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g(are)h(able)f(to)f(use)h(an)g(analog)f(of)456 1695 y(Buslae)n(v-F)o (addee)n(v-Zakharo)o(v)18 b(trace)i(formulae)f(well)g(kno)n(wn)f(for)h (one-dimensional)456 1811 y(Schr)8 b(\250)-41 b(odinger)26 b(operators.)35 b(The)26 b(multidimensional)d(case)k(is)f(reduced)h(to) f(a)g(problem)456 1927 y(for)21 b(a)h(second)f(order)h(elliptic)f (inter)n(go-dif)n(ferential)f(operator)-5 b(.)29 b(One)22 b(of)f(the)h(main)f(dif-)456 2044 y(\002culties)k(of)h(this)f(appoach)g (is)h(the)f(treatment)h(of)f(the)h(\223potential\224)f(type)g(term)h (which)456 2160 y(appears)31 b(to)f(be)h(a)g(dissipati)n(v)o(e)d(inte)o (gral)h(operator)i(depending)f(on)g(the)h(spectral)f(pa-)456 2276 y(rameter)-5 b(.)30 b(The)22 b(corresponding)g(Fredholm)h (equation)f(for)h(the)f(Jost)g(functions)g(might)456 2392 y(be)i(not)f(solv)n(able)f(for)i(a)g(discrete)g(subset)f(of)h(the) g(comple)o(x)e(upper)i(half)g(plane.)30 b(There)456 2509 y(is)k(a)i(hope)e(that)h(the)g(corresponding)f(contrib)n(ution)f(into)h 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y(the)g(follo)n(wing)f(local)i(condition)e(on)i Fs(V)46 b Fu(from)25 b([2])456 1867 y(\(2.4\))655 b Fs(V)49 b Fm(2)28 b Fs(`)1543 1825 y Fk(1)1618 1867 y Fq(\()p Fp(Z)1725 1825 y Fo(d)1762 1867 y Fq(;)17 b Fs(L)1872 1825 y Fo(q)1910 1867 y Fq(\()p Fp(Q)12 b Fq(\)\))p Fs(;)122 b(q)32 b(>)27 b(d=)p Fq(2)p Fs(;)456 2052 y Fu(which)42 b(can)i(be)f(weak)o(ened)h(by)f(using)f(the)h(characterization)h(of)f (weak)h(Hardy')-5 b(s)456 2168 y(weights)32 b(in)i(terms)f(of)h (capacities)f(obtained)g(by)g(V)-13 b(.Maz'ya)35 b(\(see)f([8]\).)57 b(Note)34 b(that)456 2284 y(if)28 b(\(2.4\))h(is)f(satis\002ed)g(then)g (the)g(operator)h Fs(H)36 b Fu(can)29 b(be)g(de\002ned)g(in)f(the)g (sense)h(of)f(qua-)456 2401 y(dratic)c(forms)h(\(see)g([2]\).)456 2660 y Fx(Theor)n(em)f(2.1.)37 b Fn(Let)23 b Fs(V)44 b Fn(be)22 b(a)g(r)l(eal)g(valued)g(function)f(on)h Fp(R)2494 2624 y Fo(d)2554 2660 y Fm(n)13 b Fq(\012)2687 2675 y Fr(1)2748 2660 y Fn(whic)o(h)22 b(obe)m(ys)h Fu(\(2.4\))456 2777 y 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b(,)17 b(N)m(ABOK)n(O)i(AND)f (SAFR)m(ONO)l(V)876 450 y Fu(3.)51 b(R)t Ft(E)t(D)t(U)t(C)t(T)t(I)t(O)t (N)35 b(T)s(O)30 b(A)f(O)t(N)t(E)t Fu(-)t Ft(D)t(I)t(M)t(E)t(N)t(S)5 b(I)g(O)t(N)r(A)g(L)35 b(P)t(R)q(O)t(B)t(L)t(E)t(M)555 624 y Fu(In)29 b(this)e(section)g(we)i(assume)e(that)h Fs(V)56 b Fm(2)34 b Fs(C)2083 588 y Fk(1)2076 649 y Fr(0)2186 624 y Fu(and)28 b(often)g(use)g(polar)g(coordinates)456 741 y Fq(\()p Fs(r)m(;)17 b(\022)s Fq(\))p Fs(;)50 b(x)c Fq(=)g Fs(r)s(\022)k Fm(2)c Fp(R)1285 705 y Fo(d)1332 741 y Fs(;)33 b(\022)49 b Fm(2)e Fp(S)1660 705 y Fo(d)p Fk(\000)p Fr(1)1785 741 y Fu(.)61 b(Denote)35 b(by)f Fm(f)p Fs(Y)2435 756 y Fo(j)2471 741 y Fm(g)2521 705 y Fk(1)2521 765 y Fo(j)t Fr(=0)2683 741 y Fu(the)g(orthonormal)g(in)456 869 y Fs(L)522 833 y Fr(2)561 869 y Fq(\()p Fp(S)661 833 y Fo(d)p Fk(\000)p Fr(1)785 869 y Fq(\))39 b Fu(basis)g(of)g (\(real\))h(spherical)e(funcions)h(and)g(let)f Fs(P)2643 884 y Fo(j)2719 869 y Fu(be)h(the)g(orthogonal)456 985 y(projection)23 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Fm(\000)23 b Fs(P)1978 1793 y Fr(0)2017 1778 y Fq(\))p Fs(;)116 b(V)2255 1793 y Fr(2)p Fo(;)p Fr(1)2377 1778 y Fq(=)28 b Fs(V)2559 1737 y Fk(\003)2538 1803 y Fr(1)p Fo(;)p Fr(2)2632 1778 y Fs(;)797 1945 y(V)854 1960 y Fr(2)921 1945 y Fq(=)f(\()p Fs(I)j Fm(\000)23 b Fs(P)1298 1960 y Fr(0)1337 1945 y Fq(\))p Fs(V)f Fq(\()p Fs(I)30 b Fm(\000)22 b Fs(P)1727 1960 y Fr(0)1767 1945 y Fq(\))p Fs(;)116 b(H)2029 1960 y Fr(0)p Fo(;)p Fr(2)2151 1945 y Fq(=)27 b(\()p Fs(I)j Fm(\000)23 b Fs(P)2528 1960 y Fr(0)2567 1945 y Fq(\))p Fs(H)2686 1960 y Fr(0)2725 1945 y Fq(\()p Fs(I)30 b Fm(\000)23 b Fs(P)2999 1960 y Fr(0)3038 1945 y Fq(\))p Fs(:)456 2095 y Fu(Then)h(the)h(operator)g Fs(H)k Fm(\000)23 b Fs(z)30 b Fu(can)25 b(be)g(represented)g(as)g(a)g (matrix:)1055 2323 y Fs(H)k Fm(\000)23 b Fs(z)32 b Fq(=)1446 2183 y Fl(\022)1519 2264 y Fs(H)1600 2279 y Fr(0)p Fo(;)p Fr(1)1716 2264 y Fq(+)22 b Fs(V)1871 2279 y Fr(1)1933 2264 y Fm(\000)h Fs(z)293 b(V)2428 2279 y Fr(1)p Fo(;)p Fr(2)1725 2380 y Fs(V)1782 2395 y Fr(2)p Fo(;)p Fr(1)2165 2380 y Fs(H)2246 2395 y Fr(0)p Fo(;)p Fr(2)2363 2380 y Fq(+)22 b Fs(V)2518 2395 y Fr(2)2579 2380 y Fm(\000)h Fs(z)2728 2183 y Fl(\023)2818 2323 y Fs(;)456 2548 y Fu(and)h(the)h(equation)1343 2678 y Fq(\()p Fs(H)30 b Fm(\000)22 b Fs(z)t Fq(\))p Fs(u)28 b Fq(=)g Fs(P)1929 2693 y Fr(0)1968 2678 y Fs(f)5 b(;)117 b Fu(Im)24 b Fs(z)33 b Fm(6)p Fq(=)27 b(0)p Fs(;)456 2827 y Fu(is)d(equi)n(v)n(alent)f(to) 456 2998 y(\(3.1\))49 b Fq(\()p Fs(H)815 3013 y Fr(0)p Fo(;)p Fr(1)918 2998 y Fq(+)9 b Fs(T)1060 3013 y Fo(z)1108 2998 y Fm(\000)g Fs(z)t Fq(\))p Fs(P)1344 3013 y Fr(0)1384 2998 y Fs(u)27 b Fq(=)g Fs(P)1633 3013 y Fr(0)1673 2998 y Fs(f)5 b(;)116 b Fq(\()p Fs(H)1988 3013 y Fr(0)p Fo(;)p Fr(2)2091 2998 y Fq(+)9 b Fs(V)2233 3013 y Fr(2)2280 2998 y Fm(\000)g Fs(z)t Fq(\))2453 2956 y Fk(\000)p Fr(1)2548 2998 y Fs(V)2605 3013 y Fr(2)p Fo(;)p Fr(1)2699 2998 y Fs(P)2762 3013 y Fr(0)2802 2998 y Fs(u)27 b Fq(=)g(\()p Fs(P)3089 3013 y Fr(0)3137 2998 y Fm(\000)9 b Fs(I)f 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3839 y Fk(\000)p Fr(\()p Fo(d)p Fk(\000)p Fr(1\))p Fo(=)p Fr(2)2280 3880 y Fs(u;)456 4030 y Fu(we)66 b(reduce)h(\(3.1\))f(to)g(the)h (problem)e(for)i(the)f(follo)n(wing)e(one-dimensional)456 4146 y(Schr)8 b(\250)-41 b(odinger)24 b(operator)h(in)g Fs(L)1493 4110 y Fr(2)1533 4146 y Fq(\(1)p Fs(;)17 b Fm(1)p Fq(\))456 4379 y Fu(\(3.2\))252 b Fs(L)965 4394 y Fo(z)1005 4379 y Fs(u)p Fq(\()p Fs(r)s Fq(\))27 b(=)h Fm(\000)1402 4312 y Fs(d)1453 4276 y Fr(2)1492 4312 y Fs(u)p 1402 4357 146 4 v 1406 4448 a(dr)1504 4419 y Fr(2)1580 4379 y Fq(+)22 b Fs(Q)1755 4394 y Fo(z)1795 4379 y Fs(u;)116 b(u)27 b Fm(2)h Fs(L)2237 4338 y Fr(2)2277 4379 y Fq(\(1)p Fs(;)17 b Fm(1)p Fq(\))p Fs(;)40 b(u)p Fq(\(1\))27 b(=)h(0)p Fs(;)456 4578 y Fu(where)456 4788 y Fs(Q)533 4803 y Fo(z)600 4788 y Fq(=)g Fs(V)761 4803 y Fr(1)815 4788 y Fq(+)916 4720 y Fs(\013)978 4735 y Fo(d)p 916 4765 103 4 v 924 4856 a Fs(r)971 4827 y Fr(2)1043 4788 y Fm(\000)15 b Fs(V)1192 4803 y Fr(1)p Fo(;)p Fr(2)1287 4788 y Fq(\()p Fs(U)1401 4747 y Fk(\003)1441 4788 y Fs(H)1522 4803 y Fr(0)p Fo(;)p Fr(2)1616 4788 y Fs(U)26 b Fq(+)15 b Fs(V)1856 4803 y Fr(2)1910 4788 y Fm(\000)g Fs(z)t Fq(\))2089 4747 y Fk(\000)p Fr(1)2184 4788 y Fs(V)2241 4803 y Fr(2)p Fo(;)p Fr(1)2335 4788 y Fs(;)117 b(\013)2541 4803 y Fo(d)2609 4788 y Fq(=)2722 4720 y(\()p Fs(d)22 b Fm(\000)h Fq(1\))3020 4684 y Fr(2)p 2722 4765 337 4 v 2866 4856 a Fq(4)3084 4788 y Fm(\000)3186 4720 y Fs(d)f Fm(\000)h Fq(1)p 3186 4765 222 4 v 3272 4856 a(2)3417 4788 y Fs(:)456 4986 y Fu(By)i(considering)f(the)g(potential)1442 5180 y Fs(V)44 b Fm(\000)1653 5113 y Fs(\013)1715 5128 y Fo(d)p 1653 5158 103 4 v 1661 5249 a Fs(r)1708 5220 y Fr(2)1865 5180 y Fq(instead)25 b(of)106 b Fs(V)p eop %%Page: 5 5 5 4 bop 1610 253 a Fi(SZEG)1811 235 y(\005)1798 253 y(O)23 b(CONDITION)1116 b(5)456 450 y Fu(without)23 b(loss)h(of)h(generality)f (we)h(can)g(assume)g(that)456 618 y(\(3.3\))593 b Fs(Q)1317 633 y Fo(z)1385 618 y Fq(=)27 b Fs(V)1545 633 y Fr(1)1607 618 y Fm(\000)22 b Fs(V)1763 633 y Fr(1)p Fo(;)p Fr(2)1858 618 y Fq(\()p Fs(S)28 b Fq(+)22 b Fs(V)2139 633 y Fr(2)2200 618 y Fm(\000)h Fs(z)t Fq(\))2387 577 y Fk(\000)p Fr(1)2482 618 y Fs(V)2539 633 y Fr(2)p Fo(;)p Fr(1)2633 618 y Fs(;)456 786 y Fu(where)456 994 y(\(3.4\))585 b Fs(S)6 b(u)27 b Fq(=)h Fm(\000)1572 926 y Fs(d)1623 890 y Fr(2)1662 926 y Fs(u)p 1572 971 146 4 v 1576 1062 a(dr)1674 1033 y Fr(2)1750 994 y Fm(\000)1860 926 y Fq(\001)1941 941 y Fo(\022)1980 926 y Fs(u)p 1860 971 176 4 v 1904 1062 a(r)1951 1033 y Fr(2)2045 994 y Fs(;)117 b(u)p Fq(\(1)p Fs(;)17 b(\022)s Fq(\))26 b(=)i(0)p Fs(:)456 1190 y Fu(Note)c(that)h (the)f(conditions)f(of)i(Theorem)g(2.1)f(on)h Fs(V)46 b Fu(w)o(on')n(t)25 b(be)g(changed.)555 1307 y(According)g(to)f (\(3.1\))h(we)g(obtain)456 1474 y(\(3.5\))586 b Fs(P)1296 1489 y Fr(0)1335 1474 y Fq(\()p Fs(H)30 b Fm(\000)23 b Fs(z)t Fq(\))1671 1433 y Fk(\000)p Fr(1)1766 1474 y Fs(P)1829 1489 y Fr(0)1896 1474 y Fq(=)k Fs(U)10 b Fq(\()p Fs(L)2179 1489 y Fo(z)2242 1474 y Fm(\000)23 b Fs(z)t Fq(\))2429 1433 y Fk(\000)p Fr(1)2524 1474 y Fs(U)2600 1433 y Fk(\003)2640 1474 y Fs(:)456 1642 y Fu(W)-8 b(e)27 b(see)h(also)f(that)f(if)i(supp)15 b Fs(V)54 b Fm(\032)33 b(f)p Fs(x)f Fm(2)h Fp(R)1948 1606 y Fo(d)2026 1642 y Fq(:)87 b Fs(c)2182 1657 y Fr(1)2254 1642 y Fs(<)32 b Fm(j)p Fs(x)p Fm(j)g Fs(<)g(c)2655 1657 y Fr(2)2694 1642 y Fm(g)p Fs(;)h(c)2846 1657 y Fr(1)2918 1642 y Fs(>)f Fq(1)p Fs(;)27 b Fu(then)g(for)456 1759 y(the)d(operator)h(\(3.3\))g (we)g(ha)n(v)o(e)1569 1927 y Fs(Q)1646 1942 y Fo(z)1713 1927 y Fq(=)j Fs(Q)1894 1942 y Fo(z)1934 1927 y Fs(\037)g Fq(=)f Fs(\037Q)2264 1942 y Fo(z)2304 1927 y Fs(;)456 2095 y Fu(where)22 b Fs(\037)g Fu(is)f(an)g(operator)h(of)g (multiplication)d(by)i(the)g(characteristic)h(function)f(of)h(the)456 2211 y(interv)n(al)h Fq(\()p Fs(c)862 2226 y Fr(1)901 2211 y Fs(;)17 b(c)987 2226 y Fr(2)1026 2211 y Fq(\))p Fs(;)33 b(c)1166 2226 y Fr(1)1234 2211 y Fs(>)27 b Fq(0)p Fu(.)j(It)25 b(is)e(impotant)g(for)h(us)g(that)g Fs(Q)2500 2226 y Fo(z)2564 2211 y Fu(is)g(an)g(analytic)f(operator)456 2327 y(v)n(alued)j(function)g(of)h Fs(z)32 b Fu(with)26 b(a)h(ne)o(gati)n(v)o(e)e(imaginary)h(part)h(in)g(the)g(upper)g(half)g (plane)456 2443 y(and)d(which)h(has)f(a)i(positi)n(v)o(e)c(imaginary)i (part)h(in)f(the)h(lo)n(wer)f(half)h(plane.)1443 2673 y(4.)52 b(G)t Ft(R)t(E)t(E)t(N)t Fu(')t Ft(S)35 b(F)t(U)t(N)t(C)t(T)t (I)t(O)t(N)t Fu(.)555 2847 y(Let)25 b(us)f(consider)h(the)f(equation) 456 3075 y(\(4.1\))299 b Fm(\000)1057 3007 y Fs(d)1108 2971 y Fr(2)p 1033 3052 137 4 v 1033 3143 a Fs(dr)1131 3114 y Fr(2)1180 3075 y Fs( )t Fq(\()p Fs(r)s Fq(\))22 b(+)g(\()p Fs(Q)1605 3090 y Fo(z)1645 3075 y Fs( )t Fq(\)\()p Fs(r)s Fq(\))27 b(=)h Fs(z)t( )t Fq(\()p Fs(r)s Fq(\))p Fs(;)116 b(r)31 b Fm(\025)d Fq(1)p Fs(;)41 b(z)33 b Fm(2)28 b Fp(C)20 b Fs(;)456 3278 y Fu(with)30 b Fs(Q)741 3293 y Fo(z)812 3278 y Fu(gi)n(v)o(en)g(by)h(\(3.3\))g(and)h(let)f Fs( )1784 3293 y Fo(k)1827 3278 y Fq(\()p Fs(r)s Fq(\))g Fu(be)g(the)g(solution)f(of)h(the)g(equation)g(\(4.1\))456 3394 y(satisfying)949 3562 y Fs( )1012 3577 y Fo(k)1055 3562 y Fq(\()p Fs(r)s Fq(\))c(=)h(exp)18 b(\()p Fs(ik)s(r)s Fq(\))o Fs(;)117 b(k)1882 3521 y Fr(2)1949 3562 y Fq(=)28 b Fs(z)t(;)42 b Fu(Im)24 b Fs(k)31 b(>)d Fq(0)p Fs(;)41 b Fm(8)p Fs(r)31 b(>)c(c)2884 3577 y Fr(2)2924 3562 y Fs(:)456 3730 y Fu(Then)d(this)g(solution)f(also)i(satis\002es)f(the)h (follo)n(wing)d(\223inte)o(gral\224)j(equation)456 3950 y(\(4.2\))290 b Fs( )1000 3965 y Fo(k)1044 3950 y Fq(\()p Fs(r)s Fq(\))27 b(=)g Fs(e)1342 3908 y Fo(ik)r(r)1465 3950 y Fm(\000)c Fs(k)1619 3908 y Fk(\000)p Fr(1)1730 3814 y Fl(Z)1829 3840 y Fk(1)1785 4040 y Fo(r)1921 3950 y Fq(sin)16 b Fs(k)s Fq(\()p Fs(r)25 b Fm(\000)e Fs(s)p Fq(\)\()p Fs(Q)2517 3965 y Fo(z)2556 3950 y Fs( )2619 3965 y Fo(k)2662 3950 y Fq(\)\()p Fs(s)p Fq(\))17 b Fs(ds:)456 4168 y Fu(According)34 b(to)g(the)h(analytic)f(Fredholm)g(theorem)g (\(see,)k(for)d(e)o(xample,)h(Theorem)456 4284 y(VI.14,)29 b([9]\))f(we)h(conclude)f(that)g(the)h(equation)e(\(4.2\))i(is)f (uniquely)f(solv)n(able)g(for)i(all)456 4400 y Fs(k)f Fu(e)o(xcept)c(perhaps)h(a)h(discret)e(sequence)h(of)h(points)d(and)i (its)f(solution)f Fs( )2977 4415 y Fo(k)3046 4400 y Fu(is)h(a)h(mero-) 456 4516 y(morphic)f(with)g(respect)h(to)f Fs(k)k Fu(function,)c(Im)h Fs(k)30 b Fm(\025)e Fq(0)p Fu(.)555 4632 y(Consider)d(the)f(resolv)o (ent)g(operator)h Fs(R)q Fq(\()p Fs(z)t Fq(\))j(=)g(\()p Fs(L)2268 4647 y Fo(z)2330 4632 y Fm(\000)22 b Fs(z)t Fq(\))2516 4596 y Fk(\000)p Fr(1)2611 4632 y Fu(,)j(where)g Fs(L)2995 4647 y Fo(z)3060 4632 y Fu(is)f(de\002ned)456 4749 y(in)19 b(\(3.2\).)29 b(If)21 b Fs(\037)946 4764 y Fo(c)977 4773 y Fh(1)1036 4749 y Fu(is)e(the)h(operator)h(of)f (multiplication)d(by)j(the)g(characteristic)g(function)456 4865 y(of)k Fq(\(1)p Fs(;)17 b(c)736 4880 y Fr(1)775 4865 y Fq(\))p Fu(.)31 b(Then)25 b Fs(R)q Fq(\()p Fs(z)t Fq(\))p Fs(\037)1360 4880 y Fo(c)1391 4889 y Fh(1)1455 4865 y Fu(is)f(an)h(inte)o(gral)f(operator)h(with)f(the)g(k)o(ernel:) 456 5124 y(\(4.3\))225 b Fs(G)949 5139 y Fo(z)989 5124 y Fq(\()p Fs(r)m(;)17 b(s)p Fq(\))27 b(=)1327 4954 y Fl(\()1418 5001 y Fo( )1464 5013 y Fg(k)1502 5001 y Fr(\()p Fo(s)p Fr(\))p 1417 5025 175 4 v 1417 5083 a Fo( )1463 5095 y Fg(k)1501 5083 y Fr(\(1\))1628 5001 y(sin)o(\()p Fo(k)r Fr(\()p Fo(r)r Fk(\000)p Fr(1\)\))p 1628 5025 359 4 v 1788 5083 a Fo(k)1997 5048 y Fs(;)116 b Fq(for)24 b Fs(r)31 b(<)c(s)h(<)f(c)2678 5063 y Fr(1)2718 5048 y Fs(;)1418 5149 y Fo( )1464 5161 y Fg(k)1502 5149 y Fr(\()p Fo(r)r Fr(\))p 1417 5174 175 4 v 1417 5231 a Fo( )1463 5243 y Fg(k)1501 5231 y Fr(\(1\))1628 5149 y(sin)o(\()p Fo(k)r Fr(\()p Fo(s)p Fk(\000)p Fr(1\)\))p 1628 5174 358 4 v 1787 5231 a Fo(k)1995 5197 y Fs(;)117 b Fq(for)24 b Fs(s)k(<)f Fq(min)o Fm(f)p Fs(c)2712 5212 y Fr(1)2751 5197 y Fs(;)17 b(r)s Fm(g)p Fs(:)p eop %%Page: 6 6 6 5 bop 456 251 a Fi(6)845 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND)f (SAFR)m(ONO)l(V)456 450 y Fu(Indeed,)26 b(assuming)e(that)h(supp)p Fq(\()p Fs(f)11 b Fq(\))28 b Fm(\032)i Fq(\(1)p Fs(;)17 b(c)1987 465 y Fr(1)2026 450 y Fq(\))25 b Fu(we)h(can)g(easily)f(check) h(that)g(the)f(func-)456 566 y(tion)456 769 y Fs(u)p Fq(\()p Fs(r)s Fq(\))h(=)866 701 y(1)p 775 746 231 4 v 775 837 a Fs( )838 852 y Fo(k)881 837 y Fq(\(1\))1016 658 y Fl(n)1082 633 y(Z)1182 659 y Fk(1)1137 859 y Fo(r)1283 701 y Fq(sin\()p Fs(k)s Fq(\()p Fs(r)e Fm(\000)f Fq(1\)\))p 1283 746 543 4 v 1527 837 a Fs(k)1836 769 y( )1899 784 y Fo(k)1942 769 y Fq(\()p Fs(s)p Fq(\))p Fs(f)11 b Fq(\()p Fs(s)p Fq(\))17 b Fs(ds)p Fq(+)2435 633 y Fl(Z)2533 659 y Fo(r)2489 859 y Fr(1)2587 769 y Fs( )2650 784 y Fo(k)2693 769 y Fq(\()p Fs(r)s Fq(\))2826 701 y(sin\()p Fs(k)s Fq(\()p Fs(s)22 b Fm(\000)g Fq(1\)\))p 2826 746 V 3070 837 a Fs(k)3378 769 y(f)11 b Fq(\()p Fs(s)p Fq(\))17 b Fs(ds)3690 658 y Fl(o)456 989 y Fu(satis\002es)24 b(the)h(equation) 456 1216 y(\(4.4\))260 b Fm(\000)1018 1149 y Fs(d)1069 1113 y Fr(2)p 994 1193 137 4 v 994 1285 a Fs(dr)1092 1256 y Fr(2)1142 1216 y Fs(u)p Fq(\()p Fs(r)s Fq(\))21 b(+)h(\()p Fs(Q)1555 1231 y Fo(z)1595 1216 y Fs(u)p Fq(\)\()p Fs(r)s Fq(\))f Fm(\000)h Fs(z)t(u)p Fq(\()p Fs(r)s Fq(\))28 b(=)g Fs(f)11 b Fq(\()p Fs(r)s Fq(\))p Fs(;)115 b(r)30 b Fm(\025)f Fq(1)p Fs(;)41 b(z)32 b Fm(2)d Fp(C)19 b Fs(;)456 1419 y Fu(and)24 b(moreo)o(v)o(er)g Fs(u)p Fq(\(1\))j(=)g(0)p Fu(.)755 1747 y(5.)51 b(W)t Ft(R)q(O)t(N)t(S)t(K)t(I)t(A)t(N)35 b(A)t(N)t(D)30 b(P)t(R)q(O)t(P)t(E)t(RT)t(I)5 b(E)t(S)36 b(O)t(F)31 b(T)t(H)t(E)d Fs(M)12 b Fu(-)t Ft(F)t(U)t(N)t(C)t(T)t(I)5 b(O)t(N)g Fu(.)555 1921 y(Let)25 b(as)g(in)f(\(3.3\))1240 2088 y Fs(Q)1317 2103 y Fo(z)1385 2088 y Fq(=)j Fs(V)1545 2103 y Fr(1)1607 2088 y Fm(\000)22 b Fs(V)1763 2103 y Fr(1)p Fo(;)p Fr(2)1858 2088 y Fq(\()p Fs(S)28 b Fq(+)22 b Fs(V)2139 2103 y Fr(2)2200 2088 y Fm(\000)h Fs(z)t Fq(\))2387 2047 y Fk(\000)p Fr(1)2482 2088 y Fs(V)2539 2103 y Fr(2)p Fo(;)p Fr(1)2633 2088 y Fs(:)456 2256 y Fu(The)h(function)1642 2417 y Fs(M)10 b Fq(\()p Fs(k)s Fq(\))28 b(=)2018 2350 y Fs( )2085 2314 y Fk(0)2081 2376 y Fo(k)2124 2350 y Fq(\(1\))p 2018 2394 231 4 v 2018 2486 a Fs( )2081 2501 y Fo(k)2124 2486 y Fq(\(1\))456 2618 y Fu(is)35 b(called)g(the)g(W)-8 b(e)o(yl)35 b Fs(M)10 b Fu(-function)36 b(of)f(the)g(operator)h(\(4.1\).)62 b(Let)36 b(us)f(consider)g(the)456 2734 y(Wronskian)456 2901 y(\(5.1\))437 b Fs(W)14 b Fq([)p 1217 2820 107 4 v Fs( )1280 2916 y Fo(k)1322 2901 y Fs(;)j( )1429 2916 y Fo(k)1472 2901 y Fq(]\()p Fs(r)s Fq(\))28 b(=)p 1753 2818 V 27 w Fs( )1820 2867 y Fk(0)1816 2929 y Fo(k)1859 2901 y Fq(\()p Fs(r)s Fq(\))p Fs( )2045 2916 y Fo(k)2088 2901 y Fq(\()p Fs(r)s Fq(\))21 b Fm(\000)p 2332 2820 V 23 w Fs( )2395 2916 y Fo(k)2438 2901 y Fq(\()p Fs(r)s Fq(\))p Fs( )2628 2860 y Fk(0)2624 2925 y Fo(k)2667 2901 y Fq(\()p Fs(r)s Fq(\))p Fs(:)456 3077 y Fu(Note)k(that)p 849 2996 V 24 w Fs( )912 3092 y Fo(k)981 3077 y Fu(satis\002es)g(the)g (equation)f(\(4.1\))h(with)g Fs(Q)p 2333 3054 40 3 v 15 x Fo(z)2398 3077 y Fu(and)p 2567 3022 50 4 v 25 w Fs(z)30 b Fu(instead)25 b(of)g Fs(Q)3135 3092 y Fo(z)3201 3077 y Fu(and)g Fs(z)t Fu(.)456 3193 y(Since)g Fs( )765 3208 y Fo(k)833 3193 y Fu(is)f(a)h(solution)e(of)i(the)g(equation)f (\(4.1\))h(we)g(\002nd)489 3343 y Fs(d)p 466 3388 98 4 v 466 3479 a(dr)573 3410 y(W)14 b Fq([)p 706 3329 107 4 v Fs( )769 3425 y Fo(k)812 3410 y Fs(;)j( )919 3425 y Fo(k)961 3410 y Fq(]\()p Fs(r)s Fq(\))28 b(=)f(\()p Fs(z)6 b Fm(\000)p 1410 3356 50 4 v 2 w Fs(z)g Fq(\))p 1499 3329 107 4 v Fs( )1562 3425 y Fo(k)1605 3410 y Fq(\()p Fs(r)s Fq(\))p Fs( )1791 3425 y Fo(k)1834 3410 y Fq(\()p Fs(r)s Fq(\))r(+)r(\()p Fs(Q)p 2152 3388 40 3 v 15 x Fo(z)p 2191 3329 107 4 v 2191 3410 a Fs( )2254 3425 y Fo(k)2297 3410 y Fq(\)\()p Fs(r)s Fq(\))p Fs( )2521 3425 y Fo(k)2563 3410 y Fq(\()p Fs(r)s Fq(\))r Fm(\000)p 2767 3329 V 2 w Fs( )2830 3425 y Fo(k)2873 3410 y Fq(\()p Fs(r)s Fq(\)\()p Fs(Q)3111 3425 y Fo(z)3151 3410 y Fs( )3214 3425 y Fo(k)3257 3410 y Fq(\)\()p Fs(r)s Fq(\))p Fs(:)456 3606 y Fu(So)25 b(we)g(obtain)456 3773 y(\(5.2\))75 b Fm(\006)p Fu(Im)25 b Fm(f)p Fs(W)14 b Fq([)p 1118 3692 V Fs( )1181 3788 y Fo(k)1224 3773 y Fs(;)j( )1331 3788 y Fo(k)1374 3773 y Fq(]\()p Fs(c)1481 3788 y Fr(2)1520 3773 y Fq(\))22 b Fm(\000)h Fs(W)14 b Fq([)p 1813 3692 V Fs( )1876 3788 y Fo(k)1919 3773 y Fs(;)j( )2026 3788 y Fo(k)2068 3773 y Fq(]\()p Fs(c)2175 3788 y Fr(1)2215 3773 y Fq(\))p Fm(g)27 b(\025)h Fq(0)p Fs(;)116 b Fq(for)55 b Fm(\006)23 b Fu(Im)i Fs(z)32 b Fm(\025)c Fq(0+)p Fs(;)456 3940 y Fu(which)c(means)g(that)h(for)g(all)f Fs(k)k Fu(we)d(ha)n(v)o(e) g(the)g(follo)n(wing)d(inequality)1680 4090 y Fs(k)p 1522 4135 371 4 v 1522 4226 a Fu(Im)j Fs(M)10 b Fq(\()p Fs(k)s Fq(\))1930 4157 y Fm(\024)28 b(j)p Fs( )2126 4172 y Fo(k)2169 4157 y Fq(\(1\))p Fm(j)2322 4116 y Fr(2)2361 4157 y Fs(:)456 4379 y Fu(Moreo)o(v)o(er)l(,)23 b(if)i(we)g(represent)g (the)g(solution)e Fs( )2064 4394 y Fo(k)2132 4379 y Fu(for)i(real)g Fs(k)j Fu(in)d(the)f(form)456 4547 y(\(5.3\))484 b Fs( )1194 4562 y Fo(k)1237 4547 y Fq(\()p Fs(x)p Fq(\))28 b(=)g Fs(a)p Fq(\()p Fs(k)s Fq(\))p Fs(e)1726 4506 y Fo(ik)r(x)1855 4547 y Fq(+)22 b Fs(b)p Fq(\()p Fs(k)s Fq(\))p Fs(e)2169 4506 y Fk(\000)p Fo(ik)r(x)2330 4547 y Fs(;)117 b(x)28 b(<)f(c)2702 4562 y Fr(1)2742 4547 y Fs(;)456 4714 y Fu(then)d(it)g(follo)n(ws)g(from)g(\(5.2\))h(that)1644 4881 y Fm(j)p Fs(a)p Fm(j)1751 4840 y Fr(2)1812 4881 y Fm(\000)d(j)p Fs(b)p Fm(j)2008 4840 y Fr(2)2075 4881 y Fm(\025)28 b Fq(1)p Fs(:)456 5048 y Fu(Then)c(for)h Fs(k)880 5012 y Fr(2)948 5048 y Fq(=)i Fs(z)456 5216 y(M)10 b Fq(\()p Fs(k)s Fq(\))28 b(=)g Fs( )889 5174 y Fk(0)885 5240 y Fo(k)928 5216 y Fq(\(1\)\()p Fs( )1154 5231 y Fo(k)1196 5216 y Fq(\(1\)\))1359 5174 y Fk(\000)p Fr(1)1481 5216 y Fq(=)f Fs(ik)s Fq(\(1)p Fm(\000)p Fs(\032)p Fq(\()p Fs(k)s Fq(\)\)\(1+)p Fs(\032)p Fq(\()p Fs(k)s Fq(\)\))2434 5174 y Fk(\000)p Fr(1)2529 5216 y Fs(;)117 b(\032)p Fq(\()p Fs(k)s Fq(\))28 b(:=)f Fs(e)3056 5174 y Fk(\000)p Fr(2)p Fo(ik)3213 5216 y Fs(b)p Fq(\()p Fs(k)s Fq(\))p Fs(a)p Fq(\()p Fs(k)s Fq(\))3565 5174 y Fk(\000)p Fr(1)3660 5216 y Fs(:)p eop %%Page: 7 7 7 6 bop 1610 253 a Fi(SZEG)1811 235 y(\005)1798 253 y(O)23 b(CONDITION)1116 b(7)456 450 y Fu(The)24 b(latter)h(implies)1215 628 y Fs(\032)p Fq(\()p Fs(k)s Fq(\))j(=)f(\()p Fs(ik)f Fm(\000)c Fs(M)10 b Fq(\()p Fs(k)s Fq(\)\)\()p Fs(ik)26 b Fq(+)c Fs(M)10 b Fq(\()p Fs(k)s Fq(\)\))2563 587 y Fk(\000)p Fr(1)2658 628 y Fs(:)456 806 y Fu(Since)25 b Fm(j)p Fs(a)p Fm(j)809 770 y Fr(2)870 806 y Fm(\000)e(j)p Fs(b)p Fm(j)1067 770 y Fr(2)1134 806 y Fm(\025)28 b Fq(1)c Fu(we)h(obtain)f(that)h(for)g(real)g Fs(k)1138 1037 y Fm(j)p Fs(a)p Fq(\()p Fs(k)s Fq(\))p Fm(j)1375 996 y Fk(\000)p Fr(2)1496 1037 y Fm(\024)j Fq(1)22 b Fm(\000)h(j)p Fs(\032)p Fq(\()p Fs(k)s Fq(\))p Fm(j)2008 996 y Fr(2)2075 1037 y Fq(=)2285 970 y(4)p Fs(k)s Fu(Im)i Fs(M)p 2188 1014 538 4 v 2188 1105 a Fm(j)p Fs(ik)g Fq(+)d Fs(M)10 b Fq(\()p Fs(k)s Fq(\))p Fm(j)2685 1077 y Fr(2)2735 1037 y Fs(:)456 1269 y Fu(Note)24 b(that)h(since)f(Im)h Fs(M)38 b Fm(\025)28 b Fq(0)p Fu(,)d(then)f(for)h(an)o(y)g Fs(k)31 b(>)c Fq(0)e Fu(we)g(ha)n(v)o(e)1064 1447 y Fm(j)p Fs(ik)h Fq(+)c Fs(M)10 b Fq(\()p Fs(k)s Fq(\))p Fm(j)1562 1406 y Fr(2)1629 1447 y Fq(=)28 b Fs(k)1787 1406 y Fr(2)1848 1447 y Fq(+)22 b Fm(j)p Fs(M)10 b Fm(j)2106 1406 y Fr(2)2168 1447 y Fq(+)22 b(2)p Fs(k)s Fu(Im)i Fs(M)39 b Fm(\025)28 b Fs(k)2796 1406 y Fr(2)456 1625 y Fu(and)c(therefore)456 1803 y(\(5.4\))604 b Fm(j)p Fs(a)p Fq(\()p Fs(k)s Fq(\))p Fm(j)1488 1762 y Fk(\000)p Fr(2)1610 1803 y Fm(\024)28 b Fq(4)p Fs(k)1818 1762 y Fk(\000)p Fr(1)1912 1722 y Fl(\000)1958 1803 y Fu(Im)d Fs(M)2198 1722 y Fl(\001)2244 1803 y Fs(;)116 b(k)31 b(>)d Fq(0)p Fs(:)456 1981 y Fu(Note)c(also)h (that)456 2159 y(\(5.5\))716 b(Im)24 b Fs(M)10 b Fq(\()p Fs(k)s Fq(\))29 b Fs(>)e Fq(0)99 b(if)31 b Fu(Im)25 b Fs(k)2291 2117 y Fr(2)2358 2159 y Fs(>)j Fq(0)p Fs(:)456 2336 y Fu(Thus,)j(there)h(are)f(constants)61 b Fs(C)1598 2351 y Fr(0)1677 2336 y Fm(2)39 b Fp(R)j Fu(and)31 b Fs(C)2130 2351 y Fr(1)2208 2336 y Fm(\025)40 b Fq(0)31 b Fu(and)g(a)g(positi)n(v)o(e)e(measure)i Fs(\026)p Fu(,)456 2453 y(such)24 b(that)1588 2489 y Fl(Z)1688 2516 y Fk(1)1644 2715 y(\0001)1812 2558 y Fs(d\026)p Fq(\()p Fs(t)p Fq(\))p 1800 2602 244 4 v 1800 2693 a(1)e(+)g Fs(t)2004 2665 y Fr(2)2081 2625 y Fs(<)28 b Fm(1)p Fs(;)456 2839 y Fu(where)456 3046 y(\(5.6\))201 b Fs(M)10 b Fq(\()p Fs(k)s Fq(\))28 b(=)g Fs(C)1284 3061 y Fr(0)1345 3046 y Fq(+)22 b Fs(C)1513 3061 y Fr(1)1553 3046 y Fs(z)k Fq(+)1722 2911 y Fl(Z)1778 3136 y Ff(R)1847 2936 y Fl(\020)1995 2979 y Fq(1)p 1916 3023 207 4 v 1916 3115 a Fs(t)c Fm(\000)h Fs(z)2155 3046 y Fm(\000)2369 2979 y Fs(t)p 2264 3023 244 4 v 2264 3115 a Fq(1)f(+)g Fs(t)2468 3086 y Fr(2)2518 2936 y Fl(\021)2577 3046 y Fs(d\026)p Fq(\()p Fs(t)p Fq(\))p Fs(;)116 b(k)2995 3005 y Fr(2)3062 3046 y Fq(=)28 b Fs(z)t(:)456 3285 y Fu(Finally)-6 b(,)27 b(note)h(that)g Fs(R)q Fq(\()p Fs(z)t Fq(\))34 b(=)f Fs(P)1569 3300 y Fr(0)1609 3285 y Fq(\()p Fs(U)1723 3249 y Fk(\003)1763 3285 y Fs(H)1844 3300 y Fr(0)1883 3285 y Fs(U)i Fq(+)25 b Fs(V)46 b Fm(\000)25 b Fs(z)t Fq(\))2377 3249 y Fk(\000)p Fr(1)2472 3285 y Fs(P)2535 3300 y Fr(0)2602 3285 y Fu(and)j(therefore)h(we)f(can)456 3401 y(write)c(formally)g(that)653 3639 y Fs(M)10 b Fq(\()p Fs(k)s Fq(\))28 b(=)1084 3572 y Fs(@)1140 3536 y Fr(2)p 1029 3616 207 4 v 1029 3707 a Fs(@)5 b(r)s(@)g(s)1245 3639 y(G)1322 3654 y Fo(z)1362 3639 y Fq(\()p Fs(r)m(;)17 b(s)p Fq(\))p Fm(j)1597 3655 y Fr(\(1)p Fo(;)p Fr(1\))1773 3639 y Fq(=)28 b(\()p Fs(P)1978 3654 y Fr(0)2017 3639 y Fq(\()p Fs(U)2131 3598 y Fk(\003)2171 3639 y Fs(H)2252 3654 y Fr(0)2292 3639 y Fs(U)k Fq(+)22 b Fs(V)44 b Fm(\000)23 b Fs(z)t Fq(\))2776 3598 y Fk(\000)p Fr(1)2871 3639 y Fs(P)2934 3654 y Fr(0)2973 3639 y Fs(\016)3020 3598 y Fk(0)3016 3664 y Fr(1)3056 3639 y Fs(;)17 b(\016)3147 3598 y Fk(0)3143 3664 y Fr(1)3182 3639 y Fq(\))p Fs(;)456 3852 y Fu(where)i Fs(\016)765 3816 y Fk(0)761 3877 y Fr(1)820 3852 y Fu(is)f(the)h(deri)n(v)n(ati)n(v)o(e)d(of)j Fs(\016)t Fq(\()p Fs(r)s Fm(\000)p Fq(1\))p Fu(.)29 b(Let)18 b Fs(\037)2118 3867 y Fo(c)2149 3876 y Fh(1)2207 3852 y Fu(be)h(the)f(characteristic)h(function)f(of)456 3968 y Fq(\(1)p Fs(;)f(c)629 3983 y Fr(1)667 3968 y Fq(\))p Fu(.)34 b(The)26 b(representation)f(\(4.3\))h(for)g(the)g(resolv)o(ent) f(operator)h(gi)n(v)o(es)e(us)h(the)h(rep-)456 4085 y(resentation)e (for)h(the)g(operator)g Fs(\037)1624 4100 y Fo(c)1655 4109 y Fh(1)1693 4085 y Fs(P)1756 4100 y Fr(0)1795 4085 y Fs(E)1867 4100 y Fo(U)1922 4081 y Fe(\003)1958 4100 y Fo(H)2016 4109 y Fh(0)2051 4100 y Fo(U)7 b Fr(+)p Fo(V)2222 4085 y Fq(\()p Fs(\016)t Fq(\))p Fs(P)2408 4100 y Fr(0)2447 4085 y Fs(\037)2508 4100 y Fo(c)2539 4109 y Fh(1)2577 4085 y Fu(,)25 b(where)g Fs(E)2967 4100 y Fo(U)3022 4081 y Fe(\003)3058 4100 y Fo(H)3116 4109 y Fh(0)3151 4100 y Fo(U)7 b Fr(+)p Fo(V)3322 4085 y Fq(\()p Fs(\016)t Fq(\))456 4201 y Fu(is)24 b(the)h(spectral)f(measure)h(of)g Fs(U)1569 4165 y Fk(\003)1609 4201 y Fs(H)1690 4216 y Fr(0)1729 4201 y Fs(U)33 b Fq(+)22 b Fs(V)g Fu(:)456 4428 y(\(5.7\))417 b Fq(\()p Fs(P)1165 4443 y Fr(0)1204 4428 y Fs(E)1276 4443 y Fo(U)1331 4424 y Fe(\003)1367 4443 y Fo(H)1425 4452 y Fh(0)1459 4443 y Fo(U)7 b Fr(+)p Fo(V)1630 4428 y Fq(\()p Fs(\016)t Fq(\))p Fs(P)1816 4443 y Fr(0)1855 4428 y Fs(f)e(;)17 b(f)11 b Fq(\))28 b(=)2180 4292 y Fl(Z)2235 4518 y Fo(\016)2290 4428 y Fm(j)p Fs(F)14 b Fq(\()p Fs(\025)p Fq(\))p Fm(j)2556 4387 y Fr(2)2594 4428 y Fs(d\026)p Fq(\()p Fs(\025)p Fq(\))456 4656 y Fu(and)24 b(where)615 4865 y Fs(F)14 b Fq(\()p Fs(\025)p Fq(\))27 b(=)968 4797 y(1)p 965 4842 55 4 v 965 4933 a Fs(k)1046 4729 y Fl(Z)1146 4755 y Fo(c)1177 4764 y Fh(1)1101 4955 y Fr(0)1232 4865 y Fq(sin)o(\()p Fs(k)s Fq(\()p Fs(r)e Fm(\000)e Fq(1\)\))p Fs(f)11 b Fq(\()p Fs(r)s Fq(\))24 b Fs(dr)m(;)141 b Fq(supp)q Fs(f)38 b Fm(\032)28 b Fq(\(1)p Fs(;)17 b(c)2806 4880 y Fr(1)2845 4865 y Fq(\))p Fs(;)67 b(k)3031 4824 y Fr(2)3098 4865 y Fq(=)27 b Fs(\025:)456 5099 y Fu(Since)d Fs(F)38 b Fu(is)23 b(a)i(boundary)e(v)n(alue)g(of)i(an)f(analytic)f(function,)g (we)i(obtain)e(that)g Fs(F)14 b Fq(\()p Fs(\025)p Fq(\))28 b Fm(6)p Fq(=)456 5216 y(0)c Fu(for)h(a.e.)32 b Fs(\025)p Fu(.)e(This)24 b(means)h(that)f Fs(E)1680 5231 y Fo(H)1747 5216 y Fq(\()p Fs(\016)t Fq(\))k Fm(6)p Fq(=)g(0)c Fu(if)h Fs(\026)2220 5179 y Fk(0)2271 5216 y Fs(>)i Fq(0)e Fu(a.e.)31 b(on)25 b Fs(\016)t Fu(.)p eop %%Page: 8 8 8 7 bop 456 251 a Fi(8)845 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND)f (SAFR)m(ONO)l(V)1417 450 y Fu(6.)52 b(A)29 b Ft(T)t(R)t(A)q(C)t(E)k(I)t (N)t(E)t(Q)s(U)q(A)t(L)t(I)t(T)t(Y)555 624 y Fu(In)45 b(this)e(section)h(we)g(assume)g(that)g Fs(V)65 b Fu(is)44 b(not)g(a)h(potential)e(b)n(ut)g(the)h(operator)456 666 y Fl(P)561 692 y Fo(n)561 770 y(j)t Fr(=0)704 741 y Fs(P)767 756 y Fo(j)803 741 y Fs(V)899 666 y Fl(P)1004 692 y Fo(n)1004 770 y(j)t Fr(=0)1147 741 y Fs(P)1210 756 y Fo(j)1246 741 y Fu(,)d(which)c(approximates)g Fs(V)59 b Fu(for)38 b(lar)n(ge)f Fs(n)p 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b(we)g(obtain)456 3521 y(\(6.1\))696 3385 y Fl(Z)796 3411 y Fr(+)p Fk(1)752 3611 y(\0001)942 3521 y Fq(log\()p Fs(a)1157 3536 y Fo(")1194 3521 y Fq(\()p Fs(k)s Fq(\))p Fs(=)p Fd(B)p Fq(\()p Fs(k)s Fq(\)\))17 b Fs(dk)30 b Fq(=)1891 3453 y Fs(\031)p 1891 3498 59 4 v 1896 3589 a Fq(2)1977 3385 y Fl(Z)2093 3521 y Fs(V)2150 3536 y Fr(1)2206 3521 y Fs(dr)12 b Fq(+)e(2)p Fs(\031)2507 3410 y Fl(\020)2566 3426 y(X)2726 3521 y Fs(\014)2781 3536 y Fo(n)2838 3521 y Fm(\000)2925 3426 y Fl(X)3085 3521 y Fu(Im)25 b Fs(\015)3272 3536 y Fo(j)3308 3410 y Fl(\021)3367 3521 y Fs(;)456 3753 y Fu(pro)o(vided)f(that)h(for) h(some)f(inte)o(ger)g Fs(l)31 b Fm(\025)f Fq(0)25 b Fu(the)g(coef)n (\002cient)h Fs(a)2571 3768 y Fo(")2608 3753 y Fq(\()p Fs(k)s Fq(\))g Fu(has)f(an)h(e)o(xpansion)456 3870 y Fs(a)507 3885 y Fo(")544 3870 y Fq(\()p Fs(k)s Fq(\))i(=)807 3795 y Fl(P)912 3899 y Fo(j)t Fk(\025\000)p Fo(l)1097 3870 y Fs(c)1139 3885 y Fo(j)1175 3870 y Fs(k)1229 3833 y Fo(j)1291 3870 y Fu(at)d(zero.)33 b(The)26 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y(Let)c(us)f(introduce)g(martices)g Fs(A)p Fq(\()p Fs(k)s Fq(\))g Fu(and)h Fs(B)5 b Fq(\()p Fs(k)s Fq(\))19 b Fu(de\002ned)h(in)f(the)h(space)f Fs(P)14 b(L)3118 1800 y Fr(2)3158 1836 y Fq(\()p Fp(S)3257 1800 y Fo(d)p Fk(\000)p Fr(1)3382 1836 y Fq(\))p Fu(,)456 1952 y(such)24 b(that)g(the)h(solution)e(of)i(the)g(equation)f(\(for)h(the)g(matrix)f (v)n(alued)g(function)g Fq(\010)p Fu(\))456 2188 y(\(6.3\))173 b Fm(\000)907 2121 y Fs(d)958 2085 y Fr(2)998 2121 y Fq(\010)p 907 2165 161 4 v 919 2257 a Fs(dr)1017 2228 y Fr(2)1101 2188 y Fm(\000)22 b Fs(\020)1243 2203 y Fo(")1290 2121 y Fq(\001)1371 2136 y Fo(\022)1410 2121 y Fq(\010)p 1290 2165 191 4 v 1342 2257 a Fs(r)1389 2228 y Fr(2)1512 2188 y Fq(+)g Fs(V)g Fq(\010)28 b(=)g Fs(k)1945 2147 y Fr(2)1984 2188 y Fq(\010)p Fs(;)117 b Fq(\010)28 b(=)g(exp)q(\()p Fs(ik)s(r)s Fq(\))p Fs(P)s(;)132 b(r)30 b(>)e(c)3204 2203 y Fr(2)3243 2188 y Fs(;)456 2399 y Fu(equals)c Fq(exp)q(\()p Fs(ik)s(r)s Fq(\))p Fs(A)p Fq(\()p Fs(k)s Fq(\))e(+)g(exp)q(\()p Fm(\000)p Fs(ik)s(r)s Fq(\))p Fs(B)5 b Fq(\()p Fs(k)s Fq(\))26 b Fu(for)f Fs(r)30 b(<)e(c)2449 2414 y Fr(1)2488 2399 y Fu(.)555 2515 y(W)-8 b(e)25 b(pro)o(v)o(e)f(in)h(the)f(appendix) g(of)h(the)g(paper)g(that)456 2737 y(\(6.4\))1058 2669 y Fq(1)p 973 2714 218 4 v 973 2805 a Fs(a)1024 2820 y Fo(")1061 2805 y Fq(\()p Fs(k)s Fq(\))1201 2737 y Fs(P)1264 2752 y Fr(0)1331 2737 y Fq(=)i Fs(P)1497 2752 y Fr(0)1537 2656 y Fl(\000)1582 2737 y Fs(A)p Fq(\()p Fs(k)s Fq(\))c(+)f(\()p Fs(I)30 b Fm(\000)22 b Fs(P)2179 2752 y Fr(0)2218 2737 y Fq(\))p Fs(e)2301 2695 y Fk(\000)p Fr(2)p Fo(ik)2458 2737 y Fs(B)5 b Fq(\()p Fs(k)s Fq(\))2667 2656 y Fl(\001)2713 2678 y Fk(\000)p Fr(1)2808 2737 y Fs(P)2871 2752 y Fr(0)2910 2737 y Fs(:)456 2973 y Fu(W)-8 b(e)33 b(shall)f(also)h(see)g(that)g Fs(A)p Fq(\()p Fs(k)s Fq(\))g Fu(and)g Fs(B)5 b Fq(\()p Fs(k)s Fq(\))33 b Fu(both)f(ha)n(v)o(e)h(at)g(most)f(a)h(simple)f(pole) g(at)456 3089 y(zero)27 b(and)g(therefore)h(by)f(\(6.4\))g Fs(a)1608 3104 y Fo(")1645 3089 y Fq(\()p Fs(k)s Fq(\))g Fu(could)g(also)g(ha)n(v)o(e)g(a)g(pole)g(at)g(zero.)39 b(Moreo)o(v)o(er)456 3205 y(we)25 b(shall)f(pro)o(v)o(e)g(that)1466 3386 y Fm(j)p Fs(a)1545 3401 y Fo(")1582 3386 y Fq(\()p Fs(k)s Fq(\))p Fm(j)e(\000)g Fq(1)28 b(=)f Fs(O)2119 3275 y Fl(\020)2238 3318 y Fq(1)p 2188 3363 149 4 v 2188 3454 a Fm(j)p Fs(k)s Fm(j)2298 3425 y Fr(2)2347 3275 y Fl(\021)2406 3386 y Fs(;)456 3605 y Fu(as)d Fs(k)31 b Fm(!)d(\0061)p Fs(;)d Fu(which,)f(in)g(particular)l(,)h(means)f(that) h Fq(log)16 b Fm(j)p Fs(a)2500 3620 y Fo(")2537 3605 y Fq(\()p Fs(k)s Fq(\))p Fm(j)27 b(2)h Fs(L)2882 3569 y Fr(1)2922 3605 y Fq(\()p Fp(R)5 b Fq(\))p Fu(.)555 3733 y(Observ)o(e)35 b(that)f(when)h Fs(")46 b Fm(!)g Fq(0)35 b Fu(the)f(eigen)l(v)n(alues)g(of)2476 3708 y Fq(^)2451 3733 y Fs(H)2532 3748 y Fo(")2598 3733 y Fq(+)c Fs(V)56 b Fu(con)l(v)o(er)n(ge)35 b(to)g(the)456 3857 y(eigen)l(v)n(alues)30 b(of)i(the)g(operator)1609 3832 y Fq(^)1583 3857 y Fs(H)j Fq(+)27 b Fs(V)22 b Fu(,)33 b(where)2240 3832 y Fq(^)2214 3857 y Fs(H)40 b Fu(is)31 b(the)g(follo)n(wing)f(operator)i(in)456 3973 y Fs(L)522 3937 y Fr(2)561 3973 y Fq(\()p Fp(R)5 b Fs(;)17 b(L)775 3937 y Fr(2)821 3973 y Fq(\()p Fp(S)920 3937 y Fo(d)p Fk(\000)p Fr(1)1045 3973 y Fq(\)\))1069 4188 y(^)1044 4213 y Fs(H)35 b Fq(=)28 b Fm(\000)1351 4145 y Fs(d)1402 4109 y Fr(2)1441 4145 y Fs(u)p 1351 4190 146 4 v 1355 4281 a(dr)1453 4252 y Fr(2)1529 4213 y Fm(\000)22 b Fs(\020)1671 4228 y Fo(")1718 4145 y Fq(\001)1799 4160 y Fo(\022)1838 4145 y Fs(u)p 1718 4190 176 4 v 1763 4281 a(r)1810 4252 y Fr(2)1903 4213 y Fs(;)117 b Fq(\()p Fs(I)30 b Fm(\000)22 b Fs(P)2320 4228 y Fr(0)2360 4213 y Fq(\))p Fs(u)p Fq(\(1)p Fs(;)17 b Fm(\001)p Fq(\))26 b(=)h(0)p Fs(:)555 4450 y Fu(Denote)e(the)f(eigen)l(v)n(alues)g(of)1636 4425 y Fq(^)1611 4450 y Fs(H)1692 4465 y Fo(")1753 4450 y Fu(by)h Fm(\000)1955 4370 y Fl(\000)2001 4450 y Fs(\014)2062 4399 y Fr(\(0\))2056 4460 y Fo(m)2156 4370 y Fl(\001)2201 4392 y Fr(2)2241 4450 y Fu(,)g(where)g Fs(\014)2620 4399 y Fr(\(0\))2614 4460 y Fo(m)2741 4450 y Fs(>)j Fq(0)p Fu(.)j(Let)24 b(us)h(pro)o(v)o(e)456 4566 y(that)f(by)g(using)g (Lieb-Thirring)g(inequalities)f([7])i(we)g(can)h(obtain)456 4792 y(\(6.5\))914 4697 y Fl(X)1075 4792 y Fs(\014)1136 4751 y Fr(\(0\))1130 4816 y Fo(m)1258 4792 y Fm(\024)i Fs(C)1440 4681 y Fl(\020)1499 4656 y(Z)1555 4882 y Ff(R)1603 4863 y Fg(d)1659 4792 y Fs(V)1738 4741 y Fr(\()p Fo(d)p Fr(+1\))p Fo(=)p Fr(2)1716 4813 y Fk(\000)1994 4792 y Fs(dx)22 b Fq(+)2220 4656 y Fl(Z)2276 4882 y Ff(R)2324 4863 y Fg(d)2381 4792 y Fs(V)2438 4807 y Fk(\000)2496 4792 y Fm(j)p Fs(x)p Fm(j)2607 4751 y Fk(\000)p Fo(d)p Fr(+1)2793 4792 y Fs(dx)2899 4681 y Fl(\021)2958 4792 y Fs(:)456 5031 y Fu(Indeed,)i(let)h Fs(W)993 5046 y Fk(\000)1080 5031 y Fq(=)1183 4958 y Fm(p)p 1266 4958 116 4 v 73 x Fs(V)1323 5046 y Fk(\000)1382 5031 y Fu(.)31 b(Then)838 5216 y Fs(W)930 5231 y Fk(\000)989 5216 y Fq(\()1052 5190 y(^)1027 5216 y Fs(H)e Fm(\000)23 b Fs(z)t Fq(\))1324 5174 y Fk(\000)p Fr(1)1419 5216 y Fs(W)1511 5231 y Fk(\000)1598 5216 y Fq(=)k Fs(W)1793 5231 y Fk(\000)1853 5216 y Fq(\()p Fs(S)g Fm(\000)c Fs(z)t Fq(\))2165 5174 y Fk(\000)p Fr(1)2260 5216 y Fs(W)2352 5231 y Fk(\000)2433 5216 y Fq(+)f Fs(W)2623 5231 y Fk(\000)2683 5216 y Fq(\002\()p Fs(z)t Fq(\))p Fs(W)2976 5231 y Fk(\000)3035 5216 y Fs(;)p eop %%Page: 10 10 10 9 bop 456 251 a Fi(10)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND)f (SAFR)m(ONO)l(V)456 450 y Fu(where)32 b Fs(S)38 b Fu(is)32 b(de\002ned)g(in)g(\(3.4\))g(and)g Fq(\002\()p Fs(z)t Fq(\))h Fu(is)f(the)g(operator)g(of)g(rank)g(one)g(with)g(the)456 569 y(inte)o(gral)23 b(k)o(ernel)i Fs(e)1101 533 y Fo(ik)r Fr(\()p Fo(r)r Fr(+)p Fo(s)p Fk(\000)p Fr(2\))1435 569 y Fs(=)p Fq(2)p Fs(ik)s Fu(,)f Fs(k)1723 533 y Fr(2)1790 569 y Fq(=)k Fs(z)t Fu(.)k(Therefore)456 779 y(\(6.6\))283 b Fm(jj)p Fs(W)1078 794 y Fk(\000)1137 779 y Fq(\002\()p Fs(z)t Fq(\))p Fs(W)1430 794 y Fk(\000)1489 779 y Fm(jj)27 b(\024)1751 712 y Fs(C)p 1687 756 205 4 v 1687 776 a Fl(p)p 1787 776 106 4 v 85 x Fm(j)p Fs(z)t Fm(j)1919 644 y Fl(Z)2035 779 y Fs(V)2092 794 y Fk(\000)2151 779 y Fm(j)p Fs(x)p Fm(j)2262 738 y Fk(\000)p Fo(d)p Fr(+1)2464 779 y Fs(dx;)116 b(z)33 b(<)27 b Fq(0)p Fs(:)456 1016 y Fu(No)n(w)57 b(for)h(an)o(y)f(compact)h(operator)g Fs(T)72 b Fu(and)58 b Fs(s)89 b(>)g Fq(0)57 b Fu(denote)h Fs(n)2984 1031 y Fr(+)3043 1016 y Fq(\()p Fs(s;)17 b(T)d Fq(\))89 b(=)456 1132 y(rank)17 b Fs(E)737 1147 y Fo(T)792 1132 y Fq(\()p Fs(s;)g Fm(1)p Fq(\))p Fu(.)30 b(Then)984 1250 y Fl(X)1145 1344 y Fs(\014)1206 1303 y Fr(\(0\))1200 1369 y Fo(m)1327 1344 y Fq(=)1431 1209 y Fl(Z)1530 1235 y Fk(1)1486 1434 y Fr(0)1622 1344 y Fs(n)1680 1359 y Fr(+)1739 1344 y Fq(\(1)p Fs(;)17 b(W)1962 1359 y Fk(\000)2021 1344 y Fq(\()2084 1319 y(^)2059 1344 y Fs(H)29 b Fq(+)22 b Fs(t)p Fq(\))2340 1303 y Fk(\000)p Fr(1)2435 1344 y Fs(W)2527 1359 y Fk(\000)2586 1344 y Fq(\))2674 1277 y Fs(dt)p 2634 1321 168 4 v 2634 1422 a Fq(2)2683 1341 y Fm(p)p 2765 1341 36 4 v 2765 1422 a Fs(t)2838 1344 y Fm(\024)584 1625 y(\024)689 1489 y Fl(Z)789 1515 y Fk(1)745 1715 y Fr(0)864 1544 y Fl(\000)909 1625 y Fs(n)967 1640 y Fr(+)1027 1625 y Fq(\(1)p Fs(=)p Fq(2)p Fs(;)17 b(W)1348 1640 y Fk(\000)1406 1625 y Fq(\()p Fs(S)28 b Fq(+)22 b Fs(t)p Fq(\))1703 1583 y Fk(\000)p Fr(1)1797 1625 y Fs(W)1889 1640 y Fk(\000)1948 1625 y Fq(\))g(+)g Fs(n)2164 1640 y Fr(+)2224 1625 y Fq(\(1)p Fs(=)p Fq(2)p Fs(;)17 b(W)2545 1640 y Fk(\000)2603 1625 y Fq(\002\()p Fm(\000)p Fs(t)p Fq(\))p Fs(W)2959 1640 y Fk(\000)3018 1625 y Fq(\))3056 1544 y Fl(\001)3152 1557 y Fs(dt)p 3112 1602 168 4 v 3112 1703 a Fq(2)3161 1622 y Fm(p)p 3244 1622 36 4 v 81 x Fs(t)3289 1625 y(:)456 1814 y Fu(No)n(w)24 b(the)g(inequality)g(\(6.5\))h(follo)n(ws)e(from)795 1887 y Fl(Z)895 1914 y Fk(1)851 2113 y Fr(0)986 2023 y Fs(n)1044 2038 y Fr(+)1104 2023 y Fq(\(1)p Fs(=)p Fq(2)p Fs(;)17 b(W)1425 2038 y Fk(\000)1483 2023 y Fq(\()p Fs(S)28 b Fq(+)22 b Fs(t)p Fq(\))1780 1982 y Fk(\000)p Fr(1)1874 2023 y Fs(W)1966 2038 y Fk(\000)2025 2023 y Fq(\))2114 1956 y Fs(dt)p 2073 2000 168 4 v 2073 2101 a Fq(2)2122 2020 y Fm(p)p 2205 2020 36 4 v 81 x Fs(t)2278 2023 y Fm(\024)28 b Fs(C)2477 1887 y Fl(Z)2532 2113 y Ff(R)2580 2094 y Fg(d)2637 2023 y Fs(V)2715 1972 y Fr(\()p Fo(d)p Fr(+1\))p Fo(=)p Fr(2)2694 2045 y Fk(\000)2971 2023 y Fs(dx;)456 2233 y Fu(which)c(is)g(the)h(classical)f(Lieb-Thirring)g (inequality)g(and)g(from)834 2306 y Fl(Z)933 2333 y Fk(1)889 2532 y Fr(0)1025 2442 y Fs(n)1083 2457 y Fr(+)1142 2442 y Fq(\(1)p Fs(=)p Fq(2)p Fs(;)17 b(W)1463 2457 y Fk(\000)1521 2442 y Fq(\002\()p Fm(\000)p Fs(t)p Fq(\))p Fs(W)1877 2457 y Fk(\000)1937 2442 y Fq(\))2025 2375 y Fs(dt)p 1985 2419 168 4 v 1985 2520 a Fq(2)2034 2439 y Fm(p)p 2116 2439 36 4 v 2116 2520 a Fs(t)2189 2442 y Fm(\024)28 b Fs(C)2388 2306 y Fl(Z)2504 2442 y Fs(V)2561 2457 y Fk(\000)2620 2442 y Fm(j)p Fs(x)p Fm(j)2731 2401 y Fk(\000)p Fo(d)p Fr(+1)2933 2442 y Fs(dx;)456 2652 y Fu(which)c(is)g(implied)f (by)h(\(6.6\).)31 b(Consequently)-6 b(,)23 b(since)h(Im)h Fs(\015)2522 2667 y Fo(j)2586 2652 y Fm(\025)j Fq(0)c Fu(the)h(trace)g(formula)456 2768 y(\(6.1\))f(together)h(with)f (\(6.5\))h(leads)f(to)h(the)f(inequality)1021 2989 y Fq(lim)17 b(sup)1101 3068 y Fo(")p Fk(!)p Fr(0)1336 2853 y Fl(Z)1436 2879 y Fr(+)p Fk(1)1391 3078 y(\0001)1582 2989 y Fq(log)g Fm(j)p Fs(a)1804 3004 y Fo(")1840 2989 y Fq(\()p Fs(k)s Fq(\))p Fm(j)g Fs(dk)30 b Fm(\024)2262 2921 y Fs(\031)p 2262 2966 59 4 v 2267 3057 a Fq(2)2347 2853 y Fl(Z)2447 2879 y Fr(+)p Fk(1)2403 3078 y(\0001)2593 2989 y Fs(V)2650 3004 y Fr(1)2706 2989 y Fs(dr)s Fq(+)1181 3258 y(+)p Fs(C)1334 3148 y Fl(\020)1393 3123 y(Z)1448 3348 y Ff(R)1496 3329 y Fg(d)1553 3258 y Fs(V)1632 3207 y Fr(\()p Fo(d)p Fr(+1\))p Fo(=)p Fr(2)1610 3280 y Fk(\000)1888 3258 y Fs(dx)22 b Fq(+)2114 3123 y Fl(Z)2170 3348 y Ff(R)2218 3329 y Fg(d)2275 3258 y Fs(V)2332 3273 y Fk(\000)2390 3258 y Fm(j)p Fs(x)p Fm(j)2501 3217 y Fk(\000)p Fo(d)p Fr(+1)2687 3258 y Fs(dx)2793 3148 y Fl(\021)2852 3258 y Fs(:)456 3118 y Fu(\(6.7\))456 3466 y(Therefore)j(for)g(an)o(y)g (pair)f(of)h(\002nite)g(numbers)f Fs(r)2111 3481 y Fr(2)2178 3466 y Fs(>)k(r)2326 3481 y Fr(1)2393 3466 y Fm(\025)g Fq(0)505 3667 y Fl(Z)604 3693 y Fo(r)636 3702 y Fh(2)560 3892 y Fo(r)592 3901 y Fh(1)701 3735 y Fq(1)p 701 3779 49 4 v 701 3871 a(2)777 3802 y(log)1112 3735 y Fs(k)p 929 3779 420 4 v 929 3871 a Fq(4)p Fu(Im)d Fs(M)10 b Fq(\()p Fs(k)s Fq(\))1375 3802 y Fs(dk)30 b Fm(\024)e Fq(lim)17 b(sup)1693 3881 y Fo(")p Fk(!)p Fr(0)1928 3667 y Fl(Z)2028 3693 y Fr(+)p Fk(1)1983 3892 y(\0001)2174 3802 y Fq(log)f Fm(j)p Fs(a)2395 3817 y Fo(")2432 3802 y Fq(\()p Fs(k)s Fq(\))p Fm(j)h Fs(dk)30 b Fm(\024)2854 3735 y Fs(\031)p 2854 3779 59 4 v 2859 3871 a Fq(2)2939 3667 y Fl(Z)3039 3693 y Fr(+)p Fk(1)2995 3892 y(\0001)3185 3802 y Fs(V)3242 3817 y Fr(1)3298 3802 y Fs(dr)1697 4074 y Fq(+)p Fs(C)1850 3964 y Fl(\020)1909 3939 y(Z)1965 4164 y Ff(R)2013 4145 y Fg(d)2069 4074 y Fs(V)2148 4023 y Fr(\()p Fo(d)p Fr(+1\))p Fo(=)p Fr(2)2126 4096 y Fk(\000)2404 4074 y Fs(dx)22 b Fq(+)2630 3939 y Fl(Z)2686 4164 y Ff(R)2734 4145 y Fg(d)2791 4074 y Fs(V)2848 4089 y Fk(\000)2906 4074 y Fm(j)p Fs(x)p Fm(j)3017 4033 y Fk(\000)p Fo(d)p Fr(+1)3203 4074 y Fs(dx)3309 3964 y Fl(\021)3368 4074 y Fs(;)456 3605 y Fu(\(6.8\))456 4282 y(where)j(the)g(\002rst)f (inequality)g(follo)n(ws)f(from)i(Corollary)g(5.3)f([4].)999 4480 y(7.)52 b(T)t Ft(H)t(E)30 b(E)t(N)t(D)h(O)t(F)f(T)t(H)t(E)h(P)t(R) q(O)t(O)t(F)i(O)t(F)e Fu(T)t Ft(H)t(E)t(O)t(R)t(E)t(M)i Fu(2)t(.)t(1)555 4737 y(Assume)k(that)g(our)g(perturbation)g Fs(V)59 b Fu(is)37 b(an)g(arbitrary)h(function)f(satisfying)f(the)456 4853 y(conditions)27 b(of)j(Theorem)f(0.1.)44 b(Then)30 b(the)f(W)-8 b(e)o(yl)29 b(function)g Fs(M)40 b Fu(can)30 b(be)f(de\002ned)h(for)456 4978 y(e)o(xample)22 b(as)h Fs(M)10 b Fq(\()p Fs(k)s Fq(\))28 b(=)1333 4938 y Fo(@)1374 4915 y Fh(2)p 1297 4955 149 4 v 1297 5012 a Fo(@)t(r)r(@)t(s)1455 4978 y Fs(G)1532 4993 y Fo(z)1572 4978 y Fq(\()p Fs(r)m(;)17 b(s)p Fq(\))p Fm(j)1807 4993 y Fr(\(1)p Fo(;)p Fr(1\))1979 4978 y Fu(where)23 b Fs(G)2322 4993 y Fo(z)2385 4978 y Fu(is)g(the)g(inte)o(gral)e(k)o(ernel)j(of)f(the)456 5099 y(operator)30 b Fs(P)881 5114 y Fr(0)920 5099 y Fq(\()p Fs(U)1034 5063 y Fk(\003)1074 5099 y Fs(H)8 b(U)36 b Fm(\000)26 b Fs(z)t Fq(\))1455 5063 y Fk(\000)p Fr(1)1551 5099 y Fs(P)1614 5114 y Fr(0)1653 5099 y Fu(.)46 b(The)30 b(ne)o(xt)f(proposition)f(allo)n(ws)h(us)g(to)h(approxi-)456 5216 y(mate)24 b Fs(V)47 b Fu(by)24 b(compactly)g(supported)g(smooth)f (functions)h Fs(V)2531 5231 y Fo(n)2578 5216 y Fu(.)p eop %%Page: 11 11 11 10 bop 1610 253 a Fi(SZEG)1811 235 y(\005)1798 253 y(O)23 b(CONDITION)1079 b(11)456 450 y Fx(Pr)n(oposition)30 b(7.1.)45 b Fn(Let)31 b Fs(V)53 b Fn(satisfy)29 b(the)i(conditions)e (of)i(Theor)l(em)g(2.1.)49 b(Then)31 b(ther)l(e)456 566 y(e)n(xists)e(a)h(sequence)g Fs(V)1237 581 y Fo(n)1314 566 y Fn(of)g(compactly)g(supported)e(smooth)h(functions)g(con)l(ver)l (ging)456 683 y(to)24 b Fs(V)46 b Fn(so)25 b(that)456 872 y Fu(\(7.1\))780 736 y Fl(Z)880 872 y Fq(\()p Fs(V)975 887 y Fo(n)1022 872 y Fq(\))1060 821 y Fr(\()p Fo(d)p Fr(+1\))p Fo(=)p Fr(2)1060 893 y Fk(\000)1332 872 y Fs(dx)j(<)g(C)7 b Fq(\()p Fs(V)21 b Fq(\))100 b(and)2174 736 y Fl(Z)2274 872 y Fq(\()p Fs(V)2369 887 y Fo(n)2415 872 y Fq(\))2453 887 y Fr(+)2529 872 y Fm(j)p Fs(x)p Fm(j)2640 830 y Fk(\000)p Fo(d)p Fr(+1)2842 872 y Fs(dx)28 b(<)f(C)7 b Fq(\()p Fs(V)22 b Fq(\))456 1078 y Fn(and)37 b(suc)o(h)f(that)h(the)g(W)-9 b(e)m(yl)37 b(functions)f Fs(M)1934 1093 y Fo(n)2019 1078 y Fn(corr)l(esponding)g(to)h Fs(V)2806 1093 y Fo(n)2890 1078 y Fn(con)l(ver)l(g)o(e)h(uni-)456 1194 y(formly)24 b(when)h Fs(k)1030 1158 y Fr(2)1094 1194 y Fn(belongs)f(to)h(any)f (compact)h(subset)f(of)g(the)h(upper)f(half)g(plane:)1606 1354 y Fs(M)1700 1369 y Fo(n)1747 1354 y Fq(\()p Fs(k)s Fq(\))k Fm(!)f Fs(M)10 b Fq(\()p Fs(k)s Fq(\))p Fs(:)456 1514 y Fn(Ther)l(efor)l(e)37 b(the)g(sequence)g(of)g(measur)l(es)f Fs(\026)2022 1529 y Fo(n)2106 1514 y Fn(con)l(ver)l(g)o(es)h(weakly)h (to)e(the)h(spectr)o(al)456 1630 y(measur)l(e)24 b Fs(\026)p Fn(.)456 1806 y(Pr)l(oof)o(.)39 b Fu(Let)29 b Fs(W)997 1821 y Fk(\006)1091 1806 y Fq(=)1201 1733 y Fm(p)p 1284 1733 116 4 v 73 x Fs(V)1341 1821 y Fk(\006)1400 1806 y Fu(.)42 b(Since)29 b(the)f(class)g Fs(C)2166 1770 y Fk(1)2159 1831 y Fr(0)2270 1806 y Fu(is)g(dense)g(in)g Fs(L)2792 1770 y Fo(p)2861 1806 y Fu(for)h(an)o(y)f Fs(p)34 b(>)h Fq(0)p Fu(,)456 1922 y(we)25 b(can)g(\002nd)g(a)g(pair)g(of)g (sequences)f Fs(W)1831 1886 y Fk(\000)1817 1947 y Fo(n)1915 1922 y Fu(and)h Fs(W)2190 1886 y Fr(+)2176 1947 y Fo(n)2276 1922 y Fm(2)j Fs(C)2447 1886 y Fk(1)2440 1947 y Fr(0)2547 1922 y Fu(satisfying)782 2090 y Fs(W)888 2049 y Fk(\000)874 2114 y Fo(n)974 2090 y Fm(!)f Fs(W)1193 2105 y Fk(\000)1302 2090 y Fq(in)16 b Fs(L)1466 2049 y Fr(\()p Fo(d)p Fr(+1\))1652 2090 y Fq(\()p Fp(R)1756 2049 y Fo(d)1802 2090 y Fq(\);)117 b Fs(W)2090 2049 y Fr(+)2076 2114 y Fo(n)2176 2090 y Fm(!)27 b Fs(W)2395 2105 y Fr(+)2471 2090 y Fq(in)16 b Fs(L)2635 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Fm(j)780 4647 y Fr(2)817 4683 y Fq(\))c Fu(when)g Fs(k)31 b Fm(!)c(\0061)p Fu(.)456 4867 y Fn(Ac)n(knowledg)o(ements.)38 b Fu(The)27 b(authors)g(are)h(grateful)f(to)g(the)g(Institute)f(Mittag-Lef)n (\003er)456 4983 y(for)20 b(its)f(hospitality)f(during)i(the)g(f)o(all) g(semester)g(2002.)28 b(A.L)20 b(and)g(O.S.)h(thank)e(a)i(partial)456 5099 y(support)36 b(by)h(the)h(SPECT)g(ESF)h(European)e(programme.)68 b(S.N.)38 b(w)o(as)g(also)f(partly)456 5216 y(supported)23 b(by)i(the)g(KBN)g(grant)f(5,)h(PO3A/026/21.)k(g1925l.)p eop %%Page: 16 16 16 15 bop 456 251 a Fi(16)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND) f(SAFR)m(ONO)l(V)1671 450 y Fu(R)t Ft(E)t(F)t(E)t(R)t(E)t(N)t(C)5 b(E)g(S)497 608 y Fw([1])40 b(Z.)27 b(S.)g(Agrano)o(vich)c(and)j(V)-11 b(.)27 b(A.)f(Marchenk)o(o,)g Fj(Re-establishment)e(of)j(the)f (potential)f(fr)l(om)i(the)635 707 y(scattering)22 b(matrix)f(for)h(a)g (system)g(of)g(dif)o(fer)m(ential)e(equations)p Fw(,)g(\(Russian\))h (Dokl.)g(Akad.)g(Nauk)635 807 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