This is a multi-part message in MIME format. ---------------0008240448471 Content-Type: text/plain; name="00-321.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-321.keywords" Spectral asymptotics, spectral gap, discrete spectrum, coupling constant ---------------0008240448471 Content-Type: application/postscript; name="lara.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lara.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: lara.dvi %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o lara.ps lara.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.08.22:1709 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 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b(+)g Fs(p)p Fu(\()p Fs(x)p Fu(\),)31 b(p)s(erturb)s(ed)e(b) m(y)g(a)f(deca)m(ying)h(p)s(oten)m(tial)d Fr(\000)p Fs(\013)q(V)2614 2031 y Fq(0)2654 2016 y Fu(,)j(whic)m(h)g(mo)s(dels)e(the)456 2133 y(impurit)m(y)38 b(\()p Fs(\013)j(>)f Fu(0)g(is)f(a)h(coupling)e (constan)m(t\).)67 b(The)41 b(basic)f(assumptions)f(are)456 2249 y(that)f Fs(H)46 b Fu(has)39 b(a)f(sp)s(ectral)h(gap)f(\003)g(in)g (its)g(essen)m(tial)h(sp)s(ectrum;)j(eigen)m(v)-5 b(alues)39 b(of)456 2365 y Fs(A)p Fu(\()p Fs(\013)q Fu(\))27 b(=)h Fs(A)14 b Fr(\000)g Fs(\013)q(V)1097 2380 y Fq(0)1166 2365 y Fu(inside)28 b(\003)g(corresp)s(ond)i(to)f(the)g(eigenstates)g (of)g(the)g(electrons)456 2481 y(observ)m(ed)j(in)e(solid)f(state)h(ph) m(ysics.)45 b(W)-8 b(e)30 b(study)i(the)f(n)m(um)m(b)s(er)g(of)e(eigen) m(v)-5 b(alues)31 b(of)456 2598 y Fs(A)p Fu(\()p Fs(\013)q Fu(\))36 b(inside)g(an)h(in)m(terv)-5 b(al)35 b(\()p Fs(\025)1576 2613 y Fq(1)1616 2598 y Fs(;)17 b(\025)1717 2613 y Fq(2)1756 2598 y Fu(\))35 b Fr(\032)g 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Fs(;)52 b Fu(then)37 b Fs(\025)1144 2331 y Fo(\000)1236 2316 y Fu(=)c Fr(\0001)p Fu(\).)54 b(Belo)m(w)36 b(w)m(e)h(p)s(erturb)f(the) g(op)s(erator)g Fs(A)g Fu(so,)h(that)456 2432 y(the)c(sp)s(ectrum)g(of) f(the)h(p)s(erturb)s(ed)g(op)s(erator)f(in)g(\003)g(is)g(discrete.)555 2548 y(Let)h Fs(V)49 b Fu(:)28 b([1)p Fs(;)17 b Fu(+)p Fr(1)p Fu(\))27 b Fr(!)g Fm(R)1445 2563 y Fq(+)1542 2548 y Fu(b)s(e)33 b(a)g(smo)s(oth)e(function)h(and)929 2729 y(lim)909 2779 y Fn(r)r Fo(!1)1101 2729 y Fs(r)1148 2688 y Fn(s)1184 2729 y Fs(V)22 b Fu(\()p Fs(r)s Fu(\))27 b(=)h Fs(v)1564 2744 y Fn(a)1606 2729 y Fs(;)114 b Fu(2)27 b Fs(<)h(s)f(<)h Fu(2)22 b(+)g(2)p Fs(=)p Fu(\()p Fs(d)f Fr(\000)i Fu(1\))p Fs(;)17 b(v)2759 2744 y Fn(a)2828 2729 y Fs(>)27 b Fu(0)p Fs(:)313 b Fu(\(1\))456 2935 y(W)-8 b(e)32 b(denote)h(b)m(y)g Fs(V)1129 2950 y Fq(0)1200 2935 y Fu(the)f(op)s(erator)g(of)f(m)m(ultiplication)c(b)m(y)33 b(the)g(function)e Fs(V)22 b Fu(\()p Fr(j)p Fs(x)p Fr(j)p Fu(\))456 3051 y(extended)44 b(b)m(y)f(zero)f(for)f Fr(j)p Fs(x)p Fr(j)j Fs(<)f Fu(1.)72 b(W)-8 b(e)42 b(put)g Fs(A)p Fu(\()p Fs(\013)q Fu(\))i(:=)f Fs(A)29 b Fr(\000)g Fs(\013)q(V)2915 3066 y Fq(0)2954 3051 y Fs(;)17 b(\013)44 b(>)g Fu(0.)71 b(It)456 3168 y(is)38 b(easy)j(to)e(see)h(that)f(the)h(sp)s(ectrum)g Fs(\033)t Fu(\()p Fs(A)p Fu(\()p Fs(\013)q Fu(\)\))e(is)h(discrete)h (in)f(\003.)63 b(Let)39 b Fs(\025)3270 3183 y Fo(\000)3369 3168 y Fs(<)456 3284 y(\025)513 3299 y Fq(1)581 3284 y Fs(<)29 b(\025)743 3299 y Fq(2)812 3284 y Fs(<)g(\025)974 3299 y Fq(+)1033 3284 y Fu(.)46 b(The)34 b(main)e(ob)5 b(ject)34 b(of)f(our)g(in)m(v)m(estigation)f(is)h(the)h(asymptotic)456 3400 y(b)s(eha)m(vior)e(of)g(the)h(v)-5 b(alue)65 b(rank)p Fs(E)1682 3416 y Fn(A)p Fq(\()p Fn(\013)p Fq(\))1839 3400 y Fu(\()p Fs(\025)1934 3415 y Fq(1)1974 3400 y Fs(;)17 b(\025)2075 3415 y Fq(2)2114 3400 y Fu(\))65 b(for)32 b Fs(\013)c Fr(!)f(1)p Fu(.)555 3516 y(Belo)m(w)e(w)m(e)h (systematically)d(use)i(the)h(follo)m(wing)c(notations)h(for)h(the)i (op)s(erators)456 3633 y(on)48 b(a)g(domain.)90 b(Let)48 b(\012)55 b Fr(\032)g Fm(R)1650 3596 y Fn(d)1745 3633 y Fu(b)s(e)49 b(an)f(op)s(en)h(domain.)89 b(W)-8 b(e)49 b(consider)g(the)456 3749 y(follo)m(wing)29 b(op)s(erator)j(in)g Fs(L)1440 3764 y Fq(2)1480 3749 y Fu(\(\012\))1166 3938 y Fs(A)1239 3897 y Fq(\()p Fn(")p Fq(\))1331 3938 y Fu(\()p Fs(\013)q(;)17 b Fu(\012\))27 b(=)h Fr(\000)p Fu(\001)22 b(+)g Fs(p)h Fr(\000)f Fs(\013)q(V)2284 3953 y Fq(0)2323 3938 y Fs(;)114 b(\013)29 b(>)e Fu(0;)456 4119 y(the)38 b(index)h Fs(")f Fu(tak)m(es)i(the)e(v)-5 b(alues)39 b Fr(D)h Fu(or)f Fr(N)52 b Fu(and)39 b(c)m(haracterizes)g(the)g(b)s (oundary)456 4235 y(\(Diric)m(hlet)g(or)j(Neumann\))g(conditions)g(on)g Fs(@)5 b Fu(\012.)73 b(In)43 b(o)s(der)f(to)f(form)m(ulate)g(the)456 4351 y(main)29 b(result)h(w)m(e)i(need)g(to)f(in)m(tro)s(duce)f(the)i (densit)m(y)f(of)g(states)g(for)g(the)g(op)s(erator)456 4467 y Fs(A)p Fu(.)62 b(Let)40 b Fs(x)855 4482 y Fq(1)894 4467 y Fs(;)17 b(x)993 4482 y Fq(2)1072 4467 y Fr(2)39 b Fm(R)1243 4431 y Fn(d)1328 4467 y Fu(and)g Fs(l)i(>)d Fu(0.)63 b(Let)39 b(also)f(\012\()p Fs(l)r Fu(\))h(:=)g(\012\()p Fs(l)r Fu(;)17 b Fs(x)2826 4482 y Fq(1)2866 4467 y Fs(;)g(x)2965 4482 y Fq(2)3004 4467 y Fu(\))39 b(=)f Fs(l)r Fu(\()p Fm(Q)45 b Fu(+)456 4584 y Fs(x)511 4599 y Fq(1)550 4584 y Fu(\))23 b(+)f Fs(x)764 4599 y Fq(2)803 4584 y Fu(.)44 b(Then)34 b(for)e(an)g(arbitrary)g Fs(\025)27 b Fr(2)h Fm(R)44 b Fu(the)33 b(follo)m(wing)d(limit)f(exists)1238 4773 y Fs(\032)p Fu(\()p Fs(\025)p Fu(\))f(=)41 b(lim)1553 4833 y Fn(l)q Fo(!1)1732 4773 y Fs(l)1763 4731 y Fo(\000)p Fn(d)1859 4773 y Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)2159 4731 y Fq(\()p Fn(")p Fq(\))2251 4773 y Fu(\(0)p Fs(;)g Fu(\012\()p Fs(l)r Fu(\)\)\))p Fs(:)456 4983 y Fu(It)37 b(do)s(es)h(not)g(dep)s(end)g(on)g Fs(x)1509 4998 y Fq(1)1549 4983 y Fs(;)54 b(x)1685 4998 y Fq(2)1762 4983 y Fu(and)38 b(on)g(the)g(v)-5 b(alues)37 b(of)g(the)h(index)g Fs(")p Fu(.)58 b(This)456 5099 y(limit)25 b(is)k(called)f(the)h(in)m(tegrated) g(densit)m(y)h(of)f(states)h(for)e(the)i(op)s(erator)e Fs(A)h Fu(at)g(the)456 5216 y(p)s(oin)m(t)i Fs(\025)p Fu(.)p eop %%Page: 3 3 3 2 bop 1513 236 a Fl(DISCRETE)33 b(SPECTR)n(UM)1019 b(3)555 425 y Fu(Let)33 b Fs(\025)787 440 y Fq(1)826 425 y Fs(;)50 b(\025)960 440 y Fq(2)1031 425 y Fu(b)s(e)33 b(\014xed,)h Fs(\025)1484 440 y Fo(\000)1571 425 y Fs(<)27 b(\025)1731 440 y Fq(1)1798 425 y Fs(<)h(\025)1959 440 y Fq(2)2026 425 y Fs(<)f(\025)2186 440 y Fq(+)2245 425 y Fu(.)44 b(W)-8 b(e)33 b(put)1039 670 y Fs(b)p Fu(\()p Fs(")p Fu(\))61 b(=)1399 496 y Fk(8)1399 571 y(<)1399 720 y(:)1514 611 y Fu(v-sup)1745 634 y Fn(x)p Fo(2)p Fj(R)1880 615 y Fi(d)1920 611 y Fu(\()p Fs(p)p Fu(\()p Fs(x)p Fu(\))22 b Fr(\000)h Fs(\025)2317 626 y Fq(1)2356 611 y Fu(\))p Fs(;)114 b(")28 b Fu(=)f Fr(D)s Fu(;)1524 727 y(v-inf)1719 746 y Fn(x)p Fo(2)p Fj(R)1854 727 y Fi(d)8 b Fu(\()p Fs(p)p Fu(\()p Fs(x)p Fu(\))22 b Fr(\000)h Fs(\025)2291 742 y Fq(2)2330 727 y Fu(\))p Fs(;)114 b(")27 b Fu(=)h Fr(N)15 b Fs(:)456 913 y Fu(W)-8 b(e)33 b(imp)s(ose)e(an)h (additional)e(condition)h(on)i(the)g(function)f Fs(V)21 b Fu(:)555 1029 y Fw(Condition)49 b(1.1.)77 b Fu(There)44 b(exists)h Fs(\021)1963 1044 y Fo(\003)2049 1029 y Fs(>)h Fu(0,)g(suc)m(h)f(that)e(for)h(all)d(0)46 b Fs(<)g(\021)51 b(<)456 1146 y(\021)504 1161 y Fo(\003)589 1146 y Fu(and)c Fs(\013)k(>)g(\021)1085 1109 y Fo(\000)p Fn(s)1223 1146 y Fu(the)c(n)m(um)m(b)s(er)f(of)g(the)h(ro)s(ots)f(of)f(the)i(equation) f(2)p Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(r)54 b Fu(=)456 1265 y Fs(\013)q Fu(\()p Fs(r)604 1229 y Fq(2)642 1265 y Fs(V)22 b Fu(\()p Fs(r)s Fu(\)\))882 1229 y Fo(0)905 1265 y Fs(;)54 b(")35 b Fu(=)h Fr(D)s Fs(;)17 b Fr(N)51 b Fu(in)37 b(the)h(in)m(terv)-5 b(al)36 b Fs(r)j Fr(2)d Fu([1)p Fs(;)17 b(\021)t(\013)2505 1229 y Fq(1)p Fn(=s)2611 1265 y Fu(])38 b(do)s(es)g(not)f(exceed)j(a)456 1381 y(\014xed)33 b(n)m(um)m(b)s(er)g Fs(l)d Fr(2)e Fm(N)9 b Fu(.)555 1497 y(W)-8 b(e)23 b(note)g(that)g(Condition)e(1.1)h(is)g (ful\014lled)f(when)j(the)f(function)f Fs(r)2964 1461 y Fo(\000)p Fq(1)3058 1497 y Fu(\()p Fs(r)3143 1461 y Fq(2)3182 1497 y Fs(V)g Fu(\()p Fs(r)s Fu(\)\))3422 1461 y Fo(0)456 1613 y Fu(decreases)39 b(monotonously)c(for)i Fs(r)g(>)e Fu(1.)56 b(In)37 b(particular)e(this)h(condition)f(is)i (ful-)456 1730 y(\014lled)31 b(for)h Fs(V)22 b Fu(\()p Fs(r)s Fu(\))27 b(=)g Fs(v)1222 1745 y Fn(a)1264 1730 y Fs(r)1311 1693 y Fo(\000)p Fn(s)1403 1730 y Fs(;)49 b(r)30 b(>)e Fu(1.)555 1846 y(The)c(main)d(result)h(of)h(the)g(presen)m (t)h(w)m(ork)g(giv)m(es)f(an)f(asymptotics)h(of)f(the)h(en)m(tire)456 1962 y(sp)s(ectrum)38 b(m)m(ultiplicit)m(y)d(of)j(the)h(op)s(erator)e Fs(A)p Fu(\()p Fs(\013)q Fu(\))h(in)g(the)h(in)m(terv)-5 b(al)37 b(\()p Fs(\025)3102 1977 y Fq(1)3141 1962 y Fs(;)17 b(\025)3242 1977 y Fq(2)3281 1962 y Fu(\))38 b(as)456 2078 y Fs(\013)28 b Fr(!)f(1)p Fu(.)594 2250 y Fw(Theorem)37 b(1.1.)49 b Fh(L)-5 b(et)40 b Fs(d)c Fr(\025)i Fu(2)h Fh(and)g Fs(\025)2006 2265 y Fo(\000)2102 2250 y Fs(<)e(\025)2272 2265 y Fq(1)2348 2250 y Fs(<)f(\025)2517 2265 y Fq(2)2593 2250 y Fs(<)h(\025)2763 2265 y Fq(+)2822 2250 y Fh(.)59 b(Assume)40 b(that)456 2367 y Fu(\(1\))34 b Fh(and)g(Condition)g(1.1)h (ar)-5 b(e)34 b(ful\014l)5 b(le)-5 b(d.)44 b(Then)1548 2531 y Fu(lim)1522 2581 y Fn(\013)p Fo(!1)1725 2531 y Fs(\013)1788 2490 y Fo(\000)p Fn(d=s)1951 2531 y Fu(rank)q Fs(E)2216 2547 y Fn(A)p Fq(\()p Fn(\013)p Fq(\))2373 2531 y Fu(\()p Fs(\025)2468 2546 y Fq(1)2507 2531 y Fs(;)17 b(\025)2608 2546 y Fq(2)2647 2531 y Fu(\))28 b(=)1083 2627 y Fk(Z)1130 2816 y Fj(R)1178 2797 y Fi(d)1218 2744 y Fu(\()p Fs(\032)p Fu(\()p Fs(\025)1401 2759 y Fq(2)1463 2744 y Fu(+)22 b Fs(v)1608 2759 y Fn(a)1649 2744 y Fr(j)p Fs(x)p Fr(j)1760 2703 y Fo(\000)p Fn(s)1852 2744 y Fu(\))g Fr(\000)g Fs(\032)p Fu(\()p Fs(\025)2156 2759 y Fq(1)2218 2744 y Fu(+)g Fs(v)2363 2759 y Fn(a)2405 2744 y Fr(j)p Fs(x)p Fr(j)2516 2703 y Fo(\000)p Fn(s)2607 2744 y Fu(\)\))p Fs(dx:)3320 2653 y Fu(\(2\))555 2954 y(No)m(w)35 b(w)m(e)g(discuss)h (the)f(condition)e Fs(s)d(<)h Fu(2)23 b(+)g(2)p Fs(=)p Fu(\()p Fs(d)f Fr(\000)i Fu(1\).)49 b(The)35 b(quasiclassical)456 3071 y(exp)s(onen)m(t)c(for)e(the)h(Laplace)f(op)s(erator)f(on)i(the)g (sphere)h(is)e(\032)f(=)g(\()p Fs(d)16 b Fr(\000)g Fu(1\))p Fs(=)p Fu(2.)41 b(The)456 3187 y("Diric)m(hlet-Neumann)g(brac)m(k)m (eting")k(metho)s(d)f(can)h(b)s(e)g(applied)f(only)g(to)g(the)456 3303 y(asymptotics)d(of)f(the)i(o)s(der)f Fs(\013)1592 3267 y Fn(q)1630 3303 y Fs(;)57 b(q)47 b(>)42 b Fu(\032.)69 b(In)42 b(Theorem)f(1.1)g Fs(q)46 b Fu(=)d Fs(d=s)p Fu(.)68 b(This)456 3419 y(leads)32 b(to)g(the)h(discussed)i(estimate)c(for)h Fs(s)p Fu(.)1459 3617 y(2.)55 b Ft(A)m(uxiliar)-7 b(y)39 b(resul)-7 b(t)555 3791 y Fw(1.)46 b Fu(Let)34 b Fr(Y)j Fu(=)29 b Fr(f)p Fs(x)g Fr(2)g Fm(R)1368 3755 y Fn(d)1444 3791 y Fu(:)62 b(1)29 b Fs(<)g Fr(j)p Fs(x)p Fr(j)g Fs(<)g(\021)t(\013) 2076 3755 y Fq(1)p Fn(=s)2182 3791 y Fr(g)p Fs(;)50 b(d)29 b Fr(\025)h Fu(2.)45 b(W)-8 b(e)34 b(write)f Fs(A)3109 3755 y Fq(\()p Fn(")p Fq(\))3201 3791 y Fu(\()p Fs(\013)q Fu(\))c(=)456 3910 y Fs(A)529 3874 y Fq(\()p Fn(")p Fq(\))620 3910 y Fu(\()p Fs(\013)q(;)17 b Fr(Y)8 b Fu(\))35 b(for)g(brevit)m(y)-8 b(.)52 b(F)-8 b(rom)34 b(the)i(inequalit)m(y)e Fs(A)2394 3874 y Fq(\()p Fo(N)10 b Fq(\))2521 3910 y Fu(\()p Fs(\013)q Fu(\))32 b Fs(<)g(A)2873 3874 y Fq(\()p Fo(D)r Fq(\))2989 3910 y Fu(\()p Fs(\013)q Fu(\))j(w)m(e)h(ob-)456 4030 y(tain)d(that)h Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)1171 3993 y Fq(\()p Fo(D)r Fq(\))1287 4030 y Fu(\()p Fs(\013)q Fu(\)\))30 b Fr(\024)h Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)1902 3993 y Fq(\()p Fo(N)10 b Fq(\))2029 4030 y Fu(\()p Fs(\013)q Fu(\)\))34 b(for)f(an)i(arbitrary)e Fs(\025)e Fr(2)g Fm(R)5 b Fu(.)54 b(The)456 4146 y(follo)m(wing)30 b(statemen)m(t)j(giv)m(es)g(an)f(estimate)g(of)g(the)h(di\013erence)h (of)e(the)h(distribu-)456 4262 y(tion)e(functions)i(of)f(the)h(sp)s (ectrum)g(of)f(the)h(op)s(erators)f Fs(A)2565 4226 y Fq(\()p Fo(D)r Fq(\))2681 4262 y Fu(\()p Fs(\013)q Fu(\))g(and)h Fs(A)3115 4226 y Fq(\()p Fo(N)10 b Fq(\))3241 4262 y Fu(\()p Fs(\013)q Fu(\).)594 4434 y Fw(Prop)s(osition)35 b(2.1.)49 b Fh(L)-5 b(et)45 b Fs(p)h Fr(2)h Fs(L)1874 4449 y Fo(1)1949 4434 y Fu(\()p Fm(R)2052 4398 y Fn(d)2099 4434 y Fu(\))p Fs(;)61 b Fu(2)46 b Fs(<)g(s)g(<)f Fu(2)30 b(+)f Fs(s=)p Fu(\()p Fs(d)g Fr(\000)h Fu(1\))44 b Fh(and)456 4550 y Fs(\025)513 4565 y Fq(1)580 4550 y Fs(<)27 b(\025)740 4565 y Fq(2)780 4550 y Fh(.)43 b(Assume)33 b(that)f Fs(V)54 b Fh(satis\014es)31 b(the)i(c)-5 b(onditions)31 b(of)h(The)-5 b(or)g(em)31 b(1.1.)43 b(Then)456 4667 y(ther)-5 b(e)34 b(is)h Fs(\021)851 4682 y Fq(0)918 4667 y Fs(>)28 b Fu(0)34 b Fh(such)h(that)g(for)g(al)5 b(l)35 b Fu(0)27 b Fs(<)h(\021)j(<)c (\021)2232 4682 y Fq(0)642 4826 y Fs(\026)p Fu(\()p Fs(\021)t Fu(\))g(:=)h(lim)17 b(sup)1043 4895 y Fn(\013)p Fo(!1)1302 4826 y Fs(\013)1365 4785 y Fo(\000)p Fn(d=s)1528 4826 y Fu(\()p Fs(N)10 b Fu(\()p Fs(\025)1749 4841 y Fq(2)1789 4826 y Fs(;)17 b(A)1906 4785 y Fq(\()p Fo(N)10 b Fq(\))2032 4826 y Fu(\()p Fs(\013)q Fu(\)\))22 b Fr(\000)h Fs(N)10 b Fu(\()p Fs(\025)2514 4841 y Fq(1)2553 4826 y Fs(;)17 b(A)2670 4785 y Fq(\()p Fo(D)r Fq(\))2786 4826 y Fu(\()p Fs(\013)q Fu(\)\)\))27 b Fs(<)g Fr(1)p Fs(:)456 5021 y Fh(A)n(lso)34 b(one)g(has)1677 5138 y Fu(lim)1673 5194 y Fn(\021)r Fo(!)p Fq(0)1833 5138 y Fs(\026)p Fu(\()p Fs(\021)t Fu(\))27 b(=)h(0)p Fs(:)p eop %%Page: 4 4 4 3 bop 456 236 a Fl(4)1180 b(O.)34 b(SAFR)n(ONO)n(V)555 425 y Fw(The)49 b(pro)s(of)43 b Fu(of)g(Prop)s(osition)d(2.1)j(is)f (based)h(on)g(the)g(elemen)m(tary)f(separa-)456 541 y(tion)k(of)h(the)g (v)-5 b(ariables.)87 b(W)-8 b(e)47 b(put)h Fs(\025)1899 556 y Fo(D)2012 541 y Fu(=)53 b Fs(\025)2198 556 y Fq(1)2284 541 y Fu(and)48 b Fs(\025)2546 556 y Fo(N)2670 541 y Fu(=)53 b Fs(\025)2856 556 y Fq(2)2895 541 y Fu(.)88 b(The)48 b(v)-5 b(alue)456 658 y Fs(N)10 b Fu(\()p Fs(\025)639 673 y Fn(")676 658 y Fs(;)17 b(A)793 622 y Fq(\()p Fn(")p Fq(\))884 658 y Fu(\()p Fs(\013)q Fu(\)\))37 b(is)g(equal)g(to)g(the)g (maximal)e(dimension)g(of)i(the)h(subspaces)i Fs(F)456 774 y Fu(on)32 b(whic)m(h)857 838 y Fk(Z)903 1026 y Fo(Y)959 955 y Fu(\()p Fr(jr)p Fs(u)p Fr(j)1192 914 y Fq(2)1253 955 y Fu(+)22 b(\()p Fs(p)p Fu(\()p Fs(x)p Fu(\))g Fr(\000)h Fs(\025)1748 970 y Fn(")1784 955 y Fu(\))p Fr(j)p Fs(u)p Fr(j)1934 914 y Fq(2)1972 955 y Fu(\))p Fs(dx)61 b(<)f(\013)2392 838 y Fk(Z)2438 1026 y Fo(Y)2511 955 y Fs(V)22 b Fu(\()p Fr(j)p Fs(x)p Fr(j)p Fu(\))p Fr(j)p Fs(u)p Fr(j)2889 914 y Fq(2)2927 955 y Fs(dx:)260 b Fu(\(3\))456 1171 y(It)44 b(is)f(su\016cien)m(t)i(to)f(consider)g(subspaces)i Fs(F)61 b Fr(\032)47 b Fs(C)2407 1135 y Fo(1)2400 1196 y Fq(0)2482 1171 y Fu(\()p Fr(Y)8 b Fu(\))43 b(\(the)i(case)f Fs(")j Fu(=)g Fr(D)s Fu(\))456 1287 y(and)41 b Fs(F)58 b Fr(\032)43 b Fs(C)972 1251 y Fo(1)1047 1287 y Fu(\()p Fr(Y)8 b Fu(\))41 b(\(the)i(case)f Fs(")h Fu(=)g Fr(N)15 b Fu(\).)71 b(Let)42 b Fs(b)p Fu(\()p Fs(")p Fu(\))f(b)s(e)h(the)h (same)e(as)h(in)f Fr(x)p Fu(1.)456 1407 y(W)-8 b(e)35 b(replace)h(the)f(function)g Fs(p)g Fu(in)g(the)g(de\014nition)g(of)g (the)g(op)s(erator)g Fs(A)3076 1370 y Fq(\()p Fn(")p Fq(\))3168 1407 y Fu(\()p Fs(\013)q Fu(\))f(b)m(y)456 1523 y Fs(b)p Fu(\()p Fs(")p Fu(\))15 b(+)g Fs(\025)782 1538 y Fn(")818 1523 y Fu(.)42 b(Since)29 b Fs(b)p Fu(\()p Fr(D)s Fu(\))15 b(+)g Fs(\025)1498 1538 y Fo(D)1586 1523 y Fr(\025)28 b Fs(p)p Fu(\()p Fs(x)p Fu(\))h(and)g Fs(b)p Fu(\()p Fr(N)15 b Fu(\))g(+)g Fs(\025)2463 1538 y Fo(N)2562 1523 y Fr(\024)28 b Fs(p)p Fu(\()p Fs(x)p Fu(\))h(the)h(di\013erence) 1193 1702 y Fs(N)10 b Fu(\()p Fs(\025)1376 1717 y Fo(N)1448 1702 y Fs(;)17 b(A)1565 1661 y Fq(\()p Fo(N)10 b Fq(\))1691 1702 y Fu(\()p Fs(\013)q Fu(\)\))54 b Fr(\000)h Fs(N)10 b Fu(\()p Fs(\025)2237 1717 y Fo(D)2299 1702 y Fs(;)17 b(A)2416 1661 y Fq(\()p Fo(D)r Fq(\))2531 1702 y Fu(\()p Fs(\013)q Fu(\)\))612 b(\(4\))456 1874 y(can)36 b(only)g(increase.)55 b(After)36 b(the)g(passage)h(to)f(the)h(spherical)f(co)s(ordinates)f (and)456 1998 y(the)40 b(substitutions)f Fs(r)j Fu(=)e Fs(e)1466 1962 y Fn(t)1496 1998 y Fs(;)56 b(u)39 b Fu(=)h Fs(v)j Fu(exp)q(\()p Fr(\000)2154 1952 y Fq(\()p Fn(d)p Fo(\000)p Fq(2\))p 2154 1975 182 4 v 2227 2033 a(2)2346 1998 y Fs(t)p Fu(\))c(and)h Fs(p)g Fr(7!)f Fs(b)p Fu(\()p Fs(")p Fu(\))27 b(+)g Fs(\025)3233 2013 y Fn(")3309 1998 y Fu(the)456 2116 y(inequalit)m(y)32 b(\(3\))h(turns)g(in)m(to)1085 2229 y Fk(Z)1168 2256 y Fn(T)1131 2418 y Fq(0)1239 2229 y Fk(Z)1286 2418 y Fn(S)1336 2346 y Fu(\()p Fr(jr)1485 2361 y Fn(\022)1524 2346 y Fs(v)t Fr(j)1603 2305 y Fq(2)1664 2346 y Fu(+)22 b Fr(j)1800 2279 y Fs(@)5 b(v)p 1800 2323 108 4 v 1808 2415 a(@)g(t)1939 2346 y Fu(+)2047 2279 y(\(2)22 b Fr(\000)h Fs(d)p Fu(\))p 2047 2323 298 4 v 2172 2415 a(2)2354 2346 y Fs(v)t Fr(j)2433 2305 y Fq(2)2472 2346 y Fu(\))p Fs(d\022)s(dt)f Fu(+)1812 2474 y Fk(Z)1895 2501 y Fn(T)1858 2663 y Fq(0)1967 2474 y Fk(Z)2013 2663 y Fn(S)2081 2592 y Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(e)2289 2550 y Fq(2)p Fn(t)2354 2592 y Fr(j)p Fs(v)t Fr(j)2461 2550 y Fq(2)2499 2592 y Fs(d\022)s(dt)28 b(<)1766 2837 y(\013)1845 2720 y Fk(Z)1928 2746 y Fn(T)1892 2908 y Fq(0)2000 2720 y Fk(Z)2046 2908 y Fn(S)2114 2837 y Fs(e)2159 2796 y Fq(2)p Fn(t)2224 2837 y Fs(V)21 b Fu(\()p Fs(e)2385 2796 y Fn(t)2415 2837 y Fu(\))p Fr(j)p Fs(u)p Fr(j)2565 2796 y Fq(2)2603 2837 y Fs(d\022)s(dt;)3320 2581 y Fu(\(5\))456 3047 y(where)33 b Fs(T)42 b Fu(=)27 b(1)p Fs(=s)17 b Fu(log)o(\()p Fs(\021)1315 3010 y Fn(s)1352 3047 y Fs(\013)q Fu(\).)555 3163 y(Let)40 b Fr(F)10 b Fu(\()p Fr(D)s Fu(\))39 b(=)g Fs(C)1206 3127 y Fo(1)1199 3187 y Fq(0)1281 3163 y Fu(\(0)p Fs(;)17 b(T)d Fu(\))p Fs(;)55 b Fr(F)10 b Fu(\()p Fr(N)15 b Fu(\))39 b(=)g Fs(C)2089 3127 y Fo(1)2164 3163 y Fu([0)p Fs(;)17 b(T)d Fu(])39 b(and)g(let)g Fs(W)2857 3178 y Fn(j)2894 3163 y Fs(;)56 b(j)46 b Fu(=)39 b(0)p Fs(;)17 b Fu(1)p Fs(;)g(:::)456 3279 y Fu(b)s(e)32 b(the)h(space)h(of)e (the)h(spherical)f(functions)h(of)f(o)s(der)g Fs(j)39 b Fu(on)32 b(the)h(sphere)h Fs(S)6 b Fu(.)43 b(The)456 3395 y(subspaces,)27 b(obtained)22 b(b)m(y)h(closing)e Fr(F)10 b Fu(\()p Fs(")p Fu(\))p Fs(W)2043 3410 y Fn(j)2079 3395 y Fu(,)24 b(dioganalize)c(the)j(quadratic)f(forms)456 3517 y(in)i(b)s(oth)h(parts)h(of)32 b(\(5\))o(.)42 b(Hence)26 b(the)g(op)s(erator)f Fs(A)2234 3481 y Fq(\()p Fn(")p Fq(\))2326 3517 y Fu(\()p Fs(\013)q Fu(\))f(is)h(decomp)s(osed)h(in)m (to)f(the)456 3651 y(orthogonal)e(sum)j(of)f(the)h(op)s(erators)f Fs(A)1898 3600 y Fq(\()p Fn(")p Fq(\))1898 3674 y Fn(j)1990 3651 y Fu(\()p Fs(\013)q Fu(\))g(op)s(erating)f(in)h(the)g(corresp)s (onding)456 3770 y("cells".)41 b(In)30 b(the)g(cell)e(with)h(the)h(n)m (um)m(b)s(er)g Fs(j)35 b Fu(the)30 b(inequalit)m(y)g(\(5\))f(is)g (transformed)456 3886 y(in)m(to)i Fs(k)704 3901 y Fn(j)768 3886 y Fu(=)d(dim)n Fs(W)1126 3901 y Fn(j)1196 3886 y Fu(copies)k(of)g(the)h(inequalities)e(for)h(scalar)g(functions)941 4003 y Fk(Z)1024 4029 y Fn(T)988 4192 y Fq(0)1080 4120 y Fu(\()p Fr(j)p Fs(y)1198 4079 y Fo(0)1242 4120 y Fr(\000)1351 4053 y Fu(\()p Fs(d)22 b Fr(\000)h Fu(2\))p 1351 4097 V 1475 4188 a(2)1658 4120 y Fs(y)t Fr(j)1738 4079 y Fq(2)1799 4120 y Fu(+)f(\(\003)2003 4135 y Fn(j)2061 4120 y Fu(+)g Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(e)2367 4079 y Fq(\(2)p Fn(t)p Fq(\))2487 4120 y Fu(\))p Fr(j)p Fs(y)t Fr(j)2633 4079 y Fq(2)2671 4120 y Fu(\))p Fs(dt)60 b(<)1400 4365 y(\013)1479 4248 y Fk(Z)1562 4274 y Fn(T)1525 4437 y Fq(0)1633 4365 y Fs(e)1678 4324 y Fq(2)p Fn(t)1744 4365 y Fs(V)21 b Fu(\()p Fs(e)1905 4324 y Fn(t)1935 4365 y Fu(\))p Fr(j)p Fs(y)t Fr(j)2081 4324 y Fq(2)2119 4365 y Fs(dt;)114 b(y)31 b Fr(2)d Fs(F)41 b Fr(\032)28 b(F)10 b Fu(\()p Fs(")p Fu(\))p Fs(;)3320 4232 y Fu(\(6\))456 4570 y(where)35 b(\003)807 4585 y Fn(j)878 4570 y Fu(are)g(the)g(eigen) m(v)-5 b(alues)35 b(of)f(the)h(op)s(erator)f(\001)2479 4585 y Fn(\022)2553 4570 y Fu(on)g(the)h(sphere)h Fs(S)6 b Fu(.)50 b(W)-8 b(e)456 4686 y(denote)49 b(b)m(y)g(\000)55 b(=)g(\000\()p Fs(\013)q Fu(\))48 b(the)h(family)d(of)i(the)h(n)m(um)m (b)s(ers)h Fs(j)k Fu(for)49 b(whic)m(h)g(\(6\))f(is)456 4802 y(ful\014lled)c(for)i Fs(")52 b Fu(=)f Fr(N)61 b Fu(at)47 b(least)f(for)g(one)h Fs(y)55 b Fr(2)d(F)10 b Fu(\()p Fr(N)15 b Fu(\).)85 b(Ob)m(viously)46 b(the)i(set)456 4918 y Fr(f)p Fs(t)d Fr(2)i Fu([0)p Fs(;)17 b(T)d Fu(])45 b(:)89 b(\003)1146 4933 y Fn(j)1212 4918 y Fu(+)29 b Fs(b)p Fu(\()p Fr(N)15 b Fu(\))p Fs(e)1576 4882 y Fq(2)p Fn(t)1730 4918 y Fr(\024)89 b Fs(\013)q(e)2004 4882 y Fq(2)p Fn(t)2069 4918 y Fs(V)21 b Fu(\()p Fs(e)2230 4882 y Fn(t)2260 4918 y Fu(\))p Fr(g)43 b Fu(is)g(not)g(empt)m(y)h(for)e Fs(j)52 b Fr(2)46 b Fu(\000.)456 5035 y(Therefore)33 b(for)f(an)h(arbitrary)e Fs(j)j Fr(2)28 b Fu(\000)795 5209 y(\003)863 5224 y Fn(j)959 5209 y Fr(\024)61 b Fs(\013)33 b Fu(sup)1339 5232 y Fn(t)p Fo(2)p Fq([0)p Fn(;)p Fq(+)p Fo(1)p Fq(\))1643 5209 y Fu(\()p Fs(e)1726 5168 y Fq(2)p Fn(t)1791 5209 y Fs(V)21 b Fu(\()p Fs(e)1952 5168 y Fn(t)1982 5209 y Fu(\)\))h(+)g Fr(j)p Fs(b)p Fu(\()p Fr(N)15 b Fu(\))p Fr(j)p Fs(\021)2500 5168 y Fq(2)2538 5209 y Fs(\013)2601 5168 y Fq(2)p Fn(=s)2708 5209 y Fs(;)115 b(j)33 b Fr(2)28 b Fu(\000)p Fs(:)215 b Fu(\(7\))p eop %%Page: 5 5 5 4 bop 1513 236 a Fl(DISCRETE)33 b(SPECTR)n(UM)1019 b(5)456 425 y Fu(By)33 b(the)g(W)-8 b(eyl)32 b(la)m(w)g(for)g(\003)1416 440 y Fn(j)1485 425 y Fu(and)h Fs(k)1726 440 y Fn(j)1795 425 y Fu(w)m(e)g(ha)m(v)m(e)h(the)f(estimate)1182 552 y Fk(X)1180 736 y Fn(j)t Fo(2)p Fq(\000)1353 635 y Fs(k)1404 650 y Fn(j)1500 635 y Fr(\024)61 b Fs(C)7 b Fu(\()p Fs(\013)1826 565 y Fi(d)p Fg(\000)p Ff(1)p 1825 579 111 4 v 1865 620 a(2)1972 635 y Fu(+)22 b(1\))p Fs(;)114 b Fu(0)27 b Fs(<)h(\021)j(<)d Fu(1)p Fs(:)583 b Fu(\(8\))456 909 y(Under)36 b(the)h(substitution)e (of)h Fr(F)10 b Fu(\()p Fr(N)15 b Fu(\))35 b(b)m(y)i Fs(C)2093 873 y Fo(1)2086 934 y Fq(0)2167 909 y Fu(\(0)p Fs(;)17 b(T)d Fu(\))35 b(the)i(maximal)c(dimension)456 1026 y(of)28 b(subspaces)k Fs(F)14 b Fu(,)30 b(on)f(whic)m(h)h(\(6\))e (is)h(ful\014lled,)f(decreases)k(at)d(most)f(b)m(y)i(t)m(w)m(o.)43 b(Ac-)456 1142 y(cording)24 b(to)i(\(8\))f(under)h(this)g(pro)s(cedure) g(the)g(di\013erence)h(\(4\))e(w)m(ould)g(b)s(e)h(c)m(hanged)456 1275 y(b)m(y)36 b(a)g(quan)m(tit)m(y)g(of)f(order)h Fs(O)s Fu(\()p Fs(\013)1633 1210 y Fi(d)p Fg(\000)p Ff(1)p 1632 1224 V 1672 1264 a(2)1756 1275 y Fu(\))g(at)f Fs(\013)f Fr(!)e(1)p Fu(.)53 b(F)-8 b(or)35 b Fr(F)10 b Fu(\()p Fs(")p Fu(\))32 b(=)h Fs(C)2960 1239 y Fo(1)2953 1300 y Fq(0)3034 1275 y Fu(\(0)p Fs(;)17 b(T)d Fu(\))35 b(the)456 1391 y(inequalit)m(y)d(\(6\))h(turns)g(in)m(to:)1381 1513 y Fk(Z)1464 1539 y Fn(T)1428 1702 y Fq(0)1520 1630 y Fu(\()p Fr(j)p Fs(y)1638 1589 y Fo(0)1660 1630 y Fr(j)1688 1589 y Fq(2)1749 1630 y Fu(+)22 b(\()p Fs(\015)1936 1645 y Fn(j)1994 1630 y Fu(+)g Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(e)2300 1589 y Fq(2)p Fn(t)2365 1630 y Fu(\))p Fr(j)p Fs(y)t Fr(j)2511 1589 y Fq(2)2549 1630 y Fu(\))p Fs(dt)28 b(<)1955 1875 y(\013)2034 1758 y Fk(Z)2118 1784 y Fn(T)2081 1947 y Fq(0)2189 1875 y Fs(e)2234 1834 y Fq(2)p Fn(t)2299 1875 y Fs(V)22 b Fu(\()p Fs(e)2461 1834 y Fn(t)2491 1875 y Fu(\))p Fr(j)p Fs(y)t Fr(j)2637 1834 y Fq(2)2674 1875 y Fs(dt;)1128 2121 y(y)31 b Fr(2)d Fs(F)42 b Fr(\032)28 b Fs(C)1588 2080 y Fo(1)1581 2145 y Fq(0)1663 2121 y Fu(\(0)p Fs(;)17 b(T)d Fu(\))p Fs(;)48 b(\015)2029 2136 y Fn(j)2093 2121 y Fu(=)27 b(\003)2264 2136 y Fn(j)2323 2121 y Fu(+)22 b(\()2469 2053 y Fs(d)f Fr(\000)i Fu(2)p 2469 2097 222 4 v 2555 2189 a(2)2700 2121 y(\))2738 2080 y Fq(2)2777 2121 y Fs(:)3320 1866 y Fu(\(9\))555 2341 y Fw(2.)41 b Fu(W)-8 b(e)23 b(denote)h(b)m(y)g Fs(Z)7 b Fu(\()p Fs(";)17 b(\013)q Fu(\))22 b(the)h(set)h(of)f(n)m(um)m(b)s (ers)h Fs(j)6 b Fu(,)25 b(for)d(whic)m(h)i(\(9\))f(is)f(ful\014lled)456 2458 y(at)32 b(least)g(for)g(one)h Fs(y)e Fr(2)d Fs(C)1381 2422 y Fo(1)1374 2482 y Fq(0)1455 2458 y Fu(\(0)p Fs(;)17 b(T)d Fu(\).)43 b(F)-8 b(or)32 b Fs(j)h Fr(2)28 b Fs(Z)7 b Fu(\()p Fs(";)17 b(\013)q Fu(\))32 b(w)m(e)h(consider)g(the)g(set)752 2644 y(\001)28 b(=)g(\001\()p Fs(";)17 b(j;)g(\013)q Fu(\))27 b(:=)g Fr(f)p Fs(t)h Fr(2)g Fu([0)p Fs(;)17 b(T)d Fu(])27 b(:)61 b Fs(\015)2107 2659 y Fn(j)2165 2644 y Fu(+)22 b Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(e)2471 2603 y Fq(2)p Fn(t)2564 2644 y Fr(\024)28 b Fs(\013)q(e)2777 2603 y Fq(2)p Fn(t)2842 2644 y Fs(V)21 b Fu(\()p Fs(e)3003 2603 y Fn(t)3033 2644 y Fu(\))p Fr(g)p Fs(:)456 2830 y Fu(Ob)m(viously)34 b(\001)c Fr(6)p Fu(=)g Fr(;)k Fu(for)f(ev)m(ery)j Fs(j)g Fr(2)30 b Fs(Z)7 b Fu(\()p Fs(";)17 b(\013)q Fu(\).)47 b(Let)34 b(us)h(clarify)d(the)j(structure)g(of)456 2946 y(the)c(set)h(\001.)44 b(If)31 b(a)g(p)s(oin)m(t)f Fs(t)e Fr(2)g Fu(\(0)p Fs(;)17 b(T)d Fu(\))31 b(lies)f(on)h(the)h(b)s(oundary) f(of)g(the)h(set)g(\001,)g(then)1430 3133 y Fs(\015)1481 3148 y Fn(j)1539 3133 y Fu(+)22 b Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(e)1845 3092 y Fq(2)p Fn(t)1938 3133 y Fu(=)27 b Fs(\013)q(e)2149 3092 y Fq(2)p Fn(t)2214 3133 y Fs(V)22 b Fu(\()p Fs(e)2376 3092 y Fn(t)2405 3133 y Fu(\))p Fs(:)801 b Fu(\(10\))456 3327 y(F)-8 b(or)31 b Fs(e)675 3290 y Fn(t)733 3327 y Fu(=)c Fs(r)36 b Fu(w)m(e)d(get)1468 3512 y Fs(\015)1519 3527 y Fn(j)1577 3512 y Fu(+)22 b Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(r)1885 3471 y Fq(2)1952 3512 y Fu(=)27 b Fs(\013)q(r)2165 3471 y Fq(2)2204 3512 y Fs(V)22 b Fu(\()p Fs(r)s Fu(\))p Fs(:)838 b Fu(\(11\))456 3698 y(According)38 b(to)g(the)g(Roll)e(theorem)i(the)h(ro)s(ot)e(of)h (this)g(equation)g(lies)f(b)s(et)m(w)m(een)456 3814 y(t)m(w)m(o)c (solutions)e(of)i(the)g(equation)1470 4048 y(2)p Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(r)d Fu(=)e Fs(\013)1956 3981 y(d)p 1933 4025 98 4 v 1933 4116 a(dr)2040 4048 y Fu(\()p Fs(r)2125 4007 y Fq(2)2164 4048 y Fs(V)21 b Fu(\()p Fs(r)s Fu(\)\))p Fs(:)841 b Fu(\(12\))456 4269 y(According)22 b(to)h(the)g(condition)e(\(1)p Fs(:)p Fu(1\))i(the)g(n)m(um)m(b)s(er)g (of)f(the)i(ro)s(ots)e(of)g(the)i(equation)456 4385 y(\(12\))39 b(do)s(es)i(not)g(exceed)h(a)e(\014xed)i(n)m(um)m(b)s(er)f Fs(l)i Fr(2)f Fm(N)9 b Fu(.)73 b(Therefore)41 b(\001)g(is)f(a)g(union) 456 4501 y(of)e(at)h(most)f Fs(l)29 b Fu(+)d(2)39 b(disjoin)m(t)e (\(probably)i(degenerated)h(to)e(a)h(p)s(oin)m(t\))f(in)m(terv)-5 b(als.)456 4618 y(Consequen)m(tly)d(,)42 b(under)d(the)h(substitution)e (of)g Fr(F)10 b Fu(\()p Fs(")p Fu(\))37 b(=)h Fs(C)2642 4581 y Fo(1)2635 4642 y Fq(0)2716 4618 y Fu(\(0)p Fs(;)17 b(T)d Fu(\))38 b(b)m(y)i Fs(C)3213 4581 y Fo(1)3206 4642 y Fq(0)3287 4618 y Fu(\(\001\))456 4734 y(the)48 b(maximal)e(dimension) h Fs(N)10 b Fu(\()p Fs(";)17 b(j;)g(\013)q Fu(\))48 b(of)f(the)i (subspaces)i Fs(F)14 b Fu(,)52 b(where)e(\(9\))e(is)456 4850 y(ful\014lled,)37 b(will)f(b)s(e)j(decreased)h(at)e(most)g(b)m(y)h (2)p Fs(l)28 b Fu(+)e(4)38 b(\(the)h(doubled)f(n)m(um)m(b)s(er)h(of)456 4966 y(the)d(in)m(terv)-5 b(als)36 b(forming)e(\001\).)54 b(Since)37 b Fs(Z)7 b Fu(\()p Fs(";)17 b(\013)q Fu(\))33 b Fr(\032)h Fu(\000)i(and)h(according)e(to)h(\(8\))g(the)456 5099 y(v)-5 b(alue)23 b(\(4\))g(will)e(b)s(e)j(c)m(hanged)h(only)e(b)m (y)i Fs(O)s Fu(\()p Fs(\013)2035 5034 y Fi(d)p Fg(\000)p Ff(1)p 2034 5048 111 4 v 2074 5089 a(2)2158 5099 y Fu(\))f(as)g Fs(\013)k Fr(!)g(1)p Fu(.)40 b(Let)24 b(\001)k(=)f Fr([)3160 5058 y Fn(l)q Fq(+2)3160 5124 y Fn(k)r Fq(=1)3293 5099 y Fu(\001)3374 5114 y Fn(k)3417 5099 y Fu(,)456 5216 y(where)k(\001)816 5231 y Fn(k)887 5216 y Fu(=)c(\001)1071 5231 y Fn(k)1114 5216 y Fu(\()p Fs(";)17 b(j;)g(\013)q Fu(\))29 b(are)h(disjoin)m(t)f(\(probably)h(empt)m(y)g(or)g (degenerated)i(to)p eop %%Page: 6 6 6 5 bop 456 236 a Fl(6)1180 b(O.)34 b(SAFR)n(ONO)n(V)456 425 y Fu(a)i(p)s(oin)m(t\))f(in)m(terv)-5 b(als.)55 b(F)-8 b(or)35 b Fs(F)48 b Fr(\032)35 b Fs(C)1756 389 y Fo(1)1749 450 y Fq(0)1831 425 y Fu(\(\001\))f(=)g Fs(C)2209 389 y Fo(1)2202 450 y Fq(0)2284 425 y Fu(\(\001)2403 440 y Fq(1)2443 425 y Fu(\))24 b(+)h Fr(\001)17 b(\001)g(\001)23 b Fu(+)i Fs(C)2925 389 y Fo(1)2918 450 y Fq(0)2999 425 y Fu(\(\001)3118 440 y Fn(l)q Fq(+2)3235 425 y Fu(\))36 b(the)456 541 y(inequalit)m(y)28 b(\(9\))f(is)g(decomp)s(osed)i(in)m (to)e Fs(l)15 b Fu(+)d(2)27 b(inequalities)f(for)h(the)h(in)m(terv)-5 b(als)27 b(\001)3374 556 y Fn(k)3417 541 y Fu(:)805 627 y Fk(Z)851 816 y Fq(\001)910 828 y Fi(k)952 744 y Fu(\()p Fr(j)p Fs(y)1070 703 y Fo(0)1092 744 y Fr(j)1120 703 y Fq(2)1182 744 y Fu(+)22 b(\()p Fs(\015)1369 759 y Fn(j)1427 744 y Fu(+)g Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(e)1733 703 y Fq(2)p Fn(t)1798 744 y Fu(\))p Fr(j)p Fs(y)t Fr(j)1944 703 y Fq(2)1982 744 y Fu(\))p Fs(dt)27 b(<)h(\013)2316 627 y Fk(Z)2362 816 y Fq(\001)2421 828 y Fi(k)2480 744 y Fs(e)2525 703 y Fq(2)p Fn(t)2590 744 y Fs(V)22 b Fu(\()p Fs(e)2752 703 y Fn(t)2781 744 y Fu(\))p Fr(j)p Fs(y)t Fr(j)2927 703 y Fq(2)2965 744 y Fs(dt;)1774 951 y(y)31 b Fr(2)d Fs(F)42 b Fr(\032)28 b Fs(C)2234 910 y Fo(1)2227 975 y Fq(0)2308 951 y Fu(\(\001)2427 966 y Fn(k)2470 951 y Fu(\))p Fs(;)82 b(k)31 b Fu(=)p 2802 872 293 4 v 27 w(1)p Fs(;)17 b(l)24 b Fu(+)e(2)3271 833 y(\(13\))456 1125 y(The)30 b(maximal)d(dimension)h Fs(N)1585 1140 y Fn(k)1655 1125 y Fu(=)g Fs(N)1837 1140 y Fn(k)1879 1125 y Fu(\()p Fs(";)17 b(j;)g(\013)q Fu(\))29 b(of)g(the)h(subspaces)i Fs(F)44 b Fu(from)28 b(\(13\))456 1241 y(coincides)37 b(with)h(the)g(en)m(tire)g(m)m(ultiplicit)m(y)c(of)k(the)g(negativ)m(e) g(sp)s(ectrum)h(of)e(the)456 1358 y(op)s(erator)456 1526 y Fr(\000)p Fs(y)585 1485 y Fo(0)o(0)637 1526 y Fu(+)10 b(\()p Fs(\015)812 1541 y Fn(j)858 1526 y Fu(+)g Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(e)1152 1485 y Fq(2)p Fn(t)1217 1526 y Fu(\))p Fs(y)k Fr(\000)c Fs(\013)q(e)1512 1485 y Fq(2)p Fn(t)1577 1526 y Fs(V)21 b Fu(\()p Fs(e)1738 1485 y Fn(t)1768 1526 y Fu(\))p Fs(y)t(;)69 b(y)31 b Fr(2)d Fs(H)2216 1485 y Fq(2)2255 1526 y Fu(\(\001)2374 1541 y Fn(k)2417 1526 y Fu(\))10 b Fr(\\)g Fs(H)2630 1485 y Fq(1)2622 1551 y(0)2670 1526 y Fu(\(\001)2789 1541 y Fn(k)2832 1526 y Fu(\))p Fs(;)69 b(k)31 b Fu(=)p 3152 1447 V 28 w(1)p Fs(;)17 b(l)24 b Fu(+)e(2)555 1695 y Fw(3.)54 b Fu(In)37 b(the)f(in)m(v)m(estigation)f(of)h(the)g(sp)s (ectrum)h(of)e(this)h(op)s(erator)g(w)m(e)h(use)g(the)456 1811 y(tec)m(hnique)d(suggested)g(in)d([3].)44 b(Let)33 b(us)g(put)750 1988 y Fs(f)39 b Fu(=)27 b Fs(f)11 b Fu(\()p Fs(";)17 b(t;)g(\013)q Fu(\))27 b(=)g(\()p Fs(\013)q(e)1583 1947 y Fq(2)p Fn(t)1648 1988 y Fs(V)21 b Fu(\()p Fs(e)1809 1947 y Fn(t)1839 1988 y Fu(\))h Fr(\000)h Fs(\015)2050 2003 y Fn(j)2108 1988 y Fr(\000)g Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(e)2416 1947 y Fq(2)p Fn(t)2503 1988 y Fu(+)f(1\))2688 1947 y Fq(1)p Fn(=)p Fq(2)2798 1988 y Fs(;)17 b(t)27 b Fr(2)i Fu(\001)3080 2003 y Fn(k)3122 1988 y Fs(:)594 2169 y Fw(Prop)s(osition)35 b(2.2.)49 b Fh(F)-7 b(or)61 b(the)h(quantity)h Fs(N)2304 2184 y Fn(k)2409 2169 y Fh(the)f(fol)5 b(lowing)60 b(ine)-5 b(quality)456 2286 y(holds)823 2464 y Fr(j)p Fs(N)929 2479 y Fn(k)971 2464 y Fu(\()p Fs(";)17 b(j;)g(\013)q Fu(\))k Fr(\000)1405 2346 y Fk(Z)1451 2535 y Fq(\001)1510 2547 y Fi(k)1548 2535 y Fq(\()p Fn(";j;\013)p Fq(\))1769 2464 y Fs(f)11 b Fu(\()p Fs(";)17 b(t;)g(\013)q Fu(\))p Fr(j)26 b(\024)i Fs(C)7 b Fu(\(1)22 b(+)g(log\(1)g(+)g Fs(\013)q Fu(\)\))p Fs(;)193 b Fu(\(14\))456 2687 y Fh(wher)-5 b(e)34 b Fs(C)g Fu(=)28 b Fs(C)7 b Fu(\()p Fs(\021)t Fu(\))34 b Fh(do)-5 b(es)35 b(not)f(dep)-5 b(end)34 b(on)h Fs(j)41 b Fh(and)34 b Fs(\013)q Fh(.)555 2886 y(Pr)-5 b(o)g(of.)59 b Fu(Let)38 b(\001)1132 2901 y Fn(k)1211 2886 y Fu(=)f([)p Fs(R)1426 2835 y Fq(\()p Fn(k)r Fq(\))1425 2908 y(1)1523 2886 y Fs(;)17 b(R)1642 2835 y Fq(\()p Fn(k)r Fq(\))1641 2908 y(2)1740 2886 y Fu(])p Fs(;)54 b(k)40 b Fu(=)p 2051 2807 V 36 w(1)p Fs(;)17 b(l)24 b Fu(+)e(2.)59 b(Let)38 b(us)g(emplo)m(y)g (the)g(rela-)456 3002 y(tion)30 b(b)s(et)m(w)m(een)k(the)e(quan)m(tit)m (y)g Fs(N)1665 3017 y Fn(k)1740 3002 y Fu(and)g(the)g(prop)s(erties)f (of)g(the)i(solution)d(of)h(the)456 3118 y(problem)1021 3199 y Fk(8)1021 3273 y(<)1021 3423 y(:)1137 3304 y Fr(\000)p Fs(\036)1272 3267 y Fo(00)1336 3304 y Fu(+)22 b(\()p Fs(\015)1523 3319 y Fn(j)1582 3304 y Fu(+)g Fs(b)p Fu(\()p Fs(")p Fu(\))p Fs(e)1888 3267 y Fq(2)p Fn(t)1953 3304 y Fu(\))p Fs(\036)g Fr(\000)g Fs(\013)q(e)2278 3267 y Fq(2)p Fn(t)2343 3304 y Fs(V)g Fu(\()p Fs(e)2505 3267 y Fn(t)2534 3304 y Fu(\))p Fs(\036)28 b Fu(=)f(0)p Fs(;)1437 3439 y(\036)p Fu(\()p Fs(R)1608 3388 y Fq(\()p Fn(k)r Fq(\))1607 3461 y(1)1706 3439 y Fu(\))g(=)h(0)p Fs(;)49 b(\036)2058 3403 y Fo(0)2081 3439 y Fu(\()p Fs(R)2194 3388 y Fq(\()p Fn(k)r Fq(\))2193 3461 y(1)2291 3439 y Fu(\))28 b(=)g(1)p Fs(:)3271 3373 y Fu(\(15\))456 3633 y(Namely)38 b(the)h(quan)m(tit)m(y)h Fs(N)1469 3648 y Fn(k)1550 3633 y Fu(coincides)f(\(see)h([3]\))f(with)f(the)i(n)m(um)m (b)s(er)f(of)f(zeros)456 3749 y Fs(\026)515 3764 y Fn(k)590 3749 y Fu(=)32 b Fs(\026)757 3764 y Fn(k)800 3749 y Fu(\()p Fs(";)17 b(j;)g(\013)q Fu(\))34 b(of)h(the)h(function)g Fs(\036)f Fu(lying)f(strictly)h(inside)g(\001)2853 3764 y Fn(k)2896 3749 y Fu(.)52 b(In)36 b(order)g(to)456 3866 y(estimate)31 b Fs(\026)905 3881 y Fn(k)980 3866 y Fu(w)m(e)j(represen) m(t)g Fs(\036)p Fu(\()p Fs(t)p Fu(\))f(in)f(the)h(p)s(olar)e(form.)42 b(W)-8 b(e)33 b(put)997 4034 y Fs(\036)p Fu(\()p Fs(t)p Fu(\))27 b(=)h Fs(\014)6 b Fu(\()p Fs(t)p Fu(\))17 b(sin)f Fs(\030)5 b Fu(\()p Fs(t)p Fu(\))p Fs(;)81 b(\036)1947 3993 y Fo(0)1970 4034 y Fu(\()p Fs(t)p Fu(\))28 b(=)f Fs(f)11 b Fu(\()p Fs(t)p Fu(\))p Fs(\014)6 b Fu(\()p Fs(t)p Fu(\))17 b(cos)g Fs(\030)5 b Fu(\()p Fs(t)p Fu(\))p Fs(:)456 4203 y Fu(Then)33 b(it)f(follo)m(ws)f(from)g(\(15\))h(\(see)i ([3]\))e(that)862 4390 y Fs(\030)910 4348 y Fo(0)961 4390 y Fu(=)27 b Fs(f)33 b Fu(+)22 b Fs(f)1302 4348 y Fo(0)1325 4390 y Fu(\(2)p Fs(f)11 b Fu(\))1509 4348 y Fo(\000)p Fq(1)1619 4390 y Fu(sin\(2)p Fs(\030)5 b Fu(\))21 b Fr(\000)i Fs(f)2092 4348 y Fo(\000)p Fq(1)2203 4390 y Fu(sin)2322 4348 y Fq(2)2379 4390 y Fs(\030)5 b(;)81 b(\030)5 b Fu(\()p Fs(R)2696 4339 y Fq(\()p Fn(k)r Fq(\))2695 4411 y(1)2793 4390 y Fu(\))27 b(=)h(0)p Fs(:)233 b Fu(\(16\))456 4564 y(W)-8 b(e)35 b(note)g(also)g(that)g Fs(\014)6 b Fu(\()p Fs(t)p Fu(\))31 b Fs(>)h Fu(0)j(for)g Fs(t)d Fr(2)g Fu(\001)2051 4579 y Fn(k)2094 4564 y Fu(.)51 b(Consequen)m(tly) 38 b(the)d(quan)m(tit)m(y)h Fs(\026)3402 4579 y Fn(k)456 4680 y Fu(coincides)k(with)g(the)h(n)m(um)m(b)s(er)h(of)e(zeros)h(of)f (the)h(function)g(sin)16 b Fs(\030)5 b Fu(\()p Fs(t)p Fu(\).)67 b(In)41 b(other)456 4797 y(w)m(ords)i Fs(\026)804 4812 y Fn(k)888 4797 y Fu(coincides)f(with)g(the)g(n)m(um)m(b)s(er)g (of)g(ro)s(ots)f(of)h(the)g(equation)g Fs(\030)5 b Fu(\()p Fs(t)p Fu(\))44 b(=)456 4913 y(0)60 b(\(mo)s(d)o Fs(\031)t Fu(\).)42 b(In)31 b(view)f(of)37 b(\(16\))29 b Fs(\030)1665 4877 y Fo(0)1716 4913 y Fu(=)e Fs(f)39 b(>)27 b Fu(0,)k(if)e Fs(\030)j Fu(=)27 b Fs(\031)t(n;)47 b(n)28 b Fr(2)g Fm(Z)p Fu(.)41 b(Therefore)31 b(for)456 5029 y(a)e(\014xed)h Fs(n)e Fr(2)g Fm(N)44 b Fu(the)30 b(function)f Fs(\030)5 b Fu(\()p Fs(t)p Fu(\))28 b(can)i(tak)m(e)g(the)g(v)-5 b(alues)29 b Fs(\031)t(n)g Fu(only)g(once.)43 b(Th)m(us)1219 5216 y Fs(\031)1278 5174 y Fo(\000)p Fq(1)1372 5216 y Fs(\030)5 b Fu(\()p Fs(R)1533 5165 y Fq(\()p Fn(k)r Fq(\))1532 5237 y(2)1630 5216 y Fu(\))22 b Fr(\000)g Fu(1)28 b Fr(\024)g Fs(\026)2030 5231 y Fn(k)2100 5216 y Fr(\024)g Fs(\031)2264 5174 y Fo(\000)p Fq(1)2358 5216 y Fs(\030)5 b Fu(\()p Fs(R)2519 5165 y Fq(\()p Fn(k)r Fq(\))2518 5237 y(2)2616 5216 y Fu(\))p Fs(:)p eop %%Page: 7 7 7 6 bop 1513 236 a Fl(DISCRETE)33 b(SPECTR)n(UM)1019 b(7)456 425 y Fu(Since)32 b Fs(N)788 440 y Fn(k)859 425 y Fu(=)27 b Fs(\026)1021 440 y Fn(k)1096 425 y Fu(according)32 b(to)g(\(16\))g(w)m(e)i(obtain)483 647 y Fr(j)p Fs(N)589 662 y Fn(k)654 647 y Fr(\000)22 b Fs(\031)812 605 y Fo(\000)p Fq(1)923 529 y Fk(Z)969 718 y Fq(\001)1028 730 y Fi(k)1087 647 y Fs(f)11 b(dt)p Fr(j)27 b(\024)h Fu(\(2)p Fs(\031)t Fu(\))1576 605 y Fo(\000)p Fq(1)1687 529 y Fk(Z)1733 718 y Fq(\001)1792 730 y Fi(k)1851 647 y Fr(j)p Fs(f)1938 605 y Fo(0)1960 647 y Fs(=f)11 b Fr(j)p Fs(dt)21 b Fu(+)h Fs(\031)2360 605 y Fo(\000)p Fq(1)2455 647 y Fr(j)p Fu(\001)2564 662 y Fn(k)2606 647 y Fr(j)g Fu(+)g(1)p Fs(;)82 b(k)30 b Fu(=)p 3097 567 293 4 v 28 w(1)p Fs(;)17 b(l)24 b Fu(+)e(2)p Fs(:)456 882 y Fu(It)54 b(is)g(easy)h(to)f(see)h(that)f(the)g(function) g Fs(f)2144 846 y Fo(0)2221 882 y Fu(c)m(hanges)i(its)d(sign)h(on)g (the)h(in-)456 998 y(terv)-5 b(al)51 b(\001)830 1013 y Fn(k)925 998 y Fu(not)g(more)h(than)g Fs(l)i Fu(times.)101 b(Therefore)2529 927 y Fk(R)2568 1024 y Fq(\001)2627 1036 y Fi(k)2686 998 y Fr(j)p Fs(f)2773 962 y Fo(0)2795 998 y Fs(=f)11 b Fr(j)p Fs(dt)60 b Fr(\024)h Fu(2\()p Fs(l)38 b Fu(+)456 1119 y(1\))30 b Fr(\002)h Fu(max)862 1134 y Fn(t)p Fo(2)p Fq(\001)993 1146 y Fi(k)1036 1119 y Fr(j)17 b Fu(log)f Fs(f)11 b Fr(j)p Fs(:)44 b Fu(It)h(remains)f(to)g (tak)m(e)i(in)m(to)e(accoun)m(t)i(that)e(1)49 b Fr(\024)g Fs(f)59 b Fr(\024)456 1239 y Fu(\()p Fs(\013)17 b Fu(sup)720 1263 y Fn(r)r(>)p Fq(1)848 1239 y Fu(\()p Fs(r)933 1203 y Fq(2)972 1239 y Fs(V)k Fu(\()p Fs(r)s Fu(\)\))h(+)g Fr(j)p Fs(b)p Fu(\()p Fr(N)15 b Fu(\))p Fr(j)p Fs(\021)1653 1203 y Fq(2)1691 1239 y Fs(\013)1754 1203 y Fq(2)p Fn(=s)1883 1239 y Fu(+)22 b(1\))2068 1203 y Fq(1)p Fn(=)p Fq(2)2211 1239 y Fu(for)32 b Fs(t)c Fr(2)g Fu(\001)2598 1254 y Fn(k)2641 1239 y Fs(:)65 b Fe(2)555 1356 y Fw(4.)44 b Fu(Let)33 b(us)g(set)931 1552 y(\010)1001 1567 y Fn(j)1038 1552 y Fu(\()p Fs(\013)q Fu(\))27 b(:=)1334 1435 y Fk(Z)1381 1624 y Fq(\001\()p Fo(N)10 b Fn(;j;\013)p Fq(\))1696 1552 y Fs(f)h Fu(\()p Fr(N)k Fs(;)i(t)p Fu(\))p Fs(dt)k Fr(\000)2213 1435 y Fk(Z)2259 1624 y Fq(\001\()p Fo(D)r Fn(;j;\013)p Fq(\))2563 1552 y Fs(f)11 b Fu(\()p Fr(D)s Fs(;)17 b(t)p Fu(\))p Fs(dt;)456 1806 y Fu(and)25 b Fs(C)708 1821 y Fq(1)775 1806 y Fu(:=)j(\()p Fs(b)p Fu(\()p Fr(D)s Fu(\))8 b Fr(\000)g Fs(b)p Fu(\()p Fr(N)15 b Fu(\)\))1486 1769 y Fq(1)p Fn(=)p Fq(2)1596 1806 y Fs(:)25 b Fu(The)i(follo)m(wing)c (statemen)m(t)j(giv)m(es)g(the)g(estimate)456 1922 y(of)32 b(the)h(quan)m(tit)m(y)g(\010)1195 1937 y Fn(j)1232 1922 y Fu(\()p Fs(\013)q Fu(\))p Fs(:)594 2113 y Fw(Prop)s(osition)i(2.3.)49 b Fh(F)-7 b(or)34 b Fu(\010)1674 2128 y Fn(j)1711 2113 y Fu(\()p Fs(\013)q Fu(\))h Fh(the)f(fol)5 b(lowing)34 b(ine)-5 b(quality)35 b(holds)1423 2338 y Fu(\010)1493 2353 y Fn(j)1530 2338 y Fu(\()p Fs(\013)q Fu(\))28 b Fr(\024)g Fs(C)1872 2353 y Fq(1)1928 2221 y Fk(Z)1974 2409 y Fq(\001\()p Fo(N)10 b Fn(;j;\013)p Fq(\))2289 2338 y Fs(e)2334 2297 y Fn(t)2364 2338 y Fs(dt:)794 b Fu(\(17\))555 2584 y Fh(Pr)-5 b(o)g(of.)126 b Fu(It)60 b(follo)m(ws)f(from)f(the)j(de\014nition)e(of)g(the)i(v)-5 b(alue)59 b Fs(f)11 b Fu(\()p Fs(";)17 b(t)p Fu(\))60 b(that)456 2701 y Fs(f)11 b Fu(\()p Fr(N)k Fs(;)i(t)p Fu(\))767 2664 y Fq(2)833 2701 y Fu(=)27 b Fs(C)1013 2664 y Fq(2)1006 2725 y(1)1053 2701 y Fs(e)1098 2664 y Fq(2)p Fn(t)1185 2701 y Fu(+)22 b Fs(f)11 b Fu(\()p Fr(D)s Fs(;)17 b(t)p Fu(\))1577 2664 y Fq(2)1648 2701 y Fu(for)32 b Fs(t)c Fr(2)g Fu(\001\()p Fr(D)s Fs(;)17 b(j;)g(\013)q Fu(\))p Fs(:)32 b Fu(Consequen)m(tly)1536 2888 y Fs(f)11 b Fu(\()p Fr(N)k Fs(;)i(t)p Fu(\))k Fr(\000)i Fs(f)11 b Fu(\()p Fr(D)s Fs(;)17 b(t)p Fu(\))26 b(=)1360 3106 y Fs(C)1437 3065 y Fq(2)1430 3131 y(1)1477 3106 y Fs(e)1522 3065 y Fq(2)p Fn(t)1587 3106 y Fs(=)p Fu(\()p Fs(f)11 b Fu(\()p Fr(N)k Fs(;)i(t)p Fu(\))k(+)h Fs(f)11 b Fu(\()p Fr(D)s Fs(;)17 b(t)p Fu(\)\))26 b Fr(\024)1139 3305 y Fs(C)1216 3263 y Fq(2)1209 3329 y(1)1255 3305 y Fs(e)1300 3263 y Fq(2)p Fn(t)1365 3305 y Fs(=f)11 b Fu(\()p Fr(N)k Fs(;)i(t)p Fu(\))27 b Fr(\024)h Fs(C)1927 3320 y Fq(1)1966 3305 y Fs(e)2011 3263 y Fn(t)2041 3305 y Fs(;)82 b(t)28 b Fr(2)g Fu(\001\()p Fr(D)s Fs(;)17 b(j;)g(\013)q Fu(\))p Fs(:)456 3471 y Fu(F)-8 b(urthermore,)71 b(the)65 b(inequalit)m(y)e Fs(f)11 b Fu(\()p Fr(N)k Fs(;)i(t)p Fu(\))80 b Fr(\024)i Fs(C)2391 3486 y Fq(1)2431 3471 y Fs(e)2476 3435 y Fn(t)2570 3471 y Fu(is)63 b(ful\014lled)g(for)g Fs(t)82 b Fr(2)456 3588 y Fu(\001\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))k Fr(n)h Fu(\001\()p Fr(D)s Fs(;)17 b(j;)g(\013)q Fu(\))p Fs(:)32 b Fu(And)h(\001\()p Fr(D)s Fs(;)17 b(j;)g(\013)q Fu(\))27 b Fr(\032)h Fu(\001\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))p Fs(:)32 b Fu(Th)m(us,)1091 3809 y(\010)1161 3824 y Fn(j)1198 3809 y Fu(\()p Fs(\013)q Fu(\))c(=)1468 3692 y Fk(Z)1514 3881 y Fq(\001\()p Fo(D)r Fn(;j;\013)p Fq(\))1802 3809 y Fu(\()p Fs(f)11 b Fu(\()p Fr(N)k Fs(;)i(t)p Fu(\))k Fr(\000)h Fs(f)11 b Fu(\()p Fr(D)s Fs(;)17 b(t)p Fu(\)\))p Fs(dt)k Fu(+)1600 3935 y Fk(Z)1646 4123 y Fq(\001\()p Fo(N)10 b Fn(;j;\013)p Fq(\))p Fo(n)p Fq(\001\()p Fo(D)r Fn(;j;\013)p Fq(\))2280 4052 y Fs(f)h Fu(\()p Fr(N)k Fs(;)i(t)p Fu(\))p Fs(dt)26 b Fr(\024)2134 4294 y Fs(C)2204 4309 y Fq(1)2259 4177 y Fk(Z)2306 4366 y Fq(\001\()p Fo(N)10 b Fn(;j;\013)p Fq(\))2621 4294 y Fs(e)2666 4253 y Fn(t)2695 4294 y Fs(dt:)3271 4062 y Fu(\(18\))456 4506 y Fe(2)555 4622 y Fu(Ab)s(o)m(v)m(e)42 b(\(see)g(Subsections)g(1,)h (2\))d(w)m(e)i(made)f(transformations)e(that)i(either)456 4741 y(increase)22 b(di\013erence)h(\(4\))e(or)h(c)m(hange)h(it)e(b)m (y)h(a)g(v)-5 b(alue)22 b(of)f(order)h Fs(o)p Fu(\()p Fs(\013)2823 4705 y Fn(d=s)2931 4741 y Fu(\))g(as)g Fs(\013)28 b Fr(!)g(1)p Fs(:)456 4858 y Fu(Finally)i(this)i(di\013erence)h(equals) 1186 5036 y Fk(X)1086 5224 y Fn(j)t Fo(2)p Fn(Z)5 b Fq(\()p Fo(N)10 b Fn(;\013)p Fq(\))1422 5119 y Fs(k)1473 5134 y Fn(j)1535 5011 y(l)q Fq(+2)1531 5036 y Fk(X)1526 5220 y Fn(k)r Fq(=1)1655 5119 y Fu(\()p Fs(N)1771 5134 y Fn(k)1814 5119 y Fu(\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))k Fr(\000)h Fs(N)2376 5134 y Fn(k)2419 5119 y Fu(\()p Fr(D)s Fs(;)17 b(j;)g(\013)q Fu(\)\))p Fs(:)440 b Fu(\(19\))p eop %%Page: 8 8 8 7 bop 456 236 a Fl(8)1180 b(O.)34 b(SAFR)n(ONO)n(V)456 425 y Fu(According)55 b(to)g(\(8\))g(and)g(\(14\))g(the)h(v)-5 b(alue)55 b(\(19\))f(coincides)i(with)f(the)g(sum)456 478 y Fk(P)543 565 y Fn(j)t Fo(2)p Fn(Z)5 b Fq(\()p Fo(N)10 b Fn(;\013)p Fq(\))883 544 y Fs(k)934 559 y Fn(j)971 544 y Fu(\010)1041 559 y Fn(j)1078 544 y Fu(\()p Fs(\013)q Fu(\))27 b(with)h(accuracy)h(of)f(order)g Fs(o)p Fu(\()p Fs(\013)2366 508 y Fn(d=s)2474 544 y Fu(\))g(when)h Fs(\013)g Fr(!)e(1)p Fs(:)h Fu(There-)456 661 y(fore)1159 808 y Fs(N)10 b Fu(\()p Fs(\025)1342 823 y Fo(N)1414 808 y Fs(;)17 b(A)1531 767 y Fq(\()p Fo(N)10 b Fq(\))1657 808 y Fu(\()p Fs(\013)q Fu(\)\))22 b Fr(\000)h Fs(N)10 b Fu(\()p Fs(\025)2139 823 y Fo(D)2200 808 y Fs(;)17 b(A)2317 767 y Fq(\()p Fo(D)r Fq(\))2432 808 y Fu(\()p Fs(\013)q Fu(\)\))27 b Fr(\024)1715 893 y Fk(X)1615 1081 y Fn(j)t Fo(2)p Fn(Z)5 b Fq(\()p Fo(N)10 b Fn(;\013)p Fq(\))1951 976 y Fs(k)2002 991 y Fn(j)2038 976 y Fu(\010)2108 991 y Fn(j)2145 976 y Fu(\()p Fs(\013)q Fu(\))22 b(+)g Fs(o)p Fu(\()p Fs(\013)2552 935 y Fn(d=s)2660 976 y Fu(\))p Fs(;)2397 1212 y(\013)28 b Fr(!)g(1)p Fs(:)3271 1009 y Fu(\(20\))456 1383 y(A)m(t)50 b(the)h(next)h(step)f(of)f(the)h(pro)s (of)f(w)m(e)h(decomp)s(ose)g Fs(Z)7 b Fu(\()p Fr(N)15 b Fs(;)i(\013)q Fu(\))50 b(in)m(to)f(subsets)456 1499 y Fs(X)537 1514 y Fn(n)639 1499 y Fu(=)56 b Fs(X)852 1514 y Fn(n)899 1499 y Fu(\()p Fs(\013)q Fu(\))p Fs(:)48 b Fu(F)-8 b(or)49 b(the)g(eac)m(h)h(of)f(these)h(subsets)h(w)m(e)f (estimate)e(the)i(v)-5 b(alue)456 1549 y Fk(P)543 1636 y Fn(j)t Fo(2)p Fn(X)681 1644 y Fi(n)744 1615 y Fs(k)795 1630 y Fn(j)831 1615 y Fu(\010)901 1630 y Fn(j)938 1615 y Fu(\()p Fs(\013)q Fu(\))32 b(separately)-8 b(.)44 b(Then)34 b(w)m(e)f(sum)g(up)g(these)g(estimates)g(o)m(v)m(er)g Fs(n)p Fu(.)555 1732 y(F)-8 b(or)33 b Fs(n)c Fr(2)g Fm(N)38 b Fr([)23 b(f)p Fu(0)p Fr(g)33 b Fu(and)g Fs(\013)d(>)f Fu(0)k(w)m(e)h(de\014ne)h(a)e(set)h Fs(X)2491 1747 y Fn(n)2567 1732 y Fu(=)29 b Fs(X)2753 1747 y Fn(n)2800 1732 y Fu(\()p Fs(\013)q Fu(\))k(of)f(n)m(um)m(b)s(ers)456 1848 y Fs(j)38 b Fu(for)32 b(whic)m(h)1402 2015 y Fs(n\013)1523 1974 y Fq(2)p Fn(=s)1658 2015 y Fr(\024)c Fu(\003)1831 2030 y Fn(j)1895 2015 y Fs(<)f Fu(\()p Fs(n)22 b Fu(+)h(1\))p Fs(\013)2365 1974 y Fq(2)p Fn(=s)2471 2015 y Fs(:)773 b Fu(\(21\))456 2180 y(Our)28 b(nearest)i(goal)d(is)i(to)f(estimate)g (the)h(sums)g(of)g(the)g(quan)m(tities)g Fs(k)2959 2195 y Fn(j)3023 2180 y Fu(=)e(dim)o Fs(W)3381 2195 y Fn(j)3417 2180 y Fu(,)456 2296 y(the)e(n)m(um)m(b)s(ers)h Fs(j)31 b Fu(of)25 b(whic)m(h)h(are)f(in)f Fs(X)1791 2311 y Fn(n)1838 2296 y Fu(\()p Fs(\013)q Fu(\).)41 b(F)-8 b(or)24 b(that)h(w)m(e)h(use) g(the)g(precise)f(v)-5 b(alues)1564 2461 y(\003)1632 2476 y Fn(j)1696 2461 y Fu(=)28 b Fs(j)6 b Fu(\()p Fs(j)28 b Fu(+)22 b Fs(d)f Fr(\000)i Fu(2\))p Fs(;)1268 2641 y(k)1319 2656 y Fn(j)1383 2641 y Fu(=)28 b(dim)n Fs(W)1741 2656 y Fn(j)1806 2641 y Fu(=)f Fs(C)1986 2600 y Fn(d)p Fo(\000)p Fq(2)1979 2666 y Fn(d)p Fq(+)p Fn(j)t Fo(\000)p Fq(2)2219 2641 y Fu(+)22 b Fs(C)2394 2600 y Fn(d)p Fo(\000)p Fq(2)2387 2666 y Fn(d)p Fq(+)p Fn(j)t Fo(\000)p Fq(3)2605 2641 y Fs(:)456 2785 y Fu(F)-8 b(rom)31 b(the)i(last)e(equation)i(it)e (follo)m(ws)h(that)1531 2950 y Fs(k)1582 2965 y Fn(j)1646 2950 y Fr(\024)d Fs(C)7 b(j)1875 2909 y Fn(d)p Fo(\000)p Fq(2)2005 2950 y Fs(;)82 b(j)33 b Fr(\025)c Fu(1)p Fs(:)902 b Fu(\(22\))594 3129 y Fw(Prop)s(osition)35 b(2.4.)49 b Fh(Ther)-5 b(e)33 b(exists)g(a)g(numb)-5 b(er)33 b Fs(n)2453 3144 y Fq(0)2521 3129 y Fr(2)28 b Fm(N)48 b Fh(such)33 b(that)h(for)f(any)456 3245 y Fs(n)28 b(>)f(n)703 3260 y Fq(0)778 3245 y Fh(and)34 b Fs(\013)28 b(>)g Fu(1)1000 3327 y Fk(X)970 3510 y Fn(j)t Fo(2)p Fn(X)1108 3518 y Fi(n)1166 3410 y Fs(k)1217 3425 y Fn(j)1281 3410 y Fr(\024)g Fs(C)7 b Fu(\()p Fs(n)1559 3369 y Fq(\()p Fn(d)p Fo(\000)p Fq(2\))p Fn(=)p Fq(2)1815 3410 y Fs(\013)1878 3369 y Fq(\()p Fn(d)p Fo(\000)p Fq(2\))p Fn(=s)2154 3410 y Fu(+)22 b Fs(n)2310 3369 y Fq(\()p Fn(d)p Fo(\000)p Fq(3\))p Fn(=)p Fq(2)2566 3410 y Fs(\013)2629 3369 y Fq(\()p Fn(d)p Fo(\000)p Fq(1\))p Fn(=s)2882 3410 y Fu(\))p Fs(;)324 b Fu(\(23\))456 3658 y Fh(wher)-5 b(e)34 b Fs(C)g(>)28 b Fu(0)35 b Fh(do)-5 b(es)34 b(not)h(dep)-5 b(end)33 b(on)i Fs(n)g Fh(and)f Fs(\013)q(:)555 3836 y Fh(Pr)-5 b(o)g(of.)43 b Fu(Let)33 b Fs(n)28 b Fr(2)g Fm(N)47 b Fu(and)33 b Fs(\013)28 b(>)g Fu(1)p Fs(:)k Fu(Than)i(for)e Fs(j)h Fr(2)28 b Fs(X)2464 3851 y Fn(n)2511 3836 y Fu(\()p Fs(\013)q Fu(\))737 4009 y Fs(n\013)858 3968 y Fq(2)p Fn(=s)993 4009 y Fr(\024)g Fu(\003)1166 4024 y Fn(j)1230 4009 y Fu(=)g Fs(j)6 b Fu(\()p Fs(j)28 b Fu(+)22 b Fs(d)g Fr(\000)g Fu(2\))28 b(=)f Fs(j)2020 3968 y Fq(2)2060 4009 y Fu(\(1)21 b(+)h Fs(d)g Fr(\000)h Fu(2)p Fs(=j)6 b Fu(\))27 b Fr(\024)h Fs(j)2799 3968 y Fq(2)2839 4009 y Fu(\()p Fs(d)21 b Fr(\000)i Fu(1\))p Fs(:)456 4174 y Fu(Comparing)31 b(the)i(utmost)f(parts)h(w)m(e)g(get)1202 4341 y Fs(j)1248 4300 y Fq(2)1315 4341 y Fr(\025)28 b Fu(\()p Fs(d)22 b Fr(\000)h Fu(1\))1718 4300 y Fo(\000)p Fq(1)1812 4341 y Fs(n\013)1933 4300 y Fq(2)p Fn(=s)2040 4341 y Fs(;)114 b Fr(8)p Fs(j)34 b Fr(2)28 b Fs(X)2485 4356 y Fn(n)2532 4341 y Fu(\()p Fs(\013)q Fu(\))p Fs(:)456 4505 y Fu(Therefore)33 b(for)f Fs(j)i Fr(2)28 b Fs(X)1295 4520 y Fn(n)1342 4505 y Fu(\()p Fs(\013)q Fu(\))722 4740 y Fs(n\013)843 4699 y Fq(2)p Fn(=s)977 4740 y Fr(\024)h Fu(\003)1151 4755 y Fn(j)1215 4740 y Fu(=)e Fs(j)1364 4699 y Fq(2)1403 4740 y Fu(\(1)22 b(+)g Fs(d)g Fr(\000)h Fu(2)p Fs(=j)6 b Fu(\))27 b Fr(\024)h Fs(j)2143 4699 y Fq(2)2182 4740 y Fu(\(1)22 b(+)2399 4673 y(\()p Fs(d)g Fr(\000)h Fu(2\)\()p Fs(d)e Fr(\000)i Fu(1\))2994 4637 y Fq(1)p Fn(=)p Fq(2)p 2399 4717 705 4 v 2582 4809 a Fs(n)2640 4780 y Fq(1)p Fn(=)p Fq(2)2750 4809 y Fs(\013)2813 4780 y Fq(1)p Fn(=s)3113 4740 y Fu(\))p Fs(:)456 4935 y Fu(T)-8 b(aking)36 b(square)j(ro)s(ot)d(of)g(the)i(utmost)f(parts)g (of)g(this)g(inequalit)m(y)f(w)m(e)i(come)f(to)456 5051 y(the)c(estimate)525 5216 y Fs(j)h Fr(\025)28 b Fu(\(1)22 b(+)g(\()p Fs(d)g Fr(\000)g Fu(2\)\()p Fs(d)g Fr(\000)h Fu(1\))1506 5174 y Fq(1)p Fn(=)p Fq(2)1615 5216 y Fs(=n)1722 5174 y Fq(1)p Fn(=)p Fq(2)1832 5216 y Fs(\013)1895 5174 y Fq(1)p Fn(=s)2002 5216 y Fu(\))2040 5174 y Fo(\000)p Fq(1)p Fn(=)p Fq(2)2205 5216 y Fs(n)2263 5174 y Fq(1)p Fn(=)p Fq(2)2373 5216 y Fs(\013)2436 5174 y Fq(1)p Fn(=s)2571 5216 y Fu(=:)28 b Fs(\030)5 b(;)81 b Fr(8)p Fs(j)34 b Fr(2)28 b Fs(X)3162 5231 y Fn(n)3209 5216 y Fu(\()p Fs(\013)q Fu(\))p Fs(:)p eop %%Page: 9 9 9 8 bop 1513 236 a Fl(DISCRETE)33 b(SPECTR)n(UM)1019 b(9)456 425 y Fu(No)m(w)26 b(let)f(us)i(deriv)m(e)g(an)e(upp)s(er)i (estimate)e(for)g Fs(j)34 b Fr(2)28 b Fs(X)2381 440 y Fn(n)2428 425 y Fu(\()p Fs(\013)q Fu(\))p Fs(:)e Fu(Since)g(for)f Fs(j)34 b Fr(2)28 b Fs(X)3259 440 y Fn(n)3306 425 y Fu(\()p Fs(\013)q Fu(\))1152 601 y Fs(j)1198 560 y Fq(2)1265 601 y Fr(\024)g Fs(j)6 b Fu(\()p Fs(j)28 b Fu(+)22 b Fs(d)g Fr(\000)h Fu(2\))k(=)h(\003)2079 616 y Fn(j)2143 601 y Fr(\024)g Fu(\()p Fs(n)22 b Fu(+)g(1\))p Fs(\013)2614 560 y Fq(2)p Fn(=s)2721 601 y Fs(;)456 769 y Fu(w)m(e)33 b(ha)m(v)m(e)1152 897 y Fs(j)h Fr(\024)28 b Fu(\()p Fs(n)22 b Fu(+)g(1\))1634 855 y Fq(1)p Fn(=)p Fq(2)1744 897 y Fs(\013)1807 855 y Fq(1)p Fn(=s)1942 897 y Fu(=:)27 b Fs(\020)8 b(;)81 b Fr(8)p Fs(j)34 b Fr(2)28 b Fs(X)2535 912 y Fn(n)2582 897 y Fu(\()p Fs(\013)q Fu(\))p Fs(:)456 1044 y Fu(According)k(to)g(\(22\))g(w)m(e)i(get)1394 1214 y Fk(X)1364 1398 y Fn(j)t Fo(2)p Fn(X)1502 1406 y Fi(n)1560 1297 y Fs(k)1611 1312 y Fn(j)1675 1297 y Fr(\024)28 b Fs(C)1916 1184 y Fn(\020)1874 1214 y Fk(X)1917 1399 y Fn(\030)2010 1297 y Fs(j)2056 1256 y Fn(d)p Fo(\000)p Fq(2)2215 1297 y Fr(\024)g Fs(C)2413 1180 y Fk(Z)2496 1206 y Fn(\020)5 b Fq(+1)2460 1369 y Fn(\030)2643 1297 y Fs(x)2698 1256 y Fn(d)p Fo(\000)p Fq(2)2829 1297 y Fs(dx)28 b Fu(=)1871 1548 y Fs(C)7 b Fu(\(\(\()p Fs(n)22 b Fu(+)g(1\))2327 1507 y Fq(1)p Fn(=)p Fq(2)2437 1548 y Fs(\013)2500 1507 y Fq(1)p Fn(=s)2629 1548 y Fu(+)g(1\))2814 1507 y Fn(d)p Fo(\000)p Fq(1)2967 1548 y Fr(\000)834 1715 y Fs(n)892 1674 y Fn(d)p Fo(\000)p Fq(1)p Fn(=)p Fq(2)1093 1715 y Fs(\013)1156 1674 y Fn(d)p Fo(\000)p Fq(1)p Fn(=s)1354 1715 y Fu(\(1)g(+)g(\()p Fs(d)g Fr(\000)h Fu(2\)\()p Fs(d)e Fr(\000)i Fu(1\))2156 1674 y Fq(1)p Fn(=)p Fq(2)2265 1715 y Fs(=n)2372 1674 y Fq(1)p Fn(=)p Fq(2)2482 1715 y Fs(\013)2545 1674 y Fq(1)p Fn(=s)2652 1715 y Fu(\))2690 1674 y Fo(\000)p Fq(\()p Fn(d)p Fo(\000)p Fq(1\))r Fn(=)p Fq(2)3001 1715 y Fu(\))p Fs(:)3271 1467 y Fu(\(24\))456 1889 y(It)32 b(leads)h(to)f(\(23\))g(for)g(large)f Fs(n)d(>)g(n)1766 1904 y Fq(0)1805 1889 y Fs(:)66 b Fe(2)555 2006 y Fw(5.)44 b Fu(Let)33 b Fs(\021)j Fu(b)s(e)d(the)g(same)f(as)h (in)f(the)h(statemen)m(t)g(of)f(the)h(Prop)s(osition)e(2.1,)1164 2174 y Fs(v)1211 2189 y Fq(+)1298 2174 y Fu(:=)c(sup)1575 2197 y Fn(r)r(>)p Fq(1)1703 2174 y Fu(\()p Fs(r)1788 2132 y Fn(s)1825 2174 y Fs(V)21 b Fu(\()p Fs(r)s Fu(\)\))p Fs(;)81 b(C)2242 2189 y Fq(2)2309 2174 y Fu(:=)28 b Fr(j)p Fs(b)p Fu(\()p Fr(N)15 b Fu(\))p Fr(j)p Fs(:)456 2342 y Fu(In)36 b(order)g(to)g(estimate)f(a)g(sum)h(of)g(the)g(kind)2159 2275 y Fk(P)2247 2362 y Fn(j)t Fo(2)p Fn(X)2385 2370 y Fi(n)2447 2342 y Fs(k)2498 2357 y Fn(j)2535 2342 y Fu(\010)2605 2357 y Fn(j)2641 2342 y Fu(\()p Fs(\013)q Fu(\))g(it)f(is)g(enough)i(to)456 2458 y(estimate)k(the)i(in)m(tegral)e (in)h(the)h(righ)m(t)e(side)i(of)49 b(\(17\))42 b(and)g(to)g(use)i (\(23\))o(.)73 b(The)456 2575 y(in)m(v)m(estigation)31 b(of)h(the)h(inequalit)m(y)g(\(17\))f(leads)g(to)g(the)h(follo)m(wing)d (result.)594 2755 y Fw(Prop)s(osition)35 b(2.5.)49 b Fh(L)-5 b(et)35 b(b)-5 b(e)35 b Fs(\021)c(>)d Fu(0)35 b Fh(and)1537 2926 y Fs(n)28 b(>)g(\021)1779 2884 y Fq(2)1818 2926 y Fs(C)1888 2941 y Fq(2)1949 2926 y Fu(+)22 b Fs(\021)2099 2884 y Fo(\000)p Fq(\()p Fn(s)p Fo(\000)p Fq(2\))2336 2926 y Fs(:)908 b Fu(\(25\))456 3094 y Fh(Then)34 b(for)g Fs(j)g Fr(2)28 b Fs(X)1114 3109 y Fn(n)1074 3274 y Fu(\010)1144 3289 y Fn(j)1181 3274 y Fu(\()p Fs(\013)q Fu(\))f Fr(\024)i Fs(C)1523 3289 y Fq(1)1562 3274 y Fs(v)1613 3223 y Fq(1)p Fn(=)p Fq(\()p Fn(s)p Fo(\000)p Fq(2\))1609 3293 y(+)1865 3274 y Fs(\013)1928 3233 y Fq(1)p Fn(=s)2035 3274 y Fu(\()p Fs(n)22 b Fr(\000)h Fs(C)2323 3289 y Fq(2)2362 3274 y Fs(\021)2414 3233 y Fq(2)2453 3274 y Fu(\))2491 3233 y Fo(\000)p Fq(1)p Fn(=)p Fq(\()p Fn(s)p Fo(\000)p Fq(2\))2799 3274 y Fs(:)445 b Fu(\(26\))555 3461 y Fh(Pr)-5 b(o)g(of.)43 b Fu(F)-8 b(or)32 b Fs(t)c Fr(2)g Fu(\001\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))32 b(w)m(e)h(ha)m(v)m(e)456 3682 y Fs(\013)q(v)566 3697 y Fq(+)624 3682 y Fs(e)669 3641 y Fo(\000)p Fq(\()p Fn(s)p Fo(\000)p Fq(2\))p Fn(t)944 3682 y Fr(\000)12 b Fu(\003)1101 3697 y Fn(j)1151 3682 y Fu(+)g Fs(C)1309 3697 y Fq(2)1348 3682 y Fs(\021)1400 3641 y Fq(2)1439 3682 y Fs(\013)1502 3641 y Fq(2)p Fn(=s)1637 3682 y Fr(\025)28 b Fs(\013)q(e)1850 3641 y Fq(2)p Fn(t)1915 3682 y Fs(V)22 b Fu(\()p Fs(e)2077 3641 y Fn(t)2106 3682 y Fu(\))12 b Fr(\000)g Fu(\003)2313 3697 y Fn(j)2363 3682 y Fr(\000)g Fu(\()2500 3615 y Fs(d)23 b Fr(\000)f Fu(2)p 2500 3659 222 4 v 2587 3751 a(2)2732 3682 y(\))2770 3641 y Fq(2)2822 3682 y Fr(\000)12 b Fs(b)p Fu(\()p Fr(N)j Fu(\))p Fs(e)3170 3641 y Fq(2)p Fn(t)3263 3682 y Fr(\025)29 b Fu(0)p Fs(:)456 3897 y Fu(Let)j(b)s(e)h Fs(j)h Fr(2)28 b Fs(X)1012 3912 y Fn(n)1059 3897 y Fs(:)k Fu(Then)i(\003)1441 3912 y Fn(j)1505 3897 y Fr(\025)28 b Fs(n\013)1731 3860 y Fq(2)p Fn(=s)1838 3897 y Fs(:)33 b Fu(Therefore)718 4076 y Fs(\013)q(v)828 4091 y Fq(+)886 4076 y Fs(e)931 4034 y Fo(\000)p Fq(\()p Fn(s)p Fo(\000)p Fq(2\))p Fn(t)1222 4076 y Fr(\025)28 b Fs(n\013)1448 4034 y Fq(2)p Fn(=s)1577 4076 y Fr(\000)23 b Fs(C)1747 4091 y Fq(2)1786 4076 y Fs(\021)1838 4034 y Fq(2)1877 4076 y Fs(\013)q Fu(2)p Fs(=s)o(;)49 b Fr(8)p Fs(t)29 b Fr(2)f Fu(\001\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))p Fs(;)g(j)32 b Fr(2)c Fs(X)3108 4091 y Fn(n)3155 4076 y Fs(:)456 4252 y Fu(Th)m(us,)38 b(if)c(\001\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))33 b Fr(6)p Fu(=)f Fr(;)p Fu(,)37 b(then)f(the)g(n)m(um)m(b)s(er)2299 4227 y(~)2280 4252 y Fs(T)47 b Fu(:=)2530 4213 y Fq(1)p 2530 4229 36 4 v 2531 4286 a Fn(s)2592 4252 y Fu(log)16 b Fs(\013)25 b Fr(\000)2977 4213 y Fq(1)p 2933 4229 123 4 v 2933 4286 a Fn(s)p Fo(\000)p Fq(2)3083 4252 y Fu(log\(\()p Fs(n)f Fr(\000)456 4393 y Fs(C)526 4408 y Fq(2)565 4393 y Fs(\021)617 4357 y Fq(2)656 4393 y Fu(\))p Fs(=v)790 4408 y Fq(+)849 4393 y Fu(\))43 b(is)g(p)s(ositiv)m(e.)76 b(Moreo)m(v)m(er)45 b(\001\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))46 b Fr(\032)g Fu([0)p Fs(;)2663 4368 y Fu(~)2644 4393 y Fs(T)14 b Fu(])p Fs(:)43 b Fu(Therefore)i(sub-)456 4517 y(stituting)50 b(\001\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))52 b(b)m(y)h([0)p Fs(;)1662 4492 y Fu(~)1643 4517 y Fs(T)13 b Fu(])53 b(w)m(e)g(come)f(to)g(\(26\))o(.)103 b(In)52 b(the)h(case)g(when)456 4633 y(\001\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))44 b(=)g Fr(;)f Fu(the)g(estimate)f(\(26\))g (is)g(ful\014lled)e(b)m(y)k(the)f(de\014nition)f(of)g(the)456 4750 y(v)-5 b(alue)32 b(\010)778 4765 y Fn(j)815 4750 y Fu(\()p Fs(\013)q Fu(\))p Fs(:)64 b Fe(2)555 4866 y Fu(No)m(w)32 b(let)f(us)h(sum)f(up)h(the)f(quan)m(tities)2001 4800 y Fk(P)2088 4887 y Fn(j)t Fo(2)p Fn(X)2226 4895 y Fi(n)2289 4866 y Fs(k)2340 4881 y Fn(j)2376 4866 y Fu(\010)2446 4881 y Fn(j)2483 4866 y Fu(\()p Fs(\013)q Fu(\))g(o)m(v)m(er)h Fs(n:)g Fu(Namely)e(w)m(e)456 4983 y(consider)i(the)g(set)h Fs(Y)1211 4998 y Fq(0)1282 4983 y Fu(-)e(the)h(union)g(of)f(all)f Fs(X)2110 4998 y Fn(n)2157 4983 y Fu(,)i(for)f(whic)m(h)i Fs(X)2724 4998 y Fn(n)2824 4983 y Fr(\\)21 b Fs(Z)7 b Fu(\()p Fr(N)15 b Fs(;)i(\013)q Fu(\))26 b Fr(6)p Fu(=)i Fr(;)456 5099 y Fu(and)35 b(\(25\))f(is)g (ful\014lled.)49 b(The)36 b(follo)m(wing)c(statemen)m(t)j(giv)m(es)h (an)e(estimate)g(of)h(the)456 5215 y(sum)662 5149 y Fk(P)750 5236 y Fn(j)t Fo(2)p Fn(Y)871 5245 y Ff(0)925 5215 y Fs(k)976 5230 y Fn(j)1013 5215 y Fu(\010)1083 5230 y Fn(j)1120 5215 y Fu(\()p Fs(\013)q Fu(\).)p eop %%Page: 10 10 10 9 bop 456 236 a Fl(10)1142 b(O.)34 b(SAFR)n(ONO)n(V)594 425 y Fw(Prop)s(osition)h(2.6.)49 b Fh(Under)25 b(the)h(c)-5 b(onditions)24 b(of)h(Pr)-5 b(op)g(osition)24 b(2.1)h(ther)-5 b(e)25 b(ex-)456 541 y(ists)34 b(a)h(function)g Fs(\026)1160 556 y Fq(1)1226 541 y Fu(:)28 b(\(0)p Fs(;)17 b Fu(1\))27 b Fr(!)g Fm(R)1719 556 y Fq(+)1784 541 y Fh(,)35 b(such)g(that)1666 729 y Fu(lim)1668 786 y Fn(t)p Fo(!)p Fq(0)1818 729 y Fs(\026)1877 744 y Fq(1)1916 729 y Fu(\()p Fs(t)p Fu(\))28 b(=)f(0)p Fs(;)787 913 y Fk(X)769 1097 y Fn(j)t Fo(2)p Fn(Y)890 1106 y Ff(0)941 996 y Fs(k)992 1011 y Fn(j)1028 996 y Fu(\010)1098 1011 y Fn(j)1135 996 y Fu(\()p Fs(\013)q Fu(\))g Fr(\024)i Fs(\026)1466 1011 y Fq(1)1505 996 y Fu(\()p Fs(\021)t Fu(\))p Fs(\013)1696 955 y Fn(d=s)1825 996 y Fu(+)22 b Fs(o)p Fu(\()p Fs(\013)2071 955 y Fn(d=s)2179 996 y Fu(\))p Fs(;)87 b(\013)28 b Fr(!)f(1)p Fs(;)17 b Fu(0)27 b Fs(<)h(\021)j(<)d Fu(1)p Fs(:)555 1281 y Fh(Pr)-5 b(o)g(of.)86 b Fu(If)46 b Fs(X)1090 1296 y Fn(n)1189 1281 y Fr(\032)53 b Fs(Y)1376 1296 y Fq(0)1415 1281 y Fu(,)d(then)d(according)f(to)h(\(25\))o(,)k Fs(n)h Fr(\025)g Fs(n)2863 1296 y Fq(1)2954 1281 y Fu(:=)g([)p Fs(\021)3188 1245 y Fq(2)3227 1281 y Fs(C)3297 1296 y Fq(2)3369 1281 y Fu(+)456 1400 y Fs(\021)508 1364 y Fo(\000)p Fq(\()p Fn(s)p Fo(\000)p Fq(2\))744 1400 y Fu(])9 b(+)g(1,)26 b(and)g(according)g(to)f(\(7\))h(and)g(\(21\))o(,)h Fs(n)h Fr(\024)g Fs(n)2505 1415 y Fq(2)2573 1400 y Fu(:=)f([)p Fs(v)2777 1415 y Fq(+)2837 1400 y Fs(\013)2900 1364 y Fq(1)p Fo(\000)p Fq(2)p Fn(=s)3105 1400 y Fu(+)9 b Fs(C)3260 1415 y Fq(2)3299 1400 y Fs(\021)3351 1364 y Fq(2)3390 1400 y Fu(])p Fs(:)456 1516 y Fu(F)-8 b(urthermore,)41 b(w)m(e)g(m)m(ultiply)c(b)s(oth)j(parts)g(of)47 b(\(26\))39 b(b)m(y)i Fs(k)2633 1531 y Fn(j)2709 1516 y Fu(and)f(sum)g(up)g(o)m(v)m (er)456 1632 y Fs(j)33 b Fr(2)28 b Fs(X)704 1647 y Fn(n)751 1632 y Fs(:)33 b Fu(T)-8 b(aking)32 b(in)m(to)g(accoun)m(t)h(\(23\))f (w)m(e)i(get)1176 1821 y Fk(X)1146 2005 y Fn(j)t Fo(2)p Fn(X)1284 2013 y Fi(n)1343 1904 y Fs(k)1394 1919 y Fn(j)1430 1904 y Fu(\010)1500 1919 y Fn(j)1537 1904 y Fu(\()p Fs(\013)q Fu(\))27 b Fr(\024)h Fs(C)1895 1837 y(n)1963 1772 y Fi(d)p Fg(\000)p Ff(2)p 1963 1786 111 4 v 2003 1826 a(2)2088 1837 y Fs(\013)2161 1772 y Fi(d)p Fg(\000)p Ff(1)p 2161 1786 V 2201 1826 a Fi(s)2307 1837 y Fu(+)22 b Fs(n)2473 1772 y Fi(d)p Fg(\000)p Ff(3)p 2473 1786 V 2513 1826 a(2)2598 1837 y Fs(\013)2671 1773 y Fi(d)p 2671 1785 33 4 v 2672 1826 a(s)p 1895 1881 822 4 v 2012 1994 a Fu(\()p Fs(n)g Fr(\000)h Fs(C)2300 2009 y Fq(2)2339 1994 y Fs(\021)2391 1965 y Fq(2)2430 1994 y Fu(\))2517 1923 y Ff(1)p 2478 1935 108 4 v 2478 1976 a Fi(s)p Fg(\000)p Ff(2)2727 1904 y Fs(:)456 2175 y Fu(Therefore)1019 2339 y Fk(X)1002 2523 y Fn(j)t Fo(2)p Fn(Y)1123 2532 y Ff(0)1173 2422 y Fs(k)1224 2437 y Fn(j)1260 2422 y Fu(\010)1330 2437 y Fn(j)1367 2422 y Fu(\()p Fs(\013)q Fu(\))28 b Fr(\024)g Fs(C)1781 2313 y Fn(n)1824 2322 y Ff(2)1760 2339 y Fk(X)1732 2514 y Fn(n)p Fq(=)p Fn(n)1873 2523 y Ff(1)1934 2355 y Fs(n)2002 2290 y Fi(d)p Fg(\000)p Ff(2)p 2002 2304 111 4 v 2042 2344 a(2)2127 2355 y Fs(\013)2200 2290 y Fi(d)p Fg(\000)p Ff(1)p 2199 2304 V 2240 2344 a Fi(s)2346 2355 y Fu(+)22 b Fs(n)2512 2290 y Fi(d)p Fg(\000)p Ff(3)p 2512 2304 V 2552 2344 a(2)2637 2355 y Fs(\013)2710 2291 y Fi(d)p 2709 2303 33 4 v 2710 2344 a(s)p 1934 2399 822 4 v 2051 2512 a Fu(\()p Fs(n)g Fr(\000)h Fs(C)2339 2527 y Fq(2)2378 2512 y Fs(\021)2430 2483 y Fq(2)2469 2512 y Fu(\))2556 2441 y Ff(1)p 2517 2453 108 4 v 2517 2494 a Fi(s)p Fg(\000)p Ff(2)2793 2422 y Fr(\024)1181 2749 y Fs(C)7 b(\013)1321 2708 y Fn(d=s)1498 2641 y Fo(1)1473 2666 y Fk(X)1446 2840 y Fn(n)p Fq(=)p Fn(n)1587 2849 y Ff(1)1869 2681 y Fs(n)1937 2616 y Fi(d)p Fg(\000)p Ff(3)p 1937 2630 111 4 v 1977 2671 a(2)p 1647 2726 636 4 v 1647 2844 a Fu(\()p Fs(n)23 b Fr(\000)f Fs(C)1935 2859 y Fq(2)1975 2844 y Fs(\021)2027 2815 y Fq(2)2066 2844 y Fu(\))2176 2768 y Ff(1)p 2114 2780 156 4 v 2114 2821 a(\()p Fi(s)p Fg(\000)p Ff(2\))2315 2749 y Fu(+)g Fs(O)s Fu(\()p Fs(\013)2592 2708 y Fn(d=)p Fq(2)p Fo(\000)p Fq(2)p Fn(=s)2860 2749 y Fu(\))3271 2587 y(\(27\))456 3023 y(as)31 b Fs(\013)d Fr(!)f(1)p Fs(:)k Fu(Since)g(in)f(the)i (conditions)e(of)g(Prop)s(osition)f(2.1)i Fs(d=)p Fu(2)18 b Fr(\000)h Fu(2)p Fs(=s)27 b(<)h(d=s)p Fu(,)456 3140 y(it)j(remains)h(to)g(note)h(that)f Fs(n)1522 3155 y Fq(1)1590 3140 y Fr(!)27 b(1)32 b Fu(when)i Fs(\021)d Fr(!)c Fu(0)p Fs(:)33 b Fe(2)555 3256 y Fw(6.)43 b Fu(Let)32 b(us)g(consider)f(the)h(set)g Fs(Y)49 b Fu(:=)28 b Fs(Z)7 b Fu(\()p Fr(N)15 b Fs(;)i(\013)q Fu(\))h Fr(n)h Fs(Y)2414 3271 y Fq(0)2453 3256 y Fs(:)32 b Fu(An)f(analog)f(of)h(Prop)s(osi-)456 3372 y(tion)g(2.6)h(for)g Fs(Y)54 b Fu(is)594 3564 y Fw(Prop)s(osition)35 b(2.7.)49 b Fh(Under)25 b(the)h(c)-5 b(onditions)24 b(of)h(Pr)-5 b(op)g(osition)24 b(2.1)h(ther)-5 b(e)25 b(ex-)456 3680 y(ists)34 b(a)h(function)g Fs(\026)1160 3695 y Fq(2)1226 3680 y Fu(:)28 b(\(0)p Fs(;)17 b Fu(1\))27 b Fr(!)g Fm(R)1719 3695 y Fq(+)1819 3680 y Fh(such)35 b(that)1666 3868 y Fu(lim)1668 3925 y Fn(t)p Fo(!)p Fq(0)1818 3868 y Fs(\026)1877 3883 y Fq(2)1916 3868 y Fu(\()p Fs(t)p Fu(\))28 b(=)f(0)p Fs(;)801 4052 y Fk(X)792 4236 y Fn(j)t Fo(2)p Fn(Y)945 4135 y Fs(k)996 4150 y Fn(j)1032 4135 y Fu(\010)1102 4150 y Fn(j)1139 4135 y Fu(\()p Fs(\013)q Fu(\))h Fr(\024)g Fs(\026)1470 4150 y Fq(2)1509 4135 y Fu(\()p Fs(\021)t Fu(\))p Fs(\013)1700 4094 y Fn(d=s)1830 4135 y Fu(+)22 b Fs(o)p Fu(\()p Fs(\013)2076 4094 y Fn(d=s)2183 4135 y Fu(\))p Fs(;)87 b(\013)28 b Fr(!)g(1)p Fs(;)17 b Fu(0)26 b Fs(<)i(\021)j(<)d Fu(1)555 4415 y Fh(Pr)-5 b(o)g(of.)42 b Fu(If)28 b Fs(X)1028 4430 y Fn(n)1088 4415 y Fr(\\)13 b Fs(Y)49 b Fr(6)p Fu(=)27 b Fr(;)p Fu(,)i(then)g(the)f (inequalit)m(y)h(\(25\))e(is)h(not)g(ful\014lled.)40 b(There-)456 4532 y(fore)32 b(according)g(to)g(\(21\))1320 4727 y(\003)1388 4742 y Fn(j)1452 4727 y Fs(<)c Fu(\(1)22 b(+)g Fs(\021)1815 4686 y Fq(2)1854 4727 y Fs(C)1924 4742 y Fq(2)1985 4727 y Fu(+)g Fs(\021)2135 4686 y Fo(\000)p Fq(\()p Fn(s)p Fo(\000)p Fq(2\))2372 4727 y Fu(\))p Fs(\013)2473 4686 y Fq(2)p Fn(=s)456 4915 y Fu(for)32 b(all)e Fs(j)k Fr(2)28 b Fs(Y)5 b(:)32 b Fu(In)h(accordance)h(with)e(the)h(W)-8 b(eyl)33 b(la)m(w)f(for)g(\003)2680 4930 y Fn(j)2749 4915 y Fu(and)g Fs(k)2989 4930 y Fn(j)3058 4915 y Fu(w)m(e)i(get)1013 5044 y Fk(X)1005 5228 y Fn(j)t Fo(2)p Fn(Y)1157 5127 y Fs(k)1208 5142 y Fn(j)1272 5127 y Fr(\024)29 b Fs(C)7 b Fu(\(1)21 b(+)h Fs(\013)1734 5057 y Fi(d)p Fg(\000)p Ff(1)p 1734 5071 111 4 v 1775 5112 a Fi(s)1858 5127 y Fu(\(1)g(+)g Fs(\021)2117 5086 y Fq(2)2156 5127 y Fs(C)2226 5142 y Fq(2)2288 5127 y Fu(+)g Fs(\021)2438 5086 y Fo(\000)p Fq(\()p Fn(s)p Fo(\000)p Fq(2\))2674 5127 y Fu(\))2722 5057 y Fi(d)p Fg(\000)p Ff(1)p 2722 5071 V 2762 5112 a(2)2847 5127 y Fu(\))p Fs(:)359 b Fu(\(28\))p eop %%Page: 11 11 11 10 bop 1513 236 a Fl(DISCRETE)33 b(SPECTR)n(UM)981 b(11)456 425 y Fu(F)-8 b(urthermore,)35 b(since)g(\001\()p Fr(N)15 b Fs(;)i(j;)g(\013)q Fu(\))32 b Fr(\032)g Fu([0)p Fs(;)17 b(T)d Fu(],)35 b(due)h(to)f(\(17\))f(w)m(e)i(ha)m(v)m(e:)50 b(\010)3159 440 y Fn(j)3196 425 y Fu(\()p Fs(\013)q Fu(\))32 b Fr(\024)456 547 y Fs(C)7 b(\021)t(\013)648 511 y Fq(1)p Fn(=s)754 547 y Fs(:)29 b Fu(Let)f(us)i(m)m(ultiply)c(b)s(oth)i(parts)h (of)f(this)g(inequalit)m(y)f(b)m(y)i Fs(k)2883 562 y Fn(j)2948 547 y Fu(and)g(sum)f(up)456 664 y(o)m(v)m(er)33 b Fs(j)h Fr(2)28 b Fs(Y)5 b(:)33 b Fu(No)m(w)g(taking)f(in)m(to)f (accoun)m(t)j(\(28\))e(w)m(e)h(get)771 794 y Fk(X)763 978 y Fn(j)t Fo(2)p Fn(Y)916 877 y Fs(k)967 892 y Fn(j)1003 877 y Fu(\010)1073 892 y Fn(j)1110 877 y Fu(\()p Fs(\013)q Fu(\))27 b Fr(\024)h Fs(C)7 b(\021)t(\013)1573 836 y Fq(1)p Fn(=s)1680 877 y Fu(\(1)22 b(+)g Fs(\013)1960 808 y Fi(d)p Fg(\000)p Ff(1)p 1959 822 111 4 v 2000 862 a Fi(s)2084 877 y Fu(\(1)g(+)g Fs(\021)2343 836 y Fq(2)2382 877 y Fs(C)2452 892 y Fq(2)2513 877 y Fu(+)g Fs(\021)2663 836 y Fo(\000)p Fq(\()p Fn(s)p Fo(\000)p Fq(2\))2900 877 y Fu(\))2948 808 y Fi(d)p Fg(\000)p Ff(1)p 2948 822 V 2988 862 a(2)3072 877 y Fu(\))p Fs(:)456 1153 y Fu(It)32 b(is)g(exactly)h(what)g(w)m(e)h(need.)g Fe(2)555 1269 y Fu(Let)g(us)g(no)m(w)f(complete)g(the)h(pro)s(of)e(of)h(Prop)s (osition)e(2.1.)46 b(Since)33 b Fs(Z)7 b Fu(\()p Fr(N)15 b Fs(;)i(\013)q Fu(\))28 b(=)456 1386 y Fs(Y)43 b Fr([)23 b Fs(Y)702 1401 y Fq(0)741 1386 y Fu(,)32 b(then)h(from)f(Prop)s (ositions)f(2.6)h(and)h(2.7)f(it)g(follo)m(ws)f(that)1173 1581 y(lim)17 b(sup)1229 1650 y Fn(\013)p Fo(!1)1488 1581 y Fs(\013)1551 1540 y Fo(\000)p Fn(d=s)1830 1498 y Fk(X)1731 1686 y Fn(j)t Fo(2)p Fn(Z)5 b Fq(\()p Fo(N)10 b Fn(;\013)p Fq(\))2067 1581 y Fs(k)2118 1596 y Fn(j)2154 1581 y Fu(\010)2224 1596 y Fn(j)2261 1581 y Fu(\()p Fs(\013)q Fu(\))27 b(=)h Fs(\026)p Fu(\()p Fs(\021)t Fu(\))p Fs(;)526 b Fu(\(29\))456 1870 y(where)42 b(0)h Fr(\024)g Fs(\026)p Fu(\()p Fs(\021)t Fu(\))f Fr(\024)i Fs(\026)1367 1885 y Fq(1)1406 1870 y Fu(\()p Fs(\021)t Fu(\))28 b(+)g Fs(\026)1725 1885 y Fq(2)1764 1870 y Fu(\()p Fs(\021)t Fu(\))p Fs(:)41 b Fu(Th)m(us)i(lim)2351 1885 y Fn(\021)r Fo(!)p Fq(0)2516 1870 y Fs(\026)p Fu(\()p Fs(\021)t Fu(\))f(=)h(0)p Fs(:)e Fu(Com)m(bining)456 1986 y(\(20\))32 b(with)g(\(29\))g(w)m(e)h(come)g (to)f(the)h(statemen)m(t)g(of)f(Prop)s(osition)f(2.1.)65 b Fe(2)1319 2286 y Fu(3.)55 b Ft(Pr)n(oof)38 b(of)g(Theorem)g(1.1)555 2460 y Fw(1.)80 b Fu(In)45 b(this)f(section)h(w)m(e)h(pro)m(v)m(e)f (the)h(relation)c(\(3\).)80 b(Necessary)46 b(auxiliary)456 2576 y(statemen)m(ts)37 b(for)e(this)h(are)h(giv)m(en)f(in)f (Subsections)j(1)e(and)g(2.)54 b(In)37 b(Subsection)g(3)456 2693 y(the)d(pro)s(of)e(of)i(Theorem)f(1.1)h(is)f(completed.)46 b(Here)34 b(w)m(e)h(follo)m(w)d(the)i(w)m(a)m(y)h(of)e(the)456 2809 y(article)e([2])h(but)h(additionally)c(w)m(e)34 b(use)f(Prop)s(osition)e(2.1.)555 2925 y(F)-8 b(or)24 b Fs(\025)j Fr(2)h Fu(\003)c(and)h Fs(\013)j(>)g Fu(0)c(w)m(e)h (de\014ne)h(a)e(quan)m(tit)m(y)h Fs(N)2383 2940 y Fq(0)2422 2925 y Fu(\()p Fs(\025;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\))24 b(as)g(a)g(n)m(um)m(b)s(er)h(of)456 3041 y(eigen)m(v)-5 b(alues)27 b(of)g(the)i(op)s(erator)e Fs(A)p Fu(\()p Fs(t)p Fu(\))h(whic)m(h)g(cross)g(the)h(p)s(oin)m(t)d Fs(\025)i Fu(with)f Fs(t)h Fu(gro)m(wing)456 3158 y(from)j(zero)i(to)f Fs(\013)h Fu(\(excluding)f Fs(\013)q Fu(\).)44 b(It)32 b(is)g(clear)g(that)h(for)f Fs(\025)2606 3173 y Fq(1)2645 3158 y Fs(;)17 b(\025)2746 3173 y Fq(2)2813 3158 y Fr(2)28 b Fu(\003)p Fs(;)17 b(\025)3076 3173 y Fq(1)3142 3158 y Fs(<)28 b(\025)3303 3173 y Fq(2)903 3350 y Fu(rank)q Fs(E)1168 3366 y Fn(A)p Fq(\()p Fn(\013)p Fq(\))1325 3350 y Fu([)p Fs(\025)1409 3365 y Fq(1)1448 3350 y Fs(;)17 b(\025)1549 3365 y Fq(2)1588 3350 y Fu(\))28 b(=)g Fs(N)1836 3365 y Fq(0)1875 3350 y Fu(\()p Fs(\025)1970 3365 y Fq(2)2009 3350 y Fs(;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\))22 b Fr(\000)h Fs(N)2503 3365 y Fq(0)2542 3350 y Fu(\()p Fs(\025)2637 3365 y Fq(1)2676 3350 y Fs(;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\))p Fs(:)456 3543 y Fu(Therefore)27 b(it)f(is)g(enough)h(to)g (obtain)e(the)j(asymptotics)e(of)g(the)h(righ)m(t-hand)f(side,)456 3659 y(as)c Fs(\013)28 b Fr(!)g(1)p Fs(:)22 b Fu(The)h(follo)m(wing)d (statemen)m(t)i(pla)m(ys)h(an)f(imp)s(ortan)m(t)f(role)g(in)g(the)i (pro)s(of)456 3775 y(of)32 b(Theorem)h(1.1.)594 3970 y Fw(Theorem)k(3.1.)49 b Fh(L)-5 b(et)32 b Fs(\025)c Fr(2)g Fu(\003)j Fh(and)h Fs(")27 b(>)h Fu(0)p Fs(:)j Fh(Assume)h(that)g Fs(\025)16 b Fr(\000)g Fs(";)h(\025)f Fu(+)g Fs(")25 b Fr(2)k Fu(\003)p Fs(:)456 4086 y Fh(Then)g(ther)-5 b(e)30 b(exist)g(p)-5 b(ositive)30 b(numb)-5 b(ers)30 b Fs(r)1939 4101 y Fq(0)1978 4086 y Fs(;)17 b(\013)2084 4101 y Fq(0)2153 4086 y Fh(and)30 b Fs(c)g Fh(such)g(that)h(for)f Fs(r)h(>)c(r)3195 4101 y Fq(0)3234 4086 y Fs(;)17 b(\013)29 b(>)456 4202 y(\013)518 4217 y Fq(0)592 4202 y Fh(the)35 b(fol)5 b(lowing)33 b(ine)-5 b(qualities)34 b(ar)-5 b(e)35 b(ful\014l)5 b(le)-5 b(d:)900 4397 y Fs(N)978 4412 y Fq(0)1018 4397 y Fu(\()p Fs(\025;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\))26 b Fr(\024)j Fs(N)10 b Fu(\(\()p Fs(\025)22 b Fu(+)g Fs(")p Fu(\))g(+)g(0)p Fs(;)17 b(A)2250 4356 y Fq(\()p Fo(N)10 b Fq(\))2376 4397 y Fu(\()p Fs(\013)q(;)17 b(\013)2584 4356 y Fq(1)p Fn(=s)2690 4397 y Fs(B)2764 4412 y Fn(r)2802 4397 y Fu(\)\))22 b Fr(\000)1236 4565 y Fs(N)10 b Fu(\()p Fs(\025)23 b Fu(+)f Fs(";)17 b(A)1703 4523 y Fq(\()p Fo(N)10 b Fq(\))1829 4565 y Fu(\(0)p Fs(;)17 b(\013)2023 4523 y Fq(1)p Fn(=s)2129 4565 y Fs(B)2203 4580 y Fn(r)2241 4565 y Fu(\)\))23 b(+)f Fs(cr)2527 4523 y Fn(d)p Fo(\000)p Fq(1)2657 4565 y Fs(\013)2720 4523 y Fq(\()p Fn(d)p Fo(\000)p Fq(1\))p Fn(=s)2973 4565 y Fs(;)3271 4480 y Fu(\(30\))1027 4855 y Fs(N)1105 4870 y Fq(0)1145 4855 y Fu(\()p Fs(\025;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\))26 b Fr(\025)j Fs(N)10 b Fu(\()p Fs(\025)22 b Fr(\000)h Fs(";)17 b(A)2134 4814 y Fq(\()p Fo(D)r Fq(\))2249 4855 y Fu(\()p Fs(\013)q(;)g(\013)2457 4814 y Fq(1)p Fn(=s)2563 4855 y Fs(B)2637 4870 y Fn(r)2675 4855 y Fu(\)\))22 b Fr(\000)1117 5023 y Fs(N)10 b Fu(\()p Fs(\025)23 b Fr(\000)f Fs(";)17 b(A)1585 4982 y Fq(\()p Fo(D)r Fq(\))1700 5023 y Fu(\(0)p Fs(;)g(\013)1894 4982 y Fq(1)p Fn(=s)2001 5023 y Fs(B)2075 5038 y Fn(r)2113 5023 y Fu(\)\))22 b Fr(\000)g Fs(cr)2399 4982 y Fn(d)p Fo(\000)p Fq(1)2530 5023 y Fs(\013)2593 4982 y Fq(\()p Fn(d)p Fo(\000)p Fq(1\))p Fn(=s)2846 5023 y Fs(;)3271 4938 y Fu(\(31\))456 5216 y Fh(wher)-5 b(e)34 b Fs(c)h Fh(do)-5 b(es)34 b(not)h(dep)-5 b(end)34 b(on)g Fs(r)k Fh(and)c Fs(\013)q(:)p eop %%Page: 12 12 12 11 bop 456 236 a Fl(12)1142 b(O.)34 b(SAFR)n(ONO)n(V)555 425 y Fu(W)-8 b(e)51 b(p)s(ostp)s(one)g(the)f(pro)s(of)g(of)g(Theorem)h (3.1)f(un)m(til)f(Subsection)i(4.)96 b(Ac-)456 541 y(tually)48 b(all)g(the)i(ideas)g(of)f(the)i(pro)s(of)e(are)h(giv)m(en)g(in)f([2].) 96 b(In)50 b Fr(x)p Fu(4)g(w)m(e)h(repro-)456 658 y(duce)56 b(them)g(in)f(the)h(exact)h(form.)112 b(Belo)m(w)55 b(w)m(e)i(use)g (the)f(follo)m(wing)d(nota-)456 774 y(tions)39 b Fr(jj)p Fs(p)p Fr(jj)863 789 y Fo(1)977 774 y Fu(=)h(sup)1240 797 y Fn(x)p Fo(2)p Fj(R)1375 779 y Fi(d)1415 774 y Fr(j)p Fs(p)p Fu(\()p Fs(x)p Fu(\))p Fr(j)p Fs(;)17 b(v)1742 789 y Fq(+)1841 774 y Fu(=)40 b(sup)2104 797 y Fn(r)r(>)p Fq(1)2232 774 y Fs(r)2279 738 y Fn(s)2316 774 y Fs(V)21 b Fu(\()p Fs(r)s Fu(\))p Fs(:)40 b Fu(F)-8 b(or)39 b(\012)i Fr(\032)g Fm(R)3061 738 y Fn(d)3147 774 y Fu(w)m(e)g(put)456 890 y Fs(L)p Fu(\(\012\))51 b(=)f(inf)963 905 y Fn(x)p Fo(2)p Fq(\012)1105 890 y Fr(j)p Fs(x)p Fr(j)p Fs(:)c Fu(F)-8 b(or)45 b Fs(L)p Fu(\(\012\))51 b Fs(>)g Fu(0)p Fs(;)17 b(\025)50 b Fr(2)g Fm(R)57 b Fu(w)m(e)47 b(in)m(tro)s(duce)f (the)h(quan)m(tit)m(y)456 1006 y Fs(g)62 b Fu(=)c Fs(g)t Fu(\()p Fs(\025;)17 b Fu(\012\))58 b(=)h Fr(jj)p Fs(p)p Fr(jj)1351 1021 y Fo(1)1459 1006 y Fu(+)34 b Fs(v)1616 1021 y Fq(+)1676 1006 y Fs(L)p Fu(\(\012\))1888 970 y Fo(\000)p Fn(s)2015 1006 y Fu(+)g Fr(j)p Fs(\025)p Fr(j)p Fs(:)50 b Fu(The)i(follo)m(wing)c(prop)s(osition)456 1126 y(giv)m(es)41 b(an)g(asymptotical)e(estimate)h(for)h(the)g(quan)m (tit)m(y)h Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)2865 1089 y Fq(\()p Fn(")p Fq(\))2957 1126 y Fu(\()p Fs(\013)q(;)g(\013)3165 1089 y Fq(1)p Fn(=s)3271 1126 y Fu(\012\)\),)456 1242 y(as)32 b Fs(\013)d Fr(!)e(1)p Fs(:)594 1418 y Fw(Prop)s(osition)35 b(3.1.)49 b Fh(L)-5 b(et)28 b Fu(\012)g Fr(\032)g Fm(R)1852 1382 y Fn(d)1926 1418 y Fh(b)-5 b(e)27 b(a)g(domain)f(with)h(a)g(pie)-5 b(c)g(ewise)26 b(smo)-5 b(oth)456 1535 y(b)g(oundary.)44 b(Assume)35 b(that)g Fs(L)p Fu(\(\012\))29 b Fs(>)e Fu(0)p Fs(:)35 b Fh(Then)f(for)h(any)f Fs(\025)28 b Fr(2)g Fm(R)738 1703 y Fu(lim)17 b(sup)795 1773 y Fn(\013)p Fo(!1)1089 1703 y Fs(\013)1152 1662 y Fo(\000)p Fn(d=s)1315 1703 y Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)1615 1662 y Fq(\()p Fn(")p Fq(\))1707 1703 y Fu(\()p Fs(\013)q(;)g(\013)1915 1662 y Fq(1)p Fn(=s)2021 1703 y Fu(\012\)\))28 b Fr(\024)g Fu(\(2)p Fs(\031)t Fu(\))2484 1662 y Fo(\000)p Fn(d)2579 1703 y Fs(!)2640 1718 y Fn(d)2680 1703 y Fs(g)2731 1662 y Fn(d=)p Fq(2)2842 1703 y Fu(mes\(\012\))p Fs(:)3271 1820 y Fu(\(32\))555 2013 y Fh(Pr)-5 b(o)g(of.)70 b Fu(F)-8 b(or)40 b Fs(x)j Fr(2)g Fs(\013)1335 1977 y Fq(1)p Fn(=s)1442 2013 y Fu(\012)f(w)m(e)g(ha)m(v)m(e)h(the)f(estimate)e Fs(\013)q(V)21 b Fu(\()p Fr(j)p Fs(x)p Fr(j)p Fu(\))43 b Fr(\024)g Fs(v)3054 2028 y Fq(+)3113 2013 y Fs(L)p Fu(\(\012\))3325 1977 y Fo(\000)p Fn(s)3417 2013 y Fs(:)456 2132 y Fu(Th)m(us)32 b Fs(\013)q(V)821 2147 y Fq(0)860 2132 y Fu(\()p Fs(x)p Fu(\))19 b(+)f Fs(\025)h Fr(\000)g Fs(p)p Fu(\()p Fs(x)p Fu(\))28 b Fr(\024)g Fs(g)t(;)47 b(x)28 b Fr(2)g Fs(\013)1954 2096 y Fq(1)p Fn(=s)2061 2132 y Fu(\012)p Fs(:)k Fu(Then)g(it)e(follo)m(ws)f(from)h(the)h(v)-5 b(ari-)456 2252 y(ational)26 b(principle)h(that)i Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)1682 2215 y Fq(\()p Fn(")p Fq(\))1774 2252 y Fu(\()p Fs(\013)q(;)g(\013)1982 2215 y Fq(1)p Fn(=s)2088 2252 y Fu(\012\)\))30 b(do)s(es)f(not)g(exceed)i (the)e(amoun)m(t)456 2371 y(of)e(the)i(negativ)m(e)g(sp)s(ectrum)f(of)g (the)h(op)s(erator)f Fr(\000)p Fu(\001)13 b Fr(\000)g Fs(g)t(I)37 b Fu(in)28 b Fs(L)2756 2386 y Fq(2)2796 2371 y Fu(\()p Fs(\013)2897 2335 y Fq(1)p Fn(=s)3004 2371 y Fu(\012\).)42 b(Let)29 b(us)456 2490 y(substitute)34 b Fs(x)29 b Fu(=)g Fs(\013)1165 2454 y Fq(1)p Fn(=s)1272 2490 y Fs(y)t(;)17 b(y)31 b Fr(2)e Fu(\012)p Fs(:)34 b Fu(As)g(a)f(result)g(w)m(e)h(obtain)e(that)h Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)3132 2454 y Fq(\()p Fn(")p Fq(\))3224 2490 y Fu(\()p Fs(\013)q(;)g Fu(\)\))456 2606 y(do)s(es)31 b(not)g(exceed)i(the)e(en)m(tire)g(m)m(ultiplicit)m(y)d (of)i(the)i(negativ)m(e)f(sp)s(ectrum)g(of)g(the)456 2722 y(op)s(erator)22 b Fr(\000)p Fu(\001)r Fr(\000)r Fs(\013)1141 2686 y Fq(2)p Fn(=s)1248 2722 y Fs(g)t(I)30 b Fu(in)22 b Fs(L)1542 2737 y Fq(2)1582 2722 y Fu(\(\012\))g(\(with)h (appropriate)e(b)s(oundary)i(conditions\).)456 2839 y(No)m(w)h(the)h (inequalit)m(y)f(\(32\))f(follo)m(ws)g(from)g(the)h(W)-8 b(eyl)24 b(la)m(w)g(for)f(the)i(eigen)m(v)-5 b(alues)24 b(of)456 2955 y(the)i(Laplace)f(op)s(erator)h(on)g(a)f(domain)g(with)g (a)h(piecewise)g(smo)s(oth)g(b)s(oundary)-8 b(.)p Fe(2)555 3071 y Fu(W)g(e)69 b(\014x)h Fs(n)89 b Fr(2)h Fm(Z)1306 3035 y Fn(d)1412 3071 y Fu(and)69 b Fs(\027)96 b(>)89 b Fu(0)p Fs(:)69 b Fu(Let)f(us)i(consider)f(the)g(op)s(erator)456 3187 y Fs(A)529 3151 y Fq(\()p Fn(")p Fq(\))620 3187 y Fu(\()p Fs(\013)q(;)17 b(\013)828 3151 y Fq(1)p 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b(\013)1830 3724 y Fq(1)p Fn(=s)1936 3765 y Fs(\027)6 b Fm(Q)2062 3780 y Fn(n)2115 3765 y Fu(\)\))28 b Fr(\024)g Fs(\027)2378 3724 y Fn(d)2419 3765 y Fs(\032)p Fu(\()p Fs(\025)22 b Fu(+)g Fs(v)2731 3780 y Fn(a)2773 3765 y Fu(\()p Fs(\027)6 b Fr(j)p Fs(x)2948 3724 y Fq(\()p Fo(N)k Fq(\))2948 3790 y Fn(n)3075 3765 y Fr(j)p Fu(\))3141 3724 y Fo(\000)p Fn(s)3232 3765 y Fu(\))p Fs(;)3271 3881 y Fu(\(33\))670 4109 y(lim)645 4159 y Fn(\013)p Fo(!1)848 4109 y Fu(inf)c Fs(\013)1029 4068 y Fo(\000)p Fn(d=s)1192 4109 y Fs(N)k Fu(\()p Fs(\025;)17 b(A)1492 4068 y Fq(\()p Fo(D)r Fq(\))1607 4109 y Fu(\()p Fs(\013)q(;)g(\013)1815 4068 y Fq(1)p Fn(=s)1922 4109 y Fs(\027)6 b Fm(Q)2048 4124 y Fn(n)2101 4109 y Fu(\)\))27 b Fr(\025)h Fs(\027)2363 4068 y Fn(d)2404 4109 y Fs(\032)p Fu(\()p Fs(\025)23 b Fu(+)f Fs(v)2717 4124 y Fn(a)2759 4109 y Fu(\()p Fs(\027)6 b Fr(j)p Fs(x)2934 4068 y Fq(\()p Fo(D)r Fq(\))2934 4134 y Fn(n)3050 4109 y Fr(j)p Fu(\))3116 4068 y Fo(\000)p Fn(s)3207 4109 y Fu(\))p Fs(:)3271 4225 y Fu(\(34\))555 4409 y Fh(Pr)-5 b(o)g(of.)44 b Fu(First)31 b(of)i(all,)d(let)i(us)i (assume)f(that)g Fs(V)21 b Fu(\()p Fs(r)s Fu(\))28 b(=)f Fs(v)2562 4424 y Fn(a)2604 4409 y Fs(r)2651 4373 y Fo(\000)p Fn(s)2742 4409 y Fs(;)17 b(r)31 b(>)c Fu(1)p Fs(:)33 b Fu(Then)h(for)456 4528 y Fs(x)28 b Fr(2)g Fs(\013)696 4492 y Fq(1)p Fn(=s)803 4528 y Fs(\027)6 b Fm(Q)929 4543 y Fn(n)982 4528 y Fs(;)17 b Fr(j)p Fs(x)p Fr(j)27 b Fs(>)g Fu(1)33 b(w)m(e)g(ha)m(v)m(e)1097 4697 y Fs(v)1144 4712 y Fn(a)1186 4697 y Fu(\()p Fs(\027)6 b Fr(j)p Fs(x)1361 4656 y Fq(\()p Fo(D)r Fq(\))1361 4722 y Fn(n)1477 4697 y Fr(j)p Fu(\))1543 4656 y Fo(\000)p Fn(s)1662 4697 y Fr(\024)28 b Fs(\013)q(V)21 b Fu(\()p Fr(j)p Fs(x)p Fr(j)p Fu(\))28 b Fr(\024)g Fs(v)2275 4712 y Fn(a)2316 4697 y Fu(\()p Fs(\027)6 b Fr(j)p Fs(x)2491 4656 y Fq(\()p Fo(N)k Fq(\))2491 4722 y Fn(n)2619 4697 y Fr(j)p Fu(\))2685 4656 y Fo(\000)p Fn(s)2776 4697 y Fs(:)456 4861 y Fu(After)86 b(the)i(substitution)e Fs(V)1641 4876 y Fq(0)1680 4861 y Fu(\()p 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b(is)f(estimated)h(from)456 5216 y(ab)s(o)m(v)m(e)c(\(the)h(case)g Fs(")d Fu(=)h Fr(N)15 b Fu(\))37 b(or)h(from)e(b)s(elo)m(w)i(\(the)g(case)h Fs(")e Fu(=)f Fr(D)s Fu(\))i(b)m(y)h Fs(M)10 b Fu(\()p Fs(";)17 b(\013)q Fu(\),)p eop %%Page: 13 13 13 12 bop 1513 236 a Fl(DISCRETE)33 b(SPECTR)n(UM)981 b(13)456 425 y Fu(whic)m(h)49 b(equals)g(the)g(amoun)m(t)f(of)g(the)i (negativ)m(e)e(sp)s(ectrum)i(of)e(the)h(op)s(erator)456 541 y Fr(\000)p Fu(\001)33 b(+)g Fs(p)g Fr(\000)g Fu(\()p Fs(\025)g Fu(+)f Fs(v)1231 556 y Fn(a)1273 541 y Fu(\()p Fs(\027)6 b Fr(j)p Fs(x)1448 505 y Fq(\()p Fn(")p Fq(\))1448 566 y Fn(n)1540 541 y Fr(j)p Fu(\))1606 505 y Fo(\000)p Fn(s)1698 541 y Fu(\))p Fs(I)56 b Fu(in)47 b Fs(L)2030 556 y Fq(2)2070 541 y Fu(\()p Fs(\013)2171 505 y Fq(1)p Fn(=s)2278 541 y Fs(\027)6 b Fm(Q)2404 556 y Fn(n)2457 541 y Fu(\))48 b(with)g Fs(")p Fu(-conditions)e(on)456 667 y Fs(\013)519 631 y Fq(1)p Fn(=s)626 667 y Fs(\027)6 b(@)f Fm(Q)809 682 y Fn(n)861 667 y Fs(:)33 b 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Fn(a)2556 3999 y Fu(\()p Fs(\027)6 b Fr(j)p Fs(x)2731 3958 y Fq(\()p Fo(N)k Fq(\))2731 4024 y Fn(n)2858 3999 y Fr(j)p Fu(\))2924 3958 y Fo(\000)p Fn(s)3015 3999 y Fu(\))p Fs(:)3271 3953 y Fu(\(36\))456 4265 y(Let)72 b(us)g(pass)h(to)f(the)g(limit)c(as)k Fs(\027)102 b Fr(!)94 b Fu(0)p Fs(:)72 b Fu(Ob)m(viously)-8 b(,)81 b(mes\(\012)3172 4280 y Fq(1)3212 4265 y Fu(\))95 b Fr(!)456 4384 y Fu(0)p Fs(;)605 4318 y Fk(P)693 4405 y Fn(n)p Fo(2J)864 4384 y Fs(\027)918 4348 y Fn(d)959 4384 y Fs(\032)p Fu(\()p Fs(\025)14 b Fu(+)g Fs(v)1255 4399 y Fn(a)1297 4384 y Fu(\()p Fs(\027)6 b Fr(j)p Fs(x)1472 4348 y Fq(\()p Fo(N)k Fq(\))1472 4409 y Fn(n)1599 4384 y Fr(j)p Fu(\))1665 4348 y Fo(\000)p Fn(s)1757 4384 y Fu(\))27 b Fr(!)1949 4313 y Fk(R)1989 4410 y Fn(\016)2020 4419 y Ff(1)2054 4410 y Fn(<)p Fo(j)p Fn(x)p Fo(j)p Fn(<\016)2275 4419 y Ff(2)2330 4384 y Fs(\032)p Fu(\()p Fs(\025)14 b Fu(+)g Fs(v)2626 4399 y Fn(a)2668 4384 y Fr(j)p Fs(x)p Fr(j)2779 4348 y Fo(\000)p Fn(s)2870 4384 y Fu(\))p Fs(dx)29 b Fu(as)g Fs(\027)34 b Fr(!)28 b Fu(0)p Fs(:)456 4504 y Fu(Therefore,)542 4716 y(lim)17 b(sup)841 4739 y Fn(\013)p Fo(!1)1032 4716 y Fs(\013)1095 4675 y Fo(\000)p Fn(d=s)1258 4716 y Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)1558 4675 y Fq(\()p Fn(")p Fq(\))1649 4716 y Fu(\()p Fs(\013)q(;)g(\013)1857 4675 y Fq(1)p Fn(=s)1964 4716 y Fu(\012\)\))28 b Fr(\024)2243 4599 y Fk(Z)2289 4787 y Fn(\016)2320 4796 y Ff(1)2355 4787 y Fn(<)p Fo(j)p Fn(x)p Fo(j)p Fn(<\016)2576 4796 y Ff(2)2630 4716 y Fs(\032)p Fu(\()p Fs(\025)22 b Fu(+)g Fs(v)2942 4731 y Fn(a)2984 4716 y Fr(j)p Fs(x)p Fr(j)3095 4675 y Fo(\000)p Fn(s)3187 4716 y Fu(\))p Fs(dx:)456 4949 y Fu(In)33 b(the)g(same)f(w)m(a)m(y)i(w)m(e)f(obtain)f(the)h(lo)m (w)m(er)g(estimate:)487 5161 y(lim)17 b(inf)757 5176 y Fn(\013)p Fo(!1)948 5161 y Fs(\013)1011 5120 y Fo(\000)p Fn(d=s)1173 5161 y Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)1473 5120 y Fq(\()p Fn(")p Fq(\))1565 5161 y Fu(\()p Fs(\013)q(;)g(\013)1773 5120 y Fq(1)p Fn(=s)1880 5161 y Fu(\012\)\))28 b Fr(\025)2159 5044 y Fk(Z)2205 5232 y Fn(\016)2236 5241 y Ff(1)2271 5232 y Fn(<)p Fo(j)p Fn(x)p Fo(j)p Fn(<\016)2492 5241 y Ff(2)2546 5161 y Fs(\032)p Fu(\()p Fs(\025)22 b Fu(+)g Fs(v)2858 5176 y Fn(a)2900 5161 y Fr(j)p Fs(x)p Fr(j)3011 5120 y Fo(\000)p Fn(s)3103 5161 y Fu(\))p Fs(dx:)65 b Fe(2)p eop %%Page: 14 14 14 13 bop 456 236 a Fl(14)1142 b(O.)34 b(SAFR)n(ONO)n(V)555 425 y Fu(Let)43 b(\012)j Fr(\032)g Fm(R)1044 389 y Fn(d)1134 425 y Fu(b)s(e)d(a)f(domain)f(with)i(a)g(piecewise)g(smo)s(oth)f(b)s (oundary)h(and)456 541 y Fs(\025)513 556 y Fn(j)577 541 y Fr(2)28 b Fm(R)5 b Fs(;)17 b(j)39 b Fu(=)28 b(1)p Fs(;)17 b Fu(2)p Fs(:)31 b Fu(W)-8 b(e)33 b(put)456 728 y Fs(n)p Fu(\()p Fs(\025)609 743 y Fq(1)648 728 y Fs(;)17 b(\025)749 743 y Fq(2)788 728 y Fs(;)g Fu(\012)p Fs(;)g(\013)q Fu(\))27 b(=)h Fs(N)10 b Fu(\()p Fs(\025)1361 743 y Fq(2)1401 728 y Fs(;)17 b(A)1518 687 y Fq(\()p Fo(N)10 b Fq(\))1644 728 y Fu(\()p Fs(\013)q(;)17 b(\013)1852 687 y Fq(1)p Fn(=s)1958 728 y Fu(\012\)\))t Fr(\000)t Fs(N)10 b Fu(\()p Fs(\025)2372 743 y Fq(1)2414 728 y Fs(;)17 b(A)2531 687 y Fq(\()p Fo(D)r Fq(\))2646 728 y Fu(\()p Fs(\013)q(;)g(\013)2854 687 y Fq(1)p Fn(=s)2961 728 y Fu(\012\)\))p Fs(;)41 b(\013)28 b(>)g Fu(0)p Fs(:)456 905 y Fu(Ob)m(viously)k(if)p 1004 827 71 4 v 32 w(\012)c(=)p 1205 827 331 4 v 27 w(\012)1275 920 y Fq(1)1337 905 y Fr([)23 b Fu(\012)1496 920 y Fq(2)1536 905 y Fu(,)32 b(then)h(the)g(follo)m(wing)d(inequalities)h(are)i (ful\014lled)1008 1081 y Fs(n)p Fu(\()p Fs(\025)1161 1096 y Fq(1)1201 1081 y Fs(;)17 b(\025)1302 1096 y Fq(2)1341 1081 y Fs(;)g Fu(\012)p Fs(;)g(\013)q Fu(\))27 b Fr(\025)h(\000)p Fs(n)p Fu(\()p Fs(\025)1962 1096 y Fq(2)2002 1081 y Fs(;)17 b(\025)2103 1096 y Fq(1)2142 1081 y Fs(;)g Fu(\012)2256 1096 y Fn(j)2293 1081 y Fs(;)g(\013)q Fu(\))p Fs(;)81 b(j)34 b Fu(=)27 b(1)p Fs(;)17 b Fu(2)p Fs(;)379 b Fu(\(37\))884 1322 y Fs(n)p Fu(\()p Fs(\025)1037 1337 y Fq(1)1076 1322 y Fs(;)17 b(\025)1177 1337 y Fq(2)1217 1322 y Fs(;)g Fu(\012)p Fs(;)g(\013)q Fu(\))27 b Fr(\024)h Fs(n)p Fu(\()p Fs(\025)1761 1337 y Fq(1)1800 1322 y Fs(;)17 b(\025)1901 1337 y Fq(2)1940 1322 y Fs(;)g Fu(\012)2054 1337 y Fq(1)2094 1322 y Fs(;)g(\013)q Fu(\))k(+)h Fs(n)p Fu(\()p Fs(\025)2511 1337 y Fq(1)2551 1322 y Fs(;)17 b(\025)2652 1337 y Fq(2)2691 1322 y Fs(;)g Fu(\012)2805 1337 y Fq(2)2845 1322 y Fs(;)g(\013)q Fu(\))p Fs(:)254 b Fu(\(38\))594 1507 y Fw(Prop)s(osition)35 b(3.4.)49 b Fh(L)-5 b(et)35 b Fs(r)c(>)c Fu(0)35 b Fh(and)f Fs(\025)2098 1522 y Fn(j)2162 1507 y Fr(2)29 b Fm(R)t Fs(;)18 b(j)39 b Fu(=)27 b(1)p Fs(;)17 b Fu(2)p Fs(:)35 b Fh(Then)1691 1693 y Fu(lim)1665 1743 y Fn(\013)p Fo(!1)1869 1693 y Fs(\013)1932 1652 y Fo(\000)p Fn(d=s)2094 1693 y Fs(n)p Fu(\()p Fs(\025)2247 1708 y Fq(1)2287 1693 y Fs(;)17 b(\025)2388 1708 y Fq(2)2427 1693 y Fs(;)g(B)2545 1708 y Fn(r)2583 1693 y Fs(;)g(\013)q Fu(\))27 b(=)1042 1789 y Fk(Z)1088 1978 y Fo(j)p Fn(x)p Fo(j)p Fn()g Fu(0)p Fs(:)h Fh(Assume)f(that)456 4781 y Fs(\025)513 4796 y Fn(j)571 4781 y Fu(+)22 b Fs(")27 b Fr(2)i Fu(\003)p Fs(;)17 b(\025)1006 4796 y Fn(j)1064 4781 y Fr(\000)22 b Fs(")28 b Fr(2)g Fu(\003)p Fs(;)17 b(j)33 b Fu(=)27 b(1)p Fs(;)17 b Fu(2)p Fs(:)34 b Fh(Then)g(ther)-5 b(e)35 b(exists)g Fs(r)30 b(>)e Fu(0)34 b Fh(such)h(that)889 4967 y Fu(lim)17 b(sup)1188 4991 y Fn(\013)p Fo(!1)1378 4967 y Fs(\013)1441 4926 y Fo(\000)p Fn(d=s)1604 4967 y Fu(\()p Fs(N)1720 4982 y Fq(0)1760 4967 y Fu(\()p Fs(\025)1855 4982 y Fq(2)1894 4967 y Fs(;)g(A)p Fu(\()p Fs(\013)q Fu(\)\))k Fr(\000)i Fs(N)2387 4982 y Fq(0)2427 4967 y Fu(\()p Fs(\025)2522 4982 y Fq(1)2561 4967 y Fs(;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\)\))27 b Fr(\024)875 5044 y Fk(Z)921 5232 y Fo(j)p Fn(x)p Fo(j)p Fn()e Fu(0)p Fs(:)i Fu(Also)f(let)h(the)g(function)g Fs( )h Fr(2)d Fs(C)2879 871 y Fo(1)2954 908 y Fu(\()p Fm(R)3058 871 y Fn(d)3104 908 y Fu(\))i(satisfy)456 1024 y(the)j(follo)m(wing)c(conditions)1000 1285 y(0)f Fr(\024)g Fs( )t Fu(\()p Fs(x)p Fu(\))g Fr(\024)g Fu(1)p Fs(;)114 b( )t Fu(\()p Fs(x)p Fu(\))28 b(=)2032 1110 y Fk(8)2032 1185 y(<)2032 1335 y(:)2148 1225 y Fu(0)64 b(for)h Fr(j)p Fs(x)p Fr(j)27 b Fs(<)h Fu(1)p Fs(=)p Fu(2)p Fs(;)2148 1341 y Fu(1)64 b(for)h Fr(j)p Fs(x)p Fr(j)27 b Fs(>)h Fu(3)p Fs(=)p Fu(4)p Fs(:)456 1551 y Fu(W)-8 b(e)34 b(denote)h(b)m(y)f (\011)1153 1566 y Fn(r)1225 1551 y Fu(the)h(m)m(ultiplication)29 b(op)s(erator)k(b)m(y)i(the)f(function)g Fs( )3169 1566 y Fn(r)3207 1551 y Fu(\()p Fs(x)p Fu(\))d(=)456 1667 y Fs( )t Fu(\()p Fs(x=r)s Fu(\))p Fs(;)17 b(r)29 b(>)f Fu(0)p Fs(;)k Fu(in)g Fs(L)1259 1682 y Fq(2)1299 1667 y Fu(\()p Fm(R)1403 1631 y Fn(d)1449 1667 y Fu(\))h(and)f(w)m(e)i(put) 1063 1825 y(~)1031 1850 y Fs(A)1104 1865 y Fn(r)1170 1850 y Fu(=)28 b Fs(A)1347 1809 y Fq(\()p Fo(D)r Fq(\))1462 1850 y Fu(\(0)p Fs(;)17 b(B)1667 1865 y Fn(r)1705 1850 y Fu(\))22 b(+)g(\()p Fs(b)1942 1809 y Fo(0)1988 1850 y Fr(\000)g Fs(a)2138 1809 y Fo(0)2162 1850 y Fu(\)\011)2276 1865 y Fn(r)2314 1850 y Fu(\005)2387 1865 y Fn(r)2425 1850 y Fu(\011)2501 1865 y Fn(r)2539 1850 y Fs(;)49 b(r)30 b(>)e Fu(0)p Fs(:)402 b Fu(\(51\))456 2047 y(In)33 b([1)o(])g(the)g (follo)m(wing)d(statemen)m(t)j(concerning)2265 2022 y(~)2234 2047 y Fs(A)2307 2062 y Fn(r)2377 2047 y Fu(is)f(pro)m(v)m(ed.)594 2231 y Fw(Prop)s(osition)j(4.1.)49 b Fh(L)-5 b(et)38 b(the)f(c)-5 b(onditions)44 b Fu(\(49\))p Fh(,)38 b Fu(\(50\))e Fh(b)-5 b(e)37 b(ful\014l)5 b(le)-5 b(d)37 b(and)f(let)487 2330 y Fu(~)456 2356 y Fs(A)529 2371 y Fn(r)602 2356 y Fh(b)-5 b(e)34 b(de\014ne)-5 b(d)34 b(as)g(in)42 b Fu(\(51\))p Fh(.)j(Then)34 b(ther)-5 b(e)34 b(exists)52 b Fu(~)-66 b Fs(r)2357 2371 y Fq(0)2424 2356 y Fs(>)28 b Fu(0)34 b Fh(such)h(that)1441 2539 y Fu([)p Fs(a;)17 b(b)p Fu(])28 b Fr(\032)g Fs(\032)p Fu(\()1884 2513 y(~)1852 2539 y Fs(A)1925 2554 y Fn(r)1964 2539 y Fu(\))p Fs(;)86 b Fr(8)p Fs(r)31 b(>)45 b Fu(~)-67 b Fs(r)2392 2554 y Fq(0)2432 2539 y Fs(:)555 2723 y Fu(W)-8 b(e)33 b(put)1581 2875 y(~)1550 2900 y Fs(A)1623 2915 y Fn(r)1661 2900 y Fu(\()p Fs(\013)q Fu(\))27 b(=)1962 2875 y(~)1931 2900 y Fs(A)2004 2915 y Fn(r)2064 2900 y Fr(\000)c Fs(\013)q(V)2284 2915 y Fq(0)2323 2900 y Fs(:)921 b Fu(\(52\))456 3081 y(The)27 b(next)g(statemen)m(t)g(allo)m(ws)e(to)h(appro)m(ximate)f(the) i(quan)m(tit)m(y)g Fs(N)2888 3096 y Fq(0)2927 3081 y Fu(\()p Fs(\025;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\))25 b(b)m(y)456 3197 y(similar)31 b(quan)m(tities)j(for)f(op)s(erators)h (on)g(a)g(ball)f Fs(B)2299 3212 y Fn(r)2337 3197 y Fs(;)17 b(r)32 b(>)f Fu(0)p Fs(:)j Fu(W)-8 b(e)34 b(refer)h(to)f([2])g(for)456 3313 y(its)e(pro)s(of.)594 3498 y Fw(Prop)s(osition)j(4.2.)49 b Fh(L)-5 b(et)38 b Fs(\025)32 b Fr(2)g Fu(\()p Fs(a;)17 b(b)p Fu(\))38 b Fh(and)e(the)i(c)-5 b(onditions)44 b Fu(\(49\))o Fh(,)38 b Fu(\(50\))f Fh(ar)-5 b(e)456 3622 y(ful\014l)5 b(le)-5 b(d.)43 b(Assume)34 b(that)h(the)f(op)-5 b(er)g(ator)1972 3597 y Fu(~)1941 3622 y Fs(A)2014 3637 y Fn(r)2086 3622 y Fh(in)41 b Fu(\(52\))34 b Fh(is)f(de\014ne)-5 b(d)33 b(in)41 b Fu(\(51\))p Fh(.)j(Then)904 3805 y Fs(N)982 3820 y Fq(0)1021 3805 y Fu(\()p Fs(\025;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\))27 b Fr(\025)h Fu(lim)17 b(sup)1604 3874 y Fn(r)r Fo(!1)1841 3805 y Fu(\()p Fs(N)10 b Fu(\()p Fs(\025;)2137 3780 y Fu(~)2106 3805 y Fs(A)2179 3820 y Fn(r)2217 3805 y Fu(\()p Fs(\013)2318 3764 y Fo(0)2341 3805 y Fu(\)\))22 b Fr(\000)h Fs(N)10 b Fu(\()p Fs(\025;)2797 3780 y Fu(~)2766 3805 y Fs(A)2839 3820 y Fn(r)2877 3805 y Fu(\)\))p Fs(;)2509 3998 y Fu(0)28 b Fs(<)f(\013)2752 3957 y Fo(0)2803 3998 y Fs(<)h(\013)q Fu(;)3271 3900 y(\(53\))918 4253 y Fs(N)996 4268 y Fq(0)1036 4253 y Fu(\()p Fs(\025;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\))26 b Fr(\024)j Fu(lim)17 b(inf)1604 4303 y Fn(r)r Fo(!1)1827 4253 y Fu(\()p Fs(N)10 b Fu(\()p Fs(\025;)2123 4228 y Fu(~)2092 4253 y Fs(A)2165 4268 y Fn(r)2203 4253 y Fu(\()p Fs(\013)2304 4212 y Fo(0)2327 4253 y Fu(\)\))22 b Fr(\000)g Fs(N)10 b Fu(\()p Fs(\025;)2783 4228 y Fu(~)2751 4253 y Fs(A)2824 4268 y Fn(r)2862 4253 y Fu(\)\))p Fs(;)2495 4427 y Fu(0)27 b Fs(<)h(\013)g(<)g(\013)2932 4386 y Fo(0)2955 4427 y Fs(:)3271 4339 y Fu(\(54\))555 4618 y(W)-8 b(e)38 b(in)m(tro)s(duce)f(the)g(selfadjoin)m(t)f(op)s(erators)g Fs(A)2320 4571 y Fn(")2353 4580 y Ff(1)2388 4571 y Fn(;")2441 4580 y Ff(2)2320 4642 y Fn(R;r)2514 4618 y Fu(=)f Fr(\000)p Fu(\001)26 b(+)f Fs(p)36 b Fu(in)h Fs(L)3180 4633 y Fq(2)3219 4618 y Fu(\()p Fs(B)3331 4633 y Fn(r)3395 4618 y Fr(n)456 4738 y Fs(B)530 4753 y Fn(R)587 4738 y Fu(\))p Fs(;)54 b(r)39 b(>)c(R)i(>)f Fu(0,)i(where)h(the)f(indexes)g Fs(")2092 4753 y Fq(1)2169 4738 y Fu(and)f Fs(")2409 4753 y Fq(2)2448 4738 y Fu(,)i(whic)m(h)f(tak)m(e)g(the)f(v)-5 b(alues)456 4855 y Fr(D)36 b Fu(or)e Fr(N)15 b Fu(,)34 b(c)m(haracterize)h(b)s(oundary)f(conditions)g(\(Diric)m(hlet)d(or)j (Neumann\))g(on)456 4979 y Fs(@)5 b(B)586 4994 y Fn(R)690 4979 y Fu(and)45 b Fs(@)5 b(B)1022 4994 y Fn(r)1106 4979 y Fu(corresp)s(ondingly)-8 b(.)81 b(W)-8 b(e)45 b(put)h(also)2485 4954 y(~)2459 4979 y Fs(A)2532 4932 y Fn(")2565 4941 y Ff(1)2600 4932 y Fn(;")2653 4941 y Ff(2)2532 5003 y Fn(R;r)2740 4979 y Fu(=)j Fs(A)2938 4932 y Fn(")2971 4941 y Ff(1)3005 4932 y Fn(;")3058 4941 y Ff(2)2938 5003 y Fn(R;r)3127 4979 y Fu(+)30 b(\()p Fs(b)3312 4943 y Fo(0)3367 4979 y Fr(\000)456 5099 y Fs(a)507 5063 y Fo(0)530 5099 y Fu(\)\011)644 5114 y Fn(r)682 5099 y Fu(\005)755 5114 y Fn(r)793 5099 y Fu(\011)869 5114 y Fn(r)907 5099 y Fs(;)46 b(r)30 b(>)d(R)i(>)f Fu(0)p Fs(:)h Fu(The)h(next)g(statemen)m (t)g(is)e(con)m(tained)i(in)e(Lemma)g(3.2)456 5216 y(and)k(Lemma)f(4.2) i(of)f([2].)p eop %%Page: 17 17 17 16 bop 1513 236 a Fl(DISCRETE)33 b(SPECTR)n(UM)981 b(17)594 425 y Fw(Prop)s(osition)35 b(4.3.)49 b Fh(L)-5 b(et)60 b Fu(~)-66 b Fs(r)1642 440 y Fq(0)1723 425 y Fh(b)-5 b(e)41 b(as)h(in)f(Pr)-5 b(op)g(osition)42 b(4.1)f(and)g Fs(\025)g Fr(2)g Fu(\()p Fs(a;)17 b(b)p Fu(\))p Fs(:)456 541 y Fh(Then)34 b(for)g Fs(r)d(>)45 b Fu(~)-67 b Fs(r)1087 556 y Fq(0)1127 541 y Fs(;)17 b Fu(1)27 b Fr(\024)h Fs(R)h Fr(\024)f Fs(r)s(=)p Fu(2)34 b Fh(the)h(fol)5 b(lowing)33 b(ine)-5 b(qualities)34 b(hold)781 715 y Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)1081 674 y Fq(\()p Fo(D)r Fq(\))1197 715 y Fu(\(0)p Fs(;)g(B)1402 730 y Fn(R)1459 715 y Fu(\)\))22 b(+)g Fs(N)10 b Fu(\()p Fs(\025;)1908 689 y Fu(~)1882 715 y Fs(A)1955 667 y Fo(D)r Fn(;)p Fo(D)1955 739 y Fn(R;r)2092 715 y Fu(\))28 b Fr(\025)g Fs(N)10 b Fu(\()p Fs(\025;)2515 689 y Fu(~)2490 715 y Fs(A)2563 730 y Fn(r)2601 715 y Fu(\))22 b Fr(\000)h Fs(C)2831 730 y Fq(1)2870 715 y Fs(R)2945 674 y Fn(d)p Fo(\000)p Fq(1)3075 715 y Fs(;)2637 866 y(C)2707 881 y Fq(1)2774 866 y Fu(=)k Fs(C)2947 881 y Fq(1)2986 866 y Fu(\()p Fs(\025)p Fu(\))3271 905 y(\(55\))771 1139 y Fs(N)10 b Fu(\()p Fs(\025;)17 b(A)1071 1098 y Fq(\()p Fo(N)10 b Fq(\))1197 1139 y Fu(\(0)p Fs(;)17 b(B)1402 1154 y Fn(R)1460 1139 y Fu(\)\))22 b(+)g Fs(N)10 b Fu(\()p Fs(\025;)1908 1114 y Fu(~)1883 1139 y Fs(A)1956 1092 y Fo(N)g Fn(;)p Fo(D)1956 1164 y Fn(R;r)2104 1139 y Fu(\))27 b Fr(\024)i Fs(N)10 b Fu(\()p Fs(\025;)2527 1114 y Fu(~)2502 1139 y Fs(A)2575 1154 y Fn(r)2613 1139 y Fu(\))22 b(+)g Fs(C)2841 1154 y Fq(2)2880 1139 y Fs(R)2955 1098 y Fn(d)p Fo(\000)p Fq(1)3086 1139 y Fs(;)2620 1290 y(C)2690 1305 y Fq(2)2757 1290 y Fu(=)27 b Fs(C)2930 1305 y Fq(2)2970 1290 y Fu(\()p Fs(\025)p Fu(\))p Fs(;)3271 1329 y Fu(\(56\))456 1493 y Fh(wher)-5 b(e)34 b Fs(C)801 1508 y Fq(1)875 1493 y Fh(and)g Fs(C)1134 1508 y Fq(2)1208 1493 y Fh(do)h(not)g(dep)-5 b(end)33 b(on)i Fs(r)i Fh(and)e Fs(R)q(:)555 1671 y Fw(2.)65 b Fu(No)m(w)41 b(let)e(us)h(complete)f (the)h(pro)s(of)f(of)h(Theorem)g(3.1.)64 b(First)39 b(of)h(all)d(w)m(e) 456 1788 y(establish)k(the)i(relation)d(\(30\))o(.)72 b(Let)42 b Fs(\013)1931 1751 y Fo(0)1998 1788 y Fs(>)h(\013)g Fu(and)f Fs(r)2465 1803 y Fq(0)2548 1788 y Fu(=)i(\()p Fs(v)2753 1803 y Fq(+)2812 1788 y Fs(=")p Fu(\))2945 1751 y Fq(1)p Fn(=s)3052 1788 y Fs(:)e Fu(W)-8 b(e)42 b(put)456 1907 y Fs(R)36 b Fu(=)f Fs(m\013)825 1871 y Fq(1)p Fn(=s)933 1907 y Fs(;)17 b(m)35 b(>)g(r)1252 1922 y Fq(0)1292 1907 y Fs(:)i Fu(W)-8 b(e)37 b(note)h(that)e Fs(\013)q(V)2085 1922 y Fq(0)2124 1907 y Fu(\()p Fs(x)p Fu(\))g Fs(<)f(")i Fu(for)g Fr(j)p Fs(x)p Fr(j)e Fs(>)g(R)q(:)i Fu(Therefore)456 2023 y(from)31 b(the)i(v)-5 b(ariational)29 b(principle)i(it)h(follo)m(ws)f(that)1134 2189 y Fs(N)10 b Fu(\()p Fs(\025;)1387 2164 y Fu(~)1361 2189 y Fs(A)1434 2204 y Fn(r)1472 2189 y Fu(\()p Fs(\013)1573 2148 y Fo(0)1596 2189 y Fu(\)\))27 b Fr(\024)i Fs(N)10 b Fu(\()p Fs(\025)22 b Fu(+)g Fs(";)17 b(A)p Fu(\()p Fs(\013)2372 2148 y Fo(0)2395 2189 y Fs(;)g(B)2513 2204 y Fn(R)2570 2189 y Fu(\)\))22 b(+)1753 2358 y Fs(N)10 b Fu(\()p Fs(\025)22 b Fu(+)g Fs(";)2171 2333 y Fu(~)2146 2358 y Fs(A)2219 2311 y Fo(N)10 b Fn(;)p Fo(D)2219 2383 y Fn(R;r)2366 2358 y Fu(\))p Fs(;)82 b(R)29 b(<)e(r)3271 2273 y Fu(\(57\))456 2528 y(Since)32 b Fs(\025)22 b Fu(+)g Fs(")28 b Fr(2)g Fu(\()p Fs(a;)17 b(b)p Fu(\),)33 b(w)m(e)g(obtain)f(from)f(\(56\))h(that)1860 2700 y Fs(N)10 b Fu(\()p Fs(\025;)2112 2675 y Fu(~)2087 2700 y Fs(A)2160 2715 y Fn(r)2198 2700 y Fu(\))27 b(=)h Fs(N)10 b Fu(\()p Fs(\025)22 b Fu(+)g Fs(";)2785 2675 y Fu(~)2760 2700 y Fs(A)2833 2715 y Fn(r)2871 2700 y Fu(\))27 b Fr(\025)858 2869 y Fs(N)10 b Fu(\()p Fs(\025)23 b Fu(+)f Fs(";)17 b(A)1325 2828 y Fq(\()p Fo(N)10 b Fq(\))1451 2869 y Fu(\(0)p Fs(;)17 b(B)1656 2884 y Fn(R)1713 2869 y Fu(\)\))22 b(+)g Fs(N)10 b Fu(\()p Fs(\025)23 b Fu(+)f Fs(";)2328 2844 y Fu(~)2303 2869 y Fs(A)2376 2822 y Fo(N)10 b Fn(;)p Fo(D)2376 2893 y Fn(R;r)2523 2869 y Fu(\))22 b Fr(\000)h Fs(C)2753 2884 y Fq(2)2792 2869 y Fs(R)2867 2828 y Fn(d)p Fo(\000)p Fq(1)2998 2869 y Fs(;)1484 3020 y(C)1554 3035 y Fq(2)1621 3020 y Fu(=)k Fs(C)1794 3035 y Fq(2)1833 3020 y Fu(\()p Fs(\025)22 b Fu(+)g Fs(")p Fu(\))p Fs(;)17 b(r)30 b(>)i Fu(~)-53 b Fs(r)2398 3035 y Fq(0)2437 3020 y Fs(;)17 b Fu(1)27 b Fr(\024)h Fs(R)h Fr(\024)f Fs(r)s(=)p Fu(2)p Fs(:)3271 2859 y Fu(\(58\))456 3189 y(Estimating)34 b(the)j(righ)m(t-hand)f(side)h(of)43 b(\(54\))36 b(with)h(the)g(help)g(of)43 b(\(57\))o(,)38 b(\(58\))e(w)m(e)456 3306 y(get)1278 3472 y Fs(N)1356 3487 y Fq(0)1395 3472 y Fu(\()p Fs(\025;)17 b(A)p Fu(\()p Fs(\013)q Fu(\)\))27 b Fr(\024)h Fs(N)10 b Fu(\()p Fs(\025)23 b Fu(+)f Fs(";)17 b(A)2383 3431 y Fq(\()p Fo(N)10 b Fq(\))2509 3472 y Fu(\()p Fs(\013)2610 3431 y Fo(0)2633 3472 y Fs(;)17 b(B)2751 3487 y Fn(R)2809 3472 y Fu(\)\))22 b Fr(\000)894 3639 y Fs(N)10 b Fu(\()p Fs(\025)22 b Fu(+)g Fs(";)17 b(A)1360 3598 y Fq(\()p Fo(N)10 b Fq(\))1486 3639 y Fu(\(0)p Fs(;)17 b(B)1691 3654 y Fn(R)1749 3639 y Fu(\)\))22 b(+)g Fs(C)2015 3654 y Fq(2)2054 3639 y Fs(R)2129 3598 y Fn(d)p Fo(\000)p Fq(1)2260 3639 y Fs(;)49 b(R)29 b Fu(=)e Fs(m\013)2690 3598 y Fq(1)p Fn(=s)2825 3639 y Fr(\025)h Fu(1)p Fs(:)3271 3554 y Fu(\(59\))456 3807 y(T)-8 b(aking)32 b Fs(\013)844 3771 y Fo(0)894 3807 y Fr(!)c Fs(\013)33 b Fu(w)m(e)h(come)e(to)g(the)h (upp)s(er)g(estimate)936 3973 y Fs(N)1014 3988 y Fq(0)1054 3973 y Fu(\()p Fs(\025;)17 b(A)p Fu(\()p Fs(E)6 b(\013)q Fu(\)\))27 b Fr(\024)h Fs(N)10 b Fu(\(\()p Fs(\025)22 b Fu(+)g Fs(")p Fu(\))g(+)g(0)p Fs(;)17 b(A)2364 3932 y Fq(\()p Fo(N)10 b Fq(\))2490 3973 y Fu(\()p Fs(\013)q(;)17 b(B)2709 3988 y Fn(R)2766 3973 y Fu(\)\))22 b Fr(\000)1033 4140 y Fs(N)10 b Fu(\()p Fs(\025)23 b Fu(+)f Fs(";)17 b(A)1500 4099 y Fq(\()p Fo(N)10 b Fq(\))1626 4140 y Fu(\(0)p Fs(;)17 b(B)1831 4155 y Fn(R)1888 4140 y Fu(\)\))22 b(+)g Fs(C)2154 4155 y Fq(2)2193 4140 y Fs(R)2268 4099 y Fn(d)p Fo(\000)p Fq(1)2399 4140 y Fs(;)49 b(R)29 b Fu(=)e Fs(m\013)2829 4099 y Fq(1)p Fn(=s)2937 4140 y Fs(;)3271 4055 y Fu(\(60\))456 4310 y(whic)m(h)35 b(coincides)f(with)h(\(30\))f(for)g Fs(r)g Fu(=)d Fs(m)k Fu(and)g Fs(c)c Fu(=)g Fs(C)2482 4325 y Fq(2)2522 4310 y Fs(:)j Fu(Moreo)m(v)m(er)j(\(60\))d(holds)456 4426 y(for)e(all)f Fs(m)e(>)f(r)1003 4441 y Fq(0)1043 4426 y Fs(;)17 b(\013)29 b(>)f(\013)1344 4441 y Fq(0)1412 4426 y Fu(=)g Fs(r)1563 4385 y Fo(\000)p Fn(s)1560 4448 y Fq(0)1655 4426 y Fs(:)33 b Fu(The)h(relation)d(\(31\))h(can)i(b)s(e)f (obtained)g(in)f(the)456 4542 y(same)g(w)m(a)m(y)i(from)d(\(53\))h(and) h(\(55\))o(.)44 b Fe(2)1660 4759 y Ft(References)497 4917 y Fv([1])d(Alama,)35 b(S.,)g(Deift,)h(P)-7 b(.A.,)36 b(Hemp)r(el,)f(R.:)49 b(Eigen)n(v)-5 b(alue)33 b(branc)n(hes)f(of)h (the)h(Sc)n(hr\177)-42 b(odinger)626 5016 y(op)r(erator)20 b Fd(H)12 b Fc(\000)6 b Fd(\025W)33 b Fv(in)21 b(a)g(gap)g(of)g Fd(\033)s Fv(\()p Fd(H)7 b Fv(\).)22 b Fb(Commun.)i(Math.)h(Phys.)e Fa(121)p Fv(,)f(291-321)c(\(1989\))497 5116 y([2])41 b(Hemp)r(el,)31 b(R.:)40 b(Eigen)n(v)-5 b(alues)27 b(in)j(gaps)e(and)h (decoupling)g(b)n(y)f(Neumann)i(b)r(oundary)e(con-)626 5216 y(ditions.)g Fb(J.)i(Math.)h(A)n(nal.)f(Appl.)f Fa(169)p Fv(,)e(no.)g(1,)g(229-259)e(\(1992\))p eop %%Page: 18 18 18 17 bop 456 236 a Fl(18)1142 b(O.)34 b(SAFR)n(ONO)n(V)497 425 y Fv([3])41 b(Sob)r(olev,)26 b(A.:)36 b(W)-7 b(eyl)25 b(asymptotics)g(for)g(the)g(discrete)g(sp)r(ectrum)g(of)h(the)f(p)r (erturb)r(ed)h(Hill)626 525 y(op)r(erator.)g Fb(A)l(dv.)k(Sov.)h(Math.) e Fa(7)p Fv(,)e(159-178,)e(\(1991\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0008240448471--