(e)e(that)i(for)f Fo(f)k Fp(2)24 b(R)p FG(\()p Ft(P)1679 4018 y Fm(1)p Fs(=)p Fm(2)1782 4048 y FG(\),)k Fo(g)e Fp(2)d(D)r FG(\()p Ft(P)2159 4018 y Fm(1)p Fs(=)p Fm(2)2263 4048 y FG(\))1150 4181 y Fr(\000)1216 4248 y Fo(f)t(;)14 b Fp(S)1354 4214 y Fi(\003)1393 4248 y Fo(g)1463 4181 y Fr(\001)1579 4248 y FG(=)1721 4181 y Fr(\000)1786 4248 y Fp(S)6 b Fo(f)t(;)14 b(g)1995 4181 y Fr(\001)2084 4248 y FG(=)2199 4181 y Fr(\000)2265 4248 y Fp(j)2301 4227 y Fo(~)2288 4248 y(B)t Fp(j)2378 4214 y Fm(1)p Fs(=)p Fm(2)2482 4248 y Fo(U)9 b(f)t(;)14 b(g)2700 4181 y Fr(\001)1579 4390 y FG(=)1721 4323 y Fr(\000)1786 4390 y Fo(U)9 b(f)t(;)14 b Fp(j)1970 4369 y Fo(~)1957 4390 y(B)t Fp(j)2047 4356 y Fm(1)p Fs(=)p Fm(2)2152 4390 y Fo(g)2222 4323 y Fr(\001)1579 4533 y FG(=)1721 4466 y Fr(\000)1786 4533 y Ft(P)1838 4499 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1993 4533 y Fo(f)t(;)g Fp(j)2111 4512 y Fo(~)2098 4533 y(B)t Fp(j)2188 4499 y Fm(1)p Fs(=)p Fm(2)2293 4533 y Fo(g)2363 4466 y Fr(\001)2401 4533 y FG(;)p eop %%Page: 10 10 10 9 bop 599 75 a FG(10)711 b FH(A.)23 b(BALINSKY)g(AND)g(W.D.)g(EV)-8 b(ANS)627 329 y FG(note)28 b(that)f Fp(j)1027 308 y Fo(~)1014 329 y(B)5 b Fp(j)1105 299 y Fm(1)p Fs(=)p Fm(2)1209 329 y Fo(g)26 b Fp(2)d(H)29 b FG(b)n(y)e(\(28\))g(and)g(since)h Fp(j)2143 308 y Fo(~)2130 329 y(B)t Fp(j)23 b(2)g Fo(L)2378 299 y Fm(3)p Fs(=)p Fm(2)2482 329 y FG(\()p Ft(R)2569 299 y Fm(3)2612 329 y FG(\).)37 b(Hence)28 b Fp(j)2987 308 y Fo(~)2974 329 y(B)t Fp(j)3064 299 y Fm(1)p Fs(=)p Fm(2)3168 329 y Fo(g)e Fp(2)599 439 y(D)r FG(\()p Ft(P)749 409 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)905 439 y FG(\))i(and)f Ft(P)1178 409 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1333 439 y Fp(j)1369 418 y Fo(~)1356 439 y(B)5 b Fp(j)1447 409 y Fm(1)p Fs(=)p Fm(2)1551 439 y Fo(g)26 b FG(=)c Fp(S)1760 409 y Fi(\003)1799 439 y Fo(g)s FG(.)37 b(In)27 b(other)g(w)n(ords)1334 653 y Fp(S)1390 618 y Fi(\003)1480 653 y FG(=)50 b Ft(P)1647 618 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1802 653 y Fp(j)1838 632 y Fo(~)1825 653 y(B)t Fp(j)1915 618 y Fm(1)p Fs(=)p Fm(2)2103 653 y FG(on)55 b Fp(D)r FG(\()p Ft(P)2396 618 y Fm(1)p Fs(=)p Fm(2)2499 653 y FG(\))p Fo(;)588 b FG(\(30\))599 857 y(whence)28 b(\(29\).)682 1033 y Fh(Lemma)i(3.1.)69 b Fp(S)6 b(S)1295 1002 y Fi(\003)1394 1033 y Fg(is)30 b(c)l(omp)l(act)g(and)1505 1237 y Fp(kS)6 b(k)1645 1202 y Fm(2)1735 1237 y Fp(\024)52 b Fo(\015)1900 1202 y Fm(2)1966 1237 y Fp(k)2021 1216 y Fo(~)2008 1237 y(B)t Fp(k)2117 1256 y Fs(L)2163 1239 y Fj(3)p Ff(=)p Fj(2)2253 1256 y Fm(\()p Fk(R)2326 1239 y Fj(3)2354 1256 y Fm(\))3142 1237 y FG(\(31\))599 1451 y Fg(wher)l(e)31 b Fo(\015)j Fg(is)c(the)g(norm)g(of)h Fo(H)1534 1421 y Fm(1)1527 1471 y(0)1594 1451 y Fo(,)-14 b Fp(!)23 b FG([)p Fo(L)1789 1421 y Fm(6)1826 1451 y FG(\()p Ft(R)1912 1421 y Fm(3)1955 1451 y FG(\)])2010 1421 y Fm(2)2048 1451 y Fg(.)682 1709 y(Pr)l(o)l(of.)79 b FG(This)24 b(is)h(quite)f(standard,)g(but)h(w)n(e)e (giv)n(e)h(the)g(short)g(pro)r(of)f(for)h(complete-)599 1808 y(ness.)36 b(W)-7 b(e)23 b(sho)n(w)g(that)h Fp(j)1359 1787 y Fo(~)1346 1808 y(B)t Fp(j)1436 1778 y Fm(1)p Fs(=)p Fm(2)1564 1808 y FG(:)f Ft(H)1668 1778 y Fm(1)1668 1831 y Fs(B)1754 1808 y Fp(\000)-14 b(!)23 b(H)i FG(is)e(compact.)35 b(Let)24 b Fp(f)p Fo(')2689 1820 y Fs(n)2734 1808 y Fp(g)f FG(b)r(e)h(a)f(sequence)599 1908 y(whic)n(h)34 b(con)n(v)n(erges)d(w)n (eakly)i(to)h(zero)f(in)h Ft(H)1953 1878 y Fm(1)1953 1931 y Fs(B)2016 1908 y FG(,)i(and)e(hence)f(in)h Fo(H)2658 1878 y Fm(1)2651 1928 y(0)2729 1908 y FG(b)n(y)g(\(20\).)55 b(Then,)599 2019 y(in)36 b(particular)f Fp(k)p Fo(')1192 2031 y Fs(n)1237 2019 y Fp(k)1279 2038 y Fs(H)1337 2017 y Fj(1)1333 2056 y(0)1410 2019 y Fp(\024)i Fo(k)s FG(,)g(sa)n(y)-7 b(.)61 b(Giv)n(en)35 b Fo(")i(>)f FG(0,)h(set)f Fp(j)2513 1998 y Fo(~)2500 2019 y(B)t Fp(j)h FG(=)f Fo(B)2791 2031 y Fm(1)2852 2019 y FG(+)24 b Fo(B)3004 2031 y Fm(2)3077 2019 y FG(where)599 2131 y Fo(B)662 2143 y Fm(1)734 2131 y Fp(2)34 b Fo(C)888 2101 y Fi(1)882 2152 y Fm(0)959 2131 y FG(\()p Ft(R)1045 2101 y Fm(3)1088 2131 y FG(\))h(with)g(supp)r (ort)f(\012)1725 2143 y Fs(")1794 2131 y FG(and)g Fo(B)2025 2143 y Fm(1)2097 2131 y Fp(\024)f Fo(k)2238 2143 y Fs(")2308 2131 y FG(sa)n(y)-7 b(,)35 b(and)f Fp(k)p Fo(B)2749 2143 y Fm(2)2786 2131 y Fp(k)2828 2151 y Fs(L)2874 2134 y Fj(3)p Ff(=)p Fj(2)2964 2151 y Fm(\()p Fk(R)3037 2134 y Fj(3)3065 2151 y Fm(\))3129 2131 y Fo(<)g(")p FG(.)599 2231 y(Then)817 2435 y Fp(k)27 b(j)922 2414 y Fo(~)909 2435 y(B)t Fp(j)999 2401 y Fm(1)p Fs(=)p Fm(2)1104 2435 y Fo(')1158 2447 y Fs(n)1231 2435 y Fp(k)1273 2401 y Fm(2)1387 2435 y Fp(\024)77 b Fo(k)1572 2447 y Fs(")1608 2435 y Fp(k)p Fo(')1704 2447 y Fs(n)1749 2435 y Fp(k)1791 2401 y Fm(2)1791 2458 y([)p Fs(L)1856 2441 y Fj(2)1887 2458 y Fm(\(\012)1960 2466 y Ff(")1993 2458 y Fm(\)])2038 2441 y Fj(2)2093 2435 y FG(+)45 b Fo(\015)2251 2401 y Fm(2)2316 2435 y Fp(k)p Fo(B)2421 2447 y Fm(2)2458 2435 y Fp(k)2500 2454 y Fs(L)2546 2438 y Fj(3)p Ff(=)p Fj(2)2636 2454 y Fm(\()p Fk(R)2709 2438 y Fj(3)2736 2454 y Fm(\))2794 2435 y Fp(k)p Fo(')2890 2447 y Fs(n)2935 2435 y Fp(k)2977 2401 y Fm(2)2977 2461 y Fs(H)3035 2441 y Fj(1)3031 2479 y(0)1387 2584 y Fp(\024)77 b Fo(k)1572 2596 y Fs(")1608 2584 y Fp(k)p Fo(')1704 2596 y Fs(n)1749 2584 y Fp(k)1791 2550 y Fm(2)1791 2607 y([)p Fs(L)1856 2590 y Fj(2)1887 2607 y Fm(\(\012)1960 2615 y Ff(")1993 2607 y Fm(\)])2038 2590 y Fj(2)2093 2584 y FG(+)45 b Fo(\015)2251 2550 y Fm(2)2316 2584 y Fo(")27 b Fp(k)p Fo(')2478 2596 y Fs(n)2523 2584 y Fp(k)2565 2602 y Fk(H)2604 2582 y Fj(1)2604 2622 y Ff(B)2663 2584 y Fo(:)637 2788 y FG(The)37 b(\014rst)g(term)g(on)g (the)h(righ)n(t-hand)e(side)h(tends)h(to)f(zero)f(as)h Fo(n)i Fp(!)g(1)e FG(b)n(y)g(the)599 2888 y(Rellic)n(h)31 b(Theorem.)46 b(Consequen)n(tly)30 b Fp(j)1825 2867 y Fo(~)1812 2888 y(B)t Fp(j)1902 2858 y Fm(1)p Fs(=)p Fm(2)2035 2888 y FG(:)e Ft(H)2145 2858 y Fm(1)2145 2911 y Fs(B)2236 2888 y Fp(\000)-14 b(!)28 b(H)k FG(is)f(compact)f(and)g(hence)599 2999 y(so)d(is)h Fp(S)h FG(=)23 b Fp(j)988 2978 y Fo(~)975 2999 y(B)t Fp(j)1065 2969 y Fm(1)p Fs(=)p Fm(2)1170 2999 y Fo(U)9 b FG(.)682 3120 y(The)28 b(inequalit)n(y)f(\(31\))h(follo)n (ws)e(from)h(\(28\).)p 2136 3136 25 75 v 1432 3435 a Fu(4.)32 b(THE)g(MAIN)f(RESUL)-8 b(T)682 3607 y FG(F)h(or)27 b Fo(t)c Fp(2)h FG(\(0)p Fo(;)14 b Fp(1)p FG(\),)27 b(replace)1536 3586 y Fo(~)1520 3607 y(A)h FG(b)n(y)f Fo(t)1771 3586 y(~)1755 3607 y(A)h FG(and)f(denote)g(the)h(corresp)r(onding)d(op)r (erators)599 3707 y(b)n(y)j Ft(P)767 3719 y Fs(t)795 3707 y FG(,)f Fp(S)895 3719 y Fs(t)953 3707 y FG(and)g Fo(F)1167 3719 y Fs(t)1197 3707 y FG(.)37 b(It)28 b(follo)n(ws)e(from)h (\(31\))h(that)1357 3921 y Fp(kS)1449 3933 y Fs(t)1478 3921 y Fp(k)1520 3886 y Fm(2)1608 3921 y Fp(\024)50 b Fo(\015)1771 3886 y Fm(2)1835 3921 y Fo(t)28 b Fp(k)1948 3900 y Fo(~)1935 3921 y(B)t Fp(k)2044 3940 y Fs(L)2090 3923 y Fj(3)p Ff(=)p Fj(2)2180 3940 y Fm(\()p Fk(R)2253 3923 y Fj(3)2280 3940 y Fm(\))2333 3921 y Fp(\000)-14 b(!)23 b FG(0)599 4125 y(as)k Fo(t)c Fp(!)g FG(0.)37 b(Hence,)28 b Fo(F)1285 4137 y Fs(t)1337 4125 y FG(=)23 b(1)18 b Fp(\000)g(S)1618 4137 y Fs(t)1648 4125 y Fp(S)1704 4095 y Fi(\003)1698 4145 y Fs(t)1770 4125 y FG(is)27 b(suc)n(h)h(that,)g(for)f(some)g Fo(t)2609 4137 y Fm(0)2669 4125 y Fo(>)c FG(0,)1487 4329 y(n)n(ul)k Fo(F)1680 4341 y Fs(t)1761 4329 y FG(=)50 b(0)p Fo(;)96 b(t)23 b Fp(2)h FG(\(0)p Fo(;)14 b(t)2310 4341 y Fm(0)2347 4329 y FG(\))p Fo(:)740 b FG(\(32\))599 4533 y(W)-7 b(e)28 b(pro)r(ceed)f(to)h(pro)n (v)n(e)e(that)i Fp(fS)1647 4545 y Fs(t)1676 4533 y Fp(S)1732 4503 y Fi(\003)1726 4554 y Fs(t)1770 4533 y Fp(g)g FG(is)f(a)g(real)g (analytic)g(family)-7 b(.)p eop %%Page: 11 11 11 10 bop 1120 75 a FH(ON)24 b(THE)f(ZER)n(O)h(MODES)f(OF)g(P)-6 b(A)n(ULI)24 b(OPERA)-6 b(TORS)438 b FG(11)682 329 y Fh(Lemma)30 b(4.1.)78 b Fg(L)l(et)37 b Fo(s)g Fp(2)h FG(\(0)p Fo(;)14 b Fp(1)p FG(\))37 b Fg(b)l(e)h(\014xe)l(d,)i(and)e (supp)l(ose)g(that)g Fp(j)2808 308 y Fo(~)2793 329 y(A)p Fp(j)g(2)f Fo(L)3065 299 y Fm(3)3102 329 y FG(\()p Ft(R)3188 299 y Fm(3)3232 329 y FG(\))p Fg(.)599 429 y(Then,)31 b(ther)l(e)f(exists)g(a)g(neighb)l(ourho)l(o)l(d)h Fo(N)9 b FG(\()p Fo(s)p FG(\))30 b Fg(of)h Fo(s)f Fg(such)f(that)1075 625 y FG(1)p 1075 662 42 4 v 1081 738 a Fo(t)1126 681 y Fp(S)1176 693 y Fs(t)1206 681 y Fp(S)1262 647 y Fi(\003)1256 702 y Fs(t)1353 681 y FG(=)1480 625 y(1)p 1480 662 V 1481 738 a Fo(s)1532 681 y Fp(S)1582 693 y Fs(s)1618 681 y Fp(S)1674 647 y Fi(\003)1668 702 y Fs(s)1731 681 y FG(+)1843 578 y Fi(1)1816 602 y Fr(X)1814 778 y Fs(n)p Fm(=1)1969 681 y FG(\()p Fo(t)18 b Fp(\000)g Fo(s)p FG(\))2203 647 y Fs(n)2249 681 y Fo(K)2320 693 y Fs(n)2364 681 y Fo(;)103 b(t)23 b Fp(2)h Fo(N)9 b FG(\()p Fo(s)p FG(\))p Fo(;)318 b FG(\(33\))599 939 y Fg(wher)l(e)31 b(the)f Fo(K)1043 951 y Fs(n)1117 939 y Fg(ar)l(e)g(b)l(ounde)l(d)g(op)l(er)l (ators)h(on)f Fp(H)q Fg(.)682 1225 y(Pr)l(o)l(of.)80 b FG(F)-7 b(or)27 b Fo( )s(;)14 b(')24 b Fp(2)1378 1158 y Fr(\002)1413 1225 y Fo(C)1478 1195 y Fi(1)1472 1246 y Fm(0)1548 1225 y FG(\()p Ft(R)1635 1195 y Fm(3)1678 1225 y FG(\))1710 1158 y Fr(\003)1745 1175 y Fm(2)654 1362 y Fr(\000)720 1429 y Ft(P)772 1441 y Fs(t)799 1429 y Fo( )s(;)14 b(')975 1362 y Fr(\001)1064 1429 y FG(=)1180 1362 y Fr(\000)27 b(\002)1308 1429 y Fp(\000)p FG(\001)18 b(+)g Fo(t)p FG(\()1618 1408 y Fo(~)1605 1429 y(D)r Fp(\001)1728 1408 y Fo(~)1713 1429 y(A)h FG(+)1892 1408 y Fo(~)1877 1429 y(A)p Fp(\001)1989 1408 y Fo(~)1976 1429 y(D)r FG(\))g(+)f Fo(t)2211 1395 y Fm(2)2248 1429 y Fp(j)2286 1408 y Fo(~)2271 1429 y(A)p Fp(j)2356 1395 y Fm(2)2412 1429 y FG(+)g Fo(t)p FG(\()n Fo(~)-40 b(\033)t Fp(\001)2658 1408 y Fo(~)2645 1429 y(B)22 b FG(+)c Fp(j)2849 1408 y Fo(~)2836 1429 y(B)t Fp(j)p FG(\))2986 1362 y Fr(\003)3021 1429 y Fo( )s(;)c(')3197 1362 y Fr(\001)599 1615 y FG(and)1235 1697 y Fr(\000)28 b(\002)1363 1764 y Ft(P)1415 1776 y Fs(t)1462 1764 y Fp(\000)18 b Ft(P)1597 1776 y Fs(s)1659 1697 y Fr(\003)1693 1764 y Fo( )s(;)c(')1869 1697 y Fr(\001)1958 1764 y FG(=)50 b(\()p Fo(t)19 b Fp(\000)f Fo(s)p FG(\))2308 1697 y Fr(\000)2374 1764 y Fo(Q )s(;)c(')2615 1697 y Fr(\001)599 1904 y FG(where)1120 2053 y Fo(Q)50 b FG(=)1364 2032 y Fo(~)1351 2053 y(D)r Fp(\001)1475 2032 y Fo(~)1459 2053 y(A)19 b FG(+)1639 2032 y Fo(~)1623 2053 y(A)p Fp(\001)1735 2032 y Fo(~)1722 2053 y(D)i FG(+)d(\()p Fo(t)g FG(+)h Fo(s)p FG(\))p Fp(j)2168 2032 y Fo(~)2153 2053 y(A)p Fp(j)2238 2019 y Fm(2)2294 2053 y FG(+)d Fo(~)-40 b(\033)s Fp(\001)2477 2032 y Fo(~)2464 2053 y(B)23 b FG(+)18 b Fp(j)2669 2032 y Fo(~)2656 2053 y(B)t Fp(j)p Fo(:)599 2193 y FG(F)-7 b(rom)28 b(\(19\))f(and)g(since) 1221 2321 y Fr(\000)1286 2388 y Fp(j)1325 2367 y Fo(~)1309 2388 y(A)q Fp(j)1395 2354 y Fm(2)1432 2388 y Fo( )s(;)14 b( )1611 2321 y Fr(\001)1726 2388 y Fp(\024)77 b(k)1925 2367 y Fo(~)1910 2388 y(A)p Fp(k)2014 2354 y Fm(2)2014 2411 y Fs(L)2060 2394 y Fj(3)2092 2411 y Fm(\()p Fk(R)2165 2394 y Fj(3)2192 2411 y Fm(\))2250 2388 y Fp(k)p Fo( )s Fp(k)2391 2354 y Fm(2)2391 2411 y([)p Fs(L)2456 2394 y Fj(6)2486 2411 y Fm(\()p Fk(R)2559 2394 y Fj(3)2587 2411 y Fm(\)])2632 2394 y Fj(2)1726 2539 y Fp(\024)g Fo(\015)1916 2504 y Fm(2)1981 2539 y Fp(k)2038 2518 y Fo(~)2023 2539 y(A)p Fp(k)2127 2504 y Fm(2)2127 2561 y Fs(L)2173 2545 y Fj(3)2204 2561 y Fm(\()p Fk(R)2277 2545 y Fj(3)2304 2561 y Fm(\))2362 2539 y Fp(k)p Fo( )s Fp(k)2503 2504 y Fm(2)2503 2564 y Fk(H)2541 2544 y Fj(1)2541 2584 y Ff(B)627 2736 y FG(and)1265 2818 y Fr(\000)1331 2886 y Fp(j)1367 2865 y Fo(~)1354 2886 y(B)t Fp(j)28 b Fo( )s(;)14 b( )1651 2818 y Fr(\001)1740 2886 y Fp(\024)50 b Fo(\015)1903 2851 y Fm(2)1940 2886 y Fp(k)1995 2865 y Fo(~)1982 2886 y(B)s Fp(k)2090 2905 y Fs(L)2136 2888 y Fj(3)p Ff(=)p Fj(2)2227 2905 y Fm(\()p Fk(R)2300 2888 y Fj(3)2327 2905 y Fm(\))2385 2886 y Fp(k)p Fo( )s Fp(k)2526 2851 y Fm(2)2526 2911 y Fk(H)2564 2891 y Fj(1)2564 2931 y Ff(B)599 3037 y FG(w)n(e)28 b(ha)n(v)n(e)1527 3186 y Fp(j)1550 3119 y Fr(\000)1616 3186 y Fo(Q )s(;)14 b( )1860 3119 y Fr(\001)1898 3186 y Fp(j)51 b(\024)f Fo(c)p Fp(k)p Fo( )s Fp(k)2264 3152 y Fm(2)2264 3212 y Fk(H)2302 3192 y Fj(1)2302 3232 y Ff(B)599 3366 y FG(for)28 b(some)f(constan)n(t)f Fo(c)p FG(,)i(and)f(so)g Fo(R)d FG(=)f Ft(P)1846 3323 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1846 3376 y Fs(s)2001 3366 y Fo(Q)p Ft(P)2119 3323 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)2119 3376 y Fs(s)2301 3366 y FG(satis\014es)1559 3552 y Fp(j)1582 3485 y Fr(\000)1647 3552 y Fo(R)q( )s(;)14 b( )1890 3485 y Fr(\001)1928 3552 y Fp(j)51 b(\024)f Fo(c)p Fp(k)p Fo( )s Fp(k)2294 3518 y Fm(2)599 3738 y FG(and)25 b(extends)f(to)h(an)f(op)r(erator)f(in)i Fp(L)p FG(\()p Fp(H)q FG(\),)h(the)f(space)f(of)h(b)r(ounded)g(linear)e(op)r (erators)599 3838 y(on)h Fp(H)q FG(.)36 b(Th)n(us,)25 b(there)f(exists)g(a)f(neigh)n(b)r(ourho)r(o)r(d)h Fo(N)9 b FG(\()p Fo(s)p FG(\))24 b(of)g Fo(s)h FG(suc)n(h)e(that)i(for)f Fo(t)f Fp(2)g Fo(N)9 b FG(\()p Fo(s)p FG(\))1198 4023 y Fr(\010)1274 4090 y FG(1)18 b(+)g(\()p Fo(t)h Fp(\000)f Fo(s)p FG(\))p Fo(R)1744 4023 y Fr(\011)1792 4040 y Fi(\000)p Fm(1)1932 4090 y FG(=)2077 3986 y Fi(1)2050 4011 y Fr(X)2047 4187 y Fs(n)p Fm(=0)2172 4090 y FG(\()p Fo(t)h Fp(\000)f Fo(s)p FG(\))2407 4056 y Fs(n)2452 4090 y FG(\()p Fp(\000)p Fo(R)q FG(\))2645 4056 y Fs(n)599 4347 y FG(in)28 b Fp(H)q FG(.)37 b(It)28 b(follo)n(ws)f(that)1084 4533 y Ft(P)1136 4545 y Fs(t)1214 4533 y FG(=)51 b Ft(P)1382 4545 y Fs(s)1434 4533 y FG(+)18 b(\()p Fo(t)h Fp(\000)f Fo(s)p FG(\))p Fo(Q)51 b FG(=)f Ft(P)2036 4499 y Fm(1)p Fs(=)p Fm(2)2036 4554 y Fs(s)2139 4533 y FG([1)18 b(+)g(\()p Fo(t)h Fp(\000)f Fo(s)p FG(\))p Fo(R)q FG(])p Ft(P)2679 4499 y Fm(1)p Fs(=)p Fm(2)2679 4554 y Fs(s)2782 4533 y Fo(;)p eop %%Page: 12 12 12 11 bop 599 75 a FG(12)711 b FH(A.)23 b(BALINSKY)g(AND)g(W.D.)g(EV)-8 b(ANS)599 320 y FG(and)810 508 y Fp(j)846 487 y Fo(~)833 508 y(B)t Fp(j)923 474 y Fm(1)p Fs(=)p Fm(2)1028 508 y Ft(P)1080 473 y Fi(\000)p Fm(1)1080 529 y Fs(t)1168 508 y Fp(j)1204 487 y Fo(~)1191 508 y(B)t Fp(j)1281 474 y Fm(1)p Fs(=)p Fm(2)1436 508 y FG(=)50 b Fp(j)1587 487 y Fo(~)1574 508 y(B)t Fp(j)1664 474 y Fm(1)p Fs(=)p Fm(2)1769 508 y Ft(P)1821 474 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1821 529 y Fs(s)1976 441 y Fr(\002)2038 508 y FG(1)18 b(+)g(\()p Fo(t)h Fp(\000)f Fo(s)p FG(\))p Fo(R)2507 441 y Fr(\003)2542 458 y Fi(\000)p Fm(1)2631 508 y Ft(P)2683 474 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)2683 529 y Fs(s)2838 508 y Fp(j)2874 487 y Fo(~)2861 508 y(B)t Fp(j)2951 474 y Fm(1)p Fs(=)p Fm(2)3056 508 y FG(;)599 721 y(note)28 b(that)g(n)n(ul)f Ft(P)1156 733 y Fs(t)1208 721 y FG(=)22 b(0)27 b(for)g(an)n(y)g Fo(t)p FG(.)37 b(F)-7 b(or)27 b Fo(f)32 b Fp(2)23 b(R)p FG(\()p Ft(P)2192 678 y Fm(1)p Fs(=)p Fm(2)2192 731 y Fs(s)2296 721 y FG(\))1048 920 y Fo(s)28 b Fp(k)f(j)1220 899 y Fo(~)1207 920 y(B)t Fp(j)1297 885 y Fm(1)p Fs(=)p Fm(2)1401 920 y Ft(P)1453 885 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1453 940 y Fs(s)1608 920 y Fo(f)37 b Fp(k)1728 885 y Fm(2)1842 920 y FG(=)77 b Fo(s)2051 852 y Fr(\000)2116 920 y Fp(j)2152 899 y Fo(~)2139 920 y(B)5 b Fp(j)p Ft(P)2282 885 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)2282 940 y Fs(s)2437 920 y Fo(f)t(;)14 b Ft(P)2571 885 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)2571 940 y Fs(s)2725 920 y Fo(f)2803 852 y Fr(\001)1842 1060 y Fp(\024)1984 993 y 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Fm(1)p Fs(=)p Fm(2)1534 1756 y FG(=)1679 1653 y Fi(1)1652 1677 y Fr(X)1649 1853 y Fs(n)p Fm(=1)1802 1756 y FG(\()p Fo(t)f Fp(\000)f Fo(s)p FG(\))2037 1722 y Fs(n)2082 1756 y Fo(K)2153 1768 y Fs(n)2272 1756 y FG(+)45 b Fp(j)2418 1735 y Fo(~)2405 1756 y(B)5 b Fp(j)2496 1722 y Fm(1)p Fs(=)p Fm(2)2600 1756 y Ft(P)2652 1722 y Fi(\000)p Fm(1)2652 1777 y Fs(s)2740 1756 y Fp(j)2776 1735 y Fo(~)2763 1756 y(B)t Fp(j)2853 1722 y Fm(1)p Fs(=)p Fm(2)2958 1756 y Fo(;)599 2017 y FG(where)1248 2166 y Fo(K)1319 2178 y Fs(n)1415 2166 y FG(=)50 b Fp(j)1566 2145 y Fo(~)1553 2166 y(B)t Fp(j)1643 2132 y Fm(1)p Fs(=)p Fm(2)1748 2166 y Ft(P)1800 2132 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1800 2187 y Fs(s)1955 2166 y FG(\()p Fp(\000)p Fo(R)q FG(\))2148 2132 y Fs(n)2193 2166 y Ft(P)2245 2132 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)2245 2187 y Fs(s)2400 2166 y Fp(j)2436 2145 y Fo(~)2423 2166 y(B)t Fp(j)2513 2132 y Fm(1)p Fs(=)p Fm(2)2617 2166 y Fo(;)599 2307 y FG(and)28 b(the)f(series)g(lies)g(in)g Fp(L)p FG(\()p Fp(H)q 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2993 y Fs(t)1409 2981 y FG(=)50 b(n)n(ul)28 b Fo(F)1718 2993 y Fs(t)1747 2981 y Fo(;)180 b(d)1993 2993 y Fm(min)2158 2981 y FG(=)78 b(min)2274 3039 y Fs(t)p Fi(2)p Fm([)p Fs(a;b)p Fm(])2508 2981 y Fo(d)2551 2993 y Fs(t)2580 2981 y Fo(:)682 3216 y Fh(Lemma)30 b(4.2.)69 b Fg(The)31 b(map)f Fo(t)23 b Fp(7!)g Fo(d)1737 3228 y Fs(t)1826 3216 y Fg(is)30 b(upp)l(er)g(semi-c)l(ontinuous.)682 3473 y(Pr)l(o)l(of.)77 b FG(The)19 b(k)n(ernel)f(of)g Fo(F)1512 3485 y Fs(t)1560 3473 y FG(is)h(\014nite-dimensional,)h(and)e (w)n(e)g(ha)n(v)n(e)g(the)h(orthogonal)599 3573 y(decomp)r(osition)1468 3722 y Fp(H)52 b FG(=)e(k)n(er)13 b Fo(F)1883 3734 y Fs(t)1959 3722 y Fp(\010)45 b FG(\(k)n(er)13 b Fo(F)2279 3734 y Fs(t)2309 3722 y FG(\))2341 3688 y Fi(?)2397 3722 y Fo(:)599 3864 y FG(With)29 b(resp)r(ect)e(to)h(this)g(decomp)r (osition,)f(w)n(e)g(can)g(represen)n(t)g Fo(F)2618 3876 y Fs(t)2675 3864 y FG(as)1607 4098 y Fo(F)1660 4110 y Fs(t)1740 4098 y FG(=)1856 3981 y Fr(\022)1942 4047 y FG(0)77 b(0)1942 4147 y(0)g Fo(D)2130 4159 y Fs(t)2184 3981 y Fr(\023)2259 4098 y Fo(;)599 4344 y FG(where)28 b Fo(D)909 4356 y Fs(t)961 4344 y FG(:)23 b(\(k)n(er)13 b Fo(F)1217 4356 y Fs(t)1246 4344 y FG(\))1278 4314 y Fi(?)1358 4344 y Fp(\000)-15 b(!)24 b FG(\(k)n(er)13 b Fo(F)1725 4356 y Fs(t)1754 4344 y FG(\))1786 4314 y Fi(?)1870 4344 y FG(and)1623 4533 y Fp(k)p Fo(D)1734 4545 y Fs(t)1762 4533 y Fp(k)51 b(\025)f Fo(c)2006 4545 y Fs(t)2086 4533 y Fo(>)g FG(0)p Fo(:)876 b FG(\(34\))p eop %%Page: 13 13 13 12 bop 1120 75 a FH(ON)24 b(THE)f(ZER)n(O)h(MODES)f(OF)g(P)-6 b(A)n(ULI)24 b(OPERA)-6 b(TORS)438 b FG(13)599 320 y(W)-7 b(e)24 b(are)e(required)g(to)g(pro)n(v)n(e)g(that,)i(for)e(an)n(y)g Fo(t)p FG(,)i(there)f(exists)g(a)f(neigh)n(b)r(ourho)r(o)r(d)g Fo(N)9 b FG(\()p Fo(t)p FG(\))599 419 y(suc)n(h)28 b(that)g Fo(d)1010 431 y Fs(t)1035 415 y Fe(0)1085 419 y Fp(\024)22 b Fo(d)1215 431 y Fs(t)1272 419 y FG(for)27 b(all)h Fo(t)1545 389 y Fi(0)1591 419 y Fp(2)23 b Fo(N)9 b FG(\()p Fo(t)p FG(\).)38 b(W)-7 b(e)28 b(can)f(write)1463 652 y Fo(F)1516 664 y Fs(t)1541 648 y Fe(0)1619 652 y FG(=)1734 535 y Fr(\022)1820 601 y Fo(L)1877 613 y Fs(t)1902 597 y Fe(0)2030 601 y Fo(M)2111 613 y Fs(t)2136 597 y Fe(0)1820 701 y Fo(M)1910 671 y Fi(\003)1901 723 y Fs(t)1926 707 y Fe(0)2030 701 y Fo(D)2099 713 y Fs(t)2146 701 y FG(+)18 b Fo(C)2288 713 y Fs(t)2313 696 y Fe(0)2365 535 y Fr(\023)599 892 y FG(where)31 b Fo(L)900 904 y Fs(t)925 888 y Fe(0)981 892 y FG(:)f(k)n(er)13 b Fo(F)1212 904 y Fs(t)1271 892 y Fp(\000)-14 b(!)30 b FG(k)n(er)12 b Fo(F)1612 904 y Fs(t)1642 892 y FG(,)96 b Fo(C)1820 904 y Fs(t)1845 888 y Fe(0)1902 892 y FG(:)30 b(\(k)n(er)13 b Fo(F)2165 904 y Fs(t)2194 892 y FG(\))2226 862 y Fi(?)2312 892 y Fp(\000)-14 b(!)30 b FG(\(k)n(er)13 b Fo(F)2686 904 y Fs(t)2715 892 y FG(\))2747 862 y Fi(?)2835 892 y FG(are)31 b(b)r(ounded)599 992 y(self-adjoin)n(t)24 b(op)r(erators)e(and)h Fo(M)1625 1004 y Fs(t)1650 988 y Fe(0)1700 992 y FG(:)g(\(k)n(er)13 b Fo(F)1956 1004 y Fs(t)1986 992 y FG(\))2018 962 y Fi(?)2097 992 y Fp(\000)-14 b(!)23 b FG(k)n(er)13 b Fo(F)2432 1004 y Fs(t)2485 992 y FG(is)24 b(b)r(ounded.)36 b(As)24 b Fo(t)3084 962 y Fi(0)3130 992 y Fp(!)f Fo(t)p FG(,)599 1092 y(w)n(e)32 b(kno)n(w)f(from)g(Lemma)h(4.1)e(that)i Fo(L)1826 1104 y Fs(t)1851 1087 y Fe(0)1878 1092 y FG(,)h Fo(M)2015 1104 y Fs(t)2040 1087 y Fe(0)2098 1092 y FG(and)e Fo(C)2322 1104 y Fs(t)2347 1087 y Fe(0)2406 1092 y Fp(!)f FG(0)h(in)h(norm.)49 b(Cho)r(ose)31 b(a)599 1191 y(neigh)n(b)r(ourho)r (o)r(d)25 b Fo(N)9 b FG(\()p Fo(t)p FG(\))25 b(of)h Fo(t)f FG(suc)n(h)g(that)h Fp(k)p Fo(C)1972 1203 y Fs(t)1997 1187 y Fe(0)2023 1191 y Fp(k)d Fo(<)f(c)2211 1203 y Fs(t)2266 1191 y FG(for)j Fo(t)2421 1161 y Fi(0)2467 1191 y Fp(2)e Fo(N)9 b FG(\()p Fo(t)p FG(\),)27 b(where)d Fo(c)3038 1203 y Fs(t)3093 1191 y FG(is)h(the)599 1291 y(constan)n(t)e(in)g (\(34\).)35 b(Then)24 b Fo(D)1510 1303 y Fs(t)1549 1291 y FG(+)10 b Fo(C)1683 1303 y Fs(t)1708 1287 y Fe(0)1757 1291 y FG(is)23 b(in)n(v)n(ertible)g(for)f(all)h Fo(t)2462 1261 y Fi(0)2509 1291 y Fp(2)g Fo(N)9 b FG(\()p Fo(t)p FG(\).)36 b(The)23 b(op)r(erator)1320 1523 y Fp(A)51 b FG(=)1552 1406 y Fr(\022)1638 1473 y Ft(I)102 b Fp(\000)p Fo(M)1931 1485 y Fs(t)1956 1468 y Fe(0)1982 1473 y FG(\()p Fo(D)2083 1485 y Fs(t)2130 1473 y FG(+)18 b Fo(C)2272 1485 y Fs(t)2297 1468 y Fe(0)2325 1473 y FG(\))2357 1442 y Fi(\000)p Fm(1)1638 1572 y FG(0)302 b(\()p Fo(D)2083 1584 y Fs(t)2130 1572 y FG(+)18 b Fo(C)2272 1584 y Fs(t)2297 1568 y Fe(0)2325 1572 y FG(\))2357 1542 y Fi(\000)p Fm(1)2471 1406 y Fr(\023)2546 1523 y Fo(;)599 1751 y FG(where)28 b Ft(I)14 b FG(is)28 b(the)g(iden)n(tit)n(y)-7 b(,)28 b(is)f(a)g(b)r(ounded)h(injection)g(on)g Fp(H)q FG(,)f(and)h(w)n(e)f (ha)n(v)n(e)1096 1979 y Fp(A\001)14 b Fo(F)1252 1991 y Fs(t)1277 1975 y Fe(0)1355 1979 y FG(=)1471 1862 y Fr(\022)1557 1928 y Fo(L)1614 1940 y Fs(t)1639 1924 y Fe(0)1683 1928 y Fp(\000)k Fo(M)1847 1940 y Fs(t)1872 1924 y Fe(0)1899 1928 y FG(\()p Fo(D)2000 1940 y Fs(t)2047 1928 y FG(+)g Fo(C)2189 1940 y Fs(t)2214 1924 y Fe(0)2241 1928 y FG(\))2273 1898 y Fi(\000)p Fm(1)2363 1928 y Fo(M)2453 1898 y Fi(\003)2444 1951 y Fs(t)2469 1934 y Fe(0)2628 1928 y FG(0)1557 2028 y(\()p Fo(D)1658 2040 y Fs(t)1705 2028 y FG(+)h Fo(C)1848 2040 y Fs(t)1873 2024 y Fe(0)1900 2028 y FG(\))1932 1998 y Fi(\000)p Fm(1)2021 2028 y Fo(M)2111 1998 y Fi(\003)2102 2051 y Fs(t)2127 2034 y Fe(0)2637 2028 y Ft(I)2694 1862 y Fr(\023)2769 1979 y Fo(:)350 b FG(\(35\))599 2207 y(It)28 b(follo)n(ws)f(that)1132 2389 y Fo(d)1175 2401 y Fs(t)1200 2385 y Fe(0)1305 2389 y FG(=)77 b(n)n(ul\()p Fp(A\001)14 b Fo(F)1748 2401 y Fs(t)1773 2385 y Fe(0)1800 2389 y FG(\))1305 2514 y(=)77 b(dim)1585 2446 y Fr(\010)1661 2514 y FG(k)n(er)o([)p Fo(L)1852 2526 y Fs(t)1877 2510 y Fe(0)1922 2514 y Fp(\000)18 b Fo(M)2086 2526 y Fs(t)2111 2510 y Fe(0)2137 2514 y FG(\()p Fo(D)2238 2526 y Fs(t)2286 2514 y FG(+)g Fo(C)2428 2526 y Fs(t)2453 2510 y Fe(0)2480 2514 y FG(\))2512 2479 y Fi(\000)p Fm(1)2601 2514 y Fo(M)2691 2479 y Fi(\003)2682 2534 y Fs(t)2707 2518 y Fe(0)2733 2514 y FG(])1640 2648 y Fp(\\)42 b FG(k)n(er)o([\()p Fo(D)1972 2660 y Fs(t)2020 2648 y FG(+)18 b Fo(C)2162 2660 y Fs(t)2187 2644 y Fe(0)2214 2648 y FG(\))2246 2613 y Fi(\000)p Fm(1)2335 2648 y Fo(M)2425 2613 y Fi(\003)2416 2668 y Fs(t)2441 2652 y Fe(0)2468 2648 y FG(])2518 2580 y Fr(\011)1305 2782 y FG(=)77 b(n)n(ul)27 b([)p Fo(L)1667 2747 y Fm(2)1667 2802 y Fs(t)1692 2786 y Fe(0)1737 2782 y FG(+)18 b Fo(M)1901 2794 y Fs(t)1926 2778 y Fe(0)1952 2782 y Fo(M)2042 2747 y Fi(\003)2033 2802 y Fs(t)2058 2786 y Fe(0)2085 2782 y FG(])1034 b(\(36\))1305 2906 y Fp(\024)77 b Fo(d)1490 2918 y Fs(t)1519 2906 y Fo(;)627 3114 y FG(whence)28 b(the)g(lemma.)p 1407 3131 25 75 v 682 3264 a Fh(Theorem)j(4.1.)105 b Fg(F)-6 b(or)34 b(any)f Fo(c)c Fp(2)g FG(\(0)p Fo(;)14 b Fp(1)p FG(\))p Fg(,)34 b Fo(d)2108 3276 y Fs(t)2200 3264 y FG(=)61 b(0)p Fg(,)34 b(and)g(henc)l(e)f FG(n)n(ul)g Ft(P)3022 3276 y Fs(tA)3129 3264 y FG(=)28 b(0)p Fg(,)599 3363 y(on)i FG([0)p Fo(;)14 b(c)p FG(])59 b Fg(exc)l(ept)30 b(at)f(a)i(\014nite)e (numb)l(er)g(of)h(p)l(oints.)682 3625 y(Pr)l(o)l(of.)101 b FG(W)-7 b(e)35 b(already)e(kno)n(w)h(from)g(Theorem)g(3.1)f(and)i (\(32\))f(that)h(n)n(ul)f Ft(P)3112 3637 y Fs(tA)3225 3625 y Fp(\024)599 3724 y Fo(d)642 3736 y Fs(t)728 3724 y FG(=)56 b(0)29 b(in)h(\(0)p Fo(;)14 b(t)1160 3736 y Fm(0)1197 3724 y FG(\).)43 b(It)31 b(is)e(therefore)g(su\016cien)n(t)h (to)f(pro)n(v)n(e)f(the)i(theorem)f(for)g([)p Fo(a;)14 b(c)p FG(],)599 3824 y(where)28 b(0)22 b Fo(<)h(a)g(<)f(t)1176 3836 y Fm(0)1214 3824 y FG(.)36 b(De\014ne)906 4007 y Fo(J)85 b FG(=)1179 3939 y Fr(\010)1255 4007 y Fo(t)23 b Fp(2)h FG([)p Fo(a;)14 b(c)p FG(])50 b(:)h(there)27 b(exists)h(a)f(neigh)n(b)r(ourho)r(o)r(d)54 b Fo(N)2847 4019 y Fs(t)1262 4151 y FG(of)h Fo(t;)70 b FG(suc)n(h)27 b(that)55 b Fo(d)1944 4163 y Fs(t)1969 4147 y Fe(0)2047 4151 y FG(=)50 b(0)55 b(in)h Fo(N)2460 4092 y Fe(0)2451 4172 y Fs(t)2509 4151 y FG(=)23 b Fo(N)2664 4163 y Fs(t)2711 4151 y Fp(n)18 b(f)p Fo(t)p Fp(g)2912 4084 y Fr(\011)2960 4151 y Fo(:)632 4334 y FG(The)33 b(theorem)f(will)h(follo)n(w)f(if)h(w) n(e)g(pro)n(v)n(e)e(that)i Fo(J)40 b FG(=)31 b([)p Fo(a;)14 b(c)p FG(],)34 b(in)f(view)f(of)h(the)g(com-)599 4433 y(pactness)h(of)g([)p Fo(a;)14 b(c)p FG(].)56 b(W)-7 b(e)34 b(shall)g(pro)n(v)n(e)e(that)i Fo(J)43 b FG(is)33 b(b)r(oth)i(op)r(en)f(and)g(closed.)55 b(Since)599 4533 y Fo(a)23 b(<)g(t)784 4545 y Fm(0)821 4533 y FG(,)28 b(w)n(e)f(kno)n(w)g(that)h Fo(J)j Fp(6)p FG(=)23 b Fp(;)p FG(.)p eop %%Page: 14 14 14 13 bop 599 75 a FG(14)711 b FH(A.)23 b(BALINSKY)g(AND)g(W.D.)g(EV)-8 b(ANS)682 320 y FG(It)27 b(is)f(clear)f(from)h(Lemma)g(4.2)f(that)h Fo(J)35 b FG(is)26 b(op)r(en.)36 b(T)-7 b(o)26 b(pro)n(v)n(e)e(that)j (it)f(is)g(closed,)g(let)599 419 y Fp(f)p Fo(t)671 431 y Fs(k)712 419 y Fp(g)31 b FG(b)r(e)h(a)f(sequence)g(in)g Fo(J)40 b FG(and)31 b(lim)14 b Fo(t)1833 431 y Fs(k)1903 419 y FG(=)29 b Fo(t)p FG(;)34 b(w)n(e)d(ma)n(y)g(assume)f(that)i Fo(d)2911 431 y Fs(t)2936 440 y Ff(k)3006 419 y FG(=)d(0.)48 b(In)599 519 y(the)28 b(notation)g(of)f(the)h(pro)r(of)f(of)h(Lemma)f (4.2,)g(set)1554 725 y Fo(Q)1620 737 y Fs(t)1645 721 y Fe(0)1722 725 y FG(=)50 b Fo(L)1894 691 y Fm(2)1894 745 y Fs(t)1919 729 y Fe(0)1964 725 y FG(+)18 b Fo(M)2128 737 y Fs(t)2153 721 y Fe(0)2179 725 y Fo(M)2269 691 y Fi(\003)2260 745 y Fs(t)2285 729 y Fe(0)2312 725 y Fo(:)599 931 y FG(Then,)28 b(from)g(\(36\))1464 1137 y Fo(d)1507 1149 y Fs(t)1614 1137 y FG(=)77 b(rank)27 b Fo(Q)2014 1149 y Fs(t)2039 1133 y Fe(0)2084 1137 y FG(+)18 b(n)n(ul)27 b Fo(Q)2373 1149 y Fs(t)2398 1133 y Fe(0)3142 1137 y FG(\(37\))1614 1261 y(=)77 b(rank)27 b Fo(Q)2014 1273 y Fs(t)2039 1257 y Fe(0)2084 1261 y FG(+)18 b Fo(d)2210 1273 y Fs(t)2235 1257 y Fe(0)3142 1261 y FG(\(38\))627 1467 y(and)1652 1673 y Fo(d)1695 1685 y Fs(t)1775 1673 y FG(=)51 b(rank)26 b Fo(Q)2148 1685 y Fs(t)2173 1694 y Ff(k)2214 1673 y Fo(:)905 b FG(\(39\))599 1879 y(If)25 b(w)n(e)f(can)f(pro)n(v)n(e)g(that)h(rank)g Fo(Q)1598 1891 y Fs(t)1623 1875 y Fe(0)1672 1879 y FG(=)f Fo(d)1803 1891 y Fs(t)1856 1879 y FG(for)h(all)g Fo(t)2122 1849 y Fi(0)2169 1879 y FG(in)g(some)g(deleted)g(neigh)n(b)r(ourho)r(o)r(d) 599 1979 y Fo(N)675 1949 y Fi(0)726 1979 y FG(of)k Fo(t)p FG(,)f(it)h(will)g(follo)n(w)f(from)g(\(38\))h(that)g Fo(t)23 b Fp(2)g Fo(J)8 b FG(,)28 b(as)f(required.)682 2078 y(Since)22 b(rank)f Fo(Q)1145 2090 y Fs(t)1170 2099 y Ff(k)1233 2078 y FG(=)i Fo(d)1364 2090 y Fs(t)1393 2078 y FG(,)g(then)f(an)n(y)f(minor)g(Min)2153 2090 y Fs(t)2178 2074 y Fe(0)2226 2078 y FG(of)g Fo(Q)2380 2090 y Fs(t)2405 2074 y Fe(0)2453 2078 y FG(of)h(order)e(greater)g(than)i Fo(d)3260 2090 y Fs(t)599 2178 y FG(m)n(ust)j(v)-5 b(anish)23 b(when)h Fo(t)1297 2148 y Fi(0)1344 2178 y FG(=)e Fo(t)1461 2190 y Fs(k)1502 2178 y FG(.)36 b(Hence,)25 b(since)f Fo(t)2058 2148 y Fi(0)2104 2178 y Fp(7!)f FG(Min)2355 2190 y Fs(t)2380 2174 y Fe(0)2431 2178 y FG(is)h(analytic,)g(Min)2992 2190 y Fs(t)3017 2174 y Fe(0)3067 2178 y FG(=)f(0)g(in)599 2278 y(some)f(neigh)n(b)r(ourho)r(o)r(d)g Fo(N)31 b FG(of)23 b Fo(t)p FG(,)g(and)g(so)e(rank)h Fo(Q)2135 2290 y Fs(t)2160 2274 y Fe(0)2209 2278 y Fp(\024)h Fo(d)2340 2290 y Fs(t)2392 2278 y FG(in)g Fo(N)9 b FG(.)35 b(By)22 b(\(39\))g(there)h(exist)599 2377 y(a)31 b(minor)f(of)g Fo(Q)1078 2389 y Fs(t)1103 2373 y Fe(0)1160 2377 y FG(of)g(order)g Fo(d)1521 2389 y Fs(t)1581 2377 y FG(whic)n(h)g(do)r(es)g(not)h(v)-5 b(anish)30 b(on)h(some)e(subsequence)h(of)599 2477 y Fp(f)p Fo(t)671 2489 y Fs(k)712 2477 y Fp(g)p FG(,)20 b(and)f(hence)g(can)g(ha)n(v)n(e)e(a)i(zero)f(only)h(at)f Fo(t)2022 2447 y Fi(0)2069 2477 y FG(=)k Fo(t)e FG(within)f(some)g (neigh)n(b)r(ourho)r(o)r(d)f Fo(N)599 2577 y FG(of)24 b Fo(t)p FG(.)35 b(Consequen)n(tly)-7 b(,)24 b Fo(d)1349 2589 y Fs(t)1401 2577 y Fp(\025)f FG(rank)f Fo(Q)1742 2589 y Fs(t)1767 2572 y Fe(0)1817 2577 y Fp(\025)g Fo(d)1947 2589 y Fs(t)2000 2577 y FG(for)h Fo(t)2153 2546 y Fi(0)2199 2577 y Fp(2)h Fo(N)2354 2546 y Fi(0)2400 2577 y FG(=)e Fo(N)d Fp(n)10 b(f)p Fo(t)p Fp(g)p FG(,)23 b(and,)h(the)g(theo-)599 2702 y(rem)k(is)f(pro)n(v)n(ed.)p 1198 2718 25 75 v 682 2881 a Fh(Theorem)k(4.2.)109 b Fg(The)35 b(set)1623 2814 y Fr(\010)1719 2860 y Fo(~)1706 2881 y(B)h FG(:)66 b(n)n(ul)35 b Ft(P)2094 2893 y Fs(A)2179 2881 y FG(=)c(0)p Fo(;)48 b FG(curl)2577 2860 y Fo(~)2561 2881 y(A)32 b FG(=)2765 2860 y Fo(~)2752 2881 y(B)73 b Fg(and)d Fp(j)3125 2860 y Fo(~)3112 2881 y(B)t Fp(j)32 b(2)599 2992 y Fo(L)656 2962 y Fm(3)p Fs(=)p Fm(2)760 2992 y FG(\()p Ft(R)847 2962 y Fm(3)890 2992 y FG(\))952 2925 y Fr(\011)1030 2992 y Fg(c)l(ontains)e(an)g(op)l(en)g(dense)g(subset)f(of)i FG([)p Fo(L)2326 2962 y Fm(3)p Fs(=)p Fm(2)2430 2992 y FG(\()p Ft(R)2516 2962 y Fm(3)2559 2992 y FG(\)])2614 2962 y Fm(3)2652 2992 y Fg(.)682 3255 y(Pr)l(o)l(of.)92 b FG(Let)32 b Fp(S)38 b FG(in)32 b(\(27\))f(b)r(e)h(denoted)g(b)n(y)f Fp(S)2116 3267 y Fs(B)2205 3255 y FG(and)h(set)f Fo(F)2557 3267 y Fs(B)2645 3255 y FG(=)e(1)21 b Fp(\000)f(S)2937 3267 y Fs(B)2995 3255 y Fp(S)3051 3225 y Fi(\003)3045 3278 y Fs(B)3102 3255 y FG(.)49 b(W)-7 b(e)599 3355 y(shall)28 b(pro)n(v)n(e)e(that)1194 3493 y Fr(\010)1283 3540 y Fo(~)1270 3561 y(B)h FG(:)51 b(n)n(ul)27 b Fo(F)1627 3573 y Fs(B)1708 3561 y FG(=)c(0)55 b(and)g Fp(j)2118 3540 y Fo(~)2105 3561 y(B)t Fp(j)23 b(2)g Fo(L)2353 3526 y Fm(3)p Fs(=)p Fm(2)2457 3561 y FG(\()p Ft(R)2543 3526 y Fm(3)2587 3561 y FG(\))2647 3493 y Fr(\011)3142 3561 y FG(\(40\))599 3778 y(is)30 b(an)f(op)r(en)h(subset)f(of)h([)p Fo(L)1438 3748 y Fm(3)p Fs(=)p Fm(2)1542 3778 y FG(\()p Ft(R)1628 3748 y Fm(3)1671 3778 y FG(\)])1726 3748 y Fm(3)1764 3778 y FG(;)g(the)g(theorem)f(will)h(then)g(follo)n(w)f(from) g(Theo-)599 3878 y(rem)f(3.1)e(since)i(the)g(densit)n(y)f(of)h(\(40\))f (is)h(a)f(consequence)f(of)i(Theorem)f(4.1.)682 3987 y(F)-7 b(or)19 b Fo(")k(>)f FG(0,)f(let)1182 3966 y Fo(~)1169 3987 y(B)t(;)1292 3966 y(~)1278 3987 y(B)1341 3999 y Fm(0)1398 3987 y FG(b)r(e)e(magnetic)g(\014elds)g(whic)n(h)g(satisfy)f Fp(k)2587 3966 y Fo(~)2574 3987 y(B)5 b Fp(\000)2721 3966 y Fo(~)2708 3987 y(B)2771 3999 y Fm(0)2808 3987 y Fp(k)2850 4006 y Fs(L)2896 3990 y Fj(3)p Ff(=)p Fj(2)2986 4006 y Fm(\()p Fk(R)3059 3990 y Fj(3)3087 4006 y Fm(\))3140 3987 y Fo(<)23 b(")p FG(.)599 4109 y(Then,)37 b(if)947 4088 y Fo(~)931 4109 y(A;)1067 4088 y(~)1051 4109 y(A)1113 4121 y Fm(0)1185 4109 y FG(are)d(the)h(asso)r(ciated)e(v)n(ector)g(p)r (oten)n(tials)h(giv)n(en)g(in)h(Lemma)f(2.2,)599 4209 y Fp(k)656 4188 y Fo(~)641 4209 y(A)6 b Fp(\000)794 4188 y Fo(~)779 4209 y(A)841 4221 y Fm(0)878 4209 y Fp(k)920 4224 y Fs(L)966 4207 y Fj(3)997 4224 y Fm(\()p Fk(R)1070 4207 y Fj(3)1098 4224 y Fm(\))1151 4209 y Fo(<)23 b(c)e(")f FG(for)h(some)f Fo(c)j(>)g FG(0.)34 b(It)22 b(follo)n(ws)e(as)g(in)i (the)f(pro)r(of)f(of)h(Lemma)g(4.1)599 4327 y(that,)28 b(with)g Ft(P)22 b FG(=)h Ft(P)1205 4339 y Fs(A)1276 4327 y FG(+)18 b Fp(j)1395 4306 y Fo(~)1382 4327 y(B)t Fp(j)28 b FG(and)g Ft(P)1714 4339 y Fm(0)1773 4327 y FG(=)22 b Ft(P)1912 4339 y Fs(A)1962 4347 y Fj(0)2016 4327 y FG(+)c Fp(j)2135 4306 y Fo(~)2122 4327 y(B)2185 4339 y Fm(0)2222 4327 y Fp(j)p FG(,)1700 4533 y Ft(P)f Fp(\000)h Ft(P)1904 4545 y Fm(0)1991 4533 y FG(=)50 b Ft(V)p Fo(;)p eop %%Page: 15 15 15 14 bop 1120 75 a FH(ON)24 b(THE)f(ZER)n(O)h(MODES)f(OF)g(P)-6 b(A)n(ULI)24 b(OPERA)-6 b(TORS)438 b FG(15)599 320 y(where,)28 b(for)f Fo(')c Fp(2)g FG([)p Fo(C)1233 290 y Fi(1)1227 340 y Fm(0)1304 320 y FG(\()p Ft(R)1390 290 y Fm(3)1434 320 y FG(\)])1489 290 y Fm(2)1526 320 y FG(,)978 444 y Fr(\000)1043 512 y Ft(V)p Fo(';)14 b(')1277 444 y Fr(\001)1392 512 y FG(=)1534 444 y Fr(\000)28 b(\002)1647 491 y Fo(~)1634 512 y(D)r Fp(\001)14 b FG(\()1790 491 y Fo(~)1774 512 y(A)19 b Fp(\000)1954 491 y Fo(~)1938 512 y(A)2000 524 y Fm(0)2038 512 y FG(\))g(+)f(\()2219 491 y Fo(~)2204 512 y(A)h Fp(\000)2383 491 y Fo(~)2368 512 y(A)2430 524 y Fm(0)2467 512 y FG(\))p Fp(\001)2549 491 y Fo(~)2536 512 y(D)2607 444 y Fr(\003)2642 512 y Fo(';)14 b(')2815 444 y Fr(\001)1562 654 y FG(+)1627 587 y Fr(\000)27 b(\002)1754 654 y Fp(j)1793 633 y Fo(~)1777 654 y(A)q Fp(j)1863 620 y Fm(2)1918 654 y Fp(\000)18 b(j)2040 633 y Fo(~)2024 654 y(A)2086 666 y Fm(0)2124 654 y Fp(j)2147 620 y Fm(2)2212 587 y Fr(\003)2247 654 y Fo(';)c(')2420 587 y Fr(\001)1562 797 y FG(+)1627 729 y Fr(\000)27 b(\002)1752 797 y Fo(~)-40 b(\033)t Fp(\001)14 b FG(\()1887 776 y Fo(~)1874 797 y(B)23 b Fp(\000)2056 776 y Fo(~)2043 797 y(B)2106 809 y Fm(0)2143 797 y FG(\))c(+)f Fp(j)2313 776 y Fo(~)2300 797 y(B)t Fp(j)g(\000)g(j)2527 776 y Fo(~)2514 797 y(B)2577 809 y Fm(0)2615 797 y Fp(j)2665 729 y Fr(\003)2700 797 y Fo(';)c(')2873 729 y Fr(\001)627 979 y FG(and)777 1091 y Fr(\014)777 1140 y(\014)832 1094 y(\000)898 1161 y Ft(V)p Fo(';)g(')1131 1094 y Fr(\001)1197 1091 y(\014)1197 1140 y(\014)1302 1161 y Fp(\024)77 b Fo(c)1507 1094 y Fr(\002)1542 1161 y Fp(k)1599 1140 y Fo(~)1584 1161 y(A)18 b Fp(\000)1763 1140 y Fo(~)1747 1161 y(A)1809 1173 y Fm(0)1847 1161 y Fp(k)1889 1176 y Fs(L)1935 1160 y Fj(3)1966 1176 y Fm(\()p Fk(R)2039 1160 y Fj(3)2067 1176 y Fm(\))2115 1161 y FG(+)g Fp(k)2253 1140 y Fo(~)2240 1161 y(B)k Fp(\000)2422 1140 y Fo(~)2408 1161 y(B)2471 1173 y Fm(0)2508 1161 y Fp(k)2550 1180 y Fs(L)2596 1164 y Fj(3)p Ff(=)p Fj(2)2687 1180 y Fm(\()p Fk(R)2760 1164 y Fj(3)2787 1180 y Fm(\))2817 1094 y Fr(\003)2879 1161 y Fp(k)p Fo(')p Fp(k)3017 1127 y Fm(2)3017 1187 y Fs(H)3075 1167 y Fj(1)3071 1205 y(0)1302 1310 y Fp(\024)77 b Fo(c)1480 1276 y Fi(0)1531 1310 y Fo(")27 b Fp(k)p Fo(')p Fp(k)1735 1276 y Fm(2)1735 1336 y Fk(H)1774 1316 y Fj(1)1774 1355 y Ff(B)1817 1367 y Fj(0)636 1516 y FG(on)37 b(using)f(Lemmas)h(2.2)f(and)h(2.3)f(and)g (H\177)-42 b(older's)36 b(inequalit)n(y)-7 b(.)65 b(Moreo)n(v)n(er,)36 b Ft(U)42 b FG(=)599 1628 y Ft(P)651 1584 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)651 1650 y(0)834 1628 y Ft(V)30 b(P)974 1584 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)974 1650 y(0)1156 1628 y FG(satis\014es)1474 1739 y Fr(\014)1474 1789 y(\014)1529 1742 y(\000)1595 1810 y Ft(U)p Fo(';)14 b(')1828 1742 y Fr(\001)1894 1739 y(\014)1894 1789 y(\014)1972 1810 y Fp(\024)50 b Fo(c)2123 1775 y Fi(0)2174 1810 y Fo(")28 b Fp(k)p Fo(')p Fp(k)2379 1775 y Fm(2)599 1992 y FG(and)858 2174 y Fp(j)894 2153 y Fo(~)881 2174 y(B)t Fp(j)971 2140 y Fm(1)p Fs(=)p Fm(2)1103 2174 y Ft(P)1155 2140 y Fi(\000)p Fm(1)1271 2174 y Fp(j)1307 2153 y Fo(~)1294 2174 y(B)t Fp(j)1384 2140 y Fm(1)p Fs(=)p Fm(2)1539 2174 y FG(=)50 b Fp(j)1690 2153 y Fo(~)1677 2174 y(B)5 b Fp(j)1768 2140 y Fm(1)p Fs(=)p Fm(2)1900 2174 y Ft(P)1952 2131 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1952 2196 y(0)2134 2107 y Fr(\002)2197 2174 y FG(1)18 b(+)g Ft(U)2427 2107 y Fr(\003)2462 2124 y Fi(\000)p Fm(1)2579 2174 y Ft(P)2631 2131 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)2631 2196 y(0)2813 2174 y Fp(j)2849 2153 y Fo(~)2836 2174 y(B)t Fp(j)2926 2140 y Fm(1)p Fs(=)p Fm(2)599 2356 y FG(for)28 b Fo(")f FG(su\016cien)n(tly)g(small.)37 b(Also,)27 b(as)g Fo(")c Fp(!)g FG(0,)1259 2550 y Fp(j)1295 2529 y Fo(~)1282 2550 y(B)t Fp(j)1372 2516 y Fm(1)p Fs(=)p Fm(2)1504 2550 y Ft(P)1556 2507 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)1556 2572 y(0)1757 2550 y Fp(\000)46 b(j)1904 2529 y Fo(~)1891 2550 y(B)1954 2562 y Fm(0)1991 2550 y Fp(j)2014 2516 y Fm(1)p Fs(=)p Fm(2)2146 2550 y Ft(P)2198 2507 y Fi(\000)p Fm(1)p Fs(=)p Fm(2)2198 2572 y(0)2404 2550 y Fp(\000)-14 b(!)50 b FG(0)599 2732 y(in)25 b Fp(L)p FG(\()p Fp(H)q FG(\).)37 b(It)25 b(follo)n(ws)e(that,)j(as)e Fo(")f Fp(!)g FG(0,)49 b Fo(F)1936 2744 y Fs(B)2017 2732 y Fp(!)23 b Fo(F)2176 2744 y Fs(B)2226 2752 y Fj(0)2288 2732 y FG(in)h Fp(L)p FG(\()p Fp(H)q FG(\),)i(and)f(that,)g(as)f(in)h (the)599 2832 y(pro)r(of)i(of)h(Lemma)f(4.2,)g(the)h(map)1653 2993 y Fo(~)1640 3014 y(B)82 b Fp(7\000)-14 b(!)79 b FG(n)n(ul)27 b Fo(F)2191 3026 y Fs(B)599 3197 y FG(is)c(upp)r(er)f (semi-con)n(tin)n(uous.)34 b(The)22 b(set)g(\(40\))g(is)g(therefore)g (op)r(en)g(and)g(the)h(theorem)f(is)599 3322 y(pro)n(v)n(ed.)p 948 3338 25 75 v 1619 3637 a Fu(REFERENCES)621 3749 y FH(1.)33 b(C.)g(Adam,)f(B.)h(Muratori)g(and)h(C.)f(Nash,)h(Zero)f(mo)r (des)g(of)g(the)i(Dirac)e(op)r(erator)h(in)f(three)709 3828 y(dimensions,)22 b Fb(Phys.)k(R)l(ev.)f(D)h(\(3\))e Fa(60)f FH(\(1999\),)i(125001.)621 3932 y(2.)33 b(C.)20 b(Adam,)f(B.)h(Muratori)g(and)h(C.)f(Nash,)g(Degeneracy)i(of)e(zero)h (mo)r(des)f(of)g(the)i(Dirac)e(op)r(erator)709 4010 y(in)k(three)g (dimensions,)e Fb(Pr)l(eprint)50 b FH(h)n(ttp://xxx.lanl.go)n (v/abs/hep-th/9910139)29 b(\(1999\).)621 4114 y(3.)k(N.)22 b(Anghel,)g(Generic)h(v)l(anishing)f(for)g(harmonic)f(spinors)g(of)h(t) n(wisted)h(Dirac)f(op)r(erators,)g Fb(Pr)l(o)l(c.)709 4193 y(A)n(mer.)k(Math.)g(So)l(c.)e Fa(124)e FH(\(1996\),)j(3555{3561.) 621 4296 y(4.)33 b(M.)26 b(Birman)g(and)i(M.)e(Solom)n(y)n(ak,)h (Estimates)f(for)h(the)h(n)n(um)n(b)r(er)e(of)h(negativ)n(e)i(eigen)n (v)l(alues)g(of)709 4375 y(the)24 b(Sc)n(hr\177)-35 b(odinger)24 b(op)r(erator)f(and)h(its)f(generalizations,)h Fb(in)e FH(\\Estimates)h(and)h(asymptotics)f(for)709 4454 y(discrete)30 b(sp)r(ectra)g(of)f(in)n(tegral)g(and)h(di\013eren)n(tial)f(equations)h (\(Leningrad,)g(1989{90\)",)h(Adv.)709 4533 y(So)n(viet)25 b(Math.,)e(7,)h(pp.)f(1{55)i(,)e(Amer.)f(Math.)h(So)r(c.,)h(Pro)n (vidence,)g(RI,)g(1991.)p eop %%Page: 16 16 16 15 bop 599 75 a FG(16)711 b FH(A.)23 b(BALINSKY)g(AND)g(W.D.)g(EV)-8 b(ANS)621 320 y(5.)33 b(H.)21 b(Cycon,)h(R.)f(F)-6 b(ro)r(ese,)21 b(W.)g(Kirsc)n(h)g(and)h(B.)f(Simon,)f(\\Sc)n(hr\177)-35 b(odinger)22 b(op)r(erators)g(with)f(applica-)709 399 y(tion)j(to)h(quan)n(tum)e(mec)n(hanics)h(and)g(global)g(geometry)-6 b(,")24 b(Springer-V)-6 b(erlag,)22 b(Berlin,)h(1987.)621 511 y(6.)33 b(D.)23 b(Elton,)h(An)f(example)h(of)f(a)h(zero)g(mo)r(de,) f Fb(Pr)l(eprint)50 b FH(\(1998\).)621 623 y(7.)33 b(L.)e(Erd)919 609 y(})919 623 y(0s)f(and)i(J.P)-6 b(.)30 b(Solo)n(v)n(ej,)h(The)h(k)n (ernel)f(of)g(Dirac)f(op)r(erators)i(on)f FD(S)2693 599 y Fz(3)2752 623 y FH(and)h FD(R)2945 599 y Fz(3)2979 623 y FH(,)f Fb(Pr)l(eprint)709 702 y FH(h)n(ttp://xxx.lanl.go)n (v/ps/math-ph/0001036)d(\(2000\).)621 814 y(8.)33 b(J.)16 b(F)-6 b(r\177)-35 b(ohlic)n(h,)16 b(E.)g(Lieb)h(and)g(M.)e(Loss,)h (Stabilit)n(y)h(of)g(Coulom)n(b)e(systems)h(with)g(magnetic)h (\014elds.)f(I.)709 892 y(The)24 b(one-electron)h(atom,)e Fb(Comm.)k(Math.)e(Phys.)f Fa(104)f FH(\(1986\),)i(251{270.)621 1005 y(9.)33 b(E.)26 b(Lieb)h(and)g(M.)f(Loss,)g(\\Analysis,")g (American)g(Mathematical)g(So)r(ciet)n(y)-6 b(,)28 b(Pro)n(vidence,)f (RI,)709 1083 y(1997.)586 1196 y(10.)33 b(E.)e(Lieb)h(and)g(M.)e(Loss,) h(Stabilit)n(y)h(of)f(Coulom)n(b)g(systems)g(with)g(magnetic)h (\014elds.)f(II.)h(The)709 1274 y(man)n(y-electron)18 b(atom)g(and)g(the)h(one-electron)g(molecule)e Fb(Comm.)k(Math.)f (Phys.)p FH(,)e Fa(104)37 b FH(\(1986\),)709 1353 y(271{282.)586 1465 y(11.)c(M.)24 b(Loss)h(and)g(H-T.)e(Y)-6 b(au,)25 b(Stabilt)n(y)g(of)g(Coulom)n(b)f(systems)g(with)g(magnetic)h (\014elds.)g(III.)g(Zero)709 1544 y(energy)18 b(b)r(ound)g(states)g(of) f(the)h(Pauli)e(op)r(erator)i Fb(Comm.)i(Math.)g(Phys.)p FH(,)d Fa(104)35 b FH(\(1986\),)19 b(283{290.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0003281023773--