%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: gatel.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips gatel.dvi -o %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1998.11.09:1459 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 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1267 y Fn(f)5 b Fo(\)\()p Fn(x)p Fo(\))13 b(=)h Fn(e)713 1247 y Fk(i)p Fh(\012\()p Fk(x;t)p Fh(\))822 1267 y Fo(\(2)p Fn(it)p Fo(\))919 1247 y Fi(\000)p Fk(d=)p Fh(2)1002 1267 y Fn(f)1026 1274 y Fi(\006)1055 1267 y Fo(\(\(2)p Fn(t)p Fo(\))1154 1247 y Fi(\000)p Fh(1)1201 1267 y Fn(x)p Fo(\))d(+)g Fn(\017)1328 1274 y Fi(\006)1357 1267 y Fo(\()p Fn(x;)d(t)p Fo(\))p Fn(;)285 b Fo(\(1.8\))-3 1371 y(where)19 b(\012\()p Fn(x;)8 b(t)p Fo(\))18 b(=)h Fm(\000)p Fo(\012\()p Fn(x;)8 b Fm(\000)p Fn(t)p Fo(\))18 b(is)h(a)h(solution)f(\(p)q(erhaps) i(appro)o(ximate\))c(of)j(the)f(time-dep)q(enden)o(t)-3 1431 y(equation)704 1479 y Fn(@)s Fo(\012)p 704 1502 64 2 v 713 1547 a Fn(@)s(t)784 1513 y Fo(+)11 b Fm(jr)p Fo(\012)p Fm(j)938 1492 y Fh(2)968 1513 y Fo(+)g Fn(V)25 b Fo(=)14 b(0)p Fn(;)602 b Fo(\(1.9\))-3 1613 y(and)23 b Fm(jj)p Fn(\017)146 1620 y Fi(\006)175 1613 y Fo(\()p Fm(\001)p Fn(;)8 b(t)p Fo(\))p Fm(jj)24 b Fo(=)h Fn(o)p Fo(\(1\))f(as)g Fn(t)h Fm(!)g(\0061)p Fo(.)41 b(Let)23 b(us)g(write)g Fn(f)1139 1620 y Fi(\006)1168 1613 y Fo(\()p Fn(x)p Fo(\))g(as)h Fn(f)1348 1620 y Fi(\006)1377 1613 y Fo(\()s(^)-27 b Fn(x;)8 b Fm(j)p Fn(x)p Fm(j)p Fo(\).)40 b(The)23 b(op)q(erator)-3 1673 y Fm(S)t Fo(\()p Fn(\025)p Fo(\))i(:)h Fn(f)186 1680 y Fi(\000)215 1673 y Fo(\()p Fm(\001)p Fn(;)270 1631 y Fm(p)p 311 1631 29 2 v 311 1673 a Fn(\025)q Fo(\))f Fm(7!)h Fn(f)484 1680 y Fh(+)513 1673 y Fo(\()p Fm(\001)p Fn(;)568 1631 y Fm(p)p 609 1631 V 609 1673 a Fn(\025)q Fo(\))d(is)g(kno)o(wn)g(as)h(the)f(scattering)g (matrix.)40 b(It)23 b(turns)h(out)f(that)-3 1733 y(the)d(op)q(erators)j Fm(S)t Fo(\()p Fn(\025)p Fo(\))e(and)h(\006\()p Fn(\025)p Fo(\))g(essen)o(tially)e(coincide.)34 b(Namely)l(,)20 b Fm(S)t Fo(\()p Fn(\025)p Fo(\))i(=)g(\006\()p Fn(\025)p Fo(\))p Fm(T)14 b Fo(,)22 b(where)e Fm(T)13 b Fo(,)-3 1793 y(\()p Fm(T)f Fn(a)p Fo(\)\()p Fn(!)r Fo(\))i(=)g Fn(a)p Fo(\()p Fm(\000)p Fn(!)r Fo(\),)g(is)g(the)g(re\015ection)g(op)q (erator)i(on)f(the)f(sphere.)21 b(Th)o(us,)15 b(in)f(the)g(long-range)i (case,)-3 1854 y(our)h(approac)o(h)g(pro)o(vides)f(a)h(w)o(a)o(y)f(to)h (construct)g(the)f(scattering)g(matrix)f(in)h(terms)f(of)i(solutions)g (of)-3 1914 y(the)f(stationary)g(equation)h(\(1.1\).)67 1993 y(W)l(e)f(emphasize)e(that)j(neither)e(relation)h(\(1.6\))h(nor)f (\(1.8\))h(de\014ne)f(the)g(scattering)g(matrix)e Fm(S)t Fo(\()p Fn(\025)p Fo(\))-3 2053 y(uniquely)l(.)33 b(Indeed,)21 b(an)g(arbitrary)g(function)f(\002\()s(^)-27 b Fn(x;)8 b(\025)p Fo(\))21 b(can)g(b)q(e)g(added)g(to)g Fn(')p Fo(\()p Fn(x;)8 b(\025)p Fo(\))21 b(in)f(de\014nition)-3 2113 y(\(1.3\))15 b(or)h(the)f(function)g Fm(\000)p Fo(sgn)9 b Fn(t)f Fo(\002\()s(^)-27 b Fn(x)o(;)8 b Fm(j)p Fn(x)p Fm(j)750 2095 y Fh(2)769 2113 y Fn(=)p Fo(\(4)p Fn(t)854 2095 y Fh(2)875 2113 y Fo(\)\))15 b(can)g(b)q(e)h(added)g(to)f(\012\()p Fn(x;)8 b(t)p Fo(\))15 b(in)g(\(1.8\).)21 b(This)16 b(corre-)-3 2173 y(sp)q(onds)j(to)f(the)g(replacemen)o(t)d(of)k Fm(S)t Fo(\()p Fn(\025)p Fo(\))f(b)o(y)g Fn(e)850 2155 y Fk(i)p Fh(\002)889 2159 y Fg(+)914 2155 y Fh(\()p Fk(\025)p Fh(\))964 2173 y Fm(S)t Fo(\()p Fn(\025)p Fo(\))p Fn(e)1087 2155 y Fk(i)p Fh(\002)1126 2159 y Ff(\000)1152 2155 y Fh(\()p Fk(\025)p Fh(\))1221 2173 y Fo(where)f(\002)1401 2180 y Fi(\006)1431 2173 y Fo(\()p Fn(\025)p Fo(\))h(is)g(m)o(ultiplic) o(ation)-3 2234 y(b)o(y)d(\002\()p Fm(\006)s Fo(^)-27 b Fn(x;)8 b(\025)p Fo(\).)67 2313 y(In)16 b(the)g(short-range)i(case)e (\(when)g Fn(')p Fo(\()p Fn(x;)8 b(\025)p Fo(\))14 b(=)932 2270 y Fm(p)p 974 2270 V 43 x Fn(\025)p Fm(j)p Fn(x)p Fm(j)d(\000)f Fo(\()p Fn(d)i Fm(\000)e Fo(3\))p Fn(\031)r(=)p Fo(4\),)17 b(the)e(results)h(form)o(ulated)-3 2373 y(ab)q(o)o(v)o(e)g (w)o(ere)f(pro)o(v)o(en)h(in)g([18],)f(where)h(the)g(tec)o(hniques)f (of)h([2],)g([3])g(w)o(ere)f(extensiv)o(ely)f(used.)67 2452 y(Our)22 b(pro)q(ofs)h(rely)e(hea)o(vily)g(on)h(the)g(w)o (ell-elab)q(orated)f(mac)o(hinery)e(of)k(long-range)g(scattering)-3 2512 y(theory)16 b(\(see)g([5],)g([15]\).)22 b(More)16 b(precisely)l(,)e(w)o(e)i(need)g(the)h(sharp)g(form)f(of)g(the)h (limiti)o(ng)e(absorption)-3 2572 y(principle)k(\([1],)i([9],)g ([12]\),)g(and)g(the)g(asymptotics)f(of)h(the)g(function)f(\(\()p Fn(H)f Fm(\000)13 b Fn(\025)i Fm(\006)f Fn(i)p Fo(0\))1637 2554 y Fi(\000)p Fh(1)1684 2572 y Fn(f)5 b Fo(\)\()p Fn(x)p Fo(\))21 b(as)-3 2632 y Fm(j)p Fn(x)p Fm(j)e(!)g(1)g Fo(\([6],)h([10]\).)31 b(A)20 b(new)f(p)q(oin)o(t,)h(ho)o(w)o(ev)o(er,) f(is)h(the)f(one-to-one)i(corresp)q(ondence)f(b)q(et)o(w)o(een)-3 2693 y(functions)c Fn(a)233 2700 y Fi(\006)276 2693 y Fm(2)e Fn(L)356 2675 y Fh(2)376 2693 y Fo(\()p Fl(S)425 2675 y Fk(d)p Fi(\000)p Fh(1)488 2693 y Fo(\))i(and)h(solutions)g(of)f (\(1.1\))h(satisfying)f(condition)g(\(1.2\).)917 2817 y(2)p eop %%Page: 3 3 3 2 bop 67 26 a Fo(T)l(o)20 b(conclude)f(the)g(in)o(tro)q(duction,)g (let)g(us)h(men)o(tion)d(the)i(recen)o(t)f(pap)q(er)i([14])f(that)h(in) f(spirit)g(is)-3 86 y(not)d(v)o(ery)e(far)i(from)e(the)i(ours.)21 b(They)16 b(get)f(in)h(the)f(general)g(con)o(text)g(of)h (asymptotically)d(Euclidean)-3 146 y(manifolds)j(a)i(result)g(of)g(t)o (yp)q(e)f(A\))g(\(but)h(not)h(of)f(t)o(yp)q(e)f(B\)\).)g(Moreo)o(v)o (er,)f(they)i(only)f(deal)h(with)f(p)q(er-)-3 207 y(turbations)h(of)g (the)f(metrics)e(whic)o(h)i(corresp)q(ond)h(to)g(v)o(ery)f(short-range) h(p)q(oten)o(tials)g(and)g(consider)-3 267 y(only)e(functions)g Fn(a)340 274 y Fi(\006)391 267 y Fm(2)22 b Fn(C)485 249 y Fi(1)522 267 y Fo(.)-3 421 y Fr(2.)55 b(PRELIMINARIES)-3 550 y(1.)31 b Fo(Let)18 b Fn(d)f Fm(\025)g Fo(2.)27 b(An)o(y)18 b Fn(x)25 b Fm(2)g Fl(R)581 532 y Fk(d)617 550 y Fo(will)17 b(b)q(e)h(written)g(as)g Fn(x)f Fo(=)g Fm(j)p Fn(x)p Fm(j)s Fo(^)-27 b Fn(x)16 b Fo(=)h Fn(r)q(!)k Fo(in)d(spherical)f(co)q (ordinates.)-3 610 y(W)l(e)g(denote)h Fn(@)267 617 y Fk(j)302 610 y Fo(=)f Fn(@)s(=@)s(x)467 617 y Fk(j)484 610 y Fn(;)26 b(j)20 b Fo(=)d(1)p Fn(;)8 b(:)g(:)g(:)f(;)h(d)p Fo(.)27 b(By)17 b Fn(C)22 b Fo(and)d Fn(c)p Fo(,)f(w)o(e)f(denote)h(p)q (ositiv)o(e)f(constan)o(ts)i(whose)-3 670 y(precise)14 b(v)m(alues)i(are)g(of)g(no)g(imp)q(ortance.)k(W)l(e)c(\014rst)g (recall)e(the)i(de\014nition)f(of)h(useful)g(spaces)g(in)f(our)-3 730 y(framew)o(ork.)20 b(Let)c Fe(S)e Fo(=)g Fe(S)p Fo(\()p Fl(R)549 712 y Fk(d)567 730 y Fo(\))i(b)q(e)h(the)f(Sc)o(h)o(w)o(artz)f (space)i(of)f(test)h(functions.)k(F)l(or)c(ev)o(ery)e Fn(G)f Fm(\032)g Fl(R)1832 712 y Fk(d)1849 730 y Fo(,)-3 791 y(let)h Fn(L)100 773 y Fh(2)100 803 y Fk(\015)122 791 y Fo(\()p Fn(G)p Fo(\))i(b)q(e)f(the)g(Hilb)q(ert)f(space)i(of)f (measurable)f(functions)i Fn(f)k Fo(on)c Fn(G)g Fo(suc)o(h)f(that)524 927 y Fm(jj)p Fn(f)5 b Fm(jj)609 906 y Fh(2)609 939 y Fk(\015)r(;G)682 927 y Fo(=)734 859 y Fj(Z)761 972 y Fk(G)791 927 y Fo(\(1)11 b(+)g Fm(j)p Fn(x)p Fm(j)950 906 y Fh(2)970 927 y Fo(\))989 906 y Fk(\015)1011 927 y Fm(j)p Fn(f)5 b Fo(\()p Fn(x)p Fo(\))p Fm(j)1134 906 y Fh(2)1153 927 y Fn(dx)14 b(<)g Fm(1)p Fn(:)-3 1064 y Fo(If)k Fn(\015)j Fo(=)e(0)g(ou)h Fn(G)f Fo(=)f 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Fk(n)1027 1250 y Fj(Z)1055 1363 y Fh(2)1073 1353 y Fd(n)1094 1363 y Fi(\024j)p Fk(x)p Fi(j\024)p Fh(2)1206 1353 y Fd(n)p Fg(+1)1276 1318 y Fm(j)p Fn(f)g Fo(\()p Fn(x)p Fo(\))p Fm(j)1399 1297 y Fh(2)1418 1318 y Fn(dx)1471 1248 y Fj(\023)1508 1259 y Fh(1)p Fk(=)p Fh(2)1577 1318 y Fn(<)14 b Fm(1)p Fn(:)-3 1471 y Fo(The)22 b(space)g Fn(B)279 1453 y Fi(\003)299 1471 y Fo(,)h(dual)f(to)h Fn(B)i Fo(with)d(resp)q(ect)g(to)h Fn(L)967 1453 y Fh(2)987 1471 y Fo(\()p Fl(R)1045 1453 y Fk(d)1062 1471 y Fo(\),)g(is)f(a)h(Banac)o(h)f(space)h(with)f(one)g (of)h(its)-3 1531 y(equiv)m(alen)o(t)14 b(norms)i(de\014ned)g(b)o(y)515 1675 y Fm(jj)p Fn(f)5 b Fm(jj)600 1682 y Fk(B)629 1673 y Ff(\003)662 1675 y Fo(=)13 b(sup)719 1715 y Fk(\032)p Fi(\025)p Fh(1)795 1604 y Fj(\022)832 1675 y Fn(\032)857 1654 y Fi(\000)p Fh(1)912 1607 y Fj(Z)940 1720 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\032)1035 1675 y Fm(j)p Fn(f)5 b Fo(\()p Fn(x)p Fo(\))p Fm(j)1158 1654 y Fh(2)1178 1675 y Fn(dx)1231 1604 y Fj(\023)1268 1615 y Fh(1)p Fk(=)p Fh(2)1331 1675 y Fn(:)417 b Fo(\(2.1\))-3 1815 y(W)l(e)16 b(denote)g(b)o(y)f Fm(h\001)p Fn(;)8 b Fm(\001i)17 b Fo(the)f(corresp)q(onding)h(dualit)o(y)e(sym)o(b)q(ol.)20 b(Clearly)l(,)243 1921 y Fe(S)14 b Fm(\032)g Fn(L)384 1900 y Fh(2)384 1933 y Fk(\015)420 1921 y Fm(\032)g Fn(B)i Fm(\032)e Fn(L)612 1900 y Fh(2)612 1933 y(1)p Fk(=)p Fh(2)681 1921 y Fm(\032)f Fn(L)766 1900 y Fh(2)786 1921 y Fo(\()p Fl(R)844 1900 y Fk(d)861 1921 y Fo(\))h Fm(\032)g Fn(L)980 1900 y Fh(2)980 1933 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)1076 1921 y Fm(\032)g Fn(B)1169 1900 y Fi(\003)1202 1921 y Fm(\032)g Fn(L)1288 1900 y Fh(2)1288 1933 y Fi(\000)p Fk(\015)1337 1921 y Fn(;)57 b Fm(8)p Fn(\015)16 b(>)d Fo(1)p Fn(=)p Fo(2)p Fn(:)-3 2032 y Fo(Let)h Fn(B)122 2014 y Fi(\003)119 2044 y Fh(0)155 2032 y Fo(b)q(e)g(the)g(closure)g (of)g Fn(L)548 2014 y Fh(2)568 2032 y Fo(\()p Fl(R)626 2014 y Fk(d)643 2032 y Fo(\))g(in)g(the)g(norm)f(of)h Fn(B)1031 2014 y Fi(\003)1050 2032 y Fo(.)21 b(Then)14 b Fn(f)19 b Fm(2)14 b Fn(B)1340 2014 y Fi(\003)1337 2044 y Fh(0)1373 2032 y Fo(if)g(and)h(only)e(if)h Fn(f)19 b Fo(satis\014es)-3 2092 y(\(1.5\).)24 b(The)18 b(spaces)g Fn(B)s(;)8 b(B)490 2074 y Fi(\003)508 2092 y Fn(;)g(B)570 2074 y Fi(\003)567 2104 y Fh(0)607 2092 y Fo(w)o(ere)16 b(\014rst)i(in)o(tro)q(duced)f(in)g(the)g(con)o(text)g(of)g(scattering) h(theory)f(in)-3 2152 y([2)o(].)67 2232 y(W)l(e)f(need)g(the)g(follo)o (wing)g(elemen)o(tary)d(prop)q(ert)o(y)j(of)h(the)f(space)g Fn(B)1313 2214 y Fi(\003)1332 2232 y Fo(.)-3 2312 y Fr(Lemm)o(a)g(2.1) 24 b Fc(If)17 b Fn(u)22 b Fm(2)g Fr(H)491 2294 y Fh(2)491 2325 y Fk(loc)537 2312 y Fc(,)17 b(then)583 2417 y Fm(jjr)p Fn(u)p Fm(jj)709 2424 y Fk(B)738 2415 y Ff(\003)770 2417 y Fm(\024)d Fn(C)t Fo(\()p Fm(jj)p Fn(u)p Fm(jj)965 2424 y Fk(B)994 2415 y Ff(\003)1022 2417 y Fo(+)d Fm(jj)p Fo(\001)p Fn(u)p Fm(jj)1196 2424 y Fk(B)1225 2415 y Ff(\003)1244 2417 y Fo(\))p Fn(:)485 b Fo(\(2.2\))-3 2543 y Fc(Pr)n(o)n(of)15 b({)j Fo(F)l(or)e(an)o(y)g(real)g Fn(\020)26 b Fm(2)d Fn(C)586 2525 y Fi(1)582 2555 y Fh(0)622 2543 y Fo(\()p Fl(R)680 2525 y Fk(d)698 2543 y Fo(\),)15 b(one)i(has)g(the)f(iden)o (tit)o(y)385 2605 y Fj(Z)413 2718 y Fb(R)440 2708 y Fd(d)465 2673 y Fm(jr)p Fn(u)p Fm(j)563 2652 y Fh(2)582 2673 y Fn(\020)t(dx)e Fo(=)g(2)750 2652 y Fi(\000)p Fh(1)805 2605 y Fj(Z)833 2718 y Fb(R)860 2708 y Fd(d)886 2673 y Fm(j)p Fn(u)p Fm(j)942 2652 y Fh(2)961 2673 y Fo(\001)p Fn(\020)t(dx)d Fm(\000)f Fo(Re)1206 2605 y Fj(Z)1234 2718 y Fb(R)1261 2708 y Fd(d)1286 2673 y Fo(\001)p Fn(u)p 1355 2645 28 2 v(u)o(\020)t(dx:)288 b Fo(\(2.3\))917 2817 y(3)p eop %%Page: 4 4 4 3 bop -3 26 a Fo(Indeed,)15 b(in)o(tegrating)h(b)o(y)f(parts)i(w)o(e) f(see)g(that)423 145 y Fm(\000)470 77 y Fj(Z)498 190 y Fb(R)525 180 y Fd(d)551 145 y Fo(\001)p Fn(u)p 620 117 28 2 v(u)o(\020)t(dx)e Fo(=)790 77 y Fj(Z)818 190 y Fb(R)845 180 y Fd(d)871 145 y Fm(jr)p Fn(u)p Fm(j)969 124 y Fh(2)987 145 y Fn(\020)t(dx)d Fo(+)1125 77 y Fj(Z)1153 190 y Fb(R)1180 180 y Fd(d)1206 145 y Fm(r)p Fn(u)p 1276 117 V(u)o Fm(r)p Fn(\020)t(dx;)325 b Fo(\(2.4\))-3 265 y(and)402 270 y Fj(Z)430 382 y Fb(R)457 373 y Fd(d)482 337 y Fm(r)p Fn(u)p 552 310 V(u)o Fm(r)p Fn(\020)t(dx)14 b Fo(=)g Fm(\000)812 270 y Fj(Z)839 382 y Fb(R)866 373 y Fd(d)892 337 y Fn(u)p Fm(r)p 962 310 V Fn(u)o Fm(r)p Fn(\020)t(dx)d Fm(\000)1169 270 y Fj(Z)1197 382 y Fb(R)1224 373 y Fd(d)1249 337 y Fm(j)p Fn(u)p Fm(j)1305 317 y Fh(2)1325 337 y Fo(\001)p Fn(\020)t(dx:)-3 443 y Fo(The)16 b(last)g(equalit)o(y)f (is)h(equiv)m(alen)o(t)f(to)536 561 y(2Re)626 493 y Fj(Z)654 606 y Fb(R)681 597 y Fd(d)706 561 y Fm(r)p Fn(u)p 776 534 V(u)o Fm(r)p Fn(\020)t(dx)f Fo(=)g Fm(\000)1036 493 y Fj(Z)1063 606 y Fb(R)1090 597 y Fd(d)1116 561 y Fm(j)p Fn(u)p Fm(j)1172 541 y Fh(2)1191 561 y Fo(\001)p Fn(\020)t(dx:)438 b Fo(\(2.5\))-3 685 y(Com)o(bining)14 b(\(2.4\))j(and)g(\(2.5\),)f(w)o (e)g(arriv)o(e)f(at)i(\(2.3\).)67 762 y(Supp)q(ose)h(no)o(w)e(that)h Fn(\020)t Fo(\()p Fn(x)p Fo(\))d(=)g(1)j(for)g Fm(j)p Fn(x)p Fm(j)c(\024)h Fo(1,)i Fn(\020)t Fo(\()p Fn(x)p Fo(\))e(=)g(0)j(for)g Fm(j)p Fn(x)p Fm(j)c(\025)h Fo(2,)i(and)h(set)g Fn(\020)1557 769 y Fk(\032)1577 762 y Fo(\()p Fn(x)p Fo(\))d(=)g Fn(\020)t Fo(\()p Fn(x=\032)p Fo(\).)-3 822 y(Then)i(\(2.3\))g(yields)157 876 y Fj(Z)185 989 y Fb(R)212 979 y Fd(d)237 944 y Fm(jr)p Fn(u)p Fm(j)335 923 y Fh(2)354 944 y Fn(\020)t Fo(\()p Fn(x=\032)p Fo(\))p Fn(dx)e Fo(=)g(2)637 923 y Fi(\000)p Fh(1)684 944 y Fn(\032)709 923 y Fi(\000)p Fh(2)765 876 y Fj(Z)793 989 y Fb(R)820 979 y Fd(d)845 944 y Fm(j)p Fn(u)p Fm(j)901 923 y Fh(2)920 944 y Fo(\(\001)p Fn(\020)t Fo(\)\()p Fn(x=\032)p Fo(\))p Fn(dx)d Fm(\000)g Fo(Re)1319 876 y Fj(Z)1347 989 y Fb(R)1374 979 y Fd(d)1399 944 y Fo(\001)p Fn(u)p 1468 916 V(u)o(\020)t Fo(\()p Fn(x=\032)p Fo(\))p Fn(dx;)-3 1064 y Fo(and)16 b(hence)483 1069 y Fj(Z)510 1181 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\032)606 1136 y Fm(jr)p Fn(u)p Fm(j)704 1116 y Fh(2)722 1136 y Fn(dx)e Fm(\024)g Fn(C)889 1069 y Fj(Z)916 1181 y Fi(j)p Fk(x)p Fi(j\024)p Fh(2)p Fk(\032)1021 1136 y Fo(\()p Fm(j)p Fo(\001)p Fn(u)p Fm(j)1137 1116 y Fh(2)1167 1136 y Fo(+)d Fm(j)p Fn(u)p Fm(j)1272 1116 y Fh(2)1291 1136 y Fo(\))p Fn(dx;)385 b Fo(\(2.6\))-3 1250 y(with)16 b Fn(C)j Fo(indep)q(enden)o(t)d(of)g Fn(\032)e Fm(\025)g Fo(1.)21 b Fa(2)-3 1357 y Fr(Corollary)c(2.2)24 b Fc(L)n(et)15 b Fn(u)e Fm(2)h Fr(H)555 1339 y Fh(2)555 1370 y Fk(loc)605 1357 y Fm(\\)5 b Fn(B)683 1339 y Fi(\003)717 1357 y Fc(b)n(e)15 b(a)f(solution)i(of)e(e)n(quation)i Fo(\(1)p Fn(:)p Fo(1\))f Fc(with)g(a)f(b)n(ounde)n(d)h(p)n(otential)-3 1417 y Fn(V)c Fc(.)22 b(Then)718 1478 y Fm(jjr)p Fn(u)p Fm(jj)844 1485 y Fk(B)873 1476 y Ff(\003)905 1478 y Fm(\024)14 b Fn(C)t Fm(jj)p Fn(u)p Fm(jj)1081 1485 y Fk(B)1110 1476 y Ff(\003)1127 1478 y Fn(:)621 b Fo(\(2.7\))67 1576 y(W)l(e)17 b(need)f(also)h(an)g(abstract)h(theorem)d(of)i(H.)f(Hahn)h(\(see)f (e.g.)22 b([19)q(],)16 b(c)o(hapter)g(7)h Fm(x)p Fo(5\))g(whic)o(h)f(w) o(e)-3 1637 y(form)o(ulate)e(in)i(terms)f(adapted)i(to)f(our)h(case.)-3 1714 y Fr(Theorem)f(2.3)24 b Fc(L)n(et)19 b Fm(F)j Fo(:)17 b Fn(B)j Fm(!)d Fn(L)662 1696 y Fh(2)682 1714 y Fo(\()p Fl(S)732 1696 y Fk(d)p Fi(\000)p Fh(1)794 1714 y Fo(\))i Fc(b)n(e)h(a)f(b)n(ounde)n(d)g(op)n(er)n(ator.)27 b(Supp)n(ose)19 b(that)h(the)f(adjoint)-3 1774 y(op)n(er)n(ator)f Fm(F)232 1756 y Fi(\003)270 1774 y Fo(:)g Fn(L)335 1756 y Fh(2)355 1774 y Fo(\()p Fl(S)404 1756 y Fk(d)p Fi(\000)p Fh(1)467 1774 y Fo(\))g Fm(!)g Fn(B)612 1756 y Fi(\003)651 1774 y Fc(has)i(a)g(close)n(d)g(r)n(ange.)30 b(Then)21 b(an)f(element)i Fn(u)c Fm(2)h Fn(B)1613 1756 y Fi(\003)1652 1774 y Fc(b)n(elongs)j(to) -3 1834 y(the)17 b(image)g Fn(R)p Fo(\()p Fm(F)315 1816 y Fi(\003)335 1834 y Fo(\))g Fc(of)g Fm(F)469 1816 y Fi(\003)506 1834 y Fc(if)f Fo(\()p Fc(and)i(only)f(if)8 b Fo(\))17 b Fm(h)p Fn(u;)8 b(f)d Fm(i)15 b Fo(=)e(0)18 b Fc(for)e(al)r(l)i Fn(f)23 b Fc(fr)n(om)15 b(the)j(kernel)g Fn(N)5 b Fo(\()p Fm(F)g Fo(\))17 b Fc(of)g(the)-3 1895 y(op)n(er)n(ator)e Fm(F)5 b Fc(.)22 b(Mor)n(e)n(over,)17 b Fn(R)p Fo(\()p Fm(F)5 b Fo(\))14 b(=)f Fn(L)710 1876 y Fh(2)730 1895 y Fo(\()p Fl(S)780 1876 y Fk(d)p Fi(\000)p Fh(1)842 1895 y Fo(\))18 b Fc(if)f Fn(N)5 b Fo(\()p Fm(F)1030 1876 y Fi(\003)1050 1895 y Fo(\))14 b(=)f Fm(f)p Fo(0)p Fm(g)p Fc(.)67 1972 y Fr(2.)25 b Fo(Let)16 b(us)h(no)o(w)f(giv)o(e)f (precise)h(conditions)g(on)h(the)f(p)q(oten)o(tial)g Fn(V)11 b Fo(.)21 b(W)l(e)16 b(alw)o(a)o(ys)g(supp)q(ose)i(that)694 2069 y Fn(V)12 b Fo(\()p Fn(x)p Fo(\))h(=)h Fn(V)893 2076 y Fk(L)920 2069 y Fo(\()p Fn(x)p Fo(\))c(+)h Fn(V)1073 2076 y Fk(S)1099 2069 y Fo(\()p Fn(x)p Fo(\))597 b(\(2.8\))-3 2166 y(where)15 b Fn(V)165 2173 y Fk(L)192 2166 y Fo(\()p Fn(x)p Fo(\))h(is)g(a)h Fn(C)403 2148 y Fh(3)422 2166 y Fo(-function)f(suc)o(h)g(that)346 2263 y Fm(j)p Fn(@)389 2243 y Fk(\013)413 2263 y Fn(V)441 2270 y Fk(L)468 2263 y Fo(\()p Fn(x)p Fo(\))p Fm(j)d(\024)h Fn(C)t Fo(\(1)d(+)g Fm(j)p Fn(x)p Fm(j)p Fo(\))831 2243 y Fi(\000)p Fk(\016)q Fi(\000j)p Fk(\013)p Fi(j)946 2263 y Fn(;)41 b(\016)15 b Fm(2)f Fo(\(0)p Fn(;)8 b Fo(1])p Fn(;)57 b Fo(0)14 b Fm(\024)f(j)p Fn(\013)p Fm(j)h(\024)g Fo(3)p Fn(;)249 b Fo(\(2.9\))-3 2361 y(while)15 b(the)h(short-range)i(part)e Fn(V)599 2368 y Fk(S)625 2361 y Fo(\()p Fn(x)p Fo(\))g(satis\014es)662 2458 y Fm(j)p Fn(V)704 2465 y Fk(S)730 2458 y Fo(\()p Fn(x)p Fo(\))p Fm(j)d(\024)h Fn(C)t Fo(\(1)d(+)g Fm(j)p Fn(x)p Fm(j)p Fo(\))1093 2437 y Fi(\000)p Fh(1)p Fi(\000)p Fk(\016)1183 2458 y Fn(:)541 b Fo(\(2.10\))-3 2555 y(Note)15 b(ho)o(w)o(ev)o(er)g(that)i(man)o(y)e(in)o(termediary)e(results)j(are)g (v)m(alid)g(under)g(w)o(eak)o(er)g(assumptions.)67 2632 y(W)l(e)f(start)g(with)g(a)h(reform)o(ulation)d(\([18]\))i(of)g(a)h (standard)g(uniqueness)f(result)f(\(see)h(e.g.)20 b([16]\))15 b(in)-3 2693 y(terms)f(con)o(v)o(enien)o(t)g(for)j(us.)917 2817 y(4)p eop %%Page: 5 5 5 4 bop -3 26 a Fr(Prop)r(osition)17 b(2.4)24 b Fc(Supp)n(ose)17 b(that)g Fn(u)f Fc(satis\014es)h Fo(\(1)p Fn(:)p Fo(1\))g Fc(and)f(for)g(some)h Fn(a)k Fm(2)i Fn(L)1452 8 y Fh(2)1472 26 y Fo(\()p Fl(S)1521 8 y Fk(d)p Fi(\000)p Fh(1)1584 26 y Fo(\))16 b(\()p Fc(and)h(one)g(of)-3 86 y(the)g(signs)i Fo(\\)11 b(+)g(")18 b Fc(or)f Fo(\\)11 b Fm(\000)g Fo("\))522 184 y Fn(u)p Fo(\()p Fn(x)p Fo(\))i(=)h Fn(a)p Fo(\()s(^)-27 b Fn(x)o Fo(\))p Fn(w)807 191 y Fi(\006)836 184 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j(+)g Fn(o)1035 191 y Fk(av)1075 184 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1150 164 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1305 184 y Fo(\))p Fn(;)400 b Fo(\(2.11\))437 287 y Fn(@)463 294 y Fk(r)481 287 y Fn(u)p Fo(\()p Fn(x)p Fo(\))14 b(=)f Fm(\006)p Fn(i)696 243 y Fm(p)p 737 243 29 2 v 737 287 a Fn(\025)q(a)p Fo(\()s(^)-27 b Fn(x)o Fo(\))p Fn(w)892 294 y Fi(\006)921 287 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j(+)g Fn(o)1120 294 y Fk(av)1160 287 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1235 267 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1390 287 y Fo(\))p Fn(;)315 b Fo(\(2.12\))-3 369 y Fc(as)17 b Fm(j)p Fn(x)p Fm(j)c(!)h(1)p Fc(.)22 b(Then)c Fn(u)13 b Fo(=)h(0)p Fc(.)67 447 y Fo(This)j(assertion)f(sho)o (ws)h(that)g(a)g(solution)f Fn(u)g Fo(of)h(equation)f(\(1.1\))h(is)f (determined)e(uniquely)h(b)o(y)g(a)-3 507 y(single)g(asymptotic)g(co)q (e\016cien)o(t)g(in)h(\(1.6\))g(and)h(\(1.7\).)67 585 y Fr(3.)k Fo(The)14 b(Sc)o(hr\177)-24 b(odinger)13 b(op)q(erator)i Fn(H)j Fo(=)c Fm(\000)p Fo(\001)6 b(+)g Fn(V)12 b Fo(\()p Fn(x)p Fo(\))h(is)h(self-adjoin)o(t)g(in)f Fn(L)1430 567 y Fh(2)1450 585 y Fo(\()p Fl(R)1508 567 y Fk(d)1525 585 y Fo(\))h(on)h(the)f(domain)-3 645 y Fr(H)41 627 y Fh(2)60 645 y Fo(.)28 b(It)18 b(follo)o(ws)h(from)e(Prop)q(osition)j (2.4)f(that)g(the)f(op)q(erator)i Fn(H)j Fo(do)q(es)c(not)g(ha)o(v)o(e) f(p)q(ositiv)o(e)g(eigen-)-3 705 y(v)m(alues.)26 b(Construction)19 b(of)f(scattering)g(theory)g(relies)f(on)h(the)g(follo)o(wing)g (fundamen)o(tal)f(result)g(on)-3 765 y(its)h(resolv)o(en)o(t)f Fn(R)p Fo(\()p Fn(z)r Fo(\))h(=)f(\()p Fn(H)g Fm(\000)12 b Fn(z)r Fo(\))618 747 y Fi(\000)p Fh(1)684 765 y Fo(kno)o(wn)18 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Ff(\003)899 1602 y Fm(\024)e Fn(C)q(;)60 b Fc(Im)8 b Fn(z)16 b Fm(6)p Fo(=)d(0)p Fn(:)481 b Fo(\(2.14\))-3 1717 y Fc(Pr)n(o)n(of)15 b({)34 b Fo(Let)16 b Fn(f)k Fm(2)14 b Fn(B)k Fo(and)f Fn(v)e Fo(=)f Fn(R)p Fo(\()p Fn(z)r Fo(\))p Fn(f)20 b Fm(2)14 b Fn(B)836 1699 y Fi(\003)855 1717 y Fo(.)21 b(Since)16 b(\001)p Fn(v)f Fo(=)e(\()p Fn(V)23 b Fm(\000)10 b Fn(z)r Fo(\))p Fn(v)j Fm(\000)e Fn(f)5 b Fo(,)16 b(w)o(e)f(ha)o(v)o(e)h(that)504 1815 y Fm(jj)p Fo(\001)p Fn(v)r Fm(jj)627 1822 y Fk(B)656 1813 y Ff(\003)688 1815 y Fm(\024)d Fn(c)p Fo(\()p Fm(jj)p Fn(v)r Fm(jj)862 1822 y Fk(B)891 1813 y Ff(\003)920 1815 y Fo(+)e Fm(jj)p Fn(f)5 b Fm(jj)1054 1822 y Fk(B)1083 1813 y Ff(\003)1102 1815 y Fo(\))14 b Fm(\024)g Fn(C)t Fm(jj)p Fn(f)5 b Fm(jj)1312 1822 y Fk(B)1341 1815 y Fn(;)-3 1913 y Fo(b)o(y)15 b(Prop)q(osition)j(2.5.)j(T)l(o)c(conclude,)e(it)h (remains)f(to)h(apply)g(Lemma)f(2.1.)21 b Fa(2)-3 2064 y Fr(3.)55 b(GENERALIZED)18 b(F)n(OURIER)h(TRANSF)n(ORMS)-3 2191 y(1.)34 b Fo(F)l(or)19 b(our)h(purp)q(ose)g(w)o(e)f(use)g(an)h (appro)o(ximate)e(solution)h(of)g(the)g(eik)o(onal)g(equation)g (\(1.4\))g(\(see)-3 2251 y(e.g.)h([10)q(]\).)-3 2328 y Fr(Lemm)o(a)c(3.1)24 b Fc(Ther)n(e)17 b(exists)h(a)g(r)n(e)n (al-value)n(d)g(function)h Fo(\010\()p Fn(x;)8 b(\025)p Fo(\))17 b Fc(satisfying)h(for)f Fm(j)p Fn(x)p Fm(j)g Fc(lar)n(ge)g(enough)517 2451 y Fo(2)541 2406 y Fm(p)p 583 2406 V 45 x Fn(\025)626 2417 y(@)s Fo(\010)p 616 2440 84 2 v 616 2485 a Fn(@)s Fm(j)p Fn(x)p Fm(j)705 2451 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))14 b(=)g Fm(jr)943 2458 y Fk(x)964 2451 y Fo(\010\()p Fn(x;)8 b(\025)p Fo(\))p Fm(j)1129 2431 y Fh(2)1160 2451 y Fo(+)j Fn(V)1237 2458 y Fk(L)1263 2451 y Fo(\()p Fn(x)p Fo(\))p Fn(;)419 b Fo(\(3.1\))-3 2581 y Fc(and)17 b(for)g(al)r(l)i Fn(x)13 b Fm(2)h Fl(R)366 2563 y Fk(d)384 2581 y Fc(,)j Fn(\025)h Fc(varying)g(on)f(any)h(c)n(omp)n(act)f(subinterval)i Fo(\003)14 b Fm(\032)f Fl(R)1373 2588 y Fh(+)372 2679 y Fm(j)p Fn(@)415 2658 y Fk(\013)412 2691 y(x)439 2679 y Fn(@)468 2655 y Fk(\014)465 2693 y(\025)491 2679 y Fo(\010\()p Fn(x;)8 b(\025)p Fo(\))p Fm(j)14 b(\024)f Fn(C)t Fo(\(1)e(+)g Fm(j)p Fn(x)p Fm(j)p Fo(\))939 2658 y Fh(1)p Fi(\000j)p Fk(\013)p Fi(j\000)p Fk(\016)1072 2679 y Fn(;)60 b Fm(j)p Fn(\013)p Fm(j)14 b(\024)g Fo(3)p Fn(;)43 b(\014)16 b Fm(\024)e Fo(2)p Fn(:)274 b Fo(\(3.2\))917 2817 y(5)p eop %%Page: 6 6 6 5 bop 67 26 a Fo(Remark)16 b(that)i(the)f(estimates)f(\(3.2\))i(for)g Fn(\014)g(>)e Fo(0)i(are)f(used)h(in)f(Section)g(6.)25 b(In)18 b(this)f(section)g(w)o(e)-3 86 y(need)e(only)h(con)o(tin)o(uit) o(y)f(of)h(\010\()p Fn(x;)8 b(\025)p Fo(\))17 b(with)f(resp)q(ect)g(to) g Fn(\025)p Fo(.)22 b(Relation)16 b(\(3.1\))g(ensures)h(that)510 201 y Fn(')p Fo(\()p Fn(x;)8 b(\025)p Fo(\))14 b(=)724 157 y Fm(p)p 765 157 29 2 v 765 201 a Fn(\025)q Fm(j)p Fn(x)p Fm(j)c(\000)h Fo(\010\()p Fn(x;)d(\025)p Fo(\))j Fm(\000)g Fn(\031)r Fo(\()p Fn(d)g Fm(\000)g Fo(3\))p Fn(=)p Fo(4)414 b(\(3.3\))-3 311 y(satis\014es)17 b(equation)h(\(1.4\)) f(with)g 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670 y Fr(Lemm)o(a)h(3.2)24 b Fc(L)n(et)f Fn(u)i Fm(2)g Fr(H)536 652 y Fh(2)536 683 y Fk(loc)605 670 y Fc(b)n(e)e(a)h(solution)g(of)f(e) n(quation)h Fo(\(1)p Fn(:)p Fo(1\))g Fc(satisfying,)h(for)e(some)g Fn(a)1775 677 y Fi(\006)1829 670 y Fm(2)-3 731 y Fn(L)30 713 y Fh(2)49 731 y Fo(\()p Fl(S)99 713 y Fk(d)p Fi(\000)p Fh(1)161 731 y Fo(\))p Fc(,)18 b(the)g(c)n(onditions)326 841 y Fn(u)p Fo(\()p Fn(x)p Fo(\))13 b(=)h Fn(a)511 848 y Fh(+)540 841 y Fo(\()s(^)-27 b Fn(x)p Fo(\))p Fn(w)641 848 y Fh(+)670 841 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j Fm(\000)g Fn(a)873 848 y Fi(\000)902 841 y Fo(\()s(^)-27 b Fn(x)p Fo(\))p Fn(w)1003 848 y Fi(\000)1032 841 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j(+)g Fn(o)1231 848 y Fk(av)1271 841 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1346 820 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1501 841 y Fo(\))p Fn(;)228 b Fo(\(3.4\))212 967 y(\()p Fn(@)257 974 y Fk(r)276 967 y Fn(u)p Fo(\)\()p Fn(x)p Fo(\))13 b(=)h Fn(i)471 922 y Fm(p)p 512 922 V 512 967 a Fn(\025)540 912 y Fj(\020)570 967 y Fn(a)596 974 y Fh(+)625 967 y Fo(\()s(^)-27 b Fn(x)p Fo(\))p Fn(w)726 974 y Fh(+)755 967 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j(+)g Fn(a)957 974 y Fi(\000)986 967 y Fo(\()s(^)-27 b Fn(x)p Fo(\))p Fn(w)1087 974 y Fi(\000)1116 967 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))1232 912 y Fj(\021)1273 967 y Fo(+)j Fn(o)1345 974 y Fk(av)1385 967 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1460 947 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1615 967 y Fo(\))p Fn(:)114 b Fo(\(3.5\))-3 1068 y Fc(as)17 b Fm(j)p Fn(x)p Fm(j)c(!)h(1)p Fc(.)22 b(Then)c Fm(jj)p Fn(a)460 1075 y Fh(+)489 1068 y Fm(jj)13 b Fo(=)h Fm(jj)p Fn(a)636 1075 y Fi(\000)665 1068 y Fm(jj)p Fc(.)-3 1162 y(Pr)n(o)n(of)h({)i Fo(It)e(follo)o(ws)h(from)g(\(1.1\))g(that)796 1272 y(Im)7 b(\001)p Fn(u)p 932 1244 28 2 v(u)13 b Fo(=)h(0)p Fn(:)699 b Fo(\(3.6\))-3 1382 y(In)o(tegration)16 b(b)o(y)f(parts)i(no) o(w)g(yields)495 1513 y(Im)562 1445 y Fj(Z)589 1558 y Fi(j)p Fk(x)p Fi(j)p Fh(=)p Fk(r)684 1513 y Fn(u)712 1520 y Fk(r)p 730 1486 V 730 1513 a Fn(u)8 b(dS)821 1520 y Fk(r)855 1513 y Fo(=)13 b(Im)981 1445 y Fj(Z)1009 1558 y Fi(j)p Fk(x)p Fi(j)p Fk()e Fo(1)p Fn(=)p Fo(2)f(and)h(for)e(suitably)g(c)o (hosen)h(sequences)-3 485 y Fm(j)p Fn(x)39 492 y Fk(n)62 485 y Fm(j)13 b(!)h(1)p Fo(.)21 b(These)16 b(tec)o(hnical)f(p)q(oin)o (ts)h(ha)o(v)o(e)g(b)q(een)g(indep)q(enden)o(tly)f(o)o(v)o(ercome)e(in) j([10)q(])f(and)i([15)q(].)67 591 y(Relation)d(\(3.9\))g(allo)o(ws)g (us)h(to)f(de\014ne)g(the)f(mapping)h Fm(F)1096 572 y Fk(L)1091 603 y Fi(\006)1122 591 y Fo(\()p Fn(\025)p Fo(\))g(:)f Fe(S)h Fm(!)g Fn(L)1381 572 y Fh(2)1401 591 y Fo(\()p Fl(S)1451 572 y Fk(d)p Fi(\000)p Fh(1)1513 591 y Fo(\))g(b)o(y)g(the)f(equalit)o(y)697 701 y(\()p Fm(F)757 680 y Fk(L)752 713 y Fi(\006)782 701 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)5 b Fo(\)\()s(^)-27 b Fn(x)q Fo(\))14 b(=)f Fn(a)1054 680 y Fk(L)1054 713 y Fi(\006)1083 701 y Fo(\()s(^)-27 b Fn(x)p Fo(\))p Fn(:)-3 811 y Fo(Set)17 b Fn(v)106 818 y Fi(\006)153 811 y Fo(=)g Fn(R)245 818 y Fk(L)271 811 y Fo(\()p Fn(\025)c Fm(\006)f Fn(i)p Fo(0\))p Fn(f)23 b Fo(for)c Fn(f)31 b Fm(2)25 b Fe(S)19 b Fo(so)g(that)f(\()p Fm(\000)p Fo(\001)12 b(+)g Fn(V)1097 818 y Fk(L)1136 811 y Fm(\000)g Fn(\025)p Fo(\))p Fn(v)1258 818 y Fi(\006)1305 811 y Fo(=)17 b Fn(f)5 b Fo(.)27 b(In)o(tegrating)18 b(b)o(y)g(parts)-3 871 y(and)e(using)h(relations)f(\(3.9\),)g (\(3.10\),)h(w)o(e)e(obtain)i(that)41 937 y Fj(Z)69 1050 y Fi(j)p Fk(x)p Fi(j)p Fk()d Fo(1)p Fn(=)p Fo(2)p Fn(:)423 b Fo(\(3.21\))-3 2624 y(Of)11 b(course,)h(the)g(de\014nition)f (of)h Fn(G)601 2631 y Fi(\006)631 2624 y Fo(\()p Fn(\025)p Fo(\))g(dep)q(ends)h(on)f Fn(\021)r Fo(,)g(but)g(as)g(w)o(e)g(will)e (see)i(b)q(elo)o(w,)g(this)f(dep)q(endence)-3 2685 y(is)16 b(inessen)o(tial.)917 2817 y(8)p eop %%Page: 9 9 9 8 bop -3 26 a Fr(Lemm)o(a)16 b(3.7)24 b Fc(L)n(et)e Fn(a)h Fm(2)g Fn(C)524 8 y Fi(1)561 26 y Fo(\()p Fl(S)610 8 y Fk(d)p Fi(\000)p Fh(1)673 26 y Fo(\))p Fc(,)g Fn(w)765 33 y Fi(\006)794 26 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))23 b Fc(b)n(e)f(de\014ne)n(d)i(by)e Fo(\(1)p Fn(:)p Fo(3\))h Fc(and)f Fn(u)1485 33 y Fi(\006)1515 26 y Fn(;)16 b(g)1568 33 y Fi(\006)1620 26 y Fc(b)n(e)23 b(given)h(by)-3 86 y Fo(\(3)p Fn(:)p Fo(18\))p Fc(,)18 b Fo(\(3)p Fn(:)p Fo(20\))p Fc(.)23 b(Then)18 b Fo(\()p Fc(as)f(elements)i(of)f Fn(B)822 68 y Fi(\003)841 86 y Fo(\))433 187 y Fm(\006)p Fo(2)p Fn(i\031)543 167 y Fh(1)p Fk(=)p Fh(2)597 187 y Fn(\025)625 167 y Fh(1)p Fk(=)p Fh(4)680 187 y Fm(F)721 167 y Fi(\003)716 199 y(\006)746 187 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)c Fo(=)f Fn(u)931 194 y Fi(\006)961 187 y Fo(\()p Fn(\025)p Fo(\))e Fm(\000)g Fn(R)p Fo(\()p Fn(\025)h Fm(\007)f Fn(i)p Fo(0\))p Fn(g)1317 194 y Fi(\006)1347 187 y Fo(\()p Fn(\025)p Fo(\))p Fn(:)311 b Fo(\(3.22\))-3 306 y Fc(Pr)n(o)n(of)16 b({)35 b Fo(W)l(e)17 b(consider)f(only)h(the)g (\\)12 b(+)f(")17 b(case,)g(pro)q(ofs)h(b)q(eing)g(similar.)j(It)16 b(su\016ces)h(to)g(c)o(hec)o(k)e(that)-3 366 y(for)h(an)o(y)g Fn(f)j Fm(2)14 b Fn(B)311 467 y Fm(h)p Fn(u)358 474 y Fh(+)388 467 y Fn(;)8 b(f)d Fm(i)11 b(\000)g(h)p Fn(g)561 474 y Fh(+)591 467 y Fn(;)d(R)p Fo(\()p Fn(\025)k Fo(+)f Fn(i)p Fo(0\))p Fn(f)5 b Fm(i)15 b Fo(=)f(2)p Fn(i\031)1004 447 y Fh(1)p Fk(=)p Fh(2)1058 467 y Fn(\025)1086 447 y Fh(1)p Fk(=)p Fh(4)1142 467 y Fo(\()p Fn(a;)8 b Fm(F)1245 474 y Fh(+)1273 467 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)d Fo(\))1387 477 y Fk(L)1411 467 y Fg(2)1429 477 y Fh(\()p Fb(S)1465 467 y Fd(d)p Ff(\000)p Fg(1)1519 477 y Fh(\))1535 467 y Fn(:)189 b Fo(\(3.23\))-3 568 y(Let)16 b(us)h(apply)f(the)g (Green)g(form)o(ula)362 623 y Fj(Z)389 736 y Fi(j)p Fk(x)p Fi(j)p Fk()e Fo(1)p Fn(=)p Fo(2)j(while)f Fm(F)343 555 y Fi(\006)372 548 y Fo(\()p Fn(\025)p Fo(\))e(:)g Fn(L)513 530 y Fh(2)513 560 y Fk(\015)535 548 y Fo(\()p Fl(R)593 530 y Fk(d)610 548 y Fo(\))g Fm(!)g Fn(L)740 530 y Fh(2)760 548 y Fo(\()p Fl(S)809 530 y Fk(d)p Fi(\000)p Fh(1)872 548 y Fo(\))i(for)g(an)o(y)g Fn(\015)h(>)d Fo(1)p Fn(=)p Fo(2.)22 b(Hence)15 b(\006)1455 555 y Fi(\006)1485 548 y Fo(\()p Fn(\025)p Fo(\))f(:)f Fn(C)1631 530 y Fi(1)1668 548 y Fo(\()p Fl(S)1718 530 y Fk(d)p Fi(\000)p Fh(1)1780 548 y Fo(\))h Fm(!)-3 608 y Fn(L)30 590 y Fh(2)49 608 y Fo(\()p Fl(S)99 590 y Fk(d)p Fi(\000)p Fh(1)161 608 y Fo(\))j(is)f(w)o(ell)f(de\014ned.)67 688 y(Com)o(bining)g(Lemma)f(3.7)j(and)g(relations)f(\(3.13\),)g (\(3.14\))h(w)o(e)f(arriv)o(e)f(at)-3 768 y Fr(Prop)r(osition)i(4.1)24 b Fc(L)n(et)17 b Fn(w)514 775 y Fi(\006)544 768 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))17 b Fc(b)n(e)h(given)h(by)e Fo(\(1)p Fn(:)p Fo(3\))p Fc(.)23 b(Then)18 b(for)f(any)g Fn(a)d Fm(2)g Fn(C)1489 750 y Fi(1)1526 768 y Fo(\()p Fl(S)1575 750 y Fk(d)p Fi(\000)p Fh(1)1638 768 y Fo(\))p Fc(,)j(we)h(have)59 890 y Fo(\()p Fm(F)119 870 y Fi(\003)114 903 y Fh(+)143 890 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()p Fn(x)p Fo(\))c(=)g Fn(\014)s Fo(\()p Fn(\025)p Fo(\))483 835 y Fj(\020)512 890 y Fn(a)p Fo(\()s(^)-27 b Fn(x)o Fo(\))p Fn(w)638 897 y Fh(+)668 890 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j Fm(\000)f Fo(\(\006)898 897 y Fi(\000)928 890 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()s(^)-27 b Fn(x)p Fo(\))p Fn(w)1140 897 y Fi(\000)1169 890 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))1285 835 y Fj(\021)1326 890 y Fo(+)j Fn(o)1398 897 y Fk(av)1438 890 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1513 870 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1667 890 y Fo(\))p Fn(;)62 b Fo(\(4.2\))39 1033 y(\()p Fm(F)99 1013 y Fi(\003)94 1045 y(\000)123 1033 y Fo(\()p Fn(\025)p Fo(\))p Fm(T)14 b Fn(a)p Fo(\)\()p Fn(x)p Fo(\))f(=)g Fn(\014)s Fo(\()p Fn(\025)p Fo(\))502 978 y Fj(\020)532 1033 y Fo(\(\006)586 1040 y Fh(+)616 1033 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()s(^)-27 b Fn(x)o Fo(\))p Fn(w)827 1040 y Fh(+)857 1033 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j Fm(\000)f Fn(a)p Fo(\()s(^)-27 b Fn(x)p Fo(\))p Fn(w)1160 1040 y Fi(\000)1189 1033 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))1305 978 y Fj(\021)1346 1033 y Fo(+)j Fn(o)1418 1040 y Fk(av)1457 1033 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1532 1013 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1687 1033 y Fo(\))p Fn(;)42 b Fo(\(4.3\))-3 1130 y Fc(and)37 1236 y Fo(\()p Fn(@)82 1243 y Fk(r)101 1236 y Fm(F)142 1215 y Fi(\003)137 1248 y Fh(+)166 1236 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()p Fn(x)p Fo(\))14 b(=)f Fn(i\025)453 1215 y Fh(1)p Fk(=)p Fh(2)508 1236 y Fn(\014)s Fo(\()p Fn(\025)p Fo(\))605 1180 y Fj(\020)635 1236 y Fn(a)p Fo(\()s(^)-27 b Fn(x)o Fo(\))p Fn(w)761 1243 y Fh(+)791 1236 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j(+)g(\(\006)1021 1243 y Fi(\000)1050 1236 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()s(^)-27 b Fn(x)p Fo(\))p Fn(w)1262 1243 y Fi(\000)1291 1236 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))1407 1180 y Fj(\021)1448 1236 y Fo(+)j Fn(o)1520 1243 y Fk(av)1560 1236 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1635 1215 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1790 1236 y Fo(\))p Fn(;)1762 1310 y Fo(\(4.4\))17 1386 y(\()p Fn(@)62 1393 y Fk(r)81 1386 y Fm(F)122 1366 y Fi(\003)117 1399 y(\000)146 1386 y Fo(\()p Fn(\025)p Fo(\))p Fm(T)i Fn(a)p Fo(\)\()p Fn(x)p Fo(\))h(=)f Fn(i\025)473 1366 y Fh(1)p Fk(=)p Fh(2)528 1386 y Fn(\014)s Fo(\()p Fn(\025)p Fo(\))625 1331 y Fj(\020)655 1386 y Fo(\(\006)709 1393 y Fh(+)739 1386 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()s(^)-27 b Fn(x)o Fo(\))p Fn(w)950 1393 y Fh(+)980 1386 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j(+)g Fn(a)p Fo(\()s(^)-27 b Fn(x)o Fo(\))p Fn(w)1282 1393 y Fi(\000)1311 1386 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))1427 1331 y Fj(\021)1468 1386 y Fo(+)j Fn(o)1540 1393 y Fk(av)1580 1386 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1655 1366 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1809 1386 y Fo(\))p Fn(;)1762 1460 y Fo(\(4.5\))-3 1520 y Fc(as)17 b Fm(j)p Fn(x)p Fm(j)c(!)h(1)p Fc(,)j(wher)n(e)g Fn(\014)s Fo(\()p Fn(\025)p Fo(\))d(=)g Fm(\000)p Fn(i)p Fo(2)654 1502 y Fi(\000)p Fh(1)701 1520 y Fn(\031)731 1502 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)813 1520 y Fn(\025)841 1502 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)924 1520 y Fc(.)-3 1612 y Fr(Lemm)o(a)i(4.2)24 b Fc(The)j(op)n(er)n(ator)d Fo(\006)635 1619 y Fi(\006)665 1612 y Fo(\()p Fn(\025)p Fo(\))j Fc(extends)h(by)f(c)n(ontinuity)g(to)f (an)h(isometric)g(op)n(er)n(ator)d(in)-3 1672 y Fn(L)30 1654 y Fh(2)49 1672 y Fo(\()p Fl(S)99 1654 y Fk(d)p Fi(\000)p Fh(1)161 1672 y Fo(\))p Fc(.)-3 1763 y(Pr)n(o)n(of)13 b({)28 b Fo(Isometricit)o(y)10 b(of)j(\006)532 1770 y Fh(+)562 1763 y Fo(\()p Fn(\025)p Fo(\))h(\(resp.)20 b(\006)817 1770 y Fi(\000)847 1763 y Fo(\()p Fn(\025)p Fo(\)\))13 b(on)h(the)f(set)g Fn(C)1203 1745 y Fi(1)1240 1763 y Fo(\()p Fl(S)1290 1745 y Fk(d)p Fi(\000)p Fh(1)1352 1763 y Fo(\))h(is)f(a)g(direct)g(consequence)-3 1823 y(of)j(Lemma)e(3.2)j(applied)e(to)i(relations)f(\(4.3\),)g(\(4.5\))h (\(resp.)k(\(4.2\),)16 b(\(4.4\)\).)21 b Fa(2)67 1927 y Fo(The)15 b(b)q(oundedness)h(of)f(the)f(op)q(erators)i Fm(F)842 1909 y Fi(\003)837 1939 y(\006)866 1927 y Fo(\()p Fn(\025)p Fo(\),)f Fn(@)987 1934 y Fk(r)1006 1927 y Fm(F)1047 1909 y Fi(\003)1042 1939 y(\006)1071 1927 y Fo(\()p Fn(\025)p Fo(\))g(and)h(\006)1281 1934 y Fi(\006)1310 1927 y Fo(\()p Fn(\025)p Fo(\))f(allo)o(ws)g(us,)g(with)f(help)g(of)-3 1987 y(Theorem)g(3.5,)j(to)f(extend)g(Prop)q(osition)h(4.1)g(to)f(an)h (arbitrary)f Fn(a)e Fm(2)g Fn(L)1308 1969 y Fh(2)1328 1987 y Fo(\()p Fl(S)1377 1969 y Fk(d)p Fi(\000)p Fh(1)1440 1987 y Fo(\).)-3 2067 y Fr(Prop)r(osition)j(4.3)24 b Fc(The)18 b(asymptotics)f Fo(\(4)p Fn(:)p Fo(2\))p Fc(-)h Fo(\(4)p Fn(:)p Fo(5\))g Fc(hold)g(true)f(for)g(al)r(l)i Fn(a)j Fm(2)g Fn(L)1504 2049 y Fh(2)1524 2067 y Fo(\()p Fl(S)1574 2049 y Fk(d)p Fi(\000)p Fh(1)1636 2067 y Fo(\))p Fc(.)67 2147 y Fo(In)16 b(fact,)g(w)o(e)g(ha)o(v)o(e)f(concluded)h(the) g(pro)q(of)h(of)g(the)f(part)g(A\))g(of)h(the)f(main)f(result)g(form)o (ulated)g(in)-3 2207 y(Section)g(1.)67 2287 y Fr(2.)24 b Fo(Our)15 b(pro)q(of)i(of)e(the)g(part)h(B\))f(relies)f(on)i(the)f (follo)o(wing)f(prop)q(erties)i(of)f(the)g(op)q(erators)i Fm(F)1767 2294 y Fi(\006)1796 2287 y Fo(\()p Fn(\025)p Fo(\))-3 2347 y(and)f Fm(F)132 2329 y Fi(\003)127 2359 y(\006)157 2347 y Fo(\()p Fn(\025)p Fo(\).)-3 2427 y Fr(Prop)r(osition)h(4.4)24 b Fc(If)17 b Fn(a)22 b Fm(2)h Fn(L)583 2409 y Fh(2)602 2427 y Fo(\()p Fl(S)652 2409 y Fk(d)p Fi(\000)p Fh(1)714 2427 y Fo(\))p Fc(,)18 b(then)g(the)g (limit)333 2557 y Fo(lim)323 2587 y Fk(\032)p Fi(!1)420 2557 y Fn(\032)445 2536 y Fi(\000)p Fh(1)501 2489 y Fj(Z)528 2602 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\032)624 2557 y Fm(j)p Fo(\()p Fm(F)698 2536 y Fi(\003)693 2569 y(\006)722 2557 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()p Fn(x)p Fo(\))p Fm(j)913 2536 y Fh(2)940 2557 y Fn(dx)c Fo(=)g(\(2)p Fn(\031)r Fo(\))1151 2536 y Fi(\000)p Fh(1)1198 2557 y Fn(\025)1226 2536 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)1309 2557 y Fm(jj)p Fn(a)p Fm(jj)1391 2536 y Fh(2)1391 2571 y Fk(L)1415 2562 y Fg(2)1431 2571 y Fh(\()p Fb(S)1466 2562 y Fd(d)p Ff(\000)p Fg(1)1520 2571 y Fh(\))1762 2557 y Fo(\(4.6\))-3 2693 y Fc(exists.)905 2817 y Fo(10)p eop %%Page: 11 11 11 10 bop -3 26 a Fc(Pr)n(o)n(of)15 b({)j Fo(According)d(to)i(\(4.2\))f (or)h(\(4.3\),)f(the)g(left-hand)g(side)g(of)h(\(4.6\))f(equals)207 154 y(4)p Fn(\031)r(\025)289 133 y Fh(1)p Fk(=)p Fh(2)363 154 y Fo(lim)353 184 y Fk(\032)p Fi(!1)450 154 y Fn(\032)475 133 y Fi(\000)p Fh(1)530 86 y Fj(Z)558 199 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\032)653 154 y Fm(j)p Fo(\()p Fm(F)727 133 y Fi(\003)722 166 y(\006)752 154 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()p Fn(x)p Fo(\))p Fm(j)943 133 y Fh(2)970 154 y Fn(dx)e Fo(=)g Fm(jj)p Fn(a)p Fm(jj)1171 133 y Fh(2)1171 168 y Fk(L)1195 158 y Fg(2)1211 168 y Fh(\()p Fb(S)1246 158 y Fd(d)p Ff(\000)p Fg(1)1300 168 y Fh(\))1327 154 y Fo(+)d Fm(jj)p Fo(\006)1439 161 y Fi(\007)1469 154 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fm(jj)1589 133 y Fh(2)1589 168 y Fk(L)1613 158 y Fg(2)1630 168 y Fh(\()p Fb(S)1665 158 y Fd(d)p Ff(\000)p Fg(1)1719 168 y Fh(\))207 296 y Fm(\000)p Fo(2)p Fn(\032)295 275 y Fi(\000)p Fh(1)343 296 y Fo(Re)417 226 y Fj(\022)454 228 y(Z)481 341 y Fb(S)503 331 y Fd(d)p Ff(\000)p Fg(1)559 296 y Fo(\(\006)613 303 y Fi(\007)643 296 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()p Fn(!)r Fo(\))p 824 268 26 2 v Fn(a)o Fo(\()p Fn(!)r Fo(\))p Fn(d!)985 228 y Fj(Z)1035 241 y Fk(\032)1013 341 y Fh(0)1064 296 y Fo(exp)o(\()p Fm(\007)p Fo(2)p Fn(i')p Fo(\()p Fn(r)q(!)r(;)d(\025)p Fo(\)\))g Fn(dr)1487 226 y Fj(\023)1536 296 y Fo(+)j Fn(o)p Fo(\(1\))p Fn(:)-3 428 y Fo(In)o(tegrating)16 b(with)g(the)g(help)f(of)i(\(3.2\))f(b)o(y)g(parts,)h(w)o(e)e(see)h (that)548 557 y(lim)546 587 y Fk(\032)p Fi(!)p Fh(0)626 557 y Fn(\032)651 536 y Fi(\000)p Fh(1)706 489 y Fj(Z)756 502 y Fk(\032)734 602 y Fh(0)784 557 y Fo(exp\()p Fm(\007)p Fo(2)p Fn(i')p Fo(\()p Fn(r)q(!)r(;)8 b(\025)p Fo(\)\))g Fn(dr)16 b Fo(=)e(0)p Fn(:)-3 687 y Fo(Therefore)h(\(4.6\))i(follo)o (ws)f(from)f(isometricit)o(y)e(of)k(the)f(op)q(erator)h(\006)1246 694 y Fi(\007)1276 687 y Fo(\()p Fn(\025)p Fo(\).)k Fa(2)67 789 y Fo(Com)o(bining)15 b(Prop)q(ositions)j(3.6)e(and)h(4.4,)f(w)o(e)g (obtain)-3 868 y Fr(Prop)r(osition)h(4.5)24 b Fc(One)19 b(has)e(the)h(two-side)n(d)g(estimate)337 972 y Fo(\(2)p Fn(\031)r Fo(\))429 951 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)511 972 y Fn(\025)539 951 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)622 972 y Fm(jj)p Fn(a)p Fm(jj)704 981 y Fk(L)728 972 y Fg(2)744 981 y Fh(\()p Fb(S)779 972 y Fd(d)p Ff(\000)p Fg(1)833 981 y Fh(\))863 972 y Fm(\024)c(jjF)985 951 y Fi(\003)980 984 y(\006)1009 972 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fm(jj)1129 979 y Fk(B)1158 970 y Ff(\003)1190 972 y Fm(\024)g Fn(C)t Fm(jj)p Fn(a)p Fm(jj)1364 981 y Fk(L)1388 972 y Fg(2)1404 981 y Fh(\()p Fb(S)1439 972 y Fd(d)p Ff(\000)p Fg(1)1493 981 y Fh(\))1509 972 y Fn(:)239 b Fo(\(4.7\))-3 1075 y Fc(Henc)n(e)22 b(the)g(kernel)g(N)p Fo(\()p Fm(F)475 1057 y Fi(\003)470 1088 y(\006)500 1075 y Fo(\()p Fn(\025)p Fo(\)\))f Fc(of)g(the)h(op)n(er)n(ator)d Fm(F)988 1057 y Fi(\003)983 1088 y(\006)1013 1075 y Fo(\()p Fn(\025)p Fo(\))h(:)42 b Fn(L)1188 1057 y Fh(2)1208 1075 y Fo(\()p Fl(S)1257 1057 y Fk(d)p Fi(\000)p Fh(1)1320 1075 y Fo(\))20 b Fm(!)h Fn(B)1470 1057 y Fi(\003)1510 1075 y Fc(is)g(trivial)g(and)h (its)-3 1136 y(image)c(R)p Fo(\()p Fm(F)235 1118 y Fi(\003)230 1148 y(\006)258 1136 y Fo(\()p Fn(\025)p Fo(\)\))g Fc(is)f(close)n(d.) 67 1215 y Fo(Let)f(us)h(return)f(to)h(equation)f(\(1.1\).)-3 1294 y Fr(Lemm)o(a)g(4.6)24 b Fc(L)n(et)13 b Fn(u)h Fm(2)g Fr(H)504 1276 y Fh(2)504 1307 y Fk(loc)552 1294 y Fm(\\)s Fn(B)628 1276 y Fi(\003)661 1294 y Fc(b)n(e)g(a)g(solution)g(of)g Fo(\(1)p Fn(:)p Fo(1\))p Fc(.)21 b(F)l(or)14 b(any)f Fn(f)19 b Fm(2)c Fn(B)h Fc(such)e(that)g Fm(F)1686 1301 y Fi(\006)1715 1294 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)20 b Fo(=)-3 1354 y(0)p Fc(,)d(we)i(have)f Fm(h)p Fn(u;)8 b(f)d Fm(i)14 b Fo(=)g(0)p Fc(.)-3 1433 y(Pr)n(o)n(of)h({)34 b Fo(By)15 b(Corollary)i(2.2,)f Fn(@)596 1440 y Fk(r)615 1433 y Fn(u)d Fm(2)h Fn(B)743 1415 y Fi(\003)779 1433 y Fo(and)j(hence)472 1490 y Fj(Z)500 1603 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\032)587 1558 y Fo(\()p Fm(j)p Fn(u)p Fo(\()p Fn(x)p Fo(\))p Fm(j)728 1538 y Fh(2)758 1558 y Fo(+)11 b Fm(j)p Fn(@)847 1565 y Fk(r)866 1558 y Fn(u)p Fo(\()p Fn(x)p Fo(\))p Fm(j)974 1538 y Fh(2)993 1558 y Fo(\))d Fn(dx)14 b Fm(\024)f Fn(C)t(\032;)40 b(\032)14 b Fm(\025)g Fo(1)p Fn(:)375 b Fo(\(4.8\))-3 1696 y(If)14 b Fn(f)19 b Fm(2)14 b Fn(B)j Fo(and)f Fm(F)318 1703 y Fi(\006)347 1696 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)k Fo(=)13 b(0)j(then,)e (according)h(to)g(Theorem)f(3.5,)h(the)f(function)h Fn(v)1539 1703 y Fi(\006)1582 1696 y Fo(=)e Fn(R)p Fo(\()p Fn(\025)8 b Fm(\006)g Fn(i)p Fo(0\))p Fn(f)-3 1757 y Fo(satis\014es)16 b(the)g(condition)521 1804 y Fj(Z)549 1917 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\032)636 1872 y Fo(\()p Fm(j)p Fn(v)693 1879 y Fi(\006)722 1872 y Fo(\()p Fn(x)p Fo(\))p Fm(j)802 1851 y Fh(2)832 1872 y Fo(+)11 b Fm(j)p Fn(@)921 1879 y Fk(r)940 1872 y Fn(v)964 1879 y Fi(\006)993 1872 y Fo(\()p Fn(x)p Fo(\))p Fm(j)1073 1851 y Fh(2)1092 1872 y Fo(\))d Fn(dx)14 b Fo(=)g Fn(o)p Fo(\()p Fn(\032)p Fo(\))p Fn(;)424 b Fo(\(4.9\))-3 2010 y(as)16 b Fn(\032)e Fm(!)g(1)p Fo(.)21 b(By)16 b(the)g(Sc)o(h)o(w)o(arz)f(inequalit)o(y)l (,)f(it)h(follo)o(ws)i(from)e(\(4.8\))h(and)h(\(4.9\))g(that)586 2070 y Fj(Z)614 2183 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\032)701 2138 y Fo(\()p Fn(@)746 2145 y Fk(r)765 2138 y Fn(u)p 793 2110 V(v)818 2145 y Fi(\006)858 2138 y Fm(\000)11 b Fn(u@)962 2145 y Fk(r)p 981 2110 V 981 2138 a Fn(v)1006 2145 y Fi(\006)1035 2138 y Fo(\))p Fn(dx)j Fo(=)g Fn(o)p Fo(\()p Fn(\032)p Fo(\))p Fn(;)-3 2273 y Fo(as)i Fn(\032)e Fm(!)g(1)p Fo(.)21 b(Consequen)o(tly)l(,)15 b(there)h(exists)f(a)i (sequence)e Fn(r)1080 2280 y Fk(n)1117 2273 y Fm(!)f(1)i Fo(suc)o(h)g(that)549 2398 y(lim)529 2428 y Fk(r)545 2432 y Fd(n)566 2428 y Fi(!1)645 2331 y Fj(Z)673 2443 y Fi(j)p Fk(x)p Fi(j)p Fh(=)p Fk(r)756 2447 y Fd(n)779 2398 y Fo(\()p Fn(@)824 2405 y Fk(r)842 2398 y Fn(u)p 870 2371 V(v)896 2405 y Fi(\006)936 2398 y Fm(\000)11 b Fn(u@)1040 2405 y Fk(r)p 1058 2371 V 1058 2398 a Fn(v)1084 2405 y Fi(\006)1113 2398 y Fo(\))p Fn(dS)1187 2405 y Fk(r)1203 2409 y Fd(n)1241 2398 y Fo(=)i(0)p Fn(:)408 b Fo(\(4.10\))-3 2536 y(No)o(w)16 b(in)o(tegrating)g(b)o(y)f(parts)i (and)g(taking)f(in)o(to)g(accoun)o(t)h(equation)f(\(1.1\))g(for)h Fn(u)p Fo(,)e(w)o(e)h(\014nd)h(that)95 2664 y Fm(h)p Fn(u;)8 b(f)d Fm(i)15 b Fo(=)34 b(lim)279 2694 y Fk(r)295 2698 y Fd(n)316 2694 y Fi(!1)395 2596 y Fj(Z)422 2709 y Fi(j)p Fk(x)p Fi(j\024)p Fk(r)505 2713 y Fd(n)537 2664 y Fn(u)p Fo(\()p Fm(\000)p Fo(\001)10 b(+)h Fn(V)22 b Fm(\000)11 b Fn(\025)p Fo(\))p 870 2637 V Fn(v)896 2671 y Fi(\006)934 2664 y Fn(dx)j Fo(=)33 b(lim)1052 2694 y Fk(r)1068 2698 y Fd(n)1090 2694 y Fi(!1)1168 2596 y Fj(Z)1196 2709 y Fi(j)p Fk(x)p Fi(j)p Fh(=)p Fk(r)1279 2713 y Fd(n)1302 2664 y Fo(\()p 1321 2637 V Fn(v)1347 2671 y Fi(\006)1376 2664 y Fn(@)1402 2671 y Fk(r)1421 2664 y Fn(u)11 b Fm(\000)f Fn(u@)1563 2671 y Fk(r)p 1582 2637 V 1582 2664 a Fn(v)1607 2671 y Fi(\006)1637 2664 y Fo(\))p Fn(dS)1711 2671 y Fk(r)1727 2675 y Fd(n)1750 2664 y Fn(:)905 2817 y Fo(11)p eop %%Page: 12 12 12 11 bop -3 26 a Fo(Therefore)15 b Fm(h)p Fn(u;)8 b(f)d Fm(i)15 b Fo(=)f(0)i(b)o(y)g(virtue)f(of)i(\(4.10\).)22 b Fa(2)67 130 y Fo(It)h(follo)o(ws)f(from)g(Prop)q(osition)i(4.5)f (that)h(Theorem)d(2.3)i(can)g(b)q(e)g(applied)g(to)g(the)f(op)q(erator) -3 190 y Fm(F)33 197 y Fi(\006)62 190 y Fo(\()p Fn(\025)p Fo(\).)28 b(Therefore,)18 b(b)o(y)g(Lemma)e(4.6,)j(ev)o(ery)e Fn(u)g Fm(2)h Fr(H)1019 172 y Fh(2)1019 203 y Fk(loc)1076 190 y Fm(\\)13 b Fn(B)1162 172 y Fi(\003)1200 190 y Fo(b)q(elongs)19 b(to)g(the)f(range)h(of)f Fm(F)1758 172 y Fi(\003)1753 202 y(\006)1783 190 y Fo(\()p Fn(\025)p Fo(\).)-3 250 y(T)l(aking)13 b(also)h(in)o(to)f(accoun)o(t)g(Prop)q(osition)h(3.9,)g (w)o(e)e(obtain)i(a)g(description)e(of)i(solutions)f(of)h(equation)-3 310 y(\(1.1\))i(in)g(terms)f(of)h(the)g(op)q(erator)i Fm(F)683 292 y Fi(\003)678 322 y(\006)707 310 y Fo(\()p Fn(\025)p Fo(\).)-3 390 y Fr(Theorem)e(4.7)24 b Fc(F)l(or)18 b Fn(u)c Fm(2)h Fr(H)556 372 y Fh(2)556 403 y Fk(loc)613 390 y Fm(\\)c Fn(B)697 372 y Fi(\003)734 390 y Fc(to)18 b(b)n(e)h(a)e(solution)i(of)f(e)n(quation)h Fo(\(1)p Fn(:)p Fo(1\))p Fc(,)f(it)g(is)g(ne)n(c)n(essary)f(and)-3 450 y(su\016cient)h(that)g Fn(u)k Fm(2)g Fc(R)p Fo(\()p Fm(F)507 432 y Fi(\003)502 463 y(\006)531 450 y Fo(\()p Fn(\025)p Fo(\)\))p Fc(.)67 530 y Fo(According)17 b(to)h(Prop)q (osition)h(4.3,)f(Theorem)f(4.7)h(leads)g(imme)o(diatel)o(y)d(to)j(the) g(part)g(B\))f(of)h(our)-3 590 y(main)10 b(result.)19 b(Let)12 b(us)g(reform)o(ulate)e(b)q(oth)i(parts)g(adding)h (informations)e(con)o(tained)g(in)g(Prop)q(ositions)-3 650 y(2.4,)20 b(4.4)g(and)h(4.5.)32 b(Note)19 b(that)h(functions)g Fn(a)852 657 y Fi(\006)901 650 y Fo(and)h Fn(u)e Fo(in)h(this)f(form)o (ulation)g(are)h(related)f(b)o(y)g(the)-3 711 y(equalities)131 817 y Fn(u)p Fo(\()p Fn(x)p Fo(\))13 b(=)h(2)p Fn(i\031)361 796 y Fh(1)p Fk(=)p Fh(2)415 817 y Fn(\025)443 796 y Fh(1)p Fk(=)p Fh(4)499 817 y Fo(\()p Fm(F)559 796 y Fi(\003)554 829 y Fh(+)583 817 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)675 824 y Fh(+)704 817 y Fo(\)\()p Fn(x)p Fo(\))p Fn(;)89 b(u)p Fo(\()p Fn(x)p Fo(\))13 b(=)h(2)p Fn(i\031)1122 796 y Fh(1)p Fk(=)p Fh(2)1177 817 y Fn(\025)1205 796 y Fh(1)p Fk(=)p Fh(4)1260 817 y Fo(\()p Fm(F)1320 796 y Fi(\003)1315 829 y(\000)1344 817 y Fo(\()p Fn(\025)p Fo(\))p Fm(T)g Fn(a)1477 824 y Fi(\000)1506 817 y Fo(\)\()p Fn(x)p Fo(\))p Fn(:)133 b Fo(\(4.11\))-3 923 y Fr(Theorem)16 b(4.8)24 b Fc(L)n(et)16 b(assumptions)g Fo(\(2)p Fn(:)p Fo(8\))p Fc(-)p Fo(\(2)p Fn(:)p Fo(10\))h Fc(hold)f(and)h(let)g Fn(w)1245 930 y Fi(\006)1274 923 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))16 b Fc(b)n(e)g(given)i(by)e Fo(\(1)p Fn(:)p Fo(3\))p Fc(.)22 b(F)l(or)-3 983 y(any)16 b Fn(a)115 990 y Fi(\000)158 983 y Fm(2)e Fn(L)238 965 y Fh(2)258 983 y Fo(\()p Fl(S)307 965 y Fk(d)p Fi(\000)p Fh(1)370 983 y Fo(\))p Fc(,)i(ther)n(e)h(exist)g (a)f(function)i Fn(a)911 990 y Fh(+)954 983 y Fm(2)c Fn(L)1034 965 y Fh(2)1054 983 y Fo(\()p Fl(S)1103 965 y Fk(d)p Fi(\000)p Fh(1)1166 983 y Fo(\))i Fc(and)g(a)h(solution)g Fn(u)d Fm(2)g Fr(H)1652 965 y Fh(2)1652 996 y Fk(loc)1705 983 y Fm(\\)9 b Fn(B)1787 965 y Fi(\003)1823 983 y Fc(of)-3 1043 y Fo(\(1)p Fn(:)p Fo(1\))18 b Fc(with)g(asymptotics)g Fo(\(1)p Fn(:)p Fo(6\))p Fn(;)f Fo(\(1)p Fn(:)p Fo(7\))h Fc(as)g Fm(j)p Fn(x)p Fm(j)d(!)f(1)p Fc(.)25 b(The)18 b(function)h Fn(a)1342 1050 y Fh(+)1390 1043 y Fc(and)f(the)g(solution) h Fn(u)f Fc(ar)n(e)-3 1103 y(determine)n(d)g(uniquely)h(by)f Fn(a)531 1110 y Fi(\000)561 1103 y Fc(.)23 b(Mor)n(e)n(over,)17 b Fm(jj)p Fn(a)883 1110 y Fi(\000)912 1103 y Fm(jj)d Fo(=)h Fm(jj)p Fn(a)1061 1110 y Fh(+)1090 1103 y Fm(jj)i Fc(and)h(the)h(r)n(oles)f(of)f Fn(a)1511 1110 y Fh(+)1558 1103 y Fc(and)h Fn(a)1679 1110 y Fi(\000)1726 1103 y Fc(in)h(this)-3 1164 y(formulation)e(c)n(an)h(b)n(e)g(inter)n(change)n (d.)67 1243 y(Conversely,)24 b(any)f Fn(u)f Fm(2)h Fr(H)585 1225 y Fh(2)585 1256 y Fk(loc)644 1243 y Fm(\\)15 b Fn(B)732 1225 y Fi(\003)774 1243 y Fc(satisfying)22 b Fo(\(1)p Fn(:)p Fo(1\))g Fc(has)g(the)h(asymptotics)f Fo(\(1)p Fn(:)p Fo(6\))p Fn(;)16 b Fo(\(1)p Fn(:)p Fo(7\))23 b Fc(as)-3 1304 y Fm(j)p Fn(x)p Fm(j)13 b(!)g(1)18 b Fc(for)e(some)i (functions)h Fn(a)637 1311 y Fi(\006)680 1304 y Fm(2)14 b Fn(L)760 1286 y Fh(2)780 1304 y Fo(\()p Fl(S)829 1286 y Fk(d)p Fi(\000)p Fh(1)892 1304 y Fo(\))p Fc(.)22 b(F)l(urthermor)n(e) 676 1410 y Fm(jj)p Fn(u)p Fm(jj)760 1417 y Fk(B)789 1408 y Ff(\003)821 1410 y Fm(\024)14 b Fn(C)t Fm(jj)p Fn(a)967 1417 y Fi(\006)995 1410 y Fm(jj)1023 1419 y Fk(L)1047 1410 y Fg(2)1064 1419 y Fh(\()p Fb(S)1099 1410 y Fd(d)p Ff(\000)p Fg(1)1154 1419 y Fh(\))1169 1410 y Fn(;)-3 1516 y Fc(and)j(the)h(fol)r(lowing)i(limit)e(exists)508 1644 y Fo(lim)498 1673 y Fk(\032)p Fi(!1)603 1644 y Fn(\032)628 1623 y Fi(\000)p Fh(1)684 1576 y Fj(Z)711 1688 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\032)807 1644 y Fm(j)p Fn(u)p Fo(\()p Fn(x)p Fo(\))p Fm(j)929 1623 y Fh(2)948 1644 y Fn(dx)c Fo(=)f(2)p Fm(jj)p Fn(a)1144 1651 y Fi(\006)1174 1644 y Fm(jj)1202 1623 y Fh(2)1202 1658 y Fk(L)1226 1648 y Fg(2)1242 1658 y Fh(\()p Fb(S)1278 1648 y Fd(d)p Ff(\000)p Fg(1)1332 1658 y Fh(\))1348 1644 y Fn(:)376 b Fo(\(4.12\))67 1804 y(It)20 b(follo)o(ws)g(from)f(\(4.11\))i(and)f(Prop)q(osition)i (4.1)e(that)h(the)f(functions)g Fn(a)1437 1811 y Fi(\006)1486 1804 y Fo(in)g(relations)g(\(1.6\),)-3 1864 y(\(1.7\))c(are)g(related)g (b)o(y)g(equalities)e Fn(a)666 1871 y Fh(+)709 1864 y Fo(=)g(\006)796 1871 y Fh(+)826 1864 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)918 1871 y Fi(\000)964 1864 y Fo(and)i Fn(a)1084 1871 y Fi(\000)1127 1864 y Fo(=)e(\006)1214 1871 y Fi(\000)1244 1864 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)1336 1871 y Fh(+)1365 1864 y Fo(.)22 b(Therefore)582 1970 y(\006)617 1977 y Fh(+)647 1970 y Fo(\()p Fn(\025)p Fo(\)\006)748 1977 y Fi(\000)778 1970 y Fo(\()p Fn(\025)p Fo(\))14 b(=)g(\006)945 1977 y Fi(\000)975 1970 y Fo(\()p Fn(\025)p Fo(\)\006)1076 1977 y Fh(+)1106 1970 y Fo(\()p Fn(\025)p Fo(\))g(=)g Fn(I)t(:)460 b Fo(\(4.13\))-3 2076 y(In)15 b(view)h(of)g(the)g (isometricit)o(y)d(of)k(\006)666 2083 y Fi(\006)696 2076 y Fo(\()p Fn(\025)p Fo(\),)f(this)g(implies)-3 2156 y Fr(Prop)r(osition)h(4.9)24 b Fc(The)18 b(op)n(er)n(ators)e Fo(\006)742 2163 y Fi(\006)771 2156 y Fo(\()p Fn(\025)p Fo(\))i Fc(ar)n(e)f(unitary)g(on)h Fn(L)1212 2138 y Fh(2)1232 2156 y Fo(\()p Fl(S)1282 2138 y Fk(d)p Fi(\000)p Fh(1)1344 2156 y Fo(\))p Fc(.)k(Mor)n(e)n(over)549 2262 y Fo(\006)584 2269 y Fh(+)614 2262 y Fo(\()p Fn(\025)p Fo(\))14 b(=)g(\006)781 2242 y Fi(\003)781 2275 y(\000)811 2262 y Fo(\()p Fn(\025)p Fo(\))p Fn(;)78 b Fo(\006)1004 2269 y Fi(\000)1033 2262 y Fo(\()p Fn(\025)p Fo(\))15 b(=)e(\006)1200 2242 y Fi(\003)1200 2275 y Fh(+)1230 2262 y Fo(\()p Fn(\025)p Fo(\))p Fn(:)-3 2388 y Fo(As)j(an)g(easy)h(consequence)e(of)i(Theorem)d(4.8,)j(w)o(e)e (obtain)-3 2468 y Fr(Prop)r(osition)i(4.10)24 b Fc(In)18 b(or)n(der)e(that)i(the)g(e)n(quation)g Fo(\(1)p Fn(:)p Fo(1\))g Fc(have)g(a)f(solution)h(with)g(asymptotics)139 2574 y Fn(u)p Fo(\()p Fn(x)p Fo(\))c(=)f Fn(a)324 2581 y Fh(+)354 2574 y Fo(\()s(^)-27 b Fn(x)o Fo(\))p Fn(w)454 2581 y Fh(+)484 2574 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j Fm(\000)f Fn(a)686 2581 y Fi(\000)716 2574 y Fo(\()s(^)-27 b Fn(x)o Fo(\))p Fn(w)816 2581 y Fi(\000)846 2574 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j(+)g Fn(o)1045 2581 y Fk(av)1084 2574 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1159 2554 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1314 2574 y Fo(\))p Fn(;)43 b(a)1416 2581 y Fi(\006)1467 2574 y Fm(2)22 b Fn(L)1555 2554 y Fh(2)1575 2574 y Fo(\()p Fl(S)1625 2554 y Fk(d)p Fi(\000)p Fh(1)1687 2574 y Fo(\))p Fn(;)-3 2680 y Fc(as)17 b Fm(j)p Fn(x)p Fm(j)c(!)h(1)p Fc(,)j(it)g(is)h(ne)n(c) n(essary)f(and)g(su\016cient)i(that)f Fn(a)1021 2687 y Fh(+)1064 2680 y Fo(=)c(\006)1151 2687 y Fh(+)1180 2680 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)1272 2687 y Fi(\000)1319 2680 y Fc(or)j Fn(a)1408 2687 y Fi(\000)1451 2680 y Fo(=)d(\006)1538 2687 y Fi(\000)1568 2680 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)1660 2687 y Fh(+)1689 2680 y Fc(.)905 2817 y Fo(12)p eop %%Page: 13 13 13 12 bop -3 26 a Fr(Corollary)17 b(4.11)24 b Fc(If)18 b Fn(u)f Fc(satis\014es)h(e)n(quation)g Fo(\(1)p Fn(:)p Fo(1\))g Fc(and)358 113 y Fn(u)p Fo(\()p Fn(x)p Fo(\))13 b(=)h Fn(a)p Fo(\()s(^)-27 b Fn(x)p Fo(\))p Fn(w)644 120 y Fi(\006)673 113 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))j(+)g Fn(o)872 120 y Fk(av)912 113 y Fo(\()p Fm(j)p Fn(x)p Fm(j)987 93 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1141 113 y Fo(\))p Fn(;)43 b(a)14 b Fm(2)g Fn(L)1337 93 y Fh(2)1357 113 y Fo(\()p Fl(S)1406 93 y Fk(d)p Fi(\000)p Fh(1)1468 113 y Fo(\))p Fn(;)237 b Fo(\(4.14\))-3 200 y Fc(as)17 b Fm(j)p Fn(x)p Fm(j)c(!)h(1)j Fo(\()p Fc(for)g(one)h(of)f(the)h(signs)g Fo(\\)12 b(+)f(")17 b Fc(or)g Fo(\\)12 b Fm(\000)f Fo("\))p Fc(,)17 b(then)i Fn(u)13 b Fo(=)h(0)p Fc(.)67 281 y Fo(Since)19 b(the)h(relation)f (\(4.14\))i(is)e(not)i Fc(a)f(priori)f Fo(supp)q(osed)i(di\013eren)o (tiable,)e(Corollary)h(4.11)g(im-)-3 341 y(pro)o(v)o(es)15 b(Prop)q(osition)j(2.4.)67 415 y(Set)h Fm(S)t Fo(\()p Fn(\025)p Fo(\))g(=)g(\006)365 422 y Fh(+)395 415 y Fo(\()p Fn(\025)p Fo(\))p Fm(T)13 b Fo(.)30 b(By)19 b(Prop)q(osition)h(4.9,)g (this)f(op)q(erator)h(is)f(unitary)g(on)h Fn(L)1614 397 y Fh(2)1634 415 y Fo(\()p Fl(S)1684 397 y Fk(d)p Fi(\000)p Fh(1)1746 415 y Fo(\))f(and)-3 475 y Fm(S)31 457 y Fi(\003)50 475 y Fo(\()p Fn(\025)p Fo(\))26 b(=)e Fm(T)13 b Fo(\006)279 482 y Fi(\000)309 475 y Fo(\()p Fn(\025)p Fo(\).)41 b(The)23 b(follo)o(wing)g(result)f(sho)o(ws)i(that)f(the)f(op)q(erator)i Fm(S)t Fo(\()p Fn(\025)p Fo(\))f(kno)o(wn)g(as)h(the)-3 536 y(scattering)16 b(matrix)e(\\relates")j Fm(F)620 543 y Fh(+)649 536 y Fo(\()p Fn(\025)p Fo(\))g(and)g Fm(F)863 543 y Fi(\000)892 536 y Fo(\()p Fn(\025)p Fo(\).)-3 610 y Fr(Prop)r(osition)g(4.12)24 b Fc(F)l(or)17 b(any)h Fn(f)h Fm(2)14 b Fn(B)s Fc(,)680 697 y Fm(F)716 704 y Fh(+)745 697 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)20 b Fo(=)14 b Fm(S)t Fo(\()p Fn(\025)p Fo(\))p Fm(F)1043 704 y Fi(\000)1072 697 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)s(:)559 b Fo(\(4.15\))-3 799 y Fc(Pr)n(o)n(of)15 b({)j Fo(Let)e Fn(a)e Fm(2)g Fn(L)377 781 y Fh(2)396 799 y Fo(\()p Fl(S)446 781 y Fk(d)p Fi(\000)p Fh(1)508 799 y Fo(\))j(b)q(e)f(arbitrary)l(.)21 b(By)16 b(Prop)q(osition)h(3.9,)f(the)g(function)473 886 y Fn(u)p Fo(\()p Fn(x)p Fo(\))e(=)f(\()p Fm(F)692 865 y Fi(\003)687 898 y Fh(+)717 886 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()p Fn(x)p Fo(\))d Fm(\000)h Fo(\()p Fm(F)1014 865 y Fi(\003)1009 898 y(\000)1038 886 y Fo(\()p Fn(\025)p Fo(\))p Fm(T)j Fo(\006)1180 893 y Fi(\000)1209 886 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(\)\()p Fn(x)p Fo(\))-3 973 y(satis\014es)i(equation)f(\(1.1\).)22 b(Moreo)o(v)o(er,) 14 b(b)o(y)h(Prop)q(osition)i(4.3)f(and)g(\(4.13\),)g Fn(u)p Fo(\()p Fn(x)p Fo(\))d(=)h Fn(o)1574 980 y Fk(av)1614 973 y Fo(\()p Fm(j)p Fn(x)p Fm(j)1689 955 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1844 973 y Fo(\))-3 1033 y(as)i Fm(j)p Fn(x)p Fm(j)e(!)g(1)p Fo(.)21 b(Therefore)16 b(Corollary)g(4.11)h(implies)d(that)i Fn(u)e Fo(=)g(0)j(or,)f(equiv)m(alen)o(tly)l(,)d(that)k Fm(F)1720 1015 y Fi(\003)1715 1046 y Fh(+)1744 1033 y Fo(\()p Fn(\025)p Fo(\))e(=)-3 1093 y Fm(F)38 1075 y Fi(\003)33 1106 y(\000)62 1093 y Fo(\()p Fn(\025)p Fo(\))p Fm(S)162 1075 y Fi(\003)182 1093 y Fo(\()p Fn(\025)p Fo(\).)21 b Fa(2)67 1186 y Fo(On)12 b(the)g(set)g Fn(C)335 1168 y Fi(1)372 1186 y Fo(\()p Fl(S)422 1168 y Fk(d)p Fi(\000)p Fh(1)484 1186 y Fo(\),)h(the)f(scattering)g(matrix)e(can)i(b) q(e)h(expressed)e(in)h(terms)f(of)h(the)g(op)q(erators)-3 1246 y Fn(G)35 1253 y Fi(\006)65 1246 y Fo(\()p Fn(\025)p Fo(\))k(de\014ned)g(b)o(y)g(\(3.18\)-\(3.20\).)23 b(Indeed,)15 b(it)h(follo)o(ws)g(from)f(\(4.1\))h(that)137 1334 y Fm(S)t Fo(\()p Fn(\025)p Fo(\))p Fn(a)e Fo(=)f Fn(\031)358 1313 y Fh(1)p Fk(=)p Fh(2)413 1334 y Fn(\025)441 1313 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)524 1334 y Fm(F)560 1341 y Fh(+)589 1334 y Fo(\()p Fn(\025)p Fo(\))p Fn(G)693 1341 y Fi(\000)723 1334 y Fo(\()p Fn(\025)p Fo(\))p Fn(a;)57 b Fm(S)920 1313 y Fi(\003)940 1334 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)13 b Fo(=)h Fn(\031)1127 1313 y Fh(1)p Fk(=)p Fh(2)1182 1334 y Fn(\025)1210 1313 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)1293 1334 y Fm(F)1329 1341 y Fi(\000)1358 1334 y Fo(\()p Fn(\025)p Fo(\))p Fn(G)1462 1341 y Fh(+)1492 1334 y Fo(\()p Fn(\025)p Fo(\))p Fn(a;)140 b Fo(\(4.16\))-3 1421 y(whic)o(h)16 b(coincides)g(with)h(the)g(represen)o(tation)f(for)i (the)f(scattering)g(matrix)e(obtained)i(in)g([16].)24 b(The)-3 1481 y(\014rst)16 b(of)h(these)f(equalities)e(sho)o(ws)j (that,)g(for)f(an)o(y)g Fn(a;)8 b(b)13 b Fm(2)h Fn(C)1087 1463 y Fi(1)1124 1481 y Fo(\()p Fl(S)1174 1463 y Fk(d)p Fi(\000)p Fh(1)1236 1481 y Fo(\),)257 1568 y(\()p Fm(S)t Fo(\()p Fn(\025)p Fo(\))p Fn(a;)8 b(b)p Fo(\))41 b(=)h Fn(\031)615 1548 y Fh(1)p Fk(=)p Fh(2)669 1568 y Fn(\025)697 1548 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)780 1568 y Fm(h)p Fn(G)837 1575 y Fi(\000)867 1568 y Fo(\()p Fn(\025)p Fo(\))p Fn(a;)8 b Fm(F)1022 1548 y Fi(\003)1017 1580 y Fh(+)1046 1568 y Fo(\()p Fn(\025)p Fo(\))p Fn(b)p Fm(i)505 1646 y Fo(=)42 b Fm(\000)p Fn(i)p Fo(2)665 1625 y Fi(\000)p Fh(1)711 1646 y Fn(\025)739 1625 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)822 1646 y Fm(h)p Fn(G)879 1653 y Fi(\000)910 1646 y Fo(\()p Fn(\025)p Fo(\))p Fn(a;)8 b(u)1052 1653 y Fh(+)1081 1646 y Fo(\()p Fn(\025)p Fo(\))j Fm(\000)g Fn(R)p Fo(\()p Fn(\025)h Fm(\000)f Fn(i)p Fo(0\))p Fn(G)1452 1653 y Fh(+)1482 1646 y Fo(\()p Fn(\025)p Fo(\))p Fn(b)p Fm(i)p Fn(:)-3 1733 y Fo(Since)19 b Fn(G)166 1740 y Fi(\000)196 1733 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)i Fo(and)h Fn(G)447 1740 y Fh(+)477 1733 y Fo(\()p Fn(\025)p Fo(\))p Fn(b)e Fo(are)h(con)o(tin)o(uous)g(functions)g(of)g Fn(\025)g Fo(in)f Fn(L)1337 1715 y Fh(2)1337 1745 y Fk(\015)1360 1733 y Fo(\()p Fl(R)1418 1715 y Fk(d)1435 1733 y Fo(\))h(for)g(some)f Fn(\015)k(>)e Fo(1)p Fn(=)p Fo(2)-3 1793 y(and)f Fn(u)124 1800 y Fh(+)153 1793 y Fo(\()p Fn(\025)p Fo(\))h(is)f(con)o(tin)o(uous) f(in)h Fn(L)638 1775 y Fh(2)638 1807 y Fi(\000)p Fk(\015)685 1797 y Ff(0)720 1793 y Fo(for)g(some)f Fn(\015)954 1775 y Fi(0)988 1793 y Fn(>)i Fo(1)p Fn(=)p Fo(2,)g(the)f(con)o(tin)o(uit)o (y)e(of)i(\()p Fm(S)t Fo(\()p Fn(\025)p Fo(\))p Fn(a;)8 b(b)p Fo(\))21 b(with)-3 1854 y(resp)q(ect)16 b(to)i Fn(\025)f Fo(follo)o(ws)g(from)f(Prop)q(osition)i(2.5.)24 b(T)l(aking)18 b(in)o(to)f(accoun)o(t)g(that)g(the)g(op)q(erators)h Fm(S)t Fo(\()p Fn(\025)p Fo(\))-3 1914 y(and)e Fm(S)125 1896 y Fi(\003)145 1914 y Fo(\()p Fn(\025)p Fo(\))h(are)f(isometric,)d (w)o(e)j(obtain)-3 1988 y Fr(Prop)r(osition)h(4.13)24 b Fc(The)c(op)n(er)n(ators)d Fm(S)t Fo(\()p Fn(\025)p Fo(\))j Fc(and)f Fm(S)988 1970 y Fi(\003)1008 1988 y Fo(\()p Fn(\025)p Fo(\))g Fc(ar)n(e)g(str)n(ongly)g(c)n(ontinuous)h (functions)h(of)-3 2048 y Fn(\025)p Fc(.)67 2123 y Fr(3.)46 b Fo(Here)21 b(w)o(e)h(form)o(ulate)f(a)i(corollary)f(of)g(our)h (considerations)g(for)g(the)f(non-homogeneous)-3 2183 y(Sc)o(hr\177)-24 b(odinger)15 b(equation.)-3 2263 y Fr(Prop)r(osition)i(4.14)24 b Fc(F)l(or)19 b(any)g Fn(a)e Fm(2)g Fn(L)737 2245 y Fh(2)757 2263 y Fo(\()p Fl(S)806 2245 y Fk(d)p Fi(\000)p Fh(1)869 2263 y Fo(\))p Fc(,)i(ther)n(e)g (exist)h(functions)h Fn(f)1398 2270 y Fi(\006)1444 2263 y Fm(2)c Fn(B)22 b Fc(and)d Fn(u)1677 2270 y Fi(\006)1723 2263 y Fm(2)f Fr(H)1818 2245 y Fh(2)1818 2276 y Fk(loc)-3 2324 y Fc(with)f(asymptotics)h Fo(\(2)p Fn(:)p Fo(11\))p Fc(-)p Fo(\(2)p Fn(:)p Fo(12\))h Fc(satisfying)f(the)f(e)n(quation)644 2411 y Fm(\000)p Fo(\001)p Fn(u)752 2418 y Fi(\006)791 2411 y Fo(+)11 b Fn(V)h(u)908 2418 y Fi(\006)951 2411 y Fo(=)i Fn(\025u)1059 2418 y Fi(\006)1099 2411 y Fo(+)d Fn(f)1172 2418 y Fi(\006)1202 2411 y Fn(:)522 b Fo(\(4.17\))-3 2512 y Fc(Pr)n(o)n(of)15 b({)i Fo(According)f(to)g(the)g(last)g (statemen)o(t)e(of)i(Theorem)f(2.3,)h(it)f(follo)o(ws)h(from)f(Prop)q (osition)i(4.5)-3 2572 y(that)f Fn(R)p Fo(\()p Fm(F)194 2579 y Fi(\006)224 2572 y Fo(\()p Fn(\025)p Fo(\)\))e(=)g Fn(L)408 2554 y Fh(2)428 2572 y Fo(\()p Fl(S)478 2554 y Fk(d)p Fi(\000)p Fh(1)540 2572 y Fo(\).)21 b(Therefore)16 b(there)g(exist)f(functions)i Fn(f)1290 2579 y Fi(\006)1333 2572 y Fm(2)d Fn(B)19 b Fo(suc)o(h)d(that)h Fm(F)1688 2579 y Fh(+)1717 2572 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)1807 2579 y Fh(=)1837 2572 y Fn(a)-3 2632 y Fo(or)e Fm(F)91 2639 y Fi(\000)120 2632 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)210 2639 y Fi(\000)254 2632 y Fo(=)f Fm(T)f Fn(a)372 2639 y Fi(\000)401 2632 y Fo(.)20 b(Clearly)l(,)14 b(functions)h Fn(u)850 2639 y Fi(\006)893 2632 y Fo(=)f Fn(R)p Fo(\()p Fn(\025)9 b Fm(\006)g Fn(i)p Fo(0\))p Fn(f)1170 2639 y Fi(\006)1214 2632 y Fo(satisfy)15 b(\(4.17\))h(and)f(ha)o(v)o(e)f (asymp-)-3 2693 y(totics)i(\(2.11\)-\(2.12\))h(b)o(y)f(virtue)f(of)i (Theorem)e(3.5.)21 b Fa(2)905 2817 y Fo(13)p eop %%Page: 14 14 14 13 bop -3 26 a Fr(5.)55 b(EXP)-5 b(ANSION)20 b(THEOREM)-3 156 y(1.)i Fo(Let)165 143 y(~)151 156 y Fm(H)15 b Fo(=)f Fn(L)293 138 y Fh(2)313 156 y Fo(\()p Fl(R)371 163 y Fh(+)398 156 y Fn(;)8 b(L)453 138 y Fh(2)472 156 y Fo(\()p Fl(S)522 138 y Fk(d)p Fi(\000)p Fh(1)584 156 y Fo(\)\))17 b(b)q(e)g(the)f(Hilb)q(ert)g(space)g(of)h Fn(L)1178 138 y Fh(2)1198 156 y Fo(\()p Fl(S)1248 138 y Fk(d)p Fi(\000)p Fh(1)1310 156 y Fo(\)-v)m(alued)g(square)g(in)o(tegrable)-3 216 y(functions)j(on)i Fl(R)323 223 y Fh(+)370 216 y Fo(with)f(the)f(Leb)q(esgue)i(measure.)33 b(Recall)20 b(that)h(the)f(op)q(erator)i Fm(F)1599 223 y Fi(\006)1629 216 y Fo(\()p Fn(\025)p Fo(\))g(:)f Fn(B)j Fm(!)-3 276 y Fn(L)30 258 y Fh(2)49 276 y Fo(\()p Fl(S)99 258 y Fk(d)p Fi(\000)p Fh(1)161 276 y Fo(\))g(w)o(as)g(de\014ned)f(b)o(y)g (equalities)f(\(3.13\),)k(\(3.14\))e(and)g(\(3.16\).)44 b(No)o(w)23 b(w)o(e)g(in)o(tro)q(duce)g(the)-3 336 y(mapping)15 b Fn(F)229 343 y Fi(\006)272 336 y Fo(:)30 b Fn(B)16 b Fm(!)447 324 y Fo(~)433 336 y Fm(H)g Fo(b)o(y)g(the)g(form)o(ula)713 445 y(\()p Fn(F)764 452 y Fi(\006)794 445 y Fn(f)5 b Fo(\)\()p Fn(\025)p Fo(\))14 b(=)g Fm(F)1010 452 y Fi(\006)1039 445 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)s(:)616 b Fo(\(5.1\))-3 553 y(In)o(tegrating)16 b(\(3.17\))g(o)o(v)o(er)g Fl(R)533 560 y Fh(+)559 553 y Fo(,)g(and)h(using)g(the)f(sp)q(ectral)g(theorem,) e(w)o(e)i(\014nd)g(that)33 689 y Fm(jj)p Fn(F)93 696 y Fi(\006)122 689 y Fn(f)5 b Fm(jj)179 668 y Fh(2)213 689 y Fo(=)265 621 y Fj(Z)314 634 y Fi(1)292 734 y Fh(0)360 689 y Fm(jjF)424 696 y Fi(\006)453 689 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)g Fm(jj)576 668 y Fh(2)576 703 y Fk(L)600 694 y Fg(2)618 703 y Fh(\()p Fb(S)653 694 y Fd(d)p Ff(\000)p Fg(1)707 703 y Fh(\))731 689 y Fn(d\025)15 b Fo(=)851 621 y Fj(Z)901 634 y Fi(1)878 734 y Fh(0)951 655 y Fn(d)p Fo(\()p Fn(E)s Fo(\()p Fn(\025)p Fo(\))p Fn(f)s(;)8 b(f)d Fo(\))p 951 678 247 2 v 1048 723 a Fn(d\025)1203 689 y(d\025)14 b Fo(=)g Fm(jj)p Fn(P)7 b(f)e Fm(jj)1445 668 y Fh(2)1465 689 y Fn(;)56 b(P)21 b Fo(=)14 b Fn(E)s Fo(\(0)p Fn(;)8 b Fm(1)p Fo(\))p Fn(;)-3 821 y Fo(for)16 b(an)o(y)h Fn(f)j Fm(2)15 b Fn(B)s Fo(.)22 b(Hence)15 b(the)i(op)q(erator)h Fn(F)791 828 y Fi(\006)836 821 y Fo(extends)f(b)o(y)f(con)o(tin)o(uit)o (y)f(to)i(a)g(b)q(ounded)g(op)q(erator)h(on)-3 881 y(the)i(en)o(tire)e (space)j Fm(H)p Fo(.)34 b(This)20 b(op)q(erator)h(is)f(isometric)e(on)j (the)f(absolutely)g(con)o(tin)o(uous)g(subspace)-3 941 y Fm(H)39 923 y Fh(\()p Fk(a)p Fh(\))101 941 y Fo(=)14 b Fn(P)7 b Fm(H)17 b Fo(of)g(the)f(op)q(erator)h Fn(H)k Fo(and)16 b(v)m(anishes)h(on)g(its)f(orthogonal)i(complem)o(en)o(t,)12 b(i.e.,)813 1050 y Fn(F)852 1029 y Fi(\003)845 1062 y(\006)874 1050 y Fn(F)906 1057 y Fi(\006)949 1050 y Fo(=)h Fn(P)q(:)716 b Fo(\(5.2\))67 1184 y(No)o(w,)17 b(denote)h(b)o(y)f(\003)g(m)o (ultiplication)e(b)o(y)i(the)g(indep)q(enden)o(t)g(v)m(ariable)g Fn(\025)h Fo(in)f(the)g(space)1739 1171 y(~)1724 1184 y Fm(H)h Fo(and)-3 1244 y(b)o(y)e Fn(E)101 1251 y Fh(\003)128 1244 y Fo(\()p Fn(X)t Fo(\))h(its)g(sp)q(ectral)g(pro)s(jection)f (acting)i(as)f(m)o(ultiplicati)o(on)e(b)o(y)i(the)f(c)o(haracteristic)g (function)-3 1304 y(of)g(the)g(Borel)f(set)i Fn(X)e Fm(\\)c Fl(R)480 1311 y Fh(+)507 1304 y Fo(.)21 b(W)l(e)16 b(shall)g(establish) g(the)g(in)o(tert)o(wining)f(prop)q(ert)o(y)684 1413 y Fn(F)716 1420 y Fi(\006)745 1413 y Fn(E)781 1420 y Fh(\003)808 1413 y Fo(\()p Fn(X)t Fo(\))f(=)g Fn(E)992 1420 y Fh(\003)1018 1413 y Fo(\()p Fn(X)t Fo(\))p Fn(F)1132 1420 y Fi(\006)1162 1413 y Fn(:)586 b Fo(\(5.3\))-3 1521 y(W)l(e)16 b(need)f(only)h(to)h(c)o(hec)o(k)d(that)685 1581 y(\()p Fn(F)736 1588 y Fi(\006)765 1581 y Fn(H)t(f)s(;)c Fo(~)-26 b Fn(g)r Fo(\))14 b(=)g(\()p Fn(F)1019 1588 y Fi(\006)1048 1581 y Fn(f)s(;)8 b Fo(\003)r(~)-26 b Fn(g)q Fo(\))-3 1668 y(for)16 b Fn(f)j Fm(2)14 b Fe(S)j Fo(and)h(~)-25 b Fn(g)r Fo(\()p Fn(\025)p Fo(\))14 b(=)g Fn(\020)t Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fo(,)i(where)g Fn(\020)h Fm(2)e Fn(C)884 1650 y Fi(1)880 1680 y Fh(0)920 1668 y Fo(\()p Fl(R)978 1675 y Fh(+)1005 1668 y Fo(\))h(and)h Fn(a)c Fm(2)h Fn(L)1254 1650 y Fh(2)1274 1668 y Fo(\()p Fl(S)1324 1650 y Fk(d)p Fi(\000)p Fh(1)1386 1668 y Fo(\),)i(i.e.,)e(to) j(v)o(erify)d(that)364 1734 y Fj(Z)414 1747 y Fi(1)392 1846 y Fh(0)451 1801 y Fm(h)p Fn(H)t(f)s(;)8 b Fm(F)604 1781 y Fi(\003)599 1814 y(\006)629 1801 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fm(i)p Fn(\020)t Fo(\()p Fn(\025)p Fo(\))g Fn(d\025)16 b Fo(=)959 1734 y Fj(Z)1009 1747 y Fi(1)987 1846 y Fh(0)1055 1801 y Fn(\025)p Fm(h)p Fn(f)s(;)8 b Fm(F)1192 1781 y Fi(\003)1187 1814 y(\006)1217 1801 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)p Fm(i)p Fn(\020)t Fo(\()p Fn(\025)p Fo(\))g Fn(d\025:)-3 1936 y Fo(This)j(equalit)o(y)f(is)i(a)g (direct)e(consequence)h(of)h(Prop)q(osition)g(3.9.)20 b(Let)12 b(us)g(\014nally)f(sho)o(w)h(that)g Fn(R)p Fo(\()p Fn(F)1762 1943 y Fi(\006)1792 1936 y Fo(\))i(=)-3 1996 y Fm(H)p Fo(.)-3 2077 y Fr(Lemm)o(a)i(5.1)24 b Fc(The)18 b(op)n(er)n(ator)e Fo(\(5)p Fn(:)p Fo(1\))h Fc(satis\014es)816 2185 y Fn(F)848 2192 y Fi(\006)877 2185 y Fn(F)916 2165 y Fi(\003)909 2198 y(\006)952 2185 y Fo(=)d Fn(I)t(:)718 b Fo(\(5.4\))-3 2314 y Fc(Pr)n(o)n(of)15 b({)34 b Fo(According)16 b(to)g(\(5.2\),)g(\(5.3\),)g(for)h(an)o(y)f Fn(f)j Fm(2)14 b(H)j Fo(and)g(an)o(y)f Fn(X)i Fm(\032)c Fl(R)1390 2321 y Fh(+)1416 2314 y Fo(,)509 2422 y(\()p Fn(E)564 2429 y Fh(\003)590 2422 y Fo(\()p Fn(X)t Fo(\))p Fn(F)704 2429 y Fi(\006)734 2422 y Fn(F)773 2402 y Fi(\003)766 2435 y(\006)796 2422 y Fo(~)-25 b Fn(g)r(;)8 b(F)874 2429 y Fi(\006)903 2422 y Fn(f)d Fo(\))14 b(=)g(\()p Fn(E)1072 2429 y Fh(\003)1098 2422 y Fo(\()p Fn(X)t Fo(\))r(~)-26 b Fn(g)s(;)8 b(F)1260 2429 y Fi(\006)1289 2422 y Fn(f)d Fo(\))p Fn(:)411 b Fo(\(5.5\))-3 2537 y(Let)84 2524 y(~)84 2537 y Fn(h)14 b Fo(=)g Fn(F)210 2544 y Fi(\006)239 2537 y Fn(F)278 2519 y Fi(\003)271 2549 y(\006)302 2537 y Fo(~)-26 b Fn(g)r Fo(.)21 b(If)16 b Fn(f)j Fm(2)14 b Fn(B)s Fo(,)i(relation)f(\(5.5\))i(is)f(equiv)m(alen)o(t)f(to)306 2605 y Fj(Z)334 2718 y Fk(X)368 2673 y Fo(\()387 2659 y(~)387 2673 y Fn(h)p Fo(\()p Fn(\025)p Fo(\))p Fn(;)8 b Fm(F)539 2680 y Fi(\006)568 2673 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)d Fo(\))682 2682 y Fk(L)706 2673 y Fg(2)724 2682 y Fh(\()p Fb(S)759 2673 y Fd(d)p Ff(\000)p Fg(1)814 2682 y Fh(\))838 2673 y Fn(d\025)14 b Fo(=)957 2605 y Fj(Z)985 2718 y Fk(X)1018 2673 y Fo(\()r(~)-26 b Fn(g)s Fo(\()p Fn(\025)p Fo(\))p Fn(;)8 b Fm(F)1187 2680 y Fi(\006)1216 2673 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)d Fo(\))1330 2682 y Fk(L)1354 2673 y Fg(2)1372 2682 y Fh(\()p Fb(S)1407 2673 y Fd(d)p Ff(\000)p Fg(1)1462 2682 y Fh(\))1486 2673 y Fn(d\025:)905 2817 y Fo(14)p eop %%Page: 15 15 15 14 bop -3 26 a Fo(Since)15 b Fn(f)k Fm(2)14 b Fn(B)19 b Fo(is)d(arbitrary)l(,)g(this)g(in)o(v)o(olv)o(es)604 83 y Fj(Z)631 195 y Fk(X)673 151 y Fm(F)714 130 y Fi(\003)709 163 y(\006)739 151 y Fo(\()p Fn(\025)p Fo(\)\()824 137 y(~)824 151 y Fn(h)p Fo(\()p Fn(\025)p Fo(\))c Fm(\000)g Fo(~)-26 b Fn(g)s Fo(\()p Fn(\025)p Fo(\)\))8 b Fn(d\025)15 b Fo(=)f(0)p Fn(:)-3 277 y Fo(and)i(th)o(us,)g(the)g(set)g Fn(X)21 b Fo(b)q(eing)16 b(arbitrary)l(,)670 380 y Fm(F)711 360 y Fi(\003)706 393 y(\006)735 380 y Fo(\()p Fn(\025)p Fo(\)\()820 367 y(~)820 380 y Fn(h)p Fo(\()p Fn(\025)p Fo(\))c Fm(\000)g Fo(~)-25 b Fn(g)r Fo(\()p Fn(\025)p Fo(\)\))14 b(=)g(0)p Fn(;)-3 489 y Fo(for)j(almost)g(all)g Fn(\025)f Fm(2)g Fl(R)432 496 y Fh(+)458 489 y Fo(.)25 b(By)17 b(Prop)q(osition)h(4.5,)g(the)f(k)o(ernel)f(of)i Fm(F)1260 471 y Fi(\003)1255 502 y(\006)1284 489 y Fo(\()p Fn(\025)p Fo(\))g(is)f(trivial,)f(hence)1715 476 y(~)1714 489 y Fn(h)q Fo(\()p Fn(\025)p Fo(\))g(=)-2 550 y(~)-25 b Fn(g)r Fo(\()p Fn(\025)p Fo(\))16 b(for)h(almost)e(all)h Fn(\025)e Fm(2)g Fl(R)532 557 y Fh(+)559 550 y Fo(.)21 b(This)16 b(implies)e(\(5.4\).)21 b Fa(2)67 652 y Fo(Let)16 b(us)h(summarize)c(the)j(results)g(obtained.)-3 730 y Fr(Theorem)g(5.2)24 b Fc(F)l(or)g(every)h Fn(f)33 b Fm(2)27 b Fn(B)g Fc(the)e(function)h Fm(F)1068 737 y Fi(\006)1097 730 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)k Fc(b)n(elongs)c(to)f(the)g(sp)n (ac)n(e)1694 718 y Fo(~)1679 730 y Fm(H)q Fc(.)43 b(The)-3 791 y(op)n(er)n(ator)18 b Fn(F)223 798 y Fi(\006)272 791 y Fc(de\014ne)n(d)j(by)f Fo(\(5)p Fn(:)p Fo(1\))g Fc(extends)h(by)f(c)n(ontinuity)h(to)f(a)f(b)n(ounde)n(d)h(op)n(er)n (ator)f(on)h(the)g(entir)n(e)-3 851 y(sp)n(ac)n(e)d Fm(H)h Fc(and)f(satis\014es)h(r)n(elations)g Fo(\(5)p Fn(:)p Fo(2\))p Fc(,)g Fo(\(5)p Fn(:)p Fo(3\))f Fc(and)h Fo(\(5)p Fn(:)p Fo(4\))p Fc(.)67 930 y Fr(2.)j Fo(This)16 b(theorem)d(is,)i(ob)o (viously)l(,)g(a)h(standard)g(\(see)f(e.g.)21 b([6]\))15 b(expansion)g(theorem.)20 b(Similarly)-3 990 y(to)d(the)f(short-range)i (case)f([18],)f(results)h(of)g(the)f(previous)g(section)h(allo)o(w)f (us)h(to)g(reform)o(ulate)e(it)h(in)-3 1050 y(terms)e(of)i(solutions)h (of)f(equation)g(\(1.1\))g(with)g(sp)q(ecial)g(asymptotics)f(at)h (in\014nit)o(y)l(.)k(By)15 b(\(5.1\),)h(\(5.2\),)-3 1110 y(the)g(equalit)o(y)444 1193 y(\()p Fn(P)7 b(f)s(;)h(g)r Fo(\))13 b(=)659 1125 y Fj(Z)709 1139 y Fi(1)687 1238 y Fh(0)746 1193 y Fm(hF)806 1173 y Fi(\003)826 1193 y Fo(\()p Fn(\025)p Fo(\))903 1180 y(~)892 1193 y Fn(f)916 1200 y Fi(\006)946 1193 y Fo(\()p Fn(\025)p Fo(\))p Fn(;)8 b(g)r Fm(i)g Fn(d\025;)1206 1180 y Fo(~)1195 1193 y Fn(f)1219 1200 y Fi(\006)1263 1193 y Fo(=)13 b Fn(F)1346 1200 y Fi(\006)1375 1193 y Fn(f)s(;)346 b Fo(\(5.6\))-3 1301 y(holds)16 b(for)h(an)o(y)f Fn(f)j Fm(2)14 b(H)j Fo(and)g Fn(g)f Fm(2)e Fn(B)s Fo(.)20 b(Prop)q(osition)e(4.3)e(implies)e(that)j (for)f(almost)g(all)f Fn(\025)23 b Fm(2)f Fl(R)1752 1308 y Fh(+)1795 1301 y Fo(the)-3 1366 y(function)16 b Fn(u)216 1341 y Fh(\()p Fi(\006)p Fh(\))216 1380 y Fk(f)272 1366 y Fo(\()p Fn(\025)p Fo(\))f(=)e Fm(F)445 1348 y Fi(\003)440 1379 y(\006)470 1366 y Fo(\()p Fn(\025)p Fo(\))547 1353 y(~)536 1366 y Fn(f)560 1373 y Fi(\006)590 1366 y Fo(\()p Fn(\025)p Fo(\))h Fm(2)g Fn(B)757 1348 y Fi(\003)792 1366 y Fo(satis\014es)j(equation)f(\(1.1\))h(and)108 1494 y Fn(u)136 1469 y Fh(\(+\))136 1508 y Fk(f)193 1494 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))41 b(=)h Fm(\000)p Fn(i)p Fo(2)510 1473 y Fi(\000)p Fh(1)556 1494 y Fn(\031)586 1473 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)668 1494 y Fn(\025)696 1473 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)779 1439 y Fj(h)813 1481 y Fo(~)803 1494 y Fn(f)5 b Fo(\()s(^)-27 b Fn(x;)8 b(\025)p Fo(\))p Fn(w)983 1501 y Fh(+)1013 1494 y Fo(\()p Fn(x;)g(\025)p Fo(\))j Fm(\000)f Fo(\()p Fm(S)1242 1473 y Fi(\003)1262 1494 y Fo(\()p Fn(\025)p Fo(\))1339 1481 y(~)1328 1494 y Fn(f)c Fo(\()p Fn(\025)p Fo(\)\)\()p Fm(\000)s Fo(^)-27 b Fn(x)o Fo(\))p Fn(w)1582 1501 y Fi(\000)1612 1494 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))1728 1439 y Fj(i)430 1588 y Fo(+)p Fn(o)491 1595 y Fk(av)530 1588 y Fo(\()p Fm(j)p Fn(x)p Fm(j)605 1568 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)760 1588 y Fo(\))983 b(\(5.7\))85 1751 y Fn(u)113 1726 y Fh(\()p Fi(\000)p Fh(\))113 1765 y Fk(f)169 1751 y Fo(\()p Fn(\025)p Fo(\)\()p Fn(x;)8 b(\025)p Fo(\))42 b(=)g Fm(\000)p Fn(i)p Fo(2)553 1731 y Fi(\000)p Fh(1)600 1751 y Fn(\031)630 1731 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)711 1751 y Fn(\025)739 1731 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)822 1696 y Fj(h)846 1751 y Fo(\()p Fm(S)t Fo(\()p Fn(\025)p Fo(\))976 1738 y(~)965 1751 y Fn(f)5 b Fo(\()p Fn(\025)p Fo(\)\)\()s(^)-27 b Fn(x)q Fo(\))p Fn(w)1181 1758 y Fh(+)1210 1751 y Fo(\()p 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Fn(u)1133 2425 y Fh(\()p Fi(\006)p Fh(\))1133 2464 y Fk(f)1191 2450 y Fo(\()p Fn(\025)p Fo(\))p Fm(jj)1285 2457 y Fk(B)1314 2448 y Ff(\003)1333 2450 y Fm(jj)p Fn(g)r Fm(jj)1414 2457 y Fk(B)864 2587 y Fm(\024)i Fn(C)t Fo(\()p Fn(\026;)8 b(\027)s Fo(\))1108 2519 y Fj(Z)1158 2533 y Fk(\027)1136 2632 y(\026)1188 2587 y Fm(jj)1226 2574 y Fo(~)1216 2587 y Fn(f)d Fo(\()p Fn(\025)p Fo(\))p Fm(jj)1339 2597 y Fk(L)1363 2587 y Fg(2)1380 2597 y Fh(\()p Fb(S)1415 2587 y Fd(d)p Ff(\000)p Fg(1)1469 2597 y Fh(\))1485 2587 y Fn(d\025)j Fm(jj)p Fn(g)r Fm(jj)1627 2594 y Fk(B)864 2693 y Fm(\024)42 b Fn(C)980 2700 y Fh(1)999 2693 y Fo(\()p Fn(\026;)8 b(\027)s Fo(\))p Fm(jj)p Fn(f)d Fm(jj)j(jj)p Fn(g)r Fm(jj)1289 2700 y Fk(B)1320 2693 y Fn(:)404 b Fo(\(5.10\))905 2817 y(15)p eop %%Page: 16 16 16 15 bop -3 26 a Fo(Since)15 b Fm(h)p Fn(u)171 1 y Fh(\()p Fi(\006)p Fh(\))171 40 y Fk(f)228 26 y Fo(\()p Fn(\025)p Fo(\))p Fn(;)8 b(g)r Fm(i)18 b Fo(is)e(an)h(in)o(tegral)f(in)g Fn(x)p Fo(,)g(\(5.6\))g(can)h(b)q(e)g(considered)f(as)h(a)g(double)f (in)o(tegral.)21 b(Th)o(us)-3 87 y(the)c(F)l(ubini)h(theorem)e(allo)o (ws)i(us)h(to)f(in)o(terc)o(hange)f(the)h(order)g(of)h(in)o(tegration)f (in)f(the)h(v)m(ariables)g Fn(\025)-3 147 y Fo(and)e Fn(x)p Fo(.)21 b(So)c(w)o(e)f(get)633 230 y Fm(h)p Fn(f)s(;)8 b(g)r Fm(i)14 b Fo(=)811 174 y Fj(\020)841 162 y(Z)891 175 y Fi(1)868 275 y Fh(0)936 230 y Fn(u)964 204 y Fh(\()p Fi(\006)p Fh(\))964 244 y Fk(f)1021 230 y Fo(\()p Fn(\025)p Fo(\))8 b Fn(d\025;)g(g)1195 174 y Fj(\021)-3 340 y Fo(and)16 b(therefore)668 413 y Fn(f)5 b Fo(\()p Fn(x)p Fo(\))13 b(=)828 346 y Fj(Z)878 359 y Fi(1)856 458 y Fh(0)924 413 y Fn(u)952 388 y Fh(\()p Fi(\006)p Fh(\))952 427 y Fk(f)1008 413 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))p Fn(d\025:)547 b Fo(\(5.11\))-3 527 y(Since)20 b(in)h(\(5.10\),)h(the)f Fn(g)i Fo(can)f(b)q(e)f(c)o(hosen)g(as)h(the)f(c)o(haracteristic)e (functions)i(of)h(the)f(balls)g Fm(f)p Fn(x)g Fm(2)-3 587 y Fl(R)36 569 y Fk(d)69 587 y Fo(:)15 b Fm(j)p Fn(x)p Fm(j)g(\024)h Fn(\032)p Fm(g)p Fo(,)h(this)g(ensures)h(that)f(the)h(in) o(tegral)e(\(5.11\))i(con)o(v)o(erges)f(\(absolutely\))g(for)h(almost)e (all)-3 647 y Fn(x)d Fm(2)h Fl(R)124 629 y Fk(d)141 647 y Fo(.)21 b(Remark)12 b(also)i(that)g(the)g(in)o(tegral)e(\(5.11\))j(b) q(elongs)f(to)g Fm(H)h Fo(although)f(its)g(in)o(tegrand)f(is)h(from)-3 708 y Fn(B)37 689 y Fi(\003)72 708 y Fo(only)l(.)67 788 y(F)l(or)i(arbitrary)h Fn(f)i Fm(2)14 b(H)p Fo(,)i(one)h(has)436 898 y Fn(P)7 b(f)19 b Fo(=)14 b Fn(s)d Fm(\000)64 b Fo(lim)653 928 y Fk(\026)p Fi(!)p Fh(0)p Fk(;\027)r Fi(!1)835 898 y Fn(f)859 905 y Fk(\026;\027)912 898 y Fn(;)122 b(f)1072 905 y Fk(\026;\027)1138 898 y Fo(=)14 b Fn(E)s Fo(\(\()p Fn(\026;)8 b(\027)s Fo(\)\))p Fn(f)s(;)-3 1040 y Fo(where)16 b Fn(s)c Fm(\000)f Fo(lim)k(means)h(the)h(strong)i(limit)14 b(in)j Fm(H)p Fo(.)24 b(By)17 b(\(5.3\),)1168 1027 y(~)1158 1040 y Fn(f)1182 1047 y Fk(\026;\027)1234 1040 y Fo(\()p Fn(\025)p Fo(\))f(=)1380 1027 y(~)1369 1040 y Fn(f)6 b Fo(\()p Fn(\025)p Fo(\))17 b(if)g Fn(\025)24 b Fm(2)g Fo(\()p Fn(\026;)8 b(\027)s Fo(\),)17 b(and)7 1087 y(~)-3 1100 y Fn(f)21 1107 y Fk(\026;\027)73 1100 y Fo(\()p Fn(\025)p Fo(\))i(=)f(0)h(if)g Fn(\025)24 b(=)-30 b Fm(2)19 b Fo(\()p Fn(\026;)8 b(\027)s Fo(\).)29 b(No)o(w)19 b(using)g(\(5.11\)) h(for)f Fn(f)1051 1107 y Fk(\026;\027)1104 1100 y Fo(,)g(w)o(e)f(can)h (reform)o(ulate)e(the)h(expansion)-3 1160 y(theorem)c(as)j(follo)o(ws) -3 1241 y Fr(Theorem)f(5.3)24 b Fc(L)n(et)d Fn(f)26 b Fm(2)20 b(H)i Fc(arbitr)n(ary,)819 1228 y Fo(~)808 1241 y Fn(f)832 1248 y Fi(\006)882 1241 y Fo(=)e Fn(F)972 1248 y Fi(\006)1001 1241 y Fn(f)27 b Fc(and)21 b(let)h Fn(u)1250 1216 y Fh(\()p Fi(\006)p Fh(\))1250 1255 y Fk(f)1307 1241 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))20 b(=)h Fm(F)1543 1223 y Fi(\003)1538 1254 y(\006)1567 1241 y Fo(\()p Fn(\025)p Fo(\))1644 1228 y(~)1633 1241 y Fn(f)1657 1248 y Fi(\006)1687 1241 y Fo(\()p Fn(\025)p Fo(\))g Fc(b)n(e)h(a)-3 1301 y(solution)c(of)f(e)n(quation)h Fo(\(1)p Fn(:)p Fo(1\))p Fc(.)23 b(Then)556 1437 y Fn(P)7 b(f)19 b Fo(=)14 b Fn(s)d Fm(\000)64 b Fo(lim)773 1467 y Fk(\026)p Fi(!)p Fh(0)p Fk(;\027)r Fi(!1)955 1369 y Fj(Z)1005 1382 y Fk(\027)983 1482 y(\026)1035 1437 y Fn(u)1063 1411 y Fh(\()p Fi(\006)p Fh(\))1063 1451 y Fk(f)1120 1437 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))p Fn(d\025;)435 b Fo(\(5.12\))-3 1577 y Fc(and)17 b(we)h(have)g(the)g(gener)n(alize)n (d)h(Parseval)f(identity)321 1713 y Fm(jj)p Fn(P)7 b(f)e Fm(jj)444 1692 y Fh(2)478 1713 y Fo(=)14 b(2)p Fn(\031)592 1645 y Fj(Z)641 1658 y Fi(1)619 1758 y Fh(0)698 1713 y Fo(lim)687 1743 y Fk(\032)p Fi(!1)784 1642 y Fj(\022)821 1713 y Fn(\032)846 1692 y Fi(\000)p Fh(1)901 1645 y Fj(Z)929 1758 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\032)1024 1713 y Fm(j)p Fn(u)1066 1687 y Fh(\()p Fi(\006)p Fh(\))1066 1727 y Fk(f)1123 1713 y Fo(\()p Fn(x;)8 b(\025)p Fo(\))p Fm(j)1253 1692 y Fh(2)1281 1713 y Fn(dx)1334 1642 y Fj(\023)1379 1713 y Fn(\025)1407 1692 y Fh(1)p Fk(=)p Fh(2)1471 1713 y Fn(d\025:)-3 1866 y Fc(Conversely,)21 b(for)353 1853 y Fo(~)343 1866 y Fn(f)367 1873 y Fi(\006)414 1866 y Fm(2)480 1854 y Fo(~)465 1866 y Fm(H)f Fc(arbitr)n(ary,)e(the)i (right-hand)h(side)e(of)h Fo(\(5)p Fn(:)p Fo(12\))g Fc(is)f(wel)r(l)j (de\014ne)n(d,)f(b)n(elongs)-3 1926 y(to)c Fm(H)97 1908 y Fh(\()p Fk(a)p Fh(\))146 1926 y Fc(,)g(and)h(applying)g(op)n(er)n (ator)e Fn(F)690 1933 y Fi(\006)736 1926 y Fc(to)i(this)f(element,)j (we)e(r)n(e)n(c)n(over)1332 1913 y Fo(~)1322 1926 y Fn(f)1346 1933 y Fi(\006)1375 1926 y Fc(.)-3 2082 y Fr(6.)55 b(TIME-DEPENDENT)17 b(SCA)-5 b(TTERING)20 b(MA)-5 b(TRIX)-3 2213 y Fo(T)l(o)16 b(\014nd)h(the)f(asymptotics)f(of)i Fn(e)599 2195 y Fi(\000)p Fk(iH)s(t)684 2213 y Fn(f)22 b Fo(as)17 b Fn(t)c Fm(!)h(\0061)p Fo(,)h(w)o(e)h(pro)q(ceed)g(from)f(the)h(equalit)o(y)371 2345 y Fn(e)394 2325 y Fi(\000)p Fk(iH)s(t)480 2345 y Fn(f)j Fo(=)575 2277 y Fj(Z)624 2291 y Fi(1)602 2390 y Fh(0)670 2345 y Fn(e)693 2325 y Fi(\000)p Fk(i\025t)768 2345 y Fm(F)809 2325 y Fi(\003)804 2358 y(\006)833 2345 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)925 2352 y Fi(\006)954 2345 y Fo(\()p Fn(\025)p Fo(\))p Fn(d\025;)42 b(a)1155 2352 y Fi(\006)1184 2345 y Fo(\()p Fn(\025)p Fo(\))14 b(=)g Fm(F)1352 2352 y Fi(\006)1381 2345 y Fo(\()p Fn(\025)p Fo(\))p Fn(f)s(;)-3 2482 y Fo(follo)o(wing)20 b(from)h(Theorem)e(5.2.) 37 b(Recall)20 b(that,)i(for)g Fn(f)k Fo(suitably)21 b(c)o(hosen,)h Fm(F)1466 2463 y Fi(\003)1461 2494 y(\006)1490 2482 y Fo(\()p Fn(\025)p Fo(\))p Fn(a)1582 2489 y Fi(\006)1612 2482 y Fo(\()p Fn(\025)p Fo(\))f(satis\014es)-3 2542 y(\(3.22\).)f(The)14 b(follo)o(wing)e(prop)q(osition)j(sho)o(ws)f(that) f(the)g(con)o(tribution)g(of)g(the)g(term)e Fn(R)p Fo(\()p Fn(\025)5 b Fm(\007)g Fn(i)p Fo(0\))p Fn(g)1766 2549 y Fi(\006)1796 2542 y Fo(\()p Fn(\025)p Fo(\))-3 2602 y(in)15 b(the)h(righ)o(t-hand)h(side)f(of)g(\(3.22\))h(disapp)q(ears)g (in)f(the)g(limit)d Fn(t)h Fm(!)f(\0061)p Fo(.)21 b(Its)16 b(pro)q(of)h(can)g(b)q(e)f(found)-3 2662 y(in)f([8].)905 2817 y(16)p eop %%Page: 17 17 17 16 bop -3 26 a Fr(Prop)r(osition)17 b(6.1)24 b Fc(L)n(et)17 b Fn(f)i Fm(2)14 b(H)611 8 y Fh(\()p Fk(a)p Fh(\))660 26 y Fc(.)22 b(Supp)n(ose)17 b(that)g Fn(a)1007 33 y Fi(\006)1036 26 y Fo(\()p Fn(\025;)8 b(!)r Fo(\))15 b(=)e(\()p Fn(F)1273 33 y Fi(\006)1302 26 y Fn(f)5 b Fo(\)\()p Fn(\025;)j(!)r Fo(\))15 b Fm(2)f Fn(C)1571 8 y Fi(1)1608 26 y Fo(\()p Fl(R)1666 8 y Fh(+)1702 26 y Fm(\002)c Fl(S)1781 8 y Fk(d)p Fi(\000)p Fh(1)1844 26 y Fo(\))-3 86 y Fc(satis\014es)18 b Fn(a)206 93 y Fi(\006)235 86 y Fo(\()p Fn(\025;)8 b(!)r Fo(\))14 b(=)g(0)k Fc(for)e Fn(\025)f Fm(\024)e Fn(\026)18 b Fc(and)g Fn(\025)c Fm(\025)g Fn(\027)20 b Fc(wher)n(e)e Fo(0)c Fn(<)g(\026)g(<)g(\027)j(<)d Fm(1)p Fc(.)22 b(Then)313 196 y Fo(\()p Fn(e)355 176 y Fi(\000)p Fk(iH)s(t)441 196 y Fn(f)5 b Fo(\)\()p Fn(x)p Fo(\))14 b(=)f Fm(\007)p Fn(i)p Fo(2)700 176 y Fi(\000)p Fh(1)747 196 y Fn(\031)777 176 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)859 196 y Fm(j)p Fn(x)p Fm(j)915 176 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(1\))p Fk(=)p Fh(2)1070 196 y Fn(\021)r Fo(\()p Fn(x)p Fo(\))p Fn(I)1184 203 y Fi(\006)1213 196 y Fo(\()p Fn(x;)8 b(t)p Fo(\))i(+)h Fn(\017)1398 203 y Fi(\006)1427 196 y Fo(\()p Fn(x;)d(t)p Fo(\))p Fn(;)215 b Fo(\(6.1\))-3 306 y Fc(wher)n(e)458 379 y Fn(I)480 386 y Fi(\006)509 379 y Fo(\()p Fn(x;)8 b(t)p Fo(\))13 b(=)680 311 y Fj(Z)730 325 y Fi(1)708 424 y Fh(0)775 379 y Fn(e)798 359 y Fi(\000)p Fk(i\025t)p Fi(\006)p Fk(i')p Fh(\()p Fk(x;\025)p Fh(\))1013 379 y Fn(\025)1041 359 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)1124 379 y Fn(a)1150 386 y Fi(\006)1179 379 y Fo(\()p Fn(\025;)8 b Fm(\006)s Fo(^)-27 b Fn(x)p Fo(\))p Fn(d\025;)361 b Fo(\(6.2\))-3 493 y Fc(and)17 b Fm(jj)p Fn(\017)139 500 y Fi(\006)168 493 y Fo(\()p Fm(\001)p Fn(;)8 b(t)p Fo(\))p Fm(jj)13 b Fo(=)h Fn(o)p Fo(\(1\))k Fc(as)f Fn(t)d Fm(!)f(\0061)p Fc(.)67 574 y Fo(If)22 b Fm(j)p Fn(x)p Fm(j)i(\024)g Fn(\013)p Fm(j)p Fn(t)p Fm(j)f Fo(or)g Fm(j)p Fn(x)p Fm(j)g(\025)i Fn(\014)s Fm(j)p Fn(t)p Fm(j)c Fo(for)i(su\016cien)o(tly) d(small)h Fn(\013)k Fo(=)f Fn(\013)p Fo(\()p Fn(\026;)8 b(\027)s Fo(\))24 b(and)f(su\016cienly)e(large)-3 634 y Fn(\014)16 b Fo(=)d Fn(\014)s Fo(\()p Fn(\026;)8 b(\027)s Fo(\),)16 b(then)g(w)o(e)g(can)g(in)o(tegrate)g(\(6.2\))g(b)o(y)g (parts)54 769 y Fn(I)76 776 y Fi(\006)105 769 y Fo(\()p Fn(x;)8 b(t)p Fo(\))13 b(=)h Fm(\000)p Fn(i)340 701 y Fj(Z)390 715 y Fi(1)367 814 y Fh(0)435 769 y Fn(e)458 749 y Fi(\000)p Fk(i\025t)p Fi(\006)p Fk(i')p Fh(\()p Fk(x;\025)p Fh(\))673 769 y Fn(@)699 776 y Fk(\025)721 714 y Fj(\020)751 769 y Fo(\()p Fn(t)d Fm(\007)f Fo(2)872 749 y Fi(\000)p Fh(1)920 769 y Fn(\025)948 749 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)1031 769 y Fm(j)p Fn(x)p Fm(j)g(\006)h Fo(\010)1182 776 y Fk(\025)1205 769 y Fo(\()p Fn(x;)d(\025)p Fo(\)\))1340 749 y Fi(\000)p Fh(1)1387 769 y Fn(a)1413 776 y Fi(\006)1442 769 y Fo(\()p Fn(\025;)g Fm(\006)s Fo(^)-27 b Fn(x)p Fo(\))p Fn(\025)1625 749 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)1708 714 y Fj(\021)1738 769 y Fn(d\025:)-3 906 y Fo(In)15 b(view)h(of)g(\(3.2\),)h(this)f(yields)f(the)h(estimate) 656 1016 y Fm(j)p Fn(I)692 1023 y Fi(\006)720 1016 y Fo(\()p Fn(x;)8 b(t)p Fo(\))p Fm(j)13 b(\024)h Fn(C)t Fo(\()p Fm(j)p Fn(x)p Fm(j)c Fo(+)h Fm(j)p Fn(t)p Fm(j)p Fo(\))1144 995 y Fi(\000)p Fh(1)1190 1016 y Fn(:)-3 1126 y Fo(Therefore)k(function)h(\(6.1\))h(satis\014es)283 1260 y(lim)265 1293 y Fi(j)p Fk(t)p Fi(j!1)368 1204 y Fj(\020)398 1192 y(Z)426 1305 y Fi(j)p Fk(x)p Fi(j\024)p Fk(\013)p Fi(j)p Fk(t)p Fi(j)558 1260 y Fm(j)p Fo(\()p Fn(e)614 1239 y Fi(\000)p Fk(iH)s(t)699 1260 y Fn(f)5 b Fo(\)\()p Fn(x)p Fo(\))p Fm(j)827 1239 y Fh(2)847 1260 y Fn(dx)11 b Fo(+)960 1192 y Fj(Z)988 1305 y Fi(j)p Fk(x)p Fi(j\025)p Fk(\014)r Fi(j)p Fk(t)p Fi(j)1119 1260 y Fm(j)p Fo(\()p Fn(e)1175 1239 y Fi(\000)p Fk(iH)s(t)1260 1260 y Fn(f)5 b Fo(\)\()p Fn(x)p Fo(\))p Fm(j)1388 1239 y Fh(2)1407 1260 y Fn(dx)1460 1204 y Fj(\021)1504 1260 y Fo(=)14 b(0)p Fn(:)67 1425 y Fo(In)k(the)g(regions)h Fn(\013)p Fm(j)p Fn(t)p Fm(j)e(\024)g(j)p Fn(x)p Fm(j)g(\024)g Fn(\014)s Fm(j)p Fn(t)p Fm(j)g Fo(the)h(asymptotics)f(of)i(in)o(tegral) e(\(6.2\))i(are)f(determined)e(b)o(y)-3 1485 y(stationary)g(p)q(oin)o (ts)h Fn(\025)403 1492 y Fk(s)438 1485 y Fo(where)f Fn(\025)607 1492 y Fk(s)642 1485 y Fo(are)g(the)g(solutions)h(of)g(equation)734 1595 y Fm(\000)p Fn(t)10 b Fm(\006)h Fn(')883 1602 y Fk(\025)906 1595 y Fo(\()p Fn(x;)d(\025)p Fo(\))14 b(=)f(0)p Fn(;)637 b Fo(\(6.3\))-3 1705 y(or)16 b(according)h(to)f(\(3.3\))538 1765 y Fn(\025)566 1745 y Fh(1)p Fk(=)p Fh(2)635 1765 y Fo(=)e(\(2)p Fm(j)p Fn(t)p Fm(j)p Fo(\))795 1745 y Fi(\000)p Fh(1)842 1765 y Fm(j)p Fn(x)p Fm(j)c(\000)h(j)p Fn(t)p Fm(j)1004 1745 y Fi(\000)p Fh(1)1050 1765 y Fo(\010)1085 1772 y Fk(\025)1108 1765 y Fo(\()p Fn(x;)d(\025)p Fo(\))p Fn(\025)1252 1745 y Fh(1)p Fk(=)p Fh(2)1308 1765 y Fn(:)440 b Fo(\(6.4\))-3 1853 y(Using)16 b(again)h(\(3.2\))f(and)h(the)f(metho)q (d)g(of)g(successiv)o(e)f(appro)o(ximations)g(w)o(e)h(obtain)-3 1934 y Fr(Lemm)o(a)g(6.2)24 b Fc(F)l(or)15 b(su\016ciently)i(lar)n(ge)f Fm(j)p Fn(x)p Fm(j)p Fc(,)g(e)n(quation)g Fo(\(6)p Fn(:)p Fo(4\))g Fc(has)f(a)h(unique)h(solution)f Fn(\025)1626 1941 y Fk(s)1659 1934 y Fo(=)e Fn(\025)1739 1941 y Fk(s)1758 1934 y Fo(\()p Fn(x;)8 b(t)p Fo(\))-3 1994 y Fc(and)609 2054 y Fn(\025)637 2061 y Fk(s)655 2054 y Fo(\()p Fn(x;)g(t)p Fo(\))13 b(=)h(\(2)p Fm(j)p Fn(t)p Fm(j)p Fo(\))934 2033 y Fi(\000)p Fh(1)981 2054 y Fm(j)p Fn(x)p Fm(j)c Fo(+)h Fn(O)q Fo(\()p Fn(t)1171 2033 y Fi(\000)p Fk(\016)1218 2054 y Fo(\))p Fn(:)511 b Fo(\(6.5\))67 2162 y(Set)612 2222 y Fn( )r Fo(\()p Fn(x;)8 b(\025;)g(t)p Fo(\))13 b(=)g Fm(\000)p Fn(\025)f Fo(+)f Fm(j)p Fn(t)p Fm(j)1040 2202 y Fi(\000)p Fh(1)1086 2222 y Fn(')p Fo(\()p Fn(x;)d(\025)p Fo(\))p Fn(:)514 b Fo(\(6.6\))-3 2309 y(Applying)13 b(no)o(w)i(to)g(in) o(tegral)f(\(6.2\))h(the)f(stationary)i(phase)f(metho)q(d)e(\(see)i (e.g.)20 b([4]\))14 b(w)o(e)g(obtain)h(that)21 2426 y Fn(I)43 2433 y Fi(\006)72 2426 y Fo(\()p Fn(x;)8 b(t)p Fo(\))13 b(=)h(\(2)p Fn(\031)r Fo(\))335 2406 y Fh(1)p Fk(=)p Fh(2)390 2426 y Fm(j)p Fn(t)p Fm(j)436 2406 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)518 2426 y Fm(j)p Fn( )564 2433 y Fk(\025\025)607 2426 y Fm(j)621 2406 y Fi(\000)p Fh(1)p Fk(=)p Fh(2)703 2426 y Fn(e)726 2406 y Fk(it )q Fi(\006)807 2392 y Fd(\031)q(i)p 807 2398 31 2 v 814 2419 a Fg(4)842 2406 y Fk(sg)q(n )920 2412 y Fd(\025\025)961 2426 y Fn(a)987 2433 y Fi(\006)1017 2426 y Fo(\()p Fn(\025)1064 2433 y Fk(s)1082 2426 y Fo(\()p Fn(x;)8 b(t)p Fo(\))p Fn(;)g Fm(\006)s Fo(^)-27 b Fn(x)o Fo(\))p Fn(\025)1323 2406 y Fi(\000)p Fh(1)p Fk(=)p Fh(4)1323 2439 y Fk(s)1406 2426 y Fo(\()p Fn(x;)8 b(t)p Fo(\))h(+)i Fn(\017)1590 2433 y Fi(\006)1619 2426 y Fo(\()p Fn(x;)d(t)p Fo(\))p Fn(;)23 b Fo(\(6.7\))-3 2536 y(where)449 2597 y Fn( )16 b Fo(=)e Fn( )r Fo(\()p Fn(x;)8 b(\025)680 2604 y Fk(s)697 2597 y Fo(\()p Fn(x;)g(t)p Fo(\))p Fn(;)g(t)p Fo(\))p Fn(;)39 b( )947 2604 y Fk(\025\025)1004 2597 y Fo(=)14 b Fn( )1088 2604 y Fk(\025\025)1131 2597 y Fo(\()p Fn(x;)8 b(\025)1228 2604 y Fk(s)1246 2597 y Fo(\()p Fn(x;)g(t)p Fo(\))p Fn(;)g(t)p Fo(\))351 b(\(6.8\))-3 2684 y(and)16 b(the)g(remainder)f Fn(\017)426 2691 y Fi(\006)455 2684 y Fo(\()p Fn(x;)8 b(t)p Fo(\))15 b(satis\014es)i Fm(jj)p Fn(\017)804 2691 y Fi(\006)833 2684 y Fo(\()p Fm(\001)p Fn(;)8 b(t)p Fo(\))p Fm(jj)13 b Fo(=)g Fn(o)p Fo(\(1\))k(as)g Fn(t)d Fm(!)f(\0061)p Fo(.)905 2817 y(17)p eop %%Page: 18 18 18 17 bop 67 26 a Fo(Di\013eren)o(tiating)16 b(\(6.6\))g(and)h(taking)f (in)o(to)g(accoun)o(t)g(\(6.5\),)h(w)o(e)e(\014nd)i(that)81 118 y Fn( )113 125 y Fk(\025\025)156 118 y Fo(\()p Fn(x;)8 b(\025)253 125 y Fk(s)271 118 y Fn(;)g(t)p Fo(\))13 b(=)h Fm(\000)p Fo(\(4)p Fm(j)p Fn(t)p Fm(j)p Fo(\))542 97 y Fi(\000)p Fh(1)589 118 y Fm(j)p Fn(x)p Fm(j)p Fn(\025)673 97 y Fi(\000)p Fh(3)p Fk(=)p Fh(2)673 130 y Fk(s)766 118 y Fm(\000)d(j)p Fn(t)p Fm(j)862 97 y Fi(\000)p Fh(1)908 118 y Fo(\010)943 125 y Fk(\025\025)987 118 y Fo(\()p Fn(x;)d(\025)1084 125 y Fk(s)1102 118 y Fo(\)\))14 b(=)g Fm(\000)p Fo(2)p Fm(j)p Fn(x)p Fm(j)1325 97 y Fi(\000)p Fh(2)1371 118 y Fm(j)p Fn(t)p Fm(j)1417 97 y Fh(2)1447 118 y Fo(+)d Fn(O)q Fo(\()p Fm(j)p Fn(t)p Fm(j)1599 97 y Fi(\000)p Fk(\016)1646 118 y Fo(\))p Fn(:)83 b Fo(\(6.9\))-3 210 y(In)15 b(particular,)h(sgn)8 b Fn( )406 217 y Fk(\025\025)464 210 y Fo(=)14 b Fm(\000)p Fo(1.)67 285 y(Let)i(us)h(set)422 346 y(\012\()p Fn(x;)8 b(t)p Fo(\))13 b(=)h Fn(t )r Fo(\()p Fn(x;)8 b(\025)777 353 y Fk(s)795 346 y Fo(\()p Fn(x;)g(t)p Fo(\)\))i Fm(\006)h Fn(\031)r Fo(\()p Fn(d)f Fm(\000)h Fo(3\))p Fn(=)p Fo(4)p Fn(;)58 b Fm(\006)p Fn(t)13 b(>)h Fo(0)p Fn(:)301 b Fo(\(6.10\))-3 425 y(Then,)15 b(according)i(again)g (to)g(\(6.5\))f(and)h(\(6.6\),)369 546 y(\012\()p Fn(x;)8 b(t)p Fo(\))13 b(=)h Fm(\000)p Fn(\025)642 553 y Fk(s)660 546 y Fn(t)d Fm(\006)g Fn(\025)767 526 y Fh(1)p Fk(=)p Fh(2)767 559 y Fk(s)822 546 y Fm(j)p Fn(x)p Fm(j)g(\007)f Fo(\010\()p Fn(x;)e(\025)1070 553 y Fk(s)1089 546 y Fo(\))14 b(=)1179 513 y Fn(x)1207 495 y Fh(2)p 1179 535 48 2 v 1182 581 a Fo(4)p Fn(t)1242 546 y Fo(+)d Fn(O)q Fo(\()p Fm(j)p Fn(t)p Fm(j)1394 526 y Fh(1)p Fi(\000)p Fk(\016)1458 546 y Fo(\))p Fn(:)247 b Fo(\(6.11\))-3 654 y(Plugging)16 b(these)g(expressions)h(in)o(to)e(\(6.7\))i(and)g(taking)f(in)o(to)g (accoun)o(t)g(\(6.1\),)g(w)o(e)g(see)g(that)48 746 y(\()p Fn(e)90 725 y Fi(\000)p Fk(iH)s(t)175 746 y Fn(f)5 b Fo(\)\()p Fn(x)p Fo(\))14 b(=)g(2)379 725 y Fh(1)p Fk(=)p Fh(2)434 746 y Fo(\(2)p Fn(it)p Fo(\))531 725 y Fi(\000)p Fk(d=)p Fh(2)613 746 y Fn(e)636 725 y Fk(i)p Fh(\012\()p Fk(x;t)p Fh(\))745 746 y Fm(j)p Fo(\(2)p Fn(t)p Fo(\))839 725 y Fi(\000)p Fh(1)886 746 y Fn(x)p Fm(j)928 725 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(2\))p Fk(=)p Fh(2)1083 746 y Fn(a)1109 753 y Fi(\006)1139 746 y Fo(\(\(4)p Fn(t)1219 725 y Fh(2)1238 746 y Fo(\))1257 725 y Fi(\000)p Fh(1)1304 746 y Fn(x)1332 725 y Fh(2)1352 746 y Fn(;)8 b Fm(\006)s Fo(^)-27 b Fn(x)o Fo(\))11 b(+)g Fn(\017)1539 753 y Fi(\006)1568 746 y Fo(\()p Fn(x;)d(t)p Fo(\))p Fn(;)50 b Fo(\(6.12\))-3 837 y(where)15 b Fm(jj)p Fn(\017)185 844 y Fi(\006)214 837 y Fo(\()p Fm(\001)p Fn(;)8 b(t)p Fo(\))p Fm(jj)13 b Fo(=)h Fn(o)p Fo(\(1\))j(as)g Fn(t)c Fm(!)h(\0061)p Fo(.)67 913 y(Let)i(us)h(c)o(hec)o(k)e(that)h(function)g(\(6.10\))h (satis\014es)g(the)f(equation)703 1005 y(\012)738 1012 y Fk(t)765 1005 y Fo(+)11 b Fm(jr)p Fo(\012)p Fm(j)919 984 y Fh(2)949 1005 y Fo(+)g Fn(V)1026 1012 y Fk(L)1066 1005 y Fo(=)j(0)p Fn(:)582 b Fo(\(6.13\))-3 1097 y(Indeed,)15 b(comparing)g(\(6.10\))i(with)f(\(6.3\))h(and)f(\(6.6\),)g(w)o(e)g(see) g(that)498 1213 y(\012)533 1220 y Fk(t)561 1213 y Fo(=)e Fm(\000)p Fn(\025)680 1220 y Fk(s)710 1213 y Fo(+)d Fn(t)782 1180 y(@)s(\025)839 1187 y Fk(s)p 782 1202 76 2 v 796 1248 a Fn(@)s(t)873 1213 y Fm(\006)g Fn(')955 1220 y Fk(\025)977 1213 y Fo(\()p Fn(x;)d(\025)1074 1220 y Fk(s)1093 1213 y Fo(\))1117 1180 y Fn(@)s(\025)1174 1187 y Fk(s)p 1117 1202 V 1131 1248 a Fn(@)s(t)1211 1213 y Fo(=)13 b Fm(\000)p Fn(\025)1329 1220 y Fk(s)1348 1213 y Fn(:)376 b Fo(\(6.14\))-3 1318 y(Similarly)l(,)336 1400 y(\012)371 1407 y Fk(x)391 1412 y Fd(i)420 1400 y Fo(=)14 b Fm(\000)p Fn(t)534 1366 y(@)s(\025)591 1373 y Fk(s)p 534 1389 V 537 1434 a Fn(@)s(x)594 1441 y Fk(i)624 1400 y Fm(\006)d Fn(')706 1407 y Fk(x)726 1412 y Fd(i)741 1400 y Fo(\()p Fn(x;)d(\025)838 1407 y Fk(s)856 1400 y Fo(\))k Fm(\006)e Fn(')968 1407 y Fk(\025)991 1400 y Fo(\()p Fn(x;)e(\025)1088 1407 y Fk(s)1106 1400 y Fo(\))1130 1366 y Fn(@)s(\025)1187 1373 y Fk(s)p 1130 1389 V 1133 1434 a Fn(@)s(x)1190 1441 y Fk(i)1224 1400 y Fo(=)14 b Fm(\006)p Fn(@)1341 1407 y Fk(x)1361 1412 y Fd(i)1376 1400 y Fo(\()p Fn(x;)8 b(\025)1473 1407 y Fk(s)1491 1400 y Fo(\))p Fn(:)214 b Fo(\(6.15\))-3 1502 y(No)o(w)16 b(it)f(follo)o(ws)i(from)e(\(6.14\))i(and)f(\(6.15\))h (that)340 1594 y(\012)375 1601 y Fk(t)390 1594 y Fo(\()p Fn(x;)8 b(t)p Fo(\))j(+)g Fm(jr)612 1601 y Fk(x)633 1594 y Fo(\012\()p Fn(x;)d(t)p Fo(\))p Fm(j)788 1574 y Fh(2)821 1594 y Fo(=)13 b Fm(\000)p Fn(\025)939 1601 y Fk(s)958 1594 y Fo(\()p Fn(x;)8 b(t)p Fo(\))i(+)h Fm(jr)1179 1601 y Fk(x)1200 1594 y Fn(')p Fo(\()p Fn(x;)d(\025)1329 1601 y Fk(s)1348 1594 y Fo(\()p Fn(x;)g(t)p Fo(\)\))p Fm(j)1487 1574 y Fh(2)1505 1594 y Fn(:)-3 1686 y Fo(Therefore)15 b(\(6.13\))i(is)f(an)h(immedi)o(ate)d(consequence)h(of)h(the)g (equation)656 1778 y Fm(jr)712 1785 y Fk(x)734 1778 y Fn(')p Fo(\()p Fn(x;)8 b(\025)p Fo(\))p Fm(j)896 1757 y Fh(2)926 1778 y Fo(+)j Fn(V)1003 1785 y Fk(L)1030 1778 y Fo(\()p Fn(x)p Fo(\))i(=)h Fn(\025:)-3 1870 y Fo(W)l(e)i(summarize)d (in)j(the)g(follo)o(wing)g(lemm)o(a)e(the)i(results)g(obtained.)-3 1945 y Fr(Lemm)o(a)g(6.3)24 b Fc(L)n(et)17 b Fn(f)i Fm(2)14 b(H)507 1927 y Fh(\()p Fk(a)p Fh(\))574 1945 y Fc(b)n(e)j(chosen)i(as)e (in)h(Pr)n(op)n(osition)e Fo(6)p Fn(:)p Fo(1)p Fc(.)23 b(Then)54 2037 y Fo(\()p Fn(e)96 2017 y Fi(\000)p Fk(iH)s(t)182 2037 y Fn(f)5 b Fo(\)\()p Fn(x)p Fo(\))14 b(=)f(2)385 2017 y Fh(1)p Fk(=)p Fh(2)441 2037 y Fo(\(2)p Fn(it)p Fo(\))538 2017 y Fi(\000)p Fk(d=)p Fh(2)620 2037 y Fn(e)643 2017 y Fk(i)p Fh(\012\()p Fk(x;t)p Fh(\))752 2037 y Fm(j)p Fo(\(2)p Fn(t)p Fo(\))846 2017 y Fi(\000)p Fh(1)893 2037 y Fn(x)p Fm(j)935 2017 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(2\))p Fk(=)p Fh(2)1090 2037 y Fn(a)1116 2044 y Fi(\006)1145 2037 y Fo(\(\(4)p Fn(t)1225 2017 y Fh(2)1245 2037 y Fo(\))1264 2017 y Fi(\000)p Fh(1)1311 2037 y Fn(x)1339 2017 y Fh(2)1359 2037 y Fn(;)8 b Fm(\006)s Fo(^)-27 b Fn(x)o Fo(\))11 b(+)g Fn(\017)1546 2044 y Fi(\006)1575 2037 y Fo(\()p Fn(x;)d(t)p Fo(\))57 b(\(6.16\))-3 2129 y Fc(wher)n(e)13 b Fo(\012\()p Fn(x;)8 b(t)p Fo(\))13 b Fc(satis\014es)h Fo(\(1)p Fn(:)p Fo(9\))g(\()p Fc(up)g(to)g(a)f(short-r)n(ange)h(term)p Fo(\))p Fc(,)g(and)g Fm(jj)p Fn(\017)1293 2136 y Fi(\006)1322 2129 y Fo(\()p Fm(\001)p Fn(;)8 b(t)p Fo(\))p Fm(jj)13 b Fo(=)g Fn(o)p Fo(\(1\))i Fc(as)e Fn(t)h Fm(!)f(\0061)p Fc(.)67 2205 y Fo(W)l(e)j(de\014ne)g(no)o(w)451 2297 y(\()p Fr(F)505 2304 y Fi(\006)535 2297 y Fn(f)5 b Fo(\)\()p Fn(\030)r Fo(\))15 b(=)f(2)735 2276 y Fh(1)p Fk(=)p Fh(2)790 2297 y Fm(j)p Fn(\030)r Fm(j)841 2276 y Fi(\000)p Fh(\()p Fk(d)p Fi(\000)p Fh(2\))p Fk(=)p Fh(2)997 2297 y Fo(\()p Fn(F)1048 2304 y Fi(\006)1077 2297 y Fn(f)5 b Fo(\)\()p Fm(j)p Fn(\030)r Fm(j)1195 2276 y Fh(2)1215 2297 y Fn(;)j Fm(\006)p Fn(\030)r(=)p Fm(j)p Fn(\030)r Fm(j)p Fo(\))p Fn(:)331 b Fo(\(6.17\))-3 2389 y(By)14 b(Theorem)g(5.2,)h Fr(F)403 2396 y Fi(\006)449 2389 y Fo(is)g(a)g(partial)h(isometry)d(on) j Fm(H)g Fo(with)f(the)g(initial)f(set)h Fm(H)1466 2371 y Fh(\()p Fk(a)p Fh(\))1530 2389 y Fo(and)h(\014nal)f(set)h Fm(H)p Fo(.)-3 2449 y(W)l(e)g(also)g(in)o(tro)q(duce)g(a)h(family)d(of) i(unitary)g(op)q(erators)i(b)o(y)e(the)g(equalit)o(y)503 2541 y(\()p Fn(U)555 2548 y Fh(0)574 2541 y Fo(\()p Fn(t)p Fo(\))p Fn(f)5 b Fo(\)\()p Fn(x)p Fo(\))14 b(=)f Fn(e)832 2520 y Fk(i)p Fh(\012\()p Fk(x;t)p Fh(\))942 2541 y Fo(\(2)p Fn(it)p Fo(\))1039 2520 y Fi(\000)p Fk(d=)p Fh(2)1121 2541 y Fn(f)5 b Fo(\(\(2)p Fn(t)p Fo(\))1249 2520 y Fi(\000)p Fh(1)1296 2541 y Fn(x)p Fo(\))p Fn(;)381 b Fo(\(6.18\))-3 2632 y(where)22 b(\012\()p Fn(x;)8 b(t)p Fo(\))22 b(is)g(c)o(hosen)h (as)g(in)f(Lemma)f(6.3.)41 b(Th)o(us)23 b(Lemma)d(6.3)j(extends)f(to)h (an)h(arbitrary)-3 2693 y Fn(f)19 b Fm(2)14 b(H)129 2675 y Fh(\()p Fk(a)p Fh(\))178 2693 y Fo(.)905 2817 y(18)p eop %%Page: 19 19 19 18 bop -3 26 a Fr(Prop)r(osition)17 b(6.4)24 b Fc(F)l(or)17 b(any)h Fn(f)h Fm(2)14 b(H)712 8 y Fh(\()p Fk(a)p Fh(\))761 26 y Fc(,)612 130 y Fo(lim)604 160 y Fk(t)p Fi(!1)695 130 y Fm(jj)p Fn(e)746 110 y Fi(\000)p Fk(iH)s(t)831 130 y Fn(f)j Fm(\000)11 b Fn(U)955 137 y Fh(0)975 130 y Fo(\()p Fn(t)p Fo(\))p Fr(F)1066 137 y Fi(\006)1095 130 y Fn(f)5 b Fm(jj)14 b Fo(=)g(0)p Fn(:)482 b Fo(\(6.19\))67 267 y(This)20 b(implies)d(the)i(existence)f(and)i(completeness)e(of)i (the)f(time-dep)q(enden)o(t)f(w)o(a)o(v)o(e)g(op)q(erators)-3 328 y(constructed)e(in)g([17].)-3 407 y Fr(Theorem)g(6.5)24 b Fc(The)18 b(str)n(ong)f(limits)663 511 y Fn(W)709 518 y Fi(\006)752 511 y Fo(=)d Fn(s)d Fm(\000)32 b Fo(lim)888 541 y Fk(t)p Fi(!\0061)1007 511 y Fn(e)1030 491 y Fk(iH)s(t)1088 511 y Fn(U)1121 518 y Fh(0)1141 511 y Fo(\()p Fn(t)p Fo(\))-3 635 y Fc(exist)14 b(and)g Fn(W)243 642 y Fi(\006)286 635 y Fo(=)g Fr(F)373 617 y Fi(\003)373 648 y(\006)403 635 y Fc(.)21 b(In)14 b(p)n(articular,)g(the)g(wave)h(op)n(er)n(ators)d Fn(W)1181 642 y Fi(\006)1224 635 y Fc(ar)n(e)h(c)n(omplete,)i(that)f (is)g Fn(R)p Fo(\()p Fn(W)1762 642 y Fi(\006)1792 635 y Fo(\))g(=)-3 695 y Fm(H)39 677 y Fh(\()p Fk(a)p Fh(\))87 695 y Fc(.)-3 775 y(Pr)n(o)n(of)f({)30 b Fo(F)l(or)15 b(arbitrary)f Fn(g)i Fm(2)e(H)p Fo(,)h(set)f Fn(f)19 b Fo(=)14 b Fr(F)832 757 y Fi(\003)832 787 y(\006)862 775 y Fn(g)h Fm(2)g(H)990 757 y Fh(\()p Fk(a)p Fh(\))1053 775 y Fo(so)f(that)h Fn(g)h Fo(=)e Fr(F)1340 782 y Fi(\006)1370 775 y Fn(f)5 b Fo(.)21 b(Then)14 b(it)g(follo)o(ws)g(from)-3 835 y(\(6.19\))i(that)603 895 y(lim)595 925 y Fk(t)p Fi(!1)687 895 y Fm(jj)p Fr(F)750 875 y Fi(\003)750 908 y(\006)779 895 y Fn(g)d Fm(\000)e Fn(e)888 875 y Fk(iH)s(t)946 895 y Fn(U)979 902 y Fh(0)999 895 y Fo(\()p Fn(t)p Fo(\))p Fn(g)r Fm(jj)i Fo(=)h(0)p Fn(:)i Fa(2)67 1083 y Fo(Usually)l(,)j(the)g (time-dep)q(enden)o(t)e(scattering)j(op)q(erator)g Fn(S)j Fo(is)c(de\014ned)g(b)o(y)g Fn(S)j Fo(=)d Fn(W)1631 1065 y Fi(\003)1624 1095 y Fh(+)1654 1083 y Fn(W)1700 1090 y Fi(\000)1729 1083 y Fo(.)31 b(In)19 b(a)-3 1143 y(diagonal)h (represen)o(tation)g(whic)o(h)f(reduces)g Fn(H)878 1150 y Fh(0)918 1143 y Fo(=)h Fm(\000)p Fo(\001)f(to)i(m)o(ultipli)o(cation) c(b)o(y)i Fn(\025)p Fo(,)i Fn(S)i Fo(reduces)c(to)-3 1203 y(m)o(ultipli)o(cation)c(b)o(y)h(the)h(op)q(erator)i(function)866 1190 y(~)854 1203 y Fm(S)t Fo(\()p Fn(\025)p Fo(\))d(:)f Fn(L)1032 1185 y Fh(2)1052 1203 y Fo(\()p Fl(S)1102 1185 y Fk(d)p Fi(\000)p Fh(1)1164 1203 y Fo(\))g Fm(!)h Fn(L)1297 1185 y Fh(2)1317 1203 y Fo(\()p Fl(S)1366 1185 y Fk(d)p Fi(\000)p Fh(1)1429 1203 y Fo(\))h(whic)o(h)g(is)g(called)f(the)-3 1263 y(time-dep)q(enden)o(t)d(scattering)k(matrix.)-3 1343 y Fr(Prop)r(osition)g(6.6)24 b Fc(The)13 b(time-dep)n(endent)i(sc) n(attering)f(matrix)1205 1330 y Fo(~)1193 1343 y Fm(S)t Fo(\()p Fn(\025)p Fo(\))f Fc(c)n(oincides)h(with)f(the)g(op)n(er)n (ator)-3 1403 y Fm(S)t Fo(\()p Fn(\025)p Fo(\))k Fc(de\014ne)n(d)i(by)e Fo(\(4)p Fn(:)p Fo(16\))p Fc(.)-3 1482 y(Pr)n(o)n(of)e(|)34 b Fo(Let)16 b(us)h(write)e(\()p Fr(F)539 1489 y Fi(\006)569 1482 y Fn(f)5 b Fo(\)\()p Fn(x)p Fo(\))14 b(=)g(\()p Fr(F)803 1489 y Fi(\006)832 1482 y Fn(f)5 b Fo(\)\()s(^)-27 b Fn(x)q(;)8 b Fm(j)p Fn(x)p Fm(j)p Fo(\).)20 b(Then)c(\(6.19\))h (implies)591 1574 y(~)579 1586 y Fm(S)t Fo(\()p Fn(\025)p Fo(\))d(:)g(\()p Fr(F)775 1593 y Fi(\000)804 1586 y Fn(f)5 b Fo(\)\()p Fm(\001)p Fn(;)j(\025)p Fo(\))15 b Fm(7!)e Fo(\()p Fr(F)1086 1593 y Fh(+)1116 1586 y Fn(f)5 b Fo(\)\()p Fm(\001)p Fn(;)j(\025)p Fo(\))p Fn(;)-3 1691 y Fo(According)15 b(to)i(\(4.15\))g(and)g(\(6.17\),)f(this)g(corresp)q(onds)i(to)e(the)g (de\014nition)g(of)h Fm(S)t Fo(\()p Fn(\025)p Fo(\).)k Fa(2)-3 1864 y Fr(REFERENCES)21 1991 y Fo([1])j(S.)11 b(Agmon,)g Fc(Some)i(new)h(r)n(esults)g(in)f(sp)n(e)n(ctr)n(al)f(and)h (sc)n(attering)h(the)n(ory)e(of)h(di\013er)n(ential)g(op)n(er)n(ators) 97 2051 y(on)18 b Fn(R)204 2033 y Fk(n)228 2051 y Fo(,)d(S)o(\023)-23 b(eminaire)14 b(Goulaouic)j(Sc)o(h)o(w)o(artz,)e(Ecole)h(P)o(olytec)o (hnique,)d(1978.)21 2152 y([2])24 b(S.)11 b(Agmon,)f(L.)i(H\177)-24 b(ormander,)10 b Fc(Asymptotic)j(pr)n(op)n(erties)f(of)g(solutions)i (of)f(di\013er)n(ential)g(e)n(quations)97 2212 y(with)18 b(simple)g(char)n(acteristics)f Fo(,)f(Journal)h(d'Analyse)e(Math)o (\023)-23 b(ematique,)14 b Fr(30)i Fo(\(1976\),)h(1-38.)21 2312 y([3])24 b(L.)17 b(H\177)-24 b(ormander,)17 b Fc(L)n(ower)h(b)n (ounds)g(at)h(in\014nity)h(for)e(solutions)h(of)g(di\013er)n(ential)g (e)n(quations)g(with)97 2372 y(c)n(onstant)f(c)n(o)n(e\016cient)p Fo(,)f(Israel)f(J.)g(Math.,)f Fr(16)h Fo(\(1973\),)i(103-116.)21 2472 y([4])24 b(L.)d(H\177)-24 b(ormander,)21 b Fc(The)h(analysis)g(of) g(Line)n(ar)f(Partial)h(Di\013er)n(ential)g(Op)n(er)n(ators)f(I)p Fo(,)g(Springer-)97 2532 y(V)l(erlag,)15 b(1985.)21 2632 y([5])24 b(L.)17 b(H\177)-24 b(ormander,)16 b Fc(The)j(analysis)g(of)f (Line)n(ar)g(Partial)g(Di\013er)n(ential)h(Op)n(er)n(ators)f(IV)p Fo(,)f(Springer-)97 2693 y(V)l(erlag,)e(1985.)905 2817 y(19)p eop %%Page: 20 20 20 19 bop 21 26 a Fo([6])24 b(T.)18 b(Ik)o(eb)q(e,)g Fc(Sp)n(e)n(ctr)n(al)i(R)n(epr)n(esentation)g(for)f(Schr\177)-25 b(odinger)20 b(Op)n(er)n(ators)f(with)h(L)n(ong-R)n(ange)g(Po-)97 86 y(tentials)p Fo(,)d(J.)f(F)l(unct.)f(Anal.,)g Fr(20)i Fo(\(1975\),)g(158-177.)21 188 y([7])24 b(T.)f(Ik)o(eb)q(e,)g(H.)f (Isozaki,)i Fc(Completeness)h(of)f(Mo)n(di\014e)n(d)f(Wave)h(Op)n(er)n (ators)e(for)h(L)n(ong-R)n(ange)97 248 y(Potentials)p Fo(,)17 b(Publ.)f(RIMS,)f(Ky)o(oto)h(Univ.,)e Fr(15)j Fo(\(1979\),)g(679-718.)21 350 y([8])24 b(T.)d(Ik)o(eb)q(e,)f(H.)g (Isozaki,)h Fc(A)h(stationary)f(appr)n(o)n(ach)g(to)g(the)i(existenc)n (e)g(and)f(c)n(ompleteness)i(of)97 410 y(long-r)n(ange)19 b(wave)g(op)n(er)n(ators)p Fo(,)14 b(In)o(t.)h(Eq.)h(Op.)f(Theory)l(,)h Fr(5)g Fo(\(1982\),)i(18-49.)21 512 y([9])24 b(T.)h(Ik)o(eb)q(e,)g(Y.)f (Saito,)j Fc(Limiting)f(absorption)f(metho)n(d)h(and)f(absolute)i(c)n (ontinuity)f(for)f(the)97 572 y(Schr\177)-25 b(odinger)18 b(op)n(er)n(ator)p Fo(,)c(J.)i(Math.)g(Ky)o(oto)g(Univ.,)f Fr(12)h Fo(\(1972\),)h(513-542.)-3 674 y([10])24 b(H.)11 b(Isozaki,)g Fc(Eikonal)j(e)n(quations)g(and)g(sp)n(e)n(ctr)n(al)e(r)n (epr)n(esentations)i(for)e(long)i(r)n(ange)g(Schr\177)-25 b(odinger)97 734 y(Hamiltonians)p Fo(,)17 b(J.)e(Math.)h(Ky)o(oto)h (Univ.,)d Fr(20)i Fo(\(1980\),)i(243-261.)-3 835 y([11])24 b(H.)16 b(Isozaki,)g(H.)g(Kitada,)h Fc(Sc)n(attering)i(matric)n(es)f (for)g(two)g(b)n(o)n(dy)f(Schr\177)-25 b(odinger)19 b(op)n(er)n(ators)p Fo(,)c(Sci.)97 896 y(P)o(ap)q(ers)i(College)e(Arts)h(and)h(Sci.,)e (Univ.)g(T)l(oky)o(o,)g Fr(35)i Fo(\(1985\),)g(81-107.)-3 997 y([12])24 b(A.)36 b(Jensen,)42 b(P)l(.)36 b(P)o(erry)l(,)41 b Fc(Commutator)36 b(Metho)n(ds)g(ans)h(Besov)g(Sp)n(ac)n(e)g (estimates)g(for)97 1057 y(Schr\177)-25 b(odinger)18 b(op)n(er)n(ators)p Fo(,)c(J.)i(Op)q(erator)h(Theory)l(,)f Fr(14)g Fo(\(1985\),)h(181-188.)-3 1159 y([13])24 b(H.)12 b(Kitada,)i Fc(Sc)n(attering)i(the)n(ory)d(for)h(Schr\177)-25 b(odinger)16 b(op)n(er)n(ators)d(with)h(long-r)n(ange)j(p)n(otentials)e (II,)97 1219 y(sp)n(e)n(ctr)n(al)i(and)g(sc)n(attering)h(the)n(ory)p Fo(,)d(J.)h(Math.)g(So)q(c.)h(Japan,)f Fr(30)h Fo(\(1978\),)g(603-632.) -3 1321 y([14])24 b(R.)d(Melrose,)g(M.)g(Zw)o(orski,)h Fc(Sc)n(attering)i(metrics)f(and)f(ge)n(o)n(desic)h(\015ow)f(at)h (in\014nity)p Fo(,)g(In)o(v)o(en)o(t.)97 1381 y(Math.,)15 b Fr(124)i Fo(\(1996\),)g(389-436.)-3 1483 y([15])24 b(Y.)16 b(Saito,)i Fc(Sp)n(e)n(ctr)n(al)g(r)n(epr)n(esentations)g(for)g (Schr\177)-25 b(odinger)19 b(op)n(er)n(ators)e(with)i(long-r)n(ange)h (p)n(oten-)97 1543 y(tials)p Fo(,)c(Lecture)g(Notes)g(in)g(Math.,)f Fr(727)p Fo(,)h(Springer,)g(Berlin,)e(1979.)-3 1645 y([16])24 b(Y.)17 b(Saito,)i Fc(On)i(the)f(S-matrix)g(for)f(Schr\177)-25 b(odinger)20 b(op)n(er)n(ators)e(with)h(long-r)n(ange)j(p)n(otentials)p Fo(,)d(J.)97 1705 y(reine)c(Angew.)h(Math.,)f Fr(314)i Fo(\(1980\),)g(99-116.)-3 1807 y([17])24 b(D.R.)14 b(Y)l(afaev,)g Fc(Wave)j(op)n(er)n(ators)e(for)g(the)i(Schr\177)-25 b(odinger)17 b(e)n(quation)p Fo(,)f(Theor.)f(Math.)f(Ph)o(ys.,)h Fr(45)97 1867 y Fo(\(1980\),)i(992-998.)-3 1969 y([18])24 b(D.R.)15 b(Y)l(afaev,)g Fc(On)j(solutions)h(of)e(the)g(Schr\177)-25 b(odinger)19 b(e)n(quation)f(with)f(r)n(adiation)g(c)n(onditions)h(at) 97 2029 y(in\014nity)p Fo(,)f(Adv)m(ances)f(in)f(So)o(v.)h(Math.,)g Fr(7)p Fo(,)f(179-204,)k(1991.)-3 2131 y([19])24 b(K.)16 b(Y)l(osida,)f Fc(F)l(unctional)20 b(analysis)p Fo(,)c(Springer-V)l (erlag,)f(1966.)905 2817 y(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF