Content-Type: multipart/mixed; boundary="-------------9906160509314" This is a multi-part message in MIME format. ---------------9906160509314 Content-Type: text/plain; name="99-227.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-227.keywords" Periodic weighted operator, inverse problem, vertical slit mapping ---------------9906160509314 Content-Type: application/postscript; name="preprint387.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="preprint387.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: ihw.dvi %%CreationDate: Sun Jun 13 12:54:08 1999 %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: C:\emtex\bin\dvips32.exe ihw -o ihw.ps -D 300 -O %+ 0mm,0mm %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.06.13:1254 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N 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Fo(f)871 1571 y Fm(0)883 1592 y Fq(\))902 1571 y Fm(0)927 1592 y Fq(=)g Fn(\000)p Fo(f)1047 1571 y Fm(00)1079 1592 y Fn(\000)d Fq(2)p Fo(q)r(f)1206 1571 y Fm(0)1218 1592 y Fo(;)122 b(q)15 b Fn(\021)1449 1558 y Fo(\032)1474 1540 y Fm(0)p 1449 1580 37 2 v 1454 1626 a Fo(\032)1490 1592 y(;)390 b Fq(\(1.1\))46 1722 y(in)21 b(the)f(Hilb)q(ert)g(space)h Fo(L)537 1704 y Fp(2)557 1722 y Fq(\()p Fl(R)p Fo(;)7 b(\032)p Fq(\()p Fo(x)p Fq(\))727 1704 y Fp(2)744 1722 y Fo(dx)p Fq(\))21 b(with)g(the)g(norm)f Fn(k)p Fo(f)5 b Fn(k)1253 1704 y Fp(2)1253 1736 y Fk(\032)1271 1727 y Fh(2)1313 1722 y Fq(=)1372 1687 y Fj(R)1400 1700 y Fp(1)1392 1735 y(0)1428 1722 y Fo(\032)1453 1704 y Fp(2)1473 1722 y Fq(\()p Fo(x)p Fq(\))p Fn(j)p Fo(f)g Fq(\()p Fo(x)p Fq(\))p Fn(j)1662 1704 y Fp(2)1681 1722 y Fo(dx;)21 b Fq(where)f Fo(\032)i Fn(2)46 1783 y Fo(W)99 1764 y Fp(2)92 1795 y(1)118 1783 y Fq(\()p Fl(R)p Fo(=)p Fl(Z)-13 b Fq(\))16 b(is)k(a)g(1-p)q(erio)q(dic)g(real)f(p)q(ositiv)o(e)f (function,)i(i.e.)30 b Fo(\032)p Fq(\()p Fo(x)p Fq(\))19 b Fo(>)g(\032)1373 1790 y Fp(0)1393 1783 y Fo(;)8 b(x)19 b Fn(2)h Fq([0)p Fo(;)8 b Fq(1])19 b(for)h(some)e Fo(\032)1860 1790 y Fp(0)1899 1783 y Fo(>)i Fq(0.)46 1843 y(Without)j(loss)h (generalit)o(y)l(,)f(assume)g Fo(\032)p Fq(\(0\))j(=)g(1.)43 b(Here)22 b(and)i(b)q(elo)o(w)f(w)o(e)g(use)g(the)g(notation)h(\()1900 1825 y Fm(0)1912 1843 y Fq(\))i(=)46 1903 y Fo(@)s(=@)s(x;)71 b(@)25 b Fq(=)e Fo(@)s(=@)s(q)r(:)d Fq(It)h(is)g(w)o(ell)f(kno)o(wn)i (\(see)f([L],)f([Kr]\))h(that)h(the)f(sp)q(ectrum)f(of)i Fo(T)28 b Fq(is)21 b(absolutely)46 1963 y(con)o(tin)o(uous)d(and)h (consists)g(of)f(in)o(terv)m(als)g Fo(\033)854 1970 y Fk(n)895 1963 y Fq(=)f([)p Fo(\025)992 1942 y Fp(+)992 1974 y Fk(n)p Fm(\000)p Fp(1)1060 1963 y Fo(;)8 b(\025)1110 1945 y Fm(\000)1110 1975 y Fk(n)1140 1963 y Fq(])p Fo(;)18 b Fq(where)g Fo(\025)1357 1942 y Fp(+)1357 1974 y Fk(n)p Fm(\000)p Fp(1)1443 1963 y Fg(6)f Fo(\025)1527 1945 y Fm(\000)1527 1975 y Fk(n)1575 1963 y Fg(6)g Fo(\025)1659 1945 y Fp(+)1659 1975 y Fk(n)1689 1963 y Fo(;)8 b(n)17 b Fn(\025)g Fq(1)p Fo(:)i Fq(These)46 2023 y(in)o(terv)m(als)e(are)i (separated)g(b)o(y)f(the)g(gaps)i Fo(\015)844 2030 y Fk(n)885 2023 y Fq(=)e(\()p Fo(\025)988 2005 y Fm(\000)988 2036 y Fk(n)1018 2023 y Fo(;)8 b(\025)1068 2005 y Fp(+)1068 2036 y Fk(n)1098 2023 y Fq(\))p Fo(;)18 b Fq(with)g(the)g(length)g Fn(j)p Fo(\015)1538 2030 y Fk(n)1562 2023 y Fn(j)g(\025)f Fq(0)p Fo(:)h Fq(If)g(a)h(gap)h Fo(\015)1920 2030 y Fk(n)1962 2023 y Fq(is)46 2083 y(degenerate,)f(i.e.)30 b Fn(j)p Fo(\015)440 2090 y Fk(n)464 2083 y Fn(j)19 b Fq(=)h(0,)g(then)f(the)h (corresp)q(onding)g(segmen)o(ts)e Fo(\033)1370 2090 y Fk(n)1393 2083 y Fo(;)8 b(\033)1443 2090 y Fk(n)p Fp(+1)1531 2083 y Fq(merge.)30 b(In)19 b(analogy)h(to)46 2144 y(the)15 b(notation)h Fo(O)q Fq(\(1)p Fo(=n)p Fq(\))h(w)o(e)d(use)i(the)f (notation)h Fo(`)944 2126 y Fk(d)964 2144 y Fq(\()p Fo(n)p Fq(\))p Fo(;)8 b(d)15 b Fn(\025)e Fq(1,)i(for)h(an)g(arbitrary)f (sequence)f(of)i(n)o(um)o(b)q(ers,)46 2204 y(whic)o(h)f(is)h(an)h (elemen)o(t)c(of)k Fo(`)558 2186 y Fk(d)595 2204 y Fq(\(see)f([PT]\).)f (F)l(or)h(instance,)310 2303 y Fo(a)336 2310 y Fk(n)373 2303 y Fq(=)e Fo(b)446 2310 y Fk(n)480 2303 y Fq(+)d Fo(`)549 2283 y Fk(d)570 2303 y Fq(\()p Fo(n)p Fq(\))48 b(is)16 b(equiv)m(alen)o(t)f(to)49 b Fo(a)1084 2310 y Fk(n)1121 2303 y Fq(=)14 b Fo(b)1194 2310 y Fk(n)1228 2303 y Fq(+)d Fo(c)1298 2310 y Fk(n)1322 2303 y Fo(;)1441 2262 y Fj(X)1509 2303 y Fn(j)p Fo(c)1544 2310 y Fk(n)1568 2303 y Fn(j)1582 2283 y Fk(d)1616 2303 y Fo(<)i Fn(1)p Fo(:)46 2402 y Fq(Let)j(us)h(recall)e(author's)i(result)f([K1])281 2525 y Fo(z)306 2504 y Fm(\006)304 2537 y Fk(n)349 2525 y Fn(\021)402 2474 y Fj(q)p 444 2474 58 2 v 51 x Fo(\025)472 2510 y Fm(\006)472 2537 y Fk(n)515 2525 y Fq(=)e Fo(\031)r(n)d Fn(\006)g(j)709 2466 y Fj(Z)750 2479 y Fp(1)732 2560 y(0)778 2525 y Fo(e)801 2504 y Fk(i)p Fp(2)p Fk(\031)q(nx)895 2525 y Fo(q)r Fq(\()p Fo(x)p Fq(\))p Fo(dx)p Fn(j)f Fq(+)h Fo(`)1131 2504 y Fk(d)1152 2525 y Fq(\()p Fo(n)p Fq(\))p Fo(;)40 b(d)15 b(>)e Fq(1)p Fo(;)74 b(as)64 b(n)14 b Fn(!)g(1)p Fo(;)134 b Fq(\(1.2\))p 46 2591 780 2 v 102 2622 a Ff(1)121 2637 y Fe(P)o(artially)13 b(supp)q(orted)j(b)o(y)f (Russian)f(F)m(und)h(of)f(F)m(undamen)o(tal)f(Researc)o(h,)j(INT)m(AS)e (and)h(Gran)o(t)f(for)h(the)g(co)q(op)q(eration)46 2687 y(b)q(et)o(w)o(een)g(Sw)o(eden)g(and)f(the)g(former)f(So)o(viet)g (Union)h(no)f(12528)1008 2811 y Fq(1)p eop %%Page: 2 2 2 1 bop 46 138 a Fq(and)12 b(these)g(asymptotics)f(giv)o(e)g(the)g (estimates)g(of)h(gap)h(lengths.)20 b(Roughly)12 b(sp)q(eaking)g(these) g(asymptotics)46 199 y(giv)o(e)22 b(the)h(main)f(di\013erence)g(b)q(et) o(w)o(een)g(the)h(sp)q(ectral)g(prop)q(erties)g(of)g(the)g(op)q(erator) h Fo(T)30 b Fq(and)24 b(the)e(Hill)46 259 y(op)q(erator:)f(the)14 b(gap)h(length)e Fn(j)p Fo(\015)617 266 y Fk(n)641 259 y Fn(j)h(!)f(1)h Fq(as)g Fo(n)g Fn(!)g(1)g Fq(for)g Fo(T)20 b Fq(while)13 b(the)h(gap)h(length)e(of)i(the)e(Hill)f(op)q(erator)46 319 y(tends)22 b(to)h(zero)f(at)h(large)f Fo(n)p Fq(.)39 b(In)o(tro)q(duce)22 b(the)g(fundamen)o(tal)f(solutions)i Fo(')p Fq(\()p Fo(x;)8 b(z)r(;)g(q)r Fq(\))p Fo(;)g(#)p Fq(\()p Fo(x;)g(z)r(;)g(q)r Fq(\))19 b(of)k(the)46 379 y(follo)o(wing)16 b(equation)742 439 y Fn(\000)p Fo(y)807 419 y Fm(00)838 439 y Fn(\000)11 b Fq(2)p Fo(q)r(y)962 419 y Fm(0)987 439 y Fq(=)j Fo(z)1064 419 y Fp(2)1083 439 y Fo(y)r(;)40 b(z)16 b Fn(2)e Fl(C)9 b Fo(;)598 b Fq(\(1.3\))46 519 y(with)18 b(the)g(conditions)g Fo(')p Fq(\(0)p Fo(;)8 b(z)r(;)g(q)r Fq(\))16 b(=)h Fo(#)767 501 y Fm(0)779 519 y Fq(\(0)p Fo(;)8 b(z)r(;)g(q)r Fq(\))16 b(=)h(0)p Fo(;)44 b(')1119 501 y Fm(0)1131 519 y Fq(\(0)p Fo(;)8 b(z)r(;)g(q)r Fq(\))16 b(=)h Fo(#)p Fq(\(0)p Fo(;)8 b(z)r(;)g(q)r Fq(\))16 b(=)h(1)p Fo(;)h Fq(here)g(and)h(b)q(elo)o(w)46 579 y(\()81 561 y Fm(0)93 579 y Fq(\))14 b(=)f Fo(@)s(=@)s(x;)8 b Fq(\()h(_)g(\))k(=)h Fo(@)s(=@)s(z)r(;)8 b(@)16 b Fq(=)d Fo(@)s(=@)s(q)r Fq(.)20 b(The)d(Ly)o(apuno)o(v)f(function)g(is)g (de\014ned)g(b)o(y)650 694 y(\001\()p Fo(z)r(;)8 b(q)r Fq(\))k(=)869 661 y(1)p 869 683 25 2 v 869 729 a(2)898 694 y(\()p Fo(')949 674 y Fm(0)961 694 y Fq(\(1)p Fo(;)c(z)r(;)g(q)r Fq(\))i(+)h Fo(#)p Fq(\(1)p Fo(;)d(z)r(;)g(q)r Fq(\)\))p Fo(:)502 b Fq(\(1.4\))46 806 y(The)22 b(sequence)g Fo(\025)389 785 y Fp(+)389 817 y(0)443 806 y Fo(<)47 b(\025)556 785 y Fm(\000)556 817 y Fp(1)611 806 y Fg(6)f Fo(\025)724 785 y Fp(+)724 817 y(1)801 806 y Fo(<)25 b(:::::)20 b Fq(is)i(the)h(sp)q(ectrum)e(of)i(Eq.)40 b(\(1.3\))22 b(with)h(the)f(p)q(erio)q(dic)46 866 y(b)q(oundary)f(conditions)g(of)f (p)q(erio)q(d)h(2,)g(i.e.)32 b Fo(f)5 b Fq(\()p Fo(x)14 b Fq(+)g(2\))21 b(=)f Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(;)j(x)20 b Fn(2)h Fl(R)p Fq(.)30 b(Here)19 b(equalit)o(y)g(means)g (that)46 926 y Fo(\025)74 908 y Fm(\000)74 939 y Fk(n)130 926 y Fq(=)26 b Fo(\025)222 908 y Fp(+)222 939 y Fk(n)275 926 y Fq(is)e(a)g(double)f(eigen)o(v)m(alue.)41 b(W)l(e)24 b(note)f(that)h(\001\()p Fo(z)1222 908 y Fm(\006)1220 939 y Fk(n)1251 926 y Fo(;)8 b(q)r Fq(\))26 b(=)g(\()p Fn(\000)p Fq(1\))1507 908 y Fk(n)1530 926 y Fo(;)31 b(n)c Fn(\025)e Fq(1)p Fo(:)f Fq(The)f(lo)o(w)o(est)46 987 y(eigen)o(v)m(alue)18 b Fo(\025)310 966 y Fp(+)310 998 y(0)359 987 y Fq(=)i(0)g(is)f(simple,)e(\001\(0)p Fo(;)8 b(q)r Fq(\))19 b(=)g(1)p Fo(;)h Fq(and)g(the)f(corresp)q(onding)h (eigenfunction)f(has)h(p)q(erio)q(d)46 1047 y(1.)42 b(The)23 b(eigenfunctions)f(corresp)q(onding)j(to)e Fo(\025)973 1029 y Fm(\006)973 1059 y Fk(n)1026 1047 y Fq(ha)o(v)o(e)f(p)q(erio)q (d)i(1)f(when)g Fo(n)h Fq(is)f(ev)o(en)e(and)j(they)f(are)46 1107 y(an)o(tip)q(erio)q(dic,)13 b Fo(f)5 b Fq(\()p Fo(x)i Fq(+)g(1\))14 b(=)g Fn(\000)p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fo(;)36 b(x)14 b Fn(2)g Fl(R)p Fq(,)d(when)j Fo(n)h Fq(is)f(o)q(dd.)21 b(F)l(or)15 b(eac)o(h)e Fo(n)h Fn(\025)g Fq(1)h(there)e(exists)h(a)g(unique)46 1167 y(p)q(oin)o(t)i Fo(z)196 1174 y Fk(n)233 1167 y Fn(2)e Fq([)p Fo(z)319 1149 y Fm(\000)317 1180 y Fk(n)348 1167 y Fo(;)8 b(z)395 1149 y Fp(+)393 1180 y Fk(n)424 1167 y Fq(])16 b(suc)o(h)g(that)683 1155 y(_)670 1167 y(\001\()p Fo(z)753 1174 y Fk(n)776 1167 y Fo(;)8 b(q)r Fq(\))13 b(=)h(0.)119 1227 y(Let)23 b Fo(\026)242 1234 y Fk(n)265 1227 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(n)24 b Fn(\025)h Fq(1)p Fo(;)d Fq(b)q(e)h(the)f(Diric)o(hlet)e(sp)q(ectrum)i(of)g Fo(q)r Fq(,)h(that)g(is)g(the)f(sp)q(ectrum)f(of)i(\(1.3\))g(with)46 1297 y(b)q(oundary)d(condition)f Fo(y)r Fq(\(0\))f(=)h Fo(y)r Fq(\(1\))f(=)g(0,)i(and)f(let)g Fo(m)1082 1304 y Fk(n)1105 1297 y Fq(\()p Fo(q)r Fq(\))f(=)1241 1247 y Fj(q)p 1282 1247 115 2 v 1282 1297 a Fo(\026)1311 1304 y Fk(n)1335 1297 y Fq(\()p Fo(q)r Fq(\))g Fo(>)g Fq(0.)30 b(It)19 b(is)f(w)o(ell)g(kno)o(wn)h(that)46 1363 y Fo(m)89 1370 y Fk(n)112 1363 y Fq(\()p Fo(q)r Fq(\))13 b Fn(2)h Fq([)p Fo(z)273 1345 y Fm(\000)271 1375 y Fk(n)302 1363 y Fo(;)8 b(z)349 1345 y Fp(+)347 1375 y Fk(n)378 1363 y Fq(])p Fo(;)g(n)14 b Fn(\025)f Fq(1)p Fo(:)119 1423 y Fq(There)20 b(are)g(v)m(arious)h(metho)q(ds)f(of)h(solving)f(in)o(v)o (erse)f(problems.)32 b(W)l(e)20 b(shortly)g(describ)q(e)g(a)g("direct) 46 1483 y(approac)o(h")h(from)d([KK1],)i(based)g(on)g(a)h(theorem)d (from)g(nonlinear)i(functional)f(analysis.)32 b(W)l(e)20 b(recall)46 1543 y(de\014nitions.)33 b(Supp)q(ose)22 b(that)e Fo(H)q(;)8 b(H)721 1550 y Fp(1)762 1543 y Fq(are)21 b(real)f(separable)g(Hilb)q(ert)f(spaces)i(with)f(norms)g Fn(k)14 b(\001)f(k)p Fo(;)8 b Fn(k)14 b(\001)g(k)1962 1550 y Fp(1)1981 1543 y Fo(:)46 1603 y Fq(The)19 b(deriv)m(ativ)o(e)f (of)i(a)g(map)e Fo(f)25 b Fq(:)19 b Fo(H)24 b Fn(!)19 b Fo(H)845 1610 y Fp(1)884 1603 y Fq(at)h(a)g(p)q(oin)o(t)g Fo(y)h Fn(2)e Fo(H)24 b Fq(is)19 b(a)h(b)q(ounded)h(linear)d(map)h (from)f Fo(H)46 1664 y Fq(in)o(to)f Fo(H)186 1671 y Fp(1)207 1664 y Fq(,)h(whic)o(h)f(w)o(e)g(denote)h(b)o(y)g Fo(d)707 1671 y Fk(y)728 1664 y Fo(f)5 b Fq(.)26 b(A)18 b(map)f Fo(f)22 b Fq(:)17 b Fo(H)k Fn(!)16 b Fo(H)1206 1671 y Fp(1)1244 1664 y Fq(is)i(compact)f(on)h Fo(H)k Fq(if)c(it)f(maps)h(w)o (eakly)46 1724 y(con)o(v)o(ergen)o(t)d(sequences)g(in)h Fo(H)k Fq(in)o(to)c(strongly)g(con)o(v)o(ergen)o(t)f(sequences)g(in)h Fo(H)1477 1731 y Fp(1)1497 1724 y Fq(.)21 b(A)16 b(map)f Fo(f)k Fq(:)14 b Fo(H)k Fn(!)c Fo(H)1926 1731 y Fp(1)1962 1724 y Fq(is)46 1784 y(a)j(real)f(analytic)g(isomorphism)e(b)q(et)o(w)o (een)i Fo(H)21 b Fq(and)c Fo(H)1038 1791 y Fp(1)1075 1784 y Fq(if)f Fo(f)22 b Fq(is)16 b(one-to-one)i(and)f(on)o(to)g(and)h (b)q(oth)f Fo(f)22 b Fq(and)46 1844 y Fo(f)75 1826 y Fm(\000)p Fp(1)139 1844 y Fq(are)17 b(real)f(analytic)g(maps)g(of)h (the)f(Hilb)q(ert)f(space.)22 b(Let)17 b Fo(H)1216 1851 y Fk(C)1263 1844 y Fq(b)q(e)f(the)h(complexi\014cation)d(of)j(the)f (real)46 1904 y(Hilb)q(ert)f(space)h Fo(H)21 b Fq(with)16 b(the)g(norm)f Fn(k)c(\001)g(k)813 1911 y Fk(C)119 1965 y Fq(W)l(e)h(form)o(ulate)e(a)j(new)f(v)o(ersion)f(of)i(a)f("basic)h (theorem")e(of)h(the)g(direct)f(metho)q(d)g(\(see)h([KK1],)g([KK2]\).) 46 2025 y Fr(Theorem)j(A.)f Fd(L)n(et)i Fo(H)q(;)8 b(H)540 2032 y Fp(1)577 2025 y Fd(b)n(e)16 b(r)n(e)n(al)g(sep)n(ar)n(able)g (Hilb)n(ert)g(sp)n(ac)n(es)g(with)g(norms)g Fn(k)8 b(\001)g(k)p Fo(;)g Fn(k)g(\001)g(k)1672 2032 y Fp(1)1691 2025 y Fd(.)22 b(Supp)n(ose)16 b(that)46 2085 y(a)h(map)g Fo(f)i Fq(:)14 b Fo(H)k Fn(!)c Fo(H)428 2092 y Fp(1)465 2085 y Fd(satis\014es)k(the)g (fol)r(lowing)i(c)n(onditions:)46 2145 y(i\))d Fo(f)23 b Fd(is)17 b(r)n(e)n(al)g(analytic,)46 2205 y(ii\))g(the)h(op)n(er)n (ator)e Fo(d)411 2212 y Fk(q)430 2205 y Fo(f)23 b Fd(has)17 b(an)h(inverse)g(for)f(al)r(l)i Fo(q)c Fn(2)f Fo(H)t Fd(,)46 2266 y(iii\))25 b(ther)n(e)h(is)f(a)h(nonde)n(cr)n(e)n(asing)g (function)h Fo(g)k Fq(:)d([0)p Fo(;)8 b Fn(1)p Fq(\))29 b Fn(!)g Fq([0)p Fo(;)8 b Fn(1)p Fq(\))p Fo(;)g(g)r Fq(\(0\))29 b(=)f(0)p Fo(;)e Fd(such)g(that)g Fn(k)p Fo(q)r Fn(k)i Fg(6)46 2326 y Fo(g)r Fq(\()p Fn(k)p Fo(f)5 b Fq(\()p Fo(q)r Fq(\))p Fn(k)231 2333 y Fp(1)250 2326 y Fq(\))18 b Fd(for)e(al)r(l)j Fo(q)c Fn(2)f 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2302 y Fq(\))17 b Fd(for)g(e)n(ach)h(\014xe)n(d)g Fo(p)c Fn(2)g Fo(H)t Fd(.)46 2362 y(R)n(emark.)22 b Fq(i\))16 b(Recall)f(that)i(in)f(analogy)i(to)f(the)f(notation)i Fo(O)q Fq(\(1)p Fo(=n)p Fq(\))g(w)o(e)e(use)g(the)h(notation)g Fo(`)1754 2344 y Fk(d)1775 2362 y Fq(\()p Fo(n)p Fq(\))p Fo(;)8 b(d)15 b Fn(\025)f Fq(1.)46 2423 y(F)l(or)g(example,)e(\(1.10\)) j(means)e(that)i(there)f(exists)g(a)g(sequence)f Fo(\014)k Fq(=)c Fn(f)p Fo(\014)1367 2430 y Fk(n)1390 2423 y Fq(\()p Fo(q)r Fq(\))p Fn(g)g(2)h Fo(`)1557 2405 y Fp(2)1592 2423 y Fq(suc)o(h)g(that)h Fn(k)p Fo(\014)s Fn(k)1885 2430 y Fk(d)1918 2423 y Fg(6)e Fq(1)46 2483 y(and)380 2543 y(\()p Fo(d)424 2550 y Fk(q)444 2543 y Fo(h)472 2550 y Fk(n)495 2543 y Fq(\()p Fo(q)r Fq(\)\)\()p Fo(x)p Fq(\))g(=)h(\(cos)8 b(2)p Fo(\031)r(nx;)g Fq(sin)g(2)p Fo(\031)r(nx)p Fq(\))j(+)g Fo(\014)1218 2550 y Fk(n)1241 2543 y Fq(\()p Fo(q)r Fq(\))p Fo(O)q Fq(\(1\))p Fo(;)73 b(n)14 b Fn(!)g(1)p Fo(;)46 2627 y Fq(uniformly)h(on)j([0)p Fo(;)8 b Fq(1])j Fn(\002)g Fo(B)534 2634 y Fk(C)564 2627 y Fq(\()p Fo(p;)d(")652 2634 y Fk(p)672 2627 y Fq(\))17 b(for)h(eac)o(h)e(\014xed)h Fo(p)f Fn(2)g Fo(H)t Fq(.)24 b(ii\))16 b(The)h(estimates)f(\(1.8\))h(w)o(ere)g(pro)o(v)o(ed)f(in)46 2687 y([K1],)e([K3].)20 b(iii\))13 b(In)h(order)g(to)h(pro)o(v)o(e)e (this)h(theorem)f(w)o(e)h(use)h(the)f(results)g(from)f([K2],)h(where)g (the)g(author)1008 2811 y(3)p eop %%Page: 4 4 4 3 bop 46 138 a Fq(solv)o(ed)20 b(the)g(same)g(problem)f(for)h(the)h (Hill)e(op)q(erator)i(\(after)g([MO1-2]\))f(b)o(y)g(the)g(direct)g (metho)q(d.)33 b(iv\))46 199 y(Roughly)19 b(sp)q(eaking)h(in)e(Theorem) g(1.1)i(the)e(result)h(of)g([MO1-2])g(is)g(extended)f(to)h(singular)h (p)q(oten)o(tials)46 259 y(\(distributions\),)f(see)g(form)o(ula)e (\(1.16\).)31 b(This)19 b(generalization)g(requires)f(essen)o(tally)g (di\013eren)o(t)g(pro)q(ofs,)46 319 y(since)g(the)h(tec)o(hnique)e (that)j(w)o(as)g(used)f(for)g(studing)h(the)f(Hill)e(op)q(erator)j (could)f(not)h(b)q(e)f(directly)e(gen-)46 379 y(eralized)f(to)h(the)g (w)o(eigh)o(ted)f(op)q(erator.)25 b(This)17 b(induced)f(us)i(to)f (reconsider)f(the)h(original)g(v)o(ersion)g(of)g(the)46 439 y(pro)q(of)f(\(Theorem)d(A\),)h(this)h(led)f(us)h(eviden)o(tlly)e (to)i(disco)o(v)o(er)e(simpli\014cations,)g(and)j(also)f(impro)o(v)o (em)o(en)n(ts)46 499 y(in)g(the)g(estimates.)20 b(The)15 b(pro)q(of)i(presen)o(ted)d(here)h(dep)q(ends)h(on)g(the)g(in)o(v)o (estigation)e(of)i(the)f(fundamen)o(tal)46 560 y(solutions)f(and)g (some)f(in)o(tegral)g(equation)h(\(see)f(\(2.2\)\))h(at)g(large)f Fn(j)p Fo(z)r Fn(j)p Fq(,)h(whic)o(h)f(seem)f(to)i(us)g(also)g(ha)o(v)o (e)f(inde-)46 620 y(p)q(enden)o(t)i(in)o(terest.)k(In)c(the)f(pro)q(of) i(of)g(asymptotics)e(estimates)f(there)h(exists)g(a)i(big)f (di\013erence)f(b)q(et)o(w)o(een)46 680 y(the)i(Hill)f(op)q(erator)j (and)f(the)g(w)o(eigh)o(ted)e(op)q(erator.)24 b(Roughly)17 b(sp)q(eaking)g(in)f(the)h(in)o(tegral)f(equation)g(for)46 740 y(the)21 b(fundamen)o(tal)f(solution)h(of)h(the)f(Hill)e(op)q (erator)k(there)e(exists)f(the)h(small)f(parameter)g(1)p Fo(=z)r Fq(!)37 b(But)46 800 y(suc)o(h)15 b(small)f(parameter)g(1)p Fo(=z)19 b Fq(is)c(absen)o(t)h(for)f(the)h(w)o(eigh)o(ted)e(op)q (erator)j(\(see)e(\(2.2\)\).)21 b(Some)15 b(asymptotics)46 861 y(estimates)f(w)o(as)j(found)g(in)f([K1],)f(but)i(the)f(main)f (part)h(of)h(needed)e(ones)i(is)f(pro)o(v)o(ed)g(here.)119 921 y(In)o(tro)q(duce)21 b(a)g(quasimomen)o(tum)c(domain)j Fo(K)26 b Fq(=)c Fl(C)k Fn(n)14 b([)p Fq(\000)1218 928 y Fk(n)1243 921 y Fq(,)21 b(where)g(\000)1454 928 y Fk(n)1500 921 y Fq(=)h(\()p Fo(\031)r(n)14 b Fq(+)g Fo(i)p Fn(j)p Fo(h)1763 928 y Fk(n)1786 921 y Fq(\()p Fo(q)r Fq(\))p Fn(j)p Fo(;)8 b(\031)r(n)13 b Fn(\000)46 981 y Fo(i)p Fn(j)p Fo(h)105 988 y Fk(n)128 981 y Fq(\()p Fo(q)r Fq(\))p Fn(j)k Fq(\))g(is)h(an)g(excised)e(slit)h(with)h(heigh)o(t)f Fn(j)p Fo(h)932 988 y Fk(n)956 981 y Fq(\()p Fo(q)r Fq(\))p Fn(j)e(\025)h Fq(0)p Fo(;)26 b(n)16 b Fn(\025)g Fq(1,)i(and)h Fo(h)1447 988 y Fp(0)1483 981 y Fq(=)d(0)p Fo(:)i Fq(Belo)o(w)e(w)o(e)i (need)f(the)46 1041 y(sp)q(ectral)h(domain)f Fn(Z)k Fq(=)16 b Fl(C)24 b Fn(n)12 b([)p Fo(g)658 1048 y Fk(n)700 1041 y Fq(where)18 b Fo(g)866 1048 y Fk(n)907 1041 y Fq(=)e(\()p Fo(z)1005 1023 y Fm(\000)1003 1053 y Fk(n)1034 1041 y Fo(;)8 b(z)1081 1023 y Fp(+)1079 1053 y Fk(n)1111 1041 y Fq(\))16 b(=)h Fn(\000)p Fo(g)1263 1048 y Fm(\000)p Fk(n)1314 1041 y Fo(;)8 b(n)17 b Fn(\025)f Fq(1)p Fo(;)i Fq(is)g(a)g(horizon)o(tal)g(slit)f(with)46 1110 y(length)f Fn(j)p Fo(g)232 1117 y Fk(n)256 1110 y Fn(j)d(\025)h Fq(0)i(and)h(w)o(e)f(recall)f Fo(z)698 1092 y Fm(\006)696 1122 y Fk(n)741 1110 y Fq(=)793 1061 y Fj(q)p 835 1061 58 2 v 49 x Fo(\025)863 1095 y Fm(\006)863 1122 y Fk(n)907 1110 y Fo(>)e Fq(0.)46 1178 y Fr(Corollary)23 b(1.2.)33 b Fd(F)l(or)21 b(e)n(ach)g Fo(h)h Fn(2)f Fo(`)753 1160 y Fp(2)787 1178 y Fn(\010)14 b Fo(`)860 1160 y Fp(2)901 1178 y Fd(ther)n(e)21 b(exist)i(a)e(unique)i(function)f Fo(q)h 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Fk(cn)1198 1676 y Fo(;)60 b(n)14 b Fn(\025)g Fq(1)p Fo(:)464 b Fq(\(1.14\))46 1763 y Fd(Pr)n(o)n(of.)25 b Fq(In)18 b(Theorem)f(1.1)h(w)o(e)g(pro)o(v)o(ed)f(that)i(for)f(eac)o (h)g Fo(h)f Fn(2)g Fo(`)1197 1745 y Fp(2)1230 1763 y Fn(\010)12 b Fo(`)1301 1745 y Fp(2)1339 1763 y Fq(there)18 b(exists)f(a)i(unique)e(p)q(oten)o(tial)46 1823 y Fo(q)g Fn(2)g Fo(H)k Fq(suc)o(h)d(that)g(the)f(iden)o(tities)f(\(1.6\))h(are)h (ful\014lled.)24 b(Moreo)o(v)o(er,)16 b(in)h([K1])g(the)g(author)i (constructed)46 1884 y(a)f(unique)f(conformal)f(mapping)h Fo(k)h Fq(:)d Fn(Z)21 b(!)15 b Fo(K)22 b Fq(with)17 b(the)h(needed)e (prop)q(erties)i(\(1.11-13\).)27 b(Then)17 b(using)46 1944 y(\(1.7\))f(w)o(e)g(obtain)h(\(1.14\).)22 b Fb(2)119 2004 y Fq(De\014ne)17 b(the)g(unitary)g(transformation)h Fo(U)j Fq(:)15 b Fo(L)979 1986 y Fp(2)999 2004 y Fq(\()p Fl(R)p Fo(;)7 b(\032)1103 1986 y Fp(2)1120 2004 y Fo(dx)p Fq(\))16 b Fn(!)g Fo(L)1307 1986 y Fp(2)1327 2004 y Fq(\()p Fl(R)p Fq(\))e(b)o(y)j(the)g(form)o(ula)f(\()p 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Fo(x)p Fq(\))1509 2462 y Fo(;)347 b Fq(\(1.16\))46 2597 y(and)17 b(hence)543 2668 y Fn(k)p Fo(V)11 b Fn(k)632 2647 y Fp(2)666 2668 y Fq(=)i Fn(k)p Fo(q)766 2647 y Fm(0)777 2668 y Fn(k)802 2647 y Fp(2)833 2668 y Fq(+)e Fn(k)p Fo(q)931 2647 y Fp(2)950 2668 y Fn(k)975 2647 y Fp(2)995 2668 y Fo(;)1098 2609 y Fj(Z)1139 2622 y Fp(2)1121 2704 y(0)1167 2668 y Fo(V)h Fq(\()p Fo(x)p Fq(\))p Fo(dx)h Fq(=)h Fn(k)p Fo(q)r Fn(k)1465 2647 y Fp(2)1484 2668 y Fo(:)372 b Fq(\(1.17\))1008 2811 y(4)p eop %%Page: 5 5 5 4 bop 46 138 a Fq(F)l(or)16 b Fo(q)i Fq(w)o(e)e(de\014ne)g(the)g(new) g(functions)784 134 y Fj(e)780 138 y Fo(q)r(;)831 134 y Fj(e)825 138 y Fo(\032)h Fq(and)g(the)f(constan)o(t)g Fo(N)1281 145 y Fk(q)1317 138 y Fq(b)o(y)f(the)h(follo)o(wing)g(form)o (ulas)76 275 y Fj(e)71 279 y Fo(\032)p Fq(\()p Fo(x)p Fq(\))d(=)h(exp)310 220 y Fj(Z)352 233 y Fk(x)333 314 y Fp(0)386 275 y Fj(e)382 279 y Fo(q)q Fq(\()p Fo(t)p Fq(\))p Fo(dt;)578 275 y Fj(e)574 279 y Fo(q)q Fq(\()p Fo(x)p Fq(\))g(=)743 225 y 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Fm(0)1076 595 y Fq(+)1130 591 y Fj(e)1125 595 y Fo(q)1149 575 y Fp(2)1168 595 y Fo(:)81 b(:)593 b Fq(\(1.19\))46 705 y(Then)456 718 y Fj(Z)498 731 y Fp(1)479 812 y(0)526 776 y Fn(j)548 760 y Fj(e)540 776 y Fo(V)11 b Fq(\()p Fo(x)p Fq(\))p Fn(j)p Fo(dx)i Fg(6)778 718 y Fj(Z)819 731 y Fp(1)801 812 y(0)847 776 y Fn(j)865 772 y Fj(e)861 776 y Fo(q)q Fq(\()p Fo(x)p Fq(\))950 756 y Fm(0)962 776 y Fn(j)p Fo(dx)e Fq(+)g Fn(k)1118 772 y Fj(e)1114 776 y Fo(q)q Fn(k)1162 756 y Fp(2)1196 776 y Fg(6)i Fq(2)p Fo(\031)r(m)p Fn(k)p Fo(q)r Fn(k)d Fq(+)h Fn(k)p Fo(q)r Fn(k)1552 756 y Fp(2)1571 776 y Fo(:)285 b Fq(\(1.20\))46 881 y(First)15 b(results)g(ab)q(out)h (p)q(erio)q(dic)f(w)o(eigh)o(ted)f(op)q(erator)j Fo(T)k Fq(w)o(ere)15 b(obtained)g(b)o(y)g(Ly)o(apuno)o(v)h([L].)e(He)g(pro)o (v)o(ed)46 941 y(that)k(the)f(sp)q(ectrum)f(of)h Fo(T)24 b Fq(has)18 b(the)f(band)h(structure.)24 b(Later)18 b(Krein)e(repro)o (v)o(ed)g(this)i(result)e(in)h(a)h(more)46 1001 y(general)g(case)g (including)f(2)c Fn(\002)f Fq(2)18 b(systems.)26 b(The)18 b(global)g(quasimomen)o(tum)c(w)o(as)19 b(in)o(tro)q(duced)e(in)o(to)h (the)46 1061 y(sp)q(ectral)13 b(theory)g(of)g(the)g(Hill)f(op)q(erator) i(b)o(y)f(Firso)o(v)m(a)g([F])f(and)i(b)o(y)f(Marc)o(henk)o(o-Ostro)o (vski)e([MO1])i(sim)o(ul-)46 1121 y(taneously)l(.)25 b(The)18 b(author)h([K1])e(obtained)h(the)f(follo)o(wing)h(basic)f (result)g(ab)q(out)i("the)f(direct)f(problem")46 1182 y(for)e(p)q(erio)q(dic)g(w)o(eigh)o(ted)g(op)q(erators:)22 b(i\))15 b(asymptotics)f(of)i(v)m(arious)g(parameters)e(\(see)h (\(1.2\)\),)g(ii\))g(double-)46 1242 y(sided)f(estimates,)e(iii\))h (the)h(global)g(quasimomen)o(tum)o(.)j(It)d(is)g(these)g(results)f (that)i(w)o(e)f(use)g(in)f(the)h(presen)o(t)46 1302 y(pap)q(er.)21 b(Note)16 b(that)f(it)g(is)h(not)g(clear)e(ho)o(w)i(to)g(get)g(the)f (asymptotics)f(of)i(the)f(fundamen)o(tal)f(solutions)i(for)46 1362 y(the)g(case)g Fo(\032)e Fn(2)g Fo(C)t Fq(\()p Fl(R)p Fo(=)p Fl(Z)-14 b Fq(\).)119 1422 y(A)22 b(great)h(n)o(um)o(b)q(ers)e (of)h(pap)q(ers)i(are)e(dev)o(oted)g(to)h(the)f(in)o(v)o(erse)f (problem)f(for)j(the)f(Hill)f(op)q(erator.)46 1483 y(Marc)o(henk)o(o)e (and)i(Ostro)o(vski)f([MO1],)h([MO2])e(pro)o(v)o(ed)h(that)h(the)f (mapping)g Fo(h)h Fq(:)g Fo(V)32 b Fn(!)21 b Fo(h)p Fq(\()p Fo(V)11 b Fq(\))21 b(is)f(con-)46 1543 y(tin)o(uous)h(isomorphism.)34 b(Garnett)22 b(and)g(T)l(rub)q(o)o(witz)f([GT])g(pro)o(v)o(ed)g(that)g (this)g(mapping)g(is)g(the)g(real)46 1603 y(analytic)13 b(isomorphism)f(for)j(the)f(case)g(of)g(ev)o(en)f(p)q(oten)o(tials.)20 b(Kargaev)15 b(and)g(the)f(author)h([KK])e(repro)o(v)o(ed)46 1663 y(the)k(result)h(of)g(Garnet)g(and)h(T)l(rub)q(o)o(witz)f([GT])f (b)o(y)h(a)g(direct)f(metho)q(d.)25 b(Note)17 b(that)i(the)e(pro)q(ofs) j(b)o(y)d(the)46 1723 y(direct)e(metho)q(d)g(are)h(short)g(but)g(this)g (approac)o(h)h(needs)e(some)g(estimates)g(\(see)g(iii\))g(in)g(Theorem) g(A)g(and)46 1784 y(\(1.8\)\).)21 b(The)13 b(author)i([K2])f(pro)o(v)o (ed)f(that)h Fo(h)g Fq(:)f Fo(V)26 b Fn(!)13 b Fo(h)p Fq(\()p Fo(V)f Fq(\))h(is)h(real)g(analytic)f(isomorphism)f(b)o(y)h (the)h(direct)46 1844 y(metho)q(d.)23 b(Double-sided)17 b(estimates)e(for)j(v)m(arious)f(parameters)f(of)i(the)e(Hill)g(op)q (erator)i(\(the)f(norm)f(of)h(a)46 1904 y(p)q(erio)q(dic)12 b(p)q(oten)o(tial,)h(e\013ectiv)o(e)e(masses,)i(gap)h(lengths,)f(heigh) o(t)f(of)h(slits)g Fn(j)p Fo(h)1407 1911 y Fk(n)1430 1904 y Fn(j)g Fq(and)g(so)h(on)f(\))g(w)o(ere)f(obtained)46 1964 y(in)h([K4-6])h(and)g(for)g(the)f(Dirac)g(op)q(erator)i(in)e ([KK3],)g([K3],)g([K5].)20 b(The)14 b(precise)e(double-sided)h (estimates)46 2024 y(for)20 b(gap)h(lengths)f(w)o(as)h(found)f(in)g ([K7].)32 b(P\177)-24 b(osc)o(hel)20 b(and)g(T)l(rub)q(o)o(witz)g([PT]) g(wrote)g(a)h(b)q(o)q(ok)g(concerning)46 2085 y(the)d(in)o(v)o(erse)e (Diric)o(hlet)g(problem.)26 b(Coleman)17 b(and)i(McLaughlin)f([CM])f (solv)o(ed)g(the)h(Diric)o(hlet)e(in)o(v)o(erse)46 2145 y(problem)h(for)h(Eq.)28 b(\(1.3\).)g(In)18 b(the)g(presen)o(t)g(pap)q (er)h(w)o(e)f(need)g(more)f(strong)i(estimates)e(than)i(in)f([CM],)46 2205 y(see)e(Lemma)e(2.1)i(and)h(Lemma)d(2.5.)119 2265 y(W)l(e)d(\014nish)g(the)g(in)o(tro)q(duction)g(b)o(y)f(brie\015y)g (explaining)g(ho)o(w)i(the)f(pro)q(of)h(b)o(y)e(a)i(direct)e(approac)o (h)h(will)f(go,)46 2325 y(i.e.,)g(ho)o(w)h(w)o(e)g(v)o(erify)e (conditions)i(i\)-v\))g(of)g(Theorem)f(A)h(for)g(the)g(mapping)f Fo(h)p Fq(.)20 b(Remark)9 b(that)i(the)g(theorem)46 2385 y(from)k([KK1],)g([KK2])g(do)q(es)i(not)g(w)o(ork)f(since)f(there)g (exists)h(a)g(di\013erence)f(b)q(et)o(w)o(een)h(the)f(asymptotics)g(of) 46 2446 y Fo(h)74 2453 y Fk(n)97 2446 y Fo(;)8 b(n)14 b Fn(!)g(1)p Fq(:)20 b Fo(h)338 2453 y Fk(n)375 2446 y Fq(=)14 b(\(2)p Fo(\031)r(n)p Fq(\))548 2428 y Fm(\000)p Fp(1)595 2446 y Fo(V)623 2453 y Fk(n)654 2446 y Fq(+)7 b Fo(O)q Fq(\()p Fo(n)785 2428 y Fm(\000)p Fp(2)832 2446 y Fq(\))p Fo(;)14 b Fq(for)g(the)g(Hill)e(op)q(erator)k(and)e Fo(h)1439 2453 y Fk(n)1477 2446 y Fq(=)g Fo(q)1551 2453 y Fk(n)1580 2446 y Fq(+)7 b Fo(`)1645 2428 y Fk(d)1666 2446 y Fo(;)h(d)14 b(>)f Fq(1)p Fo(;)8 b(n)15 b Fn(!)e(1)p Fq(,)46 2506 y(for)k(the)g(w)o(eigh)o(ted)f(op)q(erator)i(\(see)f (\(1.9\)\).)24 b(Asymptotics)15 b(for)i(the)g(Hill)e(op)q(erator)j(is)f (more)f(con)o(v)o(enien)o(t)46 2566 y(than)f(asymptotics)e(for)h(the)g (w)o(eigh)o(ted)g(op)q(erator.)22 b(Note)13 b(that)i(in)f(the)g(pro)q (of)h(w)o(e)f(only)g(use)h(some)e(results)46 2626 y(from)g([K2],)g ([PT])h(and)g(the)g(estimates)e(\(1.8\))i(from)f([K1])h(\(see)f(also)i ([K3]\).)20 b(The)14 b(c)o(hec)o(king)e(of)i(i\))f(is)h(based)46 2686 y(on)j(the)f(analyticit)o(y)f(of)h(the)g(functions)h Fo(')p Fq(\()p Fn(\001)p Fo(;)8 b(z)r(;)g(q)r Fq(\))p Fo(;)23 b(#)p Fq(\()p Fn(\001)p Fo(;)8 b(z)r(;)g(q)r Fq(\))p Fo(;)14 b Fq(\001\()p Fo(z)r(;)8 b(q)r Fq(\))15 b(of)i Fo(z)f Fn(2)e Fl(C)9 b Fo(;)f(q)19 b Fn(2)14 b Fo(H)1697 2693 y Fk(C)1727 2686 y Fq(.)22 b(In)16 b(order)g(to)1008 2811 y(5)p eop %%Page: 6 6 6 5 bop 46 138 a Fq(obtain)13 b(the)f(analyticit)o(y)e(of)j(the)f (mapping)g Fo(h)g Fq(w)o(e)g(need)g(to)g(pro)o(v)o(e)g(the)g(real)g (analyticit)o(y)e(of)j(t)o(w)o(o)f(mappings)46 199 y Fo(z)69 206 y Fk(n)92 199 y Fq(\()p Fn(\001)p Fq(\))p Fo(;)c Fn(j)p Fo(h)208 206 y Fk(n)231 199 y Fq(\()p Fn(\001)p Fq(\))p Fn(j)297 180 y Fp(2)316 199 y Fq(.)21 b(The)14 b(main)f(problem)g(is)h(connected)g(with)g(the)g(degeneracy)g(of)g(the) g(gap)i Fo(h)1724 206 y Fk(n)1747 199 y Fq(\()p Fo(q)r Fq(\))p Fo(;)d Fq(i.e.,)g(the)46 259 y(case)k Fo(z)175 241 y Fm(\000)173 271 y Fk(n)221 259 y Fq(=)f Fo(z)300 241 y Fp(+)298 271 y Fk(n)330 259 y Fo(;)h Fq(and)h(then)f(with)h(the)f (high)h(energy)f(gaps.)27 b(T)l(o)18 b(o)o(v)o(ercome)d(these)i (di\016culties)f(w)o(e)h(sho)o(w)46 319 y(the)f(analyticit)o(y)e(of)j (the)f(mappings)g(in)g(the)g(ball)f Fo(B)1004 326 y Fk(C)1034 319 y Fq(\()p Fo(q)r(;)8 b(")1122 326 y Fk(q)1140 319 y Fq(\))16 b(for)h(an)o(y)f(\014xed)g Fo(q)f Fn(2)f Fo(H)21 b Fq(and)c(all)e Fo(n)f Fn(\025)g Fq(1.)119 379 y(T)l(o)j(c)o(hec)o(k)d (ii\),)h(w)o(e)g(pro)o(v)o(e)h(that)g(the)g(F)l(r)o(\023)-23 b(ec)o(het)14 b(deriv)m(ativ)o(es)h(of)h(the)g(map)f(is)h(a)h(F)l (redholm)d(op)q(erator)j(of)46 439 y(index)c(zero)h(with)g (zero-dimensional)e(k)o(ernels)h(and,)i(therefore,)e(is)h(in)o(v)o (ertible.)k(Here)13 b(w)o(e)g(use)h(a)h(result)e(of)46 499 y(P)o(aley-Wiener)j(ab)q(out)j(en)o(tire)e(functions)h(\(see)f (Lemma)f(4.3\))i(and)g(the)g(follo)o(wing)f(fact:)24 b(for)18 b(an)o(y)g(\014xed)46 560 y Fo(q)e Fn(2)f Fo(H)21 b Fq(the)16 b(v)o(ectors)h Fn(f)p Fq(\()p Fo(d)513 567 y Fk(q)532 560 y Fo(h)560 567 y Fk(cn)599 560 y Fq(\))p Fo(;)8 b Fq(\()p Fo(d)684 567 y Fk(q)703 560 y Fo(\026)732 567 y Fk(n)756 560 y Fq(\))p Fo(;)g(n)14 b Fn(\025)h Fq(1)p Fn(g)i Fq(form)e(a)j(basis)f(of)g Fo(H)k Fq(\(see)16 b([CM]\).)g(W)l(e)g(emphasize)f(the)46 620 y(imp)q(ortan)o(t)h(role)h (of)h(the)f(en)o(tire)f(function)h Fo(@)s Fq(\001\()p Fo(z)r(;)8 b(q)r Fq(\))p Fo(:)15 b Fq(The)i(v)o(eri\014cation)f(of)i (iii\))e(is)h(based)h(on)f(estimates)46 680 y(\(1.8\).)22 b(T)l(o)16 b(c)o(hec)o(k)f(iv\),)g(w)o(e)h(use)g(the)g(compactness)g (of)g(the)g(mappings)g Fo(z)1380 662 y Fm(\006)1378 692 y Fk(n)1410 680 y Fq(\()p Fn(\001)p Fq(\))d(:)h Fo(H)k Fn(!)c Fl(R)p Fq(,)f Fo(\026)1720 687 y Fk(n)1744 680 y Fq(\()p Fn(\001)p Fq(\))g(:)h Fo(H)k Fn(!)c Fl(R)46 740 y Fq(and)j(the)f(fundamen)o(tal)f(solutions.)21 b(The)c(c)o(hec)o (king)d(of)j(v\))e(is)i(based)f(on)h(estimates)e(giv)o(en)g(in)h([K6].) 119 800 y(In)k(conclusion)f(w)o(e)h(relate)f(the)h(op)q(erator)h Fo(T)27 b Fq(with)19 b(other)i(sp)q(ectral)e(problems.)32 b(Using)19 b(a)i(unitary)46 861 y(map,)15 b(the)h(op)q(erator)h Fo(T)23 b Fq(can)16 b(b)q(e)h(transformed,)e(ev)o(en)g(with)h(nonsmo)q (oth)h(co)q(e\016cien)o(ts,)d(b)o(y)575 990 y Fo(x)f Fq(=)h Fo(x)p Fq(\()p Fo(t)p Fq(\))f(=)817 932 y Fj(Z)858 945 y Fk(t)840 1026 y Fp(0)881 990 y Fo(\032)p Fq(\()p Fo(t)p Fq(\))962 970 y Fm(\000)p Fp(2)1009 990 y Fo(dt;)41 b(c)p Fq(\()p Fo(x)p Fq(\))13 b(=)h Fo(\032)p Fq(\()p Fo(t)p Fq(\()p Fo(x)p Fq(\)\))1406 970 y Fm(\000)p Fp(2)1452 990 y Fo(;)46 1117 y Fq(where)i Fo(t)p Fq(\()p Fo(x)p Fq(\))f(is)h(the)g(in)o(v)o(erse)f(function)h Fo(x)p Fq(\()p Fo(t)p Fq(\),)f(in)o(to)h(the)g(follo)o(wing)g(w)o(eigh)o(ted)f (op)q(erator)615 1227 y Fo(T)644 1234 y Fk(c)661 1227 y Fo(f)k Fq(=)14 b Fn(\000)p Fo(c)816 1207 y Fp(2)835 1227 y Fq(\()p Fo(x)p Fq(\))p Fo(f)930 1207 y Fm(00)1000 1227 y Fq(in)49 b Fo(L)1123 1207 y Fp(2)1143 1227 y Fq(\()p Fl(R)p Fo(;)7 b(c)p Fq(\()p Fo(x)p Fq(\))1309 1207 y Fm(\000)p Fp(2)1353 1227 y Fo(dx)p Fq(\))445 b(\(1.21\))46 1337 y(where)13 b Fo(c)p Fq(\()p Fo(x)p Fq(\))h(is)f(the)h(sound)g(v)o (elo)q(cit)o(y)e(in)h(the)h(p)q(erio)q(dic)f(media.)19 b(The)14 b(op)q(erator)h Fo(T)1532 1344 y Fk(c)1562 1337 y Fq(is)f(also)g(studied)g(in)f(the)46 1397 y(photonic)i(crystal)g (problem,)f(where)h Fo(")p Fq(\()p Fo(x)p Fq(\))e(=)h Fo(c)p Fq(\()p Fo(x)p Fq(\))987 1379 y Fm(\000)p Fp(2)1049 1397 y Fq(is)h(the)g(dielectric)e(co)q(e\016cien)o(t,)g(p)q(erio)q(dic) i(in)g Fo(x)g Fq(\(see)46 1458 y([N]\).)e(Remark)h(that)i(the)f (equation)h Fn(\000)s Fq(\177)-27 b Fo(y)15 b Fq(=)f Fo(z)882 1440 y Fp(2)901 1458 y Fo(b)p Fq(\()p Fo(t)p Fq(\))978 1440 y Fp(2)997 1458 y Fo(y)j Fq(arises)f(in)f(the)g (scattering)g(of)h(the)f(few)g(particles)g(in)46 1518 y(an)i(external)e(homogeneous)i(time-p)q(erio)q(dic)d(magnetic)h (\014eld)h Fo(B)s Fq(\()p Fo(t)p Fq(\))e(=)g Fo(z)r(b)p Fq(\()p Fo(t)p Fq(\)\(0)p Fo(;)8 b Fq(0)p Fo(;)g Fq(1\))14 b Fn(2)h Fl(R)1732 1500 y Fp(3)1749 1518 y Fq(,)h(where)g Fo(z)j Fq(is)46 1578 y(the)d(amplitude)e(and)j Fo(b)p Fq(\()p Fn(\001)p Fq(\))f(is)g(1-p)q(erio)q(dic)g(\(see)g([K8]\).)46 1744 y Fi(2)81 b(Prop)r(erties)27 b(of)f(F)-7 b(undamen)n(tal)27 b(Solutions)46 1854 y Fq(De\014ne)16 b(the)g(functions)664 1928 y Fo(f)688 1935 y Fp(+)731 1928 y Fq(=)e Fo(')815 1908 y Fm(0)837 1928 y Fn(\000)d Fo(i)909 1895 y(#)938 1876 y Fm(0)p 909 1917 41 2 v 917 1962 a Fo(z)954 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1085 2547 a(2)p Fo(iz)1156 2512 y Fq(\()p Fo(f)1199 2519 y Fp(+)1229 2512 y Fq(\()p Fo(x;)c(z)r(;)g(q)r Fq(\))h Fn(\000)i Fo(f)1471 2519 y Fm(\000)1501 2512 y Fq(\()p Fo(x;)d(z)r(;)g(q)r Fq(\)\))516 2668 y(=)573 2634 y Fo(\032)p Fq(\()p Fo(x)p Fq(\))p 573 2656 91 2 v 606 2702 a Fo(z)668 2668 y Fq(\(sin)h Fo(z)r(x)h Fn(\000)869 2609 y Fj(Z)911 2622 y Fk(x)892 2704 y Fp(0)941 2668 y Fq(sin)e Fo(z)r Fq(\(2)p Fo(t)j Fn(\000)g Fo(x)p Fq(\))p Fo(q)r Fq(\()p Fo(t)p Fq(\))p Fo(dt)e Fq(+)i Fo(`)1404 2647 y Fk(d)1425 2668 y Fq(\()p Fo(n)p Fq(\)\))p Fo(;)1008 2811 y Fq(8)p eop %%Page: 9 9 9 8 bop 46 138 a Fq(whic)o(h)15 b(giv)o(es)h(\(2.14\).)22 b Fb(2)119 199 y Fq(W)l(e)11 b(need)f(some)g(results)g(on)i(Eq.)19 b(\(1.3\))11 b(with)g(p)q(oten)o(tial)f(shifted)h(b)o(y)f(parameter)g Fo(t)p Fq(.)19 b(Let)11 b Fo(')p Fq(\()p Fo(x;)d(z)r(;)g(q)r(;)g(t)p Fq(\))p Fo(;)g(#)p Fq(\()p Fo(x;)g(z)q(;)g(q)r(;)f(t)p Fq(\))46 259 y(b)q(e)16 b(the)g(solutions)h(of)g(the)f(equation)561 369 y Fn(\000)p Fo(y)626 348 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Fm(0)921 1484 y Fq(\()p Fo(t)d Fn(\000)f Fo(x;)e(z)r(;)g(q)r(;)g Fq(1)i(+)h Fo(x)p Fq(\))p Fo(')1323 1463 y Fm(0)1335 1484 y Fq(\(1)g(+)g Fo(x)g Fn(\000)g Fo(t;)d(z)r(;)g(q)r(;)g(t)p Fq(\))173 b(\(2.20\))494 1571 y Fn(\000)p Fq(2)p Fo(\037)588 1579 y Fp([)p Fk(t;)p Fp(1])650 1571 y Fq(\()p Fo(x)p Fq(\))p Fo(')748 1550 y Fm(0)759 1571 y Fq(\(1)11 b(+)g Fo(t)g Fn(\000)g Fo(x;)d(z)r(;)g(q)r (;)g Fq(1)i(+)h Fo(x)p Fq(\))p Fo(')1246 1550 y Fm(0)1257 1571 y Fq(\()p Fo(x)g Fn(\000)g Fo(t;)d(z)r(;)g(q)r(;)g(t)p Fq(\))p Fo(;)46 1658 y Fd(Pr)n(o)n(of.)31 b Fq(Let)21 b Fo(q)g Fq(=)g Fo(q)423 1665 y Fp(0)456 1658 y Fq(+)13 b Fo(v)r(;)8 b(')p Fq(\()p Fo(s;)g(t)p Fq(\))19 b(=)i Fo(')p Fq(\()p Fo(s;)8 b(z)r(;)g(q)r(;)g(t)p Fq(\))p Fo(;)g(')1046 1665 y Fp(0)1064 1658 y Fq(\()p Fo(s;)g(t)p Fq(\))19 b(=)i Fo(')p Fq(\()p Fo(s;)8 b(z)r(;)g(q)1408 1665 y Fp(0)1426 1658 y Fo(;)g(t)p Fq(\).)33 b(Then)20 b(using)g(\(2.18\))h(w)o(e)46 1718 y(obtain)421 1789 y Fo(')p Fq(\(1)p Fo(;)8 b(t)p Fq(\))14 b(=)g 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(stated)g(in)g(i\).)46 1501 y(ii\))j(Let)g Fo(')p Fq(\()p Fo(x;)8 b(t)p Fq(\))19 b(=)h Fo(')p Fq(\()p Fo(x;)8 b(z)r(;)g(q)r(;)g (t)p Fq(\))p Fo(;)g(')p Fq(\()p Fo(x)p Fq(\))17 b(=)i Fo(')p Fq(\()p Fo(x;)8 b Fq(0\))20 b(and)g Fo(#)p Fq(\()p Fo(x;)8 b(t)p Fq(\))19 b(=)g Fo(')p Fq(\()p Fo(x;)8 b(z)r(;)g(q)r(;)g (t)p Fq(\))p Fo(;)g(#)p Fq(\()p Fo(x)p Fq(\))17 b(=)j Fo(')p Fq(\()p Fo(x;)8 b Fq(0\).)31 b(W)l(e)46 1561 y(ha)o(v)o(e)15 b(the)h(iden)o(tities)227 1668 y Fo(')p Fq(\(1)11 b(+)g Fo(t)p Fq(\))i(=)h Fo(')p Fq(\(1\))p Fo(#)p Fq(\()p Fo(t)p Fq(\))d(+)g Fo(')735 1648 y Fm(0)746 1668 y Fq(\(1\))p Fo(')p Fq(\()p Fo(t)p Fq(\))p Fo(;)106 b(#)p Fq(\(1)11 b(+)g Fo(t)p Fq(\))i(=)h Fo(#)p Fq(\(1\))p Fo(#)p Fq(\()p Fo(t)p Fq(\))c(+)h Fo(#)1514 1648 y Fm(0)1526 1668 y Fq(\(1\))p Fo(')p Fq(\()p Fo(t)p Fq(\))p Fo(:)180 b Fq(\(2.28\))46 1776 y(It)16 b(is)g(easy)g(to)h(\014nd)f(the)g(follo)o(wing)g(equalit)o (y)553 1883 y Fo(')p Fq(\()p Fo(x;)8 b(t)p Fq(\))13 b(=)h Fo(\032)p Fq(\()p Fo(t)p Fq(\))837 1863 y Fp(2)856 1883 y Fq(\()p Fo(')p Fq(\()p Fo(x)d Fq(+)g Fo(t)p Fq(\))p Fo(#)p Fq(\()p Fo(t)p Fq(\))f Fn(\000)g Fo(#)p Fq(\()p Fo(x)h Fq(+)g Fo(t)p Fq(\))p Fo(')p Fq(\()p Fo(t)p Fq(\)\))p Fo(;)381 b Fq(\(2.29\))46 1991 y(since)15 b Fo(')p Fq(\(0)p Fo(;)8 b(t)p Fq(\))14 b(=)g(0)p Fo(;)8 b(')443 1972 y Fm(0)455 1991 y Fq(\(0)p Fo(;)g(t)p Fq(\))13 b(=)h(1.)22 b(Substituting)16 b(\(2.11-12\))i(in)o(to)e(\(1.4\))g(w)o(e)g(ha)o(v)o (e)193 2098 y(\()p Fo(@)s Fq(\001\()p Fo(z)r(;)8 b(q)r Fq(\)\)\()p Fo(t)p Fq(\))k(=)h Fo(\032)554 2077 y Fp(2)574 2098 y Fq(\()p Fo(t)p Fq(\)[)p Fo(')676 2077 y Fm(0)687 2098 y Fq(\()p Fo(t)p Fq(\)\()p Fo(#)791 2077 y Fm(0)802 2098 y Fq(\(1\))p Fo(')p Fq(\()p Fo(t)p Fq(\))e Fn(\000)g Fo(')1045 2077 y Fm(0)1056 2098 y Fq(\(1\))p Fo(#)p Fq(\()p Fo(t)p Fq(\)\))g(+)g Fo(#)1311 2077 y Fm(0)1322 2098 y Fq(\()p Fo(t)p Fq(\)\()p Fo(#)p Fq(\(1\))p Fo(')p Fq(\()p Fo(t)p Fq(\))f Fn(\000)h Fo(')p Fq(\(1\))p Fo(#)p Fq(\()p Fo(t)p Fq(\)\)])324 2205 y(=)j Fo(\032)401 2185 y Fp(2)421 2205 y Fq(\()p Fo(t)p Fq(\)[)p Fo(')p Fq(\()p Fo(t)p Fq(\)\()p Fo(#)627 2185 y Fm(0)637 2205 y Fq(\(1\))p Fo(')731 2185 y Fm(0)743 2205 y Fq(\()p Fo(t)p Fq(\))c(+)h Fo(#)p Fq(\(1\))p Fo(#)978 2185 y Fm(0)990 2205 y Fq(\()p Fo(t)p Fq(\)\))f Fn(\000)h Fo(#)p Fq(\()p Fo(t)p Fq(\)\()p Fo(')1261 2185 y Fm(0)1272 2205 y Fq(\(1\))p Fo(')1366 2185 y Fm(0)1378 2205 y Fq(\()p Fo(t)p Fq(\))g(+)g Fo(')p Fq(\(1\))p Fo(#)1617 2185 y Fm(0)1628 2205 y Fq(\()p Fo(t)p Fq(\)\)])46 2291 y(and)17 b(\(2.28-29\))h(imply)365 2399 y(\()p Fo(@)s Fq(\001\()p Fo(z)r(;)8 b(q)r Fq(\)\)\()p Fo(t)p Fq(\))k(=)h Fo(\032)726 2378 y Fp(2)746 2399 y Fq(\()p Fo(t)p Fq(\)[)p Fo(')p Fq(\()p Fo(t)p Fq(\))p Fo(#)933 2378 y Fm(0)943 2399 y Fq(\(1)f(+)f Fo(t)p Fq(\))f Fn(\000)h Fo(#)p Fq(\()p Fo(t)p Fq(\))p Fo(')1261 2378 y Fm(0)1272 2399 y Fq(\(1)h(+)f Fo(t)p Fq(\))i(=)h Fn(\000)p Fo(')1549 2378 y Fm(0)1560 2399 y Fq(\(1)p Fo(;)8 b(t)p Fq(\))p Fo(:)46 2506 y Fq(iii\))17 b(The)i(asymptotic)e(estimates)g (\(2.22\))i(and)h(iden)o(tit)o(y)c(\(2.23\))k(w)o(ere)d(pro)o(v)o(ed)h (in)h([K1].)28 b(Using)18 b(\(2.21\))46 2567 y(and)f(\(2.15\))g(w)o(e)e (get)i(\(2.24-25\).)46 2627 y(iv\))d(Using)h(the)f(iden)o(tit)o(y)f (\(2.8\))i(w)o(e)f(obtain)h(\001\()p Fo(z)r(;)8 b(q)r Fq(\))13 b(=)1071 2607 y Fp(1)p 1071 2615 18 2 v 1071 2644 a(2)1093 2627 y Fq([)p Fo(')1139 2609 y Fm(0)1151 2627 y Fq(\(1)p Fo(;)8 b(z)r(;)g(q)r(;)g Fq(\))g(+)g Fo(')1414 2609 y Fm(0)1424 2627 y Fq(\(1)p Fo(;)g(z)r(;)g Fn(\000)p Fo(q)r Fq(\)].)20 b(The)14 b(function)h(\001)46 2687 y(is)h(ev)o(en)f(with)h(resp)q(ect)g(to)h Fo(q)r Fq(,)e(since)g(the)h(function)g Fo(')1023 2669 y Fm(0)1035 2687 y Fq(\(1)p Fo(;)8 b(z)r(;)g(q)r(;)g Fq(\))15 b(is)h(real)g (analytic)g(with)g(resp)q(ect)g(to)h Fo(q)r Fq(.)996 2811 y(10)p eop %%Page: 11 11 11 10 bop 119 138 a Fq(By)16 b(\(2.4\),the)f(solution)i(of)f(the)g (equation)768 248 y Fn(\000)819 235 y Fq(_)807 248 y Fo(#)836 228 y Fm(00)868 248 y Fn(\000)11 b Fq(2)p Fo(q)977 235 y Fq(_)966 248 y Fo(#)f Fn(\000)h Fo(z)1080 228 y Fp(2)1112 235 y Fq(_)1100 248 y Fo(#)i Fq(=)h(2)p Fo(z)r(#)46 358 y Fq(has)j(the)f(form)535 405 y(_)524 419 y Fo(#)o Fq(\()p Fo(x)p Fq(\))e(=)g(2)p Fo(z)741 360 y Fj(Z)783 373 y Fk(x)764 454 y Fp(0)813 419 y Fo(\032)p Fq(\()p Fo(t)p Fq(\))894 398 y Fp(2)913 419 y Fo(#)p Fq(\()p Fo(t)p Fq(\)\()p Fo(#)p Fq(\()p Fo(x)p Fq(\))p Fo(')p Fq(\()p Fo(t)p Fq(\))c Fn(\000)g Fo(#)p Fq(\()p Fo(t)p Fq(\))p Fo(')p Fq(\()p Fo(x)p Fq(\)\))p Fo(dt;)46 520 y Fq(and)17 b(then)541 578 y(_)530 591 y Fo(#)p Fq(\(1\))d(=)f(2)p Fo(z)744 532 y Fj(Z)786 546 y Fp(1)767 627 y(0)814 591 y Fo(\032)p Fq(\()p Fo(t)p Fq(\))895 570 y Fp(2)914 591 y Fo(#)p Fq(\()p Fo(t)p Fq(\)\()p Fo(#)p Fq(\(1\))p Fo(')p Fq(\()p Fo(t)p Fq(\))d Fn(\000)h Fo(#)p Fq(\()p Fo(t)p Fq(\))p Fo(')p Fq(\(1\)\))p Fo(dt;)358 b Fq(\(2.30\))46 692 y(The)16 b(same)f(consideration)i(yields)522 823 y(_)-27 b Fo(')541 802 y Fm(0)553 823 y Fq(\(1\))14 b(=)g(2)p Fo(z)738 764 y Fj(Z)780 777 y Fp(1)761 859 y(0)808 823 y 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1316 y Fp(2)586 1337 y Fq([)p Fo(')p Fq(\()p Fo(t)p Fq(\))p Fo(#)p Fq(\(1)c(+)h Fo(t)p Fq(\))g Fn(\000)g Fo(#)p Fq(\()p Fo(t)p Fq(\))p Fo(')p Fq(\(1)f(+)h Fo(t)p Fq(\)\)])p Fo(dt)i Fq(=)h Fn(\000)p Fo(z)1386 1278 y Fj(Z)1427 1291 y Fp(1)1409 1372 y(0)1455 1337 y Fo(')p Fq(\(1)p Fo(;)8 b(t)p Fq(\))p Fo(dt:)81 b Fb(2)46 1464 y Fq(In)o(tro)q(duce)20 b(the)g(con)o(tour)h Fo(C)576 1471 y Fk(n)600 1464 y Fq(\()p Fo(r)q Fq(\))g(=)g Fn(f)p Fo(z)j Fq(:)c Fn(j)p Fo(z)c Fn(\000)e Fo(\031)r(n)p Fn(j)21 b Fq(=)g Fo(r)q Fn(g)p Fo(;)8 b(n)21 b Fn(\025)g Fq(0)p Fo(;)8 b(r)23 b(>)e Fq(0)p Fo(:)g Fq(Belo)o(w)f(w)o(e)g(shall)g(use)h (the)46 1524 y(follo)o(wing)16 b(results)g(on)h(the)f(Diric)o(hlet)e (problem)g(for)j(Eq.)k(\(1.3\))c(on)f(the)g(in)o(terv)m(al)f([0)p Fo(;)8 b Fq(1].)46 1584 y Fr(Lemm)o(a)18 b(2.5.)24 b Fd(i\))18 b(L)n(et)g Fo(q)f Fn(2)f Fo(H)624 1591 y Fk(C)672 1584 y Fd(and)j Fo(")791 1591 y Fk(q)825 1584 y Fq(=)d(8)903 1566 y Fm(\000)p Fp(2)959 1584 y Fq(exp)o(\()p Fn(\000)p Fq(4)p 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Fd(is)h(c)n(omp)n(act)f(and)h(r)n(e)n(al)f(analytic)i(on) f Fo(H)t Fd(.)26 b(Its)19 b(gr)n(adient)g(is)f(given)j(by)46 1885 y(the)d(formula)336 2022 y Fq(\()p Fo(d)380 2029 y Fk(q)399 2022 y Fo(m)442 2029 y Fk(n)465 2022 y Fq(\()p Fo(q)r Fq(\)\)\()p Fo(x)p Fq(\))13 b(=)h Fn(\000)924 1988 y Fo(\032)p Fq(\()p Fo(x)p Fq(\))1015 1970 y Fp(2)p 721 2010 518 2 v 721 2056 a Fq(2)p Fo(m)788 2063 y Fk(n)811 2056 y Fq(\()p Fo(q)r Fq(\))p Fn(k)p Fo(')p Fq(\()p Fn(\001)p Fo(;)8 b(m)1028 2063 y Fk(n)1050 2056 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))p Fn(k)1202 2039 y Fp(2)1202 2071 y Fk(\032)1220 2061 y Fh(2)1243 2022 y Fq(\()p Fo(')p Fq(\()p Fo(x;)g(m)1406 2029 y Fk(n)1428 2022 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))1555 2001 y Fp(2)1574 2022 y Fq(\))1593 2001 y Fm(0)1604 2022 y Fo(;)252 b Fq(\(2.32\))46 2166 y Fd(wher)n(e)453 2241 y Fn(k)p Fo(')p Fq(\()p Fn(\001)p Fo(;)8 b(m)608 2248 y Fk(n)631 2241 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))p Fn(k)783 2221 y Fp(2)783 2253 y Fk(\032)801 2244 y Fh(2)833 2241 y Fq(=)889 2207 y Fo(')921 2189 y Fm(0)933 2207 y Fq(\(1)p Fo(;)g(m)1041 2214 y Fk(n)1064 2207 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))k(_)-26 b Fo(')o Fq(\(1)p Fo(;)8 b(m)1330 2214 y Fk(n)1353 2207 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))p 889 2229 590 2 v 1108 2275 a(2)p Fo(m)1175 2282 y Fk(n)1199 2275 y Fq(\()p Fo(q)r Fq(\))1498 2241 y Fo(>)13 b Fq(0)p Fo(:)283 b Fq(\(2.33\))46 2353 y Fd(Mor)n(e)n(over,) 16 b(for)h(any)g(\014xe)n(d)h Fo(d)d(>)e Fq(1)18 b Fd(the)g(fol)r (lowing)i(asymptotic)d(estimates)h(ar)n(e)f(ful\014l)r(le)n(d:)743 2463 y Fo(m)786 2470 y Fk(n)809 2463 y Fq(\()p Fo(q)r Fq(\))c(=)h Fo(\031)r(n)c Fn(\000)h Fo(q)1077 2470 y Fk(sn)1128 2463 y Fq(+)g Fo(`)1197 2443 y Fk(d)1217 2463 y Fq(\()p Fo(n)p Fq(\))p Fo(;)572 b Fq(\(2.34\))646 2573 y(\()p Fo(d)690 2580 y Fk(q)710 2573 y Fo(m)753 2580 y Fk(n)776 2573 y Fq(\()p Fo(q)r Fq(\)\)\()p Fo(x)p Fq(\))13 b(=)g Fn(\000)8 b Fq(sin)h(2)p Fo(\031)r(nx)h Fq(+)h Fo(`)1293 2553 y Fp(2)1314 2573 y Fq(\()p Fo(n)p Fq(\))p Fo(;)475 b Fq(\(2.35\))46 2661 y Fd(as)17 b Fo(n)d Fn(!)g(1)p Fd(,)j(uniformly)g(on)h Fq([0)p Fo(;)8 b Fq(1])j Fn(\002)g Fo(B)785 2668 y Fk(C)814 2661 y Fq(\()p Fo(p;)d(")902 2668 y Fk(p)922 2661 y Fq(\))18 b Fd(for)e(e)n(ach)i(\014xe)n(d)g Fo(p)c Fn(2)g Fo(H)t Fd(.)996 2811 y Fq(11)p eop %%Page: 12 12 12 11 bop 46 138 a Fd(Pr)n(o)n(of.)19 b Fq(i\))11 b(Let)i Fo(M)19 b(>)14 b(N)j Fq(b)q(e)c(another)g(in)o(teger.)19 b(Consider)12 b(the)g(con)o(tours)h Fo(C)1421 145 y Fp(0)1441 138 y Fq(\()p Fo(\031)r Fq(\()p Fo(N)8 b Fq(+)s(\(1)p Fo(=)p Fq(2\)\)\))p Fo(;)20 b(C)1814 145 y Fp(0)1834 138 y Fq(\()p Fo(\031)r Fq(\()p Fo(M)8 b Fq(+)46 199 y(\(1)p Fo(=)p Fq(2\)\)\))p Fo(;)21 b(C)264 206 y Fk(n)288 199 y Fq(\(1\))p Fo(;)8 b Fn(j)p Fo(n)p Fn(j)14 b Fo(>)f(N)r(:)g Fq(Recall)f(that)h(for)g Fo(q)h Fq(w)o(e)e(de\014ned)g(the)h(functions) 1439 195 y Fj(e)1435 199 y Fo(q)q(;)1488 182 y Fj(e)1480 199 y Fo(V)24 b Fq(and)13 b(the)g(in)o(teger)e Fo(m)j Fn(\025)f Fq(1)46 259 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Fo(q)q Fq(\))p Fn(j)14 b Fg(6)1117 1805 y Fq(4)p Fo(")p 1114 1827 V 1114 1872 a Fn(j)p Fo(z)r Fn(j)1172 1838 y Fo(e)1195 1818 y Fm(j)5 b Fp(Im)g Fk(z)q Fm(j)p Fp(+4)p Fm(k)p Fk(q)q Fm(k)p Fp(+8)p Fk(")1446 1838 y Fo(:)410 b Fq(\(2.39\))46 1971 y(Using)16 b(\(2.36\))h(and)g(\(2.39\))g(w)o(e)e(obtain)348 2099 y Fn(j)p Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(p)p Fq(\))j Fn(\000)615 2065 y Fq(sin)d Fo(z)p 615 2088 94 2 v 649 2133 a(z)713 2099 y Fn(j)13 b Fg(6)h Fn(j)p Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(p)p Fq(\))j Fn(\000)g Fo(')p Fq(\(1)p Fo(;)d(z)r(;)1203 2095 y Fj(e)1199 2099 y Fo(q)q Fq(\))p Fn(j)j Fq(+)g Fn(j)p Fo(')p Fq(\(1)p Fo(;)d(z)r(;)1477 2095 y Fj(e)1473 2099 y Fo(q)q Fq(\))j Fn(\000)1581 2065 y Fq(sin)d Fo(z)p 1581 2088 V 1615 2133 a(z)1679 2099 y Fn(j)762 2248 y Fg(6)822 2214 y Fq(4)p Fo(")p 819 2236 53 2 v 819 2282 a Fn(j)p Fo(z)r Fn(j)877 2248 y Fo(e)900 2227 y Fm(j)p Fp(Im)t Fk(z)r Fm(j)p Fp(+8)p Fk(")1048 2248 y Fq(\(1)k(+)f Fo(e)1175 2227 y Fp(4)p Fm(k)p Fk(q)q Fm(k)1246 2248 y Fq(\))p Fo(:)46 2361 y Fq(Therefore,)k(the)h(simple)e (estimate)h(exp)7 b Fn(j)p Fq(Im)g Fo(z)r Fn(j)13 b Fo(<)h Fq(4)p Fn(j)8 b Fq(sin)h Fo(z)r Fn(j)16 b Fq(on)h(all)e(con)o(tours)i (\(see)f([PT]\))g(yields)433 2493 y Fn(j)p Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(p)p Fq(\))j Fn(\000)700 2460 y Fq(sin)d Fo(z)p 700 2482 94 2 v 734 2527 a(z)798 2493 y Fn(j)14 b Fg(6)f Fo(C)922 2460 y Fn(j)8 b Fq(sin)g Fo(z)r Fn(j)p 922 2482 130 2 v 960 2527 a(j)p Fo(z)r Fn(j)1056 2493 y Fo(;)73 b(C)17 b Fq(=)d(16)p Fo("e)1341 2473 y Fp(8)p Fk(")1377 2493 y Fq(\(1)d(+)g Fo(e)1503 2473 y Fp(4)p Fm(k)p Fk(q)q Fm(k)1575 2493 y Fq(\))p Fo(;)46 2626 y Fq(and)h Fo(C)17 b(<)d Fq(1)e(on)f(all)g(con)o(tours.)20 b(Hence,)11 b(b)o(y)g(Rouc)o(h)o(\023)-23 b(e's)10 b(theorem,)g Fo(')p Fq(\(1)p Fo(;)e(z)r(;)g(r)q Fq(\))k(has)g(as)g(man)o(y)e(ro)q (ots,)j(coun)o(ted)46 2686 y(with)19 b(m)o(ultiplic)o(iti)o(es,)e(as) 537 2667 y Fp(sin)5 b Fk(z)p 537 2675 68 2 v 561 2704 a(z)628 2686 y Fq(in)19 b(eac)o(h)f(of)i(the)f(b)q(ounded)h(domains)f (and)g(the)g(remaining)f(un)o(b)q(ounded)996 2811 y(12)p eop %%Page: 13 13 13 12 bop 46 138 a Fq(region.)36 b(Since)364 119 y Fp(sin)5 b Fk(z)p 364 127 68 2 v 388 155 a(z)457 138 y Fq(has)22 b(only)f(simple)e(ro)q(ots)k(at)f Fo(\031)r(n;)8 b(n)22 b Fn(6)p Fq(=)g(0)p Fo(;)f Fq(and)h(since)f Fo(M)28 b(>)22 b(N)27 b Fq(can)21 b(b)q(e)h(c)o(hosen)46 199 y(arbitrarily)15 b(large,)h(the)g(p)q(oin)o(t)g(i\))g(of)h(Lemma)d(2.5)i(follo)o(ws.)119 259 y(ii\))h(First)h(w)o(e)g(pro)o(v)o(e)f(\(2.33\))i(whic)o(h)e(sho)o (ws)i(that)g(the)f(zero)g Fo(m)1279 266 y Fk(n)1302 259 y Fq(\()p Fo(q)r Fq(\))f(is)h(simple)e(for)i Fo(q)h Fn(2)e Fo(H)t Fq(.)27 b(In)18 b(fact,)46 319 y(w)o(e)e(rep)q(eat)g(the)g(case) g(of)h(the)f(p)q(oten)o(tial)g(from)f([PT].)g(Di\013eren)o(tiation)h (of)g(Eq.)22 b(\(1.3\))16 b(with)g(resp)q(ect)g(to)h Fo(z)46 379 y Fq(implies)750 439 y Fn(\000)c Fq(_)-27 b Fo(')821 419 y Fm(00)853 439 y Fn(\000)10 b Fq(2)p Fo(q)16 b Fq(_)-28 b 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Fq(=)e([)p Fo(m)650 1524 y Fk(n)673 1517 y Fq(\()p Fo(q)r Fq(\))c Fn(\000)h Fo(\016)o(;)d(m)881 1524 y Fk(n)904 1517 y Fq(\()p Fo(q)r Fq(\))i(+)h Fo(\016)r Fq(])i Fn(\032)g Fq(\(0)p Fo(;)8 b Fn(1)p Fq(\))p Fo(;)41 b Fq(1)14 b Fg(6)g Fo(n)g Fg(6)f Fo(N)r(:)46 1626 y Fq(If)i Fo(\016)j Fq(is)d(su\016cien)o(tly)f(small,)g(then)i(these)f(in)o(terv)m(als)g (are)h(all)f(disjoin)o(t.)21 b(The)16 b(function)f Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)r Fq(\))15 b(c)o(hanges)46 1686 y(sign)g(on)g(eac)o(h)f(of)h(them)d(and)k Fn(j)p Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)r Fq(\))p Fn(j)k Fo(>)i Fq(2)p Fo(B)s(;)37 b(z)16 b Fn(2)e([)p Fo(I)1124 1693 y Fk(n)1147 1686 y Fq(,)g(for)h(some)f Fo(B)i(>)e Fq(0)h(since)f Fo(m)1674 1693 y Fk(n)1697 1686 y Fq(\()p Fo(q)r Fq(\))g(is)g(a)h (simple)46 1746 y(ro)q(ot.)119 1807 y(As)20 b Fo(\027)k Fn(!)d(1)p Fq(,)g(the)f(functions)g Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)869 1789 y Fk(\027)890 1807 y Fq(\))21 b Fn(!)f Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)r Fq(\))20 b(and)34 b(_)-27 b Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)1474 1789 y Fk(\027)1495 1807 y Fq(\))21 b Fn(!)33 b Fq(_)-27 b Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)r Fq(\))20 b(con)o(v)o(erge)46 1867 y(uniformly)f(on)i Fn([)p Fo(I)400 1874 y Fk(n)444 1867 y Fq(b)o(y)g(Lemma)e(2.2.)36 b(Hence,)20 b(for)h(su\016cien)o(tly) e(large)i Fo(\027;)g Fq(the)g(functions)g Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)1955 1849 y Fk(\027)1976 1867 y Fq(\))46 1927 y(also)21 b(c)o(hange)g(sign)h(and)f Fn(j)13 b Fq(_)-27 b Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)701 1909 y Fk(\027)722 1927 y Fq(\))p Fn(j)21 b Fo(>)h(B)i Fq(on)d Fn([)p Fo(I)1024 1934 y Fk(n)1047 1927 y Fo(;)g Fq(so)h(they)e(m)o(ust)g(all)g(ha)o(v)o (e)g(one)h(ro)q(ot)h(in)f(eac)o(h)f(of)46 1987 y(these)13 b(in)o(terv)m(als,)g(for)h Fo(\027)j Fq(su\016cien)o(tly)12 b(large)i(.)20 b(Hence)13 b Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)1192 1969 y Fk(\027)1212 1987 y Fq(\))14 b(has)h(exactly)d(one)i(ro)q(ot)h Fo(m)1727 1994 y Fk(n)1750 1987 y Fq(\()p Fo(q)1793 1969 y Fk(\027)1814 1987 y Fq(\))f(in)f(eac)o(h)46 2047 y(segmen)o(t)k Fo(I)259 2054 y Fk(n)282 2047 y Fo(;)8 b Fq(1)19 b Fg(6)f Fo(n)g Fg(6)g Fo(N)r(:)h Fq(This)g(yields)f Fn(j)p Fo(m)893 2054 y Fk(n)916 2047 y Fq(\()p Fo(q)959 2029 y Fk(\027)980 2047 y Fq(\))13 b Fn(\000)f Fo(m)1106 2054 y Fk(n)1130 2047 y Fq(\()p Fo(q)r Fq(\))p Fn(j)17 b Fo(<)h(\016)o(;)27 b Fq(1)19 b Fg(6)f Fo(n)g Fg(6)g Fo(N)r(;)h Fq(for)g(all)g(su\016cien)o (tly)46 2108 y(large)14 b Fo(\027:)g Fq(It)f(follo)o(ws)h(that)g Fo(m)569 2115 y Fk(n)592 2108 y Fq(\()p Fo(q)635 2089 y Fk(\027)656 2108 y Fq(\))g Fn(!)g Fo(m)796 2115 y Fk(n)819 2108 y Fq(\()p Fo(q)r Fq(\))p Fo(;)f Fq(as)h Fo(\027)j Fn(!)d(1)p Fq(,)f(since)h Fo(N)19 b Fq(and)14 b Fo(\016)h(>)f Fq(0)g(w)o(ere)f(arbitrary)l(.)21 b(Th)o(us,)46 2168 y Fo(m)89 2175 y Fk(n)112 2168 y Fq(\()p Fo(q)r Fq(\))16 b(is)g(a)g(compact)f(functions)i(of)f Fo(q)r Fq(.)119 2228 y(T)l(o)22 b(pro)o(v)o(e)g(real)f(analyticit)o(y)l(,)g(w)o(e)h (\014x)f Fo(p)j Fn(2)g Fo(H)t Fq(.)38 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Fq(\()p Fo(q)r Fq(\))f(is)h(also)g(a) h(con)o(tin)o(uous)46 2627 y(function)g(on)g Fo(W)26 b Fq(satisfying)19 b Fo(')p Fq(\(1)p Fo(;)8 b(m)742 2634 y Fk(n)765 2627 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))18 b(=)g(0)p Fo(:)h Fq(Therefore,)g(b)o(y)f(uniqueness,)1595 2623 y Fj(f)1594 2627 y Fo(m)1637 2634 y Fk(n)1660 2627 y Fq(\()p Fo(q)r Fq(\))g(=)h Fo(m)1840 2634 y Fk(n)1863 2627 y Fq(\()p Fo(q)r Fq(\))f(on)46 2687 y Fo(W)o(;)d Fq(and)i(so)g Fo(m)318 2694 y Fk(n)341 2687 y Fq(\()p Fo(q)r Fq(\))f(is)g(real)g(analytic.)996 2811 y(13)p eop %%Page: 14 14 14 13 bop 119 138 a Fq(T)l(o)17 b(calculate)e(the)h(gradien)o(t,)g(w)o (e)f(observ)o(e)h(that)h Fo(')p Fq(\(1)p Fo(;)8 b(m)1174 145 y Fk(n)1197 138 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))13 b(=)g(0)p Fo(:)k Fq(Hence)271 248 y(0)d(=)g Fo(d)386 255 y Fk(q)405 248 y Fn(f)p Fo(')p Fq(\(1)p Fo(;)8 b(m)570 255 y Fk(n)593 248 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))p Fn(g)13 b Fq(=)27 b(_)-27 b Fo(')p Fq(\(1)p Fo(;)8 b(m)950 255 y Fk(n)973 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Fo(dx)g Fq(+)g Fo(`)1192 1029 y Fk(d)1213 1049 y Fq(\()p Fo(n)p Fq(\))p Fo(;)73 b Fn(j)p Fo(z)12 b Fn(\000)f Fo(\031)r(n)p Fn(j)j Fg(6)f Fq(1)p Fo(;)73 b(n)14 b Fn(!)g(1)p Fo(:)46 1173 y Fq(Hence)417 1244 y(0)g(=)g Fo(m)550 1251 y Fk(n)573 1244 y Fo(')p Fq(\(1)p Fo(;)8 b(m)713 1251 y Fk(n)736 1244 y Fo(;)g(q)r Fq(\))13 b(=)h(sin)8 b Fo(m)977 1251 y Fk(n)1011 1244 y Fq(+)1060 1185 y Fj(Z)1102 1199 y Fp(1)1083 1280 y(0)1130 1244 y Fo(q)r Fq(\()p Fo(x)p Fq(\))g(sin)g Fo(m)1339 1251 y Fk(n)1362 1244 y Fo(xdx)j Fq(+)g Fo(`)1523 1223 y Fk(d)1543 1244 y Fq(\()p Fo(n)p Fq(\))p Fo(;)46 1348 y Fq(whic)o(h)k(implies)f(\(2.34\).)119 1408 y(The)i(asymptotic)f(estimate)g(\(2.14\))h(giv)o(es)405 1542 y Fn(k)p Fo(')p Fq(\()p Fn(\001)p Fo(;)8 b(m)560 1549 y Fk(n)582 1542 y Fo(;)g(q)r Fq(\)\))p Fn(k)691 1521 y Fp(2)691 1554 y Fk(\032)709 1545 y Fh(2)742 1542 y Fq(=)819 1508 y(1)p 798 1530 67 2 v 798 1576 a Fo(m)841 1562 y Fp(2)841 1588 y Fk(n)878 1483 y Fj(Z)919 1496 y Fp(1)901 1577 y(0)939 1542 y Fq(\(sin)1018 1521 y Fp(2)1046 1542 y Fo(xm)1117 1549 y Fk(n)1151 1542 y Fq(+)j Fo(`)1220 1521 y Fp(2)1240 1542 y Fq(\()p Fo(n)p Fq(\)\))p Fo(dx;)73 b(n)14 b Fn(!)g(1)p Fo(;)46 1677 y Fq(and)j(then)366 1748 y Fn(k)p Fo(')p Fq(\()p Fn(\001)p Fo(;)8 b(m)521 1755 y Fk(n)544 1748 y Fo(;)g(q)r Fq(\)\))p Fn(k)653 1727 y Fp(2)653 1760 y Fk(\032)671 1751 y Fh(2)703 1748 y Fq(=)781 1714 y(1)p 760 1736 V 760 1782 a Fo(m)803 1767 y Fp(2)803 1794 y Fk(n)831 1748 y Fq([)850 1714 y(1)p 850 1736 25 2 v 850 1782 a(2)890 1748 y Fn(\000)945 1714 y Fq(sin)g(2)p Fo(m)1080 1721 y Fk(n)p 945 1736 159 2 v 979 1782 a Fq(4)p Fo(m)1046 1789 y Fk(n)1120 1748 y Fq(+)j Fo(`)1189 1727 y Fp(2)1209 1748 y Fq(\()p Fo(n)p Fq(\)])i(=)1381 1714 y(1)p 1360 1736 67 2 v 1360 1782 a Fo(m)1403 1767 y Fp(2)1403 1794 y Fk(n)1431 1748 y Fq(\(1)f(+)f Fo(`)1555 1727 y Fp(2)1575 1748 y Fq(\()p Fo(n)p Fq(\)\))p Fo(:)195 b Fq(\(2.40\))46 1863 y(Using)16 b(\(2.14-15\))i(and)f(\(2.40\))g(w)o(e)e(obtain)623 2000 y Fo(d)648 2007 y Fk(q)667 2000 y Fo(m)710 2007 y Fk(n)747 2000 y Fq(=)f Fo(\032)824 1980 y Fp(2)844 2000 y Fq(\()p Fo(x)p Fq(\))915 1966 y(2)p Fo(')971 1948 y Fm(0)983 1966 y Fq(\()p Fo(x;)8 b(m)1095 1973 y Fk(n)1117 1966 y Fo(;)g(q)r Fq(\))p Fo(')p Fq(\()p Fo(x;)g(m)1326 1973 y Fk(n)1348 1966 y Fo(;)g(q)r Fq(\))p 915 1989 498 2 v 957 2034 a(2)p Fo(m)1024 2041 y Fk(n)1047 2034 y Fn(k)p Fo(')p Fq(\()p Fn(\001)p Fo(;)g(m)1202 2041 y Fk(n)1225 2034 y Fo(;)g(q)r Fq(\)\))p Fn(k)1334 2017 y Fp(2)1334 2049 y Fk(\032)1352 2040 y Fh(2)326 2183 y Fq(=)13 b Fn(\000)p Fq(2)445 2149 y([cos)c Fo(xm)604 2156 y Fk(n)638 2149 y Fq(+)i Fo(`)707 2131 y Fp(2)727 2149 y Fq(\()p Fo(n)p Fq(\)][sin)c Fo(xm)960 2156 y Fk(n)994 2149 y Fq(+)k Fo(`)1063 2131 y Fp(2)1083 2149 y Fq(\()p Fo(n)p Fq(\)])p Fo(m)1207 2131 y Fp(2)1207 2162 y Fk(n)p 445 2171 785 2 v 676 2217 a Fn(k)p Fo(')p Fq(\()p Fn(\001)p Fo(;)d(m)831 2224 y Fk(n)854 2217 y Fo(;)g(q)r Fq(\)\))p Fn(k)963 2200 y Fp(2)963 2232 y Fk(\032)981 2223 y Fh(2)1249 2183 y Fq(=)14 b Fn(\000)8 b Fq(sin)g(2)p Fo(xm)1511 2190 y Fk(n)1545 2183 y Fq(+)j Fo(`)1614 2162 y Fp(2)1634 2183 y Fq(\()p Fo(n)p Fq(\))p Fo(;)46 2307 y Fq(whic)o(h)k(implies)f (\(2.35\).)71 b Fb(2)46 2473 y Fi(3)81 b(Mappings)46 2583 y Fq(In)19 b(order)h(to)g(study)g(other)f(mappings,)h(w)o(e)f(m)o (ust)f(consider)h(the)h(function)f Fo(z)1507 2590 y Fk(n)1530 2583 y Fq(\()p Fo(q)r Fq(\))p Fo(:)g Fq(Recall)f(that)i Fo(N)1923 2590 y Fk(q)1962 2583 y Fq(is)46 2643 y(de\014ned)c(b)o(y)g (\(1.18\).)996 2811 y(14)p eop %%Page: 15 15 15 14 bop 46 138 a Fr(Lemm)o(a)15 b(3.1.)p Fd(i\))g(L)n(et)h Fo(q)f Fn(2)f Fo(H)588 145 y Fk(C)635 138 y Fd(and)i Fo(")751 145 y Fk(q)784 138 y Fq(=)e(8)860 120 y Fm(\000)p Fp(2)915 138 y Fq(exp\()p Fn(\000)p Fq(4)p Fn(k)p Fo(q)r Fn(k)p Fq(\))p Fd(.)21 b(Then)c(for)f(any)g(inte)n(ger)h Fo(N)i Fn(\025)14 b Fo(N)1806 145 y Fk(q)1841 138 y Fd(and)i(for)46 199 y(e)n(ach)g Fo(p)f Fn(2)f Fo(B)277 206 y Fk(C)306 199 y Fq(\()p Fo(q)r(;)8 b(")394 206 y Fk(q)412 199 y Fq(\))16 b Fd(the)h(function)731 186 y Fq(_)718 199 y(\001)o(\()p Fo(z)r(;)8 b(p)p Fq(\))17 b Fd(has)f(exactly)h Fq(2)p Fo(N)d Fq(+)8 b(1)17 b Fd(r)n(o)n(ots,)e(c)n(ounte)n(d)h(with)h (multiplicities,)46 259 y(in)h(the)h(disc)f Fn(f)p Fo(z)e Fq(:)f Fn(j)p Fo(z)r Fn(j)f Fo(<)h(\031)r Fq(\()p Fo(N)h Fq(+)11 b(\(1)p Fo(=)p Fq(2\)\))p Fn(g)19 b Fd(and)g(for)e(e)n(ach)h Fn(j)p Fo(n)p Fn(j)d Fo(>)f(N)r(;)k Fd(it)h(has)e(exactly)j(one)e (simple)h(r)n(o)n(ot)d(in)46 319 y(the)i(domain)f Fn(f)p Fo(z)f Fq(:)d Fn(j)p Fo(z)g Fn(\000)e Fo(\031)r(n)p Fn(j)i Fo(<)h Fq(1)p Fn(g)p Fo(:)j Fd(Ther)n(e)g(ar)n(e)g(no)h(other)f(r)n(o)n (ots.)46 379 y(ii\))i(Each)h(function)h Fo(z)457 386 y Fk(n)480 379 y Fq(\()p Fn(\001)p Fq(\))p Fo(;)8 b(n)18 b Fn(\025)g Fq(1)p Fo(;)i Fd(is)f(c)n(omp)n(act)g(and)h(r)n(e)n(al)f (analytic)i(on)f Fo(H)t Fd(.)29 b(Its)20 b(gr)n(adient)g(is)f(given)j (by)46 439 y(the)c(formula)655 532 y Fq(\()p Fo(d)699 539 y Fk(q)718 532 y Fo(z)741 539 y Fk(n)764 532 y Fq(\()p Fo(q)r Fq(\)\)\()p Fo(x)p Fq(\))13 b(=)h Fn(\000)1020 499 y Fq(\()p Fo(@)1081 486 y Fq(_)1068 499 y(\001)o(\)\()p Fo(x;)8 b(z)1219 506 y Fk(n)1241 499 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))p 1020 521 348 2 v 1086 559 a(\177)1078 571 y(\001)o(\()p Fo(z)1160 578 y Fk(n)1183 571 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))1372 532 y Fo(:)508 b Fq(\(3.1\))46 647 y Fd(Mor)n(e)n(over,)16 b(for)h(any)g(\014xe)n(d)h Fo(d)d(>)e Fq(1)18 b Fd(the)g(fol)r(lowing)i (asymptotic)d(estimates)h(ar)n(e)f(ful\014l)r(le)n(d:)690 751 y Fo(z)713 758 y Fk(n)736 751 y Fq(\()p Fo(q)r Fq(\))c(=)h Fo(\031)r(n)c Fq(+)h Fo(`)1001 730 y Fk(d)1022 751 y Fq(\()p Fo(n)p Fq(\))p Fo(;)78 b(n)14 b Fn(!)f(1)p Fo(;)543 b Fq(\(3.2\))658 855 y(\()p Fo(d)702 862 y Fk(q)721 855 y Fo(z)744 862 y Fk(n)767 855 y Fq(\()p Fo(q)r Fq(\)\)\()p Fo(x)p Fq(\))13 b(=)h Fo(`)999 834 y Fp(2)1019 855 y Fq(\()p Fo(n)p Fq(\))p Fo(;)113 b(n)14 b Fn(!)f(1)p Fo(;)511 b Fq(\(3.3\))46 939 y Fd(uniformly)17 b(on)h Fq([0)p Fo(;)8 b Fq(1])j Fn(\002)f Fo(B)533 946 y Fk(C)563 939 y Fq(\()p Fo(p;)e(")651 946 y Fk(p)671 939 y Fq(\))17 b Fd(for)g(e)n(ach)h(\014xe)n(d)g Fo(p)c Fn(2)g Fo(H)t Fd(.)46 1000 y(Pr)n(o)n(of.)39 b Fq(i\))22 b(Fix)g Fo(N)27 b Fq(and)d(let)d Fo(M)30 b(>)25 b(N)j Fq(b)q(e)23 b(another)g(in)o (teger.)39 b(Consider)23 b(the)f(con)o(tours)h Fo(C)1810 1007 y Fp(0)1830 1000 y Fq(\()p Fo(\031)r Fq(\()p Fo(N)d Fq(+)46 1060 y(\(1)p Fo(=)p Fq(2\)\)\))p Fo(;)26 b(C)269 1067 y Fp(0)288 1060 y Fq(\()p Fo(\031)r Fq(\()p Fo(M)16 b Fq(+)c(\(1)p Fo(=)p Fq(2\)\)\))p Fo(;)26 b(C)692 1067 y Fk(n)715 1060 y Fq(\(1\))p Fo(;)8 b Fn(j)p Fo(n)p Fn(j)15 b Fo(>)f(N)5 b Fq(.)24 b(Recall)15 b(that)i(for)h Fo(q)g Fq(w)o(e)e(de\014ned)h(the)f(functions)1914 1056 y Fj(e)1910 1060 y Fo(q)r(;)1964 1044 y Fj(e)1955 1060 y Fo(V)46 1120 y Fq(\(see)g(\(1.18-19\)\))i(suc)o(h)e(that)g Fn(k)p Fo(q)d Fn(\000)698 1116 y Fj(e)693 1120 y Fo(q)r Fn(k)h Fg(6)f Fo(")j Fq(and)823 1224 y(\001\()p Fo(z)r(;)934 1220 y Fj(e)930 1224 y Fo(q)q Fq(\))e(=)g Fo(D)q Fq(\()p Fo(z)r(;)1153 1208 y Fj(e)1145 1224 y Fo(V)e Fq(\))p Fo(;)46 1328 y Fq(where)17 b Fo(D)q Fq(\()p Fo(z)r(;)304 1312 y Fj(e)296 1328 y Fo(V)11 b Fq(\))17 b(is)h(the)f(Ly)o(apuno)o(v)h (function)f(for)h(the)f(Hill)f(op)q(erator.)26 b(Using)18 b(the)f(estimates)f(in)h([PT])46 1388 y(it)f(is)g(easy)g(to)h(get)348 1512 y Fn(j)379 1500 y Fq(_)362 1512 y Fo(D)r Fq(\()p Fo(z)r(;)478 1496 y Fj(e)470 1512 y Fo(V)11 b Fq(\))g Fn(\000)g Fq(sin)d Fo(z)r Fn(j)14 b Fg(6)f Fo(A)8 b Fq(exp)p Fn(f)p Fo(A)i Fq(+)h Fn(j)p Fq(Im)c Fo(z)q Fn(jg)p Fo(;)89 b(A)14 b Fq(=)1372 1479 y(1)p 1358 1501 53 2 v 1358 1546 a Fn(j)p Fo(z)r Fn(j)1423 1454 y Fj(Z)1465 1467 y Fp(1)1446 1548 y(0)1493 1512 y Fn(j)1515 1496 y Fj(e)1507 1512 y Fo(V)d Fq(\()p Fo(x)p Fq(\))p Fn(j)p Fo(dx:)201 b Fq(\(3.4\))46 1644 y(If)16 b(w)o(e)f(tak)o(e)h Fo(z)i Fq(suc)o(h)e(that)h Fn(j)p Fo(z)r Fn(j)c(\025)h Fo(\031)r Fq(\()p Fo(N)i Fq(+)11 b(\(1)p Fo(=)p Fq(2\)\),)17 b(then)f(\(1.20\))h(implies)730 1775 y Fo(A)d Fg(6)838 1742 y Fn(k)p Fo(q)r Fn(k)p 838 1764 74 2 v 848 1810 a(j)p Fo(z)r Fn(j)916 1775 y Fq(\()p Fn(k)p Fo(q)r Fn(k)d Fq(+)g(2)p Fo(\031)r(m)p Fq(\))i Fo(<)h Fq(4)p Fo(":)46 1907 y Fq(Hence,)g(using)j(\(2.27\))g(w)o(e)f (obtain)429 2034 y Fn(j)457 2022 y Fq(_)443 2034 y(\001\()p Fo(z)r(;)8 b(p)p Fq(\))j Fn(\000)668 2022 y Fq(_)654 2034 y(\001\()p Fo(z)r(;)765 2030 y Fj(e)760 2034 y Fo(q)r Fq(\))p Fn(j)i Fq(=)h Fn(j)p Fo(z)929 1976 y Fj(Z)971 1989 y Fp(1)952 2070 y(0)990 2034 y Fq(\()p Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(q)r(;)g(t)p Fq(\))i Fn(\000)h Fo(')p Fq(\(1)p Fo(;)d(z)r(;)1444 2030 y Fj(e)1440 2034 y Fo(q)q(;)g(t)p Fq(\)\))p Fo(dt)p Fn(j)p Fo(;)282 b Fq(\(3.5\))46 2153 y(and)291 2224 y Fo(')p Fq(\(1)p Fo(;)8 b(z)r(;)g(p;)g(t)p Fq(\))j Fn(\000)g Fo(')p Fq(\(1)p Fo(;)d(z)r(;)727 2220 y Fj(e)723 2224 y Fo(q)q(;)g(t)p Fq(\))14 b(=)870 2165 y Fj(Z)912 2178 y Fp(1)893 2259 y(0)932 2224 y Fq(\()p Fo(@)s(')p Fq(\(1)p Fo(;)8 b(z)r(;)1127 2220 y Fj(e)1123 2224 y Fo(q)k Fq(+)f Fo(y)r(v)r(;)d(t)p Fq(\))p Fo(;)g(v)r Fq(\))p Fo(dy)r(;)87 b(v)15 b Fq(=)f Fo(p)d Fn(\000)1717 2220 y Fj(e)1712 2224 y Fo(q)r(;)144 b Fq(\(3.6\))46 2325 y(where)16 b(\()p Fn(\001)p Fo(;)8 b Fn(\001)p Fq(\))15 b(is)i(the)f(scalar)g(pro)q(duct)h(in)f Fo(H)842 2332 y Fk(C)872 2325 y Fq(.)21 b(The)16 b(iden)o(tities)f(\(2.19\))h(and)h (the)f(estimate)f(\(2.9\))h(yield)412 2451 y Fn(j)p Fo(@)s(')p Fq(\(1)p Fo(;)8 b(z)r(;)602 2447 y Fj(e)598 2451 y Fo(q)13 b Fq(+)e Fo(y)r(v)r(;)d(t)p Fq(\))p Fn(j)k Fg(6)890 2418 y Fq(2)p 876 2440 53 2 v 876 2485 a Fn(j)p Fo(z)r Fn(j)934 2451 y Fo(e)957 2431 y Fm(j)5 b Fp(Im)g Fk(z)q Fm(j)p Fp(+4)p Fm(k)p Fj(e)-23 b Fk(q)q Fp(+)p Fk(y)q(v)q Fm(k)1225 2451 y Fg(6)1297 2418 y Fq(2)p 1283 2440 V 1283 2485 a Fn(j)p Fo(z)r Fn(j)1340 2451 y Fo(e)1363 2431 y Fm(j)6 b Fp(Im)f Fk(z)q Fm(j)p Fp(+4)p Fm(k)p Fk(q)q Fm(k)p Fp(+8)p Fk(")1615 2451 y Fo(;)265 b Fq(\(3.7\))46 2583 y(since)15 b Fn(k)p Fo(v)r Fn(k)e Fg(6)h Fn(k)p Fo(p)d Fn(\000)g Fo(q)r Fn(k)g Fq(+)g Fn(k)p Fo(q)h Fn(\000)639 2579 y Fj(e)635 2583 y Fo(q)q Fn(k)i Fg(6)g Fq(2)p Fo(")p Fq(.)21 b(Substituting)16 b(this)g(estimate)f(in)o(to)h(\(3.5\))g(w)o (e)g(ha)o(v)o(e)626 2687 y Fn(j)653 2674 y Fq(_)640 2687 y(\001)o(\()p Fo(z)r(;)8 b(p)p Fq(\))k Fn(\000)864 2674 y Fq(_)850 2687 y(\001\()p Fo(z)r(;)961 2683 y Fj(e)957 2687 y Fo(q)q Fq(\))p Fn(j)i Fg(6)f Fq(4)p Fo("e)1149 2666 y Fm(j)6 b Fp(Im)f Fk(z)q Fm(j)p Fp(+4)p Fm(k)p Fk(q)q Fm(k)p Fp(+8)p Fk(")1401 2687 y Fo(:)479 b Fq(\(3.8\))996 2811 y(15)p eop %%Page: 16 16 16 15 bop 46 138 a Fq(Using)16 b(\(3.4\))g(and)h(\(3.7\))g(w)o(e)f (deduce)f(that)433 246 y Fn(j)461 233 y Fq(_)447 246 y(\001\()p Fo(z)r(;)8 b(p)p Fq(\))j(+)g(sin)d Fo(z)r Fn(j)14 b Fg(6)f Fn(j)858 233 y Fq(_)844 246 y(\001\()p Fo(z)r(;)8 b(p)p Fq(\))j Fn(\000)1068 233 y Fq(_)1055 246 y(\001\()p Fo(z)r(;)1166 242 y Fj(e)1161 246 y Fo(q)r Fq(\))p Fn(j)g Fq(+)g Fn(j)1305 233 y Fq(_)1292 246 y(\001)o(\()p Fo(z)r(;)1402 242 y Fj(e)1398 246 y Fo(q)q Fq(\))g(+)g(sin)d Fo(z)r Fn(j)769 353 y Fg(6)14 b Fq(4)p Fo("e)892 332 y Fm(j)p Fp(Im)5 b Fk(z)q Fm(j)p Fp(+8)p Fk(")1041 353 y Fq(\(1)11 b(+)g Fo(e)1167 332 y Fp(4)p Fm(k)p Fk(q)q Fm(k)1239 353 y Fq(\))p Fo(:)46 439 y Fq(Therefore,)k(the)h(simple)e (estimate)h(exp)7 b Fn(j)p Fq(Im)g Fo(z)r Fn(j)13 b Fo(<)h Fq(4)p Fn(j)8 b Fq(sin)h Fo(z)r Fn(j)16 b Fq(on)h(all)e(con)o(tours)i (\(see)f([PT]\))g(yields)462 546 y Fn(j)490 534 y Fq(_)476 546 y(\001\()p Fo(z)r(;)8 b(p)p Fq(\))j(+)g(sin)d Fo(z)r Fn(j)14 b Fg(6)f Fo(C)t Fn(j)8 b Fq(sin)g Fo(z)r Fn(j)p Fo(;)73 b(C)17 b Fq(=)d(16)p Fo("e)1312 526 y Fp(8)p Fk(")1348 546 y Fq(\(1)d(+)g Fo(e)1474 526 y Fp(4)p Fm(k)p Fk(q)q Fm(k)1546 546 y Fq(\))p Fo(:)46 653 y Fq(By)17 b(assumptions,)h Fo(C)i(<)d Fq(1)h(on)h(all)e(con)o(tours.)27 b(Hence,)16 b(b)o(y)i(Rouc)o(h)o(\023)-23 b(e's)17 b(theorem,)1585 641 y(_)1571 653 y(\001\()p Fn(\001)p Fo(;)8 b(p)p Fq(\))18 b(has)h(as)f(man)o(y)46 714 y(ro)q(ots,)c(coun)o(ted)f(with)g(m)o (ultipli)o(citi)o(es,)d(as)k(sin)8 b Fo(z)15 b Fq(in)d(eac)o(h)h(of)g (the)g(b)q(ounded)h(domains)e(and)i(the)e(remaining)46 774 y(un)o(b)q(ounded)20 b(region.)29 b(Since)18 b(sin)8 b Fo(z)21 b Fq(has)f(only)e(simple)f(ro)q(ots)j Fo(\031)r(n;)8 b(n)18 b Fn(2)h Fl(Z)-11 b Fq(,)16 b(and)k(since)e Fo(M)24 b(>)18 b(N)24 b Fq(can)c(b)q(e)46 834 y(c)o(hosen)c(arbitrarily)f (large,)h(the)g(p)q(oin)o(t)g(i\))g(of)h(Lemma)d(3.1)i(follo)o(ws.)119 894 y(ii\))k(In)g(order)h(to)g(pro)o(v)o(e)f(compactness)g(of)h Fo(z)953 901 y Fk(n)977 894 y Fq(\()p Fo(q)r Fq(\),)f(supp)q(ose)i(the) f(sequence)f Fo(q)1581 876 y Fk(\027)1602 894 y Fo(;)8 b(\027)24 b Fn(\025)e Fq(1,)g(con)o(v)o(erges)46 954 y(w)o(eakly)17 b(to)h Fo(q)h Fq(in)f Fo(H)t Fq(.)26 b(Let)18 b Fo(z)568 936 y Fp(0)566 967 y Fk(n)589 954 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(n)16 b Fn(\025)h Fq(1,)h(b)q(e)g(the)g(zeros)g (of)g(the)f(function)h(\001\()p Fo(z)r(;)8 b(q)r Fq(\))p Fo(;)g(z)17 b(>)g Fq(0.)26 b(De\014ne)18 b(the)46 1015 y(segmen)o(ts)d Fo(I)280 997 y Fp(0)276 1027 y Fk(n)314 1015 y Fq(=)g([)p Fo(z)406 997 y Fp(0)404 1027 y Fk(n)427 1015 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(z)536 997 y Fp(0)534 1027 y Fk(n)p Fp(+1)601 1015 y Fq(\()p Fo(q)r Fq(\)])p Fo(;)24 b(n)15 b Fn(\025)f Fq(1.)23 b(In)17 b(eac)o(h)f Fo(I)1071 997 y Fp(0)1067 1027 y Fk(n)1090 1015 y Fo(;)8 b(n)15 b Fn(\025)g Fq(1,)i(there)f(exists)g(exactly)g(one)h(zero)f Fo(z)1910 1022 y Fk(n)1933 1015 y Fq(\()p Fo(q)r Fq(\))46 1080 y(of)g(the)g(function)390 1068 y(_)376 1080 y(\001\()p Fo(z)r(;)8 b(q)r Fq(\),)15 b(and)678 1187 y Fo(z)703 1167 y Fp(0)701 1200 y Fk(n)724 1187 y Fq(\()p Fo(q)r Fq(\))f Fo(<)f(z)874 1194 y Fk(n)898 1187 y Fq(\()p Fo(q)r Fq(\))g Fo(<)h(z)1050 1167 y Fp(0)1048 1200 y Fk(n)p Fp(+1)1116 1187 y Fq(\()p Fo(q)r Fq(\)])p Fo(;)23 b(n)14 b Fn(\025)f Fq(1)p Fo(;)532 b Fq(\(3.9\))46 1295 y(since)15 b(the)g(function)h(\001\()p Fn(\001)p Fo(;)8 b(q)r Fq(\))14 b(is)i(en)o(tire)e(with)i(real)f(zero)q(es)h(\(see)f([K1]\).)21 b(Using)15 b(Theorem)g(2.1)h(w)o(e)f(obtain)46 1355 y(\001\()p Fo(z)r(;)8 b(q)177 1337 y Fk(\027)197 1355 y Fq(\))16 b Fn(!)f Fq(\001\()p Fo(z)r(;)8 b(q)r Fq(\))16 b(and)573 1342 y(_)559 1355 y(\001\()p Fo(z)r(;)8 b(q)690 1337 y Fk(\027)710 1355 y Fq(\))16 b Fn(!)824 1342 y Fq(_)810 1355 y(\001\()p Fo(z)r(;)8 b(q)r Fq(\))16 b(as)i Fo(\027)h Fn(!)d(1)p Fq(,)h(uniformly)e(on)j(b)q(ounded)g(subsets)g(of)f Fl(C)9 b Fq(.)46 1415 y(Then)16 b(b)o(y)f(\(3.9\),)h(in)f(eac)o(h)h Fo(I)562 1397 y Fp(0)558 1427 y Fk(n)581 1415 y Fo(;)8 b Fq(1)14 b Fg(6)f Fo(n)h Fg(6)g Fo(N)5 b Fq(,)16 b(for)g(an)o(y)f Fo(N)20 b Fn(\025)13 b Fq(2,)j(there)f(exists)g(exactly)g(one)h(zero)f Fo(z)1833 1422 y Fk(n)1857 1415 y Fq(\()p Fo(q)1900 1397 y Fk(\027)1921 1415 y Fq(\))g(of)46 1475 y(the)h(function)334 1463 y(_)321 1475 y(\001\()p Fo(z)r(;)8 b(q)452 1457 y Fk(\027)472 1475 y Fq(\))16 b(for)h(large)f Fo(\027)s Fq(.)22 b(F)l(or)16 b(small)f Fo(\016)g(>)f Fq(0)i(w)o(e)g(in)o(tro)q (duce)g(the)g(in)o(terv)m(als)547 1583 y Fo(I)569 1590 y Fk(n)606 1583 y Fq(=)d([)p Fo(z)694 1590 y Fk(n)717 1583 y Fq(\()p Fo(q)r Fq(\))d Fn(\000)h Fo(\016)o(;)d(z)905 1590 y Fk(n)928 1583 y Fq(\()p Fo(q)r Fq(\))i(+)h Fo(\016)r Fq(])i Fn(\032)h Fo(I)1179 1562 y Fp(0)1175 1595 y Fk(n)1198 1583 y Fo(;)40 b Fq(1)15 b Fg(6)e Fo(n)h Fg(6)g Fo(N)r(:)46 1690 y Fq(If)j Fo(\016)i Fq(is)e(su\016cien)o(tly)e(small,)h(then)h (these)h(in)o(terv)m(als)e(are)i(all)f(disjoin)o(t)g(and)h Fo(I)1455 1697 y Fk(n)1494 1690 y Fn(\032)d Fo(I)1574 1672 y Fp(0)1570 1702 y Fk(n)1593 1690 y Fo(;)26 b Fq(1)16 b Fg(6)g Fo(n)g Fg(6)g Fo(N)5 b Fq(.)25 b(The)46 1750 y(function)250 1737 y(_)237 1750 y(\001\()p Fo(z)r(;)8 b(q)r Fq(\))15 b(c)o(hanges)h(sign)h(on)g(eac)o(h)e(of)i(them,)d(since) i Fo(z)1197 1757 y Fk(n)1220 1750 y Fq(\()p Fo(q)r Fq(\))f(is)h(a)h (simple)d(ro)q(ot.)119 1810 y(As)j Fo(\027)h Fn(!)d(1)p Fq(,)i(the)f(functions)689 1798 y(_)675 1810 y(\001\()p Fo(z)r(;)8 b(q)806 1792 y Fk(\027)827 1810 y Fq(\))17 b(con)o(v)o(erge)f(to)1137 1798 y(_)1123 1810 y(\001\()p Fo(z)r(;)8 b(q)r Fq(\))16 b(uniformly)f(on)i Fn([)p Fo(I)1635 1817 y Fk(n)1675 1810 y Fq(b)o(y)g(Lemma)e(2.4.)46 1870 y(Hence,)c(for)i(su\016cien)o(tly)e(large)h Fo(\027;)h Fq(they)f(also)i(c)o(hange)e(sign)h(on)g Fn([)p Fo(I)1253 1877 y Fk(n)1276 1870 y Fo(;)g Fq(so)g(they)f(m)o(ust)f(all)h(ha)o(v)o (e)g(at)h(least)g(one)46 1931 y(ro)q(ot)19 b(in)f(eac)o(h)g(of)g(these) g(in)o(terv)m(als.)26 b(The)19 b(functions)f Fn(j)p Fq(\001\()p Fo(z)r(;)8 b(q)r Fq(\))p Fn(j)16 b(\025)h Fq(1)p Fo(=)p Fq(2)i(and)g Fn(j)p Fq(\001\()p Fo(z)r(;)8 b(q)1625 1913 y Fk(\027)1645 1931 y Fq(\))p Fn(j)17 b(\025)g Fq(1)p Fo(=)p Fq(4)i(for)g(an)o(y)46 1991 y Fo(z)d Fn(2)f([)p Fo(I)188 1998 y Fk(n)228 1991 y Fq(and)i(for)g(su\016cien)o(tly)e (large)i Fo(\027)s Fq(.)22 b(Hence)16 b(b)o(y)g(\(3.9\),)1184 1978 y(_)1171 1991 y(\001)o(\()p Fo(z)r(;)8 b(q)1301 1973 y Fk(\027)1322 1991 y Fq(\))17 b(has)g(exactly)f(one)g(ro)q(ot)i Fo(z)1831 1998 y Fk(n)1854 1991 y Fq(\()p Fo(q)1897 1973 y Fk(\027)1918 1991 y Fq(\))f(in)46 2051 y(eac)o(h)d(segmen)o(t)g Fo(I)364 2058 y Fk(n)387 2051 y Fo(;)8 b Fq(1)14 b Fg(6)g Fo(n)f Fg(6)h Fo(N)r(:)h Fq(This)g(yields)f Fn(j)p Fo(z)948 2058 y Fk(n)971 2051 y Fq(\()p Fo(q)1014 2033 y Fk(\027)1035 2051 y Fq(\))9 b Fn(\000)g Fo(z)1134 2058 y Fk(n)1157 2051 y Fq(\()p Fo(q)r Fq(\))p Fn(j)k Fo(<)g(\016)o(;)38 b Fq(1)15 b Fg(6)e Fo(n)h Fg(6)g Fo(N)r(;)h Fq(for)g(all)g(su\016cien)o (tly)46 2111 y(large)h Fo(\027:)h Fq(It)f(follo)o(ws)h(that)g Fo(z)563 2118 y Fk(n)586 2111 y Fq(\()p Fo(q)629 2093 y Fk(\027)650 2111 y Fq(\))d Fn(!)h Fo(z)771 2118 y Fk(n)794 2111 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g Fq(as)i Fo(\027)h Fn(!)c(1)p Fq(,)i(since)g Fo(N)22 b Fq(and)17 b Fo(")d(>)h Fq(0)i(w)o(ere)f(arbitrary)l(.)22 b(Th)o(us,)46 2171 y Fo(z)69 2178 y Fk(n)92 2171 y Fq(\()p Fo(q)r Fq(\))16 b(is)g(a)g(compact)f(functions)i(of)f Fo(q)r Fq(.)119 2232 y(T)l(o)g(pro)o(v)o(e)g(real)f(analyticit)o(y)l(,)f(w)o(e)h(\014x) h Fo(r)f Fn(2)f Fo(H)t Fq(.)21 b(Then)1112 2219 y(_)1098 2232 y(\001\()p Fo(z)1181 2239 y Fk(n)1204 2232 y Fq(\()p Fo(r)q Fq(\))p Fo(;)8 b(r)q Fq(\))14 b(=)g(0)j(and)1538 2219 y(\177)1530 2232 y(\001\()p Fo(z)1613 2239 y Fk(n)1636 2232 y Fq(\()p Fo(r)q Fq(\))p Fo(;)8 b(r)q Fq(\))14 b Fn(6)p Fq(=)g(0.)21 b(No)o(w,)46 2292 y(the)12 b(implicit)e(function)i (theorem)g(guaran)o(tees)h(the)g(existence)e(of)i(a)g(function)1480 2288 y Fj(e)1477 2292 y Fo(z)1500 2299 y Fk(n)1536 2292 y Fq(de\014ned)f(on)i(some)d(small)46 2352 y(neigh)o(b)q(orho)q(o)q(d) 18 b Fo(W)j Fn(\032)13 b Fo(H)21 b Fq(of)16 b Fo(r)i Fq(and)f(suc)o(h)f(that)673 2447 y(_)660 2459 y(\001)o(\()723 2455 y Fj(e)719 2459 y Fo(z)742 2466 y Fk(n)765 2459 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))13 b(=)h(0)p Fo(;)1104 2455 y Fj(e)1100 2459 y Fo(z)1123 2466 y Fk(n)1147 2459 y Fq(\()p Fo(r)q Fq(\))g(=)f Fo(z)1296 2466 y Fk(n)1320 2459 y Fq(\()p Fo(r)q Fq(\))46 2567 y(on)21 b Fo(W)o(:)e Fq(F)l(urthermore,)501 2563 y Fj(e)497 2567 y Fo(z)520 2574 y Fk(n)543 2567 y Fq(\()p Fo(q)r Fq(\))h(is)g(real)g(analytic.)33 b(On)20 b(the)g(other)h(hand,)g Fo(z)1462 2574 y Fk(n)1485 2567 y Fq(\()p Fo(q)r Fq(\))f(is)g(also)h(a)g(con)o(tin)o(uous)46 2627 y(function)13 b(on)i Fo(W)20 b Fq(satisfying)595 2614 y(_)581 2627 y(\001\()p Fo(z)664 2634 y Fk(n)687 2627 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))k(=)i(0)p Fo(:)g Fq(Therefore,)f(b)o(y)g(uniqueness,)1488 2623 y Fj(e)1485 2627 y Fo(z)1508 2634 y Fk(n)1531 2627 y Fq(\()p Fo(q)r Fq(\))g(=)h Fo(z)1681 2634 y Fk(n)1704 2627 y Fq(\()p Fo(q)r Fq(\))f(on)h Fo(W)o(;)f Fq(and)46 2687 y(so)k Fo(z)129 2694 y Fk(n)152 2687 y Fq(\()p Fo(q)r Fq(\))e(is)i(real)e(analytic.)996 2811 y(16)p eop %%Page: 17 17 17 16 bop 119 138 a Fq(T)l(o)17 b(calculate)e(the)h(gradien)o(t,)g(w)o (e)f(observ)o(e)h(that)1047 126 y(_)1034 138 y(\001)o(\()p Fo(z)1116 145 y Fk(n)1140 138 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))k(=)i(0)p Fo(:)i Fq(Hence)493 237 y(0)f(=)e Fo(d)608 244 y Fk(q)628 237 y Fn(f)666 225 y Fq(_)653 237 y(\001)o(\()p Fo(z)735 244 y Fk(n)758 237 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))p Fn(g)13 b Fq(=)983 225 y(\177)975 237 y(\001)o(\()p Fo(z)1057 244 y Fk(n)1080 237 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))p Fo(d)1232 244 y Fk(q)1250 237 y Fo(z)1273 244 y Fk(n)1307 237 y Fq(+)j Fo(@)1398 225 y Fq(_)1385 237 y(\001\()p Fo(z)r(;)d(q)r Fq(\))p Fo(;)46 336 y Fq(whic)o(h)15 b(implies)f(\(3.1\).)119 396 y(Using)f(i\),)g(w)o(e)g(obtain)g Fn(j)p Fo(z)567 403 y Fk(n)590 396 y Fq(\()p Fo(q)r Fq(\))5 b Fn(\000)g Fo(\031)r(n)p Fn(j)12 b Fg(6)i Fq(1.)20 b(W)l(e)13 b(can)g(impro)o(v)o(e)e(i\).)20 b(The)13 b(asymptotic)f(estimates)f (\(2.22\))46 456 y(yield)180 444 y(_)166 456 y(\001\()p Fo(z)249 463 y Fk(n)272 456 y Fq(\()p Fo(q)r Fq(\))p Fo(;)d(q)r Fq(\))17 b(=)i Fn(\000)8 b Fq(sin)g Fo(z)611 463 y Fk(n)634 456 y Fq(\()p Fo(q)r Fq(\))k(+)h Fo(`)779 438 y Fk(d)800 456 y Fq(\()p Fo(n)p Fq(\).)29 b(Then)19 b(sin\()p Fo(z)1142 463 y Fk(n)1165 456 y Fq(\()p Fo(q)r Fq(\))13 b Fn(\000)f Fo(\031)r(n)p Fq(\))18 b(=)h Fo(`)1464 438 y Fk(d)1484 456 y Fq(\()p Fo(n)p Fq(\))g(whic)o(h)g(implies)d (\(3.2\).)46 517 y(Hence)f(from)g(\(2.22\),)h(\(2.25\))h(w)o(e)f (conclude)377 651 y Fo(d)402 658 y Fk(q)421 651 y Fo(z)444 658 y Fk(n)467 651 y Fq(\()p Fo(q)r Fq(\))d(=)h Fn(\000)638 618 y Fo(@)680 605 y Fq(_)667 618 y(\001)o(\()p Fo(z)749 625 y Fk(n)772 618 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))p 638 640 260 2 v 660 678 a(\177)652 690 y(\001\()p Fo(z)735 697 y Fk(n)758 690 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))916 651 y(=)1094 618 y Fo(`)1114 600 y Fp(2)1134 618 y Fq(\()p Fo(n)p Fq(\))p 973 640 349 2 v 973 686 a(cos)h Fo(z)1070 693 y Fk(n)1093 686 y Fq(\()p Fo(q)r Fq(\))h(+)h Fo(`)1234 671 y Fp(2)1254 686 y Fq(\()p Fo(n)p Fq(\))1340 651 y(=)j Fo(`)1412 631 y Fp(2)1432 651 y Fq(\()p Fo(n)p Fq(\))p Fo(:)114 b Fb(2)119 785 y Fq(W)l(e)16 b(need)g(the)g(follo)o(wing)g(results)g(concerning)g(the) g(function)g Fo(r)1303 792 y Fk(n)1340 785 y Fq(=)e Fo(h)1420 767 y Fp(2)1420 797 y Fk(n)1443 785 y Fq(.)46 845 y Fr(Lemm)o(a)i(3.2.) 21 b Fd(Each)d(function)g Fo(r)689 852 y Fk(n)713 845 y Fq(\()p Fn(\001)p Fq(\))p Fo(;)8 b(n)13 b Fn(\025)h Fq(1)p Fo(;)j Fd(is)g(c)n(omp)n(act)g(and)g(r)n(e)n(al)g(analytic)h(on) g Fo(H)t Fd(.)k(Its)17 b(gr)n(adient)h(is)46 906 y(given)h(by)e(the)h (formula)499 1029 y Fq(\()p Fo(d)543 1036 y Fk(q)562 1029 y Fo(r)584 1036 y Fk(n)607 1029 y Fq(\()p Fo(q)r Fq(\)\)\()p Fo(x)p Fq(\))13 b(=)h(\()p Fn(\000)p Fq(1\))920 1009 y Fk(n)949 995 y Fq(\()p Fo(@)s Fq(\001\()p Fo(z)1080 1002 y Fk(n)1102 995 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\)\)\()p Fo(x)p Fq(\))p 949 1017 364 2 v 953 1063 a(\()p Fo(d)g Fq(cosh)1106 1031 y Fn(p)p 1148 1031 46 2 v 32 x Fo(r)1170 1070 y Fk(n)1193 1063 y Fo(=dr)1264 1070 y Fk(n)1288 1063 y Fq(\))1317 1029 y Fo(;)77 b(n)14 b Fn(\025)g Fq(1)p Fo(:)328 b Fq(\(3.10\))46 1157 y Fd(Mor)n(e)n(over,) 16 b(for)h(any)g(\014xe)n(d)h Fo(d)d(>)e Fq(1)18 b Fd(the)g(fol)r (lowing)i(estimates)e(ar)n(e)f(ful\014l)r(le)n(d:)659 1256 y Fo(r)681 1263 y Fk(n)705 1256 y Fq(\()p Fo(q)r Fq(\))c(=)h Fo(q)856 1235 y Fp(2)854 1268 y Fk(n)888 1256 y Fq(+)d(\()p Fo(`)976 1235 y Fk(d)997 1256 y Fq(\()p Fo(n)p Fq(\)\))1083 1235 y Fp(3)1102 1256 y Fo(;)95 b(n)14 b Fn(!)g(1)p Fo(;)488 b Fq(\(3.11\))643 1354 y(\()p Fo(d)687 1361 y Fk(q)706 1354 y Fo(r)728 1361 y Fk(n)752 1354 y Fq(\()p Fo(q)r Fq(\)\)\()p Fo(x)p Fq(\))13 b(=)g Fo(d)988 1361 y Fk(q)1008 1354 y Fo(q)1032 1334 y Fp(2)1030 1367 y Fk(n)1064 1354 y Fq(+)e Fo(`)1133 1334 y Fk(d)1154 1354 y Fo(;)60 b(n)14 b Fn(!)f(1)p Fo(;)472 b Fq(\(3.12\))46 1437 y Fd(uniformly)17 b(on)h Fq([0)p Fo(;)8 b Fq(1])j Fn(\002)f Fo(B)533 1444 y Fk(C)563 1437 y Fq(\()p Fo(p;)e(")651 1444 y Fk(p)671 1437 y Fq(\))17 b Fd(for)g(e)n(ach)h(\014xe)n(d)g Fo(p)c Fn(2)g Fo(H)t Fd(.)46 1497 y(Pr)n(o)n(of.)30 b Fq(Since)18 b(\001\()p Fo(z)419 1504 y Fk(n)442 1497 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))18 b(is)i(compact,)e(so)j (is)e Fo(r)990 1504 y Fk(n)1013 1497 y Fq(\()p Fo(q)r Fq(\).)31 b(In)19 b(order)h(to)g(pro)o(v)o(e)e(real)i(analyticit)o(y)l (,)e(w)o(e)h(\014x)46 1557 y Fo(v)c Fn(2)f Fo(H)t Fq(.)21 b(Then)13 b(\()p Fn(\000)p Fq(1\))436 1539 y Fk(n)459 1557 y Fq(\001\()p Fo(z)542 1564 y Fk(n)565 1557 y Fq(\()p Fo(v)r Fq(\))p Fo(;)8 b(v)r Fq(\))k(=)i(cosh)861 1525 y Fn(p)p 902 1525 V 902 1557 a Fo(r)924 1564 y Fk(n)961 1557 y Fq(and)f(\(cosh)1172 1525 y Fn(p)p 1214 1525 V 32 x Fo(r)1236 1564 y Fk(n)1259 1557 y Fq(\))1278 1539 y Fm(0)1278 1570 y Fk(r)1294 1574 y Fc(n)1331 1557 y Fn(6)p Fq(=)h(0.)20 b(No)o(w,)13 b(the)g(implicit)d(function)46 1617 y(theorem)15 b(guaran)o(tees)i(the)f(existence)f(of)i(a)g(unique)f (con)o(tin)o(uous)g(function)1465 1613 y Fj(e)1462 1617 y Fo(r)1484 1624 y Fk(n)1524 1617 y Fq(de\014ned)h(on)g(some)e(small)46 1678 y(neigh)o(b)q(orho)q(o)q(d)j Fo(W)j Fn(\032)13 b Fo(H)21 b Fq(of)16 b Fo(v)i Fq(and)f(suc)o(h)f(that)477 1786 y(\()p Fn(\000)p Fq(1\))578 1765 y Fk(n)602 1786 y Fq(\001\()p Fo(z)685 1793 y Fk(n)707 1786 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))13 b(=)g(cosh)999 1733 y Fj(q)p 1041 1733 107 2 v 1044 1782 a(e)1041 1786 y Fo(r)1063 1793 y Fk(n)1086 1786 y Fq(\()p Fo(q)r Fq(\))o Fo(;)1270 1782 y Fj(e)1267 1786 y Fo(r)1289 1793 y Fk(n)1312 1786 y Fq(\()p Fo(v)r Fq(\))g(=)h Fo(r)1463 1793 y Fk(n)1487 1786 y Fq(\()p Fo(v)r Fq(\))p Fo(;)46 1885 y Fq(on)21 b Fo(W)o(:)e Fq(F)l(urthermore,)500 1881 y Fj(e)498 1885 y Fo(r)520 1892 y Fk(n)543 1885 y Fq(\()p Fo(q)r Fq(\))h(is)g(real)g (analytic.)33 b(On)20 b(the)h(other)f(hand,)i Fo(r)1462 1892 y Fk(n)1485 1885 y Fq(\()p Fo(q)r Fq(\))e(is)g(also)h(a)g(con)o (tin)o(uous)46 1955 y(function)14 b(on)g Fo(W)21 b Fq(satisfying)14 b(\()p Fn(\000)p Fq(1\))683 1937 y Fk(n)707 1955 y Fq(\001\()p Fo(z)790 1962 y Fk(n)813 1955 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))k(=)i(cosh)1104 1905 y Fj(q)p 1146 1905 V 50 x Fo(r)1168 1962 y Fk(n)1191 1955 y Fq(\()p Fo(q)r Fq(\))p Fo(:)f Fq(Therefore,)h(b)o(y)g(uniqueness,)1839 1951 y Fj(e)1836 1955 y Fo(r)1858 1962 y Fk(n)1882 1955 y Fq(\()p Fo(q)r Fq(\))f(=)46 2019 y Fo(r)68 2026 y Fk(n)91 2019 y Fq(\()p Fo(q)r Fq(\))j(on)h Fo(W)o(;)e Fq(and)i(so)g Fo(r)488 2026 y Fk(n)511 2019 y Fq(\()p Fo(q)r Fq(\))f(is)g(real)g (analytic.)119 2089 y(T)l(o)21 b(calculate)e(the)g(gradien)o(t,)i(w)o (e)e(observ)o(e)h(that)g(\()p Fn(\000)p Fq(1\))1162 2071 y Fk(n)1186 2089 y Fq(\001\()p Fo(z)1269 2096 y Fk(n)1292 2089 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))18 b(=)j(cosh)1596 2039 y Fj(q)p 1638 2039 V 50 x Fo(r)1660 2096 y Fk(n)1683 2089 y Fq(\()p Fo(q)r Fq(\))p Fo(:)e Fq(Hence)g(the)46 2149 y(implici)o(t)14 b(function)i(theorem)e(yields)345 2248 y(\()p Fn(\000)p Fq(1\))446 2228 y Fk(n)469 2248 y Fq(\()p Fo(d)8 b Fq(cosh)623 2214 y Fn(p)p 664 2214 46 2 v 664 2248 a Fo(r)686 2255 y Fk(n)710 2248 y Fo(=dr)781 2255 y Fk(n)805 2248 y Fq(\))p Fo(d)849 2255 y Fk(q)868 2248 y Fo(r)890 2255 y Fk(n)927 2248 y Fq(=)14 b Fo(@)s Fq(\001\()p Fo(z)1091 2255 y Fk(n)1113 2248 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))i(+)1313 2236 y(_)1299 2248 y(\001\()p Fo(z)1382 2255 y Fk(n)1405 2248 y Fq(\()p Fo(q)r Fq(\))p Fo(;)e(q)r Fq(\))p Fo(d)1557 2255 y Fk(q)1575 2248 y Fo(z)1598 2255 y Fk(n)1621 2248 y Fq(\()p Fo(q)r Fq(\))p Fo(;)46 2347 y Fq(and)17 b(the)f(iden)o(tit)o(y)418 2335 y(_)405 2347 y(\001\()p Fo(z)488 2354 y Fk(n)511 2347 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))k(=)i(0)i(implies)e (\(3.10\).)119 2407 y(W)l(e)19 b(need)g(a)h(simple)d(estimate)g (concerning)i(the)g(analytic)g(function)g Fo(w)h Fq(=)f Fo(w)q Fq(\()p Fo(z)r Fq(\))h(=)f(cosh)1825 2372 y Fn(p)p 1867 2372 25 2 v 35 x Fo(z)c Fn(\000)e Fq(1)p Fo(;)46 2468 y Fq(where)19 b Fo(z)j Fn(2)e Fl(C)9 b Fq(.)34 b(There)19 b(exists)g(a)h(domain)f Fo(B)j Fn(\032)e Fl(C)9 b Fo(;)f Fq(0)23 b Fn(2)d Fo(B)s(;)e Fq(suc)o(h)i(that)g(the)f(new)h(analytic)f (function)46 2528 y Fo(w)81 2535 y Fp(1)100 2528 y Fq(\()p Fo(z)r Fq(\))g Fn(\021)f Fo(w)q Fq(\()p Fo(z)r Fq(\))p Fo(;)8 b(z)20 b Fn(2)f Fo(B)s(;)f Fq(has)h(an)h(in)o(v)o(erse)d (function)i Fo(g)r Fq(\()p Fo(w)q Fq(\),)g(analytic)g(in)f Fn(j)p Fo(w)q Fn(j)g Fo(<)h Fq(2)p Fo(;)g Fq(and)g(the)g(follo)o(wing) 46 2588 y(estimates)14 b(are)j(ful\014lled:)588 2687 y Fo(z)f Fq(=)e Fo(g)r Fq(\()p Fo(w)q Fq(\))g(=)g(2)p Fo(w)q Fq(\(1)e Fn(\000)f Fo(w)q(=)p Fq(6)h(+)f Fo(:)d(:)g(:)o Fq(\))p Fo(;)57 b(w)15 b Fn(!)e Fq(0)p Fo(:)418 b Fq(\(3.13\))996 2811 y(17)p eop %%Page: 18 18 18 17 bop 46 138 a Fq(No)o(w)18 b(w)o(e)g(let)g Fo(n)g Fn(!)f(1)p Fq(.)28 b(The)19 b(function)f Fo(t)826 145 y Fk(n)849 138 y Fq(\()p Fo(q)r Fq(\))f Fn(\021)h Fq(\()p Fn(\000)p Fq(1\))1086 120 y Fk(n)1109 138 y Fq(\001\()p Fo(z)1192 145 y Fk(n)1215 138 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))j Fn(\000)i Fq(1)18 b(is)h(real)f(analytic)g(and)h (\(3.2\),)46 199 y(\(2.22\))e(imply)d(the)i(follo)o(wing)g(estimates) 422 337 y Fo(t)440 344 y Fk(n)477 337 y Fq(=)d([\()p Fn(\000)p Fq(1\))643 317 y Fk(n)675 337 y Fq(cos)8 b Fo(z)771 344 y Fk(n)806 337 y Fq(+)j Fo(q)879 317 y Fp(2)877 350 y Fk(n)900 337 y Fo(=)p Fq(2)h(+)f(\()p Fo(`)1048 317 y Fk(d)1068 337 y Fq(\()p Fo(n)p Fq(\)\))1154 317 y Fp(2)1174 337 y Fq(])g Fn(\000)f Fq(1)15 b(=)1343 304 y Fo(q)1367 285 y Fp(2)1365 316 y Fk(n)p 1343 326 46 2 v 1354 371 a Fq(2)1404 337 y(+)d(\()p Fo(`)1493 317 y Fk(d)1513 337 y Fq(\()p Fo(n)p Fq(\)\))1599 317 y Fp(2)1870 337 y Fq(\(3.14\))46 462 y(uniformly)i(on)j Fo(B)373 469 y Fk(C)402 462 y Fq(\()p Fo(p;)8 b(")490 469 y Fk(p)510 462 y Fq(\).)21 b(Hence)15 b(the)h(function)g Fo(t)1002 469 y Fk(n)1025 462 y Fq(\()p Fo(r)1066 469 y Fk(n)1090 462 y Fq(\))e(=)f(cosh)1275 430 y Fn(p)p 1317 430 V 32 x Fo(r)1339 469 y Fk(n)1373 462 y Fn(\000)e Fq(1)16 b(has)h(an)g(in)o (v)o(erse)d Fo(r)1801 469 y Fk(n)1839 462 y Fq(=)g Fo(g)r Fq(\()p Fo(t)1953 469 y Fk(n)1976 462 y Fq(\))46 522 y(for)i Fn(j)p Fo(t)152 529 y Fk(n)175 522 y Fn(j)e Fo(<)g Fq(2)p Fo(;)i Fq(and)h(\(3.13-14\))h(yield)592 631 y Fo(r)614 638 y Fk(n)651 631 y Fq(=)c(2)p Fo(t)745 638 y Fk(n)768 631 y Fq(\(1)e Fn(\000)f Fo(t)891 638 y Fk(n)914 631 y Fo(=)p Fq(6)h(+)f Fo(:)d(:)g(:)o Fq(\))14 b(=)f Fo(q)1188 611 y Fp(2)1186 643 y Fk(n)1221 631 y Fq(+)e(\()p Fo(`)1309 611 y Fk(d)1329 631 y Fq(\()p Fo(n)p Fq(\)\))1415 611 y Fp(2)1435 631 y Fo(:)46 740 y Fq(Substituting)16 b(the)g(asymptotic)f(estimates)g(\(3.2\),)h(\(3.11\),)g(\(2.25\))h(in)o (to)f(\(3.10\))h(w)o(e)f(obtain)363 881 y Fo(d)388 888 y Fk(q)407 881 y Fo(r)429 888 y Fk(n)466 881 y Fq(=)e(\()p Fn(\000)p Fq(1\))619 860 y Fk(n)696 847 y Fo(@)s Fq(\001\()p Fo(z)808 854 y Fk(n)830 847 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\))p 648 869 356 2 v 648 915 a(\(sinh)762 883 y Fn(p)p 803 883 46 2 v 803 915 a Fo(r)825 922 y Fk(n)849 915 y Fo(=)p Fq(2)897 883 y Fn(p)p 939 883 V 32 x Fo(r)961 922 y Fk(n)985 915 y Fq(\))1022 881 y(=)1079 847 y Fo(d)1104 854 y Fk(q)1124 847 y Fo(q)1148 829 y Fp(2)1146 859 y Fk(n)1180 847 y Fq(+)j Fo(`)1249 829 y Fk(d)1269 847 y Fq(\()p Fo(n)p Fq(\))p 1079 869 258 2 v 1105 915 a(1)g(+)g Fo(O)q Fq(\()p Fo(r)1268 922 y Fk(n)1292 915 y Fq(\))1355 881 y(=)j Fo(d)1432 888 y Fk(q)1451 881 y Fo(q)1475 860 y Fp(2)1473 893 y Fk(n)1508 881 y Fq(+)d Fo(`)1577 860 y Fk(d)1597 881 y Fq(\()p Fo(n)p Fq(\))p 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Fk(n)218 597 y Fq(\()p Fo(q)r Fq(\))p Fo(=f)328 604 y Fk(n)351 597 y Fq(\()p Fo(q)r Fq(\))h(on)h Fo(H)q(:)g Fq(Then)g(w)o(e)f(can)h (de\014ne)f(a)h(compact,)e(real)h(analytic)g(and)h(p)q(ositiv)o(e)f (function)46 657 y Fo(b)67 664 y Fk(n)104 657 y Fq(=)156 627 y Fn(p)p 197 627 48 2 v 197 657 a Fo(y)221 664 y Fk(n)261 657 y Fq(on)f Fo(H)q(:)h Fq(Using)f(\(4.11-12\))i(w)o(e)e (obtain)g(the)g(asymptotic)f(estimates)g(\(4.8\).)119 718 y(The)h(iden)o(tities)f(\(1.5\),)h(\(1.7\))g(yield)135 828 y Fo(\020)160 807 y Fp(2)156 840 y Fk(n)180 828 y Fo(F)212 835 y Fk(n)235 828 y Fo(=)p Fq(2)e(=)g(\()p Fn(\000)p Fq(1\))450 807 y Fk(n)474 828 y Fq(\(\001\()p Fo(z)576 835 y Fk(n)598 828 y Fq(\))d Fn(\000)g Fq(\001\()p Fo(m)781 835 y Fk(n)804 828 y Fq(\)\))j(=)g(cosh)1008 793 y Fn(p)p 1050 793 46 2 v 35 x Fo(r)1072 835 y Fk(n)1106 828 y Fn(\000)d Fq(cosh)1257 793 y Fn(p)p 1298 793 61 2 v 1298 828 a Fo(r)1320 835 y Fk(cn)1373 828 y Fq(=)j(\()p Fo(r)1466 835 y Fk(n)1500 828 y Fn(\000)d Fo(r)1572 835 y Fk(cn)1611 828 y Fq(\))p Fo(f)1654 835 y Fk(n)1677 828 y Fq(\()p Fo(r)1718 835 y Fk(n)1742 828 y Fo(;)d(r)1786 835 y Fk(cn)1824 828 y Fq(\))p Fo(=)p Fq(2)p Fo(:)46 938 y Fq(Therefore,)15 b(w)o(e)h(obtain)h Fo(r)526 945 y Fk(n)560 938 y Fn(\000)11 b Fo(r)632 945 y Fk(cn)685 938 y Fq(=)j Fo(\020)762 920 y Fp(2)758 950 y Fk(n)781 938 y Fo(y)805 945 y Fk(n)829 938 y Fo(;)h Fq(whic)o(h)h(implies)d (\(4.6\),)k(indeed:)199 1048 y Fo(h)227 1055 y Fk(sn)267 1048 y Fq(\()p Fo(q)r Fq(\))c(=)h Fn(j)p Fo(r)430 1055 y Fk(n)464 1048 y Fn(\000)d Fo(r)536 1055 y Fk(cn)575 1048 y Fn(j)589 1027 y Fp(1)p Fk(=)p Fp(2)644 1048 y Fq(sign\()p Fo(z)770 1055 y Fk(n)793 1048 y Fq(\()p Fo(q)r Fq(\))f Fn(\000)h Fo(m)958 1055 y Fk(n)981 1048 y Fq(\()p Fo(q)r Fq(\)\))i(=)h(\()p Fo(z)1169 1055 y Fk(n)1192 1048 y Fq(\()p Fo(q)r Fq(\))d Fn(\000)g Fo(m)1358 1055 y Fk(n)1381 1048 y Fq(\()p Fo(q)r Fq(\)\))p Fo(b)1483 1055 y Fk(n)1505 1048 y Fq(\()p Fo(q)r Fq(\))p Fo(;)121 b(q)15 b Fn(2)g Fo(H)q(;)46 1158 y Fq(therefore,)g(the)g(function)h Fo(h)566 1165 y Fk(sn)606 1158 y Fq(\()p Fo(u)p Fq(\))g(is)f(compact)g (and)i(real)e(analytic)g(on)i Fo(H)q(:)f Fq(Iden)o(tit)o(y)e(\(4.6\))i (yields)f(\(4.7\).)119 1218 y(The)h(iden)o(tit)o(y)e(\(4.6\))j(and)g (the)f(estimates)e(\(4.8\),)j(\(2.34-35\))g(and)g(\(3.2-3\))g(imply)d (\(4.9-10\).)23 b Fb(2)119 1278 y Fq(No)o(w)16 b(w)o(e)g(recall)f(the)h (w)o(ell)f(kno)o(wn)h(P)o(aley-Wiener)f(result)h(\()g(see)g([Le].)46 1338 y Fr(Lemm)o(a)g(4.3.)21 b Fd(Supp)n(ose)d(that)f Fo(f)5 b Fq(\()p Fo(z)r Fq(\))18 b Fd(is)f(an)h(entir)n(e)g(function)h (with)510 1463 y Fq(sup)513 1504 y Fk(z)q Fm(2)p Fa(C)592 1463 y Fn(j)p Fo(f)5 b Fq(\()p Fo(z)r Fq(\))p Fo(e)721 1443 y Fm(\000j)p Fk(z)q Fm(j)787 1463 y Fn(j)14 b Fo(<)g Fn(1)p Fo(;)1148 1405 y Fj(Z)1171 1499 y Fa(R)1206 1463 y Fn(j)p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)1329 1443 y Fp(2)1349 1463 y Fo(dx)13 b(<)h Fn(1)p Fo(;)46 1601 y Fd(and)k Fo(f)5 b Fq(\()p Fo(z)212 1608 y Fk(n)235 1601 y Fq(\))15 b(=)g(0)j Fd(for)f(some)h(se)n(quenc)n(e)i Fn(f)p Fo(z)814 1608 y Fk(n)837 1601 y Fn(g)862 1583 y Fm(1)862 1614 y(\0001)945 1601 y Fd(of)d(distinct)i(r)n(e)n(al)e (numb)n(ers)i(such)f(that)g Fo(z)1699 1608 y Fk(n)1737 1601 y Fq(=)d Fo(\031)r(n)c Fq(+)g Fo(o)p Fq(\(1\))46 1662 y Fd(as)17 b Fn(j)p Fo(n)p Fn(j)d(!)f(1)p Fd(.)23 b(Then)18 b Fo(f)h Fn(\021)13 b Fq(0)p Fo(:)53 b Fb(2)119 1722 y Fq(W)l(e)16 b(need)h(the)f(follo)o(wing)g(result)h(from)e (analysis)i(and)g(w)o(e)g(pro)o(v)o(e)e(Theorem)h(A.)g(In)g(fact,)g(w)o (e)h(rep)q(eat)46 1782 y(the)k(corresp)q(onding)g(pro)q(of)h(of)g (Theorem)d(5.1)i(from)f([KK1-2].)35 b(W)l(e)21 b(giv)o(e)f(the)h (detail)f(pro)q(of)i(since)e(in)46 1842 y([KK1])15 b(there)h(w)o(as)h (a)f(mistak)o(e,)e(corrected)i(in)f([KK2].)46 1902 y Fr(Theorem)g(A.)f Fd(L)n(et)i Fo(H)q(;)8 b(H)540 1909 y Fp(1)577 1902 y Fd(b)n(e)16 b(r)n(e)n(al)g(sep)n(ar)n(able)g(Hilb)n (ert)g(sp)n(ac)n(es)g(with)g(norms)g Fn(k)8 b(\001)g(k)p Fo(;)g Fn(k)g(\001)g(k)1672 1909 y Fp(1)1691 1902 y Fd(.)22 b(Supp)n(ose)16 b(that)46 1963 y(a)h(map)g Fo(f)i Fq(:)14 b Fo(H)k Fn(!)c Fo(H)428 1970 y Fp(1)465 1963 y Fd(satis\014es)k(the)g (fol)r(lowing)i(c)n(onditions:)46 2023 y(i\))d Fo(f)23 b Fd(is)17 b(r)n(e)n(al)g(analytic,)46 2083 y(ii\))g(the)h(op)n(er)n (ator)e Fo(d)411 2090 y Fk(q)430 2083 y Fo(f)23 b Fd(has)17 b(an)h(inverse)g(for)f(al)r(l)i Fo(q)c Fn(2)f Fo(H)t Fd(,)46 2143 y(iii\))25 b(ther)n(e)h(is)f(a)h(nonde)n(cr)n(e)n(asing)g (function)h Fo(g)k Fq(:)d([0)p Fo(;)8 b Fn(1)p Fq(\))29 b Fn(!)g Fq([0)p Fo(;)8 b Fn(1)p Fq(\))p Fo(;)g(g)r Fq(\(0\))29 b(=)f(0)p Fo(;)e Fd(such)g(that)g Fn(k)p Fo(q)r Fn(k)i Fg(6)46 2203 y Fo(g)r Fq(\()p Fn(k)p Fo(f)5 b Fq(\()p Fo(q)r Fq(\))p Fn(k)231 2210 y Fp(1)250 2203 y Fq(\))18 b Fd(for)e(al)r(l)j Fo(q)c Fn(2)f Fo(H)t Fd(,)46 2263 y(iv\))19 b(ther)n(e)f(exists)i(a)e(b)n(asis)h Fn(f)p Fo(e)589 2270 y Fk(n)612 2263 y Fn(g)637 2245 y Fm(1)637 2276 y Fp(1)693 2263 y Fd(of)f Fo(H)791 2270 y Fp(1)830 2263 y Fd(such)h(that)g(for)f(al)r(l)i Fo(n)c 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Fq(\))943 436 y Fp(1)977 429 y Fq(=)13 b(0)33 b(for)g Fo(n)14 b(>)g(N)5 b Fn(g)14 b(\032)f Fo(H)1446 436 y Fp(1)46 539 y Fq(and)j Fo(M)187 546 y Fk(N)235 539 y Fq(=)d Fo(f)315 521 y Fm(\000)p Fp(1)363 539 y Fq(\()p Fo(K)423 546 y Fk(N)457 539 y Fq(\))h Fn(\032)f Fo(H)q(:)j Fq(The)g(map)e Fo(f)21 b Fq(is)15 b(a)h(smo)q(oth)g(lo)q(cal)f(isomorphism)f(on)i Fo(H)q(;)f Fq(so)i(that)f Fo(M)1873 546 y Fk(N)1922 539 y Fq(is)f(a)46 599 y(real)h(smo)q(oth)g(submanifold)f(of)i Fo(H)j Fq(\(of)d(dimension)e Fo(N)5 b Fq(\).)119 659 y(W)l(e)14 b(sho)o(w)h(that)f(w)o(eak)g(con)o(v)o(ergence)f(in)h Fo(M)913 666 y Fk(N)961 659 y Fq(implies)d(strong)k(con)o(v)o(ergence.) k(Indeed,)14 b(let)f Fo(q)1820 666 y Fk(m)1867 659 y Fn(2)h Fo(M)1961 666 y Fk(N)46 719 y Fq(and)19 b Fo(q)165 726 y Fk(m)217 719 y Fn(!)f Fo(q)i Fq(w)o(eakly)e(as)i Fo(m)e Fn(!)g(1)p Fq(.)29 b(Then)19 b Fo(h)934 726 y Fk(m)986 719 y Fq(=)g Fo(f)5 b Fq(\()p Fo(q)1113 726 y Fk(m)1146 719 y Fq(\))18 b Fn(2)h Fo(K)1276 726 y Fk(N)1329 719 y Fq(and)g(b)o(y)g(Condition)g(iv\),)g Fo(h)1847 726 y Fk(m)1898 719 y Fn(!)g Fo(h)46 780 y Fq(strongly)l(.)24 b(Hence)398 746 y Fj(P)450 780 y Fo(n)479 762 y Fp(2)499 780 y Fq(\()p Fo(f)5 b Fq(\()p Fo(q)588 787 y Fk(m)621 780 y Fq(\))p Fo(;)j(e)685 787 y Fk(n)708 780 y Fq(\))727 762 y Fp(2)727 792 y(1)762 780 y Fo(<)15 b(C)850 787 y Fp(0)885 780 y Fo(<)g Fn(1)i Fq(where)g(the)g(constan)o(t)g Fo(C)1464 787 y Fp(0)1501 780 y Fq(do)q(es)h(not)g(dep)q(end)f(on)g Fo(m)p Fq(.)46 840 y(Then)g(b)o(y)g(Condition)h(v\),)f(the)g(sequence)f Fo(q)858 847 y Fk(m)908 840 y Fq(is)h(compact.)24 b(Hence)16 b(the)h(top)q(ology)h(on)g Fo(M)1724 847 y Fk(N)1758 840 y Fo(;)f Fq(generated)46 900 y(b)o(y)g(the)g(norm)f Fn(k)c(\001)f(k)p Fo(;)17 b Fq(coincides)f(with)i(the)f(top)q(ology)h (on)g Fo(M)1168 907 y Fk(N)1219 900 y Fq(induced)f(b)o(y)g(the)g(w)o (eak)g(top)q(ology)h(of)g Fo(H)t Fq(,)46 960 y(that)f(is,)e(w)o(eak)h (con)o(v)o(ergence)f(implies)e(strong)18 b(con)o(v)o(ergence.)119 1020 y(Denoting)d(b)o(y)g Fo(E)431 1027 y Fk(N)480 1020 y Fq(the)f(set)h(of)h(p)q(oin)o(ts)f(in)g Fo(K)934 1027 y Fk(N)983 1020 y Fq(that)g(ha)o(v)o(e)f(more)g(than)i(one)f (pre-image,)e(w)o(e)i(see)f(that)46 1081 y Fo(E)82 1088 y Fk(N)132 1081 y Fq(is)i(op)q(en)h(b)q(ecause)g Fo(f)22 b Fq(is)16 b(a)h(lo)q(cal)f(isomorphism.)k(But)c Fo(E)1169 1088 y Fk(N)1219 1081 y Fq(is)h(also)g(closed.)k(Indeed,)16 b(supp)q(ose)h(there)46 1141 y(are)h(distinct)f(p)q(oin)o(ts)h Fo(q)477 1148 y Fk(j)513 1141 y Fq(and)g Fo(w)644 1148 y Fk(j)680 1141 y Fq(in)f Fo(M)785 1148 y Fk(N)837 1141 y Fq(suc)o(h)h(that)g Fo(f)5 b Fq(\()p Fo(q)1126 1148 y Fk(j)1144 1141 y Fq(\))17 b(=)f Fo(f)5 b Fq(\()p Fo(w)1317 1148 y Fk(j)1336 1141 y Fq(\))16 b Fn(!)g Fo(h)i Fq(as)h Fo(j)g Fn(!)e(1)p Fq(.)25 b(Then,)18 b(using)46 1201 y(Condition)g(iii\),)f(w)o(e)g(obtain)h(t)o(w)o(o)g(subsequences)g(suc) o(h)f(that)i Fo(q)1225 1208 y Fk(j)1239 1212 y Fc(m)1286 1201 y Fn(!)e Fo(q)r(;)25 b(w)1451 1208 y Fk(j)1465 1212 y Fc(m)1513 1201 y Fn(!)17 b Fo(w)q(;)g Fq(as)i Fo(m)d Fn(!)g(1)p Fq(,)i(and)46 1261 y(Condition)f(iv\))f(implies)e Fo(q)r(;)8 b(w)15 b Fn(2)f Fo(M)705 1268 y Fk(N)739 1261 y Fq(.)22 b(If)17 b Fo(q)e Fq(=)g Fo(w)q(;)h Fq(then)h Fo(q)1115 1268 y Fk(j)1129 1272 y Fc(m)1174 1261 y Fq(=)e Fo(w)1262 1268 y Fk(j)1276 1272 y Fc(m)1324 1261 y Fq(for)i(large)f Fo(m)p Fq(,)g(b)q(ecause)h(the)g(map)e Fo(f)46 1321 y Fq(is)j(a)g(lo)q(cal)h(homeomorphism)o(.)25 b(Hence)17 b Fo(q)h Fn(6)p Fq(=)f Fo(w)j Fq(and)e Fo(E)1077 1328 y Fk(N)1129 1321 y Fq(is)g(closed.)27 b(But)18 b(0)23 b Fo(=)-29 b Fn(2)17 b Fo(E)1576 1328 y Fk(N)1610 1321 y Fo(;)h Fq(whence)f Fo(E)1850 1328 y Fk(N)1901 1321 y Fq(=)g Fn(;)p Fq(.)46 1382 y(Th)o(us,)f Fo(f)j Fq(:)14 b Fo(M)301 1389 y Fk(N)348 1382 y Fn(!)g Fo(K)453 1389 y Fk(N)503 1382 y Fq(is)i(an)h(isomorphism.)119 1442 y(Supp)q(ose)k(that)g Fo(f)26 b Fq(:)20 b Fo(H)25 b Fn(!)c Fo(H)684 1449 y Fp(1)724 1442 y Fq(is)f(not)h(an)g(injection.)32 b(Then)20 b(some)g(p)q(oin)o(t)g Fo(h)h Fn(2)g Fo(H)1702 1449 y Fp(1)1742 1442 y Fq(has)g(at)g(least)46 1502 y(t)o(w)o(o)16 b(pre-images.)21 b(Since)15 b Fo(f)22 b Fq(is)16 b(a)h(lo)q(cal)g (homeomorphism)o(,)c(the)k(same)e(is)h(true)g(of)h(ev)o(ery)e(p)q(oin)o (t)i(in)f(some)46 1562 y(neigh)o(b)q(ourho)q(o)q(d)22 b(of)e Fo(h)f Fq(and,)i(consequen)o(tly)l(,)e(for)h(some)e Fo(h)1136 1544 y Fp(0)1176 1562 y Fn(2)i Fo(K)1270 1569 y Fk(N)1324 1562 y Fq(suc)o(h)f(that)h(\()p Fo(h)1593 1544 y Fp(0)1613 1562 y Fo(;)8 b(e)1658 1569 y Fk(k)1679 1562 y Fq(\))1698 1569 y Fp(1)1737 1562 y Fq(=)20 b(\()p Fo(h;)8 b(e)1887 1569 y Fk(k)1908 1562 y Fq(\))1927 1569 y Fp(1)1966 1562 y Fq(if)46 1622 y(1)18 b Fg(6)f Fo(k)j Fg(6)e Fo(N)24 b Fq(and)19 b(\()p Fo(h)453 1604 y Fp(0)473 1622 y Fo(;)8 b(e)518 1629 y Fk(k)538 1622 y Fq(\))557 1629 y Fp(1)595 1622 y Fq(=)17 b(0)i(if)f Fo(k)i(>)e(N)r(;)h Fq(where)f Fo(N)24 b Fq(is)18 b(su\016cien)o(tly)e(large.)28 b(But)19 b(this)f(con)o(tradicts)46 1683 y(the)e(fact)g(that)h Fo(f)i Fq(:)13 b Fo(M)449 1690 y Fk(N)497 1683 y Fn(!)h Fo(K)602 1690 y Fk(N)652 1683 y Fq(is)i(an)h(isomorphism.)51 b Fb(2)119 1743 y Fq(W)l(e)16 b(are)g(no)o(w)h(ready)f(to)h(pro)o(v)o (e)e(the)h(our)h(main)e(theorem.)46 1803 y Fr(Pro)r(of)k(of)h(Theorem)e (1.1.)23 b Fq(W)l(e)17 b(c)o(hec)o(k)e(all)h(conditions)h(of)h(Theorem) d(A)i(for)g(the)g(mapping)f Fo(h)g Fq(:)e Fo(H)20 b Fn(!)46 1863 y Fo(`)66 1845 y Fp(2)97 1863 y Fn(\010)11 b Fo(`)167 1845 y Fp(2)187 1863 y Fq(.)119 1923 y(i\))17 b(By)g(Lemma)f(4.1-2,)j (eac)o(h)e(function)h Fo(h)890 1930 y Fk(n)913 1923 y Fq(\()p Fn(\001)p Fq(\))p Fo(;)8 b(n)17 b Fn(\025)f Fq(1)p Fo(;)i Fq(is)f(compact)g(and)h(real)g(analytic)f(on)h Fo(H)t Fq(,)g(and)46 1983 y(the)i(estimates)f(\(1.9-10\))j(are)e (ful\014lled.)32 b(The)21 b(relations)f(\(1.9\))h(sho)o(w)g(that)g(the) f(map)f Fo(h)p Fq(\()p Fo(q)r Fq(\))h(is)g(lo)q(cally)46 2044 y(b)q(ounded;)d(b)o(y)e(the)h(uniform)f(b)q(oundedness)j (principle,)c Fo(h)p Fq(\()p Fo(q)r Fq(\))h(is)h(real)g(analytic.)119 2104 y(ii\))d(Using)h(the)g(estimate)e(\(1.10\),)i(w)o(e)g(see)f(that)i Fo(d)1016 2111 y Fk(q)1035 2104 y Fo(h)f Fq(is)g(the)g(sum)f(of)h(the)g (op)q(erator)h(\010)f(and)g(a)h(compact)46 2164 y(op)q(erator)k(for)f (all)g Fo(q)g Fn(2)f Fo(H)q(:)h Fq(That)g(is)g Fo(d)752 2171 y Fk(q)772 2164 y Fo(h)12 b Fn(\000)g Fq(\(1)p Fo(=)930 2123 y Fn(p)p 972 2123 25 2 v 41 x Fq(2\)\010)19 b(is)e(a)i(compact)d (op)q(erator;)k(consequen)o(tly)l(,)c Fo(d)1896 2171 y Fk(q)1916 2164 y Fo(h)i Fq(is)46 2224 y(a)i(F)l(redholm)f(op)q (erator.)34 b(W)l(e)20 b(pro)o(v)o(e)f(that)i(the)e(op)q(erator)j Fo(d)1185 2231 y Fk(q)1204 2224 y Fo(h)e Fq(is)g(in)o(v)o(ertible)e(b)o (y)h(con)o(tradiction.)33 b(Let)46 2284 y Fo(g)16 b Fn(2)e Fo(H)20 b Fq(b)q(e)d(a)f(solution)h(of)g(the)f(equation)494 2394 y(\()p Fo(d)538 2401 y Fk(q)558 2394 y Fo(h)p Fq(\()p Fo(q)r Fq(\)\))p Fo(g)f Fq(=)f(0)p Fo(;)73 b Fq(or)98 b Fn(f)p Fq(\()p Fo(d)1078 2401 y Fk(q)1097 2394 y Fo(h)1125 2401 y Fk(n)1149 2394 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(g)r Fq(\))13 b(=)h(0)p Fo(;)8 b(n)14 b Fn(\025)f Fq(1)p Fn(g)p Fo(;)324 b Fq(\(4.13\))46 2504 y(for)16 b(some)g(\014xed)f Fo(q)h Fn(2)e Fo(H)q(:)i Fq(W)l(e)g(in)o(tro)q(duce)g(the)g(function) 604 2635 y Fo(f)5 b Fq(\()p Fo(z)r Fq(\))14 b(=)761 2576 y Fj(Z)803 2589 y Fp(1)784 2671 y(0)823 2635 y Fq(\()p Fo(@)s Fq(\001\)\()p Fo(x;)8 b(z)r(;)g(q)r Fq(\))p Fo(g)r Fq(\()p Fo(x)p Fq(\))p Fo(dx;)54 b(z)15 b Fn(2)f Fl(C)9 b Fo(:)996 2811 y Fq(21)p eop %%Page: 22 22 22 21 bop 46 138 a Fq(The)16 b(function)g Fo(r)359 145 y Fk(n)396 138 y Fq(=)e Fo(h)476 120 y Fp(2)476 151 y Fk(cn)526 138 y Fq(+)d Fo(h)603 120 y Fp(2)603 151 y Fk(sn)659 138 y Fq(is)16 b(analytic)g(and)h(\(4.13\))g(implies)c(\()p Fo(d)1339 145 y Fk(q)1359 138 y Fo(r)1381 145 y Fk(n)1404 138 y Fo(;)8 b(g)r Fq(\))14 b(=)g(0)p Fo(:)i Fq(Then)g(\(3.10\))h (yields)549 258 y Fo(f)5 b Fq(\()p Fo(z)620 265 y Fk(n)644 258 y Fq(\))13 b(=)h(\()p Fn(\000)p Fq(1\))829 238 y Fk(n)853 258 y Fq(\()p Fo(d)8 b Fq(cosh)1006 205 y Fj(q)p 1047 205 52 2 v 1047 258 a Fo(h)1075 265 y Fk(n)1099 258 y Fo(=dr)1170 265 y Fk(n)1194 258 y Fq(\)\()p Fo(d)1257 265 y Fk(q)1277 258 y Fo(r)1299 265 y Fk(n)1322 258 y Fo(;)g(g)r Fq(\))14 b(=)g(0)p Fo(:)46 363 y Fq(Hence)i Fo(f)5 b Fq(\()p Fo(z)263 370 y Fk(n)286 363 y Fq(\))16 b(=)g(0)i(for)g(an)o(y)f Fo(n)f Fn(\025)f Fq(1,)j(and)g(\(3.2\))g (implies)d Fo(z)1145 370 y Fk(n)1184 363 y Fq(=)g Fo(\031)r(n)d Fq(+)g Fo(o)p Fq(\(1\))18 b(as)g Fo(n)e Fn(!)f(1)p Fo(:)i Fq(No)o(w)g(w)o(e)g(ha)o(v)o(e)46 423 y(to)f(sho)o(w)h(that)g Fo(f)i Fn(2)14 b Fo(L)455 405 y Fp(2)475 423 y Fq(\()p Fl(R)p Fq(\).)k(The)f(asymptotic)d(estimate)h(\(2.24\))i(yields)222 552 y Fo(f)5 b Fq(\()p Fo(z)r Fq(\))14 b(=)380 493 y Fj(Z)421 506 y Fp(1)403 587 y(0)441 552 y Fq(\()p Fo(@)s Fq(\001\()p Fo(z)r(;)8 b(q)r Fq(\)\)\()p Fo(t)p Fq(\))p Fo(g)r Fq(\()p Fo(t)p Fq(\))p Fo(dt)j Fq(=)901 493 y Fj(Z)942 506 y Fp(1)924 587 y(0)970 493 y Fj(Z)1012 506 y Fp(1)993 587 y(0)1040 552 y Fq(cos)e Fo(z)r Fq(\(2)p Fo(y)j Fn(\000)f Fq(1\))p Fo(q)r Fq(\()p Fo(y)i Fq(+)e Fo(t)p Fq(\))p Fo(g)r Fq(\()p Fo(t)p Fq(\))p Fo(dtdy)h Fq(+)f Fo(`)1731 531 y Fk(d)1752 552 y Fq(\()p Fo(n)p Fq(\))266 703 y(=)317 645 y Fj(Z)359 658 y Fp(1)341 739 y(0)387 703 y Fq(cos)e Fo(z)r Fq(\(2)p Fo(y)j Fn(\000)f Fq(1\))p Fo(G)p Fq(\()p Fo(y)r Fq(\))p Fo(dy)j Fq(+)d Fo(`)892 683 y Fk(d)912 703 y Fq(\()p Fo(n)p Fq(\)\))p Fo(;)73 b Fn(j)p Fo(z)13 b Fn(\000)e Fo(\031)r(n)p Fn(j)i Fg(6)h Fo(\031)r(;)39 b Fn(j)p Fo(n)p Fn(j)14 b(!)g(1)p Fo(;)40 b(d)14 b(>)g Fq(1)p Fo(;)46 812 y Fq(where)i Fo(G)p Fq(\()p Fo(y)r Fq(\))e(=)354 776 y Fj(R)382 789 y Fp(1)374 825 y(0)410 812 y Fo(q)r Fq(\()p Fo(y)e Fq(+)f Fo(t)p Fq(\))p Fo(g)r Fq(\()p Fo(t)p Fq(\))p Fo(dt)p Fq(.)21 b(Hence)14 b Fo(f)20 b Fn(2)14 b Fo(L)1002 794 y Fp(2)1022 812 y Fq(\()p Fl(R)p Fq(\).)119 872 y(So,)i(using)h (\(2.5\))f(w)o(e)g(see)g(that)h(all)f(conditions)g(of)g(Lemma)e(4.3)j (are)f(ful\014lled,)f(and)i(w)o(e)e(get)i Fo(f)i Fn(\021)13 b Fq(0)p Fo(:)119 932 y Fq(F)l(or)25 b(\014xed)e Fo(q)30 b Fn(2)d Fo(H)i Fq(w)o(e)24 b(ha)o(v)o(e)f(3)i(cases.)46 b(First,)25 b(let)f Fo(h)1185 939 y Fk(sn)1253 932 y Fq(=)j(0)p Fo(:)d Fq(Then)h(b)o(y)f(\(4.7\),)i(\()p Fo(d)1776 939 y Fk(q)1795 932 y Fo(h)1823 939 y Fk(sn)1863 932 y Fo(;)8 b(g)r Fq(\))28 b(=)46 992 y Fo(b)67 999 y Fk(n)90 992 y Fq(\()p Fo(d)134 999 y Fk(q)153 992 y Fo(z)176 999 y Fk(n)206 992 y Fn(\000)6 b Fo(d)276 999 y Fk(q)295 992 y Fo(m)338 999 y Fk(n)361 992 y Fo(;)i(g)r Fq(\))14 b(=)g(0)g(and)g(b)o(y)g(\(3.1\))g(w)o(e)f(deduce)g(that)h(\()p Fo(d)1179 999 y Fk(q)1199 992 y Fo(z)1222 999 y Fk(n)1245 992 y Fo(;)8 b(g)r Fq(\))14 b(=)f(0)i(and)f(therefore)f(\()p Fo(d)1753 999 y Fk(q)1773 992 y Fo(m)1816 999 y Fk(n)1839 992 y Fo(;)8 b(g)r Fq(\))14 b(=)f(0)119 1053 y(Second,)j(if)g Fo(h)371 1060 y Fk(cn)423 1053 y Fn(6)p Fq(=)e(0)p Fo(;)8 b(h)549 1060 y Fk(sn)603 1053 y Fn(6)p Fq(=)14 b(0)p Fo(;)i Fq(then)g(w)o(e)g(ha)o(v)o(e)f(b)o(y)h(relation)g(\(4.2\))512 1158 y(\()p Fn(\000)p Fq(1\))613 1137 y Fk(n)645 1158 y Fq(sinh)8 b Fo(h)768 1165 y Fk(cn)807 1158 y Fq(\()p Fo(d)851 1165 y Fk(q)870 1158 y Fo(h)898 1165 y Fk(cn)937 1158 y Fo(;)g(g)r Fq(\))14 b(=)1082 1145 y(_)1068 1158 y(\001\()p Fo(m)1171 1165 y Fk(n)1194 1158 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(q)r Fq(\)\()p Fo(d)1365 1165 y Fk(q)1383 1158 y Fo(m)1426 1165 y Fk(n)1449 1158 y Fo(;)g(g)r Fq(\))p Fo(;)46 1263 y Fq(and)17 b(then)f(w)o(e)g(ha)o(v)o (e)f(\()p Fo(d)480 1270 y Fk(q)499 1263 y Fo(m)542 1270 y Fk(n)566 1263 y Fo(;)8 b(g)r Fq(\))13 b(=)h(0)p Fo(;)i Fq(since)g Fo(f)j Fn(\021)14 b Fq(0)i(and)1116 1250 y(_)1102 1263 y(\001\()p Fo(m)1205 1270 y Fk(n)1228 1263 y Fq(\))e Fn(6)p Fq(=)f(0)p Fo(:)119 1323 y Fq(In)19 b(the)h(last)f(case,)h(let)f Fo(h)589 1330 y Fk(cn)648 1323 y Fq(=)g(0)p Fo(;)8 b(h)779 1330 y Fk(sn)839 1323 y Fn(6)p Fq(=)19 b(0)p Fo(;)h Fq(then)f(b)o(y)h (\(1.7\),)g Fo(#)p Fq(\(1)p Fo(;)8 b(m)1411 1330 y Fk(n)1434 1323 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))18 b(=)i Fo(')1669 1305 y Fm(0)1680 1323 y Fq(\(1)p Fo(;)8 b(m)1788 1330 y Fk(n)1812 1323 y Fq(\()p Fo(q)r Fq(\))p Fo(;)g(q)r Fq(\))18 b(=)46 1383 y(\()p Fn(\000)p Fq(1\))147 1365 y Fk(n)170 1383 y Fo(:)e Fq(Therefore,)g(using)g(the)g(relation)g (\(2.11\),)h(w)o(e)e(get)76 1488 y Fo(f)5 b Fq(\()p Fo(m)167 1495 y Fk(n)190 1488 y Fq(\()p Fo(q)r Fq(\)\))14 b(=)f(\()p Fo(@)s Fq(\001\()p Fo(m)487 1495 y Fk(n)510 1488 y Fq(\()p Fo(q)r Fq(\)\))p Fo(;)8 b(g)r Fq(\))13 b(=)g Fo(A)758 1495 y Fk(n)782 1488 y Fq(\()p Fo(d)826 1495 y Fk(q)845 1488 y Fo(m)888 1495 y Fk(n)911 1488 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(g)r Fq(\))64 b Fo(A)1140 1495 y Fk(n)1177 1488 y Fq(=)14 b Fn(\000)p Fo(#)1297 1468 y Fm(0)1308 1488 y Fq(\()p Fo(m)1370 1495 y Fk(n)1393 1488 y Fq(\()p Fo(q)r Fq(\)\))p Fo(')1506 1468 y Fm(0)1517 1488 y Fq(\()p Fo(m)1579 1495 y Fk(n)1602 1488 y Fq(\()p Fo(q)r Fq(\)\))f(_)-27 b Fo(')p Fq(\()p Fo(m)1777 1495 y Fk(n)1800 1488 y Fq(\()p Fo(q)r Fq(\))13 b Fn(6)p Fq(=)h(0)p Fo(;)46 1593 y Fq(since)h Fo(f)20 b Fq(=)13 b(0)k(and)396 1558 y Fj(R)423 1571 y Fp(1)415 1606 y(0)452 1593 y Fo(g)r Fq(\()p Fo(x)p Fq(\))p Fo(dx)c Fq(=)h(0.)22 b(Then)16 b(w)o(e)g(ha)o(v)o(e)f(\()p Fo(d)1076 1600 y Fk(q)1096 1593 y Fo(m)1139 1600 y Fk(n)1162 1593 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(g)r Fq(\))13 b(=)h(0)p Fo(:)119 1653 y Fq(Hence)j(w)o(e)h(obtain)g(\()p Fo(d)537 1660 y Fk(q)557 1653 y Fo(m)600 1660 y Fk(n)623 1653 y Fq(\()p Fo(q)r Fq(\))p Fo(;)8 b(g)r Fq(\))16 b(=)h(0)p Fo(;)8 b Fq(\()p Fo(d)912 1660 y Fk(q)932 1653 y Fo(h)960 1660 y Fk(cn)999 1653 y Fo(;)g(g)r Fq(\))17 b(=)g(0)p Fo(;)8 b(n)18 b Fn(\025)e Fq(1)p Fo(:)j Fq(Note)e(that)i(for)g(an)o(y)f (\014xed)f Fo(q)i Fn(2)e Fo(H)46 1714 y Fq(the)j(v)o(ectors)f Fn(f)p Fq(\()p Fo(d)372 1721 y Fk(q)391 1714 y Fo(h)419 1721 y Fk(cn)458 1714 y Fq(\))p Fo(;)8 b Fq(\()p Fo(d)543 1721 y Fk(q)562 1714 y Fo(m)605 1721 y Fk(n)628 1714 y Fq(\))p Fo(;)g(n)21 b Fn(\025)f Fq(1)p Fn(g)g Fq(form)f(a)h(basis)h (of)f Fo(H)k Fq(\(see)c([CML]\).)f(Then)h Fo(g)j Fq(=)d(0)g(and)h(the) 46 1774 y(op)q(erator)c Fo(d)267 1781 y Fk(q)287 1774 y Fo(h)f Fq(is)g(in)o(v)o(ertible.)119 1834 y(iii\))f(The)h(estimates)f (\(1.8\))h(w)o(ere)g(pro)o(v)o(ed)f(in)h([K1].)119 1894 y(iv\))g(By)f(Lemma)f(4.1-2,)j(eac)o(h)f(mapping)f Fo(h)915 1901 y Fk(n)953 1894 y Fq(:)e Fo(H)18 b Fn(!)c Fl(R)1141 1876 y Fp(2)1157 1894 y Fo(;)8 b(n)14 b Fn(\025)g Fq(1)p Fo(;)i Fq(is)g(compact.)119 1954 y(v\))g(W)l(e)g(need)f(some)g(results) h(ab)q(out)h(the)f(Hill)f(op)q(erator)i Fo(T)1196 1961 y Fp(1)1229 1954 y Fq(=)d Fn(\000)1334 1935 y Fk(d)1352 1923 y Fh(2)p 1324 1943 56 2 v 1324 1972 a Fk(dx)1362 1962 y Fh(2)1395 1954 y Fq(+)d Fo(V)g Fq(\()p Fo(x)p Fq(\))16 b(in)g Fo(L)1655 1936 y Fp(2)1675 1954 y Fq(\()p Fl(R)p Fq(\))d(with)j(a)g(real)46 2015 y(1-p)q(erio)q(dic)g(p)q(oten)o (tial)g Fo(V)26 b Fn(2)14 b Fo(L)613 1996 y Fp(2)632 2015 y Fq(\(0)p Fo(;)8 b Fq(1\).)22 b(In)16 b(the)g(pap)q(er)h([K6])f (the)g(follo)o(wing)g(estimate)e(has)j(b)q(een)f(pro)o(v)o(ed)429 2120 y Fn(k)p Fo(V)11 b Fn(k)j Fg(6)g Fq(2\(1)e(+)f Fn(k)p Fo(h)p Fn(k)791 2127 y Fp(\(1\))838 2120 y Fq(\))p Fn(k)p Fo(h)p Fn(k)935 2127 y Fp(\(1\))982 2120 y Fo(;)105 b Fn(k)p Fo(h)p Fn(k)1179 2099 y Fp(2)1179 2132 y(\(1\))1240 2120 y Fq(=)1295 2078 y Fj(X)1292 2169 y Fk(n)p Fm(\025)p Fp(1)1358 2120 y Fq(\(2)p Fo(\031)r(n)p Fq(\))1479 2099 y Fp(2)1499 2120 y Fn(j)p Fo(h)1541 2127 y Fk(n)1564 2120 y Fn(j)1578 2099 y Fp(2)1598 2120 y Fo(:)46 2266 y Fq(Then)20 b(the)g(iden)o(tit)o(y)e(\(1.17\))i(yields)f Fn(k)p Fo(q)783 2248 y Fm(0)794 2266 y Fn(k)h Fg(6)g Fq(2\(1)14 b(+)g Fn(k)p Fo(h)p Fn(k)1109 2273 y Fp(\(1\))1156 2266 y Fq(\))p Fn(k)p Fo(h)p Fn(k)1253 2273 y Fp(\(1\))1300 2266 y Fq(.)32 b(Hence,)19 b(for)h(eac)o(h)g Fo(\030)j(>)d Fq(0)g(the)g(set)46 2326 y Fn(f)p Fo(q)15 b Fq(:)e Fn(k)p Fo(h)p Fn(k)213 2334 y Fp(\(1\))274 2326 y Fo(<)h(\030)r Fn(g)j Fq(is)f(compact.)119 2386 y(Therefore,)i(all)g(conditions)g(of)h (Theorem)e(A)h(are)h(ful\014lled)d(and)j(then)g Fo(h)f Fq(is)g(the)h(real)e(analytic)h(iso-)46 2446 y(morphism)c(b)q(et)o(w)o (een)h Fo(H)21 b Fq(and)16 b Fo(`)637 2428 y Fp(2)669 2446 y Fn(\010)10 b Fo(`)738 2428 y Fp(2)759 2446 y Fq(.)21 b Fb(2)46 2506 y Fr(Ac)n(kno)n(wledgmen)n(ts.)h Fq(The)17 b(v)m(arious)g(parts)g(of)g(this)g(pap)q(er)g(w)o(ere)f(written)g(at)h (FIM)f(P)o(otsdam)h(Univ.,)46 2567 y(and)g(at)h(KTH)e(\(Sto)q(c)o (kholm\).)22 b(The)17 b(author)h(is)f(grateful)g(to)g(M.)f(Klein)g(\()h (P)o(otsdam)g(Univ.\),)e(and)i(to)h(A.)46 2627 y(Laptev)e(\(KTH\))g (for)h(their)e(hospitalit)o(y)l(.)46 2687 y Fr(References)996 2811 y Fq(22)p eop %%Page: 23 23 23 22 bop 46 138 a Fq([CM])18 b(Coleman)g(C.,)g(McLaughlin)h(J.:)26 b(Solution)19 b(of)g(the)g(in)o(v)o(erse)e(problem)g(for)i(an)h(imp)q (edance)d(with)46 199 y(in)o(tegrable)e(deriv)m(ativ)o(e,)f(Comm)o(un.) 19 b(Pure)d(Appl.)21 b(Math.)g(1993,)d(46,)e(145-184;)j(185-212)46 259 y([CMK])g(Constan)o(tin)i(A.,)g(McKean)e(H.:)29 b(A)20 b(shallo)o(w)h(w)o(ater)f(equation)g(on)h(the)g(circle,)e(to)i(app)q (ear)g(in)46 319 y(Comm)o(un.)e(Pure)d(Appl.)21 b(Math.)46 379 y([F])12 b(Firso)o(v)m(a)i(N.)f(:)19 b(Riemann)12 b(surface)i(of)g(quasimomen)o(tum)9 b(and)14 b(scattering)g(theory)f (for)h(the)f(p)q(erturb)q(ed)46 439 y(Hill)h(op)q(erator,)j(J.)f(So)o (viet)g(Math.)21 b(11,)16 b(487-497)j(\(1979\))46 499 y([GT])g(Garnett)i(J.,)f(T)l(rub)q(o)o(witz)g(E.:)29 b(Gaps)21 b(and)f(bands)h(of)g(one)f(dimensional)e(p)q(erio)q(dic)i(Sc) o(hr\177)-24 b(odinger)46 560 y(op)q(erators.)23 b(Commen)o(t.)18 b(Math.)j(Helv.)f(59,)d(258-312)i(\(1984\))46 620 y([KK1])j(Kargaev)i (P)l(.,)g(Korot)o(y)o(aev)f(E.:)35 b(The)24 b(in)o(v)o(erse)d(problem)h (for)h(the)g(Hill)f(op)q(erator,)k(the)d(direct)46 680 y(approac)o(h.)f(In)o(v)o(en)o(t.)d(Math.)j(129,)17 b(no)f(3,)g (567-593)j(\(1997\))46 740 y([KK2])j(Kargaev)i(P)l(.,)g(Korot)o(y)o (aev)f(E.:)35 b(The)24 b(in)o(v)o(erse)d(problem)h(for)h(the)g(Hill)f (op)q(erator,)k(the)d(direct)46 800 y(approac)o(h,)16 b(Correction.)22 b(\(will)15 b(b)q(e)h(published)g(in)g(In)o(v)o(en)o (t.)j(Math.\))46 861 y([KK3])k(Kargaev)g(P)l(.,)i(Korot)o(y)o(aev)d (E.:)36 b(E\013ectiv)o(e)21 b(masses)i(and)h(conformal)e(mappings.)42 b(Comm)o(un.)46 921 y(Math.)21 b(Ph)o(ys.)g(169,)c(597-625)i(\(1995\).) 46 981 y([K1])d(Korot)o(y)o(aev)f(E.)h(:)21 b(P)o(erio)q(dic)16 b(w)o(eigh)o(ted)f(op)q(erator.)46 1041 y([K2])20 b(E.)h(Korot)o(y)o (aev:)31 b(The)21 b(in)o(v)o(erse)e(problem)g(for)j(the)f(Hill)e(op)q (erator)j(I.)e(In)o(ter.)35 b(Math.)h(Researc)o(h.)46 1101 y(Notes.)21 b(1997,)c(3,)f(113-125.)46 1162 y([K3])j(Korot)o(y)o (aev)h(E.)g(:)29 b(Metric)19 b(prop)q(erties)h(of)g(conformal)g (mappings)f(on)i(the)f(complex)e(plane)i(with)46 1222 y(parallel)15 b(slits.)21 b(In)o(ter.)f(Math.)i(Reseac)o(h.)e(Notes.)h (1996,)d(10,)e(493-503.)46 1282 y([K4])i(Korot)o(y)o(aev)g(E.)g(:)26 b(The)19 b(propagation)h(of)f(the)g(w)o(a)o(v)o(es)e(in)i(the)f (one-dimensional)f(p)q(erio)q(dic)i(media.)46 1342 y(Asymptotic)14 b(Analysis,)h(15,)i(1-24,)g(\(1997\))46 1402 y([K5])10 b(E.)h(Korot)o(y)o(aev:)18 b(The)11 b(estimate)f(of)h(p)q(erio)q(dic)g (p)q(oten)o(tials)g(in)f(terms)g(of)h(e\013ectiv)o(e)e(masses.)19 b(Comm)o(un.)46 1463 y(Math.)i(Ph)o(ys.)g(1997,)d(183,)f(383-400)46 1523 y([K6])f(Korot)o(y)o(aev)h(E.)g(Estimate)e(for)j(the)f(Hill)e(op)q (erator.I,)i(preprin)o(t)f(FIM)g(ETH,)h(Zuric)o(h,)f(No)o(v.)23 b(1997,)46 1583 y(will)15 b(b)q(e)h(published)g(in)g(J.)g(Di\013.)21 b(Eq.)46 1643 y([K7])12 b(Korot)o(y)o(aev)h(E.:)20 b(Estimate)11 b(of)j(p)q(erio)q(dic)f(p)q(oten)o(tials)g(in)g(terms)e(of)j(gap)g (lengths.)20 b(Comm)o(un.)e(Math.)46 1703 y(Ph)o(ys.)j(1998,)c(197,)g (no.)22 b(3,)16 b(521-526)46 1763 y([K8])h(Korot)o(y)o(aev)g(E.:)25 b(On)18 b(the)f(scattering)h(in)f(an)i(external,)e(homogeneous)h (time-p)q(erio)q(dic)d(magnetic)46 1824 y(\014eld.)21 b(Mat.)g(Sb.)g(180\(1989\),)e(no.)i(4,)c(491-512,)h(English)e(transl)h (in)f(Math.)21 b(USSR)16 b(Sb.)21 b(66\(1990\).)46 1884 y([Kr])c(Krein)h(M.)g(On)g(the)g(c)o(haracteristic)f(function)h Fo(A)p Fq(\()p Fo(\025)p Fq(\))h(of)f(a)h(linear)f(canonical)g(system)f (di\013eren)o(tial)46 1944 y(equation)j(of)g(the)g(second)h(order)f (with)g(p)q(erio)q(dic)g(co)q(e\016cien)o(ts)f(\(Russian\))h(,)h (Prikl.)32 b(Mat.)h(Meh.)f(21,)46 2004 y(320-329)19 b(\(1957\))46 2064 y([Le])14 b(Levinson)h(N.)f(Gap)i(and)f(densit)o(y)f(theorem.)19 b(Amer.)g(Math.)i(So)q(c.)g(Coll.)f(Publ.,)15 b(New)f(Y)l(ork,)g(1940) 46 2125 y([L])j(Ly)o(apuno)o(v,A:)g(Sur)i(une)f(equation)g(transcendan) o(te)g(et)f(les)h(equations)g(di\013eren)o(tielles)d(lineares)i(du)46 2185 y(second)f(ordre)h(a)f(co)q(e\016cien)o(ts)f(p)q(erio)q(diques,)h (C.)g(R.,)f(t.)21 b(CXXVI)q(I)q(I,)14 b(18,)j(1085-1088)j(\(1899\).)46 2245 y([MO1])15 b(Marc)o(henk)o(o)g(V.,)f(Ostro)o(vski)i(I.:)k(A)15 b(c)o(haracterization)g(of)i(the)e(sp)q(ectrum)g(of)h(the)g(Hill)e(op)q (erator.)46 2305 y(Math.)21 b(USSR)16 b(Sb.)22 b(26,)16 b(493-554)j(\(1975\).)46 2365 y([MO2])j(Marc)o(henk)o(o)g(V.,)i(Ostro)o (vski)f(I.:)34 b(Appro)o(ximation)21 b(of)j(p)q(erio)q(dic)f(b)o(y)f (\014nite-zone)h(p)q(oten)o(tials.)46 2426 y(Selecta)15 b(Math.)22 b(So)o(vietica.)d(1987,)f(6,)e(No)g(2,)g(101-136.)46 2486 y([N])e(No)s(jima)g(S.)h(Exitonic)g(p)q(olaritons)i(in)e (one-dimensional)g(photonic)h(crystal,)e(Ph)o(ys.)21 b(Rev.)g(B,)15 b(1998,)46 2546 y(54,)h(no.)22 b(4,)16 b(R2057-R2060)46 2606 y([PT])g(P\177)-24 b(osc)o(hel,)15 b(P)l(.,)g(T)l(rub)q(o)o(witz)i(E.:)k(In)o(v)o(erse)14 b(Sp)q(ectral)i(Theory)l(.)22 b(Boston:)f(Academic)14 b(Press,)i(1987.)46 2666 y([T])f(Titc)o(hmarsh)g(E.:)21 b(The)16 b(theory)h(of)f(functions.)21 b(Sec.)g(ed.,)15 b(Univ.)20 b(Press,)c(London,)i(1975)996 2811 y(23)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9906160509314--