Content-Type: multipart/mixed; boundary="-------------0402091934421" This is a multi-part message in MIME format. ---------------0402091934421 Content-Type: text/plain; name="04-32.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-32.comments" MSC: Primary 34A55, 81U40; Secondary 34L25, 34L40, 47A40, 8105 ---------------0402091934421 Content-Type: text/plain; name="04-32.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-32.keywords" Schroedinger equation, inverse scattering, potential recovery, incomplete data ---------------0402091934421 Content-Type: application/postscript; name="newpasi4.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="newpasi4.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: newpasi4.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -f 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y Fn(1)514 1279 y(1)550 1259 y Fp(\()p Fq(R)p Fp(\))p Fo(;)g Fp(the)g(class)f(of)g(measurable)g(functions)g (on)h(the)g(real)e(axis)h Fq(R)g Fp(suc)n(h)h(that)0 1292 y Fi(R)55 1313 y Fj(1)39 1389 y(\0001)175 1359 y Fo(dx)14 b Fp(\(1)21 b(+)f Fm(j)p Fo(x)p Fm(j)p Fp(\))14 b Fm(j)p Fo(V)19 b Fp(\()p Fo(x)p Fp(\))p Fm(j)32 b Fp(is)e(\014nite.) 45 b(The)30 b(prime)g(is)g(used)h(for)e(the)i(deriv)-5 b(ativ)n(e)29 b(with)i(resp)r(ect)0 1462 y(to)g(the)h(spatial)f(co)r (ordinate)g Fo(x:)h Fp(The)f(Jost)g(solutions)g Fo(f)1786 1474 y Fn(l)1840 1462 y Fp(and)g Fo(f)2046 1474 y Fn(r)2076 1462 y Fo(;)h Fp(from)f(the)h(left)g(and)g(righ)n(t,)0 1561 y(resp)r(ectiv)n(ely)-7 b(,)27 b(satisfy)g(the)h(resp)r(ectiv)n(e) f(b)r(oundary)g(conditions)287 1721 y Fo(e)326 1687 y Fj(\000)p Fh(ik)q(x)480 1721 y Fo(f)521 1733 y Fn(l)543 1721 y Fp(\()p Fo(k)s(;)14 b(x)p Fp(\))24 b(=)f(1)18 b(+)g Fo(o)p Fp(\(1\))p Fo(;)97 b(e)1297 1687 y Fj(\000)p Fh(ik)q(x)1451 1721 y Fo(f)1501 1687 y Fj(0)1492 1742 y Fn(l)1524 1721 y Fp(\()p Fo(k)s(;)14 b(x)p Fp(\))23 b(=)g Fo(ik)e Fp(+)d Fo(o)p Fp(\(1\))p Fo(;)180 b(x)24 b Fm(!)f Fp(+)p Fm(1)p Fo(;)303 1891 y(e)342 1857 y Fh(ik)q(x)444 1891 y Fo(f)485 1903 y Fn(r)515 1891 y Fp(\()p Fo(k)s(;)14 b(x)p Fp(\))24 b(=)e(1)c(+)g Fo(o)p Fp(\(1\))p Fo(;)97 b(e)1268 1857 y Fh(ik)q(x)1370 1891 y Fo(f)1420 1857 y Fj(0)1411 1912 y Fn(r)1443 1891 y Fp(\()p Fo(k)s(;)14 b(x)p Fp(\))24 b(=)e Fm(\000)p Fo(ik)f Fp(+)d Fo(o)p Fp(\(1\))p Fo(;)180 b(x)24 b Fm(!)f(\0001)p Fo(;)0 2013 y Fp(and)h(the)g(transmission)f(co)r(e\016cien)n(t)h Fo(T)7 b(;)23 b Fp(and)h(the)h(re\015ection)e(co)r(e\016cien)n(ts)h Fo(L)f Fp(and)h Fo(R)q(;)g Fp(from)f(the)0 2112 y(left)28 b(and)g(righ)n(t,)f(resp)r(ectiv)n(ely)-7 b(,)26 b(are)h(obtained)g (from)h(the)g(spatial)f(asymptotics)568 2324 y Fo(f)609 2336 y Fn(l)632 2324 y Fp(\()p Fo(k)s(;)14 b(x)p Fp(\))24 b(=)962 2268 y Fo(e)1001 2238 y Fh(ik)q(x)p 947 2305 171 4 v 947 2381 a Fo(T)12 b Fp(\()p Fo(k)s Fp(\))1146 2324 y(+)1239 2268 y Fo(L)p Fp(\()p 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b(p)r(oles.)71 b(Note)39 b(that)h Fq(I)1052 2942 y Fn(+)1150 2972 y Fp(:=)h Fo(i)p Fp(\(0)p Fo(;)14 b Fp(+)p Fm(1)p Fp(\))39 b(denotes)g(the)g(p)r(ositiv)n(e)g(imaginary)f(axis.)0 3078 y(Later)29 b(w)n(e)h(will)g(let)p 631 3008 125 4 v 30 w Fq(C)700 3054 y Fn(+)782 3078 y Fp(:=)c Fq(C)965 3048 y Fn(+)1040 3078 y Fm([)21 b Fq(R)29 b Fp(and)h Fq(I)1417 3048 y Fj(\000)1500 3078 y Fp(:=)d Fo(i)p Fp(\()p Fm(\0001)p Fo(;)14 b Fp(0\))p Fo(:)29 b Fp(The)h(b)r(eha)n(vior)e(at)i Fo(k)g Fp(=)c(0)k(tells)0 3178 y(us)c(whether)g(the)h(p)r(oten)n(tial)f (in)h(\(1.1\))f(is)g(generic)f(or)g(exceptional:)36 b(The)26 b(generic)f(case)h(o)r(ccurs)0 3277 y(if)j Fo(T)12 b Fp(\(0\))24 b(=)g(0)k(and)h(the)g(exceptional)f(case)f(o)r(ccurs)h(if)h Fo(T)12 b Fp(\(0\))24 b Fm(6)p Fp(=)g(0)p Fo(:)k Fp(F)-7 b(or)28 b(a)h(review)e(of)i(scattering)0 3377 y(and)d(b)r(ound)h (states)f(of)g(\(1.1\),)g(the)h(reader)e(is)h(referred)f(to)i([3-9])e (and)h(the)h(references)e(therein.)149 3477 y(Our)30 b(pap)r(er)g(is)g(organized)e(as)i(follo)n(ws:)41 b(In)30 b(Section)h(2)e(w)n(e)h(brie\015y)g(review)g(the)g(e\013ect)h(of)0 3576 y(adding)24 b(a)g(b)r(ound)h(state)f(to)g(a)g(p)r(oten)n(tial)h (and)f(sho)n(w)g(that)g(certain)g(in)n(tegrals)f(of)i(the)f(resulting)0 3676 y(p)r(oten)n(tial)g(remain)g(una\013ected)h(b)n(y)f(the)h(b)r (ound-state)f(norming)g(constan)n(t)g(but)h(a\013ected)f(only)0 3775 y(b)n(y)i(the)h(b)r(ound-state)f(energy)f(and)h(in)g(a)g(rather)f (simple)i(manner.)36 b(In)26 b(Section)g(3)g(w)n(e)g(analyze)0 3875 y(a)e(consequence)f(of)h(the)g(result)g(of)g(Section)g(2)g(in)h (the)f(reco)n(v)n(ery)d(of)k(a)e(real-v)-5 b(alued,)24 b(compactly-)0 3975 y(supp)r(orted,)36 b(square-in)n(tegrable)c(p)r (oten)n(tial)j(in)g(terms)f(of)h(the)g(data)f Fo(L)p Fp(\()p Fo(k)s Fp(\))p Fo(=T)12 b Fp(\()p Fo(k)s Fp(\))p Fo(:)34 b Fp(W)-7 b(e)35 b(sho)n(w)0 4074 y(that,)27 b(corresp)r(onding)d(to)i(that)g(data,)g(there)g(are)f(a)h(discrete)f 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Fh(j)2251 4829 y Fo(;)g Fp(where)g(w)n(e)f(ha)n(v)n(e)g(the)0 4928 y(ordering)22 b(0)h Fo(<)f(\024)524 4940 y Fn(1)585 4928 y Fo(<)g Fm(\001)14 b(\001)g(\001)23 b Fo(<)g(\024)928 4940 y Fh(N)991 4928 y Fo(:)h Fp(The)f(sup)r(erscript)h([)p Fo(j)5 b Fp(])24 b(refers)f(to)g(quan)n(tities)h(asso)r(ciated)e(with)0 5031 y(the)g(p)r(oten)n(tial)g Fo(V)549 5001 y Fn([)p Fh(j)s Fn(])621 5031 y Fp(;)g(for)g(example,)g Fo(T)1192 5001 y Fn([)p Fh(j)s Fn(])1264 5031 y Fo(;)g(R)1373 5001 y Fn([)p Fh(j)s Fn(])1445 5031 y Fo(;)g Fp(and)g Fo(L)1703 5001 y Fn([)p Fh(j)s Fn(])1797 5031 y Fp(denote)f(the)h(scattering)f (co)r(e\016cien)n(ts,)p eop %%Page: 3 3 3 2 bop 179 42 a Fn(INVERSE)28 b(SCA)-5 b(TTERING)30 b(ON)f(THE)g(LINE)g(WITH)g(INCOMPLETE)f(SCA)-5 b(TTERING)30 b(D)n(A)-5 b(T)g(A)146 b(3)0 244 y Fp(and)38 b Fo(f)222 201 y Fn([)p Fh(j)s Fn(])213 269 y(l)332 244 y Fp(and)g Fo(f)554 201 y Fn([)p Fh(j)s Fn(])545 254 y(r)664 244 y Fp(denote)g(the)h(left)f(and)g(righ)n(t)f(Jost)h(solutions.)67 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4647 y Fo(n)p Fp(!)p 1616 4684 346 4 v 1616 4760 a(\(2)p Fo(n)c Fp(+)g(1\)!!)1971 4703 y Fo(;)184 b(n)23 b Fm(\025)f Fp(0)p Fo(;)73 b Fp(1)23 b Fm(\024)f Fo(j)28 b Fm(\024)23 b Fo(N)t(;)0 4947 y Ff(wher)l(e)30 b Fp(\(2)p Fo(n)19 b Fp(+)f(1\)!!)23 b(:=)f(\(1\)\(3\)\(5\))14 b Fm(\001)g(\001)g(\001)g Fp(\(2)p Fo(n)k Fp(+)g(1\))p Fo(:)p eop %%Page: 4 4 4 3 bop 0 42 a Fn(4)1138 b(TUNCA)-5 b(Y)29 b(AKTOSUN)149 241 y Fg(Pr)n(oof.)46 b Fp(T)-7 b(aking)32 b(the)j Fo(n)p Fp(th)f(p)r(o)n(w)n(er)e(in)i(\(2.5\))g(and)f(expanding)g(the)i (result,)g(from)e(\(2.3\))0 340 y(and)27 b(\(2.5\))h(w)n(e)f(get)0 562 y(\(2.8\))313 b Fo(I)520 574 y Fh(j;n)613 562 y Fp(\()p Fo(x)p Fp(\))24 b(=)f Fm(\000)p Fp(2)943 528 y Fh(n)p Fn(+1)1105 506 y Fo(d)p 1081 543 91 4 v 1081 619 a(dx)1235 458 y Fh(n)1195 483 y Fi(X)1196 659 y Fh(p)p Fn(=0)1315 562 y Fp(\()p Fm(\000)p Fp(1\))1486 528 y Fh(n)p Fj(\000)p Fh(p)1617 562 y Fo(\024)1665 519 y Fn(2\()p Fh(n)p Fj(\000)p Fh(p)p Fn(\))1665 585 y Fh(j)1895 445 y Fi(\022)1970 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Fp(=)g(0)k(and)h Fo(n)22 b Fp(=)h(1)p Fo(;)k Fp(resp)r(ectiv)n(ely)-7 b(,)27 b(from)g(\(2.7\))h(w)n(e)f(get)0 2981 y(\(2.12\))511 2868 y Fi(Z)594 2889 y Fj(1)557 3057 y(\0001)693 2981 y Fo(dx)811 2889 y Fi(h)851 2981 y Fo(V)918 2947 y Fn([)p Fh(j)s Fn(])990 2981 y Fp(\()p Fo(x)p Fp(\))20 b Fm(\000)e Fo(V)1270 2947 y Fn([)p Fh(j)s Fj(\000)p Fn(1])1428 2981 y Fp(\()p Fo(x)p Fp(\))1539 2889 y Fi(i)1602 2981 y Fp(=)23 b Fm(\000)p Fp(4)p Fo(\024)1845 2993 y Fh(j)1879 2981 y Fo(;)180 b Fp(1)22 b Fm(\024)h Fo(j)28 b Fm(\024)23 b Fo(N)t(;)0 3301 y Fp(\(2.13\))467 3188 y Fi(Z)550 3208 y Fj(1)513 3377 y(\0001)649 3301 y Fo(dx)768 3209 y Fi(h)807 3301 y Fo(V)874 3267 y Fn([)p Fh(j)s Fn(])946 3301 y Fp(\()p Fo(x)p Fp(\))1057 3267 y Fn(2)1114 3301 y Fm(\000)18 b Fo(V)1264 3267 y Fn([)p Fh(j)s Fj(\000)p Fn(1])1422 3301 y Fp(\()p Fo(x)p Fp(\))1533 3267 y Fn(2)1571 3209 y Fi(i)1633 3301 y Fp(=)1731 3245 y(16)p 1731 3282 84 4 v 1751 3358 a(3)1838 3301 y Fo(\024)1886 3267 y Fn(3)1886 3321 y Fh(j)1923 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y(\(a.i\))50 b(If)23 b Fm(D)r Fp(\()p Fo(k)s Fp(\))h(is)f(b)r(ounded)g(at)f Fo(k)k Fp(=)d(0)p Fo(;)f Fp(then)i(there)e(is)h(no)f(restriction)g(on)h Fo(N)31 b Fp(and)23 b(hence)415 1893 y Fo(N)45 b Fm(2)37 b(f)p Fp(0)p Fo(;)14 b Fp(1)p Fo(;)g Fp(2)p Fo(;)g(:)g(:)g(:)d Fm(g)p Fo(:)36 b Fp(Note)f(that)h(this)g(case)f(corresp)r(onds)e(to)j (the)g(exceptional)415 1993 y(case)27 b(for)g(\(1.1\).)190 2093 y(\(a.ii\))50 b(If)22 b Fm(D)r Fp(\()p Fo(k)s Fp(\))g(is)f(un)n(b) r(ounded)h(at)f Fo(k)26 b Fp(=)d(0)p Fo(;)e Fp(then)g(lim)1826 2105 y Fh(k)q Fj(!)p Fn(0)1966 2093 y Fp([2)p Fo(ik)16 b Fm(D)r Fp(\()p Fo(k)s Fp(\)])22 b(is)f(either)g(a)g(p)r(ositiv)n(e) 415 2192 y(constan)n(t)29 b(or)g(a)h(negativ)n(e)f(constan)n(t.)43 b(Th)n(us,)31 b(either)e Fm(D)r Fp(\()p Fo(k)s Fp(\))f Fm(!)f(\0001)j Fp(or)f Fm(D)r Fp(\()p Fo(k)s Fp(\))f Fm(!)415 2292 y Fp(+)p Fm(1)36 b Fp(as)f Fo(k)k Fm(!)e Fp(0)f(on)f Fq(I)1148 2262 y Fn(+)1204 2292 y Fo(:)h Fp(In)g(the)g(former)f(case)g Fo(N)44 b Fp(m)n(ust)36 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y(at)i Fo(x)d Fp(=)g(0)j(and)g Fo(\032)544 4641 y Fn(1)581 4629 y Fp(\()p Fo(x)p Fp(\))e(=)e(0)k(for)g Fo(x)d Fm(\024)g Fp(0)p Fo(;)j Fp(whic)n(h)g(is)g(satis\014ed)g(when)g Fo(V)2197 4586 y Fn([0])2178 4651 y(1)2298 4629 y Fm(\021)c Fp(0)k(for)g Fo(x)d(<)g Fp(0)p Fo(:)j Fp(The)0 4728 y(second)e (statemen)n(t)h(in)f(\(iii\))i(is)e(obtained)h(in)g(a)f(similar)g (manner.)852 b Fe(\003)149 4932 y Fp(Note)37 b(that)h(Prop)r(osition)d (3.1\(i\))i(holds)f(ev)n(en)h(when)g Fo(n)g Fp(is)g(a)f(nonin)n(teger.) 64 b(By)37 b(letting)0 5031 y Fo(n)27 b Fp(=)g(0)i(and)h Fo(n)d Fp(=)g(1)j(there)g(and)g(using)f(the)i(fact)f(that)h(a)e(p)r (oten)n(tial)h(in)h(the)f(F)-7 b(addeev)30 b(class)f(is)p eop %%Page: 8 8 8 7 bop 0 42 a Fn(8)1138 b(TUNCA)-5 b(Y)29 b(AKTOSUN)0 241 y Fp(in)n(tegrable,)d(w)n(e)i(obtain)0 458 y(\(3.5\))264 345 y Fi(Z)347 366 y Fj(1)310 534 y(\0001)446 458 y Fo(dx)14 b(V)617 415 y Fn([0])598 480 y(1)692 458 y Fp(\()p Fo(x)p Fp(\))24 b(=)915 345 y Fi(Z)998 366 y Fj(1)961 534 y(\0001)1097 458 y Fo(dx)14 b(V)1268 415 y Fn([0])1249 480 y(2)1343 458 y Fp(\()p Fo(x)p Fp(\))p Fo(;)1658 345 y Fi(Z)1741 366 y Fj(1)1704 534 y(\0001)1840 458 y Fo(dx)1959 366 y Fi(h)1998 458 y Fo(V)2065 415 y Fn([0])2046 480 y(1)2140 458 y Fp(\()p Fo(x)p Fp(\))2251 424 y Fn(2)2307 458 y Fm(\000)k Fo(V)2457 415 y Fn([0])2438 480 y(2)2532 458 y Fp(\()p Fo(x)p Fp(\))2643 424 y Fn(2)2681 366 y Fi(i)2744 458 y Fp(=)k(0)p Fo(:)0 687 y Fp(F)-7 b(or)31 b(smo)r(oth)g(p)r(oten)n (tials,)i(w)n(e)e(refer)g(the)h(reader)e(to)h(\(2.101\))g(of)g([10])g (for)g(results)g(similar)g(to)0 787 y(\(3.5\))c(and)h(their)f (generalizations.)149 945 y Fq(Case)32 b(\(b\):)42 b(Reco)m(v)m(ery)33 b(of)f Fo(V)51 b Fq(from)30 b Fm(D)k Fq(with)e(Half-line)e(Supp)s(ort.) 149 1044 y Fp(If)c Fm(D)r Fp(\()p Fo(k)s Fp(\))g(is)f(giv)n(en)f(for)h Fo(k)h Fm(2)d Fq(R)i Fp(and)g(if)h(it)f(is)g(also)g(kno)n(wn)f(that)h Fo(V)42 b Fm(\021)23 b Fp(0)i(for)f Fo(x)g(<)f Fp(0)p Fo(;)h Fp(then,)j(in)0 1144 y(addition)k(to)f(all)h(the)g(results)f (giv)n(en)g(in)h(case)f(\(a\),)i(in)f(particular,)f(in)h(addition)f(to) h(\(a.i\))g(and)0 1244 y(\(a.ii\),)d(w)n(e)f(ha)n(v)n(e)g(the)h(follo)n (wing)e(impro)n(v)n(emen)n(ts:)162 1343 y(\(b.iii\))51 b Fm(D)r Fp(\()p Fo(k)s Fp(\))28 b(has)f(a)f(unique)i(analytic)e (extension)h(to)g Fo(k)f Fm(2)d Fq(C)2104 1313 y Fn(+)2187 1343 y Fp(and)k(suc)n(h)f(an)h(extension)415 1443 y(is)k(uniquely)g (determined)g(b)n(y)f(our)g(data)h Fm(D)r Fp(\()p Fo(k)s Fp(\))g(kno)n(wn)f(for)h Fo(k)g Fm(2)e Fq(R)p Fo(:)h Fp(The)h(v)-5 b(alue)415 1542 y(of)29 b Fo(N)38 b Fp(m)n(ust)29 b(satisfy)f Fo(N)34 b Fm(\024)25 b Fo(Z)g Fp(+)19 b(1)p Fo(;)29 b Fp(where)f Fo(Z)35 b Fp(denotes)28 b(the)i(n)n(um)n(b)r(er)e (of)h(zeros)f(of)415 1642 y Fm(D)r Fp(\()p Fo(k)s Fp(\))k(on)f Fq(I)778 1612 y Fn(+)833 1642 y Fo(:)g Fp(In)h(fact,)g(from)f(the)g (pro)r(of)g(of)g(Prop)r(osition)e(3.1)h(in)i([2])e(it)i(follo)n(ws)415 1742 y(that)27 b(if)g Fm(D)r Fp(\()p Fo(k)s Fp(\))g(has)f(m)n(ultiple)h (zeros)e(on)h Fq(I)1700 1712 y Fn(+)1755 1742 y Fo(;)h Fp(then)g Fo(Z)32 b Fp(is)26 b(actually)g(the)h(n)n(um)n(b)r(er)f(of) 415 1841 y(distinct)i(zeros)f(of)g(o)r(dd)h(m)n(ultiplicit)n(y)-7 b(,)28 b(without)g(coun)n(ting)f(the)h(m)n(ultiplicities.)165 1941 y(\(b.iv\))50 b(F)-7 b(or)18 b(eac)n(h)g Fo(N)9 b Fp(-v)-5 b(alue)19 b(resulting)f(from)h(restrictions)e(\(a.i\),)k (\(a.ii\),)g(and)e(\(b.iii\),)j(giv)n(en)415 2041 y Fm(D)r Fp(\()p Fo(k)s Fp(\))27 b(for)e Fo(k)h Fm(2)e Fq(R)p Fo(;)h Fp(there)h(corresp)r(onds)e(an)h Fo(N)9 b Fp(-parameter)24 b(family)i(of)g(p)r(oten)n(tials)415 2140 y(where)h(the)h(parameter)f (set)h(is)f Fm(f)p Fo(\024)1499 2152 y Fh(j)1533 2140 y Fm(g)1575 2110 y Fh(N)1575 2162 y(j)s Fn(=1)1694 2140 y Fo(:)h Fp(The)g Fo(\024)1964 2152 y Fh(j)2026 2140 y Fp(satisfy)g(the)g(restrictions)e(0)d Fo(<)415 2251 y(\024)463 2263 y Fn(1)535 2251 y Fo(<)35 b Fm(\001)14 b(\001)g(\001)35 b Fo(<)f(\024)914 2263 y Fh(N)1012 2251 y Fp(and)g(\()p Fm(\000)p Fp(1\))1351 2221 y Fh(N)6 b Fj(\000)p Fh(j)1497 2251 y Fm(D)r Fp(\()p Fo(i\024)1672 2263 y Fh(j)1707 2251 y Fp(\))35 b Fo(>)f Fp(0)h(for)f Fo(j)40 b Fp(=)34 b(1)p Fo(;)14 b(:)g(:)g(:)g(;)g(N)t(:)34 b Fp(The)h(latter)415 2351 y(restriction)24 b(con\014nes)h(the)g Fo(\024)1310 2363 y Fh(j)1371 2351 y Fp(to)g(subin)n(terv)-5 b(als)24 b(whose)g(endp)r(oin)n(ts)i(are)e(uniquely)415 2450 y(determined)32 b(b)n(y)g(the)g(zeros)f(of)h Fm(D)r Fp(\()p Fo(k)s Fp(\))h(on)f Fq(I)1797 2420 y Fn(+)1852 2450 y Fo(:)g Fp(The)g(dep)r(endency)h(constan)n(ts)e Fo(\015)2954 2462 y Fh(j)415 2550 y Fp(are)k(uniquely)h(determined)f (as)g Fo(\015)1501 2562 y Fh(j)1573 2550 y Fp(=)h Fm(D)r Fp(\()p Fo(i\024)1849 2562 y Fh(j)1884 2550 y Fp(\))g(and)f(hence)h (they)g(are)e(not)i(free)415 2650 y(parameters.)46 b(The)31 b(left)h(re\015ection)e(co)r(e\016cien)n(t)h Fo(L)p Fp(\()p Fo(k)s Fp(\))h(giv)n(en)e(in)h(\(3.2\))g(b)r(ecomes)415 2749 y(meromorphic)37 b(in)i Fq(C)1106 2719 y Fn(+)1199 2749 y Fp(with)g(simple)g(p)r(oles)f(at)g Fo(k)44 b Fp(=)d Fo(i\024)2272 2761 y Fh(j)2345 2749 y Fp(for)d Fo(j)46 b Fp(=)41 b(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(N)t(:)415 2858 y Fp(Th)n(us,)42 b(\(3.2\))d(no)n(w)f(holds)h(for)f Fo(k)46 b Fm(2)p 1610 2788 125 4 v 42 w Fq(C)1679 2834 y Fn(+)1773 2858 y Fp(and)39 b(the)h(left)g(re\015ection)e(co)r(e\016cien)n(t)415 2961 y Fo(L)472 2931 y Fn([0])546 2961 y Fp(\()p Fo(k)s Fp(\))29 b(giv)n(en)d(in)i(\(3.3\))f(b)r(ecomes)h(analytic)f(in)h Fq(C)2009 2931 y Fn(+)2064 2961 y Fo(:)149 3119 y Fq(Case)k(\(c\):)43 b(Reco)m(v)m(ery)33 b(of)e Fo(V)51 b Fq(from)31 b Fm(D)j Fq(with)d(Compact)g(Supp)s(ort.)149 3218 y Fp(If)25 b Fm(D)r Fp(\()p Fo(k)s Fp(\))f(for)g Fo(k)i Fm(2)d Fq(R)h Fp(is)f(giv)n(en)h(and)f(if)i(it)f(is)g(also)f(kno)n(wn)g(that)h Fo(V)42 b Fm(\021)22 b Fp(0)i(for)f Fo(x)33 b(=)-51 b Fm(2)23 b Fp([0)p Fo(;)14 b Fp(1])p Fo(;)23 b Fp(then,)0 3318 y(in)28 b(addition)f(to)g(all)g(the)h(results)f(in)h(cases)e (\(a\))i(and)f(\(b\),)h(in)g(particular,)e(in)i(addition)f(to)h (\(a.i\))0 3418 y(and)f(\(a.ii\),)h(w)n(e)g(ha)n(v)n(e)e(the)i(follo)n (wing)f(impro)n(v)n(emen)n(ts:)172 3517 y(\(c.iii\))50 b(The)26 b(quan)n(tit)n(y)g Fo(k)17 b Fm(D)r Fp(\()p Fo(k)s Fp(\))27 b(has)e(a)h(unique)g(analytic)g(extension)f(to)h(the)h (en)n(tire)f(com-)415 3617 y(plex)31 b(plane,)i(and)e(suc)n(h)g(an)g (extension)g(is)g(uniquely)g(determined)h(b)n(y)f(our)f(data)415 3717 y Fm(D)r Fp(\()p Fo(k)s Fp(\))e(kno)n(wn)f(for)g Fo(k)f Fm(2)d Fq(R)p Fo(:)k Fp(Moreo)n(v)n(er,)e(as)i(in)g(\(b.iii\))i (the)e(v)-5 b(alue)28 b(of)f Fo(N)36 b Fp(m)n(ust)28 b(sat-)415 3816 y(isfy)i Fo(N)36 b Fm(\024)27 b Fo(Z)f Fp(+)19 b(1)p Fo(;)30 b Fp(where)f Fo(Z)36 b Fp(is)30 b(the)g(n)n(um)n(b)r(er)g(of)g(zeros)f(of)h Fm(D)r Fp(\()p Fo(k)s Fp(\))h(on)f Fq(I)2664 3786 y Fn(+)2749 3816 y Fp(ha)n(ving)415 3916 y(o)r(dd)e(m)n(ultiplicities,)g(without)g(coun)n (ting)f(the)h(m)n(ultiplicities.)174 4015 y(\(c.iv\))50 b(F)-7 b(or)19 b(eac)n(h)g Fo(N)9 b Fp(-v)-5 b(alue)20 b(resulting)f(from)g(restrictions)g(\(a.i\),)j(\(a.ii\),)g(and)d (\(c.iii\),)k(giv)n(en)415 4115 y Fm(D)r Fp(\()p Fo(k)s Fp(\))j(for)e Fo(k)i Fm(2)d Fq(R)p Fo(;)h Fp(there)h(corresp)r(ond)e(a) h(discrete)g(n)n(um)n(b)r(er)g(of)h(p)r(oten)n(tials)f(where)415 4215 y(the)g(discrete)f(parameter)f(set)i(is)f Fm(f)p Fo(\024)1543 4227 y Fh(j)1578 4215 y Fm(g)1620 4185 y Fh(N)1620 4236 y(j)s Fn(=1)1738 4215 y Fo(:)h Fp(The)g(set)f Fm(f)p Fo(\024)2167 4227 y Fh(j)2202 4215 y Fm(g)2244 4185 y Fh(N)2244 4236 y(j)s Fn(=1)2386 4215 y Fp(m)n(ust)h(b)r(e)g(a)f (subset)415 4318 y(of)30 b Fm(f)p Fo(\014)601 4330 y Fh(m)664 4318 y Fm(g)f Fp(and)h(satisfy)g(the)g(additional)g (restrictions)f(0)d Fo(<)h(\024)2342 4330 y Fn(1)2406 4318 y Fo(<)g Fm(\001)14 b(\001)g(\001)27 b Fo(<)g(\024)2762 4330 y Fh(N)2855 4318 y Fp(and)415 4418 y(\()p Fm(\000)p Fp(1\))586 4388 y Fh(N)6 b Fj(\000)p Fh(j)731 4418 y Fm(D)r Fp(\()p Fo(i\024)906 4430 y Fh(j)942 4418 y Fp(\))23 b Fo(>)f Fp(0)k(for)f Fo(j)j Fp(=)23 b(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(N) t(:)26 b Fp(Here,)f(eac)n(h)g Fo(k)h Fp(=)d Fm(\000)p Fo(i\014)2473 4430 y Fh(m)2561 4418 y Fp(corresp)r(onds)415 4524 y(to)k(a)g(zero)f(of)h(1)p Fo(=T)999 4493 y Fn([0])1072 4524 y Fp(\()p Fo(k)s Fp(\))h(on)f Fq(I)1361 4493 y Fj(\000)1417 4524 y Fo(;)h Fp(where)e Fo(T)1768 4493 y Fn([0])1842 4524 y Fp(\()p Fo(k)s Fp(\))i(is)f(the)h(quan)n(tit)n(y)e(in)i (\(3.1\),)f(and)415 4626 y Fo(k)s(=T)564 4596 y Fn([0])637 4626 y Fp(\()p Fo(k)s Fp(\))34 b(is)f(no)n(w)g(en)n(tire)g(on)g Fq(C)g Fp(and)g(uniquely)h(constructed)e(via)h(\(3.1\))g(from)415 4726 y(our)e(data)h Fm(D)r Fp(\()p Fo(k)s Fp(\))g(kno)n(wn)f(for)h Fo(k)h Fm(2)d Fq(R)p Fo(:)i Fp(The)g(v)-5 b(alues)31 b Fo(k)i Fp(=)d Fm(\000)p Fo(i\014)2394 4738 y Fh(m)2488 4726 y Fp(corresp)r(ond)g(to)415 4829 y(the)25 b(\(real\))e(resonances) f(of)j Fo(V)1343 4799 y Fn([0])1418 4829 y Fo(:)f Fp(F)-7 b(or)24 b(an)g(answ)n(er)e(to)i(the)h(question)f(whether)f(the)415 4928 y(set)35 b Fm(f)p Fo(\014)641 4940 y Fh(m)704 4928 y Fm(g)f Fp(is)h(a)g(\014nite)g(set)g(or)f(an)h(in\014nite)h(set,)h(w)n (e)d(refer)h(the)g(reader)f(to)h([15].)415 5031 y(Informally)26 b(sp)r(eaking,)f(if)i Fo(V)1323 5001 y Fn([0])1421 5031 y Fm(2)c Fo(C)1564 5001 y Fj(1)1558 5052 y Fn(0)1635 5031 y Fp([0)p Fo(;)14 b Fp(1])25 b(and)h(the)h(order)e(of)h(the)h (zero)e(of)h Fo(V)2914 5001 y Fn([0])p eop %%Page: 9 9 9 8 bop 179 42 a Fn(INVERSE)28 b(SCA)-5 b(TTERING)30 b(ON)f(THE)g(LINE)g(WITH)g(INCOMPLETE)f(SCA)-5 b(TTERING)30 b(D)n(A)-5 b(T)g(A)146 b(9)415 241 y Fp(at)36 b Fo(x)i Fp(=)g(0)e(or)f(at)h Fo(x)i Fp(=)g(1)e(is)g(in\014nite,)j(then)e(the)g (set)f Fm(f)p Fo(\014)2261 253 y Fh(m)2324 241 y Fm(g)g Fp(ma)n(y)f(b)r(e)i(in\014nite;)415 340 y(otherwise,)27 b(it)h(is)f(a)h(\014nite)g(set.)149 498 y Fq(Case)45 b(\(d\):)69 b(Reco)m(v)m(ery)47 b(of)e Fo(V)64 b Fq(from)44 b Fm(D)j Fq(with)e(Compact)f(Supp)s(ort)i(and)f Fo(L)2920 468 y Fn(2)2957 498 y Fq(-)0 598 y(norm.)149 697 y Fp(Let)34 b(us)f(assume)g(that)g Fm(D)r Fp(\()p Fo(k)s Fp(\))h(is)f(giv)n(en)g (for)f Fo(k)k Fm(2)c Fq(R)h Fp(and)g(it)h(is)f(kno)n(wn)g(that)g Fo(V)52 b Fm(\021)32 b Fp(0)g(for)0 797 y Fo(x)51 b(=)-51 b Fm(2)41 b Fp([0)p Fo(;)14 b Fp(1])p Fo(;)38 b(V)59 b Fm(2)42 b Fo(L)674 767 y Fn(2)711 797 y Fp([0)p Fo(;)14 b Fp(1])p Fo(;)37 b Fp(and)h Fm(jj)p Fo(V)19 b Fm(jj)42 b(\024)e Fo(C)q(;)g Fp(where)d(w)n(e)h(denote)h(the)g Fo(L)2412 767 y Fn(2)2448 797 y Fp(-norm)f(of)g Fo(V)58 b Fp(as)0 925 y Fm(jj)p Fo(V)19 b Fm(jj)30 b Fp(:=)306 833 y Fi(q)p 389 833 496 4 v 26 x(R)444 879 y Fj(1)428 955 y(\0001)564 925 y Fo(dx)14 b(V)20 b Fp(\()p Fo(x)p Fp(\))847 901 y Fn(2)885 925 y Fo(:)32 b Fp(W)-7 b(e)32 b(will)f(determine)h(the)g(precise)f(v)-5 b(alues)31 b(of)g Fo(C)38 b Fp(that)32 b(assure)e(a)0 1045 y(unique)e(or)f(non)n (unique)g(determination)g(of)h Fo(V)46 b Fp(from)27 b Fm(D)r Fo(:)149 1145 y Fp(As)40 b(outlined)f(b)r(elo)n(w)g(\(3.3\))g (in)g(case)f(\(a\),)43 b(giv)n(en)38 b Fm(D)r Fp(\()p Fo(k)s Fp(\))i(for)f Fo(k)45 b Fm(2)d Fq(R)p Fo(;)d Fp(w)n(e)g(are)f (able)h(to)0 1247 y(uniquely)23 b(determine)f Fo(V)781 1217 y Fn([0])879 1247 y Fp(when)h Fm(D)i Fp(is)d(singular)g(at)g Fo(k)k Fp(=)d(0)p Fo(;)f Fp(and)g(w)n(e)h(determine)f(t)n(w)n(o)g (distinct)0 1358 y(p)r(oten)n(tials)33 b Fo(V)456 1315 y Fn([0])437 1381 y(1)564 1358 y Fp(and)g Fo(V)798 1315 y Fn([0])779 1381 y(2)906 1358 y Fp(if)h Fm(D)h Fp(is)e(\014nite)h(at)f Fo(k)i Fp(=)d(0)p Fo(:)h Fp(In)h(the)f(latter)g(case,)h(w)n(e)f(kno)n (w)f(from)0 1476 y(\(3.5\))f(that)g Fm(jj)p Fo(V)498 1432 y Fn([0])479 1498 y(1)573 1476 y Fm(jj)e Fp(=)f Fm(jj)p Fo(V)854 1432 y Fn([0])835 1498 y(2)929 1476 y Fm(jj)p Fo(:)j Fp(Th)n(us,)g Fm(D)r Fp(\()p Fo(k)s Fp(\))h(uniquely)f(determines)g(the)g Fo(L)2441 1445 y Fn(2)2478 1476 y Fp(-norm)f(of)h Fo(V)2891 1445 y Fn([0])2966 1476 y Fo(;)0 1580 y Fp(ev)n(en)e(though)f(there)h(are)f(t)n(w)n(o)g (distinct)i(c)n(hoices)e(for)h Fo(V)1763 1550 y Fn([0])1867 1580 y Fp(in)h(the)f(exceptional)f(case.)41 b(Let)29 b(us)0 1679 y(denote)f(that)f(unique)h(v)-5 b(alue)28 b(b)n(y)f Fm(jj)p Fo(V)1160 1649 y Fn([0])1235 1679 y Fm(jj)p Fo(:)149 1779 y Fp(As)42 b(seen)f(from)g(\(c.iv\),)k(for)c(eac) n(h)g(allo)n(w)n(ed)f(allo)n(w)n(ed)g(in)n(teger)g Fo(N)t(;)i Fm(D)r Fp(\()p Fo(k)s Fp(\))g(uniquely)f([2])0 1879 y(determines)30 b(a)f(discrete)h(n)n(um)n(b)r(er)f(of)h(ordered)f(sets)g Fm(f)p Fo(\024)1760 1891 y Fh(j)1795 1879 y Fm(g)1837 1849 y Fh(N)1837 1900 y(j)s Fn(=1)1985 1879 y Fp(with)i(the)f(ordering) e(0)f Fo(<)f(\024)2860 1891 y Fn(1)2924 1879 y Fo(<)0 1994 y Fm(\001)14 b(\001)g(\001)23 b Fo(<)g(\024)256 2006 y Fh(N)346 1994 y Fp(related)k(to)g(the)h(b)r(ound)g(states)g(of)f Fo(V)1521 1963 y Fn([)p Fh(N)6 b Fn(])1622 1994 y Fo(:)27 b Fp(Let)h(us)g(de\014ne)594 2283 y Fo(C)653 2295 y Fn(0)714 2283 y Fp(:=)22 b Fm(jj)p Fo(V)937 2248 y Fn([0])1012 2283 y Fm(jj)p Fp(;)97 b Fo(C)1237 2295 y Fh(N)1324 2283 y Fp(:=)1434 2116 y Fi(2)1434 2265 y(4)1490 2283 y Fm(jj)p Fo(V)1603 2248 y Fn([0])1678 2283 y Fm(jj)1724 2248 y Fn(2)1779 2283 y Fp(+)1872 2226 y(16)p 1872 2264 84 4 v 1893 2340 a(3)2010 2179 y Fh(N)1979 2204 y Fi(X)1982 2381 y Fh(j)s Fn(=1)2113 2283 y Fo(\024)2161 2248 y Fn(3)2161 2303 y Fh(j)2198 2116 y Fi(3)2198 2265 y(5)2253 2133 y Fn(1)p Fh(=)p Fn(2)2372 2283 y Fo(:)0 2552 y Fp(Th)n(us,)35 b(for)f(eac)n(h)f Fo(N)t(;)h(C)755 2564 y Fh(N)852 2552 y Fp(consists)g(of)g(a)f(sequence)h(of)f(v)-5 b(alues.)56 b(Clearly)-7 b(,)35 b Fo(C)2463 2564 y Fn(0)2534 2552 y Fp(consists)e(of)h(a)0 2652 y(single)22 b(n)n(um)n(b)r(er.)35 b(By)22 b(listing)h(all)f(the)h(elemen)n(ts)g(in)g Fo(C)1663 2664 y Fh(N)1749 2652 y Fp(for)f(all)g(allo)n(w)n(ed)f Fo(N)9 b Fp(-v)-5 b(alues,)23 b(w)n(e)f(obtain)0 2752 y(a)31 b(discrete)g(set)g(of)g(ordered)f(p)r(ositiv)n(e)h(n)n(um)n(b)r (ers)g(consisting)f(of)h(v)-5 b(arious)30 b Fo(\024)2388 2764 y Fh(j)2454 2752 y Fp(v)-5 b(alues,)32 b(and)f(w)n(e)0 2851 y(denote)21 b(that)h(discrete)e(set)h(b)n(y)g Fm(f)p Fo(C)1066 2863 y Fh(N)1129 2851 y Fm(g)p Fo(:)g Fp(This)g(set)g(is)g(a) g(subset)g(of)h Fm(f)p Fo(\014)2087 2863 y Fh(m)2149 2851 y Fm(g)p Fo(;)f Fp(as)g(indicated)g(in)g(\(c.iv\).)0 2951 y(The)31 b(smallest)f(n)n(um)n(b)r(er)g(in)h(the)g(ordered)e(set)i Fm(f)p Fo(C)1584 2963 y Fh(N)1646 2951 y Fm(g)g Fp(is)f(strictly)g (less)g(than)h(the)g(next)g(larger)0 3050 y(n)n(um)n(b)r(er)25 b(due)h(to)f(the)h(fact)g(that)g(eac)n(h)f(set)g Fm(f)p Fo(\024)1437 3062 y Fh(j)1471 3050 y Fm(g)1513 3020 y Fh(N)1513 3072 y(j)s Fn(=1)1658 3050 y Fp(with)h(the)f(largest)g(allo)n (w)n(able)e Fo(N)35 b Fp(consists)0 3150 y(of)g(distinct)h(p)r(ositiv)n (e)g(elemen)n(ts.)60 b(This)35 b(allo)n(ws)f(us)i(to)f(determine)g(the) h(v)-5 b(alue)36 b(of)f Fo(C)42 b Fp(in)35 b(the)0 3250 y(inequalit)n(y)30 b Fm(jj)p Fo(V)19 b Fm(jj)29 b(\024)f Fo(C)37 b Fp(in)31 b(order)f(to)g(determine)h(a)g(unique)g(p)r(oten)n (tial)f Fo(V)50 b Fp(corresp)r(onding)29 b(to)0 3349 y(our)e(data)g Fm(D)r Fo(:)h Fp(By)f(c)n(ho)r(osing)f Fo(C)34 b Fp(as)26 b(greater)g(than)i(or)e(equal)h(to)h(the)f(smallest) h(n)n(um)n(b)r(er)f(in)g(the)0 3449 y(set)e Fm(f)p Fo(C)228 3461 y Fh(N)291 3449 y Fm(g)g Fp(but)h(strictly)f(less)g(than)h(the)g (next)f(larger)f(elemen)n(t,)i(w)n(e)f(will)g(uniquely)h(determine)0 3549 y(the)g(p)r(oten)n(tial)f Fo(V)5 b(:)25 b Fp(Next)h(w)n(e)f (illustrate)g(this)g(determination)g(with)h(some)f(explicit)g (examples.)149 3705 y(As)30 b(our)f(scattering)g(data)g(let)h(us)f(use) h Fm(D)r Fp(\()p Fo(k)s Fp(\))d(=)1681 3649 y Fm(\000)p Fo(\017)14 b(e)1833 3619 y Fh(ik)1910 3649 y Fp(sin)2026 3580 y Fm(p)p 2095 3580 219 4 v 69 x Fo(k)2141 3625 y Fn(2)2197 3649 y Fp(+)k Fo(\017)p 1681 3686 633 4 v 1789 3772 a Fp(2)p Fo(ik)1918 3703 y Fm(p)p 1988 3703 219 4 v 1988 3772 a Fo(k)2034 3748 y Fn(2)2089 3772 y Fp(+)g Fo(\017)2323 3705 y(;)30 b Fp(where)f Fo(\017)h Fp(is)f(a)g(p)r(os-)0 3848 y(itiv)n(e)g(parameter.)41 b(In)30 b(fact,)g(one)f(corresp)r (onding)f(p)r(oten)n(tial)h(is)g(the)h(square)e(w)n(ell)i(of)f(depth)h Fo(\017)0 3948 y Fp(supp)r(orted)d(on)g(the)h(in)n(terv)-5 b(al)27 b([0)p Fo(;)14 b Fp(1])p Fo(:)26 b Fp(F)-7 b(or)27 b(eac)n(h)f(v)-5 b(alue)27 b(of)h Fo(\017;)f Fp(let)g(us)h(obtain)f (all)g(the)g(p)r(oten)n(tials)0 4047 y(corresp)r(onding)h(to)i Fm(D)r Fp(\()p Fo(k)s Fp(\))h(with)g(supp)r(ort)f(con\014ned)g(to)g([0) p Fo(;)14 b Fp(1])30 b(and)g(sp)r(ecify)g(their)g Fo(L)2679 4017 y Fn(2)2716 4047 y Fp(-norms.)0 4147 y(W)-7 b(e)36 b(ha)n(v)n(e)f(lim)467 4159 y Fh(k)q Fj(!)p Fn(0)607 4147 y Fp([2)p Fo(ik)16 b Fm(D)r Fp(\()p Fo(k)s Fp(\)])37 b(=)g Fm(\000)1163 4087 y(p)p 1232 4087 34 4 v 60 x Fo(\017)27 b Fp(sin)1409 4087 y Fm(p)p 1478 4087 V 60 x Fo(\017;)36 b Fp(and)g(hence)g(the)g(exceptional)g(case)f(o)r(ccurs)0 4247 y(when)220 4187 y Fm(p)p 289 4187 V 60 x Fo(\017=\031)e Fp(is)d(a)g(p)r(ositiv)n(e)h(in)n(teger)e(and)h(the)h(generic)f(case)f (o)r(ccurs)h(otherwise.)45 b(The)30 b(zeros)0 4352 y(of)k Fm(D)r Fp(\()p Fo(k)s Fp(\))i(on)e Fq(I)471 4321 y Fn(+)560 4352 y Fp(o)r(ccur)g(when)h(sin)1130 4282 y Fm(p)p 1200 4282 219 4 v 1200 4352 a Fo(k)1246 4328 y Fn(2)1301 4352 y Fp(+)18 b Fo(\017)34 b Fp(=)g(0)p Fo(;)g Fp(and)g(hence)h(these)f (are)g(all)g(simple)g(zeros)0 4458 y(o)r(ccurring)25 b(at)i Fo(k)f Fp(=)c Fo(i)652 4387 y Fi(p)p 735 4387 507 4 v 71 x Fo(\017)c Fm(\000)g Fp(\()p Fo(j)24 b Fm(\000)18 b Fp(1\))1117 4434 y Fn(2)1154 4458 y Fo(\031)1204 4434 y Fn(2)1268 4458 y Fp(for)26 b Fo(j)i Fp(=)23 b(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(Z)q(;)27 b Fp(with)g Fo(Z)33 b Fp(b)r(eing)26 b(equal)h(to)f Fm(b)2734 4398 y(p)p 2803 4398 34 4 v 60 x Fo(\017=\031)s Fm(c)p Fo(;)0 4558 y Fp(i.e.)59 b(the)35 b(greatest)f(in)n(teger)g(less)g(than)h(or)f(equal)g(to)1724 4499 y Fm(p)p 1793 4499 V 59 x Fo(\017=\031)s(:)h Fp(As)g Fo(k)j Fm(!)d(1)g Fp(on)f Fq(I)2582 4528 y Fn(+)2638 4558 y Fo(;)h Fp(w)n(e)f(ha)n(v)n(e)0 4658 y Fm(D)r Fp(\()p Fo(k)s Fp(\))24 b Fm(!)f Fp(0)348 4628 y Fn(+)403 4658 y Fo(:)j Fp(As)h Fo(k)f Fm(!)d Fp(0)j(on)g Fq(I)967 4628 y Fn(+)1023 4658 y Fo(;)g Fp(w)n(e)h(get)f(\()p Fm(\000)p Fp(1\))1502 4628 y Fh(Z)1555 4658 y Fm(D)r Fp(\()p Fo(k)s Fp(\))e Fm(!)f Fp(0)1903 4628 y Fn(+)1984 4658 y Fp(in)k(the)g (exceptional)f(case,)g(and)0 4758 y(\()p Fm(\000)p Fp(1\))171 4727 y Fh(Z)224 4758 y Fm(D)r Fp(\()p Fo(k)s Fp(\))e Fm(!)f Fp(+)p Fm(1)k Fp(in)h(the)g(generic)f(case.)36 b(De\014ne)588 4929 y(1)p 531 4966 157 4 v 531 5042 a Fo(\034)9 b Fp(\()p Fo(k)s Fp(\))720 4985 y(:=)23 b Fo(e)870 4951 y Fh(ik)948 4868 y Fi(\024)991 4985 y Fp(cos)1116 4904 y Fi(p)p 1199 4904 219 4 v 81 x Fo(k)1245 4961 y Fn(2)1301 4985 y Fp(+)18 b Fo(\017)g Fp(+)1601 4929 y(2)p Fo(k)1689 4899 y Fn(2)1744 4929 y Fp(+)g Fo(\017)p 1529 4966 404 4 v 1529 5052 a Fp(2)p Fo(ik)1646 4983 y Fm(p)p 1714 4983 219 4 v 1714 5052 a Fo(k)1760 5028 y Fn(2)1816 5052 y Fp(+)g Fo(\017)1970 4985 y Fp(sin)2086 4904 y Fi(p)p 2169 4904 V 81 x Fo(k)2215 4961 y Fn(2)2270 4985 y Fp(+)g Fo(\017)2387 4868 y Fi(\025)2445 4985 y Fo(:)p eop %%Page: 10 10 10 9 bop 0 42 a Fn(10)1105 b(TUNCA)-5 b(Y)29 b(AKTOSUN)0 241 y Fp(Note)34 b(that)g Fo(\034)9 b Fp(\()p Fo(k)s Fp(\))36 b(corresp)r(onds)c(to)i(the)g(transmission)f(co)r(e\016cien)n (t)h(of)g(the)g(square-w)n(ell)e(p)r(o-)0 340 y(ten)n(tial)i(of)f (depth)h Fo(\017)g Fp(supp)r(orted)f(on)g([0)p Fo(;)14 b Fp(1])p Fo(:)33 b Fp(It)h(is)g(kno)n(wn)f(that)g(1)p Fo(=\034)9 b Fp(\()p Fo(k)s Fp(\))34 b(has)f(exactly)g Fo(Z)28 b Fp(+)22 b(1)0 440 y(\(simple\))j(zeros)d(on)i Fq(I)672 410 y Fn(+)728 440 y Fo(;)g Fp(whic)n(h)g(w)n(e)f(denote)h(b)n (y)g Fo(\030)1539 452 y Fh(j)1599 440 y Fp(with)g(the)g(ordering)f(0)f Fo(<)h(\030)2436 452 y Fn(1)2497 440 y Fo(<)f Fm(\001)14 b(\001)g(\001)23 b Fo(<)g(\030)2828 452 y Fh(Z)t Fn(+1)2966 440 y Fo(:)0 540 y Fp(The)28 b(quan)n(tit)n(y)f(in)h(\(3.1\))f(is)g (obtained)h(as)0 773 y(\(3.6\))1105 717 y(1)p 1003 754 246 4 v 1003 832 a Fo(T)1064 808 y Fn([0])1138 832 y Fp(\()p Fo(k)s Fp(\))1281 773 y(=)1436 717 y(1)p 1379 754 157 4 v 1379 830 a Fo(\034)9 b Fp(\()p Fo(k)s Fp(\))1559 670 y Fh(Z)t Fn(+1)1572 694 y Fi(Y)1568 871 y Fh(j)s Fn(=1)1716 717 y Fo(k)21 b Fp(+)d Fo(i\030)1928 729 y Fh(j)p 1716 754 248 4 v 1716 830 a Fo(k)j Fm(\000)d Fo(i\030)1928 842 y Fh(j)1973 773 y Fo(:)149 1010 y Fg(Example)40 b(3.1.)45 b Fp(When)37 b Fo(\017)g Fp(=)f(5)p Fo(;)g Fp(w)n(e)g(are)f(in)h(the)h (generic)e(case)g(and)h Fo(Z)43 b Fp(=)37 b(0)p Fo(:)f Fp(Hence,)0 1109 y Fo(N)42 b Fm(\024)33 b Fp(1)p Fo(;)g Fp(but)h Fm(D)r Fp(\()p Fo(k)s Fp(\))g Fm(!)f Fp(+)p Fm(1)h Fp(on)f Fq(I)1128 1079 y Fn(+)1217 1109 y Fp(indicates)h(that)g Fo(N)42 b Fp(m)n(ust)34 b(b)r(e)g(o)r(dd.)55 b(Th)n(us,)35 b Fo(N)42 b Fp(=)33 b(1)g(is)0 1212 y(the)f(only)f(allo)n(w)n(ed)f(v)-5 b(alue.)49 b(In)32 b(this)g(case,)f(1)p Fo(=T)1513 1182 y Fn([0])1586 1212 y Fp(\()p Fo(k)s Fp(\))h(giv)n(en)f(in)h(\(3.6\))f (has)g(t)n(w)n(o)g(zeros)f(on)i Fq(I)2933 1182 y Fj(\000)0 1312 y Fp(at)h Fo(k)h Fp(=)e Fm(\000)p Fo(i\014)422 1324 y Fh(j)489 1312 y Fp(with)h Fo(\014)730 1324 y Fn(1)799 1312 y Fp(=)f(1)p Fo(:)p Fp(5433)p 1129 1248 42 4 v(4)e(and)i Fo(\014)1414 1324 y Fn(2)1483 1312 y Fp(=)g(1)p Fo(:)p Fp(585)p 1771 1248 V(7)m Fo(:)h Fp(W)-7 b(e)34 b(use)e(an)h(o)n(v)n (erline)e(to)i(indicate)0 1411 y(roundo\013.)k(In)28 b(\(3.6\))f(w)n(e)h(ha)n(v)n(e)e Fo(\030)1015 1423 y Fn(1)1076 1411 y Fp(=)d Fo(\014)1211 1423 y Fn(2)1248 1411 y Fo(:)28 b Fp(Corresp)r(onding)e(to)i Fm(D)r Fp(\()p Fo(k)s Fp(\))g(w)n(e)g(ha)n(v)n(e)e(t)n(w)n(o)h(p)r(oten)n(tials)0 1527 y Fo(V)67 1484 y Fn([1])48 1549 y(1)169 1527 y Fp(and)g Fo(V)397 1484 y Fn([1])378 1549 y(2)472 1527 y Fo(;)g Fp(ha)n(ving)f(b)r(ound)i(states)e(at)h Fo(k)f Fp(=)d Fo(i\014)1614 1539 y Fn(1)1678 1527 y Fp(and)k Fo(k)f Fp(=)c Fo(i\014)2071 1539 y Fn(2)2108 1527 y Fo(;)27 b Fp(resp)r(ectiv)n(ely)-7 b(.)36 b(Note)28 b(that)0 1644 y Fo(V)67 1601 y Fn([1])48 1666 y(2)167 1644 y Fp(is)d(the)g (square)f(w)n(ell)g(of)h(depth)h Fo(\017:)f Fp(W)-7 b(e)25 b(ha)n(v)n(e)f Fm(jj)p Fo(V)1661 1601 y Fn([1])1642 1666 y(1)1736 1644 y Fm(jj)f Fp(=)g(4)p Fo(:)p Fp(8312)p 2126 1581 V(6)f(and)j Fm(jj)p Fo(V)2462 1601 y Fn([1])2443 1666 y(2)2537 1644 y Fm(jj)e Fp(=)f(5)p Fo(:)j Fp(Th)n(us,)0 1761 y(kno)n(wledge)e(of)g(an)n(y)h Fo(C)30 b Fp(satisfying)23 b Fm(jj)p Fo(V)1210 1718 y Fn([1])1191 1784 y(1)1285 1761 y Fm(jj)g(\024)g Fo(C)29 b(<)23 b Fm(jj)p Fo(V)1731 1718 y Fn([1])1712 1784 y(2)1805 1761 y Fm(jj)i Fp(helps)e(us)h(to)g (iden)n(tify)g Fo(V)2650 1718 y Fn([1])2631 1784 y(1)2749 1761 y Fp(or)f Fo(V)2914 1718 y Fn([1])2895 1784 y(2)0 1879 y Fp(as)30 b(the)h(unique)g(p)r(oten)n(tial)f(corresp)r(onding)f (to)i Fm(D)r Fp(\()p Fo(k)s Fp(\))p Fo(:)g Fp(The)g(left)g (re\015ection)f(co)r(e\016cien)n(ts)g Fo(L)2914 1835 y Fn([1])2914 1901 y(1)0 1996 y Fp(and)d Fo(L)218 1953 y Fn([1])218 2018 y(2)320 1996 y Fp(corresp)r(onding)f(to)i Fo(V)1022 1953 y Fn([1])1004 2018 y(1)1125 1996 y Fp(and)f Fo(V)1353 1953 y Fn([1])1334 2018 y(2)1428 1996 y Fo(;)h Fp(resp)r(ectiv)n(ely)-7 b(,)27 b(are)f(obtained)i(from)f(\(3.2\))g(as) 707 2193 y Fo(L)764 2150 y Fn([1])764 2216 y Fh(j)838 2193 y Fp(\()p Fo(k)s Fp(\))c(=)g Fm(D)r Fp(\()p Fo(k)s Fp(\))14 b Fo(T)1310 2159 y Fn([0])1385 2193 y Fp(\()p Fo(k)s Fp(\))1519 2137 y Fo(k)21 b Fp(+)d Fo(i\014)1742 2149 y Fh(j)p 1519 2174 258 4 v 1519 2250 a Fo(k)j Fm(\000)d Fo(i\014)1742 2262 y Fh(j)1787 2193 y Fo(;)180 b(j)28 b Fp(=)22 b(1)p Fo(;)14 b Fp(2)p Fo(:)0 2421 y Fp(Note)24 b(that)g Fo(V)440 2378 y Fn([1])421 2444 y(1)539 2421 y Fp(and)g Fo(V)764 2378 y Fn([1])745 2444 y(2)863 2421 y Fp(can)g(uniquely)g(b)r(e)g(constructed)g([11])f(from)h Fo(L)2302 2378 y Fn([1])2302 2444 y(1)2400 2421 y Fp(and)g Fo(L)2615 2378 y Fn([1])2615 2444 y(2)2690 2421 y Fo(;)g Fp(resp)r(ec-)0 2521 y(tiv)n(ely)-7 b(,)27 b(b)r(ecause)h(they)f(v)-5 b(anish)28 b(for)f Fo(x)c(<)g Fp(0)p Fo(:)149 2671 y Fg(Example)k(3.2.)44 b Fp(When)24 b Fo(\017)f Fp(=)f Fo(\031)1159 2641 y Fn(2)1197 2671 y Fo(;)h Fp(w)n(e)g(are)f(in)i(the)f (exceptional)f(case)h(and)g Fo(Z)28 b Fp(=)23 b(0)p Fo(:)g Fp(Hence,)0 2771 y(b)r(oth)30 b Fo(N)35 b Fp(=)26 b(0)j(and)g Fo(N)35 b Fp(=)25 b(1)30 b(are)e(allo)n(w)n(ed.)41 b(In)30 b(this)f(case)g(1)p Fo(=T)1954 2741 y Fn([0])2027 2771 y Fp(\()p Fo(k)s Fp(\))h(giv)n(en)e(in)i(\(3.6\))f(has)g(only)0 2871 y(one)i(zero)f(on)g Fq(I)489 2841 y Fj(\000)577 2871 y Fp(at)h Fo(k)g Fp(=)e Fm(\000)p Fo(i\014)991 2883 y Fn(1)1058 2871 y Fp(with)j Fo(\014)1298 2883 y Fn(1)1364 2871 y Fp(=)c(2)p Fo(:)p Fp(52258)p 1732 2807 42 4 v(8)m Fo(:)j Fp(Th)n(us,)h(in)f(\(3.6\))g(w)n(e)g(ha)n(v)n(e)f Fo(\030)2722 2883 y Fn(1)2788 2871 y Fp(=)f Fo(\014)2929 2883 y Fn(1)2966 2871 y Fo(:)0 2974 y Fp(Corresp)r(onding)i(to)h Fm(D)r Fp(\()p Fo(k)s Fp(\))h(w)n(e)f(ha)n(v)n(e)g(t)n(w)n(o)g(p)r (oten)n(tials)g Fo(V)1818 2943 y Fn([0])1925 2974 y Fp(and)g Fo(V)2158 2943 y Fn([1])2233 2974 y Fo(;)h Fp(the)g(former)e(with)i(no) 0 3076 y(b)r(ound)j(states)f(and)g(the)h(latter)g(with)g(one)f(b)r (ound)h(state)f(at)h Fo(k)j Fp(=)d Fo(i\014)2264 3088 y Fn(1)2300 3076 y Fo(:)g Fp(Note)g(that)g Fo(V)2823 3046 y Fn([1])2933 3076 y Fp(is)0 3176 y(the)i(square)e(w)n(ell)h(of)g (depth)h Fo(\017:)g Fp(W)-7 b(e)37 b(ha)n(v)n(e)g Fm(jj)p Fo(V)1513 3146 y Fn([0])1587 3176 y Fm(jj)j Fp(=)f(3)p Fo(:)p Fp(3853)p 2010 3112 V(7)34 b(and)k Fm(jj)p Fo(V)2371 3146 y Fn([1])2446 3176 y Fm(jj)h Fp(=)g Fo(\031)2685 3146 y Fn(2)2723 3176 y Fo(:)e Fp(Th)n(us,)0 3279 y(kno)n(wledge)25 b(of)i(an)n(y)f Fo(C)33 b Fp(satisfying)26 b Fm(jj)p Fo(V)1224 3249 y Fn([0])1299 3279 y Fm(jj)d(\024)g Fo(C)29 b(<)23 b Fm(jj)p Fo(V)1745 3249 y Fn([1])1820 3279 y Fm(jj)k Fp(helps)g(us)f(to)h(iden)n(tify)g(either)g Fo(V)2914 3249 y Fn([0])0 3381 y Fp(or)20 b Fo(V)162 3351 y Fn([1])258 3381 y Fp(as)g(the)i(unique)f(p)r(oten)n(tial)g(corresp)r(onding)e(to)i Fm(D)r Fp(\()p Fo(k)s Fp(\))p Fo(:)h Fp(The)f(left)h(re\015ection)e(co) r(e\016cien)n(ts)0 3484 y Fo(L)57 3454 y Fn([0])162 3484 y Fp(and)30 b Fo(L)383 3454 y Fn([1])488 3484 y Fp(corresp)r(onding)f (to)h Fo(V)1196 3454 y Fn([0])1301 3484 y Fp(and)h Fo(V)1532 3454 y Fn([1])1607 3484 y Fo(;)g Fp(resp)r(ectiv)n(ely)-7 b(,)30 b(are)g(obtained)g(from)g(\(3.2\))0 3584 y(as)488 3707 y Fo(L)545 3673 y Fn([0])619 3707 y Fp(\()p Fo(k)s Fp(\))23 b(=)g Fm(D)r Fp(\()p Fo(k)s Fp(\))14 b Fo(T)1091 3673 y Fn([0])1166 3707 y Fp(\()p Fo(k)s Fp(\))p Fo(;)97 b(L)1453 3673 y Fn([1])1527 3707 y Fp(\()p Fo(k)s Fp(\))24 b(=)e Fm(D)r Fp(\()p Fo(k)s Fp(\))14 b Fo(T)1999 3673 y Fn([0])2074 3707 y Fp(\()p Fo(k)s Fp(\))2208 3651 y Fo(k)21 b Fp(+)d Fo(i\014)2431 3663 y Fn(1)p 2208 3688 261 4 v 2208 3764 a Fo(k)j Fm(\000)d Fo(i\014)2431 3776 y Fn(1)2478 3707 y Fo(:)0 3881 y Fp(Ha)n(ving)35 b Fo(L)349 3851 y Fn([0])459 3881 y Fp(and)g Fo(L)685 3851 y Fn([1])795 3881 y Fp(at)g(hand,)j(the)e(p)r(oten)n(tials)f Fo(V)1754 3851 y Fn([0])1864 3881 y Fp(and)h Fo(V)2100 3851 y Fn([1])2211 3881 y Fp(can)f(b)r(e)h(uniquely)g(con-)0 3980 y(structed.)149 4131 y Fg(Example)i(3.3.)45 b Fp(When)34 b Fo(\017)f Fp(=)g(20)p Fo(;)g Fp(w)n(e)g(are)g(in)h(the)g(generic)f(case)g(and)g Fo(Z)39 b Fp(=)33 b(1)p Fo(:)h Fp(Hence,)0 4230 y Fo(N)h Fm(\024)26 b Fp(2)p Fo(;)j Fp(but)h Fm(D)r Fp(\()p Fo(k)s Fp(\))e Fm(!)e(\0001)j Fp(on)g Fq(I)1084 4200 y Fn(+)1169 4230 y Fp(indicates)h(that)g Fo(N)38 b Fp(m)n(ust)30 b(b)r(e)g(ev)n(en.)42 b(Th)n(us,)30 b Fo(N)35 b Fp(=)26 b(0)j(and)0 4333 y Fo(N)j Fp(=)22 b(2)27 b(are)g(the)g(only)g(p)r (ossibilities.)37 b(In)27 b(this)h(case)e(1)p Fo(=T)1782 4303 y Fn([0])1855 4333 y Fp(\()p Fo(k)s Fp(\))i(giv)n(en)e(in)i (\(3.6\))f(has)g(t)n(w)n(o)f(zeros)0 4433 y(on)j Fq(I)153 4403 y Fj(\000)238 4433 y Fp(at)f Fo(k)g Fp(=)d Fm(\000)p Fo(i\014)642 4445 y Fn(1)707 4433 y Fp(with)30 b Fo(\014)945 4445 y Fn(1)1007 4433 y Fp(=)25 b Fo(\030)1133 4445 y Fn(1)1195 4433 y Fp(=)g(1)p Fo(:)p Fp(9302)p 1518 4369 42 4 v(1)h(and)j Fo(k)f Fp(=)d Fm(\000)p Fo(i\014)2051 4445 y Fn(2)2116 4433 y Fp(with)k Fo(\014)2353 4445 y Fn(2)2416 4433 y Fp(=)24 b Fo(\030)2541 4445 y Fn(2)2604 4433 y Fp(=)h(3)p Fo(:)p Fp(9255)p 2927 4369 V(6)m Fo(:)0 4532 y Fp(When)38 b Fo(N)49 b Fp(=)39 b(2)p Fo(;)f Fp(the)g(only)f(p)r (oten)n(tial)h Fo(V)1348 4502 y Fn([2])1460 4532 y Fp(corresp)r(onding) e(to)i Fm(D)r Fp(\()p Fo(k)s Fp(\))g(is)g(the)g(square)e(w)n(ell)0 4635 y(of)f(depth)g Fo(\017)g Fp(with)g(supp)r(ort)g([0)p Fo(;)14 b Fp(1])p Fo(:)34 b Fp(When)h Fo(N)44 b Fp(=)35 b(0)p Fo(;)f Fp(the)i(corresp)r(onding)d(p)r(oten)n(tial)h Fo(V)2823 4605 y Fn([0])2933 4635 y Fp(is)0 4738 y(uniquely)26 b(determined)g(from)g Fm(D)r Fp(\()p Fo(k)s Fp(\))g(and)g(its)g(left)h (re\015ection)e(co)r(e\016cien)n(t)h Fo(L)2383 4708 y Fn([0])2457 4738 y Fp(\()p Fo(k)s Fp(\))g(is)g(obtained)0 4838 y(from)h(\(3.2\))h(as)747 4987 y Fo(L)804 4952 y Fn([0])878 4987 y Fp(\()p Fo(k)s Fp(\))c(=)f Fm(D)r Fp(\()p Fo(k)s Fp(\))14 b Fo(T)1351 4952 y Fn([0])1425 4987 y Fp(\()p Fo(k)s Fp(\))1559 4930 y(\()p Fo(k)22 b Fm(\000)c Fo(i\014)1815 4942 y Fn(1)1852 4930 y Fp(\)\()p Fo(k)k Fm(\000)c Fo(i\014)2140 4942 y Fn(2)2176 4930 y Fp(\))p 1559 4968 650 4 v 1559 5044 a(\()p Fo(k)k Fp(+)c Fo(i\014)1815 5056 y Fn(1)1852 5044 y Fp(\)\()p Fo(k)k Fp(+)c Fo(i\014)2140 5056 y Fn(2)2176 5044 y Fp(\))2219 4987 y Fo(:)p eop %%Page: 11 11 11 10 bop 179 42 a Fn(INVERSE)28 b(SCA)-5 b(TTERING)30 b(ON)f(THE)g(LINE)g(WITH)g(INCOMPLETE)f(SCA)-5 b(TTERING)30 b(D)n(A)-5 b(T)g(A)113 b(11)0 241 y Fp(In)38 b(this)h(case)e(w)n(e)h (ha)n(v)n(e)f Fm(jj)p Fo(V)921 211 y Fn([0])996 241 y Fm(jj)k Fp(=)g(6)p Fo(:)p Fp(2463)p 1422 177 42 4 v(5)35 b(and)j Fm(jj)p Fo(V)1784 211 y Fn([2])1859 241 y Fm(jj)j Fp(=)f(20)p Fo(:)e Fp(Th)n(us,)j(an)d(appropriate)0 340 y(sp)r(eci\014cation)29 b(of)h(the)f(upp)r(er)h(limit)g(on)g(the)f Fo(L)1475 310 y Fn(2)1512 340 y Fp(-norm)g(of)g(the)h(p)r(oten)n(tial)g (allo)n(ws)e(the)i(unique)0 440 y(iden)n(ti\014cation)d(of)h Fo(V)662 410 y Fn([0])764 440 y Fp(or)f Fo(V)933 410 y Fn([2])1035 440 y Fp(from)h Fm(D)r Fp(\()p Fo(k)s Fp(\))p Fo(:)149 589 y Fg(Example)h(3.4.)44 b Fp(When)26 b Fo(\017)d Fp(=)f(130)p Fo(;)i 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y(2)180 1710 y Fp(has)38 b Fm(f\000)p Fo(\014)497 1679 y Fn(2)493 1730 y(1)533 1710 y Fo(;)14 b Fm(\000)p Fo(\014)686 1679 y Fn(2)682 1730 y(3)723 1710 y Fo(;)g Fm(\000)p Fo(\014)876 1679 y Fn(2)872 1730 y(4)914 1710 y Fo(;)g Fm(\000)p Fo(\014)1067 1679 y Fn(2)1063 1730 y(6)1104 1710 y Fm(g)p Fo(;)37 b(V)1273 1666 y Fn([4])1254 1732 y(3)1387 1710 y Fp(has)g Fm(f\000)p Fo(\014)1703 1679 y Fn(2)1699 1730 y(1)1740 1710 y Fo(;)14 b Fm(\000)p Fo(\014)1893 1679 y Fn(2)1889 1730 y(2)1930 1710 y Fo(;)g Fm(\000)p Fo(\014)2083 1679 y Fn(2)2079 1730 y(5)2120 1710 y Fo(;)g Fm(\000)p Fo(\014)2273 1679 y Fn(2)2269 1730 y(6)2310 1710 y Fm(g)p Fo(;)38 b Fp(and)g(\014nally)g Fo(V)2914 1666 y Fn([4])2895 1732 y(4)0 1809 y Fp(has)30 b Fm(f\000)p Fo(\014)309 1779 y Fn(2)305 1830 y(1)346 1809 y Fo(;)14 b Fm(\000)p Fo(\014)499 1779 y Fn(2)495 1830 y(3)536 1809 y Fo(;)g Fm(\000)p Fo(\014)689 1779 y Fn(2)685 1830 y(5)726 1809 y Fo(;)g Fm(\000)p Fo(\014)879 1779 y Fn(2)875 1830 y(6)916 1809 y Fm(g)p Fo(:)31 b 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(inverse)g(sc)l(attering)h(on)g(the)f(r)l(e)l(al)i(line)p Fs(,)c(Comm.)e(P)n(artial)i(Di\013eren)n(tial)145 3348 y(Equations)25 b Fb(10)e Fs(\(1985\),)i(677{766.)0 3431 y([7])70 b(V.)23 b(A.)g(Marc)n(henk)n(o,)h Fl(Sturm-Liouvil)t(le)h(op)l (er)l(ators)j(and)f(applic)l(ations)p Fs(,)e(Birkh\177)-35 b(auser,)23 b(Basel,)g(1986.)0 3514 y([8])70 b(K.)33 b(Chadan)i(and)g(P)-6 b(.)33 b(C.)h(Sabatier,)j Fl(Inverse)e(pr)l (oblems)h(in)f(quantum)h(sc)l(attering)f(the)l(ory)p Fs(,)h(2nd)f(ed.,)145 3597 y(Springer,)23 b(New)h(Y)-6 b(ork,)23 b(1989.)0 3680 y([9])70 b(T.)27 b(Aktosun)i(and)f(M.)f (Klaus,)h Fl(Chapter)i(2.2.4,)h(Inverse)e(the)l(ory:)41 b(pr)l(oblem)30 b(on)g(the)f(line)p Fs(,)f(Scattering,)145 3763 y(ed.)c(b)n(y)g(E.)f(R.)g(Pik)n(e)h(and)g(P)-6 b(.)23 b(C.)g(Sabatier,)h(Academic)g(Press,)e(London,)j(2001,)f(pp.)f (770{785.)0 3846 y([10])35 b(A.)26 b(Degasp)r(eris)h(and)g(P)-6 b(.)27 b(C.)f(Sabatier,)i Fl(Extension)g(of)h(the)f(one-dimensional)i (sc)l(attering)e(the)l(ory,)i(and)145 3929 y(ambiguities)p Fs(,)23 b(In)n(v)n(erse)i(Problems)d Fb(3)h Fs(\(1987\),)i(73{109.)0 4012 y([11])35 b(T.)27 b(Aktosun,)h(M.)e(Klaus,)i(and)f(C.)g(v)l(an)h (der)f(Mee,)g Fl(On)i(the)g(R)n(iemann-Hilb)l(ert)f(pr)l(oblem)i(for)f (the)g(one-)145 4096 y(dimensional)e(Schr\177)-36 b(odinger)27 b(e)l(quation)p Fs(,)c(J.)h(Math.)f(Ph)n(ys.)h Fb(34)f Fs(\(1993\),)i(2651{2690.)0 4179 y([12])35 b(P)-6 b(.)31 b(D.)g(Lax,)i Fl(Inte)l(gr)l(als)h(of)f(nonline)l(ar)h(e)l(quations)f (of)g(evolution)g(and)h(solitary)f(waves)p Fs(,)g(Comm.)c(Pure)145 4262 y(Appl.)23 b(Math.)h Fb(21)f Fs(\(1968\),)i(467{490.)0 4345 y([13])35 b(G.)24 b(L.)f(Lam)n(b,)f(Jr.,)h Fl(Elements)j(of)f (soliton)i(the)l(ory)p Fs(,)c(Wiley,)h(New)f(Y)-6 b(ork,)24 b(1980.)0 4428 y([14])35 b(M.)20 b(J.)g(Ablo)n(witz)h(and)h(H.)e (Segur,)h Fl(Solitons)j(and)g(the)e(inverse)h(sc)l(attering)f(tr)l (ansform)p Fs(,)h(SIAM,)d(Philadel-)145 4511 y(phia,)j(1981.)0 4594 y([15])35 b(M.)25 b(Zw)n(orski,)g Fl(Distribution)j(of)f(p)l(oles) i(for)f(sc)l(attering)f(on)h(the)g(r)l(e)l(al)h(line)p Fs(,)d(J.)f(F)-6 b(unct.)27 b(Anal.)f Fb(73)f Fs(\(1987\),)145 4677 y(277{296.)149 4859 y Ft(Dep)l(ar)l(tment)i(of)e(Ma)l(thema)l (tics)g(and)g(St)l(a)l(tistics,)f(Mississippi)i(St)l(a)l(te)e (University,)j(Missis-)0 4942 y(sippi)g(St)l(a)l(te,)e(Mississippi)i (39762)149 5025 y Fl(E-mail)g(addr)l(ess)5 b Fs(:)37 b Fa(aktosun@math.msstate.edu)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0402091934421--