Content-Type: multipart/mixed; boundary="-------------0512230156223" This is a multi-part message in MIME format. ---------------0512230156223 Content-Type: text/plain; name="05-437.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-437.keywords" Feshbach decomposition, optical potential, Lax-Phillips scattering theory, dissipative scattering theory, scattering matrix, characteristic function, dissipative operators ---------------0512230156223 Content-Type: application/postscript; name="block-preprint.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="block-preprint.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: block-preprint.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Myriad-CnSemibold %%EndComments %DVIPSCommandLine: /usr/local/bin/dvips -o block-preprint.ps %+ block-preprint.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.12.06:1559 %%BeginProcSet: texc.pro %! 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FG(\))28 b(=)1853 2237 y Fp(\022)1927 2317 y Fx(S)1987 2332 y Fw(11)2144 2317 y Fx(S)2204 2332 y Fw(12)1927 2437 y Fx(S)1987 2452 y Fw(21)2144 2437 y Fx(S)2204 2452 y Fw(22)2279 2237 y Fp(\023)2369 2378 y Fx(:)1055 b FG(\(1.3\))109 2655 y(In)27 b(ph)m(ysics)h(the)f(decomp)s(osition)e(\(1.2\))i(is)f(often)h (motiv)-5 b(ated)25 b(either)i(b)m(y)g(the)g(exclusiv)m(e)h(in)m (terest)109 2775 y(to)37 b(scattering)h(data)g(in)f(a)g(certain)h(c)m (hannel)g(or)g(b)m(y)h(the)f(limited)d(measuring)i(pro)s(cess)i(whic)m (h)109 2896 y(allo)m(ws)45 b(to)h(measure)g(the)h(scattering)f(data)f (only)h(of)g(a)g(certain)g(c)m(hannel,)k(sa)m(y)d Fv(H)3260 2911 y Fw(1)3299 2896 y FG(.)84 b(Th)m(us,)109 3016 y(let)40 b(us)h(assume)g(that)g(only)f(the)h(c)m(hannel)h(scattering)e(op)s (erator)g Fx(S)2637 3031 y Fw(11)2753 3016 y FG(:)i Fv(H)2894 2980 y Fu(ac)2894 3041 y Fw(1)3008 3016 y Ft(\000)-16 b(!)41 b Fv(H)3282 2980 y Fu(ac)3282 3041 y Fw(1)3394 3016 y FG(in)f(the)109 3136 y(scattering)35 b(c)m(hannel)i Fv(H)993 3151 y Fw(1)1068 3136 y FG(is)e(kno)m(wn.)55 b(This)36 b(rises)h(the)f(follo)m(wing)d(problem:)49 b(Is)37 b(it)e(p)s(ossible)g(to)109 3257 y(replace)28 b(the)h(full)e(Hamiltonian)e Fx(L)k FG(b)m(y)g(an)g(e\013ectiv)m(e)g (one)g Fx(H)36 b FG(acting)28 b(only)g(in)f Fv(H)3025 3272 y Fw(1)3093 3257 y FG(suc)m(h)j(that)e(the)109 3377 y(scattering)34 b(op)s(erator)f(of)h(the)g(scattering)g(system)h Ft(f)p Fx(H)r(;)17 b(H)2272 3392 y Fw(1)2311 3377 y Ft(g)34 b FG(coincides)g(with)f Fx(S)3092 3392 y Fw(11)3167 3377 y FG(?)48 b(Since)34 b Fx(S)3577 3392 y Fw(11)109 3498 y FG(is)i(a)g(con)m(traction,)h(in)f(general,)h(this)f(implies)f(that)h (either)g(the)h(scattering)g(system)g Ft(f)p Fx(H)r(;)17 b(H)3563 3513 y Fw(1)3602 3498 y Ft(g)109 3618 y FG(cannot)32 b(b)s(e)h(complete)f(or)g Fx(H)40 b FG(is)32 b(not)h(self-adjoin)m(t.) 109 3786 y(The)39 b(problem)f(has)h(a)f(solution)f(within)h(the)h(scop) s(e)g(of)f(dissipativ)m(e)g(scattering)h(systems)h(de-)109 3906 y(v)m(elop)s(ed)34 b(in)f([18)o(,)h(19,)g(20)o(])g(for)f(pairs)g Ft(f)p Fx(H)r(;)17 b(H)1730 3921 y Fw(1)1769 3906 y Ft(g)34 b FG(of)f(dissipativ)m(e)g(and)h(self-adjoin)m(t)e(op)s(erators)i(in) 109 4026 y(some)40 b(separable)h(Hilb)s(ert)f(space.)70 b(F)-8 b(or)40 b(suc)m(h)i(pairs)e(the)i(w)m(a)m(v)m(e)h(op)s(erators)d Fx(W)3094 3990 y Fu(D)3080 4051 y Fs(\006)3158 4026 y FG(\()p Fx(H)r(;)17 b(H)3404 4041 y Fw(1)3443 4026 y FG(\))41 b(are)109 4147 y(de\014ned)34 b(b)m(y)957 4267 y Fx(W)1063 4226 y Fu(D)1049 4292 y Fw(+)1127 4267 y FG(\()p Fx(H)r(;)17 b(H)1373 4282 y Fw(1)1412 4267 y FG(\))28 b(:=)f Fx(s)22 b Ft(\000)66 b FG(lim)1776 4327 y Fu(t)p Fs(!)p Fw(+)p Fs(1)2014 4267 y Fx(e)2059 4226 y Fu(itH)2171 4202 y Fn(\003)2212 4267 y Fx(e)2257 4226 y Fs(\000)p Fu(itH)2419 4235 y Fr(1)2458 4267 y Fx(P)2535 4226 y Fu(ac)2607 4267 y FG(\()p Fx(H)2726 4282 y Fw(1)2765 4267 y FG(\))109 4475 y(and)962 4595 y Fx(W)1068 4554 y Fu(D)1054 4620 y Fs(\000)1132 4595 y FG(\()p Fx(H)r(;)17 b(H)1378 4610 y Fw(1)1417 4595 y FG(\))27 b(:=)h Fx(s)22 b Ft(\000)66 b FG(lim)1780 4655 y Fu(t)p Fs(!)p Fw(+)p Fs(1)2019 4595 y Fx(e)2064 4554 y Fs(\000)p Fu(itH)2235 4595 y Fx(e)2280 4554 y Fu(itH)2387 4563 y Fr(1)2427 4595 y Fx(P)2504 4554 y Fu(ac)2576 4595 y FG(\()p Fx(H)2695 4610 y Fw(1)2734 4595 y FG(\))p Fx(;)109 4808 y FG(and)48 b(the)h(notion)e(of)h(completeness)h(is)f(generalized,)k(cf.)91 b([18,)48 b(19,)g(20].)91 b(The)49 b(scattering)109 4929 y(op)s(erator)32 b(of)g(a)g(dissipativ)m(e)g(scattering)g(system)i Ft(f)p Fx(H)r(;)17 b(H)2204 4944 y Fw(1)2243 4929 y Ft(g)32 b FG(is)g(de\014ned)i(b)m(y)1213 5148 y Fx(S)1273 5163 y Fu(D)1365 5148 y FG(:=)28 b Fx(W)1602 5107 y Fu(D)1588 5173 y Fw(+)1665 5148 y FG(\()p Fx(H)r(;)17 b(H)1911 5163 y Fw(1)1950 5148 y FG(\))1988 5107 y Fs(\003)2028 5148 y Fx(W)2134 5107 y Fu(D)2120 5173 y Fs(\000)2197 5148 y FG(\()p Fx(H)r(;)g(H)2443 5163 y Fw(1)2482 5148 y FG(\))p Fx(:)109 5367 y FG(It)34 b(turns)i(out)e(that)g Fx(S)922 5382 y Fu(D)1021 5367 y FG(is)g(a)g(con)m(traction)h(acting)e (on)i(the)g(absolutely)f(con)m(tin)m(uous)h(subspace)109 5487 y Fv(H)181 5451 y Fu(ac)181 5512 y Fw(1)278 5487 y FG(of)25 b Fx(H)463 5502 y Fw(1)528 5487 y FG(whic)m(h)i(comm)m(utes) e(with)g Fx(H)1548 5502 y Fw(1)1588 5487 y FG(.)41 b(In)26 b([17)o(,)g(18])f(it)g(w)m(as)i(sho)m(wn)g(that)e(for)g(an)m(y)i (self-adjoin)m(t)p Black 1856 5728 a(2)p Black eop %%Page: 3 5 3 4 bop Black Black 109 171 a FG(op)s(erator)47 b Fx(H)598 186 y Fw(1)685 171 y FG(in)g Fv(H)886 186 y Fw(1)972 171 y FG(and)h(an)m(y)h(con)m(traction)e Fx(S)1964 186 y Fu(D)2076 171 y FG(acting)f(on)i(the)g(absolutely)f(con)m(tin)m(uous) 109 291 y(subspace)c Fv(H)597 255 y Fu(ac)597 316 y Fw(1)710 291 y FG(and)e(comm)m(uting)e(with)i Fx(H)1735 306 y Fw(1)1815 291 y FG(there)h(is)e(a)h(maximal)d(dissipativ)m(e)i(op)s (erator)g Fx(H)109 411 y FG(on)45 b Fv(H)329 426 y Fw(1)413 411 y FG(suc)m(h)i(that)e Ft(f)p Fx(H)r(;)17 b(H)1128 426 y Fw(1)1167 411 y Ft(g)45 b FG(p)s(erforms)f(a)h(complete)g (scattering)g(system)h(with)f(scattering)109 532 y(op)s(erator)32 b(giv)m(en)g(b)m(y)i Fx(S)952 547 y Fu(D)1016 532 y FG(.)43 b(In)33 b(particular,)e(this)h(holds)g(for)g(the)h(self-adjoin)m(t)e (op)s(erator)h Fx(H)3423 547 y Fw(1)3495 532 y FG(and)109 652 y(the)42 b(c)m(hannel)h(scattering)f(op)s(erator)g Fx(S)1574 667 y Fw(11)1649 652 y FG(.)73 b(That)42 b(means,)j(there)e (is)f(a)g(maximal)d(dissipativ)m(e)109 772 y(op)s(erator)g Fx(H)48 b FG(on)40 b Fv(H)853 787 y Fw(1)933 772 y FG(suc)m(h)i(that)e (the)h(c)m(hannel)g(scattering)f(op)s(erator)f Fx(S)2837 787 y Fw(11)2952 772 y FG(is)h(the)h(scattering)109 893 y(op)s(erator)k(of)h(the)g(complete)f(dissipativ)m(e)h(scattering)f (system)i Ft(f)p Fx(H)r(;)17 b(H)2796 908 y Fw(1)2835 893 y Ft(g)p FG(.)84 b(Hence,)51 b(roughly)109 1013 y(sp)s(eaking,)25 b(the)f(scattering)g(op)s(erator)e Fx(S)1574 1028 y Fw(11)1673 1013 y FG(can)i(b)s(e)f(alw)m(a)m(ys)i(view)m(ed)f(as)g(the)g (scattering)g(op)s(erator)109 1134 y(of)36 b(a)g(suitable)g(c)m(hosen)i (dissipativ)m(e)e(scattering)g(system)i(on)e Fv(H)2468 1149 y Fw(1)2508 1134 y FG(.)55 b(The)38 b(disadv)-5 b(an)m(tage)36 b(of)g(this)109 1254 y(fact)c(is)g(that)h Fx(H)40 b FG(is)32 b(not)g(kno)m(wn)i(explicitly)-8 b(.)109 1422 y(Another)38 b(approac)m(h)g(to)f(this)g(problem)g(w)m(as)h (suggested)i(b)m(y)e(F)-8 b(esh)m(bac)m(h)39 b(in)e([10,)g(11],)i(see)g (also)109 1542 y([6,)24 b(9].)40 b(He)25 b(prop)s(oses)g(a)f(concrete)i (dissipativ)m(e)d(p)s(erturbation)h Fx(V)2451 1557 y Fw(1)2514 1542 y FG(of)g(the)h(c)m(hannel)g(Hamiltonian)109 1662 y Fx(H)190 1677 y Fw(1)229 1662 y FG(,)j(called)d(\\optical)g(p)s (oten)m(tial",)h(suc)m(h)i(that)f(the)g(scattering)f(op)s(erator)g Fx(S)2873 1677 y Fw(1)2939 1662 y FG(of)g(the)h(dissipativ)m(e)109 1783 y(scattering)21 b(system)i Ft(f)p Fx(H)992 1798 y Fw(1)1031 1783 y FG(+)p Fx(V)1164 1798 y Fw(1)1203 1783 y Fx(;)17 b(H)1328 1798 y Fw(1)1367 1783 y Ft(g)22 b FG(appro)m(ximates)f Fx(S)2090 1798 y Fw(11)2187 1783 y FG(with)g(a)h(certain)f(accuracy)-8 b(.)41 b(T)-8 b(o)22 b(explain)109 1903 y(this)34 b(approac)m(h)g(in)g(more)g(detail)e(let)i (us)h(assume)g(that)f(the)h(full)d(Hamiltonian)f Fx(L)k FG(is)e(obtained)109 2023 y(from)e Fx(L)405 2038 y Fw(0)477 2023 y FG(b)m(y)j(an)e(additiv)m(e)g(p)s(erturbation,)g Fx(L)c FG(=)g Fx(L)1985 2038 y Fw(0)2047 2023 y FG(+)22 b Fx(V)f FG(,)33 b(where)h Fx(V)54 b FG(is)32 b(giv)m(en)g(b)m(y)1193 2348 y Fx(V)49 b FG(=)1403 2208 y Fp(\022)1510 2287 y FG(0)117 b Fx(G)1476 2408 y(G)1553 2372 y Fs(\003)1690 2408 y FG(0)1753 2208 y Fp(\023)1854 2348 y FG(:)1950 2227 y Fv(H)2022 2242 y Fw(1)1967 2348 y Ft(\010)1950 2468 y Fv(H)2022 2483 y Fw(2)2130 2348 y Ft(\000)-16 b(!)2360 2227 y Fv(H)2432 2242 y Fw(1)2377 2348 y Ft(\010)2360 2468 y Fv(H)2432 2483 y Fw(2)2540 2348 y Fx(:)884 b FG(\(1.4\))109 2673 y(In)m(tro)s(ducing)32 b(the)h(\\optical)e(p)s(oten)m(tial")1005 2885 y Fx(V)1062 2900 y Fw(1)1102 2885 y FG(\()p Fx(\025)p FG(\))c(:=)h Ft(\000)p Fx(G)p FG(\()p Fx(H)1666 2900 y Fw(2)1727 2885 y Ft(\000)23 b Fx(\025)f Ft(\000)h Fx(i)p FG(0\))2126 2844 y Fs(\000)p Fw(1)2220 2885 y Fx(G)2297 2844 y Fs(\003)2336 2885 y Fx(;)115 b(\025)27 b Ft(2)h Fm(R)5 b Fx(;)702 b FG(\(1.5\))109 3098 y(it)33 b(w)m(as)i(sho)m(wn)h (in)d([8,)i(Theorem)f(4.4.4])g(that)g(under)h(strong)g(assumptions)f (indeed)g(the)h(scat-)109 3218 y(tering)c(op)s(erator)h Fx(S)843 3233 y Fw(1)883 3218 y FG([)p Fx(\025)p FG(])g(of)h(the)g (\(in)e(general)h(dissipativ)m(e\))g(scattering)h(system)g Ft(f)p Fx(H)3214 3233 y Fw(1)3253 3218 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)3511 3233 y Fw(1)3550 3218 y Ft(g)p FG(,)1230 3430 y Fx(H)1311 3445 y Fw(1)1351 3430 y FG(\()p Fx(\025)p FG(\))27 b(:=)h Fx(H)1723 3445 y Fw(1)1784 3430 y FG(+)22 b Fx(V)1939 3445 y Fw(1)1978 3430 y FG(\()p Fx(\025)p FG(\))p Fx(;)115 b(\025)27 b Ft(2)h Fm(R)5 b Fx(;)927 b FG(\(1.6\))109 3642 y(coincides)48 b(with)g(the)g (scattering)g(op)s(erator)g Fx(S)1893 3657 y Fw(11)2016 3642 y FG(with)g(an)g(error)g(of)g(second)i(order)e(in)f(the)109 3763 y(coupling)31 b(constan)m(t.)109 3930 y(W)-8 b(e)32 b(sho)m(w)g(that)g(F)-8 b(esh)m(bac)m(h's)34 b(prop)s(osal)c(can)i(b)s (e)g(made)f(precise)h(in)f(another)h(sense.)45 b(Note)31 b(\014rst)109 4051 y(that)i(the)h(decomp)s(osition)e(\(1.2\))h(leads)g (not)h(only)f(to)g(the)h(decomp)s(osition)e(\(1.3\))h(of)g(the)h(scat-) 109 4171 y(tering)d(op)s(erator)h Fx(S)39 b FG(but)33 b(also)e(to)h(a)h(decomp)s(osition)d(of)j(the)g(scattering)f(matrix)f Ft(f)p Fx(S)6 b FG(\()p Fx(\026)p FG(\))p Ft(g)3447 4186 y Fu(\026)p Fs(2)p Fq(R)3588 4171 y FG(,)1295 4444 y Fx(S)g FG(\()p Fx(\026)p FG(\))27 b(:=)1654 4304 y Fp(\022)1727 4383 y Fx(S)1787 4398 y Fw(11)1862 4383 y FG(\()p Fx(\026)p FG(\))82 b Fx(S)2139 4398 y Fw(12)2214 4383 y FG(\()p Fx(\026)p FG(\))1727 4504 y Fx(S)1787 4519 y Fw(21)1862 4504 y FG(\()p Fx(\026)p FG(\))g Fx(S)2139 4519 y Fw(22)2214 4504 y FG(\()p Fx(\026)p FG(\))2349 4304 y Fp(\023)2439 4444 y Fx(;)109 4715 y FG(where)42 b Ft(f)p Fx(S)509 4730 y Fu(ij)569 4715 y FG(\()p Fx(\026)p FG(\))p Ft(g)754 4730 y Fu(\026)p Fs(2)p Fq(R)936 4715 y FG(are)f(called)f(the)h(c)m (hannel)h(scattering)f(matrices.)68 b(Denoting)39 b(the)j(scat-)109 4835 y(tering)31 b(matrix)g(of)h(the)h(dissipativ)m(e)f(scattering)g (system)i Ft(f)p Fx(H)2369 4850 y Fw(1)2408 4835 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)2666 4850 y Fw(1)2705 4835 y Ft(g)32 b FG(b)m(y)h Ft(f)p Fx(S)3032 4850 y Fw(1)3071 4835 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))p Ft(g)3367 4850 y Fu(\026)p Fs(2)p Fq(R)3541 4835 y FG(w)m(e)109 4955 y(pro)m(v)m(e)h(that)1509 5076 y Fx(S)1569 5091 y Fw(11)1644 5076 y FG(\()p Fx(\025)p FG(\))27 b(=)h Fx(S)1968 5091 y Fw(1)2007 5076 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))1200 b(\(1.7\))109 5247 y(holds)40 b(for)h(a.e.)69 b Fx(\025)42 b Ft(2)h Fm(R)5 b FG(.)75 b(This)41 b(sho)m(ws,)k(that)c (F)-8 b(esh)m(bac)m(h's)43 b(prop)s(osal)d(giv)m(es)i(in)e(fact)h(a)g (go)s(o)s(d)109 5367 y(appro)m(ximation)36 b(of)j(the)g(c)m(hannel)g (scattering)g(matrix)e Ft(f)p Fx(S)2308 5382 y Fw(11)2382 5367 y FG(\()p Fx(\026)p FG(\))p Ft(g)2567 5382 y Fu(\026)p Fs(2)p Fq(R)2747 5367 y FG(in)h(a)h(neigh)m(b)s(orho)s(o)s(d)e(of)109 5487 y(the)c(c)m(hosen)h(sp)s(ectral)e(parameter)g Fx(\025)h FG(of)f(the)h(optical)e(p)s(oten)m(tial)f Fx(V)2578 5502 y Fw(1)2618 5487 y FG(\()p Fx(\025)p FG(\).)p Black 1856 5728 a(3)p Black eop %%Page: 4 6 4 5 bop Black Black 109 171 a FG(Moreo)m(v)m(er,)42 b(F)-8 b(esh)m(bac)m(h's)41 b(prop)s(osal)c(implies)g(a)h(second)i(problem.)61 b(Similarly)35 b(to)k(the)g(optical)109 291 y(p)s(oten)m(tial)31 b Fx(V)578 306 y Fw(1)617 291 y FG(\()p Fx(\025)p FG(\))h(in)g(the)h (\014rst)g(c)m(hannel)g Fv(H)1692 306 y Fw(1)1764 291 y FG(one)g(can)g(in)m(tro)s(duce)g(an)f(optical)f(p)s(oten)m(tial)g Fx(V)3480 306 y Fw(2)3519 291 y FG(\()p Fx(\025)p FG(\))109 411 y(in)g(the)i(second)h(c)m(hannel,)1005 631 y Fx(V)1062 646 y Fw(2)1102 631 y FG(\()p Fx(\025)p FG(\))27 b(:=)h Ft(\000)p Fx(G)1547 590 y Fs(\003)1587 631 y FG(\()p Fx(H)1706 646 y Fw(1)1767 631 y Ft(\000)22 b Fx(\025)h Ft(\000)f Fx(i)p FG(0\))2165 590 y Fs(\000)p Fw(1)2259 631 y Fx(G;)115 b(\025)27 b Ft(2)h Fm(R)5 b Fx(;)702 b FG(\(1.8\))109 850 y(and)32 b(de\014ne)i(a)e(p)s(erturb)s(ed)i(op)s (erator)e Fx(H)1591 865 y Fw(2)1630 850 y FG(\()p Fx(\025)p FG(\),)1230 1070 y Fx(H)1311 1085 y Fw(2)1351 1070 y FG(\()p Fx(\025)p FG(\))27 b(:=)h Fx(H)1723 1085 y Fw(2)1784 1070 y FG(+)22 b Fx(V)1939 1085 y Fw(2)1978 1070 y FG(\()p Fx(\025)p FG(\))p Fx(;)115 b(\025)27 b Ft(2)h Fm(R)5 b Fx(;)927 b FG(\(1.9\))109 1289 y(in)42 b Fv(H)305 1304 y Fw(2)344 1289 y FG(.)75 b(W)-8 b(e)43 b(sho)m(w)h(b)s(elo)m(w)f(that) f(the)i(c)m(haracteristic)e(function)h(\002)2641 1304 y Fw(2)2680 1289 y FG([)p Fx(\025)p FG(]\()p Fx(\030)5 b FG(\),)45 b Fx(\030)50 b Ft(2)45 b Fm(C)3257 1304 y Fs(\000)3322 1289 y FG(,)h(of)c(the)109 1409 y(dissipativ)m(e)32 b(op)s(erator)g Fx(H)1061 1424 y Fw(2)1100 1409 y FG(\()p Fx(\025)p FG(\))g(and)h(the)g(scattering)f(matrix)f Ft(f)p Fx(S)2500 1424 y Fw(11)2575 1409 y FG(\()p Fx(\025)p FG(\))p Ft(g)2758 1424 y Fu(\025)p Fs(2)p Fq(R)2930 1409 y FG(are)i(related)f(b)m(y)1482 1629 y Fx(S)1542 1644 y Fw(11)1616 1629 y FG(\()p Fx(\025)p FG(\))c(=)f(\002)1956 1644 y Fw(2)1996 1629 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))2240 1588 y Fs(\003)3403 1629 y FG(\(1.10\))109 1848 y(for)40 b(a.e.)68 b Fx(\025)41 b Ft(2)h Fm(R)5 b FG(.)74 b(By)41 b([1]-[4)o(])g(the)g(last)f(relation)e(also)i(yields) g(that)h(the)g(scattering)f(matrix)109 1969 y Ft(f)p Fx(S)219 1984 y Fw(11)293 1969 y FG(\()p Fx(\025)p FG(\))p Ft(g)476 1984 y Fu(\025)p Fs(2)p Fq(R)656 1969 y FG(can)h(b)s(e)f (regarded)g(as)h(the)f(scattering)g(matrix)f Fx(S)2537 1984 y Fu(LP)2643 1969 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))h(of)g(a)g(Lax-Phillips)109 2089 y(scattering)32 b(system)i(at)e(the)h(p)s(oin)m(t)f Fx(\025)p FG(.)109 2257 y(Belo)m(w)22 b(w)m(e)h(restrict)f(ourself)g(to)g(a)g(complete)f (scattering)h(system)h Ft(f)p Fx(L;)17 b(L)2708 2272 y Fw(0)2748 2257 y Ft(g)p FG(,)24 b Fx(L)k FG(=)g Fx(L)3113 2272 y Fw(0)3175 2257 y FG(+)22 b Fx(V)f FG(,)k(where)109 2377 y(the)40 b(p)s(erturbation)f Fx(V)61 b FG(is)40 b(a)f(self-adjoin)m(t)f(trace)i(class)g(op)s(erator.)65 b(The)41 b(assumption)e(that)g Fx(V)109 2498 y FG(is)j(a)g(trace)h (class)f(op)s(erator)g(is)g(made)g(for)g(simplicit)m(y)-8 b(.)71 b(Indeed,)46 b(it)c(w)m(ould)g(b)s(e)h(su\016cien)m(t)g(to)109 2618 y(assume)34 b(that)f(the)g(resolv)m(en)m(t)i(di\013erence)f(\()p Fx(L)23 b Ft(\000)g Fx(z)t FG(\))1989 2582 y Fs(\000)p Fu(p)2107 2618 y Ft(\000)g FG(\()p Fx(L)2311 2633 y Fw(0)2373 2618 y Ft(\000)h Fx(z)t FG(\))2561 2582 y Fs(\000)p Fu(p)2689 2618 y FG(is)33 b(n)m(uclear)g(for)g(a)g(certain)109 2738 y Fx(p)27 b Ft(2)h Fm(N)43 b FG(or,)28 b(more)e(generally)-8 b(,)28 b(that)f(the)g(conditions)g(of)f(the)i(so-called)e (\\stationary")g(scattering)109 2859 y(theory)35 b(are)g(satis\014ed,)g (cf.)50 b([5,)36 b(Section)e(14].)49 b(Ho)m(w)m(ev)m(er,)38 b(w)m(e)e(emphasize)f(that)f(in)g(con)m(trast)h(to)109 2979 y([8])29 b(the)i(smallness)e(of)g(the)i(p)s(erturbation)e Fx(V)51 b FG(is)29 b(not)h(assumed.)43 b(F)-8 b(ollo)m(wing)27 b(the)k(lines)e(of)g([5])h(w)m(e)109 3099 y(sho)m(w)g(in)e(Section)g(2) h(ho)m(w)g(the)g(scattering)g(matrix)e(of)h(the)h(scattering)g(system)h Ft(f)p Fx(L;)17 b(L)3259 3114 y Fw(0)3298 3099 y Ft(g)29 b FG(can)g(b)s(e)109 3220 y(calculated.)57 b(Under)38 b(the)g(additional)c(assumptions)k(\(1.2\))e(and)i(\(1.4\))f(w)m(e)h (\014nd)g(in)f(Section)g(3)109 3340 y(the)i(c)m(hannel)g(scattering)f (matrices)g Ft(f)p Fx(S)1607 3355 y Fu(ij)1667 3340 y FG(\()p Fx(\025)p FG(\))p Ft(g)1850 3355 y Fu(\025)p Fs(2)p Fq(R)1990 3340 y FG(.)62 b(In)39 b(Section)g(4)f(w)m(e)i(pro)m (v)m(e)g(relation)c(\(1.7\).)109 3461 y(Section)i(5)h(is)g(dev)m(oted)h (to)f(the)g(pro)s(of)f(of)46 b(\(1.10\))o(.)62 b(Moreo)m(v)m(er,)43 b(the)c(Lax-Phillips)d(scattering)109 3581 y(theory)d(for)f(whic)m(h)h Ft(f)p FG(\002)969 3596 y Fw(2)1008 3581 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))1254 3545 y Fs(\003)1293 3581 y Ft(g)1343 3596 y Fu(\026)p Fs(2)p Fq(R)1517 3581 y FG(is)f(the)h(scattering)g(matrix)e(is)h(indicated.)109 3961 y FE(2)161 b(Scattering)53 b(matrix)109 4227 y FG(In)36 b(this)g(section)h(w)m(e)g(brie\015y)f(recall)f(the)i(notion)e(of)h (the)h(scattering)f(matrix)f Ft(f)p Fx(S)6 b FG(\()p Fx(\025)p FG(\))p Ft(g)3313 4242 y Fu(\025)p Fs(2)p Fq(R)3488 4227 y FG(of)36 b(a)109 4348 y(scattering)31 b(system)h Ft(f)p Fx(L;)17 b(L)1106 4363 y Fw(0)1146 4348 y Ft(g)p FG(,)31 b(where)i(it)d(is)h(assumed)h(that)f(the)h(unp)s(erturb)s(ed)h (op)s(erator)d Fx(L)3515 4363 y Fw(0)3586 4348 y FG(is)109 4468 y(self-adjoin)m(t)37 b(in)g(the)j(separable)e(Hilb)s(ert)f(space)j Fv(L)f FG(and)g(the)g(p)s(erturb)s(ed)g(op)s(erator)f Fx(L)h FG(di\013ers)109 4588 y(from)31 b Fx(L)405 4603 y Fw(0)477 4588 y FG(b)m(y)j(a)e(self-adjoin)m(t)f(trace)i(class)g(op)s (erator)e Fx(V)50 b Ft(2)28 b(B)2327 4603 y Fw(1)2367 4588 y FG(\()p Fv(L)p FG(\),)1156 4808 y Fx(L)g FG(=)g Fx(L)1420 4823 y Fw(0)1482 4808 y FG(+)22 b Fx(V)5 b(;)212 b(V)49 b FG(=)28 b Fx(V)2169 4767 y Fs(\003)2236 4808 y Ft(2)g(B)2395 4823 y Fw(1)2435 4808 y FG(\()p Fv(L)p FG(\))p Fx(:)847 b FG(\(2.1\))109 5027 y(Let)40 b Fx(E)363 5042 y Fw(0)403 5027 y FG(\()p Ft(\001)p FG(\))g(b)s(e)g(the)h(sp)s (ectral)f(measure)g(of)g Fx(L)1809 5042 y Fw(0)1889 5027 y FG(and)h(denote)g(b)m(y)g Fv(B)p FG(\()p Fm(R)5 b FG(\))46 b(the)41 b(set)g(of)f(all)e(Borel)109 5148 y(subsets)43 b(of)e(the)h(real)e(axis)h Fm(R)5 b FG(.)76 b(Without)41 b(loss)g(of)g(generalit)m(y)f(w)m(e)i(assume)g(throughout)f(the)109 5268 y(pap)s(er)32 b(that)h(the)g(condition)996 5487 y Fv(L)28 b FG(=)f(clospan)p Ft(f)p Fx(E)1630 5502 y Fw(0)1669 5487 y FG(\(\001\)ran)17 b(\()p Ft(j)p Fx(V)k Ft(j)p FG(\))28 b(:)f(\001)h Ft(2)g Fv(B)p FG(\()p Fm(R)5 b FG(\))p Ft(g)692 b FG(\(2.2\))p Black 1856 5728 a(4)p Black eop %%Page: 5 7 5 6 bop Black Black 109 173 a FG(is)31 b(satis\014ed,)h(where)h Ft(j)p Fx(V)21 b Ft(j)27 b FG(:=)h(\()p Fx(V)1296 137 y Fs(\003)1336 173 y Fx(V)21 b FG(\))1452 137 y Fw(1)p Fu(=)p Fw(2)1562 173 y FG(.)43 b(By)32 b(Theorem)g(X.4.4)g(of)f([13)o (])h(the)g(scattering)f(system)109 294 y Ft(f)p Fx(L;)17 b(L)335 309 y Fw(0)374 294 y Ft(g)41 b FG(is)f(complete,)i(that)f(is,)i (the)e(ranges)g(of)f(the)i(w)m(a)m(v)m(e)g(op)s(erators)f Fx(W)2938 309 y Fs(\006)2997 294 y FG(\()p Fx(L;)17 b(L)3211 309 y Fw(0)3251 294 y FG(\))40 b(in)g(\(1.1\))109 414 y(coincide)46 b(with)g(the)i(absolutely)e(con)m(tin)m(uous)i(subspace)g Fv(L)2384 378 y Fu(ac)2457 414 y FG(\()p Fx(L)p FG(\))f(of)g Fx(L)p FG(.)86 b(The)48 b(op)s(erator)e Fx(V)109 535 y FG(admits)31 b(the)i(represen)m(tation)762 746 y Fx(V)50 b FG(=)27 b Ft(j)p Fx(V)21 b Ft(j)1106 705 y Fw(1)p Fu(=)p Fw(2)1216 746 y Fx(C)7 b Ft(j)p Fx(V)21 b Ft(j)1427 705 y Fw(1)p Fu(=)p Fw(2)1537 746 y Fx(;)114 b Ft(j)p Fx(V)21 b Ft(j)28 b FG(=)f(\()p Fx(V)2060 705 y Fs(\003)2099 746 y Fx(V)22 b FG(\))2216 705 y Fw(1)p Fu(=)p Fw(2)2326 746 y Fx(;)114 b(C)35 b FG(=)27 b(sgn)q(\()p Fx(V)21 b FG(\))p Fx(;)453 b FG(\(2.3\))109 957 y(where)41 b Ft(j)p Fx(V)21 b Ft(j)532 921 y Fw(1)p Fu(=)p Fw(2)681 957 y FG(b)s(elongs)40 b(to)f(the)h(Hilb)s(ert-Sc)m(hmidt)e(class)i Ft(B)2367 972 y Fw(2)2406 957 y FG(\()p Fv(L)p FG(\))g(and)g(sgn)q(\()p Ft(\001)p FG(\))f(is)g(the)i(sign)m(um)109 1077 y(function.)i(By)33 b(Prop)s(osition)e(3.14)h(of)g([5])g(the)h(limits)610 1289 y Ft(j)p Fx(V)21 b Ft(j)744 1248 y Fw(1)p Fu(=)p Fw(2)854 1289 y FG(\()p Fx(L)h Ft(\000)h Fx(\025)f Ft(\006)g Fx(i)p FG(0\))1378 1248 y Fs(\000)p Fw(1)1473 1289 y Ft(j)p Fx(V)f Ft(j)1607 1248 y Fw(1)p Fu(=)p Fw(2)1744 1289 y FG(=)55 b(lim)1848 1349 y Fu(\017)p Fs(!)p Fw(+0)2054 1289 y Ft(j)p Fx(V)21 b Ft(j)2188 1248 y Fw(1)p Fu(=)p Fw(2)2297 1289 y FG(\()p Fx(L)i Ft(\000)g Fx(\025)f Ft(\006)g Fx(i\017)p FG(\))2812 1248 y Fs(\000)p Fw(1)2907 1289 y Ft(j)p Fx(V)f Ft(j)3041 1248 y Fw(1)p Fu(=)p Fw(2)3451 1289 y FG(\(2.4\))109 1539 y(exist)33 b(in)e Ft(B)518 1554 y Fw(2)558 1539 y FG(\()p Fv(L)p FG(\))i(for)f(a.e.)44 b Fx(\025)27 b Ft(2)h Fm(R)5 b FG(.)50 b(The)33 b(same)g(holds)f(for)g (the)h(limits)1294 1751 y Ft(j)p Fx(V)21 b Ft(j)1428 1709 y Fw(1)p Fu(=)p Fw(2)1538 1751 y FG(\()p Fx(L)1642 1766 y Fw(0)1703 1751 y Ft(\000)i Fx(\025)f Ft(\006)h Fx(i)p FG(0\))2102 1709 y Fs(\000)p Fw(1)2196 1751 y Ft(j)p Fx(V)e Ft(j)2330 1709 y Fw(1)p Fu(=)p Fw(2)2440 1751 y Fx(:)109 1962 y FG(Moreo)m(v)m(er)34 b(b)m(y)f(Prop)s(osition)e (3.13)h(of)g([5])h(the)g(deriv)-5 b(ativ)m(e)1176 2239 y Fx(M)1270 2254 y Fw(0)1309 2239 y FG(\()p Fx(\025)p FG(\))28 b(:=)1611 2172 y Ft(j)p Fx(V)21 b Ft(j)1745 2136 y Fw(1)p Fu(=)p Fw(2)1854 2172 y Fx(E)1926 2187 y Fw(0)1966 2172 y FG(\()p Fx(d\025)p FG(\))p Ft(j)p Fx(V)g Ft(j)2284 2136 y Fw(1)p Fu(=)p Fw(2)p 1611 2216 783 4 v 1948 2308 a Fx(d\025)2431 2239 y Ft(\025)28 b FG(0)866 b(\(2.5\))109 2481 y(exists)33 b(in)f Ft(B)557 2496 y Fw(1)597 2481 y FG(\()p Fv(L)p FG(\))g(for)g(a.e.)44 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(.)49 b(W)-8 b(e)33 b(set)1256 2693 y Fv(Q)1338 2708 y Fu(\025)1412 2693 y FG(:=)27 b(clo)1662 2612 y Fp(\010)1720 2693 y FG(ran)16 b(\()p Fx(M)2009 2708 y Fw(0)2049 2693 y FG(\()p Fx(\025)p FG(\)\))2220 2612 y Fp(\011)2306 2693 y Ft(\022)28 b Fv(L)p Fx(:)109 2904 y FG(By)i Ft(f)p Fx(Q)p FG(\()p Fx(\025)p FG(\))p Ft(g)569 2919 y Fu(\025)p Fs(2)p Fq(R)738 2904 y FG(w)m(e)h(denote)f(the)g(family)d(of)i(orthogonal)f(pro)5 b(jections)30 b(from)e Fv(L)i FG(on)m(to)f Fv(Q)3363 2919 y Fu(\025)3409 2904 y FG(.)43 b(One)109 3024 y(v)m(eri\014es)38 b(that)f Ft(f)p Fx(Q)p FG(\()p Fx(\025)p FG(\))p Ft(g)965 3039 y Fu(\025)p Fs(2)p Fq(R)1142 3024 y FG(is)f(measurable.)57 b(Let)37 b(us)h(consider)f(the)h(standard)f(Hilb)s(ert)f(space)109 3145 y Fx(L)175 3109 y Fw(2)214 3145 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(L)p FG(\).)49 b(On)33 b Fx(L)923 3109 y Fw(2)962 3145 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(L)p FG(\))38 b(w)m(e)c(in)m(tro)s(duce)e(the)h(pro)5 b(jection)33 b Fx(Q)800 3356 y FG(\()p Fx(Qf)11 b FG(\)\()p Fx(\025)p FG(\))28 b(:=)f Fx(Q)p FG(\()p Fx(\025)p FG(\))p Fx(f)11 b FG(\()p Fx(\025)p FG(\))p Fx(;)114 b(\025)28 b Ft(2)g Fm(R)5 b Fx(;)120 b(f)38 b Ft(2)28 b Fx(L)2484 3315 y Fw(2)2524 3356 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(L)p FG(\))p Fx(;)109 3567 y FG(and)31 b(set)g Fv(Q)d FG(=)g(ran)16 b(\()p Fx(Q)p FG(\).)43 b(F)-8 b(urther,)32 b(in)e Fx(L)1599 3531 y Fw(2)1639 3567 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(L)p FG(\))36 b(w)m(e)c(de\014ne)g(the)f(m)m (ultiplication)c(op)s(erator)109 3688 y Fx(M)203 3703 y Fl(L)286 3688 y FG(b)m(y)641 3800 y(\()p Fx(M)773 3815 y Fl(L)824 3800 y Fx(f)11 b FG(\)\()p Fx(\025)p FG(\))82 b(:=)h Fx(\025f)11 b FG(\()p Fx(\025)p FG(\))p Fx(;)114 b(\025)27 b Ft(2)i Fm(R)t Fx(;)632 3920 y FG(dom)16 b(\()p Fx(M)965 3935 y Fl(L)1015 3920 y FG(\))83 b(:=)1322 3840 y Fp(\010)1380 3920 y Fx(f)39 b Ft(2)28 b Fx(L)1627 3884 y Fw(2)1667 3920 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(L)p FG(\))32 b(:)c Fx(\025f)11 b FG(\()p Fx(\025)p FG(\))27 b Ft(2)h Fx(L)2594 3884 y Fw(2)2634 3920 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(L)p FG(\))3038 3840 y Fp(\011)3102 3920 y Fx(:)109 4084 y FG(Ob)m(viously)-8 b(,)32 b(the)h(m)m(ultiplication)28 b(op)s(erator)k Fx(M)1867 4099 y Fl(L)1950 4084 y FG(and)h(the)g(pro)5 b(jection)32 b Fx(Q)h FG(comm)m(ute.)43 b(W)-8 b(e)33 b(set)1282 4295 y Fx(M)1376 4310 y Fl(Q)1465 4295 y FG(:=)28 b Fx(M)1690 4310 y Fl(L)1768 4295 y Fk(\026)f FG(dom)16 b(\()p Fx(M)2170 4310 y Fl(L)2220 4295 y FG(\))23 b Ft(\\)f Fv(Q)p Fx(:)109 4507 y FG(F)-8 b(rom)45 b(Section)i(4.5)f(of)h([5])g(one)g(gets)h(that) e(the)i(absolutely)e(con)m(tin)m(uous)i(part)e Fx(L)3271 4471 y Fu(ac)3391 4507 y FG(of)g(the)109 4627 y(p)s(erturb)s(ed)26 b(op)s(erator)g Fx(L)g FG(and)g(the)h(op)s(erator)e Fx(M)1861 4642 y Fl(Q)1949 4627 y FG(are)h(unitarily)e(equiv)-5 b(alen)m(t.)41 b(In)26 b(the)h(follo)m(wing)109 4747 y(w)m(e)43 b(denote)g(the)g(subspace)h Fv(Q)f FG(b)m(y)g Fx(L)1517 4711 y Fw(2)1557 4747 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(Q)1939 4762 y Fu(\025)1990 4747 y FG(\))42 b(whic)m(h)h(can)f(b)s(e)h(regarded)g(as)f(the)h(direct)109 4868 y(in)m(tegral)34 b(of)i(the)g(family)e(of)h(subspaces)k Ft(f)p Fv(Q)1751 4883 y Fu(\025)1796 4868 y Ft(g)1846 4883 y Fu(\025)p Fs(2)p Fq(R)2022 4868 y FG(with)d(resp)s(ect)h(to)f (the)g(Leb)s(esgue)i(measure)109 4988 y Fx(d\025)32 b FG(on)g Fm(R)5 b FG(,)39 b(cf.)k([5].)109 5156 y(Since)32 b(the)h(scattering)g(op)s(erator)e Fx(S)j FG(=)27 b Fx(W)1663 5171 y Fw(+)1722 5156 y FG(\()p Fx(L;)17 b(L)1936 5171 y Fw(0)1976 5156 y FG(\))2014 5120 y Fs(\003)2054 5156 y Fx(W)2146 5171 y Fs(\000)2205 5156 y FG(\()p Fx(L;)g(L)2419 5171 y Fw(0)2459 5156 y FG(\))32 b(acts)h(on)f Fx(L)2931 5120 y Fu(ac)2931 5180 y Fw(0)3036 5156 y FG(and)h(comm)m(utes)109 5276 y(with)f Fx(L)397 5240 y Fu(ac)397 5301 y Fw(0)502 5276 y FG(there)h(is)f(a)h(measurable)f(family)e Ft(f)p Fx(S)6 b FG(\()p Fx(\025)p FG(\))p Ft(g)2038 5291 y Fu(\025)p Fs(2)p Fq(R)2210 5276 y FG(of)32 b(op)s(erators)1504 5487 y Fx(S)6 b FG(\()p Fx(\025)p FG(\))27 b(:)h Fv(Q)1867 5502 y Fu(\025)1941 5487 y Ft(\000)-17 b(!)28 b Fv(Q)2211 5502 y Fu(\025)p Black 1856 5728 a FG(5)p Black eop %%Page: 6 8 6 7 bop Black Black 109 171 a FG(suc)m(h)34 b(that)e Fx(S)38 b FG(is)32 b(unitarily)f(equiv)-5 b(alen)m(t)32 b(to)g(the)h(m)m(ultiplication)28 b(op)s(erator)1166 382 y(\()p Fx(M)1298 397 y Fl(Q)1360 382 y FG(\()p Fx(S)6 b FG(\))p Fx(f)11 b FG(\)\()p Fx(\025)p FG(\))82 b(:=)176 b Fx(S)6 b FG(\()p Fx(\025)p FG(\))p Fx(f)11 b FG(\()p Fx(\025)p FG(\))p Fx(;)1157 502 y FG(dom)16 b(\()p Fx(M)1490 517 y Fl(Q)1551 502 y FG(\()p Fx(S)6 b FG(\)\))83 b(:=)g Fx(L)2066 466 y Fw(2)2106 502 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(Q)2488 517 y Fu(\025)2538 502 y FG(\))p Fx(:)109 721 y FG(The)23 b(family)d Ft(f)p Fx(S)6 b FG(\()p Fx(\025)p FG(\))p Ft(g)886 736 y Fu(\025)p Fs(2)p Fq(R)1048 721 y FG(is)22 b(called)g(the)h Fj(sc)-5 b(attering)25 b(matrix)d FG(of)g(the)h(scattering)g(system)g Ft(f)p Fx(L;)17 b(L)3535 736 y Fw(0)3575 721 y Ft(g)p FG(.)109 842 y(Since)32 b(the)g(scattering)g(system)h Ft(f)p Fx(L;)17 b(L)1529 857 y Fw(0)1568 842 y Ft(g)32 b FG(is)f(complete)h(the)g(op)s (erator)f Fx(S)6 b FG(\()p Fx(\025)p FG(\))32 b(is)f(unitary)g(on)h Fv(Q)3606 857 y Fu(\025)109 962 y FG(for)g(a.e.)43 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(.)109 1130 y(The)39 b(follo)m(wing)c (represen)m(tation)k(theorem)e(of)h(the)g(scattering)g(matrix)e(is)i(a) f(consequence)k(of)109 1250 y(Corollary)31 b(18.9)g(of)i([5)o(],)g(see) h(also)d([5,)i(Section)f(18.2.2].)p Black 109 1493 a Fi(Theorem)37 b(2.1)p Black 49 w Fj(L)-5 b(et)43 b Fx(L)p Fj(,)i Fx(L)1159 1508 y Fw(0)1241 1493 y Fj(and)d Fx(V)64 b Fj(b)-5 b(e)42 b(self-adjoint)g(op)-5 b(er)g(ators)42 b(in)g Fv(L)h Fj(as)g(in)49 b FG(\(2.1\))p Fj(.)68 b(Then)109 1606 y Ft(f)p Fx(L;)17 b(L)335 1621 y Fw(0)374 1606 y Ft(g)31 b Fj(is)f(a)g(c)-5 b(omplete)30 b(sc)-5 b(attering)30 b(system)g(and)g(the)g(c)-5 b(orr)g(esp)g(onding)29 b(sc)-5 b(attering)30 b(matrix)h(ma-)109 1719 y(trix)k Ft(f)p Fx(S)6 b FG(\()p Fx(\025)p FG(\))p Ft(g)591 1734 y Fu(\025)p Fs(2)p Fq(R)765 1719 y Fj(admits)34 b(the)h(r)-5 b(epr)g(esentation)434 1948 y Fh(S)5 b FC(\()p Fh(\025)p FC(\))26 b(=)f Fh(I)780 1963 y Fl(Q)837 1975 y Fo(\025)902 1948 y Fg(\000)20 b FC(2)p Fh(\031)s(i)15 b(M)1237 1900 y Fw(1)p Fu(=)p Fw(2)1227 1974 y(0)1348 1948 y FC(\()p Fh(\025)p FC(\))1486 1847 y Ff(n)1547 1948 y Fh(C)27 b Fg(\000)20 b Fh(C)7 b Fg(j)p Fh(V)20 b Fg(j)1925 1911 y Fw(1)p Fu(=)p Fw(2)2035 1948 y FC(\()p Fh(L)h Fg(\000)e Fh(\025)i Fg(\000)f Fh(i)p FC(0\))2519 1911 y Fs(\000)p Fw(1)2614 1948 y Fg(j)p Fh(V)h Fg(j)2738 1911 y Fw(1)p Fu(=)p Fw(2)2848 1948 y Fh(C)2920 1847 y Ff(o)2995 1948 y Fh(M)3093 1900 y Fw(1)p Fu(=)p Fw(2)3083 1974 y(0)3203 1948 y FC(\()p Fh(\025)p FC(\))144 2176 y Fj(for)34 b(a.e.)44 b Fx(\025)28 b Ft(2)g Fm(R)5 b Fj(.)109 2556 y FE(3)161 b(Channel)53 b(scattering)g(matrices)109 2822 y FG(Let)30 b(us)g(no)m(w)h(assume)f (that)g(the)g(Hilb)s(ert)e(space)j Fv(L)f FG(is)g(the)g(orthogonal)e (sum)i(of)f(t)m(w)m(o)h(subspaces)109 2943 y Fv(H)181 2958 y Fw(1)252 2943 y FG(and)j Fv(H)514 2958 y Fw(2)553 2943 y FG(,)g Fv(L)28 b FG(=)f Fv(H)882 2958 y Fw(1)944 2943 y Ft(\010)22 b Fv(H)1115 2958 y Fw(2)1155 2943 y FG(,)32 b(that)h Fx(L)1492 2958 y Fw(0)1564 2943 y FG(is)f(a)g (diagonal)e(blo)s(c)m(k)j(op)s(erator)f(matrix)f(of)h(the)h(form)1170 3275 y Fx(L)1236 3290 y Fw(0)1303 3275 y FG(=)1407 3135 y Fp(\022)1480 3214 y Fx(H)1561 3229 y Fw(1)1719 3214 y FG(0)1516 3335 y(0)118 b Fx(H)1764 3350 y Fw(2)1804 3135 y Fp(\023)1905 3275 y FG(:)2001 3154 y Fv(H)2073 3169 y Fw(1)2018 3275 y Ft(\010)2001 3395 y Fv(H)2073 3410 y Fw(2)2181 3275 y Ft(\000)-16 b(!)2411 3154 y Fv(H)2483 3169 y Fw(1)2428 3275 y Ft(\010)2411 3395 y Fv(H)2483 3410 y Fw(2)2564 3275 y Fx(;)860 b FG(\(3.1\))109 3614 y(cf.)59 b(\(1.2\),)39 b(where)h Fx(H)904 3629 y Fw(1)981 3614 y FG(and)e Fx(H)1257 3629 y Fw(2)1334 3614 y FG(are)g(self-adjoin) m(t)e(op)s(erators)i(in)f Fv(H)2637 3629 y Fw(1)2714 3614 y FG(and)h Fv(H)2981 3629 y Fw(2)3058 3614 y FG(and)g(that)g Fx(V)59 b Ft(2)109 3734 y(B)174 3749 y Fw(1)214 3734 y FG(\()p Fv(L)p FG(\))32 b(is)g(a)h(self-adjoin)m(t)d(trace)j(class)g (op)s(erator)f(of)g(the)h(form)1207 4072 y Fx(V)49 b FG(=)1417 3932 y Fp(\022)1524 4011 y FG(0)117 b Fx(G)1490 4132 y(G)1567 4096 y Fs(\003)1704 4132 y FG(0)1766 3932 y Fp(\023)1867 4072 y FG(:)1964 3951 y Fv(H)2036 3966 y Fw(1)1981 4072 y Ft(\010)1964 4192 y Fv(H)2036 4207 y Fw(2)2144 4072 y Ft(\000)-16 b(!)2374 3951 y Fv(H)2446 3966 y Fw(1)2391 4072 y Ft(\010)2374 4192 y Fv(H)2446 4207 y Fw(2)2527 4072 y Fx(;)897 b FG(\(3.2\))109 4411 y(see)36 b(\(1.4\))o(.)50 b(The)36 b(op)s(erator)e Fx(G)e FG(:)f Fv(H)1383 4426 y Fw(2)1454 4411 y Ft(\000)-16 b(!)31 b Fv(H)1718 4426 y Fw(1)1792 4411 y FG(describ)s(es)36 b(the)f(in)m(teraction)f(b)s(et)m(w)m(een)i(the)g(c)m(han-)109 4531 y(nels.)43 b(Since)33 b Fx(V)54 b FG(is)32 b(a)g(trace)h(class)g (op)s(erator)f(w)m(e)h(ha)m(v)m(e)1544 4751 y Fx(G)28 b Ft(2)g(B)1808 4766 y Fw(1)1848 4751 y FG(\()p Fv(H)1958 4766 y Fw(2)1997 4751 y Fx(;)17 b Fv(H)2113 4766 y Fw(1)2152 4751 y FG(\))p Fx(:)109 4971 y FG(The)33 b(p)s(erturb)s(ed)g(or)g(full) d(Hamiltonian)f Fx(L)k FG(has)g(the)g(form)944 5304 y Fx(L)28 b FG(:=)g Fx(L)1235 5319 y Fw(0)1297 5304 y FG(+)22 b Fx(V)49 b FG(=)1605 5163 y Fp(\022)1678 5243 y Fx(H)1759 5258 y Fw(1)1903 5243 y Fx(G)1680 5363 y(G)1757 5327 y Fs(\003)1881 5363 y Fx(H)1962 5378 y Fw(2)2002 5163 y Fp(\023)2103 5304 y FG(:)2199 5183 y Fv(H)2271 5198 y Fw(1)2216 5303 y Ft(\010)2199 5423 y Fv(H)2271 5438 y Fw(2)2379 5304 y Ft(\000)-16 b(!)2609 5183 y Fv(H)2681 5198 y Fw(1)2626 5303 y Ft(\010)2609 5423 y Fv(H)2681 5438 y Fw(2)2789 5304 y Fx(:)635 b FG(\(3.3\))p Black 1856 5728 a(6)p Black eop %%Page: 7 9 7 8 bop Black Black 109 171 a FG(The)25 b(follo)m(wing)d(lemma)g(is)i (kno)m(wn)i(as)f(the)g(F)-8 b(esh)m(bac)m(h)26 b(decomp)s(osition)d(in) g(ph)m(ysics,)28 b(cf.)41 b([10,)24 b(11].)109 291 y(W)-8 b(e)33 b(use)g(the)g(notation)565 497 y Fx(H)646 512 y Fw(1)686 497 y FG(\()p Fx(z)t FG(\))28 b(=)g Fx(H)1024 512 y Fw(1)1085 497 y FG(+)22 b Fx(V)1240 512 y Fw(1)1279 497 y FG(\()p Fx(z)t FG(\))131 b(and)f Fx(H)1903 512 y Fw(2)1942 497 y FG(\()p Fx(z)t FG(\))29 b(=)e Fx(H)2280 512 y Fw(2)2342 497 y FG(+)22 b Fx(V)2497 512 y Fw(2)2536 497 y FG(\()p Fx(z)t FG(\))p Fx(;)115 b(z)32 b Ft(2)c Fm(C)20 b Ft(n)p Fm(R)11 b Fx(;)262 b FG(\(3.4\))109 704 y(where)572 824 y Fx(V)629 839 y Fw(1)668 824 y FG(\()p Fx(z)t FG(\))29 b(=)e Ft(\000)p Fx(G)p FG(\()p Fx(H)1198 839 y Fw(2)1260 824 y Ft(\000)c Fx(z)t FG(\))1447 783 y Fs(\000)p Fw(1)1542 824 y Fx(G)1619 783 y Fs(\003)1788 824 y FG(and)130 b Fx(V)2132 839 y Fw(2)2172 824 y FG(\()p Fx(z)t FG(\))28 b(=)g Ft(\000)p Fx(G)2583 783 y Fs(\003)2622 824 y FG(\()p Fx(H)2741 839 y Fw(1)2803 824 y Ft(\000)22 b Fx(z)t FG(\))2989 783 y Fs(\000)p Fw(1)3084 824 y Fx(G;)263 b FG(\(3.5\))109 992 y(see)33 b(\(1.6\),)f(\(1.9\),)h(\(1.5\))f(and)g (\(1.8\).)p Black 109 1224 a Fi(Lemma)37 b(3.1)p Black 49 w Fj(L)-5 b(et)30 b Fx(L)p Fj(,)i Fx(H)1070 1239 y Fw(1)1109 1224 y FG(\()p Fx(z)t FG(\))f Fj(and)f Fx(H)1531 1239 y Fw(2)1570 1224 y FG(\()p Fx(z)t FG(\))p Fj(,)i Fx(z)g Ft(2)d Fm(C)19 b Ft(n)p Fm(R)11 b Fj(,)37 b(b)-5 b(e)30 b(given)g(by)39 b FG(\(3.3\))30 b Fj(and)40 b FG(\(3.4\))p Fj(,)31 b(r)-5 b(esp)g(e)g(c-)109 1337 y(tively.)45 b(Then)34 b(we)g(have)g Fx(z)f Ft(2)28 b FG(res)q(\()p Fx(H)1441 1352 y Fu(i)1468 1337 y FG(\()p Fx(z)t FG(\)\))p Fj(,)36 b Fx(i)28 b FG(=)f(1)p Fx(;)17 b FG(2)p Fj(,)34 b(for)h(al)5 b(l)34 b Fx(z)f Ft(2)28 b Fm(C)20 b Ft(n)p Fm(R)52 b Fj(and)247 1609 y FC(\()p Fh(L)20 b Fg(\000)g Fh(z)t FC(\))536 1571 y Fs(\000)p Fw(1)656 1609 y FC(=)752 1453 y Ff( )1134 1529 y FC(\()p Fh(H)1245 1543 y Fw(1)1284 1529 y FC(\()p Fh(z)t FC(\))h Fg(\000)f Fh(z)t FC(\))1593 1496 y Fs(\000)p Fw(1)2081 1529 y Fg(\000)p FC(\()p Fh(H)2263 1543 y Fw(1)2322 1529 y Fg(\000)g Fh(z)t FC(\))2494 1496 y Fs(\000)p Fw(1)2589 1529 y Fh(G)p FC(\()p Fh(H)2772 1543 y Fw(2)2811 1529 y FC(\()p Fh(z)t FC(\))h Fg(\000)f Fh(z)t FC(\))3120 1496 y Fs(\000)p Fw(1)824 1689 y Fg(\000)p FC(\()p Fh(H)1006 1703 y Fw(2)1045 1689 y FC(\()p Fh(z)t FC(\))h Fg(\000)f Fh(z)t FC(\))1354 1656 y Fs(\000)p Fw(1)1449 1689 y Fh(G)1521 1656 y Fs(\003)1560 1689 y FC(\()p Fh(H)1671 1703 y Fw(1)1731 1689 y Fg(\000)g Fh(z)t FC(\))1903 1656 y Fs(\000)p Fw(1)2370 1689 y FC(\()p Fh(H)2481 1703 y Fw(2)2521 1689 y FC(\()p Fh(z)t FC(\))h Fg(\000)f Fh(z)t FC(\))2830 1656 y Fs(\000)p Fw(1)3215 1453 y Ff(!)3302 1609 y Fh(:)138 b FC(\(3.6\))109 1927 y Fi(Pro)s(of.)43 b FG(F)-8 b(rom)558 2133 y(Im)691 2053 y Fp(\000)737 2133 y FG(\()p Fx(H)856 2148 y Fw(1)895 2133 y FG(\()p Fx(z)t FG(\))23 b Ft(\000)f Fx(z)t FG(\))p Fx(h)1285 2148 y Fw(1)1326 2133 y Fx(;)17 b(h)1426 2148 y Fw(1)1465 2053 y Fp(\001)1538 2133 y FG(=)28 b(Im)p 1775 2079 50 4 v 16 w Fx(z)5 b Ft(k)p Fx(h)1931 2148 y Fw(1)1970 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Fw(1)p Fu(=)p Fw(2)2719 3955 y Fx(C)7 b Ft(j)p Fx(V)22 b Ft(j)2931 3919 y Fw(1)p Fu(=)p Fw(2)3040 3955 y FG(,)30 b(cf.)43 b(\(2.3\))o(.)g(Let)109 4076 y Fx(E)181 4091 y Fw(1)220 4076 y FG(\()p Ft(\001)p FG(\))33 b(and)h Fx(E)620 4091 y Fw(2)660 4076 y FG(\()p Ft(\001)p FG(\))f(b)s(e)h(the)g(sp)s(ectral)g(measures)g(of)f Fx(H)2080 4091 y Fw(1)2153 4076 y FG(and)h Fx(H)2425 4091 y Fw(2)2464 4076 y FG(,)g(resp)s(ectiv)m(ely)-8 b(.)48 b(The)35 b(op)s(erator)109 4196 y(function)d Fx(M)585 4211 y Fw(0)625 4196 y FG(\()p Ft(\001)p FG(\))g(from)f(\(2.5\))h(here) h(admits)f(the)h(represen)m(tation)1300 4472 y Fx(M)1394 4487 y Fw(0)1434 4472 y FG(\()p Fx(\025)p FG(\))27 b(=)1698 4332 y Fp(\022)1771 4411 y Fx(M)1865 4426 y Fw(1)1905 4411 y FG(\()p Fx(\025)p FG(\))192 b(0)1880 4532 y(0)g Fx(M)2215 4547 y Fw(2)2255 4532 y FG(\()p Fx(\025)p FG(\))2387 4332 y Fp(\023)3403 4472 y FG(\(3.14\))109 4743 y(for)32 b(a.e)g Fx(\025)c Ft(2)g Fm(R)5 b FG(,)39 b(where)33 b(the)g(deriv)-5 b(ativ)m(es)390 5021 y Fx(M)484 5036 y Fw(1)524 5021 y FG(\()p Fx(\025)p FG(\))28 b(=)798 4954 y Ft(j)p Fx(G)903 4917 y Fs(\003)942 4954 y Ft(j)970 4917 y Fw(1)p Fu(=)p Fw(2)1080 4954 y Fx(E)1152 4969 y Fw(1)1192 4954 y FG(\()p Fx(d\025)p FG(\))p Ft(j)p Fx(G)1481 4917 y Fs(\003)1519 4954 y Ft(j)1547 4917 y Fw(1)p Fu(=)p Fw(2)p 798 4998 859 4 v 1174 5089 a Fx(d\025)1717 5021 y FG(and)49 b Fx(M)2017 5036 y Fw(2)2057 5021 y FG(\()p Fx(\025)p FG(\))28 b(=)2331 4954 y Ft(j)p Fx(G)p Ft(j)2464 4917 y Fw(1)p Fu(=)p Fw(2)2574 4954 y Fx(E)2646 4969 y Fw(2)2685 4954 y FG(\()p Fx(d\025)p FG(\))p Ft(j)p Fx(G)p Ft(j)3002 4917 y Fw(1)p Fu(=)p Fw(2)p 2331 4998 780 4 v 2667 5089 a Fx(d\025)3403 5021 y FG(\(3.15\))109 5270 y(exist)33 b(in)e Ft(B)518 5285 y Fw(1)558 5270 y FG(\()p Fv(H)668 5285 y Fw(1)707 5270 y FG(\))i(and)f Ft(B)1032 5285 y Fw(1)1072 5270 y FG(\()p Fv(H)1182 5285 y Fw(2)1221 5270 y FG(\))h(for)f(a.e.)44 b Fx(\025)27 b Ft(2)h Fm(R)5 b FG(,)39 b(resp)s(ectiv)m(ely)-8 b(.)44 b(Setting)1104 5487 y Fv(Q)1186 5502 y Fu(j;\025)1307 5487 y FG(:=)28 b(clo)1557 5407 y Fp(\010)1615 5487 y FG(ran)17 b(\()p Fx(M)1905 5502 y Fu(j)1941 5487 y FG(\()p Fx(\025)p FG(\)\))2112 5407 y Fp(\011)2170 5487 y Fx(;)114 b(j)34 b FG(=)28 b(1)p Fx(;)17 b FG(2)p Fx(;)p Black 1856 5728 a FG(8)p Black eop %%Page: 9 11 9 10 bop Black Black 109 171 a FG(and)1494 291 y Fv(Q)1576 306 y Fu(\025)1649 291 y FG(:=)28 b Fv(Q)1862 306 y Fw(1)p Fu(;\025)1985 291 y Ft(\010)22 b Fv(Q)2166 306 y Fw(2)p Fu(;\025)3403 291 y FG(\(3.16\))109 448 y(for)32 b(a.e.)43 b Fx(\025)28 b Ft(2)g Fm(R)44 b FG(w)m(e)33 b(obtain)f(the)h(decomp)s (osition)821 628 y Fx(L)887 587 y Fw(2)927 628 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(Q)1309 643 y Fu(\025)1359 628 y FG(\))28 b(=)f Fx(L)1594 587 y Fw(2)1634 628 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(Q)2016 643 y Fw(1)p Fu(;\025)2122 628 y FG(\))22 b Ft(\010)g Fx(L)2347 587 y Fw(2)2387 628 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(Q)2769 643 y Fw(2)p Fu(;\025)2875 628 y FG(\))p Fx(;)109 808 y FG(cf.)43 b(Section)33 b(2.)43 b(F)-8 b(rom)31 b(\(2.2\))h(the)h(conditions)885 985 y Fv(H)957 1000 y Fw(1)1080 985 y FG(=)82 b(clospan)p Ft(f)p Fx(E)1675 1000 y Fw(1)1715 985 y FG(\(\001\)ran)16 b(\()p Ft(j)p Fx(G)2172 949 y Fs(\003)2211 985 y Ft(j)p FG(\))28 b(:)f(\001)h Ft(2)h Fv(B)p FG(\()p Fm(R)t FG(\))p Ft(g)p Fx(;)885 1105 y Fv(H)957 1120 y Fw(2)1080 1105 y FG(=)82 b(clospan)p Ft(f)p Fx(E)1675 1120 y Fw(2)1715 1105 y FG(\(\001\)ran)16 b(\()p Ft(j)p Fx(G)p Ft(j)p FG(\))27 b(:)h(\001)g Ft(2)g Fv(B)p FG(\()p Fm(R)5 b FG(\))p Ft(g)3403 1046 y FG(\(3.17\))109 1284 y(follo)m(w.)58 b(Moreo)m(v)m(er,)41 b(the)d(con)m(v)m(erse)j(is)c (also)g(true,)j(that)e(is,)h(condition)e(\(3.17\))g(implies)e(\(2.2\).) 109 1404 y(Hence,)28 b(without)d(loss)h(of)f(generalit)m(y)f(w)m(e)j (assume)f(that)f(condition)g(\(3.17\))f(is)i(satis\014ed.)41 b(There-)109 1524 y(fore)32 b(the)h(reduced)h(m)m(ultiplication)28 b(op)s(erators)k Fx(M)1983 1539 y Fl(Q)2040 1549 y Fo(j)2078 1524 y FG(,)959 1704 y Fx(M)1053 1719 y Fl(Q)1110 1729 y Fo(j)1174 1704 y FG(:=)c Fx(M)1399 1719 y Fl(H)1449 1729 y Fo(j)1513 1704 y Fk(\026)f FG(dom)16 b(\()p Fx(M)1915 1719 y Fl(H)1965 1729 y Fo(j)2002 1704 y FG(\))22 b Ft(\\)h Fx(L)2217 1663 y Fw(2)2256 1704 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(Q)2638 1719 y Fu(j;\025)2737 1704 y FG(\))p Fx(;)109 1884 y FG(where)581 2036 y(\()p Fx(M)713 2051 y Fl(H)763 2061 y Fo(j)800 2036 y Fx(f)11 b FG(\)\()p Fx(\025)p FG(\))82 b(:=)h Fx(\025f)11 b FG(\()p Fx(\025)p FG(\))p Fx(;)114 b(\025)28 b Ft(2)g Fm(R)5 b Fx(;)572 2157 y FG(dom)16 b(\()p Fx(M)905 2172 y Fl(H)955 2182 y Fo(j)991 2157 y FG(\))83 b(:=)1298 2076 y Fp(\010)1357 2157 y Fx(f)38 b Ft(2)28 b Fx(L)1603 2121 y Fw(2)1643 2157 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(H)2015 2172 y Fu(j)2056 2157 y FG(\))28 b(:)f Fx(\025f)11 b FG(\()p Fx(\025)p FG(\))27 b Ft(2)i Fx(L)2613 2121 y Fw(2)2652 2157 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(H)3024 2172 y Fu(j)3066 2157 y FG(\))3104 2076 y Fp(\011)3162 2157 y Fx(:)109 2330 y FG(are)35 b(unitary)g(equiv)-5 b(alen)m(t)36 b(to)f(the)h(absolutely)f(con)m(tin)m(uous)h(parts)g Fx(H)2681 2294 y Fu(ac)2673 2355 y(j)2788 2330 y FG(of)f(the)h(op)s (erators)g Fx(H)3589 2345 y Fu(j)3625 2330 y FG(,)109 2450 y Fx(j)d FG(=)28 b(1)p Fx(;)17 b FG(2.)109 2618 y(With)37 b(resp)s(ect)i(to)e(the)h(decomp)s(osition)e(\(3.16\))h(the)h (scattering)g(matrix)e Ft(f)p Fx(S)6 b FG(\()p Fx(\025)p FG(\))p Ft(g)3187 2633 y Fu(\025)p Fs(2)p Fq(R)3364 2618 y FG(admits)109 2738 y(the)33 b(decomp)s(osition)918 3031 y Fx(S)6 b FG(\()p Fx(\025)p FG(\))28 b(=)1248 2891 y Fp(\022)1321 2970 y Fx(S)1381 2985 y Fw(11)1456 2970 y FG(\()p Fx(\025)p FG(\))83 b Fx(S)1732 2985 y Fw(12)1807 2970 y FG(\()p Fx(\025)p FG(\))1321 3090 y Fx(S)1381 3105 y Fw(21)1456 3090 y FG(\()p Fx(\025)p FG(\))g Fx(S)1732 3105 y Fw(22)1807 3090 y FG(\()p Fx(\025)p FG(\))1939 2891 y Fp(\023)2040 3031 y FG(:)2137 2910 y Fv(Q)2219 2925 y Fw(1)p Fu(;\025)2189 3030 y Ft(\010)2137 3151 y Fv(Q)2219 3166 y Fw(2)p Fu(;\025)2389 3031 y Ft(\000)-17 b(!)2618 2910 y Fv(Q)2700 2925 y Fw(1)p Fu(;\025)2671 3030 y Ft(\010)2618 3151 y Fv(Q)2700 3166 y Fw(2)p Fu(;\025)3403 3031 y FG(\(3.18\))109 3329 y(for)39 b(a.e.)66 b Fx(\025)40 b Ft(2)g Fm(R)5 b FG(.)72 b(The)41 b(en)m(tries)f Ft(f)p Fx(S)1482 3344 y Fu(ij)1542 3329 y FG(\()p Fx(\025)p FG(\))p Ft(g)1725 3344 y Fu(\025)p Fs(2)p Fq(R)1865 3329 y FG(,)i Fx(i;)17 b(j)46 b FG(=)40 b(1)p Fx(;)17 b FG(2,)41 b(are)f(called)f Fj(channel)h(sc)-5 b(attering)109 3450 y(matric)g(es)p FG(.)41 b(W)-8 b(e)28 b(note)g(that)g(the)h(m)m (ultiplication)23 b(op)s(erators)28 b(induced)g(b)m(y)h(the)f(c)m (hannel)h(scatter-)109 3570 y(ing)e(matrices)h(are)h(unitary)f(equiv)-5 b(alen)m(t)28 b(to)h(the)g(c)m(hannel)g(scattering)f(op)s(erators)h Fx(S)3177 3585 y Fu(ij)3265 3570 y FG(=)f Fx(P)3432 3585 y Fu(i)3460 3570 y Fx(S)6 b(P)3589 3585 y Fu(j)3625 3570 y FG(,)109 3690 y Fx(i;)17 b(j)33 b FG(=)28 b(1)p Fx(;)17 b FG(2,)30 b(where)h Fx(P)904 3705 y Fu(i)962 3690 y 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b(=)g Ft(\000)p FG(2)p Fx(\031)t(iM)1258 5018 y Fw(1)1298 5003 y FG(\()p Fx(\025)p FG(\))1431 4967 y Fw(1)p Fu(=)p Fw(2)1541 5003 y Ft(f)p Fx(U)32 b FG(+)22 b Fx(U)10 b(F)1926 5018 y Fw(2)1967 5003 y FG(\()p Fx(\025)p FG(\))p Fx(U)2176 4967 y Fs(\003)2216 5003 y Fx(N)2294 5018 y Fw(1)2333 5003 y FG(\()p Fx(\025)p FG(\))p Fx(U)g Ft(g)p Fx(M)2686 5018 y Fw(2)2726 5003 y FG(\()p Fx(\025)p FG(\))2859 4967 y Fw(1)p Fu(=)p Fw(2)2969 5003 y Fx(;)436 5149 y(S)496 5164 y Fw(21)571 5149 y FG(\()p Fx(\025)p FG(\))83 b(=)g Ft(\000)p FG(2)p Fx(\031)t(iM)1258 5164 y Fw(2)1298 5149 y FG(\()p Fx(\025)p FG(\))1431 5113 y Fw(1)p Fu(=)p Fw(2)1541 5149 y Ft(f)p Fx(U)1667 5113 y Fs(\003)1729 5149 y FG(+)22 b Fx(U)1903 5113 y Fs(\003)1943 5149 y Fx(N)2021 5164 y Fw(1)2061 5149 y FG(\()p Fx(\025)p FG(\))p Fx(U)10 b(F)2333 5164 y Fw(1)2373 5149 y FG(\()p Fx(\025)p FG(\))p Fx(U)2582 5113 y Fs(\003)2621 5149 y Ft(g)p Fx(M)2765 5164 y Fw(1)2805 5149 y FG(\()p Fx(\025)p FG(\))2938 5113 y Fw(1)p Fu(=)p Fw(2)3048 5149 y Fx(;)436 5295 y(S)496 5310 y Fw(22)571 5295 y FG(\()p Fx(\025)p FG(\))83 b(=)g Fx(I)989 5310 y Fl(Q)1046 5322 y Fr(2)p Fo(;\025)1162 5295 y FG(+)22 b(2)p Fx(\031)t(iM)1495 5310 y Fw(2)1535 5295 y FG(\()p Fx(\025)p FG(\))1668 5259 y Fw(1)p Fu(=)p Fw(2)1778 5295 y Fx(U)1854 5259 y Fs(\003)1894 5295 y Fx(F)1957 5310 y Fw(1)1997 5295 y FG(\()p Fx(\025)p FG(\))p Fx(U)10 b(M)2300 5310 y Fw(2)2340 5295 y FG(\()p Fx(\025)p FG(\))2473 5259 y Fw(1)p Fu(=)p Fw(2)2583 5295 y Fx(:)3403 5076 y FG(\(3.19\))109 5468 y Fj(for)34 b(a.e.)45 b Fx(\025)27 b Ft(2)h Fm(R)5 b Fj(.)p Black 1856 5728 a FG(9)p Black eop %%Page: 10 12 10 11 bop Black Black 109 187 a FE(4)161 b(Dissipativ)l(e)54 b(c)l(hannel)f(scattering)109 453 y FG(In)36 b(this)g(section)h(w)m(e)g (consider)g(the)g(\(dissipativ)m(e\))e(scattering)i(system)g Ft(f)p Fx(H)2931 468 y Fw(1)2970 453 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)3228 468 y Fw(1)3267 453 y Ft(g)36 b FG(for)g(a.e.)109 574 y Fx(\025)27 b Ft(2)h Fm(R)5 b FG(,)39 b(where)1440 694 y Fx(H)1521 709 y Fw(1)1560 694 y FG(\()p Fx(\025)p FG(\))28 b(=)f Fx(H)1905 709 y Fw(1)1967 694 y FG(+)22 b Fx(V)2122 709 y Fw(1)2161 694 y FG(\()p Fx(\025)p FG(\))p Fx(;)1130 b FG(\(4.1\))109 864 y(is)44 b(de\014ned)i(for)f(a.e.)80 b Fx(\025)49 b Ft(2)g Fm(R)5 b FG(,)54 b(and)45 b Fx(H)1606 879 y Fw(1)1690 864 y FG(is)f(the)h(self-adjoin)m(t)e(op)s(erator)i(in)f Fv(H)3098 879 y Fw(1)3182 864 y FG(from)f(\(3.1\).)109 984 y(The)33 b(limit)28 b Fx(V)598 999 y Fw(1)638 984 y FG(\()p Fx(\025)p FG(\))f(=)h(lim)1037 999 y Fu(\017)p Fs(!)p Fw(+0)1247 984 y Fx(V)1304 999 y Fw(1)1343 984 y FG(\()p Fx(\025)21 b FG(+)g Fx(i\017)p FG(\))33 b(\(see)g(Lemma)e (4.1\))g(is)h(called)f(the)h Fj(optic)-5 b(al)34 b(p)-5 b(otential)109 1105 y FG(of)41 b(the)h(c)m(hannel)f Fv(H)841 1120 y Fw(1)881 1105 y FG(.)70 b(In)42 b(Theorem)f(4.4)g(b)s(elo)m(w)h (w)m(e)g(establish)f(a)g(connection)h(b)s(et)m(w)m(een)h(the)109 1225 y(scattering)g(matrices)g(corresp)s(onding)h(to)g(the)g (scattering)g(systems)h Ft(f)p Fx(H)2886 1240 y Fw(1)2925 1225 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)3183 1240 y Fw(1)3222 1225 y Ft(g)43 b FG(and)h(the)109 1345 y(c)m(hannel)33 b(scattering)f(matrix)f Fx(S)1291 1360 y Fw(11)1366 1345 y FG(\()p Fx(\025)p FG(\))h(from)f(\(3.18\))h(and)h(\(3.19\))o(.)p Black 109 1586 a Fi(Lemma)k(4.1)p Black 49 w Fj(L)-5 b(et)48 b Fx(V)936 1601 y Fw(1)975 1586 y FG(\()p Fx(z)t FG(\))k(=)f Ft(\000)p Fx(G)p FG(\()p Fx(H)1552 1601 y Fw(2)1623 1586 y Ft(\000)32 b Fx(z)t FG(\))1819 1550 y Fs(\000)p Fw(1)1914 1586 y Fx(G)1991 1550 y Fs(\003)2031 1586 y Fj(,)50 b Fx(z)57 b Ft(2)51 b Fm(C)20 b Ft(n)p Fm(R)11 b Fj(,)57 b(b)-5 b(e)47 b(de\014ne)-5 b(d)46 b(by)57 b FG(\(3.5\))47 b Fj(with)109 1707 y Fx(G)42 b Ft(2)g(B)401 1722 y Fw(1)441 1707 y FG(\()p Fv(H)551 1722 y Fw(2)590 1707 y Fx(;)17 b Fv(H)706 1722 y Fw(1)745 1707 y FG(\))p Fj(.)68 b(Then)41 b(the)i(limit)f Fx(V)1608 1722 y Fw(1)1648 1707 y FG(\()p Fx(\025)p FG(\))f(=)h(lim)2076 1722 y Fu(\017)p Fs(!)p Fw(+0)2286 1707 y Fx(V)2343 1722 y Fw(1)2382 1707 y FG(\()p Fx(\025)28 b FG(+)f Fx(i\017)p FG(\))44 b Fj(exists)e(in)g Ft(B)3228 1722 y Fw(1)3268 1707 y FG(\()p Fv(H)3378 1722 y Fw(1)3417 1707 y FG(\))h Fj(and)109 1827 y Fx(V)166 1842 y Fw(1)205 1827 y FG(\()p Fx(\025)p FG(\))35 b Fj(is)f(dissip)-5 b(ative)34 b(for)h(a.e.)44 b Fx(\025)27 b Ft(2)i Fm(R)5 b Fj(.)109 2068 y Fi(Pro)s(of.)43 b FG(Using)32 b(the)h(factorizations)e(\(3.12\))h(of)g Fx(G)g FG(and)h Fx(G)2271 2032 y Fs(\003)2343 2068 y FG(w)m(e)g(\014nd)898 2277 y Fx(V)955 2292 y Fw(1)995 2277 y FG(\()p Fx(z)t FG(\))28 b(=)g Ft(\000j)p Fx(G)1434 2236 y Fs(\003)1473 2277 y Ft(j)1501 2236 y Fw(1)p Fu(=)p Fw(2)1611 2277 y Fx(U)10 b(N)1765 2292 y Fw(2)1805 2277 y FG(\()p Fx(z)t FG(\))p Fx(U)2006 2236 y Fs(\003)2047 2277 y Ft(j)p Fx(G)2152 2236 y Fs(\003)2191 2277 y Ft(j)2219 2236 y Fw(1)p Fu(=)p Fw(2)2329 2277 y Fx(;)114 b(z)32 b Ft(2)c Fm(C)20 b Ft(n)p Fm(R)11 b Fx(;)595 b FG(\(4.2\))109 2486 y(where)30 b Fx(N)465 2501 y Fw(2)505 2486 y FG(\()p Fx(z)t FG(\))f(is)g(giv)m(en)g(b)m(y)j(\(3.7\))o(.)43 b(According)29 b(to)f(Lemma)g(3.2)h(the)h(limit)25 b(lim)3013 2501 y Fu(\017)p Fs(!)p Fw(+0)3223 2486 y Fx(N)3301 2501 y Fw(2)3340 2486 y FG(\()p Fx(\025)15 b FG(+)g Fx(i\017)p FG(\))109 2606 y(exists)33 b(in)f Ft(B)557 2621 y Fw(2)597 2606 y FG(\()p Fv(H)707 2621 y Fw(1)746 2606 y FG(\))g(and)h(since)g Ft(j)p Fx(G)1350 2570 y Fs(\003)1389 2606 y Ft(j)1417 2570 y Fw(1)p Fu(=)p Fw(2)1554 2606 y Ft(2)28 b(B)1713 2621 y Fw(2)1753 2606 y FG(\()p Fv(H)1863 2621 y Fw(1)1902 2606 y FG(\))33 b(w)m(e)h(conclude)f(that)f(the)h(limit)579 2815 y Fx(V)636 2830 y Fw(1)676 2815 y FG(\()p Fx(\025)p FG(\))27 b(=)54 b(lim)940 2875 y Fu(\017)p Fs(!)p Fw(+0)1145 2815 y Fx(V)1202 2830 y Fw(1)1242 2815 y FG(\()p Fx(\025)22 b FG(+)g Fx(i\017)p FG(\))28 b(=)54 b(lim)1699 2875 y Fu(\017)p Fs(!)p Fw(+0)1904 2815 y Ft(\000j)p Fx(G)2086 2774 y Fs(\003)2126 2815 y Ft(j)2154 2774 y Fw(1)p Fu(=)p Fw(2)2264 2815 y Fx(U)10 b(N)2418 2830 y Fw(2)2458 2815 y FG(\()p Fx(\025)22 b FG(+)g Fx(i\017)p FG(\))p Fx(U)2859 2774 y Fs(\003)2900 2815 y Ft(j)p Fx(G)3005 2774 y Fs(\003)3044 2815 y Ft(j)3072 2774 y Fw(1)p Fu(=)p Fw(2)109 3065 y FG(exists)28 b(in)f Ft(B)547 3080 y Fw(1)587 3065 y FG(\()p Fv(H)697 3080 y Fw(1)736 3065 y FG(\))g(for)g(a.e.)42 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(.)48 b(It)28 b(is)f(not)g(di\016cult) g(to)g(see)i(that)e(Im)16 b Fx(V)2831 3080 y Fw(1)2871 3065 y FG(\()p Fx(z)t FG(\))28 b Ft(\024)g FG(0)f(for)g Fx(z)33 b Ft(2)28 b Fm(C)3587 3029 y Fw(+)109 3185 y FG(and)k(therefore)i(also)d(the)i(limit)c Fx(V)1361 3200 y Fw(1)1400 3185 y FG(\()p Fx(\025)p FG(\))k(is)f(dissipativ)m(e)g(for) g(a.e.)44 b Fx(\025)27 b Ft(2)h Fm(R)5 b FG(.)822 b Fk(\003)109 3446 y FG(It)34 b(follo)m(ws)f(from)g(Lemma)g(4.1)g(that)h(for)g(a.e.) 48 b Fx(\025)31 b Ft(2)g Fm(R)45 b FG(the)34 b(op)s(erator)g Fx(H)2772 3461 y Fw(1)2811 3446 y FG(\()p Fx(\025)p FG(\))c(=)h Fx(H)3162 3461 y Fw(1)3224 3446 y FG(+)23 b Fx(V)3380 3461 y Fw(1)3419 3446 y FG(\()p Fx(\025)p FG(\))34 b(is)109 3567 y(maximal)e(dissipativ)m(e)i(and)i(therefore)g Ft(f)p Fx(H)1726 3582 y Fw(1)1765 3567 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)2023 3582 y Fw(1)2061 3567 y Ft(g)35 b FG(is)g(a)g Fj(dissip)-5 b(ative)36 b(sc)-5 b(attering)38 b(system)d FG(in)109 3687 y(the)e(sense)h(of)e([19,)g(20].)44 b(By)33 b(Theorem)f(4.3)h(of)f([20)o(])h(the)g(corresp)s(onding)f(w)m(a)m(v)m (e)j(op)s(erators)823 3896 y Fx(W)929 3855 y Fu(D)915 3921 y Fw(+)993 3896 y FG(\()p Fx(H)1112 3911 y Fw(1)1151 3896 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)1409 3911 y Fw(1)1448 3896 y FG(\))27 b(=)h Fx(s)22 b Ft(\000)66 b FG(lim)1785 3956 y Fu(t)p Fs(!)p Fw(+)p Fs(1)2023 3896 y Fx(e)2068 3855 y Fu(itH)2175 3864 y Fr(1)2210 3855 y Fw(\()p Fu(\025)p Fw(\))2305 3831 y Fn(\003)2346 3896 y Fx(e)2391 3855 y Fs(\000)p Fu(itH)2553 3864 y Fr(1)2593 3896 y Fx(P)2670 3855 y Fu(ac)2741 3896 y FG(\()p Fx(H)2860 3911 y Fw(1)2899 3896 y FG(\))109 4139 y(and)841 4259 y Fx(W)947 4218 y Fu(D)933 4284 y Fs(\000)1011 4259 y FG(\()p Fx(H)1130 4274 y Fw(1)1169 4259 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)1427 4274 y Fw(1)1466 4259 y FG(\))27 b(=)h Fx(s)22 b Ft(\000)66 b FG(lim)1803 4319 y Fu(t)p Fs(!)p Fw(+)p Fs(1)2041 4259 y Fx(e)2086 4218 y Fs(\000)p Fu(itH)2248 4227 y Fr(1)2283 4218 y Fw(\()p Fu(\025)p Fw(\))2383 4259 y Fx(e)2428 4218 y Fu(itH)2535 4227 y Fr(1)2575 4259 y Fx(P)2652 4218 y Fu(ac)2723 4259 y FG(\()p Fx(H)2842 4274 y Fw(1)2881 4259 y FG(\))109 4468 y(exist)22 b(and)h(are)f(complete)g(whic)m(h)h(yields)f(that)h Ft(f)p Fx(H)1929 4483 y Fw(1)1968 4468 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)2226 4483 y Fw(1)2264 4468 y Ft(g)23 b FG(p)s(erforms)f(a)g (complete)f(dissipativ)m(e)109 4588 y(scattering)28 b(system)i(for)e (a.e.)42 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(.)48 b(The)29 b(asso)s(ciated)g(scattering)f(op)s(erators)g(are)h(de\014ned)h(b)m(y) 1001 4797 y Fx(S)1061 4812 y Fu(D)1125 4797 y FG([)p Fx(\025)p FG(])e(:=)g Fx(W)1501 4756 y Fu(D)1487 4822 y Fw(+)1564 4797 y FG(\()p Fx(H)1683 4812 y Fw(1)1723 4797 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)1981 4812 y Fw(1)2019 4797 y FG(\))2057 4756 y Fs(\003)2097 4797 y Fx(W)2203 4756 y Fu(D)2189 4822 y Fs(\000)2266 4797 y FG(\()p Fx(H)2385 4812 y Fw(1)2425 4797 y FG(\()p Fx(\025)p FG(\))p Fx(;)g(H)2683 4812 y Fw(1)2721 4797 y FG(\))109 5006 y(and)23 b(act)h(on)g(the)g(absolutely)f(con)m(tin)m(uous)h (subspaces)j Fv(H)2173 4970 y Fu(ac)2173 5031 y Fw(1)2244 5006 y FG(\()p Fx(H)2363 5021 y Fw(1)2403 5006 y FG(\).)40 b(Since)24 b Fx(S)2814 5021 y Fu(D)2878 5006 y FG([)p Fx(\025)p FG(])g(comm)m(utes)f(with)109 5126 y Fx(H)190 5141 y Fw(1)262 5126 y FG(the)33 b(scattering)g(op)s(erator)f(is)h (unitary)f(equiv)-5 b(alen)m(t)33 b(to)g(a)g(m)m(ultiplication)28 b(op)s(erator)k(in)g(the)109 5247 y(sp)s(ectral)42 b(represen)m(tation) i Fx(L)1199 5211 y Fw(2)1239 5247 y FG(\()p Fm(R)5 b Fx(;)17 b(d\025;)g Fv(Q)1621 5262 y Fw(1)p Fu(;\026)1728 5247 y FG(\))42 b(of)h Fx(H)2011 5262 y Fw(1)2093 5247 y FG(induced)g(b)m(y)h(a)f(family)d(of)j(con)m(tractions)109 5367 y Ft(f)p Fx(S)219 5382 y Fu(D)282 5367 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))p Ft(g)578 5382 y Fu(\026)p Fs(2)p Fq(R)720 5367 y FG(.)87 b(The)49 b(family)44 b Ft(f)p Fx(S)1472 5382 y Fu(D)1536 5367 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))p Ft(g)1832 5382 y Fu(\026)p Fs(2)p Fq(R)2021 5367 y FG(is)j(called)f(the)h Fj(sc)-5 b(attering)49 b(matrix)e FG(of)f(the)109 5487 y(complete)32 b(dissipativ)m(e)g(scattering)g(system)h Ft(f)p Fx(H)1903 5502 y Fw(1)1943 5487 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)2201 5502 y Fw(1)2239 5487 y Ft(g)p FG(.)p Black 1832 5728 a(10)p Black eop %%Page: 11 13 11 12 bop Black Black 109 171 a FG(Using)24 b(the)h(fact)g(that)g(ev)m (ery)h(maximal)c(dissipativ)m(e)i(op)s(erator)g(admits)g(a)h (self-adjoin)m(t)e(dilation,)109 291 y(i.e.,)40 b(there)f(exists)h(a)e (self-adjoin)m(t)g(op)s(erator)g(in)g(a)g(\(in)g(general\))h(larger)e (Hilb)s(ert)h(space)h(suc)m(h)109 411 y(that)34 b(its)g(compressed)i (resolv)m(en)m(t)f(coincides)g(with)f(the)h(resolv)m(en)m(ts)h(of)e (the)h(maximal)c(dissipa-)109 532 y(tiv)m(e)41 b(op)s(erator)g(for)f (all)g Fx(z)47 b Ft(2)c Fm(C)1278 496 y Fw(+)1343 532 y FG(,)g(cf.)70 b([7,)43 b(Section)e(7],)j(see)e(also)e([12],)j(one)f (concludes)g(from)109 652 y(Prop)s(osition)31 b(3.14)g(of)i([5)o(])g (that)f(the)h(limit)1269 872 y Fv(F)1330 887 y Fw(1)1370 872 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))27 b(=)54 b(lim)1747 932 y Fu(\017)p Fs(!)p Fw(+0)1952 872 y Fv(F)2013 887 y Fw(1)2053 872 y FG([)p Fx(\025)p FG(]\()p Fx(\026)22 b FG(+)g Fx(i\017)p FG(\))109 1131 y(exist)33 b(in)e Ft(B)518 1146 y Fw(1)558 1131 y FG(\()p Fv(H)668 1146 y Fw(1)707 1131 y FG(\))i(for)f(a.e.)44 b Fx(\026)27 b Ft(2)h Fm(R)5 b FG(,)39 b(where)818 1351 y Fv(F)879 1366 y Fw(1)918 1351 y FG([)p Fx(\025)p FG(]\()p Fx(z)t FG(\))29 b(:=)e Ft(j)p Fx(G)1418 1310 y Fs(\003)1458 1351 y Ft(j)1486 1310 y Fw(1)p Fu(=)p Fw(2)1595 1351 y FG(\()p Fx(H)1714 1366 y Fw(1)1753 1351 y FG(\()p Fx(\025)p FG(\))22 b Ft(\000)h Fx(z)t FG(\))2095 1310 y Fs(\000)p Fw(1)2190 1351 y Ft(j)p Fx(G)2295 1310 y Fs(\003)2334 1351 y Ft(j)2362 1310 y Fw(1)p Fu(=)p Fw(2)2472 1351 y Fx(;)114 b(z)32 b Ft(2)c Fm(C)2850 1366 y Fw(+)2915 1351 y Fx(;)109 1571 y FG(is)33 b(de\014ned)i(for)e(a.e.)46 b Fx(\025)30 b Ft(2)f Fm(R)5 b FG(.)53 b(The)34 b(next)h(prop)s (osition)c(is)i(a)h(direct)f(consequence)k(of)c(Theorem)109 1692 y(2.2)f(of)g([16],)g(see)i(also)e([15)o(].)p Black 109 1935 a Fi(Prop)s(osition)j(4.2)p Black 49 w Fj(L)-5 b(et)40 b Fx(G)c Ft(2)g(B)1359 1950 y Fw(1)1399 1935 y FG(\()p Fv(H)1509 1950 y Fw(2)1548 1935 y Fx(;)17 b Fv(H)1664 1950 y Fw(1)1703 1935 y FG(\))40 b Fj(and)f Fx(H)2056 1950 y Fw(1)2095 1935 y FG(\()p Fx(\025)p FG(\))g Fj(b)-5 b(e)39 b(given)g(by)48 b FG(\(4.1\))o Fj(.)59 b(Then)38 b(for)i(a.e.)109 2048 y Fx(\025)47 b Ft(2)h Fm(R)56 b Fj(the)46 b(sc)-5 b(attering)45 b(matrix)g Ft(f)p Fx(S)1505 2063 y Fu(D)1569 2048 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))p Ft(g)1865 2063 y Fu(\026)p Fs(2)p Fq(R)2052 2048 y Fj(of)g(the)g(c)-5 b(omplete)45 b(dissip)-5 b(ative)44 b(sc)-5 b(attering)109 2161 y(system)34 b Ft(f)p Fx(H)558 2176 y Fw(1)598 2161 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)856 2176 y Fw(1)894 2161 y Ft(g)35 b Fj(admits)f(the)h(r)-5 b(epr)g(esentation)249 2390 y Fh(S)305 2404 y Fu(D)369 2390 y FC([)p Fh(\025)p FC(]\()p Fh(\026)p FC(\))26 b(=)f Fh(I)759 2405 y Fl(Q)816 2414 y Fr(1)p Fo(;\026)931 2390 y FC(+)20 b(2)p Fh(\031)s(iM)1241 2404 y Fw(1)1281 2390 y FC(\()p Fh(\026)p FC(\))1406 2352 y Fw(1)p Fu(=)p Fw(2)1517 2390 y Fh(U)1589 2289 y Ff(n)1649 2390 y Fh(N)1722 2404 y Fw(2)1762 2390 y FC(\()p Fh(\025)p FC(\))h(+)f Fh(N)2070 2404 y Fw(2)2109 2390 y FC(\()p Fh(\025)p FC(\))p Fh(U)2304 2352 y Fs(\003)2344 2390 y Fa(F)2400 2404 y Fw(1)2439 2390 y FC([)p Fh(\025)p FC(]\()p Fh(\026)p FC(\))p Fh(U)10 b(N)2812 2404 y Fw(2)2853 2390 y FC(\()p Fh(\025)p FC(\))2976 2289 y Ff(o)3037 2390 y Fh(U)3109 2352 y Fs(\003)3149 2390 y Fh(M)3237 2404 y Fw(1)3276 2390 y FC(\()p Fh(\026)p FC(\))3401 2352 y Fw(1)p Fu(=)p Fw(2)144 2628 y Fj(for)34 b(a.e.)44 b FG(\()p Fx(\026;)17 b(\025)p FG(\))27 b Ft(2)h Fm(R)921 2591 y Fw(2)1002 2628 y Fj(with)34 b(r)-5 b(esp)g(e)g(ct)35 b(to)g(the)g(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e)35 b(in)f Fm(R)2775 2591 y Fw(2)2820 2628 y Fj(.)109 2878 y FG(In)f(the)g(next)g(lemma)d(w)m(e)k(sho)m(w)g(that)e(the)h(limit)c Fv(F)1991 2893 y Fw(1)2030 2878 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\),)309 3098 y Fv(F)370 3113 y Fw(1)409 3098 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))f(=)54 b(lim)784 3158 y Fu(\017)p Fs(!)p Fw(+0)990 3098 y Fv(F)1051 3113 y Fw(1)1091 3098 y FG([)p Fx(\025)p FG(]\()p Fx(\025)22 b FG(+)g Fx(i\017)p FG(\))28 b(=)54 b(lim)1658 3158 y Fu(\017)p Fs(!)p Fw(+0)1864 3098 y Ft(j)p Fx(G)1969 3057 y Fs(\003)2008 3098 y Ft(j)2036 3057 y Fw(1)p Fu(=)p Fw(2)2146 3098 y FG(\()p Fx(H)2265 3113 y Fw(1)2304 3098 y FG(\()p Fx(\025)p FG(\))22 b Ft(\000)h Fx(\025)f Ft(\000)g Fx(i\017)p FG(\))2847 3057 y Fs(\000)p Fw(1)2942 3098 y Ft(j)p Fx(G)3047 3057 y Fs(\003)3086 3098 y Ft(j)3114 3057 y Fw(1)p Fu(=)p Fw(2)3224 3098 y Fx(;)200 b FG(\(4.3\))109 3357 y(exist)33 b(in)e Ft(B)518 3372 y Fw(2)558 3357 y FG(\()p Fv(H)668 3372 y Fw(1)707 3357 y FG(\))i(for)f(a.e.)44 b Fx(\025)27 b Ft(2)h Fm(R)5 b FG(.)p Black 109 3608 a Fi(Lemma)37 b(4.3)p Black 49 w Fj(L)-5 b(et)36 b Fx(L)933 3623 y Fw(0)973 3608 y Fj(,)h Fx(V)58 b Fj(and)35 b Fx(L)i Fj(b)-5 b(e)36 b(given)g(by)45 b FG(\(3.1\))o Fj(,)37 b FG(\(3.2\))f Fj(and)46 b FG(\(3.3\))o Fj(,)37 b(r)-5 b(esp)g(e)g(ctively,)36 b(with)109 3728 y Fx(G)41 b Ft(2)g(B)399 3743 y Fw(1)439 3728 y FG(\()p Fv(H)549 3743 y Fw(2)588 3728 y Fx(;)17 b Fv(H)704 3743 y Fw(1)743 3728 y FG(\))p Fj(.)67 b(F)-7 b(urther,)43 b(let)f Fx(F)1476 3743 y Fw(1)1516 3728 y FG(\()p Fx(\025)p FG(\))g Fj(b)-5 b(e)42 b(as)f(in)h(L)-5 b(emma)42 b FG(3.2)f Fj(and)h(let)g Fx(H)3017 3743 y Fw(1)3056 3728 y FG(\()p Fx(\025)p FG(\))g Fj(b)-5 b(e)42 b(de\014ne)-5 b(d)109 3849 y(by)47 b FG(\(4.1\))o Fj(.)55 b(Then)37 b(the)h(limit)g Fv(F)1251 3864 y Fw(1)1290 3849 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))g Fj(in)46 b FG(\(4.3\))37 b Fj(exists)h(in)g Ft(B)2399 3864 y Fw(2)2439 3849 y FG(\()p Fv(H)2549 3864 y Fw(1)2588 3849 y FG(\))g Fj(for)g(a.e.)54 b Fx(\025)33 b Ft(2)h Fm(R)49 b Fj(and)38 b(the)109 3969 y(r)-5 b(elation)1525 4090 y Fv(F)1586 4105 y Fw(1)1625 4090 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))28 b(=)f Fx(F)2063 4105 y Fw(1)2103 4090 y FG(\()p Fx(\025)p FG(\))1215 b(\(4.4\))109 4264 y Fj(holds)34 b(for)g(a.e.)45 b Fx(\025)27 b Ft(2)h Fm(R)5 b Fj(.)109 4515 y Fi(Pro)s(of.)43 b FG(W)-8 b(e)33 b(ha)m(v)m(e)390 4714 y Fx(F)453 4729 y Fw(1)492 4714 y FG(\()p Fx(z)t FG(\))23 b Ft(\000)g Fv(F)801 4729 y Fw(1)840 4714 y FG([)p Fx(\025)p FG(]\()p Fx(z)t FG(\))28 b(=)g Ft(j)p Fx(G)1313 4672 y Fs(\003)1352 4714 y Ft(j)1380 4672 y Fw(1)p Fu(=)p Fw(2)1490 4633 y Fp(\000)1535 4714 y FG(\()p Fx(H)1654 4729 y Fw(1)1693 4714 y FG(\()p Fx(z)t FG(\))23 b Ft(\000)g Fx(z)t FG(\))2028 4672 y Fs(\000)p Fw(1)2145 4714 y Ft(\000)g FG(\()p Fx(H)2364 4729 y Fw(1)2403 4714 y FG(\()p Fx(\025)p FG(\))f Ft(\000)g Fx(z)t FG(\))2744 4672 y Fs(\000)p Fw(1)2839 4633 y Fp(\001)2885 4714 y Ft(j)p Fx(G)2990 4672 y Fs(\003)3029 4714 y Ft(j)3057 4672 y Fw(1)p Fu(=)p Fw(2)645 4874 y FG(=)28 b Ft(j)p Fx(G)854 4833 y Fs(\003)893 4874 y Ft(j)921 4833 y Fw(1)p Fu(=)p Fw(2)1031 4874 y FG(\()p Fx(H)1150 4889 y Fw(1)1189 4874 y FG(\()p Fx(z)t FG(\))23 b Ft(\000)f Fx(z)t FG(\))1523 4833 y Fs(\000)p Fw(1)1618 4874 y FG(\()p Fx(V)1713 4889 y Fw(1)1753 4874 y FG(\()p Fx(\025)p FG(\))f Ft(\000)i Fx(V)2064 4889 y Fw(1)2103 4874 y FG(\()p Fx(z)t FG(\)\)\()p Fx(H)2385 4889 y Fw(1)2425 4874 y FG(\()p Fx(\025)p FG(\))f Ft(\000)h Fx(z)t FG(\))2767 4833 y Fs(\000)p Fw(1)2862 4874 y Ft(j)p Fx(G)2967 4833 y Fs(\003)3006 4874 y Ft(j)3034 4833 y Fw(1)p Fu(=)p Fw(2)3143 4874 y Fx(:)3451 4792 y FG(\(4.5\))109 5093 y(F)-8 b(rom)31 b(\(4.2\))h(w)m(e)h(obtain)797 5313 y Fx(V)854 5328 y Fw(1)893 5313 y FG(\()p Fx(\025)p FG(\))22 b Ft(\000)h Fx(V)1205 5328 y Fw(1)1244 5313 y FG(\()p Fx(z)t FG(\))28 b(=)g Ft(j)p Fx(G)1606 5271 y Fs(\003)1645 5313 y Ft(j)1673 5271 y Fw(1)p Fu(=)p Fw(2)1783 5313 y Fx(U)1859 5232 y Fp(\000)1905 5313 y Fx(N)1983 5328 y Fw(2)2023 5313 y FG(\()p Fx(z)t FG(\))22 b Ft(\000)h Fx(N)2348 5328 y Fw(2)2387 5313 y FG(\()p Fx(\025)p FG(\))2520 5232 y Fp(\001)2566 5313 y Fx(U)2642 5271 y Fs(\003)2682 5313 y Ft(j)p Fx(G)2787 5271 y Fs(\003)2826 5313 y Ft(j)2854 5271 y Fw(1)p Fu(=)p Fw(2)p Black 1832 5728 a FG(11)p Black eop %%Page: 12 14 12 13 bop Black Black 109 171 a FG(and)39 b(inserting)g(this)g (expression)i(in)m(to)d(\(4.5\))h(and)h(using)f(the)h(de\014nitions)f (of)g Fx(F)3126 186 y Fw(1)3166 171 y FG(\()p Fx(z)t FG(\))h(in)e(\(3.8\))109 291 y(and)32 b Fv(F)359 306 y Fw(1)399 291 y FG([)p Fx(\025)p FG(]\()p Fx(z)t FG(\))h(yields)733 487 y Fx(F)796 502 y Fw(1)836 487 y FG(\()p Fx(z)t FG(\))22 b Ft(\000)h Fv(F)1144 502 y Fw(1)1183 487 y FG([)p Fx(\025)p FG(]\()p Fx(z)t FG(\))29 b(=)e Fx(F)1614 502 y Fw(1)1654 487 y FG(\()p Fx(z)t FG(\))p Fx(U)10 b FG(\()p Fx(N)1971 502 y Fw(2)2012 487 y FG(\()p Fx(z)t FG(\))22 b Ft(\000)h Fx(N)2337 502 y Fw(2)2377 487 y FG(\()p Fx(\025)p FG(\)\))p Fx(U)2624 446 y Fs(\003)2663 487 y Fv(F)2724 502 y Fw(1)2764 487 y FG([)p Fx(\025)p FG(]\()p Fx(z)t FG(\))p Fx(:)109 683 y FG(Hence)792 804 y Fx(F)855 819 y Fw(1)895 804 y FG(\()p Fx(z)t FG(\))28 b(=)g Ft(f)o Fx(I)1244 819 y Fl(H)1294 828 y Fr(1)1355 804 y FG(+)22 b Fx(F)1516 819 y Fw(1)1556 804 y FG(\()p Fx(z)t FG(\))p Fx(U)10 b FG(\()p Fx(N)1873 819 y Fw(2)1913 804 y FG(\()p Fx(z)t FG(\))23 b Ft(\000)g Fx(N)2239 819 y Fw(2)2278 804 y FG(\()p Fx(\025)p FG(\)\))p Fx(U)2525 763 y Fs(\003)2565 804 y Ft(g)16 b Fv(F)2692 819 y Fw(1)2732 804 y FG([)p Fx(\025)p FG(]\()p Fx(z)t FG(\))109 968 y(and)32 b(for)g Fx(z)h FG(=)27 b Fx(\025)22 b FG(+)g Fx(i\017)p FG(,)34 b Fx(\017)28 b(>)f FG(0)33 b(su\016cien)m(tly)g(small,)d(the)j(op)s (erator)1149 1164 y Ft(f)o Fx(I)1241 1179 y Fl(H)1291 1188 y Fr(1)1352 1164 y FG(+)22 b Fx(F)1513 1179 y Fw(1)1553 1164 y FG(\()p Fx(z)t FG(\))p Fx(U)10 b FG(\()p Fx(N)1870 1179 y Fw(2)1910 1164 y FG(\()p Fx(z)t FG(\))23 b Ft(\000)g Fx(N)2236 1179 y Fw(2)2275 1164 y FG(\()p Fx(\025)p FG(\)\))p Fx(U)2522 1123 y Fs(\003)2562 1164 y Ft(g)109 1360 y FG(is)32 b(in)m(v)m(ertible.)43 b(Therefore)33 b(w)m(e)h(conclude)331 1557 y Ft(f)p Fx(I)424 1572 y Fl(H)474 1581 y Fr(1)535 1557 y FG(+)22 b Fx(F)696 1572 y Fw(1)735 1557 y FG(\()p Fx(\025)g FG(+)g Fx(i\017)p FG(\))p Fx(U)10 b FG(\()p Fx(N)1252 1572 y Fw(2)1293 1557 y FG(\()p Fx(\025)22 b FG(+)g Fx(i\017)p FG(\))h Ft(\000)g Fx(N)1819 1572 y Fw(2)1858 1557 y FG(\()p Fx(\025)p FG(\)\))p Fx(U)2105 1516 y Fs(\003)2145 1557 y Ft(g)2195 1508 y Fs(\000)p Fw(1)2306 1557 y Fx(F)2369 1572 y Fw(1)2408 1557 y FG(\()p Fx(\025)f FG(+)g Fx(i\017)p FG(\))28 b(=)g Fv(F)2926 1572 y Fw(1)2965 1557 y FG([)p Fx(\025)p FG(]\()p Fx(\025)22 b FG(+)g Fx(i\017)p FG(\))p Fx(:)109 1753 y FG(F)-8 b(rom)27 b(this)h(represen)m(tation)h(w)m(e)h(get)e(the)h(existence)i(of)d Fv(F)2222 1768 y Fw(1)2261 1753 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))g(in)g Ft(B)2708 1768 y Fw(2)2748 1753 y FG(\()p Fv(H)2858 1768 y Fw(1)2897 1753 y FG(\))h(and)f(the)h (equalit)m(y)109 1873 y(\(4.4\))j(for)g(a.e.)43 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(.)2622 b Fk(\003)109 2127 y FG(The)45 b(next)h(theorem)e(is)h(the)g(main)e(result)h(of)h(this)f(section.)80 b(W)-8 b(e)45 b(sho)m(w)h(ho)m(w)f(the)h(c)m(hannel)109 2247 y(scattering)h(matrix)f Fx(S)966 2262 y Fw(11)1041 2247 y FG(\()p Fx(\025)p FG(\))h(of)g(the)h(scattering)f(system)h Ft(f)p Fx(L;)17 b(L)2559 2262 y Fw(0)2599 2247 y Ft(g)47 b FG(is)g(connected)i(with)e(the)109 2368 y(scattering)32 b(matrices)g Fx(S)1010 2383 y Fu(D)1074 2368 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))g(of)g(the)h(dissipativ)m(e)f (scattering)g(systems)i Ft(f)p Fx(H)3052 2383 y Fw(1)3091 2368 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)3349 2383 y Fw(1)3388 2368 y Ft(g)p FG(.)p Black 109 2598 a Fi(Theorem)37 b(4.4)p Black 49 w Fj(L)-5 b(et)28 b Ft(f)p Fx(L;)17 b(L)1163 2613 y Fw(0)1202 2598 y Ft(g)27 b Fj(b)-5 b(e)28 b(the)f(sc)-5 b(attering)27 b(system)h(fr)-5 b(om)27 b(Se)-5 b(ction)26 b FG(3)p Fj(,)j(wher)-5 b(e)27 b Fx(L)3293 2613 y Fw(0)3333 2598 y Fj(,)i Fx(V)49 b Fj(and)109 2718 y Fx(L)42 b Fj(ar)-5 b(e)42 b(given)f(by)51 b FG(\(3.1\))o Fj(,)44 b FG(\(3.2\))e Fj(and)51 b FG(\(3.3\))p Fj(,)44 b(r)-5 b(esp)g(e)g(ctively,)43 b(and)e Fx(G)g Ft(2)g(B)2826 2733 y Fw(1)2866 2718 y FG(\()p Fv(H)2976 2733 y Fw(2)3015 2718 y Fx(;)17 b Fv(H)3131 2733 y Fw(1)3170 2718 y FG(\))p Fj(.)66 b(F)-7 b(urther,)109 2839 y(let)39 b Ft(f)p Fx(S)360 2854 y Fu(ij)420 2839 y FG(\()p Fx(\025)p FG(\))p Ft(g)p Fj(,)g Fx(i;)17 b(j)41 b FG(=)35 b(1)p Fx(;)17 b FG(2)p Fj(,)40 b(b)-5 b(e)38 b(the)h(c)-5 b(orr)g(esp)g(onding)37 b(sc)-5 b(attering)39 b(matrix)f(fr)-5 b(om)46 b FG(\(3.18\))39 b Fj(and)f(let)109 2959 y Fx(S)169 2974 y Fu(D)233 2959 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))27 b Fj(b)-5 b(e)26 b(the)i(sc)-5 b(attering)27 b(matric)-5 b(es)26 b(of)h(the)g(dissip)-5 b(ative)27 b(sc)-5 b(attering)26 b(systems)h Ft(f)p Fx(H)3236 2974 y Fw(1)3276 2959 y FG(\()p Fx(\025)p FG(\))p Fx(;)17 b(H)3534 2974 y Fw(1)3572 2959 y Ft(g)p Fj(.)109 3080 y(Then)34 b(the)h(sc)-5 b(attering)34 b(matrix)h Fx(S)1342 3095 y Fu(D)1406 3080 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))g Fj(exists)f(for)h(a.e)f Fx(\025)28 b Ft(2)g Fm(R)46 b Fj(and)34 b(satis\014es)g(the)h(r)-5 b(elation)1497 3276 y Fx(S)1557 3291 y Fu(D)1621 3276 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))28 b(=)f Fx(S)2056 3291 y Fw(11)2131 3276 y FG(\()p Fx(\025)p FG(\))109 3472 y Fj(for)34 b(a.e.)45 b Fx(\025)27 b Ft(2)h Fm(R)5 b Fj(.)109 3695 y Fi(Pro)s(of.)42 b FG(F)-8 b(rom)28 b(Prop)s(osition)h(4.2)g(and)h(Lemma)f(4.3)g(w)m(e)i(obtain)e(that)h Fx(S)2782 3710 y Fu(D)2846 3695 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))g(exists)g(for)g(a.e)109 3808 y Fx(\025)d Ft(2)h Fm(R)44 b FG(and)32 b(has)h(the)g(form)207 3992 y Fh(S)263 4006 y Fu(D)326 3992 y FC([)p Fh(\025)p FC(]\()p Fh(\026)p FC(\))27 b(=)e Fh(I)717 4007 y Fl(Q)774 4016 y Fr(1)p Fo(;\026)889 3992 y FC(+)20 b(2)p Fh(\031)s(iM)1199 4006 y Fw(1)1239 3992 y FC(\()p Fh(\026)p FC(\))1364 3954 y Fw(1)p Fu(=)p Fw(2)1475 3992 y Fh(U)1547 3891 y Ff(n)1607 3992 y Fh(N)1680 4006 y Fw(2)1720 3992 y FC(\()p Fh(\025)p FC(\))h(+)f Fh(N)2028 4006 y Fw(2)2067 3992 y FC(\()p Fh(\025)p FC(\))p Fh(U)2262 3954 y Fs(\003)2302 3992 y Fh(F)2360 4006 y Fw(1)2400 3992 y FC(\()p Fh(\025)p FC(\))p Fh(U)10 b(N)2668 4006 y Fw(2)2708 3992 y FC(\()p Fh(\025)p FC(\))2831 3891 y Ff(o)2893 3992 y Fh(U)2965 3954 y Fs(\003)3004 3992 y Fh(M)3092 4006 y Fw(1)3132 3992 y FC(\()p Fh(\026)p FC(\))3257 3954 y Fw(1)p Fu(=)p Fw(2)3465 3992 y FC(\(4.6\))141 4199 y FG(A)33 b(similar)c(calculation) h(as)j(in)f(the)h(pro)s(of)f(of)g(Lemma)f(3.1)h(sho)m(ws)830 4395 y Fx(F)893 4410 y Fw(2)932 4395 y FG(\()p Fx(z)t FG(\))d(=)e Fx(N)1267 4410 y Fw(2)1307 4395 y FG(\()p Fx(z)t FG(\))c(+)f Fx(N)1631 4410 y Fw(2)1670 4395 y FG(\()p Fx(z)t FG(\))p Fx(U)1871 4354 y Fs(\003)1912 4395 y Fx(F)1975 4410 y Fw(1)2014 4395 y FG(\()p Fx(z)t FG(\))p Fx(U)10 b(N)2293 4410 y Fw(2)2334 4395 y FG(\()p Fx(z)t FG(\))p Fx(;)115 b(z)33 b Ft(2)28 b Fm(C)2839 4410 y Fw(+)2904 4395 y Fx(:)109 4592 y FG(If)k Fx(z)37 b FG(tends)d(to)e Fx(\025)c Ft(2)g Fm(R)5 b FG(,)39 b(then)33 b(w)m(e)g(get)1047 4788 y Fx(F)1110 4803 y Fw(2)1150 4788 y FG(\()p Fx(\025)p FG(\))27 b(=)h Fx(N)1492 4803 y Fw(2)1532 4788 y FG(\()p Fx(\025)p FG(\))21 b(+)h Fx(N)1862 4803 y Fw(2)1902 4788 y FG(\()p Fx(\025)p FG(\))p Fx(U)2111 4747 y Fs(\003)2151 4788 y Fx(F)2214 4803 y Fw(1)2253 4788 y FG(\()p Fx(\025)p FG(\))p Fx(U)10 b(N)2540 4803 y Fw(2)2580 4788 y FG(\()p Fx(\025)p FG(\))738 b(\(4.7\))109 4984 y(for)32 b(a.e.)43 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(.)49 b(Inserting)34 b(\(4.7\))f(in)m(to)e(\(4.6\))h(w)m(e)i (obtain)812 5180 y Fx(S)872 5195 y Fu(D)936 5180 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))28 b(=)f Fx(I)1354 5195 y Fl(Q)1411 5207 y Fr(1)p Fo(;\025)1528 5180 y FG(+)22 b(2)p Fx(\031)t(iM)1861 5195 y Fw(1)1901 5180 y FG(\()p Fx(\025)p FG(\))2034 5139 y Fw(1)p Fu(=)p Fw(2)2144 5180 y Fx(U)10 b(F)2283 5195 y Fw(2)2323 5180 y FG(\()p Fx(\025)p FG(\))p Fx(U)2532 5139 y Fs(\003)2572 5180 y Fx(M)2666 5195 y Fw(1)2705 5180 y FG(\()p Fx(\025)p FG(\))2838 5139 y Fw(1)p Fu(=)p Fw(2)109 5377 y FG(and)32 b(b)m(y)i(Prop)s (osition)d(3.3)h(this)g(coincides)g(with)h Fx(S)2001 5392 y Fw(11)2075 5377 y FG(\()p Fx(\025)p FG(\))g(for)f(a.e.)43 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(.)723 b Fk(\003)p Black 1832 5728 a FG(12)p Black eop %%Page: 13 15 13 14 bop Black Black 109 187 a FE(5)161 b(Lax-Phillips)57 b(c)l(hannel)52 b(scattering)109 453 y FG(Similarly)25 b(to)k(Lemma)f(4.1)g(one)i(v)m(eri\014es)g(that)f Fx(V)1897 468 y Fw(2)1937 453 y FG(\()p Fx(\025)p FG(\))e(=)h(lim)2336 468 y Fu(\017)p Fs(!)p Fw(+0)2546 453 y Fx(V)2603 468 y Fw(2)2642 453 y FG(\()p Fx(\025)15 b FG(+)g Fx(i\017)p FG(\))31 b(exists)f(in)e Ft(B)3425 468 y Fw(1)3465 453 y FG(\()p Fv(H)3575 468 y Fw(2)3614 453 y FG(\))109 574 y(for)36 b(a.e.)58 b Fx(\025)35 b Ft(2)h Fm(R)5 b FG(.)63 b(The)38 b(limit)c Fx(V)1316 589 y Fw(2)1355 574 y FG(\()p Fx(\025)p FG(\),)k(whic)m(h)g(is)e(called)g(the)i(optical)d(p)s(oten)m (tial)g(of)i(the)h(c)m(han-)109 694 y(nel)k Fv(H)348 709 y Fw(2)387 694 y FG(,)j(is)d(dissipativ)m(e)g(for)f(a.e.)74 b Fx(\025)44 b Ft(2)h Fm(R)5 b FG(.)79 b(The)43 b(optical)d(p)s(oten)m (tial)h(de\014nes)j(the)f(maximal)109 815 y(dissipativ)m(e)32 b(op)s(erator)1440 935 y Fx(H)1521 950 y Fw(2)1560 935 y FG(\()p Fx(\025)p FG(\))c(:=)f Fx(H)1932 950 y Fw(2)1994 935 y FG(+)22 b Fx(V)2149 950 y Fw(2)2188 935 y FG(\()p Fx(\025)p FG(\))109 1109 y(for)g(a.e.)40 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(.)46 b(The)24 b(op)s(erator)e Fx(H)1407 1124 y Fw(2)1447 1109 y FG(\()p Fx(\025)p FG(\))g(decomp)s(oses)i(for)e (a.e.)41 b Fx(\025)27 b Ft(2)h Fm(R)34 b FG(in)m(to)22 b(a)h(self-adjoin)m(t)e(part)109 1230 y(and)34 b(a)g(completely)g (non-self-adjoin)m(t)e(part.)49 b(Let)35 b(\002)2077 1245 y Fw(2)2116 1230 y FG([)p Fx(\025)p FG(]\()p Fx(\030)5 b FG(\),)34 b Fx(\030)h Ft(2)d Fm(C)2654 1245 y Fs(\000)2719 1230 y FG(,)j(b)s(e)f(the)h(c)m(haracteristic)109 1350 y(function,)40 b(cf.)65 b([12)o(],)42 b(of)d(the)h(completely)e (non-self-adjoin)m(t)g(part)h(of)g Fx(H)2817 1365 y Fw(2)2856 1350 y FG(\()p Fx(\025)p FG(\).)64 b(W)-8 b(e)40 b(are)f(going)109 1470 y(to)45 b(v)m(erify)i Fx(S)586 1485 y Fw(11)661 1470 y FG(\()p Fx(\025)p FG(\))j(=)h(\002)1047 1485 y Fw(2)1086 1470 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))1330 1434 y Fs(\003)1415 1470 y FG(for)46 b(a.e.)84 b Fx(\025)51 b Ft(2)g Fm(R)57 b FG(whic)m(h)47 b(sho)m(ws)g(that)f(for) g(a.e.)84 b Fx(\025)51 b Ft(2)g Fm(R)109 1591 y FG(the)33 b(scattering)f(matrix)g Fx(S)1105 1606 y Fw(11)1179 1591 y FG(\()p Fx(\025)p FG(\))h(can)g(b)s(e)g(regarded)g(as)g(the)h(result) e(of)h(a)f(certain)h(Lax-Phillips)109 1711 y(scattering)f(theory)-8 b(,)33 b(cf.)44 b([1,)32 b(2,)h(3)o(,)g(4,)f(14].)109 1879 y(There)h(is)g(an)f(orthogonal)f(decomp)s(osition)1505 2099 y Fv(H)1577 2114 y Fw(2)1644 2099 y FG(=)c Fv(H)1819 2058 y Fu(cns)1819 2123 y Fw(2)p Fu(;\025)1952 2099 y Ft(\010)22 b Fv(H)2123 2052 y Fu(sel)q(f)2123 2127 y Fw(2)p Fu(;\025)109 2341 y FG(for)38 b(a.e.)64 b Fx(\025)39 b Ft(2)h Fm(R)50 b FG(suc)m(h)41 b(that)e Fv(H)1304 2305 y Fu(cns)1304 2367 y Fw(2)p Fu(;\025)1453 2341 y FG(and)h Fv(H)1722 2294 y Fu(sel)q(f)1722 2369 y Fw(2)p Fu(;\025)1893 2341 y FG(reduce)h Fx(H)2290 2356 y Fw(2)2329 2341 y FG(\()p Fx(\025)p FG(\))e(in)m(to)g(a)g(completely)f(non-self-)109 2483 y(adjoin)m(t)31 b(op)s(erator)h Fx(H)921 2447 y Fu(cns)913 2508 y Fw(2)1031 2483 y FG(\()p Fx(\025)p FG(\))g(and)h(a)f(self-adjoin)m(t)f(op)s(erator)h Fx(H)2451 2436 y Fu(sel)q(f)2443 2508 y Fw(2)2583 2483 y FG(\()p Fx(\025)p FG(\),)1271 2703 y Fx(H)1352 2718 y Fw(2)1391 2703 y FG(\()p Fx(\025)p FG(\))c(=)f Fx(H)1744 2662 y Fu(cns)1736 2728 y Fw(2)1854 2703 y FG(\()p Fx(\025)p FG(\))22 b Ft(\010)h Fx(H)2198 2656 y Fu(sel)q(f)2190 2728 y Fw(2)2330 2703 y FG(\()p Fx(\025)p FG(\))p Fx(:)109 2923 y FG(T)-8 b(aking)32 b(in)m(to)g(accoun)m(t)h(Prop)s(osition)e (3.14)h(of)g([5])g(w)m(e)i(get)f(that)1037 3143 y Ft(=)p FG(m)16 b(\()p Fx(V)1302 3158 y Fw(2)1341 3143 y FG(\()p Fx(\025)p FG(\)\))28 b(=)f Ft(\000)p Fx(\031)t Ft(j)p Fx(G)p Ft(j)1912 3102 y Fw(1)p Fu(=)p Fw(2)2022 3143 y Fx(U)2098 3102 y Fs(\003)2138 3143 y Fx(M)2232 3158 y Fw(1)2272 3143 y FG(\()p Fx(\025)p FG(\))p Fx(U)10 b Ft(j)p Fx(G)p Ft(j)2614 3102 y Fw(1)p Fu(=)p Fw(2)109 3363 y FG(for)32 b(a.e.)43 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(.)49 b(Let)33 b(us)g(in)m(tro)s(duce)g(the)g(op)s(erator)1280 3583 y Fx(\013)q FG(\()p Fx(\025)p FG(\))27 b(:=)1633 3493 y Fp(p)p 1733 3493 375 4 v 90 x FG(2)p Fx(\031)t(M)1935 3598 y Fw(1)1975 3583 y FG(\()p Fx(\025)p FG(\))g Fx(U)10 b Ft(j)p Fx(G)p Ft(j)2344 3542 y Fw(1)p Fu(=)p Fw(2)2454 3583 y Fx(:)970 b FG(\(5.1\))109 3803 y(Notice)32 b(that)1400 3924 y(clo)o Ft(f)p FG(ran)16 b(\()p Fx(\013)q FG(\()p Fx(\025)p FG(\)\))p Ft(g)27 b FG(=)g Fv(Q)2260 3939 y Fw(1)p Fu(;\025)109 4098 y FG(for)38 b(a.e.)63 b Fx(\025)38 b Ft(2)h Fm(R)5 b FG(.)69 b(With)38 b(the)i(completely)e (non-self-adjoin)m(t)e(part)j Fx(H)2769 4062 y Fu(cns)2879 4098 y FG(\()p Fx(\025)p FG(\))g(one)g(asso)s(ciates)109 4218 y(the)33 b(c)m(haracteristic)f(function)g(\002)1334 4233 y Fw(2)1373 4218 y FG([)p Fx(\025)p FG(]\()p Ft(\001)p FG(\))c(:)g Fv(Q)1753 4233 y Fw(1)p Fu(;\025)1881 4218 y Ft(\000)-16 b(!)27 b Fv(Q)2151 4233 y Fw(1)p Fu(;\025)2284 4218 y FG(de\014ned)34 b(b)m(y)907 4438 y(\002)983 4453 y Fw(2)1022 4438 y FG([)p Fx(\025)p FG(]\()p Fx(\030)5 b FG(\))27 b(:=)h Fx(I)1458 4453 y Fl(Q)1515 4465 y Fr(1)p Fo(;\025)1631 4438 y Ft(\000)23 b Fx(i\013)q FG(\()p Fx(\025)p FG(\)\()p Fx(H)2079 4453 y Fw(2)2118 4438 y FG(\()p Fx(\025)p FG(\))2251 4397 y Fs(\003)2312 4438 y Ft(\000)g Fx(\030)5 b FG(\))2498 4397 y Fs(\000)p Fw(1)2592 4438 y Fx(\013)q FG(\()p Fx(\025)p FG(\))2788 4397 y Fs(\003)2827 4438 y Fx(;)109 4658 y(\030)32 b Ft(2)c Fm(C)344 4673 y Fs(\000)409 4658 y FG(.)42 b(The)30 b(c)m (haracteristic)f(function)g(is)f(a)h(con)m(traction-v)-5 b(alued)28 b(holomorphic)f(function)i(in)109 4779 y Fm(C)175 4794 y Fs(\000)240 4779 y FG(.)43 b(F)-8 b(rom)31 b([12,)h(Section)h (V.2])f(w)m(e)i(get)f(that)f(the)h(b)s(oundary)g(v)-5 b(alues)1156 4999 y(\002)1232 5014 y Fw(2)1271 4999 y FG([)p Fx(\025)p FG(]\()p Fx(\026)p FG(\))28 b(:=)f Fx(s)c Ft(\000)49 b FG(lim)1843 5058 y Fu(\017)p Fs(!)p Fw(+0)2049 4999 y FG(\002)2125 5014 y Fw(2)2164 4999 y FG([)p Fx(\025)p FG(]\()p Fx(\026)22 b Ft(\000)h Fx(i\017)p FG(\))109 5252 y(exist)33 b(for)f(a.e.)43 b Fx(\026)28 b Ft(2)g Fm(R)5 b FG(.)p Black 1832 5728 a(13)p Black eop %%Page: 14 16 14 15 bop Black Black Black 109 171 a Fi(Theorem)37 b(5.1)p Black 49 w Fj(L)-5 b(et)34 b Fx(L)1009 186 y Fw(0)1049 171 y Fj(,)h Fx(V)55 b Fj(and)34 b Fx(L)h Fj(b)-5 b(e)34 b(given)g(by)43 b FG(\(3.1\))o Fj(,)35 b FG(\(3.2\))f Fj(and)44 b FG(\(3.3\))o Fj(.)h(If)34 b(the)g(c)-5 b(ondition)109 291 y Fx(G)27 b Ft(2)h(B)372 306 y Fw(1)412 291 y FG(\()p Fv(H)522 306 y Fw(2)561 291 y Fx(;)17 b Fv(H)677 306 y Fw(1)716 291 y FG(\))35 b Fj(is)g(satis\014e)-5 b(d,)34 b(then)g(the)h(limit)g FG(\002)1981 306 y Fw(2)2020 291 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))p Fj(,)1158 504 y FG(\002)1234 519 y Fw(2)1273 504 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))28 b(:=)f Fx(s)c Ft(\000)49 b FG(lim)1843 564 y Fu(\017)p Fs(!)p Fw(+0)2049 504 y FG(\002)2125 519 y Fw(2)2164 504 y FG([)p Fx(\025)p FG(]\()p Fx(\025)22 b Ft(\000)h Fx(i\017)p FG(\))109 750 y Fj(exists)34 b(for)h(a.e.)44 b Fx(\025)28 b Ft(2)g Fm(R)46 b Fj(and)34 b(the)h(r)-5 b(elation)1482 963 y Fx(S)1542 978 y Fw(11)1616 963 y FG(\()p Fx(\025)p FG(\))28 b(=)f(\002)1956 978 y Fw(2)1996 963 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))2240 922 y Fs(\003)109 1176 y Fj(holds)34 b(for)g(a.e.)45 b Fx(\025)27 b Ft(2)h Fm(R)5 b Fj(.)109 1421 y Fi(Pro)s(of.)43 b FG(W)-8 b(e)33 b(set)681 1634 y(\002)757 1593 y Fs(\003)757 1658 y Fw(2)796 1634 y FG([)p Fx(\025)p FG(]\()p Fx(\030)5 b FG(\))27 b(:=)h(\002)1265 1649 y Fw(2)1304 1634 y FG([)p Fx(\025)p FG(]\()p 1453 1553 48 4 v Fx(\030)5 b FG(\))1539 1593 y Fs(\003)1606 1634 y FG(=)27 b Fx(I)1752 1649 y Fl(Q)1809 1661 y Fr(1)p Fo(;\025)1926 1634 y FG(+)22 b Fx(i\013)q FG(\()p Fx(\025)p FG(\)\()p Fx(H)2372 1649 y Fw(2)2411 1634 y FG(\()p Fx(\025)p FG(\))g Ft(\000)g Fx(\030)5 b FG(\))2751 1593 y Fs(\000)p Fw(1)2845 1634 y Fx(\013)q FG(\()p Fx(\025)p FG(\))3041 1593 y Fs(\003)109 1847 y Fx(\030)32 b Ft(2)c Fm(C)344 1862 y Fw(+)409 1847 y FG(.)43 b(Using)34 b(\(5.1\))e(w)m(e)h(get)768 2070 y(\002)844 2029 y Fs(\003)844 2095 y Fw(2)884 2070 y FG([)p Fx(\025)p FG(]\()p Fx(\030)5 b FG(\))27 b(=)g Fx(I)1292 2085 y Fl(Q)1349 2097 y Fr(1)p Fo(;\025)1466 2070 y FG(+)22 b(2)p Fx(\031)t(i)1705 1980 y Fp(p)p 1804 1980 267 4 v 1804 2070 a Fx(M)1898 2085 y Fw(1)1938 2070 y FG(\()p Fx(\025)p FG(\))28 b Fx(U)10 b Fv(F)2236 2085 y Fw(2)2276 2070 y FG([)p Fx(\025)p FG(]\()p Fx(\030)5 b FG(\))p Fx(U)2587 2029 y Fs(\003)2626 1980 y Fp(p)p 2726 1980 V 90 x Fx(M)2820 2085 y Fw(1)2860 2070 y FG(\()p Fx(\025)p FG(\))109 2283 y(for)32 b(a.e.)43 b Fx(\025)28 b Ft(2)g Fm(R)5 b FG(,)39 b(where)861 2496 y Fv(F)922 2511 y Fw(2)961 2496 y FG([)p Fx(\025)p FG(]\()p Fx(\030)5 b FG(\))27 b(:=)h Ft(j)p Fx(G)p Ft(j)1487 2455 y Fw(1)p Fu(=)p Fw(2)1596 2496 y FG(\()p Fx(H)1715 2511 y Fw(2)1755 2496 y FG(\()p Fx(\025)p FG(\))21 b Ft(\000)i Fx(\030)5 b FG(\))2095 2455 y Fs(\000)p Fw(1)2189 2496 y Ft(j)p Fx(G)p Ft(j)2322 2455 y Fw(1)p Fu(=)p Fw(2)2431 2496 y Fx(;)114 b(\030)32 b Ft(2)d Fm(C)2807 2511 y Fw(+)2873 2496 y Fx(:)109 2709 y FG(Similar)g(to)j(the)h(pro)s(of)f(of)g(Lemma)f (4.3)h(one)h(v)m(eri\014es)h(that)e(the)h(limit)c Fv(F)2781 2724 y Fw(2)2820 2709 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))1271 2922 y Fv(F)1332 2937 y Fw(2)1371 2922 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))f(=)54 b(lim)1747 2982 y Fu(\017)p Fs(!)p Fw(+0)1952 2922 y Fv(F)2013 2937 y Fw(2)2053 2922 y FG([)p Fx(\025)p FG(]\()p Fx(\025)22 b FG(+)g Fx(i\017)p FG(\))109 3174 y(exist)30 b(in)f Ft(B)513 3189 y Fw(2)553 3174 y FG(\()p Fv(H)663 3189 y Fw(2)702 3174 y FG(\))h(for)f(a.e.)43 b Fx(\025)27 b Ft(2)h Fm(R)41 b FG(and)30 b(satis\014es)h(the)f(relation)e Fv(F)2510 3189 y Fw(2)2549 3174 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))g(=)f Fx(F)2987 3189 y Fw(2)3027 3174 y FG(\()p Fx(\025)p FG(\).)42 b(Hence)31 b(the)109 3295 y(limit)e(\002)418 3259 y Fs(\003)418 3319 y Fw(2)457 3295 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))f(=)f Fx(s)22 b Ft(\000)h FG(lim)1135 3310 y Fu(\017)p Fs(!)p Fw(+0)1345 3295 y FG(\002)1421 3310 y Fw(2)1461 3295 y FG([)p Fx(\025)p FG(]\()p Fx(\025)f Ft(\000)g Fx(i\017)p FG(\))1898 3259 y Fs(\003)1971 3295 y FG(exists)33 b(for)f(a.e.)44 b Fx(\025)28 b Ft(2)g Fm(R)43 b FG(and)33 b(the)g(relation)759 3519 y(\002)835 3477 y Fs(\003)835 3543 y Fw(2)874 3519 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))28 b(=)f Fx(I)1292 3534 y Fl(Q)1349 3546 y Fr(1)p Fo(;\025)1466 3519 y FG(+)22 b(2)p Fx(\031)t(i)1705 3428 y Fp(p)p 1804 3428 V 1804 3519 a Fx(M)1898 3534 y Fw(1)1938 3519 y FG(\()p Fx(\025)p FG(\))28 b Fx(U)10 b Fv(F)2236 3534 y Fw(2)2276 3519 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))p Fx(U)2596 3477 y Fs(\003)2636 3428 y Fp(p)p 2735 3428 V 2735 3519 a Fx(M)2829 3534 y Fw(1)2869 3519 y FG(\()p Fx(\025)p FG(\))109 3731 y(holds)34 b(for)h(a.e)g Fx(\025)d Ft(2)g Fm(R)5 b FG(.)56 b(F)-8 b(rom)34 b(\(3.19\))g(w)m(e)i (obtain)e(that)h Fx(S)2277 3746 y Fw(11)2351 3731 y FG(\()p Fx(\025)p FG(\))d(=)g(\002)2700 3695 y Fs(\003)2700 3756 y Fw(2)2739 3731 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))j(for)f(a.e.)51 b Fx(\025)32 b Ft(2)g Fm(R)5 b FG(.)109 3852 y(Since)32 b(the)h(limit)c(\002)840 3816 y Fs(\003)840 3876 y Fw(2)880 3852 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))j(exists)h(for)f(a.e.)44 b Fx(\025)28 b Ft(2)g Fm(R)43 b FG(one)33 b(concludes)h(that)1158 4065 y(\002)1234 4080 y Fw(2)1273 4065 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))28 b(:=)f Fx(s)c Ft(\000)49 b FG(lim)1843 4124 y Fu(\017)p Fs(!)p Fw(+0)2049 4065 y FG(\002)2125 4080 y Fw(2)2164 4065 y FG([)p Fx(\025)p FG(]\()p Fx(\025)22 b Ft(\000)h Fx(i\017)p FG(\))109 4317 y(exists)38 b(for)f(a.e.)58 b Fx(\025)36 b Ft(2)g Fm(R)48 b FG(and)38 b(\002)1316 4332 y Fw(2)1355 4317 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))1599 4281 y Fs(\003)1675 4317 y FG(=)d(\002)1862 4281 y Fs(\003)1862 4342 y Fw(2)1902 4317 y FG([)p Fx(\025)p FG(]\()p Fx(\025)p FG(\))i(is)g(v)-5 b(alid.)56 b(This)38 b(completes)f(the)h(pro)s(of)109 4437 y(Theorem)33 b(5.1.)2901 b Fk(\003)109 4701 y FG(The)42 b(last)e(theorem)g(admits)g(an)h(in)m(terpretation)f(of)g(the)i (scattering)f(matrix)e Fx(S)3140 4716 y Fw(11)3215 4701 y FG(\()p Fx(\025)p FG(\))h(as)h(the)109 4821 y(result)30 b(of)h(a)f(Lax-Phillips)f(scattering.)42 b(Indeed,)33 b(let)d(us)i(in)m(tro)s(duce)f(the)g(minimal)c(self-adjoin)m(t)109 4941 y(dilation)i Fx(K)549 4956 y Fw(2)589 4941 y FG(\()p Fx(\025)p FG(\))j(of)g(the)h(maximal)c(dissipativ)m(e)j(op)s(erator)g Fx(H)2383 4956 y Fw(2)2423 4941 y FG(\()p Fx(\025)p FG(\).)43 b(W)-8 b(e)33 b(set)1338 5154 y Fv(K)1405 5169 y Fw(2)p Fu(;\025)1532 5154 y FG(=)28 b Fv(D)1719 5169 y Fs(\000)p Fu(;\025)1861 5154 y Ft(\010)22 b Fv(H)2032 5169 y Fw(2)2094 5154 y Ft(\010)g Fv(D)2276 5169 y Fw(+)p Fu(;\025)2396 5154 y Fx(;)109 5367 y FG(where)1342 5487 y Fv(D)1425 5502 y Fs(\006)p Fu(;\025)1572 5487 y FG(:=)28 b Fx(L)1769 5446 y Fw(2)1809 5487 y FG(\()p Fm(R)1913 5502 y Fs(\006)1978 5487 y Fx(;)17 b(dx;)g Fv(Q)2254 5502 y Fw(1)p Fu(;\025)2354 5487 y FG(\))p Fx(:)p Black 1832 5728 a FG(14)p Black eop %%Page: 15 17 15 16 bop Black Black 109 171 a FG(F)-8 b(urther,)32 b(w)m(e)i(de\014ne)570 488 y Fx(K)653 503 y Fw(2)692 488 y FG(\()p Fx(\025)p FG(\))842 288 y Fp(0)842 467 y(@)970 367 y Fx(f)1018 382 y Fs(\000)994 487 y Fx(f)970 608 y(f)1018 623 y Fw(+)1119 288 y Fp(1)1119 467 y(A)1234 488 y FG(:=)1364 288 y Fp(0)1364 467 y(@)2121 367 y Ft(\000)p Fx(i)2261 328 y Fu(d)p 2241 344 77 4 v 2241 401 a(dx)2327 367 y Fx(f)2375 382 y Fs(\000)1493 487 y Ft(<)p FG(e)17 b(\()p Fx(H)1744 502 y Fw(2)1783 487 y FG(\()p Fx(\025)p FG(\)\))p Fx(f)33 b Ft(\000)2144 448 y Fw(1)p 2144 464 36 4 v 2144 522 a(2)2189 487 y Fx(\013)q FG(\()p Fx(\025)p FG(\))2385 451 y Fs(\003)2424 487 y FG([)p Fx(f)2499 502 y Fw(+)2559 487 y FG(\(0\))21 b(+)h Fx(f)2851 502 y Fs(\000)2911 487 y FG(\(0\)])2121 609 y Ft(\000)p Fx(i)2261 570 y Fu(d)p 2241 586 77 4 v 2241 643 a(dx)2327 609 y Fx(f)2375 624 y Fw(+)3104 288 y Fp(1)3104 467 y(A)109 806 y FG(for)32 b(elemen)m(ts)h(of)f(the)h(domain)619 1103 y(dom)16 b(\()p Fx(K)941 1118 y Fw(2)980 1103 y FG(\()p Fx(\025)p FG(\)\))28 b(:=)1309 899 y Fp(8)1309 988 y(<)1309 1168 y(:)1398 903 y(0)1398 1082 y(@)1527 982 y Fx(f)1575 997 y Fs(\000)1551 1102 y Fx(f)1527 1222 y(f)1575 1237 y Fw(+)1675 903 y Fp(1)1675 1082 y(A)1790 1103 y FG(:)2072 982 y Fx(f)38 b Ft(2)28 b FG(dom)16 b(\()p Fx(H)2572 997 y Fw(2)2611 982 y FG(\()p Fx(\025)p FG(\)\))1921 1102 y Fx(f)1969 1117 y Fs(\006)2056 1102 y Ft(2)28 b Fx(W)2256 1066 y Fw(1)p Fu(;)p Fw(2)2350 1102 y FG(\()p Fm(R)2454 1117 y Fs(\006)2519 1102 y Fx(;)17 b(dx;)g Fv(Q)2795 1117 y Fw(1)p Fu(;\025)2895 1102 y FG(\))1886 1222 y Fx(f)1934 1237 y Fw(+)1993 1222 y FG(\(0\))22 b Ft(\000)h Fx(f)2288 1237 y Fs(\000)2347 1222 y FG(\(0\))k(=)h Ft(\000)p Fx(i\013)q FG(\()p Fx(\025)p FG(\))p Fx(f)3009 899 y Fp(9)3009 988 y(=)3009 1168 y(;)3114 1103 y Fx(:)109 1425 y FG(The)i(op)s(erator)f Fx(K)779 1440 y Fw(2)819 1425 y FG(\()p Fx(\025)p FG(\))g(is)g(self-adjoin)m(t)f(and)h(is)h(a)f (minimal)c(self-adjoin)m(t)j(dilation)f(of)i(the)h(max-)109 1545 y(imal)f(dissipativ)m(e)j(op)s(erator)g Fx(H)1277 1560 y Fw(2)1317 1545 y FG(\()p Fx(\025)p FG(\),)g(that)g(is,)1006 1764 y(\()p Fx(H)1125 1779 y Fw(2)1165 1764 y FG(\()p Fx(\025)p FG(\))22 b Ft(\000)g Fx(z)t FG(\))1506 1723 y Fs(\000)p Fw(1)1629 1764 y FG(=)27 b Fx(P)1809 1708 y Fl(K)1855 1720 y Fr(2)p Fo(;\025)1795 1791 y Fl(H)1845 1800 y Fr(2)1949 1764 y FG(\()p Fx(K)2070 1779 y Fw(2)2110 1764 y FG(\()p Fx(\025)p FG(\))22 b Ft(\000)g Fx(z)t FG(\))2451 1723 y Fs(\000)p Fw(1)2574 1764 y Fk(\026)27 b Fv(H)2715 1779 y Fw(2)109 1967 y FG(for)32 b Fx(z)g Ft(2)c Fm(C)495 1982 y Fw(+)593 1967 y FG(and)984 2088 y Fv(K)1051 2103 y Fw(2)p Fu(;\025)1178 2088 y FG(=)g(clospan)1597 2007 y Fp(\010)1655 2088 y Fx(E)1727 2103 y Fu(K)1787 2112 y Fr(2)1821 2103 y Fw(\()p Fu(\025)p Fw(\))1921 2088 y FG(\(\001\))p Fv(H)2150 2103 y Fw(2)2217 2088 y FG(:)g(\001)g Ft(2)g(B)s FG(\()p Fm(R)6 b FG(\))2686 2007 y Fp(\011)2750 2088 y Fx(;)109 2255 y FG(where)25 b Fx(E)454 2271 y Fu(K)514 2280 y Fr(2)548 2271 y Fw(\()p Fu(\025)p Fw(\))648 2255 y FG(\()p Ft(\001)p FG(\))e(is)h(the)g(sp)s (ectral)f(measure)h(of)f Fx(K)1937 2270 y Fw(2)1977 2255 y FG(\()p Fx(\025)p FG(\).)40 b(It)24 b(turns)g(out)g(that)f Fv(D)2970 2270 y Fs(\006)p Fu(;\025)3113 2255 y FG(are)h(incoming)109 2376 y(and)32 b(outgoing)f(subspaces)k(with)e(resp)s(ect)g(to)g Fx(K)1904 2391 y Fw(2)1943 2376 y FG(\()p Fx(\025)p FG(\),)f(i.e.,)1247 2579 y Fx(e)1292 2538 y Fs(\000)p Fu(itK)1456 2547 y Fr(2)1490 2538 y Fw(\()p Fu(\025)p Fw(\))1591 2579 y Fv(D)1674 2594 y Fw(+)p Fu(;\025)1821 2579 y Ft(\022)c Fv(D)2009 2594 y Fw(+)p Fu(;\025)2129 2579 y Fx(;)114 b(t)28 b Ft(\025)g FG(0)p Fx(;)109 2783 y FG(and)1247 2903 y Fx(e)1292 2862 y Fs(\000)p Fu(itK)1456 2871 y Fr(2)1490 2862 y Fw(\()p Fu(\025)p Fw(\))1591 2903 y Fv(D)1674 2918 y Fs(\000)p Fu(;\025)1821 2903 y Ft(\022)g Fv(D)2009 2918 y Fs(\000)p Fu(;\025)2129 2903 y Fx(;)114 b(t)28 b Ft(\024)g FG(0)p Fx(:)109 3071 y FG(Ho)m(w)m(ev)m(er,)35 b(w)m(e)e(remark)g(that)f(the)h(completeness)g(condition)1087 3275 y Fv(K)1154 3290 y Fw(2)p Fu(;\025)1281 3275 y FG(=)28 b(clospan)1700 3194 y Fp(\010)1758 3275 y Fx(e)1803 3233 y Fs(\000)p Fu(itK)1967 3242 y Fr(2)2001 3233 y Fw(\()p Fu(\025)p Fw(\))2102 3275 y Fv(D)2185 3290 y Fs(\006)p Fu(;\025)2332 3275 y FG(:)g Fx(t)g Ft(2)g Fm(R)2610 3194 y Fp(\011)3451 3275 y FG(\(5.2\))109 3478 y(is)22 b(in)g(general)h(not) f(satis\014ed.)41 b(Condition)22 b(\(5.2\))g(holds)h(if)e(and)i(only)g (if)e(the)j(maximal)19 b(dissipativ)m(e)109 3599 y(op)s(erator)41 b Fx(H)592 3614 y Fw(2)631 3599 y FG(\()p Fx(\025)p FG(\))h(is)f (completely)g(non-selfadjoin)m(t)g(and)h Fx(H)2362 3614 y Fw(2)2443 3599 y FG(is)f(singular,)i(that)f(means,)i(the)109 3719 y(absolutely)32 b(con)m(tin)m(uous)h(part)f Fx(H)1361 3683 y Fu(ac)1353 3744 y Fw(2)1465 3719 y FG(of)h Fx(H)1658 3734 y Fw(2)1729 3719 y FG(is)f(trivial.)109 3887 y(On)g(the)h (subspace)i Fv(D)930 3902 y Fu(\025)975 3887 y FG(,)943 4090 y Fv(D)1026 4105 y Fu(\025)1099 4090 y FG(=)28 b Fv(D)1286 4105 y Fs(\000)p Fu(;\025)1427 4090 y Ft(\010)23 b Fv(D)1610 4105 y Fw(+)p Fu(;\025)1757 4090 y FG(=)28 b Fx(L)1927 4049 y Fw(2)1967 4090 y FG(\()p Fm(R)5 b Fx(;)17 b(dx;)g Fv(Q)2347 4105 y Fw(1)p Fu(;\025)2453 4090 y FG(\))27 b Ft(\022)h Fv(K)2690 4105 y Fw(2)p Fu(;\025)2790 4090 y Fx(;)109 4294 y FG(let)k(us)h(de\014ne)h(the)f(op)s(erator)e Fx(K)1300 4309 y Fw(0)1340 4294 y FG(\()p Fx(\025)p FG(\),)544 4547 y(\()p Fx(K)665 4562 y Fw(0)705 4547 y FG(\()p Fx(\025)p FG(\))p Fx(g)t FG(\)\()p Fx(x)p FG(\))c(:=)g Ft(\000)p Fx(i)1364 4480 y(d)p 1335 4524 107 4 v 1335 4616 a(dx)1452 4547 y(g)t FG(\()p Fx(x)p FG(\))p Fx(;)114 b FG(dom)16 b(\()p Fx(K)2097 4562 y Fw(0)2136 4547 y FG(\()p Fx(\025)p FG(\)\))28 b(:=)f Fx(W)2571 4506 y Fw(1)p Fu(;)p Fw(2)2665 4547 y FG(\()p Fm(R)5 b Fx(;)17 b(dx;)g Fv(Q)3045 4562 y Fw(1)p Fu(;\025)3151 4547 y FG(\))p Fx(:)109 4782 y FG(The)33 b(self-adjoin)m(t)e(op)s(erator)h Fx(K)1287 4797 y Fw(0)1326 4782 y FG(\()p Fx(\025)p FG(\))h(generates)g(the)g (shift)f(group,)h(i.e,)1110 4986 y(\()p Fx(e)1193 4945 y Fs(\000)p Fu(itK)1357 4954 y Fr(0)1392 4945 y Fw(\()p Fu(\025)p Fw(\))1492 4986 y Fx(g)t FG(\)\()p Fx(x)p FG(\))27 b(=)h Fx(g)t FG(\()p Fx(x)22 b Ft(\000)g Fx(t)p FG(\))p Fx(;)115 b(g)31 b Ft(2)d Fv(D)2578 5001 y Fu(\025)2623 4986 y Fx(:)109 5190 y FG(Using)k(the)h(F)-8 b(ourier)31 b(transform)h Ft(F)37 b FG(:)28 b Fx(L)1566 5153 y Fw(2)1606 5190 y FG(\()p Fm(R)t Fx(;)18 b(dx;)f Fv(Q)1986 5205 y Fw(1)p Fu(;\025)2092 5190 y FG(\))27 b Ft(\000)-16 b(!)27 b Fx(L)2411 5153 y Fw(2)2451 5190 y FG(\()p Fm(R)5 b Fx(;)17 b(d\026;)g Fv(Q)2835 5205 y Fw(1)p Fu(;\025)2940 5190 y FG(\),)1183 5445 y(\()p Ft(F)10 b Fx(f)h FG(\)\()p Fx(\026)p FG(\))26 b(=)1746 5377 y(1)p 1675 5422 191 4 v 1675 5442 a Ft(p)p 1758 5442 108 4 v 82 x FG(2)p Fx(\031)1892 5309 y Fp(Z)1947 5535 y Fq(R)2016 5445 y Fx(dx)i(e)2195 5404 y Fs(\000)p Fu(i\026x)2360 5445 y Fx(f)11 b FG(\()p Fx(x)p FG(\))p Fx(;)p Black 1832 5728 a FG(15)p Black eop %%Page: 16 18 16 17 bop Black Black 109 171 a FG(the)29 b(op)s(erator)f Fx(K)745 186 y Fw(0)784 171 y FG(\()p Fx(\025)p FG(\))g(transforms)h 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(Birkh)1411 415 y(\177)1411 411 y(auser,)h(Basel,)f(1984.)p Black 1832 5728 a(18)p Black eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0512230156223--