Content-Type: multipart/mixed; boundary="-------------0105180811828" This is a multi-part message in MIME format. ---------------0105180811828 Content-Type: text/plain; name="01-183.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-183.keywords" symmetric operators, self-adjoint extensions, abstract boundary conditions, Weyl function ---------------0105180811828 Content-Type: application/postscript; name="w-spec13.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="w-spec13.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: w-spec13.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips w-spec13.dvi -o %+ /home/unix/ima23/neidhard/w-spec13.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.05.18:1433 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S 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-118 100 a Fs(sense)39 b(self-adjoin)m(t)d(extensions)j(of)e Fq(A)h Fs(and)f(W)-8 b(eyl)38 b(functions)f(are)h(iden)m(tical)e(ob)5 b(jects)38 b(the)g(sp)s(ectral)g(in-)-118 224 y(formation)33 b(has)i(to)g(b)s(e)h(enco)s(ded)g(in)e(the)i(W)-8 b(eyl)35 b(function.)51 b(Indeed,)38 b(Lemma)c(3.2)g(sho)m(ws)j(that)e(it)f(is)h (so.)-118 349 y(Ho)m(w)m(ev)m(er,)40 b(under)e(practical)e(asp)s(ects)i (this)f(corresp)s(ondence)i(is)d(useless)j(since)e(the)h (reconstruction)f(of)-118 473 y(the)c(sp)s(ectral)f(prop)s(erties)g(is) h(v)m(ery)h(implicit.)255 598 y(Our)40 b(aim)e(is)i(to)f(describ)s(e)i (sp)s(ectral)f(subsets)h(b)m(y)g(the)g(b)s(oundary)f(v)-5 b(alues)40 b(of)f(W)-8 b(eyl)40 b(function)-118 722 y(at)46 b(the)h(real)f(axis.)85 b(So)47 b(the)g(problem)e(is)h(the)i(follo)m (wing:)68 b(Let)47 b Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))47 b Fs(b)s(e)g(the)g(W)-8 b(eyl)47 b(function)f(of)g(a)-118 847 y(certain)i(self-adjoin)m(t)e(extension)k Fq(A)1261 862 y Fl(0)1349 847 y Fs(of)e Fq(A)p Fs(.)91 b(In)m(tro)s(ducing)48 b(the)h(asso)s(ciated)f(scalar)g(W)-8 b(eyl)48 b(function)-118 971 y Fq(M)-24 986 y Fk(h)21 971 y Fn(\()p Fm(\001)p Fn(\))42 b(:=)h(\()p Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Fq(h;)17 b(h)p Fn(\))p Fs(,)43 b Fq(h)g Fm(2)g(H)q Fs(,)h(is)c(it)h(p)s (ossible)f(to)h(lo)s(calize)e(the)i(di\033eren)m(t)h(sp)s(ectral)f (subsets)i(of)e Fq(A)3740 986 y Fl(0)-118 1096 y Fs(kno)m(wing)33 b(the)h(b)s(oundary)f(v)-5 b(alues)33 b Fq(M)1260 1111 y Fk(h)1305 1096 y Fn(\()p Fq(x)23 b Fn(+)f Fq(i)p Fn(0\))p Fs(,)34 b Fq(x)29 b Fm(2)g Fj(R)5 b Fs(,)39 b(of)33 b(the)g(asso)s (ciated)g(scalar)f(W)-8 b(eyl)34 b(function)e(for)-118 1220 y(a)40 b(suitable)f(set)h Fm(T)66 b Fs(of)40 b(elemen)m(ts)g(of)f Fm(H)q Fs(.)66 b(It)41 b(turns)f(out)g(that)g(the)g(p)s(oin)m(t)f(and)i (absolutely)e(con)m(tin)m(uous)-118 1345 y(sp)s(ectrum)i(can)g(b)s(e)g (exactly)g(lo)s(calized)e(in)h(this)g(w)m(a)m(y)-8 b(,)44 b(see)e(Theorem)f(3.8)f(and)h(Prop)s(osition)e(4.2.)68 b(W)-8 b(e)-118 1469 y(also)37 b(men)m(tion)h(Theorem)g(4.3)g(and)g (Corollary)f(4.4)h(whic)m(h)g(indicate)g(su\036cien)m(t)h(conditions)e (for)h Fq(A)3615 1484 y Fl(0)3693 1469 y Fs(to)-118 1594 y(b)s(e)33 b(purely)f(absolutely)g(con)m(tin)m(uous)h(within)f(an)g (arbitrary)g(in)m(terv)-5 b(al)31 b Fn(\()p Fq(a;)17 b(b)p Fn(\))p Fs(.)255 1719 y(With)34 b(resp)s(ect)h(to)e(the)i (singular)d(con)m(tin)m(uous)j(sp)s(ectrum)f(w)m(e)h(get)f(in)f (general)g(an)h(incomplete)-118 1843 y(information.)50 b(That)37 b(means,)g(w)m(e)g(are)f(only)f(able)g(to)h(indicate)f(a)h (subset)h(in)e(whic)m(h)i(the)f(singular)f(con-)-118 1968 y(tin)m(uous)g(sp)s(ectrum)h(is)f(con)m(tained,)h(see)h(Theorem)f (3.8.)52 b(Ho)m(w)m(ev)m(er,)38 b(under)e(certain)f(circumstances)h(w)m (e)-118 2092 y(are)k(able)g(to)g(lo)s(calize)e(the)i(singular)f(con)m (tin)m(uous)i(sp)s(ectrum)g(to)s(o,)g(see)g(Theorem)g(4.3)f(and)g (Corollary)-118 2217 y(4.4.)54 b(W)-8 b(e)37 b(note)f(in)g(this)g (connection)g(that)g(in)g(our)g(pap)s(er)g([8])h(applying)d(Theorem)j (4.3)f(and)g(Corollary)-118 2341 y(4.4)27 b(w)m(e)h(construct)g (self-adjoin)m(t)e(extensions)j Fq(A)1630 2356 y Fl(0)1696 2341 y Fs(with)e(arbitrary)g(pure)g(singular)f(con)m(tin)m(uous)i(sp)s (ectrum)-118 2466 y(within)j(the)i(gap)f Fn(\()p Fq(\013)q(;)17 b(\014)6 b Fn(\))32 b Fs(of)g(a)g(symmetric)g(op)s(erator)g Fq(A)p Fs(.)255 2590 y(The)43 b(pap)s(er)e(is)g(organized)f(as)h(follo) m(ws.)69 b(In)41 b(Section)g(2)g(w)m(e)h(de\034ne)h(and)e(recall)f(the) h(notions)-118 2715 y(of)c Fq(R)q Fs(-function,)h(b)s(oundary)g (triple,)g(W)-8 b(eyl)38 b(function)f(and)h Fq(\015)5 b Fs(-\034eld)37 b(whic)m(h)h(are)g(imp)s(ortan)m(t)e(ingredien)m(ts) -118 2839 y(for)d(the)h(next)h(sections.)47 b(In)34 b(Section)g(3)f(w)m (e)i(in)m(v)m(estigate)f(the)g(relation)e(b)s(et)m(w)m(een)j(the)g(b)s (oundary)f(v)-5 b(alues)-118 2964 y(of)46 b(the)i(W)-8 b(eyl)47 b(function)f(and)h(the)g(sp)s(ectral)g(prop)s(erties)g(of)f (the)h(self-adjoin)m(t)e(extensions.)88 b(Finally)-8 b(,)-118 3088 y(in)42 b(Section)g(4)h(w)m(e)h(pro)m(v)m(e)g(a)e (re\034nemen)m(t)i(of)e(the)h(results)g(for)g(the)g(absolutely)f(con)m (tin)m(uous)h(sp)s(ectrum)-118 3213 y(and)i(indicate)g(su\036cien)m(t)h (conditions)e(whic)m(h)i(allo)m(w)e(to)h(lo)s(calize)e(exactly)i(the)h (singular)e(con)m(tin)m(uous)-118 3337 y(sp)s(ectrum.)60 b(In)39 b(Section)e(5)h(w)m(e)h(giv)m(e)f(examples)g(whic)m(h)h (illustrate)d(the)i(abstract)h(observ)-5 b(ations)37 b(of)h(the)-118 3462 y(sections)e(ab)s(o)m(v)m(e.)51 b(In)36 b(particular)d(w)m(e)j(consider)f(here)h(an)f (in\034nite-dimensional)c(Dirac)k(op)s(erator)f Fq(A)3632 3477 y Fl(0)3704 3462 y Fn(=)-118 3635 y Fq(A)-45 3599 y Fo(\003)-45 3660 y Fl(0)22 3635 y Fn(=)126 3495 y Fi(\022)242 3574 y Fn(0)83 b Fm(\000)p Fq(I)241 3694 y(I)130 b Fn(0)544 3495 y Fi(\023)626 3635 y Fm(\012)741 3596 y Fk(d)p 721 3612 77 4 v 721 3669 a(dx)815 3635 y Fn(+)899 3495 y Fi(\022)1014 3574 y Fq(T)147 b Fn(0)1025 3694 y(0)94 b Fm(\000)p Fq(T)1358 3495 y Fi(\023)1457 3635 y Fs(in)24 b Fq(L)1629 3599 y Fl(2)1669 3635 y Fn(\()p Fj(R)5 b Fq(;)17 b Fm(H)q Fn(\))p Fs(,)33 b Fm(H)c Fn(=)e Fm(K)10 b(\010)e(K)q Fs(,)28 b(with)d(an)h(arbitrary)e(nonnegativ)m(e)-118 3807 y(b)s(ounded)33 b(op)s(erator)g Fq(T)46 b Fs(in)32 b Fm(K)q Fs(.)45 b(Applying)32 b(Theorem)h(4.6)f(and)h(Corollary)e(4.7) i(w)m(e)h(establish)e(that)h Fq(A)3642 3822 y Fl(0)3714 3807 y Fs(is)-118 3932 y(purely)g(absolutely)e(con)m(tin)m(uous.)44 b(W)-8 b(e)33 b(\034nish)g(the)g(pap)s(er)g(with)f(some)g(general)g (remarks)h(in)f(Section)g(6.)255 4056 y(In)27 b(a)e(forthcoming)f(pap)s (er)i(w)m(e)h(apply)f(our)f(general)g(results)i(from)d(Sections)i(3)g (and)g(4)g(to)f(in)m(v)m(erse)-118 4181 y(sp)s(ectral)32 b(theory)h(\(in)f(the)h(sense)h(of)e([2],)h([6]-[11)o(]\))f(of)h(a)f (symmetric)f(op)s(erator)h(with)g(sev)m(eral)i(gaps.)255 4305 y(In)g(conclusion)f(the)h(reader)g(is)f(referred)h(to)f(recen)m(t) h(publications)e([18,)h(19])g(whic)m(h)h(are)f(closed)-118 4430 y(to)25 b(our)h(pap)s(er.)41 b(In)26 b([18,)g(19)o(])g(a)f(sp)s (ectral)h(analysis)f(of)g(matrix-v)-5 b(alued)23 b(Herglotz)i(\(Nev)-5 b(anlinna\))25 b(functions)-118 4555 y(as)33 b(w)m(ell)e(as)i(their)f (application)e(to)i(the)h(w)m(ell-kno)m(wn)g(Aronsza)5 b(jn-Donogh)m(ue)34 b(theory)f(is)f(considered.)p eop %%Page: 3 3 3 2 bop 255 100 a Fr(2)156 b(Preliminaries)255 348 y Fh(2.1)131 b(R-functions)255 537 y Fs(Let)42 b Fm(H)h Fs(b)s(e)f(a)f(separable)h(Hilb)s(ert)e(space.)72 b(Recall)40 b(that)h(an)h(op)s(erator)f(function)g Fq(F)14 b Fn(\()p Fm(\001)p Fn(\))41 b Fs(with)-118 661 y(v)-5 b(alues)32 b(in)g Fn([)p Fm(H)q Fn(])h Fs(is)f(said)f(to)h(b)s(e)h(a)f(Herglotz)g (or)g(Nev)-5 b(anlinna)31 b(function)h([1,)g(5,)h(18)o(,)g(19)o(,)g(21) o(])g(or)f(R-function)-118 786 y([24])27 b(if)f(it)g(is)h(holomorphic)e (in)i Fj(C)1058 801 y Fl(+)1150 786 y Fs(and)g(for)g(ev)m(ery)i Fq(z)k Fm(2)28 b Fj(C)1968 801 y Fl(+)2060 786 y Fs(the)g(op)s(erator)f Fq(F)14 b Fn(\()p Fq(z)t Fn(\))27 b Fs(in)g Fm(H)h Fs(is)f(dissipativ)m (e,)h(i.e.)-118 910 y Fm(=)p Fs(m)o Fn(\()p Fq(F)14 b Fn(\()p Fq(z)t Fn(\)\))28 b(:=)g(\()p Fq(F)14 b Fn(\()p Fq(z)t Fn(\))21 b Fm(\000)h Fq(F)14 b Fn(\()p Fq(z)t Fn(\))1034 874 y Fo(\003)1074 910 y Fn(\))p Fq(=)p Fn(2)p Fq(i)27 b Fm(\025)h Fn(0)p Fs(.)44 b(In)32 b(the)h(follo)m(wing)c(w)m (e)k(prefer)g(the)f(notion)g(R-function.)42 b(The)-118 1035 y(class)33 b(of)f(R-functions)g(with)g(v)-5 b(alues)33 b(in)f Fn([)p Fm(H)q Fn(])h Fs(is)g(denoted)g(b)m(y)h Fn(\()p Fq(R)2261 1050 y Fo(H)2325 1035 y Fn(\))p Fs(.)44 b(If)33 b Fq(F)14 b Fn(\()p Fm(\001)p Fn(\))28 b Fm(2)g Fn(\()p Fq(R)2947 1050 y Fo(H)3012 1035 y Fn(\))p Fs(,)k(then)i(there)f (exist)-118 1159 y(b)s(ounded)e(self-adjoin)m(t)d(op)s(erators)i Fq(C)1276 1174 y Fl(0)1315 1159 y Fs(,)h Fq(C)1443 1174 y Fl(1)1510 1159 y Fm(\025)d Fn(0)p Fs(,)i(a)g(Hilb)s(ert)f(space)i Fm(K)e(\023)f(H)q Fs(,)j(a)e(self-adjoin)m(t)g(op)s(erator)g Fq(L)-118 1284 y Fs(in)j Fm(K)q Fs(,)h(a)f(b)s(ounded)h(non-negativ)m (e)f(op)s(erator)g Fq(R)d Fm(\025)f Fn(0)k Fs(with)h Fq(R)q Fm(jK)23 b(\011)f(H)29 b Fn(=)f(0)k Fs(suc)m(h)i(that)522 1508 y Fq(F)14 b Fn(\()p Fq(z)t Fn(\))28 b(=)g Fq(C)926 1523 y Fl(0)987 1508 y Fn(+)22 b Fq(C)1155 1523 y Fl(1)1194 1508 y Fq(z)27 b Fn(+)22 b Fq(R)1439 1467 y Fl(1)p Fk(=)p Fl(2)1549 1508 y Fn(\()p Fq(I)1630 1523 y Fo(K)1711 1508 y Fn(+)g Fq(z)t(L)p Fn(\)\()p Fq(L)h Fm(\000)g Fq(z)t Fn(\))2276 1467 y Fo(\000)p Fl(1)2371 1508 y Fq(R)2446 1467 y Fl(1)p Fk(=)p Fl(2)2556 1508 y Fm(jH)q Fq(;)114 b(z)32 b Fm(2)c Fj(C)3047 1523 y Fl(+)3112 1508 y Fq(;)440 b Fs(\(2.1\))-118 1732 y(see)33 b([25)o(].)44 b(Denoting)31 b(b)m(y)i Fq(E)888 1747 y Fk(L)940 1732 y Fn(\()p Fm(\001)p Fn(\))e Fs(the)h(sp)s(ectral)g(measure)g(of)f(the)h(self-adjoin)m(t)e (op)s(erator)h Fq(L)h Fs(one)g(immedi-)-118 1857 y(ately)g(obtains)g (from)f(\(2.1\))h(the)h(represen)m(tation)540 2134 y Fq(F)14 b Fn(\()p Fq(z)t Fn(\))28 b(=)g Fq(C)944 2149 y Fl(0)1005 2134 y Fn(+)22 b Fq(C)1173 2149 y Fl(1)1213 2134 y Fq(z)k Fn(+)1382 1998 y Fi(Z)1482 2025 y Fo(1)1438 2224 y(\0001)1584 1993 y Fi(\022)1746 2066 y Fn(1)p 1667 2111 207 4 v 1667 2202 a Fq(t)d Fm(\000)f Fq(z)1906 2134 y Fm(\000)2120 2066 y Fq(t)p 2016 2111 244 4 v 2016 2202 a Fn(1)g(+)g Fq(t)2220 2173 y Fl(2)2269 1993 y Fi(\023)2359 2134 y Fq(d)p Fn(\006)2480 2149 y Fk(F)2539 2134 y Fn(\()p Fq(t)p Fn(\))p Fq(;)114 b(z)33 b Fm(2)28 b Fj(C)3029 2149 y Fl(+)3094 2134 y Fq(;)458 b Fs(\(2.2\))-118 2413 y(where)34 b Fn(\006)234 2428 y Fk(F)293 2413 y Fn(\()p Fm(\001)p Fn(\))e Fs(is)g(an)g(op)s(erator-v)-5 b(alued)31 b(Borel)h(measure)h(on)f Fj(R)44 b Fs(giv)m(en)32 b(b)m(y)888 2638 y Fq(d)p Fn(\006)1009 2653 y Fk(F)1068 2638 y Fn(\()p Fq(t)p Fn(\))c(=)f(\(1)22 b(+)g Fq(t)1552 2596 y Fl(2)1592 2638 y Fn(\))p Fq(R)1705 2596 y Fl(1)p Fk(=)p Fl(2)1815 2638 y Fq(dE)1938 2653 y Fk(L)1990 2638 y Fn(\()p Fm(\001)p Fn(\))p Fq(R)2169 2596 y Fl(1)p Fk(=)p Fl(2)2278 2638 y Fq(;)212 b(t)28 b Fm(2)g Fj(R)5 b Fq(:)812 b Fs(\(2.3\))-118 2862 y(Ob)m(viously)-8 b(,)32 b(the)h(measure)g Fn(\006)977 2877 y Fk(F)1036 2862 y Fn(\()p Fm(\001)p Fn(\))f Fs(is)g(self-adjoin)m (t)f(and)i(ob)s(eys)1287 3008 y Fi(Z)1386 3035 y Fl(+)p Fo(1)1342 3234 y(\0001)1640 3077 y Fn(1)p 1542 3121 V 1542 3212 a(1)22 b(+)g Fq(t)1746 3184 y Fl(2)1796 3144 y Fq(d)p Fn(\006)1917 3159 y Fk(F)1976 3144 y Fn(\()p Fq(t)p Fn(\))27 b Fm(2)h Fn([)p Fm(H)q Fn(])p Fq(:)1205 b Fs(\(2.4\))-118 3417 y(In)49 b(con)m(trast)g(to)g(sp)s(ectral)f (measures)i(of)e(self-adjoin)m(t)f(op)s(erators)i(it)e(is)i(not)f (necessarily)i(true)f(that)-118 3542 y(ran)p Fn(\(\006\()p Fq(\016)212 3557 y Fl(1)252 3542 y Fn(\)\))24 b Fs(is)h(orthogonal)d (to)j(ran)p Fn(\(\006\()p Fq(\016)1364 3557 y Fl(2)1404 3542 y Fn(\)\))f Fs(for)g(disjoin)m(t)g(Borel)g(sets)i Fq(\016)2462 3557 y Fl(1)2526 3542 y Fs(and)f Fq(\016)2751 3557 y Fl(2)2791 3542 y Fs(.)40 b(Ho)m(w)m(ev)m(er,)29 b(the)c(measure)-118 3666 y Fn(\006)-48 3681 y Fk(F)11 3666 y 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Fm(\001)p Fn(\))g Fs(is)g(called)f(the)i(sp)s(ectral)g(function)f(of)g Fq(F)14 b Fn(\()p Fm(\001)p Fn(\))p Fs(.)p eop %%Page: 4 4 4 3 bop 255 100 a Fs(W)-8 b(e)38 b(note)g(that)f(the)h(sp)s(ectral)f (function)g Fn(\006)1867 115 y Fk(F)1926 100 y Fn(\()p Fm(\001)p Fn(\))g Fs(of)g Fq(F)14 b Fn(\()p Fm(\001)p Fn(\))36 b Fs(can)i(b)s(e)f(obtained)g(b)m(y)i(the)e(Stieltjes)-118 224 y(transformation:)388 396 y Fn(1)p 388 441 49 4 v 388 532 a(2)446 464 y(\(\006)554 479 y Fk(F)613 464 y Fn(\()p Fq(t)23 b Fn(+)f(0\))g(+)g(\006\()p Fq(t)p Fn(\)\))g Fm(\000)1365 396 y Fn(1)p 1365 441 V 1365 532 a(2)1423 464 y(\(\006\()p Fq(s)h Fn(+)f(0\))g(+)g(\006\()p Fq(s)p Fn(\)\))28 b(=)1302 b Fs(\(2.7\))1016 715 y Fq(w)25 b Fm(\000)d Fn(lim)1224 778 y Fk(y)r Fo(#)p Fl(0)1377 648 y Fn(1)p 1372 692 59 4 v 1372 784 a Fq(\031)1458 580 y Fi(Z)1557 606 y Fk(t)1513 805 y(s)1604 715 y Fm(=)p Fs(m)o Fn(\()p Fq(F)14 b Fn(\()p Fq(x)22 b Fn(+)g Fq(iy)t Fn(\)\))p Fq(dx;)114 b(t;)17 b(s)27 b Fm(2)h Fj(R)-118 982 y Fs(where)34 b(it)d(is)h(used)i(that)e(the)h(sp)s(ectral)f (function)g(is)h(strongly)f(left)f(con)m(tin)m(uous.)255 1271 y Fh(2.2)131 b(Boundary)45 b(triples)g(and)e(self-adjoin)l(t)i (relations)255 1460 y Fs(In)34 b(the)f(follo)m(wing)e Fq(A)i Fs(will)d(alw)m(a)m(ys)k(denote)g(a)e(closed)h(symmetric)g(op)s (erator)f(with)g(equal)h(de\034-)-118 1584 y(ciency)i(indices)f Fq(n)559 1599 y Fl(+)619 1584 y Fn(\()p Fq(A)p Fn(\))d(=)f Fq(n)963 1599 y Fo(\000)1023 1584 y Fn(\()p Fq(A)p Fn(\))p Fs(.)49 b(Without)34 b(loss)g(of)g(generalit)m(y)g(w)m(e)h(can)g (assume)g(that)f Fq(A)h Fs(is)f(simple.)-118 1709 y(This)f(means)f (that)g Fq(A)h Fs(has)g(no)f(self-adjoin)m(t)f(parts.)255 1833 y(Our)38 b(approac)m(h)h(to)f(the)g(in)m(v)m(erse)i(sp)s(ectral)d (theory)i(of)f(self-adjoin)m(t)e(extensions)j(is)f(based)h(on)-118 1958 y(the)29 b(concept)h(of)f(b)s(oundary)g(triples)f(\(see)i([21])f (and)g(references)i(therein\))e(whic)m(h)h(ma)m(y)e(b)s(e)h(considered) h(as)-118 2082 y(an)i(abstract)h(v)m(ersion)g(of)f(the)h(second)h (Green's)f(form)m(ula.)255 2290 y Ft(DEFINITION)j(2.1)49 b Fs(A)44 b(triple)f Fn(\005)k(=)g Fm(fH)q Fq(;)17 b Fn(\000)2049 2305 y Fl(0)2088 2290 y Fq(;)g Fn(\000)2193 2305 y Fl(1)2233 2290 y Fm(g)43 b Fs(consisting)h(of)f(an)h(auxiliary)e (Hilb)s(ert)-118 2414 y(space)33 b Fm(H)g Fs(and)g(linear)d(mappings)i Fn(\000)1220 2429 y Fk(i)1275 2414 y Fn(:)c Fs(dom)o Fn(\()p Fq(A)1625 2378 y Fo(\003)1665 2414 y Fn(\))f Fm(\000)-16 b(!)28 b(H)q Fq(;)114 b(i)28 b Fn(=)f(0)p Fq(;)17 b Fn(1)p Fq(;)32 b Fs(is)g(called)f(a)h(b)s(oundary)g(triple)f (for)-118 2539 y(the)i(adjoin)m(t)e(op)s(erator)h Fq(A)846 2503 y Fo(\003)918 2539 y Fs(of)g Fq(A)h Fs(if:)-26 2746 y(\(i\))48 b(The)33 b(second)h(Green's)f(form)m(ula)e(tak)m(es)j(place) 601 2971 y Fn(\()p Fq(A)712 2929 y Fo(\003)751 2971 y Fq(f)5 b(;)17 b(g)t Fn(\))k Fm(\000)i Fn(\()p Fq(f)5 b(;)17 b(A)1266 2929 y Fo(\003)1306 2971 y Fq(g)t Fn(\))26 b(=)i(\(\000)1624 2986 y Fl(1)1663 2971 y Fq(f)5 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3842 y(triple)31 b Fm(fH)q Fq(;)17 b Fn(\000)382 3857 y Fl(0)421 3842 y Fq(;)g Fn(\000)526 3857 y Fl(1)565 3842 y Fm(g)32 b Fs(suc)m(h)i(that)1104 3817 y Fn(~)1079 3842 y Fq(A)27 b Fn(=)h Fq(A)1356 3806 y Fo(\003)1396 3842 y Fm(j)17 b Fn(k)m(er\(\000)1669 3857 y Fl(0)1708 3842 y Fn(\))33 b Fs(is)f(v)-5 b(alid.)255 3967 y(\(3\))22 b(Moreo)m(v)m(er)h(\(see)g([15)o(]\))f(for)f(eac)m(h)i(transv)m(ersal)f (pair)f Fq(A)2308 3982 y Fl(0)2347 3967 y Fq(;)c(A)2464 3982 y Fl(1)2531 3967 y Fm(2)28 b Fq(E)6 b(xt)2793 3982 y Fk(A)2873 3967 y Fs(\(that)21 b(means)h(dom)o Fn(\()p Fq(A)3694 3982 y Fl(0)3734 3967 y Fn(\))p Fm(\\)-118 4091 y Fs(dom)o Fn(\()p Fq(A)177 4106 y Fl(1)217 4091 y Fn(\))42 b(=)g Fs(dom)p Fn(\()p Fq(A)p Fn(\))f Fs(and)g(dom)p Fn(\()p Fq(A)1284 4106 y Fl(0)1323 4091 y Fn(\))28 b(+)g Fs(dom)o Fn(\()p Fq(A)1788 4106 y Fl(1)1828 4091 y Fn(\))42 b(=)h Fs(dom)o Fn(\()p Fq(A)2322 4055 y Fo(\003)2361 4091 y Fn(\))p Fs(\))e(there)h(exists)g(a)f(b)s(oundary)h(triple)-118 4216 y Fm(fH)q 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(self-adjoin)m(t)f(relations:)p eop %%Page: 5 5 5 4 bop 255 100 a Ft(DEFINITION)36 b(2.3)141 b Fs(\(i\))48 b(A)37 b(closed)g(linear)e(relation)g Fn(\002)h Fs(in)g Fm(H)i Fs(is)f(a)f(closed)h(subspace)i Fn(\002)126 224 y Fs(of)32 b Fm(H)23 b(\010)g(H)q Fs(.)-53 432 y(\(ii\))47 b(The)22 b(closed)g(linear)e(relation)g Fn(\002)i Fs(is)f(symmetric)g (if)f Fn(\()p Fq(g)2011 447 y Fl(1)2050 432 y Fq(;)d(f)2142 447 y Fl(2)2182 432 y Fn(\))p Fm(\000)p Fn(\()p Fq(f)2383 447 y Fl(1)2422 432 y Fq(;)g(g)2513 447 y Fl(2)2552 432 y Fn(\))28 b(=)f(0)p Fs(,)d(for)d(all)f Fm(f)p Fq(f)3182 447 y Fl(1)3221 432 y Fq(;)d(g)3312 447 y Fl(1)3351 432 y Fm(g)p Fq(;)g Fm(f)p Fq(f)3543 447 y Fl(2)3582 432 y Fq(;)g(g)3673 447 y Fl(2)3712 432 y Fm(g)27 b(2)126 556 y Fn(\002)p Fq(:)-80 764 y Fs(\(iii\))46 b(The)33 b(closed)g(linear)e(relation)f Fn(\002)j Fs(is)f(self-adjoin)m(t)f(if)g (it)h(is)g(maximal)d(symmetric.)255 996 y Ft(EXAMPLE)36 b(2.4)49 b Fs(\(1\))22 b(Let)f Fq(B)27 b Fs(b)s(e)22 b(a)g(self-adjoin)m(t)d(op)s(erator)i(in)g Fm(H)q Fs(,)j(not)e (necessarily)g(b)s(ounded.)-118 1121 y(Then)34 b(the)f(graph)f Fq(G)p Fn(\()p Fq(B)5 b Fn(\))32 b Fs(of)g Fq(B)38 b Fs(is)32 b(a)g(self-adjoin)m(t)f(relation)g(in)g Fm(H)q Fs(.)255 1245 y(\(2\))43 b(The)i(subspaces)h Fn(\002)1168 1260 y Fl(0)1253 1245 y Fn(:=)g Fm(f)p Fn(0)p Fm(g)29 b(\002)h(H)q Fq(;)115 b Fn(\002)1990 1260 y Fl(1)2075 1245 y Fn(:=)46 b Fm(H)31 b(\002)f(f)p Fn(0)p Fm(g)p Fq(;)43 b Fs(of)g Fm(H)30 b(\002)g(H)45 b Fs(are)e(self-adjoin)m(t)-118 1370 y(relations)31 b(in)h Fm(H)q Fs(.)43 b(Ob)m(viously)-8 b(,)33 b Fn(\002)1101 1385 y Fl(0)1172 1370 y Fs(is)f(not)h(the)g (graph)f(of)g(an)m(y)h(op)s(erator.)-118 1602 y(The)g(follo)m(wing)d (notation)h(is)h(\034xed.)255 1810 y Ft(DEFINITION)k(2.5)49 b Fs(Let)33 b Fm(fH)q Fq(;)17 b Fn(\000)1592 1825 y Fl(0)1631 1810 y Fq(;)g Fn(\000)1736 1825 y Fl(1)1775 1810 y Fm(g)32 b Fs(b)s(e)h(a)f(b)s(oundary)h(triple)e(for)h Fq(A)2992 1773 y Fo(\003)3032 1810 y Fs(.)-26 2017 y(\(i\))48 b(F)-8 b(or)31 b(ev)m(ery)k(self-adjoin)m(t)c(relation)f Fn(\002)j Fs(in)e Fm(H)j Fs(w)m(e)g(put)753 2241 y Fq(D)837 2200 y Fl(\002)923 2241 y Fn(:=)28 b Fm(f)p Fq(f)38 b Fm(2)28 b Fs(dom)p Fn(\()p Fq(A)1580 2200 y Fo(\003)1619 2241 y Fn(\))g(:)f Fm(f)p Fn(\000)1850 2256 y Fl(0)1890 2241 y Fq(f)5 b(;)17 b Fn(\000)2048 2256 y Fl(1)2087 2241 y Fq(f)11 b Fm(g)27 b(2)h Fn(\002)p Fm(g)p Fq(;)82 b(A)2625 2200 y Fl(\002)2712 2241 y Fn(:=)27 b Fq(A)2915 2200 y Fo(\003)2955 2241 y Fm(j)p Fq(D)3067 2200 y Fl(\002)3125 2241 y Fq(:)427 b Fs(\(2.9\))-53 2507 y(\(ii\))47 b(In)33 b(particular,)d(w)m(e)k(set)f Fq(A)1093 2522 y Fk(i)1149 2507 y Fn(:=)28 b Fq(A)1353 2471 y Fl(\002)1408 2481 y Fg(i)1438 2507 y Fs(,)33 b Fq(i)28 b Fn(=)g(0)p Fq(;)17 b Fn(1)p Fs(,)31 b(if)h Fn(\002)2029 2522 y Fk(i)2057 2507 y Fs(,)g Fq(i)c Fn(=)g(0)p Fq(;)17 b Fn(1)p Fs(,)32 b(are)g(giv)m(en)h(b)m(y)g(Example)f(2.4\(2\).)-80 2714 y(\(iii\))46 b(If)32 b Fn(\002)c(=)f Fq(G)p Fn(\()p Fq(B)5 b Fn(\))p Fs(,)33 b(where)h Fq(B)j Fs(is)32 b(an)h(op)s(erator)f(on)g Fm(H)q Fs(,)h(then)g(w)m(e)h(set)f Fq(A)2613 2678 y Fk(B)2701 2714 y Fn(:=)28 b Fq(A)2905 2678 y Fl(\002)2964 2714 y Fs(.)255 2947 y Ft(REMARK)36 b(2.6)49 b Fs(If)27 b Fq(B)32 b Fs(b)s(e)c(is)e(a)h(self-adjoin)m(t)e(op)s(erator)i(on)g Fm(H)q Fs(,)h(then)g(a)f(simple)e(computation)-118 3071 y(sho)m(ws)34 b(that)e Fq(A)446 3035 y Fk(B)535 3071 y Fn(=)27 b Fq(A)711 3035 y Fo(\003)751 3071 y Fm(j)17 b Fn(k)m(er\(\000)1024 3086 y Fl(1)1086 3071 y Fm(\000)22 b Fq(B)5 b Fn(\000)1325 3086 y Fl(0)1365 3071 y Fn(\))27 b(=)h Fq(A)1607 3035 y Fk(B)1607 3096 y Fl(0)1668 3071 y Fs(.)255 3304 y Ft(PR)m(OPOSITION)36 b(2.7)49 b Fs(\([14,)27 b(21]\))j Ff(L)-5 b(et)31 b Fm(fH)q Fq(;)17 b Fn(\000)2096 3319 y Fl(0)2135 3304 y Fq(;)g Fn(\000)2240 3319 y Fl(1)2279 3304 y Fm(g)31 b Ff(b)-5 b(e)30 b(a)g(b)-5 b(oundary)30 b(triple)g(for)h Fq(A)3448 3267 y Fo(\003)3487 3304 y Ff(.)43 b(Then)-118 3428 y(for)f(every)h(self-adjoint)e(r)-5 b(elation)43 b Fn(\002)f Ff(in)h Fm(H)h Ff(the)f(op)-5 b(er)g(ator)42 b Fq(A)2198 3392 y Fl(\002)2257 3428 y Ff(,)j(given)d(by)h(De\034nition)e(2.5,)j(is)f(a)f(self-)-118 3553 y(adjoint)e(extension)g(of)g Fq(A)p Ff(.)63 b(The)40 b(mapping)f Fn(\002)g Fm(7!)f Fq(A)1872 3516 y Fl(\002)1972 3553 y Ff(fr)-5 b(om)40 b(the)h(set)g(of)f(self-adjoint)g(r)-5 b(elations)40 b(in)g Fm(H)-118 3677 y Ff(onto)35 b(the)f(set)h Fq(E)6 b(xt)585 3692 y Fk(A)678 3677 y Ff(of)34 b(self-adjoint)g (extensions)f(of)i Fq(A)g Ff(is)g(bije)-5 b(ctive.)255 3966 y Fh(2.3)131 b(W)-11 b(eyl)44 b(functions)255 4155 y Fs(It)29 b(is)f(w)m(ell)g(kno)m(wn)i(that)e(W)-8 b(eyl)29 b(functions)g(are)f(an)h(imp)s(ortan)m(t)d(to)s(ol)h(in)h(the)h(direct) f(and)h(in)m(v)m(erse)-118 4279 y(sp)s(ectral)39 b(theory)i(of)e (singular)f(Sturm-Liouville)e(op)s(erators.)64 b(In)40 b([13,)g(14)o(,)g(15])f(the)i(concept)f(of)g(W)-8 b(eyl)-118 4404 y(function)23 b(w)m(as)i(generalized)e(to)g(an)h(arbitrary)f (symmetric)g(op)s(erator)g Fq(A)g Fs(with)h(in\034nite)f(de\034ciency)i (indices)-118 4528 y Fq(n)-60 4543 y Fl(+)-1 4528 y Fn(\()p Fq(A)p Fn(\))j(=)f Fq(n)337 4543 y Fo(\000)397 4528 y Fn(\()p Fq(A)p Fn(\))p Fs(.)43 b(In)33 b(this)f(subsection)i(w)m(e)f (recall)e(basic)i(facts)f(ab)s(out)g(W)-8 b(eyl)33 b(functions.)255 4736 y Ft(DEFINITION)j(2.8)49 b Fs(\([13,)24 b(14)o(]\))g(Let)h Fm(fH)q Fq(;)17 b Fn(\000)1984 4751 y Fl(0)2023 4736 y Fq(;)g Fn(\000)2128 4751 y Fl(1)2167 4736 y Fm(g)23 b Fs(b)s(e)i(a)e(b)s(oundary)i(triple)d(for)i(the)g(op)s(erator)-118 4860 y Fq(A)-45 4824 y Fo(\003)-6 4860 y Fs(.)44 b(The)34 b(W)-8 b(eyl)32 b(function)g(of)g Fq(A)h Fs(corresp)s(onding)f(to)g (the)h(b)s(oundary)g(triple)f Fm(fH)q Fq(;)17 b Fn(\000)2964 4875 y Fl(0)3003 4860 y Fq(;)g Fn(\000)3108 4875 y Fl(1)3147 4860 y Fm(g)32 b Fs(is)g(the)h(unique)-118 4985 y(mapping)e Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))28 b(:)g Fq(\032)p Fn(\()p Fq(A)735 5000 y Fl(0)774 4985 y Fn(\))g Fm(\000)-16 b(!)27 b Fn([)p Fm(H)q Fn(])33 b Fs(satisfying)936 5209 y Fn(\000)997 5224 y Fl(1)1037 5209 y Fq(f)1085 5224 y Fk(z)1152 5209 y Fn(=)28 b Fq(M)10 b Fn(\()p Fq(z)t Fn(\)\000)1546 5224 y Fl(0)1586 5209 y Fq(f)1634 5224 y Fk(z)1674 5209 y Fq(;)115 b(f)1864 5224 y Fk(z)1931 5209 y Fm(2)28 b(N)2107 5224 y Fk(z)2147 5209 y Fq(;)114 b(z)32 b Fm(2)c Fq(\032)p Fn(\()p Fq(A)2620 5224 y Fl(0)2660 5209 y Fn(\))p Fq(;)805 b Fs(\(2.10\))-118 5433 y(where)34 b Fm(N)246 5448 y Fk(z)313 5433 y Fn(=)27 b(k)m(er)q(\()p Fq(A)657 5397 y Fo(\003)719 5433 y Fm(\000)c Fq(z)t(I)8 b Fn(\))p Fs(.)p eop %%Page: 6 6 6 5 bop -118 100 a Fs(It)25 b(is)f(w)m(ell)g(kno)m(wn)i(\(see)g([13)o (,)f(14]\))g(that)f(the)h(ab)s(o)m(v)m(e)h(implicit)21 b(de\034nition)i(of)i(the)g(W)-8 b(eyl)24 b(function)h(is)f(correct) -118 224 y(and)33 b(the)f(W)-8 b(eyl)33 b(function)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))33 b Fs(is)f(a)g(R-function)f(ob)s (eying)h Fn(0)c Fm(2)g Fq(\032)p Fn(\()p Fm(=)p Fs(m)o Fn(\()p Fq(M)10 b Fn(\()p Fq(i)p Fn(\)\)\))p Fs(.)255 349 y(The)25 b(W)-8 b(eyl)23 b(function)g(immediately)d(pro)m(vides)k (some)g(information)c(ab)s(out)j(the)g(\020sp)s(ectral)g(prop-)-118 473 y(erties\021)44 b(of)35 b(self-adjoin)m(t)g(extensions)i(of)f(the)g (symmetric)g(op)s(erator)f Fq(A)p Fs(.)54 b(T)-8 b(o)37 b(explain)e(this)h(let)f(us)i(recall)-118 598 y(the)c(notion)e(of)h (the)h(in)m(v)m(erse)h(of)e(a)h(closed)f(linear)f(relation.)255 785 y Ft(DEFINITION)36 b(2.9)49 b Fs(A)32 b(closed)f(linear)f(relation) f Fn(\002)j Fs(in)e Fm(H)j Fs(is)e(called)f(b)s(oundedly)i(inv)m (ertible)-118 909 y(if)40 b(the)h(in)m(v)m(erse)h(relation)d Fn(\002)929 873 y Fo(\000)p Fl(1)1065 909 y Fn(:=)i Fm(ff)p Fq(g)t(;)17 b(f)11 b Fm(g)40 b(2)i(H)29 b(\002)f(H)43 b Fn(:)f Fm(f)p Fq(f)5 b(;)17 b(g)t Fm(g)40 b(2)j Fn(\002)p Fm(g)d Fs(is)g(the)h(graph)g(of)f(a)h(b)s(ounded)-118 1034 y(op)s(erator)33 b(de\034ned)i(on)f Fm(H)q Fs(.)47 b(In)34 b(particular,)e(w)m(e)j(sa)m(y)f Fq(\025)c Fm(2)g Fj(C)60 b Fs(b)s(elongs)33 b(to)g(the)i(resolv)m(en)m(t)f(set)h Fq(\032)p Fn(\(\002\))e 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Fm(H)h Ff(and)e Fq(\025)27 b Fm(2)i Fq(\032)p Fn(\()p Fq(A)2364 1797 y Fl(0)2403 1782 y Fn(\))p Ff(.)45 b(Then)34 b(the)h(fol)5 b(lowing)33 b(holds:)-26 1969 y Fs(\(i\))48 b Fq(\025)27 b Fm(2)h Fq(\032)p Fn(\()p Fq(A)465 1933 y Fl(\002)525 1969 y Fn(\))35 b Ff(if)f(and)h(only)f(if)h Fn(0)27 b Fm(2)h Fq(\032)p Fn(\(\002)23 b Fm(\000)f Fq(M)10 b Fn(\()p Fq(\025)p Fn(\)\))p Ff(.)-53 2170 y Fs(\(ii\))47 b Fq(\025)27 b Fm(2)h Fq(\033)359 2185 y Fk(\034)403 2170 y Fn(\()p Fq(A)514 2133 y Fl(\002)573 2170 y Fn(\))35 b Ff(if)g(and)f(only)g(if)h Fn(0)28 b Fm(2)g Fq(\033)1462 2185 y Fk(\034)1505 2170 y Fn(\(\002)22 b Fm(\000)h Fq(M)10 b Fn(\()p Fq(\025)p Fn(\)\))p Ff(,)35 b Fq(\034)k Fn(=)28 b Fq(p;)17 b(c)p Ff(.)-118 2357 y Fs(If)48 b Fq(A)h Fs(is)f(a)g(simple) f(symmetric)h(op)s(erator,)k(then)d(the)g(W)-8 b(eyl)48 b(function)g Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))49 b Fs(determines)g (the)g(pair)-118 2481 y Fm(f)p Fq(A;)17 b(A)122 2496 y Fl(0)161 2481 y Fm(g)42 b Fs(up)f(to)h(unitary)f(equiv)-5 b(alence)42 b(\(see)g([15,)g(23)o(]\).)71 b(This)42 b(fact)f(allo)m(ws) g(one)h(to)f(reform)m(ulate)f(the)-118 2606 y(sp)s(ectral)32 b(prop)s(erties)h(of)f Fq(A)891 2621 y Fl(0)963 2606 y Fs(in)f(terms)i(of)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Fs(.)255 2730 y(W)-8 b(e)31 b(shall)f(often)g(sa)m(y)i(that)e Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))31 b Fs(is)f(the)h(W)-8 b(eyl)31 b(function)f(of)g(the)h(pair)e Fm(f)p Fq(A;)17 b(A)3107 2745 y Fl(0)3146 2730 y Fm(g)31 b Fs(or,)f(simply)-8 b(,)30 b(of)-118 2855 y Fq(A)-45 2870 y Fl(0)-6 2855 y Fs(.)51 b(Though)35 b(the)g(W)-8 b(eyl)35 b(function)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))36 b Fs(of)e(a)g(b)s(oundary)i (triple)d(is)h(unique)i(there)f(are)g(a)g(lot)e(of)i(W)-8 b(eyl)-118 2979 y(functions)34 b(corresp)s(onding)h(in)f(the)h(ab)s(o)m (v)m(e)g(sense)i(to)d(a)g(pair)g Fm(f)p Fq(A;)17 b(A)2397 2994 y Fl(0)2436 2979 y Fm(g)p Fs(.)50 b(Ho)m(w)m(ev)m(er)37 b(it)c(is)i(not)f(di\036cult)g(to)-118 3104 y(pro)m(v)m(e)25 b([14])f(that)g(t)m(w)m(o)g(suc)m(h)i(functions)e Fq(M)1408 3119 y Fl(1)1448 3104 y Fn(\()p Fm(\001)p Fn(\))f Fs(and)h Fq(M)1850 3119 y Fl(2)1890 3104 y Fn(\()p Fm(\001)p Fn(\))f Fs(with)h(v)-5 b(alues)24 b(in)f Fn([)p Fm(H)2729 3119 y Fl(1)2769 3104 y Fn(])h Fs(and)g Fn([)p Fm(H)3112 3119 y Fl(2)3152 3104 y Fn(])g Fs(are)g(connected)-118 3228 y(via)1294 3353 y Fq(M)1388 3368 y Fl(2)1428 3353 y Fn(\()p Fq(z)t Fn(\))k(=)g Fq(K)1775 3312 y Fo(\003)1814 3353 y Fq(M)1908 3368 y Fl(1)1948 3353 y Fn(\()p Fq(z)t Fn(\))p Fq(K)i Fn(+)22 b Fq(D)1165 b Fs(\(2.11\))-118 3521 y(where)34 b Fq(D)c Fn(=)e Fq(D)463 3485 y Fo(\003)529 3521 y Fm(2)g Fn([)p Fm(H)734 3536 y Fl(2)774 3521 y Fn(])33 b Fs(and)f Fq(K)j Fm(2)28 b Fn([)p Fm(H)1346 3536 y Fl(2)1386 3521 y Fq(;)17 b Fm(H)1514 3536 y Fl(1)1553 3521 y Fn(])33 b Fs(is)f(b)s(oundedly)h(in)m(v)m(ertible.)255 3807 y Fh(2.4)131 b Fe(\015)7 b Fh(-\034elds)255 3995 y Fs(With)32 b(eac)m(h)i(b)s(oundary)f(triple)e(w)m(e)i(can)g(asso)s(ciate)f(a)h 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Fq(A)1620 4923 y Fl(0)1682 4908 y Fm(\000)g Fq(z)t Fn(\))1868 4867 y Fo(\000)p Fl(1)1963 4908 y Fq(\015)5 b Fn(\()p Fq(z)2102 4923 y Fl(0)2142 4908 y Fn(\))p Fq(;)114 b(z)t(;)17 b(z)2459 4923 y Fl(0)2527 4908 y Fm(2)28 b Fq(\032)p Fn(\()p Fq(A)2782 4923 y Fl(0)2822 4908 y Fn(\))670 b Fs(\(2.13\))-118 5108 y(and)29 b(consequen)m(tly)j Fq(\015)5 b Fn(\()p Fm(\001)p Fn(\))29 b Fs(is)f(a)i Fq(\015)5 b Fs(-\034eld)28 b(in)h(the)h(sense)h(of)e([23)o(].)43 b(The)30 b Fq(\015)5 b Fs(-\034eld)29 b Fq(\015)5 b Fn(\()p Fm(\001)p Fn(\))29 b Fs(and)g(the)h(W)-8 b(eyl)29 b(function)-118 5233 y Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))33 b Fs(are)f(related)g(b)m(y)698 5433 y Fq(M)10 b Fn(\()p Fq(z)t Fn(\))24 b Fm(\000)e Fq(M)10 b Fn(\()p Fq(z)1237 5448 y Fl(0)1278 5433 y Fn(\))1316 5392 y Fo(\003)1383 5433 y Fn(=)27 b(\()p Fq(z)g Fm(\000)i Fn(\026)-55 b Fq(z)1741 5448 y Fl(0)1781 5433 y Fn(\))p Fq(\015)5 b Fn(\()p Fq(z)1958 5448 y Fl(0)1997 5433 y Fn(\))2035 5392 y Fo(\003)2075 5433 y Fq(\015)g Fn(\()p Fq(z)t Fn(\))p Fq(;)114 b(z)t(;)17 b(z)2535 5448 y Fl(0)2603 5433 y Fm(2)28 b Fq(\032)p Fn(\()p Fq(A)2858 5448 y Fl(0)2898 5433 y Fn(\))p Fq(:)567 b Fs(\(2.14\))p eop %%Page: 7 7 7 6 bop -118 100 a Fs(The)33 b(relation)e(\(2.14\))h(means)g(the)h Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))33 b Fs(is)f(a)g Fq(Q)p Fs(-function)g(of)g(a)g(pair)g Fm(f)p Fq(A;)17 b(A)2733 115 y Fl(0)2772 100 y Fm(g)32 b Fs([23].)255 224 y(F)-8 b(urther,)32 b(w)m(e)f(note)g(that)g(if)e Fq(A)i Fs(is)g(simple,)e (then)j Fm(N)2121 239 y Fk(z)2160 224 y Fs(,)f Fq(z)i Fm(2)28 b Fq(\032)p Fn(\()p Fq(A)2551 239 y Fl(0)2591 224 y Fn(\))p Fs(,)j(is)f(generating)g(with)g(resp)s(ect)-118 349 y(to)37 b Fq(A)79 364 y Fl(0)119 349 y Fs(,)h(otherwise,)i Fq(A)d Fs(w)m(ould)h(not)f(b)s(e)h(simple.)57 b(Since)37 b(ran)p Fn(\()p Fq(\015)5 b Fn(\()p Fq(z)t Fn(\)\))37 b(=)f Fm(N)2613 364 y Fk(z)2652 349 y Fs(,)j Fq(z)i Fm(2)36 b Fq(\032)p Fn(\()p Fq(A)3067 364 y Fl(0)3107 349 y Fn(\))p Fs(,)j(one)f(gets)g(that)-118 473 y(ran)p Fn(\()p 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b(called)g(orthogonal)e(if)h(some)i(of)f(their)g(supp)s(orts)-118 1527 y(are)32 b(disjoin)m(t.)42 b(The)33 b(top)s(ological)28 b(supp)s(ort)k Fq(S)6 b Fn(\()p Fq(\026)p Fn(\))31 b Fs(of)h Fq(\026)f Fs(is)h(the)g(smallest)e(closed)i(set)h(whic)m(h)f (is)g(a)f(supp)s(ort)-118 1652 y(of)k Fq(\026)p Fs(.)54 b(It)36 b(is)f(clear)h(that)f Fq(S)6 b Fn(\()p Fq(\026)p Fn(\))36 b Fs(is)f(the)h(in)m(tersection)g(of)g(all)d(closed)k(supp)s (orts)f(of)g Fq(\026)p Fs(.)53 b(It)36 b(is)g(ob)m(vious)g(that)-118 1776 y(t)m(w)m(o)29 b(equiv)-5 b(alen)m(t)29 b(measures)g(ha)m(v)m(e)h (the)f(same)g(top)s(ological)c(supp)s(ort)k(but)g(the)g(con)m(v)m(erse) i(is)d(not)h(generally)-118 1901 y(true.)62 b(A)m(ccording)38 b(to)h(the)g(Leb)s(esgue-Jordan)g(decomp)s(osition)e Fq(\026)g Fn(=)h Fq(\026)2582 1916 y Fk(s)2645 1901 y Fn(+)26 b Fq(\026)2806 1916 y Fk(ac)2878 1901 y Fq(;)212 b(\026)3176 1916 y Fk(s)3250 1901 y Fn(=)38 b Fq(\026)3423 1916 y Fk(pp)3524 1901 y Fn(+)26 b Fq(\026)3685 1916 y Fk(sc)3752 1901 y Fq(;)-118 2025 y Fs(where)f Fq(\026)214 2040 y Fk(s)250 2025 y Fs(,)h Fq(\026)362 2040 y Fk(pp)437 2025 y Fs(,)f Fq(\026)548 2040 y Fk(sc)639 2025 y Fs(and)f Fq(\026)879 2040 y Fk(ac)975 2025 y Fs(are)g(the)g(corresp)s(onding)f (singular,)i(pure)f(p)s(oin)m(t,)h(singular)d(con)m(tin)m(uous)i(and) -118 2150 y(absolutely)32 b(con)m(tin)m(uous)h(measures)g(of)f Fq(\026)p Fs(,)h(resp)s(ectiv)m(ely)-8 b(.)44 b(W)-8 b(e)33 b(set)g Fq(S)2423 2165 y Fk(\034)2466 2150 y Fn(\()p Fq(\026)p Fn(\))27 b(:=)h Fq(S)6 b Fn(\()p Fq(\026)2922 2165 y Fk(\034)2965 2150 y Fn(\))p Fs(,)32 b Fq(\034)39 b Fn(=)28 b Fq(s;)17 b(pp;)g(sc;)g(ac)p Fs(.)255 2274 y(The)28 b(sets)f Fq(S)694 2289 y Fk(s)731 2274 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)g Fq(S)980 2289 y Fk(pp)1055 2274 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)g Fq(S)1304 2289 y Fk(sc)1371 2274 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)g Fq(S)1620 2289 y Fk(ac)1693 2274 y Fn(\()p Fq(\026)p Fn(\))e Fs(are)h(closed)h(and)f (called)f(singular,)h(pure)h(p)s(oin)m(t,)f(sin-)-118 2399 y(gular)i(con)m(tin)m(uous)i(and)f(absolutely)g(con)m(tin)m(uous)h (supp)s(orts)g(of)f Fq(\026)p Fs(.)42 b(W)-8 b(e)30 b(note)f(that)g (the)h(closed)g(supp)s(orts)-118 2523 y Fq(S)-58 2538 y Fk(s)-21 2523 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)42 b Fq(S)243 2538 y Fk(pp)318 2523 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)h Fq(S)583 2538 y Fk(ac)655 2523 y Fn(\()p Fq(\026)p Fn(\))d Fs(and)h Fq(S)1088 2538 y Fk(sc)1156 2523 y Fn(\()p Fq(\026)p Fn(\))f Fs(are)h(not)g(generally)e(m)m(utually)h(disjoin)m (t.)67 b(T)-8 b(o)41 b(obtain)e(m)m(utually)-118 2648 y(disjoin)m(t)31 b(supp)s(orts)j(w)m(e)f(in)m(tro)s(duce)g(the)g(follo) m(wing)c(\(not)k(necessarily)g(closed\))f(sets:)366 2871 y Fq(S)432 2830 y Fo(0)426 2896 y Fk(s)463 2871 y Fn(\()p Fq(\026)p Fn(\))82 b(:=)h Fm(f)p Fq(t)28 b Fm(2)g Fj(R)39 b Fn(:)27 b Fq(d\026)p Fn(\()p Fq(t)p Fn(\))p Fq(=dt)g Fs(exists)33 b(and)28 b Fq(d\026)p Fn(\()p Fq(t)p Fn(\))p Fq(=dt)f Fn(=)h Fm(1g)878 b Fs(\(3.1\))327 3021 y Fq(S)393 2980 y Fo(0)387 3045 y Fk(pp)463 3021 y Fn(\()p Fq(\026)p Fn(\))82 b(:=)h Fm(f)p Fq(t)28 b Fm(2)g Fj(R)39 b Fn(:)27 b Fq(\026)p Fn(\()p Fm(f)p Fq(t)p Fm(g)p Fn(\))g Fm(6)p Fn(=)h(0)p Fm(g)p Fq(;)1825 b Fs(\(3.2\))335 3170 y Fq(S)401 3129 y Fo(0)395 3195 y Fk(sc)463 3170 y Fn(\()p Fq(\026)p Fn(\))82 b(:=)h Fm(f)p Fq(t)28 b Fm(2)g Fj(R)39 b Fn(:)27 b Fq(d\026)p Fn(\()p Fq(t)p Fn(\))p Fq(=dt)g Fs(exists)q Fq(;)44 b(d\026)p Fn(\()p Fq(t)p Fn(\))p Fq(=dt)27 b Fn(=)h Fm(1)f Fs(and)h Fq(\026)p Fn(\()p Fm(f)p Fq(t)p Fm(g)p Fn(\))f(=)g(0)p Fm(g)363 b Fs(\(3.3\))330 3320 y Fq(S)396 3279 y Fo(0)390 3344 y Fk(ac)463 3320 y Fn(\()p Fq(\026)p Fn(\))82 b(:=)h Fm(f)p Fq(t)28 b Fm(2)g Fj(R)39 b Fn(:)27 b Fq(d\026)p Fn(\()p Fq(t)p Fn(\))p Fq(=dt)g Fs(exists)33 b(and)28 b Fn(0)g Fq(<)f(d\026)p Fn(\()p Fq(t)p Fn(\))p Fq(=dt)g(<)h Fm(1g)698 b Fs(\(3.4\))-118 3543 y(where)40 b(the)g(distribution)d(function)i Fq(\026)p Fn(\()p Fm(\001)p Fn(\))f Fs(is)h(similar)d(to)i(\(2.5\))h(de\034ned.) 65 b(By)39 b(De)h(la)e(V)-8 b(all\351e)38 b(P)m(oussin's)-118 3668 y(Decomp)s(osition)27 b(Theorem)i(\(see)h([28]\))f(it)f(turns)h (out)g(that)g Fq(S)2149 3631 y Fo(0)2143 3692 y Fk(s)2180 3668 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)g Fq(S)2437 3631 y Fo(0)2431 3692 y Fk(pp)2506 3668 y Fn(\()p Fq(\026)p Fn(\))g Fs(and)g Fq(S)2922 3631 y Fo(0)2916 3692 y Fk(sc)2983 3668 y Fn(\()p Fq(\026)p Fn(\))f Fs(are)h(supp)s(orts)h(of)-118 3792 y Fq(\026)-59 3807 y Fk(s)-23 3792 y Fs(,)37 b Fq(\026)100 3807 y Fk(pp)174 3792 y Fs(,)g Fq(\026)297 3807 y Fk(sc)399 3792 y Fs(and)f Fq(\026)651 3807 y Fk(ac)723 3792 y Fs(.)52 b(Ob)m(viously)-8 b(,)36 b(the)g(sets)g Fq(S)1713 3756 y Fo(0)1707 3817 y Fk(s)1744 3792 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)g Fq(S)2008 3756 y Fo(0)2002 3817 y Fk(ac)2074 3792 y Fn(\()p Fq(\026)p Fn(\))f Fs(as)h(w)m(ell)e(as)i Fq(S)2756 3756 y Fo(0)2750 3817 y Fk(pp)2825 3792 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)g Fq(S)3089 3756 y Fo(0)3083 3817 y Fk(sc)3150 3792 y Fn(\()p Fq(\026)p Fn(\))f Fs(and)h Fq(S)3579 3756 y Fo(0)3573 3817 y Fk(ac)3645 3792 y Fn(\()p Fq(\026)p Fn(\))-118 3917 y Fs(are)e(m)m(utually)e(disjoin)m(t.)46 b(Since)34 b(the)h(sets)g Fq(S)1532 3881 y Fo(0)1526 3941 y Fk(\034)1569 3917 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)f Fq(\034)41 b Fn(=)30 b Fq(s;)17 b(pp;)g(sc)p Fs(,)33 b(are)h(of)g(Leb)s(esgue)h(measure)f(zero)g(and)-118 4041 y(m)m(utually)d(disjoin)m(t)g(w)m(e)j(\034nd)f(that)f(for)g(an)m (y)h(Borel)f(set)h Fm(X)42 b(\022)29 b Fj(R)43 b Fs(one)33 b(has)945 4265 y Fq(\026)p Fn(\()p Fm(X)j(\\)23 b Fq(S)1304 4224 y Fo(0)1298 4289 y Fk(\034)1341 4265 y Fn(\()p Fq(\026)p Fn(\)\))k(=)g Fq(\026)1703 4280 y Fk(\034)1746 4265 y Fn(\()p Fm(X)15 b Fn(\))p Fq(;)114 b(\034)39 b Fn(=)28 b Fq(s;)17 b(pp;)g(sc;)g(ac:)861 b Fs(\(3.5\))-118 4488 y(Therefore)29 b(w)m(e)g(call)d(in)h(the)h(sequel)h(the)f(sets)h Fq(S)1605 4452 y Fo(0)1599 4513 y Fk(s)1636 4488 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)f Fq(S)1892 4452 y Fo(0)1886 4513 y Fk(pp)1961 4488 y Fn(\()p Fq(\026)p Fn(\))p Fs(,)h Fq(S)2218 4452 y Fo(0)2212 4513 y Fk(sc)2279 4488 y Fn(\()p Fq(\026)p Fn(\))e Fs(and)h Fq(S)2692 4452 y Fo(0)2686 4513 y Fk(ac)2758 4488 y Fn(\()p Fq(\026)p Fn(\))g Fs(singular,)f(pure) h(p)s(oin)m(t,)-118 4613 y(singular)j(con)m(tin)m(uous)i(and)g (absolutely)e(con)m(tin)m(uous)j(supp)s(orts)f(of)f Fq(\026)p Fs(,)g(resp)s(ectiv)m(ely)-8 b(.)45 b(W)-8 b(e)33 b(note)f(that)550 4836 y Fq(S)610 4851 y Fk(pp)685 4836 y Fn(\()p Fq(\026)p Fn(\))27 b(=)p 951 4749 270 4 v 28 w Fq(S)1017 4807 y Fo(0)1011 4861 y Fk(pp)1086 4836 y Fn(\()p Fq(\026)p Fn(\))97 b Fs(and)h Fq(S)1633 4851 y Fk(\034)1676 4836 y Fn(\()p Fq(\026)p Fn(\))27 b Fm(\022)p 1943 4749 238 4 v 28 w Fq(S)2009 4807 y Fo(0)2003 4861 y Fk(\034)2046 4836 y Fn(\()p Fq(\026)p Fn(\))h Fm(\022)g Fq(S)6 b Fn(\()p Fq(\026)p Fn(\))p Fq(;)43 b(\034)d Fn(=)27 b Fq(s;)17 b(sc;)g(ac:)467 b Fs(\(3.6\))-118 5059 y(In)33 b(general,)f(it)f(is)h (not)h(p)s(ossible)f(to)g(replace)g(inclusions)g(b)m(y)h(equalities.) 255 5184 y(Let)26 b(no)m(w)g Fn(\006\()p Fm(\001)p Fn(\))g Fs(b)s(e)g(a)f(measure)h(with)f(v)-5 b(alues)25 b(in)g Fn([)p Fm(H)q Fn(])p Fs(.)41 b(The)27 b(measure)f Fn(\006\()p Fm(\001)p Fn(\))f Fs(admits)f(a)i(Leb)s(esgue-)-118 5309 y(Jordan)i(decomp)s(osition)e Fn(\006)i(=)f(\006)1113 5272 y Fk(s)1163 5309 y Fn(+)13 b(\006)1322 5272 y Fk(ac)1394 5309 y Fs(,)29 b Fn(\006)1520 5272 y Fk(s)1585 5309 y Fn(=)f(\006)1759 5272 y Fk(pp)1847 5309 y Fn(+)13 b(\006)2006 5272 y Fk(sc)2073 5309 y Fs(.)42 b(As)29 b(ab)s(o)m(v)m(e)f(the)h (notation)d Fq(S)3162 5324 y Fk(s)3199 5309 y Fn(\(\006\))i(:=)g Fq(S)6 b Fn(\(\006)3678 5272 y Fk(s)3715 5309 y Fn(\))p Fs(,)-118 5433 y Fq(S)-58 5448 y Fk(pp)17 5433 y Fn(\(\006\))48 b(:=)f Fq(S)6 b Fn(\(\006)535 5397 y Fk(pp)610 5433 y Fn(\))p Fs(,)47 b Fq(S)782 5448 y Fk(sc)850 5433 y Fn(\(\006\))h(:=)f Fq(S)6 b Fn(\(\006)1368 5397 y Fk(sc)1435 5433 y Fn(\))44 b Fs(and)h Fq(S)1779 5448 y Fk(ac)1851 5433 y Fn(\(\006\))i(:=)h Fq(S)6 b Fn(\(\006)2369 5448 y Fk(ac)2441 5433 y Fn(\))44 b Fs(stands)h(for)f(the)g(singular,)i(pure)p eop %%Page: 8 8 8 7 bop -118 100 a Fs(p)s(oin)m(t,)39 b(singular)f(con)m(tin)m(uous)h (and)g(absolutely)f(con)m(tin)m(uous)h(closed)g(supp)s(orts)h(of)e(the) h(measure)g Fn(\006\()p Fm(\001)p Fn(\))p Fs(,)-118 224 y(resp)s(ectiv)m(ely)-8 b(.)44 b(Moreo)m(v)m(er,)34 b(w)m(e)g(set)1192 448 y Fq(S)1252 463 y Fk(p)1292 448 y Fn(\(\006\))28 b(=)g Fm(f)p Fq(t)f Fm(2)h Fj(R)39 b Fn(:)28 b(\006\()p Fm(f)p Fq(t)p Fm(g)p Fn(\))g Fm(6)p Fn(=)f(0)p Fm(g)p Fq(:)1110 b Fs(\(3.7\))-118 672 y(Ob)m(viously)-8 b(,)32 b(w)m(e)i(ha)m(v)m(e)g Fq(S)788 687 y Fk(p)827 672 y Fn(\(\006\))28 b Fm(\022)g Fq(S)1166 687 y Fk(pp)1242 672 y Fn(\(\006\))33 b Fs(and)p 1610 586 246 4 v 32 w Fq(S)1670 687 y Fk(p)1710 672 y Fn(\(\006\))28 b(=)f Fq(S)2047 687 y Fk(pp)2123 672 y Fn(\(\006\))p Fs(.)255 797 y(With)h(eac)m(h)g(op)s(erator-v)-5 b(alued)26 b(measure)i Fn(\006\()p Fm(\001)p Fn(\))g Fs(w)m(e)h(can)f(asso)s(ciate)f(a)h (scalar)f(measure)h Fn(\006)3500 812 y Fk(h)3545 797 y Fn(\()p Fm(\001)p Fn(\))g(:=)-118 922 y(\(\006\()p Fm(\001)p Fn(\))p Fq(h;)17 b(h)p Fn(\))p Fs(,)40 b Fq(h)d Fm(2)h(H)q Fs(.)61 b(In)38 b(the)h(follo)m(wing)c(w)m(e)40 b(are)e(in)m(terested)h(in)f(the)h(problem)d(whether)k(the)f(sp)s (ectral)-118 1046 y(prop)s(erties)h(of)g(the)g(op)s(erator-v)-5 b(alued)39 b(measure)i Fn(\006\()p Fm(\001)p Fn(\))f Fs(can)g(b)s(e)h(c)m(haracterized)g(b)m(y)g(a)f(family)d(of)j(scalar) -118 1171 y(measures.)64 b(T)-8 b(o)39 b(this)g(end)h(let)f Fm(T)65 b Fn(=)38 b Fm(f)p Fq(h)1383 1186 y Fk(k)1426 1171 y Fm(g)1476 1134 y Fk(N)1476 1197 y(k)r Fl(=1)1609 1171 y Fs(,)j Fn(1)e Fm(\024)g Fq(N)50 b Fm(\024)39 b Fn(+)p Fm(1)p Fs(,)i(b)s(e)e(a)g(total)f(set)i(in)e Fm(H)q Fs(.)64 b(With)39 b Fm(T)64 b Fs(w)m(e)-118 1295 y(asso)s(ciate)32 b(the)h(family)d Fm(f)p Fn(\006)878 1310 y Fk(h)919 1322 y Fg(k)961 1295 y Fn(\()p Fm(\001)p Fn(\))p Fm(g)1115 1259 y Fk(N)1115 1321 y(k)r Fl(=1)1280 1295 y Fs(of)i(scalar)g (measures.)44 b(Let)33 b(us)g(in)m(tro)s(duce)f(the)h(follo)m(wing)d (sets:)1045 1519 y Fq(S)1111 1478 y Fo(0)1105 1544 y Fk(s)1141 1519 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))83 b(:=)g Fm([)1746 1478 y Fk(N)1746 1544 y(k)r Fl(=1)1879 1519 y Fq(S)1945 1478 y Fo(0)1939 1544 y Fk(s)1976 1519 y Fn(\(\006)2084 1534 y Fk(h)2125 1546 y Fg(k)2167 1519 y Fn(\))p Fq(;)1347 b Fs(\(3.8\))1006 1669 y Fq(S)1072 1628 y Fo(0)1066 1693 y Fk(pp)1141 1669 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))83 b(:=)g Fm([)1746 1628 y Fk(N)1746 1693 y(k)r Fl(=1)1879 1669 y Fq(S)1945 1628 y Fo(0)1939 1693 y Fk(pp)2014 1669 y Fn(\(\006)2122 1684 y Fk(h)2163 1696 y Fg(k)2206 1669 y Fn(\))p Fq(;)1308 b Fs(\(3.9\))1014 1818 y Fq(S)1080 1777 y Fo(0)1074 1843 y Fk(sc)1141 1818 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))83 b(:=)g Fm([)1746 1777 y Fk(N)1746 1843 y(k)r Fl(=1)1879 1818 y Fq(S)1945 1777 y Fo(0)1939 1843 y Fk(sc)2007 1818 y Fn(\(\006)2115 1833 y Fk(h)2156 1845 y Fg(k)2198 1818 y Fn(\))22 b Fm(n)g Fq(S)2396 1777 y Fo(0)2390 1843 y Fk(pp)2465 1818 y Fn(\(\006\))p Fq(;)892 b Fs(\(3.10\))1009 1968 y Fq(S)1075 1926 y Fo(0)1069 1992 y Fk(ac)1141 1968 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))83 b(:=)g Fm([)1746 1926 y Fk(N)1746 1992 y(k)r Fl(=1)1879 1968 y Fq(S)1945 1926 y Fo(0)1939 1992 y Fk(ac)2011 1968 y Fn(\(\006)2119 1983 y Fk(h)2160 1995 y Fg(k)2203 1968 y Fn(\))22 b Fm(n)g Fq(S)2401 1926 y Fo(0)2395 1992 y Fk(s)2432 1968 y Fn(\(\006\))p Fq(:)925 b Fs(\(3.11\))-118 2192 y(By)43 b(de\034nition)f(the)h(sets)h Fq(S)934 2156 y Fo(0)928 2216 y Fk(s)965 2192 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))43 b Fs(and)g Fq(S)1543 2156 y Fo(0)1537 2216 y Fk(ac)1609 2192 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))43 b Fs(are)f(m)m(utually)f(disjoin)m(t.)73 b(The)44 b(same)f(holds)f(for) -118 2316 y Fq(S)-52 2280 y Fo(0)-58 2341 y Fk(pp)17 2316 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Fs(,)32 b Fq(S)412 2280 y Fo(0)406 2341 y Fk(sc)474 2316 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))33 b Fs(and)f Fq(S)1031 2280 y Fo(0)1025 2341 y Fk(ac)1098 2316 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))p Fs(.)255 2524 y Ft(LEMMA)38 b(3.1)49 b Ff(L)-5 b(et)30 b Fm(H)g Ff(b)-5 b(e)30 b(a)f(sep)-5 b(ar)g(able)28 b(Hilb)-5 b(ert)30 b(sp)-5 b(ac)g(e)29 b(and)g(let)h Fn(\006\()p Fm(\001)p Fn(\))f Ff(b)-5 b(e)29 b(an)h(op)-5 b(er)g(ator-value)g(d) -118 2648 y(me)g(asur)g(e)35 b(in)h Fm(H)q Ff(.)49 b(F)-7 b(urther,)36 b(let)g Fm(T)56 b Fn(=)30 b Fm(f)p Fq(h)1389 2663 y Fk(k)1431 2648 y Fm(g)1481 2612 y Fk(N)1481 2674 y(k)r Fl(=1)1614 2648 y Ff(,)36 b Fn(1)30 b Fm(\024)h Fq(N)40 b Fm(\024)31 b Fn(+)p Fm(1)p Ff(,)k(b)-5 b(e)36 b(a)g(total)h(set)f(in)g Fm(H)q Ff(.)48 b(Then)36 b(the)g(sets)-118 2773 y Fq(S)-52 2737 y Fo(0)-58 2798 y Fk(s)-21 2773 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))p Ff(,)41 b Fq(S)385 2737 y Fo(0)379 2798 y Fk(pp)454 2773 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))p Ff(,)41 b Fq(S)860 2737 y Fo(0)854 2798 y Fk(sc)921 2773 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))39 b Ff(and)g Fq(S)1490 2737 y Fo(0)1484 2798 y Fk(ac)1556 2773 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))39 b Ff(ar)-5 b(e)39 b(singular,)h(pur)-5 b(e)39 b(p)-5 b(oint,)40 b(singular)f(c)-5 b(ontinuous)-118 2897 y(and)34 b(absolutely)h(c)-5 b(ontinuous)34 b(supp)-5 b(orts)35 b(of)g Fn(\006\()p Fm(\001)p Fn(\))p Ff(,)f(r)-5 b(esp)g(e)g(ctively,)34 b(i.e.)864 3122 y Fn(\006\()p Fm(X)j(\\)23 b Fq(S)1235 3080 y Fo(0)1229 3146 y Fk(\034)1272 3122 y Fn(\(\006;)17 b Fm(T)26 b Fn(\)\))h(=)h(\006)1781 3080 y Fk(\034)1824 3122 y Fn(\()p Fm(X)15 b Fn(\))p Fq(;)116 b(\034)39 b Fn(=)28 b Fq(s;)17 b(pp;)g(sc;)g(ac;)732 b Fs(\(3.12\))-118 3346 y Ff(for)35 b(any)f(Bor)-5 b(el)34 b(set)h Fm(X)43 b(\022)28 b Fj(R)5 b Ff(.)50 b(In)35 b(p)-5 b(articular,)34 b(the)h(fol)5 b(lowing)33 b(r)-5 b(elations)35 b(hold)472 3570 y Fq(S)532 3585 y Fk(p)572 3570 y Fn(\(\006\))28 b(=)f Fq(S)915 3529 y Fo(0)909 3595 y Fk(pp)985 3570 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))j Ff(and)55 b Fq(S)1552 3585 y Fk(\034)1595 3570 y Fn(\(\006\))28 b Fm(\022)p 1874 3483 373 4 v 28 w Fq(S)1940 3541 y Fo(0)1934 3595 y Fk(\034)1977 3570 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))h Fm(\022)h Fq(S)6 b Fn(\(\006\))p Fq(;)45 b(\034)39 b Fn(=)28 b Fq(s;)17 b(sc;)g(ac:)340 b Fs(\(3.13\))-118 3794 y Ff(Pr)-5 b(o)g(of:)43 b Fs(By)33 b(the)g(Leb)s(esgue-Jordan)g (decomp)s(osition)e(one)h(easily)g(gets)h(that)f(for)g(eac)m(h)i Fq(h)28 b Fm(2)g(H)33 b Fs(w)m(e)h(ha)m(v)m(e)939 4018 y Fn(\(\006)1047 3977 y Fk(\034)1091 4018 y Fn(\()p Fm(X)15 b Fn(\))p Fq(h;)i(h)p Fn(\))27 b(=)h(\006)1648 4033 y Fk(h;\034)1751 4018 y Fn(\()p Fm(X)15 b Fn(\))p Fq(;)114 b(\034)39 b Fn(=)28 b Fq(s;)17 b(pp;)g(sc;)g(ac;)807 b Fs(\(3.14\))-118 4242 y(for)37 b(an)m(y)h(Borel)f(set)h Fm(X)51 b(\022)36 b Fj(R)5 b Fs(,)45 b(where)39 b Fn(\006)1376 4257 y Fk(h;\034)1480 4242 y Fn(\()p Fm(\001)p Fn(\))e Fs(arises)g(from)f(the)i(Leb)s(esgue-Jordan)g(decomp)s(osition)e(of) -118 4367 y(the)d(scalar)f(measure)h Fn(\006)777 4382 y Fk(h)822 4367 y Fn(\()p Fm(\001)p Fn(\))p Fs(.)255 4491 y(Let)g Fq(\034)39 b Fn(=)28 b Fq(s)p Fs(.)43 b(Since)33 b(mes)p Fn(\()p Fq(S)1253 4455 y Fo(0)1247 4516 y Fk(s)1284 4491 y Fn(\(\006;)17 b Fm(T)25 b Fn(\)\))j(=)g(0)k Fs(w)m(e)h(get)244 4716 y Fn(\(\006\()p Fm(X)k(\\)22 b Fq(S)652 4675 y Fo(0)646 4740 y Fk(s)683 4716 y Fn(\(\006;)17 b Fm(T)26 b Fn(\)\))p Fq(h)1047 4731 y Fk(k)1089 4716 y Fq(;)17 b(h)1189 4731 y Fk(k)1232 4716 y Fn(\))28 b(=)f(\006)1471 4731 y Fk(h)1512 4743 y Fg(k)1555 4716 y Fn(\()p Fm(X)36 b(\\)23 b Fq(S)1855 4675 y Fo(0)1849 4740 y Fk(s)1886 4716 y Fn(\(\006;)17 b Fm(T)25 b Fn(\)\))j(=)f(\006)2394 4731 y Fk(h)2435 4743 y Fg(k)2474 4731 y Fk(;s)2530 4716 y Fn(\()p Fm(X)37 b(\\)22 b Fq(S)2830 4675 y Fo(0)2824 4740 y Fk(s)2861 4716 y Fn(\(\006;)17 b Fm(T)26 b Fn(\)\))361 b Fs(\(3.15\))-118 4940 y(for)32 b(an)m(y)h Fq(h)271 4955 y Fk(k)342 4940 y Fm(2)28 b(T)d Fs(.)44 b(Using)32 b(\(3.14\),)g(\(3.15\))f(and)390 5164 y Fn(\006)460 5179 y Fk(h)501 5191 y Fg(k)540 5179 y Fk(;s)596 5164 y Fn(\()p Fm(X)37 b(\\)22 b Fq(S)896 5123 y Fo(0)890 5189 y Fk(s)927 5164 y Fn(\(\006;)17 b Fm(T)26 b Fn(\)\))h(=)2192 b Fs(\(3.16\))557 5313 y Fn(\006)627 5328 y Fk(h)668 5340 y Fg(k)706 5328 y Fk(;s)762 5313 y Fn(\()p Fm(X)37 b(\\)22 b Fq(S)1062 5272 y Fo(0)1056 5338 y Fk(s)1093 5313 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))c Fm(\\)h Fq(S)1540 5272 y Fo(0)1534 5338 y Fk(s)1570 5313 y Fn(\(\006)1678 5328 y Fk(h)1719 5340 y Fg(k)1762 5313 y Fn(\)\))k(=)h(\006)2039 5328 y Fk(h)2080 5340 y Fg(k)2118 5328 y Fk(;s)2174 5313 y Fn(\()p Fm(X)37 b(\\)23 b Fq(S)2475 5272 y Fo(0)2469 5338 y Fk(s)2505 5313 y Fn(\(\006)2613 5328 y Fk(h)2654 5340 y Fg(k)2697 5313 y Fn(\)\))k(=)h(\006)2974 5328 y Fk(h)3015 5340 y Fg(k)3053 5328 y Fk(;s)3109 5313 y Fn(\()p Fm(X)15 b Fn(\))p eop %%Page: 9 9 9 8 bop -118 100 a Fs(w)m(e)32 b(\034nd)f Fn(\(\006\()p Fm(X)j(\\)19 b Fq(S)619 63 y Fo(0)613 124 y Fk(s)649 100 y Fn(\(\006;)e Fm(T)26 b Fn(\)\))p Fq(h)1013 115 y Fk(k)1056 100 y Fq(;)17 b(h)1156 115 y Fk(k)1198 100 y Fn(\))28 b(=)f(\(\006)1475 63 y Fk(s)1513 100 y Fn(\()p Fm(X)15 b Fn(\))p Fq(h)1731 115 y Fk(k)1773 100 y Fq(;)i(h)1873 115 y Fk(k)1916 100 y Fn(\))30 b Fs(for)h(an)m(y)g Fq(h)2370 115 y Fk(k)2440 100 y Fm(2)e(T)c Fs(.)43 b(Since)31 b Fm(T)56 b Fs(is)31 b(total)e(w)m(e)j(\034nally)-118 224 y(obtain)23 b Fn(\006\()p Fm(X)d(\\)5 b Fq(S)513 188 y Fo(0)507 249 y Fk(s)544 224 y Fn(\(\006;)17 b Fm(T)26 b Fn(\)\))h(=)h(\006)1053 188 y Fk(s)1090 224 y Fn(\()p Fm(X)15 b Fn(\))24 b Fs(for)g(an)m(y)g(Borel)g(set)h Fm(X)42 b(\022)28 b Fj(R)5 b Fs(.)47 b(Similarly)-8 b(,)21 b(w)m(e)26 b(pro)m(v)m(e)f(the)g(statemen)m(ts)-118 349 y(for)32 b Fq(\034)39 b Fn(=)28 b Fq(pp;)17 b(sc;)g(ac)p Fs(.)255 473 y(Let)26 b Fq(x)j Fm(2)f Fq(S)667 437 y Fo(0)661 498 y Fk(pp)736 473 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))p Fs(.)42 b(Then)26 b(there)h(is)e Fq(h)1711 488 y Fk(k)1782 473 y Fm(2)j(T)51 b Fs(suc)m(h)27 b(that)f Fq(x)i Fm(2)g Fq(S)2642 437 y Fo(0)2636 498 y Fk(pp)2711 473 y Fn(\(\006)2819 488 y Fk(h)2860 500 y Fg(k)2903 473 y Fn(\))p Fs(.)41 b(Hence)27 b Fn(\006)3362 488 y Fk(h)3403 500 y Fg(k)3445 473 y Fn(\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))h(=)-118 598 y(\(\006\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))p Fq(h)277 613 y Fk(k)320 598 y Fq(;)17 b(h)420 613 y Fk(k)463 598 y Fn(\))38 b Fm(6)p Fn(=)h(0)g Fs(whic)m(h)g(yields)g Fn(\006\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))g Fm(6)p Fn(=)g(0)g Fs(or)g Fq(x)g Fm(2)g Fq(S)2236 613 y Fk(p)2276 598 y Fn(\(\006\))p Fs(,)i(i.e.)63 b Fq(S)2744 562 y Fo(00)2738 622 y Fk(pp)2813 598 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))39 b Fm(\022)h Fq(S)3298 613 y Fk(p)3337 598 y Fn(\(\006\))p Fs(.)64 b(Con-)-118 722 y(v)m(ersely)-8 b(,)50 b(if)45 b Fq(x)51 b Fm(2)g Fq(S)626 737 y Fk(p)665 722 y Fn(\(\006\))q Fs(,)e(then)e(there)f(is)g(a)f Fq(h)51 b Fm(2)f(H)d Fs(suc)m(h)h(that)d Fn(\006)2474 737 y Fk(h)2520 722 y Fn(\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))50 b Fm(6)p Fn(=)g(0)p Fs(.)83 b(Hence)48 b(there)e(ex-)-118 847 y(ists)g(a)h Fq(h)222 862 y Fk(k)316 847 y Fm(2)52 b(T)72 b Fs(suc)m(h)48 b(that)e Fn(\006)1089 862 y Fk(h)1130 874 y Fg(k)1173 847 y Fn(\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))51 b(=)h(\(\006\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))p Fq(h)1978 862 y Fk(k)2021 847 y Fq(;)17 b(h)2121 862 y Fk(k)2163 847 y Fn(\))52 b Fm(6)p Fn(=)f(0)p Fs(.)86 b(If)46 b(this)g(is)g(not)h (the)g(case,)k(then)-118 971 y(for)46 b(eac)m(h)i Fq(h)335 986 y Fk(k)429 971 y Fm(2)53 b(T)72 b Fs(one)47 b(has)g Fn(\006)1125 986 y Fk(h)1166 998 y Fg(k)1208 971 y Fn(\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))52 b(=)g(\(\006\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))p Fq(h)2014 986 y Fk(k)2057 971 y Fq(;)17 b(h)2157 986 y Fk(k)2200 971 y Fn(\))51 b(=)h(0)p Fs(.)86 b(Since)47 b Fm(T)72 b Fs(is)46 b(total)g(this)g(yields)-118 1096 y Fn(\(\006\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))p Fq(h;)17 b(h)p Fn(\))30 b(=)f(\006)620 1111 y Fk(h)666 1096 y Fn(\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))g(=)h(0)j Fs(for)g(eac)m(h)i Fq(h)30 b Fm(2)g(H)q Fs(,)k(con)m(trary)g(to)g(the)g(assumption.)46 b(Ho)m(w)m(ev)m(er,)36 b(if)d(there)-118 1220 y(is)39 b(a)g Fq(h)131 1235 y Fk(k)212 1220 y Fm(2)h(T)65 b Fs(suc)m(h)40 b(that)f Fn(\006)951 1235 y Fk(h)992 1247 y Fg(k)1035 1220 y Fn(\()p Fm(f)p Fq(x)p Fm(g)p Fn(\))g Fm(6)p Fn(=)f(0)p Fs(,)j(then)f Fq(x)f Fm(2)h Fq(S)2031 1184 y Fo(0)2025 1245 y Fk(pp)2100 1220 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Fs(,)41 b(i.e.)62 b Fq(S)2685 1235 y Fk(p)2725 1220 y Fn(\(\006\))39 b Fm(\022)h Fq(S)3093 1184 y Fo(00)3087 1245 y Fk(pp)3162 1220 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Fs(.)63 b(Hence)-118 1345 y Fq(S)-58 1360 y Fk(p)-18 1345 y Fn(\(\006\))28 b(=)f Fq(S)325 1309 y Fo(00)319 1370 y Fk(pp)394 1345 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Fs(.)255 1485 y(F)-8 b(urther,)38 b(from)e(\(3.12\))g(w)m(e)h(get)g Fq(S)1536 1500 y Fk(\034)1579 1485 y Fn(\(\006\))e Fm(\022)p 1872 1399 373 4 v 35 w Fq(S)1938 1457 y Fo(0)1932 1510 y Fk(\034)1976 1485 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))p Fs(,)38 b Fq(\034)46 b Fn(=)35 b Fq(s;)17 b(pp;)g(sc;)g(ac)p Fs(.)55 b(T)-8 b(aking)36 b(\(3.6\))g(in)m(to)-118 1610 y(accoun)m(t)g(w)m(e)g(get)f Fq(S)622 1574 y Fo(0)616 1635 y Fk(\034)659 1610 y Fn(\(\006)767 1625 y Fk(h)808 1637 y Fg(k)851 1610 y Fn(\))d Fm(\022)h Fq(S)6 b Fn(\(\006)1205 1625 y Fk(h)1246 1637 y Fg(k)1288 1610 y Fn(\))p Fs(,)36 b Fq(\034)43 b Fn(=)32 b Fq(s;)17 b(sc;)g(ac)p Fs(,)36 b(for)f(eac)m(h)h Fq(h)2390 1625 y Fk(k)2465 1610 y Fm(2)c(T)26 b Fs(.)51 b(Since)36 b Fq(S)6 b Fn(\(\006)3153 1625 y Fk(h)3194 1637 y Fg(k)3236 1610 y Fn(\))32 b Fm(\022)h Fq(S)6 b Fn(\(\006\))35 b Fs(for)-118 1735 y(eac)m(h)e Fq(h)157 1750 y Fk(k)228 1735 y Fm(2)28 b(T)58 b Fs(w)m(e)34 b(get)e Fq(S)806 1698 y Fo(0)800 1759 y Fk(\034)843 1735 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))i Fm(\022)g Fq(S)6 b Fn(\(\006\))p Fq(;)114 b(\034)39 b Fn(=)28 b Fq(s;)17 b(sc;)g(ac;)32 b Fs(whic)m(h)h(immediately)c(pro)m(v)m(es)34 b(\(3.13\))148 b Fm(4)255 1859 y Fs(Let)28 b Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))27 b Fs(b)s(e)g(the)g(W)-8 b(eyl)27 b(function)f(of)h Fq(A)g Fs(with)f(resp)s(ect)i(to)f(the)g(b)s (oundary)g(triple)f Fm(fH)q Fq(;)17 b Fn(\000)3520 1874 y Fl(0)3559 1859 y Fq(;)g Fn(\000)3664 1874 y Fl(1)3703 1859 y Fm(g)p Fs(.)-118 1984 y(Since)36 b(eac)m(h)g(W)-8 b(eyl)35 b(function)g(is)h(an)f(R-function)f(one)i(gets)g(the)g (represen)m(tation)g(\(2.2\).)52 b(T)-8 b(aking)35 b(\(2.13\))-118 2108 y(and)e(\(2.14\))e(in)m(to)h(accoun)m(t)h(w)m(e)h(obtain)d(that)i Fq(C)1640 2123 y Fl(1)1707 2108 y Fn(=)27 b(0)32 b Fs(whic)m(h)h(leads) g(to)f(the)h(represen)m(tation)764 2364 y Fq(M)10 b Fn(\()p Fq(z)t Fn(\))29 b(=)e Fq(C)1195 2379 y Fl(0)1257 2364 y Fn(+)1355 2229 y Fi(Z)1410 2454 y Fd(R)1479 2224 y Fi(\022)1641 2297 y Fn(1)p 1562 2341 207 4 v 1562 2433 a Fq(t)22 b Fm(\000)h Fq(z)1801 2364 y Fm(\000)2015 2297 y Fq(t)p 1910 2341 244 4 v 1910 2433 a Fn(1)f(+)g Fq(t)2114 2404 y Fl(2)2164 2224 y Fi(\023)2254 2364 y Fq(d)p Fn(\006\()p Fq(t)p Fn(\))p Fq(;)114 b(z)33 b Fm(2)28 b Fj(C)20 b Fq(:)638 b Fs(\(3.17\))-118 2614 y(In)30 b(the)g(follo)m(wing)d(w)m(e)j (are)g(in)m(terested)g(in)f(sp)s(ectral)g(prop)s(erties)h(of)f Fq(A)2430 2629 y Fl(0)2499 2614 y Fs(c)m(haracterized)h(b)m(y)h(the)f (op)s(erator-)-118 2739 y(v)-5 b(alued)32 b(measures)h Fn(\006\()p Fm(\001)p Fn(\))p Fs(.)44 b(Imp)s(ortan)m(t)31 b(is)h(the)h(follo)m(wing)d(lemma.)255 2928 y Ft(LEMMA)38 b(3.2)49 b Ff(L)-5 b(et)42 b Fq(A)f Ff(b)-5 b(e)41 b(a)g(simple)g (densely)f(de\034ne)-5 b(d)40 b(close)-5 b(d)41 b(symmetric)g(op)-5 b(er)g(ator)41 b(on)g(a)-118 3053 y(sep)-5 b(ar)g(able)35 b(Hilb)-5 b(ert)37 b(sp)-5 b(ac)g(e)35 b Fp(H)h Ff(with)h Fq(n)1255 3068 y Fl(+)1314 3053 y Fn(\()p Fq(A)p Fn(\))30 b(=)h Fq(n)1658 3068 y Fo(\000)1717 3053 y Fn(\()p Fq(A)p Fn(\))p Ff(.)49 b(F)-7 b(urther,)37 b(let)f Fn(\005)31 b(=)f Fm(fH)q Fq(;)17 b Fn(\000)2918 3068 y Fl(0)2957 3053 y Fq(;)g Fn(\000)3062 3068 y Fl(1)3101 3053 y Fm(g)37 b Ff(b)-5 b(e)36 b(a)g(b)-5 b(oundary)-118 3177 y(triple)35 b(of)f Fq(A)327 3141 y Fo(\003)402 3177 y Ff(with)g(W)-7 b(eyl)35 b(function)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Ff(.)255 3302 y(If)33 b Fq(E)428 3317 y Fk(A)481 3326 y Fc(0)520 3302 y Fn(\()p Fm(\001)p Fn(\))f Ff(is)h(the)g(sp)-5 b(e)g(ctr)g(al)32 b(me)-5 b(asur)g(e)33 b(of)f Fq(A)1825 3317 y Fl(0)1893 3302 y Fn(:=)27 b Fq(A)2096 3266 y Fo(\003)2136 3302 y Fm(j)17 b Fn(k)m(er\(\000)2409 3317 y Fl(0)2449 3302 y Fn(\))32 b(\()p Fm(2)d Fq(E)6 b(xt)2820 3317 y Fk(A)2877 3302 y Fn(\))33 b Ff(and)g Fn(\006\()p Fm(\001)p Fn(\))g Ff(that)g(of)g(the)-118 3426 y(inte)-5 b(gr)g(al)33 b(r)-5 b(epr)g(esentation)33 b(\(3.17\))g(of)g(the)h(W)-7 b(eyl)34 b(function)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Ff(,)34 b(then)g(the)f(me)-5 b(asur)g(es)33 b Fq(E)3188 3441 y Fk(A)3241 3450 y Fc(0)3280 3426 y Fn(\()p Fm(\001)p Fn(\))g Ff(and)h Fn(\006\()p Fm(\001)p Fn(\))-118 3551 y Ff(ar)-5 b(e)34 b(e)-5 b(quivalent.)45 b(In)34 b(p)-5 b(articular,)34 b(one)h(has)f Fq(\033)1557 3566 y Fk(\034)1600 3551 y Fn(\()p Fq(A)1711 3566 y Fl(0)1751 3551 y Fn(\))28 b(=)f Fq(S)1980 3566 y Fk(\034)2023 3551 y Fn(\(\006\))p Fq(;)117 b(\034)39 b Fn(=)28 b Fq(s;)17 b(p;)g(pp;)g(sc;)g(ac:)-118 3740 y Ff(Pr)-5 b(o)g(of:)43 b Fs(By)33 b(\(2.14\))f(one)g(obtains)442 3944 y Fm(=)p Fs(m)o Fn(\()p Fq(M)10 b Fn(\()p Fq(x)23 b Fn(+)f Fq(iy)t Fn(\))p Fq(h;)17 b(h)p Fn(\))27 b(=)2159 b Fs(\(3.18\))962 4093 y Fq(y)t Fn(\()p Fq(\015)1108 4052 y Fo(\003)1147 4093 y Fn(\()p Fq(x)22 b Fn(+)g Fq(iy)t Fn(\))p Fq(\015)5 b Fn(\()p Fq(x)22 b Fn(+)g Fq(iy)t Fn(\))p Fq(h;)17 b(h)p Fn(\))27 b(=)h Fq(y)t Fm(k)p Fq(\015)5 b Fn(\()p Fq(x)21 b Fn(+)h Fq(iy)t Fn(\))p Fq(h)p Fm(k)2799 4052 y Fl(2)2838 4093 y Fq(:)-118 4296 y Fs(F)-8 b(urther,)32 b(it)g(follo)m(ws)f(from)h(\(2.13\))f(that)680 4499 y Fq(\015)5 b Fn(\()p Fq(x)23 b Fn(+)f Fq(iy)t Fn(\))p Fq(h)27 b Fn(=)g([)p Fq(I)j Fn(+)22 b(\()p Fq(x)h Fn(+)f Fq(i)p Fn(\()p Fq(y)j Fm(\000)e Fn(1\)\)\()p Fq(A)2151 4514 y Fl(0)2212 4499 y Fm(\000)g Fq(x)g Fm(\000)f Fq(iy)t Fn(\))2612 4458 y Fo(\000)p Fl(1)2706 4499 y Fn(])p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h:)549 b Fs(\(3.19\))-118 4702 y(Com)m(bining)31 b(\(3.18\))g(and)i(\(3.19\))f(one)h(gets)470 4905 y Fm(=)p Fs(m)p Fn(\()p Fq(M)10 b Fn(\()p Fq(x)23 b Fn(+)f Fq(iy)t Fn(\))p Fq(h;)17 b(h)p Fn(\))27 b(=)g Fq(y)t Fm(k)p Fn(\()p Fq(A)1640 4920 y Fl(0)1701 4905 y Fm(\000)22 b Fq(i)p Fn(\)\()p Fq(A)1982 4920 y Fl(0)2044 4905 y Fm(\000)h Fq(x)g Fm(\000)f Fq(iy)t Fn(\))2444 4864 y Fo(\000)p Fl(1)2538 4905 y Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)p Fm(k)2809 4864 y Fl(2)3530 4905 y Fs(\(3.20\))991 5110 y Fn(=)27 b Fq(y)1162 4974 y Fi(Z)1262 5000 y Fo(1)1217 5200 y(\0001)1488 5042 y Fm(j)p Fq(t)22 b Fm(\000)g Fq(i)p Fm(j)1733 5006 y Fl(2)p 1373 5087 514 4 v 1373 5178 a Fm(j)p Fq(t)h Fm(\000)f Fq(x)h Fm(\000)f Fq(iy)t Fm(j)1848 5149 y Fl(2)1897 5110 y Fq(d)p Fn(\()p Fq(E)2058 5125 y Fk(A)2111 5134 y Fc(0)2149 5110 y Fn(\()p Fq(t)p Fn(\))p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h;)17 b(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)p Fn(\))991 5386 y(=)27 b Fq(y)1162 5250 y Fi(Z)1262 5277 y Fo(1)1217 5476 y(\0001)1521 5319 y Fn(1)22 b(+)g Fq(t)1725 5282 y Fl(2)p 1373 5363 539 4 v 1373 5454 a Fn(\()p Fq(t)h Fm(\000)f Fq(x)p Fn(\))1661 5426 y Fl(2)1723 5454 y Fn(+)g Fq(y)1873 5426 y Fl(2)1922 5386 y Fq(d)p Fn(\()p Fq(E)2083 5401 y Fk(A)2136 5410 y Fc(0)2174 5386 y Fn(\()p Fq(t)p Fn(\))p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h;)17 b(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)p Fn(\))p Fq(:)p eop %%Page: 10 10 10 9 bop -118 100 a Fs(On)32 b(the)h(other)g(hand)g(w)m(e)g(obtain)f (from)f(\(3.17\))h(that)925 374 y Fm(=)p Fs(m)o Fn(\()p Fq(M)10 b Fn(\()p Fq(x)23 b Fn(+)f Fq(iy)t Fn(\))p Fq(h;)17 b(h)p Fn(\))27 b(=)h Fq(y)1950 238 y Fi(Z)2049 265 y Fo(1)2005 464 y(\0001)2198 307 y Fq(d)p Fn(\(\006\()p Fq(t)p Fn(\))p Fq(h;)17 b(h)p Fn(\))p 2161 351 539 4 v 2161 442 a(\()p Fq(t)22 b Fm(\000)h Fq(x)p Fn(\))2449 414 y Fl(2)2511 442 y Fn(+)f Fq(y)2661 414 y Fl(2)2709 374 y Fq(:)794 b Fs(\(3.21\))-118 653 y(Applying)31 b(Stieltjes)h(in)m (v)m(ersion)h(form)m(ula)d(\(2.7\))i(to)h(\(3.20\))e(and)i(\(3.21\))f (one)g(gets)698 876 y Fq(d)p Fn(\006)819 891 y Fk(h)864 876 y Fn(\()p Fq(t)p Fn(\))c(:=)g Fq(d)p Fn(\(\006\()p Fq(t)p Fn(\))p Fq(h;)17 b(h)p Fn(\))27 b(=)h(\(1)22 b(+)g Fq(t)1971 835 y Fl(2)2010 876 y Fn(\))p Fq(d)p Fn(\()p Fq(E)2209 891 y Fk(A)2262 900 y Fc(0)2301 876 y Fn(\()p Fq(t)p Fn(\))p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h;)17 b(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)p Fn(\))p Fq(:)567 b Fs(\(3.22\))-118 1100 y(Therefore,)34 b(the)f(measures)g Fn(\006)1007 1115 y Fk(h)1052 1100 y Fn(\()p Fm(\001)p Fn(\))f Fs(and)h Fn(\()p Fq(E)1488 1115 y Fk(A)1541 1124 y Fc(0)1580 1100 y Fn(\()p Fm(\001)p Fn(\))p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h;)17 b(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)p Fn(\))32 b Fs(are)g(equiv)-5 b(alen)m(t)33 b(for)f(eac)m(h)h Fq(h)28 b Fm(2)g(H)q Fs(.)255 1225 y(If)38 b Fq(E)430 1240 y Fk(A)483 1249 y Fc(0)522 1225 y Fn(\()p Fm(X)15 b Fn(\))37 b(=)g(0)g Fs(for)h(some)g(Borel)f(set)h Fm(X)52 b(\022)37 b Fj(R)5 b Fs(,)46 b(then)38 b Fn(\()p Fq(E)2459 1240 y Fk(A)2512 1249 y Fc(0)2551 1225 y Fn(\()p Fm(X)15 b Fn(\))p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h;)17 b(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)p Fn(\))37 b(=)g(0)g Fs(for)h(an)m(y)-118 1349 y Fq(h)50 b Fm(2)g(H)q Fs(.)83 b(Hence)47 b Fn(\(\006\()p Fm(X)15 b Fn(\))p Fq(h;)i(h)p Fn(\))50 b(=)g(0)45 b Fs(for)g(an)m(y)h Fq(h)k Fm(2)h(H)46 b Fs(whic)m(h)g(yields)g Fn(\006\()p Fm(X)15 b Fn(\))50 b(=)f(0)p Fs(.)83 b(Con)m(v)m(ersely)-8 b(,)51 b(if)-118 1474 y Fn(\006\()p Fm(X)15 b Fn(\))48 b(=)g(0)d Fs(for)f(some)g(Borel)g(set)h Fm(X)63 b(\022)49 b Fj(R)5 b Fs(,)53 b(then)46 b Fn(\(\006\()p Fm(X)15 b Fn(\))p Fq(h;)i(h)p Fn(\))48 b(=)g(0)c Fs(for)g(an)m(y)i Fq(h)i Fm(2)h(H)q Fs(.)79 b(Therefore)-118 1598 y Fn(\()p Fq(E)-8 1613 y Fk(A)45 1622 y Fc(0)84 1598 y Fn(\()p Fm(X)15 b Fn(\))p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h;)17 b(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)p Fn(\))27 b(=)h(0)j Fs(for)g(an)m(y)h Fq(h)c Fm(2)g(H)q Fs(.)43 b(Hence)32 b Fq(E)2005 1613 y Fk(A)2058 1622 y Fc(0)2097 1598 y Fn(\()p Fm(X)15 b Fn(\))p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)28 b Fn(=)f(0)p Fs(.)43 b(On)31 b(the)h(other)g(hand,)g (from)-118 1723 y(\(2.13\))g(one)g(gets)485 1946 y Fq(E)557 1961 y Fk(A)610 1970 y Fc(0)648 1946 y Fn(\()p Fm(X)15 b Fn(\)\))p Fq(\015)5 b Fn(\()p Fq(z)t Fn(\))p Fq(h)28 b Fn(=)g Fq(E)1289 1961 y Fk(A)1342 1970 y Fc(0)1381 1946 y Fn(\()p Fm(X)15 b Fn(\)[)p Fq(I)29 b Fn(+)22 b(\()p Fq(z)27 b Fm(\000)c Fq(i)p Fn(\)\()p Fq(A)2132 1961 y Fl(0)2194 1946 y Fm(\000)f Fq(z)t Fn(\))2380 1905 y Fo(\000)p Fl(1)2475 1946 y Fn(])p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)1005 2096 y Fn(=)28 b([)p Fq(I)i Fn(+)22 b(\()p Fq(z)k Fm(\000)d Fq(i)p Fn(\)\()p Fq(A)1698 2111 y Fl(0)1760 2096 y Fm(\000)g Fq(z)t Fn(\))1947 2055 y Fo(\000)p Fl(1)2042 2096 y Fn(])p Fq(E)2141 2111 y Fk(A)2194 2120 y Fc(0)2233 2096 y Fn(\()p Fm(X)15 b Fn(\))p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)27 b Fn(=)h(0)p Fq(:)-118 2319 y Fs(for)42 b Fq(z)50 b Fm(2)c Fj(C)314 2334 y Fl(+)408 2319 y Fm([)30 b Fj(C)570 2334 y Fo(\000)635 2319 y Fs(.)75 b(Since)43 b(the)g(op)s(erator)f Fq(A)h Fs(is)g(simple)e(the)j(set)f Fm(f)p Fq(\015)5 b Fn(\()p Fq(z)t Fn(\))p Fm(H)47 b Fn(:)f Fq(z)k Fm(2)45 b Fj(C)3170 2334 y Fl(+)3265 2319 y Fm([)29 b Fj(C)3426 2334 y Fo(\000)3491 2319 y Fm(g)46 b Fn(=)f Fp(H)-118 2444 y Fs(is)40 b(generating)f(with)h(resp)s(ect)i(to)e Fq(A)1244 2459 y Fl(0)1284 2444 y Fs(.)66 b(Therefore)42 b Fq(E)1899 2459 y Fk(A)1952 2468 y Fc(0)1991 2444 y Fn(\()p Fm(X)15 b Fn(\))p Fp(H)40 b Fn(=)h(0)p Fs(.)67 b(The)41 b(remaining)d(assertions)i(are)-118 2568 y(ob)m(vious.)3462 b Fm(4)255 2800 y Ft(REMARK)36 b(3.3)49 b Fs(A)m(ctually)26 b(w)m(e)i(ha)m(v)m(e)h(pro)m(v)m(ed)f(a)f(little)d(bit)i(more.)41 b(In)27 b(fact,)h(the)g(pro)s(of)e(sho)m(ws)-118 2925 y(that)32 b(for)g(eac)m(h)i Fq(h)27 b Fm(2)i(H)k Fs(the)g(measures)g Fn(\()p Fq(E)1453 2940 y Fk(A)1506 2949 y Fc(0)1545 2925 y Fn(\()p Fm(\001)p Fn(\))p Fq(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h;)17 b(\015)5 b Fn(\()p Fq(i)p Fn(\))p Fq(h)p Fn(\))32 b Fs(and)h Fn(\006)2465 2940 y Fk(h)2510 2925 y Fn(\()p Fm(\001)p Fn(\))f Fs(are)h(equiv)-5 b(alen)m(t.)-118 3156 y(Using)40 b(Lemma)g(3.1)g(and)h(3.2)f(w)m(e)i(can)f(made)f(the)h 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Fq(E)434 3901 y Fk(A)487 3910 y Fc(0)526 3886 y Fn(\()p Fm(\001)p Fn(\))f Ff(is)g(the)h(sp)-5 b(e)g(ctr)g(al)39 b(me)-5 b(asur)g(e)38 b(of)g Fq(A)1866 3901 y Fl(0)1941 3886 y Fn(:=)d Fq(A)2152 3850 y Fo(\003)2192 3886 y Fm(j)17 b Fn(k)m(er)g(\000)2444 3901 y Fl(0)2483 3886 y Fn(\()p Fm(2)35 b Fq(E)6 b(xt)2790 3901 y Fk(A)2848 3886 y Fn(\))39 b Ff(and)f Fn(\006\()p Fm(\001)p Fn(\))h Ff(that)g(of)g(the)-118 4011 y(inte)-5 b(gr)g(al)27 b(r)-5 b(epr)g(esentation)27 b(\(3.17\))f(of)i(the)f(W)-7 b(eyl)28 b(function)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Ff(,)29 b(then)f(for)f(e)-5 b(ach)27 b(total)h(set)g Fm(T)53 b Fn(=)27 b Fm(f)p Fq(h)3524 4026 y Fk(k)3567 4011 y Fm(g)3617 3975 y Fk(N)3617 4037 y(k)r Fl(=1)3750 4011 y Ff(,)-118 4135 y Fn(1)g Fm(\024)h Fq(N)39 b Fm(\024)28 b Fn(+)p Fm(1)p Ff(,)j(in)f Fm(H)i Ff(the)e(sets)h Fq(S)1164 4099 y Fo(0)1158 4160 y Fk(s)1194 4135 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Ff(,)31 b Fq(S)1591 4099 y Fo(0)1585 4160 y Fk(pp)1661 4135 y Fn(\(\006;)17 b Fm(T)25 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Fk(h)2137 5008 y Fg(k)2179 4981 y Fn(\))p Fq(;)1286 b Fs(\(3.36\))1006 5130 y Fq(S)1072 5089 y Fo(00)1066 5155 y Fk(pp)1141 5130 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))83 b(:=)g Fm([)1746 5089 y Fk(N)1746 5155 y(k)r Fl(=1)1879 5130 y Fq(S)1945 5089 y Fo(00)1939 5155 y Fk(pp)2014 5130 y Fn(\(\006)2122 5145 y Fk(h)2163 5157 y Fg(k)2206 5130 y Fn(\))p Fq(;)1259 b Fs(\(3.37\))1014 5280 y Fq(S)1080 5239 y Fo(00)1074 5305 y Fk(sc)1141 5280 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))83 b(:=)g Fm([)1746 5239 y Fk(N)1746 5305 y(k)r Fl(=1)1879 5280 y Fq(S)1945 5239 y Fo(00)1939 5305 y Fk(sc)2007 5280 y Fn(\(\006)2115 5295 y Fk(h)2156 5307 y Fg(k)2198 5280 y Fn(\))22 b Fm(n)g Fq(S)2396 5239 y Fo(00)2390 5305 y Fk(pp)2465 5280 y Fn(\(\006\))p Fq(;)892 b Fs(\(3.38\))1009 5429 y Fq(S)1075 5388 y Fo(00)1069 5454 y Fk(ac)1141 5429 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))83 b(:=)g Fm([)1746 5388 y Fk(N)1746 5454 y(k)r Fl(=1)1879 5429 y Fq(S)1945 5388 y Fo(00)1939 5454 y Fk(ac)2011 5429 y Fn(\(\006)2119 5444 y Fk(h)2160 5456 y Fg(k)2203 5429 y Fn(\))22 b Fm(n)g Fq(S)2401 5388 y Fo(00)2395 5454 y Fk(s)2443 5429 y Fn(\(\006\))941 b Fs(\(3.39\))p eop %%Page: 13 13 13 12 bop -118 100 a Fs(By)39 b(de\034nition)e(the)i(sets)h Fq(S)917 63 y Fo(00)911 124 y Fk(s)959 100 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))38 b Fs(and)h Fq(S)1529 63 y Fo(00)1523 124 y Fk(ac)1595 100 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))38 b Fs(are)h(disjoin)m(t.)60 b(The)39 b(same)g(holds)f(for)g Fq(S)3414 63 y Fo(00)3408 124 y Fk(pp)3483 100 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Fs(,)-118 224 y Fq(S)-52 188 y Fo(00)-58 249 y Fk(sc)9 224 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))32 b Fs(and)g Fq(S)566 188 y Fo(00)560 249 y Fk(ac)632 224 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))p Fs(.)44 b(F)-8 b(urthermore,)31 b(w)m(e)i(denote)f(that)g(the)g(sets)h Fq(S)2653 188 y Fo(00)2647 249 y Fk(\034)2696 224 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))p Fs(,)32 b Fq(\034)39 b Fn(=)28 b Fq(s;)17 b(pp;)g(sc)p Fs(,)31 b(ha)m(v)m(e)-118 349 y(Leb)s(esgue)45 b(measure)f(zero,)j(i.e)d(mes)p Fn(\()p Fq(S)1362 313 y Fo(00)1356 373 y Fk(\034)1404 349 y Fn(\(\006;)17 b Fm(T)26 b Fn(\)\))47 b(=)f(0)p Fs(,)h Fq(\034)58 b Fn(=)47 b Fq(s;)17 b(pp;)g(sc)p Fs(.)77 b(It)44 b(turns)h(out)e(that)h(the)g (sets)-118 473 y Fq(S)-52 437 y Fo(0)-58 498 y Fk(\034)-15 473 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))32 b Fs(in)g(Theorem)h(3.4)f(can) h(b)s(e)f(replaced)h(b)m(y)h(the)e(sets)i Fq(S)2226 437 y Fo(00)2220 498 y Fk(\034)2268 473 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Fs(.)255 680 y Ft(THEOREM)37 b(3.6)48 b Ff(L)-5 b(et)36 b Fq(A)g Ff(b)-5 b(e)35 b(a)h(simple)f(densely)f (de\034ne)-5 b(d)35 b(close)-5 b(d)35 b(symmetric)g(op)-5 b(er)g(ator)35 b(on)-118 804 y(a)30 b(sep)-5 b(ar)g(able)29 b(Hilb)-5 b(ert)30 b(sp)-5 b(ac)g(e)29 b Fp(H)h Ff(with)g Fq(n)1303 819 y Fl(+)1362 804 y Fn(\()p Fq(A)p Fn(\))e(=)g Fq(n)1701 819 y Fo(\000)1760 804 y Fn(\()p Fq(A)p Fn(\))p Ff(.)43 b(F)-7 b(urther,)31 b(let)f Fn(\005)e(=)f Fm(fH)q Fq(;)17 b Fn(\000)2937 819 y Fl(0)2976 804 y Fq(;)g Fn(\000)3081 819 y Fl(1)3120 804 y Fm(g)30 b Ff(b)-5 b(e)30 b(a)g(b)-5 b(oundary)-118 929 y(triple)35 b(of)f Fq(A)327 893 y Fo(\003)402 929 y Ff(with)g(W)-7 b(eyl)35 b(function)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Ff(.)255 1053 y(If)39 b Fq(E)434 1068 y Fk(A)487 1077 y Fc(0)526 1053 y Fn(\()p Fm(\001)p Fn(\))f Ff(is)g(the)h(sp)-5 b(e)g(ctr)g(al)39 b(me)-5 b(asur)g(e)38 b(of)g Fq(A)1866 1068 y Fl(0)1941 1053 y Fn(:=)d Fq(A)2152 1017 y Fo(\003)2192 1053 y Fm(j)17 b Fn(k)m(er)g(\000)2444 1068 y Fl(0)2483 1053 y Fn(\()p Fm(2)35 b Fq(E)6 b(xt)2790 1068 y Fk(A)2848 1053 y Fn(\))39 b Ff(and)f Fn(\006\()p Fm(\001)p Fn(\))h Ff(that)g(of)g(the)-118 1178 y(inte)-5 b(gr)g(al)37 b(r)-5 b(epr)g(esentation)36 b(\(3.17\))g(of)h Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Ff(,)38 b(then)e(for)h(e)-5 b(ach)37 b(total)g(set)g Fm(T)58 b Fn(=)32 b Fm(f)p Fq(h)2862 1193 y Fk(k)2904 1178 y Fm(g)2954 1142 y Fk(N)2954 1204 y(k)r Fl(=1)3087 1178 y Ff(,)37 b Fn(1)32 b Fm(\024)g Fq(N)43 b Fm(\024)32 b Fn(+)p Fm(1)p Ff(,)-118 1302 y(in)d Fm(H)i Ff(the)e(sets)h Fq(S)520 1266 y Fo(00)514 1327 y Fk(s)562 1302 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Ff(,)k Fq(S)958 1266 y Fo(00)952 1327 y Fk(pp)1027 1302 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Ff(,)k Fq(S)1423 1266 y Fo(00)1417 1327 y Fk(sc)1485 1302 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))30 b Ff(and)f Fq(S)2034 1266 y Fo(00)2028 1327 y Fk(ac)2100 1302 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))j Ff(ar)-5 b(e)29 b(singular,)h(pur)-5 b(e)30 b(p)-5 b(oint,)30 b(singular)-118 1427 y(c)-5 b(ontinuous)34 b(and)h(absolutely)f(c)-5 b(ontinuous)35 b(supp)-5 b(orts)35 b(of)f Fq(E)2069 1442 y Fk(A)2122 1451 y Fc(0)2161 1427 y Fn(\()p Fm(\001)p Fn(\))p Ff(,)g(r)-5 b(esp)g(e)g(ctively,)34 b(i.e,)h(we)f(have)790 1628 y Fq(E)862 1643 y Fk(A)915 1652 y Fc(0)954 1628 y Fn(\()p Fm(X)i(\\)23 b Fq(S)1254 1587 y Fo(00)1248 1653 y Fk(\034)1296 1628 y Fn(\(\006;)17 b Fm(T)26 b Fn(\)\))h(=)h Fq(E)1813 1587 y Fk(\034)1807 1653 y(A)1860 1662 y Fc(0)1899 1628 y Fn(\()p Fm(X)15 b Fn(\))p Fq(;)116 b(\034)39 b Fn(=)27 b Fq(s;)17 b(pp;)g(sc;)g(ac;)658 b Fs(\(3.40\))-118 1830 y Ff(for)51 b(e)-5 b(ach)50 b(Bor)-5 b(el)50 b(set)h Fm(X)72 b(\022)58 b Fj(R)5 b Ff(.)99 b(In)50 b(p)-5 b(articular,)55 b(it)c(holds)f Fq(\033)2277 1845 y Fk(p)2317 1830 y Fn(\()p Fq(A)2428 1845 y Fl(0)2468 1830 y Fn(\))57 b(=)g Fq(S)2762 1794 y Fo(00)2756 1854 y Fk(pp)2832 1830 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))51 b Ff(and)g Fq(\033)3413 1845 y Fk(\034)3456 1830 y Fn(\()p Fq(A)3567 1845 y Fl(0)3607 1830 y Fn(\))57 b Fm(\022)p -118 1884 378 4 v -118 1970 a Fq(S)-52 1941 y Fo(00)-58 1995 y Fk(\034)-10 1970 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))h Fm(\022)i Fq(\033)t Fn(\()p Fq(A)563 1985 y Fl(0)602 1970 y Fn(\))p Ff(,)35 b Fq(\034)k Fn(=)27 b Fq(s;)17 b(sc;)g(ac)p Ff(.)-118 2177 y(Pr)-5 b(o)g(of:)56 b Fs(W)-8 b(e)40 b(note)f(that)g(mes)p Fn(\()p Fq(S)1078 2141 y Fo(00)1072 2201 y Fk(s)1120 2177 y Fn(\(\006)1228 2192 y Fk(h)1273 2177 y Fn(\)\))g(=)g(0)g Fs(for)f(eac)m(h)j Fq(h)e Fm(2)g(H)q Fs(.)63 b(This)40 b(is)e(a)h(consequence)j(of)d(the)h (fact)-118 2301 y(that)e(for)g(a.e.)62 b Fq(x)39 b Fm(2)f Fj(R)50 b Fs(the)39 b(symmetric)f(deriv)-5 b(ativ)m(e)38 b Fq(D)s Fn(\006)2033 2316 y Fk(h)2078 2301 y Fn(\()p Fq(x)p Fn(\))g Fs(exists)i(and)f(is)f(\034nite.)61 b(By)39 b(Prop)s(osition)-118 2426 y(3.5\(i\))i(this)g(yields)h(that)g(for)g (a.e.)72 b Fq(x)45 b Fm(2)g Fj(R)53 b Fs(the)42 b(limit)d Fq(V)2035 2441 y Fk(h)2080 2426 y Fn(\()p Fq(x)29 b Fn(+)f Fq(i)p Fn(0\))42 b Fs(exists)h(and)g(is)e(\034nite.)72 b(Therefore)-118 2550 y(mes)p Fn(\()p Fq(S)149 2514 y Fo(00)143 2575 y Fk(s)191 2550 y Fn(\(\006;)17 b Fm(T)26 b Fn(\)\))i(=)g(0)p Fs(.)45 b(Using)32 b(this)h(and)g(follo)m(wing)d (the)k(line)d(of)i(reasoning)f(of)h(Theorem)g(3.4)f(w)m(e)i(easily)-118 2675 y(pro)m(v)m(e)g(\(3.40\).)255 2799 y(Since)39 b Fq(S)582 2763 y Fo(0)576 2824 y Fk(pp)652 2799 y Fn(\(\006)760 2814 y Fk(h)801 2826 y Fg(k)843 2799 y Fn(\))f(=)g Fq(S)1099 2763 y Fo(00)1093 2824 y Fk(pp)1168 2799 y Fn(\(\006)1276 2814 y Fk(h)1317 2826 y Fg(k)1360 2799 y Fn(\))g Fs(for)g(eac)m(h)i Fq(h)1873 2814 y Fk(k)1954 2799 y Fm(2)f(T)64 b Fs(one)39 b(gets)g Fq(S)2635 2763 y Fo(0)2629 2824 y Fk(pp)2704 2799 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))38 b(=)g Fq(S)3192 2763 y Fo(00)3186 2824 y Fk(pp)3261 2799 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Fs(.)62 b(Ap-)-118 2924 y(plying)32 b(Theorem)h(3.4)g(w)m(e)h(\034nd)g Fq(\033)1143 2939 y Fk(p)1183 2924 y Fn(\()p Fq(A)1294 2939 y Fl(0)1333 2924 y Fn(\))29 b(=)f Fq(S)1570 2888 y Fo(00)1564 2949 y Fk(pp)1640 2924 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))p Fs(.)45 b(Since)34 b Fq(S)2303 2888 y Fo(0)2297 2949 y Fk(\034)2340 2924 y Fn(\(\006)2448 2939 y Fk(h)2489 2951 y Fg(k)2531 2924 y Fn(\))29 b Fm(\022)g Fq(S)2770 2888 y Fo(00)2764 2949 y Fk(\034)2812 2924 y Fn(\(\006)2920 2939 y Fk(h)2961 2951 y Fg(k)3003 2924 y Fn(\))p Fs(,)34 b Fq(\034)40 b Fn(=)28 b Fq(s;)17 b(sc;)g(ac)p Fs(,)33 b(for)-118 3048 y(eac)m(h)g Fq(h)157 3063 y Fk(k)233 3048 y Fs(one)f(gets)896 3173 y Fm([)962 3132 y Fk(N)962 3198 y(k)r Fl(=1)1096 3173 y Fq(S)1162 3132 y Fo(0)1156 3198 y Fk(\034)1199 3173 y Fn(\(\006)1307 3188 y Fk(h)1348 3200 y Fg(k)1390 3173 y Fn(\))c Fm(\022)g([)1627 3132 y Fk(N)1627 3198 y(k)r Fl(=1)1760 3173 y Fq(S)1826 3132 y Fo(00)1820 3198 y Fk(\034)1868 3173 y Fn(\(\006)1976 3188 y Fk(h)2017 3200 y Fg(k)2060 3173 y Fn(\))p Fq(;)114 b(\034)39 b Fn(=)27 b Fq(s;)17 b(sc;)g(ac;)765 b Fs(\(3.41\))-118 3342 y(whic)m(h)34 b(yields)g Fq(S)504 3306 y Fo(0)498 3366 y Fk(\034)541 3342 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))k Fm(\022)g Fq(S)1014 3306 y Fo(00)1008 3366 y Fk(\034)1057 3342 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))p Fs(,)35 b Fq(\034)41 b Fn(=)30 b Fq(s;)17 b(sc;)g(ac)p Fs(.)47 b(Applying)33 b(Theorem)i(3.4)e(w)m(e)i(\034nd)g Fq(\033)3359 3357 y Fk(\034)3402 3342 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))k Fm(\022)p -118 3380 V -118 3466 a Fq(S)-52 3437 y Fo(00)-58 3491 y Fk(\034)-10 3466 y Fn(\(\006;)17 b Fm(T)26 b Fn(\))p Fs(,)34 b Fq(\034)40 b Fn(=)29 b Fq(s;)17 b(sc;)g(ac)p Fs(.)46 b(Otherwise,)35 b(w)m(e)f(ha)m(v)m(e)h Fq(S)1822 3430 y Fo(00)1816 3491 y Fk(\034)1864 3466 y Fn(\(\006)1972 3481 y Fk(h)2013 3493 y Fg(k)2056 3466 y Fn(\))29 b Fm(\022)h Fq(S)6 b Fn(\(\006)2404 3481 y Fk(h)2445 3493 y Fg(k)2487 3466 y Fn(\))29 b Fm(\022)h Fq(S)6 b Fn(\(\006\))33 b Fs(or)p 3026 3380 V 33 w Fq(S)3092 3437 y Fo(00)3086 3491 y Fk(\034)3135 3466 y Fn(\(\006;)17 b Fm(T)25 b Fn(\))30 b Fm(\022)f Fq(S)6 b Fn(\(\006\))p Fs(,)-118 3591 y Fq(\034)39 b Fn(=)28 b Fq(s;)17 b(sc;)g(ac)p Fs(.)43 b(Using)32 b(Lemma)f(3.2)h(w)m(e)h(complete)f(the)h(pro)s(of.)1487 b Fm(4)255 3715 y Fs(Theorem)35 b(3.6)e(c)m(haracterizes)i(the)g(sp)s (ectral)f(prop)s(erties)f(of)h Fq(A)2589 3730 y Fl(0)2662 3715 y Fs(b)m(y)h(b)s(oundary)g(v)-5 b(alues)34 b(of)f(the)-118 3840 y(imaginary)43 b(part)i(of)g(the)h(asso)s(ciated)f(scalar)f(W)-8 b(eyl)46 b(function.)81 b(Our)46 b(next)g(aim)d(is)i(to)g(use)i(for)e (that)-118 3964 y(purp)s(ose)38 b(the)f(b)s(oundary)g(v)-5 b(alues)37 b(of)g(the)g(asso)s(ciated)g(scalar)f(W)-8 b(eyl)37 b(function)g(themselv)m(es.)57 b(This)38 b(step)-118 4089 y(relies)32 b(on)g(the)h(follo)m(wing)d(prop)s(osition)g(for)i (scalar)g(R-functions.)255 4295 y Ft(PR)m(OPOSITION)k(3.7)49 b Ff(L)-5 b(et)35 b Fn(\010\()p Fm(\001)p Fn(\))g Ff(b)-5 b(e)34 b(a)g(sc)-5 b(alar)34 b(R-function.)44 b(Then)34 b(for)g(almost)g(al)5 b(l)34 b Fq(x)28 b Fm(2)h Fj(R)-118 4420 y Ff(the)40 b(limit)f Fn(\010\()p Fq(x)27 b Fn(+)e Fq(i)p Fn(0\))37 b(:=)f(lim)1008 4435 y Fk(y)r Fo(#)p Fl(0)1136 4420 y Fn(\010\()p Fq(x)27 b Fn(+)f Fq(i)p Fn(0\))39 b Ff(exists)g(and)h(is)f(\034nite)g(and,)h(mor)-5 b(e)g(over,)40 b(in)g(this)f(c)-5 b(ase)39 b(one)-118 4544 y(has)34 b Fn(\010\()p Fq(x)23 b Fn(+)f Fq(i)p Fn(0\))28 b(=)f(lim)727 4559 y Fk(z)s Fo(!\037)p Fk(x)948 4544 y Fn(\010\()p Fq(z)t Fn(\))p Ff(.)-118 4751 y Fs(Let)33 b(us)g(in)m(tro)s(duce)f(the)h(follo)m(wing)d(sets:)325 4952 y Fn(\012)395 4967 y Fk(s)433 4952 y Fn(\(\010\))83 b(:=)g Fm(f)p Fq(x)28 b Fm(2)g Fj(R)39 b Fn(:)27 b Fm(j)p Fn(\010\()p Fq(z)t Fn(\))p Fm(j)h(!)g Fn(+)p Fm(1)f Fs(as)h Fq(z)k Fm(!\037)c Fq(x)p Fm(g)p Fq(;)1162 b Fs(\(3.42\))287 5102 y Fn(\012)357 5117 y Fk(pp)433 5102 y Fn(\(\010\))83 b(:=)g Fm(f)p Fq(x)28 b Fm(2)g Fj(R)39 b Fn(:)60 b(lim)1229 5162 y Fk(z)s Fo(!\037)p Fk(x)1430 5102 y Fn(\()p Fq(z)27 b Fm(\000)c Fq(x)p Fn(\)\010\()p Fq(z)t Fn(\))29 b Fm(6)p Fn(=)e(0)p Fm(g)p Fq(;)1344 b Fs(\(3.43\))295 5284 y Fn(\012)365 5299 y Fk(sc)433 5284 y Fn(\(\010\))83 b(:=)g Fm(f)p Fq(x)28 b Fm(2)g Fj(R)39 b Fn(:)27 b Fm(j)p Fn(\010\()p Fq(z)t Fn(\))p Fm(j)h(!)g Fn(+)p Fm(1)f Fs(and)h Fn(\()p Fq(z)f Fm(\000)22 b Fq(x)p Fn(\)\010\()p Fq(z)t Fn(\))29 b Fm(!)f Fn(0)f Fs(as)h Fq(z)k Fm(!\037)c Fq(x)p Fm(g)274 b Fs(\(3.44\))290 5433 y Fn(\012)360 5448 y Fk(ac)433 5433 y Fn(\(\010\))83 b(:=)g Fm(f)p Fq(x)28 b Fm(2)g Fj(R)39 b Fn(:)27 b(0)h Fq(<)f Fm(=)p Fs(m)p Fn(\010\()p Fq(x)22 b Fn(+)h Fq(i)p Fn(0\))k Fq(<)h Fn(+)p Fm(1g)1206 b Fs(\(3.45\))p eop %%Page: 14 14 14 13 bop -118 100 a Fs(In)33 b(accordance)g(with)f(\(2.2\))g(the)h (scalar)f(R-function)f Fn(\010\()p Fm(\001)p Fn(\))i Fs(admits)e(the)i(represen)m(tation)557 356 y Fn(\010\()p Fq(z)t Fn(\))28 b(=)g Fq(C)954 371 y Fl(0)1015 356 y Fn(+)22 b Fq(C)1183 371 y Fl(1)1223 356 y Fq(z)k Fn(+)1392 220 y Fi(Z)1492 246 y Fl(+)p Fo(1)1448 446 y(\0001)1638 215 y Fi(\022)1801 288 y Fn(1)p 1722 333 207 4 v 1722 424 a Fq(t)c Fm(\000)h Fq(z)1960 356 y Fm(\000)2174 288 y Fq(t)p 2070 333 244 4 v 2070 424 a Fn(1)f(+)g Fq(t)2274 395 y Fl(2)2324 215 y Fi(\023)2413 356 y Fq(d\026)p Fn(\()p Fq(t)p Fn(\))p Fq(;)114 b(z)33 b Fm(2)28 b Fj(C)3013 371 y Fl(+)3078 356 y Fq(;)425 b Fs(\(3.46\))-118 609 y(where)42 b Fq(C)242 624 y Fl(0)281 609 y Fq(;)17 b(C)395 624 y Fl(1)476 609 y Fm(2)43 b Fj(R)5 b Fs(,)49 b Fq(C)797 624 y Fl(1)878 609 y Fm(\025)42 b Fn(0)f Fs(and)g(the)g(Borel)g (measure)g Fq(\026)p Fn(\()p Fm(\001)p Fn(\))f Fs(ob)s(eys)i(\(3.24\).) 68 b(Therefore,)44 b(one)d(gets)-118 733 y Fq(V)21 b Fn(\()p Fq(z)t Fn(\))32 b(=)f Fm(=)p Fs(m)o Fn(\010\()p Fq(z)t Fn(\))p Fs(,)37 b Fq(z)e Fm(2)d Fj(C)880 748 y Fl(+)945 733 y Fs(,)j(where)h Fq(V)21 b Fn(\()p Fm(\001)p Fn(\))34 b Fs(is)g(de\034ned)i(b)m(y)g(\(3.25\).)49 b(Using)34 b(that)g(and)h(Prop)s(ositions)e(3.5)-118 858 y(and)g(3.7)f(one)g (immediately)e(gets)j(that)593 1056 y Fq(S)659 1015 y Fo(00)653 1080 y Fk(pp)729 1056 y Fn(\()p Fq(\026)p Fn(\))27 b(=)g(\012)1064 1071 y Fk(pp)1140 1056 y Fn(\(\010\))98 b Fs(and)g Fq(S)1705 1015 y Fo(00)1699 1080 y Fk(\034)1747 1056 y Fn(\()p Fq(\026)p Fn(\))27 b Fm(\022)h Fn(\012)2084 1071 y Fk(\034)2128 1056 y Fn(\(\010\))g Fm(\022)g Fq(S)6 b Fn(\()p Fq(\026)p Fn(\))p Fq(;)44 b(\034)39 b Fn(=)27 b Fq(s;)17 b(sc;)462 b Fs(\(3.47\))-118 1254 y(but)33 b Fn(\012)131 1269 y Fk(ac)203 1254 y Fn(\(\010\))28 b Fm(\022)h Fq(S)549 1217 y Fo(00)543 1278 y Fk(ac)615 1254 y Fn(\()p Fq(\026)p Fn(\))e Fm(\022)h Fq(S)6 b Fn(\()p Fq(\026)p Fn(\))p Fq(:)32 b Fs(Ho)m(w)m(ev)m(er)j(w)m(e)e(men)m(tioned) g(that)f Fn(\012)2434 1269 y Fk(ac)2506 1254 y Fn(\(\010\))c(=)g Fq(S)2850 1217 y Fo(00)2844 1278 y Fk(ac)2916 1254 y Fn(\()p Fq(\026)p Fn(\))22 b Fm(\\)g Fn(\012)3231 1269 y Fk(f)3277 1254 y Fn(\(\010\))33 b Fs(where)802 1451 y Fn(\012)872 1466 y Fk(f)918 1451 y Fn(\(\010\))28 b(=)g Fm(f)p Fq(x)f Fm(2)i Fj(R)38 b Fn(:)28 b(\010\()p Fq(x)23 b Fn(+)f Fq(i)p Fn(0\))28 b Fs(exists)33 b(and)g(is)f(\034nite)o Fm(g)p Fq(:)671 b Fs(\(3.48\))-118 1649 y(Ob)m(viously)35 b(the)h(sets)g Fn(\012)777 1664 y Fk(s)814 1649 y Fn(\(\010\))p Fs(,)h Fn(\012)1094 1664 y Fk(pp)1170 1649 y Fn(\(\010\))p Fs(,)f Fn(\012)1449 1664 y Fk(sc)1517 1649 y Fn(\(\010\))f Fs(and)h Fn(\012)1961 1664 y Fk(ac)2033 1649 y Fn(\(\010\))g Fs(are)f(m)m(utually)f(disjoin)m(t)g(and,)i(moreo)m(v)m(er,)-118 1774 y(are)31 b(singular,)e(pure)j(p)s(oin)m(t,)e(singular)f(con)m(tin) m(uous)i(and)g(absolutely)f(con)m(tin)m(uous)i(supp)s(orts)f(of)g Fq(\026)p Fn(\()p Fm(\001)p Fn(\))p Fs(,)f(i.e.)933 1972 y Fq(\026)p Fn(\()p Fm(X)37 b(\\)22 b Fn(\012)1296 1987 y Fk(\034)1340 1972 y Fn(\(\010\)\))28 b(=)f Fq(\026)1714 1987 y Fk(\034)1757 1972 y Fn(\()p Fm(X)15 b Fn(\))p Fq(;)114 b(\034)39 b Fn(=)28 b Fq(s;)17 b(pp;)g(sc;)g(ac;)801 b Fs(\(3.49\))-118 2170 y(for)32 b(an)m(y)h(Borel)f(set)h Fm(X)42 b(\022)29 b Fj(R)t Fs(.)255 2294 y(Let)k Fn(\005)27 b(=)h Fm(fH)q Fq(;)17 b Fn(\000)874 2309 y Fl(0)913 2294 y Fq(;)g Fn(\000)1018 2309 y Fl(1)1057 2294 y Fm(g)32 b Fs(b)s(e)g(a)g(b)s(oundary)g(triple)e(of)i Fq(A)2233 2258 y Fo(\003)2304 2294 y Fs(with)g(W)-8 b(eyl)32 b(function)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Fs(.)43 b(F)-8 b(urther,)-118 2419 y(let)32 b Fm(T)53 b Fn(=)28 b Fm(f)p Fq(h)340 2434 y Fk(k)382 2419 y Fm(g)432 2382 y Fk(N)432 2445 y(k)r Fl(=1)565 2419 y Fs(,)k Fn(1)c Fm(\024)g Fn(+)p Fm(1)p Fs(,)k(b)s(e)h(a)f(total)f(set)i(in)f Fm(H)q Fs(.)44 b(W)-8 b(e)33 b(set)936 2616 y Fn(\012)1006 2631 y Fk(s)1044 2616 y Fn(\()p Fq(M)10 b Fn(;)17 b Fm(T)26 b Fn(\))82 b(:=)h Fm([)1682 2575 y Fk(N)1682 2641 y(k)r Fl(=1)1816 2616 y Fn(\012)1886 2631 y Fk(s)1923 2616 y Fn(\()p Fq(M)2055 2631 y Fk(h)2096 2643 y Fg(k)2138 2616 y Fn(\))p Fq(;)1327 b Fs(\(3.50\))898 2766 y Fn(\012)968 2781 y Fk(pp)1044 2766 y Fn(\()p Fq(M)10 b Fn(;)17 b Fm(T)26 b Fn(\))82 b(:=)h Fm([)1682 2725 y Fk(N)1682 2791 y(k)r Fl(=1)1816 2766 y Fn(\012)1886 2781 y Fk(pp)1961 2766 y Fn(\()p Fq(M)2093 2781 y Fk(h)2134 2793 y Fg(k)2177 2766 y Fn(\))p Fq(;)1288 b Fs(\(3.51\))906 2915 y Fn(\012)976 2930 y Fk(sc)1044 2915 y Fn(\()p Fq(M)10 b Fn(;)17 b Fm(T)26 b Fn(\))82 b(:=)h Fm([)1682 2874 y Fk(N)1682 2940 y(k)r Fl(=1)1816 2915 y Fn(\012)1886 2930 y Fk(s)1923 2915 y Fn(\()p Fq(M)2055 2930 y Fk(h)2096 2942 y Fg(k)2138 2915 y Fn(\))22 b Fm(n)g Fn(\012)2340 2930 y Fk(pp)2416 2915 y Fn(\()p Fq(M)10 b Fn(;)17 b Fm(T)26 b Fn(\))p Fq(;)783 b Fs(\(3.52\))901 3065 y Fn(\012)971 3080 y Fk(ac)1044 3065 y Fn(\()p Fq(M)10 b Fn(;)17 b Fm(T)26 b Fn(\))82 b(:=)h Fm([)1682 3024 y Fk(N)1682 3089 y(k)r Fl(=1)1816 3065 y Fn(\012)1886 3080 y Fk(ac)1958 3065 y Fn(\()p Fq(M)2090 3080 y Fk(h)2131 3092 y Fg(k)2174 3065 y Fn(\))22 b Fm(n)g 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Fl(0)642 4982 y Fn(\))35 b Ff(for)f Fq(\034)39 b Fn(=)28 b Fq(s;)17 b(sc;)g(ac)p Ff(.)-118 5184 y Fs(The)41 b(pro)s(of)e(is)h(similar)c(to)k(that)g(of)f(Theorem)h Fn(3)p Fq(:)p Fn(6)p Fs(.)66 b(So)40 b(w)m(e)h(omit)d(it.)65 b(W)-8 b(e)40 b(note)g(that)g(the)h(inclusions)-118 5309 y Fq(\033)-63 5324 y Fk(s)-26 5309 y Fn(\()p Fq(A)85 5324 y Fl(0)125 5309 y Fn(\))30 b Fm(\022)p 301 5222 412 4 v 31 w Fn(\012)371 5324 y Fk(s)408 5309 y Fn(\()p Fq(M)10 b Fn(;)17 b Fm(T)26 b Fn(\))35 b Fs(and)f Fq(\033)993 5324 y Fk(sc)1061 5309 y Fn(\()p Fq(A)1172 5324 y Fl(0)1211 5309 y Fn(\))d Fm(\022)p 1388 5222 442 4 v 31 w Fn(\012)1458 5324 y Fk(sc)1526 5309 y Fn(\()p Fq(M)10 b Fn(;)17 b Fm(T)26 b Fn(\))34 b Fs(of)g(Theorem)g(3.8)g(ma)m(y)g(b)s(e)g(strict)g (ev)m(en)i(if)d Fq(\033)3523 5324 y Fk(sc)3591 5309 y Fn(\()p Fq(A)3702 5324 y Fl(0)3742 5309 y Fn(\))-118 5433 y Fs(is)f(empt)m(y)-8 b(.)p eop %%Page: 15 15 15 14 bop 255 100 a Fr(4)156 b(Singular)54 b(and)e(absolutely)j(con)l 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Fq(\017;)d(x)j Fn(+)f Fq(\017)p Fn(\))h Fm(\\)f Fn(\012)3314 1824 y Fk(ac)3387 1809 y Fn(\(\010\)\))28 b(=)g Fm(;)p Fs(.)-118 1933 y(T)-8 b(aking)32 b(\(3.49\))g(in)m(to)f(accoun)m(t)j(w) m(e)f(\034nd)704 2158 y Fq(\026)763 2173 y Fk(ac)835 2158 y Fn(\(\()p Fq(x)22 b Fm(\000)h Fq(\017;)17 b(x)22 b Fn(+)g Fq(\017)p Fn(\)\))28 b(=)g Fq(\026)1652 2173 y Fk(ac)1724 2158 y Fn(\(\()p Fq(x)22 b Fm(\000)h Fq(\017;)17 b(x)23 b Fn(+)f Fq(\017)p Fn(\))g Fm(\\)h Fn(\012)2494 2173 y Fk(ac)2566 2158 y Fn(\(\010\)\))28 b(=)g(0)p Fq(:)621 b Fs(\(4.2\))-118 2382 y(Hence)39 b Fq(x)e Fm(62)h Fq(S)6 b Fn(\()p Fq(\026)536 2397 y Fk(ac)607 2382 y Fn(\))37 b(=)g Fq(S)855 2397 y Fk(ac)927 2382 y Fn(\()p Fq(\026)p Fn(\))h Fs(whic)m(h)g(yields)g Fq(S)1724 2397 y Fk(ac)1796 2382 y Fn(\()p Fq(\026)p Fn(\))e Fm(\022)i Fs(cl)2152 2397 y Fk(ac)2224 2382 y Fn(\(\012)2332 2397 y Fk(ac)2405 2382 y Fn(\(\010\)\))p Fs(.)60 b(Con)m(v)m(ersely)-8 b(,)41 b(if)c Fq(x)g Fm(62)h Fq(S)3546 2397 y Fk(ac)3618 2382 y Fn(\()p 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Fn(0\)\))27 b Fq(>)g Fn(0)22 b Fs(for)f(all)f Fq(t)28 b Fm(2)g Fn(\012)1786 3917 y Fk(ac)1859 3902 y Fn(\(\010\))22 b Fs(whic)m(h)g(implies)e(mes)p Fn(\(\()p Fq(x)q Fm(\000)q Fq(\017;)d(x)q Fn(+)q Fq(\017)p Fn(\))q Fm(\\)q Fn(\012)3420 3917 y Fk(ac)3492 3902 y Fn(\(\010\)\))28 b(=)-118 4027 y(0)p Fs(.)43 b(Hence)34 b Fq(x)28 b Fm(62)g Fs(cl)539 4042 y Fk(ac)611 4027 y Fn(\(\012)719 4042 y Fk(ac)791 4027 y Fn(\(\010\)\))33 b Fs(or,)f(equiv)-5 b(alen)m(tly)d(,)32 b(cl)1786 4042 y Fk(ac)1858 4027 y Fn(\(\012)1966 4042 y Fk(ac)2038 4027 y Fn(\(\010\)\))c Fm(\022)h Fq(S)2416 4042 y Fk(ac)2488 4027 y Fn(\()p Fq(\026)p Fn(\))p Fs(.)1041 b Fm(4)255 4151 y Fs(Using)41 b(this)g(lemma)e(Theorem)i(3.8)g(admits)e(with)i (resp)s(ect)h(to)f(the)h(absolutely)e(con)m(tin)m(uous)-118 4276 y(part)32 b(the)h(follo)m(wing)d(re\034nemen)m(t.)255 4483 y Ft(PR)m(OPOSITION)36 b(4.2)49 b Ff(L)-5 b(et)28 b Fq(A)g Ff(b)-5 b(e)28 b(a)f(simple)g(densely)g(de\034ne)-5 b(d)27 b(close)-5 b(d)27 b(symmetric)g(op)-5 b(er)g(ator)-118 4608 y(on)45 b(a)g(sep)-5 b(ar)g(able)44 b(Hilb)-5 b(ert)46 b(sp)-5 b(ac)g(e)44 b Fp(H)h Ff(with)g Fq(n)1544 4623 y Fl(+)1604 4608 y Fn(\()p Fq(A)p Fn(\))i(=)g Fq(n)1981 4623 y Fo(\000)2040 4608 y Fn(\()p Fq(A)p Fn(\))p Ff(.)76 b(F)-7 b(urther,)48 b(let)d Fn(\005)i(=)g Fm(fH)q Fq(;)17 b Fn(\000)3321 4623 y Fl(0)3360 4608 y Fq(;)g Fn(\000)3465 4623 y Fl(1)3505 4608 y Fm(g)45 b Ff(b)-5 b(e)45 b(a)-118 4732 y(b)-5 b(oundary)34 b(triple)h(of)g Fq(A)747 4696 y Fo(\003)821 4732 y Ff(with)g(W)-7 b(eyl)35 b(function)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Ff(.)255 4857 y(If)37 b Fm(T)58 b Fn(=)32 b Fm(f)p Fq(h)686 4872 y Fk(k)728 4857 y Fm(g)778 4821 y Fk(N)778 4883 y(k)r Fl(=1)911 4857 y Ff(,)38 b Fn(1)32 b Fm(\024)g Fq(N)43 b Fm(\024)33 b Fn(+)p Fm(1)p Ff(,)k(is)g(a)g(total)g(set)h(in)f Fm(H)q Ff(,)g(then)g(the)h(absolutely)f(c)-5 b(ontinuous)-118 4981 y(sp)g(e)g(ctrum)35 b(of)f(the)h(self-adjoint)e(extension)h Fq(A)1577 4996 y Fl(0)1652 4981 y Ff(of)g Fq(A)h Ff(is)g(given)f(by) 1187 5205 y Fq(\033)1242 5220 y Fk(ac)1314 5205 y Fn(\()p Fq(A)1425 5220 y Fl(0)1465 5205 y Fn(\))27 b(=)p 1634 5114 814 4 v 28 w Fm([)1700 5171 y Fk(N)1700 5233 y(k)r Fl(=1)1833 5205 y Fs(cl)1904 5220 y Fk(ac)1976 5205 y Fn(\(\012)2084 5220 y Fk(ac)2156 5205 y Fn(\()p Fq(M)2288 5220 y Fk(h)2329 5232 y Fg(k)2372 5205 y Fn(\)\))p Fq(:)1104 b Fs(\(4.5\))p eop %%Page: 16 16 16 15 bop -118 100 a Ff(Pr)-5 b(o)g(of:)38 b Fs(By)25 b(Lemma)d(4.1)h(one)i(has)f(cl)1215 115 y Fk(ac)1287 100 y Fn(\(\012)1395 115 y Fk(ac)1468 100 y Fn(\()p Fq(M)1600 115 y Fk(h)1641 127 y Fg(k)1683 100 y Fn(\)\))k(=)f Fq(S)1950 115 y Fk(ac)2022 100 y Fn(\(\006)2130 115 y Fk(h)2171 127 y Fg(k)2214 100 y Fn(\))c Fs(for)h Fq(k)30 b Fn(=)e(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(N)10 b Fs(.)40 b(Since)24 b Fq(S)3373 115 y Fk(ac)3445 100 y Fn(\(\006)3553 115 y Fk(h)3594 127 y Fg(k)3636 100 y Fn(\))k Fm(\022)-118 224 y Fq(S)-58 239 y Fk(ac)14 224 y Fn(\(\006\))g(=)g Fq(\033)347 239 y Fk(ac)419 224 y Fn(\()p Fq(A)530 239 y Fl(0)570 224 y Fn(\))k Fs(w)m(e)i(\034nd)f Fq(S)1039 239 y Fk(ac)1111 224 y Fn(\(\006)1219 239 y Fk(h)1260 251 y Fg(k)1302 224 y Fn(\))28 b Fm(\022)g Fq(\033)1528 239 y Fk(ac)1601 224 y Fn(\()p Fq(A)1712 239 y Fl(0)1751 224 y Fn(\))33 b Fs(for)f Fq(k)e Fn(=)e(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(N)10 b Fs(.)43 b(Hence)p 1186 357 814 4 v 1186 449 a Fm([)1252 414 y Fk(N)1252 476 y(k)r Fl(=1)1385 449 y Fs(cl)1456 464 y Fk(ac)1528 449 y Fn(\(\012)1636 464 y Fk(ac)1708 449 y Fn(\()p Fq(M)1840 464 y Fk(h)1881 476 y Fg(k)1924 449 y Fn(\)\))27 b Fm(\022)h Fq(\033)2187 464 y Fk(ac)2260 449 y Fn(\()p Fq(A)2371 464 y Fl(0)2410 449 y Fn(\))p Fq(:)1104 b Fs(\(4.6\))-118 674 y(Con)m(v)m(ersely)-8 b(,)51 b(if)44 b Fq(x)51 b Fm(62)p 735 583 V 50 w([)801 640 y Fk(N)801 702 y(k)r Fl(=1)934 674 y Fs(cl)1005 689 y Fk(ac)1077 674 y Fn(\(\012)1185 689 y Fk(ac)1258 674 y Fn(\()p Fq(M)1390 689 y Fk(h)1431 701 y Fg(k)1473 674 y Fn(\)\))p Fs(,)e(then)d(there)g(is)g(a)f Fq(\017)50 b(>)g Fn(0)c Fs(suc)m(h)h(that)e Fn(\()p Fq(x)32 b Fm(\000)f Fq(\017;)17 b(x)32 b Fn(+)f Fq(\017)p Fn(\))g Fm(\\)-118 799 y Fs(cl)-48 814 y Fk(ac)25 799 y Fn(\(\012)133 814 y Fk(ac)205 799 y Fn(\()p Fq(M)337 814 y Fk(h)378 826 y Fg(k)420 799 y Fn(\)\))42 b(=)h Fm(;)d Fs(for)h(eac)m(h)h Fq(k)j Fn(=)d(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(N)10 b Fs(.)69 b(Hence)42 b Fn(\()p Fq(x)29 b Fm(\000)f Fq(\017;)17 b(x)29 b Fn(+)e Fq(\017)p Fn(\))i Fm(\\)f Fq(S)2897 814 y Fk(ac)2969 799 y Fn(\(\006)3077 814 y Fk(h)3118 826 y Fg(k)3161 799 y Fn(\))42 b(=)g Fm(;)f Fs(for)f Fq(k)46 b Fn(=)-118 923 y(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(N)40 b Fs(or,)30 b(equiv)-5 b(alen)m(tly)d(,)30 b Fn(\006)1084 938 y Fk(h)1125 950 y Fg(k)1164 938 y Fk(;ac)1255 923 y Fn(\(\()p Fq(x)18 b Fm(\000)h Fq(\017;)e(x)h Fn(+)f Fq(\017)p Fn(\)\))28 b(=)g(0)i Fs(for)g Fq(k)h Fn(=)c(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(M)10 b Fs(.)43 b(Since)30 b Fm(T)56 b Fs(is)30 b(a)g(total)f(set)-118 1048 y(one)37 b(easily)g(gets)h(that)f Fn(\006)831 1063 y Fk(h;ac)963 1048 y Fn(\(\()p Fq(x)26 b Fm(\000)g Fq(\017;)17 b(x)26 b Fn(+)f Fq(\017)p Fn(\)\))36 b(=)f(0)i Fs(for)g(eac)m(h)h Fq(h)e Fm(2)g(H)q Fs(.)58 b(Therefore)38 b Fq(x)e Fm(62)g Fq(S)3277 1063 y Fk(ac)3349 1048 y Fn(\(\006\))i Fs(whic)m(h)-118 1184 y(yields)32 b Fq(x)c Fm(62)g Fq(S)393 1199 y Fk(ac)465 1184 y Fn(\()p Fq(E)575 1199 y Fk(A)628 1208 y Fc(0)667 1184 y Fn(\))p Fs(.)44 b(Hence)33 b Fq(\033)1120 1199 y Fk(ac)1193 1184 y Fn(\()p Fq(A)1304 1199 y Fl(0)1344 1184 y Fn(\))27 b Fm(\022)p 1514 1092 742 4 v 28 w([)1580 1149 y Fk(N)1580 1211 y(k)r Fl(=1)1714 1184 y Fs(cl)1784 1199 y Fk(ac)1856 1184 y Fn(\(\012\()p Fq(M)2096 1199 y Fk(h)2137 1211 y Fg(k)2180 1184 y Fn(\)\))32 b Fs(whic)m(h)h(pro)m(v) m(es)h(\(4.5\).)595 b Fm(4)255 1308 y Fs(Theorem)36 b(3.8)f(and)g(Prop) s(osition)e(4.2)i(immediately)d(ha)m(v)m(e)37 b(sev)m(eral)e (implications)d(whic)m(h)j(are)-118 1433 y(summarized)c(in)h(the)h (follo)m(wing)d(theorem.)255 1654 y Ft(THEOREM)37 b(4.3)48 b Ff(L)-5 b(et)36 b Fq(A)g Ff(b)-5 b(e)35 b(a)h(simple)f(densely)f (de\034ne)-5 b(d)35 b(close)-5 b(d)35 b(symmetric)g(op)-5 b(er)g(ator)35 b(on)-118 1779 y(a)30 b(sep)-5 b(ar)g(able)29 b(Hilb)-5 b(ert)30 b(sp)-5 b(ac)g(e)29 b Fp(H)h Ff(with)g Fq(n)1303 1794 y Fl(+)1362 1779 y Fn(\()p Fq(A)p Fn(\))e(=)g Fq(n)1701 1794 y Fo(\000)1760 1779 y Fn(\()p Fq(A)p Fn(\))p Ff(.)43 b(F)-7 b(urther,)31 b(let)f Fn(\005)e(=)f Fm(fH)q Fq(;)17 b Fn(\000)2937 1794 y Fl(0)2976 1779 y Fq(;)g Fn(\000)3081 1794 y Fl(1)3120 1779 y Fm(g)30 b Ff(b)-5 b(e)30 b(a)g(b)-5 b(oundary)-118 1903 y(triple)35 b(of)f Fq(A)327 1867 y Fo(\003)402 1903 y Ff(with)g(W)-7 b(eyl)35 b(function)f Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))p Ff(.)255 2028 y(If)27 b Fm(T)53 b Fn(=)28 b Fm(f)p Fq(h)667 2043 y Fk(k)709 2028 y Fm(g)759 1991 y Fk(N)759 2053 y(k)r Fl(=1)892 2028 y Ff(,)g Fn(1)g Fm(\024)g Fq(N)38 b Fm(\024)28 b Fn(+)p Fm(1)p Ff(,)g(is)f(a)g(total)g(set)g(in)f Fm(H)q Ff(,)j(then)d(for)h(the)g(self-adjoint)f(extension)-118 2152 y Fq(A)-45 2167 y Fl(0)29 2152 y Ff(of)35 b Fq(A)g Ff(the)g(fol)5 b(lowing)33 b(c)-5 b(onclusions)33 b(ar)-5 b(e)35 b(valid:)-26 2351 y Fs(\(i\))48 b Ff(The)30 b(self-adjoint)g (extension)h Fq(A)1324 2366 y Fl(0)1395 2351 y Ff(of)g Fq(A)g Ff(has)g(no)g(p)-5 b(oint)31 b(sp)-5 b(e)g(ctrum)31 b(within)g(the)g(interval)g Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(,)32 b(i.e.)126 2476 y Fq(\033)181 2491 y Fk(pp)256 2476 y Fn(\()p Fq(A)367 2491 y Fl(0)407 2476 y Fn(\))22 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))27 b(=)h Fm(;)p Ff(,)35 b(if)f(and)g(only)h(if)g(for)f(e)-5 b(ach)34 b Fq(k)d Fn(=)d(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(N)44 b Ff(one)34 b(has)1504 2690 y Fn(lim)1518 2753 y Fk(y)r Fo(#)p Fl(0)1657 2690 y Fq(y)t(M)1803 2705 y Fk(h)1844 2717 y Fg(k)1885 2690 y Fn(\()p Fq(x)22 b Fn(+)g Fq(iy)t Fn(\))28 b(=)f(0)1178 b Fs(\(4.7\))126 2949 y Ff(for)34 b(al)5 b(l)35 b Fq(x)28 b Fm(2)g Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(.)45 b(In)34 b(this)h(c)-5 b(ase)34 b(the)h(fol)5 b(lowing)33 b(r)-5 b(elation)35 b(holds)225 3186 y Fq(\033)t Fn(\()p Fq(A)395 3201 y Fl(0)435 3186 y Fn(\))22 b Fm(\\)g Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)g Fq(\033)982 3201 y Fk(c)1017 3186 y Fn(\()p Fq(A)1128 3201 y Fl(0)1167 3186 y Fn(\))22 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)1659 3076 y Fi(\020)p 1719 3095 591 4 v 1719 3186 a Fm([)1785 3152 y Fk(N)1785 3214 y(k)r Fl(=1)1918 3186 y Fn(\012)1988 3201 y Fk(sc)2056 3186 y Fn(\()p Fq(M)2188 3201 y Fk(h)2229 3213 y Fg(k)2271 3186 y Fn(\))50 b Fm([)p 2475 3095 596 4 v 50 w([)2541 3152 y Fk(N)2541 3214 y(k)r Fl(=1)2674 3186 y Fn(\012)2744 3201 y Fk(ac)2817 3186 y Fn(\()p Fq(M)2949 3201 y Fk(h)2990 3213 y Fg(k)3032 3186 y Fn(\))3070 3076 y Fi(\021)3152 3186 y Fm(\\)22 b Fn(\()p Fq(a;)17 b(b)p Fn(\))p Fq(:)100 b Fs(\(4.8\))-53 3464 y(\(ii\))47 b Ff(The)f(self-adjoint)f(extension)g Fq(A)1369 3479 y Fl(0)1455 3464 y Ff(of)h Fq(A)h Ff(has)e(no)i (singular)e(c)-5 b(ontinuous)46 b(sp)-5 b(e)g(ctrum)47 b(within)f(the)126 3589 y(interval)25 b Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(,)28 b(i.e.)42 b Fq(\033)976 3604 y Fk(sc)1044 3589 y Fn(\()p Fq(A)1155 3604 y Fl(0)1194 3589 y Fn(\))s Fm(\\)s Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)f Fm(;)p Ff(,)h(if)e(for)g(e)-5 b(ach)25 b Fq(k)31 b Fn(=)c(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(N)36 b Ff(the)26 b(set)g Fn(\012)3174 3604 y Fk(sc)3242 3589 y Fn(\()p Fq(M)3374 3604 y Fk(h)3415 3616 y Fg(k)3457 3589 y Fn(\))s Fm(\\)s Fn(\()p Fq(a;)17 b(b)p Fn(\))126 3713 y Ff(is)34 b(c)-5 b(ountable,)35 b(in)f(p)-5 b(articular,)35 b(if)f Fn(\()p Fq(a;)17 b(b)p Fn(\))23 b Fm(n)f Fn(\012)1755 3728 y Fk(ac)1827 3713 y Fn(\()p Fq(M)1959 3728 y Fk(h)2000 3740 y Fg(k)2042 3713 y Fn(\))35 b Ff(is)g(c)-5 b(ountable.)-80 3918 y Fs(\(iii\))46 b Ff(The)39 b(self-adjoint)g(extension)h Fq(A)1351 3933 y Fl(0)1430 3918 y Ff(of)g Fq(A)h Ff(has)f(no)g (absolutely)g(c)-5 b(ontinuous)40 b(sp)-5 b(e)g(ctrum)40 b(within)g(the)126 4043 y(interval)45 b Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(,)49 b(i.e.)77 b Fq(\033)1052 4058 y Fk(ac)1124 4043 y Fn(\()p Fq(A)1235 4058 y Fl(0)1275 4043 y Fn(\))30 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))48 b(=)g Fm(;)p Ff(,)g(if)e(and)f(only)g(if)h(for)f(e)-5 b(ach)45 b Fq(k)51 b Fn(=)d(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(N)56 b Ff(the)126 4167 y(c)-5 b(ondition)1493 4292 y Fm(=)p Fs(m)o Fn(\()p Fq(M)1778 4307 y Fk(h)1819 4319 y Fg(k)1861 4292 y Fn(\()p Fq(x)23 b Fn(+)f Fq(i)p Fn(0\)\))27 b(=)h(0)1166 b Fs(\(4.9\))126 4466 y Ff(holds)34 b(for)g(a.e.)45 b Fq(x)28 b Fm(2)g Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(.)45 b(In)34 b(his)g(c)-5 b(ase)34 b(we)h(have)854 4680 y Fq(\033)t Fn(\()p Fq(A)1024 4695 y Fl(0)1063 4680 y Fn(\))22 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)f Fq(\033)1610 4695 y Fk(s)1648 4680 y Fn(\()p Fq(A)1759 4695 y Fl(0)1798 4680 y Fn(\))22 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)p 2290 4594 412 4 v 27 w(\012)2360 4695 y Fk(s)2398 4680 y Fn(\()p Fq(M)10 b Fn(;)17 b Fm(T)26 b Fn(\))c Fm(\\)g Fn(\()p Fq(a;)17 b(b)p Fn(\))p Fq(:)479 b Fs(\(4.10\))-118 4935 y Ff(Pr)-5 b(o)g(of:)69 b Fs(\(i\))44 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Fq(A)3109 488 y Fl(0)3149 473 y Fn(\))26 b Fm([)g Fq(\033)3360 488 y Fk(c)3396 473 y Fn(\()p Fq(A)3507 488 y Fl(0)3546 473 y Fn(\))38 b Fs(and)-118 598 y Fq(\033)-63 613 y Fk(pp)13 598 y Fn(\()p Fq(A)124 613 y Fl(0)163 598 y Fn(\))12 b Fm(\\)g Fn(\()p Fq(a;)17 b(b)p Fn(\))29 b(=)e Fm(;)p Fs(.)42 b(The)28 b(second)h(part)f(of)f(relation)e(\(4.8\))j(is)f(a)g (consequence)k(of)c(Theorem)h(3.8)f(whic)m(h)-118 722 y(sho)m(ws)34 b(that)598 847 y Fq(\033)653 862 y Fk(\034)696 847 y Fn(\()p Fq(A)807 862 y Fl(0)847 847 y Fn(\))28 b Fm(\022)p 1018 760 418 4 v 28 w Fn(\012)1088 862 y Fk(\034)1131 847 y Fn(\()p Fq(M)10 b Fn(;)17 b Fm(T)26 b Fn(\))i(=)p 1566 755 567 4 v 27 w Fm([)1632 813 y Fk(N)1632 875 y(k)r Fl(=1)1766 847 y Fn(\012)1836 862 y Fk(\034)1879 847 y Fn(\()p Fq(M)2011 862 y Fk(h)2052 874 y Fg(k)2095 847 y Fn(\))f Fm(\022)h Fq(\033)t Fn(\()p Fq(A)2435 862 y Fl(0)2475 847 y Fn(\))p Fq(;)114 b(\034)39 b Fn(=)28 b Fq(sc;)17 b(ac)466 b Fs(\(4.11\))-118 1007 y(and)33 b Fq(\033)127 1022 y Fk(c)162 1007 y Fn(\()p Fq(A)273 1022 y Fl(0)312 1007 y Fn(\))28 b(=)f Fq(\033)536 1022 y Fk(sc)604 1007 y Fn(\()p Fq(A)715 1022 y Fl(0)755 1007 y Fn(\))22 b Fm([)g Fq(\033)958 1022 y Fk(ac)1031 1007 y Fn(\()p Fq(A)1142 1022 y Fl(0)1181 1007 y Fn(\))p Fs(.)44 b(Both)32 b(facts)h(imply)e(that)133 1189 y Fq(\033)188 1204 y Fk(c)223 1189 y Fn(\()p Fq(A)334 1204 y Fl(0)374 1189 y Fn(\))22 b Fm(\\)g Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b Fm(\022)2691 b Fs(\(4.12\))299 1252 y Fi(\020)p 359 1271 591 4 v 359 1362 a Fm([)425 1328 y Fk(N)425 1390 y(k)r Fl(=1)558 1362 y Fn(\012)628 1377 y Fk(sc)696 1362 y Fn(\()p Fq(M)828 1377 y Fk(h)869 1389 y Fg(k)911 1362 y Fn(\))50 b Fm([)p 1115 1271 596 4 v 50 w([)1181 1328 y Fk(N)1181 1390 y(k)r Fl(=1)1314 1362 y Fn(\012)1384 1377 y Fk(ac)1457 1362 y Fn(\()p Fq(M)1589 1377 y Fk(h)1630 1389 y Fg(k)1672 1362 y Fn(\))1710 1252 y Fi(\021)1792 1362 y Fm(\\)22 b Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b Fm(\022)g Fq(\033)t Fn(\()p Fq(A)2395 1377 y Fl(0)2435 1362 y Fn(\))22 b Fm(\\)g Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)g Fq(\033)2982 1377 y Fk(c)3017 1362 y Fn(\()p Fq(A)3128 1377 y Fl(0)3167 1362 y Fn(\))23 b Fm(\\)f Fn(\()p Fq(a;)17 b(b)p Fn(\))-118 1568 y Fs(whic)m(h)33 b(pro)m(v)m(es)h(\(4.8\).)255 1692 y(\(ii\))29 b(By)h(\(3.32\))g(and)g(\(3.47\))g(w)m(e)h(get)f(that)g Fq(S)1889 1656 y Fo(0)1883 1717 y Fk(sc)1950 1692 y Fn(\(\006)2058 1707 y Fk(h)2099 1719 y Fg(k)2142 1692 y Fn(\))d(:=)h Fq(S)2404 1656 y Fo(0)2427 1692 y Fn(\(\006)2535 1707 y Fk(h)2576 1719 y Fg(k)2614 1707 y Fk(;sc)2701 1692 y Fn(\))g Fm(\022)g Fq(S)2938 1656 y Fo(00)2932 1717 y Fk(sc)2999 1692 y Fn(\(\006)3107 1707 y Fk(h)3148 1719 y Fg(k)3191 1692 y Fn(\))f Fm(\022)h Fn(\012)3431 1707 y Fk(sc)3499 1692 y Fn(\()p Fq(M)3631 1707 y Fk(h)3672 1719 y Fg(k)3715 1692 y Fn(\))p Fs(.)-118 1817 y(Therefore,)33 b(if)e Fn(\012)509 1832 y Fk(sc)577 1817 y Fn(\()p Fq(M)709 1832 y Fk(h)750 1844 y Fg(k)792 1817 y Fn(\))22 b Fm(\\)f Fn(\()p Fq(a;)c(b)p Fn(\))33 b 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Ff(The)28 b(sp)-5 b(e)g(ctrum)29 b(of)g Fq(A)905 4056 y Fl(0)974 4041 y Ff(is)f(pur)-5 b(ely)30 b(singular)e(c)-5 b(ontinuous)29 b(within)g(the)g(interval)f Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(,)31 b(i.e.)42 b Fq(\033)t Fn(\()p Fq(A)3626 4056 y Fl(0)3666 4041 y Fn(\))9 b Fm(\\)126 4165 y Fn(\()p Fq(a;)17 b(b)p Fn(\))52 b(=)h Fq(\033)574 4180 y Fk(sc)641 4165 y Fn(\()p Fq(A)752 4180 y Fl(0)792 4165 y Fn(\))32 b Fm(\\)g Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(,)52 b(if)c(for)g(e)-5 b(ach)47 b Fq(k)55 b Fn(=)e(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(N)57 b Ff(c)-5 b(ondition)47 b(\(4.7\))h(is)g(valid)f(for)h(al)5 b(l)126 4290 y Fq(x)28 b Fm(2)g Fn(\()p Fq(a;)17 b(b)p Fn(\))35 b Ff(and)f(c)-5 b(ondition)34 b(\(4.9\))g(is)h(satis\034e)-5 b(d)34 b(for)g(a.e.)45 b Fq(x)28 b Fm(2)g Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(.)45 b(In)34 b(this)h(c)-5 b(ase)34 b(the)h(r)-5 b(elation)775 4472 y Fq(\033)t Fn(\()p Fq(A)945 4487 y Fl(0)984 4472 y Fn(\))22 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)f Fq(\033)1531 4487 y Fk(sc)1599 4472 y Fn(\()p Fq(A)1710 4487 y Fl(0)1749 4472 y Fn(\))22 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)p 2242 4380 567 4 v 28 w Fm([)2308 4438 y Fk(N)2308 4500 y(k)r Fl(=1)2441 4472 y Fn(\012)2511 4487 y Fk(sc)2579 4472 y Fn(\(\006)2687 4487 y Fk(h)2728 4499 y Fg(k)2770 4472 y Fn(\))22 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))399 b Fs(\(4.13\))126 4654 y Ff(holds.)-50 4850 y Fs(\(v\))49 b Ff(The)j(sp)-5 b(e)g(ctrum)53 b(of)g Fq(A)977 4865 y Fl(0)1070 4850 y Ff(is)g(pur)-5 b(ely)53 b(absolutely)h(c)-5 b(ontinuous)52 b(within)h(the)g(interval)g Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(,)58 b(i.e.)126 4974 y Fq(\033)t Fn(\()p Fq(A)296 4989 y Fl(0)335 4974 y Fn(\))29 b Fm(\\)g Fn(\()p Fq(a;)17 b(b)p Fn(\))44 b(=)f Fq(\033)927 4989 y Fk(ac)1000 4974 y Fn(\()p Fq(A)1111 4989 y Fl(0)1150 4974 y Fn(\))29 b Fm(\\)g Fn(\()p Fq(a;)17 b(b)p Fn(\))p Ff(,)46 b(if)d(for)g(e)-5 b(ach)43 b Fq(k)k Fn(=)c(1)p Fq(;)17 b Fn(2)p Fq(;)g(:::;)g(:)g(:)g(:)42 b Ff(c)-5 b(ondition)42 b(\(4.7\))h(is)g(valid)126 5099 y(for)33 b(al)5 b(l)32 b Fq(x)c Fm(2)h Fn(\()p Fq(a;)17 b(b)p Fn(\))33 b Ff(and)g(additional)5 b(ly)32 b(the)h(set)g Fn(\012)1939 5114 y Fk(sc)2007 5099 y Fn(\()p Fq(M)2139 5114 y Fk(k)2182 5099 y Fn(\))18 b Fm(\\)h Fn(\()p Fq(a;)e(b)p Fn(\))33 b Ff(is)g(c)-5 b(ountable,)33 b(in)g(p)-5 b(articular,)33 b(if)126 5223 y Fn(\()p Fq(a;)17 b(b)p Fn(\))22 b Fm(n)g Fn(\012)502 5238 y Fk(ac)575 5223 y Fn(\()p Fq(M)707 5238 y Fk(k)750 5223 y Fn(\))35 b Ff(is)f(c)-5 b(ountable.)44 b(W)-7 b(e)35 b(have)770 5405 y Fq(\033)t Fn(\()p Fq(A)940 5420 y Fl(0)979 5405 y Fn(\))22 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)f Fq(\033)1526 5420 y Fk(ac)1599 5405 y Fn(\()p Fq(A)1710 5420 y Fl(0)1749 5405 y Fn(\))22 b Fm(\\)h Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b(=)p 2242 5314 572 4 v 28 w Fm([)2308 5371 y Fk(N)2308 5433 y(k)r Fl(=1)2441 5405 y Fn(\012)2511 5420 y Fk(ac)2583 5405 y Fn(\(\006)2691 5420 y Fk(h)2732 5432 y Fg(k)2775 5405 y Fn(\))22 b Fm(\\)g Fn(\()p Fq(a;)17 b(b)p Fn(\))395 b Fs(\(4.14\))p eop %%Page: 18 18 18 17 bop -118 100 a Ff(Pr)-5 b(o)g(of:)67 b Fs(\(iv\))44 b(Since)h Fq(\033)t Fn(\()p Fq(A)842 115 y Fl(0)881 100 y Fn(\))j(=)g Fq(\033)1146 115 y Fk(pp)1222 100 y Fn(\()p Fq(A)1333 115 y Fl(0)1372 100 y Fn(\))31 b Fm([)f Fq(\033)1592 115 y Fk(sc)1660 100 y Fn(\()p Fq(A)1771 115 y Fl(0)1811 100 y Fn(\))g Fm([)h Fq(\033)2031 115 y Fk(ac)2103 100 y Fn(\()p Fq(A)2214 115 y Fl(0)2254 100 y Fn(\))44 b Fs(w)m(e)i(obtain)e(from)f(\(i\))g(and)i(\(iii\))d(that)-118 235 y Fq(\033)t Fn(\()p Fq(A)52 250 y Fl(0)91 235 y Fn(\))t Fm(\\)t Fn(\()p Fq(a;)17 b(b)p Fn(\))29 b(=)f Fq(\033)603 250 y Fk(sc)671 235 y Fn(\()p Fq(A)782 250 y Fl(0)821 235 y Fn(\))t Fm(\\)t Fn(\()p Fq(a;)17 b(b)p Fn(\))p Fs(.)42 b(F)-8 b(urther,)26 b(Theorem)e(3.8)f(implies)e(that)j Fq(\033)2719 250 y Fk(sc)2787 235 y Fn(\()p Fq(A)2898 250 y Fl(0)2937 235 y Fn(\))k Fm(\022)p 3108 144 567 4 v 28 w([)3174 201 y Fk(N)3174 263 y(k)r Fl(=1)3307 235 y Fn(\012)3377 250 y Fk(sc)3445 235 y 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b Fn(\()p Fq(a;)17 b(b)p Fn(\))p Fs(.)255 1514 y Ft(LEMMA)38 b(4.5)49 b Ff(L)-5 b(et)42 b Fq(A)f Ff(b)-5 b(e)41 b(a)g(simple)g(densely)f(de\034ne)-5 b(d)40 b(close)-5 b(d)41 b(symmetric)g(op)-5 b(er)g(ator)41 b(on)g(a)-118 1638 y(sep)-5 b(ar)g(able)35 b(Hilb)-5 b(ert)37 b(sp)-5 b(ac)g(e)35 b Fp(H)h Ff(with)h Fq(n)1255 1653 y Fl(+)1314 1638 y Fn(\()p Fq(A)p Fn(\))30 b(=)h Fq(n)1658 1653 y Fo(\000)1717 1638 y Fn(\()p Fq(A)p Fn(\))p Ff(.)49 b(F)-7 b(urther,)37 b(let)f Fn(\005)31 b(=)f Fm(fH)q Fq(;)17 b Fn(\000)2918 1653 y Fl(0)2957 1638 y Fq(;)g Fn(\000)3062 1653 y Fl(1)3101 1638 y Fm(g)37 b Ff(b)-5 b(e)36 b(a)g(b)-5 b(oundary)-118 1763 y(triple)37 b(of)g Fq(A)332 1727 y Fo(\003)409 1763 y Ff(with)h(W)-7 b(eyl)37 b(function)g Fq(M)10 b Fn(\()p Fm(\001)p Fn(\))38 b Ff(and)f Fq(\015)5 b Ff(-\034eld)36 b Fq(\015)5 b Fn(\()p Fm(\001)p Fn(\))p Ff(.)52 b(Assume)38 b(that)f(ther)-5 b(e)38 b(is)f(a)g(non-ne)-5 b(gative)-118 1888 y(non-de)g(cr)g(e)g(asing)29 b(function)i Fn(\011\()p Fm(\001)p Fn(\))c(:)h Fj(R)1245 1903 y Fl(+)1337 1888 y Fm(!)g Fj(R)1531 1903 y Fl(+)1627 1888 y Ff(such)j(that)h Fn(lim)2176 1903 y Fk(x)p Fo(!)p Fl(+)p Fo(1)2433 1888 y Fn(\011\()p Fq(x)p Fn(\))c(=)f(+)p Fm(1)p Ff(.)43 b(If)31 b(for)g(some)g Fq(h)c Fm(2)h(H)-118 2012 y Ff(the)35 b(c)-5 b(ondition)742 2176 y Fq(c)784 2191 y Fl(\011)843 2176 y Fn(\()p Fq(h)p Fn(\))27 b(:=)63 b(sup)1133 2255 y Fl(0)p Fk()f Fn(0)k Fs(and)h Fn(\()p Fm(\001)p Fn(\))718 762 y Fk(t)779 798 y Fs(means)f(that)h(one)f(has)h(to)g(tak)m(e)g(the)g (transp)s(osed)g(expression.)45 b(It)31 b(is)g(easily)g(seen)-118 922 y(\(and)h(w)m(ell-kno)m(wn\))h(that)f Fq(n)923 937 y Fo(\006)983 922 y Fn(\()p Fq(A)p Fn(\))27 b(=)h(2)k Fs(and)h(dom)o Fn(\()p Fq(A)1829 886 y Fo(\003)1868 922 y Fn(\))28 b(=)g Fq(W)2144 886 y Fl(1)2130 947 y(2)2183 922 y Fn(\()p Fj(R)2287 937 y Fo(\000)2352 922 y Fq(;)17 b Fj(C)2461 886 y Fl(2)2507 922 y Fn(\))22 b Fm(\010)g Fq(W)2772 886 y Fl(1)2758 947 y(2)2811 922 y Fn(\()p Fj(R)2915 937 y Fl(+)2980 922 y Fq(;)17 b Fj(C)3090 886 y Fl(2)3135 922 y Fn(\))p Fs(.)44 b(Setting)605 1125 y Fn(\000)666 1140 y Fl(0)719 1099 y Fq(~)705 1125 y(f)39 b Fn(=)27 b(\()p Fq(f)981 1140 y Fl(2)1021 1125 y Fn(\()p Fm(\000)p Fn(0\))p Fq(;)49 b(f)1347 1140 y Fl(1)1387 1125 y Fn(\(+0\)\))1626 1084 y Fk(t)1752 1125 y Fs(and)98 b Fn(\000)2068 1140 y Fl(1)2121 1099 y Fq(~)2107 1125 y(f)39 b Fn(=)27 b(\()p Fq(f)2383 1140 y Fl(1)2423 1125 y Fn(\()p Fm(\000)p Fn(0\))p Fq(;)49 b(f)2749 1140 y Fl(2)2788 1125 y Fn(\(+0\)\))3027 1084 y Fk(t)3579 1125 y Fs(\(5.6\))-118 1328 y(for)53 1302 y Fq(~)39 1328 y(f)k Fm(2)43 b Fs(dom)o Fn(\()p Fq(A)544 1292 y Fo(\003)583 1328 y Fn(\))e Fs(one)g(has)h(that)f Fn(\005)h(=)g Fm(f)p Fj(C)1600 1292 y Fl(2)1646 1328 y Fq(;)17 b Fn(\000)1751 1343 y Fl(0)1790 1328 y Fq(;)g Fn(\000)1895 1343 y Fl(1)1934 1328 y Fm(g)41 b Fs(p)s(erforms)f(a)h(b)s(oundary)g(triple)f(for)g Fq(A)3476 1292 y Fo(\003)3516 1328 y Fs(.)69 b(The)-118 1453 y(self-adjoin)m(t)31 b(extension)i Fq(A)888 1468 y Fl(0)955 1453 y Fn(:=)28 b Fq(A)1159 1416 y Fo(\003)1198 1453 y Fm(j)17 b Fn(k)m(er)q(\(\000)1472 1468 y Fl(0)1511 1453 y Fn(\))32 b Fs(has)h(the)g(domain)432 1655 y(dom)o Fn(\()p Fq(A)727 1670 y Fl(0)766 1655 y Fn(\))28 b(=)g Fm(f)1000 1629 y Fq(~)986 1655 y(f)38 b Fm(2)28 b Fq(W)1272 1614 y Fl(1)1258 1680 y(2)1311 1655 y Fn(\()p Fj(R)1415 1670 y Fo(\000)1480 1655 y Fn(\))22 b Fm(\010)h Fq(W)1746 1614 y Fl(1)1732 1680 y(2)1785 1655 y Fn(\()p Fj(R)1889 1670 y Fl(+)1954 1655 y Fn(\))k(:)h Fq(f)2122 1670 y Fl(2)2161 1655 y Fn(\()p Fm(\000)p Fn(0\))g(=)g(0)p Fq(;)114 b(f)2733 1670 y Fl(1)2772 1655 y Fn(\(+0\))28 b(=)f(0)p Fm(g)p Fq(:)349 b Fs(\(5.7\))-118 1858 y(By)41 1787 y Fm(p)p 124 1787 50 4 v 71 x Fq(z)5 b Fs(,)40 b Fq(z)i Fm(2)c Fj(C)20 b Fs(,)46 b(w)m(e)39 b(denote)h(the)e(branc)m(h)i(of)e (the)g(complex)g(square)i(ro)s(ot)d(with)h(cut)h(along)e(the)i(non-) -118 1983 y(negativ)m(e)i(real)g(axis.)70 b(W)-8 b(e)42 b(set)g Fq(k)s Fn(\()p Fq(\025)p Fn(\))h(:=)1451 1902 y Fm(p)p 1534 1902 230 4 v 81 x Fq(\025)22 b Fm(\000)g Fq(a)1763 1902 y Fm(p)p 1846 1902 229 4 v 81 x Fq(\025)h Fn(+)f Fq(a)p Fs(,)44 b Fq(\025)e Fm(2)i Fj(C)19 b Fs(.)77 b(Ob)m(viously)-8 b(,)43 b Fq(k)s Fn(\()p Fm(\001)p Fn(\))e Fs(is)g(analytic)f(in)-118 2107 y Fj(C)53 b Fm(n)26 b Fn([)p Fm(\000)p Fq(a;)17 b(a)p Fn(])p Fs(.)65 b(One)39 b(has)h Fm(=)p Fs(m)o Fn(\()p Fq(k)s Fn(\()p Fq(\025)p Fn(\)\))f Fm(\025)h Fn(0)f Fs(for)g Fq(\025)g Fm(2)g Fj(C)1902 2122 y Fl(+)2007 2107 y Fs(and)g Fm(=)p Fs(m)o Fn(\()p Fq(k)s Fn(\()p Fq(\025)p Fn(\)\))g Fm(\024)h Fn(0)f Fs(for)g Fq(\025)g Fm(2)h Fj(C)3286 2122 y Fo(\000)3352 2107 y Fs(.)63 b(W)-8 b(e)40 b(\034nd)-118 2232 y Fq(k)s Fn(\()p Fq(x)p Fn(\))50 b(=)242 2150 y Fm(p)p 325 2150 308 4 v 82 x Fq(x)380 2203 y Fl(2)442 2232 y Fm(\000)23 b Fq(a)593 2203 y Fl(2)678 2232 y Fs(for)45 b Fq(x)50 b Fm(\025)g Fq(a)45 b Fs(and)h Fq(k)s Fn(\()p Fq(x)p Fn(\))k(=)f Fm(\000)1808 2150 y(p)p 1892 2150 V 1892 2232 a Fq(x)1947 2203 y Fl(2)2009 2232 y Fm(\000)22 b Fq(a)2159 2203 y Fl(2)2244 2232 y Fs(for)45 b Fq(x)50 b Fm(\024)g(\000)p Fq(a)p Fs(.)83 b(Solving)44 b(the)i(equation)-118 2356 y Fq(A)-45 2320 y Fo(\003)8 2330 y Fq(~)-6 2356 y(f)39 b Fn(=)27 b Fq(\025)255 2330 y(~)241 2356 y(f)44 b Fs(one)32 b(gets)848 2611 y Fq(~)834 2637 y(f)893 2596 y Fo(\007)882 2665 y Fk(\025)952 2637 y Fn(\()p Fq(x)p Fn(\))c(:=)1242 2466 y Fi( )1362 2576 y Fm(\006)p Fq(i)1482 2481 y Fo(p)p 1542 2481 134 3 v 1542 2535 a Fk(\025)p Fl(+)p Fk(a)p 1482 2553 193 4 v 1482 2563 a Fo(p)p 1542 2563 134 3 v 1542 2616 a Fk(\025)p Fo(\000)p Fk(a)1685 2576 y Fq(e)1730 2540 y Fo(\007)p Fk(ik)r Fl(\()p Fk(\025)p Fl(\))p Fk(x)1524 2712 y Fq(e)1569 2676 y Fo(\007)p Fk(ik)r Fl(\()p Fk(\025)p Fl(\))p Fk(x)2029 2466 y Fi(!)2125 2637 y Fq(\037)2186 2652 y Fo(\007)2245 2637 y Fn(\()p Fq(x)p Fn(\))p Fq(;)114 b(\025)28 b Fm(2)g Fj(C)2762 2652 y Fl(+)3579 2637 y Fs(\(5.8\))-118 2917 y(Using)k(the)h (de\034nition)e(of)i(the)f(W)-8 b(eyl)33 b(function)f(w)m(e)i(\034nd) 854 3179 y Fq(M)10 b Fn(\()p Fq(\025)p Fn(\))28 b(=)f Fs(diag)1418 3039 y Fi(\022)1491 3179 y Fq(i)1535 3031 y Fm(p)p 1618 3031 229 4 v 81 x Fq(\025)22 b Fn(+)g Fq(a)p 1534 3156 313 4 v 1534 3176 a Fm(p)p 1617 3176 230 4 v 81 x Fq(\025)g Fm(\000)h Fq(a)1874 3179 y(i)1917 3031 y Fm(p)p 2000 3031 V 81 x Fq(\025)f Fm(\000)h Fq(a)p 1917 3156 313 4 v 1918 3176 a Fm(p)p 2001 3176 229 4 v 81 x Fq(\025)f Fn(+)g Fq(a)2240 3039 y Fi(\023)2330 3179 y Fq(;)114 b(\025)28 b Fm(2)g Fj(C)2716 3194 y Fl(+)2781 3179 y Fq(:)771 b Fs(\(5.9\))-118 3439 y(Since)33 b Fm(H)d Fn(=)f Fj(C)421 3403 y Fl(2)500 3439 y Fs(is)k(t)m(w)m(o-dimensional)d (w)m(e)35 b(c)m(ho)s(ose)f Fm(T)54 b Fn(=)28 b Fm(f)p Fq(h)2093 3454 y Fl(1)2133 3439 y Fq(;)17 b(h)2233 3454 y Fl(2)2272 3439 y Fm(g)33 b Fs(with)g Fq(h)2634 3454 y Fl(1)2702 3439 y Fn(=)c(\(1)p Fq(;)17 b Fn(0\))3025 3403 y Fk(t)3087 3439 y Fs(and)33 b Fq(h)3333 3454 y Fl(2)3401 3439 y Fn(=)c(\(0)p Fq(;)17 b Fn(1\))3724 3403 y Fk(t)3752 3439 y Fs(.)-118 3573 y(This)40 b(leads)f(to)g(the)i(asso)s (ciated)e(scalar)g(W)-8 b(eyl)40 b(functions)f Fq(M)2195 3588 y Fk(h)2236 3597 y Fc(1)2275 3573 y Fn(\()p Fq(\025)p Fn(\))g(=)h Fq(i)2606 3479 y Fo(p)p 2665 3479 134 3 v 53 x Fk(\025)p Fl(+)p Fk(a)p 2606 3550 193 4 v 2606 3560 a Fo(p)p 2665 3560 134 3 v 53 x Fk(\025)p Fo(\000)p Fk(a)2848 3573 y Fs(and)g Fq(M)3139 3588 y Fk(h)3180 3597 y Fc(2)3219 3573 y Fn(\()p Fq(\025)p Fn(\))f(=)h Fq(i)3550 3479 y Fo(p)p 3609 3479 V 53 x Fk(\025)p Fo(\000)p Fk(a)p 3550 3550 193 4 v 3550 3560 a Fo(p)p 3609 3560 134 3 v 53 x Fk(\025)p Fl(+)p Fk(a)3752 3573 y Fs(,)-118 3706 y Fq(\025)53 b Fm(2)h Fj(C)177 3721 y Fl(+)243 3706 y Fs(.)88 b(One)48 b(immediately)c(gets:)74 b Fn(\012)1488 3721 y Fk(s)1525 3706 y Fn(\()p Fq(M)1657 3721 y Fk(h)1698 3730 y Fc(1)1737 3706 y Fn(\))53 b(=)g Fm(f)p Fq(a)p Fm(g)p Fs(,)e Fn(\012)2256 3721 y Fk(s)2293 3706 y Fn(\()p Fq(M)2425 3721 y Fk(h)2466 3730 y Fc(2)2505 3706 y Fn(\))i(=)g Fm(f\000)p Fq(a)p Fm(g)p Fs(,)f Fn(\012)3102 3721 y Fk(sc)3170 3706 y Fn(\()p Fq(M)3302 3721 y Fk(h)3343 3730 y Fc(1)3381 3706 y Fn(\))i(=)f Fm(f)p Fq(a)p Fm(g)p Fs(,)-118 3831 y Fn(\012)-48 3846 y Fk(sc)20 3831 y Fn(\()p Fq(M)152 3846 y Fk(h)193 3855 y Fc(2)231 3831 y Fn(\))28 b(=)g Fm(f\000)p Fq(a)p Fm(g)p Fs(,)k Fn(\012)758 3846 y Fk(pp)834 3831 y Fn(\()p Fq(M)966 3846 y Fk(h)1007 3855 y Fc(1)1046 3831 y Fn(\))27 b(=)h(\012)1285 3846 y Fk(pp)1361 3831 y Fn(\()p Fq(M)1493 3846 y Fk(h)1534 3855 y Fc(2)1572 3831 y Fn(\))g(=)g Fm(;)k Fs(and)566 4034 y Fn(\012)636 4049 y Fk(ac)709 4034 y Fn(\()p Fq(M)841 4049 y Fk(h)882 4058 y Fc(1)921 4034 y Fn(\))27 b(=)h(\012)1160 4049 y Fk(ac)1232 4034 y Fn(\()p Fq(M)1364 4049 y Fk(h)1405 4058 y Fc(2)1444 4034 y Fn(\))g(=)f(\012)1683 4049 y Fk(ac)1756 4034 y Fn(\()p Fq(M)5 b(;)17 b Fm(T)26 b Fn(\))h(=)h(\()p Fm(\0001)p Fq(;)17 b Fm(\000)p Fq(a)p Fn(\))22 b Fm([)h Fn(\()p Fq(a;)17 b Fn(+)p Fm(1)p Fn(\))p Fq(:)434 b Fs(\(5.10\))-118 4237 y(By)48 b(Theorem)f(3.8)g(w)m(e)h(\034nd)g Fq(\033)1071 4252 y Fk(p)1111 4237 y Fn(\()p Fq(A)1222 4252 y Fl(0)1261 4237 y Fn(\))53 b(=)f Fm(;)p Fs(.)88 b(F)-8 b(urther,)50 b(from)c(Prop)s(osition)g(4.2)h(w)m(e)h(get)f Fq(\033)3390 4252 y Fk(ac)3463 4237 y Fn(\()p Fq(A)3574 4252 y Fl(0)3613 4237 y Fn(\))53 b(=)-118 4361 y(\()p Fm(\0001)p Fq(;)17 b Fm(\000)p Fq(a)p Fn(])h Fm([)f Fn([)p Fq(a;)g Fn(+)p Fm(1)p Fn(\))p Fs(.)42 b(Since)31 b(the)f(sets)h Fn(\012)1478 4376 y Fk(sc)1546 4361 y Fn(\()p Fq(M)1678 4376 y Fk(h)1719 4385 y Fc(1)1758 4361 y Fn(\))f Fs(and)g Fn(\012)2083 4376 y Fk(sc)2151 4361 y Fn(\()p Fq(M)2283 4376 y Fk(h)2324 4385 y Fc(2)2363 4361 y Fn(\))g Fs(are)g(\034nite)g(Theorem)g (4.3\(ii\))e(sho)m(ws)-118 4486 y(that)43 b Fq(\033)159 4501 y Fk(sc)226 4486 y Fn(\()p Fq(A)337 4501 y Fl(0)377 4486 y Fn(\))i(=)g Fm(;)p Fs(.)74 b(So)43 b(the)g(op)s(erator)g Fq(A)1533 4501 y Fl(0)1615 4486 y Fs(is)f(absolutely)h(con)m(tin)m (uous)g(with)f Fq(\033)t Fn(\()p Fq(A)3098 4501 y Fl(0)3138 4486 y Fn(\))j(=)g Fq(\033)3397 4501 y Fk(ac)3470 4486 y Fn(\()p Fq(A)3581 4501 y Fl(0)3620 4486 y Fn(\))h(=)-118 4610 y(\()p Fm(\0001)p Fq(;)17 b Fm(\000)p Fq(a)p Fn(])23 b Fm([)f Fn([)p Fq(a;)17 b Fn(+)p Fm(1)p Fn(\))p Fs(.)255 4818 y Ft(EXAMPLE)36 b(5.3)49 b Fs(Let)44 b Fq(T)61 b Fn(=)47 b Fq(T)1549 4782 y Fo(\003)1632 4818 y Fs(b)s(e)e(a)e(b)s (ounded)i(nonnegativ)m(e)f(op)s(erator)g(on)f(a)h(Hilb)s(ert)-118 4943 y(space)34 b Fm(K)q Fs(,)e Fn(dim)16 b Fm(K)29 b Fn(=)e Fm(1)p Fs(.)43 b(Let)33 b Fq(A)g Fs(b)s(e)f(a)h(symmetric)e (Dirac)h(t)m(yp)s(e)i(op)s(erator)e(de\034ned)i(on)e(dom)o Fn(\()p Fq(A)p Fn(\))c(=)-118 4980 y Fi(n)-38 5064 y Fq(~)-52 5091 y(f)39 b Fn(=)27 b(\()p Fq(f)224 5106 y Fl(1)264 5091 y Fq(;)49 b(f)388 5106 y Fl(2)427 5091 y Fn(\))465 5055 y Fk(t)523 5091 y Fm(2)28 b Fq(W)723 5055 y Fl(1)709 5115 y(2)762 5091 y Fn(\()p Fj(R)5 b Fq(;)17 b Fm(H)q Fn(\))33 b(:)1134 5064 y Fq(~)1121 5091 y(f)11 b Fn(\(0\))27 b(=)h(0)1485 4980 y Fi(o)1583 5091 y Fs(b)m(y)448 5387 y Fn(\()p Fq(A)573 5361 y(~)559 5387 y(f)11 b Fn(\)\()p Fq(x)p Fn(\))28 b(=)918 5247 y Fi(\022)1034 5326 y Fn(0)84 b Fm(\000)p Fq(I)1033 5447 y(I)131 b Fn(0)1337 5247 y Fi(\023)1464 5320 y Fq(d)p 1437 5364 107 4 v 1437 5456 a(dx)1566 5361 y(~)1553 5387 y(f)11 b Fn(\()p Fq(x)p Fn(\))22 b(+)1863 5247 y Fi(\022)1978 5326 y Fq(T)146 b Fn(0)1989 5447 y(0)94 b Fm(\000)p Fq(T)2321 5247 y Fi(\023)2425 5361 y Fq(~)2411 5387 y(f)11 b Fn(\()p Fq(x)p Fn(\))p Fq(;)2686 5361 y(~)2673 5387 y(f)38 b Fm(2)28 b Fs(dom)o Fn(\()p Fq(A)p Fn(\))p Fq(;)317 b Fs(\(5.11\))p eop %%Page: 22 22 22 21 bop -118 100 a Fs(where)37 b Fm(H)f Fn(=)e Fm(K)26 b(\010)f(K)38 b Fs(and)f Fq(I)44 b Fs(stands)37 b(for)f(the)h(iden)m (tit)m(y)f(op)s(erator)g(on)g Fm(K)q Fs(.)55 b(One)37 b(has)g Fq(n)3134 115 y Fo(\006)3193 100 y Fn(\()p Fq(A)p Fn(\))d(=)g Fm(1)i Fs(and)-118 224 y(dom)o Fn(\()p Fq(A)177 188 y Fo(\003)217 224 y Fn(\))27 b(=)h Fq(W)492 188 y Fl(1)478 249 y(2)531 224 y Fn(\()p Fj(R)635 239 y Fo(\000)700 224 y Fq(;)17 b Fm(H)q Fn(\))22 b Fm(\010)g Fq(W)1094 188 y Fl(1)1080 249 y(2)1133 224 y Fn(\()p Fj(R)1237 239 y Fl(+)1302 224 y Fq(;)17 b Fm(H)q Fn(\))p Fs(.)255 349 y(Let)39 b(us)h(c)m(hec)m(k)h(that)d(the)i(op)s(erator)e Fq(A)h Fs(is)f(a)h(simple)e(symmetric)h(op)s(erator.)61 b(Indeed,)42 b(in)c(the)-118 473 y(opp)s(osite)27 b(case)h Fq(A)g Fs(admits)e(a)i(decomp)s(osition)d Fq(A)j Fn(=)g Fq(C)1871 488 y Fl(1)1922 473 y Fm(\010)12 b Fq(C)2081 488 y Fl(2)2149 473 y Fs(with)27 b(self-adjoin)m(t)f Fq(C)2933 488 y Fl(1)2999 473 y Fn(=)i Fq(C)3180 437 y Fo(\003)3173 498 y Fl(1)3247 473 y Fs(and)g(a)f(simple)-118 598 y(symmetric)47 b(op)s(erator)g Fq(C)850 613 y Fl(2)889 598 y Fs(.)89 b(By)49 b(Gelfand-K)m(ost)m(yuc)m(henk)m(o)g(Theorem)f (\(see[5]\))h Fq(C)3028 613 y Fl(1)3115 598 y Fs(has)f(a)f(complete) -118 722 y(system)37 b Fm(f)p Fq(')323 737 y Fk(\025)368 722 y Fm(g)418 737 y Fk(\025)p Fo(2)p Fl(\003)596 722 y Fs(of)f(generalized)g(eigenfunctions)g(\(distributions\).)54 b(Hence)38 b(eac)m(h)f Fq(')3137 737 y Fk(\025)3178 746 y Fc(0)3217 722 y Fn(\()p Fq(\025)3312 737 y Fl(0)3386 722 y Fm(2)e Fn(\003\))h Fs(is)g(a)-118 847 y(\(generalized\))f (eigenfunction)g(of)h Fq(A)p Fs(.)54 b(It)36 b(follo)m(ws)f(from)g(the) h(corresp)s(onding)g(fact)g(on)g(scalar)f(ordinary)-118 971 y(di\033eren)m(tial)41 b(equations)i(\(see[5]\))h(applied)e(to)h Fn(\()p Fq(')1758 986 y Fk(\025)1799 995 y Fc(0)1838 971 y Fn(\()p Fq(x)p Fn(\))p Fq(;)17 b(e)p Fn(\))43 b Fs(with)f Fq(e)k Fm(2)g(H)e Fs(that)f Fq(')2988 986 y Fk(\025)3029 995 y Fc(0)3111 971 y Fs(is)f(w)m(eakly)i Fq(C)3629 935 y Fo(1)3747 971 y Fs(-)-118 1096 y(solution)31 b(of)h(the)h(homogeneous)f(initial)d(v)-5 b(alue)32 b(problem)339 1224 y Fi(\022)455 1303 y Fn(0)84 b Fm(\000)p Fq(I)454 1424 y(I)131 b Fn(0)758 1224 y Fi(\023)848 1364 y Fq(')912 1323 y Fo(0)912 1389 y Fk(\025)953 1398 y Fc(0)992 1364 y Fn(\()p Fq(x)p Fn(\))22 b(+)1243 1224 y Fi(\022)1358 1303 y Fq(T)36 b Fm(\000)22 b Fq(\025I)256 b Fn(0)1484 1424 y(0)208 b Fm(\000)p Fq(T)36 b Fm(\000)23 b Fq(\025I)2160 1224 y Fi(\023)2250 1364 y Fq(')2314 1379 y Fk(\025)2355 1388 y Fc(0)2394 1364 y Fn(\()p Fq(x)p Fn(\))28 b(=)g(0)p Fq(;)114 b(')2911 1379 y Fk(\025)2952 1388 y Fc(0)2990 1364 y Fn(\(0\))28 b(=)f(0)p Fq(:)-118 1632 y Fs(By)33 b(Cauc)m(h)m(y)h(Uniqueness)h(Theorem)e(w)m(e)g(get)g Fq(')1677 1647 y Fk(\025)1718 1656 y Fc(0)1784 1632 y Fn(=)28 b(0)p Fs(.)43 b(Th)m(us)34 b Fq(A)f Fs(is)f(simple.)255 1756 y(Next)42 b(w)m(e)h(de\034ne)f Fn(\000)1006 1771 y Fl(0)1087 1756 y Fs(and)f Fn(\000)1346 1771 y Fl(1)1426 1756 y Fs(in)g(accordance)h(with)f(\(5.6\))f(whic)m(h)i(leads)f(to)g (the)g(b)s(oundary)-118 1881 y(triple)35 b Fn(\005)f(=)g Fm(fH)q Fq(;)17 b Fn(\000)603 1896 y Fl(0)642 1881 y Fq(;)g Fn(\000)747 1896 y Fl(1)786 1881 y Fm(g)36 b Fs(for)g Fq(A)1098 1845 y Fo(\003)1137 1881 y Fs(.)55 b(F)-8 b(ollo)m(wing)33 b(the)k(line)e(of)h(reasoning)f(of)h(Example)g(5.2)g(one)g(gets)h(the) -118 2006 y(W)-8 b(eyl)32 b(function)956 2130 y Fq(M)10 b Fn(\()p Fq(\025)p Fn(\))28 b(=)f Fs(diag)o Fn(\()p Fq(M)1635 2145 y Fl(+)1695 2130 y Fn(\()p Fq(\025)p Fn(\))p Fq(;)49 b(M)1998 2145 y Fo(\000)2057 2130 y Fn(\()p Fq(\025)p Fn(\)\))p Fq(;)114 b(\025)28 b Fm(2)g Fj(C)2614 2145 y Fl(+)2679 2130 y Fq(;)824 b Fs(\(5.12\))-118 2316 y(with)32 b Fq(M)198 2331 y Fl(+)257 2316 y Fn(\()p Fq(\025)p Fn(\))c(:=)g Fq(i)592 2221 y Fo(p)p 651 2221 147 3 v 54 x Fk(\025)p Fl(+)p Fk(T)p 592 2293 206 4 v 592 2303 a Fo(p)p 651 2303 147 3 v 53 x Fk(\025)p Fo(\000)p Fk(T)840 2316 y Fs(and)33 b Fq(M)1124 2331 y Fo(\000)1183 2316 y Fn(\()p Fq(\025)p Fn(\))27 b(:=)h Fq(i)1517 2221 y Fo(p)p 1576 2221 V 54 x Fk(\025)p Fo(\000)p Fk(T)p 1517 2293 206 4 v 1517 2303 a Fo(p)p 1576 2303 147 3 v 53 x Fk(\025)p Fl(+)p Fk(T)1733 2316 y Fs(.)44 b(Here)2034 2235 y Fm(p)p 2117 2235 248 4 v 81 x Fq(\025)22 b Fn(+)g Fq(T)46 b Fs(and)2587 2235 y Fm(p)p 2670 2235 250 4 v 81 x Fq(\025)22 b Fm(\000)g Fq(T)47 b Fs(are)32 b(de\034ned)i(b)m(y)1158 2543 y Fm(p)p 1241 2543 V 86 x Fq(\025)22 b Fm(\006)g Fq(T)42 b Fn(=)1621 2493 y Fi(Z)1721 2520 y Fo(k)p Fk(T)10 b Fo(k)1677 2719 y Fl(0)1863 2543 y Fm(p)p 1946 2543 214 4 v 86 x Fq(\025)22 b Fm(\006)h Fq(t)28 b(dE)2311 2644 y Fk(T)2366 2629 y Fn(\()p Fq(t)p Fn(\))p Fq(:)1026 b Fs(\(5.13\))-118 2902 y(and)33 b(the)f(branc)m(h)i(of)673 2831 y Fm(p)p 756 2831 50 4 v 71 x Fq(z)j Fs(is)32 b(c)m(hosen)i(lik)m (e)e(in)g(Example)g(5.2.)255 3027 y(If)h Fq(x)28 b Fm(2)g Fq(\032)p Fn(\()p Fq(T)14 b Fn(\))28 b(:=)f Fj(R)33 b Fm(n)22 b Fq(\033)t Fn(\()p Fq(T)14 b Fn(\))33 b Fs(and)f Fq(h)c Fn(=)g Fq(k)1718 3042 y Fl(+)1798 3027 y Fm(\010)23 b Fn(0)p Fs(,)32 b(then)i(w)m(e)f(easily)f(\034nd)h(that)427 3247 y Fq(M)521 3262 y Fk(h)566 3247 y Fn(\()p Fq(x)22 b Fn(+)g Fq(i)p Fn(0\))28 b(=)g(\()p Fq(M)1163 3262 y Fl(+)1222 3247 y Fn(\()p Fq(x)22 b Fn(+)g Fq(i)p Fn(0\))p Fq(k)1606 3262 y Fl(+)1665 3247 y Fq(;)17 b(k)1760 3262 y Fl(+)1819 3247 y Fn(\))27 b(=)1570 b Fs(\(5.14\))593 3467 y Fq(i)643 3332 y Fi(Z)742 3358 y Fo(j)p Fk(x)p Fo(j)698 3557 y Fl(0)853 3313 y Fi(p)p 952 3313 267 4 v 952 3398 a Fm(j)p Fq(x)22 b Fn(+)h Fq(t)p Fm(j)p 852 3444 368 4 v 852 3464 a Fi(p)p 951 3464 268 4 v 951 3549 a Fm(j)p Fq(x)g Fm(\000)f Fq(t)p Fm(j)1229 3467 y Fq(d)p Fn(\()p Fq(E)1390 3482 y Fk(T)1445 3467 y Fn(\()p Fq(t)p Fn(\))p Fq(k)1607 3482 y Fl(+)1666 3467 y Fq(;)17 b(k)1761 3482 y Fl(+)1819 3467 y Fn(\))22 b(+)1977 3332 y Fi(Z)2077 3358 y Fo(k)p Fk(T)10 b Fo(k)2033 3557 y(j)p Fk(x)p Fo(j)2230 3313 y Fi(p)p 2330 3313 267 4 v 85 x Fm(j)p Fq(x)22 b Fn(+)g Fq(t)p Fm(j)p 2229 3444 368 4 v 2229 3464 a Fi(p)p 2329 3464 268 4 v 85 x Fm(j)p Fq(x)g Fm(\000)h Fq(t)p Fm(j)2607 3467 y Fq(d)p Fn(\()p Fq(E)2768 3482 y Fk(T)2823 3467 y Fn(\()p Fq(t)p Fn(\))p Fq(k)2985 3482 y Fl(+)3043 3467 y Fq(;)17 b(k)3138 3482 y Fl(+)3197 3467 y Fn(\))-118 3752 y Fs(whic)m(h)33 b(yields)419 4037 y Fn(0)28 b Fm(\024)g Fq(V)658 4052 y Fk(h)703 4037 y Fn(\()p Fq(x)22 b Fn(+)g Fq(i)p Fn(0\))28 b(=)f Fm(=)p Fs(m)o Fn(\()p Fq(M)1452 4052 y Fk(h)1498 4037 y Fn(\()p Fq(x)22 b Fn(+)g Fq(i)p Fn(0\)\))28 b(=)2000 3902 y Fi(Z)2100 3928 y Fo(j)p Fk(x)p Fo(j)2056 4127 y Fl(0)2210 3883 y Fi(p)p 2310 3883 267 4 v 85 x Fm(j)p Fq(x)22 b Fn(+)g Fq(t)p Fm(j)p 2210 4015 368 4 v 2210 4035 a Fi(p)p 2309 4035 268 4 v 2309 4120 a Fm(j)p Fq(x)g Fm(\000)h Fq(t)p Fm(j)2587 4037 y Fq(d)p Fn(\()p Fq(E)2748 4052 y Fk(T)2803 4037 y Fn(\()p Fq(t)p Fn(\))p Fq(k)2965 4052 y Fl(+)3024 4037 y Fq(;)17 b(k)3119 4052 y Fl(+)3177 4037 y Fn(\))p Fq(:)288 b Fs(\(5.15\))-118 4328 y(Let)43 b Fq(\017)k Fn(:=)f(inf)22 b Fq(\033)t Fn(\()p Fq(T)14 b Fn(\))46 b Fm(\025)g Fn(0)p Fs(.)76 b(If)43 b Fq(x)j Fm(2)h Fn(\()p Fm(\000)p Fq(\017;)17 b(\017)p Fn(\))p Fs(,)46 b(then)e Fq(V)1923 4343 y Fk(h)1968 4328 y Fn(\()p Fq(x)30 b Fn(+)f Fq(i)p Fn(0\))46 b(=)g(0)p Fs(.)75 b(If)43 b Fq(x)k Fm(2)f Fq(\032)p Fn(\()p Fq(T)14 b Fn(\))29 b Fm(n)h Fn([)p Fm(\000)p Fq(\017;)17 b(\017)p Fn(])p Fs(,)47 b(then)-118 4453 y(there)40 b(is)g(alw)m(a)m(ys)g(a)g Fq(k)703 4468 y Fl(+)802 4453 y Fm(2)h(K)g Fs(suc)m(h)g(that)f Fq(V)1529 4468 y Fk(h)1574 4453 y Fn(\()p Fq(x)27 b Fn(+)g Fq(i)p Fn(0\))40 b Fq(>)g Fn(0)p Fs(.)66 b(Cho)s(osing)39 b Fq(h)h Fn(=)g(0)27 b Fm(\010)h Fq(k)3092 4468 y Fo(\000)3151 4453 y Fs(,)42 b Fq(k)3271 4468 y Fo(\000)3370 4453 y Fm(2)e(K)q Fs(,)i(and)-118 4578 y(rep)s(eating)36 b(the)i (considerations)f(w)m(e)h(get)f Fq(V)1502 4593 y Fk(h)1547 4578 y Fn(\()p Fq(x)26 b Fn(+)f Fq(i)p Fn(0\))36 b(=)f(0)i Fs(for)g Fq(x)f Fm(2)g Fn(\()p Fm(\000)p Fq(\017;)17 b(\017)p Fn(\))p Fs(.)59 b(If)37 b Fm(\000)p Fq(x)g Fm(2)f Fq(\032)p Fn(\()p Fq(T)14 b Fn(\))25 b Fm(n)g Fn([)p Fm(\000)p Fq(\017;)17 b(\017)p Fn(])p Fs(,)-118 4702 y(then)37 b(there)h(is)e(alw)m(a)m(ys)h(a)g Fq(k)917 4717 y Fo(\000)1011 4702 y Fm(2)e(K)j Fs(suc)m(h)g(that)f Fq(V)1723 4717 y Fk(h)1768 4702 y Fn(\()p Fq(x)25 b Fn(+)g Fq(i)p Fn(0\))35 b Fq(>)g Fn(0)p Fs(.)55 b(Applying)36 b(Prop)s(osition)f(4.2)i(w)m(e)g(get)-118 4827 y(that)32 b Fq(\033)148 4842 y Fk(ac)221 4827 y Fn(\()p Fq(A)332 4842 y Fl(0)371 4827 y Fn(\))c(=)f(\()p Fm(\0001)p Fq(;)17 b(\017)p Fn(])23 b Fm([)f Fn([)p Fq(\017;)17 b Fn(+)p Fm(1)p Fn(\))p Fq(:)33 b Fs(In)g(particular,)d(w)m(e)k(ha)m(v)m(e)g Fq(\033)2382 4842 y Fk(ac)2454 4827 y Fn(\()p Fq(A)2565 4842 y Fl(0)2605 4827 y Fn(\))28 b(=)f Fj(R)44 b Fs(if)31 b Fq(\017)d Fn(=)g(0)p Fs(.)255 4951 y(Finally)-8 b(,)30 b(let)i(us)h(sho)m(w)h(that)e Fq(A)1398 4966 y Fl(0)1470 4951 y Fs(is)g(purely)h(absolutely)f(con)m(tin)m(uous.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)468 5171 y Fq(V)525 5186 y Fk(h)570 5171 y Fn(\()p Fq(x)23 b Fn(+)f Fq(iy)t Fn(\))27 b(=)2520 b Fs(\(5.16\))635 5388 y Fm(=)p Fs(m)o Fn(\()p Fq(M)920 5403 y Fk(h)965 5388 y Fn(\()p Fq(x)22 b Fn(+)g Fq(iy)t Fn(\)\))27 b(=)1470 5253 y Fi(Z)1570 5279 y Fo(k)p Fk(T)10 b Fo(k)1525 5478 y Fl(0)1712 5388 y Fm(=)p Fs(m)1882 5248 y Fi(\022)h Fm(p)p 2049 5248 416 4 v 73 x Fq(x)22 b Fn(+)g Fq(iy)k Fn(+)c Fq(t)p 1965 5365 501 4 v 1965 5385 a Fm(p)p 2048 5385 417 4 v 74 x Fq(x)g Fn(+)g Fq(iy)k Fm(\000)d Fq(t)2475 5248 y Fi(\023)2565 5388 y Fq(d)p Fn(\()p Fq(E)2726 5403 y Fk(T)2781 5388 y Fn(\()p Fq(t)p Fn(\))p Fq(k)2943 5403 y Fl(+)3002 5388 y Fq(;)17 b(k)3097 5403 y Fl(+)3155 5388 y Fn(\))p eop %%Page: 23 23 23 22 bop -118 100 a Fs(where)34 b Fq(h)27 b Fn(=)h Fq(k)402 115 y Fl(+)483 100 y Fm(\010)23 b Fn(0)p Fs(.)43 b(Applying)31 b(the)i(H\366lder)f(inequalit)m(y)g(one)g(gets)h(from)f(\(5.16\))637 390 y Fm(j)p Fq(V)722 405 y Fk(h)767 390 y Fn(\()p Fq(x)22 b Fn(+)g Fq(iy)t Fn(\))p Fm(j)27 b(\024)1263 255 y Fi(Z)1363 281 y Fo(k)p Fk(T)10 b Fo(k)1319 480 y Fl(0)1505 246 y Fi(\014)1505 306 y(\014)1505 365 y(\014)1505 425 y(\014)1549 250 y Fm(p)p 1632 250 416 4 v 73 x Fq(x)23 b Fn(+)f Fq(iy)j Fn(+)d Fq(t)p 1548 367 501 4 v 1548 387 a Fm(p)p 1631 387 417 4 v 74 x Fq(x)h Fn(+)f Fq(iy)j Fm(\000)e Fq(t)2058 246 y Fi(\014)2058 306 y(\014)2058 365 y(\014)2058 425 y(\014)2108 390 y Fq(d)p Fn(\()p Fq(E)2269 405 y Fk(T)2324 390 y Fn(\()p Fq(t)p Fn(\))p Fq(k)2486 405 y Fl(+)2545 390 y Fq(;)17 b(k)2640 405 y Fl(+)2698 390 y Fn(\))794 b Fs(\(5.17\))804 677 y Fm(\024)28 b(k)p Fq(k)1010 692 y Fl(+)1068 677 y Fm(k)1118 636 y Fl(2\()p Fk(p)p Fo(\000)p Fl(1\))1355 542 y Fi(Z)1454 568 y Fo(k)p Fk(T)10 b Fo(k)1410 767 y Fl(0)1607 610 y Fn(\(\()p Fq(x)23 b Fn(+)f Fq(t)p Fn(\))1932 574 y Fl(2)1993 610 y Fn(+)h Fq(y)2144 574 y Fl(2)2182 610 y Fn(\))2220 574 y Fk(p=)p Fl(4)p 1606 654 725 4 v 1606 746 a Fn(\(\()p Fq(x)g Fm(\000)g Fq(t)p Fn(\))1933 717 y Fl(2)1994 746 y Fn(+)f Fq(y)2144 717 y Fl(2)2183 746 y Fn(\))2221 717 y Fk(p=)p Fl(4)2369 677 y Fq(d)p Fn(\()p Fq(E)2530 692 y Fk(T)2585 677 y Fn(\()p Fq(t)p Fn(\))p Fq(k)2747 692 y Fl(+)2805 677 y Fq(;)17 b(k)2900 692 y Fl(+)2959 677 y Fn(\))p Fq(;)-118 952 y Fn(1)27 b Fq(<)h(p)g(<)f Fn(+)p Fm(1)p Fs(.)43 b(Let)33 b Fn(\()p Fq(a;)17 b(b)p Fn(\))28 b Fm(\022)g Fj(R)43 b Fs(b)s(e)33 b(an)f(arbitrary)g(\034nite)g(in)m(terv)-5 b(al.)42 b(One)33 b(has)37 1107 y Fi(Z)137 1133 y Fk(b)92 1332 y(a)187 1242 y Fq(dx)28 b Fm(j)p Fq(V)406 1257 y Fk(h)451 1242 y Fn(\()p Fq(x)22 b Fn(+)g Fq(iy)t Fn(\))p Fm(j)815 1201 y Fk(p)882 1242 y Fm(\024)28 b(k)p Fq(k)1088 1257 y Fl(+)1146 1242 y Fm(k)1196 1201 y Fl(2\()p Fk(p)p Fo(\000)p Fl(1\))1433 1107 y Fi(Z)1532 1133 y Fo(k)p Fk(T)10 b Fo(k)1488 1332 y Fl(0)1675 1242 y Fq(d)p Fn(\()p Fq(E)1836 1257 y Fk(T)1890 1242 y Fn(\()p Fq(t)p Fn(\))p Fq(k)2052 1257 y Fl(+)2111 1242 y Fq(;)17 b(k)2206 1257 y Fl(+)2265 1242 y Fn(\))2320 1107 y Fi(Z)2419 1133 y Fk(b)2375 1332 y(a)2470 1242 y Fq(dx)2614 1175 y Fn(\(\()p Fq(x)23 b Fn(+)f Fq(t)p Fn(\))2939 1139 y Fl(2)3001 1175 y Fn(+)g Fq(y)3151 1139 y Fl(2)3189 1175 y Fn(\))3227 1139 y Fk(p=)p Fl(4)p 2614 1220 V 2614 1311 a Fn(\(\()p Fq(x)g Fm(\000)h Fq(t)p Fn(\))2940 1282 y Fl(2)3001 1311 y Fn(+)f Fq(y)3151 1282 y Fl(2)3190 1311 y Fn(\))3228 1282 y Fk(p=)p Fl(4)3348 1242 y Fq(:)155 b Fs(\(5.18\))-118 1521 y(If)35 b Fn(0)e Fq(<)g(y)j Fm(\024)d Fn(1)p Fs(,)j(then)g(there)g (is)f(a)h(constan)m(t)g Fq(C)1608 1536 y Fl(1)1680 1521 y Fq(>)d Fn(0)i Fs(suc)m(h)i(that)e Fn(\()p Fq(x)25 b Fn(+)f Fq(t)p Fn(\))2601 1485 y Fl(2)2665 1521 y Fn(+)g Fq(y)2817 1485 y Fl(2)2888 1521 y Fm(\024)34 b Fq(C)3069 1536 y Fl(1)3143 1521 y Fs(for)h Fq(t)e Fm(2)h Fn([0)p Fq(;)17 b Fm(k)p Fq(T)d Fm(k)p Fn(])-118 1646 y Fs(and)33 b Fq(x)28 b Fm(2)g Fn([)p Fq(a;)17 b(b)p Fn(])p Fs(.)44 b(Hence)117 1801 y Fi(Z)216 1827 y Fk(b)172 2026 y(a)267 1936 y Fq(dx)28 b Fm(j)p Fq(V)486 1951 y Fk(h)530 1936 y Fn(\()p Fq(x)23 b Fn(+)f Fq(iy)t Fn(\))p Fm(j)895 1895 y Fk(p)961 1936 y Fm(\024)28 b Fq(C)1143 1885 y Fk(p=)p Fl(4)1136 1961 y(1)1254 1936 y Fm(k)p Fq(k)1355 1951 y Fl(+)1413 1936 y Fm(k)1463 1895 y Fl(2\()p Fk(p)p Fo(\000)p Fl(1\))1700 1801 y Fi(Z)1799 1827 y Fo(k)p Fk(T)10 b Fo(k)1755 2026 y Fl(0)1941 1936 y Fq(d)p Fn(\()p Fq(E)2102 1951 y Fk(T)2157 1936 y Fn(\()p Fq(t)p Fn(\))p Fq(k)2319 1951 y Fl(+)2378 1936 y Fq(;)17 b(k)2473 1951 y Fl(+)2532 1936 y Fn(\))2587 1801 y Fi(Z)2686 1827 y Fk(b)2642 2026 y(a)2737 1936 y Fq(dx)3045 1869 y Fn(1)p 2881 1913 379 4 v 2881 2005 a Fm(j)p Fq(x)22 b Fm(\000)g Fq(t)p Fm(j)3148 1976 y Fk(p=)p Fl(2)3268 1936 y Fq(:)235 b Fs(\(5.19\))-118 2236 y(If)32 b Fn(1)c Fq(<)f(p)h(<)g Fn(2)p Fs(,)k(then)h(w)m(e)h(ha)m (v)m(e)g Fq(C)1109 2251 y Fk(p)1176 2236 y Fn(:=)27 b(sup)1453 2260 y Fk(t)p Fo(2)p Fl([0)p Fk(;)p Fo(k)p Fk(T)10 b Fo(k)p Fl(])1762 2156 y Fi(R)1828 2182 y Fk(b)1809 2271 y(a)1879 2236 y Fq(dx)28 b Fm(j)p Fq(x)22 b Fm(\000)h Fq(t)p Fm(j)2281 2200 y Fo(\000)p Fk(p=)p Fl(2)2474 2236 y Fq(<)k Fn(+)p Fm(1)p Fq(;)32 b Fs(whic)m(h)h(yields)984 2538 y Fn(sup)948 2618 y Fl(0)p Fk(