Content-Type: multipart/mixed; boundary="-------------0308130325702" This is a multi-part message in MIME format. ---------------0308130325702 Content-Type: text/plain; name="03-365.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-365.comments" AMS Code: 47A40, 35P25, 81U ---------------0308130325702 Content-Type: text/plain; name="03-365.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-365.keywords" Lax-Phillips approach, wave equation, free and perturbed Lax-Phillips evolutions. ---------------0308130325702 Content-Type: application/postscript; name="BER6.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BER6.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: BER6.dvi %%CreationDate: Wed Aug 13 10:45:15 2003 %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: 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Ft(L)3165 3895 y Fs(k)3208 3880 y Fx(\))p Ft(:)-3037 b Fx(\(5\))383 4082 y(Consider)34 b(the)h(follo)m(wing)d (problem:)45 b Fj(whether)36 b(we)h(c)-5 b(an)35 b(c)-5 b(onsider)36 b(simultane)-5 b(ously)36 b(e)-5 b(quations)36 b(\(2\))236 4239 y(and)f(\(4\))f(as,)g(r)-5 b(esp)g(e)g(ctively,)35 b(p)-5 b(erturb)g(e)g(d)34 b(and)h(fr)-5 b(e)g(e)34 b(e)-5 b(quations)35 b(in)f(the)h(L)-5 b(ax{Phil)5 b(lips)34 b(fr)-5 b(amework?)383 4395 y Fx(W)d(e)37 b(recall)f([5])h(that)g (equation)g(\(4\))f(can)i(de\014ne)g(a)f(free)h(ev)m(olution)e(in)g (the)i(Lax{Phillips)d(sense)k(if)236 4552 y(and)i(only)g(if,)h(for)f (the)g(group)g Ft(W)1495 4567 y Fs(L)1543 4579 y Fi(k)1586 4552 y Fx(\()p Ft(t)p Fx(\))g(of)g(solutions)f(of)g(the)i(Cauc)m(h)m(y) h(problem)d(of)g(\(4\),)j(there)f(exist)236 4708 y(orthogonal)31 b(subspaces)k Ft(D)1251 4723 y Fo(\006)1342 4708 y Fx(\(so-called)c (outgoing)g(and)i(incoming)d(subspaces\))35 b(of)d(the)h(energy)g (space)236 4865 y Ft(H)317 4880 y Fs(L)365 4892 y Fi(k)440 4865 y Fx(suc)m(h)h(that)390 5100 y(\()p Ft(a)p Fx(\))65 b Ft(W)674 5115 y Fs(L)722 5127 y Fi(k)764 5100 y Fx(\()p Fr(\006)q Ft(t)p Fx(\))p Ft(D)1034 5115 y Fo(\006)1121 5100 y Fr(\032)28 b Ft(D)1307 5115 y Fo(\006)1431 5100 y Fx(\()p Ft(t)g Fr(\025)g Fx(0\);)114 b(\()p Ft(b)p Fx(\))2066 5006 y Fp(\\)2064 5216 y Fs(t)p Fo(\025)p Fv(0)2196 5100 y Ft(W)14 b Fx(\()p Fr(\006)p Ft(t)p Fx(\))p Ft(D)2571 5115 y Fo(\006)2658 5100 y Fx(=)28 b Fr(f)p Fx(0)p Fr(g)p Fx(;)81 b(\()p Ft(c)p Fx(\))65 b Ft(D)3283 5115 y Fo(\000)3364 5100 y Fr(\010)22 b Ft(D)3544 5115 y Fv(+)3631 5100 y Fx(=)28 b Ft(H)3816 5115 y Fs(L)3864 5127 y Fi(k)3906 5100 y Ft(:)236 5394 y Fx(Here,)34 b(the)f(energy)g (space)h Ft(H)1315 5409 y Fs(L)1363 5421 y Fi(k)1437 5394 y Fx(is)f(de\014ned)h(as)e(the)h(closure)g(of)f(the)h(set)g(of)g (initial)28 b(data)1319 5519 y Fp(\()q( )1520 5617 y Ft(u)1522 5774 y(v)1617 5519 y Fp(!)1745 5689 y Fr(j)k(8)p Ft(u)c Fr(2)g Ft(D)s Fx(\()p Ft(L)2226 5704 y Fs(k)2269 5689 y Fx(\))o Ft(;)50 b Fr(8)p Ft(v)31 b Fr(2)e Ft(L)2677 5704 y Fv(2)2716 5689 y Fx(\()p Fq(R)2820 5704 y Fv(+)2885 5689 y Fx(\))2923 5519 y Fp(\))2137 6037 y Fx(2)p eop %%Page: 3 4 3 3 bop 236 454 a Fx(with)32 b(resp)s(ect)i(to)e(the)h(energy)h(norm) 1481 605 y Fp(\015)1481 665 y(\015)1481 724 y(\015)1481 784 y(\015)1481 844 y(\015)1536 609 y( )1656 707 y Ft(u)1659 864 y(v)1753 609 y Fp(!)1832 605 y(\015)1832 665 y(\015)1832 724 y(\015)1832 784 y(\015)1832 844 y(\015)1888 624 y Fv(2)1888 915 y Fs(H)1946 926 y Fi(L)1988 944 y(k)2062 779 y Fx(=)27 b(\()p Ft(L)2269 794 y Fs(k)2312 779 y Ft(u;)17 b(u)p Fx(\))k(+)h Fr(k)p Ft(v)t Fr(k)2776 738 y Fv(2)2815 779 y Ft(:)236 1126 y Fx(Moreo)m(v)m(er,)32 b(if)c(equation)i(\(2\))f(also)g(de\014nes)i(a)e(p)s(erturb)s(ed)h(ev)m (olution)f(in)g(the)h(Lax{Phillips)d(sense,)32 b(then)236 1283 y(the)39 b(corresp)s(onding)f(group)h Ft(W)1425 1293 y Fv(~)1417 1310 y Fs(L)1465 1322 y Fi(k)1507 1283 y Fx(\()p Ft(t)p Fx(\))g(of)f(solutions)f(of)h(the)h(Cauc)m(h)m(y)h (problem)d(for)h(\(2\))g(p)s(ossesses)k Fj(the)236 1439 y(same)34 b(subsp)-5 b(ac)g(es)32 b Ft(D)995 1454 y Fo(\006)1086 1439 y Fx(with)h(prop)s(erties)f(\()p Ft(a)p Fx(\))h(and)f(\()p Ft(b)p Fx(\).)383 1596 y(The)i(conditions)e(of)h(applicabilit)m(y)c(of) k(the)h(Lax{Phillips)d(approac)m(h)i(w)m(ere)i(discussed)g(in)d([6])h (for)g(a)236 1752 y(general)39 b(case)h(of)f(the)h(abstract)g(w)m(a)m (v)m(e)h(equation.)63 b(It)40 b(follo)m(ws)e(from)g([6])h(that)g (conditions)g(\()p Ft(a)p Fx(\))26 b Fr(\000)i Fx(\()p Ft(c)p Fx(\),)236 1909 y(whic)m(h)38 b(guaran)m(tee)g(that)f(equation)g (\(4\))g(is)g(free)h(in)e(the)i(Lax{Phillips)d(sense,)41 b(are)c(equiv)-5 b(alen)m(t)37 b(to)g(the)236 2065 y(follo)m(wing)30 b(conditions)i(on)g(the)h(op)s(erator)f Ft(L)1877 2080 y Fs(k)1920 2065 y Fx(:)383 2272 y(\()p Ft(i)p Fx(\))60 b Fj(in)32 b(the)h(sp)-5 b(ac)g(e)32 b Ft(L)1147 2287 y Fv(2)1187 2272 y Fx(\()p Fq(R)1291 2287 y Fv(+)1356 2272 y Fx(\))p Fj(,)h(ther)-5 b(e)32 b(exists)h(a)f(simple)g(maximal)f (symmetric)i(op)-5 b(er)g(ator)32 b Ft(B)38 b Fj(such)32 b(that)236 2428 y(the)j(op)-5 b(er)g(ator)35 b Ft(L)848 2443 y Fs(k)926 2428 y Fj(is)f(a)h(p)-5 b(ositive)34 b(self-adjoint)g(extension)g(of)g(the)h(symmetric)g(op)-5 b(er)g(ator)34 b Ft(B)3616 2392 y Fv(2)3656 2428 y Fj(;)383 2634 y Fx(\()p Ft(ii)p Fx(\))163 b(\()p Ft(L)792 2649 y Fs(k)835 2634 y Ft(f)5 b(;)17 b(f)11 b Fx(\))27 b(=)g Fr(k)p Ft(B)1288 2598 y Fo(\003)1328 2634 y Ft(f)11 b Fr(k)1437 2598 y Fv(2)1573 2634 y Fj(for)35 b(any)g Ft(f)j Fr(2)28 b Ft(D)s Fx(\()p Ft(L)2284 2649 y Fs(k)2327 2634 y Fx(\).)383 2841 y(F)-8 b(urthermore,)28 b(it)g(is)g(easy)h(to)f(v)m (erify)h(that)f(if,)g(sim)m(ultaneously)-8 b(,)28 b(equation)g(\(2\))g (is)g(p)s(erturb)s(ed)h(in)f(the)236 2997 y(Lax{Phillips)g(sense,)33 b(then,)f Fj(for)h(the)g(same)f(choic)-5 b(e)32 b(of)h Ft(B)5 b Fx(,)31 b(the)g(op)s(erator)3012 2972 y(~)2982 2997 y Ft(L)3048 3012 y Fs(k)3122 2997 y Fx(satis\014es)g(only)f (condition)236 3154 y(\()p Ft(i)p Fx(\).)383 3310 y(W)-8 b(e)44 b(note)h(that)f(incoming)e Ft(D)1524 3325 y Fo(\000)1627 3310 y Fx(and)i(outgoing)f Ft(D)2322 3325 y Fv(+)2425 3310 y Fx(subspaces)k(in)d(\()p Ft(a)p Fx(\))30 b Fr(\000)g Fx(\()p Ft(c)p Fx(\))44 b(are)h(determined)236 3467 y(uniquely)33 b(b)m(y)g(the)g(op)s(erator)f Ft(B)38 b Fx(and)33 b(coincide)f(with)g (the)h(closures)g(of)f(the)h(sets)708 3605 y Fp(\( )1003 3703 y Ft(f)908 3860 y Fr(\000)p Ft(iB)5 b(f)1198 3605 y Fp(!)1326 3775 y Fr(j)59 b(8)p Ft(f)39 b Fr(2)28 b Ft(D)s Fx(\()p Ft(B)1850 3734 y Fv(2)1889 3775 y Fx(\))1927 3605 y Fp(\))2122 3775 y Fx(and)2393 3605 y Fp(\( )2650 3703 y Ft(f)2594 3860 y(iB)5 b(f)2806 3605 y Fp(!)2934 3775 y Fr(j)59 b(8)p Ft(f)39 b Fr(2)28 b Ft(D)s Fx(\()p Ft(B)3458 3734 y Fv(2)3497 3775 y Fx(\))3535 3605 y Fp(\))236 3775 y Fx(\(6\))236 4094 y(in)k(the)h(energy)h(space)f Ft(H)1171 4109 y Fs(L)1219 4121 y Fi(k)1261 4094 y Fx(,)g(resp)s(ectiv) m(ely)-8 b(.)1838 4058 y Fv(2)383 4250 y Fx(Th)m(us,)51 b(w)m(e)c(can)f(form)m(ulate)f(the)i(problem)e(giv)m(en)h(ab)s(o)m(v)m (e)h(as)f(follo)m(ws:)70 b Fj(whether)46 b(ther)-5 b(e)48 b(exists)f(a)236 4407 y(simple)31 b(maximal)f(symmetric)h(op)-5 b(er)g(ator)31 b Ft(B)37 b Fj(in)31 b Ft(L)2071 4422 y Fv(2)2111 4407 y Fx(\()p Fq(R)2215 4422 y Fv(+)2280 4407 y Fx(\))g Fj(for)g(which)g(the)h(op)-5 b(er)g(ators)31 b Ft(L)3417 4422 y Fs(k)3491 4407 y Fj(and)3689 4382 y Fx(~)3677 4407 y Ft(L)3743 4422 y Fs(k)3818 4407 y Fj(satisfy)236 4563 y(c)-5 b(onditions)34 b Fx(\()p Ft(i)p Fx(\))p Ft(;)17 b Fx(\()p Ft(ii)p Fx(\))35 b Fj(and)f Fx(\()p Ft(i)p Fx(\))p Fj(,)h(r)-5 b(esp)g(e)g(ctively?)383 4720 y Fw(Lemma)38 b(1.)47 b Fj(The)36 b(op)-5 b(er)g(ators)35 b Ft(L)1606 4735 y Fs(k)1685 4720 y Fj(and)1887 4695 y Fx(~)1876 4720 y Ft(L)1942 4735 y Fs(k)2021 4720 y Fj(satisfy)h(simultane)-5 b(ously)35 b(c)-5 b(ondition)35 b Fx(\()p Ft(i)p Fx(\))h Fj(for)g(a)g(c)-5 b(ertain)236 4876 y(choic)g(e)34 b(of)h Ft(B)40 b Fj(if)34 b(and)h(only)f(if)1360 4851 y Fx(~)1349 4876 y Ft(L)1415 4891 y Fs(k)1486 4876 y Fx(=)27 b Ft(L)1655 4891 y Fs(k)1698 4876 y Fj(.)383 5033 y(Pr)-5 b(o)g(of.)44 b Fx(If)33 b Ft(L)848 5048 y Fs(k)924 5033 y Fx(and)1125 5008 y(~)1114 5033 y Ft(L)1180 5048 y Fs(k)1256 5033 y Fx(satisfy)g(condition)f(\()p Ft(i)p Fx(\),)h(then)2393 5008 y(~)2381 5033 y Ft(L)2447 5048 y Fs(k)2491 5033 y Ft(f)39 b Fx(=)28 b Ft(L)2748 5048 y Fs(k)2791 5033 y Ft(f)44 b Fx(for)32 b(an)m(y)i(function)e Ft(f)40 b Fr(2)28 b Ft(D)s Fx(\()p Ft(B)3982 4997 y Fv(2)4022 5033 y Fx(\).)236 5189 y(On)33 b(the)h(other)f(hand,)h(b)m(y)g(virtue)f (of)g(\(3\))g(and)g(\(5\),)g(the)h(last)e(equalit)m(y)h(is)g(true)g (only)g(on)g(the)h(subspace)236 5346 y Fh(H)308 5310 y Fo(0)360 5346 y Fx(:=)28 b Ft(L)557 5361 y Fv(2)597 5346 y Fx(\()p Fq(R)701 5361 y Fv(+)766 5346 y Fx(\))p Fr(\011)g Ft(<)g(p)p Fx(\()p Ft(s)p Fx(\))h Ft(>)j Fx(of)h(the)g(space) h Ft(L)1928 5361 y Fv(2)1968 5346 y Fx(\()p Fq(R)2072 5361 y Fv(+)2137 5346 y Fx(\).)44 b(Therefore,)35 b Fh(H)2788 5310 y Fo(0)2839 5346 y Fr(\033)29 b Ft(D)s Fx(\()p Ft(B)3146 5310 y Fv(2)3185 5346 y Fx(\),)k(and,)g(in)f(view)i(of)e(the)236 5502 y(densit)m(y)i(of)e Ft(D)s Fx(\()p Ft(B)885 5466 y Fv(2)924 5502 y Fx(\))h(in)e Ft(L)1174 5517 y Fv(2)1214 5502 y Fx(\()p Fq(R)1318 5517 y Fv(+)1383 5502 y Fx(\),)i(w)m(e)g(get)g (that)f Fh(H)2070 5466 y Fo(0)2121 5502 y Fx(=)27 b Ft(L)2290 5517 y Fv(2)2330 5502 y Fx(\()p Fq(R)2434 5517 y Fv(+)2499 5502 y Fx(\).)44 b(Hence,)34 b Ft(p)p Fx(\()p Ft(s)p Fx(\))27 b(=)h(0)k(and)3509 5477 y(~)3498 5502 y Ft(L)3564 5517 y Fs(k)3634 5502 y Fx(=)c Ft(L)3804 5517 y Fs(k)3847 5502 y Fx(.)p 236 5595 1541 4 v 348 5677 a Fn(2)386 5708 y Fm(Without)g(less)f(of)h(generalit)n(y)-7 b(,)26 b(w)n(e)h(assume)g (that)h Fg(\000)p Ff(i)22 b Fg(2)i Ff(\032)p Fm(\()p Ff(B)t Fm(\).)2137 6037 y Fx(3)p eop %%Page: 4 5 4 4 bop 383 454 a Fx(Con)m(v)m(ersely)-8 b(,)47 b(in)c(the)g(case)h (where)1732 429 y(~)1721 454 y Ft(L)1787 469 y Fs(k)1876 454 y Fx(=)h Ft(L)2063 469 y Fs(k)2106 454 y Fx(,)h(a)d(simple)e (maximal)f(symmetric)j(op)s(erator)f Ft(B)48 b Fx(in)236 611 y Ft(L)302 626 y Fv(2)342 611 y Fx(\()p Fq(R)446 626 y Fv(+)511 611 y Fx(\))31 b(for)f(whic)m(h)h(the)g(op)s(erator)f Ft(L)1627 626 y Fs(k)1701 611 y Fx(satis\014es)h(condition)f(\()p Ft(i)p Fx(\))h(\(and,)g(moreo)m(v)m(er,)g(condition)f(\(ii\))e(to)s (o\))236 767 y(w)m(as)34 b(constructed)g(in)e([7])g(as)h(follo)m(ws:) 1335 1051 y Ft(B)g Fx(=)28 b Ft(T)1617 1010 y Fo(\000)p Fv(1)1711 1051 y Fr(B)s Ft(T)8 b(;)147 b(D)s Fx(\()p Ft(B)5 b Fx(\))28 b(=)f Ft(T)2459 1010 y Fo(\000)p Fv(1)2636 947 y(0)2581 1051 y Ft(W)2687 1010 y Fv(1)2673 1076 y(2)2754 1051 y Fx(\()p Fq(R)2857 1066 y Fv(+)2923 1051 y Fx(\))p Ft(:)-2752 b Fx(\(7\))236 1330 y(Here,)40 b Ft(T)50 b Fx(=)37 b Ft(F)783 1345 y Fs(sin)886 1330 y Fx(\000)947 1346 y Fs(k)r Fv(+1)p Fs(=)p Fv(2)1188 1330 y Fx(is)h(an)f(unitary)h (op)s(erator)f(in)g Ft(L)2365 1345 y Fv(2)2404 1330 y Fx(\()p Fq(R)2508 1345 y Fv(+)2573 1330 y Fx(\),)i Fr(B)h Fx(=)d Ft(i)2954 1291 y Fs(d)p 2938 1307 69 4 v 2938 1365 a(ds)3092 1330 y Fx(\()p Ft(D)s Fx(\()p Fr(B)s Fx(\))g(=)3525 1231 y Fv(0)3471 1330 y Ft(W)3577 1294 y Fv(1)3563 1355 y(2)3652 1330 y Fx(\()p Fq(R)3756 1345 y Fv(+)3821 1330 y Fx(\)\))h(is)f(a)236 1487 y(simple)31 b(maximal)f(symmetric)h(op)s (erator)h(in)g Ft(L)1986 1502 y Fv(2)2026 1487 y Fx(\()p Fq(R)2130 1502 y Fv(+)2195 1487 y Fx(\),)g(and)486 1779 y(\()p Ft(F)587 1794 y Fs(sin)691 1779 y Ft(u)p Fx(\)\()p Ft(x)p Fx(\))c(=)1047 1622 y Fp(r)p 1147 1622 79 4 v 1162 1711 a Fx(2)p 1157 1756 59 4 v 1157 1847 a Ft(\031)1242 1643 y Fp(Z)1342 1669 y Fo(1)1298 1869 y Fv(0)1433 1779 y Ft(u)p Fx(\()p Ft(s)p Fx(\))17 b(sin)e Ft(sx)q(ds;)114 b Fx(\(\000)2202 1794 y Fs(k)r Fv(+1)p Fs(=)p Fv(2)2405 1779 y Ft(u)p Fx(\)\()p Ft(x)p Fx(\))27 b(=)2761 1643 y Fp(Z)2861 1669 y Fo(1)2816 1869 y Fv(0)2952 1712 y Fr(p)p 3035 1712 98 4 v 67 x Ft(sy)s(u)p Fx(\()p Ft(s)p Fx(\))p Ft(J)3364 1794 y Fs(k)r Fv(+1)p Fs(=)p Fv(2)3567 1779 y Fx(\()p Ft(sy)t Fx(\))p Ft(ds)236 2040 y Fx(are)33 b(the)g(sine-F)-8 b(ourier)31 b(and)h(Hank)m(el)h(transformations,)e (resp)s(ectiv)m(ely)-8 b(.)45 b(Lemma)31 b(1)h(is)g(pro)m(v)m(ed.)383 2196 y(Lemma)j(1)h(states)i(that)e(w)m(e)i(cannot)f(consider)g(the)g (op)s(erators)2792 2171 y(~)2780 2196 y Ft(L)2846 2211 y Fs(k)2926 2196 y Fx(and)f Ft(L)3185 2211 y Fs(k)3265 2196 y Fx(in)g(the)h(Lax{Phillips)236 2353 y(framew)m(ork)g(sim)m (ultaneously)-8 b(.)54 b(A)m(t)37 b(the)f(same)h(time,)f(the)h (assumption)f(that)g(equation)g(\(4\))h(de\014nes)h(a)236 2509 y(free)e(ev)m(olution)f(is)g(natural)g(and,)h(as)g(w)m(as)h(sho)m (wn)g(in)e([7],)h(equation)g(\(4\))f(is)g(free)h(in)f(the)h (Lax{Phillips)236 2666 y(sense.)45 b(Therefore,)33 b(to)f(a)m(v)m(oid)g (the)g(trivial)d(situation)2242 2641 y(~)2230 2666 y Ft(L)2296 2681 y Fs(k)2367 2666 y Fx(=)f Ft(L)2537 2681 y Fs(k)2580 2666 y Fx(,)k(w)m(e)h(m)m(ust)f(sligh)m(tly)e(mo)s(dify)g (condition)236 2822 y(\()p Ft(i)p Fx(\))j(under)g(whic)m(h)g(equation)g 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b(ax{Phil)5 b(lips)28 b(p)-5 b(erturb)g(e)g(d)30 b(op)-5 b(er)g(ator)30 b(if)f(ther)-5 b(e)30 b(exists)236 4849 y(an)41 b(op)-5 b(er)g(ator)41 b Ft(B)851 4812 y Fo(0)915 4849 y Fr(2)f(L)1090 4864 y Fs(B)1191 4849 y Fj(such)i(that)1637 4823 y Fx(~)1625 4849 y Ft(L)1691 4864 y Fs(k)1776 4849 y Fj(is)f(an)g(extension)g (\(with)g(the)g(exit)g(in)g(the)h(mor)-5 b(e)41 b(wide)f(sp)-5 b(ac)g(e)236 5005 y Ft(L)302 5020 y Fv(2)342 5005 y Fx(\()p Fq(R)446 5020 y Fv(+)511 5005 y Fx(\))p Fj(\))35 b(of)f(the)h (symmetric)g(op)-5 b(er)g(ator)34 b Ft(B)1837 4969 y Fo(0)1861 4958 y Fv(2)1900 5005 y Fj(.)383 5162 y Fx(Replacing)k(in)h (\(6\))g Ft(B)45 b Fx(b)m(y)40 b Ft(B)1466 5125 y Fo(0)1490 5162 y Fx(,)h(it)e(is)g(easy)h(to)g(see)g(that)g(the)g(condition)e(of)h (De\014nition)f(1)i(ensures)236 5318 y(the)33 b(existence)i(of)d (subspaces)j Ft(D)1461 5282 y Fo(0)1484 5333 y(\006)1576 5318 y Fx(with)e(prop)s(erties)f(\()p Ft(a)p Fx(\))h(and)g(\()p Ft(b)p Fx(\))g(for)f(the)h(group)g Ft(W)3451 5328 y Fv(~)3443 5345 y Fs(L)3491 5357 y Fi(k)3533 5318 y Fx(\()p Ft(t)p Fx(\))g(whic)m(h)h(are)236 5475 y Fj(p)-5 b(arts)33 b Fx(of)f(the)h(initial)c(subspaces)35 b Ft(D)1565 5490 y Fo(\006)1656 5475 y Fx(constructed)f(b)m(y)g Ft(B)5 b Fx(.)383 5631 y(No)m(w,)34 b(w)m(e)g(establish)e(necessary)k(and)d (su\016cien)m(t)h(conditions)f(on)g(the)g(function)g Ft(p)p Fx(\()p Ft(s)p Fx(\))g(in)f(\(3\))h(under)236 5787 y(whic)m(h)g(the)g(op)s(erator)1088 5762 y(~)1076 5787 y Ft(L)1142 5802 y Fs(k)1218 5787 y Fx(is)f(Lax{Phillips)e(p)s (erturb)s(ed.)2137 6037 y(4)p eop %%Page: 5 6 5 5 bop 383 454 a Fx(In)42 b Ft(L)580 469 y Fv(2)620 454 y Fx(\()p Fq(R)724 469 y Fv(+)789 454 y Fx(\),)j(w)m(e)f(consider)f (the)g(isometric)e(op)s(erator)g Ft(G)k Fx(=)g(\()p Ft(B)34 b Fr(\000)29 b Ft(iI)8 b Fx(\)\()p Ft(B)34 b Fx(+)29 b Ft(iI)8 b Fx(\))3441 418 y Fo(\000)p Fv(1)3535 454 y Fx(,)46 b(where)d Ft(B)48 b Fx(is)236 611 y(de\014ned)34 b(b)m(y)g(\(7\).)43 b(A)32 b(function)h Ft(p)p Fx(\()p Ft(s)p Fx(\))27 b Fr(2)h Ft(L)1749 626 y Fv(2)1789 611 y Fx(\()p Fq(R)1893 626 y Fv(+)1958 611 y Fx(\))k(is)g(called)g Fj(noncyclic)f Fx(for)h Ft(G)3057 574 y Fo(\003)3129 611 y Fx(if)g(the)h(set)1741 906 y Ft(E)1813 922 y Fs(p)p Fv(\()p Fs(s)p Fv(\))1968 906 y Fx(:=)2130 782 y Fo(1)2110 812 y Fp(_)2098 1021 y Fs(n)p Fv(=0)2248 906 y Ft(G)2325 865 y Fo(\003)2364 861 y Fs(n)2411 906 y Ft(p)p Fx(\()p Ft(s)p Fx(\))236 1214 y(is)f(con)m(tained)h(strictly)f(in)g Ft(L)1290 1229 y Fv(2)1329 1214 y Fx(\()p Fq(R)1433 1229 y Fv(+)1498 1214 y Fx(\),)h(i.e.,)f Ft(E)1852 1229 y Fs(p)p Fv(\()p Fs(s)p Fv(\))2007 1214 y Fr(6)p Fx(=)c Ft(L)2177 1229 y Fv(2)2217 1214 y Fx(\()p Fq(R)2321 1229 y Fv(+)2386 1214 y Fx(\).)383 1370 y Fw(Theorem)45 b(1.)64 b Fj(The)40 b(op)-5 b(er)g(ator)1624 1345 y Fx(~)1612 1370 y Ft(L)1678 1385 y Fs(k)1763 1370 y Fj(is)41 b(a)g(L)-5 b(ax{Phil)5 b(lips)40 b(p)-5 b(erturb)g(e)g(d)41 b(op)-5 b(er)g(ator)41 b(if)g(and)g(only)g(if)g(the)236 1527 y(function)35 b Ft(p)p Fx(\()p Ft(s)p Fx(\))f Fj(in)h(\(3\))f(is)h (noncyclic)f(for)h Ft(G)1878 1490 y Fo(\003)1917 1527 y Fj(.)383 1683 y(Pr)-5 b(o)g(of.)45 b Fx(Assume)34 b(that)1271 1658 y(~)1260 1683 y Ft(L)1326 1698 y Fs(k)1402 1683 y Fx(is)f(a)g(Lax{Phillips)e(p)s(erturb)s(ed)j(op)s(erator.)46 b(By)33 b(virtue)h(of)f(condition)f Fw(I)p Fx(,)236 1840 y(there)i(exists)f(a)f(nonzero)h(subspace)i Fh(H)1679 1803 y Fo(0)1734 1840 y Fx(suc)m(h)f(that)1183 2071 y(~)1172 2096 y Ft(L)1238 2111 y Fs(k)1281 2096 y Ft(u)27 b Fx(=)h Ft(B)1547 2055 y Fo(0)1570 2044 y Fv(2)1609 2096 y Ft(u)g Fx(=)f Ft(B)1875 2055 y Fv(2)1915 2096 y Ft(u)g Fx(=)h Ft(L)2168 2111 y Fs(k)2211 2096 y Ft(u;)113 b Fr(8)p Ft(u)28 b Fr(2)g Ft(D)s Fx(\()p Ft(B)2841 2055 y Fv(2)2880 2096 y Fx(\))22 b Fr(\\)h Fh(H)3101 2055 y Fo(0)3124 2096 y Ft(:)236 2352 y Fx(Therefore,)34 b(equalities)d(\(3\))i(and)f (\(5\))g(imply)f(that)h Ft(p)p Fx(\()p Ft(s)p Fx(\))c Fr(2)g Fh(H)2490 2367 y Fv(1)2557 2352 y Fx(=)f Ft(L)2726 2367 y Fv(2)2766 2352 y Fx(\()p Fq(R)2870 2367 y Fv(+)2935 2352 y Fx(\))22 b Fr(\011)h Fh(H)3167 2316 y Fo(0)3190 2352 y Fx(.)383 2508 y(Ob)m(viously)-8 b(,)39 b(the)g(subspace)h Fh(H)1526 2472 y Fo(0)1587 2508 y Fx(is)d(in)m(v)-5 b(arian)m(t)37 b(with)h(resp)s(ect)h(to)f Ft(G)p Fx(.)60 b(Hence,)41 b(the)e(subspace)h Fh(H)3944 2523 y Fv(1)4021 2508 y Fx(is)236 2665 y(in)m(v)-5 b(arian)m(t)31 b(with)i(resp)s(ect)g(to)g Ft(G)1398 2629 y Fo(\003)1437 2665 y Fx(.)43 b(Therefore,)34 b(the)f(relation)e Ft(p)p Fx(\()p Ft(s)p Fx(\))c Fr(2)h Fh(H)2866 2680 y Fv(1)2938 2665 y Fx(yields)1703 2921 y Ft(E)1775 2937 y Fs(p)p Fv(\()p Fs(s)p Fv(\))1930 2921 y Fr(\022)h Fh(H)2108 2936 y Fv(1)2175 2921 y Fr(\032)f Ft(L)2346 2936 y Fv(2)2385 2921 y Fx(\()p Fq(R)2489 2936 y Fv(+)2554 2921 y Fx(\))p Ft(:)236 3177 y Fx(Th)m(us,)34 b(the)f(function)g Ft(p)p Fx(\()p Ft(s)p Fx(\))f(is)g(noncyclic)g(for)g Ft(G)2013 3141 y Fo(\003)2053 3177 y Fx(.)383 3334 y(Con)m(v)m(ersely) -8 b(,)46 b(if)40 b(the)j(function)e Ft(p)p Fx(\()p Ft(s)p 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y(The)k(condition)d (that)i(the)h(function)f Ft(p)p Fx(\()p Ft(s)p Fx(\))f(m)m(ust)i(b)s(e) f(noncyclic)g(from)f(Theorem)h(1)g(is)g(not)g(alw)m(a)m(ys)236 4472 y(con)m(v)m(enien)m(t)25 b(for)e(the)g(v)m(eri\014cation.)40 b(Equiv)-5 b(alen)m(t)23 b(conditions)f(can)i(b)s(e)f(form)m(ulated)f (with)g(the)i(use)g(of)f(w)m(ell-)236 4628 y(dev)m(elop)s(ed)33 b(theory)f(of)f(cyclic)g(elemen)m(ts)h(in)f(Hardy)h(spaces)h(\(see)g ([8])e(and)h(the)g(bibliograph)m(y)e(therein\).)236 4785 y(W)-8 b(e)38 b(recall)d(some)i(results)g(in)f(the)i(form)d(con)m(v)m (enien)m(t)k(for)d(our)h(considerations.)56 b(Similarly)33 b(to)k([8],)h(w)m(e)236 4941 y(denote)29 b(b)m(y)h Fr(R)e Fx(the)h(set)g(of)f(rational)e(functions)j(from)e(the)i(Hardy)f(space)i Ft(H)3014 4905 y Fv(2)3006 4967 y(\005)3091 4941 y Fx(in)e(the)h(upp)s (er)g(half-plane.)236 5098 y(It)g(is)f(clear)h(that)f(an)m(y)i (function)e Ft(r)s Fx(\()p Ft(z)t Fx(\))g Fr(2)g(R)h Fx(has)h(a)e(\014nite)h(n)m(um)m(b)s(er)g(of)f(p)s(oles)h Ft(z)3076 5113 y Fs(j)3141 5098 y Fx(in)f(the)i(lo)m(w)m(er)f (half-plane)236 5254 y Fr(=)p Ft(m)k(z)f(<)c Fx(0.)383 5411 y Fw(De\014nition)36 b(2.)87 b([8])33 b Fj(The)h(c)-5 b(ap)g(acity)35 b(of)f(a)h(function)f Ft(r)d Fr(2)d(R)35 b Fj(is)g(c)-5 b(al)5 b(le)-5 b(d)34 b(the)h(quantity)1413 5688 y Ft(cap)1555 5703 y Fs(z)1595 5688 y Fx(\()p Ft(r)s Fx(\))27 b(=)1848 5593 y Fp(X)2009 5688 y Ft(k)s Fx(\()p Ft(z)2146 5703 y Fs(j)2183 5688 y Fx(\))2237 5548 y Fp(\022)2311 5688 y Fx(1)21 b Fr(\000)2492 5621 y(j)p Ft(z)2565 5636 y Fs(j)2623 5621 y Fx(+)h Ft(i)p Fr(j)p 2491 5665 293 4 v 2491 5756 a(j)p Ft(z)2564 5771 y Fs(j)2623 5756 y Fr(\000)g Ft(i)p Fr(j)2793 5548 y Fp(\023)2883 5688 y Ft(;)2137 6037 y Fx(5)p eop %%Page: 6 7 6 6 bop 236 454 a Fj(wher)-5 b(e)40 b(the)g(summation)f(is)h(taken)g (over)g(al)5 b(l)39 b(p)-5 b(oles)40 b Ft(z)2224 469 y Fs(j)2301 454 y Fj(of)f(the)i(function)e Ft(r)s Fx(\()p Ft(z)t Fx(\))i Fj(and)e Ft(k)s Fx(\()p Ft(z)3520 469 y Fs(j)3557 454 y Fx(\))h Fj(is)g(or)-5 b(der)40 b(of)236 611 y(the)35 b(p)-5 b(ole)34 b(at)h(the)g(p)-5 b(oint)35 b Ft(z)1168 626 y Fs(j)1205 611 y Fj(.)383 767 y Fx(W)-8 b(e)41 b(denote)h(b)m(y)h Fb(G)e Fx(the)h(op)s(erator)e(of)h(m)m (ultiplication)c(b)m(y)2620 728 y Fs(i\016)r Fv(+1)p 2620 744 148 4 v 2620 801 a Fs(i\016)r Fo(\000)p Fv(1)2819 767 y Fx(in)k(the)h(space)g Ft(H)3477 731 y Fv(2)3469 792 y(\005)3526 767 y Fx(.)70 b(A)41 b(function)236 924 y Ft( )t Fx(\()p Ft(\016)t Fx(\))28 b Fr(2)g Ft(H)637 887 y Fv(2)629 949 y(\005)718 924 y Fx(is)k(called)g Fj(noncyclic)f Fx(for)h Fb(G)1735 887 y Fo(\003)1807 924 y Fx(if)1582 1193 y Ft(E)1654 1209 y Fs( )r Fv(\()p Fs(\016)r Fv(\))1823 1193 y Fx(:=)1985 1069 y Fo(1)1965 1098 y Fp(_)1954 1308 y Fs(n)p Fv(=0)2103 1193 y Fb(G)2168 1152 y Fo(\003)2208 1147 y Fs(n)2255 1193 y Ft( )t Fx(\()p Ft(\016)t Fx(\))27 b Fr(6)p Fx(=)h Ft(H)2665 1152 y Fv(2)2657 1218 y(\005)2714 1193 y Ft(:)383 1474 y Fx(It)k(is)g(w)m(ell)g(kno)m (wn)i(\(see.,)g(for)e(example,)g([8,)h(p.)43 b(327]\))32 b(that)g(the)h(op)s(erator)657 1737 y(\010)28 b(:)g Ft(f)11 b Fx(\()p Ft(e)952 1696 y Fs(i\022)1015 1737 y Fx(\))27 b Fr(!)h Ft( )t Fx(\()p Ft(\016)t Fx(\))f(=)1598 1588 y Fr(p)p 1681 1588 49 4 v 82 x Fx(2)p 1539 1715 251 4 v 1539 1806 a(1)22 b Fr(\000)g Ft(i\016)1799 1737 y(f)1875 1597 y Fp(\022)1959 1670 y Ft(i\016)k Fx(+)c(1)p 1958 1715 V 1958 1806 a Ft(i\016)k Fr(\000)d Fx(1)2219 1597 y Fp(\023)2374 1737 y Fx(\()p Ft(\016)31 b Fr(2)d Fq(R)5 b Fx(\))p Ft(;)153 b Fr(8)p Ft(f)11 b Fx(\()p Ft(e)3061 1696 y Fs(i\022)3124 1737 y Fx(\))28 b Fr(2)g Ft(L)3350 1752 y Fv(2)3390 1737 y Fx(\(0)p Ft(;)17 b Fx(2)p Ft(\031)t Fx(\))-3431 b(\(8\))236 1992 y(maps)40 b(isometrically)d(the)j(space)i Ft(L)1589 2007 y Fv(2)1628 1992 y Fx(\(0)p Ft(;)17 b Fx(2)p Ft(\031)t Fx(\))40 b(on)m(to)g Ft(L)2238 2007 y Fv(2)2277 1992 y Fx(\()p Fq(R)5 b Fx(\).)72 b(F)-8 b(urthermore,)42 b(the)e(Hardy)h(class)f Ft(H)3926 1956 y Fv(2)4005 1992 y Fx(in)236 2149 y(the)30 b(unit)f(disk)g(is)g(mapp)s (ed)g(on)m(to)h(the)f(Hardy)h(class)g Ft(H)2258 2113 y Fv(2)2250 2174 y(\005)2336 2149 y Fx(in)f(upp)s(er)g(half-plane.)41 b(Under)30 b(this)f(mapping,)236 2305 y(the)37 b(op)s(erator)f Fr(S)44 b Fx(of)36 b(m)m(ultiplication)c(b)m(y)38 b Ft(e)1839 2269 y Fs(i\022)1939 2305 y Fx(in)d Ft(H)2145 2269 y Fv(2)2221 2305 y Fx(transforms)h(in)m(to)g(the)h(op)s(erator)f Fb(G)p Fx(.)55 b(Hence,)39 b(the)236 2462 y(set)c(of)f(noncyclic)g (elemen)m(ts)h(for)f Fb(G)1547 2426 y Fo(\003)1620 2462 y Fx(is)g(the)h(image)d(of)i(the)h(set)g(of)f(noncyclic)g(elemen)m(ts)g (for)g Fr(S)3768 2426 y Fo(\003)3843 2462 y Fx(under)236 2618 y(the)c(mapping)e(\010.)43 b(W)-8 b(e)30 b(note)g(that)f(the)h (set)g Fr(R)g Fx(is)f(the)h(image)e(of)h(the)h(set)g(of)f(rational)e (functions)j Ft(f)40 b Fx(from)236 2775 y(the)33 b(Hardy)h(class)f Ft(H)1021 2739 y Fv(2)1092 2775 y Fx(under)h(the)f(mapping)f(\010.)44 b(F)-8 b(urthermore,)33 b(the)g(capacit)m(y)g Ft(cap)3368 2790 y Fs(w)3457 2775 y Fx(of)g(the)g(function)236 2931 y Ft(f)42 b Fx(de\014ned)32 b(in)e([8,)h(p.)43 b(57])30 b(either)h(is)f(equal)h(to)f(the)h(capacit)m(y)g Ft(cap)2620 2946 y Fs(z)2660 2931 y Fx(\(\010)p Ft(f)11 b Fx(\))31 b(of)f(the)h(function)f(\010)p Ft(f)42 b Fx(or)31 b(di\013ers)236 3088 y(from)h(this)g(capacit)m(y)h(b)m(y)g(one.)383 3244 y(Using)g(prop)s(erties)g(of)g(the)i(op)s(erator)e(\010)h(men)m(tioned) f(ab)s(o)m(v)m(e,)i(w)m(e)f(can)g(form)m(ulate)e(assertion)i(\(3\))f (of)236 3401 y(the)g(theorem)g(on)f(\014nite-dimensional)d(appro)m (ximation)i(\([8,)h(p.)44 b(57]\))32 b(as)h(follo)m(ws:)383 3557 y Fw(Lemma)50 b(2.)78 b Fj(The)45 b(function)g Ft( )t Fx(\()p Ft(\016)t Fx(\))i Fr(2)h Ft(H)2002 3521 y Fv(2)1994 3582 y(\005)2096 3557 y Fj(is)d(noncyclic)g(for)g Fb(G)2884 3521 y Fo(\003)2969 3557 y Fj(if)g(and)g(only)g(if)h(ther)-5 b(e)45 b(exists)236 3714 y(a)50 b(se)-5 b(quenc)g(e)49 b(of)g(r)-5 b(ational)49 b(functions)h Ft(r)1736 3729 y Fs(n)1782 3714 y Fx(\()p Ft(\016)t Fx(\))55 b Fr(2)h(R)50 b Fj(such)g(that)g Fx(lim)2802 3729 y Fs(n)p Fo(!1)3007 3714 y Fr(k)p Ft( )37 b Fr(\000)c Ft(r)3311 3729 y Fs(n)3358 3714 y Fr(k)3408 3729 y Fs(L)3456 3738 y Fl(2)3491 3729 y Fv(\()p Fk(R)p Fv(\))3653 3714 y Fx(=)55 b(0)99 b Fj(and)236 3870 y Fx(sup)383 3894 y Fs(n)p Fo(\025)p Fv(1)537 3870 y Ft(cap)p Fx(\()p Ft(r)761 3885 y Fs(n)808 3870 y Fx(\))27 b Ft(<)h Fr(1)p Ft(:)383 4027 y Fx(T)-8 b(aking)32 b(relations)f(\(7\)) h(in)m(to)g(accoun)m(t,)h(it)f(is)g(easy)i(to)e(see)i(that)e(the)h(op)s (erator)1033 4271 y([)p Fb(Y)r Ft(p)p Fx(]\()p Ft(\016)t Fx(\))28 b(:=)1565 4203 y(1)p 1494 4248 191 4 v 1494 4268 a Fr(p)p 1577 4268 108 4 v 82 x Fx(2)p Ft(\031)1711 4135 y Fp(Z)1811 4161 y Fo(1)1766 4361 y Fv(0)1902 4271 y Ft(e)1947 4230 y Fs(i\016)r(s)2042 4271 y Fx(\()p Ft(T)14 b(p)p Fx(\)\()p Ft(s)p Fx(\))p Ft(ds;)146 b Fr(8)p Ft(p)p Fx(\()p Ft(s)p Fx(\))27 b Fr(2)h Ft(L)3043 4286 y Fv(2)3083 4271 y Fx(\()p Fq(R)3187 4286 y Fv(+)3252 4271 y Fx(\))-3054 b(\(9\))236 4524 y(maps)31 b(isometrically)c Ft(L)1126 4539 y Fv(2)1166 4524 y Fx(\()p Fq(R)1270 4539 y Fv(+)1335 4524 y Fx(\))j(on)m(to)h Ft(H)1710 4487 y Fv(2)1702 4549 y(\005)1789 4524 y Fx(in)f(suc)m(h)i(a)f(w)m(a)m(y)h(that)e(the)h(op)s (erator)f Ft(B)36 b Fx(transforms)30 b(in)m(to)g(the)236 4680 y(op)s(erator)i(of)g(m)m(ultiplication)c(b)m(y)34 b Ft(\016)i Fx(in)c Ft(H)1784 4644 y Fv(2)1776 4705 y(\005)1865 4680 y Fx(and)h(the)g(op)s(erator)f Ft(G)g Fx(transforms)g(in)m(to)g Fb(G)p Fx(.)383 4837 y Fw(Theorem)45 b(2.)66 b Fj(The)41 b(op)-5 b(er)g(ator)1626 4811 y Fx(~)1615 4837 y Ft(L)1681 4852 y Fs(k)1765 4837 y Fj(is)42 b(a)f(L)-5 b(ax{Phil)5 b(lips)41 b(p)-5 b(erturb)g(e)g(d)41 b(op)-5 b(er)g(ator)42 b(if)f(and)g(only)g(if)h(for)236 4993 y(the)d(function)f Ft( )t Fx(\()p Ft(\016)t Fx(\))33 b(=)h([)p Fb(Y)r Ft(p)p Fx(]\()p Ft(\016)t Fx(\))p Fj(,)40 b(wher)-5 b(e)38 b Fb(Y)j Fj(is)d(de\014ne)-5 b(d)37 b(by)i(\(9\),)f(ther)-5 b(e)39 b(exists)f(a)g(se)-5 b(quenc)g(e)37 b(of)h(r)-5 b(ational)236 5150 y(functions)35 b Ft(r)704 5165 y Fs(n)751 5150 y Fx(\()p Ft(\016)t Fx(\))27 b Fr(2)h(R)35 b Fj(such)g(that)1161 5379 y Fx(lim)1137 5439 y Fs(n)p Fo(!1)1338 5379 y Fr(k)p Ft( )26 b Fr(\000)c Ft(r)1620 5394 y Fs(n)1667 5379 y Fr(k)1717 5395 y Fs(L)1765 5404 y Fl(2)1800 5395 y Fv(\()p Fk(R)p Fv(\))1934 5379 y Fx(=)28 b(0)97 b(and)114 b(sup)2462 5459 y Fs(n)p Fo(\025)p Fv(1)2619 5379 y Ft(cap)p Fx(\()p Ft(r)2843 5394 y Fs(n)2890 5379 y Fx(\))27 b Ft(<)h Fr(1)p Ft(:)383 5631 y Fj(Pr)-5 b(o)g(of.)42 b Fx(Since)29 b(the)h(mapping)f Fb(Y)j Fx(transfers)f Ft(G)e Fx(in)m(to)g Fb(G)p Fx(,)h(w)m(e)h(get)f (that)f Fb(Y)k Fx(maps)c(the)h(set)h(of)e(noncyclic)236 5787 y(elemen)m(ts)e(for)f Ft(G)849 5751 y Fo(\003)914 5787 y Fx(in)g(the)h(space)g Ft(L)1504 5802 y Fv(2)1544 5787 y Fx(\()p Fq(R)1648 5802 y Fv(+)1713 5787 y Fx(\))f(on)m(to)g(the) h(set)g(of)f(noncyclic)g(elemen)m(ts)h(for)f Fb(G)3423 5751 y Fo(\003)3488 5787 y Fx(in)g Ft(H)3685 5751 y Fv(2)3677 5813 y(\005)3734 5787 y Fx(.)41 b(Hence,)2137 6037 y(6)p eop %%Page: 7 8 7 7 bop 236 454 a Fx(taking)34 b(in)m(to)g(accoun)m(t)i(Lemma)d(2,)i(w) m(e)h(obtain)e(that)g(Theorem)h(2)g(is)f(a)g(direct)h(corollary)e(of)h (Theorem)236 611 y(1.)383 767 y Fw(Corollary)g(1.)43 b Fj(The)33 b(set)h(of)g(functions)f Ft(p)p Fx(\()p Ft(s)p Fx(\))27 b Fr(2)h Ft(L)2252 782 y Fv(2)2292 767 y Fx(\()p Fq(R)2396 782 y Fv(+)2461 767 y Fx(\))34 b Fj(that)g(determine)f(\(by)h (me)-5 b(ans)33 b(of)g(r)-5 b(elation)236 924 y(\(3\)\))27 b(L)-5 b(ax{Phil)5 b(lips)27 b(p)-5 b(erturb)g(e)g(d)28 b(op)-5 b(er)g(ators)1824 898 y Fx(~)1813 924 y Ft(L)1879 939 y Fs(k)1949 924 y Fj(is)28 b(a)g(dense)f(line)-5 b(ar)27 b(set)h(of)g(the)g(\014rst)g(c)-5 b(ate)g(gory)27 b(in)h Ft(L)3810 939 y Fv(2)3850 924 y Fx(\()p Fq(R)3954 939 y Fv(+)4019 924 y Fx(\))p Fj(.)383 1080 y(Pr)-5 b(o)g(of.)43 b Fx(Corollary)30 b(1)j(follo)m(ws)e(from)g(Theorem)i(2)f(and)h(the)g (Corollary)e(in)h([8,)g(p.)44 b(58].)236 1464 y Fu(3)161 b(The)54 b(Construction)d(of)j(the)f(Scattering)g(Matrix)236 1720 y Fx(It)40 b(follo)m(ws)e(from)g(\(3\))h(and)h(\(5\))f(that)g(the) h(op)s(erator)2242 1694 y(~)2230 1720 y Ft(L)2296 1735 y Fs(k)2379 1720 y Fx(can)g(b)s(e)f(considered)h(as)g(a)f (one-dimensional)236 1876 y(p)s(erturbation)e(of)g Ft(L)995 1891 y Fs(k)1038 1876 y Fx(.)59 b(Hence,)41 b(in)c(view)h(of)f(w)m (ell-kno)m(wn)h(results)g(of)f(the)h(Kato{Birman)d(theory)k([1],)236 2033 y(the)33 b(w)m(a)m(v)m(e)i(op)s(erators)1408 2189 y Ft(Q)1485 2204 y Fo(\006)1544 2189 y Fx(\()1594 2164 y(~)1582 2189 y Ft(L)1648 2204 y Fs(k)1691 2189 y Ft(;)17 b(L)1801 2204 y Fs(k)1844 2189 y Fx(\))27 b(=)71 b(lim)2013 2249 y Fs(t)p Fo(!\0061)2251 2189 y Ft(W)2351 2199 y Fv(~)2343 2216 y Fs(L)2391 2228 y Fi(k)2433 2189 y Fx(\()p Fr(\000)p Ft(t)p Fx(\))p Ft(W)2713 2204 y Fs(L)2761 2216 y Fi(k)2804 2189 y Fx(\()p Ft(t)p Fx(\))236 2390 y(exist)33 b(and)g(are)f(de\014ned)i(on)f(the)g(whole)f(space)i Ft(H)2077 2405 y Fs(L)2125 2417 y Fi(k)2167 2390 y Fx(.)43 b(Th)m(us,)35 b(the)e(scattering)f(op)s(erator)1460 2624 y Ft(S)1520 2651 y Fv(\()1555 2634 y(~)1547 2651 y Fs(L)1595 2663 y Fi(k)1633 2651 y Fs(;L)1701 2663 y Fi(k)1739 2651 y Fv(\))1798 2624 y Fx(=)c(\012)1972 2583 y Fo(\003)1972 2649 y Fv(+)2031 2624 y Fx(\()2081 2599 y(~)2069 2624 y Ft(L)2135 2639 y Fs(k)2178 2624 y Ft(;)17 b(L)2288 2639 y Fs(k)2331 2624 y Fx(\)\012)2439 2639 y Fo(\000)2498 2624 y Fx(\()2548 2599 y(~)2536 2624 y Ft(L)2602 2639 y Fs(k)2645 2624 y Ft(;)g(L)2755 2639 y Fs(k)2798 2624 y Fx(\))p Ft(:)-2627 b Fx(\(10\))236 2858 y(is)38 b(w)m(ell)f(p)s(osed) i(as)f(an)g(op)s(erator)f(in)h Ft(H)1688 2873 y Fs(L)1736 2885 y Fi(k)1778 2858 y Fx(.)60 b(The)39 b(aim)d(of)h(the)i(presen)m(t) h(section)e(is)f(to)h(in)m(v)m(estigate)g(the)236 3015 y(structure)c(of)e Ft(S)825 3042 y Fv(\()860 3025 y(~)852 3042 y Fs(L)900 3054 y Fi(k)939 3042 y Fs(;L)1007 3054 y Fi(k)1044 3042 y Fv(\))1109 3015 y Fx(in)f(the)i(case)h(where)1909 2989 y(~)1879 3015 y Ft(L)1945 3030 y Fs(k)2020 3015 y Fx(is)e(a)h(Lax{Phillips)d(p)s(erturb)s(ed)j(op)s(erator.)383 3171 y(W)-8 b(e)24 b(start)g(from)f(the)i(more)f(detail)e(in)m(v)m (estigation)h(of)h(the)h(structure)g(of)f(the)h(set)g Fr(L)3359 3186 y Fs(B)3443 3171 y Fx(from)e(De\014nition)236 3328 y(1.)383 3484 y(Let)32 b Fh(A)g Fx(b)s(e)g(the)g(set)h(of)f(all)d (isometric)i(op)s(erators)g Ft(A)h Fx(in)f Ft(L)2488 3499 y Fv(2)2528 3484 y Fx(\()p Fq(R)2632 3499 y Fv(+)2697 3484 y Fx(\))h(that)g(admit)e(the)i(follo)m(wing)d(repre-)236 3641 y(sen)m(tation:)1596 3797 y Ft(A)f Fx(=)f Ft(g)t Fx(\()p Ft(B)5 b Fx(\))27 b(:=)2164 3662 y Fp(Z)2264 3688 y Fo(1)2219 3887 y(\0001)2366 3797 y Ft(g)t Fx(\()p Ft(\030)5 b Fx(\))p Ft(dE)2664 3812 y Fs(\030)2700 3797 y Ft(;)-2491 b Fx(\(11\))236 4022 y(where)40 b Ft(E)596 4037 y Fs(\030)672 4022 y Fx(is)d(the)i(sp)s(ectral)f(function)f(of)h (the)h(op)s(erator)e Ft(B)43 b Fx(and)c Ft(g)t Fx(\()p Ft(\025)p Fx(\))e(is)h(an)g(inner)f(function,)j(i.e.,)f(a)236 4178 y(b)s(ounded)e(analytic)d(function)h(in)g(the)h(upp)s(er)h (half-plane)c(and)j(suc)m(h)h(that)f Fr(j)p Ft(g)t Fx(\()p Ft(\016)t Fx(\))p Fr(j)31 b Fx(=)i(1)71 b(\()p Ft(\016)37 b Fr(2)c Fq(R)5 b Fx(\).)59 b(W)-8 b(e)236 4335 y(note)31 b(that,)f(b)m(y)h(virtue)f(of)g(the)g(Plesner's)i(results)e([9,)h(10],) f(the)h(righ)m(t-hand)e(side)h(of)f(\(11\))h(is)g(w)m(ell)f(p)s(osed) 236 4491 y(and)k(determines)g(an)f(isometric)f(op)s(erator)h(in)g Ft(L)2046 4506 y Fv(2)2085 4491 y Fx(\()p Fq(R)2189 4506 y Fv(+)2254 4491 y Fx(\).)383 4648 y(No)m(w,)j(w)m(e)h(sho)m(w)f(that)f (the)h(set)g Fr(L)1629 4663 y Fs(B)1724 4648 y Fx(admits)e(the)i(follo) m(wing)d(simple)h(description)h(where)h(elemen)m(ts)236 4804 y(from)d Fh(A)h Fx(are)f(considered)i(as)e(parameters:)383 4961 y Fw(Lemma)k(3.)43 b Fj(A)n(n)34 b(op)-5 b(er)g(ator)34 b Ft(B)1530 4924 y Fo(0)1588 4961 y Fj(b)-5 b(elongs)33 b(to)i Fr(L)2107 4976 y Fs(B)2202 4961 y Fj(if)f(and)g(only)g(if)g (ther)-5 b(e)34 b(exists)g(an)g(op)-5 b(er)g(ator)34 b Ft(A)28 b Fr(2)g Fh(A)236 5117 y Fj(such)35 b(that)1459 5273 y Ft(B)1538 5232 y Fo(0)1589 5273 y Fx(=)28 b Ft(AB)5 b(A)1918 5232 y Fo(\003)1958 5273 y Ft(;)146 b(D)s Fx(\()p Ft(B)2332 5232 y Fo(0)2356 5273 y Fx(\))27 b(=)h Ft(AD)s Fx(\()p Ft(B)5 b Fx(\))p Ft(:)-2628 b Fx(\(12\))383 5475 y Fj(Pr)-5 b(o)g(of.)42 b Fx(By)32 b(virtue)f(of)g(Prop)s(osition)f(1)h (in)f([11],)i(an)f(op)s(erator)g Ft(A)d Fr(2)g Fh(A)j Fx(comm)m(utes)h(with)f Ft(B)5 b Fx(.)43 b(There-)236 5631 y(fore,)32 b(c)m(ho)s(osing)g Fh(H)923 5595 y Fo(0)977 5631 y Fx(as)g Ft(AL)1235 5646 y Fv(2)1275 5631 y Fx(\()p Fq(R)1379 5646 y Fv(+)1444 5631 y Fx(\),)g(it)f(is)g(easy)i(to)f(v)m (erify)g(that)g(relations)e(\(12\))h(de\014ne)i(an)f(op)s(erator)f Ft(B)4036 5595 y Fo(0)4060 5631 y Fx(,)236 5787 y(whic)m(h)i (satis\014es)h(condition)d Fw(I)h Fx(and,)h(hence,)h Ft(B)1971 5751 y Fo(0)2022 5787 y Fr(2)28 b(L)2185 5802 y Fs(B)2245 5787 y Fx(.)2137 6037 y(7)p eop %%Page: 8 9 8 8 bop 383 454 a Fx(Con)m(v)m(ersely)-8 b(,)36 b(let)e Ft(B)1119 418 y Fo(0)1172 454 y Fr(2)d(L)1338 469 y Fs(B)1398 454 y Fx(.)48 b(By)35 b(virtue)f(of)f(condition)g Fw(I)p Fx(,)h Ft(B)2638 418 y Fo(0)2695 454 y Fx(acts)h(in)e(a)h(subspace)i Fh(H)3577 418 y Fo(0)3634 454 y Fx(of)e Ft(L)3813 469 y Fv(2)3853 454 y Fx(\()p Fq(R)3957 469 y Fv(+)4022 454 y Fx(\),)236 611 y(whic)m(h)41 b(is)g(in)m(v)-5 b(arian)m(t)39 b(with)h(resp)s(ect)i(to)f Ft(B)5 b Fx(.)67 b(Therefore,)44 b(taking)c(in)m(to)g(accoun)m(t)i(the)f(de\014nition)f(of)g Fb(Y)236 767 y Fx(\(relation)30 b(\(9\)\),)h(w)m(e)i(get)f(that)f(the)i (subspace)g Fh(L)28 b Fx(=)g Fb(Y)r Fh(H)2277 731 y Fo(0)2332 767 y Fx(of)k Ft(H)2532 731 y Fv(2)2524 792 y(\005)2612 767 y Fx(is)f(in)m(v)-5 b(arian)m(t)31 b(with)g(resp)s(ect)i(to)e Fb(G)p Fx(.)43 b(But)236 924 y(then,)35 b(b)m(y)f(virtue)g(of)f(the)h (Beurling)f(theorem)g([12],)h(there)g(exists)h(an)e(inner)h(function)f Ft(g)t Fx(\()p Ft(\025)p Fx(\))g(suc)m(h)i(that)236 1080 y Fh(L)28 b Fx(=)g Ft(g)t Fx(\()p Ft(\016)t Fx(\))p 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h(problems)e(for)g(equations)236 5631 y(\(14\))c(and)g(\(2\))f Fj(simultane)-5 b(ously)26 b Fx(can)g(b)s(e)g(considered)h(in)e(the)h (Lax{Phillips)e(framew)m(ork,)j(moreo)m(v)m(er,)h(the)236 5787 y(\014rst)39 b(of)f(them)g(de\014nes)i(a)e(free)h(ev)m(olution)f (and)g(the)h(second)h(group)e(de\014nes)i(a)e(p)s(erturb)s(ed)h(ev)m (olution)2137 6037 y(8)p eop %%Page: 9 10 9 9 bop 236 454 a Fx(in)40 b(the)i(Lax{Phillips)c(sense.)71 b(In)41 b(this)g(case,)j(in)c(addition)f(to)i(a)g(similar)d(statemen)m (t)j(\(10\))f(that)h(the)236 611 y(corresp)s(onding)33 b(scattering)f(op)s(erator)1439 857 y Ft(S)1499 884 y Fv(\()1534 867 y(~)1526 884 y Fs(L)1574 896 y Fi(k)1613 884 y Fs(;L)1681 895 y Fi(A)1731 884 y Fv(\))1790 857 y Fx(=)27 b(\012)1963 816 y Fo(\003)1963 882 y Fv(+)2023 857 y Fx(\()2073 832 y(~)2061 857 y Ft(L)2127 872 y Fs(k)2170 857 y Ft(;)17 b(L)2280 872 y Fs(A)2337 857 y Fx(\)\012)2445 872 y Fo(\000)2504 857 y Fx(\()2554 832 y(~)2542 857 y Ft(L)2608 872 y Fs(k)2651 857 y 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Fs(L)2749 3007 y Fi(A)2804 2981 y Fx(\()p Ft(t)p Fx(\))p Ft(;)-2706 b Fx(\(17\))236 3187 y(whic)m(h)28 b(are)f(de\014ned)i(on)e(the)h (whole)f(space)h Ft(H)1898 3202 y Fs(L)1946 3213 y Fi(A)2000 3187 y Fx(.)42 b(In)28 b(what)f(follo)m(ws,)g(w)m(e)h(assume)g(that)f (these)i(op)s(erators)236 3344 y(satisfy)k(the)g(follo)m(wing)d (condition)1661 3590 y Ft(Q)1738 3605 y Fo(\006)1797 3590 y Fx(\()p Ft(L)1901 3605 y Fs(k)1944 3590 y Ft(;)17 b(L)2054 3605 y Fs(A)2111 3590 y Fx(\))p Ft(H)2230 3605 y Fs(L)2278 3616 y Fi(A)2360 3590 y Fx(=)28 b Ft(H)2545 3605 y Fs(L)2593 3617 y Fi(k)2635 3590 y Ft(:)-2426 b Fx(\(18\))383 3837 y(W)-8 b(e)41 b(note)g(that)f(condition)f(\(18\))h (is)h(natural)e(in)h(the)h(sense)i(that)d(it)g(ensures)i(an)f(equiv)-5 b(alence)41 b(of)236 3993 y(Lax{Phillips)30 b(free)j(ev)m(olutions,)g (whic)m(h)g(are)f(describ)s(ed)i(b)m(y)f(the)g(groups)g Ft(W)3059 4008 y Fs(L)3107 4020 y Fi(k)3149 3993 y Fx(\()p Ft(t)p Fx(\))g(and)g Ft(W)3575 4008 y Fs(L)3623 4019 y Fi(A)3677 3993 y Fx(\()p Ft(t)p Fx(\).)383 4150 y(It)21 b(follo)m(ws)g(from)f(\(18\))h(and)h(the)g(c)m(hain)g(la)m(w)f([1,)j (p.)40 b(29])21 b(that)h(\012)2614 4165 y Fo(\006)2673 4150 y Fx(\()2723 4124 y(~)2711 4150 y Ft(L)2777 4165 y Fs(k)2820 4150 y Ft(;)17 b(L)2930 4165 y Fs(A)2988 4150 y Fx(\))27 b(=)h(\012)3227 4165 y Fo(\006)3286 4150 y Fx(\()3336 4124 y(~)3324 4150 y Ft(L)3390 4165 y Fs(k)3433 4150 y Ft(;)17 b(L)3543 4165 y Fs(k)3586 4150 y Fx(\)\012)3694 4165 y Fo(\006)3753 4150 y Fx(\()p Ft(L)3857 4165 y Fs(k)3900 4150 y Ft(;)g(L)4010 4165 y Fs(A)4067 4150 y Fx(\))p Ft(;)236 4306 y Fx(Multiplying)30 b(this)j(equalit)m(y)f(b)m(y)h(\012) 1534 4270 y Fo(\003)1534 4331 y(\006)1594 4306 y Fx(\()p Ft(L)1698 4321 y Fs(k)1741 4306 y Ft(;)17 b(L)1851 4321 y Fs(A)1908 4306 y Fx(\))32 b(and)h(passing)f(to)h(adjoin)m(t)e(op)s (erators,)i(w)m(e)g(obtain)1367 4552 y(\012)1437 4511 y Fo(\003)1437 4577 y(\006)1497 4552 y Fx(\()1547 4527 y(~)1535 4552 y Ft(L)1601 4567 y Fs(k)1644 4552 y Ft(;)17 b(L)1754 4567 y Fs(k)1796 4552 y Fx(\))28 b(=)g(\012)2036 4567 y Fo(\006)2095 4552 y Fx(\()p Ft(L)2199 4567 y Fs(k)2242 4552 y Ft(;)17 b(L)2352 4567 y Fs(A)2409 4552 y Fx(\)\012)2517 4511 y Fo(\003)2517 4577 y(\006)2577 4552 y Fx(\()2627 4527 y(~)2615 4552 y Ft(L)2681 4567 y Fs(k)2724 4552 y Ft(;)g(L)2834 4567 y Fs(A)2891 4552 y Fx(\))p Ft(:)236 4799 y Fx(Substituting)33 b(the)h(obtained)g(equalities)e(in)m(to)h (\(10\))h(and)g(taking)f(in)m(to)g(accoun)m(t)h(\(16\),)g(w)m(e)h (arriv)m(e)e(at)h(a)236 4955 y(conclusion)e(that)1284 5112 y Ft(S)1344 5139 y Fv(\()1379 5122 y(~)1371 5139 y Fs(L)1419 5151 y Fi(k)1457 5139 y Fs(;L)1525 5151 y Fi(k)1563 5139 y Fv(\))1622 5112 y Fx(=)c(\012)1796 5127 y Fv(+)1855 5112 y Fx(\()p Ft(L)1959 5127 y Fs(k)2002 5112 y Ft(;)17 b(L)2112 5127 y Fs(A)2170 5112 y Fx(\))p Ft(S)2268 5139 y Fv(\()2303 5122 y(~)2295 5139 y Fs(L)2343 5151 y Fi(k)2381 5139 y Fs(;L)2449 5150 y Fi(A)2499 5139 y Fv(\))2530 5112 y Fx(\012)2600 5071 y Fo(\003)2600 5136 y(\000)2660 5112 y Fx(\()p Ft(L)2764 5127 y Fs(k)2807 5112 y Ft(;)g(L)2917 5127 y Fs(A)2974 5112 y Fx(\))p Ft(:)-2803 b Fx(\(19\))383 5318 y(As)35 b(w)m(as)h(sho)m(wn)g(in)e ([7],)i Ft(L)1365 5333 y Fs(k)1439 5318 y Fx(=)c Ft(B)1626 5282 y Fo(\003)1665 5318 y Ft(B)5 b Fx(,)36 b(where)g Ft(B)k Fx(is)34 b(de\014ned)j(b)m(y)e(\(7\).)50 b(Therefore,)37 b(the)e(translation)236 5475 y(represen)m(tation)i(for)e(the)i(group)e Ft(W)1572 5490 y Fs(L)1620 5502 y Fi(k)1663 5475 y Fx(\()p Ft(t)p Fx(\))h(whic)m(h)g(is)g(asso)s(ciated)f(with)h(subspaces)i Ft(D)3417 5490 y Fo(\006)3512 5475 y Fx(of)e(form)f(\(6\))g(is)236 5631 y(de\014ned)42 b(b)m(y)f(form)m(ula)d(4.18)i(in)f([14,)j(p.)66 b(221].)g(Applying)39 b(to)h(this)g(form)m(ula)e(the)j(F)-8 b(ourier)38 b(transform)236 5787 y(in)i Ft(L)424 5802 y Fv(2)464 5787 y Fx(\()p Fq(R)5 b Fx(\))47 b(and)41 b(taking)f(in)m(to)h(accoun)m(t)g(the)h(de\014nition)e(\(9\))g(of)h Fb(Y)r Fx(,)j(w)m(e)e(conclude)g(that)e(an)h(isometric)2137 6037 y(9)p eop %%Page: 10 11 10 10 bop 236 454 a Fx(op)s(erator)40 b Fr(F)50 b Fx(:)41 b Ft(H)908 469 y Fs(L)956 481 y Fi(k)1038 454 y Fr(!)g Ft(L)1245 469 y Fv(2)1285 454 y Fx(\()p Fq(R)t Fx(\))46 b(that,)c(for)e(an)m(y)h Ft(u)f Fr(2)h Ft(D)s Fx(\()p Ft(L)2460 469 y Fs(k)2503 454 y Fx(\))f(and)g Ft(v)45 b Fr(2)c Ft(L)3043 469 y Fv(2)3083 454 y Fx(\()p Fq(R)3187 469 y Fv(+)3252 454 y Fx(\),)h(is)d(de\014ned)j(b)m(y)f(the)236 611 y(form)m(ula)1136 800 y Fr(F)1234 630 y Fp( )1354 728 y Ft(u)1357 885 y(v)1451 630 y Fp(!)1558 800 y Fx(=)1671 733 y(1)p 1671 777 49 4 v 1671 869 a(2)1747 800 y(\()p Fb(Y)r Fx([)p Ft(iB)5 b(u)23 b Fx(+)f Ft(v)t Fx(]\()p Ft(\016)t Fx(\))f(+)h Fb(Y)r Fx([)p Ft(iB)5 b(u)23 b Fr(\000)g Ft(v)t Fx(]\()p Fr(\000)p Ft(\016)t Fx(\)\))-2951 b(\(20\))236 1069 y(determines)33 b(the)g(sp)s(ectral)f(represen)m (tation)i(for)e Ft(W)2143 1084 y Fs(L)2191 1096 y Fi(k)2233 1069 y Fx(\()p Ft(t)p Fx(\))h(asso)s(ciated)f(with)g Ft(D)3144 1084 y Fo(\006)3203 1069 y Fx(.)383 1225 y(Similarly)-8 b(,)36 b(the)k(sp)s(ectral)f(represen)m(tation)h(for)e(the)i(group)f Ft(W)2721 1240 y Fs(L)2769 1251 y Fi(A)2823 1225 y Fx(\()p Ft(t)p Fx(\))g(asso)s(ciated)g(with)g(subspaces)236 1382 y Ft(D)320 1346 y Fs(A)317 1406 y Fo(\006)377 1382 y Fx(,)51 b(whic)m(h)e(are)e(de\014ned)j(b)m(y)e(\(6\))g(with)f(the)h (replacemen)m(t)g(of)f Ft(B)53 b Fx(b)m(y)c Ft(B)3060 1397 y Fs(A)3117 1382 y Fx(,)i(is)c(determined)h(b)m(y)h(an)236 1538 y(isometric)35 b(op)s(erator)h Fr(F)1127 1553 y Fs(A)1217 1538 y Fx(:)f Ft(H)1360 1553 y Fs(L)1408 1564 y Fi(A)1496 1538 y Fr(!)f Ft(L)1696 1553 y Fv(2)1736 1538 y Fx(\()p Fq(R)t Fx(\))6 b(,)38 b(whic)m(h,)g(for)e(an)m(y)h Ft(u)c Fr(2)i Ft(D)s Fx(\()p Ft(L)2978 1553 y Fs(A)3035 1538 y Fx(\))h(and)h Ft(v)h Fr(2)c Fh(H)3560 1553 y Fs(A)3617 1538 y Fx(,)j(is)f(de\014ned)236 1695 y(b)m(y)e(the)f(form)m(ula)984 2004 y Fr(F)1056 2019 y Fs(A)1129 1833 y Fp( )1249 1932 y Ft(u)1252 2088 y(v)1346 1833 y Fp(!)1453 2004 y Fx(=)1566 1936 y(1)p 1566 1981 V 1566 2072 a(2)1642 2004 y(\()p Fb(Y)1744 2019 y Fs(A)1801 2004 y Fx([)p Ft(iB)1935 2019 y Fs(A)1993 2004 y Ft(u)21 b Fx(+)h Ft(v)t Fx(]\()p Ft(\016)t Fx(\))g(+)g Fb(Y)2553 2019 y Fs(A)2611 2004 y Fx([)p Ft(iB)2745 2019 y Fs(A)2802 2004 y Ft(u)g Fr(\000)g Ft(v)t Fx(]\()p Fr(\000)p Ft(\016)t Fx(\)\))17 b Ft(:)-3103 b Fx(\(21\))236 2312 y(Here,)1041 2477 y([)p Fb(Y)1132 2492 y Fs(A)1190 2477 y Ft(p)p Fx(]\()p Ft(\016)t Fx(\))27 b(:=)1628 2410 y(1)p 1557 2454 191 4 v 1557 2474 a Fr(p)p 1640 2474 108 4 v 83 x Fx(2)p Ft(\031)1774 2342 y Fp(Z)1874 2368 y Fo(1)1829 2567 y Fv(0)1965 2477 y Ft(e)2010 2436 y Fs(i\016)r(s)2105 2477 y Fx(\()p Ft(T)14 b(A)2287 2436 y Fo(\003)2326 2477 y Ft(p)p Fx(\)\()p Ft(s)p Fx(\))p Ft(ds;)146 b Fr(8)p Ft(p)p Fx(\()p Ft(s)p Fx(\))28 b Fr(2)g Fh(H)3225 2492 y Fs(A)236 2707 y Fx(and)33 b Ft(T)46 b Fx(is)32 b(the)h(same)g(as)f(in)g(\(7\).)383 2863 y(Equalities)f (\(18\))h(and)h(\(19\))f(imply)e(that)1175 3120 y Fh(S)1258 3147 y Fv(\()1293 3130 y(~)1285 3147 y Fs(L)1333 3159 y Fi(k)1371 3147 y Fs(;L)1439 3159 y Fi(k)1477 3147 y Fv(\))1536 3120 y Fx(:=)e Fr(F)10 b Ft(S)1809 3147 y Fv(\()1844 3130 y(~)1836 3147 y Fs(L)1884 3159 y Fi(k)1922 3147 y Fs(;L)1990 3159 y Fi(k)2027 3147 y Fv(\))2059 3120 y Fr(F)2141 3078 y Fo(\000)p Fv(1)2262 3120 y Fx(=)28 b Fh(K)2433 3135 y Fv(+)2492 3120 y Fh(S)2575 3147 y Fv(\()2610 3130 y(~)2602 3147 y Fs(L)2650 3159 y Fi(k)2688 3147 y Fs(;L)2756 3158 y Fi(A)2806 3147 y Fv(\))2837 3120 y Fx(\()p Ft(\016)t Fx(\))p Fh(K)3027 3078 y Fo(\000)p Fv(1)3027 3144 y Fo(\000)3121 3120 y Ft(;)-2912 b Fx(\(22\))236 3376 y(where)34 b Fh(K)585 3391 y Fo(\006)671 3376 y Fx(=)28 b Fr(F)10 b Fx(\012)927 3391 y Fo(\006)986 3376 y Fx(\()p Ft(L)1090 3391 y Fs(k)1133 3376 y Ft(;)17 b(L)1243 3391 y Fs(A)1300 3376 y Fx(\))p Fr(F)1420 3334 y Fo(\000)p Fv(1)1410 3403 y Fs(A)1546 3376 y Fx(and)33 b Fh(S)1819 3403 y Fv(\()1854 3386 y(~)1846 3403 y Fs(L)1894 3415 y Fi(k)1932 3403 y Fs(;L)2000 3414 y Fi(A)2050 3403 y Fv(\))2081 3376 y Fx(\()p Ft(\016)t Fx(\))28 b(=)f Fr(F)2407 3391 y Fs(A)2464 3376 y Ft(S)2524 3403 y Fv(\()2559 3386 y(~)2551 3403 y Fs(L)2599 3415 y Fi(k)2638 3403 y Fs(;L)2706 3414 y Fi(A)2755 3403 y Fv(\))2787 3376 y Fr(F)2869 3334 y Fo(\000)p Fv(1)2859 3403 y Fs(A)2963 3376 y Fx(.)383 3532 y Fw(Lemma)42 b(4.)57 b Fj(The)38 b(op)-5 b(er)g(ators)39 b Fh(K)1627 3547 y Fv(+)1725 3532 y Fj(and)f Fh(K)1985 3547 y Fo(\000)2083 3532 y Fj(act)h(in)g Ft(L)2439 3547 y Fv(2)2478 3532 y Fx(\()p Fq(R)5 b Fx(\))45 b Fj(as)39 b(the)g(op)-5 b(er)g(ators)39 b(of)f(multiplic)-5 b(ation)236 3689 y(by)35 b(the)g(functions)g Ft(g)t Fx(\()p Ft(\016)t Fx(\))f Fj(and)g Ft(g)t Fx(\()p Fr(\000)p Ft(\016)t Fx(\))p Fj(,)g(r)-5 b(esp)g(e)g(ctively.)383 3845 y(Pr)g(o)g(of.)41 b Fx(T)-8 b(aking)29 b(in)m(to)f(accoun)m(t)h(\(14\),)g(it)f(is)g(easy) i(to)e(see)i(that)f(the)g(closure)g(in)f Ft(H)3336 3860 y Fs(L)3384 3872 y Fi(k)3455 3845 y Fx(of)g(the)h(op)s(erator)236 4002 y Fq(A)56 b Fx(:)28 b Ft(H)472 4017 y Fs(L)520 4029 y Fi(k)589 4002 y Fr(!)g Ft(H)798 4017 y Fs(L)846 4028 y Fi(A)932 4002 y Fx(de\014ned)34 b(b)m(y)g(the)f(form)m(ula)1086 4318 y Fq(A)1169 4147 y Fp( )1295 4246 y Ft(u)1298 4402 y(v)1392 4147 y Fp(!)1499 4318 y Fx(=)1602 4147 y Fp( )1723 4246 y Ft(Au)1725 4402 y(Av)1893 4147 y Fp(!)1988 4318 y Ft(;)147 b Fr(8)p Ft(u)28 b Fr(2)g Ft(D)s Fx(\()p Ft(L)2583 4333 y Fs(k)2625 4318 y Fx(\))p Ft(;)82 b Fr(8)p Ft(v)32 b Fr(2)c Ft(L)3066 4333 y Fv(2)3106 4318 y Fq(R)3172 4333 y Fv(+)236 4626 y Fx(is)k(an)h(isometric)e(op)s(erator)h(from)f Ft(H)1592 4641 y Fs(L)1640 4653 y Fi(k)1714 4626 y Fx(on)m(to)i Ft(H)2015 4641 y Fs(L)2063 4652 y Fi(A)2117 4626 y Fx(.)383 4782 y(In)e(view)g(of)f(\(20\))g(and)h(\(21\),)g(taking)f(in)m(to)g (accoun)m(t)h(the)h(de\014nitions)e(of)g Ft(B)3132 4797 y Fs(A)3220 4782 y Fx(and)h Fh(H)3480 4797 y Fs(A)3537 4782 y Fx(,)g(it)f(is)g(easy)i(to)236 4939 y(establish)g(that)1941 5095 y Fr(F)2013 5110 y Fs(A)2070 5095 y Fq(A)55 b Fx(=)28 b Fr(F)10 b Ft(:)-2146 b Fx(\(23\))236 5306 y(Hence,)36 b(the)e(op)s(erators)g Fh(K)1224 5321 y Fo(\006)1316 5306 y Fx(in)f(\(22\))g(can)h(b)s(e)g(represen)m(ted)j(in)c(the)h(form) f Fh(K)3055 5321 y Fo(\006)3143 5306 y Fx(=)d Fr(F)10 b Fx(\012)3401 5321 y Fo(\006)3460 5306 y Fx(\()p Ft(L)3564 5321 y Fs(k)3607 5306 y Ft(;)17 b(L)3717 5321 y Fs(A)3774 5306 y Fx(\))p Fq(A)22 b Fr(F)3960 5270 y Fo(\000)p Fv(1)4060 5306 y Fx(.)236 5462 y(Twice)33 b(using)g(\(23\),)f(w)m(e)h(obtain)f (that)806 5718 y Fr(F)878 5733 y Fs(A)935 5718 y Fq(A)22 b Ft(W)1093 5733 y Fs(L)1141 5745 y Fi(k)1189 5718 y Fx(\()p Ft(t)p Fx(\))p Ft(d)27 b Fx(=)h Ft(e)1527 5677 y Fs(i\016)r(t)1615 5718 y Fr(F)10 b Ft(d)26 b Fx(=)i Ft(e)1923 5677 y Fs(i\016)r(t)2011 5718 y Fr(F)2083 5733 y Fs(A)2139 5718 y Fq(A)22 b Ft(d)34 b Fx(=)27 b Fr(F)2465 5733 y Fs(A)2522 5718 y Ft(W)2614 5733 y Fs(L)2662 5744 y Fi(A)2716 5718 y Fx(\()p Ft(t)p Fx(\))p Fq(A)22 b Ft(d)71 b Fx(\()p Fr(8)p Ft(d)28 b Fr(2)g Ft(H)3362 5733 y Fs(L)3410 5745 y Fi(k)3452 5718 y Fx(\))p Ft(:)2113 6037 y Fx(10)p eop %%Page: 11 12 11 11 bop 236 454 a Fx(Therefore,)38 b Ft(W)801 469 y Fs(L)849 480 y Fi(A)903 454 y Fx(\()p Ft(t)p Fx(\))p Fq(A)61 b Fx(=)33 b Fq(A)22 b Ft(W)1386 469 y Fs(L)1434 481 y Fi(k)1482 454 y Fx(\()p Ft(t)p Fx(\))p Ft(:)36 b Fx(The)h(last)e(equalit)m(y)g(and)h(relations)e(\(17\),)i(\(18\))f (imply)f(that)h(the)236 611 y(op)s(erators)28 b(\012)733 626 y Fo(\006)792 611 y Fx(\()p Ft(L)896 626 y Fs(k)939 611 y Ft(;)17 b(L)1049 626 y Fs(A)1106 611 y Fx(\))p Fq(A)56 b Fx(are)28 b(unitary)f(op)s(erators)g(in)g Ft(H)2357 626 y Fs(L)2405 638 y Fi(k)2475 611 y Fx(that)h(comm)m(ute)f(with)g Ft(W)3406 626 y Fs(L)3458 611 y Fx(\()p Ft(t)p Fx(\).)42 b(Hence)29 b(\(see)236 767 y(Corollary)20 b(4.1)i(in)f([5,)j(Chap.)40 b(2]\),)24 b(the)e(images)f Fh(K)2026 782 y Fo(\006)2106 767 y Fx(of)g(these)i(op)s(erators)f(in)f(the)h(sp)s(ectral)g(represen) m(tation)236 924 y(of)32 b Ft(W)439 939 y Fs(L)487 951 y Fi(k)530 924 y Fx(\()p Ft(t)p Fx(\))g(can)h(b)s(e)g(written)f(as)h (op)s(erators)f(of)h(m)m(ultiplication)28 b(b)m(y)33 b(scalar)f(functions)g Fh(K)3514 939 y Fo(\006)3573 924 y Fx(\()p Ft(\016)t Fx(\).)383 1140 y(Let)g Ft(d)608 1155 y Fv(+)695 1140 y Fx(=)798 970 y Fp( )975 1068 y Ft(u)919 1225 y(iB)5 b(u)1128 970 y Fp(!)1207 1140 y Fx(,)33 b(where)g Ft(u)28 b Fr(2)g Ft(B)1805 1104 y Fv(2)1844 1140 y Fx(.)44 b(Clearly)-8 b(,)31 b Ft(d)2323 1155 y Fv(+)2410 1140 y Fr(2)d Ft(D)2585 1155 y Fv(+)2676 1140 y Fx(and)621 1561 y Fq(A)22 b Ft(d)738 1576 y Fv(+)831 1561 y Fx(=)934 1390 y Fp( )1111 1489 y Ft(Au)1055 1645 y(iAB)5 b(u)1337 1390 y Fp(!)1444 1561 y Fx(=)1547 1390 y Fp( )1750 1489 y Ft(v)1668 1645 y(iB)1775 1660 y Fs(A)1832 1645 y Ft(v)1924 1390 y Fp(!)2031 1561 y Fr(2)28 b Ft(D)2206 1576 y Fs(A)2393 1561 y Fx(\()p Ft(v)j Fx(=)d Ft(Au)f Fr(2)h Ft(D)s Fx(\()p Ft(B)3064 1520 y Fv(2)3059 1585 y Fs(A)3116 1561 y Fx(\))g(=)f Ft(AD)s Fx(\()p Ft(B)3559 1520 y Fv(2)3599 1561 y Fx(\)\))p Ft(:)236 1882 y Fx(Here,)47 b Ft(D)591 1846 y Fs(A)588 1907 y Fv(+)691 1882 y Fx(is)c(de\014ned)h (b)m(y)g(\(6\))f(with)g(the)h(replacemen)m(t)f(of)g Ft(B)48 b Fx(b)m(y)c Ft(B)2891 1897 y Fs(A)2949 1882 y Fx(.)75 b(It)43 b(follo)m(ws)f(from)g(the)i(\014rst)236 2038 y(equalit)m(y)39 b(\(for)819 2013 y(~)808 2038 y Ft(L)g Fx(=)g Ft(L)1094 2053 y Fs(k)1177 2038 y Fx(and)g Ft(L)g Fx(=)g Ft(L)1659 2053 y Fs(A)1717 2038 y Fx(\))g(of)g(the)h(pro)s(of)e (of)h(Prop)s(osition)e(5.2)i(in)g([14,)h(Chap.)65 b(4])39 b(that)236 2195 y(\012)306 2210 y Fv(+)366 2195 y Fx(\()p Ft(L)470 2210 y Fs(k)513 2195 y Ft(;)17 b(L)623 2210 y Fs(A)680 2195 y Fx(\))p Fq(A)22 b Ft(d)835 2210 y Fv(+)927 2195 y Fx(=)28 b Fq(A)22 b Ft(d)1148 2210 y Fv(+)1212 2195 y Fx(.)44 b(Hence,)34 b(taking)e(\(20\))f(and)i(\(13\))f(in)m(to)g (accoun)m(t,)h(w)m(e)h(obtain)360 2451 y(\()p Fh(K)465 2466 y Fv(+)524 2451 y Fx(\()p Ft(\016)t Fx(\)\()p Fr(F)10 b Ft(d)818 2466 y Fv(+)876 2451 y Fx(\)\()p Ft(\016)t Fx(\))27 b(=)g([)p Fr(F)10 b Fq(A)22 b Ft(d)1393 2466 y Fv(+)1458 2451 y Fx(\)]\()p Ft(\016)t Fx(\))27 b(=)h Ft(i)1810 2364 y Fr(p)p 1893 2364 49 4 v 87 x Fx(2[)p Fb(Y)r Ft(AB)5 b(u)p Fx(]\()p Ft(\016)t Fx(\))28 b(=)g Ft(i)2558 2364 y Fr(p)p 2641 2364 V 87 x Fx(2)p Ft(g)t Fx(\()p Ft(\016)t Fx(\)[)p Fb(Y)r Ft(B)5 b(u)p Fx(]\()p Ft(\016)t Fx(\))27 b(=)h Ft(g)t Fx(\()p Ft(\016)t Fx(\)\()p Fr(F)10 b Ft(d)3718 2466 y Fv(+)3775 2451 y Fx(\)\()p Ft(\016)t Fx(\))p Ft(:)236 2707 y Fx(Th)m(us)34 b Fh(K)550 2722 y Fv(+)609 2707 y Fx(\()p Ft(\016)t Fx(\))27 b(=)h Ft(g)t Fx(\()p Ft(\016)t Fx(\).)43 b(Reasoning)32 b(similarly)-8 b(,)28 b(w)m(e)34 b(obtain)d Fh(K)2506 2722 y Fo(\000)2565 2707 y Fx(\()p Ft(\016)t Fx(\))d(=)f Ft(g)t Fx(\()p Fr(\000)p Ft(\016)t Fx(\).)43 b(Lemma)31 b(4)i(is)f(pro)m(v)m(ed.)383 2864 y Fw(Theorem)41 b(3)p Fx(.)55 b Fj(Assume)39 b(that)1582 2838 y Fx(~)1570 2864 y Ft(L)1636 2879 y Fs(k)1718 2864 y Fj(is)f(a)g(L)-5 b(ax{Phil)5 b(lips)37 b(p)-5 b(erturb)g(e)g(d)39 b(op)-5 b(er)g(ator)38 b(and)g(c)-5 b(ondition)37 b(\(18\))236 3020 y(is)32 b(true.)45 b(Then)31 b(the)i(image)e Fh(S)1358 3047 y Fv(\()1393 3030 y(~)1385 3047 y Fs(L)1433 3059 y Fi(k)1471 3047 y Fs(;L)1539 3059 y Fi(k)1577 3047 y Fv(\))1641 3020 y Fj(of)h(the)h(sc)-5 b(attering)32 b(op)-5 b(er)g(ator)32 b Ft(S)2796 3047 y Fv(\()2831 3030 y(~)2823 3047 y Fs(L)2871 3059 y Fi(k)2909 3047 y Fs(;L)2977 3059 y Fi(k)3015 3047 y Fv(\))3079 3020 y Fj(in)g(the)h(sp)-5 b(e)g(ctr)g(al)32 b(r)-5 b(epr)g(esen-)236 3177 y(tation)32 b(for)f(the)g(fr)-5 b(e)g(e)31 b(gr)-5 b(oup)32 b Ft(W)1366 3192 y Fs(L)1414 3204 y Fi(k)1456 3177 y Fx(\()p Ft(t)p Fx(\))g Fj(acts)f(as)g(the)g(op)-5 b(er)g(ator)31 b(of)g(multiplic)-5 b(ation)31 b(by)h(the)f(sc)-5 b(alar)31 b(function)236 3333 y(\(sc)-5 b(attering)35 b(matrix\))1255 3508 y Fh(S)1338 3535 y Fv(\()1373 3519 y(~)1365 3535 y Fs(L)1413 3547 y Fi(k)1452 3535 y Fs(;L)1520 3547 y Fi(k)1557 3535 y Fv(\))1589 3508 y Fx(\()p Ft(\016)t Fx(\))27 b(=)1892 3441 y Ft(g)t Fx(\()p Ft(\016)t Fx(\))p 1853 3486 251 4 v 1853 3577 a Ft(g)t Fx(\()p Fr(\000)p Ft(\016)t Fx(\))2113 3508 y Fh(S)2196 3535 y Fv(\()2231 3519 y(~)2223 3535 y Fs(L)2271 3547 y Fi(k)2309 3535 y Fs(;L)2377 3546 y Fi(A)2427 3535 y Fv(\))2459 3508 y Fx(\()p Ft(\016)t Fx(\))p Ft(;)191 b(\016)31 b Fr(2)d Fq(R)5 b Ft(;)236 3739 y Fj(wher)-5 b(e)38 b Ft(g)t Fx(\()p Ft(\016)t Fx(\))g Fj(is)h(the)g(r)-5 b(estriction)38 b(on)h(the)g(r)-5 b(e)g(al)38 b(axis)g(of)h(the)g(inner)f(function)g Ft(g)t Fx(\()p Ft(\025)p Fx(\))g Fj(in)h(the)g(upp)-5 b(er)38 b(half-)236 3895 y(plane)c(and)f Fh(S)766 3922 y Fv(\()801 3905 y(~)793 3922 y Fs(L)841 3934 y Fi(k)879 3922 y Fs(;L)947 3933 y Fi(A)997 3922 y Fv(\))1029 3895 y Fx(\()p Ft(\016)t Fx(\))g Fj(is)h(the)g(r)-5 b(estriction)34 b(on)g(the)g(r)-5 b(e)g(al)34 b(axis)g(of)g(the)g(Heisenb)-5 b(er)g(g)33 b(sc)-5 b(attering)34 b(matrix)236 4052 y Fh(S)319 4079 y Fv(\()354 4062 y(~)346 4079 y Fs(L)394 4091 y Fi(k)432 4079 y Fs(;L)500 4090 y Fi(A)550 4079 y Fv(\))582 4052 y Fx(\()p Ft(z)t Fx(\))h Fj(holomorphic)e(in)i(the)g(lower)f (half-plane.)383 4208 y(Pr)-5 b(o)g(of.)43 b Fx(Theorem)32 b(3)h(follo)m(ws)e(directly)h(from)f(relation)g(\(22\))h(and)h(Lemma)e (4.)236 4596 y Fu(4)161 b(Examples)84 b(of)g(Lax{Phillips)i(p)t(erturb) t(ed)c(op)t(erators)h(in)478 4833 y(the)53 b(case)g(of)h Fa(s)p Fu(-w)l(a)l(v)l(e.)236 5088 y Fx(In)33 b(the)g(case)g(of)f Ft(s)p Fx(-w)m(a)m(v)m(e,)i(the)e(parameter)g Ft(k)k Fx(in)31 b(\(3\))h(and)h(\(5\))f(is)g(equal)g(to)g(zero.)44 b(Hence,)33 b(the)g(op)s(erator)236 5245 y Ft(T)46 b Fx(in)32 b(\(7\))g(is)h(the)g(iden)m(tit)m(y)f(op)s(erator)g(and)h Ft(B)f Fx(=)c Fr(B)s Fx(.)383 5401 y(In)k(the)h(space)h Ft(L)999 5416 y Fv(2)1039 5401 y Fx(\()p Fq(R)1143 5416 y Fv(+)1208 5401 y Fx(\),)e(w)m(e)i(consider)f(the)g(orthonormal)d (basis)j(of)f(Laguerre)h(functions)1305 5688 y Ft(q)1348 5703 y Fs(n)1395 5688 y Fx(\()p Ft(s)p Fx(\))28 b(=)1658 5620 y Ft(e)1703 5584 y Fs(s=)p Fv(2)p 1658 5665 153 4 v 1692 5756 a Ft(n)p Fx(!)1854 5620 y Ft(d)1905 5584 y Fs(n)p 1831 5665 144 4 v 1831 5756 a Ft(ds)1928 5727 y Fs(n)1985 5688 y Fx(\()p Ft(s)2069 5646 y Fs(n)2115 5688 y Ft(e)2160 5646 y Fo(\000)p Fs(s)2252 5688 y Fx(\))163 b(\()p Ft(n)28 b Fx(=)f(0)p Ft(;)17 b Fx(1)g Ft(:)g(:)g(:)o Fx(\))p Ft(:)2113 6037 y Fx(11)p eop %%Page: 12 13 12 12 bop 383 454 a Fx(Using)33 b(\(9\))g(\(as)h Ft(T)43 b Fx(=)29 b Ft(I)8 b Fx(\))33 b(and)h(relation)d(\(25\))i(in)g([15,)h (p.)46 b(158],)33 b(for)g(an)m(y)i(Laguerre)e(function)g Ft(q)3890 469 y Fs(n)3938 454 y Fx(\()p Ft(s)p Fx(\),)236 611 y(w)m(e)h(obtain)1275 783 y Ft( )t Fx(\()p Ft(\016)t Fx(\))27 b(=)h([)p Fb(Y)r Ft(q)1732 798 y Fs(n)1780 783 y Fx(]\()p Ft(\016)t Fx(\))f(=)h(\()p Fr(\000)p Fx(1\))2263 742 y Fs(n)2310 627 y Fp(r)p 2410 627 79 4 v 2425 716 a Fx(2)p 2420 760 59 4 v 2420 852 a Ft(\031)2544 716 y Fx(\(1)22 b(+)g(2)p Ft(i\016)t Fx(\))2918 680 y Fs(n)p 2499 760 513 4 v 2499 852 a Fx(\(1)f Fr(\000)i Fx(2)p Ft(i\016)t Fx(\))2874 823 y Fs(n)p Fv(+1)3021 783 y Ft(:)236 1014 y Fx(It)31 b(is)f(clear)g(that)h(the)g(function)f Ft( )t Fx(\()p Ft(\016)t Fx(\))g(b)s(elongs)h(to)f(the)h(class)g Fr(R)p Fx(.)43 b(Hence,)33 b(Theorem)d(2)h(yields)f(that,)h(for)236 1170 y(an)m(y)i(partial)e(F)-8 b(ourier)31 b(sum)1220 1460 y Ft(Q)1297 1475 y Fs(m)1364 1460 y Fx(\()p Ft(s)p Fx(\))c(=)1657 1336 y Fs(m)1617 1366 y Fp(X)1622 1575 y Fs(n)p Fv(=0)1777 1460 y Ft(\013)1839 1475 y Fs(n)1886 1460 y Ft(q)1929 1475 y Fs(n)1976 1460 y Fx(\()p Ft(s)p Fx(\))163 b(\()p Ft(\013)2361 1475 y Fs(n)2435 1460 y Fx(=)28 b(\()p Ft(p;)17 b(q)2713 1475 y Fs(n)2760 1460 y Fx(\))2798 1476 y Fs(L)2846 1485 y Fl(2)2880 1476 y Fv(\()p Fk(R)2955 1485 y Fl(+)3006 1476 y Fv(\))3038 1460 y Fx(\))p Ft(;)236 1768 y Fx(of)32 b(an)h(arbitrary)e(function)h Ft(p)p Fx(\()p Ft(s)p Fx(\))c Fr(2)g Ft(L)1638 1783 y Fv(2)1678 1768 y Fx(\()p Fq(R)1782 1783 y Fv(+)1847 1768 y Fx(\),)k(the)h(op)s(erator)1277 2022 y(~)1265 2047 y Ft(L)28 b Fx(=)g Fr(\000)1573 1979 y Ft(d)1624 1943 y Fv(2)p 1550 2024 137 4 v 1550 2115 a Ft(ds)1647 2086 y Fv(2)1696 2047 y Ft(f)11 b Fx(\()p Ft(s)p Fx(\))22 b(+)g Ft(Q)2074 2062 y Fs(m)2141 2047 y Fx(\()p Ft(s)p Fx(\))2280 1911 y Fp(Z)2379 1938 y Fo(1)2335 2137 y Fv(0)2470 2047 y Ft(Q)2547 2062 y Fs(m)2614 2047 y Fx(\()p Ft(\034)11 b Fx(\))p Ft(f)g Fx(\()p Ft(\034)g Fx(\))p Ft(d\034)6 b(;)236 2336 y(D)s Fx(\()370 2311 y(~)358 2336 y Ft(L)p Fx(\))28 b(=)f Fr(f)p Ft(f)11 b Fx(\()p Ft(s)p Fx(\))27 b Fr(2)i Ft(W)1052 2300 y Fv(2)1038 2361 y(2)1091 2336 y Fx(\()p Fq(R)1194 2351 y Fv(+)1260 2336 y Fx(\))j Fr(j)g Ft(f)11 b Fx(\(0\))27 b(=)h(0)p Fr(g)k Fx(is)g(a)g(Lax{Phillips)e(p)s (erturb)s(ed)k(op)s(erator.)383 2493 y(No)m(w,)f(w)m(e)g(presen)m(t)i (an)d(example)g(of)g Ft(p)p Fx(\()p Ft(s)p Fx(\),)h(for)f(whic)m(h)h (the)g(corresp)s(onding)f(op)s(erator)1491 2747 y(~)1479 2772 y Ft(L)c Fx(=)g Fr(\000)1787 2704 y Ft(d)1838 2668 y Fv(2)p 1764 2749 V 1764 2840 a Ft(ds)1861 2811 y Fv(2)1910 2772 y Ft(f)11 b Fx(\()p Ft(s)p Fx(\))22 b(+)g Ft(p)p Fx(\()p Ft(s)p Fx(\)\()p Ft(f)5 b(;)17 b(p)p Fx(\))2604 2787 y Fs(L)2652 2796 y Fl(2)2686 2787 y Fv(\()p Fk(R)2761 2796 y Fl(+)2812 2787 y Fv(\))236 2772 y Fx(\(24\))236 3028 y(is)32 b(not)g(a)g(Lax{Phillips)e(p)s(erturb)s(ed)i(op)s(erator)g (\(under)h(the)g(c)m(hoice)f(of)g(the)g(initial)d(op)s(erator)i Ft(B)38 b Fx(as)32 b Fr(B)s Fx(\).)236 3184 y(In)j(this)g(case,)h(as)f (follo)m(ws)f(from)f(the)i(pro)s(of)f(of)h(Theorem)g(2,)g(the)g (function)f Ft( )t Fx(\()p Ft(\016)t Fx(\))e(=)f([)p Fb(Y)r Ft(p)p Fx(]\()p Ft(\016)t Fx(\))k(m)m(ust)g(b)s(e)236 3341 y(a)i(cyclic)h(function)f(for)f Fb(G)1201 3305 y Fo(\003)1278 3341 y Fx(in)h Ft(H)1486 3305 y Fv(2)1478 3366 y(\005)1572 3341 y Fx(or,)i(whic)m(h)f(is)f(equiv)-5 b(alen)m(t,)38 b(the)g(function)f(\010)3238 3305 y Fo(\000)p Fv(1)3333 3341 y Fx([)p Ft( )3423 3356 y Fs(k)3466 3341 y Fx(\()p Ft(\016)t Fx(\)])h(m)m(ust)f(b)s(e)h(a)236 3497 y(cyclic)32 b(function)h(for)f Fr(S)1103 3461 y Fo(\003)1175 3497 y Fx(in)g Ft(H)1378 3461 y Fv(2)1417 3497 y Fx(,)h(where)h(the)f(transformation)d(\010)j(is)f(de\014ned)i(b) m(y)g(\(8\).)383 3654 y(F)-8 b(or)42 b(example,)k(the)d(function)g Ft(f)11 b Fx(\()p Ft(e)1704 3618 y Fs(i\022)1767 3654 y Fx(\))46 b(=)g(\(1)29 b Fr(\000)h Ft(e)2241 3618 y Fs(i\022)2304 3654 y Fx(\))2342 3618 y Fv(1)p Fs(=)p Fv(2)2495 3654 y Fx(is)43 b(cyclic)g(for)g Fr(S)3110 3618 y Fo(\003)3193 3654 y Fx(\([8,)j(p.)76 b(50]\).)f(Hence,)236 3810 y(it)46 b(follo)m(ws)g(from)g(the)h(de\014nition)f(of)g(\010,)51 b(equalit)m(y)c(\(9\),)j(and)d(relation)e(\(3\))i(in)f([15,)k(p.)87 b(133])93 b(\(for)236 3967 y Ft(\013)31 b Fx(=)g(1)p Ft(;)17 b(p)29 b Fx(=)i Fr(\000)p Ft(i\016)n(;)17 b(\027)37 b Fx(=)31 b(3)p Ft(=)p Fx(2\))i(that)h(the)h(function)e Ft(p)p Fx(\()p Ft(s)p Fx(\))d(=)h Fb(Y)2459 3931 y Fo(\000)p Fv(1)2554 3967 y Fx([\010)p Ft(f)11 b Fx(\()p Ft(e)2793 3931 y Fs(i\022)2856 3967 y Fx(\)])31 b(=)3058 3895 y Fr(p)p 3141 3895 46 4 v 72 x Ft(se)3232 3931 y Fo(\000)p Fs(s)3358 3967 y Fx(do)s(es)j(not)g(de\014ne)i(a)236 4123 y(Lax{Phillips)30 b(p)s(erturb)s(ed)k(op)s(erator)d(in)h(\(24\).) 383 4280 y(Let)j(us)g(consider)h(the)f(inner)g(function)f Ft(g)t Fx(\()p Ft(\025)p Fx(\))d(=)h([\()p Ft(\025)24 b Fr(\000)g Ft(i)p Fx(\)\()p Ft(\025)g Fx(+)f Ft(i)p Fx(\))2806 4244 y Fo(\000)p Fv(1)2901 4280 y Fx(])2928 4244 y Fs(m)3065 4280 y Fx(\()p Ft(m)32 b Fr(2)g Fq(N)9 b Fx(\))41 b(and,)36 b(b)m(y)g(means)236 4436 y(of)c(\(11\),)g (de\014ne)i(the)f(corresp)s(onding)g(isometric)d(op)s(erator)1589 4692 y Ft(A)e Fx(=)f([\()p Fr(B)f(\000)c Ft(iI)8 b Fx(\)\()p Fr(B)26 b Fx(+)c Ft(iI)8 b Fx(\))2519 4651 y Fo(\000)p Fv(1)2613 4692 y Fx(])2640 4651 y Fs(m)2707 4692 y Ft(:)-2498 b Fx(\(25\))236 4949 y(Using)29 b(the)g(w)m(ell-kno)m(wn)h(\([16,)f(p.) 42 b(365]\))29 b(relation)e Ft(Aq)2228 4964 y Fs(n)2275 4949 y Fx(\(2)p Ft(s)p Fx(\))h(=)f Ft(q)2620 4964 y Fs(n)p Fv(+)p Fs(m)2785 4949 y Fx(\(2)p Ft(s)p Fx(\))h(for)h(the)g(Laguerre)h (functions)236 5105 y Ft(q)279 5120 y Fs(n)326 5105 y Fx(\(2)p Ft(s)p Fx(\))65 b(\()p Ft(n)28 b Fr(2)g Fq(N)37 b Fr([)23 b(f)p Fx(0)p Fr(g)p Fx(\))p Ft(;)32 b Fx(w)m(e)h(get)g(that) 1161 5361 y Ft(<)28 b(q)1308 5376 y Fv(0)1347 5361 y Fx(\(2)p Ft(s)p Fx(\))p Ft(;)17 b(:)g(:)g(:)f(;)h(q)1780 5376 y Fs(m)p Fo(\000)p Fv(1)1937 5361 y Fx(\(2)p Ft(s)p Fx(\))27 b Ft(>)p Fx(=)g Ft(L)2380 5376 y Fv(2)2420 5361 y Fx(\()p Fq(R)2524 5376 y Fv(+)2589 5361 y Fx(\))22 b Fr(\011)h Ft(AL)2888 5376 y Fv(2)2928 5361 y Fx(\()p Fq(R)3032 5376 y Fv(+)3097 5361 y Fx(\))o Ft(:)236 5617 y Fx(It)g(follo)m(ws)f(from)g(the)h(last)f(relation)f(and)i(the)h (de\014nition)e(of)g(the)i(Laguerre)f(functions)g(that)g(an)g (arbitrary)236 5774 y(function)38 b Ft(p)p Fx(\()p Ft(s)p Fx(\))g(=)h Ft(e)993 5738 y Fo(\000)p Fs(s)1084 5774 y Ft(P)1147 5789 y Fs(m)p Fo(\000)p Fv(1)1304 5774 y Fx(\()p Ft(s)p Fx(\),)h(where)g Ft(P)1844 5789 y Fs(m)p Fo(\000)p Fv(1)2001 5774 y Fx(\()p Ft(s)p Fx(\))e(is)h(a)f(real)g(p)s (olynomial)d(of)j(order)h Ft(m)27 b Fr(\000)g Fx(1,)40 b(b)s(elongs)2113 6037 y(12)p eop %%Page: 13 14 13 13 bop 236 454 a Fx(to)35 b(the)h(subspace)h Ft(L)1005 469 y Fv(2)1044 454 y Fx(\()p Fq(R)1148 469 y Fv(+)1213 454 y Fx(\))24 b Fr(\011)g Ft(AL)1515 469 y Fv(2)1555 454 y Fx(\()p Fq(R)1659 469 y Fv(+)1724 454 y Fx(\).)51 b(Therefore,)37 b(if)d(w)m(e)i(c)m(ho)s(ose)g Ft(A)f Fx(in)g(the)g(form)f(\(25\),)h(then)h(the)236 611 y(op)s(erator)846 759 y(~)834 785 y Ft(L)28 b Fx(=)g Fr(\000)1142 717 y Ft(d)1193 681 y Fv(2)p 1119 762 137 4 v 1119 853 a Ft(ds)1216 824 y Fv(2)1265 785 y Ft(f)11 b Fx(\()p Ft(s)p Fx(\))22 b(+)g Ft(e)1611 743 y Fo(\000)p Fs(s)1703 785 y Ft(P)1766 800 y Fs(m)p Fo(\000)p Fv(1)1922 785 y Fx(\()p Ft(s)p Fx(\))2061 649 y Fp(Z)2160 675 y Fo(1)2116 875 y Fv(0)2252 785 y Ft(e)2297 743 y Fo(\000)p Fs(\034)2395 785 y Ft(P)2458 800 y Fs(m)p Fo(\000)p Fv(1)2615 785 y Fx(\()p Ft(\034)11 b Fx(\))p Ft(f)g Fx(\()p Ft(\034)g Fx(\))p Ft(d\034)109 b Fx(\()p Ft(m)28 b Fr(2)g Fq(N)9 b Fx(\))-3247 b(\(26\))236 1021 y(is)32 b(an)h(extension)g(of)f Ft(B)1086 1036 y Fs(A)1143 980 y Fv(2)1183 1021 y Fx(,)g(where)i Ft(B)1598 1036 y Fs(A)1688 1021 y Fx(is)e(de\014ned)i(b)m(y)f(\(12\))f(\(for)g Ft(B)h Fx(=)28 b Fr(B)s Fx(\).)383 1178 y(W)-8 b(e)38 b(note)g(that)g(condition)f(\(18\))g(is)h(equiv)-5 b(alen)m(t)38 b(to)g(the)g(condition)f(of)h(quasi-equiv)-5 b(alence)37 b(of)h(the)236 1334 y(subspaces)i Ft(D)768 1349 y Fo(\006)864 1334 y Fx(and)e Ft(D)1143 1298 y Fs(A)1140 1359 y Fo(\006)1237 1334 y Fx(prop)s(osed)g(in)f([11].)57 b(Therefore,)40 b(using)d(Prop)s(osition)f(2)h(in)g([11],)h(it)f(is)g(easy)236 1491 y(to)31 b(v)m(erify)h(that)g(equalit)m(y)f(\(18\))g(is)g(true)h (if)e(w)m(e)j(c)m(ho)s(ose)f Ft(A)g Fx(as)f(\(25\).)43 b(Th)m(us,)33 b(w)m(e)g(can)f(use)g(Theorem)g(3)f(for)236 1647 y(the)i(determination)e(of)h(the)h(scattering)f(matrix)f Fh(S)2162 1674 y Fv(\()2197 1658 y(~)2189 1674 y Fs(L;L)2305 1683 y Fl(0)2340 1674 y Fv(\))2371 1647 y Fx(\()p Ft(\016)t Fx(\),)i(where)2847 1622 y(~)2835 1647 y Ft(L)g Fx(is)f(de\014ned)i(b)m (y)g(\(26\).)383 1804 y(Let)k(for)g(de\014niteness)j Ft(m)d Fx(=)f(2.)62 b(Then)39 b(the)g(p)s(olynomial)c Ft(P)2627 1819 y Fv(1)2667 1804 y Fx(\()p Ft(s)p Fx(\))j(in)g(\(26\))g (has)g(the)h(form)f Ft(P)3812 1819 y Fv(1)3851 1804 y Fx(\()p Ft(s)p Fx(\))g(=)236 1960 y Ft(p)28 b Fx(+)g Ft(p)466 1975 y Fv(1)505 1960 y Ft(s)82 b Fr(f)p Ft(p;)17 b(p)825 1975 y Fv(1)864 1960 y Fr(g)42 b(2)g Fq(R)5 b Fx(.)75 b(Using)40 b(the)i(algorithm)37 b(for)k(the)g(construction)g (of)g(Heisen)m(b)s(erg)h(scattering)236 2117 y(matrices)32 b(prop)s(osed)h(in)f([7,)g(13],)h(w)m(e)g(obtain)572 2396 y Fh(S)655 2423 y Fv(\()690 2406 y(~)682 2423 y Fs(L;L)798 2434 y Fi(A)848 2423 y Fv(\))879 2396 y Fx(\()p Ft(\016)t Fx(\))28 b(=)1145 2329 y(4\()p Ft(\016)e Fx(+)c Ft(i)p Fx(\))1470 2292 y Fv(4)1531 2329 y Fr(\000)h Fx(\()p Ft(p)f Fx(+)g Ft(p)1887 2344 y Fv(1)1926 2329 y Fx(\))1964 2292 y Fv(2)2004 2329 y Fx(\()p Ft(\016)k Fx(+)c(2)p Ft(i)p Fx(\))2329 2292 y Fv(2)2390 2329 y Fr(\000)h Ft(p)2539 2292 y Fv(2)2578 2329 y Fx(\()p Ft(\016)2663 2292 y Fv(2)2725 2329 y Fx(+)f(1\))g(+)g(2)p Ft(p)3128 2292 y Fv(2)3128 2353 y(1)3189 2329 y Fr(\000)h Fx(\()p Ft(p)f Fx(+)g Ft(p)3545 2344 y Fv(1)3584 2329 y Fx(\))3622 2292 y Fv(2)p 1143 2373 2520 4 v 1143 2464 a Fx(4\()p Ft(\016)k Fr(\000)d Ft(i)p Fx(\))1470 2436 y Fv(4)1531 2464 y Fr(\000)g Fx(\()p Ft(p)f Fx(+)g Ft(p)1887 2479 y Fv(1)1926 2464 y Fx(\))1964 2436 y Fv(2)2004 2464 y Fx(\()p Ft(\016)k Fr(\000)c Fx(2)p Ft(i)p Fx(\))2330 2436 y Fv(2)2392 2464 y Fr(\000)h Ft(p)2541 2436 y Fv(2)2580 2464 y Fx(\()p Ft(\016)2665 2436 y Fv(2)2726 2464 y Fx(+)g(1\))e(+)h(2)p Ft(p)3129 2430 y Fv(2)3129 2489 y(1)3191 2464 y Fr(\000)g Fx(\()p Ft(p)g Fx(+)g Ft(p)3546 2479 y Fv(1)3586 2464 y Fx(\))3624 2436 y Fv(2)3673 2396 y Ft(I)8 b(:)236 2672 y Fx(Therefore,)34 b(b)m(y)f(virtue)g(of)f (Theorem)h(3)f(and)h(relation)e(\(25\),)h(w)m(e)h(get)1491 2951 y Fh(S)1574 2978 y Fv(\()1609 2961 y(~)1601 2978 y Fs(L;L)1717 2987 y Fl(0)1751 2978 y Fv(\))1783 2951 y Fx(\()p Ft(\016)t Fx(\))28 b(=)2047 2884 y(\()p Ft(\016)e Fr(\000)d Ft(i)p Fx(\))2325 2848 y Fv(4)p 2047 2928 318 4 v 2048 3019 a Fx(\()p Ft(\016)j Fx(+)c Ft(i)p Fx(\))2324 2991 y Fv(4)2374 2951 y Fh(S)2457 2978 y Fv(\()2492 2961 y(~)2484 2978 y Fs(L;L)2600 2989 y Fi(A)2650 2978 y Fv(\))2682 2951 y Fx(\()p Ft(\016)t Fx(\))p Ft(:)2113 6037 y Fx(13)p eop %%Page: 14 15 14 14 bop 355 454 a Fx(1.)49 b(Reed,)c(M.)d(and)g(Simon)e(B.,)k Fj(Metho)-5 b(ds)43 b(of)g(mo)-5 b(dern)43 b(mathematic)-5 b(al)42 b(physics.)69 b(III:)42 b(Sc)-5 b(attering)480 611 y(the)g(ory)p Fx(,)33 b(Academic)f(Press,)i(New)g(Y)-8 b(ork,)33 b(1979.)355 846 y(2.)49 b(Phillips)34 b(R.,)j Fj(Sc)-5 b(attering)39 b(the)-5 b(ory)38 b(for)g(the)h(wave)f(e)-5 b(quation)38 b(with)g(a)g(short)h(r)-5 b(ange)37 b(p)-5 b(erturb)g(ation)p Fx(,)480 1003 y(Indiana)32 b(Univ.)43 b(Math.)h(Journ.,)33 b Fw(31)g Fx(\(1982\),)e(no.)44 b(5,)32 b(602{639.)355 1238 y(3.)49 b(Phillips)34 b(R.,)j Fj(Sc)-5 b(attering)38 b(the)-5 b(ory)38 b(for)g(the)g(wave)g(e)-5 b(quation)38 b(with)g(a)g(short)g(r)-5 b(ange)38 b(p)-5 b(erturb)g(ation.)480 1395 y(II)p Fx(,)32 b(Indiana)g(Univ.)43 b(Math.)h(Journ.,)33 b Fw(33)f Fx(\(1984\),)g(no.)43 b(6,)33 b(832{846.)355 1630 y(4.)49 b(P)m(a)m(vlo)m(v,)32 b(B.)f(S.,)h Fj(Partial)i(sc)-5 b(attering)33 b(matrix)g(its)g (factorization)g(and)g(analyticity)p Fx(,)f(Dokl.)41 b(Ak)-5 b(ad.)480 1787 y(Nauk)33 b(SSSR,)g Fw(244)g Fx(\(1979\),)e(no.) 44 b(2,)32 b(291{295.)42 b(\(Russian\))355 2023 y(5.)49 b(Lax,)33 b(P)-8 b(.)33 b(and)f(Phillips,)f(R.,)h Fj(Sc)-5 b(attering)35 b(the)-5 b(ory)p Fx(,)33 b(Academic)f(Press,)i(New)f(Y)-8 b(ork,)33 b(1967.)355 2258 y(6.)49 b(Kuzhel,)39 b(S.)e(A.,)i Fj(On)f(the)h(determination)g(of)f(fr)-5 b(e)g(e)39 b(evolution)g(in)g (the)g(L)-5 b(ax{Phil)5 b(lips)38 b(sc)-5 b(attering)480 2415 y(scheme)29 b(for)g(se)-5 b(c)g(ond-or)g(der)28 b(op)-5 b(er)g(ator-di\013er)g(ential)29 b(e)-5 b(quations)p Fx(,)27 b(Mat.)42 b(Zametku)27 b Fw(68)f Fx(\(2000\),)i(no.)480 2571 y(6,)k(854{861.)42 b(\(Russian\))355 2807 y(7.)49 b(Kuzhel,)36 b(S.)f(A.,)h Fj(On)g(the)h(form)g(of)f(sc)-5 b(attering)37 b(matrix)g(for)g Ft(\032)p Fj(-p)-5 b(erturb)g(ations)37 b(of)f(abstr)-5 b(act)37 b(wave)480 2963 y(e)-5 b(quation)p Fx(,)32 b(Ukr.)44 b(Mat.)g(Zh.)f Fw(51)33 b Fx(\(1999\),)e(no.)44 b(4,)32 b(445{457.)42 b(\(Ukrainian\))355 3199 y(8.)49 b(Nik)m(ol'skii,)31 b(N.)i(K.,)f Fj(T)-7 b(r)i(e)g(atise)34 b(on)g(the)h(shift)g(op)-5 b(er)g(ator)p Fx(,)32 b(Nauk)-5 b(a,)33 b(Mosco)m(w,)h(1980.)43 b(\(Russian\))355 3435 y(9.)49 b(Plesner,)g(A.)c(I.,)k Fj(F)-7 b(unctions)45 b(of)h(maximal)f(op)-5 b(er)g(ator,)47 b Fx(Dokl.)80 b(Ak)-5 b(ad.)81 b(Nauk)45 b(SSSR,)h Fw(XXI)s(I)s(I)480 3591 y Fx(\(1939\),)32 b(no.)43 b(4,)32 b(327{330.)42 b(\(Russian\))307 3827 y(10.)48 b(Plesner,)30 b(A.)f(I.,)h Fj(On)h(semiunitary)g(op)-5 b(er)g(ators,)29 b Fx(Dokl.)41 b(Ak)-5 b(ad.)43 b(Nauk)29 b(SSSR,)g Fw(XXV)f Fx(\(1939\),)h(no.)480 3983 y(9,)j(708{710.)42 b(\(Russian\))307 4219 y(11.)48 b(Kuzhel,)29 b(S.)e(A.,)i Fj(On)h(dep)-5 b(endenc)g(e)28 b(of)i(L)-5 b(ax{Phil)5 b(lips)29 b(sc)-5 b(attering)29 b(matrix)h(on)g(choic)-5 b(e)29 b(of)h(inc)-5 b(oming)480 4375 y(and)34 b(outgoing)h(subsp)-5 b(ac)g(es,)31 b Fx(Meth.)45 b(F)-8 b(unc.)43 b(Anal.)g(and)33 b(T)-8 b(op)s(ology)g(,)31 b Fw(7)i Fx(\(2001\),)e(no.)44 b(1,)32 b(45{53.)307 4611 y(12.)48 b(Garnett,)32 b(J.,)h Fj(Bounde)-5 b(d)34 b(analytic)h (functions)p Fx(,)d(Academic)g(Press,)i(New)g(Y)-8 b(ork,)32 b(1981.)307 4847 y(13.)48 b(Kuzhel,)38 b(S.)f(A.,)g Fj(On)i(inverse)e (pr)-5 b(oblem)38 b(in)g(the)h(L)-5 b(ax{Phil)5 b(lips)37 b(sc)-5 b(attering)38 b(scheme)g(for)g(a)g(class)480 5003 y(of)d(op)-5 b(er)g(ator-di\013er)g(ential)33 b(e)-5 b(quations)p Fx(,)32 b(Algebra)g(Analiz)f Fw(13)i Fx(\(2001\),)e(no.)44 b(1,)32 b(60{83.)43 b(\(Russian\))307 5239 y(14.)48 b(Kuzhel,)31 b(A.)f(and)g(Kuzhel,)g(S.,)h Fj(R)-5 b(e)g(gular)32 b(extensions)f(of)i (Hermitian)f(op)-5 b(er)g(ators,)30 b Fx(VSP)-8 b(,)30 b(Utrec)m(h)m(t,)480 5395 y(1998.)307 5631 y(15.)48 b(Bateman,)41 b(H.)e(and)h(Erdelyi,)g(A.,)i Fj(T)-7 b(ables)39 b(of)i(inte)-5 b(gr)g(al)41 b(tr)-5 b(ansforms.)62 b(I.)p Fx(,)41 b(McGra)m(w-Hill)c (Bo)s(ok)480 5787 y(Compan)m(y)-8 b(,)33 b(New)g(Y)-8 b(ork,)33 b(1954.)2113 6037 y(14)p eop %%Page: 15 16 15 15 bop 307 454 a Fx(16.)48 b(Akhiezer,)26 b(N.)d(I.)h(and)f (Glazman,)g(I.)g(M.,)j Fj(The)-5 b(ory)26 b(of)g(line)-5 b(ar)25 b(op)-5 b(er)g(ators)26 b(in)g(Hilb)-5 b(ert)27 b(sp)-5 b(ac)g(e,)24 b Fx(Nauk)-5 b(a,)480 611 y(Mosco)m(w,)34 b(1966.)43 b(\(Russian\))2113 6037 y(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0308130325702--