Content-Type: multipart/mixed; boundary="-------------0110300901949" This is a multi-part message in MIME format. ---------------0110300901949 Content-Type: text/plain; name="01-402.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-402.comments" 30 pages, no figures. 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b(getting)f(that)f Fx(\020)1302 4937 y FC(2)1342 4915 y Fy(\()p Fx(k)s Fy(\))d Fu(\020)f Fx(k)1642 4882 y Fr(\000)p FC(3)1769 4915 y Fy(and)k Fx(\020)1995 4937 y FC(2)2034 4915 y Fy(\()p Fx(k)s Fy(\))e Fu(\020)e Fx(k)2335 4882 y Fr(\000)p FC(4)2462 4915 y Fy(as)k Fx(k)g Fu(!)c Fy(+)p Fu(1)p Fy(.)80 b(F)-8 b(or)34 b(the)f(\014rst)f(term,)-9 5028 y(w)m(e)61 b(notice)30 b(that)502 5264 y Fx(k)552 5227 y FC(3)591 5264 y Fx(\020)638 5286 y FC(1)677 5264 y Fy(\()p Fx(k)s Fy(\))c(=)f Fu(\000)p Fx(i)1036 5140 y Fw(Z)1127 5167 y FC(+)p Fr(1)1087 5347 y FC(0)1272 5136 y Fw(\022)1375 5203 y Fx(d)1422 5170 y FC(2)p 1349 5243 139 4 v 1349 5327 a Fx(dx)1448 5300 y FC(2)1498 5264 y Fx(e)1540 5227 y Fv(ik)r(x)1646 5136 y Fw(\023)1729 5163 y(\020)1783 5264 y Fx( )s Fy(\()p Fx(x)p Fy(\))c Fu(\000)f Fx( )t Fy(\(0\))31 b Fx(e)2330 5227 y Fr(\000)p Fv(x)2425 5203 y Fq(2)2484 5264 y Fu(\000)20 b Fx( )2637 5227 y Fr(0)2661 5264 y Fy(\(0\))31 b Fx(xe)2901 5227 y Fr(\000)p Fv(x)2996 5203 y Fq(2)3035 5163 y Fw(\021)3134 5264 y Fx(dx:)-9 5525 y Fy(By)36 b(in)m(tegrating)g(b)m(y)f(parts)h(t)m (wice)g(it)g(follo)m(ws)e(that)j Fu(j)p Fx(\020)1892 5547 y FC(1)1931 5525 y Fy(\()p Fx(k)s Fy(\))p Fu(j)f(\024)e Fx(C)7 b(k)2339 5492 y Fr(\000)p FC(3)2433 5525 y Fy(.)57 b(This)34 b(pro)m(v)m(es)i(\(5\))h(for)f Fx(m)e Fy(=)g(0.)58 b(By)-9 5638 y(di\013eren)m(tiating)h(explicitly)28 b(\(6\))j(and)f (using)f(similar)f(argumen)m(ts,)j(one)f(obtains)g(\(5\))i(for)e(an)m (y)g Fx(m)25 b Fu(2)g Ft(N)7 b Fy(.)p 3584 5638 61 61 v 1858 5887 a(4)p eop %%Page: 5 5 5 4 bop 132 444 a Fy(Using)30 b(this)h(lemma,)h(w)m(e)g(can)f(pro)m(v)m (e)i(the)f(F)-10 b(AS)31 b(theorem)h(for)f(the)h(p)s(oin)m(t-in)m (teraction)f(scattering)h(with-)-9 557 y(out)j(making)f Fs(ad)j(ho)-5 b(c)41 b Fy(assumptions)33 b(on)i(the)g(asymptotic)g (outgoing)g(state.)55 b(Although)34 b(the)h(statemen)m(t)i(is)-9 670 y(iden)m(tical)30 b(to)i([PT,)f(Theorem)g(1])h(w)m(e)g(presen)m(t)f (a)h(completely)f(di\013eren)m(t)g(pro)s(of,)g(whic)m(h)f(can)i(b)s(e)e (generalized)-9 783 y(to)h(the)f(case)i(of)e(p)s(oten)m(tial)g (scattering.)41 b(With)30 b(the)h(previous)e(notation,)i(our)e(result)h (is)f(the)i(follo)m(wing.)-9 994 y Fn(Theorem)i(2.2)47 b Fs(L)-5 b(et)44 b(us)h(\014x)g Fy(\011)1122 1008 y FC(0)1208 994 y Fu(2)h(S)7 b Fy(\()p Ft(R)1472 961 y FC(3)1517 994 y Fy(\))30 b Fu(\\)e(H)1748 1008 y Fv(ac)1820 994 y Fy(\()p Fx(H)1931 1008 y Fv(\015)t(;)p FC(y)2033 994 y Fy(\))p Fs(.)78 b(Then)45 b Fy(\011)2494 1008 y Fv(t)2570 994 y Fy(:=)i Fx(e)2755 961 y Fr(\000)p Fv(iH)2892 969 y Fo(\015)t(;)p Fq(y)2984 961 y Fv(t)3013 994 y Fy(\011)3084 1008 y FC(0)3168 994 y Fs(is)e(c)-5 b(ontinuously)-9 1107 y(di\013er)g(entiable)33 b(in)g Ft(R)704 1074 y FC(3)750 1107 y Fu(nf)p Fy(y)q Fu(g)h Fs(and)f(r)-5 b(elation)35 b(\(F)-9 b(AS\))32 b(holds)j(true,)d(for)h(every)g Fx(T)2644 1121 y FC(1)2709 1107 y Fu(2)25 b Ft(R)r Fs(.)-9 1319 y Fy(W)-8 b(e)32 b(fo)s(cus)f(on)g(the)g(pro)s(of)f(in)g(the)i(resonan) m(t)f(case)i Fx(\015)e Fy(=)c(0)k(\(see)i([PT)o(])f(for)f Fx(\015)g Fu(6)p Fy(=)c(0\).)44 b(T)-8 b(o)31 b(clarify)f(the)h (structure)-9 1432 y(of)g(the)g(pro)s(of,)g(w)m(e)g(will)e(decomp)s (ose)i(it)g(in)e(some)j(steps.)43 b(T)-8 b(o)31 b(streamline)f(the)h (exp)s(osition,)f(it)h(is)f(con)m(v)m(enien)m(t)-9 1545 y(to)h(in)m(tro)s(duce)e(the)i(follo)m(wing)d(notation.)-9 1757 y Fn(De\014nition)35 b(2.3)46 b Fs(Fix)28 b Fx(\027)j Fu(2)25 b Ft(R)1036 1771 y FC(+)1129 1757 y Fy(\()p Fs(We)j(ar)-5 b(e)29 b(inter)-5 b(este)g(d)29 b(in)f(c)-5 b(ases)28 b Fx(\027)j Fy(=)2396 1721 y FC(1)p 2396 1736 36 4 v 2396 1788 a(2)2469 1757 y Fs(and)e Fx(\027)i Fy(=)25 b(1\))p Fs(.)41 b(Consider)29 b(an)f(interval)-9 1891 y Fy([)p Fx(a;)15 b(b)p Fy(])28 b Fu(\022)f Ft(R)s Fs(.)51 b(We)34 b(say)g(that)h Fx(F)41 b Fy(:)28 b Ft(R)1159 1858 y FC(3)1226 1891 y Fu(\002)21 b Ft(R)33 b Fu(!)26 b Ft(C)57 b Fs(is)34 b(of)g(typ)-5 b(e)35 b Fu(O)2095 1909 y FC([)p Fv(a;b)p FC(])2226 1891 y Fy(\()2271 1846 y Fr(j)p FC(x)p Fr(j)p 2271 1870 77 4 v 2277 1922 a Fv(t)2302 1903 y Fo(\027)2357 1891 y Fy(\))g Fs(if)e(ther)-5 b(e)35 b(exist)f Fx(T)3010 1905 y FC(0)3077 1891 y Fx(>)27 b Fy(0)p Fs(,)34 b Fx(R)3351 1905 y FC(0)3418 1891 y Fx(>)28 b Fy(0)34 b Fs(such)-9 2004 y(that)1336 2146 y Fy(Sup)1214 2229 y Fr(j)p FC(x)p Fr(j\025)p Fv(R)1399 2238 y Fq(0)1433 2229 y Fv(;t)p Fr(\025)p Fv(T)1574 2238 y Fq(0)1634 2018 y Fw(\022)1711 2085 y Fu(j)p Fy(x)p Fu(j)p 1711 2125 99 4 v 1722 2209 a Fx(t)1755 2182 y Fv(\027)1820 2018 y Fw(\023)1887 2041 y Fv(\034)1945 2146 y Fu(j)p Fx(F)13 b Fy(\(x)p Fx(;)i(t)p Fy(\))p Fu(j)27 b(\024)e Fx(C)2445 2160 y Fv(\027)t(;\034)3654 2146 y Fy(\(7\))-9 2411 y Fs(for)33 b(e)-5 b(ach)33 b Fx(\034)i Fu(2)25 b Fy([)p Fx(a;)15 b(b)p Fy(])p Fs(.)43 b(In)33 b(this)g(c)-5 b(ase)33 b(we)g(write)g Fx(F)38 b Fy(=)25 b Fu(O)1867 2429 y FC([)p Fv(a;b)p FC(])1998 2411 y Fy(\()2043 2366 y Fr(j)p FC(x)p Fr(j)p 2043 2390 77 4 v 2049 2442 a Fv(t)2074 2423 y Fo(\027)2130 2411 y Fy(\))p Fs(.)132 2747 y Fy(If)k([)p Fx(a;)15 b(b)p Fy(])26 b(=)f([0)p Fx(;)15 b(n)p Fy(])31 b(\(with)d Fx(n)d Fu(2)g Ft(N)7 b Fy(\))36 b(w)m(e)30 b(write,)f(with)f(a)j(harmless)d(abuse)h(of)h(notation,)g Fx(F)39 b Fy(=)25 b Fu(O)3334 2761 y Fv(n)3381 2747 y Fy(\()3426 2702 y Fr(j)p FC(x)p Fr(j)p 3426 2726 V 3432 2778 a Fv(t)3457 2759 y Fo(\027)3513 2747 y Fy(\).)41 b(The)-9 2860 y(previous)28 b(de\014nitions)g(are)j(trivially)d (extended)i(to)h(the)g(case)g(of)g Ft(C)2297 2827 y Fv(d)2344 2860 y Fy(-v)-5 b(alued)30 b(functions.)-9 3072 y Fn(Preliminaries.)40 b Fy(Since)29 b Fu(F)969 3086 y FC(+)1059 3072 y Fy(sp)s(ectralize)h (the)g(hamiltonian)e Fx(H)2244 3086 y Fv(\015)t(;)p FC(y)2376 3072 y Fy(and)i Fu(F)2618 3086 y FC(+)2677 3072 y Fy(\011)2748 3086 y FC(0)2813 3072 y Fy(=)2919 3049 y Fw(b)2909 3072 y Fy(\011)2980 3086 y FC(out)3116 3072 y Fy(w)m(e)h(get)179 3321 y(\011)250 3335 y Fv(t)279 3321 y Fy(\(x\))84 b(=)635 3197 y Fw(Z)685 3403 y Fm(R)733 3384 y Fq(3)787 3321 y Fx(e)829 3283 y Fr(\000)p Fv(i)p FC(k)945 3260 y Fq(2)980 3283 y Fv(t)1020 3298 y Fw(b)1010 3321 y Fy(\011)1081 3335 y FC(out)1186 3321 y Fy(\(k\)\010)1370 3335 y FC(+)1430 3321 y Fy(\(x)p Fx(;)15 b Fy(k\))g(\(2)p Fx(\031)5 b Fy(\))1823 3283 y Fr(\000)p FC(3)p Fv(=)p FC(2)1988 3321 y Fx(dk)481 3581 y Fy(=)635 3457 y Fw(Z)685 3663 y Fm(R)733 3644 y Fq(3)787 3581 y Fx(e)829 3543 y Fr(\000)p Fv(i)p FC(k)945 3520 y Fq(2)980 3543 y Fv(t)1020 3558 y Fw(b)1010 3581 y Fy(\011)1081 3595 y FC(out)1186 3581 y Fy(\(k\))p Fx(e)1346 3543 y Fv(i)p FC(k)p Fr(\001)q FC(x)1485 3581 y Fy(\(2)p Fx(\031)t Fy(\))1656 3543 y Fr(\000)p FC(3)p Fv(=)p FC(2)1821 3581 y Fx(dk)24 b Fy(+)2030 3457 y Fw(Z)2080 3663 y Fm(R)2128 3644 y Fq(3)2182 3581 y Fx(e)2224 3543 y Fr(\000)p Fv(i)p FC(k)2340 3520 y Fq(2)2375 3543 y Fv(t)2415 3558 y Fw(b)2405 3581 y Fy(\011)2476 3595 y FC(out)2582 3581 y Fy(\(k\))2752 3519 y(1)p 2710 3560 130 4 v 2710 3643 a Fx(i)p Fu(j)p Fy(k)p Fu(j)2860 3519 y Fx(e)2902 3486 y Fr(\000)p Fv(i)p Fr(j)p FC(k)p Fr(jj)p FC(x)p Fr(j)p 2860 3560 279 4 v 2950 3643 a Fu(j)p Fy(x)p Fu(j)3148 3581 y Fy(\(2)p Fx(\031)t Fy(\))3319 3543 y Fr(\000)p FC(3)p Fv(=)p FC(2)3484 3581 y Fx(dk)481 3767 y Fu(\021)83 b Fx(\013)p Fy(\(x)p Fx(;)15 b(t)p Fy(\))21 b(+)f Fx(\014)5 b Fy(\(x)p Fx(;)15 b(t)p Fy(\))g Fx(:)-9 3971 y Fy(Then)29 b(the)h(probabilit)m(y)e(densit)m(y)i(curren)m(t)g (is)1025 4174 y(j)1053 4136 y FC(\011)1108 4144 y Fo(t)1165 4174 y Fy(=)25 b(Im)o(\()p Fx(\013)1463 4136 y Fr(\003)1503 4174 y Fu(r)o Fx(\013)c Fy(+)f Fx(\013)1807 4136 y Fr(\003)1846 4174 y Fu(r)p Fx(\014)25 b Fy(+)20 b Fx(\014)2145 4136 y Fr(\003)2185 4174 y Fu(r)o Fx(\013)h Fy(+)f Fx(\014)2486 4136 y Fr(\003)2526 4174 y Fu(r)o Fx(\013)q Fy(\))p Fx(:)934 b Fy(\(8\))-9 4377 y(The)43 b(\014rst)g(term)h(j)651 4391 y FC(0)738 4377 y Fy(=)j(Im\()p Fx(\013)1058 4344 y Fr(\003)1098 4377 y Fu(r)p Fx(\013)p Fy(\))d(corresp)s(onds)f(to)h (the)g(free)g(ev)m(olution)f(of)h(\011)2901 4391 y FC(out)3007 4377 y Fy(,)k(so)c(using)e(the)i(free)-9 4490 y(\015ux-across-surfaces) 30 b(theorem)h([DDGZ)1425 4504 y FC(1)1465 4490 y Fy(])g(one)f(has)887 4749 y(lim)853 4810 y Fv(R)p Fr(!1)1063 4626 y Fw(Z)1154 4652 y FC(+)p Fr(1)1113 4832 y Fv(T)1298 4749 y Fx(dt)1393 4626 y Fw(Z)1444 4832 y FC(\006)1495 4843 y Fo(R)1565 4749 y Fy(j)1593 4763 y FC(0)1632 4749 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))p Fu(\001)q Fy(n)g Fx(d\033)29 b Fy(=)2139 4626 y Fw(Z)2189 4832 y Fv(C)5 b FC(\(\006\))2369 4749 y Fu(j)2404 4726 y Fw(b)2394 4749 y Fy(\011)2465 4763 y FC(out)2571 4749 y Fy(\(k\))p Fu(j)2714 4712 y FC(2)2770 4749 y Fx(dk)18 b(:)-9 5010 y Fy(Therefore,)30 b(to)h(pro)m(v)m(e)g (Theorem)f(2.2)i(what)e(remains)f(to)i(sho)m(w)g(is)e(that)1216 5264 y(lim)1182 5325 y Fv(R)p Fr(!1)1392 5141 y Fw(Z)1483 5167 y FC(+)p Fr(1)1442 5347 y Fv(T)1627 5264 y Fx(dt)1722 5141 y Fw(Z)1773 5347 y FC(\006)1824 5358 y Fo(R)1894 5264 y Fu(j)p Fy(j)1947 5278 y FC(1)1986 5264 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))p Fu(\001)q Fy(n)p Fu(j)g Fx(d\033)29 b Fy(=)c(0)1091 b(\(9\))-9 5525 y(where)34 b(j)286 5539 y FC(1)358 5525 y Fy(:=)f(Im)o(\()p Fx(\013)689 5492 y Fr(\003)728 5525 y Fu(r)p Fx(\014)c Fy(+)23 b Fx(\014)1034 5488 y Fr(\003)1073 5525 y Fu(r)p Fx(\013)h Fy(+)f Fx(\014)1381 5488 y Fr(\003)1420 5525 y Fu(r)p Fx(\014)5 b Fy(\).)55 b(In)34 b(order)g(to)i(pro)m(v)m(e)g(\(9\))f(w)m(e)h(need)e(estimates)i (on)f Fx(\013;)15 b(\014)41 b Fy(and)-9 5638 y(their)29 b(gradien)m(ts.)1858 5887 y(5)p eop %%Page: 6 6 6 5 bop -9 444 a Fn(Estimates)44 b(on)j Fx(\013)g Fn(and)g Fu(r)p Fx(\013)p Fn(.)91 b Fy(First)40 b(of)h(all,)h(w)m(e)f(decomp)s (ose)g Fx(\013)g Fy(as)g Fx(\013)2615 458 y FC(reg)2741 444 y Fy(+)27 b Fx(\013)2898 458 y FC(sing)3065 444 y Fy(b)m(y)41 b(extracting)g(the)-9 567 y(singular)28 b(part)i(of)646 545 y Fw(b)636 567 y Fy(\011)707 581 y FC(out)812 567 y Fy(,)h(whic)m(h)e(can)i(b)s(e)e(read)i(from)f(\(4\).)41 b(More)31 b(precisely)-8 b(,)30 b(w)m(e)h(p)s(ose)606 811 y Fx(f)651 825 y FC(1)690 811 y Fy(\(k\))26 b(:=)965 788 y Fw(b)955 811 y Fy(\011)1026 825 y FC(out)1132 811 y Fy(\(k\))21 b Fu(\000)1399 749 y Fx(r)p 1372 790 99 4 v 1372 873 a Fu(j)p Fy(k)p Fu(j)1480 811 y Fx(e)1522 773 y Fr(\000j)p FC(k)p Fr(j)1654 750 y Fq(2)1874 811 y Fy(where)182 b Fx(r)28 b Fy(:=)d Fx(i)2525 687 y Fw(Z)2576 893 y Fm(R)2624 874 y Fq(3)2714 749 y Fy(1)p 2688 790 V 2688 873 a Fu(j)p Fy(x)p Fu(j)2796 811 y Fy(\011)2867 825 y FC(0)2906 811 y Fy(\(x\))31 b Fx(dx)454 b Fy(\(10\))-9 1055 y(and)29 b(then)h(w)m(e)h(de\014ne)886 1276 y Fx(\013)944 1290 y FC(reg)1044 1276 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))83 b(=)1472 1152 y Fw(Z)1523 1358 y Fm(R)1571 1339 y Fq(3)1625 1276 y Fx(e)1667 1238 y Fv(i)p FC(k)p Fr(\001)p FC(x)1789 1276 y Fx(e)1831 1238 y Fr(\000)p Fv(i)p FC(k)1947 1215 y Fq(2)1982 1238 y Fv(t)2027 1276 y Fx(f)2072 1290 y FC(1)2111 1276 y Fy(\(k\)\(2)p Fx(\031)5 b Fy(\))2401 1238 y Fr(\000)p FC(3)p Fv(=)p FC(2)2566 1276 y Fx(dk)858 1515 y(\013)916 1529 y FC(sing)1044 1515 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))83 b(=)g Fu(\000)1558 1391 y Fw(Z)1609 1597 y Fm(R)1657 1578 y Fq(3)1710 1515 y Fx(e)1752 1477 y Fv(i)p FC(k)p Fr(\001)q FC(x)1875 1515 y Fx(e)1917 1477 y Fr(\000)p Fv(i)p FC(k)2033 1454 y Fq(2)2068 1477 y Fv(t)2135 1453 y Fx(r)p 2108 1494 V 2108 1577 a Fu(j)p Fy(k)p Fu(j)2216 1515 y Fx(e)2258 1477 y Fr(\000j)p FC(k)p Fr(j)2390 1454 y Fq(2)2444 1515 y Fy(\(2)p Fx(\031)t Fy(\))2615 1477 y Fr(\000)p FC(3)p Fv(=)p FC(2)2780 1515 y Fx(dk)s(:)-9 1759 y Fy(Moreo)m(v)m(er,)954 1908 y Fu(r)p Fx(\013)p Fy(\(x)p Fx(;)15 b(t)p Fy(\))27 b(=)e Fx(i)1448 1785 y Fw(Z)1499 1991 y Fm(R)1547 1972 y Fq(3)1600 1908 y Fx(e)1642 1871 y Fv(i)p FC(k)p Fr(\001)q FC(x)1765 1908 y Fx(e)1807 1871 y Fr(\000)p Fv(ik)1925 1847 y Fq(2)1959 1871 y Fv(t)1989 1908 y Fy(k)2047 1885 y Fw(b)2037 1908 y Fy(\011)2108 1922 y FC(out)2214 1908 y Fy(\(k\))15 b(\(2)p Fx(\031)t Fy(\))2518 1871 y Fr(\000)p FC(3)p Fv(=)p FC(2)2684 1908 y Fx(dk)s(:)-9 2115 y Fy(The)26 b(prop)s(erties)g(of)i Fx(\013)758 2129 y FC(reg)885 2115 y Fy(and)f Fu(r)p Fx(\013)g Fy(are)h(giv)m(en)g(in)e(the)h(follo)m (wing)f(lemma,)i(whic)m(h)e(will)f(b)s(e)i(useful)e(also)j(in)e(the)-9 2228 y(general)k(case)h(of)g(p)s(oten)m(tial)f(scattering.)-9 2535 y Fn(Lemma)i(2.4)k(\(F)-9 b(ree)34 b(ev)m(olution)i(of)f(a)f(slo)m (w-decreasing)i(state\))44 b Fs(L)-5 b(et)24 b(us)f(supp)-5 b(ose)25 b(that)g Fx(f)34 b Fu(2)25 b Fx(C)3501 2502 y FC(5)3540 2535 y Fy(\()p Ft(R)3635 2502 y FC(3)3681 2535 y Fu(nf)p Fy(0)p Fu(g)p Fy(\))-9 2648 y Fs(satis\014es)33 b(the)g(fol)5 b(lowing)34 b(assumptions:)89 2857 y(a\))46 b Fl(r)-5 b(e)g(gularity)41 b(in)h(a)f(neighb)-5 b(orho)g(o)g(d)40 b(of)h(the)g(origin:)49 b Fs(ther)-5 b(e)37 b(exists)f(a)g(suitable)g (punctur)-5 b(e)g(d)37 b(neigh-)218 2970 y(b)-5 b(orho)g(o)g(d)36 b Fx(U)611 2984 y FC(0)683 2970 y Fs(of)d(the)g(origin)g(such)g(that) 1707 3171 y Fx(@)1760 3134 y Fv(\026)1807 3171 y Fx(f)h Fu(2)25 b Fx(L)2034 3134 y Fr(1)2109 3171 y Fy(\()p Fx(U)2206 3185 y FC(0)2246 3171 y Fy(\))1327 b(\(11\))218 3372 y Fs(for)33 b(every)f(multi-index)66 b Fx(\026)25 b Fu(2)g Ft(N)1340 3339 y FC(3)1418 3372 y Fs(with)34 b Fy(1)26 b Fu(\024)f(j)p Fx(\026)p Fu(j)g(\024)g Fy(5;)93 3559 y Fs(b\))46 b Fl(de)-5 b(cr)g(e)g(ase)34 b(at)h(in\014nity:)42 b Fs(for)30 b(every)g(multi-index)60 b Fx(\026)26 b Fu(2)e Ft(N)2266 3526 y FC(3)2341 3559 y Fs(with)31 b Fy(1)26 b Fu(\024)f(j)p Fx(\026)p Fu(j)g(\024)g Fy(5)31 b Fs(ther)-5 b(e)31 b(exists)f(p)-5 b(ositive)218 3672 y Fx(C)283 3686 y Fv(\026)362 3672 y Fs(and)33 b Fx(K)615 3686 y Fv(\026)695 3672 y Fs(such)g(that)1267 3831 y Fu(j)p Fx(@)1345 3793 y Fv(\026)1392 3831 y Fx(f)10 b Fy(\(k\))p Fu(j)26 b(\024)1803 3769 y Fx(C)1868 3783 y Fv(\026)p 1722 3810 275 4 v 1722 3897 a Fu(j)p Fy(k)p Fu(j)1820 3871 y FC(3+)p Fr(j)p Fv(\026)p Fr(j)2192 3831 y Fs(for)48 b Fu(j)p Fy(k)p Fu(j)26 b(\025)f Fx(K)2649 3845 y Fv(\026)2696 3831 y Fx(:)887 b Fy(\(12\))132 4088 y Fs(Then)899 4228 y Fx(\013)957 4243 y Fv(f)1003 4228 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))26 b Fu(\021)1316 4104 y Fw(Z)1366 4310 y Fm(R)1414 4291 y Fq(3)1468 4228 y Fx(e)1510 4190 y Fv(i)p FC(k)p Fr(\001)p FC(x)1633 4228 y Fx(e)1675 4190 y Fr(\000)p Fv(i)p FC(k)1791 4167 y Fq(2)1826 4190 y Fv(t)1855 4228 y Fx(f)10 b Fy(\(k\))33 b Fx(dk)c Fy(=)2338 4166 y(1)p 2290 4207 143 4 v 2290 4294 a Fx(t)2323 4268 y FC(3)p Fv(=)p FC(2)2442 4228 y Fu(O)2514 4242 y FC(1)2569 4100 y Fw(\022)2646 4166 y Fu(j)p Fy(x)p Fu(j)p 2646 4207 99 4 v 2679 4290 a Fx(t)2755 4100 y Fw(\023)2837 4228 y Fx(:)746 b Fy(\(13\))-9 4495 y Fn(Pro)s(of)44 b(of)g(lemma.)62 b Fy(Since)37 b Fx(f)47 b Fu(2)38 b Fx(L)1309 4462 y FC(2)1348 4495 y Fy(\()p Ft(R)1443 4462 y FC(3)1489 4495 y Fy(\))g(as)g(a)h(consequence)g(of)f(h)m(yp)s(otheses)g(\(a\))2983 4509 y Fv(\026)r FC(=0)3159 4495 y Fy(and)f(\(b\))3464 4509 y Fv(\026)q FC(=0)3601 4495 y Fy(,)k(w)m(e)-9 4608 y(obtain)29 b(that)771 4794 y Fx(\013)829 4809 y Fv(f)875 4794 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))26 b(=)f(\()p Fx(e)1265 4756 y Fr(\000)p Fv(iH)1402 4765 y Fq(0)1437 4756 y 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5352 y(\014)1212 5375 y Fu(\024)1336 5314 y Fx(C)1401 5328 y Fv(m)p 1318 5354 167 4 v 1318 5437 a Fu(j)p Fy(y)q Fu(j)1417 5411 y Fv(m)1676 5375 y Fy(for)30 b Fx(m)c Fy(=)e(2)p Fx(;)15 b(:)g(:)g(:)j(;)d Fy(5)31 b(and)e(y)e Fu(2)e Ft(R)2736 5338 y FC(3)2782 5375 y Fx(;)801 b Fy(\(15\))1080 5638 y Fx(@)1128 5652 y Fv(j)1215 5614 y Fy(\024)1195 5638 y Fx(f)35 b Fu(2)24 b Fx(L)1422 5600 y FC(1)1462 5638 y Fy(\()p Ft(R)1557 5600 y FC(3)1602 5638 y Fy(\))182 b(for)30 b(ev)m(ery)i Fx(j)f Fy(=)25 b(1)p Fx(;)15 b(:)g(:)g(:)i(;)e Fy(3)p Fx(:)928 b Fy(\(16\))1858 5887 y(6)p eop %%Page: 7 7 7 6 bop -9 444 a Fy(The)29 b(b)s(ound)g(\(15\))j(can)e(b)s(e)g(pro)m(v) m(ed)h(b)m(y)f(observing)f(that)1175 686 y Fx(e)1217 649 y Fv(i)p FC(k)p Fr(\001)p FC(y)1366 686 y Fy(=)c(\()p Fu(\000)p Fx(i)p Fy(\))1634 649 y Fv(m)1771 625 y Fy(1)p 1711 665 167 4 v 1711 748 a Fu(j)p Fy(y)q Fu(j)1810 722 y Fv(m)1902 558 y Fw(\022)2004 625 y Fy(y)p 1979 665 100 4 v 1979 748 a Fu(j)p Fy(y)q Fu(j)2109 686 y(\001)20 b(r)2230 701 y FC(k)2272 558 y Fw(\023)2339 581 y Fv(m)2420 686 y Fx(e)2462 649 y Fv(i)p FC(k)p Fr(\001)p FC(y)3608 686 y Fy(\(17\))-9 926 y(so)30 b(that)h(\(b)m(y)g(in)m(tegration)f(b)m (y)g(parts)g(and)g(a)h(standard)f(densit)m(y)f(argumen)m(t\))i(w)m(e)g (get)1027 1144 y(\024)1008 1168 y Fx(f)9 b Fy(\(y\))26 b(=)f Fx(i)1333 1130 y Fv(m)1470 1106 y Fy(1)p 1410 1147 167 4 v 1410 1230 a Fu(j)p Fy(y)q Fu(j)1509 1204 y Fv(m)1601 1044 y Fw(Z)1652 1250 y Fm(R)1700 1231 y Fq(3)1753 1168 y Fx(e)1795 1130 y Fv(i)p FC(k)p Fr(\001)p FC(y)1934 1040 y Fw(\022)2036 1106 y Fy(y)p 2011 1147 100 4 v 2011 1230 a Fu(j)p Fy(y)q Fu(j)2141 1168 y(\001)20 b(r)2262 1183 y FC(k)2304 1040 y Fw(\023)2371 1062 y Fv(m)2452 1168 y Fx(f)10 b Fy(\(k\))31 b Fx(dk)858 b Fy(\(18\))-9 1446 y(Our)29 b(assumptions)g(imply)g(that)1148 1345 y Fw(\020)1232 1405 y FC(y)p 1212 1425 78 4 v 1212 1479 a Fr(j)p FC(y)q Fr(j)1320 1446 y Fu(\001)20 b(r)1441 1461 y FC(k)1483 1345 y Fw(\021)1537 1368 y Fv(m)1619 1446 y Fx(f)35 b Fu(2)26 b Fx(L)1848 1413 y FC(1)1887 1446 y Fy(\()p Ft(R)1982 1413 y FC(3)2027 1446 y Fy(\))31 b(for)g Fx(m)26 b Fy(=)g(2)p Fx(;)15 b(:)g(:)g(:)i(;)e Fy(5,)32 b(so)f(that)g(\(15\))i(follo)m(ws)c(from)-9 1577 y(the)h(Riemann-Leb)s(esgue)f(lemma.)132 1690 y(A)j(similar)d (argumen)m(t)j(sho)m(ws)f(that)1401 1613 y Fw(\014)1401 1667 y(\014)1431 1690 y Fx(@)1479 1704 y Fv(j)1565 1666 y Fy(\024)1546 1690 y Fx(f)9 b Fy(\(y\))1718 1613 y Fw(\014)1718 1667 y(\014)1777 1690 y Fu(\024)1913 1654 y Fv(C)p 1885 1669 113 4 v 1885 1724 a Fr(j)p FC(y)q Fr(j)1963 1705 y Fq(4)2038 1690 y Fy(for)32 b(ev)m(ery)g(y)d Fu(2)e Ft(R)2645 1657 y FC(3)2690 1690 y Fx(:)64 b Fy(As)32 b(for)f(the)h(lo)s(cal)f(b)s(eha)m(vior)-9 1830 y(the)f(fact)h(that)g Fx(k)570 1844 y Fv(j)607 1830 y Fx(f)k Fu(2)25 b Fx(L)835 1797 y FC(2)904 1830 y Fy(implies)j(that)j Fx(@)1458 1844 y Fv(j)1514 1806 y Fy(\024)1495 1830 y Fx(f)j Fu(2)25 b Fx(L)1722 1797 y FC(2)1787 1830 y Fu(\022)f Fx(L)1944 1797 y FC(1)1944 1857 y(lo)r(c)2067 1830 y Fx(:)31 b Fy(This)d(completes)j(the)f(pro)s(of)g(of)h(\(16\).)132 1987 y(The)f(claim)f(\(13\))j(means)e(that)h(for)f(ev)m(ery)h Fx(\034)36 b Fu(2)25 b Fy([0)p Fx(;)15 b Fy(1])32 b(there)e(exists)g Fx(C)2529 2001 y Fv(\034)2598 1987 y Fx(>)25 b Fy(0)30 b(suc)m(h)g(that)1397 2101 y Fw(\022)1474 2168 y Fu(j)p Fy(x)p Fu(j)p 1474 2209 99 4 v 1507 2292 a Fx(t)1583 2101 y Fw(\023)1650 2124 y Fv(\034)1708 2230 y Fu(j)p Fx(\013)1792 2245 y Fv(f)1837 2230 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))p Fu(j)26 b(\024)2203 2168 y Fx(C)2268 2182 y Fv(\034)p 2185 2209 143 4 v 2185 2296 a Fx(t)2218 2269 y FC(3)p Fv(=)p FC(2)2338 2230 y Fx(:)1245 b Fy(\(19\))-9 2469 y(In)29 b(the)i(case)g Fx(\034)k Fy(=)25 b(0,)31 b(b)m(y)g(using)e(the)h(represen)m(tation)h(\(14\))h(and)d(the)i(b)s (ound)d(\(15\),)k(w)m(e)f(get)638 2701 y Fu(j)p Fx(\013)722 2716 y Fv(f)767 2701 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))p Fu(j)84 b(\024)1266 2640 y Fx(C)p 1231 2680 V 1231 2767 a(t)1264 2741 y FC(3)p Fv(=)p FC(2)1399 2577 y Fw(Z)1449 2783 y Fm(R)1497 2765 y Fq(3)1551 2624 y Fw(\014)1551 2678 y(\014)1601 2677 y Fy(\024)1581 2701 y Fx(f)10 b Fy(\(y\))1754 2624 y Fw(\014)1754 2678 y(\014)1830 2701 y Fx(dy)1067 2976 y Fu(\024)1266 2914 y Fx(C)p 1231 2955 V 1231 3042 a(t)1264 3016 y FC(3)p Fv(=)p FC(2)1399 2820 y Fw(\()1472 2852 y(Z)1522 3058 y Fv(B)1575 3067 y Fq(1)1610 3058 y FC(\(0\))1747 2914 y Fx(C)1812 2928 y FC(2)p 1730 2955 140 4 v 1730 3038 a Fu(j)p Fy(y)q Fu(j)1829 3012 y FC(2)1909 2976 y Fx(dy)24 b Fy(+)2115 2852 y Fw(Z)2166 3058 y Fm(R)2214 3039 y Fq(3)2249 3058 y Fr(n)p Fv(B)2337 3067 y Fq(1)2372 3058 y FC(\(0\))2508 2914 y Fx(C)2573 2928 y FC(4)p 2491 2955 V 2491 3038 a Fu(j)p Fy(y)q Fu(j)2590 3012 y FC(4)2670 2976 y Fx(dy)2765 2820 y Fw(\))2864 2976 y Fu(\024)2994 2914 y Fx(C)3066 2881 y Fr(0)p 2970 2955 143 4 v 2970 3042 a Fx(t)3003 3016 y FC(3)p Fv(=)p FC(2)-9 3243 y Fy(In)29 b(the)i(case)g Fx(\034)k Fy(=)25 b(1,)31 b(b)m(y)g(using)e(\(14\))i(and)f(\(17\))i(w)m(e)f(obtain)786 3467 y Fu(j)p Fy(x)p Fu(j)p 786 3507 99 4 v 819 3591 a Fx(t)895 3528 y(\013)953 3543 y Fv(f)999 3528 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))26 b(=)1365 3467 y Fu(\000)p Fx(ie)1509 3431 y Fv(i)1543 3404 y Fq(x)1575 3383 y(2)p 1543 3416 67 3 v 1549 3457 a(2)p Fo(t)p 1322 3507 346 4 v 1322 3594 a Fy(\(4)p Fx(\031)t(it)p Fy(\))1557 3568 y FC(3)p Fv(=)p FC(2)1692 3404 y Fw(Z)1743 3611 y Fm(R)1791 3592 y Fq(3)1875 3528 y Fx(e)1917 3491 y Fr(\000)p Fv(i)2006 3460 y Fq(x)p Fk(\001)p Fq(y)p 2006 3476 84 3 v 2036 3517 a Fo(t)2119 3400 y Fw(\022)2221 3467 y Fy(x)p 2196 3507 99 4 v 2196 3591 a Fu(j)p Fy(x)p Fu(j)2325 3528 y(\001)20 b(r)2446 3400 y Fw(\023)2528 3528 y Fx(e)2570 3491 y Fv(i)2604 3460 y Fq(y)2637 3439 y(2)p 2604 3476 68 3 v 2611 3517 a(2)p Fo(t)2705 3504 y Fy(\024)2686 3528 y Fx(f)9 b Fy(\(y\))31 b Fx(dy)-9 3765 y Fy(and)e(then)747 3878 y Fu(j)p Fy(x)p Fu(j)p 747 3918 99 4 v 780 4002 a Fx(t)871 3939 y Fu(j)p Fx(\013)955 3954 y Fv(f)1000 3939 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))p Fu(j)26 b(\024)1384 3878 y Fx(C)p 1348 3918 143 4 v 1348 4006 a(t)1381 3979 y FC(3)p Fv(=)p FC(2)1564 3826 y(3)1516 3853 y Fw(X)1520 4047 y Fv(r)r FC(=1)1663 3816 y Fw(Z)1713 4022 y Fm(R)1761 4003 y Fq(3)1815 3808 y Fw(\014)1815 3862 y(\014)1815 3917 y(\014)1815 3971 y(\014)1855 3878 y Fy(1)p 1855 3918 46 4 v 1861 4002 a Fx(t)1911 3939 y(y)1956 3953 y Fv(r)1993 3939 y Fx(e)2035 3902 y Fv(i)2069 3871 y Fq(y)2102 3850 y(2)p 2070 3887 68 3 v 2077 3928 a(2)p Fo(t)2171 3915 y Fy(\024)2151 3939 y Fx(f)10 b Fy(\(y\))20 b(+)g Fx(e)2477 3902 y Fv(i)2511 3871 y Fq(y)2544 3850 y(2)p 2512 3887 V 2519 3928 a(2)p Fo(t)2593 3939 y Fx(@)2641 3953 y Fv(r)2699 3915 y Fy(\024)2679 3939 y Fx(f)10 b Fy(\(y\))2852 3808 y Fw(\014)2852 3862 y(\014)2852 3917 y(\014)2852 3971 y(\014)2928 3939 y Fx(dy)588 b Fy(\(20\))-9 4193 y(As)31 b(a)h(consequence)g(of)g(\(15\))h(and)e(\(16\))i(w)m(e)f (ha)m(v)m(e)h(that)f Fx(y)1975 4207 y Fv(r)2032 4169 y Fy(\024)2012 4193 y Fx(f)41 b Fy(and)31 b Fx(@)2324 4207 y Fv(r)2382 4169 y Fy(\024)2362 4193 y Fx(f)41 b Fy(b)s(elong)31 b(to)h 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y(')2127 5463 y Fw(\022)2310 5530 y Fx(i)p Fu(j)p Fy(x)p Fu(j)p 2203 5571 342 4 v 2203 5661 a Fy(2)2248 5589 y Fu(p)p 2325 5589 221 4 v 2325 5661 a Fy(1)g(+)g Fx(it)2555 5463 y Fw(\023)1858 5887 y Fy(7)p eop %%Page: 8 8 8 7 bop -9 444 a Fy(where)29 b Fx(')p Fy(\()p Fx(z)t Fy(\))e(:=)e Fx(e)618 411 y Fv(z)654 388 y Fq(2)708 444 y Fy(\(erfc\()p Fx(z)t Fy(\))c Fu(\000)f Fy(erfc)q(\()p Fu(\000)p Fx(z)t Fy(\)\))30 b(for)g Fx(z)g Fu(2)24 b Ft(C)18 b Fx(:)36 b Fy(T)-8 b(o)31 b(pro)m(v)m(e)f(\(22\))i(it)d(is)g (then)h(su\016cien)m(t)f(to)i(sho)m(w)e(that)1509 638 y(Sup)1416 730 y FC(arg)q(\()p Fv(z)s FC(\))p Fv(<)1670 703 y Fq(3)p 1670 715 31 3 v 1670 756 a(4)1711 730 y Fv(\031)1779 638 y Fy(\()p Fx(z)1860 601 y Fv(\034)1904 638 y Fx(')p Fy(\()p Fx(z)t Fy(\)\))e Fu(\024)e Fx(C)2302 652 y Fv(\034)3608 638 y Fy(\(23\))-9 938 y(for)37 b(ev)m(ery)i Fx(\034)48 b Fu(2)38 b Fy([)p Fu(\000)p Fy(1)p Fx(;)15 b Fy(1].)913 905 y FC(\()5 b(2)t(\))1054 938 y Fy(F)-8 b(or)39 b Fx(\034)48 b Fu(2)38 b Fy([)p Fu(\000)p Fy(1)p Fx(;)15 b Fy(0])39 b(the)f(b)s(ound)e(\(23\))k(is)d(trivially)f(true,)k (since)d Fx(')h Fy(has)g(a)h(\014rst)-9 1051 y(order)29 b(zero)h(in)e Fx(z)i Fy(=)25 b(0)30 b(and)e(is)h(b)s(ounded)e(at)k (in\014nit)m(y)c(in)h(the)i(sp)s(eci\014ed)e(region.)40 b(F)-8 b(or)30 b Fx(\034)35 b Fu(2)25 b Fy(\(0)p Fx(;)15 b Fy(1],)32 b(one)e(notices)-9 1164 y(that)j(the)f(asymptotic)h (expansion)f(of)g(the)h(error)f(function)f(\(in)h(the)h(sp)s(eci\014ed) e(region\))h(assure)g(that)h Fx(z)t(')q Fy(\()p Fx(z)t Fy(\))-9 1277 y(is)c(b)s(ounded)f(at)j(in\014nit)m(y)-8 b(.)-9 1515 y Fn(Estimates)46 b(on)i Fx(\014)53 b Fn(and)48 b Fu(r)o Fx(\014)6 b(:)91 b Fy(W)-8 b(e)43 b(turn)d(no)m(w)i(to)g(the)g (estimates)g(on)g Fx(\014)k Fy(and)c Fu(r)o Fx(\014)5 b Fy(.)75 b(As)42 b(b)s(efore,)i(it)d(is)-9 1628 y(con)m(v)m(enien)m(t) 31 b(to)g(extract)h(the)f(singular)d(part)i(of)1674 1605 y Fw(b)1664 1628 y Fy(\011)1735 1642 y FC(out)1840 1628 y Fy(.)41 b(W)-8 b(e)32 b(p)s(ose)1073 1855 y Fx(f)1118 1869 y FC(3)1157 1855 y Fy(\(k\))26 b(:=)1459 1794 y(1)p 1432 1834 99 4 v 1432 1918 a Fu(j)p Fy(k)p Fu(j)1551 1832 y Fw(b)1541 1855 y Fy(\011)1612 1869 y FC(out)1717 1855 y Fy(\(k\))21 b Fu(\000)2004 1794 y Fx(r)p 1957 1834 138 4 v 1957 1918 a Fu(j)p Fy(k)p Fu(j)2055 1891 y FC(2)2105 1855 y Fx(e)2147 1818 y Fr(\000)p Fv(k)2241 1794 y Fq(2)2300 1855 y Fu(\000)2432 1794 y Fx(c)p 2401 1834 103 4 v 2401 1918 a Fu(j)p Fy(k)p Fu(j)2513 1855 y Fx(e)2555 1818 y Fr(\000)p Fv(k)2649 1794 y Fq(2)3608 1855 y Fy(\(24\))-9 2098 y(where)41 b Fx(r)48 b Fu(2)d Ft(C)66 b Fy(has)42 b(b)s(een)f(de\014ned)g(in)g(\(10\),)48 b(and)41 b Fx(c)46 b Fu(2)f Ft(C)65 b Fy(is)42 b(the)g(zeroth)h(order)f (term)g(in)f(the)i(Lauren)m(t)-9 2211 y(expansion)29 b(of)557 2188 y Fw(b)547 2211 y Fy(\011)618 2225 y FC(out)723 2211 y Fy(;)i(then)f(w)m(e)h(decomp)s(ose)f Fx(\014)36 b Fy(as)31 b Fx(\014)1832 2233 y FC(sing)q Fv(;)p FC(2)2034 2211 y Fy(+)20 b Fx(\014)2181 2233 y FC(sing)q Fv(;)p FC(1)2383 2211 y Fy(+)g Fx(\014)2530 2233 y FC(reg)2660 2211 y Fy(where)769 2537 y Fx(\014)826 2559 y FC(sing)p Fv(;)p FC(2)1007 2537 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))84 b Fu(\021)1446 2476 y(\000)p Fx(i)p 1446 2516 V 1448 2600 a Fu(j)p Fy(x)p Fu(j)1573 2414 y Fw(Z)1624 2620 y Fm(R)1672 2601 y Fq(3)1726 2537 y Fx(e)1768 2500 y Fr(\000)p Fv(i)p FC(k)1884 2476 y Fq(2)1919 2500 y Fv(t)2005 2476 y Fx(r)p 1958 2516 138 4 v 1958 2600 a Fu(j)p Fy(k)p Fu(j)2056 2573 y FC(2)2106 2537 y Fx(e)2148 2500 y Fr(\000)p FC(k)2240 2476 y Fq(2)2279 2537 y Fx(e)2321 2500 y Fr(\000)p Fv(i)p Fr(j)p FC(k)p Fr(jj)p FC(x)p Fr(j)2558 2537 y Fy(\(2)p Fx(\031)t Fy(\))2729 2500 y Fr(\000)p FC(3)p Fv(=)p FC(2)2894 2537 y Fx(dk)620 b Fy(\(25\))769 2779 y Fx(\014)826 2801 y FC(sing)p Fv(;)p FC(1)1007 2779 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))84 b Fu(\021)1446 2718 y(\000)p Fx(i)p 1446 2758 103 4 v 1448 2842 a Fu(j)p Fy(x)p Fu(j)1573 2656 y Fw(Z)1624 2862 y Fm(R)1672 2843 y Fq(3)1726 2779 y Fx(e)1768 2742 y Fr(\000)p Fv(i)p FC(k)1884 2718 y 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(exactly)g(b)m(y)f(gaussian)g(in)m(tegration,)g(obtaining)633 3498 y Fx(\014)690 3520 y FC(sing)q Fv(;)p FC(2)897 3498 y Fy(=)1029 3437 y(1)p 1003 3477 99 4 v 1003 3561 a Fu(j)p Fy(x)p Fu(j)1153 3437 y Fy(1)p 1121 3477 109 4 v 1121 3496 a Fu(p)p 1197 3496 33 4 v 73 x Fx(t)1240 3498 y Fu(O)1312 3512 y FC(1)1367 3370 y Fw(\022)1449 3437 y Fu(j)p Fy(x)p Fu(j)p 1444 3477 109 4 v 1444 3496 a(p)p 1519 3496 33 4 v 1519 3569 a Fx(t)1562 3370 y Fw(\023)1826 3498 y Fy(and)181 b Fx(\014)2211 3520 y FC(sing)p Fv(;)p FC(1)2418 3498 y Fy(=)2550 3437 y(1)p 2524 3477 99 4 v 2524 3561 a Fu(j)p Fy(x)p Fu(j)2642 3437 y Fy(1)p 2642 3477 46 4 v 2648 3561 a Fx(t)2697 3498 y Fu(O)2769 3512 y FC(2)2824 3370 y Fw(\022)2906 3437 y Fu(j)p Fy(x)p Fu(j)p 2901 3477 109 4 v 2901 3496 a(p)p 2977 3496 33 4 v 73 x Fx(t)3020 3370 y Fw(\023)3102 3498 y Fx(:)481 b Fy(\(28\))132 3737 y(As)30 b(for)g(the)h(third)d(term,)j(w)m(e)g (will)c(sho)m(w)k(that)957 3968 y Fx(\014)1013 3990 y FC(reg)1113 3968 y Fy(\(x)p Fx(;)15 b(t)p 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Fy(\()2015 4448 y Fv(k)2054 4425 y Fq(2)2088 4448 y Fv(t)p FC(+)p Fv(k)r Fr(j)p FC(x)p Fr(j)2284 4454 y Fy(\))2342 4463 y(~)2323 4487 y Fx(f)9 b Fy(\()p Fx(k)s Fy(\))31 b Fx(k)2578 4449 y FC(2)2618 4487 y Fx(dk)s(:)868 b Fy(\(30\))-9 4725 y(In)29 b(order)h(to)h(apply)e(a)i(stationary)g (phase)f(metho)s(d,)g(w)m(e)g(de\014ne)1043 4969 y Fx(\021)f Fy(:=)c Fu(j)p Fy(x)p Fu(j)c Fy(+)f Fx(t)182 b Fy(and)f Fx(\037)p Fy(\()p Fx(k)s Fy(\))26 b(:=)2325 4908 y Fx(k)2375 4875 y FC(2)2415 4908 y Fx(t)20 b Fy(+)g Fx(k)s Fu(j)p Fy(x)p Fu(j)p 2325 4948 383 4 v 2395 5031 a(j)p Fy(x)p Fu(j)h Fy(+)f Fx(t)3608 4969 y Fy(\(31\))-9 5207 y(observing)29 b(moreo)m(v)m(er)j(that)1077 5384 y(1)p 999 5425 202 4 v 999 5508 a Fx(\037)1056 5482 y Fr(0)1079 5508 y Fy(\()p Fx(k)s Fy(\))1236 5446 y Fu(\024)25 b Fy(Max\(1)p Fx(;)15 b(k)1678 5408 y Fr(\000)p FC(1)1774 5446 y Fy(\))182 b(and)2340 5384 y Fx(\037)2397 5351 y Fr(00)2439 5384 y Fy(\()p Fx(k)s Fy(\))p 2329 5425 241 4 v 2329 5508 a Fx(\037)2386 5482 y Fr(0)2410 5508 y Fy(\()p Fx(k)s Fy(\))2530 5482 y FC(2)2605 5446 y Fu(\024)2713 5384 y Fy(1)p 2711 5425 51 4 v 2711 5508 a Fx(k)3608 5446 y Fy(\(32\))p -9 5543 1512 4 v 94 5606 a Fq(2)129 5638 y Fp(Notice)26 b(that,)g(for)g Fj(z)e Fp(=)882 5602 y Fo(i)p Fk(j)p Fq(x)p Fk(j)p 826 5621 204 4 v 826 5667 a Fq(2)856 5629 y Fk(p)p 907 5629 123 3 v 907 5667 a Fq(1+)p Fo(it)1040 5638 y Fp(,)h(one)h(has)g(arg)r(\()p Fj(z)s Fp(\))21 b Fj(<)1683 5607 y Fq(3)p 1683 5621 31 4 v 1683 5663 a(4)1724 5638 y Fj(\031)28 b Fp(for)e(ev)n(ery)f(v)l (alue)h(of)g(x)f(and)h Fj(t)p Fp(.)1858 5887 y Fy(8)p eop %%Page: 9 9 9 8 bop -9 444 a Fy(where)29 b Fx(\037)310 411 y Fr(0)363 444 y Fy(indicate)g(the)h(deriv)-5 b(ativ)m(e)29 b(of)h Fx(\037)f Fy(with)g(resp)s(ect)h(to)g Fx(k)s Fy(.)41 b(F)-8 b(rom)30 b(\(30\))i(and)d(de\014nition)e(\(31\))32 b(it)d(follo)m(ws)-9 557 y(that)707 809 y Fx(\014)763 831 y FC(reg)863 809 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))26 b(=)1352 747 y Fx(C)1424 714 y Fr(0)p 1186 788 427 4 v 1186 871 a Fu(j)p Fy(x)p Fu(j)15 b Fy(\()q Fu(j)p Fy(x)p Fu(j)20 b Fy(+)g Fx(t)p Fy(\))1637 685 y Fw(Z)1728 711 y FC(+)p Fr(1)1688 891 y FC(0)1873 681 y Fw(\022)1975 747 y Fx(d)p 1950 788 98 4 v 1950 871 a(dk)2058 809 y(e)2100 771 y Fr(\000)p Fv(i\021)s(\037)o FC(\()p Fv(k)r FC(\))2358 681 y Fw(\023)2528 747 y Fy(1)p 2450 788 202 4 v 2450 871 a Fx(\037)2507 845 y Fr(0)2530 871 y Fy(\()p Fx(k)s Fy(\))2681 785 y(~)2661 809 y Fx(f)9 b Fy(\()p Fx(k)s Fy(\))31 b Fx(k)2916 771 y FC(2)2956 809 y Fx(dk)-9 1113 y Fy(By)25 b(recalling)f(de\014nition)f(\(24\))k(and)d(using)g(\(4\))j (and)d(Lemma)i(2.1)g(it)f(is)f(easy)i(to)g(sho)m(w)f(that)3156 1089 y(~)3136 1113 y Fx(f)35 b Fy(and)3397 1072 y Fv(d)3448 1054 y FC(~)3433 1072 y Fv(f)p 3397 1092 78 4 v 3398 1144 a(dk)3510 1113 y Fy(b)s(elong)-9 1226 y(to)c Fx(C)174 1193 y FC(1)213 1226 y Fy(\(0)p Fx(;)15 b Fy(+)p Fu(1)p Fy(\),)31 b(are)g(b)s(ounded)d(in)h(a)i(neigh)m(b)s(orho)s(o)s(d)d(of)j (zero)g(and)f(satisfy)f(the)i(b)s(ound)1099 1485 y(~)1079 1509 y Fx(f)10 b Fy(\()p Fx(k)s Fy(\))26 b Fu(\024)f Fx(C)7 b(k)1498 1471 y Fr(\000)p FC(3)1773 1509 y Fy(and)2112 1447 y Fx(d)2179 1423 y Fy(~)2159 1447 y Fx(f)p 2112 1488 102 4 v 2114 1571 a(dk)2223 1509 y Fy(\()p Fx(k)s Fy(\))26 b Fu(\024)f Fx(C)7 b(k)2587 1471 y Fr(\000)p FC(4)3608 1509 y Fy(\(33\))-9 1749 y(for)26 b Fx(k)i Fu(!)d Fy(+)p Fu(1)p Fy(.)39 b(These)27 b(facts)g(imply)d(that)j(in)m (tegration)g(b)m(y)f(part)h(is)e(p)s(ossible)g(and)h(that)h(the)f(b)s (oundary)f(term)-9 1861 y(is)k(zero,)i(so)g(w)m(e)g(get)776 2054 y Fw(\014)776 2108 y(\014)807 2131 y Fx(\014)863 2153 y FC(reg)962 2131 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))1153 2054 y Fw(\014)1153 2108 y(\014)1210 2131 y Fu(\024)1482 2069 y Fx(C)1554 2036 y Fr(0)p 1316 2110 427 4 v 1316 2193 a Fu(j)p Fy(x)p Fu(j)g Fy(\()q Fu(j)p Fy(x)p Fu(j)20 b Fy(+)g Fx(t)p Fy(\))1767 2007 y Fw(Z)1858 2034 y FC(+)p Fr(1)1818 2213 y FC(0)2003 1999 y Fw(\014)2003 2054 y(\014)2003 2108 y(\014)2003 2163 y(\014)2069 2069 y Fx(d)p 2043 2110 98 4 v 2043 2193 a(dk)2166 2003 y Fw(\022)2321 2069 y Fy(1)p 2243 2110 202 4 v 2243 2193 a Fx(\037)2300 2167 y Fr(0)2323 2193 y Fy(\()p Fx(k)s Fy(\))2474 2107 y(~)2454 2131 y Fx(f)9 b Fy(\()p Fx(k)s Fy(\))31 b Fx(k)2709 2093 y FC(2)2749 2003 y Fw(\023)2816 1999 y(\014)2816 2054 y(\014)2816 2108 y(\014)2816 2163 y(\014)2861 2131 y Fx(dk)s(:)625 b Fy(\(34\))-9 2400 y(F)-8 b(rom)28 b(\(33\))h(and)f (\(32\))h(it)e(follo)m(ws)g(that)i(the)f(in)m(tegral)g(app)s(earing)e (on)i(the)g(righ)m(t-hand)f(side)f(of)i(\(34\))i(is)d(\014nite.)-9 2513 y(This)h(pro)m(v)m(es)j(\(29\).)132 2626 y(Finally)-8 b(,)29 b(w)m(e)i(giv)m(e)f(an)h(estimate)g(on)f Fu(r)p Fx(\014)5 b Fy(.)41 b(By)30 b(direct)g(computation)h(w)m(e)f(obtain)712 2881 y Fu(r)p Fx(\014)5 b Fy(\(x)p Fx(;)15 b(t)p Fy(\))84 b(=)f Fu(\000)1400 2820 y Fy(1)p 1354 2860 138 4 v 1354 2944 a Fu(j)p Fy(x)p Fu(j)1452 2917 y FC(2)1516 2758 y Fw(Z)1567 2964 y Fm(R)1615 2945 y Fq(3)1669 2881 y Fx(e)1711 2844 y Fr(\000)p Fv(i)p FC(k)1827 2820 y Fq(2)1862 2844 y Fv(t)1891 2881 y Fx(e)1933 2844 y Fr(\000)p Fv(i)p Fr(j)p FC(k)p Fr(jj)p FC(x)p Fr(j)2180 2858 y Fw(b)2170 2881 y Fy(\011)2241 2895 y FC(out)2347 2881 y Fy(\(k\))2517 2820 y(1)p 2475 2860 130 4 v 2475 2944 a Fx(i)p Fu(j)p Fy(k)p Fu(j)2615 2881 y Fy(\(2)p Fx(\031)t Fy(\))2786 2844 y Fr(\000)p FC(3)p Fv(=)p FC(2)2951 2881 y Fx(dk)1273 3123 y Fy(+)1380 3062 y(1)p 1354 3102 99 4 v 1354 3186 a Fu(j)p Fy(x)p Fu(j)1497 3062 y Fy(x)p 1472 3102 V 1472 3186 a Fu(j)p Fy(x)p Fu(j)1595 3000 y Fw(Z)1646 3206 y Fm(R)1694 3187 y Fq(3)1748 3123 y Fx(e)1790 3086 y Fr(\000)p Fv(i)p FC(k)1906 3062 y Fq(2)1941 3086 y Fv(t)1970 3123 y Fx(e)2012 3086 y Fr(\000)p Fv(i)p Fr(j)p FC(k)p Fr(jj)p FC(x)p Fr(j)2259 3100 y Fw(b)2249 3123 y Fy(\011)2320 3137 y FC(out)2426 3123 y Fy(\(k\)\(2)p Fx(\031)t Fy(\))2715 3086 y Fr(\000)p FC(3)p Fv(=)p FC(2)2880 3123 y Fx(dk)1119 3362 y Fu(\021)g(\000)1380 3300 y Fy(1)p 1354 3341 V 1354 3424 a Fu(j)p Fy(x)p Fu(j)1462 3362 y Fx(\014)5 b Fy(\(x)p Fx(;)15 b(t)p Fy(\))21 b(+)f(\()p Fu(r)p Fx(\014)5 b Fy(\))2024 3389 y FC(r)2071 3362 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))-9 3629 y(The)39 b(term)g(\()q Fu(r)o Fx(\014)6 b Fy(\))616 3656 y FC(r)687 3629 y Fy(can)40 b(b)s(e)f(treated)i (exactly)f(as)g Fx(\014)5 b Fy(;)45 b(ho)m(w)m(ev)m(er,)e(since)c(the)h (second)g(order)f(p)s(ole)f(do)s(es)i(not)-9 3742 y(app)s(ear,)30 b(w)m(e)g(get)i(the)e(b)s(ound)1157 4006 y(\()q Fu(r)p Fx(\014)5 b Fy(\))1360 4033 y FC(r)1417 4006 y Fu(\024)1549 3945 y Fy(1)p 1523 3985 V 1523 4068 a Fu(j)p Fy(x)p Fu(j)1641 3945 y Fy(1)p 1641 3985 46 4 v 1647 4068 a Fx(t)1697 4006 y Fu(O)1769 4020 y FC(2)1824 3878 y Fw(\022)1906 3945 y Fu(j)p Fy(x)p Fu(j)p 1901 3985 109 4 v 1901 4003 a(p)p 1976 4003 33 4 v 1976 4077 a Fx(t)2019 3878 y Fw(\023)2106 4006 y Fy(+)2234 3945 y(1)p 2207 3985 99 4 v 2207 4068 a Fu(j)p Fy(x)p Fu(j)2411 3945 y Fx(C)p 2326 3985 243 4 v 2326 4068 a Fu(j)p Fy(x)p Fu(j)20 b Fy(+)g Fx(t)2578 4006 y(:)1005 b Fy(\(35\))-9 4457 y Fn(Pro)s(of)41 b(of)g(Theorem)f (2.2.)57 b Fy(W)-8 b(e)37 b(remark)m(ed)e(that)i(in)d(order)h(to)h(pro) m(v)m(e)h(Theorem)e(2.2)i(it)e(is)f(su\016cien)m(t)h(to)-9 4570 y(pro)m(v)m(e)c(\(9\).)42 b(T)-8 b(o)30 b(ac)m(hiev)m(e)i(the)f (pro)s(of,)f(w)m(e)g(notice)h(that)g(the)g(singular)d(term)1356 4827 y Fu(\000)1463 4766 y Fy(1)p 1437 4806 99 4 v 1437 4889 a Fu(j)p Fy(x)p Fu(j)1545 4827 y Fx(\014)1601 4789 y Fr(\003)1601 4849 y FC(sing)q Fv(;)p FC(2)1783 4827 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))p Fx(\014)2031 4849 y FC(sing)q Fv(;)p FC(2)2213 4827 y Fy(\(x)p Fx(;)g(t)p Fy(\))-9 5095 y(app)s(earing)25 b(in)g Fx(\014)5 b Fu(r)p Fx(\014)31 b Fy(is)26 b Fs(r)-5 b(e)g(al)p Fy(,)29 b(so)g(it)d(do)s(es) g(not)h(con)m(tribute)f(to)h(Im\()p Fx(\014)2302 5058 y Fr(\003)2342 5095 y Fu(r)o Fx(\014)6 b Fy(\))p Fx(:)27 b Fy(As)f(for)g(all)g(the)g(remaining)f(terms,)-9 5207 y(they)k(can)h(b)s(e)f(sho)m(wn)g(to)h(v)-5 b(anish)28 b(as)i Fx(R)c Fu(!)f Fy(+)p Fu(1)k Fy(b)m(y)g(using)f(estimates)j (\(21\),)g(\(22\),)h(\(28\),)f(\(29\))g(and)e(\(35\).)42 b(As)-9 5320 y(an)30 b(example)g(w)m(e)h(sho)m(w)f(ho)m(w)g(to)h(pro)m (v)m(e)h(that)1121 5590 y(lim)1087 5650 y Fv(R)p Fr(!1)1297 5466 y Fw(Z)1387 5492 y FC(+)p Fr(1)1347 5672 y Fv(T)1532 5590 y Fx(dt)1627 5466 y Fw(Z)1678 5672 y FC(\006)1729 5683 y Fo(R)1799 5590 y Fu(j)15 b Fy(Im\()p Fx(\014)2039 5552 y Fr(\003)2079 5590 y Fu(r)o Fx(\013)q Fy(\))p Fu(\001)p Fy(n)p Fu(j)g Fx(d\033)29 b Fy(=)c(0)15 b Fx(:)935 b Fy(\(36\))1858 5887 y(9)p eop %%Page: 10 10 10 9 bop -9 444 a Fy(W)-8 b(e)31 b(observ)m(e)g(that)135 544 y Fw(Z)185 750 y FC(\006)236 761 y Fo(R)306 667 y Fu(j)15 b Fy(Im\()p Fx(\014)546 630 y Fr(\003)586 667 y Fu(r)p Fx(\013)p Fy(\))p Fu(\001)q Fy(n)p Fu(j)g Fx(d\033)86 b Fu(\024)d Fy(4)p Fx(\031)s(R)1381 630 y FC(2)1436 590 y Fw(\014)1436 645 y(\014)1466 667 y Fx(\014)1522 689 y FC(sing)q Fv(;)p FC(2)1724 667 y Fy(+)20 b Fx(\014)1872 689 y FC(sing)p Fv(;)p FC(1)2074 667 y Fy(+)g Fx(\014)2221 689 y FC(reg)2320 590 y Fw(\014)2320 645 y(\014)2366 667 y Fu(jr)p Fx(\013)p 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b(one)g(mak)m(es)g(use)f(of)h(prop)s(ert)m (y)f(\(7\))-9 1293 y(with)f(suitable)g(c)m(hoices)j(of)f Fx(\034)44 b Fy(and)33 b Fx(\027)39 b Fy(in)33 b(order)g(to)h(con)m (trol)h(the)f(previous)e(expression.)50 b(F)-8 b(or)34 b(example,)h(the)-9 1406 y(\014rst)29 b(term)i(is)987 1485 y Fx(R)p 968 1525 109 4 v 968 1543 a Fu(p)p 1044 1543 33 4 v 74 x Fx(t)1087 1546 y Fu(O)1159 1560 y FC(1)1214 1418 y Fw(\022)1310 1485 y Fx(R)p 1290 1525 109 4 v 1290 1543 a Fu(p)p 1366 1543 33 4 v 74 x Fx(t)1409 1418 y Fw(\023)1647 1485 y Fy(1)p 1501 1525 337 4 v 1501 1612 a Fx(R)1571 1586 y Fv(")1608 1612 y Fx(t)1641 1586 y FC(3)p Fv(=)p FC(2)p Fr(\000)p Fv(")1858 1485 y Fx(R)1928 1452 y Fv(")p 1858 1525 107 4 v 1876 1608 a Fx(t)1909 1582 y Fv(")1974 1546 y Fu(O)2046 1560 y FC(1)2101 1418 y Fw(\022)2178 1485 y Fx(R)p 2178 1525 70 4 v 2196 1608 a(t)2258 1418 y Fw(\023)2350 1546 y Fu(\024)2588 1485 y Fx(C)p 2456 1525 337 4 v 2456 1612 a(R)2526 1586 y Fv(")2562 1612 y Fx(t)2595 1586 y FC(3)p Fv(=)p FC(2)p Fr(\000)p Fv(")-9 1773 y Fy(for)f(ev)m(ery)h Fx(")25 b Fu(2)g Fy(\(0)p Fx(;)655 1738 y FC(1)p 655 1753 36 4 v 655 1805 a(2)700 1773 y Fy(\).)41 b(By)30 b(similar)e(computations) i(w)m(e)h(get)1050 1912 y Fw(Z)1100 2118 y FC(\006)1151 2129 y Fo(R)1221 2036 y Fu(j)15 b Fy(Im\()p Fx(\014)1461 1998 y Fr(\003)1501 2036 y Fu(r)p Fx(\013)p Fy(\))p Fu(\001)q Fy(n)p Fu(j)g Fx(d\033)28 b Fu(\024)2153 1974 y Fx(C)p 2020 2015 337 4 v 2020 2102 a(R)2090 2076 y Fv(")2127 2102 y Fx(t)2160 2076 y FC(3)p Fv(=)p FC(2)p Fr(\000)p Fv(")2387 2036 y Fy(+)2547 1974 y Fx(C)2619 1941 y Fr(0)p 2488 2015 213 4 v 2488 2102 a Fx(R)q(t)2591 2076 y FC(3)p Fv(=)p FC(2)-9 2297 y Fy(This)38 b(b)s(ound)g(is)h(su\016cien)m(t)h(to) g(pro)m(v)m(e)h(the)g(v)-5 b(anishing)37 b(of)k(the)f(left-hand)f(side) g(of)h(\(36\))i(b)m(y)e(applying)e(the)-9 2410 y(dominated)29 b(con)m(v)m(ergence)k(theorem.)41 b(This)29 b(completes)h(the)h(pro)s (of)e(of)i(Theorem)f(2.2)p 3018 2410 61 61 v 1835 5887 a(10)p eop %%Page: 11 11 11 10 bop -9 444 a Fz(3)134 b(Zero)67 b(energy)g(resonances)f(and)g (the)g(F)-15 b(AS)65 b(theorem)i(in)f(p)t(oten)l(tial)192 593 y(scattering)-9 801 y Fy(In)29 b(this)g(section)h(and)f(the)h (follo)m(wing)e(one)j(w)m(e)f(will)d(consider)i(the)h(scattering)h (theory)f(for)f(the)h(pair)f(\()p Fx(H)r(;)15 b(H)3695 815 y FC(0)3735 801 y Fy(\))-9 914 y(where)30 b Fx(H)330 928 y FC(0)395 914 y Fy(=)c Fu(\000)p Fy(\001)k(on)h(the)g(domain)f Fu(D)s Fy(\()p Fx(H)1461 928 y FC(0)1500 914 y Fy(\))c(=)g Fx(H)1741 881 y FC(2)1780 914 y Fy(\()p Ft(R)1876 881 y FC(3)1921 914 y Fy(\))31 b(and)f Fx(H)j Fy(=)26 b Fx(H)2446 928 y FC(0)2506 914 y Fy(+)20 b Fx(V)g Fy(.)43 b(Later)31 b(on,)g(w)m(e)h(will)c(fo)s(cus)i(on)-9 1027 y(p)s(oten)m(tials)f (satisfying)g(the)i(follo)m(wing)e(assumptions.)-9 1258 y Fn(De\014nition)35 b(3.1)46 b Fs(We)33 b(say)h(that)h(a)f(me)-5 b(asur)g(able)36 b(function)e Fx(V)47 b Fy(:)28 b Ft(R)2288 1225 y FC(3)2361 1258 y Fu(!)f Ft(R)42 b Fs(b)-5 b(elongs)34 b(to)h(the)41 b Fy(Ik)m(eb)s(e)31 b(class)j(\(I\))3723 1285 y Fv(n)-9 1371 y Fy(\()p Fs(with)f Fx(n)25 b Fu(2)g Ft(N)7 b Fy(\))p Fs(,)38 b(if:)76 1603 y Fy(\(i\))46 b Fx(V)53 b Fs(is)32 b(lo)-5 b(c)g(al)5 b(ly)35 b(H\177)-46 b(older)33 b(c)-5 b(ontinuous)34 b(exc)-5 b(ept)33 b(that)h(in)e(a)h (\014nite)g(numb)-5 b(er)33 b(of)g(p)-5 b(oints)51 1809 y Fy(\(ii\))45 b Fx(V)g Fu(2)25 b Fx(L)464 1776 y FC(2)503 1809 y Fy(\()p Ft(R)598 1776 y FC(3)644 1809 y Fy(\))26 2016 y(\(iii\))44 b Fs(ther)-5 b(e)33 b(exist)g Fx(R)725 2030 y FC(0)790 2016 y Fx(>)25 b Fy(0)33 b Fs(and)g Fx(")26 b(>)f Fy(0)33 b Fs(such)g(that)h Fu(j)p Fx(V)20 b Fy(\(x\))p Fu(j)26 b(\024)2203 1979 y Fv(C)2253 1988 y Fq(0)p 2148 1995 195 4 v 2148 2050 a Fr(j)p FC(x)p Fr(j)2225 2031 y Fo(n)p Fq(+)p Fo(")2385 2016 y Fs(for)34 b Fu(j)p Fy(x)p Fu(j)25 b(\025)g Fx(R)2819 2030 y FC(0)2859 2016 y Fs(.)-9 2261 y(Mor)-5 b(e)g(over,)33 b(we)g(de\014ne)g Fy(\(I\))921 2288 y Fr(1)1021 2261 y Fy(=)1117 2193 y Fw(T)1193 2288 y Fv(n)p Fr(2)p Fm(N)1350 2261 y Fy(\(I\))1454 2288 y Fv(n)1501 2261 y Fs(.)-9 2493 y Fy(The)j(terminology)g(follo)m(ws)g (from)g(the)h(fact)g(that,)i(for)e Fx(n)e Fy(=)g(2)p Fx(;)j Fy(these)f(are)g(the)g(h)m(yp)s(otheses)f(under)f(whic)m(h)-9 2606 y(Ik)m(eb)s(e's)25 b(eigenfunction)e(expansion)h(theorem)h([Ik])h (has)e(b)s(een)g(pro)m(v)m(ed.)40 b(Under)24 b(these)h(assumptions,)g (the)g(op-)-9 2719 y(erator)g Fx(H)32 b Fy(is)24 b(self-adjoin)m(t)g (on)h Fu(D)s Fy(\()p Fx(H)1214 2733 y FC(0)1253 2719 y Fy(\).)39 b(Moreo)m(v)m(er)27 b Fx(H)32 b Fy(has)24 b(neither)g(p)s(ositiv)m(e)g(eigen)m(v)-5 b(alues)24 b(nor)h(singular)d(con-)-9 2832 y(tin)m(uous)i(sp)s(ectrum)g(and)h Fx(\033)927 2846 y FC(ac)997 2832 y Fy(\()p Fx(H)7 b Fy(\))26 b(=)f([0)p Fx(;)15 b Fy(+)p Fu(1)p Fy(\).)40 b(Finally)-8 b(,)26 b(the)f(w)m(a)m(v)m(e)j(op)s(erators)d Fx(W)2820 2846 y Fr(\006)2905 2832 y Fy(=)f(lim)3127 2846 y Fv(t)p Fr(!\0061)3368 2832 y Fx(e)3410 2799 y Fv(iH)5 b(t)3527 2832 y Fx(e)3569 2799 y Fr(\000)p Fv(iH)3706 2808 y 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FC(k)785 4853 y Fy(=)25 b Fx(\030)925 4875 y FC(k)997 4853 y Fy(and)30 b(this)f(pro)m(v)m(es)i(our)f(claim.)p 2062 4853 61 61 v -9 5076 a Fn(Remark.)45 b Fy(In)32 b(the)g(follo)m(wing)f(w)m(e)i (will)d(not)i(distinguish)d(an)m(ymore)k(b)s(et)m(w)m(een)h(~)-46 b Fx(\021)2796 5098 y FC(k)2870 5076 y Fy(and)31 b Fx(\021)3097 5098 y FC(k)3138 5076 y Fy(.)47 b(The)31 b(p)s(oin)m(t)m(wise)-9 5189 y(v)-5 b(alues)29 b(of)i Fx(\021)413 5211 y FC(k)455 5189 y Fy(will)c(b)s(e)j(denoted)g(as)h Fx(\021)1253 5211 y FC(k)1295 5189 y Fy(\(y\))26 b Fu(\021)f Fx(\021)s Fy(\(y)q Fx(;)15 b Fy(k)q(\))31 b(for)f(ev)m(ery)h(y)c Fu(2)d Ft(R)2421 5156 y FC(3)2467 5189 y Fy(.)132 5412 y(W)-8 b(e)34 b(turn)d(no)m(w)i(to)g(study)f(the)h(b)s(eha)m(vior)f(of) h(the)f(generalized)h(eigenfunctions)e(in)g(presence)i(of)g(a)g(zero-) -9 5525 y(energy)28 b(resonance.)69 b(Although)28 b(the)g(follo)m(wing) f(de\014nition)f(could)h(app)s(ear)h(quite)g Fs(ad)k(ho)-5 b(c)p Fy(,)30 b(w)m(e)e(will)e(sho)m(w)i(in)-9 5638 y(the)i(App)s (endix)e(that)j(it)f(is)f(completely)h(equiv)-5 b(alen)m(t)30 b(to)h(the)g(most)f(common)h(ones)g(\(see)g(Prop.)40 b(4.3\).)1835 5887 y(13)p eop %%Page: 14 14 14 13 bop -9 444 a Fn(De\014nition)35 b(3.4)46 b Fs(We)34 b(say)g(that)i(ther)-5 b(e)35 b(is)f(a)h Fl(zer)-5 b(o-ener)g(gy)39 b(r)-5 b(esonanc)g(e)34 b Fs(for)h(the)f(p)-5 b(air)36 b Fy(\()p Fx(H)r(;)15 b(H)3343 458 y FC(0)3383 444 y Fy(\))34 b Fs(if)68 b(ther)-5 b(e)-9 570 y(exists)32 b(a)h Fx( )c Fu(2)c Fx(H)575 534 y FC(1)p Fv(;)p Fr(\000)695 507 y Fq(1)p 695 519 31 3 v 695 560 a(2)735 534 y Fr(\000)p FC(0)829 570 y Fy(\()p Ft(R)924 537 y FC(3)970 570 y Fy(\))33 b Fs(such)g(that)g Fy(\()q(1)20 b(+)g Fx(G)1693 584 y FC(0)1733 570 y Fx(V)g Fy(\))c Fx( )29 b Fy(=)c(0)33 b Fs(but)f Fx( )39 b(=)-55 b Fu(2)25 b Fx(L)2508 537 y FC(2)2547 570 y Fy(\()p Ft(R)2642 537 y FC(3)2687 570 y Fy(\))p Fx(:)-9 785 y Fy(Suc)m(h)f(a)h Fx( )t Fy(,)h(if)e(it)g (exists,)i(is)e(unique)f(up)h(to)i(a)g(complex)e(phase.)39 b(It)25 b(will)d(b)s(e)j(called)f(the)h Fn(resonance)k(function)-9 898 y Fy(and)g(denoted)i(with)e Fx( )780 920 y FC(res)871 898 y Fy(.)132 1011 y(Roughly)35 b(sp)s(eaking,)h(if)f(the)h(pair)f(\() p Fx(H)r(;)15 b(H)1579 1025 y FC(0)1618 1011 y Fy(\))36 b(admits)f(a)i(zero-energy)g(resonance,)h(then)e(the)g(map)f Fx(\024)g Fu(7!)-9 1124 y Fy(\(1)25 b(+)f Fx(G)262 1138 y Fv(\024)307 1124 y Fx(V)c Fy(\))415 1091 y Fr(\000)p FC(1)547 1124 y Fy(has)36 b(a)h(singular)d(b)s(eha)m(vior)i(as)h Fx(\024)f Fu(!)f Fy(0.)97 b(This)34 b(b)s(eha)m(vior)i(is)g(describ)s (ed)e(b)m(y)j(the)g(follo)m(wing)-9 1237 y(theorem,)31 b(whic)m(h)e(has)h(b)s(een)f(pro)m(v)m(ed)i(in)e([JK,)h(Lemmas)h(4.2)g (and)f(4.4].)-9 1451 y Fn(Theorem)j(3.5)j(\(Jensen-Kato,)e(1979\))46 b Fs(Assume)39 b(that)h Fx(V)59 b Fs(satis\014es)39 b(c)-5 b(ondition)41 b(\()p Fx(V)5 b(:)p Fy(1)p Fx(:\014)h Fs(\))39 b(with)g Fx(\014)j(>)36 b Fy(7)p Fs(.)-9 1564 y(L)-5 b(et)32 b Fx(s)h Fs(satisfy)g Fy(7)p Fx(=)p Fy(2)27 b Fx(<)e(s)f(<)h(\014)h Fu(\000)20 b Fy(7)p Fx(=)p Fy(2)p Fx(:)132 1677 y Fs(Assume)26 b(that)i(ther)-5 b(e)28 b(is)e(a)h(zer)-5 b(o-ener)g(gy)28 b(r)-5 b(esonanc)g(e)28 b(for)f(the)g(p)-5 b(air)29 b Fy(\()p Fx(H)r(;)15 b(H)2611 1691 y FC(0)2650 1677 y Fy(\))27 b Fs(and)h(that)g Fy(0)35 b Fx(=)-55 b Fu(2)25 b Fx(\033)3274 1691 y FC(p)3318 1677 y Fy(\()p Fx(H)7 b Fy(\))p Fs(.)67 b(Then)-9 1790 y(for)33 b Fx(\024)25 b Fu(!)h Fy(0)p Fx(;)15 b(\024)26 b Fu(2)f Ft(R)639 1804 y FC(+)704 1790 y Fs(,)32 b(we)h(have)g(in)f Fu(B)s Fy(\()p Fx(H)1400 1757 y FC(1)p Fv(;)p Fr(\000)p Fv(s)1547 1790 y Fy(\))h Fs(the)g(exp)-5 b(ansion)805 2036 y Fy(\(1)21 b(+)f Fx(G)1068 2050 y Fv(\024)1113 2036 y Fx(V)g Fy(\))1221 1998 y Fr(\000)p FC(1)1341 2036 y Fy(=)25 b Fu(\000)1528 1974 y Fx(i)p 1518 2015 53 4 v 1518 2098 a(\024)1595 2036 y Fu(h)q(\001)p Fx(;)15 b(V)21 b( )1832 2058 y FC(res)1923 2036 y Fu(i)15 b Fx( )2036 2058 y FC(res)2147 2036 y Fy(+)20 b Fx(C)2303 2050 y FC(0)2363 2036 y Fy(+)f Fx(\024)q(C)2571 2050 y FC(1)2630 2036 y Fy(+)h Fx(O)s Fy(\()p Fx(\024)2881 1998 y FC(2)2920 2036 y Fy(\))653 b(\(40\))-9 2275 y Fs(wher)-5 b(e)33 b Fx(C)312 2289 y FC(0)384 2275 y Fs(and)h Fx(C)626 2289 y FC(1)698 2275 y Fs(ar)-5 b(e)33 b(explicitly)h(c)-5 b(omputable)34 b(op)-5 b(er)g(ators)36 b(in)c Fu(B)s Fy(\()p Fx(H)2390 2242 y FC(1)p Fv(;)p Fr(\000)p Fv(s)2537 2275 y Fy(\))p Fs(.)132 2490 y Fy(An)j(explicit)f(expression)h(for)h (the)g(op)s(erators)g Fx(C)1836 2504 y FC(0)1911 2490 y Fy(and)f Fx(C)2158 2504 y FC(1)2234 2490 y Fy(can)h(b)s(e)f(found)g (in)f([JK,)j(Lemma)f(4.3].)59 b(As)-9 2603 y(p)s(oin)m(ted)30 b(out)i(in)e([JK,)i(Remark)g(4.6],)i(b)m(y)d(assuming)g(su\016cien)m (tly)f(large)i Fx(\014)37 b Fy(and)31 b Fx(s)g Fy(it)g(is)g(p)s (ossible)e(to)k(obtain)-9 2716 y(an)27 b(expansion)g(of)h(\(1)15 b(+)g Fx(G)887 2730 y Fv(\024)933 2716 y Fx(V)20 b Fy(\))1041 2683 y Fr(\000)p FC(1)1164 2716 y Fy(to)28 b(an)m(y)h(order)e(in)f Fx(\024)q Fy(.)40 b(Ho)m(w)m(ev)m(er,)30 b(the)e(actual)g(computation)g (of)g(co)s(e\016cien)m(ts)-9 2829 y(b)s(ecomes)i(rather)g(di\016cult.) -9 3044 y Fn(Remark)k(3.6)46 b Fs(If)33 b(ther)-5 b(e)34 b(is)f(a)h(zer)-5 b(o-ener)g(gy)34 b(r)-5 b(esonanc)g(e)35 b(for)f(the)f(p)-5 b(air)35 b Fy(\()p Fx(H)r(;)15 b(H)2681 3058 y FC(0)2720 3044 y Fy(\))34 b Fs(and)g(mor)-5 b(e)g(over)35 b Fy(0)27 b Fu(2)f Fx(\033)3573 3058 y FC(p)3616 3044 y Fy(\()p Fx(H)7 b Fy(\))-9 3157 y Fs(then)33 b(exp)-5 b(ansion)34 b(\(40\))g(is)f(r)-5 b(eplac)g(e)g(d)35 b(by)431 3402 y Fy(\(1)21 b(+)f Fx(G)694 3416 y Fv(\024)739 3402 y Fx(V)g Fy(\))847 3365 y Fr(\000)p FC(1)967 3402 y Fy(=)25 b Fu(\000)1167 3341 y Fy(1)p 1144 3381 92 4 v 1144 3465 a Fx(\024)1196 3438 y FC(2)1246 3402 y Fx(P)1304 3416 y FC(0)1343 3402 y Fx(V)41 b Fu(\000)1548 3341 y Fx(i)p 1538 3381 53 4 v 1538 3465 a(\024)1615 3402 y Fy(\()q Fu(h\001)p Fx(;)15 b(V)21 b( )1887 3424 y FC(res)1978 3402 y Fu(i)16 b Fx( )2091 3424 y FC(res)2202 3402 y Fu(\000)k Fx(P)2351 3416 y FC(0)2391 3402 y Fx(V)g(C)7 b(V)20 b(P)2667 3416 y FC(0)2707 3402 y Fx(V)g Fy(\))g(+)2947 3379 y(~)2926 3402 y Fx(C)2991 3416 y FC(0)3051 3402 y Fy(+)g Fx(O)s Fy(\(1\))279 b(\(41\))-9 3661 y Fs(wher)-5 b(e)34 b Fx(P)306 3675 y FC(0)379 3661 y Fs(is)f(the)g(pr)-5 b(oje)g(ctor)36 b(on)e(the)f(eigensp)-5 b(ac)g(e)34 b(r)-5 b(elative)34 b(to)g(zer)-5 b(o)34 b(\(natur)-5 b(al)5 b(ly)36 b(extende)-5 b(d)34 b(to)g Fx(H)3342 3625 y Fr(\000)p FC(1)p Fv(;)p Fr(\000)3517 3598 y Fq(1)p 3516 3610 31 3 v 3516 3651 a(2)3557 3625 y FC(+0)3684 3661 y Fs(by)-9 3787 y(using)28 b(the)h(fact)f(that)i(the)f(eigenfunctions)f(b)-5 b(elongs)29 b(to)g Fx(H)1966 3751 y FC(1)p Fv(;)2031 3724 y Fq(1)p 2031 3736 V 2031 3777 a(2)2072 3751 y Fr(\000)p FC(0)2166 3787 y Fs(\))f(and)i Fx(C)35 b Fs(is)28 b(the)57 b(c)-5 b(onvolution)30 b(op)-5 b(er)g(ator)32 b(with)-9 3900 y(kernel)g Fx(C)7 b Fy(\(x)p Fx(;)15 b Fy(y\))26 b(=)710 3864 y FC(1)p 671 3879 114 4 v 671 3931 a(24)p Fv(\031)794 3900 y Fu(j)p Fy(x)21 b Fu(\000)f Fy(y)q Fu(j)1053 3867 y FC(3)1093 3900 y Fs(.)-9 4115 y Fn(Remark)34 b(3.7)46 b Fs(A)-5 b(c)g(c)g(or)g(ding)37 b(to)f(Jensen)f(and)i(Kato)f Fy(\()p Fs(se)-5 b(e)36 b([JK,)g(R)-5 b(emark)37 b(6.7])p Fy(\))f Fs(the)g(asymptotic)i(exp)-5 b(ansion)-9 4228 y(\(40\))33 b(c)-5 b(an)34 b(b)-5 b(e)32 b(di\013er)-5 b(entiate)g(d)35 b(any)e(numb)-5 b(er)33 b(of)g(times,)g(in)f(the)h (sense)g(that)h(for)f(every)f Fx(r)c Fu(2)d Ft(N)1042 4477 y Fx(d)1089 4444 y Fv(r)p 1016 4517 138 4 v 1016 4601 a Fx(d\024)1116 4574 y Fv(r)1179 4355 y Fw(2)1179 4519 y(4)1239 4538 y Fy(\(1)c(+)f Fx(G)1502 4552 y Fv(\024)1548 4538 y Fx(V)g Fy(\))1656 4501 y Fr(\000)p FC(1)1771 4538 y Fu(\000)1929 4425 y Fv(n)1885 4452 y Fw(X)1862 4647 y Fv(j)t FC(=)p Fr(\000)p FC(1)2054 4538 y Fx(\024)2107 4501 y Fv(j)2143 4538 y Fx(C)2208 4552 y Fv(j)2245 4355 y Fw(3)2245 4519 y(5)2330 4538 y Fy(=)25 b Fx(o)p Fy(\()p Fx(\024)2558 4501 y Fv(n)p Fr(\000)p Fv(r)2694 4538 y Fy(\))p Fx(:)854 b Fy(\(42\))-9 4844 y Fs(However,)38 b(these)g(asymptotic)i(exp)-5 b(ansions)39 b(r)-5 b(e)g(quir)g(e)38 b(lar)-5 b(ger)39 b(values)f(of)f Fx(s)g Fs(and)i Fx(\014)j Fs(than)d(for)f Fx(r)f Fy(=)c(0)p Fs(.)57 b(T)-7 b(o)38 b(\014x)-9 4957 y(notation,)i(for)e(every)f Fx(n;)15 b(r)36 b Fu(2)e Ft(N)50 b Fs(with)38 b Fx(r)e(<)e(n)j Fs(ther)-5 b(e)38 b(exists)g(a)g(r)-5 b(e)g(al)38 b(numb)-5 b(er)2709 4933 y Fy(\026)2703 4957 y Fx(\014)39 b Fy(=)2903 4933 y(\026)2898 4957 y Fx(\014)5 b Fy(\()p Fx(n;)15 b(r)s Fy(\))38 b Fs(such)f(that)i(\(41\))-9 5084 y(and)33 b(\(42\))h(holds)g(true,)f(pr)-5 b(ovide)g(d)35 b(that)f Fx(\014)c(>)1546 5060 y Fy(\026)1541 5084 y Fx(\014)38 b Fs(and)33 b Fx(s)f Fs(satis\014es)2237 5026 y FC(\026)2233 5043 y Fv(\014)p 2233 5063 44 4 v 2237 5116 a FC(2)2311 5084 y Fx(<)25 b(s)g(<)g(\014)g Fu(\000)2752 5026 y FC(\026)2748 5043 y Fv(\014)p 2748 5063 V 2752 5116 a FC(2)2801 5084 y Fs(.)132 5299 y Fy(It)f(w)m(ould)f(b)s(e)g(tempting)g(to)i(rephrase)e (the)i(previous)d(result)h(b)m(y)h(sa)m(ying)g(that)g(\(1)7 b(+)g Fx(G)3016 5313 y Fv(\024)3063 5299 y Fx(V)20 b Fy(\))3171 5266 y Fr(\000)p FC(1)3290 5299 y Fy(has)k(a)g(\\p)s(ole")-9 5412 y(in)31 b Fx(\024)d Fy(=)h(0,)k(but)f(this)f(term)h(is)g(usually)e (reserv)m(ed)i(for)g(meromorphic)f(functions)g(\(whic)m(h)g(are)i (de\014ned)e(in)g(an)-9 5525 y(op)s(en)f(neigh)m(b)s(orho)s(o)s(d)e(of) j(the)g(p)s(oin)m(t)f Fx(\024)d Fy(=)e(0\))32 b(while)d(w)m(e)i(are)g (facing)g(with)f(a)h(function)e(with)h(an)h(asymptotic)-9 5638 y(expansion)k(only)i(on)g(\012)f(=)g Fu(f)p Fx(\024)h Fu(2)f Ft(C)60 b Fy(:)37 b Fx(\024)1415 5605 y FC(2)1491 5638 y Fu(2)p 1588 5564 66 4 v 36 w Ft(C)1653 5652 y FC(+)1712 5638 y Fu(g)p Fy(.)62 b(In)36 b(suc)m(h)h(a)g(case)h(w)m(e)g (prefer)e(to)i(use)f(the)g(term)g Fn(p)s(olar)1835 5887 y Fy(14)p eop %%Page: 15 15 15 14 bop -9 444 a Fn(singularit)m(y)p Fy(.)47 b(The)32 b(usual)f(concept)j(of)e(complex)g(analysis)f(\(simple)g(p)s(ole,)i (residue,...\))46 b(extends)33 b(trivially)-9 557 y(to)e(this)e(case.) -9 670 y(W)-8 b(e)33 b(emphasize)f(that,)i(when)d(zero)j(is)d(a)i (resonance)g(but)e(not)i(an)f(eigen)m(v)-5 b(alue,)34 b(the)e(exp)s(ected)h(p)s(olar)e(singu-)-9 783 y(larit)m(y)e(for)g(\(1) 20 b(+)f Fx(G)632 797 y Fv(\024)677 783 y Fx(V)h Fy(\))785 750 y Fr(\000)p FC(1)910 783 y Fy(m)m(ust)29 b(b)s(e)g Fs(simple)68 b Fy(and)29 b(the)h(residue)e(corresp)s(onding)g(to)i(the) g(p)s(ole)f(is)g(a)h Fs(r)-5 b(ank-one)-9 909 y(op)g(er)g(ator)p Fy(,)33 b(pro)5 b(jecting)30 b(on)h(the)f(subspace)g(generated)h(b)m(y) g(the)f(resonance)h(function)e Fx( )2997 931 y FC(res)3113 909 y Fu(2)c Fx(H)3282 873 y FC(1)p Fv(;)p Fr(\000)3402 846 y Fq(1)p 3401 858 31 3 v 3401 899 a(2)3442 873 y Fr(\000)p FC(0)3536 909 y Fy(\()p Ft(R)3631 876 y FC(3)3677 909 y Fy(\))p Fx(:)132 1133 y Fy(By)31 b(recalling)e(that)i(the)g (solution)e Fx(\021)1386 1155 y FC(k)1458 1133 y Fy(of)i(the)f (Lippman-Sc)m(h)m(winger)e(equation)j(\(37\))h(is)e(giv)m(en)h(b)m(y)61 b Fx(\021)3632 1155 y FC(k)3699 1133 y Fy(=)-9 1246 y(\(1)21 b(+)e Fx(G)253 1260 y Fv(\024)299 1246 y Fx(V)h Fy(\))407 1213 y Fr(\000)p FC(1)502 1246 y Fx(g)545 1261 y FC(k)617 1246 y Fy(w)m(e)31 b(can)f(pro)m(v)m(e)h(the)g(follo)m(wing)e(result.) -9 1470 y Fn(Prop)s(osition)36 b(3.8)46 b Fs(Assume)36 b(that)i Fx(V)57 b Fs(satis\014es)37 b(c)-5 b(ondition)38 b Fy(\()p Fs(V.)p Fy(1)p Fx(:\014)6 b Fy(\))37 b Fs(for)g Fx(\014)h(>)2800 1446 y Fy(\026)2795 1470 y Fx(\014)f Fu(\021)2992 1446 y Fy(\026)2987 1470 y Fx(\014)5 b Fy(\(6)p Fx(;)15 b Fy(5\))38 b Fs(and)g(that)g(the)-9 1597 y(p)-5 b(air)34 b Fy(\()p Fx(H)r(;)15 b(H)407 1611 y FC(0)446 1597 y Fy(\))33 b Fs(has)h(a)f(zer)-5 b(o-ener)g(gy)33 b(r)-5 b(esonanc)g(e.)44 b(Fix)33 b Fx(s)f Fs(so)h(that)2244 1539 y FC(\026)2240 1556 y Fv(\014)p 2240 1577 44 4 v 2244 1629 a FC(2)2318 1597 y Fx(<)25 b(s)g(<)g(\014)g Fu(\000)2759 1539 y FC(\026)2755 1556 y Fv(\014)p 2755 1577 V 2759 1629 a FC(2)2808 1597 y Fs(.)-9 1710 y(If)34 b Fy(0)39 b Fx(=)-55 b Fu(2)29 b Fx(\033)307 1724 y FC(p)350 1710 y Fy(\()p Fx(H)7 b Fy(\))p Fx(;)36 b Fs(then)f(the)g(solution)h Fx(\021)1315 1732 y FC(k)1391 1710 y Fs(of)f(the)g(Lippman-Schwinger)h (e)-5 b(quation)36 b(\(38\))g(c)-5 b(an)35 b(b)-5 b(e)34 b(de)-5 b(c)g(omp)g(ose)g(d)-9 1823 y(as)1221 1945 y Fx(\021)t Fy(\(x)p Fx(;)15 b Fy(k\))26 b(=)1617 1883 y Fx(r)1658 1897 y FC(0)p 1608 1924 99 4 v 1608 2007 a Fu(j)p Fy(k)p Fu(j)1716 1945 y Fx( )1779 1967 y FC(res)1870 1945 y Fy(\(x\))21 b(+)e Fx(\032)p Fy(\(x)p Fx(;)c Fy(k\))1256 b(\(43\))-9 2178 y Fs(wher)-5 b(e)44 b Fx(r)299 2192 y FC(0)383 2178 y Fy(=)g Fu(\000)p Fx(i)15 b Fu(h)p Fx(g)693 2192 y FC(0)733 2178 y Fx(;)g(V)21 b( )909 2200 y FC(res)1000 2178 y Fu(i)45 b Fy(=)f Fx(i)15 b Fu(h)q Fx(V)5 b(;)15 b( )1438 2200 y FC(res)1529 2178 y Fu(i)43 b Fs(and)h(the)g(map)g Fy(k)h Fu(7!)f Fx(\032)2441 2200 y FC(k)2526 2178 y Fs(fr)-5 b(om)44 b Ft(R)2811 2145 y FC(3)2884 2178 y Fu(n)28 b(f)p Fy(0)p Fu(g)45 b Fs(to)f Fx(H)3340 2145 y FC(1)p Fv(;)p Fr(\000)p Fv(s)3486 2178 y Fy(\()p Ft(R)3581 2145 y FC(3)3627 2178 y Fy(\))f Fs(is)-9 2291 y(b)-5 b(ounde)g(d)37 b(\(with)g(al)5 b(l)37 b(its)f(derivatives)h(until)f(or)-5 b(der)38 b Fx(r)c Fy(=)d(5)37 b Fs(at)f(le)-5 b(ast\))38 b(in)e(a)g(punctur)-5 b(e)g(d)38 b(neighb)-5 b(orho)g(o)g(d)39 b(of)e(the)-9 2404 y(origin.)-9 2517 y(If)j Fy(0)g Fu(2)f Fx(\033)334 2531 y FC(p)377 2517 y Fy(\()p Fx(H)7 b Fy(\))41 b Fs(and)756 2443 y Fw(\010)809 2517 y Fx( )871 2539 y Fv(j)908 2443 y Fw(\011)961 2466 y Fv(p)961 2556 y(j)t FC(=1)1127 2517 y Fu(\022)e Fx(H)1320 2481 y FC(1)p Fv(;)1385 2454 y Fq(1)p 1385 2466 31 3 v 1385 2507 a(2)1425 2481 y Fr(\000)p FC(0)1559 2517 y Fu(\022)g Fx(L)1731 2484 y FC(2)1811 2517 y Fs(ar)-5 b(e)41 b(the)g(c)-5 b(orr)g(esp)g(onding)44 b(eigenfunctions,)e(then)f Fx(\021)3609 2539 y FC(k)3691 2517 y Fs(b)-5 b(e)-9 2640 y(de)g(c)g(omp)g(ose)g(d)36 b(as)1017 2831 y Fx(\021)t Fy(\(x)p Fx(;)15 b Fy(k\))26 b(=)1413 2769 y Fx(r)1454 2783 y FC(0)p 1404 2810 99 4 v 1404 2893 a Fu(j)p Fy(k)p Fu(j)1512 2831 y Fx( )1575 2853 y FC(res)1666 2831 y 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Fx(V)20 b(g)1586 4693 y FC(0)1651 4679 y Fy(=)1795 4560 y Fv(p)1747 4593 y Fw(X)1751 4788 y Fv(j)t FC(=1)1893 4605 y Fw(\012)1936 4679 y Fx( )1999 4701 y Fv(j)2035 4679 y Fx(;)15 b(V)21 b(g)2192 4693 y FC(0)2232 4605 y Fw(\013)2290 4679 y Fx( )2352 4701 y Fv(j)-9 4986 y Fy(Ho)m(w)m(ev)m(er,)36 b(this)d(term)h(is)f(iden)m(tically)f(zero)i (since)f(\(w)m(e)i(use)e(the)h(fact)h(that)f(\(1)24 b(+)e Fx(G)2891 5000 y FC(0)2931 4986 y Fx(V)e Fy(\))p Fx( )3102 5008 y Fv(j)3169 4986 y Fy(=)31 b(0)j(since)f Fx( )3639 5008 y Fv(j)3709 4986 y Fy(is)-9 5099 y(an)d(eigenfunction,)f(see)i (the)g(App)s(endix\))d(one)i(has)583 5238 y Fw(\012)626 5312 y Fx( )688 5334 y Fv(j)725 5312 y Fx(;)15 b(V)21 b(g)882 5326 y FC(0)921 5238 y Fw(\013)990 5312 y Fy(=)1086 5238 y Fw(\012)1128 5312 y Fx( )1191 5334 y Fv(j)1227 5312 y Fx(;)15 b(V)21 b(G)1412 5326 y FC(0)1452 5312 y Fx(V)40 b Fu(\001)21 b Fy(1)1636 5238 y Fw(\013)1705 5312 y Fy(=)1801 5238 y Fw(\012)1843 5312 y Fx(G)1914 5326 y FC(0)1954 5312 y Fx(V)g( 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y Fx( )1033 727 y Fv(j)1070 706 y Fx(;)15 b(V)20 b Fy(\()p Fx(g)1261 721 y FC(k)1323 706 y Fu(\000)g Fx(g)1457 720 y FC(0)1497 706 y Fy(\))1532 632 y Fw(\013)1591 706 y Fx( )1653 727 y Fv(j)1710 706 y Fu(\000)1844 644 y Fx(i)p 1811 685 99 4 v 1811 768 a Fu(j)p Fy(k)p Fu(j)1934 706 y(h)q Fx(V)g( )2105 727 y FC(res)2196 706 y Fx(;)15 b(g)2279 721 y FC(k)2321 706 y Fu(i)h Fx( )2434 727 y FC(res)2545 706 y Fy(+)2657 683 y(~)2636 706 y Fx(C)2701 720 y FC(0)2741 706 y Fx(g)2784 721 y FC(k)2846 706 y Fy(+)k Fu(\001)15 b(\001)g(\001)21 b Fy(+)f Fx(O)s Fy(\()p Fu(j)p Fy(k)p Fu(j)3359 668 y FC(5)3399 706 y Fy(\))-9 988 y(and)29 b(the)i(claim)e(follo)m(ws)h(as)g (in)f(the)i(previous)e(case.)p 1859 988 61 61 v 132 1192 a(The)f(previous)g(lemma)h(giv)m(es)h(relev)-5 b(an)m(t)29 b(information)f(ab)s(out)h(the)h(b)s(eha)m(vior)e(of)h(the)h (generalized)f(eigen-)-9 1305 y(functions)d(in)g(a)i(neigh)m(b)s(orho)s (o)s(d)d(of)j(the)g(p)s(oin)m(t)e(k)f(=)g(0)q(.)67 b(What)28 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y(\014)1005 1072 y Fx(@)1058 1039 y Fv(m)p 956 1112 219 4 v 956 1195 a Fx(@)5 b Fu(j)p Fy(k)p Fu(j)1107 1169 y Fv(m)1194 1110 y Fw(b)1184 1133 y Fy(\011)1255 1147 y FC(out)1361 1133 y Fy(\(k\))1479 1001 y Fw(\014)1479 1056 y(\014)1479 1110 y(\014)1479 1165 y(\014)1535 1133 y Fu(\024)1747 1072 y Fx(C)1812 1086 y Fv(m)p 1641 1112 343 4 v 1641 1195 a Fu(j)p Fy(k)p Fu(j)1739 1169 y FC(3+)p Fv(")p FC(+)p Fv(m)2176 1133 y Fy(for)197 b Fu(j)p Fy(k)p Fu(j)25 b(\025)g Fx(K)2778 1147 y Fv(m)3564 1133 y Fy(\(D)m(A\))-9 1365 y(where)k Fx(C)318 1379 y Fv(m)385 1365 y Fy(,)i Fx(K)518 1379 y Fv(m)615 1365 y Fy(and)e Fx(")i Fy(are)g(suitable)e(p)s (ositiv)m(e)g(constan)m(ts.)-9 1478 y(Although)35 b(this)f(condition)h (is)f(stronger)i(than)g(what)g(w)m(e)g(pro)m(v)m(ed)g(in)f(the)h(case)h (of)e(p)s(oin)m(t)g(in)m(teraction)h(\(see)-9 1591 y(Lemma)f(2.1\))h (the)g(estimate)f(\(47\))i(can)e(b)s(e)g(pro)m(v)m(ed)g(to)h(hold)e (for)g(some)i(solv)-5 b(able)34 b(smo)s(oth)h(p)s(oten)m(tial.)54 b(F)-8 b(or)-9 1704 y(example,)30 b(the)g Fs(Bar)-5 b(gmann)35 b(p)-5 b(otential)1232 1942 y Fx(V)1285 1957 y Fv(b)1320 1942 y Fy(\(x\))26 b(=)f Fu(\000)1788 1881 y Fy(2)p Fx(b)1872 1848 y FC(2)p 1641 1921 420 4 v 1641 2012 a Fy(cosh)1812 1975 y FC(2)1852 2012 y Fy(\()p Fx(b)p Fu(j)p Fy(x)p Fu(j)p Fy(\))2252 1942 y(\()p Fx(b)h(>)e Fy(0\))-9 2180 y(admits)42 b(a)88 b(zero-energy)45 b(resonance)f(and,)i(b)m(y)e(using) e(the)h(explicit)f(form)h(of)h(the)g(Lippman-Sc)m(h)m(winger)-9 2293 y(radial)30 b(w)m(a)m(v)m(efunctions)i(\(see,)i(for)e(example,)g ([CS)o(]\))1770 2260 y FC(\()5 b(4)g(\))1905 2293 y Fy(it)32 b(is)f(p)s(ossible)e(to)k(pro)m(v)m(e)g(that)f(for)g(an)m(y)g(\011)3377 2307 y FC(0)3444 2293 y Fu(2)27 b(S)7 b Fy(\()p Ft(R)3689 2260 y FC(3)3735 2293 y Fy(\))-9 2406 y(the)31 b(corresp)s(onding)744 2383 y Fw(b)734 2406 y Fy(\011)805 2420 y FC(out)942 2406 y Fy(decreases)h(at)h(in\014nit)m(y)-8 b(,)30 b(with)g(its)h (deriv)-5 b(ativ)m(es,)32 b(faster)g(than)f(the)h(in)m(v)m(erse)f(of)h (an)m(y)-9 2519 y(p)s(olynomial.)-9 2715 y Fn(Theorem)h(3.10)47 b Fs(Assume)35 b Fx(V)52 b Fu(2)30 b Fy(\(I\))1308 2742 y Fr(1)1454 2715 y Fs(and)37 b(that)g(the)f(hamiltonian)j Fx(H)f Fy(=)30 b Fx(H)2776 2729 y FC(0)2838 2715 y Fy(+)22 b Fx(V)92 b Fs(has)37 b(a)f(zer)-5 b(o-ener)g(gy)-9 2828 y(r)g(esonanc)g(e)35 b(or/and)g(eigenvalue.)44 b(L)-5 b(et)34 b(b)-5 b(e)33 b Fy(\011)1539 2842 y FC(0)1605 2828 y Fu(2)26 b(H)1769 2842 y Fv(ac)1841 2828 y Fy(\()p Fx(H)7 b Fy(\))22 b Fu(\\)e(S)7 b Fy(\()p Ft(R)2254 2795 y FC(3)2300 2828 y Fy(\))33 b Fs(such)h(that)g(the)g(c)-5 b(orr)g(esp)g(onding)37 b(asymp-)-9 2941 y(totic)e(outgoing)h(state)g Fy(\011)864 2955 y FC(out)1000 2941 y Fy(=)29 b Fx(W)1199 2903 y Fr(\000)p FC(1)1186 2964 y(+)1293 2941 y Fy(\011)1364 2955 y FC(o)1438 2941 y Fs(satis\014es)37 b(the)e(assumption)i(\(D)n (A\).)e(Then)g(r)-5 b(elation)37 b(\(F)-9 b(AS\))35 b(holds,)-9 3054 y(for)e(every)f Fx(T)429 3068 y FC(1)494 3054 y Fu(2)25 b Ft(R)s Fs(.)-9 3249 y Fn(Remark.)74 b Fy(W)-8 b(e)43 b(emphasize)e(that)i(the)f(h)m(yp)s(othesis)e Fx(V)65 b Fu(2)44 b Fy(\(I\))2234 3276 y Fr(1)2350 3249 y Fy(has)e(b)s(een)f(assumed)g(only)g(for)h(sak)m(e)h(of)-9 3362 y(simplicit)m(y)-8 b(.)38 b(The)30 b(pro)s(of)f(w)m(orks)i (assuming)e(that)i Fx(V)45 b Fu(2)25 b Fy(\(I\))1996 3389 y Fv(n)2073 3362 y Fy(for)30 b Fx(n)g Fy(su\016cien)m(tly)f (large.)-9 3564 y(W)-8 b(e)31 b(k)m(eep)g(the)g(notation)f(as)h(close)g (as)f(p)s(ossible)e(to)j(the)g(notation)g(used)e(in)g(Section)h(2)h (and)f(in)f([TDMB)q(].)-9 3765 y Fn(Pro)s(of.)39 b Fy(The)23 b(\014rst)g(part)g(of)h(the)g(pro)s(of)46 b(follo)m(ws)23 b(closely)g(the)h(pro)s(of)e(of)i(Theorem)f(2.2.)40 b(Using)23 b(the)h(prop)s(erties)-9 3878 y(of)30 b Fu(F)159 3892 y FC(+)249 3878 y Fy(and)g(\(3\))h(w)m(e)g(obtain)275 4107 y(\011)346 4121 y Fv(t)375 4107 y Fy(\(x\))84 b(=)731 3984 y Fw(Z)781 4190 y Fm(R)829 4171 y Fq(3)883 4107 y Fx(e)925 4070 y Fr(\000)p Fv(i)p FC(k)1041 4046 y Fq(2)1076 4070 y Fv(t)1116 4084 y Fw(b)1106 4107 y Fy(\011)1177 4121 y FC(out)1283 4107 y Fy(\(k\)\010)1467 4121 y FC(+)1526 4107 y Fy(\(x)p Fx(;)15 b Fy(k\))g(\(2)p Fx(\031)5 b Fy(\))1919 4070 y Fr(\000)p FC(3)p Fv(=)p FC(2)2084 4107 y Fx(dk)577 4347 y Fy(=)731 4223 y Fw(Z)781 4429 y Fm(R)829 4410 y Fq(3)883 4347 y Fx(e)925 4309 y Fr(\000)p Fv(i)p FC(k)1041 4286 y Fq(2)1076 4309 y Fv(t)1116 4324 y Fw(b)1106 4347 y Fy(\011)1177 4361 y FC(out)1283 4347 y Fy(\(k\))p Fx(e)1443 4309 y Fv(i)p FC(k)p Fr(\001)q FC(x)1581 4347 y Fy(\(2)p Fx(\031)t Fy(\))1752 4309 y Fr(\000)p FC(3)p Fv(=)p FC(2)1917 4347 y Fx(dk)24 b Fy(+)2126 4223 y Fw(Z)2176 4429 y Fm(R)2224 4410 y Fq(3)2278 4347 y Fx(e)2320 4309 y Fr(\000)p Fv(i)p FC(k)2436 4286 y Fq(2)2471 4309 y Fv(t)2511 4324 y Fw(b)2501 4347 y Fy(\011)2572 4361 y FC(out)2678 4347 y Fy(\(k\))p Fx(\021)t Fy(\(x)p Fx(;)15 b Fy(k\)\(2)p Fx(\031)5 b Fy(\))3223 4309 y Fr(\000)p FC(3)p Fv(=)p FC(2)3388 4347 y Fx(dk)577 4531 y Fu(\021)83 b Fx(\013)p Fy(\(x)p Fx(;)15 b(t)p Fy(\))21 b(+)f Fx(\014)6 b Fy(\(x)p 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(zero)i(is)f(not)g(an)g(eigen)m(v)-5 b(alue.)p -9 5370 1512 4 v 94 5423 a Fq(4)129 5455 y Fp(It)28 b(is)h(w)n(ell-kno)n(wn)g (that,)g(for)h(a)f(spherically)g(simmetric)f(p)r(oten)n(tial,)i(a)f (zero-energy)g(resonance)h(a\013ects)f(only)f(the)h Fj(s)p Fp(-w)n(a)n(v)n(e)-9 5547 y(comp)r(onen)n(t)19 b(of)i(the)f(scattering) i(op)r(erator)f(\(see,)h(e.g.)34 b([Ba)q(]\).)f(Then)20 b(one)h(needs)f(only)g(the)g Fj(s)p Fp(-w)n(a)n(v)n(e)g(comp)r(onen)n (t)g(of)h(the)f(Lippman-)-9 5638 y(Sc)n(h)n(winger)25 b(w)n(a)n(v)n(efunction.)1835 5887 y Fy(17)p eop %%Page: 18 18 18 17 bop -9 444 a Fn(Estimates)33 b(on)i Fx(\013)g Fn(and)g Fu(r)p Fx(\013)91 b Fy(One)30 b(decomp)s(oses)h Fx(\013)f Fy(as)h Fx(\013)2083 458 y FC(reg)2202 444 y Fy(+)20 b Fx(\013)2351 458 y FC(sing)2509 444 y Fy(where)655 796 y Fx(\013)714 810 y FC(reg)813 796 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))84 b(=)1242 673 y Fw(Z)1292 879 y Fm(R)1340 860 y Fq(3)1394 796 y Fx(e)1436 759 y Fv(i)p FC(k)p Fr(\001)p FC(x)1559 796 y Fx(e)1601 759 y Fr(\000)p Fv(ik)1719 735 y Fq(2)1753 759 y Fv(t)1813 668 y Fw(\022)1890 773 y(b)1880 796 y Fy(\011)1951 810 y FC(out)2057 796 y Fy(\(k\))21 b Fu(\000)2324 735 y Fx(r)p 2297 775 99 4 v 2297 859 a Fu(j)p Fy(k)p Fu(j)2405 796 y Fx(e)2447 759 y Fr(\000j)p FC(k)p Fr(j)2579 735 y Fq(2)2617 668 y Fw(\023)2699 796 y Fy(\(2)p Fx(\031)t Fy(\))2870 759 y Fr(\000)p FC(3)p Fv(=)p FC(2)3036 796 y Fx(dk)628 1040 y(\013)686 1054 y FC(sing)813 1040 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))84 b(=)1242 916 y Fw(Z)1292 1122 y Fm(R)1340 1103 y Fq(3)1394 1040 y Fx(e)1436 1002 y Fv(i)p FC(k)p Fr(\001)p FC(x)1559 1040 y Fx(e)1601 1002 y Fr(\000)p Fv(ik)1719 979 y Fq(2)1753 1002 y Fv(t)1820 978 y Fx(r)p 1793 1019 V 1793 1102 a Fu(j)p Fy(k)p Fu(j)1901 1040 y Fx(e)1943 1002 y Fr(\000j)p FC(k)p Fr(j)2075 979 y Fq(2)2129 1040 y Fy(\(2)p Fx(\031)t Fy(\))2300 1002 y Fr(\000)p FC(3)p Fv(=)p FC(2)2465 1040 y Fx(dk)-9 1295 y Fy(and)29 b(from)h(the)h(prop)s(erties)e(of)h (distributional)d(F)-8 b(ourier)29 b(transform)h(w)m(e)h(get)954 1544 y Fu(r)p Fx(\013)p Fy(\(x)p Fx(;)15 b(t)p Fy(\))27 b(=)e Fx(i)1448 1420 y Fw(Z)1499 1627 y Fm(R)1547 1608 y Fq(3)1600 1544 y Fx(e)1642 1507 y Fv(i)p FC(k)p Fr(\001)q FC(x)1765 1544 y Fx(e)1807 1507 y Fr(\000)p Fv(ik)1925 1483 y Fq(2)1959 1507 y Fv(t)1989 1544 y Fy(k)2047 1521 y Fw(b)2037 1544 y Fy(\011)2108 1558 y FC(out)2214 1544 y Fy(\(k\))15 b(\(2)p Fx(\031)t Fy(\))2518 1507 y Fr(\000)p FC(3)p Fv(=)p FC(2)2684 1544 y Fx(dk)s(:)-9 1802 y Fy(Decomp)s(osition) 29 b(\(47\))j(and)e(Prop)s(osition)f(3.8)i(sho)m(w)f(that)h(the)g (functions)1292 2033 y Fx(f)1337 2047 y FC(1)1376 2033 y Fy(\(k\))84 b(=)1741 2010 y Fw(b)1731 2033 y Fy(\011)1802 2047 y FC(out)1908 2033 y Fy(\(k\))21 b Fu(\000)2175 1972 y Fx(r)p 2148 2012 V 2148 2096 a Fu(j)p Fy(k)p Fu(j)2256 2033 y Fx(e)2298 1996 y Fr(\000j)p FC(k)p Fr(j)2430 1972 y Fq(2)3608 2033 y Fy(\(49\))1292 2240 y Fx(f)1337 2254 y FC(2)1376 2240 y Fy(\(k\))84 b(=)e(k)1789 2217 y Fw(b)1779 2240 y Fy(\011)1850 2254 y FC(out)1956 2240 y Fy(\(k\))1534 b(\(50\))-9 2452 y(satisfy)36 b(the)h(condition)e(\(11\).)61 b(Moreo)m(v)m(er,assumption)38 b(\(D)m(A\))h(implies)34 b(that)j(they)g(satisfy)f(also)h(\(12\).)61 b(So)-9 2565 y(from)29 b(Lemma)i(2.4)g(it)f(follo)m(ws)g(that)808 2826 y Fx(\013)867 2840 y FC(reg)992 2826 y Fy(=)1146 2765 y(1)p 1097 2805 143 4 v 1097 2892 a Fx(t)1130 2866 y FC(3)p Fv(=)p FC(2)1250 2826 y Fu(O)1322 2840 y FC(1)1377 2698 y Fw(\022)1454 2765 y Fu(j)p Fy(x)p Fu(j)p 1454 2805 99 4 v 1487 2888 a Fx(t)1563 2698 y Fw(\023)1826 2826 y Fy(and)182 b Fu(r)o Fx(\013)26 b Fy(=)2469 2765 y(1)p 2420 2805 143 4 v 2420 2892 a Fx(t)2453 2866 y FC(3)p Fv(=)p FC(2)2573 2826 y Fu(O)2645 2840 y FC(1)2700 2698 y Fw(\022)2777 2765 y Fu(j)p Fy(x)p Fu(j)p 2777 2805 99 4 v 2810 2888 a Fx(t)2885 2698 y Fw(\023)3608 2826 y Fy(\(51\))-9 3047 y(Finally)-8 b(,)29 b(as)h(in)f(the)i(pro)s (of)e(of)i(Theorem)f(2.2)i(one)e(pro)m(v)m(es)h(that)1297 3304 y Fx(\013)1355 3318 y FC(sing)1508 3304 y Fy(=)1640 3243 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5245 y Fy(and)820 5204 y Fv(d)p FC(~)-35 b Fv(\032)p 819 5224 75 4 v 819 5276 a(dk)938 5245 y Fy(are)34 b(b)s(ounded)f(in)g(ev)m(ery)i(neigh)m (b)s(orho)s(o)s(d)d(of)i(the)h(origin)d(\(see)k(Prop.)51 b(3.8\).)k(The)-9 5358 y(condition)29 b(\002)461 5372 y FC(1)525 5358 y Fu(2)c Fx(H)694 5325 y FC(1)p Fv(;)p Fr(\000)p Fv(s)840 5358 y Fy(\()p Ft(R)935 5325 y FC(3)981 5358 y Fy(\))30 b(it)g(is)g(su\016cien)m(t)g(to)h(pro)m(v)m(e)g(that) 1094 5475 y Fw(Z)1145 5681 y Fm(R)1193 5662 y Fq(3)1265 5537 y Fu(j)p Fx(V)20 b Fy(\(y\))p Fu(j)p 1256 5578 259 4 v 1256 5661 a(j)p Fy(x)h Fu(\000)f Fy(y)q Fu(j)1627 5537 y(j)p Fy(\002)1723 5551 y FC(1)1763 5537 y Fy(\(y\))p Fu(j)p 1565 5578 403 4 v 1565 5661 a(j)p Fy(x)h Fu(\000)f Fy(y)q Fu(j)h Fy(+)e Fx(t)1978 5598 y(dy)29 b Fu(\024)2382 5537 y Fx(C)p 2205 5578 427 4 v 2205 5661 a Fu(j)p Fy(x)p Fu(j)15 b Fy(\()q Fu(j)p Fy(x)p Fu(j)20 b Fy(+)g Fx(t)p Fy(\))2641 5598 y Fx(:)1835 5887 y Fy(19)p eop %%Page: 20 20 20 19 bop -9 444 a Fy(As)30 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Fy(in)34 b(order)g(to)i(pro)m(v)m(e)g(\(48\).)56 b(Notice)36 b(that,)h(in)d(the)-9 1895 y(presen)m(t)28 b(case,)i(the)f(most)g(singular)e(part)h(of)57 b Fx(\014)1621 1859 y Fr(\003)1661 1895 y Fu(r)p Fx(\014)33 b Fy(is)28 b Fs(not)k(r)-5 b(e)g(al)39 b Fy(as)29 b(w)m(as)g(in)e(the)i(case)g(of) g(p)s(oin)m(t)f(in)m(teraction.)-9 2008 y(The)e(most)g(singular)f(part) h(\(the)h(only)f(one)h(that)g(cannot)g(b)s(e)f(con)m(trolled)g(b)m(y)g (using)f(the)i(previous)e(estimates\))-9 2121 y(is)k(explicitly)f(giv)m (en)i(b)m(y)115 2374 y(j)143 2388 y FC(cr)206 2374 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))84 b(=)f(Im)758 2246 y Fw(\022)825 2250 y(Z)876 2456 y Fm(R)924 2437 y Fq(3)1021 2312 y Fx(V)20 b Fy(\(y\))p 988 2353 259 4 v 988 2436 a Fu(j)p Fy(x)g Fu(\000)g Fy(y)q Fu(j)1256 2374 y Fy(\007)1327 2388 y FC(sing)q Fv(;)p FC(2)1509 2374 y Fy(\(x)p Fx(;)15 b Fy(y)q Fx(;)g(t)p Fy(\))32 b Fx(dy)1916 2246 y Fw(\023)1983 2268 y Fr(\003)2038 2246 y Fw(\022)2104 2374 y Fu(\000)2210 2312 y Fy(x)p 2185 2353 99 4 v 2185 2436 a Fu(j)p Fy(x)p Fu(j)2309 2250 y Fw(Z)2359 2456 y Fm(R)2407 2437 y Fq(3)2524 2312 y Fx(V)20 b Fy(\(y)2681 2279 y Fr(0)2705 2312 y Fy(\))p 2471 2353 322 4 v 2471 2436 a Fu(j)p Fy(x)g Fu(\000)g Fy(y)2705 2409 y Fr(0)2728 2436 y Fu(j)2753 2410 y FC(2)2803 2374 y Fy(\007)2874 2388 y FC(sing)p Fv(;)p FC(2)3055 2374 y Fy(\(x)p Fx(;)15 b Fy(y)3227 2336 y Fr(0)3251 2374 y Fx(;)g(t)p Fy(\))31 b Fx(dy)3485 2336 y Fr(0)3509 2246 y Fw(\023)481 2625 y Fy(=)83 b Fx(C)741 2563 y Fy(x)p 716 2604 99 4 v 716 2687 a Fu(j)p Fy(x)p Fu(j)840 2501 y Fw(Z)890 2707 y Fm(R)938 2688 y Fq(3)992 2501 y Fw(Z)1042 2707 y Fm(R)1090 2688 y Fq(3)1154 2563 y Fx(V)21 b Fy(\(y\))p Fx( )1409 2585 y FC(res)1500 2563 y Fy(\(y\))p 1154 2604 464 4 v 1257 2687 a Fu(j)p Fy(x)f Fu(\000)g Fy(y)q Fu(j)1638 2563 y Fx(V)g Fy(\(y)1795 2530 y Fr(0)1819 2563 y Fy(\))p Fx( )1917 2585 y FC(res)2008 2563 y Fy(\(y)2092 2530 y Fr(0)2116 2563 y Fy(\))p 1638 2604 514 4 v 1734 2687 a Fu(j)p Fy(x)g Fu(\000)g Fy(y)1967 2660 y Fr(0)1991 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4252 y FC([)p Fr(\000)p FC(1)p Fv(;)p FC(0])833 4234 y Fy(of)h(the)g(second)g(term)g (can)g(b)s(e)f(no)m(w)h(used)f(to)i(impro)m(v)m(e)f(the)g(decrease)h (in)d(time)i(of)g(the)-9 4347 y(corresp)s(onding)g(terms,)i(leading)g (to)h(the)f(usual)f(v)-5 b(anishing)28 b(argumen)m(t.)41 b(The)30 b(only)g(di\016cult)e(term)i(is)g(then)347 4595 y(Im\(exp\()p Fx(z)711 4557 y FC(2)771 4595 y Fy(+)862 4521 y Fw(\000)904 4595 y Fx(z)950 4557 y Fr(0)973 4521 y Fw(\001)1015 4544 y FC(2)1055 4595 y Fy(\)\))c(=)f Fx(e)1289 4544 y Fr(\000)1406 4517 y Fq(1)p 1354 4529 136 3 v 1354 4578 a(1+)p Fo(t)1455 4564 y Fq(2)1499 4550 y Fy(\()1534 4544 y Fr(j)p FC(x)p Fr(\000)p FC(y)1684 4520 y Fk(0)1707 4544 y Fr(j)1727 4520 y Fq(2)1761 4544 y FC(+)p Fr(j)p FC(x)p Fr(\000)p FC(y)q Fr(j)1986 4520 y Fq(2)2020 4550 y Fy(\))2075 4595 y(sin)2201 4467 y Fw(\022)2376 4533 y Fx(t)p 2278 4574 229 4 v 2278 4657 a Fy(1)c(+)f Fx(t)2468 4631 y FC(2)2532 4521 y Fw(\000)2574 4595 y Fu(j)p Fy(x)g Fu(\000)g Fy(y)2807 4557 y 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Fq(3)770 5454 y Fw(\000)812 5527 y Fu(j)p Fy(x)20 b Fu(\000)g Fy(y)1045 5494 y Fr(0)1069 5527 y Fu(j)1094 5494 y FC(2)1154 5527 y Fy(+)f Fu(j)p Fy(x)i Fu(\000)f Fy(y)q Fu(j)1503 5494 y FC(2)1543 5454 y Fw(\001)1584 5476 y FC(1)p Fv(=)p FC(4)p 770 5574 925 4 v 942 5657 a Fu(j)p Fy(x)h Fu(\000)e Fy(y)r Fu(jj)p Fy(x)h Fu(\000)g Fy(y)1434 5630 y Fr(0)1458 5657 y Fu(j)1483 5631 y FC(2)1704 5595 y Fx(V)h Fy(\(y\))p Fx( )1959 5616 y FC(res)2050 5595 y Fy(\(y\))p Fx(V)g Fy(\(y)2326 5557 y Fr(0)2349 5595 y Fy(\))p Fx( )2447 5616 y FC(res)2538 5595 y Fy(\(y)2622 5557 y Fr(0)2646 5595 y Fy(\))31 b Fx(dy)i(dy)2932 5557 y Fr(0)2981 5595 y Fu(\024)3155 5533 y Fx(C)p 3087 5574 209 4 v 3087 5661 a Fu(j)p Fy(x)p Fu(j)3185 5634 y FC(5)p Fv(=)p FC(2)1835 5887 y Fy(20)p eop %%Page: 21 21 21 20 bop -9 444 a Fy(and)29 b(then)h(it)g(follo)m(ws)g(that)1008 692 y Fu(j)p Fy(j)1061 706 y FC(cr)1124 692 y Fy(\(x)p Fx(;)15 b(t)p Fy(\))p Fu(j)26 b(\024)1595 630 y Fx(C)1660 644 y FC(1)p 1472 671 352 4 v 1472 758 a Fu(j)p Fy(x)p Fu(j)1570 732 y FC(5)p Fv(=)p FC(2)1680 758 y Fx(t)1713 732 y FC(5)p Fv(=)p FC(4)1853 692 y Fy(+)1987 630 y Fx(C)2052 644 y FC(2)p 1954 671 171 4 v 1954 754 a Fu(j)p Fy(x)p Fu(j)2052 728 y FC(3)2092 754 y Fx(t)2135 692 y Fu(O)2207 710 y FC([)p Fr(\000)p FC(1)p Fv(;)p FC(0])2411 564 y Fw(\022)2493 630 y Fu(j)p Fy(x)p Fu(j)p 2488 671 109 4 v 2488 689 a(p)p 2564 689 33 4 v 74 x Fx(t)2606 564 y Fw(\023)2673 586 y FC(2)2728 692 y Fx(:)-9 944 y Fy(The)k(time)h (decreasing)f(of)i(the)f(\014rst)f(term)h(and)g(the)g(prop)s(ert)m(y)f Fu(O)2273 962 y FC([)p Fr(\000)p FC(1)p Fv(;)p FC(0])2493 944 y Fy(of)h(the)g(second)g(term)g(are)h(su\016cien)m(t)-9 1057 y(to)41 b(emplo)m(y)g(the)g(dominated)f(con)m(v)m(ergence)k (theorem)d(and)f(pro)m(v)m(e)i(\(48\).)74 b(This)39 b(concludes)h(the)i (pro)s(of)e(of)-9 1169 y(Theorem)30 b(3.10)i(in)d(the)h(case)i(in)d (whic)m(h)g(zero)i(is)e(not)i(an)f(eigen)m(v)-5 b(alue)31 b(of)f Fx(H)7 b Fy(.)132 1282 y(If)39 b(zero)i(is)d(an)i(eigen)m(v)-5 b(alue,)42 b(the)e(pro)s(of)g(follo)m(ws)e(a)j(similar)c(line,)j(the)g (main)f(ingredien)m(t)g(b)s(eing)f(again)-9 1395 y(Prop.)i(3.8)p 426 1395 61 61 v 1835 5887 a(21)p eop %%Page: 22 22 22 21 bop -9 444 a Fz(4)134 b(App)t(endix:)59 b(some)46 b(remarks)f(on)g(zero-energy)h(resonances)-9 650 y Fy(F)-8 b(rom)31 b(a)g(phenomenological)f(p)s(oin)m(t)g(of)h(view,)f(the)h (idea)g(of)g Fs(quantum)i(r)-5 b(esonanc)g(e)39 b Fy(is)30 b(related)h(to)h(a)f(\\bump")-9 763 y(of)k(an)h(observ)-5 b(able)35 b(quan)m(tit)m(y)h(as)g(a)g(function)f(of)h(some)g(exp)s (erimen)m(tal)f(parameter,)i(lik)m(e)e({)i(for)e(example)h({)-9 876 y(the)30 b(scattering)h(cross)g(section)f(as)h(a)f(function)g(of)g (the)h(energy)f(of)h(the)f(incoming)f(particles.)3197 843 y FC(\()t(5)5 b(\))-9 989 y Fy(The)23 b(app)s(earing)g(of)h(these)g (\\bumps")f(in)g(the)h(scattering)g(cross-section)h(can)f(b)s(e)f (related,)j(from)d(a)i(theoretical)-9 1101 y(p)s(oin)m(t)k(of)h(view,)g 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5426 y Fp(\()p Fj(\024)p Fp(\))p Ff(j)p Fj(V)17 b Ff(j)1937 5390 y Fq(1)p Fo(=)p Fq(2)2188 5426 y Fp(\()p Fj(\024)22 b Ff(2)f Fe(C)2404 5434 y Fq(+)2461 5426 y Fp(\))p Fj(:)-9 5547 y Fp(This)e(symmetric)d(decomp)r(osition)j(can)g (b)r(e)f(useful)h(in)f(order)h(to)g(de\014ne)f(a)g(con)n(v)n(enien)n(t) g(meromorphic)f(extension)i(of)g Fj(R)q Fp(\()p Fj(\024)p Fp(\))s Ff(\000)s Fj(R)3633 5555 y Fq(0)3666 5547 y Fp(\()p Fj(\024)p Fp(\))-9 5638 y(to)25 b(a)h(region)h(\012)f(larger)h(than)e Fe(C)905 5646 y Fq(+)962 5638 y Fp(.)1835 5887 y Fy(23)p eop %%Page: 24 24 24 23 bop -9 444 a Fy(is)31 b(a)h Fs(zer)-5 b(o-ener)g(gy)35 b(r)-5 b(esonanc)g(e)41 b Fy(and)32 b Fx( )j Fy(is)c(called)h Fs(r)-5 b(esonanc)g(e)36 b(function)p Fy(.)46 b(Notice)33 b(that)g Fx(\024)3048 458 y FC(0)3115 444 y Fy(=)28 b(0)33 b(can)f(b)s(e)g(b)s(oth)-9 557 y(\(and)e(sim)m(ultaneously\))e(an)i (eigen)m(v)-5 b(alue)31 b(for)f Fx(H)37 b Fy(and)30 b(a)h(resonance)g (for)f(the)g(pair)f(\()p Fx(H)r(;)15 b(H)3047 571 y FC(0)3087 557 y Fy(\))31 b(\(see)g(Lemma)g(3\).)132 670 y(F)-8 b(or)25 b(exp)s(onen)m(tially)d(deca)m(ying)j(p)s(oten)m(tials,)g(it)f (is)g(p)s(ossible)e(to)j(\014nd)47 b(a)25 b(relationship)c(b)s(et)m(w)m (een)k(to)h(p)s(oles)d(of)-9 783 y(the)h(resolv)m(en)m(t)h(and)e(the)i (p)s(oles)e(of)h(the)h(analytically)d(con)m(tin)m(ued)i(scattering)h (matrix)f(and)f(then)h(a)h(connection)-9 896 y(with)k(observ)-5 b(able)29 b(quan)m(tities)h(is)f(p)s(ossible)f(\(see)k(again)e([Ba)q(]) h(for)f(details\).)-9 1145 y Fi(4.2)112 b(General)37 b(theory)h(of)f(zero-energy)h(resonances)-9 1318 y Fy(In)27 b(the)i(previous)e(analysis,)h(w)m(e)h(in)m(tro)s(duced)e(the)h Fs(exp)-5 b(onential)5 b(ly)33 b(weighte)-5 b(d)39 b Fy(Sob)s(olev)28 b(spaces)h Fx(H)3336 1285 y Fv(m)3329 1340 y Fr(\006)p Fv(\033)3458 1318 y Fy(in)f(order)-9 1431 y(to)f(obtain)f(an)g Fs(analytic)33 b Fy(con)m(tin)m(uation)26 b(of)h(the)f(free)h(resolv)m(en)m(t)g(to)g(the)g(op)s(en)f(region)g (\012)2975 1445 y Fv(a)3041 1431 y Fu(\023)f Ft(C)3197 1445 y FC(+)3262 1431 y Fy(.)39 b(This)25 b(forced)-9 1544 y(us)d(to)i(consider)e(exp)s(onen)m(tially)g(decreasing)h(p)s (oten)m(tials,)h(to)g(mak)m(e)g Fx(G)2382 1558 y FC(0)2422 1544 y Fx(V)43 b Fy(a)24 b(w)m(ell-de\014ned)d(compact)k(op)s(erator)-9 1657 y(on)30 b Fx(H)200 1624 y FC(2)193 1679 y Fr(\000)p Fv(a)289 1657 y Fy(.)132 1770 y(Ho)m(w)m(ev)m(er,)i(w)m(e)f(are)g(only) e(in)m(terested)h(in)f Fs(zer)-5 b(o-ener)g(gy)34 b(r)-5 b(esonanc)g(es)p Fy(,)32 b(since)d(only)h(the)g(p)s(oles)f(of)i(the)f (resol-)-9 1882 y(v)m(en)m(t)37 b(on)g(the)g Fs(r)-5 b(e)g(al)40 b(axis)45 b Fy(\(in)36 b(momen)m(tum)g(complex)h(plane\))f (can)h(a\013ect)h(the)f(b)s(eha)m(vior)f(of)h(the)g(Lippman-)-9 1995 y(Sc)m(h)m(winger)32 b(generalized)h(eigenfunctions.)81 b(T)-8 b(o)34 b(de\014ne)e(zero-energy)j(resonances,)g(w)m(e)e(need)g (only)g(to)h Fs(c)-5 b(on-)-9 2108 y(tinuously)34 b Fy(extend)27 b(the)f(free)g(resolv)m(en)m(t)h(to)f(the)h(closed)f(upp)s(er)e 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(to)i(asso)s(ciate)g(the)f(\(p)s(ossible\))f(complemen)m(tary)h (subspace)f(to)i(a)g(zero-energy)g(resonance.)p -9 5459 1512 4 v 94 5515 a Fq(8)129 5547 y Fp(In)23 b(other)h(w)n(ords,)h (while)f Ff(\000)p Fp(\001)g(has)g(only)g(a)g Fh(right)31 b Fp(in)n(v)n(erse)24 b Fj(G)1871 5555 y Fq(0)1929 5547 y Fp(on)g Ff(S)2087 5515 y Fk(0)2109 5547 y Fp(\()p Fe(R)2190 5515 y Fo(d)2229 5547 y Fp(\),)g(the)f(restriction)35 b Ff(\000)p Fp(\001)p Ff(j)2955 5569 y Fc(M)3047 5547 y Fp(has)24 b(a)g Fh(bilater)l(al)32 b Fp(in)n(v)n(erse)1 5638 y Fj(G)61 5646 y Fq(0)95 5638 y Ff(j)117 5661 y Fc(N)1835 5887 y Fy(24)p eop %%Page: 25 25 25 24 bop -9 444 a Fy(Ho)m(w)m(ev)m(er,)35 b(b)s(efore)c(form)m (ulating)h(a)g(precise)g(de\014nition,)f(w)m(e)h(relate)h(this)e (approac)m(h)i(to)g(other)f(p)s(ossible)e(def-)-9 557 y(initions)j(of)j(zero)g(energy-resonances,)j(in)34 b(particular)g(to)j (the)f(ones)g(used)f(in)f(Sec.)57 b(3)37 b(\(in)d(particular)h([Ik],)-9 670 y([TDMB)q(],)30 b([JK],)h([Ag)q(]\).)41 b(A)31 b(connection)f(with) f(the)i(approac)m(h)f(outlined)f(in)g(Note)j(7)e(is)g(p)s(ossible)e(to) s(o.)-9 872 y Fn(Prop)s(osition)36 b(4.3)46 b Fs(Assume)32 b Fx(V)46 b Fu(2)25 b Fy(\(I\))1362 899 y FC(3)1402 872 y Fs(.)41 b(Then)33 b(the)g(fol)5 b(lowing)34 b(pr)-5 b(op)g(ositions)36 b(ar)-5 b(e)34 b(e)-5 b(quivalent:)76 1075 y Fy(\(i\))46 b(\()p Fl(standar)-5 b(d)33 b(de\014nition)p Fy(\))f Fs(ther)-5 b(e)31 b(exists)f(a)37 b Fy(distributional)26 b Fs(solution)31 b Fx( )2705 1097 y FC(1)2770 1075 y Fu(2)24 b Fx(L)2917 1042 y FC(2)2917 1103 y(lo)r(c)3010 1075 y Fy(\()p Ft(R)3105 1042 y FC(3)3150 1075 y Fy(\))30 b Fs(of)h(the)f(station-)218 1199 y(ary)36 b(Schr\177)-46 b(odinger)36 b(e)-5 b(quation)36 b Fy(\()p Fu(\000)p Fy(\001)20 b(+)f Fx(V)i Fy(\))15 b Fx( )1732 1221 y FC(1)1800 1199 y Fy(=)29 b(0)36 b Fs(such)f(that)h Fy(\(1)22 b(+)g Fu(j)g(\001)g(j)2692 1166 y FC(2)2732 1199 y Fy(\))2767 1163 y Fr(\000)2832 1132 y Fo(\015)p 2832 1148 36 3 v 2835 1189 a Fq(2)2882 1199 y Fx( )2944 1221 y FC(1)3013 1199 y Fu(2)29 b Fx(L)3165 1166 y FC(2)3204 1199 y Fy(\()p Ft(R)3299 1166 y FC(3)3345 1199 y Fy(\))35 b Fs(for)g(every)218 1312 y Fx(\015)30 b(>)25 b Fy(1)p Fx(=)p Fy(2)34 b Fs(but)e(not)i(for)f Fx(\015)d Fy(=)25 b(0)p Fs(.)51 1496 y Fy(\(ii\))45 b(\()p Fl(A)-5 b(gmon)56 b(de\014nition)p Fy(\))49 b Fs(ther)-5 b(e)50 b(exists)f(a)g Fx( )1813 1518 y FC(2)1907 1496 y Fu(2)54 b Fx(H)2105 1463 y FC(2)p Fv(;)p Fr(\000)p Fv(\015)2290 1496 y Fu(\\)32 b Fx(C)2448 1510 y Fr(1)2571 1496 y Fy(\()p Fs(for)49 b(every)g Fx(\015)59 b(>)3265 1460 y FC(1)p 3265 1475 36 4 v 3265 1528 a(2)3310 1496 y Fy(\))49 b Fs(such)g(that)218 1609 y Fy(\(1)21 b(+)f Fx(G)481 1623 y FC(0)521 1609 y Fx(V)g Fy(\))15 b Fx( )707 1631 y FC(2)772 1609 y Fy(=)25 b(0)32 b Fs(but)h Fx( )1161 1631 y FC(2)1236 1609 y Fx(=)-55 b Fu(2)25 b Fx(L)1374 1576 y FC(2)1413 1609 y Fs(.)26 1793 y Fy(\(iii\))44 b(\()p Fl(Jensen-Kato)57 b(de\014nition)p Fy(\))50 b Fs(ther)-5 b(e)50 b(exists)g(a)f Fx( )2071 1815 y FC(3)2166 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y Fy(\(v\))46 b(\()p Fl(Ikeb)-5 b(e)39 b(de\014nition)p Fy(\))34 b Fs(ther)-5 b(e)33 b(exists)g(a)g Fx( )1642 2410 y FC(5)1706 2388 y Fu(2)25 b Fx(C)1857 2402 y Fr(1)1932 2388 y Fy(\()p Ft(R)2027 2355 y FC(3)2072 2388 y Fy(\))33 b Fs(such)g(that)h Fy(\(1)21 b(+)f Fx(G)2796 2402 y FC(0)2836 2388 y Fx(V)g Fy(\))15 b Fx( )3022 2410 y FC(5)3087 2388 y Fy(=)25 b(0)33 b Fs(but)f Fx( )3476 2410 y FC(5)3551 2388 y Fx(=)-55 b Fu(2)25 b Fx(L)3689 2355 y FC(2)3728 2388 y Fs(.)-9 2590 y(In)42 b(addition,)k(al)5 b(l)42 b(the)g(functions)h Fx( )1280 2612 y Fv(i)1350 2590 y Fy(\()p Fx(i)g Fy(=)f(1)p Fx(;)15 b(:)g(:)g(:)i(;)e Fy(5\))43 b Fs(ar)-5 b(e)43 b(the)f(same)h(element)f(of)84 b Fu(S)3074 2557 y Fr(0)3097 2590 y Fy(\()p Ft(R)3192 2557 y FC(3)3237 2590 y Fy(\))43 b Fs(that)g(wil)5 b(l)42 b(b)-5 b(e)-9 2703 y(denote)g(d)47 b(by)e Fx( )530 2725 y FC(res)621 2703 y Fs(.)81 b(Mor)-5 b(e)g(over,)50 b(if)45 b(the)h(p)-5 b(otential)48 b(satis\014es)e (\(61\))h(\(and)g(is)f(a)g(multiplic)-5 b(ation)48 b(op)-5 b(er)g(ator\))-9 2816 y(then)39 b Fx( )261 2838 y FC(res)388 2816 y Fu(2)d Fx(H)568 2783 y FC(2)561 2839 y Fr(\000)p Fv(a)657 2816 y Fy(\()p Ft(R)752 2783 y FC(3)797 2816 y Fy(\))j Fs(and)h(e)-5 b(ach)39 b(one)g(of)g(the)g(pr)-5 b(evious)40 b(pr)-5 b(op)g(ositions)42 b(implies)d(that)h(the)f(mer)-5 b(omorphic)-9 2929 y(c)g(ontinuation)27 b(of)f(the)g(r)-5 b(esolvent,)28 b(se)-5 b(en)25 b(as)i(a)e Fu(B)s Fy(\()p Fx(H)1704 2952 y Fv(a)1745 2929 y Fx(;)15 b(H)1868 2896 y FC(2)1861 2952 y Fr(\000)p Fv(a)1958 2929 y Fy(\))p Fs(-value)-5 b(d)26 b(function,)h(has)g(a)f(c)-5 b(omplex)27 b(p)-5 b(ole)27 b(in)e Fx(\024)h Fy(=)f(0)p Fs(.)-9 3132 y Fn(De\014nition)35 b(4.4)g(\(Zero-energy)g(resonances\))46 b Fs(We)33 b(say)g(that)i(ther)-5 b(e)34 b(is)e(a)i(zer)-5 b(o-ener)g(gy)34 b(r)-5 b(esonanc)g(e)34 b(for)-9 3245 y(the)f(p)-5 b(air)34 b Fy(\()p Fx(H)r(;)15 b(H)558 3259 y FC(0)597 3245 y Fy(\))33 b Fs(if)65 b(one)33 b(of)g(the)g(c)-5 b(onditions)34 b(of)f(Pr)-5 b(op)g(osition)35 b(4.3)f(holds)g(true.)-9 3447 y Fn(Remark.)d Fy(F)-8 b(rom)33 b(the)f(fact)h(that)f Fu(\000)p Fy(\001)p Fx(G)1410 3461 y FC(0)1478 3447 y Fy(=)27 b(1)33 b(it)e(follo)m(ws)g(straigh)m(tforw)m(ardly)g(that)i(if) e Fx( )g Fu(2)d(S)3286 3414 y Fr(0)3309 3447 y Fy(\()p Ft(R)3404 3414 y FC(3)3449 3447 y Fy(\))33 b(satisfy)-9 3560 y(the)d(equation)1537 3786 y(\()q(1)20 b(+)g Fx(G)1800 3800 y FC(0)1840 3786 y Fx(V)g Fy(\))c Fx( )29 b Fy(=)c(0)1415 b(\(64\))-9 3948 y(then)30 b(it)g(satis\014es)f(also)i(the)f(equation) 1569 4061 y Fu(\000)p Fy(\001)p Fx( )23 b Fy(+)d Fx(V)g( )29 b Fy(=)c(0)1417 b(\(65\))-9 4222 y(pro)m(vided)20 b(that)j Fx(V)d( )26 b Fy(is)21 b(a)i(w)m(ell-de\014ned)d(distribution.)1872 4189 y FC(\()t(9)5 b(\))1996 4222 y Fy(The)22 b(con)m(v)m(erse)i (statemen)m(t)g(is)d(not)i(true)f(in)f(general,)-9 4335 y(since)29 b(a)i Fs(left)39 b Fy(in)m(v)m(erse)30 b(of)h Fu(\000)p Fy(\001)f(in)f Fu(S)1201 4302 y Fr(0)1284 4335 y Fy(do)s(es)i(not)f(exist.)-9 4541 y Fn(Pro)s(of)35 b(of)h(prop)s(osition.)48 b Fy(W)-8 b(e)31 b(will)d(sho)m(w)j(that)g (\(i\))f Fu(\))p Fy(\(ii\))g Fu(\))p Fy(\(\(iii\))f(or)h(\(v\)\))p Fu(\))p Fy(\(i\).)73 b(Moreo)m(v)m(er,)33 b(\(iii\))p Fu(,)p Fy(\(iv\))-9 4654 y(as)d(a)h(consequence)g(of)g(Lemma)f(3.ii)g (.)132 4767 y Fn(\(i\))23 b Fu(\))p Fn(\(ii\).)37 b Fy(Supp)s(ose)j (that)21 b Fx( )1181 4789 y FC(1)1245 4767 y Fu(\021)k Fx( )k Fu(2)c Fx(L)1577 4734 y FC(2)1577 4795 y(lo)r(c)1669 4767 y Fy(\()p Ft(R)1764 4734 y FC(3)1809 4767 y Fy(\))d(satisfy)e(the) h(Sc)m(hr\177)-45 b(odinger)20 b(equation)g(in)g(distributional)-9 4880 y(sense.)49 b(F)-8 b(rom)34 b(the)g(fact)g(that)g(the)g(k)m(ernel) f(of)g Fu(\000)p Fy(\001)g(is)g(closed)g(in)65 b Fu(S)2339 4847 y Fr(0)2363 4880 y Fy(\()p Ft(R)2458 4847 y FC(3)2503 4880 y Fy(\))34 b(it)f(follo)m(ws,)g(b)m(y)h(a)f(decomp)s(osition)-9 4993 y(argumen)m(t,)e(that)g(the)f(general)h(solution)d(of)j(equation)f (\(65\))i(m)m(ust)e(b)s(e)g(in)f(the)i(form)1580 5187 y Fx( )e Fy(=)c Fx(G)1835 5201 y FC(0)1874 5187 y Fx(V)c( )i Fy(+)d Fx(')1428 b Fy(\(66\))-9 5381 y(where)42 b Fx(')i Fy(is)e(an)h(harmonic)g(distribution,)g(i.e.)79 b(\001)p Fx(')47 b Fy(=)f(0.)123 b(W)-8 b(e)45 b(are)e(going)h(to)g(sho)m(w)f (that)h(from)f(our)-9 5494 y(assumptions)28 b(it)i(follo)m(ws)g(that)61 b Fx(')26 b Fy(=)f(0.)p -9 5552 1512 4 v 94 5606 a Fq(9)129 5638 y Fp(This)h(condition)g(is)g(satis\014ed)h(for)f(our)g(class)h(of) f(p)r(oten)n(tials)h(if)f Fj( )e Ff(2)e Fj(L)2159 5606 y Fq(2)2159 5653 y(lo)r(c)1835 5887 y Fy(25)p eop %%Page: 26 26 26 25 bop 132 444 a Fy(Indeed)31 b Fx(')e Fu(2)f Fx(L)668 411 y FC(2)668 466 y Fr(\000)p Fv(\015)768 444 y Fy(\()p Ft(R)863 411 y FC(3)908 444 y Fy(\))33 b(\(in)e(the)i(follo)m(wing)d(w) m(e)j(understand)e Fx(\015)i Fu(2)c Fy(\()2492 408 y FC(1)p 2492 423 36 4 v 2492 475 a(2)2537 444 y Fx(;)15 b Fy(+)p Fu(1)p Fy(\)\).)48 b(This)30 b(follo)m(ws)i(from)f(\(66\))-9 571 y(b)m(y)j(noticing)g(that)i Fx( )g Fu(2)d Fx(L)927 538 y FC(2)927 593 y Fr(\000)p Fv(\015)1026 571 y Fy(\()p Ft(R)1121 538 y FC(3)1166 571 y Fy(\))j(b)m(y)e(h)m(yp)s(othesis)1820 538 y FC(\()5 b(10)t(\))1993 571 y Fy(and)34 b(that)i Fx(G)2447 585 y FC(0)2487 571 y Fx(V)20 b( )38 b Fy(also)d(b)s(elongs)f (to)i Fx(L)3353 538 y FC(2)3353 593 y Fr(\000)p Fv(\015)3452 571 y Fy(\()p Ft(R)3547 538 y FC(3)3592 571 y Fy(\),)h(as)-9 683 y(can)30 b(b)s(e)g(pro)m(v)m(ed)h(b)m(y)f(using)f(the)h(explicit)f (form)h(of)h Fx(G)1808 697 y FC(0)1878 683 y Fy(as)f(con)m(v)m(olution) h(op)s(erator)f(and)g(Jensen)g(inequalit)m(y)-8 b(.)132 796 y(In)31 b(particular)f Fx(')i Fy(is)f(an)g(harmonic)g(distribution) d(that)k(b)s(elongs)f(to)h Fx(L)2577 763 y FC(1)2577 824 y(lo)r(c)2669 796 y Fy(,)g(and)f(then)h(it)f(is)g(\(iden)m (ti\014able)-9 909 y(with\))e(a)i(smo)s(oth)f(function)f(in)g Fx(C)1164 876 y Fr(1)1238 909 y Fy(\()p Ft(R)1333 876 y FC(3)1379 909 y Fy(\))h(\(see)i([LL],)e(Th.)40 b(9.3\).)132 1022 y(Moreo)m(v)m(er,)c Fx(')e Fy(m)m(ust)f(b)s(e)g Fs(b)-5 b(ounde)g(d)p Fy(.)51 b(In)32 b(order)h(to)h(see)g(this,)f (supp)s(ose)f(that)i Fx(')g Fy(is)e(un)m(b)s(ounded.)47 b(Then)32 b(it)-9 1135 y(is)d(p)s(ossible)g(to)i(\014nd)e(a)j(sequence) f Fu(f)p Fy(x)1265 1149 y Fv(n)1312 1135 y Fu(g)1358 1162 y Fv(n)p Fr(2)p Fm(N)1530 1135 y Fy(suc)m(h)g(that)62 b Fu(j)p Fx(')q Fy(\(x)2132 1149 y Fv(n)2179 1135 y Fy(\))p Fu(j)26 b(\025)g Fx(n)p Fy(.)41 b(The)31 b(con)m(tin)m(uit)m(y)f(of)h Fx(')g Fy(implies)d(that)-9 1248 y(this)36 b(sequence)i(cannot)g(ha)m (v)m(e)h(accum)m(ulation)e(p)s(oin)m(ts.)62 b(Then)36 b(it)h(is)g(p)s(ossible)e(to)k(extract)g(a)f(subsequence)-9 1287 y Fw(\010)44 1361 y Fy(x)92 1375 y Fv(n)135 1385 y Fo(j)196 1361 y Fu(\021)25 b Fy(w)358 1375 y Fv(j)394 1287 y Fw(\011)447 1398 y Fv(j)t Fr(2)p Fm(N)609 1361 y Fy(suc)m(h)30 b(that:)76 1593 y(\(i\))46 b Fu(j)p Fx(')q Fy(\(w)404 1607 y Fv(j)440 1593 y Fy(\))p Fu(j)26 b(\025)f Fx(n)677 1607 y Fv(j)738 1593 y Fu(\025)g Fx(j)66 b Fy(and)61 b Fu(j)p Fy(w)1236 1607 y Fv(j)1272 1593 y Fu(j)25 b(\025)g Fy(2)1463 1560 y Fv(j)51 1780 y Fy(\(ii\))45 b(if)29 b Fx(\016)345 1794 y Fv(j)407 1780 y Fy(=)513 1744 y FC(1)p 513 1759 V 513 1811 a(2)558 1780 y Fu(j)p Fy(w)649 1794 y Fv(j)685 1780 y Fu(j)61 b Fy(then)g Fx(B)1078 1795 y Fv(\016)1111 1805 y Fo(j)1148 1780 y Fy(\(w)1249 1794 y Fv(j)1285 1780 y Fy(\))21 b Fu(\\)50 b Fx(B)1521 1795 y Fv(\016)1555 1807 y Fo(l)1583 1780 y Fy(\(w)1684 1795 y Fv(l)1710 1780 y Fy(\))26 b(=)f Fu(;)30 b Fy(if)g Fx(j)h Fu(6)p Fy(=)25 b Fx(l)r Fy(.)-9 1990 y(By)30 b(using)f(the)i (disjoin)m(tedness)d(of)i(the)h(balls)59 b Fx(B)1677 2004 y Fv(j)1739 1990 y Fu(\021)25 b Fx(B)1904 2005 y Fv(\016)1937 2015 y Fo(j)1974 1990 y Fy(\(w)2075 2004 y Fv(j)2112 1990 y Fy(\))30 b(w)m(e)h(get)h(that)864 2245 y Fu(k)p Fx(')q Fu(k)1014 2272 y Fv(L)1062 2249 y Fq(2)1062 2293 y Fk(\000)p Fo(\015)1237 2245 y Fy(=)1391 2121 y Fw(Z)1442 2327 y Fm(R)1490 2308 y Fq(3)1543 2245 y Fu(j)q Fx(')p Fy(\(y\))p Fu(j)1772 2203 y FC(2)1827 2245 y Fy(\(1)21 b(+)f Fu(j)p Fy(y)q Fu(j)2118 2207 y FC(2)2158 2245 y Fy(\))2193 2207 y Fr(\000)p Fv(\015)2323 2245 y Fx(d)p Fy(y)27 b Fu(\025)1237 2484 y(\025)1391 2398 y Fw(X)1393 2595 y Fv(j)t Fr(2)p Fm(N)1538 2360 y Fw(Z)1588 2567 y Fv(B)1641 2577 y Fo(j)1693 2484 y Fu(j)p Fx(')q Fy(\(y\))p Fu(j)1922 2442 y FC(2)1976 2484 y Fy(\(1)21 b(+)f Fu(j)p Fy(y)q Fu(j)2267 2447 y FC(2)2307 2484 y Fy(\))2342 2447 y Fr(\000)p Fv(\015)2472 2484 y Fx(d)p Fy(y)27 b Fu(\025)1237 2774 y(\025)1391 2688 y Fw(X)1393 2885 y Fv(j)t Fr(2)p Fm(N)1578 2774 y Fy(inf)1548 2835 y FC(y)q Fr(2)p Fv(B)1686 2845 y Fo(j)1744 2700 y Fw(\010)1797 2774 y Fy(\(1)21 b(+)f Fu(j)p Fy(y)q Fu(j)2088 2737 y FC(2)2128 2774 y Fy(\))2163 2737 y Fr(\000)p Fv(\015)2263 2700 y Fw(\011)2331 2650 y(Z)2381 2856 y Fv(B)2434 2866 y Fo(j)2486 2774 y Fu(j)q Fx(')p Fy(\(y\))p Fu(j)2715 2732 y FC(2)2800 2774 y Fx(d)p Fy(y)-9 3088 y(>F)-8 b(rom)26 b(the)h(fact)g(that)g(\001)p Fx(')f Fy(=)f(0)h(it)g(follo)m(ws)g(that)h Fu(j)p Fx(')p Fu(j)1801 3047 y FC(2)1867 3088 y Fy(is)f(a)g (subharmonic)e(function,)i(i.e.)40 b(\001)15 b Fu(j)p Fx(')p Fu(j)3293 3047 y FC(2)3357 3088 y Fu(\025)25 b Fy(0.)40 b(Then)-9 3201 y(w)m(e)30 b(can)h(apply)e(the)i(mean)f(v)-5 b(alue)30 b(inequalit)m(y)f(getting)869 3436 y Fu(k)p Fx(')q Fu(k)1019 3463 y Fv(L)1067 3440 y Fq(2)1067 3484 y Fk(\000)p Fo(\015)1242 3436 y Fu(\025)1396 3349 y Fw(X)1398 3547 y Fv(j)t Fr(2)p Fm(N)1583 3436 y Fy(inf)1552 3496 y FC(y)q Fr(2)p Fv(B)1690 3506 y Fo(j)1749 3362 y Fw(\010)1802 3436 y Fy(\(1)21 b(+)f Fu(j)p Fy(y)q Fu(j)2093 3398 y FC(2)2133 3436 y Fy(\))2168 3398 y Fr(\000)p Fv(\015)2267 3362 y Fw(\011)2346 3374 y Fy(4)p 2346 3415 46 4 v 2346 3498 a(3)2401 3436 y Fx(\031)s(\016)2500 3398 y FC(3)2500 3458 y Fv(j)2554 3436 y Fu(j)q Fx(')p Fy(\(w)2740 3450 y Fv(j)2777 3436 y Fy(\))p Fu(j)2837 3394 y FC(2)1242 3752 y Fu(\025)1396 3666 y Fw(X)1398 3863 y Fv(j)t Fr(2)p Fm(N)1542 3752 y Fx(C)1859 3682 y(\016)1902 3645 y FC(3)1902 3704 y Fv(j)2033 3682 y Fx(j)2075 3649 y FC(2)p 1624 3732 726 4 v 1624 3815 a Fy(\(1)h(+)f(\()p Fu(j)p Fy(w)1942 3829 y Fv(j)1978 3815 y Fu(j)h Fy(+)f Fx(\016)2159 3829 y Fv(j)2195 3815 y Fy(\))2230 3789 y FC(2)2270 3815 y Fy(\))2305 3789 y Fv(\015)-9 4050 y Fy(By)45 b(using)f(the)h(claimed)f (prop)s(erties)g(of)h(the)g(sequence)h Fu(f)p Fy(w)2155 4064 y Fv(j)2191 4050 y Fu(g)2237 4081 y Fv(j)t Fr(2)p Fm(N)2413 4050 y Fy(one)g(gets)g(that)g(the)f(previous)f(series)-9 4178 y(div)m(erges,)30 b(against)h(the)f(fact)i(that)f Fx(')25 b Fu(2)g Fx(L)1444 4145 y FC(2)1444 4200 y Fr(\000)p Fv(\015)1543 4178 y Fy(\()p Ft(R)1638 4145 y FC(3)1684 4178 y Fy(\).)41 b(Then)29 b Fx(')i Fy(is)e(b)s(ounded.)132 4301 y(By)43 b(Liouville's)e(theorem,)47 b(a)d(b)s(ounded)d(harmonic)h (function)g(on)h Ft(R)2582 4268 y Fv(d)2671 4301 y Fy(is)g(constan)m(t) h(and)f(a)h(constan)m(t)-9 4414 y(function)29 b(that)i(b)s(elongs)e(to) i Fx(L)1043 4381 y FC(2)1043 4436 y Fr(\000)p Fv(\015)1142 4414 y Fy(\()p Ft(R)1237 4381 y FC(3)1283 4414 y Fy(\))f(\(for)h(ev)m (ery)g Fx(\015)f(>)1946 4378 y FC(1)p 1946 4393 36 4 v 1946 4445 a(2)1991 4414 y Fy(\))h(m)m(ust)f(b)s(e)g(iden)m(tically)e (zero.)42 b(Then)29 b Fx(')d Fy(=)f(0.)132 4527 y(This)35 b(pro)m(v)m(es)i(that)h Fx( )i Fy(satis\014es)c(equation)h(\(64\))h(in) e(distributional)c(sense.)60 b(It)37 b(remains)f(to)h(b)s(e)g(sho)m(wn) -9 4640 y(that)31 b Fx( )d Fu(2)d Fx(H)444 4607 y FC(2)p Fv(;)p Fr(\000)p Fv(\015)618 4640 y Fu(\\)20 b Fx(C)764 4654 y Fr(1)839 4640 y Fy(.)132 4752 y(Since)i Fx( )29 b Fy(=)c Fx(G)617 4766 y FC(0)657 4752 y Fx(V)20 b( )t Fy(,)25 b(an)e(iterativ)m(e)h(or)g(\\b)s(o)s(otstrap")g(argumen)m(t)g (based)f(up)s(on)f(the)i(smo)s(othing)f(prop)s(erties)-9 4865 y(of)36 b(the)72 b(op)s(erator)36 b Fx(G)741 4879 y FC(0)817 4865 y Fy(\(see,)i(for)e(example,)h([LL])f(Sec.)58 b(10.2\))38 b(sho)m(ws)d(that)i Fx( )h Fu(2)c Fx(C)2944 4832 y FC(0)p Fv(;\013)3048 4865 y Fy(\()p Ft(R)3143 4832 y FC(3)3188 4865 y Fy(\),)k(the)e(space)h(of)-9 4978 y(uniformly)27 b(H\177)-45 b(older-con)m(tin)m(uous)30 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Fq(2)1557 5638 y Fp(\))1597 5587 y Fb(s)p 1597 5595 29 3 v 1597 5628 a Fa(2)1640 5638 y Fj(f)29 b Ff(2)22 b Fj(L)1831 5606 y Fq(2)1891 5638 y Fp(.)1835 5887 y Fy(26)p eop %%Page: 27 27 27 26 bop -9 444 a Fy(Cho)s(ose)24 b Fx(\015)353 466 y FC(0)418 444 y Fy(=)524 408 y FC(1)p 524 423 36 4 v 524 475 a(2)578 444 y Fy(+)670 408 y Fv(")p 668 423 V 668 475 a FC(2)738 444 y Fy(\(with)g Fx(")h Fy(from)g(\(67\)\).)41 b(Fix)24 b(a)h(compact)h(set)g Fx(K)31 b Fy(suc)m(h)25 b(that)g Fx(K)32 b Fu(\023)25 b Fx(B)3008 458 y FC(2)p Fv(R)3101 444 y Fy(.)64 b(The)24 b(con)m(tin)m(uit)m(y)-9 557 y(of)29 b Fx( )k Fy(implies)26 b(that)k(it)f(is)f(b)s(ounded)f(o)m (v)m(er)j(the)g(compact)g(set)g Fx(K)7 b Fy(.)69 b(T)-8 b(o)30 b(sho)m(w)f(the)g(b)s(oundness)e(on)i Ft(R)3407 524 y FC(3)3470 557 y Fu(n)18 b Fx(K)q(;)31 b Fy(w)m(e)-9 670 y(use)f(the)g(explicit)f(form)h(of)g(the)h(distributional)26 b(k)m(ernel)k(of)h Fx(G)2109 684 y FC(0)2179 670 y Fy(getting)424 1013 y Fu(j)p Fx( )s Fy(\(x\))p Fu(j)84 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b(the)i(last)f(b)s(ound)f(follo)m(ws)g(b)m(y)h(c)m(ho)s(osing)h Fx(\015)k Fu(2)c Fy(\()1948 4757 y FC(1)p 1948 4772 36 4 v 1948 4825 a(2)1993 4793 y Fx(;)15 b Fy(2\).)66 b(F)-8 b(rom)39 b(estimates)g(\(68\),)j(\(69\))79 b(and)38 b(\(70\))h(it)-9 4906 y(follo)m(ws)29 b(our)h(claim)f Fx( )g Fu(2)c Fx(L)929 4873 y Fr(1)1004 4906 y Fy(\()p Ft(R)1099 4873 y FC(3)1144 4906 y Fy(\).)132 5019 y(It)30 b(has)h(b)s(een)f(pro)m(v)m(ed)h(b)m(y)f (Ik)m(eb)s(e)h(\(see)g([Ik],)62 b(Lemma)31 b(3.1\))h(that)f(a)g(b)s (ounded)d(and)i(con)m(tin)m(uous)g(solution)-9 5132 y(of)g(the)h (equation)f(\(64\))i(with)d Fx(V)45 b Fu(2)25 b Fy(\(I\))1309 5159 y FC(2)1378 5132 y Fy(m)m(ust)31 b(v)-5 b(anish)29 b(at)i(in\014nit)m(y)-8 b(.)39 b(Then)29 b Fx( )g Fu(2)h Fx(C)2818 5146 y Fr(1)2892 5132 y Fy(\()p Ft(R)2987 5099 y FC(3)3033 5132 y Fy(\).)132 5245 y(Finally)-8 b(,)25 b(w)m(e)h(notice)g(that)h(\001)p Fx( )h Fy(=)d Fx(V)20 b( )29 b Fu(2)c Fx(L)1608 5212 y FC(2)1608 5273 y(lo)r(c)1726 5245 y Fy(and)g(hence,)i(b)m(y)f(Sob)s(olev)f(inequalities,)f(w)m(e)i (get)h Fx( )i Fu(2)c Fx(H)3660 5212 y FC(2)3653 5273 y(lo)r(c)3745 5245 y Fy(.)-9 5358 y(Since,)h(b)m(y)f(h)m(yp)s(othesis,) h Fx( )j Fy(b)s(elongs)c(also)h(to)52 b Fx(L)1623 5325 y FC(2)1623 5380 y Fr(\000)p Fv(\015)1748 5358 y Fy(a)26 b(theorem)g(b)m(y)g(Agmon)g(\(see)h([Ag)q(])f(Lemma)g(5.1\))53 b(implies)-9 5471 y(that)31 b Fx( )d Fu(2)d Fx(H)444 5438 y FC(2)p Fv(;)p Fr(\000)p Fv(\015)598 5471 y Fy(.)132 5638 y Fn(\(ii\))34 b Fu(\))p Fn(\(iii\))g(or)h(\(v\).)40 b Fy(T)-8 b(rivial.)1835 5887 y(27)p eop %%Page: 28 28 28 27 bop 132 444 a Fn(\(iii\))32 b Fu(\))p Fn(\(i\).)45 b Fy(F)-8 b(rom)30 b(the)f(fact)g(that)h Fx(G)1446 458 y FC(0)1514 444 y Fy(is)e(the)h(righ)m(t)g(in)m(v)m(erse)f(of)h Fu(\000)p Fy(\001)g(it)f(follo)m(ws)g(that)h Fx( )3199 466 y FC(3)3267 444 y Fy(is)f(a)h(distribu-)-9 557 y(tional)k(solution) f(of)i(the)h(stationary)f(Sc)m(hr\177)-45 b(odinger)32 b(equation)i(\(65\).)53 b(Since)33 b Fx( )2721 579 y FC(3)2794 557 y Fy(b)s(elongs)g(to)i Fx(H)3322 524 y FC(2)p Fv(;)p Fr(\000)p Fv(\015)3507 557 y Fu(\032)c Fx(L)3671 524 y FC(2)3671 579 y Fr(\000)p Fv(\015)-9 670 y Fy(the)f(claim)g(follo)m(ws.)132 783 y Fn(\(v\))i Fu(\))p Fn(\(i\).)39 b Fy(As)29 b(b)s(efore,)g(w)m(e)h(kno)m(w)e(that)i Fx( )1627 805 y FC(5)1692 783 y Fu(\021)25 b Fx( )32 b Fy(is)c(a)h(distributional)c(solution)i(of)i(\(65\).)42 b(The)29 b(iden)m(tit)m(y)-9 896 y Fx( )i Fy(=)c Fx(G)250 910 y FC(0)290 896 y Fx(V)20 b( )35 b Fy(and)c(a)h(result)f(b)m(y)h(Ik) m(eb)s(e)f(\([Ik)q(],)h(Lemma)g(3.2;)i(see)e(also)g([TDMB)q(],)g(Lemma) g(3.3\))h(implies)c(that)-9 1009 y Fu(j)p Fx( )s Fy(\(x\))p Fu(j)k(\024)27 b Fx(C)38 b Fu(j)p Fy(x)p Fu(j)553 976 y Fr(\000)p FC(1)680 1009 y Fy(as)32 b Fu(j)p Fy(x)p Fu(j)c(!)f(1)p Fy(,)33 b(pro)m(vided)d(that)i Fx(V)48 b Fu(2)27 b Fy(\(I\))2052 1035 y FC(3)2091 1009 y Fy(.)45 b(T)-8 b(aking)32 b(in)m(to)f(accoun)m(t)j(the)e(con)m(tin)m(uit)m(y)f (of)h Fx( )t Fy(,)-9 1121 y(it)d(follo)m(ws)h(that)h Fx( )e Fu(2)h Fx(L)818 1088 y FC(2)818 1144 y Fr(\000)p Fv(\015)917 1121 y Fy(\()p Ft(R)1012 1088 y FC(3)1057 1121 y Fy(\))h(for)f(ev)m(ery)h Fx(\015)f(>)1685 1086 y FC(1)p 1685 1101 36 4 v 1685 1153 a(2)1731 1121 y Fy(.)p 1786 1121 61 61 v -9 1334 a Fn(Remark.)43 b Fy(W)-8 b(e)25 b(used)e(the)h(h)m(yp)s(othesis)f Fx(V)45 b Fu(2)25 b Fy(\(I\))1656 1361 y FC(3)1720 1334 y Fy(only)e(to)h(pro)m(v)m(e)h (that)f(\(v\))p Fu(\))p Fy(\(i\))p Fx(:)h Fy(All)d(the)i(remaining)e (results)-9 1447 y(hold)29 b(true)h(under)f(the)h(w)m(eak)m(er)i(h)m (yp)s(othesis)d(that)i Fx(V)45 b Fu(2)25 b Fy(\(I\))2033 1474 y FC(2)2072 1447 y Fy(.)-9 1659 y Fn(Remark.)77 b Fy(F)-8 b(rom)41 b(the)e(pro)s(of)g(of)h(the)g(previous)e(prop)s (osition)f(it)i(is)g(p)s(ossible)e(to)k(extract)g(the)f(follo)m(wing)-9 1772 y(result)34 b(concerning)h(the)h(asymptotic)g(b)s(eha)m(vior)f (and)g(the)h(lo)s(cal)f(regularit)m(y)f(of)72 b(distributional)32 b(solutions)-9 1885 y(of)37 b(the)g(stationary)g(Sc)m(hr\177)-45 b(odinger)36 b(equation:)54 b(if)36 b Fx(V)57 b Fu(2)36 b Fy(\(I\))2028 1912 y FC(2)2105 1885 y Fy(and)73 b Fx( )40 b Fu(2)d Fx(L)2584 1852 y FC(2)2584 1908 y Fr(\000)p Fv(\015)2683 1885 y Fy(\()p Ft(R)2778 1852 y FC(3)2823 1885 y Fy(\))74 b(\(with)37 b Fx(\015)k(>)3387 1849 y FC(1)p 3387 1864 36 4 v 3387 1917 a(2)3432 1885 y Fy(\))75 b(solv)m(es)-9 1998 y(equation)30 b(\(65\))i(then)e Fx( )823 2020 y FC(2)888 1998 y Fu(2)25 b Fx(H)1057 1965 y FC(2)p Fv(;)p Fr(\000)p Fv(\015)1231 1998 y Fu(\\)20 b Fx(C)1377 2012 y Fr(1)1452 1998 y Fy(.)132 2211 y(The)36 b(\\equiv)-5 b(alence)37 b(theorem")h(\(Prop.)60 b(4.3\))39 b(sho)m(ws)d(that)i({)f (as)g(exp)s(ected)h(from)e(the)h(ph)m(ysical)f(p)s(oin)m(t)-9 2324 y(of)29 b(view)f({)i(the)f(concept)i(of)e(\(zero-energy\))i (resonance)f(is)e(largely)h(indep)s(enden)m(t)e(from)i(the)g(tec)m (hnical)g(to)s(ols)-9 2437 y(needed)g(to)i(de\014ne)e(it.)40 b(In)29 b(this)g(spirit,)f(as)i(p)s(oin)m(ted)f(out)h(b)m(y)f(Enss)g ([En],)h(it)g(w)m(ould)e(b)s(e)h(in)m(teresting)h(to)g(giv)m(e)h(a)-9 2549 y(c)m(haracterization)f(of)f(resonances)h(whic)m(h)e(in)m(v)m(olv) m(es)h(only)f(the)i(time)f(ev)m(olution)f(and)h(the)g(p)s(osition)f(op) s(erator)-9 2662 y({)i(analogous)h(to)g(the)g(RA)m(GE)g(c)m (haracterization)g(of)g(b)s(ound)d(states)k(and)e(scattering)h(states.) 1835 5887 y(28)p eop %%Page: 29 29 29 28 bop -9 444 a Fz(References)-9 648 y Fy([)p Fn(Sec.1)p Fy(])93 b Fn(The)34 b(F)-12 b(AS)35 b(problem)-9 841 y Fy([AP])196 b(W.O.)24 b(Amrein,)f(D.B.)i(P)m(earson,)h(Flux)c(and)h (scattering)h(in)m(to)f(cones)h(for)g(long)f(range)g(and)g(singular)367 954 y(p)s(oten)m(tials,)30 b Fs(Journal)j(of)g(Physics)g(A)p Fy(,)d Fn(30)h Fy(\(1997\),)i(5361-5379.)-9 1147 y([AZ])202 b(W.O.)25 b(Amrein,)g(J.L.)h(Zuleta,)f(Flux)g(and)f(scattering)i(in)m (to)f(cones)g(in)f(p)s(oten)m(tial)h(scattering,)i Fs(Helv.)367 1260 y(Phys.)33 b(A)-5 b(cta)37 b Fn(70)31 b Fy(\(1997\),)i(1-15.)-9 1453 y([CNS])141 b(M.)32 b(Com)m(b)s(es,)h(R.G.)g(Newton,)g(R.)f(Sh)m (tokhamer,)h(Scattering)g(in)m(to)f(cones)g(and)g(\015ux)f(across)i (sur-)367 1566 y(faces,)e Fs(Phys.)i(R)-5 b(ev.)33 b(D)38 b Fn(11)p Fy(\(1975\),)c(366{372.)-9 1758 y([DDGZ)281 1772 y FC(1)321 1758 y Fy(])46 b(M.)26 b(Daumer,)h(D.)f(D)s(\177)-48 b(urr,)26 b(S.)g(Goldstein,)f(N.)i(Zangh)2149 1737 y(\022)2159 1758 y(i,)e(On)g(the)h(Flux-Across-Surfaces)f(Theorem,)367 1871 y Fs(L)-5 b(etters)33 b(in)g(Mathematic)-5 b(al)34 b(Physics)39 b Fn(38)c Fy(\(1996\),)e(103{116.)-9 2064 y([DDGZ)281 2078 y FC(2)321 2064 y Fy(])46 b(M.)g(Daumer,)j(D.)d(D)s (\177)-48 b(urr,)49 b(S.)c(Goldstein,)j(N.)e(Zangh)2295 2043 y(\022)2305 2064 y(i,)i(On)d(the)g(quan)m(tum)g(probabilit)m(y)e (\015ux)367 2177 y(through)29 b(surfaces,)i Fs(Journal)j(of)e(Stat.)i (Phys.)46 b Fn(88)31 b Fy(\(1997\),)i(967{977.)-9 2370 y([Do])212 b(J.D.)42 b(Dollard,)i(Scattering)e(in)m(to)g(cones)g(I,)g (P)m(oten)m(tial)h(scattering,)j Fs(Comm.)e(Math.)f(Phys.)g Fn(12)367 2483 y Fy(\(1969\),)33 b(193{203.)-9 2676 y([DT])191 b(D.)31 b(D)s(\177)-48 b(urr,)29 b(S.)i(T)-8 b(eufel,)30 b(On)f(the)i(role)f(of)g(\015ux)f(in)g(scattering)i(theory)-8 b(,)32 b(in)d([IDSA],)i(v)m(ol.)g(I,)f(123-137.)-9 2869 y([JLN])154 b(J.)39 b(M.)g(Jauc)m(h,)j(R.)d(La)m(vine,)i(R.)e(G.)h (Newton,)h(Scattering)f(in)m(to)f(cones,)j Fs(Helv.)d(Phys.)i(A)-5 b(cta)47 b Fn(45)367 2982 y Fy(\(1972\),)33 b(325-330.)-9 3175 y([PT])198 b(G.)34 b(P)m(anati,)h(A.)f(T)-8 b(eta,)35 b(The)e(Flux-Across-Surfaces)g(theorem)h(for)g(a)g(p)s(oin)m(t)e(in)m (teraction)i(hamil-)367 3287 y(tonian,)c(in)f([IDSA],)i(v)m(ol.)g(I)s (I,)e(547-557.)-9 3480 y([TDMB])46 b(S.)37 b(T)-8 b(eufel,)38 b(D.)g(D)s(\177)-48 b(urr,)38 b(K.)f(Munc)m(h-Berndl,)g(The)g(Flux)f (Across)h(Surfaces)f(Theorem)h(for)g(Short)367 3593 y(Range)42 b(P)m(oten)m(tials)h(and)e(W)-8 b(a)m(v)m(e)45 b(F)-8 b(unctions)41 b(without)g(Energy)h(Cuto\013s,)i Fs(J.)f(Math.)h(Phys.) 58 b Fn(40)367 3706 y Fy(\(1999\),)33 b(1901{1922.)-9 3899 y([IDSA])105 b(F.)25 b(Gesztesy)g Fs(et)i(al.)e Fy(\(Eds.\),)h Fs(Sto)-5 b(chastic)29 b(Pr)-5 b(o)g(c)g(esses,)29 b(Physics)e(and)h(Ge)-5 b(ometry:)41 b(New)27 b(Interplays)p Fy(,)367 4012 y(CMS)j(Conference)g(Pro)s(ceedings)g(Series,)g(AMS,)h (Pro)m(vidence,)f(Rho)s(de)g(Island,)f(2000.)k(Pro)s(ceed-)367 4125 y(ings)k(of)i(the)g(conference)g(on)g Fs(In\014nite)h(Dimensional) h(\(Stho)-5 b(chastic\))43 b(A)n(nalysis)e(and)g(Quantum)367 4238 y(Physics)p Fy(,)31 b(Leipzig,)e(1999.)-9 4536 y([)p Fn(Sec.)35 b(2)p Fy(])58 b Fn(P)m(oin)m(t)35 b(in)m(teraction)-9 4728 y Fy([A)m(GHH])54 b(S.)34 b(Alb)s(ev)m(erio,)h(F.)f(Gesztesy)-8 b(,)38 b(R.)d(H\034egh-Krohn,)g(H.)g(Holden,)g Fs(Solvable)i(Mo)-5 b(dels)37 b(in)g(Quantum)367 4841 y(Me)-5 b(chanics)p Fy(,)31 b(Springer)d(V)-8 b(erlag,)31 b(New)g(Y)-8 b(ork,)31 b(1988.)-9 5139 y([)p Fn(Sec.)k(3)p Fy(])58 b Fn(Theory)35 b(of)g(resonances)-9 5332 y Fy([AFS])148 b(S.)27 b(Alb)s(ev)m(erio,)g (L.)g(S.)g(F)-8 b(erreira,)28 b(L.)f(Streit,)g Fs(R)-5 b(esonanc)g(es)32 b(-)d(Mo)-5 b(dels)31 b(and)f(Phenomena)p Fy(,)f(LNP)e(211,)367 5445 y(Springer)h(V)-8 b(erlag,)31 b(Berlin,)e(1984.)-9 5638 y([Ba])217 b(E.)24 b(Baslev,)i(Resonances,)g (resonance)f(functions)d(and)i(sp)s(ectral)f(deformations,)i(in)e ([AFS],)j(27-63.)1835 5887 y(29)p eop %%Page: 30 30 30 29 bop -9 444 a Fy([En])213 b(V.)30 b(Enss,)g(Summary)f(of)i(the)f (conference)h(and)f(some)h(op)s(en)f(problems,)e(in)h([AFS)q(],)i (351-359.)-9 632 y([Ge])215 b(F.)27 b(Gesztesy)-8 b(,)29 b(P)m(erturbation)d(theory)h(for)f(resonances)h(in)e(terms)i(of)g(F)-8 b(redholm)25 b(determinan)m(ts,)i(in)367 745 y([AFS],)k(78-104.)-9 932 y([JK])208 b(A.)38 b(Jensen,)i(T.)e(Kato,)j(Sp)s(ectral)c(prop)s (erties)f(of)i(Sc)m(hr\177)-45 b(odinger)37 b(op)s(erators)h(and)g (time-deca)m(y)h(of)367 1045 y(the)30 b(w)m(a)m(v)m(e)j(function,)c Fs(Duke)i(Math.)i(Journal)41 b Fn(46)31 b Fy(\(1979\),)i(583-611.)-9 1233 y([)p Fn(Sec.4)p Fy(])93 b Fn(General)35 b(references)g(on)g (scattering)g(theory)-9 1420 y Fy([Ag])213 b(S.)30 b(Agmon,)h(Sp)s (ectral)f(prop)s(erties)f(of)i(Sc)m(hr\177)-45 b(odinger)29 b(op)s(erators)i(and)f(scattering)h(theory)-8 b(,)32 b Fs(A)n(nnali)367 1533 y(Scuola)h(Normale)h(Sup)-5 b(erior)g(e)p Fy(,)32 b(Serie)e(IV,)g(V)-8 b(ol.)31 b(I)s(I,)f(n.)g(2)h(\(1975\).)-9 1721 y([AJS])160 b(W.)36 b(O.)g(Amrein,)f(J.)h(M.)g(Jauc)m(h,)h(K.)f (B.)g(Sinha,)g Fs(Sc)-5 b(attering)38 b(the)-5 b(ory)39 b(in)f(Quantum)f(Me)-5 b(chanics)p Fy(,)367 1834 y(Benjamin,)29 b(Reading,)h(Massac)m(h)m(ussets,)j(1977.)-9 2021 y([CS])209 b(K.)25 b(Chadan,)h(Sabatier)f(P)-8 b(.)26 b(C.,)g Fs(Inverse)i(pr)-5 b(oblems)30 b(in)e(quantum)h(sc)-5 b(attering)29 b(the)-5 b(ory)p Fy(,)28 b(2)3363 1988 y Fv(nd)3472 2021 y Fy(edition,)367 2134 y(Springer)g(V)-8 b(erlag,)31 b(New)g(Y)-8 b(ork,)31 b(1989.)-9 2322 y([Ik])245 b(T.)25 b(Ik)m(eb)s(e,)i(Eigenfunction)c (expansion)i(asso)s(ciated)h(with)e(the)h(Sc)m(hr\177)-45 b(odinger)24 b(op)s(erators)i(and)f(their)367 2435 y(applications)j(to) j(scattering)g(theory)-8 b(,)32 b Fs(A)n(r)-5 b(ch.)32 b(R)-5 b(ational)35 b(Me)-5 b(ch.)33 b(A)n(nal.)d Fn(5)p Fy(,)h(\(1960\),)i(1{34.)-9 2623 y([LL])212 b(E.)40 b(H.)g(Lieb,)h (Loss)f(M.,)j Fs(A)n(nalysis)p Fy(,)f(American)e(Mathematical)g(So)s (ciet)m(y)-8 b(,)44 b(Pro)m(vidence,)e(Rho)s(de)367 2735 y(Island,)29 b(1996.)-9 2923 y([P)m(o])222 b(A.)36 b(J.)g(P)m(o)m (vzner,)j(The)d(expansion)f(of)h(arbitrary)f(functions)g(in)g(terms)h (of)g(eigenfunctions)f(of)h(te)367 3036 y(op)s(erator)28 b Fu(\000)p Fy(\001)p Fx(u)d Fy(=)g Fx(cu)p Fy(,)k Fs(Math.)i(Sb)-5 b(ornik)39 b Fn(32)28 b Fy(\(1953\),)j(109-156;)h Fs(A.M.S.)e(T)-7 b(r)i(anslations)p Fy(,)31 b(Series)c(2,)367 3149 y Fn(60)35 b Fy(\(1967\),)e(1-49.)-9 3336 y([RS3])163 b(M.)41 b(Reed,)k(B.)c (Simon,)i Fs(Metho)-5 b(ds)44 b(of)f(Mo)-5 b(dern)43 b(Mathematic)-5 b(al)45 b(Physics)k Fy(,)44 b(V)-8 b(ol.I)s(I)s(I,)40 b Fs(Sc)-5 b(attering)367 3449 y(The)g(ory)p Fy(,)32 b(Academic)f(Press,)f(London,)g(1979.)-9 3637 y([RS4])163 b(M.)37 b(Reed,)i(B.)f(Simon,)f Fs(Metho)-5 b(ds)41 b(of)e(Mo)-5 b(dern)39 b(Mathematic)-5 b(al)41 b(Physics)k Fy(,)39 b(V)-8 b(ol.IV,)38 b Fs(A)n(nalysis)h(of)367 3750 y(op)-5 b(er)g(ators)p Fy(,)34 b(Academic)c(Press,)h(London,)e(1978.)-9 3938 y([Y)-8 b(a])221 b(K.)26 b(Y)-8 b(a)5 b(jima,)28 b(The)e Fx(W)1109 3905 y Fv(k)r(;p)1206 3938 y Fy(-con)m(tin)m(uit)m(y) h(of)g(w)m(a)m(v)m(e)h(op)s(erators)f(for)f(Sc)m(hr\177)-45 b(odinger)25 b(op)s(erators,)j Fs(J.)h(Math.)367 4050 y(So)-5 b(c.)33 b(Jap)-5 b(an.)p Fy(,)32 b Fn(47)f Fy(\(1995\),)i (551{581.)1835 5887 y(30)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0110300901949--