Content-Type: multipart/mixed; boundary="-------------0110180904276" This is a multi-part message in MIME format. ---------------0110180904276 Content-Type: text/plain; name="01-385.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-385.keywords" absolutely continuous spectrum, Jacobi matrices, trace formulae ---------------0110180904276 Content-Type: application/postscript; name="lns.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lns.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: lns.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman Times-Italic Helvetica %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o lns.ps lns.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.10.18:1547 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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b(may)f(re)n(write)h (this)f(equation)g(as)764 1955 y Fq(v)811 1970 y Fk(n)886 1955 y Fo(=)k Fq(\034)33 b Fm(\000)1280 1888 y Fq(k)p 1175 1932 265 4 v 1175 2024 a(k)1229 1995 y Fp(2)1291 2024 y Fm(\000)22 b Fo(1)1510 1831 y Fn(1)1474 1861 y Fg(X)1466 2070 y Fk(m)p Fp(=)p Fk(n)1626 1955 y Fo(\(1)g Fm(\000)g Fq(k)1888 1914 y Fp(2\()p Fk(n)p Fn(\000)p Fk(m)p Fp(\))2143 1955 y Fo(\))2181 1875 y Fg(\000)2226 1955 y Fq(\025)2283 1914 y Fp(0)2283 1980 y Fk(m)2350 1955 y Fq(v)2397 1970 y Fk(m)2486 1955 y Fo(+)2594 1888 y(2)p Fq(\025)2700 1903 y Fk(m)2788 1888 y Fo(+)g Fq(\025)2943 1852 y Fp(2)2943 1913 y Fk(m)p 2594 1932 416 4 v 2774 2024 a Fq(k)3019 1955 y(v)3066 1970 y Fk(m)p Fp(+1)3223 1845 y Fg(\021)3283 1955 y Fq(;)-2854 b Fs(\(2.2\))456 2221 y(where)25 b Fq(v)771 2236 y Fk(n)846 2221 y Fo(=)i Fq(k)1003 2185 y Fk(n)1050 2221 y Fq(\036)1108 2236 y Fk(n)1180 2221 y Fs(and)e Fq(\034)39 b Fo(=)1533 2146 y Fg(Q)1627 2173 y Fn(1)1627 2250 y Fp(1)1702 2221 y Fo(\(1)22 b(+)g Fq(\025)1966 2236 y Fk(k)2009 2221 y Fo(\))2047 2185 y Fn(\000)p Fp(1)2141 2221 y Fs(.)555 2337 y(F)o(or)i Fm(j)p Fq(k)s Fm(j)k Fq(>)f Fo(1)d Fs(this)f(equation)h (can)g(be)h(solv)o(ed)e(by)h(repeated)g(substitution)e(\(or)i(Neu-)456 2453 y(mann)g(series\),)h(from)f(which)h(we)g(conclude)f(that)930 2669 y Fq(a)k Fo(=)79 b(lim)1112 2729 y Fk(n)p Fn(!\0001)1367 2669 y Fq(v)1414 2684 y Fk(n)1489 2669 y Fo(=)28 b Fq(\034)1646 2558 y Fg(\020)1706 2669 y Fo(1)22 b Fm(\000)1886 2601 y Fo(tr\003)2030 2616 y Fp(0)p 1886 2646 184 4 v 1951 2737 a Fq(k)2101 2669 y Fo(+)2232 2601 y(1)p 2209 2646 94 4 v 2209 2737 a Fq(k)2263 2708 y Fp(2)2313 2669 y Fm(f)p Fo(tr\()p Fq(I)30 b Fm(\000)22 b Fo(\()p Fq(I)30 b Fo(+)22 b(\003\))2964 2628 y Fp(2)3003 2669 y Fo(\))g(+)948 2842 y Fg(X)17 b(X)1279 2869 y Fo(\(1)22 b Fm(\000)h Fq(k)1542 2833 y Fp(2)p Fn(j)p Fk(n)p Fn(\000)p Fk(m)p Fn(j)1780 2869 y Fo(\))p 1279 2913 540 4 v 1524 3005 a(2)1828 2936 y Fq(\025)1885 2895 y Fp(0)1885 2961 y Fk(m)1952 2936 y Fq(\025)2009 2895 y Fp(0)2009 2961 y Fk(n)2056 2936 y Fm(g)2106 2826 y Fg(\021)2187 2936 y Fo(+)f Fq(O)2363 2826 y Fg(\020)2454 2869 y Fo(1)p 2432 2913 94 4 v 2432 3005 a Fq(k)2486 2976 y Fp(3)2535 2826 y Fg(\021)2595 2936 y Fq(;)91 b Fo(as)25 b Fq(k)31 b Fm(!)d(1)p Fq(:)456 2794 y Fs(\(2.3\))456 3135 y(Therefore)467 3461 y Fq(\034)520 3419 y Fn(\000)p Fp(1)615 3461 y Fq(a)f Fo(=)h(1)22 b Fm(\000)978 3393 y Fo(tr)o(\003)1121 3408 y Fp(0)p 978 3438 184 4 v 1042 3529 a Fq(k)1193 3461 y Fm(\000)1325 3393 y Fo(1)p 1302 3438 94 4 v 1302 3529 a Fq(k)1356 3500 y Fp(2)1406 3461 y Fm(f)p Fo(tr)o(\(\()p Fq(I)30 b Fo(+)22 b(\003\))1884 3419 y Fp(2)1946 3461 y Fm(\000)g Fq(I)8 b Fo(\))22 b(+)2264 3393 y(\(tr\003)2446 3357 y Fp(2)2446 3418 y(0)2507 3393 y Fm(\000)h Fo(\(tr\003)2789 3408 y Fp(0)2828 3393 y Fo(\))2866 3357 y Fp(2)2905 3393 y Fo(\))p 2264 3438 679 4 v 2579 3529 a(2)2953 3461 y Fm(g)f Fo(+)g Fq(O)s Fo(\()3270 3393 y(1)p 3249 3438 94 4 v 3249 3529 a Fq(k)3303 3500 y Fp(3)3352 3461 y Fo(\))p Fq(;)456 3267 y Fs(\(2.4\))456 3660 y(as)i Fq(k)31 b Fm(!)d(1)p Fs(.)i(The)25 b(crucual)g(circumstance)f(is)h(that)694 3830 y Fq(\034)747 3789 y Fn(\000)p Fp(1)842 3830 y Fq(a)j Fo(=)f(det)q(\()p Fq(I)j Fo(+)22 b Fq(Q)p Fo(\()p Fq(H)1565 3845 y Fp(0)1626 3830 y Fm(\000)h Fq(z)t Fo(\))1813 3789 y Fn(\000)p Fp(1)1908 3830 y Fo(\))p Fq(;)91 b(H)2145 3845 y Fp(0)2212 3830 y Fo(=)28 b Fq(S)g Fo(+)22 b Fq(S)2568 3789 y Fn(\003)2607 3830 y Fq(;)41 b(z)33 b Fo(=)27 b Fq(k)f Fo(+)c Fq(k)3085 3789 y Fn(\000)p Fp(1)3179 3830 y Fq(:)456 4000 y Fs(Here,)28 b(as)e(in)h(the)g(pre)n(vious)e(section,) i Fq(S)33 b Fs(and)26 b Fq(Q)i Fs(are)f(operators)g(in)f Fq(l)2816 3964 y Fp(2)2856 4000 y Fo(\()p Ff(Z)p Fo(\))p Fs(.)34 b(It)27 b(follo)n(ws)456 4117 y(from)d(this)g(relation)g(that) 664 4343 y Fo(log\()p Fq(a)p Fo(\()p Fq(k)s Fo(\)\))k(=)f(log\()p Fq(\034)11 b Fo(\))23 b(+)1590 4218 y Fn(1)1554 4248 y Fg(X)1559 4458 y Fk(n)p Fp(=1)1724 4276 y Fo(\()p Fm(\000)p Fo(1\))1926 4239 y Fk(n)p Fp(+1)p 1724 4320 340 4 v 1865 4411 a Fq(n)2073 4343 y Fo(tr\()p Fq(Q)p Fo(\()p Fq(H)2383 4358 y Fp(0)2445 4343 y Fm(\000)f Fq(z)t Fo(\))2631 4302 y Fn(\000)p Fp(1)2726 4343 y Fo(\))2764 4302 y Fk(n)2811 4343 y Fq(;)67 b(z)32 b Fm(!)27 b(1)p Fq(:)555 4598 y Fw(2.)k Fs(Let)25 b(us)f(introduce)g(the)h(operator)1555 4768 y Fq(M)39 b Fo(=)27 b(\()p Fq(I)j Fo(+)22 b(\003\))2106 4727 y Fp(2)2167 4768 y Fm(\000)h Fq(I)8 b(:)456 4939 y Fs(Then)1176 5124 y Fo(log\()p Fq(a)p Fo(\()p Fq(k)s Fo(\)\))27 b(=)h Fm(\000)1821 5000 y Fn(1)1784 5030 y Fg(X)1789 5239 y Fk(n)p Fp(=0)1944 5124 y Fq(A)2017 5139 y Fk(n)2065 5124 y Fq(k)2119 5083 y Fn(\000)p Fk(n)2220 5124 y Fq(;)142 b(k)30 b Fm(!)e(1)p Fq(;)p eop %%Page: 4 4 4 3 bop 456 235 a Fi(4)920 b(LAPTEV)-10 b(,)23 b(N)m(ABOK)n(O,)h(SAFR)m (ONO)l(V)456 425 y Fs(where)h(\(compare)g(with)f([4],)h(p.)31 b(155\))24 b Fq(A)1879 440 y Fk(n)1954 425 y Fo(=)j Fq(A)2130 440 y Fk(n)2178 425 y Fo(\(\003)p Fq(;)17 b Fo(\003)2396 440 y Fp(0)2434 425 y Fo(\))p Fs(,)1402 587 y Fq(A)1475 602 y Fp(0)1542 587 y Fo(=)28 b(log\()p Fq(\034)11 b Fo(\))p Fq(;)116 b(A)2117 602 y Fp(1)2185 587 y Fo(=)27 b(tr\003)2432 602 y Fp(0)2471 587 y Fq(;)1500 802 y(A)1573 817 y Fp(2)1641 802 y Fo(=)1754 734 y(1)p 1754 779 49 4 v 1754 870 a(2)1813 802 y(tr\003)1957 761 y Fp(2)1957 826 y(0)2018 802 y Fo(+)22 b(tr\()p Fq(M)10 b Fo(\))p Fq(;)1090 1040 y(A)1163 1055 y Fp(3)1230 1040 y Fo(=)28 b(tr)o(\()1457 973 y(1)p 1457 1017 V 1457 1108 a(3)1516 1040 y(\003)1584 999 y Fp(3)1584 1065 y(0)1645 1040 y Fo(+)22 b(\(\003)1849 1055 y Fp(0)1911 1040 y Fo(+)g Fq(S)2075 999 y Fn(\003)2114 1040 y Fo(\003)2182 1055 y Fp(0)2221 1040 y Fq(S)6 b Fo(\)\()2373 973 y(1)p 2373 1017 V 2373 1108 a(2)2431 1040 y Fq(I)30 b Fo(+)22 b Fq(M)10 b Fo(\)\))p Fq(;)639 1278 y(A)712 1293 y Fp(4)779 1278 y Fo(=)893 1211 y(1)p 893 1255 V 893 1346 a(4)951 1278 y(tr\()p Fm(f)p Fo(\(\003)1221 1237 y Fp(2)1221 1303 y(0)1282 1278 y Fo(+)22 b Fq(M)33 b Fo(+)22 b Fq(S)6 b(M)k(S)1841 1237 y Fn(\003)1881 1278 y Fo(\))1919 1237 y Fp(2)1980 1278 y Fo(+)22 b(2\(\003)2233 1293 y Fp(0)2294 1278 y Fo(+)g Fq(S)2458 1237 y Fn(\003)2498 1278 y Fo(\003)2566 1293 y Fp(0)2605 1278 y Fq(S)6 b Fo(\))2709 1237 y Fp(2)2748 1278 y Fo(\()p Fq(M)33 b Fo(+)22 b(1\))p Fm(g)p Fo(\)+)1135 1516 y(+)1221 1449 y(1)p 1221 1493 V 1221 1584 a(4)1280 1516 y(tr)p Fm(f)p Fo(\()p Fq(M)33 b Fo(+)22 b Fq(S)6 b(M)k(S)1905 1475 y Fn(\003)1944 1516 y Fo(\))1982 1475 y Fp(2)2044 1516 y Fm(\000)22 b Fo(2\()p Fq(M)2334 1475 y Fp(2)2397 1516 y Fm(\000)g Fo(2)p Fq(M)10 b Fo(\))p Fm(g)p Fq(:)555 1693 y Fs(The)28 b(function)f Fq(a)p Fo(\()p Fq(k)s Fo(\))h Fs(v)n(anishes)e(when)i Fq(k)f Fo(+)d(1)p Fq(=k)31 b Fs(is)c(an)h(eigen)l(v)n(alue)f(of)g(\(2.1\).)40 b(Let)456 1809 y Fm(f)p Fq(\014)561 1824 y Fk(n)607 1809 y Fm(g)32 b Fs(be)f(the)g(zeros)h(of)f Fq(a)p Fo(\()p Fq(k)s Fo(\))h Fs(lying)e(in)h(the)g(domain)f Fm(j)p Fq(k)s Fm(j)40 b Fq(>)f Fo(1)p Fs(.)50 b(W)-8 b(e)32 b(introduce)f(the)456 1925 y(Blaschk)o(e)25 b(product)1120 2137 y Fq(B)33 b Fo(=)1331 2042 y Fg(Y)1373 2251 y Fk(n)1509 2070 y Fq(\014)1564 2085 y Fk(n)1633 2070 y Fm(\000)23 b Fq(k)p 1485 2114 327 4 v 1485 2205 a Fo(1)f Fm(\000)g Fq(\014)1710 2220 y Fk(n)1757 2205 y Fq(k)1859 2070 y(\014)1914 2085 y Fk(n)p 1831 2114 158 4 v 1831 2205 a Fm(j)p Fq(\014)1914 2220 y Fk(n)1961 2205 y Fm(j)1999 2137 y Fq(;)315 b Fo(\()p 2379 2056 103 4 v Fq(\014)2434 2152 y Fk(n)2509 2137 y Fo(=)27 b Fq(\014)2667 2152 y Fk(n)2714 2137 y Fo(\))p Fq(:)456 2397 y Fs(On)f(the)g(unit)g(circle,)h(when)f Fq(k)34 b Fo(=)d Fq(e)1686 2361 y Fk(i\022)1749 2397 y Fs(,)c(we)g(ha)n(v)o(e)f Fm(j)p Fq(B)5 b Fm(j)30 b Fo(=)h(1)26 b Fs(and)h(Im[)p Fo(log\()p Fq(B)5 b(=a)p Fo(\))p Fs(])26 b(is)g(an)456 2514 y(odd)e(function)g(of)h Fq(\022)j Fs(and)d(so,)f(by)h(\(2.4\))662 2668 y Fo(2)p 657 2712 59 4 v 657 2804 a Fq(\031)742 2600 y Fg(Z)842 2626 y Fk(\031)798 2825 y Fn(\000)p Fk(\031)916 2735 y Fo(log)17 b Fm(j)p Fq(a)p Fm(j)g Fo(sin)1302 2694 y Fp(2)1341 2735 y Fo(\()p Fq(m\022)s Fo(\))p Fq(d\022)31 b Fo(=)1837 2668 y(1)p 1791 2712 141 4 v 1791 2804 a(2)p Fq(\031)t(i)1958 2600 y Fg(Z)2013 2825 y Fn(j)p Fk(k)r Fn(j)p Fp(=1)2202 2735 y Fo(log\()p Fq(B)5 b(=a)p Fo(\))2593 2668 y(\()p Fq(k)2685 2632 y Fp(2)p Fk(m)2809 2668 y Fm(\000)23 b Fo(1\))2996 2632 y Fp(2)3035 2668 y Fq(dk)p 2593 2712 547 4 v 2743 2804 a(k)2797 2775 y Fp(2)p Fk(m)p Fp(+1)3177 2735 y Fo(=)1339 3005 y(2)17 b(log)o(\()p Fq(\034)11 b Fo(\))23 b(+)f Fq(A)1853 3020 y Fp(2)p Fk(m)1977 3005 y Fm(\000)2076 2910 y Fg(X)2127 3120 y Fk(n)2237 3005 y Fq(f)11 b Fo(\()p Fq(\014)2395 2964 y Fp(2)p Fk(m)2389 3030 y(n)2496 3005 y Fo(\))p Fq(;)456 3242 y Fs(where)25 b Fq(f)11 b Fo(\()p Fq(t)p Fo(\))27 b(=)h(1)p Fq(=)p Fo(2\()p Fq(t)22 b Fm(\000)g Fq(t)1401 3206 y Fn(\000)p Fp(1)1496 3242 y Fo(\))g Fm(\000)g Fo(log\()p Fm(j)p Fq(t)p Fm(j)p Fo(\))28 b Fq(>)f Fo(0)e Fs(for)g Fq(t)j(>)f Fo(1)p Fs(.)k(Finally)-6 b(,)24 b(we)h(obtain)788 3391 y Fo(2)p 783 3435 59 4 v 783 3527 a Fq(\031)868 3323 y Fg(Z)968 3349 y Fk(\031)924 3548 y Fn(\000)p Fk(\031)1042 3458 y Fo(log)17 b Fm(j)p Fq(a)p Fm(j)g Fo(sin)1428 3417 y Fp(2)1467 3458 y Fo(\()p Fq(m\022)s Fo(\))p Fq(d\022)25 b Fo(+)1895 3364 y Fg(X)1946 3573 y Fk(n)2056 3458 y Fq(f)11 b Fo(\()p Fq(\014)2214 3417 y Fp(2)p Fk(m)2208 3483 y(n)2315 3458 y Fo(\))28 b(=)f(2)17 b(log\()p Fq(\034)11 b Fo(\))22 b(+)g Fq(A)2998 3473 y Fp(2)p Fk(m)3100 3458 y Fq(:)555 3705 y Fs(Which)j(leads)f(to)h(the)f(inequality:)1076 3850 y Fo(2)p 1071 3895 V 1071 3986 a Fq(\031)1156 3782 y Fg(Z)1256 3809 y Fk(\031)1212 4008 y Fn(\000)p Fk(\031)1330 3918 y Fo(log)17 b Fm(j)p Fq(a)p Fm(j)g Fo(sin)1716 3877 y Fp(2)1755 3918 y Fo(\()p Fq(m\022)s Fo(\))p Fq(d\022)31 b Fm(\024)d Fo(2)17 b(log\()p Fq(\034)11 b Fo(\))22 b(+)g Fq(A)2710 3933 y Fp(2)p Fk(m)2812 3918 y Fq(:)-2383 b Fs(\(2.5\))456 4136 y(In)24 b(particular)h(for)g Fq(m)j Fo(=)g(4)c Fs(we)i(can)f(estimate)f(the)g(right)g(hand)h(side)f(by)1009 4343 y Fo(2)17 b(log)o(\()p Fq(\034)11 b Fo(\))23 b(+)f Fq(A)1523 4358 y Fp(4)1590 4343 y Fm(\024)1705 4276 y Fo(1)p 1705 4320 49 4 v 1705 4412 a(4)1764 4343 y(tr\()p Fm(f)p Fo(\(\003)2034 4302 y Fp(2)2034 4368 y(0)2095 4343 y Fo(+)g Fq(M)33 b Fo(+)22 b Fq(S)6 b(M)k(S)2654 4302 y Fn(\003)2694 4343 y Fo(\))2732 4302 y Fp(2)2793 4343 y Fo(+)1664 4533 y(2\(\003)1819 4548 y Fp(0)1880 4533 y Fo(+)22 b Fq(S)2044 4492 y Fn(\003)2083 4533 y Fo(\003)2151 4548 y Fp(0)2190 4533 y Fq(S)6 b Fo(\))2294 4492 y Fp(2)2334 4533 y Fo(\()p Fq(M)32 b Fo(+)22 b(1\))p Fm(g)p Fo(\))g(+)1650 4741 y(+)1736 4673 y(1)p 1736 4718 V 1736 4809 a(4)1795 4741 y(tr)o(\()p Fq(M)33 b Fo(+)22 b Fq(S)6 b(M)k(S)2369 4700 y Fn(\003)2409 4741 y Fo(\))2447 4700 y Fp(2)2509 4741 y Fo(+)22 b Fq(R)2681 4756 y Fp(3)2720 4741 y Fo(\(\003\))p Fq(:)456 4535 y Fs(\(2.6\))456 4938 y(where)j Fq(R)798 4953 y Fp(3)838 4938 y Fo(\(\003\))f Fs(is)g(\002nite)h(if)g Fo(\003)i Fm(2)h Fl(S)1685 4953 y Fp(3)1724 4938 y Fs(.)555 5054 y Fw(3.)j Fs(Assume)24 b(for)h(a)g(moment)e(that)1302 5216 y Fq(Q)28 b Fo(=)f(\003)1578 5231 y Fp(0)1617 5216 y Fq(;)67 b Fo(\003)1779 5231 y Fp(0)1846 5216 y Fo(=)27 b(\003)2017 5174 y Fn(\003)2017 5240 y Fp(0)2084 5216 y Fo(=)h(diag)p Fm(f)p Fq(\025)2475 5174 y Fp(0)2475 5240 y Fk(n)2521 5216 y Fm(g)p Fq(:)p eop %%Page: 5 5 5 4 bop 1246 235 a Fi(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 751 b(5)456 425 y Fs(Since)563 644 y Fo(\()p Fq(H)682 659 y Fp(0)743 644 y Fm(\000)23 b Fq(z)t Fo(\))930 603 y Fn(\000)p Fp(1)1053 644 y Fo(=)k Fm(\000)1246 577 y Fo(1)p 1243 621 55 4 v 1243 713 a Fq(k)1308 644 y Fo(\()p Fq(I)j Fm(\000)22 b Fq(S=k)s Fo(\))1719 603 y Fn(\000)p Fp(1)1814 644 y Fo(\()p Fq(I)30 b Fm(\000)22 b Fq(S)2090 603 y Fn(\003)2130 644 y Fq(=k)s Fo(\))2271 603 y Fn(\000)p Fp(1)2392 644 y Fo(=)28 b Fm(\000)2586 577 y Fo(1)p 2583 621 V 2583 713 a Fq(k)2701 520 y Fn(1)2664 550 y Fg(X)2675 760 y Fk(j)t Fp(=0)2866 520 y Fn(1)2829 550 y Fg(X)2825 759 y Fk(m)p Fp(=0)3004 577 y Fq(S)3070 541 y Fk(j)t Fn(\000)p Fk(m)p 3004 621 220 4 v 3010 713 a Fq(k)3064 684 y Fk(j)t Fp(+)p Fk(m)3261 644 y Fo(=)1426 1006 y(=)g Fm(\000)1661 881 y Fn(1)1624 911 y Fg(X)1629 1120 y Fk(n)p Fp(=0)1866 938 y Fo(1)p 1794 983 192 4 v 1794 1074 a Fq(k)1848 1045 y Fk(n)p Fp(+1)2067 881 y Fk(n)2016 911 y Fg(X)2012 1120 y Fk(m)p Fp(=0)2181 1006 y Fq(S)2247 965 y Fk(n)p Fn(\000)p Fp(2)p Fk(m)2447 1006 y Fq(;)456 1233 y Fs(we)d(obtain)1321 1355 y Fo(tr)p Fq(Q)p Fo(\()p Fq(H)1593 1370 y Fp(0)1655 1355 y Fm(\000)d Fq(z)t Fo(\))1841 1313 y Fn(\000)p Fp(1)1936 1355 y Fq(Q)p Fo(\()p Fq(H)2132 1370 y Fp(0)2194 1355 y Fm(\000)g Fq(z)t Fo(\))2380 1313 y Fn(\000)p Fp(1)2503 1355 y Fo(=)802 1474 y Fn(1)765 1503 y Fg(X)774 1713 y Fk(p)p Fp(=0)1003 1531 y Fo(1)p 936 1575 184 4 v 936 1666 a Fq(k)990 1638 y Fk(p)p Fp(+2)1130 1598 y Fo(tr)1276 1469 y Fk(p)1222 1503 y Fg(X)1228 1713 y Fk(n)p Fp(=0)1383 1598 y Fq(S)1449 1557 y Fn(\000)p Fk(n)1550 1598 y Fo(\003)p Fq(S)1684 1557 y Fk(n)1802 1474 y(n)1752 1503 y Fg(X)1747 1713 y Fk(m)p Fp(=0)1917 1598 y Fq(S)1983 1557 y Fn(\000)p Fp(2)p Fk(m)2139 1598 y Fo(\003)2229 1469 y Fk(p)p Fn(\000)p Fk(n)2223 1503 y Fg(X)2234 1713 y Fk(j)t Fp(=0)2384 1598 y Fq(S)2450 1557 y Fk(p)p Fn(\000)p Fp(2)p Fk(j)2640 1598 y Fo(=)2780 1474 y Fn(1)2743 1503 y Fg(X)2752 1713 y Fk(p)p Fp(=0)2959 1531 y Fq(J)3013 1546 y Fk(p)p 2914 1575 V 2914 1666 a Fq(k)2968 1638 y Fk(p)p Fp(+2)3108 1598 y Fq(;)456 1838 y Fs(where)877 2062 y Fq(J)931 2077 y Fp(2\()p Fk(p)p Fp(+1\))1178 2062 y Fo(=)28 b(tr)1429 1933 y Fk(p)1374 1967 y Fg(X)1380 2176 y Fk(n)p Fp(=0)1535 2062 y Fq(S)1601 2021 y Fn(\000)p Fk(n)1703 2062 y Fo(\003)1771 2077 y Fp(0)1810 2062 y Fq(S)1876 2021 y Fk(n)1994 1937 y(n)1943 1967 y Fg(X)1939 2176 y Fk(m)p Fp(=0)2108 2062 y Fq(S)2174 2021 y Fn(\000)p Fp(2)p Fk(m)2331 2062 y Fo(\003)2399 2077 y Fp(0)2547 1933 y Fk(p)2493 1967 y Fg(X)2455 2177 y Fk(j)t Fp(=)p Fk(n)p Fn(\000)p Fk(p)2692 2062 y Fq(S)2758 2021 y Fk(p)p Fn(\000)p Fp(2)p Fk(j)2947 2062 y Fo(=)464 2441 y(tr)611 2313 y Fk(p)556 2347 y Fg(X)562 2556 y Fk(n)p Fp(=0)717 2441 y Fq(S)783 2400 y Fn(\000)p Fk(n)885 2441 y Fo(\003)953 2456 y Fp(0)992 2441 y Fq(S)1058 2400 y Fk(n)1176 2317 y(n)1125 2347 y Fg(X)1121 2556 y Fk(m)p Fp(=0)1290 2441 y Fq(S)1356 2400 y Fn(\000)p Fp(2)p Fk(m)1513 2441 y Fo(\003)1581 2456 y Fp(0)1620 2441 y Fq(S)1686 2400 y Fp(2)p Fk(m)1810 2441 y Fo(+)22 b(tr)2077 2313 y Fp(2)p Fk(p)2040 2347 y Fg(X)2000 2556 y Fk(n)p Fp(=)p Fk(p)p Fp(+1)2240 2441 y Fq(S)2306 2400 y Fn(\000)p Fk(n)2408 2441 y Fo(\003)2476 2456 y Fp(0)2515 2441 y Fq(S)2581 2400 y Fk(n)2752 2313 y(p)2698 2347 y Fg(X)2644 2556 y Fk(m)p Fp(=)p Fk(n)p Fn(\000)p Fk(p)2912 2441 y Fq(S)2978 2400 y Fn(\000)p Fp(2)p Fk(m)3134 2441 y Fo(\003)3202 2456 y Fp(0)3241 2441 y Fq(S)3307 2400 y Fp(2)p Fk(m)3409 2441 y Fq(:)456 2681 y Fs(On)i(the)h(other)g(side,)1148 2931 y Fo(tr)1316 2803 y Fp(2)p Fk(p)1280 2837 y Fg(X)1240 3046 y Fk(n)p Fp(=)p Fk(p)p Fp(+1)1480 2931 y Fq(S)1546 2890 y Fn(\000)p Fk(n)1648 2931 y Fo(\003)1716 2946 y Fp(0)1755 2931 y Fq(S)1821 2890 y Fk(n)1992 2803 y(p)1938 2837 y Fg(X)1884 3046 y Fk(m)p Fp(=)p Fk(n)p Fn(\000)p Fk(p)2151 2931 y Fq(S)2217 2890 y Fn(\000)p Fp(2)p Fk(m)2374 2931 y Fo(\003)2442 2946 y Fp(0)2481 2931 y Fq(S)2547 2890 y Fp(2)p Fk(m)2676 2931 y Fo(=)946 3310 y(tr)1114 3182 y Fp(2)p Fk(p)1078 3216 y Fg(X)1038 3425 y Fk(n)p Fp(=)p Fk(p)p Fp(+1)1278 3310 y Fq(S)1344 3269 y Fn(\000)p Fk(n)1446 3310 y Fo(\003)1514 3325 y Fp(0)1553 3310 y Fq(S)1619 3269 y Fk(n)1682 3182 y Fp(2)p Fk(p)p Fn(\000)p Fk(n)1695 3216 y Fg(X)1690 3425 y Fk(m)p Fp(=0)1867 3310 y Fq(S)1933 3269 y Fn(\000)p Fp(2\()p Fk(m)p Fp(+)p Fk(n)p Fn(\000)p Fk(p)p Fp(\))2333 3310 y Fo(\003)2401 3325 y Fp(0)2440 3310 y Fq(S)2506 3269 y Fp(2\()p Fk(m)p Fp(+)p Fk(n)p Fn(\000)p Fk(p)p Fp(\))2878 3310 y Fo(=)456 3669 y(tr)624 3540 y Fp(2)p Fk(p)588 3574 y Fg(X)548 3784 y Fk(n)p Fp(=)p Fk(p)p Fp(+1)788 3669 y Fq(S)854 3628 y Fk(n)p Fn(\000)p Fp(2)p Fk(p)1027 3669 y Fo(\003)1095 3684 y Fp(0)1134 3669 y Fq(S)1200 3628 y Fp(2)p Fk(p)p Fn(\000)p Fk(n)1389 3540 y Fp(2)p Fk(p)p Fn(\000)p Fk(n)1401 3574 y Fg(X)1397 3784 y Fk(m)p Fp(=0)1574 3669 y Fq(S)1640 3628 y Fn(\000)p Fp(2)p Fk(m)1796 3669 y Fo(\003)1864 3684 y Fp(0)1904 3669 y Fq(S)1970 3628 y Fp(2)p Fk(m)2099 3669 y Fo(=)i(tr)2304 3540 y Fk(p)p Fn(\000)p Fp(1)2295 3574 y Fg(X)2300 3784 y Fk(n)p Fp(=0)2455 3669 y Fq(S)2521 3628 y Fn(\000)p Fk(n)2623 3669 y Fo(\003)2691 3684 y Fp(0)2730 3669 y Fq(S)2796 3628 y Fk(n)2914 3544 y(n)2864 3574 y Fg(X)2860 3784 y Fk(m)p Fp(=0)3029 3669 y Fq(S)3095 3628 y Fn(\000)p Fp(2)p Fk(m)3251 3669 y Fo(\003)3319 3684 y Fp(0)3358 3669 y Fq(S)3424 3628 y Fp(2)p Fk(m)3526 3669 y Fq(:)456 3909 y Fs(Therefore)456 4261 y Fq(J)510 4276 y Fp(2\()p Fk(p)p Fp(+1\))757 4261 y Fo(=)h(tr)1007 4132 y Fk(p)953 4166 y Fg(X)959 4375 y Fk(n)p Fp(=0)1114 4261 y Fq(S)1180 4220 y Fn(\000)p Fk(n)1281 4261 y Fo(\003)1349 4276 y Fp(0)1389 4261 y Fq(S)1455 4220 y Fk(n)1573 4136 y(n)1522 4166 y Fg(X)1518 4375 y Fk(m)p Fp(=0)1687 4261 y Fq(S)1753 4220 y Fn(\000)p Fp(2)p Fk(m)1910 4261 y Fo(\003)1978 4276 y Fp(0)2017 4261 y Fq(S)2083 4220 y Fp(2)p Fk(m)2184 4261 y Fo(+tr)2362 4132 y Fk(p)p Fn(\000)p Fp(1)2353 4166 y Fg(X)2358 4375 y Fk(n)p Fp(=0)2513 4261 y Fq(S)2579 4220 y Fn(\000)p Fk(n)2681 4261 y Fo(\003)2749 4276 y Fp(0)2788 4261 y Fq(S)2854 4220 y Fk(n)2972 4136 y(n)2922 4166 y Fg(X)2917 4375 y Fk(m)p Fp(=0)3087 4261 y Fq(S)3153 4220 y Fn(\000)p Fp(2)p Fk(m)3309 4261 y Fo(\003)3377 4276 y Fp(0)3416 4261 y Fq(S)3482 4220 y Fp(2)p Fk(m)3584 4261 y Fq(:)456 4508 y Fs(So)d(we)g(observ)o(e,)f (that)456 4758 y Fq(J)510 4774 y Fp(2\()p Fk(p)p Fp(+1\))757 4758 y Fo(=)k Fq(J)915 4773 y Fp(2)p Fk(p)990 4758 y Fo(+tr)o Fq(S)1207 4717 y Fn(\000)p Fk(p)1302 4758 y Fo(\003)1370 4773 y Fp(0)1409 4758 y Fq(S)1475 4717 y Fk(p)1589 4629 y(p)1535 4663 y Fg(X)1531 4873 y Fk(m)p Fp(=0)1700 4758 y Fq(S)1766 4717 y Fn(\000)p Fp(2)p Fk(m)1923 4758 y Fo(\003)1991 4773 y Fp(0)2030 4758 y Fq(S)2096 4717 y Fp(2)p Fk(m)2197 4758 y Fo(+tr)p Fq(S)2415 4717 y Fn(\000)p Fk(p)p Fp(+1)2600 4758 y Fo(\003)2668 4773 y Fp(0)2707 4758 y Fq(S)2773 4717 y Fk(p)p Fn(\000)p Fp(1)2933 4629 y Fk(p)p Fn(\000)p Fp(1)2923 4663 y Fg(X)2919 4873 y Fk(m)p Fp(=0)3088 4758 y Fq(S)3154 4717 y Fn(\000)p Fp(2)p Fk(m)3311 4758 y Fo(\003)3379 4773 y Fp(0)3418 4758 y Fq(S)3484 4717 y Fp(2)p Fk(m)1191 5124 y Fo(=)g Fq(J)1349 5139 y Fp(2)p Fk(p)1446 5124 y Fo(+)22 b(tr)p Fq(S)1686 5083 y Fn(\000)p Fk(p)1780 5124 y Fo(\003)1848 5139 y Fp(0)1887 5124 y Fq(S)1953 5083 y Fk(p)2050 4996 y Fp(2)p Fk(p)2014 5030 y Fg(X)2009 5239 y Fk(m)p Fp(=0)2178 5124 y Fq(S)2244 5083 y Fn(\000)p Fk(m)2366 5124 y Fo(\003)2434 5139 y Fp(0)2473 5124 y Fq(S)2539 5083 y Fk(m)2633 5124 y Fo(=)p eop %%Page: 6 6 6 5 bop 456 235 a Fi(6)920 b(LAPTEV)-10 b(,)23 b(N)m(ABOK)n(O,)h(SAFR)m (ONO)l(V)1226 509 y Fo(=)j Fq(J)1383 524 y Fp(2)p Fk(p)1480 509 y Fo(+)22 b(2tr\003)1846 381 y Fk(p)1792 415 y Fg(X)1788 624 y Fk(m)p Fp(=1)1957 509 y Fq(S)2023 468 y Fn(\000)p Fk(m)2144 509 y Fo(\003)p Fq(S)2278 468 y Fk(m)2366 509 y Fo(+)g(tr\003)2608 468 y Fp(2)2647 509 y Fq(:)456 739 y Fs(Since)1733 863 y Fq(J)1787 878 y Fp(2)1854 863 y Fo(=)27 b(tr\003)2101 822 y Fp(2)2101 888 y(0)2140 863 y Fq(;)456 1008 y Fs(by)d(induction)f(we)i(obtain)f(that)1364 1255 y Fq(J)1418 1270 y Fp(2\()p Fk(p)p Fp(+1\))1666 1255 y Fo(=)j(tr\()1937 1126 y Fk(p)1883 1160 y Fg(X)1888 1369 y Fk(n)p Fp(=0)2044 1255 y Fq(S)2110 1213 y Fn(\000)p Fk(n)2211 1255 y Fo(\003)2279 1270 y Fp(0)2319 1255 y Fq(S)2385 1213 y Fk(n)2431 1255 y Fo(\))2469 1213 y Fp(2)2509 1255 y Fq(:)555 1504 y Fs(As)e(a)g(result)f(we)h(ha)n(v)o(e)1157 1726 y Fo(tr)o Fq(Q)p Fo(\()p Fq(H)1428 1741 y Fp(0)1490 1726 y Fm(\000)e Fq(z)t Fo(\))1677 1685 y Fn(\000)p Fp(1)1772 1726 y Fq(Q)p Fo(\()p Fq(H)1968 1741 y Fp(0)2029 1726 y Fm(\000)g Fq(z)t Fo(\))2216 1685 y Fn(\000)p Fp(1)2339 1726 y Fo(=)2479 1602 y Fn(1)2442 1632 y Fg(X)2451 1841 y Fk(p)p Fp(=2)2613 1659 y Fq(J)2667 1674 y Fk(p)p 2613 1703 94 4 v 2613 1795 a Fq(k)2667 1766 y Fk(p)2716 1726 y Fq(;)456 1989 y Fs(where)1360 2174 y Fq(J)1414 2190 y Fp(2\()p Fk(p)p Fp(+1\))1661 2174 y Fo(=)28 b(2tr)o(\()1982 2046 y Fk(p)1927 2080 y Fg(X)1932 2289 y Fk(n)p Fp(=0)2088 2174 y Fq(S)2154 2133 y Fn(\000)p Fk(n)2255 2174 y Fo(\003)p Fq(S)2389 2133 y Fk(n)2436 2174 y Fo(\))2474 2133 y Fp(2)2513 2174 y Fq(:)456 2405 y Fw(4.)i Fs(Let)25 b(us)f(mak)o(e)h(another)g (temporary)f(assumption,)f(that)1419 2570 y Fq(Q)28 b Fo(=)g Fq(S)6 b Fo(\003)21 b(+)h(\003)p Fq(S)2015 2529 y Fn(\003)2054 2570 y Fq(;)67 b Fo(\003)27 b(=)h(\003)2415 2529 y Fn(\003)2454 2570 y Fq(:)456 2735 y Fs(Then)677 2900 y Fo(tr)p Fq(Q)p Fo(\()p Fq(H)949 2915 y Fp(0)1011 2900 y Fm(\000)22 b Fq(z)t Fo(\))1197 2859 y Fn(\000)p Fp(1)1292 2900 y Fq(Q)p Fo(\()p Fq(H)1488 2915 y Fp(0)1550 2900 y Fm(\000)h Fq(z)t Fo(\))1737 2859 y Fn(\000)p Fp(1)1859 2900 y Fo(=)28 b(2tr)o(\003\()p Fq(H)2274 2915 y Fp(0)2336 2900 y Fm(\000)22 b Fq(z)t Fo(\))2522 2859 y Fn(\000)p Fp(1)2617 2900 y Fo(\003\()p Fq(H)2804 2915 y Fp(0)2865 2900 y Fm(\000)h Fq(z)t Fo(\))3052 2859 y Fn(\000)p Fp(1)3147 2900 y Fo(+)532 3075 y(+tr)p Fq(S)6 b Fo(\003\()p Fq(H)937 3090 y Fp(0)998 3075 y Fm(\000)22 b Fq(z)t Fo(\))1184 3034 y Fn(\000)p Fp(1)1279 3075 y Fq(S)6 b Fo(\003\()p Fq(H)1532 3090 y Fp(0)1593 3075 y Fm(\000)23 b Fq(z)t Fo(\))1780 3034 y Fn(\000)p Fp(1)1897 3075 y Fo(+)f(tr\003)p Fq(S)2205 3034 y Fn(\003)2244 3075 y Fo(\()p Fq(H)2363 3090 y Fp(0)2424 3075 y Fm(\000)h Fq(z)t Fo(\))2611 3034 y Fn(\000)p Fp(1)2706 3075 y Fo(\003)p Fq(S)2840 3034 y Fn(\003)2879 3075 y Fo(\()p Fq(H)2998 3090 y Fp(0)3059 3075 y Fm(\000)g Fq(z)t Fo(\))3246 3034 y Fn(\000)p Fp(1)3341 3075 y Fq(:)555 3220 y Fs(Consider)i(the)g(last)f(term)g(in)h(the)f (sum)456 3385 y Fo(tr)o(\003)p Fq(S)665 3344 y Fn(\003)704 3385 y Fo(\()p Fq(H)823 3400 y Fp(0)871 3385 y Fm(\000)9 b Fq(z)t Fo(\))1044 3344 y Fn(\000)p Fp(1)1139 3385 y Fo(\003)p Fq(S)1273 3344 y Fn(\003)1312 3385 y Fo(\()p Fq(H)1431 3400 y Fp(0)1479 3385 y Fm(\000)g Fq(z)t Fo(\))1652 3344 y Fn(\000)p Fp(1)1775 3385 y Fo(=)27 b(tr)p Fq(S)6 b Fo(\003)p Fq(S)2154 3344 y Fn(\003)2193 3385 y Fo(\()p Fq(H)2312 3400 y Fp(0)2360 3385 y Fm(\000)j Fq(z)t Fo(\))2533 3344 y Fn(\000)p Fp(1)2628 3385 y Fo(\003)p Fq(S)2762 3344 y Fn(\003)2801 3385 y Fo(\()p Fq(H)2920 3400 y Fp(0)2968 3385 y Fm(\000)g Fq(z)t Fo(\))3141 3344 y Fn(\000)p Fp(1)3236 3385 y Fq(S)3302 3344 y Fn(\003)3369 3385 y Fo(=)456 3657 y(=:)623 3533 y Fn(1)586 3563 y Fg(X)595 3772 y Fk(p)p Fp(=2)757 3590 y Fq(D)838 3605 y Fk(p)p 757 3635 121 4 v 770 3726 a Fq(k)824 3697 y Fk(p)887 3657 y Fq(;)93 b Fo(where)77 b Fq(D)1413 3673 y Fp(2\()p Fk(p)p Fp(+1\))1660 3657 y Fo(=)28 b(tr)1893 3529 y Fp(2)p Fk(p)1856 3563 y Fg(X)1862 3772 y Fk(n)p Fp(=0)2017 3657 y Fq(S)2083 3616 y Fn(\000)p Fk(n)p Fp(+1)2275 3657 y Fo(\003)p Fq(S)2409 3616 y Fk(n)p Fn(\000)p Fp(1)2617 3533 y Fk(n)2566 3563 y Fg(X)2562 3772 y Fk(m)p Fp(=0)2731 3657 y Fq(S)2797 3616 y Fn(\000)p Fp(2)p Fk(m)2954 3657 y Fo(\003)3131 3529 y Fk(p)3076 3563 y Fg(X)3039 3773 y Fk(j)t Fp(=)p Fk(n)p Fn(\000)p Fk(p)3275 3657 y Fq(S)3341 3616 y Fp(2)p Fk(j)t Fn(\000)p Fp(2)3531 3657 y Fo(=)1029 4022 y(tr)1158 3894 y Fp(2)p Fk(p)1121 3928 y Fg(X)1127 4137 y Fk(n)p Fp(=0)1282 4022 y Fq(S)1348 3981 y Fn(\000)p Fk(n)p Fp(+1)1540 4022 y Fo(\003)p Fq(S)1674 3981 y Fk(n)p Fn(\000)p Fp(1)1882 3898 y Fk(n)1831 3928 y Fg(X)1827 4137 y Fk(m)p Fp(=0)1996 4022 y Fq(S)2062 3981 y Fn(\000)p Fp(2)p Fk(m)2219 4022 y Fo(\003)2395 3894 y Fk(p)p Fn(\000)p Fp(1)2386 3928 y Fg(X)2304 4138 y Fk(j)t Fp(=)p Fk(n)p Fn(\000)p Fk(p)p Fn(\000)p Fp(1)2630 4022 y Fq(S)2696 3981 y Fp(2)p Fk(j)2795 4022 y Fo(=)456 4387 y(tr)557 4259 y Fk(p)p Fn(\000)p Fp(1)548 4293 y Fg(X)554 4502 y Fk(n)p Fp(=0)709 4387 y Fq(S)775 4346 y Fn(\000)p Fk(n)p Fp(+1)966 4387 y Fo(\003)p Fq(S)1100 4346 y Fk(n)p Fn(\000)p Fp(1)1309 4263 y Fk(n)1258 4293 y Fg(X)1254 4502 y Fk(m)p Fp(=0)1423 4387 y Fq(S)1489 4346 y Fn(\000)p Fp(2)p Fk(m)1645 4387 y Fo(\003)p Fq(S)1779 4346 y Fp(2)p Fk(m)1881 4387 y Fo(+tr)2125 4259 y Fp(2)p Fk(p)2089 4293 y Fg(X)2049 4502 y Fk(n)p Fp(=)p Fk(p)p Fp(+1)2289 4387 y Fq(S)2355 4346 y Fn(\000)p Fk(n)p Fp(+1)2547 4387 y Fo(\003)p Fq(S)2681 4346 y Fk(n)p Fn(\000)p Fp(1)2942 4259 y Fk(p)p Fn(\000)p Fp(1)2933 4293 y Fg(X)2834 4502 y Fk(m)p Fp(=)p Fk(n)p Fn(\000)p Fk(p)p Fn(\000)p Fp(1)3191 4387 y Fq(S)3257 4346 y Fn(\000)p Fp(2)p Fk(m)3414 4387 y Fo(\003)p Fq(S)3548 4346 y Fp(2)p Fk(m)3649 4387 y Fo(+)1358 4752 y Fq(S)1424 4711 y Fn(\000)p Fk(p)p Fp(+1)1608 4752 y Fo(\003)p Fq(S)1742 4711 y Fk(p)p Fn(\000)p Fp(1)1902 4623 y Fk(p)p Fn(\000)p Fp(1)1892 4657 y Fg(X)1888 4866 y Fk(m)p Fp(=0)2057 4752 y Fq(S)2123 4711 y Fn(\000)p Fp(2)p Fk(m)2280 4752 y Fo(\003)p Fq(S)2414 4711 y Fp(2)p Fk(m)2515 4752 y Fq(:)1052 5111 y Fo(tr)1221 4983 y Fp(2)p Fk(p)1184 5017 y Fg(X)1144 5226 y Fk(n)p Fp(=)p Fk(p)p Fp(+1)1384 5111 y Fq(S)1450 5070 y Fn(\000)p Fk(n)p Fp(+1)1642 5111 y Fo(\003)p Fq(S)1776 5070 y Fk(n)p Fn(\000)p Fp(1)2037 4983 y Fk(p)p Fn(\000)p Fp(1)2028 5017 y Fg(X)1929 5226 y Fk(m)p Fp(=)p Fk(n)p Fn(\000)p Fk(p)p Fn(\000)p Fp(1)2287 5111 y Fq(S)2353 5070 y Fn(\000)p Fp(2)p Fk(m)2509 5111 y Fo(\003)p Fq(S)2643 5070 y Fp(2)p Fk(m)2772 5111 y Fo(=)p eop %%Page: 7 7 7 6 bop 1246 235 a Fi(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 751 b(7)1040 524 y Fo(tr)1208 395 y Fp(2)p Fk(p)1172 429 y Fg(X)1132 638 y Fk(n)p Fp(=)p Fk(p)p Fp(+1)1372 524 y Fq(S)1438 482 y Fk(n)p Fn(\000)p Fp(2)p Fk(p)p Fn(\000)p Fp(1)1701 524 y Fo(\003)p Fq(S)1835 482 y Fp(2)p Fk(p)p Fn(\000)p Fk(n)p Fp(+1)2114 395 y(2)p Fk(p)p Fn(\000)p Fk(n)2126 429 y Fg(X)2122 638 y Fk(m)p Fp(=0)2299 524 y Fq(S)2365 482 y Fn(\000)p Fp(2)p Fk(m)2521 524 y Fo(\003)p Fq(S)2655 482 y Fp(2)p Fk(m)2784 524 y Fo(=)1224 881 y(tr)1325 753 y Fk(p)p Fn(\000)p Fp(1)1316 787 y Fg(X)1322 996 y Fk(n)p Fp(=0)1477 881 y Fq(S)1543 840 y Fn(\000)p Fk(n)p Fn(\000)p Fp(1)1735 881 y Fo(\003)p Fq(S)1869 840 y Fk(n)p Fp(+1)2077 757 y Fk(n)2026 787 y Fg(X)2022 996 y Fk(m)p Fp(=0)2191 881 y Fq(S)2257 840 y Fn(\000)p Fp(2)p Fk(m)2414 881 y Fo(\003)p Fq(S)2548 840 y Fp(2)p Fk(m)2649 881 y Fq(:)456 1108 y Fs(Thus,)456 1357 y Fq(D)537 1373 y Fp(2\()p Fk(p)p Fp(+1\))784 1357 y Fo(=)28 b Fq(D)969 1372 y Fp(2)p Fk(p)1044 1357 y Fo(+tr)p Fq(S)1262 1316 y Fn(\000)p Fk(p)p Fp(+2)1446 1357 y Fo(\003)p Fq(S)1580 1316 y Fk(p)p Fn(\000)p Fp(2)1739 1229 y Fk(p)p Fn(\000)p Fp(1)1730 1263 y Fg(X)1726 1472 y Fk(m)p Fp(=0)1895 1357 y Fq(S)1961 1316 y Fn(\000)p Fp(2)p Fk(m)2118 1357 y Fo(\003)p Fq(S)2252 1316 y Fp(2)p Fk(m)2353 1357 y Fm(\000)p Fo(tr)p Fq(S)2572 1316 y Fn(\000)p Fk(p)p Fp(+2)2757 1357 y Fo(\003)p Fq(S)2891 1316 y Fk(p)p Fn(\000)p Fp(2)3050 1229 y Fk(p)p Fn(\000)p Fp(2)3041 1263 y Fg(X)3037 1472 y Fk(m)p Fp(=0)3206 1357 y Fq(S)3272 1316 y Fn(\000)p Fp(2)p Fk(m)3429 1357 y Fo(\003)p Fq(S)3563 1316 y Fp(2)p Fk(m)3664 1357 y Fo(+)657 1723 y(+tr)p Fq(S)875 1681 y Fn(\000)p Fk(p)969 1723 y Fo(\003)p Fq(S)1103 1681 y Fk(p)1172 1594 y(p)p Fn(\000)p Fp(1)1163 1628 y Fg(X)1159 1837 y Fk(m)p Fp(=0)1328 1723 y Fq(S)1394 1681 y Fn(\000)p Fp(2)p Fk(m)1551 1723 y Fo(\003)p Fq(S)1685 1681 y Fp(2)p Fk(m)1808 1723 y Fo(+)22 b(tr)p Fq(S)2048 1681 y Fn(\000)p Fk(p)p Fp(+1)2232 1723 y Fo(\003)p Fq(S)2366 1681 y Fk(p)p Fn(\000)p Fp(1)2526 1594 y Fk(p)p Fn(\000)p Fp(1)2517 1628 y Fg(X)2512 1837 y Fk(m)p Fp(=0)2682 1723 y Fq(S)2748 1681 y Fn(\000)p Fp(2)p Fk(m)2904 1723 y Fo(\003)p Fq(S)3038 1681 y Fp(2)p Fk(m)3167 1723 y Fo(=)571 2068 y Fq(D)652 2083 y Fp(2)p Fk(p)749 2068 y Fo(+)g(tr)p Fq(S)989 2026 y Fn(\000)p Fk(p)1083 2068 y Fo(\003)p Fq(S)1217 2026 y Fk(p)1286 1939 y(p)p Fn(\000)p Fp(1)1277 1973 y Fg(X)1273 2182 y Fk(m)p Fp(=0)1442 2068 y Fq(S)1508 2026 y Fn(\000)p Fp(2)p Fk(m)1665 2068 y Fo(\003)p Fq(S)1799 2026 y Fp(2)p Fk(m)1922 2068 y Fo(+)g(tr)p Fq(S)2162 2026 y Fn(\000)p Fk(p)2256 2068 y Fo(\003)p Fq(S)2390 2026 y Fk(p)2459 1939 y(p)p Fn(\000)p Fp(1)2450 1973 y Fg(X)2446 2182 y Fk(m)p Fp(=0)2615 2068 y Fq(S)2681 2026 y Fn(\000)p Fp(2)p Fk(m)p Fn(\000)p Fp(1)2928 2068 y Fo(\003)p Fq(S)3062 2026 y Fp(2)p Fk(m)p Fp(+1)3253 2068 y Fo(+)961 2412 y(+tr)o Fq(S)1178 2371 y Fk(p)1218 2412 y Fo(\003)p Fq(S)1352 2371 y Fn(\000)p Fk(p)1474 2412 y Fo(=)27 b Fq(D)1658 2427 y Fp(2)p Fk(p)1755 2412 y Fo(+)22 b(tr)p Fq(S)1995 2371 y Fn(\000)p Fk(p)2089 2412 y Fo(\003)p Fq(S)2223 2371 y Fk(p)2320 2284 y Fp(2)p Fk(p)2284 2318 y Fg(X)2279 2527 y Fk(m)p Fp(=0)2448 2412 y Fq(S)2514 2371 y Fn(\000)p Fk(m)2636 2412 y Fo(\003)p Fq(S)2770 2371 y Fk(m)2863 2412 y Fo(=)1212 2743 y(=)28 b Fq(D)1397 2758 y Fp(2)p Fk(p)1494 2743 y Fo(+)22 b(2tr\003)1859 2615 y Fk(p)1805 2649 y Fg(X)1802 2858 y Fk(m)p Fp(=1)1970 2743 y Fq(S)2036 2702 y Fn(\000)p Fk(m)2157 2743 y Fo(\003)p Fq(S)2291 2702 y Fk(m)2380 2743 y Fo(+)g(tr)o(\003)2621 2702 y Fp(2)2661 2743 y Fq(:)456 2969 y Fs(Since)1477 3091 y Fq(D)1558 3106 y Fp(2)1625 3091 y Fo(=)27 b(0)h(=)f(tr\003)2052 3050 y Fp(2)2114 3091 y Fm(\000)22 b Fo(tr\003)2357 3050 y Fp(2)2396 3091 y Fq(;)456 3233 y Fs(we)j(obtain)1218 3415 y Fq(D)1299 3430 y Fp(2\()p Fk(p)p Fp(+1\))1547 3415 y Fo(=)i(tr\()1818 3286 y Fk(p)1764 3320 y Fg(X)1769 3529 y Fk(n)p Fp(=0)1925 3415 y Fq(S)1991 3374 y Fn(\000)p Fk(n)2092 3415 y Fo(\003)p Fq(S)2226 3374 y Fk(n)2273 3415 y Fo(\))2311 3374 y Fp(2)2372 3415 y Fm(\000)c Fo(tr\003)2616 3374 y Fp(2)2655 3415 y Fq(:)456 3641 y Fs(As)h(a)h(result)g(we)g(ha)n (v)o(e)1124 3860 y Fo(tr)p Fq(Q)p Fo(\()p Fq(H)1396 3875 y Fp(0)1457 3860 y Fm(\000)e Fq(z)t Fo(\))1644 3819 y Fn(\000)p Fp(1)1739 3860 y Fq(Q)p Fo(\()p Fq(H)1935 3875 y Fp(0)1997 3860 y Fm(\000)f Fq(z)t Fo(\))2183 3819 y Fn(\000)p Fp(1)2306 3860 y Fo(=)27 b(2)2511 3735 y Fn(1)2475 3765 y Fg(X)2484 3974 y Fk(p)p Fp(=2)2651 3792 y Fq(I)2694 3807 y Fk(p)p 2645 3837 94 4 v 2645 3928 a Fq(k)2699 3899 y Fk(p)2749 3860 y Fq(;)456 4119 y Fs(where)1213 4301 y Fq(I)1256 4317 y Fp(2\()p Fk(p)p Fp(+1\))1503 4301 y Fo(=)h(2tr\()1824 4173 y Fk(p)1770 4207 y Fg(X)1775 4416 y Fk(n)p Fp(=0)1930 4301 y Fq(S)1996 4260 y Fn(\000)p Fk(n)2098 4301 y Fo(\003)p Fq(S)2232 4260 y Fk(n)2278 4301 y Fo(\))2316 4260 y Fp(2)2378 4301 y Fm(\000)22 b Fo(tr\003)2621 4260 y Fp(2)2660 4301 y Fq(:)456 4529 y Fw(5.)30 b Fs(Finally)24 b(we)i(consider)e(the)h(general)g(case,)g (when)g Fo(\003)i Fm(6)p Fo(=)h(0)p Fq(;)41 b Fo(\003)2683 4544 y Fp(0)2750 4529 y Fm(6)p Fo(=)27 b(0)p Fs(.)k(Since)869 4768 y Fo(tr)p Fq(Q)p Fo(\()p Fq(H)1141 4783 y Fp(0)1203 4768 y Fm(\000)22 b Fq(z)t Fo(\))1389 4727 y Fn(\000)p Fp(1)1512 4768 y Fo(=)28 b Fm(\000)p Fo(2tr\003)1943 4643 y Fn(1)1906 4673 y Fg(X)1903 4882 y Fk(m)p Fp(=0)2135 4700 y Fo(1)p 2081 4745 156 4 v 2081 4836 a Fq(k)2135 4807 y Fp(2)p Fk(m)2269 4768 y Fm(\000)23 b Fo(tr\003)2513 4783 y Fp(0)2609 4643 y Fn(1)2573 4673 y Fg(X)2568 4882 y Fk(m)p Fp(=0)2846 4700 y Fo(1)p 2748 4745 247 4 v 2748 4836 a Fq(k)2802 4807 y Fp(2)p Fk(m)p Fp(+1)3004 4768 y Fq(;)456 5014 y Fs(we)i(obtain:)948 5177 y Fq(A)1021 5192 y Fp(2)p Fk(m)1150 5177 y Fo(=)j(2tr\003)21 b(+)i Fq(J)1621 5192 y Fp(2)p Fk(m)1722 5177 y Fq(=)p Fo(2)f(+)g Fq(I)1983 5192 y Fp(2)p Fk(m)2107 5177 y Fo(+)g Fq(F)2268 5192 y Fp(2)p Fk(m)2370 5177 y Fo(\()p Fq(Q)p Fo(\))p Fq(;)92 b(m)28 b Fm(\025)g Fo(3)p Fq(;)-2480 b Fs(\(2.7\))p eop %%Page: 8 8 8 7 bop 456 235 a Fi(8)920 b(LAPTEV)-10 b(,)23 b(N)m(ABOK)n(O,)h(SAFR)m (ONO)l(V)456 425 y Fs(where)h Fq(F)787 440 y Fp(2)p Fk(m)889 425 y Fo(\()p Fq(Q)p Fo(\))g Fs(is)f(\002nite)h(if)g(one)g(of)f(the)h (tw)o(o)f(follo)n(wing)f(conditions)g(hold)1512 587 y Fq(i)p Fo(\))50 b(\003)1701 602 y Fp(0)1767 587 y Fm(2)28 b Fl(S)1944 602 y Fp(3)1983 587 y Fq(;)67 b Fo(\003)27 b Fm(2)h Fl(S)2349 602 y Fp(3)2127 733 y Fo(or)1515 884 y Fq(ii)p Fo(\))50 b(\003)1737 899 y Fp(0)1804 884 y Fm(2)28 b Fl(S)1981 899 y Fp(4)2020 884 y Fq(;)67 b Fo(\003)27 b(=)h(0)p Fq(:)456 743 y Fs(\(2.8\))1520 1120 y(3.)51 b(W)t Fr(E)t(Y)t(L)32 b(F)t(U)t(N)t(C)t(T)t(I)t(O)t(N)555 1294 y Fs(W)l(ithout)26 b(loss)g(of)h(generality)g(we)g(may)g(assume)g (that)f Fq(\025)2519 1309 y Fk(n)2598 1294 y Fo(=)32 b Fq(\025)2763 1258 y Fp(0)2763 1319 y Fk(n)2842 1294 y Fo(=)g(0)27 b Fs(for)g Fq(n)32 b Fm(\024)h Fo(2)p Fq(:)456 1410 y Fs(Let)c(us)h(de\002ne)g(the)f(W)-8 b(e)o(yl)29 b(function)g(for)h(the)g(operator)f(in)h Fq(l)2557 1374 y Fp(2)2596 1410 y Fo(\()p Ff(N)9 b Fo(\))36 b Fs(\(on)30 b(the)f(so)g(called)456 1526 y(semi-axis\))1392 1656 y Fq(m)p Fo(\()p Fq(z)t Fo(\))g(:=)f(\(\()p Fq(H)h Fm(\000)23 b Fq(z)t Fo(\))2135 1615 y Fn(\000)p Fp(1)2230 1656 y Fq(e)2275 1671 y Fp(0)2314 1656 y Fq(;)17 b(e)2403 1671 y Fp(0)2443 1656 y Fo(\))p Fq(:)456 1807 y Fs(From)24 b(the)h(e)o(xplicit)e(representation)i(of)g(the)f(v)o(ector)h Fq(y)30 b Fo(=)e(\()p Fq(H)h Fm(\000)23 b Fq(z)t Fo(\))2799 1770 y Fn(\000)p Fp(1)2894 1807 y Fq(e)2939 1822 y Fp(0)1410 1977 y Fq(y)1458 1992 y Fk(n)1532 1977 y Fo(=)28 b Fm(\000)p Fq( )1776 1992 y Fk(n)1824 1977 y Fq(= )1936 1992 y Fn(\000)p Fp(1)2030 1977 y Fq(;)166 b(n)28 b Fm(\025)g Fo(0)p Fq(;)456 2148 y Fs(we)19 b(obtain)g(an)h(important)e(relation)h(which)g(reco)o (v)o(ers)g Fq(m)i Fs(from)e(the)g(solution)f(of)i(\(2.1\):)456 2264 y Fq(m)p Fo(\()p Fq(k)27 b Fo(+)d(1)p Fq(=k)s Fo(\))33 b(=)g Fm(\000)p Fq( )1229 2279 y Fp(0)1269 2264 y Fq(= )1381 2279 y Fn(\000)p Fp(1)1503 2264 y Fs(for)28 b Fm(j)p Fq(k)s Fm(j)33 b Fq(>)g Fo(1)p Fs(.)39 b(Since)28 b Fq( )2324 2279 y Fk(n)2404 2264 y Fo(=)33 b Fq(a)p Fo(\()p Fq(k)s Fo(\))p Fq(k)2748 2228 y Fn(\000)p Fk(n)2874 2264 y Fo(+)25 b Fq(b)p Fo(\()p Fq(k)s Fo(\))p Fq(k)3200 2228 y Fk(n)3247 2264 y Fs(,)j(this)456 2380 y(relation)c(leads)h(to)1054 2586 y Fq(m)p Fo(\()p Fq(z)t Fo(\))k(=)e Fm(\000)1585 2519 y Fq(a)p Fo(\()p Fq(k)s Fo(\))22 b(+)g Fq(b)p Fo(\()p Fq(k)s Fo(\))p 1483 2563 676 4 v 1483 2654 a Fq(a)p Fo(\()p Fq(k)s Fo(\))p Fq(k)k Fo(+)c Fq(b)p Fo(\()p Fq(k)s Fo(\))p Fq(k)2064 2626 y Fn(\000)p Fp(1)2169 2586 y Fq(;)116 b(z)33 b Fo(=)27 b Fq(k)e Fo(+)d(1)p Fq(=k)s(:)456 2810 y Fs(Therefore)1457 2975 y Fq(r)30 b Fo(:=)e Fq(b=a)g Fo(=)1944 2908 y Fm(\000)p Fo(\(1)23 b(+)f Fq(k)s(m)p Fo(\))p 1944 2952 462 4 v 1981 3043 a Fq(k)s Fo(\()p Fq(m)g Fo(+)g Fq(k)s Fo(\))2416 2975 y Fq(:)456 3179 y Fs(Introducing)h(the)i(Wronskian)1178 3352 y Fq(W)1270 3367 y Fk(n)1344 3352 y Fo(=)j Fq(W)1540 3367 y Fk(n)1587 3352 y Fo(\()p Fq( )t(;)p 1736 3271 67 4 v 17 w( )t Fo(\))f(=)h Fq( )2035 3367 y Fk(n)p 2082 3271 V 2082 3352 a Fq( )2149 3375 y Fk(n)p Fn(\000)p Fp(1)2308 3352 y Fm(\000)23 b Fq( )2471 3367 y Fk(n)p Fn(\000)p Fp(1)p 2609 3271 V 2609 3352 a Fq( )2676 3375 y Fk(n)456 3527 y Fs(we)29 b(observ)o(e)f(that)h(for)g Fq(k)39 b Fo(=)c Fq(e)1508 3491 y Fk(i\022)1572 3527 y Fq(;)46 b(\022)38 b Fm(2)e Fo(\(0)p Fq(;)17 b(\031)t Fo(\))29 b Fs(the)g(relation)2571 3453 y Fg(P)2676 3527 y Fo(\()p Fq(W)2806 3542 y Fk(n)p Fp(+1)2969 3527 y Fm(\000)d Fq(W)3164 3542 y Fk(n)3211 3527 y Fo(\))35 b(=)h(0)456 3644 y Fs(leads)24 b(to)1644 3773 y Fm(j)p Fq(a)p Fm(j)1751 3732 y Fp(2)1813 3773 y Fm(\000)e(j)p Fq(b)p Fm(j)2009 3732 y Fp(2)2076 3773 y Fo(=)28 b(1)p Fq(:)456 3924 y Fs(Therefore)37 b(for)g(v)n(alues)f Fq(k)54 b Fo(=)c Fq(e)1603 3888 y Fk(i\022)1666 3924 y Fq(;)j(\022)h Fm(2)c Fo(\(0)p Fq(;)17 b(\031)t Fo(\))p Fs(,)39 b(corresponding)d(to)h Fq(k)d Fo(+)d(1)p Fq(=k)53 b Fo(=)456 4040 y(2)17 b(cos\()p Fq(\022)s Fo(\))22 b(+)g Fq(i)p Fo(0)p Fs(,)j(we)g(ha)n(v)o(e:)1006 4270 y Fm(j)p Fq(a)p Fm(j)1113 4228 y Fn(\000)p Fp(2)1234 4270 y Fo(=)j(1)22 b Fm(\000)g(j)p Fq(r)s Fm(j)1611 4228 y Fp(2)1678 4270 y Fo(=)1791 4202 y(4)17 b(sin)o(\()p Fq(\022)s Fo(\)Im\()p Fq(m)p Fo(\))p 1791 4247 587 4 v 1907 4338 a Fm(j)p Fq(m)22 b Fo(+)g Fq(k)s Fm(j)2222 4309 y Fp(2)2416 4270 y Fm(\024)2531 4202 y Fo(4Im)o(\()p Fq(m)p Fo(\))p 2531 4247 327 4 v 2572 4338 a(sin\()p Fq(\022)s Fo(\))2867 4270 y Fq(:)456 4500 y Fs(The)e(last)f(inequality)g(holds)g(since)h(Im)p Fo(\()p Fq(m)p Fo(\()p Fq(\025)5 b Fo(+)g Fq(i)p Fo(0\)\))28 b Fm(\025)g Fo(0)20 b Fs(for)g Fq(\025)28 b Fm(2)g Ff(R)5 b Fs(.)35 b(F)o(or)20 b(a)h(compact)456 4616 y(set)j Fq(K)35 b Fm(\032)28 b Fo([)p Fm(\000)p Fo(2)p Fq(;)17 b Fo(2])25 b Fs(put)1443 4800 y Fq(w)s Fo(\()p Fq(K)7 b Fo(\))27 b(:=)1839 4664 y Fg(Z)1895 4890 y Fk(K)1980 4713 y Fm(p)p 2063 4713 267 4 v 87 x Fo(4)22 b Fm(\000)g Fq(\025)2290 4771 y Fp(2)2340 4733 y Fq(d\025)p 2340 4777 108 4 v 2340 4868 a Fo(8)p Fq(\031)456 5000 y Fs(and)1323 5159 y Fq(\031)t(\026)p Fo(\()p Fq(K)7 b Fo(\))27 b(:=)1765 5023 y Fg(Z)1820 5249 y Fk(K)1905 5159 y Fo(Im)p Fq(m)p Fo(\()p Fq(\025)22 b Fo(+)g Fq(i)p Fo(0\))p Fq(d\025;)p eop %%Page: 9 9 9 8 bop 1246 235 a Fi(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 751 b(9)456 425 y Fs(then)24 b(according)h(to)f([3])862 658 y Fq(w)s Fo(\()p Fq(K)7 b Fo(\))17 b(log)o(\()1291 591 y Fq(w)s Fo(\()p Fq(K)7 b Fo(\))p 1291 636 239 4 v 1298 727 a Fq(\026)p Fo(\()p Fq(K)g Fo(\))1540 658 y(\))27 b Fm(\024)1693 591 y(\000)p Fo(1)p 1693 636 127 4 v 1702 727 a(2)p Fq(\031)1846 523 y Fg(Z)1901 748 y Fk(K)1961 757 y Fe(0)2016 658 y Fo(log\()2190 591 y(4Im)o(\()p Fq(m)p Fo(\))p 2190 636 327 4 v 2231 727 a(sin\()p Fq(\022)s Fo(\))2526 658 y(\))17 b(sin)2700 617 y Fp(2)2740 658 y Fo(\()p Fq(\022)s Fo(\))p Fq(d\022)1710 931 y Fm(\024)1831 863 y Fo(1)p 1826 908 59 4 v 1826 999 a Fq(\031)1911 795 y Fg(Z)1966 1021 y Fk(K)2026 1030 y Fe(0)2081 931 y Fo(log\()p Fm(j)p Fq(a)p Fo(\()p Fq(e)2407 889 y Fk(i\022)2470 931 y Fo(\))p Fm(j)p Fo(\))g(sin)2710 889 y Fp(2)2750 931 y Fo(\()p Fq(\022)s Fo(\)\))p Fq(d\022)s(:)456 798 y Fs(\(3.1\))456 1174 y(where)25 b Fq(K)807 1189 y Fp(0)874 1174 y Fo(=)j Fm(f)p Fq(\022)i Fm(2)e Fo([)p Fm(\000)p Fq(\031)t(;)17 b(\031)t Fo(])28 b(:)78 b(2)17 b(cos\()p Fq(\022)s Fo(\))27 b Fm(2)i Fq(K)7 b Fm(g)p Fs(.)555 1290 y(Assume)24 b(no)n(w)g(that)g Fq(K)32 b Fs(is)25 b(a)g(compact)f(set)h(with)f(the)h(property:)1615 1468 y Fo(inf)1627 1530 y Fk(K)1687 1539 y Fe(0)1749 1468 y Fo(sin\()p Fq(m\022)s Fo(\))j Fq(>)g Fo(0)p Fq(:)-1830 b Fs(\(3.2\))456 1686 y(Then)24 b(the)h(inte)o(gral)1496 1739 y Fg(Z)1596 1765 y Fk(\031)1552 1965 y Fn(\000)p Fk(\031)1670 1875 y Fo(log)17 b Fm(j)p Fq(a)p Fm(j)g Fo(sin)2056 1834 y Fp(2)2095 1875 y Fo(\()p Fq(m\022)s Fo(\))p Fq(d\022)456 2089 y Fs(can)25 b(be)g(estimated)f(from)g(belo)n (w)g(by)1496 2318 y Fq(c)1555 2182 y Fg(Z)1654 2209 y Fk(\031)1610 2408 y Fn(\000)p Fk(\031)1729 2318 y Fo(log)16 b Fm(j)p Fq(a)p Fm(j)h Fo(sin)2114 2277 y Fp(2)2154 2318 y Fo(\()p Fq(\022)s Fo(\))p Fq(d\022)s(:)456 2552 y Fs(with)24 b Fq(c)j(>)h Fo(0)d Fs(independent)e(of)i Fo(\003)g Fs(and)g Fo(\003)1854 2567 y Fp(0)1893 2552 y Fs(.)31 b(F)o(or)24 b(such)h Fq(K)32 b Fs(we)25 b(obtain:)1142 2786 y Fq(w)s Fo(\()p Fq(K)7 b Fo(\))17 b(log)o(\()1571 2718 y Fq(w)s Fo(\()p Fq(K)7 b Fo(\))p 1571 2763 239 4 v 1578 2854 a Fq(\026)p Fo(\()p Fq(K)g Fo(\))1819 2786 y(\))28 b Fm(\024)g Fq(C)7 b Fo(\()p Fq(A)2178 2801 y Fp(2)p Fk(m)2302 2786 y Fo(+)22 b(2)17 b(log)o(\()p Fq(\034)11 b Fo(\)\))-2302 b Fs(\(3.3\))456 3016 y(with)24 b(a)h(constant)f Fq(C)32 b Fs(independent)23 b(of)i Fo(\003)g Fs(and)g Fo(\003)2135 3031 y Fp(0)2174 3016 y Fs(.)467 3273 y(4.)51 b(P)t Fr(E)t(R)o(T)t(U)t (R)t(B)s(A)-5 b(T)5 b(I)t(O)g(N)t(S)36 b(O)t(F)30 b(A)g(C)t(L)t(A)t(S)t (S)j(W)t(I)t(D)t(E)t(R)g(T)t(H)t(A)t(N)e(T)t(H)t(E)f Fs(H)t Fr(I)t(L)t(B)t(E)t(R)o(T)l Fs(-)5 b(S)h Fr(C)f(H)t(M)g(I)g(D)t (T)1878 3389 y(C)t(L)t(A)t(S)t(S)555 3563 y Fs(The)25 b(spectral)g(signi\002cance)g(of)g(the)g(W)-8 b(e)o(yl)24 b(function)h(is)f(represented)h(by)g(the)g(rela-)456 3680 y(tion)1461 3816 y Fq(\026)p Fo(\()p Fq(K)7 b Fo(\))28 b(=)f(\()p Fq(E)1927 3831 y Fk(H)1995 3816 y Fo(\()p Fq(K)7 b Fo(\))p Fq(e)2206 3831 y Fp(0)2245 3816 y Fq(;)17 b(e)2334 3831 y Fp(0)2374 3816 y Fo(\))p Fq(;)456 3974 y Fs(where)34 b Fq(E)805 3989 y Fk(H)907 3974 y Fs(denotes)g(the)g (spectral)g(measure)h(of)f Fq(H)8 b Fs(.)59 b(So)34 b(f)o(ar)h(we)g(ha) n(v)o(e)f(only)f(con-)456 4090 y(sidered)g(\002nite)h(rank)h(operators) e Fo(\003)p Fq(;)17 b Fo(\003)1812 4105 y Fp(0)1851 4090 y Fs(.)58 b(Suppose)34 b(no)n(w)f(that)h Fo(\003)p Fq(;)17 b Fo(\003)2873 4105 y Fp(0)2946 4090 y Fs(are)34 b(arbitrary)456 4206 y(compact)24 b(self)o(adjoint)g(operators)g(such)h(that)1513 4384 y Fq(A)1586 4399 y Fp(2)p Fk(m)1710 4384 y Fo(+)d(2)17 b(log\()p Fq(\034)11 b Fo(\))28 b Fq(<)f Fm(1)p Fq(:)-1931 b Fs(\(4.1\))456 4567 y(Let)28 b Fo(\003\()p Fq(n)p Fo(\))p Fq(;)17 b Fo(\003)931 4582 y Fp(0)970 4567 y Fo(\()p Fq(n)p Fo(\))29 b Fs(be)g(sequences)g(of)g(\002nite)g(rank)g(operators) g(not)f(only)g(con)l(v)o(er)n(ging)456 4683 y(to)c Fo(\003)h Fs(and)f Fo(\003)887 4698 y Fp(0)951 4683 y Fs(in)h Fl(S)1137 4698 y Fn(1)1211 4683 y Fs(,)g(b)n(ut)f(also)h(satisfying)e(the)i (condition)e(that)1159 4861 y Fq(A)1232 4876 y Fp(2)p Fk(m)1334 4861 y Fo(\(\003\()p Fq(n)p Fo(\))p Fq(;)17 b Fo(\003)1686 4876 y Fp(0)1725 4861 y Fo(\()p Fq(n)p Fo(\)\))22 b(+)g(2)p Fq(A)2139 4876 y Fp(0)2178 4861 y Fo(\(\003\()p Fq(n)p Fo(\))p Fq(;)17 b Fo(\003)2530 4876 y Fp(0)2569 4861 y Fo(\()p Fq(n)p Fo(\)\))456 5038 y Fs(con)l(v)o(er)n(ges)24 b(to)1142 5216 y Fq(A)1215 5231 y Fp(2)p Fk(m)1317 5216 y Fo(\(\003)p Fq(;)17 b Fo(\003)1535 5231 y Fp(0)1574 5216 y Fo(\))22 b(+)g(2)p Fq(A)1854 5231 y Fp(0)1893 5216 y Fo(\(\003)p Fq(;)17 b Fo(\003)2111 5231 y Fp(0)2149 5216 y Fo(\))p Fq(;)92 b Fo(as)25 b Fq(n)j Fm(!)f(1)p Fq(:)p eop %%Page: 10 10 10 9 bop 456 235 a Fi(10)883 b(LAPTEV)-10 b(,)23 b(N)m(ABOK)n(O,)h (SAFR)m(ONO)l(V)456 425 y Fs(As)29 b(al)o(w)o(ays)h(we)h(assume)e(that) h Fq(I)k Fo(+)25 b(\003)38 b Fq(>)f(cI)8 b(;)17 b(c)37 b(>)g Fo(0)p Fs(.)46 b(Let)30 b Fq(Q)2637 440 y Fk(n)2722 425 y Fo(=)37 b(2Re\()p Fq(S)6 b Fo(\003\()p Fq(n)p Fo(\)\))26 b(+)456 541 y(\003)524 556 y Fp(0)563 541 y Fo(\()p Fq(n)p Fo(\))p Fs(,)822 732 y Fq(H)903 747 y Fk(n)977 732 y Fo(=)i Fq(S)g Fo(+)22 b Fq(S)1333 691 y Fn(\003)1394 732 y Fo(+)g Fq(Q)1569 747 y Fk(n)1616 732 y Fq(;)167 b Fo(and)49 b Fq(\026)2075 747 y Fk(n)2122 732 y Fo(\()p Fq(K)7 b Fo(\))28 b(=)f(\()p Fq(E)2529 747 y Fk(H)2587 755 y Fd(n)2634 732 y Fo(\()p Fq(K)7 b Fo(\))p Fq(e)2845 747 y Fp(0)2885 732 y Fq(;)17 b(e)2974 747 y Fp(0)3013 732 y Fo(\))p Fq(:)456 923 y Fs(Then)1314 1072 y Fq(\026)1373 1087 y Fk(n)1420 1072 y Fo(\()p Fq(K)7 b Fo(\))28 b Fm(!)f Fq(\026)p Fo(\()p Fq(K)7 b Fo(\))p Fq(;)116 b Fo(as)50 b Fq(n)28 b Fm(!)f(1)p Fq(;)456 1243 y Fs(since)32 b Fo(\()p Fq(H)812 1258 y Fk(n)887 1243 y Fm(\000)d Fq(z)t Fo(\))1080 1207 y Fn(\000)p Fp(1)1207 1243 y Fs(con)l(v)o(er)n(ges)k (to)f Fo(\()p Fq(H)k Fm(\000)28 b Fq(z)t Fo(\))2094 1207 y Fn(\000)p Fp(1)2222 1243 y Fs(uniformly)k(on)g(compact)h(subsets)456 1359 y(of)i(the)h(upper)f(half-plane.)63 b(Indeed,)38 b(the)e(dif)n(ference)g(between)f(resolv)o(ents)g(is)g(the)456 1475 y(operator)724 1666 y Fo(\()p Fq(H)843 1681 y Fk(n)912 1666 y Fm(\000)23 b Fq(z)t Fo(\))1099 1625 y Fn(\000)p Fp(1)1216 1666 y Fm(\000)f Fo(\()p Fq(H)30 b Fm(\000)23 b Fq(z)t Fo(\))1651 1625 y Fn(\000)p Fp(1)1773 1666 y Fo(=)28 b(\()p Fq(H)1996 1681 y Fk(n)2065 1666 y Fm(\000)22 b Fq(z)t Fo(\))2251 1625 y Fn(\000)p Fp(1)2346 1666 y Fo(\()p Fq(Q)h Fm(\000)f Fq(Q)2660 1681 y Fk(n)2708 1666 y Fo(\)\()p Fq(H)29 b Fm(\000)23 b Fq(z)t Fo(\))3081 1625 y Fn(\000)p Fp(1)456 1856 y Fs(whose)j(norm)h(can)g(be)g (estimated)f(by)h Fq(C)1867 1871 y Fp(0)1906 1856 y Fm(jj)p Fq(Q)d Fm(\000)g Fq(Q)2241 1871 y Fk(n)2288 1856 y Fm(jj)p Fs(,)j(with)g Fq(C)2671 1871 y Fp(0)2737 1856 y Fs(independent)f(of)h Fq(n)p Fs(.)456 1972 y(Thus)c(despite)g(the)h(f)o(act)g(that)f (\(2.5\),)h(\(3.1\))g(and)g(\(3.3\))g(were)g(obtained)f(for)i(\002nite) e(rank)456 2089 y(perturbations)j Fq(Q)p Fs(,)j(we)e(come)h(to)f(the)g (conclusion)g(that)g(under)g(the)h(restriction)f(\(3.2\))456 2205 y(the)d(inequality)1048 2451 y Fq(w)s Fo(\()p Fq(K)7 b Fo(\))17 b(log)o(\()1477 2384 y Fq(w)s Fo(\()p Fq(K)7 b Fo(\))p 1477 2428 239 4 v 1484 2520 a Fq(\026)p Fo(\()p Fq(K)g Fo(\))1726 2451 y(\))27 b Fm(\024)h Fq(C)7 b Fo(\()p Fq(A)2084 2466 y Fp(2)p Fk(m)2208 2451 y Fm(\000)23 b Fo(2)17 b(log)o(\()p Fq(I)30 b Fo(+)22 b(\003\)\))456 2696 y Fs(holds)36 b(for)j(general)f Fo(\003)p Fq(;)54 b Fo(\003)1412 2711 y Fp(0)1503 2696 y Fm(2)e Fl(S)1704 2711 y Fn(1)1817 2696 y Fs(satisfying)36 b(\(4.1\))i(with)f Fq(I)j Fo(+)31 b(\003)52 b Fq(>)g Fo(0)p Fs(.)69 b(This)456 2812 y(ineqality)36 b(holds)g(\(with)h(a)h(constant)e(independant)h(of) g 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