Content-Type: multipart/mixed; boundary="-------------0302210546245" This is a multi-part message in MIME format. ---------------0302210546245 Content-Type: text/plain; name="03-69.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-69.keywords" anisotropy, cartesian, asymptotic completeness, spectral theory, scattering theory, asymptotic velocity, minimal velocity, maximal velocity, N-body, algebra, propagation estimates, Schrodinger operators, Schroedinger operators ---------------0302210546245 Content-Type: application/postscript; name="comp.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="comp.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: comp.dvi %%Pages: 29 1 %%BoundingBox: 0 0 596 843 %%EndComments %%BeginProcSet: tex.pro /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ 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Ft(+)2601 959 y(2)2660 933 y Fx(,)34 b(and)f(th)m(us)h(this)f(state)g(will)g(b)s(e)83 1046 y(asymptotically)f(free.)41 b(But)31 b(there)g(migh)m(t)h(also)e(exist)h (some)f(in\034nite)i(v)-5 b(alley)31 b(parallel)g(to)g(one)g(of)f(the)h(axis) 83 1159 y(whic)m(h)k(could)f(trap)g(some)g(scattering)h(states.)49 b(And)34 b(suc)m(h)g(states)f(w)m(ould)i(then)f(b)s(eha)m(v)m(e)h (asymptotically)83 1272 y(lik)m(e)c(guided)g(w)m(a)m(v)m(es.)249 1385 y(This)36 b(v)-5 b(ariet)m(y)37 b(of)f(p)s(ossible)g(outcomes)h(for)f (the)g(asymptotic)h(ev)m(olution)h(is)e(one)g(of)g(the)h(reasons)f(for)83 1498 y(the)43 b(complexit)m(y)g(of)f(the)g(analysis)g(of)g(anisotropic)i (systems.)75 b(In)42 b(order)h(to)f(predict)i(the)e(asymptotic)83 1611 y(b)s(eha)m(viour)32 b(of)f(a)g(giv)m(en)g(scattering)i(state,)e(one)g (has)g(to)h(kno)m(w)f(roughly)h(its)f(asymptotic)g(lo)s(calization.)44 b(It)83 1724 y(seems)27 b(to)g(us)g(that)h(the)f(righ)m(t)i(concept)f(for)f (obtaining)h(this)g(information)g(is)f(the)g Fr(asymptotic)j(velo)-5 b(city)p Fx(.)40 b(In)83 1836 y(the)c(previous)f(example,)j(the)d(asymptotic) i(v)m(elo)s(cit)m(y)f(of)f(the)g(asymptotically)h(free)g(state)f(p)s(oin)m (ts)h(out)g(in)83 1949 y(the)c(p)s(ositiv)m(e)h(quadran)m(t,)h(while)e(for)g (the)h(asymptotically)f(guided)h(state,)g(the)g(asymptotic)f(v)m(elo)s(cit)m (y)h(has)83 2062 y(a)d(zero)h(comp)s(onen)m(t.)42 b(Suc)m(h)32 b(c)m(haracteristics)g(will)f(b)s(e)f(used)g(for)g(classifying)g(the)h (scattering)g(states.)249 2175 y(Let)j(us)g(brie\035y)g(describ)s(e)g(our)g (mathematical)h(to)s(ols.)50 b(F)-8 b(or)35 b(the)e(sp)s(ectral)h(analysis,)h (w)m(e)f(mainly)g(use)83 2288 y(the)29 b(metho)s(d)g(of)f(the)g(conjugate)i (op)s(erator)f(in)f(the)h(algebraic)g(framew)m(ork)h(dev)m(elop)s(ed)f(b)m(y) g(W.O.)f(Amrein,)83 2401 y(A.)33 b(Boutet)i(de)f(Mon)m(v)m(el)g(and)h(V.)e (Georgescu)h([1)q(].)50 b(In)33 b(this)h(approac)m(h,)j(the)d(main)g(ob)5 b(ject)34 b(of)f(the)h(theory)83 2514 y(is)i(a)g(C)326 2481 y FC(\003)366 2514 y Fx(-subalgebra)h Fo(C)49 b Fx(of)36 b(the)g(set)g(of)g (b)s(ounded)h(linear)g(op)s(erators)g(in)f(some)h(Hilb)s(ert)f(space)g Fu(H)q Fx(.)58 b(This)83 2627 y(C)148 2594 y FC(\003)188 2627 y Fx(-algebra)30 b(is)e(closely)h(related)h(to)f(the)g(anisotrop)m(y)-8 b(.)42 b(The)29 b(op)s(erators)g Fw(H)36 b Fx(under)30 b(consideration)g(are) f(then)83 2740 y(self-adjoin)m(t)35 b(op)s(erators)f(in)h Fu(H)f Fx(a\036liated)i(to)e Fo(C)13 b Fx(,)35 b Fr(i.e.)e Fx(the)i(resolv)m(en)m(t) g Fv(\()p Fw(H)30 b Fu(\000)23 b Fw(z)t Fv(\))2812 2707 y FC(\000)p Ft(1)2940 2740 y Fx(b)s(elongs)35 b(to)f Fo(C)47 b Fx(for)33 b(an)m(y)83 2853 y(complex)42 b(n)m(um)m(b)s(er)h Fw(z)i Fx(with)d(non-zero)g (imaginary)g(part.)74 b(W)-8 b(e)41 b(also)g(rely)g(up)s(on)h(the)f(recen)m (t)i(idea)e(that)83 2966 y(a)d(class)f(of)h(functions)g(de\034ned)h(on)f Fv(I)-18 b(R)1427 2930 y Fs(n)1511 2966 y Fx(ha)m(ving)39 b(a)f(certain)h(t)m (yp)s(e)f(of)f(anisotrop)m(y)i(is)f(asso)s(ciated)f(with)i(a)83 3078 y(compacti\034cation)g(of)e Fv(I)-18 b(R)958 3043 y Fs(n)1005 3078 y Fx(,)38 b(the)g(one)f(on)f(whic)m(h)i(all)f(these)g(functions)g(admit) h(a)e(con)m(tin)m(uous)j(extension.)83 3191 y(W)-8 b(e)27 b(refer)f(to)h ([2],)g([8)q(])f(and)h(to)f([12)q(],)h([13)q(])f(of)g(M.)g(M)p Fv(\025)-45 b Fx(an)m(toiu)28 b(for)e(motiv)-5 b(ations)27 b(and)g(some)f(general)i(principles.)83 3304 y(F)-8 b(or)29 b(the)g(scattering)h(theory)-8 b(,)30 b(the)f(strategy)g(of)g(J.)g (Derezi\253ski)g(and)g(C.)g(G\351rard)g(exp)s(osed)g(in)g([4],)g(Sections)83 3417 y(6.6)42 b(and)h(6.7,)i(is)d(follo)m(w)m(ed.)77 b(V)-8 b(arious)43 b(propagation)g(estimates)g(are)f(pro)m(v)m(ed)i(with)e(the)h (help)f(of)g(some)83 3530 y(propagation)32 b(observ)-5 b(ables)30 b(and)g(with)g(a)g(partition)h(of)e(unit)m(y)i(inspired)f(b)m(y)g(the)g(pap)s (er)g(of)g(G.M.)f(Graf)g([9)q(].)83 3643 y(The)i(notions)g(of)f(minimal)h (and)g(maximal)g(v)m(elo)s(cities)f(are)h(in)m(tro)s(duced)h(and)f(the)f (asymptotic)h(v)m(elo)s(cit)m(y)g(is)83 3756 y(used)f(for)h(the)f (de\034nition)i(of)e(the)h(w)m(a)m(v)m(e)h(op)s(erators)e(and)h(the)g(pro)s (of)f(of)g(asymptotic)h(completeness.)249 3869 y(In)j(the)h(sequel)g(w)m(e)g (shall)g(consider)g(p)s(oten)m(tials)h Fw(V)54 b Fx(suc)m(h)36 b(that)f Fv(lim)2573 3883 y Fs(x)2613 3893 y Fn(j)2645 3883 y FC(!\0061)2861 3869 y Fw(V)20 b Fv(\()p Fu(\001)p Fv(\))35 b Fx(exist)f(for)h(eac)m(h)g Fw(j)j Fu(2)83 3982 y(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)33 b(;)15 b(n)p Fu(g)41 b Fx(in)g(a)g(suitable)g(sense,)i (and)f(call)f(them)g Fr(c)-5 b(artesian)43 b(p)-5 b(otentials)p Fx(.)72 b(This)41 b(leads)g(to)g(a)g(natural)83 4095 y Fw(n)p Fx(-dimensional)e(generalization)g(of)e(certain)i(situations)f(considered)g (in)g([3])g(and)g([7].)62 b(The)38 b(underlying)83 4208 y(compacti\034cation) c(of)d Fv(I)-18 b(R)947 4172 y Fs(n)1025 4208 y Fx(is)31 b(the)g(cartesian)h (pro)s(duct)g(of)f Fw(n)f Fx(copies)h(of)g(the)h(t)m(w)m(o-p)s(oin)m(t)h (compacti\034cation)p 83 4247 82 4 v 83 4320 a Fv(I)-18 b(R)33 b(:=)h Fu(f\0001g)24 b(t)f Fv(I)-18 b(R)23 b Fu(t)g(f)p Fv(+)p Fu(1g)36 b Fx(of)f Fv(I)-18 b(R)o Fx(.)56 b(Hence)35 b(let)h(us)f(de\034ne)p 2231 4247 V 36 w Fv(I)-18 b(R)2322 4266 y Fs(n)2402 4320 y Fv(:=)p 2532 4247 V 34 w(I)g(R)2613 4334 y Ft(1)2676 4320 y Fu(\002)23 b(\001)15 b(\001)g(\001)25 b(\002)p 2994 4247 V 23 w Fv(I)-18 b(R)3076 4334 y Fs(n)3158 4320 y Fx(\(the)36 b(indexation)83 4433 y(corresp)s(onds)22 b(to)g(that)g(of)f(the)h(v)-5 b(ariables\))23 b(endo)m(w)m(ed)h(with)e(the)g(pro)s(duct)g(top)s(ology)-8 b(,)25 b(and)d(let)g Fw(C)7 b Fv(\()p 3313 4360 V(I)-18 b(R)3403 4379 y Fs(n)3450 4433 y Fv(\))22 b Fx(denote)83 4546 y(the)32 b(algebra)g(of)f(con)m(tin)m(uous)i(complex)f(functions)f(on)p 1986 4473 V 32 w Fv(I)-18 b(R)2076 4492 y Fs(n)2123 4546 y Fx(.)43 b(This)31 b(algebra)h(is)f(naturally)h(iden)m(ti\034ed)h(with)83 4659 y(a)e(subalgebra)h(of)f Fw(B)5 b(C)855 4673 y Fs(u)899 4659 y Fv(\(I)-18 b(R)1016 4623 y Fs(n)1063 4659 y 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y(the)g(h)m(yp)s(ercub)s(e,)g(w)m(e)g(tak)m(e)g(b)m(y)f(con)m(v)m(en)m (tion)j Fu(H)1684 1103 y Fs(\013)1759 1089 y Fv(=)25 b Fu(H)1933 1056 y Ft(2)1932 1111 y Fs(\013)2006 1089 y Fv(=)g Fk(C)18 b Fx(.)249 1202 y(F)-8 b(or)43 b(an)m(y)f(function)h Fw(V)65 b Fu(2)44 b Fw(C)7 b Fv(\()p 1301 1129 V(I)-18 b(R)1392 1148 y Fs(n)1439 1202 y Fv(\))p Fx(,)45 b(its)d(restriction)h Fw(V)2199 1169 y Fs(\013)2290 1202 y Fx(to)f(the)h(h)m(yp)s(ersurface)p 3119 1129 V 42 w Fv(I)-18 b(R)3210 1148 y Fs(\013)3301 1202 y Fx(is)42 b(iden)m(ti\034ed)83 1314 y(with)j(an)f(elemen)m(t)i(of)d Fw(B)5 b(C)1045 1328 y Fs(u)1090 1314 y Fv(\(I)-18 b(R)1207 1279 y Fs(\013)1256 1314 y Fv(\))p Fx(.)82 b(One)45 b(notices)f(that)h(the)f (expression)g Fw(V)2820 1281 y Fs(\013)2869 1314 y Fv(\()p Fw(x)p Fv(\))h Fx(with)f Fw(x)49 b Fu(2)e Fv(I)-18 b(R)3546 1279 y Fs(n)3637 1314 y Fx(has)83 1427 y(an)46 b(unam)m(biguous)h(meaning.)87 b(Indeed,)50 b(the)c(algebra)g Fw(B)5 b(C)2210 1441 y Fs(u)2255 1427 y Fv(\(I)-18 b(R)2372 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1846 y Fs(\013)609 1879 y Fv(+)25 b Fw(V)779 1846 y Fs(\013)828 1879 y Fx(,)40 b(the)f(former)f(b)s(eing)g(a)h(self-adjoin)m(t)f(op)s(erator) h(in)g Fu(H)f Fx(with)h(domain)g Fu(H)3382 1846 y Ft(2)3459 1879 y Fx(and)g(the)83 1992 y(latter)33 b(a)e(self-adjoin)m(t)i(op)s(erator)f (in)g Fu(H)1423 2006 y Fs(\013)1504 1992 y Fx(with)g(domain)h Fu(H)2112 1959 y Ft(2)2111 2014 y Fs(\013)2160 1992 y Fx(.)45 b(Let)32 b Fw(\033)2445 2006 y Fs(p)2485 1992 y Fv(\()p Fu(\001)p Fv(\))g Fx(denote)h(the)f(p)s(oin)m(t)g(sp)s(ectrum)g(of)83 2105 y(an)m(y)c(self-adjoin)m(t)h(op)s(erator.)41 b(With)28 b(the)g Fr(c)-5 b(artesian)31 b(Hamiltonian)c Fw(H)32 b Fu(\021)25 b Fw(H)2685 2072 y Fs(o)2748 2105 y Fv(=)g Fu(\000)p Fv(\001)15 b(+)g Fw(V)21 b Fx(,)28 b(one)g(asso)s(ciates)83 2218 y(t)m(w)m(o)35 b(sp)s(ecial)e(sets:)47 b(the)33 b(set)h(of)f(thresholds)h Fw(\034)10 b Fv(\()p Fw(H)d Fv(\))31 b(=)f Fu([)2014 2233 y Fs(\013)p FC(6)p Ft(=)p Fs(o)2182 2218 y Fw(\033)2234 2232 y Fs(p)2274 2218 y Fv(\()p Fw(H)2392 2185 y Fs(\013)2442 2218 y Fv(\))p Fx(,)k(and)g Fw(\024)p Fv(\()p Fw(H)7 b Fv(\))32 b(=)e Fu([)3114 2232 y Fs(\013)p FC(2L)3289 2218 y Fw(\033)3341 2232 y Fs(p)3380 2218 y Fv(\()p Fw(H)3498 2185 y Fs(\013)3548 2218 y Fv(\))p Fx(,)35 b(the)83 2331 y(critical)c(set)g(of)f Fw(H)7 b Fx(.)249 2444 y(In)32 b(order)i(to)f(giv)m(e)g(a)g(precise)g (description)h(of)e(the)h(sp)s(ectrum)g Fw(\033)s Fv(\()p Fw(H)7 b Fv(\))34 b Fx(of)e Fw(H)7 b Fx(,)33 b(some)g(regularit)m(y)h(of)f(the)83 2556 y(p)s(oten)m(tial)f(with)f(resp)s(ect)g(to)g(the)g(generator)h Fw(A)f Fx(of)f(dilations)i(has)e(to)h(b)s(e)f(imp)s(osed.)42 b(W)-8 b(e)31 b(refer)g(to)g(Section)83 2669 y(2)k(for)g(the)g(description)h (of)e(this)h(regularit)m(y)1638 2596 y Fl(\000)1679 2669 y Fx(including)i(the)e(de\034nition)h(of)f(the)g(class)f Fw(C)3198 2636 y Ft(1)p Fs(;)p Ft(1)3292 2669 y Fv(\()p Fw(A)p Fv(\))3430 2596 y Fl(\001)3507 2669 y Fx(and)i(to)83 2782 y(Section)25 b(5)g(for)f(its)g(compatibilit)m(y)j(with)e(the)f(cartesian)i(anisotrop)m(y) -8 b(.)40 b(If)23 b Fu(G)30 b Fx(is)24 b(a)g(Banac)m(h)i(space,)g(its)e(norm) h(is)83 2895 y(written)i Fu(k)12 b(\001)g(k)534 2909 y FC(G)585 2895 y Fx(.)39 b(The)26 b(w)m(eigh)m(ted)j(Sob)s(olev)d(space)g Fu(H)1850 2862 y Fs(s)1849 2918 y(t)1913 2895 y Fx(is)g(the)g(closure)h(of)e (the)i(Sc)m(h)m(w)m(artz)i(space)d(on)h Fv(I)-18 b(R)3519 2859 y Fs(n)3592 2895 y Fx(with)83 3008 y(resp)s(ect)32 b(to)h(the)f(norm)h Fu(k)22 b(\001)g(k)1061 3022 y FC(H)1121 2999 y Fn(s)1121 3042 y(t)1187 3008 y Fv(=)28 b Fu(k)p Fv(\(1)23 b(+)e Fw(P)1597 2975 y Ft(2)1637 3008 y Fv(\))1672 2975 y Fs(s=)p Ft(2)1779 3008 y Fv(\(1)i(+)e Fw(Q)2046 2975 y Ft(2)2085 3008 y Fv(\))2120 2975 y Fs(t=)p Ft(2)2242 3008 y Fu(\001)h(k)p Fx(,)33 b(where)g Fw(P)2713 3022 y Fs(j)2778 3008 y Fv(:=)c Fu(\000)p Fw(i)p Fu(r)3081 3022 y Fs(j)3117 3008 y Fx(,)k Fw(j)h Fu(2)28 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)33 b(;)15 b(n)p Fu(g)p Fx(,)83 3121 y(are)36 b(the)h(comp)s(onen)m(ts)g(of)f(the)g(momen)m(tum)j(op)s(erator)d (and)h Fw(Q)2278 3135 y Fs(j)2350 3121 y Fx(is)f(the)g(op)s(erator)h(of)f(m)m (ultiplication)j(b)m(y)83 3234 y(the)31 b(v)-5 b(ariable)31 b Fw(x)630 3248 y Fs(j)666 3234 y Fx(.)41 b(If)29 b Fw(t)c Fv(=)g(0)p Fx(,)31 b(w)m(e)g(simply)f(omit)h(this)f(index.)83 3446 y Fj(Theorem)35 b(1.1)44 b Fr(L)-5 b(et)33 b Fw(H)f Fv(=)25 b Fu(\000)p Fv(\001)20 b(+)g Fw(V)52 b Fr(with)32 b Fw(V)53 b Fr(a)32 b(c)-5 b(artesian)33 b(p)-5 b(otential.)42 b(Then)199 3634 y(i\))j Fw(\033)361 3648 y Ft(ess)452 3634 y Fv(\()p Fw(H)7 b Fv(\))26 b(=)727 3560 y Fl(\002)780 3634 y Fv(min)932 3653 y FC(j)p Fs(\013)p FC(j)p Ft(=)p Fs(n)p FC(\000)p Ft(1)1256 3634 y Fv(inf)20 b Fw(\033)s Fv(\()p Fw(H)1554 3601 y Fs(\013)1604 3634 y Fv(\))p Fw(;)15 b Fu(1)1770 3560 y Fl(\001)1813 3634 y Fr(.)83 3822 y(F)-7 b(urthermor)i(e,)34 b(if)d Fw(V)53 b Fr(is)32 b(of)g(class)g Fw(C)1316 3789 y Ft(1)p Fs(;)p Ft(1)1410 3822 y Fv(\()p Fw(A)p Fv(\))p Fr(,)h(with)f Fw(A)g Fr(the)h(gener)-5 b(ator)34 b(of)e(dilations,)g(then)172 4009 y(ii\))44 b Fw(\034)10 b Fv(\()p Fw(H)d Fv(\))39 b Fr(and)f Fw(\024)p Fv(\()p Fw(H)7 b Fv(\))39 b Fr(ar)-5 b(e)40 b(close)-5 b(d)39 b(c)-5 b(ountable)41 b(sets,)f(the)e(eigenvalues)j(of)d Fw(H)45 b Fr(not)38 b(b)-5 b(elonging)40 b(to)e Fw(\034)10 b Fv(\()p Fw(H)d Fv(\))309 4122 y Fr(ar)-5 b(e)33 b(of)f(\034nite)g(multiplicity)g(and)g(c)-5 b(an)33 b(ac)-5 b(cumulate)35 b(only)d(at)g(p)-5 b(oints)33 b(of)f Fw(\034)10 b Fv(\()p Fw(H)d Fv(\))p Fr(,)144 4310 y(iii\))44 b Fw(H)39 b Fr(has)33 b(no)f(singularly)g(c)-5 b(ontinuous)33 b(sp)-5 b(e)g(ctrum,)158 4498 y(iv\))45 b(for)31 b(e)-5 b(ach)33 b Fw(\016)c(>)c Fv(0)p Fr(,)32 b(ther)-5 b(e)33 b(exists)f Fw(c)26 b(<)f Fu(1)31 b Fr(such)h(that)g Fu(jh)p Fw(';)15 b Fv(\()p Fw(H)27 b Fu(\000)18 b Fw(\025)h Fu(\006)f Fw(i\026)p Fv(\))2733 4465 y FC(\000)p Ft(1)2828 4498 y Fw(')p Fu(ij)26 b(\024)f Fw(c)p Fu(k)p Fw(')p Fu(k)3257 4465 y Ft(2)3257 4541 y FC(H)3317 4511 y Fq(\000)p 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Fs(iH)5 b(t)1015 5292 y(Q)p 1013 5312 61 4 v 1013 5365 a Ft(2)p Fs(t)1083 5333 y Fw(e)1125 5300 y FC(\000)p Fs(iH)g(t)1338 5333 y Fx(in)42 b(a)f(suitable)h(sense.)72 b(Since)42 b(the)g(limit)g Fw(t)h Fu(!)g(\0001)e Fx(is)g(completely)83 5446 y(similar,)28 b(w)m(e)f(do)f(not)h(consider)g(it.)39 b(W)-8 b(e)26 b(denote)h(b)m(y)g Fu(P)1904 5460 y Fs(\013)1980 5446 y Fx(the)f(asymptotic)h(v)m(elo)s(cit)m(y) g(obtained)g(for)f Fw(H)3499 5460 y Fs(\013)3548 5446 y Fx(.)39 b(The)1903 5836 y(3)p eop %%Page: 4 4 bop 83 482 a Fx(follo)m(wing)30 b(partition)g(of)f Fv(I)-18 b(R)1022 446 y Fs(n)1097 482 y Fx(is)29 b(useful)f(for)h(the)g(description)h (of)f(the)g(di\033eren)m(t)h(p)s(ossible)e(outcomes)i(of)f(the)83 594 y(asymptotic)i(ev)m(olution.)42 b(F)-8 b(or)31 b(eac)m(h)g Fw(\013)26 b Fu(2)f(L)p Fx(,)30 b(w)m(e)h(de\034ne)768 797 y Fw(Z)830 811 y Fs(\013)904 797 y Fv(:=)26 b Fu(f)p Fw(x)f Fu(2)g Fv(I)-18 b(R)1316 760 y Fs(n)1393 720 y Fl(\014)1393 774 y(\014)1453 797 y Fw(x)1505 811 y Fs(j)1567 797 y Fv(=)25 b(0)31 b Fx(if)e Fw(\013)1879 811 y Fs(j)1941 797 y Fv(=)c(0)31 b Fx(and)g Fw(\013)2347 811 y Fs(j)2383 797 y Fw(x)2435 811 y Fs(j)2497 797 y Fw(>)25 b Fv(0)31 b Fx(if)f Fw(\013)2810 811 y Fs(j)2871 797 y Fu(6)p Fv(=)25 b(0)p Fu(g)p Fw(:)83 1000 y Fx(W)-8 b(e)36 b(shall)g(pro)m(v)m(e)g(that)g(for)g Fw(\013)e Fu(6)p Fv(=)g Fw(o)p Fx(,)j(the)f(elemen)m(ts)h(of)e Fu(H)h Fx(with)g(supp)s(ort)g(of)f(their)h(asymptotic)g(v)m(elo)s(cit)m(y)83 1113 y(on)e Fw(Z)274 1127 y Fs(\013)356 1113 y Fx(ha)m(v)m(e)h(an)f (asymptotic)g(ev)m(olution)h(go)m(v)m(erned)g(b)m(y)f(the)g(Hamiltonian)g Fw(H)2828 1127 y Fs(\013)2877 1113 y Fx(.)50 b(F)-8 b(or)34 b(this)g(purp)s(ose,)g(w)m(e)83 1226 y(roughly)e(imp)s(ose)f(that)h(the)f(p)s (oten)m(tial)h Fw(V)52 b Fx(approac)m(hes)32 b(its)f(limits)h(at)f(in\034nit) m(y)h(in)g(a)f(short-range)h(w)m(a)m(y)-8 b(.)44 b(A)83 1339 y(more)31 b(precise)g(condition)h(is)e(giv)m(en)h(in)f(Section)h(7,)g (equation)g(\(22\).)249 1452 y(If)g Fw(C)38 b Fx(is)32 b(an)g Fw(m)p Fx(-tuple)h(of)f(comm)m(uting)i(self-adjoin)m(t)f(op)s(erators)f(\()p Fw(m)g Fx(a)g(p)s(ositiv)m(e)g(in)m(teger\),)i(w)m(e)f(denote)83 1564 y(b)m(y)44 b Fw(E)289 1589 y Fm(\004)337 1564 y Fv(\()p Fw(C)7 b Fv(\))43 b Fx(its)h(sp)s(ectral)f(pro)5 b(jection)44 b(corresp)s(onding)h(to)e(the)h(subset)f Fv(\004)k Fu(\032)g Fv(I)-18 b(R)2940 1529 y Fs(m)3006 1564 y Fx(.)79 b(W)-8 b(e)44 b(also)f(use)h(the)83 1677 y(notation)31 b Fw(E)511 1691 y Ft(p)555 1677 y Fv(\()p Fw(B)5 b Fv(\))29 b Fx(for)h(the)g(orthogonal)i(pro)5 b(jection)31 b(on)f(the)g(subspace)g(spanned)g(b)m(y)g(the)h(eigen)m(v)m (ectors)g(of)83 1790 y(a)f(self-adjoin)m(t)h(op)s(erator)h Fw(B)5 b Fx(.)83 2001 y Fj(Theorem)35 b(1.2)44 b Fr(Assume)30 b(that)f Fw(V)49 b Fr(is)29 b(a)g(c)-5 b(artesian)30 b(p)-5 b(otential)30 b(of)f(class)g Fw(C)2624 1968 y Ft(1)p Fs(;)p Ft(1)2718 2001 y Fv(\()p Fw(A)p Fv(\))h Fr(satisfying)e(\(22\),)h(with)g Fw(A)83 2114 y Fr(the)k(gener)-5 b(ator)34 b(of)e(dilations.)40 b(Then)33 b(for)f(e)-5 b(ach)33 b Fw(\013)26 b Fu(2)f(L)p Fr(,)199 2301 y(i\))45 b(the)28 b(op)-5 b(er)g(ator)28 b Fv(\012)869 2268 y Ft(+)869 2323 y Fs(\013)953 2301 y Fv(:=)e Fr(s)8 b Fu(\000)h Fv(lim)1326 2315 y Fs(t)p FC(!)p Ft(+)p FC(1)1567 2301 y Fw(e)1609 2268 y Fs(iH)c(t)1726 2301 y Fw(e)1768 2268 y FC(\000)p Fs(iH)1905 2276 y Fn(\013)1949 2268 y Fs(t)1979 2301 y Fw(E)2046 2325 y Fn(Z)2088 2333 y(\013)2140 2301 y Fv(\()p Fu(P)2238 2315 y Fs(\013)2288 2301 y Fv(\))28 b Fr(exists,)g(and)f(its)g(r)-5 b(ange)28 b Fx(Ran)14 b Fv(\012)3395 2268 y Ft(+)3395 2323 y Fs(\013)3480 2301 y Fr(is)27 b(e)-5 b(qual)309 2414 y(to)32 b Fw(E)484 2438 y Fn(Z)526 2446 y(\013)579 2414 y Fv(\()p Fu(P)7 b Fv(\))p Fu(H)q Fr(,)172 2601 y(ii\))44 b(if)30 b Fw(\014)g Fu(6)p Fv(=)25 b Fw(\013)p Fr(,)31 b(then)f Fx(Ran)14 b Fv(\012)1129 2562 y Ft(+)1129 2630 y Fs(\014)1218 2601 y Fr(is)30 b(ortho)-5 b(gonal)31 b(to)f Fx(Ran)14 b Fv(\012)2100 2568 y Ft(+)2100 2623 y Fs(\013)2159 2601 y Fr(,)30 b(and)g(the)h(dir)-5 b(e)g(ct)32 b(sum)2973 2532 y Fl(L)3074 2628 y Fs(\014)s FC(6)p Ft(=)p Fs(o)3225 2601 y Fx(Ran)14 b Fv(\012)3467 2562 y Ft(+)3467 2630 y Fs(\014)3556 2601 y Fr(sp)-5 b(ans)309 2714 y(the)33 b(absolutely)g(c)-5 b(ontinuous)34 b(subsp)-5 b(ac)g(e)34 b(of)f Fu(H)f Fr(with)g(r)-5 b(esp)g(e)g(ct)35 b(to)e Fw(H)7 b Fr(,)144 2918 y(iii\))44 b(if)25 b Fu(H)h Fr(is)f(identi\034e)-5 b(d)26 b(with)1155 2817 y Fl(\020)1225 2850 y(N)1326 2945 y FC(f)p Fs(j)t FC(j)p Fs(\013)1459 2955 y Fn(j)1491 2945 y FC(6)p Ft(=0)p FC(g)1636 2918 y Fw(L)1698 2885 y Ft(2)1737 2918 y Fv(\(I)-18 b(R)1854 2932 y Fs(j)1890 2918 y Fv(\))1925 2817 y Fl(\021)1985 2918 y Fu(\012)5 b(H)2138 2932 y Fs(\013)2188 2918 y Fr(,)26 b(the)g(sp)-5 b(e)g(ctr)g(al)28 b(pr)-5 b(oje)g(ction)27 b Fw(E)3179 2943 y Fn(Z)3221 2951 y(\013)3273 2918 y Fv(\()p Fu(P)3371 2932 y Fs(\013)3421 2918 y Fv(\))f Fr(is)f(e)-5 b(qual)309 3090 y(to)417 2989 y Fl(\020)487 3022 y(N)588 3117 y FC(f)p Fs(j)t FC(j)p Fs(\013)721 3127 y Fn(j)753 3117 y FC(6)p Ft(=0)p FC(g)898 3090 y Fw(E)965 3109 y FC(f)p Fs(y)r FC(2)p Ft(I)-18 b(R)p FC(j)p Fs(\013)1208 3119 y Fn(j)1241 3109 y Fs(y)r(>)p Ft(0)p FC(g)1408 3090 y Fv(\()p Fw(P)1501 3104 y Fs(j)1538 3090 y Fv(\))1573 2989 y Fl(\021)1648 3090 y Fu(\012)20 b Fw(E)1806 3104 y Ft(p)1849 3090 y Fv(\()p Fw(H)1967 3057 y Fs(\013)2017 3090 y Fv(\))p Fr(.)249 3329 y Fx(Let)36 b(us)f(notice)g(that)h(the)g(pro)5 b(jections)35 b Fw(E)1705 3354 y Fn(Z)1747 3362 y(\013)1800 3329 y Fv(\()p Fu(P)7 b Fv(\))36 b Fx(corresp)s(ond)g(to)f(the)h(pro)5 b(jections)35 b Fw(P)3257 3296 y Ft(+)3316 3329 y Fv(\()p Fw(E)5 b Fv(\))36 b Fx(conjec-)83 3442 y(tured)g(in)f(the)h(In)m(tro)s(duction)h(of) e([3].)55 b(In)35 b(relation)i(with)f(this)f(result,)i(w)m(e)f(men)m(tion)h (the)f(recen)m(t)g(w)m(ork)g(of)83 3555 y(Y.)d(Dermenjian)g(and)h(V.)e (Iftimie)h(in)g(the)g(case)g(of)g(p)s(erturb)s(ed)g(strati\034ed)h(media)g ([5].)48 b(Their)34 b(results)f(are)83 3668 y(comparable)e(but)e(the)h (anisotrop)m(y)g(they)f(consider)h(is)f(less)f(general)i(than)g(ours)f(since) h(it)f(is)f(a)i(short-range)83 3780 y(p)s(erturbation)39 b(of)d(a)i Fw(L)874 3748 y FC(1)948 3780 y Fx(-function)g(whic)m(h)h(dep)s(ends)e(only)g (on)g(the)h(v)-5 b(ariable)38 b Fw(x)2859 3794 y Fs(n)2943 3780 y Fx(and)f(admits)h(limits)f(as)83 3893 y Fw(x)135 3907 y Fs(n)207 3893 y Fu(!)25 b(\0061)p Fx(.)249 4006 y(In)31 b(Section)h(2)g(w)m (e)g(describ)s(e)f(the)h(algebraic)g(framew)m(ork)h(and)e(some)h (generalities)g(on)g(the)f(regularit)m(y)83 4119 y(of)g Fw(H)38 b Fx(with)32 b(resp)s(ect)f(to)g(the)h(conjugate)g(op)s(erator.)44 b(The)32 b(algebra)g(related)h(to)e(the)h(cartesian)g(anisotrop)m(y)83 4232 y(is)k(in)m(tro)s(duced)i(in)f(Section)g(3,)h(where)f(its)f(ric)m(h)i (in)m(ternal)g(structure)f(is)f(in)m(v)m(estigated.)61 b(It)36 b(already)h(giv)m(es)83 4345 y(some)h(informations)g(on)f(the)h(essen)m(tial) g(sp)s(ectrum.)62 b(In)37 b(order)h(to)g(deal)g(with)g(non-smo)s(oth)g(p)s (oten)m(tials,)83 4458 y(some)23 b(tec)m(hnicalities)i(are)e(needed.)39 b(Section)24 b(4)e(is)h(dev)m(oted)h(to)f(this)f(purp)s(ose.)38 b(De\034nition)24 b(4.3)g(con)m(tains)g(the)83 4571 y(description)h(of)f(a)g (generalized)i(class)d(of)h(cartesian)h(p)s(oten)m(tials,)h(whic)m(h)f (includes)g Fw(C)7 b Fv(\()p 2981 4498 82 4 v(I)-18 b(R)3071 4517 y Fs(n)3118 4571 y Fv(\))p Fx(.)39 b(The)24 b(a\036liation)83 4684 y(of)29 b(the)h(corresp)s(onding)h(cartesian)g(Hamiltonians)f(to)g(the)g (men)m(tioned)i(algebra)e(is)g(pro)m(v)m(ed.)41 b(The)30 b(Mourre)83 4797 y(estimate)h(and)f(the)g(limiting)h(absorption)g(principle)g(are)f(elab) s(orated)h(in)f(Section)h(5,)f(where)h(the)f(pro)s(of)f(of)83 4910 y(Theorem)37 b(1.1)e(is)h(giv)m(en.)56 b(The)36 b(last)f(t)m(w)m(o)i (sections)f(are)f(dedicated)i(to)f(the)g(scattering)g(theory)-8 b(.)57 b(Section)83 5023 y(6)32 b(deals)h(with)g(the)f(asymptotic)i(v)m(elo)s (cit)m(y)f Fu(P)39 b Fx(and)33 b(some)g(of)f(its)g(prop)s(erties.)47 b(In)32 b(Section)h(7,)g(w)m(e)g(use)f(it)h(to)83 5135 y(construct)e(the)g(w) m(a)m(v)m(e)h(op)s(erators)f(and)f(to)h(pro)m(v)m(e)h(Theorem)f(1.2.)249 5248 y(W)-8 b(e)35 b(end)g(the)g(In)m(tro)s(duction)h(with)g(t)m(w)m(o)g (observ)-5 b(ations.)54 b(The)35 b(\034rst)g(one)g(concerns)g(the)g (relationship)83 5361 y(b)s(et)m(w)m(een)42 b(cartesian)g(and)f(N-b)s(o)s(dy) e(Hamiltonians.)73 b(Although)41 b(our)g(approac)m(h)h(for)f(the)g(sp)s (ectral)f(and)83 5474 y(scattering)f(theory)g(of)f(the)g(former)h(is)f (similar)g(to)h(that)f(dev)m(elop)s(ed)i(for)e(the)g(latter,)j(p)s(oten)m (tials)f(whic)m(h)83 5587 y(are)e(b)s(oth)g(cartesian)g(and)g(of)f(N-b)s(o)s (dy)g(t)m(yp)s(e)h(are)f(v)m(ery)h(sp)s(ecial)g(cases)f(of)g(cartesian)i(p)s (oten)m(tials)f(and)g(of)1903 5836 y(4)p eop %%Page: 5 5 bop 83 482 a Fx(N-b)s(o)s(dy)26 b(p)s(oten)m(tials.)40 b(Indeed,)28 b(in)f(the)h(formalism)f(of)g(generalized)h(N-b)s(o)s(dy)e(systems)g(\(see)h (Section)h(5.1)f(of)83 594 y([4]\))33 b(suc)m(h)h(p)s(oten)m(tials)g(corresp) s(ond)f(to)g(a)g(system)g(related)h(to)f(a)g(\034nite)g(semilattice)h(of)f (subspaces)g(of)f Fv(I)-18 b(R)3721 559 y Fs(n)83 707 y Fx(whic)m(h)39 b(satisfy)d(some)h(orthogonalit)m(y)j(relations;)h(on)d(the)g(other)g(hand,)h (as)e(cartesian)i(p)s(oten)m(tials,)h(they)83 820 y(m)m(ust)32 b(con)m(v)m(erge)i(to)e(zero)g(\(in)g(a)g(suitable)g(sense\))g(except)g(in)g (the)g(vicinit)m(y)h(of)e(some)h(subspaces)g(of)f Fv(I)-18 b(R)3617 784 y Fs(n)3695 820 y Fx(of)83 933 y(lo)m(w)m(er)28 b(dimensions.)40 b(The)27 b(second)f(observ)-5 b(ation)27 b(is)f(that)h(the)g (di\036culties)g(due)g(to)f(the)h(anisotrop)m(y)g(already)83 1046 y(app)s(ear)35 b(in)f(t)m(w)m(o)i(dimensions,)h(a)d(situation)h(whic)m (h)h(is)e(easily)g(visualized.)54 b(Ho)m(w)m(ev)m(er,)37 b(since)d(our)h(w)m (ork)g(is)83 1159 y Fw(n)p Fx(-dimensional,)c(w)m(e)f(ha)m(v)m(e)g(included)g (some)g(relev)-5 b(an)m(t)30 b(examples)f(of)g(cartesian)h(p)s(oten)m(tials)h (in)e(Sections)h(4,)83 1272 y(5)g(and)h(7.)83 1558 y Fy(2)131 b(The)44 b(Algebraic)i(F)-11 b(ramew)l(ork)83 1761 y Fx(Let)26 b(us)e(consider)i(a)f(self-adjoin)m(t)h(op)s(erator)g Fw(H)32 b Fx(in)25 b(a)g(Hilb)s(ert)g(space)g Fu(H)q Fx(.)38 b(The)26 b(sp)s(ectrum)f(and)h(the)f(essen)m(tial)83 1874 y(sp)s(ectrum)31 b(of)f Fw(H)37 b Fx(can)31 b(b)s(e)f(expressed)g(in)g(terms)h(of)f(its)g(con) m(tin)m(uous)j(functional)e(calculus:)3187 1841 y Fi(1)594 2078 y Fw(\033)s Fv(\()p Fw(H)7 b Fv(\))84 b(=)f Fu(f)p Fw(\025)26 b Fu(2)e Fv(I)-18 b(R)1361 2001 y Fl(\014)1361 2056 y(\014)1422 2078 y Fx(if)30 b Fw(\021)e Fu(2)d Fw(C)1729 2092 y Ft(0)1768 2078 y Fv(\(I)-18 b(R\))31 b Fx(and)g Fw(\021)s Fv(\()p Fw(\025)p Fv(\))26 b Fu(6)p Fv(=)f(0)p Fw(;)46 b Fx(then)31 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))26 b Fu(6)p Fv(=)f(0)p Fu(g)p Fw(;)506 2216 y(\033)558 2230 y Ft(ess)649 2216 y Fv(\()p Fw(H)7 b Fv(\))84 b(=)f Fu(f)p Fw(\025)26 b Fu(2)e Fv(I)-18 b(R)1361 2139 y Fl(\014)1361 2193 y(\014)1422 2216 y Fx(if)30 b Fw(\021)e Fu(2)d Fw(C)1729 2230 y Ft(0)1768 2216 y Fv(\(I)-18 b(R\))31 b Fx(and)g Fw(\021)s Fv(\()p Fw(\025)p Fv(\))26 b Fu(6)p Fv(=)f(0)p Fw(;)46 b Fx(then)31 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))26 b Fu(62)f(K)q Fv(\()p Fu(H)q Fv(\))p Fu(g)p Fw(:)83 2420 y Fx(If)40 b Fo(C)54 b Fx(is)41 b(a)g(C)547 2387 y FC(\003)586 2420 y Fx(-subalgebra)i(of)d Fu(B)s Fv(\()p Fu(H)q Fv(\))p Fx(,)k(then)e Fw(H)47 b Fx(is)41 b(said)g(to)g(b)s(e)g Fr(a\036liate)-5 b(d)43 b(to)e Fo(C)53 b Fx(if)41 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))44 b Fu(2)f Fo(C)54 b Fx(for)41 b(all)83 2533 y Fw(\021)29 b Fu(2)c Fw(C)308 2547 y Ft(0)347 2533 y Fv(\(I)-18 b(R\))p Fx(.)41 b(A)30 b(su\036cien)m(t)h (condition)h(is)e(that)h Fv(\()p Fw(H)c Fu(\000)20 b Fw(z)t Fv(\))2036 2500 y FC(\000)p Ft(1)2156 2533 y Fu(2)25 b Fo(C)43 b Fx(for)30 b(some)h Fw(z)e Fu(2)c Fk(C)44 b Fu(n)21 b Fw(\033)s Fv(\()p Fw(H)7 b Fv(\))p Fx(.)249 2646 y(The)31 b(ab)s(o)m(v)m(e)g(situation) g(is)f(a)g(sp)s(ecial)h(case)f(of)g(the)h(follo)m(wing)h(more)e(abstract)i (framew)m(ork:)83 2834 y Fj(De\034nition)i(2.1)161 b Fr(i\))45 b(A)d Fx(self-adjoin)m(t)g(observ)-5 b(able)42 b(a\036liated)g(to)f(a)h(C) 2672 2801 y FC(\003)2711 2834 y Fx(-algebra)i Fo(C)55 b Fr(is)42 b(a)g(functional)309 2947 y(c)-5 b(alculus)36 b(taking)f(value)h(in)d Fo(C)13 b Fr(,)35 b(i.e.)f(a)1637 2914 y FC(\003)1676 2947 y Fr(-morphism)g Fw(H)i Fv(:)29 b Fw(C)2362 2961 y Ft(0)2401 2947 y Fv(\(I)-18 b(R\))29 b Fu(!)g Fo(C)13 b Fr(.)47 b(The)34 b(notation)f Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))36 b Fr(wil)5 b(l)309 3060 y(b)-5 b(e)34 b(use)-5 b(d)33 b(inste)-5 b(ad)33 b(of)f Fw(H)7 b Fv(\()p Fw(\021)s Fv(\))p Fr(.)172 3247 y(ii\))44 b(The)35 b Fx(sp)s(ectrum)g Fw(\033)s Fv(\()p Fw(H)7 b Fv(\))35 b Fr(of)g(the)g(observable)i Fw(H)42 b Fr(is)34 b(the)h(set)g(of)f(r)-5 b(e)g(al)36 b(values)g Fw(\025)f Fr(such)g(that,)g(whenever)309 3360 y Fw(\021)29 b Fu(2)c Fw(C)534 3374 y Ft(0)573 3360 y Fv(\(I)-18 b(R\))32 b Fr(and)g Fw(\021)s Fv(\()p Fw(\025)p Fv(\))27 b Fu(6)p Fv(=)e(0)p Fr(,)32 b(then)h Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))26 b Fu(6)p Fv(=)f(0)p Fr(.)144 3548 y(iii\))44 b(If)25 b Fw(\031)j Fv(:)e Fo(C)38 b Fu(!)25 b Fo(C)807 3515 y FC(0)856 3548 y Fr(is)h(a)1018 3515 y FC(\003)1058 3548 y Fr(-morphism)f(b)-5 b(etwe)g(en)28 b(two)e(C)2063 3515 y FC(\003)2102 3548 y Fr(-algebr)-5 b(as)28 b(and)d Fw(H)33 b Fr(is)25 b(a)h(self-adjoint)f (observable)309 3661 y(a\036liate)-5 b(d)29 b(to)g Fo(C)13 b Fr(,)29 b(then)h Fw(\031)s Fv(\()p Fw(H)7 b Fv(\))26 b(:)f Fw(C)1451 3675 y Ft(0)1491 3661 y Fv(\(I)-18 b(R)o(\))26 b Fu(!)f Fo(C)1853 3628 y FC(0)1876 3661 y Fr(,)30 b(given)f(by)g Fw(\021)2327 3587 y Fl(\000)2369 3661 y Fw(\031)s Fv(\()p Fw(H)7 b Fv(\))2577 3587 y Fl(\001)2645 3661 y Fv(:=)25 b Fw(\031)2821 3587 y Fl(\000)2863 3661 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))3064 3587 y Fl(\001)3107 3661 y Fr(,)29 b(is)f(a)h(self-adjoint)309 3774 y(observable)35 b(a\036liate)-5 b(d)33 b(to)f Fo(C)1284 3741 y FC(0)1307 3774 y Fr(.)41 b(W)-7 b(e)33 b(c)-5 b(al)5 b(l)33 b(it)f Fx(the)f(image)g(of)f Fw(H)39 b Fr(thr)-5 b(ough)34 b Fw(\031)s Fr(.)83 3961 y Fx(In)c(the)h(sequel)f(w)m(e)h(shall)f(simply)h (write)g(morphism)g(for)2046 3928 y FC(\003)2086 3961 y Fx(-morphism)g(b)s (et)m(w)m(een)h(t)m(w)m(o)g(C)3123 3928 y FC(\003)3163 3961 y Fx(-algebras.)249 4074 y(W)-8 b(e)27 b(recall)h(some)f(de\034nitions)h (related)f(to)g(the)h(Mourre)f(estimate)h(and)f(refer)g(to)g([1])g(for)f (details)i(and)f(a)83 4187 y(self-con)m(tained)e(presen)m(tation.)41 b(Let)24 b Fu(f)p Fw(W)1479 4201 y Fs(t)1509 4187 y Fu(g)1554 4201 y Fs(t)p FC(2)p Ft(I)-18 b(R)1714 4187 y Fx(b)s(e)23 b(the)h(unitary)g (group)h(in)e Fu(H)h Fx(generated)i(b)m(y)d(a)h(self-adjoin)m(t)83 4300 y(op)s(erator)f Fw(A)p Fx(.)38 b(F)-8 b(or)22 b(an)m(y)h Fw(B)30 b Fu(2)25 b(B)s Fv(\()p Fu(H)q Fv(\))p Fx(,)e(w)m(e)g(write)g Fw(B)30 b Fu(2)24 b Fw(C)1934 4267 y Ft(1)1973 4300 y Fv(\()p Fw(A)p Fv(\))f Fx(if)f(the)g(mapping)i Fv(I)-18 b(R)25 b Fu(3)f Fw(t)i Fu(7!)f Fw(W)3173 4314 y FC(\000)p Fs(t)3257 4300 y Fw(B)5 b(W)3417 4314 y Fs(t)3471 4300 y Fu(2)25 b(B)s Fv(\()p Fu(H)q Fv(\))83 4413 y Fx(is)43 b(strongly)h Fw(C)622 4380 y Ft(1)661 4413 y Fx(.)79 b(If)43 b(this)g(mapping)i(is)e Fw(C)1619 4380 y Ft(1)1701 4413 y Fx(in)g(norm)h(w)m(e)h(write)f Fw(B)51 b Fu(2)c Fw(C)2759 4380 y Ft(1)2752 4435 y Fs(u)2798 4413 y Fv(\()p Fw(A)p Fv(\))p Fx(.)81 b(By)43 b(assuming)g(that)83 4526 y Fw(B)30 b Fu(2)25 b Fw(C)340 4493 y Ft(1)379 4526 y Fv(\()p Fw(A)p Fv(\))p Fx(,)31 b(w)m(e)g(giv)m(e)g(a)f(rigorous)h(sense)f(to) h(the)f(comm)m(utator)j Fv([)p Fw(B)5 b(;)15 b(iA)p Fv(])26 b Fu(2)f(B)s Fv(\()p Fu(H)q Fv(\))p Fx(.)249 4639 y(A)37 b(self-adjoin)m(t)h (op)s(erator)g Fw(H)44 b Fx(in)37 b Fu(H)h Fx(is)f Fr(of)h(class)f Fw(C)2047 4606 y Ft(1)2086 4639 y Fv(\()p Fw(A)p Fv(\))2262 4565 y Fl(\000)2304 4639 y Fx(resp.)61 b Fw(C)2623 4606 y Ft(1)2616 4661 y Fs(u)2662 4639 y Fv(\()p Fw(A)p Fv(\))2800 4565 y Fl(\001)2880 4639 y Fx(if)36 b Fv(\()p Fw(H)d Fu(\000)24 b Fw(z)t Fv(\))3289 4606 y FC(\000)p Ft(1)3421 4639 y Fu(2)36 b Fw(C)3590 4606 y Ft(1)3629 4639 y Fv(\()p Fw(A)p Fv(\))83 4678 y Fl(\000)125 4752 y Fx(resp.)41 b Fv(\()p Fw(H)28 b Fu(\000)20 b Fw(z)t Fv(\))663 4719 y FC(\000)p Ft(1)784 4752 y Fu(2)26 b Fw(C)943 4719 y Ft(1)936 4774 y Fs(u)982 4752 y Fv(\()p Fw(A)p Fv(\))1120 4678 y Fl(\001)1193 4752 y Fx(for)k(some,)h(and)g(then)h(for)e(all,)h Fw(z)g Fu(2)25 b Fk(C)44 b Fu(n)21 b Fw(\033)s Fv(\()p Fw(H)7 b Fv(\))p Fx(.)43 b(Let)31 b Fu(G)36 b Fx(b)s(e)30 b(the)h(domain)h(of)83 4865 y Fw(H)40 b Fx(endo)m(w)m(ed)35 b(with)f(the)f(graph)h(norm)g(and)f (assume)h(that)f(it)g(is)g(left)g(in)m(v)-5 b(arian)m(t)35 b(b)m(y)e(the)h(group)g Fu(f)p Fw(W)3532 4879 y Fs(t)3562 4865 y Fu(g)3607 4879 y Fs(t)p FC(2)p Ft(I)-18 b(R)3743 4865 y Fx(.)83 4977 y(W)-8 b(e)32 b(denote)h(b)m(y)f Fu(G)719 4944 y FC(\003)790 4977 y Fx(its)g(dual)g(space)g(and)g(b)m(y)g Fu(f)p Fw(W)1811 4944 y FC(\003)1798 5000 y Fs(t)1851 4977 y Fu(g)1896 4991 y Fs(t)p FC(2)p Ft(I)-18 b(R)2064 4977 y Fx(the)32 b(standard)g(C)2664 4991 y Ft(0)2704 4977 y Fx(-group)h(obtained)g(b)m(y)f(dualit)m(y)83 5090 y(from)40 b(the)f(action)i(of)e(the)h(group)g(restricted)g(to)g Fu(G)5 b Fx(.)68 b(Then)40 b Fw(H)47 b Fx(is)39 b(of)g(class)g Fw(C)2849 5057 y Ft(1)2888 5090 y Fv(\()p Fw(A)p Fv(\))h Fx(if)f(and)h(only)f (if)h(the)p 83 5175 1474 4 v 187 5228 a Fh(1)221 5260 y Fp(If)25 b Fg(m)f Fp(and)g Fg(k)i Fp(are)f(p)r(ositiv)n(e)g(in)n(tegers,)h(w)n(e)e (denote)g(b)n(y)g Fg(C)1836 5268 y Fm(0)1870 5260 y Ff(\(I)-15 b(R)1969 5228 y Fn(m)2028 5260 y Ff(\))24 b Fp(the)g(set)g(of)i(all)f(con)n (tin)n(uous)f(complex)f(functions)i(on)g Ff(I)-15 b(R)3709 5228 y Fn(m)83 5351 y Fp(con)n(v)n(erging)33 b(to)f(zero)h(at)f(in\034nit)n (y)-6 b(,)33 b(and)f(b)n(y)f Fg(C)1458 5320 y Fn(k)1453 5365 y(c)1496 5351 y Ff(\(I)-15 b(R)1595 5320 y Fn(m)1654 5351 y Ff(\))32 b Fp(the)g(subset)g(of)h Fg(C)2246 5359 y Fm(0)2280 5351 y Ff(\(I)-15 b(R)2379 5320 y Fn(m)2438 5351 y Ff(\))32 b Fp(of)h Fg(k)h Fp(times)e(con)n(tin)n(uously)g(di\033eren)n(tiable)83 5443 y(functions)24 b(of)h(compact)f(supp)r(ort.)33 b(F)-6 b(or)24 b(an)n(y)g(Hilb)r(ert)f(spaces)i Fe(H)g Fp(and)e Fe(G)t Fp(,)i Fe(B)r Ff(\()p Fe(H)p Fg(;)14 b Fe(G)t Ff(\))24 b Fp(denotes)g(the)f (Banac)n(h)i(space)f(of)h(b)r(ounded)83 5534 y(linear)h(op)r(erators)h(from)f Fe(H)g Fp(to)g Fe(G)t Fp(,)g Fe(B)r Ff(\()p Fe(H)p Ff(\))c(:=)f Fe(B)r Ff(\()p Fe(H)p Fg(;)14 b Fe(H)p Ff(\))26 b Fp(and)f Fe(K)q Ff(\()p Fe(H)p Ff(\))i Fp(is)f(the)f(ideal)i(of)f(compact)f(op)r (erators)i(in)f Fe(H)p Fp(.)1903 5836 y Fx(5)p eop %%Page: 6 6 bop 83 482 a Fx(mapping)35 b Fv(I)-18 b(R)31 b Fu(3)f Fw(t)h Fu(7!)g Fw(W)947 449 y FC(\003)934 504 y Fs(t)986 482 y Fw(H)7 b(W)1155 496 y Fs(t)1215 482 y Fu(2)31 b(B)s Fv(\()p Fu(G)5 b Fw(;)15 b Fu(G)1563 449 y FC(\003)1604 482 y Fv(\))33 b Fx(is)h(strongly)g Fw(C)2192 449 y Ft(1)2264 482 y Fx(\(see)g([1],)h(Theorem)g(6.3.4\).)52 b(In)33 b(this)h(case,)83 594 y(the)d(comm)m(utator)h Fv([)p Fw(H)r(;)15 b(iA)p Fv(])32 b Fx(b)s(elongs)e(unam)m(bigously)i(to)f Fu(B)s Fv(\()p Fu(G)5 b Fw(;)15 b Fu(G)2311 561 y FC(\003)2351 594 y Fv(\))p Fx(.)249 707 y(With)35 b(an)m(y)f Fw(H)42 b Fx(of)34 b(class)g Fw(C)1174 674 y Ft(1)1212 707 y Fv(\()p Fw(A)p Fv(\))p Fx(,)j(one)d(asso)s(ciates)g(the)h(functions)g Fw(\045)2602 674 y Fs(A)2602 734 y(H)2703 707 y Fx(and)43 b Fv(~)-53 b Fw(\045)2930 674 y Fs(A)2930 734 y(H)3031 707 y Fx(de\034ned)36 b(on)e Fv(I)-18 b(R)34 b Fx(with)83 820 y(v)-5 b(alues)30 b(in)h Fv(\()p Fu(\0001)p Fw(;)15 b Fu(1)p Fv(])30 b Fx(b)m(y)281 1014 y Fw(\045)328 981 y Fs(A)328 1041 y(H)395 1014 y Fv(\()p Fw(\025)p Fv(\))c(=)f(sup)o Fu(f)p Fw(a)h Fu(2)e Fv(I)-18 b(R)1093 937 y Fl(\014)1093 991 y(\014)1154 1014 y Fu(9)17 b Fw(\021)29 b Fu(2)c Fw(C)1454 981 y FC(1)1447 1037 y Fs(c)1528 1014 y Fv(\(I)-18 b(R\))30 b Fx(s.t.)41 b Fw(\021)s Fv(\()p Fw(\025)p Fv(\))26 b Fu(6)p Fv(=)f(0)31 b Fx(and)f Fw(a\021)2512 981 y Ft(2)2552 1014 y Fv(\()p Fw(H)7 b Fv(\))26 b Fu(\024)f Fw(\021)s Fv(\()p Fw(H)7 b Fv(\)[)p Fw(H)r(;)15 b(iA)p Fv(])p Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fu(g)p Fw(;)648 1152 y Fv(~)-54 b Fw(\045)686 1119 y Fs(A)686 1179 y(H)754 1152 y Fv(\()p Fw(\025)p Fv(\))26 b(=)f(sup)1151 1078 y Fl(\010)1204 1152 y Fw(a)g Fu(2)g Fv(I)-18 b(R)1475 1075 y Fl(\014)1475 1129 y(\014)1535 1152 y Fu(9)18 b Fw(\021)28 b Fu(2)d Fw(C)1835 1119 y FC(1)1828 1174 y Fs(c)1909 1152 y Fv(\(I)-18 b(R\))31 b Fx(and)g Fw(K)h Fu(2)25 b(K)q Fv(\()p Fu(H)q Fv(\))31 b Fx(s.t.)40 b Fw(\021)s Fv(\()p Fw(\025)p Fv(\))26 b Fu(6)p Fv(=)f(0)1536 1290 y Fx(and)31 b Fw(a\021)1808 1257 y Ft(2)1848 1290 y Fv(\()p Fw(H)7 b Fv(\))21 b(+)f Fw(K)32 b Fu(\024)25 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\)[)p Fw(H)r(;)15 b(iA)p Fv(])p Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))2987 1216 y Fl(\011)3043 1290 y Fw(:)83 1483 y Fx(Some)25 b(prop)s(erties)f(of)f(these)h (functions)h(will)f(b)s(e)f(quoted)i(later)f(on)g(\(Prop)s(osition)h(5.3\).) 39 b(The)24 b Fr(Mourr)-5 b(e)28 b(set)e(of)83 1596 y Fw(H)34 b Fr(with)27 b(r)-5 b(esp)g(e)g(ct)30 b(to)e Fw(A)d Fx(is)f Fw(\026)1010 1563 y Fs(A)1067 1596 y Fv(\()p Fw(H)7 b Fv(\))26 b(:=)f Fu(f)p Fw(\025)h Fu(2)f Fv(I)-18 b(R)24 b Fu(j)h Fw(\045)1780 1563 y Fs(A)1780 1623 y(H)1847 1596 y Fv(\()p Fw(\025)p Fv(\))h Fw(>)f Fv(0)p Fu(g)p Fx(.)40 b(Since)25 b(the)h(w)m(ork)f(of)g(Mourre)g (\([14)q(],)h([15)q(]\),)g(it)83 1709 y(is)i(kno)m(wn)i(that)f Fw(H)35 b Fx(has)29 b(nice)g(sp)s(ectral)g(prop)s(erties)g(on)g(this)g(set.) 39 b(In)29 b(particular)h Fw(H)35 b Fx(has)29 b(no)f(eigen)m(v)-5 b(alue)31 b(in)83 1822 y Fw(\026)138 1789 y Fs(A)195 1822 y Fv(\()p Fw(H)7 b Fv(\))28 b Fx(and,)i(under)f(an)f(additional)i(regularit)m (y)g(assumption,)f(a)f(limiting)i(absorption)f(principle)h(can)e(b)s(e)83 1935 y(stated)i(on)h(it.)40 b(This)30 b(additional)i(condition)f(is)f(as)g (follo)m(ws:)41 b(for)29 b(some,)i(and)f(then)h(for)f(all,)g Fw(z)g Fu(2)25 b Fk(C)43 b Fu(n)20 b Fw(\033)s Fv(\()p Fw(H)7 b Fv(\))p Fx(,)83 1975 y Fl(R)144 2001 y Ft(1)126 2080 y(0)198 2048 y Fu(k)p Fw(W)329 2062 y FC(\000)p Fs(t)414 2048 y Fv(\()p Fw(H)25 b Fu(\000)17 b Fw(z)t Fv(\))719 2015 y FC(\000)p Ft(1)814 2048 y Fw(W)900 2062 y Fs(t)947 2048 y Fv(+)g Fw(W)1121 2062 y Fs(t)1151 2048 y Fv(\()p Fw(H)25 b Fu(\000)17 b Fw(z)t Fv(\))1456 2015 y FC(\000)p Ft(1)1551 2048 y Fw(W)1637 2062 y FC(\000)p Fs(t)1739 2048 y Fu(\000)g Fv(2\()p Fw(H)25 b Fu(\000)17 b Fw(z)t Fv(\))2177 2015 y FC(\000)p Ft(1)2272 2048 y Fu(k)2327 2012 y Fs(dt)p 2327 2027 62 4 v 2328 2082 a(t)2353 2063 y Fm(2)2425 2048 y Fw(<)25 b Fu(1)p Fx(.)39 b(If)29 b(this)f(condition)j(is)d (satis\034ed,)83 2161 y Fw(H)39 b Fx(is)31 b(said)h(to)g(b)s(e)g(of)f(class)h Fw(C)1105 2128 y Ft(1)p Fs(;)p Ft(1)1199 2161 y Fv(\()p Fw(A)p Fv(\))p Fx(.)46 b(Assuming)32 b(the)g(in)m(v)-5 b(ariance)33 b(of)f Fu(G)37 b Fx(under)c(eac)m(h)g Fw(W)3155 2175 y Fs(t)3184 2161 y Fx(,)g(an)f(equiv)-5 b(alen)m(t)83 2274 y(requiremen)m(t)33 b(\(see)f(Theorem)g(6.3.4)h(of)e([1]\))g(is)h(that)1923 2201 y Fl(R)1984 2227 y Ft(1)1966 2306 y(0)2039 2274 y Fu(k)p Fw(W)2183 2241 y FC(\003)2170 2296 y Fs(t)2222 2274 y Fw(H)7 b(W)2391 2288 y Fs(t)2441 2274 y Fv(+)21 b Fw(W)2632 2241 y FC(\003)2619 2296 y(\000)p Fs(t)2703 2274 y Fw(H)7 b(W)2872 2288 y FC(\000)p Fs(t)2978 2274 y Fu(\000)20 b Fv(2)p Fw(H)7 b Fu(k)3242 2288 y FC(G)t(!G)3405 2269 y Fq(\003)3455 2238 y Fs(dt)p 3455 2253 V 3456 2308 a(t)3481 2289 y Fm(2)3554 2274 y Fw(<)27 b Fu(1)p Fx(,)83 2387 y(where)k Fu(k)21 b(\001)f(k)500 2401 y FC(G)t(!G)663 2382 y Fq(\003)733 2387 y Fx(is)30 b(the)h(norm)g(of)f Fu(B)s Fv(\()p Fu(G)5 b Fw(;)15 b Fu(G)1575 2354 y FC(\003)1615 2387 y Fv(\))p Fx(.)249 2500 y(In)30 b(our)h(applications,)i Fw(H)k Fx(is)30 b(equal)h(to)g Fu(\000)p Fv(\001)20 b(+)g Fw(V)51 b Fx(in)31 b Fu(H)26 b Fv(=)g Fw(L)2336 2467 y Ft(2)2375 2500 y Fv(\(I)-18 b(R)2492 2464 y Fs(n)2539 2500 y Fv(\))31 b Fx(with)g(domain)h Fu(H)3211 2467 y Ft(2)3250 2500 y Fx(,)f(the)g(Sob)s(olev)83 2613 y(space)f(of)g(order)h(t)m(w)m(o)g(on)f Fv(I)-18 b(R)1040 2577 y Fs(n)1087 2613 y Fx(.)40 b(The)31 b(unitary)f(group)h Fu(f)p Fw(W)2043 2627 y Fs(t)2073 2613 y Fu(g)2118 2627 y Fs(t)p FC(2)p Ft(I)-18 b(R)2284 2613 y Fx(is)30 b(the)g(group)h(of)f(dilations,)h (whic)m(h)g(lea)m(v)m(es)83 2725 y Fu(H)161 2693 y Ft(2)227 2725 y Fx(in)m(v)-5 b(arian)m(t.)41 b(Since)28 b Fw(W)972 2693 y FC(\003)959 2748 y Fs(t)1011 2725 y Fv(\001)p Fw(W)1173 2739 y Fs(t)1227 2725 y Fv(=)d Fw(e)1365 2693 y Ft(2)p Fs(t)1430 2725 y Fv(\001)p Fx(,)j(an)f(easy)f(calculation)j(sho)m(ws)e(that)h(the)f(op) s(erator)h Fu(\000)p Fv(\001)e Fx(satis\034es)83 2838 y(the)h Fw(C)307 2805 y Ft(1)p Fs(;)p Ft(1)401 2838 y Fv(\()p Fw(A)p Fv(\))p Fx(-condition.)41 b(Hence)27 b Fw(H)33 b Fx(is)27 b(of)f(class)g Fw(C)1841 2805 y Ft(1)p Fs(;)p Ft(1)1935 2838 y Fv(\()p Fw(A)p Fv(\))h Fx(if)f Fw(V)47 b Fx(is)26 b Fv(\001)p Fx(-b)s(ounded)h(with)g (relativ)m(e)h(b)s(ound)e(less)83 2951 y(than)34 b(one)g(and)g(is)g(of)f (class)g Fw(C)1133 2918 y Ft(1)p Fs(;)p Ft(1)1227 2951 y Fv(\()p Fw(A)p Fv(\))p Fx(.)51 b(W)-8 b(e)34 b(still)g(recall)g(some)g(de\034nitions) h(related)f(to)g(this)g(condition)h(in)83 3064 y(suc)m(h)c(a)f(setting.)83 3266 y Fj(De\034nition)k(2.2)45 b Fr(L)-5 b(et)32 b Fw(U)k Fv(:)25 b Fu(H)1127 3233 y Ft(2)1192 3266 y Fu(!)g(H)33 b Fr(b)-5 b(e)33 b(a)g(line)-5 b(ar)32 b(symmetric)g(op)-5 b(er)g(ator.)199 3445 y(i\))45 b(W)-7 b(e)36 b(say)f(that)g Fw(U)46 b Fr(is)34 b(a)i Fx(Mourre)e(p)s(oten)m(tial)j Fr(if)d(the)i(sesquiline)-5 b(ar)37 b(form)2771 3372 y Fl(\002)2808 3445 y Fv([)p Fw(U;)15 b(A)p Fv(])p Fw(;)g(A)3136 3372 y Fl(\003)3211 3445 y Fr(de\034ne)-5 b(d)36 b(on)f(the)309 3558 y(Schwartz)d(sp)-5 b(ac)g(e)35 b(on)d Fv(I)-18 b(R)1136 3522 y Fs(n)1215 3558 y Fr(is)32 b(c)-5 b(ontinuous)33 b(for)f(the)h(top)-5 b(olo)g(gy)34 b(induc)-5 b(e)g(d)34 b(by)e Fu(H)2926 3525 y Ft(2)2966 3558 y Fr(.)172 3742 y(ii\))44 b(W)-7 b(e)42 b(say)f(that)g Fw(U)51 b Fr(is)41 b(a)g Fx(long-range)h(p)s(oten)m (tial)h Fr(if)d Fv([)p Fw(U;)15 b(A)p Fv(])43 b Fu(2)e(B)s Fv(\()p Fu(H)2624 3709 y Ft(2)2663 3742 y Fw(;)15 b Fu(H)2781 3709 y FC(\000)p Ft(1)2875 3742 y Fv(\))42 b Fr(and)f(if)g(ther)-5 b(e)42 b(exists)g(a)309 3855 y(function)32 b Fw(\030)d Fu(2)c Fw(C)891 3822 y FC(1)965 3855 y Fv(\(I)-18 b(R)1082 3819 y Fs(n)1129 3855 y Fv(\))33 b Fr(with)e Fw(\030)t Fv(\()p Fw(x)p Fv(\))26 b(=)f(0)33 b Fr(if)f Fu(j)p Fw(x)p Fu(j)25 b(\024)g Fv(1)33 b Fr(and)f Fw(\030)t Fv(\()p Fw(x)p Fv(\))26 b(=)f(1)33 b Fr(if)e Fu(j)p Fw(x)p Fu(j)26 b(\025)f Fv(2)33 b Fr(such)g(that)1299 3975 y Fl(Z)1390 4002 y FC(1)1350 4181 y Ft(1)1480 3967 y Fl(\015)1480 4022 y(\015)1480 4076 y(\015)1480 4131 y(\015)1531 4099 y Fw(\030)1590 3971 y Fl(\022)1667 4037 y Fw(Q)p 1667 4078 72 4 v 1681 4161 a(r)1749 3971 y Fl(\023)1831 4099 y Fv([)p Fw(U;)15 b(A)p Fv(])2051 3967 y Fl(\015)2051 4022 y(\015)2051 4076 y(\015)2051 4131 y(\015)2102 4189 y FC(H)2162 4171 y Fm(2)2197 4189 y FC(!H)2328 4171 y Fq(\000)p Fm(1)2439 4037 y Fw(dr)p 2439 4078 91 4 v 2463 4161 a(r)2565 4099 y(<)25 b Fu(1)p Fw(:)144 4419 y Fr(iii\))44 b(W)-7 b(e)37 b(say)e(that)h Fw(U)46 b Fr(is)36 b(a)g Fx(short-range)f(p)s (oten)m(tial)j Fr(if)2071 4346 y Fl(R)2132 4372 y FC(1)2114 4451 y Ft(1)2222 4314 y Fl(\015)2222 4369 y(\015)2222 4423 y(\015)2272 4419 y Fw(\030)2331 4318 y Fl(\020)2395 4378 y Fs(Q)p 2395 4398 56 4 v 2406 4450 a(r)2461 4318 y Fl(\021)2530 4419 y Fw(U)2602 4314 y Fl(\015)2602 4369 y(\015)2602 4423 y(\015)2653 4482 y FC(H)2713 4463 y Fm(2)2748 4482 y FC(!H)2898 4419 y Fw(dr)c(<)e Fu(1)k Fr(for)f(some)i Fw(\030)e Fu(2)309 4559 y Fw(C)381 4526 y FC(1)455 4559 y Fv(\(I)-18 b(R)572 4523 y Fs(n)619 4559 y Fv(\))32 b Fr(such)i(that)e Fw(\030)t Fv(\()p Fw(x)p Fv(\))26 b(=)f(0)33 b Fr(if)e Fu(j)p Fw(x)p Fu(j)26 b(\024)f Fv(1)33 b Fr(and)f Fw(\030)t Fv(\()p Fw(x)p Fv(\))26 b(=)f(1)32 b Fr(if)g Fu(j)p Fw(x)p Fu(j)26 b(\025)f Fv(2)p Fr(.)83 4761 y Fx(It)31 b(is)g(sho)m(wn)h(in)f([1])g(that)h(in)g(all)f(three) h(cases,)f Fw(U)41 b Fx(is)31 b(of)g(class)g Fw(C)2280 4728 y Ft(1)p Fs(;)p Ft(1)2373 4761 y Fv(\()p 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Fw(\013)k Fx(if)p 940 535 82 4 v 33 w Fv(I)-18 b(R)1031 554 y Fs(\014)1109 608 y Fu(\032)p 1212 535 V 32 w Fv(I)g(R)1302 554 y Fs(\013)1352 608 y Fx(.)51 b Fw(\014)37 b(<)31 b(\013)j Fx(means)h(strict)f(ordering,)i(and)f(w)m(e)g(write)f Fw(\014)28 b Fk(l)23 b Fw(\013)34 b Fx(if)g Fw(\014)i(<)c(\013)83 734 y Fx(and)p 260 661 V 32 w Fv(I)-18 b(R)351 680 y Fs(\014)426 734 y Fu(\032)p 524 661 V 27 w Fv(I)g(R)615 680 y Fs(\015)687 734 y Fu(\032)p 785 661 V 27 w Fv(I)g(R)876 680 y Fs(\013)957 734 y Fx(implies)32 b(that)h(either)f Fw(\015)h Fv(=)28 b Fw(\014)36 b Fx(or)c Fw(\015)h Fv(=)28 b Fw(\013)p Fx(.)45 b(F)-8 b(or)32 b Fw(j)h Fu(2)28 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)i(;)e(n)p Fx(},)33 b(let)f Fv(\()p Fw(\014)26 b Fu(\000)21 b Fw(\013)p Fv(\))3606 748 y Fs(j)3675 734 y Fx(b)s(e)83 847 y(equal)35 b(to)h Fw(\014)493 861 y Fs(j)553 847 y Fu(\000)23 b Fw(\013)705 861 y Fs(j)742 847 y Fx(.)54 b(One)36 b(has)f(equiv)-5 b(alen)m(tly)36 b(that)f Fw(\014)k Fu(\024)33 b Fw(\013)i Fx(if,)h(whenev)m(er)g Fw(\013)2750 861 y Fs(j)2820 847 y Fu(6)p Fv(=)d(0)p Fx(,)k(then)f Fw(\014)3293 861 y Fs(j)3363 847 y Fv(=)d Fw(\013)3525 861 y Fs(j)3561 847 y Fx(,)j(and)83 960 y(that)28 b Fw(\014)19 b Fk(l)14 b Fw(\013)26 b Fx(if)h(and)g(only)h(if)e Fw(\014)31 b Fu(\024)25 b Fw(\013)i Fx(and)h(there)f(is)g(exactly)g(one)h(v)-5 b(alue)27 b(of)g Fw(j)32 b Fx(suc)m(h)c(that)g Fv(\()p Fw(\014)19 b Fu(\000)14 b Fw(\013)p Fv(\))3340 974 y Fs(j)3402 960 y Fu(6)p Fv(=)25 b(0)p Fx(.)39 b(One)83 1072 y(also)30 b(notices)h(that)g Fu(j)p Fw(\013)p Fu(j)g Fx(is)f(equal)h(to)f Fw(n)20 b Fu(\000)1507 1004 y Fl(P)1603 1031 y Fs(n)1603 1099 y(j)t Ft(=1)1745 1072 y Fu(j)p Fw(\013)1828 1086 y Fs(j)1865 1072 y Fu(j)p Fx(.)249 1203 y(In)j(the)h(sequel,)h(w)m(e)f(shall)f(mak)m(e)h(some)g(abuses)f(of)g (notation:)p 2341 1130 V 38 w Fv(I)-18 b(R)2432 1149 y Fs(\013)2505 1203 y Fx(will)23 b(denote)i(either)f(a)f(h)m(yp)s(ersurface)83 1316 y(of)p 182 1243 V 26 w Fv(I)-18 b(R)273 1262 y Fs(n)346 1316 y Fx(or)27 b(the)g(isomorphic)h(cartesian)g(pro)s(duct)f(of)p 1872 1243 V 26 w Fv(I)-18 b(R)1954 1330 y Fs(j)2017 1316 y Fx(for)26 b(all)h Fw(\013)2331 1330 y Fs(j)2393 1316 y Fv(=)e(0)i Fx(\(a)g Fu(j)p Fw(\013)p Fu(j)p Fx(-dimensional)i(h)m(yp)s(ercub)s(e\).)83 1429 y(Similarly)-8 b(,)27 b Fw(C)7 b Fv(\()p 587 1356 V(I)-18 b(R)677 1375 y Fs(\013)727 1429 y Fv(\))23 b Fx(will)i(b)s(e)e(view)m(ed)i (either)f(as)f(a)h(C)1843 1396 y FC(\003)1883 1429 y Fx(-algebra)h(on)f(its)f (o)m(wn,)j(or)e(as)f(a)h(subalgebra)h(of)e Fw(C)7 b Fv(\()p 3595 1356 V(I)-18 b(R)3686 1375 y Fs(n)3733 1429 y Fv(\))83 1542 y Fx(with)34 b(elemen)m(ts)h(dep)s(ending)f(only)f(on)h(the)g(v)-5 b(ariables)33 b Fw(x)2020 1556 y Fs(j)2090 1542 y Fx(for)g(whic)m(h)i Fw(\013)2551 1556 y Fs(j)2618 1542 y Fv(=)30 b(0)p Fx(.)50 b(Ho)m(w)m(ev)m(er,)36 b(in)d(ev)m(ery)h(case,)83 1655 y(the)d(con)m(text)g (should)g(suppress)f(the)h(am)m(biguit)m(y)-8 b(.)249 1768 y(Before)42 b(de\034ning)h Fo(C)13 b Fx(,)45 b(w)m(e)d(summarize)h(some)f (easy)g(prop)s(erties)g(of)g(the)g(ab)s(elian)g(algebra)h Fw(C)7 b Fv(\()p 3570 1695 V(I)-18 b(R)3661 1714 y Fs(n)3708 1768 y Fv(\))p Fx(.)83 1881 y(F)-8 b(or)34 b(eac)m(h)g Fw(\013)d Fu(2)e(L)p Fx(,)34 b(let)f(us)g(sho)m(w)h(the)f(in)m(v)-5 b(ariance)35 b(of)e(the)g(h)m(yp)s(ersurface)p 2617 1808 V 34 w Fv(I)-18 b(R)2708 1827 y Fs(\013)2790 1881 y Fx(under)34 b(the)g(natural)g(action)83 1994 y Fu(U)149 1961 y Fs(o)231 1994 y Fx(of)43 b Fv(I)-18 b(R)429 1958 y Fs(n)519 1994 y Fx(on)p 658 1921 V 44 w Fv(I)g(R)749 1940 y Fs(n)840 1994 y Fx(b)m(y)44 b(translations.)81 b(F)-8 b(or)45 b Fw(y)50 b Fu(2)e Fv(I)-18 b(R)o Fx(,)48 b(let)c Fw(U)2287 2008 y Fs(y)2376 1994 y Fv(:)p 2449 1921 V 48 w(I)-18 b(R)47 b Fu(!)p 2717 1921 V 48 w Fv(I)-18 b(R)43 b Fx(with)i Fw(U)3124 2008 y Fs(y)3165 1994 y Fv(\()p Fw(z)t Fv(\))k(=)e Fw(z)34 b Fv(+)29 b Fw(y)46 b Fx(if)83 2106 y Fw(z)i Fu(2)c Fv(I)-18 b(R)41 b Fx(and)h Fw(U)650 2120 y Fs(y)691 2106 y Fv(\()p Fu(\0061)p Fv(\))i(=)g Fu(\0061)p Fx(,)g(b)s(e)d(the)h(extension)g(to)p 2147 2033 V 42 w Fv(I)-18 b(R)41 b Fx(of)g(the)g(translation)i(b)m(y)f Fw(y)i Fx(on)e Fv(I)-18 b(R)o Fx(.)74 b(Since)p 83 2146 V 83 2219 a Fv(I)-18 b(R)174 2165 y Fs(n)260 2219 y Fx(equals)p 544 2146 V 40 w Fv(I)g(R)626 2233 y Ft(1)692 2219 y Fu(\002)27 b Fw(:)15 b(:)g(:)27 b Fu(\002)p 1020 2146 V 27 w Fv(I)-18 b(R)1101 2233 y Fs(n)1148 2219 y Fx(,)43 b(the)d(action)h(of)e(the)i(group)g (on)p 2341 2146 V 40 w Fv(I)-18 b(R)2431 2165 y Fs(n)2518 2219 y Fx(can)41 b(b)s(e)e(de\034ned)i(comp)s(onen)m(t)m(wise:)83 2332 y Fv([)p Fu(U)174 2299 y Fs(o)165 2355 y(x)212 2332 y Fv(\()p Fw(z)t Fv(\)])353 2346 y Fs(j)431 2332 y Fv(=)f Fw(U)604 2346 y Fs(x)644 2356 y Fn(j)680 2332 y Fv(\()p Fw(z)757 2346 y Fs(j)794 2332 y Fv(\))g Fx(for)f(an)m(y)g Fw(z)44 b Fu(2)p 1382 2259 V 40 w Fv(I)-18 b(R)1473 2278 y Fs(n)1559 2332 y Fx(and)39 b Fw(x)h Fu(2)g Fv(I)-18 b(R)2017 2296 y Fs(n)2064 2332 y Fx(.)67 b(But)39 b Fu(f\0001g)p Fx(,)j Fu(f)p Fv(+)p 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3596 y Fv(:)39 b Fw(C)7 b Fv(\()p 2905 3523 V(I)-18 b(R)2996 3542 y Fs(n)3043 3596 y Fv(\))39 b Fu(!)f(\010)3317 3610 y Fs(\013)p Fd(l)p Fs(o)3486 3596 y Fw(C)7 b Fv(\()p 3593 3523 V(I)-18 b(R)3683 3542 y Fs(\013)3733 3596 y Fv(\))83 3708 y Fx(is)35 b(a)h(co)m(v)-5 b(arian)m(t)38 b(morphism)f(with)f(k)m(ernel)g(equal)g(to)g Fw(C)1990 3722 y Ft(0)2030 3708 y Fv(\(I)-18 b(R)2147 3673 y Fs(n)2194 3708 y Fv(\))p Fx(.)57 b(Th)m(us)36 b(there)g(exists)g(a)f (natural)i(injectiv)m(e)83 3821 y(morphism)1196 4026 y Fw(\031)29 b Fv(:)c Fw(C)7 b Fv(\()p 1434 3953 V(I)-18 b(R)1525 3972 y Fs(n)1572 4026 y Fv(\))p Fw(=C)1717 4040 y Ft(0)1757 4026 y Fv(\(I)g(R)1874 3988 y Fs(n)1921 4026 y Fv(\))26 b Fw(,)-15 b Fu(!)25 b(\010)2179 4040 y Fs(\013)p Fd(l)p Fs(o)2347 4026 y Fw(C)7 b Fv(\()p 2454 3953 V(I)-18 b(R)2545 3972 y Fs(\013)2594 4026 y Fv(\))p Fw(:)999 b Fx(\(1\))249 4242 y(W)-8 b(e)42 b(no)m(w)g(iden)m (tify)g Fw(C)7 b Fv(\()p 1062 4169 V(I)-18 b(R)1152 4188 y Fs(\013)1202 4242 y Fv(\))41 b Fx(with)h(the)f(subalgebra)i(of)e Fu(B)s Fv(\()p Fu(H)2414 4256 y Fs(\013)2463 4242 y Fv(\))g Fx(consisting)h(of)f(all)g(m)m(ultiplication)83 4355 y(op)s(erators)35 b Fw(f)10 b Fv(\()p Fw(Q)p Fv(\))34 b Fx(with)h Fw(f)41 b Fu(2)32 b Fw(C)7 b Fv(\()p 1214 4282 V(I)-18 b(R)1304 4301 y Fs(\013)1354 4355 y Fv(\))p Fx(.)53 b Fw(C)1532 4369 y Ft(0)1571 4355 y Fv(\(I)-18 b(R)1688 4319 y Fs(\013)p FC(\003)1773 4355 y Fv(\))35 b Fx(denotes)g(the)f(set)g(of)g(op)s(erators)h Fw(h)p Fv(\()p Fw(P)13 b Fv(\))33 b(:=)f Fu(F)3417 4322 y FC(\003)3408 4377 y Fs(\013)3458 4355 y Fw(h)p Fv(\()p Fw(Q)p Fv(\))p Fu(F)3717 4369 y Fs(\013)83 4468 y Fx(with)i Fw(h)e Fu(2)f Fw(C)533 4482 y Ft(0)572 4468 y Fv(\(I)-18 b(R)689 4432 y Fs(\013)739 4468 y Fv(\))34 b Fx(and)g(where)g Fu(F)1316 4482 y Fs(\013)1400 4468 y Fx(is)f(the)h(F)-8 b(ourier)36 b(transform)e(in)g Fu(H)2574 4482 y Fs(\013)2657 4468 y Fx(\(w)m(e)g(ha)m(v)m(e)h(iden)m(ti\034ed)h(the)e (dual)83 4580 y(of)h Fv(I)-18 b(R)273 4545 y Fs(\013)357 4580 y Fx(with)36 b Fv(I)-18 b(R)650 4545 y Fs(\013)735 4580 y Fx(itself)7 b(\).)55 b(A)35 b(few)g(elemen)m(ts)i(from)e(the)h(theory)g(of)f(crossed)g (pro)s(ducts)h(are)g(used)f(in)h(the)83 4693 y(sequel.)74 b(W)-8 b(e)41 b(refer)h(to)f([8)q(],)j(Sections)e(3)f(and)h(4)f(for)h(an)f(o)m(v)m (erview)i(on)e(this)h(sub)5 b(ject)41 b(in)h(relation)g(with)83 4806 y(sp)s(ectral)29 b(analysis.)40 b(This)29 b(reference)g(includes)h(some) f(precise)g(de\034nitions)h(and)g(all)f(the)g(required)h(results.)249 4919 y(One)40 b(de\034nes)f Fo(C)819 4886 y Fs(\013)908 4919 y Fv(:=)1044 4846 y Fl(\012)1087 4919 y Fw(C)7 b Fv(\()p 1194 4846 V(I)-18 b(R)1285 4865 y Fs(\013)1334 4919 y Fv(\))26 b Fu(\001)h Fw(C)1512 4933 y Ft(0)1551 4919 y Fv(\(I)-18 b(R)1668 4883 y Fs(\013)p FC(\003)1753 4919 y Fv(\))1788 4846 y Fl(\013)1831 4919 y Fx(,)41 b(the)f(norm)f(closure)h(in)f Fu(B)s Fv(\()p Fu(H)2906 4933 y Fs(\013)2955 4919 y Fv(\))h Fx(of)e(the)i(set)e(of)h (\034nite)83 5042 y(sums)31 b(of)g(the)h(form)f Fw(f)831 5056 y Ft(1)870 5042 y Fv(\()p Fw(Q)p Fv(\))p Fw(h)1064 5056 y Ft(1)1105 5042 y Fv(\()p Fw(P)13 b Fv(\))21 b(+)g Fw(:)15 b(:)g(:)22 b Fv(+)f Fw(f)1623 5056 y Fs(N)1689 5042 y Fv(\()p Fw(Q)p Fv(\))p Fw(h)1883 5056 y Fs(N)1952 5042 y Fv(\()p Fw(P)13 b Fv(\))31 b Fx(with)h Fw(f)2376 5057 y Fs(k)2446 5042 y Fu(2)26 b Fw(C)7 b Fv(\()p 2640 4969 V(I)-18 b(R)2731 4988 y Fs(\013)2780 5042 y Fv(\))32 b Fx(and)g Fw(h)3076 5057 y Fs(k)3146 5042 y Fu(2)26 b Fw(C)3298 5056 y Ft(0)3338 5042 y Fv(\(I)-18 b(R)3455 5006 y Fs(\013)3504 5042 y Fv(\))p Fx(.)44 b(It)31 b(is)83 5155 y(sho)m(wn)i(in)g([8],)g(Theorem)h(4.1,)f(that)g Fo(C)1440 5122 y Fs(\013)1522 5155 y Fx(is)f(a)g(C)1757 5122 y FC(\003)1797 5155 y Fx(-algebra;)j(the)e(stabilit)m(y)g(of)f Fw(C)7 b Fv(\()p 2896 5082 V(I)-18 b(R)2986 5101 y Fs(\013)3036 5155 y Fv(\))32 b Fx(under)i Fu(U)3428 5122 y Fs(\013)3509 5155 y Fx(is)e(here)83 5268 y(essen)m(tial.)43 b(Moreo)m(v)m(er,)33 b(this)e(algebra)h(is)e (isomorphic)i(to)g(the)f(crossed)g(pro)s(duct)h Fw(C)7 b Fv(\()p 2974 5195 V(I)-18 b(R)3064 5214 y Fs(\013)3114 5268 y Fv(\))21 b Fk(o)f Fv(I)-18 b(R)3342 5232 y Fs(\013)3392 5268 y Fx(,)31 b(whic)m(h)h(is)83 5381 y(de\034ned)f(abstractly)f(in)g(terms)f(of)h(the)g (action)g(of)f(translations)i(on)e Fw(C)7 b Fv(\()p 2527 5308 V(I)-18 b(R)2618 5327 y Fs(\013)2667 5381 y Fv(\))p Fx(.)41 b(In)29 b(the)h(sp)s(ecial)g(case)f Fw(\013)d Fv(=)f Fw(o)p Fx(,)83 5494 y(w)m(e)32 b(simply)e(set)h Fo(C)39 b Fv(:=)26 b Fo(C)930 5461 y Fs(o)968 5494 y Fx(.)42 b(W)-8 b(e)32 b(shall)f(giv)m(e)g (in)g(Lemma)h(4.1)f(another)h(description)h(of)d(this)h(C)3306 5461 y FC(\003)3346 5494 y Fx(-algebra)h(in)1903 5836 y(7)p eop %%Page: 8 8 bop 83 482 a Fx(terms)31 b(of)f(suitable)h(limits)f(at)h(in\034nit)m(y)-8 b(.)42 b(W)-8 b(e)30 b(also)h(men)m(tion)h(the)e(follo)m(wing)i(kno)m(wn)f (relations:)1013 686 y Fu(K)q Fv(\()p Fu(H)q Fv(\))c(=)1353 612 y Fl(\012)1396 686 y Fw(C)1461 700 y Ft(0)1501 686 y Fv(\(I)-18 b(R)1618 648 y Fs(n)1665 686 y Fv(\))20 b Fu(\001)h Fw(C)1831 700 y Ft(0)1870 686 y Fv(\(I)-18 b(R)1987 648 y Fs(n)p FC(\003)2070 686 y Fv(\))2105 612 y Fl(\013)2173 661 y Fu(\030)2173 690 y Fv(=)2269 686 y Fw(C)2334 700 y Ft(0)2373 686 y Fv(\(I)g(R)2490 648 y Fs(n)2537 686 y Fv(\))21 b Fk(o)f Fv(I)-18 b(R)2766 648 y Fs(n)2813 686 y Fw(:)815 b Fx(\(2\))249 902 y(Due)36 b(to)g(the)g(em)m(b)s (edding)i(of)d Fw(C)7 b Fv(\()p 1399 829 82 4 v(I)-18 b(R)1490 848 y Fs(\013)1539 902 y Fv(\))36 b Fx(in)m(to)h Fw(C)7 b Fv(\()p 1906 829 V(I)-18 b(R)1996 848 y Fs(n)2043 902 y Fv(\))36 b Fx(and)g(its)g(stabilit)m(y)g(under)h Fu(U)3112 869 y Fs(o)3150 902 y Fx(,)g(one)f(ma)m(y)h(form)83 953 y Fl(\012)126 1027 y Fw(C)7 b Fv(\()p 233 954 V(I)-18 b(R)323 973 y Fs(\013)373 1027 y Fv(\))5 b Fu(\001)g Fw(C)508 1041 y Ft(0)547 1027 y Fv(\(I)-18 b(R)664 991 y Fs(n)p FC(\003)746 1027 y Fv(\))781 953 y Fl(\013)825 1027 y Fx(,)24 b(whic)m(h)g(is)e(a)g(C)1340 994 y FC(\003)1380 1027 y Fx(-subalgebra)i(of)e Fo(C)35 b Fx(isomorphic)24 b(to)f Fw(C)7 b Fv(\()p 2701 954 V(I)-18 b(R)2791 973 y Fs(\013)2841 1027 y Fv(\))5 b Fk(o)g Fv(I)-18 b(R)3037 991 y Fs(n)3084 1027 y Fx(.)38 b(Let)23 b Fv(I)-18 b(R)3383 991 y Fs(\013)3428 967 y Fq(?)3507 1027 y Fx(denote)83 1149 y(the)36 b(orthogonal)i(complemen)m(t)g (of)e Fv(I)-18 b(R)1403 1114 y Fs(\013)1488 1149 y Fx(in)36 b Fv(I)-18 b(R)1681 1114 y Fs(n)1728 1149 y Fx(.)57 b(Prop)s(osition)37 b(2.4)f(of)g([19])g(asserts)g(that)g Fw(C)7 b Fv(\()p 3345 1076 V(I)-18 b(R)3436 1095 y Fs(\013)3485 1149 y Fv(\))25 b Fk(o)e Fv(I)-18 b(R)3721 1114 y Fs(n)83 1274 y Fx(is)34 b(isomorphic)h(to)g Fv([)p Fk(C)47 b(o)22 b Fv(I)-18 b(R)1043 1239 y Fs(\013)1088 1215 y Fq(?)1145 1274 y Fv(])23 b Fu(\012)f Fv([)p Fw(C)7 b Fv(\()p 1418 1201 V(I)-18 b(R)1509 1220 y Fs(\013)1559 1274 y Fv(\))23 b Fk(o)g Fv(I)-18 b(R)1792 1239 y Fs(\013)1842 1274 y Fv(])p Fx(.)52 b(Hence,)36 b(if)e Fu(H)h Fx(is)f(iden)m(ti\034ed)i(with)f Fw(L)3207 1241 y Ft(2)3246 1274 y Fv(\(I)-18 b(R)3363 1239 y Fs(\013)3408 1215 y Fq(?)3465 1274 y Fv(\))23 b Fu(\012)g(H)3694 1288 y Fs(\013)3743 1274 y Fx(,)83 1387 y(the)28 b(C)301 1354 y FC(\003)341 1387 y Fx(-algebra)684 1314 y Fl(\012)727 1387 y Fw(C)7 b Fv(\()p 834 1314 V(I)-18 b(R)925 1333 y Fs(\013)974 1387 y Fv(\))14 b Fu(\001)g Fw(C)1127 1401 y Ft(0)1168 1387 y Fv(\(I)-18 b(R)1285 1351 y Fs(n)p FC(\003)1367 1387 y Fv(\))1402 1314 y Fl(\013)1473 1387 y Fx(of)27 b(b)s(ounded)h(op)s(erators)g(in)g Fu(H)g Fx(is)f(isomorphic)h(to)g(the)g(C)3412 1354 y FC(\003)3452 1387 y Fx(-algebra)83 1521 y Fw(C)148 1535 y Ft(0)187 1521 y Fv(\(I)-18 b(R)304 1485 y Fs(\013)349 1462 y Fq(?)402 1485 y FC(\003)441 1521 y Fv(\))21 b Fu(\012)f Fo(C)653 1488 y Fs(\013)732 1521 y Fx(of)30 b(b)s(ounded)h(op)s(erators)g(in)f Fw(L)1771 1488 y Ft(2)1811 1521 y Fv(\(I)-18 b(R)1928 1485 y Fs(\013)1973 1462 y Fq(?)2029 1521 y Fv(\))21 b Fu(\012)f(H)2253 1535 y Fs(\013)2302 1521 y Fx(.)249 1634 y(Since)46 b(the)h(morphism)f Fw(\031)1161 1648 y Fs(\013)1256 1634 y Fx(is)f(co)m(v)-5 b(arian)m(t,)52 b(there)46 b(exists)f(a)h(unique)g(morphism)h Fv(\005)3213 1648 y Fs(\013)3313 1634 y Fv(:)3390 1561 y Fl(\012)3432 1634 y Fw(C)7 b Fv(\()p 3539 1561 V(I)-18 b(R)3630 1580 y Fs(n)3677 1634 y Fv(\))31 b Fu(\001)83 1757 y Fw(C)148 1771 y Ft(0)187 1757 y Fv(\(I)-18 b(R)304 1721 y Fs(n)p FC(\003)387 1757 y Fv(\))422 1683 y Fl(\013)492 1757 y Fu(!)610 1683 y Fl(\012)653 1757 y Fw(C)7 b Fv(\()p 760 1684 V(I)-18 b(R)851 1703 y Fs(\013)900 1757 y Fv(\))22 b Fu(\001)f Fw(C)1068 1771 y Ft(0)1107 1757 y Fv(\(I)-18 b(R)1224 1721 y Fs(n)p FC(\003)1307 1757 y Fv(\))1342 1683 y Fl(\013)1416 1757 y Fx(suc)m(h)33 b(that)f Fv(\005)1887 1771 y Fs(\013)1936 1757 y Fv([)p Fw(f)10 b Fv(\()p Fw(Q)p Fv(\))p Fw(h)p Fv(\()p Fw(P)j Fv(\)])29 b(=)e Fw(f)2558 1724 y Fs(\013)2607 1757 y Fv(\()p Fw(Q)p Fv(\))p Fw(h)p Fv(\()p Fw(P)13 b Fv(\))33 b Fx(for)e(eac)m(h)i Fw(f)j Fu(2)27 b Fw(C)7 b Fv(\()p 3595 1684 V(I)-18 b(R)3686 1703 y Fs(n)3733 1757 y Fv(\))83 1880 y Fx(and)39 b(eac)m(h)g Fw(h)g Fu(2)f Fw(C)733 1894 y Ft(0)773 1880 y Fv(\(I)-18 b(R)890 1844 y Fs(n)937 1880 y Fv(\))p Fx(.)64 b(F)-8 b(urthermore,)43 b(since)38 b Fw(C)1914 1894 y Ft(0)1954 1880 y Fv(\(I)-18 b(R)2071 1844 y Fs(n)2118 1880 y Fv(\))38 b Fx(is)g(a)h(stable)f(ideal)h(of)f Fw(C)7 b Fv(\()p 3085 1807 V(I)-18 b(R)3176 1826 y Fs(n)3223 1880 y Fv(\))p Fx(,)40 b(the)f(general)83 1993 y(theory)29 b(of)f(crossed)g(pro)s (ducts)h(giv)m(es)f(the)h(canonical)g(isomorphism:)41 b Fv([)p Fw(C)7 b Fv(\()p 2611 1920 V(I)-18 b(R)2702 1939 y Fs(n)2749 1993 y Fv(\))16 b Fk(o)g Fv(I)-18 b(R)2969 1957 y Fs(n)3016 1993 y Fv(])p Fw(=)p Fv([)p Fw(C)3176 2007 y Ft(0)3216 1993 y Fv(\(I)g(R)3333 1957 y Fs(n)3380 1993 y Fv(\))16 b Fk(o)g Fv(I)-18 b(R)3600 1957 y Fs(n)3647 1993 y Fv(])3697 1968 y Fu(\030)3697 1997 y Fv(=)83 2032 y Fl(\002)121 2105 y Fw(C)7 b Fv(\()p 228 2032 V(I)-18 b(R)318 2051 y Fs(n)365 2105 y Fv(\))p Fw(=C)510 2119 y Ft(0)551 2105 y Fv(\(I)g(R)668 2070 y Fs(n)715 2105 y Fv(\))750 2032 y Fl(\003)797 2105 y Fk(o)9 b Fv(I)-18 b(R)958 2070 y Fs(n)1005 2105 y Fx(.)39 b(Using)24 b(\(1\),)i(\(2\))f(and)h (some)e(isomorphisms)i(in)m(tro)s(duced)g(ab)s(o)m(v)m(e,)h(one)e(obtains:) 697 2320 y Fo(C)13 b Fw(=)p Fu(K)q Fv(\()p Fu(H)q Fv(\))1129 2294 y Fu(\030)1129 2324 y Fv(=)1298 2246 y Fl(\002)1336 2320 y Fw(C)7 b Fv(\()p 1443 2246 V(I)-18 b(R)1533 2266 y Fs(n)1580 2320 y Fv(\))p Fw(=C)1725 2334 y Ft(0)1765 2320 y Fv(\(I)g(R)1882 2282 y Fs(n)1929 2320 y Fv(\))1964 2246 y Fl(\003)2023 2320 y Fk(o)20 b Fv(I)-18 b(R)2195 2282 y Fs(n)1114 2467 y Fw(,)j Fu(!)1298 2394 y Fl(\002)1336 2467 y Fu(\010)1407 2481 y Fs(\013)p Fd(l)p Fs(o)1575 2467 y Fw(C)7 b Fv(\()p 1682 2394 V(I)-18 b(R)1772 2413 y Fs(\013)1822 2467 y Fv(\))1857 2394 y Fl(\003)1915 2467 y Fk(o)20 b Fv(I)-18 b(R)2088 2430 y Fs(n)2190 2442 y Fu(\030)2190 2471 y Fv(=)2316 2467 y Fu(\010)2387 2481 y Fs(\013)p Fd(l)p Fs(o)2541 2394 y Fl(\002)2579 2467 y Fw(C)7 b Fv(\()p 2686 2394 V(I)-18 b(R)2776 2413 y Fs(\013)2826 2467 y Fv(\))20 b Fk(o)g Fv(I)-18 b(R)3054 2430 y Fs(n)3101 2394 y Fl(\003)3653 2467 y Fx(\(3\))1129 2590 y Fu(\030)1129 2619 y Fv(=)1298 2615 y Fu(\010)1369 2629 y Fs(\013)p Fd(l)p Fs(o)1537 2541 y Fl(\012)1580 2615 y Fw(C)7 b Fv(\()p 1687 2542 V(I)-18 b(R)1777 2561 y Fs(\013)1827 2615 y Fv(\))21 b Fu(\001)f Fw(C)1993 2629 y Ft(0)2032 2615 y Fv(\(I)-18 b(R)2149 2577 y Fs(n)p FC(\003)2232 2615 y Fv(\))2267 2541 y Fl(\013)2310 2615 y Fw(:)83 2819 y Fx(The)33 b(resulting)h(injectiv)m (e)g(morphism)g(is)e(denoted)i(b)m(y)f Fv(\005)p Fx(.)48 b(But)33 b(if)g Fv(\002)f Fx(is)h(the)g(canonical)h(surjection)g Fo(C)42 b Fu(!)83 2932 y Fo(C)13 b Fw(=)p Fu(K)q Fv(\()p Fu(H)q Fv(\))p Fx(,)30 b(then)f Fv(\005)15 b Fu(\016)i Fv(\002)25 b(=)g Fu(\010)1082 2946 y Fs(\013)p Fd(l)p Fs(o)1250 2932 y Fv(\005)1318 2946 y Fs(\013)1368 2932 y Fx(.)39 b(Assume)28 b(no)m(w)h(that)g Fw(H)35 b Fx(is)27 b(a)i(self-adjoin)m(t)f(observ)-5 b(able)29 b(a\036liated)h(to)83 3045 y Fo(C)13 b Fx(.)50 b(Then)34 b Fw(\033)518 3059 y Ft(ess)609 3045 y Fv(\()p Fw(H)7 b Fv(\))34 b Fx(is)f(equal)h(to)g Fw(\033)s Fv([\002\()p Fw(H)7 b Fv(\)])p Fx(,)35 b(where)g Fv(\002\()p Fw(H)7 b Fv(\))33 b Fx(is)g(the)h(image)h(of)e Fw(H)40 b Fx(in)34 b(the)g(Calkin)g(algebra.)83 3158 y(Since)d(an)g(injectiv) m(e)g(morphism)g(preserv)m(es)g(the)g(sp)s(ectrum,)f(w)m(e)h(ha)m(v)m(e)h(:) 454 3380 y Fw(\033)506 3394 y Ft(ess)597 3380 y Fv(\()p Fw(H)7 b Fv(\))26 b(=)f Fw(\033)927 3306 y Fl(\000)969 3380 y Fv(\002\()p Fw(H)7 b Fv(\))1193 3306 y Fl(\001)1260 3380 y Fv(=)25 b Fw(\033)1411 3279 y Fl(\020)1465 3380 y Fv(\005)1533 3306 y Fl(\000)1575 3380 y Fv(\002\()p Fw(H)7 b Fv(\))1799 3306 y Fl(\001)1841 3279 y(\021)1921 3380 y Fv(=)25 b Fw(\033)2072 3306 y Fl(\000)2134 3380 y Fu(\010)2205 3394 y Fs(\013)p Fd(l)p Fs(o)2363 3380 y Fv(\005)2431 3394 y Fs(\013)2481 3380 y Fv(\()p Fw(H)7 b Fv(\))2634 3306 y Fl(\001)2702 3380 y Fv(=)2814 3293 y Fl([)2798 3484 y Fs(\013)p Fd(l)p Fs(o)2947 3380 y Fw(\033)3002 3306 y Fl(\000)3044 3380 y Fv(\005)3112 3394 y Fs(\013)3162 3380 y Fv(\()p Fw(H)g Fv(\))3315 3306 y Fl(\001)3357 3380 y Fw(;)271 b Fx(\(4\))83 3660 y(the)30 b(last)f(equalit)m(y)h(b)s(eing)f(v)-5 b(alid)30 b(b)s(ecause)f(the)h(sp)s(ectrum)f(of)g(an)h(observ)-5 b(able)30 b(a\036liated)g(to)g(a)f(\034nite)h(direct)83 3773 y(sum)k(is)f(the)h(union)g(of)f(the)h(sp)s(ectra)g(of)f(its)g(comp)s(onen)m (ts.)51 b(Let)35 b(us)e(men)m(tion)i(that)f(some)g(similar)g(results)83 3885 y(w)m(ere)d(already)g(obtained)h(in)e([12)q(].)83 4172 y Fy(4)131 b(Cartesian)44 b(Hamiltonians)83 4375 y Fx(Sc)m(hr\366)s(dinger)33 b(op)s(erators)g Fu(\000)p Fv(\001)20 b(+)h Fw(V)51 b Fx(in)32 b Fw(L)1515 4342 y Ft(2)1554 4375 y Fv(\(I)-18 b(R)1672 4339 y Fs(n)1718 4375 y Fv(\))32 b Fx(a\036liated)h(to)f(the)g(C)2492 4342 y FC(\003)2532 4375 y Fx(-algebra)g Fo(C)45 b Fx(are)32 b(called)g Fr(c)-5 b(artesian)p Fx(.)83 4488 y(If)35 b Fw(V)56 b Fx(is)36 b(a)f(real)i(elemen)m(t)g(of)f Fw(C)7 b Fv(\()p 1202 4415 V(I)-18 b(R)1293 4434 y Fs(n)1339 4488 y Fv(\))p Fx(,)38 b(the)e(corresp)s(onding)i(Hamiltonian)f(is)e(cartesian.)59 b(This)36 b(is)f(easily)83 4601 y(seen)28 b(b)m(y)f(using)h(the)g(Neumann)h (series)e Fv(\()p Fu(\000)p Fv(\001)15 b(+)g Fw(V)34 b Fu(\000)15 b Fw(z)t Fv(\))1969 4568 y FC(\000)p Ft(1)2088 4601 y Fv(=)2184 4532 y Fl(P)2280 4559 y FC(1)2280 4628 y Fs(k)r Ft(=0)2413 4601 y Fv(\()p Fu(\000)p Fv(\001)g Fu(\000)g Fw(z)t Fv(\))2777 4568 y FC(\000)p Ft(1)2871 4601 y Fv([)p Fw(V)20 b Fv(\()p Fw(z)f Fv(+)c(\001\))3262 4568 y FC(\000)p Ft(1)3356 4601 y Fv(])3381 4568 y Fs(k)3451 4601 y Fx(whic)m(h)29 b(is)83 4714 y(norm)34 b(con)m(v)m(ergen)m(t)h(for)e Fu(j=)p Fw(z)t Fu(j)g Fx(large)g(enough.)49 b(In)33 b(order)g(to)g(deal)h(with)f(non-smo)s(oth)h(p) s(oten)m(tials,)g(sev)m(eral)83 4826 y(tec)m(hnical)e(results)f(ha)m(v)m(e)g (to)f(b)s(e)g(obtained.)42 b(This)31 b(section)g(is)e(en)m(tirely)j(dev)m (oted)f(to)g(this)f(question.)249 4939 y(In)39 b(the)g(sequel,)j(w)m(e)d (shall)h(often)f(use)g(some)g(non-decreasing)i(functions)f Fw(\030)j Fx(in)c Fw(C)3130 4906 y FC(1)3204 4939 y Fv(\(I)-18 b(R\))39 b Fx(satisfying)83 5052 y Fw(\030)t Fv(\()p Fw(y)s Fv(\))f(=)f(0)g Fx(if)g Fw(y)j Fu(\024)d Fv(1)h Fx(and)g Fw(\030)t Fv(\()p Fw(y)s Fv(\))g(=)e(1)i Fx(if)f Fw(y)j Fu(\025)d Fv(2)p Fx(.)62 b(F)-8 b(or)38 b(reasons)g(that)g(will)g(b)s(ecome)g(ob)m(vious)g (already)g(in)83 5165 y(the)e(next)g(lemma,)i(w)m(e)f(call)f(them)h Fr(asymptotic)g(lo)-5 b(c)g(alization)38 b(functions)p Fx(.)56 b(Let)37 b(us)e(sa)m(y)h(that)g(a)g(b)s(ounded)83 5278 y(op)s(erator)d Fw(B)k Fx(is)32 b Fr(semi-c)-5 b(omp)g(act)35 b Fx(if)d Fw(\020)7 b Fv(\()p Fw(Q)p Fv(\))p Fw(B)37 b Fx(is)32 b(compact)i(for)e(all)h Fw(\020)j Fu(2)28 b Fw(C)2544 5292 y Ft(0)2584 5278 y Fv(\(I)-18 b(R)2701 5242 y Fs(n)2748 5278 y Fv(\))p Fx(.)47 b(W)-8 b(e)33 b(recall)g(that)g(for)g(eac)m(h)83 5391 y Fw(\013)25 b Fk(l)e Fw(o)p Fx(,)38 b(there)f(exists)e(exactly)i(one)f Fw(j)42 b Fx(suc)m(h)36 b(that)h Fw(\013)1896 5405 y Fs(j)1968 5391 y Fu(6)p Fv(=)d(0)p Fx(.)59 b(Hence)36 b Fw(\013)25 b Fu(\001)f Fw(Q)36 b Fx(means)g Fw(\013)3056 5405 y Fs(j)3093 5391 y Fw(Q)3165 5405 y Fs(j)3237 5391 y Fx(and)h(therefore)83 5504 y Fw(\030)t Fv(\()p Fw(\013)21 b Fu(\001)f Fw(Q)p Fv(\))31 b Fx(is)f(w)m(ell-de\034ned)i (for)f(an)m(y)f(function)h Fw(\030)k Fx(on)30 b Fv(I)-18 b(R)p Fx(.)40 b(W)-8 b(e)31 b(start)f(with)h(a)f(new)h(description)h(of)e Fo(C)13 b Fx(.)1903 5836 y(8)p eop %%Page: 9 9 bop 83 482 a Fj(Lemma)35 b(4.1)161 b Fr(i\))45 b(Each)33 b(op)-5 b(er)g(ator)34 b(in)e Fo(C)45 b Fr(is)32 b(semi-c)-5 b(omp)g(act.)172 669 y(ii\))44 b(A)f(semi-c)-5 b(omp)g(act)44 b(op)-5 b(er)g(ator)44 b Fw(B)j Fr(b)-5 b(elongs)45 b(to)d Fo(C)56 b Fr(if)42 b(and)g(only)h(if)f (ther)-5 b(e)44 b(exist)f(an)f(asymptotic)h(lo-)309 782 y(c)-5 b(alization)42 b(function)g Fw(\030)k Fr(and)c(a)h(family)e Fu(f)p Fw(B)1845 796 y Fs(\013)1895 782 y Fu(g)1940 796 y Fs(\013)p Fd(l)p Fs(o)2121 782 y Fr(such)i(that)f Fw(B)2599 796 y Fs(\013)2692 782 y Fu(2)2796 708 y Fl(\012)2839 782 y Fw(C)7 b Fv(\()p 2946 709 82 4 v(I)-18 b(R)3036 728 y Fs(\013)3086 782 y Fv(\))28 b Fu(\001)f Fw(C)3266 796 y Ft(0)3306 782 y Fv(\(I)-18 b(R)3422 746 y Fs(n)p FC(\003)3505 782 y Fv(\))3540 708 y Fl(\013)3625 782 y Fr(and)309 927 y Fv(lim)435 941 y Fs(r)r FC(!1)630 822 y Fl(\015)630 877 y(\015)630 931 y(\015)680 927 y Fw(\030)739 826 y Fl(\020)793 927 y Fw(\013)21 b Fu(\001)928 886 y Fs(Q)p 928 906 56 4 v 939 958 a(r)993 826 y Fl(\021)1062 927 y Fv(\()p Fw(B)26 b Fu(\000)19 b Fw(B)1351 941 y Fs(\013)1401 927 y Fv(\))1436 822 y Fl(\015)1436 877 y(\015)1436 931 y(\015)1518 927 y Fv(=)30 b(0)p Fr(.)51 b(Mor)-5 b(e)g(over,)38 b(e)-5 b(ach)37 b(op)-5 b(er)g(ator)37 b Fw(B)2804 941 y Fs(\013)2889 927 y Fr(is)e(unique)h(and)g(e) -5 b(qual)37 b(to)309 1062 y Fv(\005)377 1076 y Fs(\013)427 1062 y Fv(\()p Fw(B)5 b Fv(\))p Fr(.)249 1275 y(Pr)-5 b(o)g(of:)52 b Fx(a\))36 b(By)f(using)i(\(2\))f(one)g(observ)m(es)g(that)h(the)f(pro)s (duct)h Fw(\020)7 b Fv(\()p Fw(Q)p Fv(\)[)p Fw(f)j Fv(\()p Fw(Q)p Fv(\))p Fw(h)p Fv(\()p Fw(P)j Fv(\)])37 b Fx(b)s(elongs)f(to)g Fu(K)q Fv(\()p Fu(H)q Fv(\))83 1388 y Fx(for)d(an)m(y)h Fw(\020)7 b(;)15 b(h)31 b Fu(2)f Fw(C)724 1402 y Ft(0)763 1388 y Fv(\(I)-18 b(R)880 1352 y Fs(n)927 1388 y Fv(\))34 b Fx(and)g(an)m(y)g Fw(f)40 b Fu(2)30 b Fw(C)7 b Fv(\()p 1632 1315 82 4 v(I)-18 b(R)1722 1334 y Fs(n)1769 1388 y Fv(\))p Fx(.)50 b(Since)34 b Fo(C)46 b Fx(is)33 b(the)h(norm)g(closure)g(of)f(the)h(v)m(ector)h(space)83 1501 y(generated)25 b(b)m(y)f(pro)s(ducts)g(of)f(the)h(form)f Fw(f)10 b Fv(\()p Fw(Q)p Fv(\))p Fw(h)p Fv(\()p Fw(P)j Fv(\))25 b Fx(and)f(since)f Fu(K)q Fv(\()p Fu(H)q Fv(\))i Fx(is)e(norm)h(closed,)i Fw(\020)7 b Fv(\()p Fw(Q)p Fv(\))p Fw(B)27 b Fx(is)d(compact)83 1614 y(for)30 b(an)m(y)h Fw(\020)h Fu(2)24 b Fw(C)614 1628 y Ft(0)654 1614 y Fv(\(I)-18 b(R)771 1578 y Fs(n)818 1614 y Fv(\))30 b Fx(and)h(an)m(y)g Fw(B)f 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(ersurface)p 2202 1879 V 25 w Fv(I)-18 b(R)2293 1898 y Fs(\013)2343 1952 y Fx(.)38 b(So,)27 b(one)e(has)g Fw(C)7 b Fv(\()p 2976 1879 V(I)-18 b(R)3067 1898 y Fs(n)3114 1952 y Fv(\))25 b(=)g Fw(C)7 b Fv(\()p 3377 1879 V(I)-18 b(R)3468 1898 y Fs(\013)3517 1952 y Fv(\))10 b(+)g Fw(J)3693 1966 y Fs(\013)3743 1952 y Fx(,)83 2065 y(and)27 b Fw(J)305 2079 y Fs(\013)382 2065 y Fx(is)g(nothing)h(but)f(the)g(k)m(ernel)h(of)f(the)g(morphism)h Fw(\031)2098 2079 y Fs(\013)2174 2065 y Fx(of)f(the)g(previous)g(section.)40 b(By)27 b(Corollary)h(3.1)83 2178 y(of)k([8)q(],)h(one)g(gets)f Fo(C)42 b Fv(=)29 b Fu(h)p Fw(C)7 b Fv(\()p 1038 2105 V(I)-18 b(R)1129 2124 y Fs(\013)1178 2178 y Fv(\))22 b Fu(\001)g Fw(C)1347 2192 y Ft(0)1387 2178 y Fv(\(I)-18 b(R)1504 2142 y Fs(n)p FC(\003)1586 2178 y Fv(\))p Fu(i)22 b Fv(+)g Fu(h)p Fw(J)1856 2192 y Fs(\013)1928 2178 y Fu(\001)f Fw(C)2039 2192 y Ft(0)2079 2178 y Fv(\(I)-18 b(R)2196 2142 y Fs(n)p FC(\003)2278 2178 y Fv(\))p Fu(i)p Fx(,)34 b(where)f Fu(h)p Fw(J)2755 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Fx(suc)m(h)h(that)f(for)g (all)g Fw(r)e Fu(\025)e Fw(r)3569 3055 y FC(0)3622 3088 y Fx(and)83 3248 y(all)h Fw(\013)17 b Fk(l)f Fw(o)p Fx(,)465 3143 y Fl(\015)465 3198 y(\015)465 3252 y(\015)516 3248 y Fw(\030)575 3147 y Fl(\020)629 3248 y Fw(\013)21 b Fu(\001)763 3207 y Fs(Q)p 763 3227 V 774 3279 a(r)829 3147 y Fl(\021)898 3248 y Fv(\()p Fw(B)k Fu(\000)20 b Fw(B)1187 3262 y Fs(\013)1237 3248 y Fv(\))1272 3143 y Fl(\015)1272 3198 y(\015)1272 3252 y(\015)1348 3248 y Fw(<)1566 3212 y Fs(")p 1454 3227 259 4 v 1454 3280 a Ft(2\(3)1551 3261 y Fn(n)1594 3280 y FC(\000)p Ft(1\))1722 3248 y Fx(.)39 b(F)-8 b(or)29 b(eac)m(h)g Fw(\013)17 b Fk(l)f Fw(o)p Fx(,)28 b(there)h(exists)f Fw(N)2956 3262 y Fs(\013)3030 3248 y Fw(<)d Fu(1)p Fx(,)k Fw(f)3326 3215 y Fs(\013)3316 3275 y(k)3400 3248 y Fu(2)c Fw(C)7 b Fv(\()p 3593 3175 82 4 v(I)-18 b(R)3683 3194 y Fs(\013)3733 3248 y Fv(\))83 3402 y Fx(and)37 b Fw(h)317 3369 y Fs(\013)317 3430 y(k)402 3402 y Fu(2)e Fw(C)563 3416 y Ft(0)602 3402 y Fv(\(I)-18 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Fx(and)h(since)f Fu(L)83 3641 y Fx(con)m(tains)e Fv(3)483 3608 y Fs(n)560 3641 y Fx(elemen)m(ts,)g(one)f(obtains)741 3759 y Fl(\015)741 3813 y(\015)741 3868 y(\015)741 3922 y(\015)741 3977 y(\015)741 4032 y(\015)791 3945 y Fw(B)25 b Fu(\000)20 b Fw(\030)1016 3959 y Fs(o)1069 3817 y Fl(\022)1146 3884 y Fw(Q)p 1146 3924 72 4 v 1149 4008 a(r)1193 3981 y FC(0)1228 3817 y Fl(\023)1310 3945 y Fw(B)25 b Fu(\000)1495 3859 y Fl(X)1495 4056 y Fs(\014)s FC(6)p Ft(=)p Fs(o)1642 3813 y(N)1698 3827 y Fn(\013)p Fm(\()p Fn(\014)s Fm(\))1669 3859 y Fl(X)1670 4056 y Fs(k)r Ft(=1)1842 3945 y Fw(\030)1882 3960 y Fs(\014)1944 3817 y Fl(\022)2021 3884 y Fw(Q)p 2021 3924 V 2024 4008 a(r)2068 3981 y FC(0)2103 3817 y Fl(\023)2185 3945 y Fw(f)2240 3897 y Fs(\013)p Ft(\()p Fs(\014)s Ft(\))2230 3975 y Fs(k)2387 3945 y Fv(\()p Fw(Q)p Fv(\))p Fw(h)2581 3897 y Fs(\013)p Ft(\()p Fs(\014)s Ft(\))2581 3975 y Fs(k)2729 3945 y Fv(\()p Fw(P)13 b Fv(\))2870 3759 y Fl(\015)2870 3813 y(\015)2870 3868 y(\015)2870 3922 y(\015)2870 3977 y(\015)2870 4032 y(\015)2947 3945 y Fw(<)25 b(":)83 4290 y Fx(By)k(semi-compactness)i(of)e Fw(B)5 b Fx(,)29 b Fw(\030)1223 4304 y Fs(o)1277 4189 y Fl(\020)1341 4249 y Fs(Q)p 1341 4269 57 4 v 1341 4322 a(r)1375 4303 y Fq(0)1407 4189 y Fl(\021)1476 4290 y Fw(B)34 b Fx(b)s(elongs)c(to)g Fu(K)q Fv(\()p Fu(H)q Fv(\))p Fx(,)g(and)g(hence)h(to)e Fo(C)13 b Fx(;)30 b(and)g(eac)m(h)h(term)f (in)g(the)83 4421 y(sum)g(b)s(elongs)h(to)f Fo(C)43 b Fx(b)m(y)31 b(construction.)42 b(Since)31 b Fo(C)43 b Fx(is)30 b(norm)h(closed,)g(one)f (gets)h(that)g Fw(B)e Fu(2)c Fo(C)13 b Fx(.)249 4533 y(d\))33 b(F)-8 b(or)34 b(eac)m(h)f Fw(\013)23 b Fk(l)e Fw(o)p Fx(,)34 b(the)f(uniqueness)g(of)g Fw(B)1804 4547 y Fs(\013)1886 4533 y Fx(is)f(sho)m(wn)i(b)m(y)f(pro)m(ving)h(the)f(follo)m(wing)h(statemen)m(t:) 47 b(if)83 4669 y Fw(C)36 b Fx(b)s(elongs)30 b(to)h Fu(h)p Fw(C)7 b Fv(\()p 761 4596 82 4 v(I)-18 b(R)852 4615 y Fs(\013)901 4669 y Fv(\))20 b Fu(\001)g Fw(C)1066 4683 y Ft(0)1105 4669 y Fv(\(I)-18 b(R)1222 4633 y Fs(n)p FC(\003)1304 4669 y Fv(\))p Fu(i)31 b Fx(and)f(satis\034es)g Fv(lim)2039 4683 y Fs(r)r FC(!1)2234 4564 y Fl(\015)2234 4619 y(\015)2234 4674 y(\015)2284 4669 y Fw(\030)2343 4568 y Fl(\020)2398 4669 y Fw(\013)20 b Fu(\001)2532 4628 y Fs(Q)p 2532 4648 56 4 v 2543 4700 a(r)2597 4568 y Fl(\021)2667 4669 y Fw(C)2739 4564 y Fl(\015)2739 4619 y(\015)2739 4674 y(\015)2814 4669 y Fv(=)25 b(0)p Fx(,)30 b(then)h Fw(C)h Fv(=)25 b(0)p Fx(.)40 b(T)-8 b(o)31 b(see)83 4805 y(this,)37 b(assume)e(that)h Fw(C)41 b Fu(6)p Fv(=)33 b(0)j Fx(and)g(for)f(simplicit)m (y)i(let)e(us)h(\034x)f Fw(\013)f Fv(=)g(\(1)p Fw(;)15 b Fv(0)p Fw(;)g(:)g(:)g(:)33 b(;)15 b Fv(0\))p Fx(.)57 b(Cho)s(ose)36 b Fw(')e Fu(2)f(H)j Fx(suc)m(h)83 4940 y(that)29 b Fu(k)p Fw(')p Fu(k)e Fv(=)e(1)j Fx(and)h Fw(C)7 b(')26 b Fu(6)p Fv(=)e(0)p Fx(.)41 b(By)28 b(h)m(yp)s(othesis,)h(there)g(exists)f Fw(r)g Fu(\025)c Fv(1)29 b Fx(suc)m(h)g(that)2879 4836 y Fl(\015)2879 4890 y(\015)2879 4945 y(\015)2930 4940 y Fw(\030)2989 4839 y Fl(\020)3053 4899 y Fs(Q)3109 4908 y Fm(1)p 3053 4919 91 4 v 3081 4972 a Fs(r)3153 4839 y Fl(\021)3223 4940 y Fw(C)3295 4836 y Fl(\015)3295 4890 y(\015)3295 4945 y(\015)3370 4940 y Fw(<)3476 4904 y Ft(1)p 3476 4919 36 4 v 3476 4972 a(2)3521 4940 y Fu(k)p Fw(C)7 b(')p Fu(k)p Fx(,)83 5112 y(and)39 b(so)386 5008 y Fl(\015)386 5062 y(\015)386 5117 y(\015)437 5112 y Fw(\030)496 5011 y Fl(\020)560 5071 y Fs(Q)616 5080 y Fm(1)p 560 5091 91 4 v 588 5143 a Fs(r)660 5011 y Fl(\021)729 5112 y Fw(C)7 b(e)843 5079 y FC(\000)p Fs(iy)r(P)1004 5088 y Fm(1)1043 5112 y Fw(')1102 5008 y Fl(\015)1102 5062 y(\015)1102 5117 y(\015)1192 5112 y Fw(<)1311 5076 y Ft(1)p 1311 5091 36 4 v 1311 5143 a(2)1357 5112 y Fu(k)p Fw(C)g(')p Fu(k)39 b Fx(for)f(eac)m(h)i Fw(y)h Fu(2)e Fv(I)-18 b(R)p Fx(.)64 b(But,)41 b(since)e(all)g(elemen)m(ts)g(of)f Fu(h)p Fw(C)7 b Fv(\()p 3541 5039 82 4 v(I)-18 b(R)3632 5058 y Fs(\013)3682 5112 y Fv(\))26 b Fu(\001)83 5284 y Fw(C)148 5298 y Ft(0)187 5284 y Fv(\(I)-18 b(R)304 5248 y Fs(n)p FC(\003)387 5284 y Fv(\))p Fu(i)32 b Fx(comm)m(ute)h(with)f(the)g(unitary)g(op)s(erator)h Fw(e)1971 5251 y FC(\000)p Fs(iy)r(P)2132 5260 y Fm(1)2202 5284 y Fx(\()p Fw(y)d Fu(2)d Fv(I)-18 b(R)p Fx(\),)32 b(one)g(has)2903 5180 y Fl(\015)2903 5234 y(\015)2903 5289 y(\015)2954 5284 y Fw(\030)3013 5183 y Fl(\020)3077 5243 y Fs(Q)3133 5252 y Fm(1)p 3077 5263 91 4 v 3105 5315 a Fs(r)3177 5183 y Fl(\021)3247 5284 y Fw(C)7 b(e)3361 5251 y FC(\000)p Fs(iy)r(P)3522 5260 y Fm(1)3560 5284 y Fw(')3619 5180 y Fl(\015)3619 5234 y(\015)3619 5289 y(\015)3697 5284 y Fv(=)83 5351 y Fl(\015)83 5406 y(\015)83 5461 y(\015)134 5456 y Fw(\030)193 5355 y Fl(\020)257 5415 y Fs(Q)313 5424 y Fm(1)p 257 5435 V 285 5487 a Fs(r)377 5456 y Fv(+)20 b Fw(y)516 5355 y Fl(\021)585 5456 y Fw(C)7 b(')716 5351 y Fl(\015)716 5406 y(\015)716 5461 y(\015)800 5456 y Fu(!)33 b(k)p Fw(C)7 b(')p Fu(k)35 b Fx(as)g Fw(y)g Fu(!)e(1)p Fx(,)j(a)f(con)m (tradiction)j(with)d(the)g(preceding)i(inequalit)m(y)-8 b(.)56 b(Hence)83 5586 y Fw(C)32 b Fv(=)25 b(0)p Fx(.)3351 b Fc(\004)1903 5836 y Fx(9)p eop %%Page: 10 10 bop 83 482 a Fj(Lemma)35 b(4.2)45 b Fr(L)-5 b(et)37 b Fw(H)44 b Fr(b)-5 b(e)39 b(a)e(self-adjoint)f(op)-5 b(er)g(ator)39 b(in)e Fu(H)g Fr(with)g(domain)f Fw(D)s Fv(\()p Fw(H)7 b Fv(\))38 b Fr(and)f(assume)g(that)g(for)83 594 y(e)-5 b(ach)32 b Fw(\013)16 b Fk(l)g Fw(o)p Fr(,)30 b(ther)-5 b(e)32 b(exists)f(a)f(self-adjoint)g(op)-5 b(er)g(ator)32 b Fw(H)1986 608 y Fs(\013)2066 594 y Fr(in)d Fu(H)q Fr(,)h(a\036liate)-5 b(d)31 b(to)2782 521 y Fl(\012)2825 594 y Fw(C)7 b Fv(\()p 2932 521 82 4 v(I)-18 b(R)3023 540 y Fs(\013)3072 594 y Fv(\))16 b Fu(\001)g Fw(C)3229 608 y Ft(0)3269 594 y Fv(\(I)-18 b(R)3385 559 y Fs(n)p FC(\003)3468 594 y Fv(\))3503 521 y Fl(\013)3546 594 y Fr(,)30 b(with)83 707 y(domain)e(e)-5 b(qual)30 b(to)e Fw(D)s Fv(\()p Fw(H)7 b Fv(\))p Fr(.)40 b(Assume)29 b(also)f(that)h(for)f(some)g(asymptotic)g(lo)-5 b(c)g(alization)29 b(function)f Fw(\030)k Fr(and)c(e)-5 b(ach)83 843 y Fw(\013)7 b Fk(l)g Fw(o)p Fr(,)28 b Fv(lim)452 857 y Fs(r)r FC(!1)646 738 y Fl(\015)646 793 y(\015)646 847 y(\015)697 843 y Fw(\030)756 742 y Fl(\020)810 843 y Fw(\013)21 b Fu(\001)944 802 y Fs(Q)p 944 822 56 4 v 955 874 a(r)1010 742 y Fl(\021)1079 843 y Fv(\()p Fw(H)28 b Fu(\000)20 b Fw(H)1385 857 y Fs(\013)1434 843 y Fv(\))1469 738 y Fl(\015)1469 793 y(\015)1469 847 y(\015)1520 906 y Fs(D)r Ft(\()p Fs(H)5 b Ft(\))p FC(!H)1857 843 y Fv(=)25 b(0)i Fr(.)39 b(Then)27 b Fw(H)33 b Fr(is)26 b(a\036liate)-5 b(d)26 b(to)h Fo(C)39 b Fr(and)26 b Fv(\005)3318 857 y Fs(\013)3368 843 y Fv(\()p Fw(H)7 b Fv(\))26 b(=)f Fw(H)3719 857 y Fs(\013)83 999 y Fr(\(in)32 b(the)g(sense)h(of)f(De\034nition)g(2.1\).)249 1207 y(Pr)-5 b(o)g(of:)40 b Fx(Set)29 b Fw(R)d Fv(:=)f(\()p Fw(H)e Fu(\000)17 b Fw(z)t Fv(\))1199 1174 y FC(\000)p Ft(1)1322 1207 y Fx(and)29 b Fw(R)1565 1221 y Fs(\013)1639 1207 y Fv(:=)d(\()p Fw(H)1872 1221 y Fs(\013)1937 1207 y Fu(\000)16 b Fw(z)t Fv(\))2105 1174 y FC(\000)p Ft(1)2229 1207 y Fx(for)28 b(an)m(y)h(\034xed)f Fw(z)i Fu(2)25 b Fk(C)40 b Fu(n)17 b Fv(I)-18 b(R)28 b Fx(and)g(eac)m(h)i Fw(\013)17 b Fk(l)f Fw(o)p Fx(.)83 1320 y(Since)34 b Fw(H)398 1334 y Fs(\013)481 1320 y Fx(is)f(a\036liated)i(to)f(the)g(subalgebra)1679 1247 y Fl(\012)1722 1320 y Fw(C)7 b Fv(\()p 1829 1247 82 4 v(I)-18 b(R)1919 1266 y Fs(\013)1969 1320 y Fv(\))23 b Fu(\001)f Fw(C)2139 1334 y Ft(0)2179 1320 y Fv(\(I)-18 b(R)2296 1284 y Fs(n)p FC(\003)2378 1320 y Fv(\))2413 1247 y Fl(\013)2490 1320 y Fx(of)33 b Fo(C)13 b Fx(,)34 b Fw(R)2793 1334 y Fs(\013)2876 1320 y Fx(b)s(elongs)g(to)g Fo(C)46 b Fx(and)34 b(th)m(us)83 1433 y(is)k(semi-compact,)k Fr(cf.)e(i\))e Fx(of)g(Lemma)i(4.1.)65 b(Then)39 b Fw(R)g Fx(is)f(semi-compact)i(since)f(for)f(an)m(y)h Fw(\020)46 b Fu(2)38 b Fw(C)3504 1447 y Ft(0)3544 1433 y Fv(\(I)-18 b(R)3661 1397 y Fs(n)3708 1433 y Fv(\))p Fx(,)83 1546 y Fw(\020)7 b Fv(\()p Fw(Q)p Fv(\))p Fw(R)341 1560 y Fs(\013)416 1546 y Fu(2)25 b(K)q Fv(\()p Fu(H)q Fv(\))31 b Fx(and)g Fw(R)997 1513 y FC(\000)p Ft(1)996 1569 y Fs(\013)1091 1546 y Fw(R)f Fx(is)g(b)s(ounded)h (b)m(y)g(the)g(closed)f(graph)i(theorem)f([11)q(].)249 1659 y(F)-8 b(urthermore,)364 1772 y Fl(\015)364 1826 y(\015)364 1881 y(\015)364 1936 y(\015)414 1904 y Fw(\030)474 1776 y Fl(\022)540 1904 y Fw(\013)21 b Fu(\001)675 1842 y Fw(Q)p 675 1883 72 4 v 689 1966 a(r)756 1776 y Fl(\023)838 1904 y Fv(\()p Fw(R)h Fu(\000)e Fw(R)1124 1918 y Fs(\013)1173 1904 y Fv(\))1208 1772 y Fl(\015)1208 1826 y(\015)1208 1881 y(\015)1208 1936 y(\015)1284 1904 y Fu(\024)25 b Fw(c)1434 1776 y Fl(\032)1503 1772 y(\015)1503 1826 y(\015)1503 1881 y(\015)1503 1936 y(\015)1553 1904 y Fw(\030)1613 1776 y Fl(\022)1679 1904 y Fw(\013)c Fu(\001)1814 1842 y Fw(Q)p 1814 1883 V 1828 1966 a(r)1895 1776 y Fl(\023)1977 1904 y Fv(\()p Fw(H)28 b Fu(\000)20 b Fw(H)2283 1918 y Fs(\013)2332 1904 y Fv(\))p Fw(R)2437 1772 y Fl(\015)2437 1826 y(\015)2437 1881 y(\015)2437 1936 y(\015)2508 1904 y Fv(+)2599 1772 y Fl(\015)2599 1826 y(\015)2599 1881 y(\015)2599 1936 y(\015)2649 1776 y(\024)2697 1904 y Fw(\030)2756 1776 y Fl(\022)2823 1904 y Fw(\013)h Fu(\001)2957 1842 y Fw(Q)p 2957 1883 V 2971 1966 a(r)3039 1776 y Fl(\023)3121 1904 y Fw(;)15 b(R)3230 1918 y Fs(\013)3280 1776 y Fl(\025)3328 1772 y(\015)3328 1826 y(\015)3328 1881 y(\015)3328 1936 y(\015)3379 1776 y(\033)3462 1904 y Fw(;)83 2153 y Fx(where)43 b Fw(c)k Fv(=)e(max)p Fu(fk)p Fw(R)886 2167 y Fs(\013)937 2153 y Fu(k)p Fw(;)15 b Fu(k)p Fv(\()p Fw(H)37 b Fu(\000)28 b Fw(H)1390 2167 y Fs(\013)1439 2153 y Fv(\))p Fw(R)q Fu(kg)p Fx(.)78 b(The)43 b(\034rst)f(term)i(on)e(the)h(r.h.s.)g(go)s(es)f(to)h Fv(0)g Fx(as)f Fw(r)48 b Fu(!)e(1)83 2266 y Fx(b)m(y)35 b(h)m(yp)s(othesis.)55 b(F)-8 b(or)36 b(the)f(second)h(term,)h(one)e(has)g(to)g(use)g(the)h (isomorphism)2885 2193 y Fl(\012)2928 2266 y Fw(C)7 b Fv(\()p 3035 2193 82 4 v(I)-18 b(R)3125 2212 y Fs(\013)3175 2266 y Fv(\))24 b Fu(\001)f Fw(C)3347 2280 y Ft(0)3387 2266 y Fv(\(I)-18 b(R)3504 2231 y Fs(n)p FC(\003)3586 2266 y Fv(\))3621 2193 y Fl(\013)3697 2241 y Fu(\030)3697 2271 y Fv(=)83 2400 y Fw(C)148 2414 y Ft(0)187 2400 y Fv(\(I)g(R)304 2365 y Fs(\013)349 2341 y Fq(?)402 2365 y FC(\003)441 2400 y Fv(\))27 b Fu(\012)f Fo(C)665 2367 y Fs(\013)754 2400 y Fx(in)m(tro)s(duced)42 b(in)d(Section)i(3)f(and)g (either)h(Lemma)g(3.4)f(of)f([7)q(])g(or)h(a)g(comm)m(utator)i(ex-)83 2536 y(pansion)36 b(for)f(terms)h(of)f(the)g(form)1307 2435 y Fl(h)1350 2536 y Fw(\030)1409 2435 y Fl(\020)1463 2536 y Fw(\013)21 b Fu(\001)1597 2495 y Fs(Q)p 1597 2515 56 4 v 1608 2567 a(r)1663 2435 y Fl(\021)1732 2536 y Fw(;)15 b(g)s Fv(\()p Fw(\013)22 b Fu(\001)f Fw(P)13 b Fv(\))2085 2435 y Fl(i)2163 2536 y Fx(with)36 b Fw(g)h Fu(2)c Fw(C)2613 2550 y Ft(0)2653 2536 y Fv(\(I)-18 b(R)2770 2500 y Fs(\013)2815 2477 y Fq(?)2871 2536 y Fv(\))p Fx(.)56 b(It)35 b(then)h(follo)m(ws)f(that)83 2708 y Fv(lim)209 2722 y Fs(r)r FC(!1)404 2603 y Fl(\015)404 2658 y(\015)404 2712 y(\015)454 2607 y(h)497 2708 y Fw(\030)556 2607 y Fl(\020)610 2708 y Fw(\013)21 b Fu(\001)744 2667 y Fs(Q)p 744 2687 V 755 2739 a(r)810 2607 y Fl(\021)879 2708 y Fw(;)15 b(R)988 2722 y Fs(\013)1038 2607 y Fl(i)1081 2603 y(\015)1081 2658 y(\015)1081 2712 y(\015)1157 2708 y Fv(=)25 b(0)p Fx(,)k(and)g(the)f (a\036liation)i(of)e Fw(H)35 b Fx(to)28 b Fo(C)41 b Fx(is)28 b(obtained)i(with)e(Lemma)i(4.1)e(and)83 2838 y(the)j(observ)-5 b(ation)31 b(made)g(b)s(efore)f(De\034nition)i(2.1.)249 2951 y(W)-8 b(e)37 b(ha)m(v)m(e)g(th)m(us)g(obtained)h(that)f Fv(\005)1478 2965 y Fs(\013)1527 2877 y Fl(\000)1569 2951 y Fv(\()p Fw(H)31 b Fu(\000)24 b Fw(z)t Fv(\))1887 2918 y FC(\000)p Ft(1)1982 2877 y Fl(\001)2059 2951 y Fv(=)35 b(\()p Fw(H)2276 2965 y Fs(\013)2350 2951 y Fu(\000)24 b Fw(z)t Fv(\))2526 2918 y FC(\000)p Ft(1)2657 2951 y Fx(for)36 b(all)h Fw(z)j Fu(2)34 b Fk(C)48 b Fu(n)25 b Fv(I)-18 b(R)p Fx(.)58 b(The)37 b(last)83 3064 y(statemen)m(t)f(of)f(the)g(lemma)g(follo)m(ws)h(from)f(the)g(densit)m(y)g (in)g Fw(C)2241 3078 y Ft(0)2280 3064 y Fv(\(I)-18 b(R\))35 b Fx(of)g(the)g(v)m(ector)g(space)g(generated)i(b)m(y)83 3177 y(the)31 b(set)f(of)g(functions)h Fu(f)p Fv(\()p Fu(\001)21 b(\000)f Fw(z)t Fv(\))1171 3144 y FC(\000)p Ft(1)1296 3177 y Fu(j)30 b Fw(z)g Fu(2)25 b Fk(C)43 b Fu(n)21 b Fv(I)-18 b(R)p Fu(g)p Fx(.)1885 b Fc(\004)249 3290 y Fx(W)-8 b(e)40 b(no)m(w)g(giv)m(e)g (the)g(general)g(de\034nition)h(of)e(the)h(p)s(oten)m(tials)g(under)g (consideration,)k(and)c(w)m(e)g(shall)83 3402 y(pro)m(v)m(e)29 b(in)f(Prop)s(osition)h(4.5)f(the)h(a\036liation)g(to)f Fo(C)41 b Fx(for)27 b(the)h(corresp)s(onding)i(Sc)m(hr\366)s(dinger)f(op)s(erators.) 40 b(One)83 3515 y(notices)c(that)f(if)g Fw(V)55 b Fx(b)s(elongs)35 b(to)g Fw(C)7 b Fv(\()p 1337 3442 82 4 v(I)-18 b(R)1427 3461 y Fs(n)1474 3515 y Fv(\))p Fx(,)37 b(the)e(functions)g Fw(V)2199 3482 y Fs(\013)2283 3515 y Fx(in)m(tro)s(duced)i(in)e(the)g(follo)m(wing)h (de\034nition)83 3628 y(are)31 b(nothing)g(but)g(the)g(restrictions)g(of)f Fw(V)50 b Fx(to)31 b(the)f(h)m(yp)s(ersurfaces)p 2391 3555 V 31 w Fv(I)-18 b(R)2481 3574 y Fs(\013)2531 3628 y Fx(.)83 3836 y Fj(De\034nition)34 b(4.3)45 b Fr(A)31 b(Bor)-5 b(el)32 b(function)f Fw(V)45 b Fv(:)26 b(I)-18 b(R)1666 3801 y Fs(n)1738 3836 y Fu(!)25 b Fv(I)-18 b(R)30 b Fr(is)h Fx(a)e(cartesian)g(p)s(oten)m (tial)i(\(relativ)m(e)f(to)f Fu(L)p Fx(\))h Fr(if)h(ther)-5 b(e)83 3949 y(exists)37 b(a)f(c)-5 b(ol)5 b(le)-5 b(ction)38 b(of)e(Bor)-5 b(el)38 b(functions)e Fu(f)p Fw(V)1687 3916 y Fs(\013)1737 3949 y Fu(g)1782 3963 y Fs(\013)p FC(2L)1960 3949 y Fr(,)h(with)e Fw(V)2298 3916 y Fs(o)2369 3949 y Fu(\021)d Fw(V)56 b Fr(and)37 b Fw(V)2834 3916 y Fs(\013)2916 3949 y Fv(:)c(I)-18 b(R)3055 3913 y Fs(\013)3137 3949 y Fu(!)32 b Fv(I)-18 b(R)p Fr(,)37 b(such)g(that)83 4062 y(for)32 b(e)-5 b(ach)34 b Fw(\013)26 b Fu(2)f(L)32 b Fr(:)199 4246 y(i\))45 b Fw(V)382 4213 y Fs(\013)432 4246 y Fv(\()p Fw(Q)p Fv(\))33 b Fr(is)e Fv(\001)779 4213 y Fs(\013)829 4246 y Fr(-b)-5 b(ounde)g(d)34 b(with)e(r)-5 b(elative)34 b(b)-5 b(ound)34 b(less)e(than)g(one,)172 4455 y(ii\))44 b Fv(lim)435 4469 y Fs(r)r FC(!1)630 4350 y Fl(\015)630 4405 y(\015)630 4460 y(\015)680 4455 y Fw(\030)739 4354 y Fl(\020)793 4455 y Fv(\()p Fw(\014)26 b Fu(\000)20 b Fw(\013)p Fv(\))h Fu(\001)1166 4414 y Fs(Q)p 1166 4434 56 4 v 1177 4486 a(r)1231 4354 y Fl(\021)1300 4381 y(\000)1342 4455 y Fw(V)1415 4422 y Fs(\013)1465 4455 y Fv(\()p Fw(Q)p Fv(\))g Fu(\000)f Fw(V)1792 4422 y Fs(\014)1839 4455 y Fv(\()p Fw(Q)p Fv(\))1981 4381 y Fl(\001)2023 4350 y(\015)2023 4405 y(\015)2023 4460 y(\015)2074 4518 y FC(H)2134 4499 y Fm(2)2134 4535 y Fn(\013)2178 4518 y FC(!H)2309 4526 y Fn(\013)2382 4455 y Fv(=)25 b(0)g Fr(for)g(e)-5 b(ach)26 b Fw(\014)9 b Fk(l)t Fw(\013)25 b Fr(and)g(some)g (asymp-)309 4606 y(totic)32 b(lo)-5 b(c)g(alization)33 b(function)f Fw(\030)t Fr(.)249 4814 y Fx(The)40 b(second)f(condition)i(means)f(that)f (for)g(eac)m(h)i Fw(\013)f Fu(2)g(L)p Fx(,)h(the)f(function)g Fw(V)2929 4781 y Fs(\013)3018 4814 y Fx(de\034ned)g(on)f Fv(I)-18 b(R)3554 4779 y Fs(\013)3643 4814 y Fx(ap-)83 4927 y(proac)m(hes)37 b(its)e(asymptotic)h(limits)f Fw(V)1388 4894 y Fs(\014)1471 4927 y Fx(with)g Fw(\014)29 b Fk(l)23 b Fw(\013)36 b Fx(in)f(the)h(norm)g Fu(k)24 b(\001)g(k)2624 4947 y FC(H)2684 4928 y Fm(2)2684 4964 y Fn(\013)2728 4947 y FC(!H)2859 4955 y Fn(\013)2908 4927 y Fx(.)55 b(Let)36 b(us)e(observ)m(e)i(that)83 5040 y(Lemma)e(9.4.8)e(of)g([1)q (])g(implies)g(that)h(if)f Fw(V)52 b Fx(is)32 b(a)g(cartesian)h(p)s(oten)m (tial,)i(then)d Fw(V)2799 5007 y Fs(\013)2848 5040 y Fv(\()p Fw(Q)p Fv(\))h Fx(is)e Fv(\001)p Fx(-b)s(ounded)i(with)83 5153 y(relativ)m(e)i(b)s(ound)g(less)e(than)i(one,)h(and)f Fr(ii\))e Fx(is)h(also)g(ful\034lled)h(with)f(the)h(norm)g Fu(k)23 b(\001)g(k)2956 5173 y FC(H)3016 5154 y Fm(2)3051 5173 y FC(!H)3220 5153 y Fx(instead)35 b(of)f(the)83 5266 y(norm)d Fu(k)21 b(\001)f(k)475 5286 y FC(H)535 5267 y Fm(2)535 5302 y Fn(\013)580 5286 y FC(!H)711 5294 y Fn(\013)759 5266 y Fx(.)40 b(W)-8 b(e)31 b(giv)m(e)g(a)f(rather)h (general)h(example)f(of)f(suc)m(h)h(p)s(oten)m(tials.)83 5474 y Fj(Example)k(4.4)45 b Fx(Let)28 b Fw(V)902 5488 y Ft(1)969 5474 y Fx(b)s(e)f(a)h(b)s(ounded)h(real)f(function)g(on)g Fv(I)-18 b(R)2258 5438 y Fs(n)2333 5474 y Fx(suc)m(h)28 b(that)g(for)g(eac)m(h)h Fw(\013)15 b Fk(l)g Fw(o)p Fx(,)28 b(there)h(exists)83 5587 y Fw(V)156 5554 y Fs(\013)231 5587 y Fu(2)c Fw(L)379 5554 y FC(1)453 5587 y Fv(\(I)-18 b(R)570 5551 y Fs(\013)620 5587 y Fv(\))23 b Fx(satisfying)f Fv(lim)1199 5601 y Fs(r)r FC(!1)1393 5587 y Fv(sup)1530 5609 y Fs(x)p FC(2)p Ft(I)-18 b(R)1676 5584 y Fn(n)1738 5510 y Fl(\014)1738 5564 y(\014)1768 5587 y Fw(\030)1827 5513 y Fl(\000)1869 5587 y Fw(\013)21 b Fu(\001)2003 5551 y Fs(x)p 2003 5566 40 4 v 2006 5618 a(r)2053 5513 y Fl(\001)15 b(\000)2151 5587 y Fw(V)2204 5601 y Ft(1)2244 5587 y Fv(\()p Fw(x)p Fv(\))21 b Fu(\000)f Fw(V)2551 5554 y Fs(\013)2600 5587 y Fv(\()p Fw(x)p Fv(\))2722 5513 y Fl(\001)2765 5510 y(\014)2765 5564 y(\014)2820 5587 y Fv(=)25 b(0)e Fx(for)f(some)h(asymptotic)1880 5836 y(10)p eop %%Page: 11 11 bop 83 482 a Fx(lo)s(calization)29 b(function)e Fw(\030)t Fx(.)40 b(Let)27 b Fw(V)1230 496 y Ft(2)1270 482 y Fv(\()p Fw(Q)p Fv(\))h Fx(b)s(e)e(a)i Fv(\001)p Fx(-b)s(ounded)f(op)s(erator)h(with)g(relativ)m(e)h (b)s(ound)e(less)g(than)g(one)83 617 y(and)g(suc)m(h)f(that)h Fv(lim)772 631 y Fs(r)r FC(!1)967 513 y Fl(\015)967 567 y(\015)967 622 y(\015)1017 617 y Fw(\030)1076 516 y Fl(\020)1140 572 y FC(j)p Fs(Q)p FC(j)p 1140 596 95 4 v 1171 648 a Fs(r)1245 516 y Fl(\021)1315 617 y Fw(V)1368 631 y Ft(2)1407 617 y Fv(\()p Fw(Q)p Fv(\))1549 513 y Fl(\015)1549 567 y(\015)1549 622 y(\015)1600 680 y FC(H)1660 661 y Fm(2)1695 680 y FC(!H)1855 617 y Fv(=)e(0)p Fx(.)39 b(Then)27 b Fw(V)2365 584 y Fs(o)2429 617 y Fv(:=)e Fw(V)2603 631 y Ft(1)2654 617 y Fv(+)12 b Fw(V)2790 631 y Ft(2)2855 617 y Fx(is)26 b(a)g(cartesian)h(p)s(oten)m(tial.)83 759 y(Indeed,)f(let)f Fw(\014)30 b Fu(2)25 b(L)f Fx(with)g Fu(j)p Fw(\014)5 b Fu(j)26 b Fv(=)f Fw(n)8 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y Fu(j)1748 999 y Fs(x)1788 1018 y Fn(j)1816 1004 y Fq(0)1843 999 y Ft(=)p Fs(\014)1938 1018 y Fn(j)1966 1004 y Fq(0)1992 999 y Fs(n)2039 984 y Fu(g)2084 998 y Fs(n)p FC(2)p Fd(N)2259 984 y Fx(are)g(t)m(w)m(o)h(Cauc)m(h)m(y)h(sequences)d(in)h Fw(L)3494 951 y FC(1)3569 984 y Fv(\(I)-18 b(R)3686 949 y Fs(\014)3733 984 y Fv(\))83 1115 y Fx(whic)m(h)37 b(con)m(v)m(erge)h(to)d(the)i(same)e (elemen)m(t)j(\(denoted)f Fw(V)2025 1082 y Fs(\014)2072 1115 y Fx(\).)57 b(Both)36 b(requiremen)m(ts)h(of)e(De\034nition)i(4.3)g(are)83 1228 y(no)m(w)d(clearly)g(satis\034ed)f(for)g Fw(\013)d Fv(=)g Fw(o)j Fx(and)g(for)g(eac)m(h)h Fw(\013)23 b Fk(l)e Fw(o)p Fx(.)49 b(The)34 b(same)f(pro)s(cedure)h(can)f(then)h(b)s(e)f(applied)83 1341 y(again)g(in)g(order)g(to)g(construct)g(successiv)m(ely)g Fw(V)1751 1308 y Fs(\014)1830 1341 y Fx(for)g(all)g Fw(\014)h Fu(2)28 b(L)k Fx(and)h(to)g(c)m(hec)m(k)h(that)f(the)g(conditions)g(of)83 1454 y(De\034nition)f(4.3)e(are)h(satis\034ed.)249 1645 y(F)-8 b(or)21 b(the)g(next)f(pro)s(of)g(and)h(some)g(later)g(uses,)h(w)m(e)f(in)m (tro)s(duce)h(the)e(semilattice)i Fu(L)2925 1612 y Fs(\013)2999 1645 y Fv(:=)j Fu(f)p Fw(\014)31 b Fu(2)25 b(L)20 b(j)h Fw(\014)30 b Fu(\024)25 b Fw(\013)p Fu(g)p Fx(.)83 1835 y Fj(Prop)s(osition)33 b(4.5)45 b Fr(Assume)34 b(that)g Fw(V)53 b Fr(is)33 b(a)h(c)-5 b(artesian)34 b(p)-5 b(otential.)45 b(Then)33 b Fw(H)i Fv(=)27 b Fu(\000)p Fv(\001)20 b(+)h Fw(V)54 b Fr(is)33 b(a)g(c)-5 b(artesian)83 1948 y(Hamiltonian)31 b(and)h Fv(\005)847 1962 y Fs(\013)897 1948 y Fv(\()p Fw(H)7 b Fv(\))26 b(=)f Fu(\000)p Fv(\001)20 b(+)f Fw(V)1503 1915 y Fs(\013)1584 1948 y Fr(for)32 b(e)-5 b(ach)34 b Fw(\013)26 b Fu(2)f(L)p Fr(.)249 2139 y(Pr)-5 b(o)g(of:)43 b Fx(Since)32 b(the)f(pro)s(of)g(is)g(p)s(erformed)h(b)m(y)g (induction)g(o)m(v)m(er)h(the)e(lattice)h Fu(L)p Fx(,)g(let)f(us)g(\034rst)g (in)m(tro)s(duce)83 2265 y(some)37 b(notations.)61 b(F)-8 b(or)38 b(eac)m(h)g Fw(\013)f Fu(2)f(L)g Fx(and)h(eac)m(h)h Fw(\014)k Fu(\024)36 b 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Fs(\013)2453 2591 y(\014)2503 2563 y Fv([)p Fw(f)10 b Fv(\()p Fw(Q)p Fv(\))p Fw(h)p Fv(\()p Fw(P)j Fv(\)])27 b(=)e Fw(\031)3121 2530 y Fs(\013)3118 2591 y(\014)3170 2563 y Fv(\()p Fw(f)10 b Fv(\)\()p Fw(Q)p Fv(\))p Fw(h)p Fv(\()p Fw(P)j Fv(\))30 b Fx(for)83 2695 y(eac)m(h)37 b Fw(f)45 b Fu(2)35 b Fw(C)7 b Fv(\()p 585 2622 V(I)-18 b(R)676 2641 y Fs(\013)725 2695 y Fv(\))37 b Fx(and)g(eac)m(h)g Fw(h)f Fu(2)f Fw(C)1437 2709 y Ft(0)1476 2695 y Fv(\(I)-18 b(R)1593 2659 y Fs(\013)1643 2695 y Fv(\))p Fx(.)59 b(If)35 b Fw(\013)h Fv(=)f Fw(o)p Fx(,)j(then)f Fv(\005)2445 2662 y Fs(\013)2445 2722 y(\014)2531 2695 y Fx(is)f(just)g Fv(\005)2881 2710 y Fs(\014)2928 2695 y Fx(.)59 b(In)36 b(this)g(setting,)j (the)83 2807 y(statemen)m(t)f(of)d(the)i(prop)s(osition)g(reads:)52 b(if)36 b Fw(V)1705 2774 y Fs(\013)1790 2807 y Fx(is)g(a)g(cartesian)h(p)s (oten)m(tial)h(relativ)m(e)f(to)f(the)h(lattice)g Fu(L)3694 2774 y Fs(\013)3743 2807 y Fx(,)83 2920 y(then)31 b Fw(H)372 2887 y Fs(\013)447 2920 y Fv(=)25 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Fx(is)e(clearly)h (a\036liated)i(to)83 3259 y Fo(C)148 3226 y Fs(\013)197 3259 y Fx(,)35 b(whic)m(h)g(is)f(simply)g Fk(C)17 b Fx(.)57 b(F)-8 b(or)35 b(an)m(y)f(\034xed)g Fw(\013)p Fx(,)h Fu(j)p Fw(\013)p Fu(j)e(6)p Fv(=)e(0)p Fx(,)k(w)m(e)f(ma)m(y)h(no)m(w)g(assume)f(that)g(the)g (assertions)g(of)83 3372 y(the)g(prop)s(osition)g(are)f(pro)m(v)m(ed)i(for)e (eac)m(h)i Fw(H)1601 3339 y Fs(\014)1681 3372 y Fx(with)f Fw(\014)27 b Fk(l)22 b Fw(\013)34 b Fx(and)g(w)m(e)g(pro)m(v)m(e)g(it)g(for)f Fw(H)3032 3339 y Fs(\013)3081 3372 y Fx(.)49 b(With)34 b(no)f(loss)g(of)83 3485 y(generalit)m(y)f(and)f(for)f(simplicit)m(y)h(of)f(notations,)i(w)m(e)f (c)m(ho)s(ose)f Fw(\013)c Fv(=)f Fw(o)p Fx(.)249 3598 y(So)43 b(assume)g(that)g(for)f(eac)m(h)i Fw(\014)34 b Fk(l)28 b Fw(o)p Fx(,)46 b Fw(H)1668 3565 y Fs(\014)1757 3598 y Fx(is)c(a\036liated)i(to)f Fo(C)2432 3565 y Fs(\014)2479 3598 y Fx(.)77 b(Let)44 b Fw(j)51 b Fu(2)46 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)i(;)e(n)p Fu(g)43 b Fx(suc)m(h)g(that)83 3711 y Fw(\014)134 3725 y Fs(j)198 3711 y Fu(6)p Fv(=)27 b(0)p Fx(.)45 b(Then)32 b(the)g(op)s(erator)h Fu(\000)p Fv(\001)1318 3725 y Fs(j)1375 3711 y Fu(\012)21 b Fw(I)28 b Fv(+)20 b Fw(I)29 b Fu(\012)20 b Fw(H)1869 3678 y Fs(\014)1948 3711 y Fx(in)32 b Fw(L)2117 3678 y Ft(2)2156 3711 y Fv(\(I)-18 b(R)2273 3725 y Fs(j)2309 3711 y Fv(\))22 b Fu(\012)f(H)2535 3726 y Fs(\014)2613 3711 y Fx(is)31 b(a\036liated)i(to)f(the)g(C)3412 3678 y FC(\003)3452 3711 y Fx(-algebra)83 3824 y Fw(C)148 3838 y Ft(0)187 3824 y Fv(\(I)-18 b(R)304 3788 y FC(\003)304 3846 y Fs(j)344 3824 y Fv(\))8 b Fu(\012)g Fo(C)531 3791 y Fs(\014)602 3824 y Fx(\(see)25 b([1],)h(Section)f(8.2.3\).)39 b(F)-8 b(urthermore,)28 b(if)c(w)m(e)h(iden)m(tify)g Fw(L)2651 3791 y Ft(2)2691 3824 y Fv(\(I)-18 b(R)2808 3838 y Fs(j)2844 3824 y Fv(\))8 b Fu(\012)g(H)3043 3839 y Fs(\014)3115 3824 y Fx(with)24 b Fu(H)q Fx(,)i(then)f(the)83 3968 y(op)s(erator)h Fu(\000)p Fv(\001)589 3982 y Fs(j)636 3968 y Fu(\012)10 b Fw(I)18 b Fv(+)10 b Fw(I)17 b 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Fx(.)54 b(But)35 b Fw(H)42 b Fx(is)35 b(also)g(a)g(self-adjoin)m(t)h(op)s(erator)g(with)f(domain)h Fu(H)3548 4184 y Ft(2)3622 4217 y Fx(and)83 4257 y Fl(\015)83 4311 y(\015)83 4366 y(\015)134 4361 y Fw(\030)193 4260 y Fl(\020)247 4361 y Fw(\014)25 b Fu(\001)379 4320 y Fs(Q)p 379 4340 56 4 v 390 4393 a(r)444 4260 y Fl(\021)514 4361 y Fv(\()p Fw(H)i Fu(\000)20 b Fw(H)819 4376 y Fs(\014)866 4361 y Fv(\))901 4257 y Fl(\015)901 4311 y(\015)901 4366 y(\015)952 4425 y FC(H)1012 4406 y Fm(2)1046 4425 y FC(!H)1220 4361 y Fv(=)1330 4257 y Fl(\015)1330 4311 y(\015)1330 4366 y(\015)1380 4361 y Fw(\030)1439 4260 y Fl(\020)1494 4361 y Fw(\014)25 b Fu(\001)1625 4320 y Fs(Q)p 1625 4340 V 1636 4393 a(r)1691 4260 y Fl(\021)1760 4288 y(\000)1802 4361 y Fw(V)20 b Fv(\()p Fw(Q)p Fv(\))h Fu(\000)f Fw(V)2202 4328 y Fs(\014)2249 4361 y Fv(\()p Fw(Q)p Fv(\))2391 4288 y Fl(\001)2433 4257 y(\015)2433 4311 y(\015)2433 4366 y(\015)2484 4425 y FC(H)2544 4406 y Fm(2)2579 4425 y FC(!H)2752 4361 y Fu(!)39 b Fv(0)g 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h(in)f(this)f(title)i(is)e(similar)h(to)g(that)g(dev)m(elop)s(ed)h(for)83 5587 y(the)e(N-b)s(o)s(dy)e(problem.)42 b(The)31 b(analogy)g(is)f(p)s (ossible)g(mainly)h(b)s(ecause)g(of)f(the)g(ric)m(h)i(in)m(ternal)g (structure)f(of)1880 5836 y(11)p eop %%Page: 12 12 bop 83 482 a Fo(C)40 b Fx(and)29 b(its)e(compatibilit)m(y)j(with)e(the)g (unitary)g(group)h Fu(f)p Fw(W)2083 496 y Fs(t)2113 482 y Fu(g)2158 496 y Fs(t)p FC(2)p Ft(I)-18 b(R)2322 482 y Fx(of)27 b(dilations)i(in)e Fu(H)h Fx(\(Lemma)i(5.1\).)40 b(More)83 594 y(precisely)-8 b(,)29 b(w)m(e)f(consider)g Fu(f)p Fw(W)1078 608 y Fs(t)1133 594 y Fv(=)d Fw(e)1271 561 y Ft(2)p Fs(iAt)1413 594 y Fu(g)1458 608 y Fs(t)p FC(2)p Ft(I)-18 b(R)1621 594 y Fx(with)28 b(self-adjoin)m(t)g (generator)h Fw(A)c Fv(=)2884 559 y Ft(1)p 2884 574 36 4 v 2884 626 a(4)2945 526 y Fl(P)3040 553 y Fs(n)3040 621 y(j)t Ft(=1)3167 594 y Fv(\()p Fw(P)3260 608 y Fs(j)3297 594 y Fw(Q)3369 608 y Fs(j)3420 594 y Fv(+)14 b Fw(Q)3577 608 y Fs(j)3613 594 y Fw(P)3671 608 y Fs(j)3708 594 y Fv(\))p Fx(.)83 718 y(Let)39 b(us)g(observ)m(e)g(that)g Fv([\001)p Fw(;)15 b(iA)p Fv(])41 b(=)e(\001)p Fx(,)h(and)g(recall)f(that)g(this)g(group)h(lea)m(v)m(es)f Fu(H)2902 685 y Ft(2)2980 718 y Fx(in)m(v)-5 b(arian)m(t.)67 b(Eac)m(h)41 b Fw(W)3739 732 y Fs(t)83 831 y Fx(induces)31 b(an)g(automorphism)h Fu(W)1219 845 y Fs(t)1278 831 y Fx(of)e Fu(B)s Fv(\()p Fw(H)7 b Fv(\))p Fx(,)31 b(namely)f Fu(W)2056 845 y Fs(t)2086 831 y Fv([)p Fw(B)5 b Fv(])25 b(=)g Fw(W)2417 845 y FC(\000)p Fs(t)2501 831 y Fw(B)5 b(W)2661 845 y Fs(t)2721 831 y Fx(for)30 b Fw(B)g Fu(2)24 b(B)s Fv(\()p Fu(H)q Fv(\))p Fx(.)83 1044 y Fj(Lemma)35 b(5.1)45 b Fr(F)-7 b(or)26 b(e)-5 b(ach)27 b Fw(t)e Fu(2)g Fv(I)-18 b(R)o Fr(,)26 b Fu(W)1356 1058 y Fs(t)1411 1044 y Fr(le)-5 b(aves)28 b(the)e(C)1880 1011 y FC(\003)1919 1044 y Fr(-algebr)-5 b(a)28 b Fo(C)39 b Fr(invariant;)27 b(for)e(e)-5 b(ach)27 b Fw(\013)f Fu(2)f(L)p Fr(,)h Fu(W)3469 1058 y Fs(t)3504 1044 y Fu(\016)5 b Fv(\005)3622 1058 y Fs(\013)3697 1044 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Fv(\()p Fw(Q)p Fv(\))p Fw(h)p Fv(\()p Fw(P)j Fv(\)])p Fx(.)42 b(Since)30 b Fo(C)42 b Fx(is)29 b(the)g(norm)h(closure)83 1595 y(of)k(the)h(v)m(ector)h(space)f(generated)h(b)m(y)f(suc)m(h)g(pro)s(ducts,)h Fo(C)47 b Fx(is)35 b(in)m(v)-5 b(arian)m(t)36 b(and)f Fu(W)2883 1609 y Fs(t)2947 1595 y Fx(and)g Fv(\005)3195 1609 y Fs(\013)3279 1595 y Fx(comm)m(ute)i(on)83 1708 y Fo(C)13 b Fx(.)3520 b Fc(\004)249 1821 y Fx(The)37 b(next)g(lemma)g(con)m(tains)h(t)m(w)m(o)g(results)f(whic)m (h)h(are)f(analogous)h(to)e(the)h(statemen)m(ts)h(of)f(Lemma)83 1934 y(9.4.3)26 b(and)g(Theorem)h(8.4.3)f(of)f([1)q(].)39 b(The)26 b(compatibilit)m(y)h(of)f(the)f(structure)i(of)e Fo(C)38 b Fx(with)26 b(the)g(dilation)h(group)83 2047 y(is)g(essen)m(tial.)41 b(One)28 b(observ)m(es)g(that)h Fw(A)c Fv(=)1502 2011 y Ft(1)p 1502 2026 36 4 v 1502 2078 a(4)1563 1978 y Fl(P)1658 2074 y FC(f)p Fs(j)t FC(j)p Fs(\013)1791 2084 y Fn(j)1824 2074 y Ft(=0)p FC(g)1953 2047 y Fv(\()p Fw(P)2046 2061 y Fs(j)2083 2047 y Fw(Q)2155 2061 y Fs(j)2207 2047 y Fv(+)15 b Fw(Q)2365 2061 y Fs(j)2401 2047 y Fw(P)2459 2061 y Fs(j)2496 2047 y Fv(\))h(+)2643 2011 y Ft(1)p 2643 2026 V 2643 2078 a(4)2703 1978 y Fl(P)2799 2074 y FC(f)p Fs(j)t FC(j)p Fs(\013)2932 2084 y Fn(j)2964 2074 y FC(6)p Ft(=0)p FC(g)3094 2047 y Fv(\()p Fw(P)3187 2061 y Fs(j)3224 2047 y Fw(Q)3296 2061 y Fs(j)3348 2047 y Fv(+)e Fw(Q)3505 2061 y Fs(j)3542 2047 y Fw(P)3600 2061 y Fs(j)3637 2047 y Fv(\))25 b Fu(\021)83 2196 y Fw(A)151 2163 y Fs(\013)221 2196 y Fv(+)20 b Fw(A)380 2163 y Fs(\013)425 2139 y Fq(?)482 2196 y Fx(,)30 b(where)h Fw(A)866 2163 y Fs(\013)946 2196 y Fx(\(resp.)40 b Fw(A)1275 2163 y Fs(\013)1320 2139 y Fq(?)1377 2196 y Fx(\))31 b(is)e(the)i(generator)h(of)e(dilations)h(in)g Fv(I)-18 b(R)2748 2160 y Fs(\013)2828 2196 y Fx(\(resp.)41 b Fv(I)-18 b(R)3171 2160 y Fs(\013)3216 2136 y Fq(?)3273 2196 y Fx(\).)83 2408 y Fj(Lemma)35 b(5.2)45 b Fr(L)-5 b(et)31 b Fw(V)51 b Fr(b)-5 b(e)33 b(a)e(c)-5 b(artesian)31 b(p)-5 b(otential)32 b(such)g(that)f Fw(H)h Fv(=)25 b Fu(\000)p Fv(\001)17 b(+)g Fw(V)51 b Fr(is)30 b(of)h(class)g Fw(C)3260 2375 y Ft(1)3253 2431 y Fs(u)3299 2408 y Fv(\()p Fw(A)p Fv(\))p Fr(.)42 b(Then)30 b(:)199 2596 y(i\))45 b(for)32 b(e)-5 b(ach)34 b Fw(\013)26 b Fu(2)f(L)p Fr(,)31 b Fw(H)1024 2610 y Fs(\013)1105 2596 y Fr(is)h(of)g(class)h Fw(C)1597 2563 y Ft(1)1590 2618 y Fs(u)1636 2596 y Fv(\()p Fw(A)p Fv(\))g Fr(and)f Fw(H)2065 2563 y Fs(\013)2147 2596 y Fr(is)g(of)g(class)h Fw(C)2639 2563 y Ft(1)2632 2618 y Fs(u)2678 2596 y Fv(\()p Fw(A)2781 2563 y Fs(\013)2831 2596 y Fv(\))p Fr(,)172 2783 y(ii\))44 b(one)33 b(has)40 b Fv(~)-53 b Fw(\045)688 2750 y Fs(A)688 2810 y(H)780 2783 y Fv(=)25 b(min)1028 2797 y Fs(\013)p Fd(l)p Fs(o)1182 2783 y Fw(\045)1229 2750 y Fs(A)1229 2810 y(H)1287 2818 y Fn(\013)1335 2783 y Fr(.)249 2996 y Fx(The)41 b(pro)s(of)f(of)g Fr(i\))f Fx(can)i(b)s(e)f(p)s(erformed)h(b)m(y)f(rewriting) i(the)e(pro)s(of)g(of)g(Lemma)i(9.4.3)f(of)f([1])g(in)g(our)83 3109 y(formalism.)49 b(The)34 b(statemen)m(t)g Fr(ii\))e Fx(is)h(obtained)h (b)m(y)f(taking)h(in)m(to)g(accoun)m(t)g(the)g(injectiv)m(e)f(morphism)h (\(3\),)83 3222 y(together)e(with)f(our)f(Lemma)i(5.1)e(and)h(Prop)s(osition) h(8.3.5)f(of)f([1].)249 3335 y(Let)f(us)g(no)m(w)g(recall)h(t)m(w)m(o)g(imp)s (ortan)m(t)g(results)e(in)h(Mourre)g(theory)g(whic)m(h)h(are)f(expressed)g (in)f(terms)h(of)83 3448 y(the)i(functions)f Fw(\045)h Fx(and)39 b Fv(~)-54 b Fw(\045)p Fx(.)41 b(Pro)s(ofs)30 b(can)g(b)s(e)g(found)h(in)g (Section)g(7.2)g(of)f([1].)83 3660 y Fj(Prop)s(osition)j(5.3)45 b Fr(L)-5 b(et)33 b Fw(H)39 b Fr(b)-5 b(e)34 b(a)e(self-adjoint)g(op)-5 b(er)g(ator)34 b(of)e(class)h Fw(C)2500 3627 y Ft(1)2539 3660 y Fv(\()p Fw(A)p Fv(\))p Fr(.)199 3848 y(i\))45 b(If)e Fv(~)-53 b Fw(\045)454 3815 y Fs(A)454 3875 y(H)521 3848 y Fv(\()p Fw(\025)p Fv(\))32 b Fw(>)f Fv(0)36 b Fr(for)g(some)g Fw(\025)31 b Fu(2)g Fv(I)-18 b(R)o Fr(,)36 b(then)g Fw(\025)g Fr(has)g(a)f(neighb)-5 b(ourho)g(o)g(d)39 b(in)c(which)h(ther)-5 b(e)37 b(is)e(at)h(most)f(a)309 3961 y(\034nite)d(numb)-5 b(er)33 b(of)f(eigenvalues)j(of)d Fw(H)7 b Fr(,)32 b(e)-5 b(ach)34 b(of)e(\034nite)g(multiplicity.)172 4148 y(ii\))44 b(One)36 b(has)g Fw(\045)718 4115 y Fs(A)718 4175 y(H)785 4148 y Fv(\()p Fw(\025)p Fv(\))c(=)39 b(~)-53 b Fw(\045)1089 4115 y Fs(A)1089 4175 y(H)1156 4148 y Fv(\()p Fw(\025)p Fv(\))36 b Fr(unless)g Fw(\025)g Fr(is)f(an)h(eigenvalue)i(of)d Fw(H)43 b Fr(and)g Fv(~)-53 b Fw(\045)2802 4115 y Fs(A)2802 4175 y(H)2869 4148 y Fv(\()p Fw(\025)p Fv(\))32 b Fw(>)f Fv(0)p Fr(,)37 b(in)e(which)h(c)-5 b(ase)309 4261 y Fw(\045)356 4228 y Fs(A)356 4288 y(H)423 4261 y Fv(\()p Fw(\025)p Fv(\))26 b(=)f(0)p Fr(.)249 4474 y Fx(The)h(next)g(statemen)m(t)h(con)m(tains)g(the)f(Mourre)g (estimate,)h(whic)m(h)g(is)e(the)h(main)h(result)e(of)h(this)f(section.)83 4686 y Fj(Prop)s(osition)33 b(5.4)45 b Fr(Assume)37 b(that)g Fw(V)57 b Fr(is)36 b(a)h(c)-5 b(artesian)37 b(p)-5 b(otential)38 b(and)e(that)h Fw(H)j Fv(=)33 b Fu(\000)p Fv(\001)23 b(+)g Fw(V)57 b Fr(is)36 b(of)h(class)83 4799 y Fw(C)155 4766 y Ft(1)148 4822 y Fs(u)194 4799 y Fv(\()p Fw(A)p Fv(\))p Fr(.)k(Then)29 b Fw(\034)10 b Fv(\()p Fw(H)d Fv(\))29 b Fr(and)g Fw(\024)p Fv(\()p Fw(H)7 b Fv(\))31 b Fr(ar)-5 b(e)30 b(close)-5 b(d)30 b(c)-5 b(ountable)31 b(sets,)f(the)g(eigenvalues)h(of)e Fw(H)36 b Fr(outside)29 b Fw(\034)10 b Fv(\()p Fw(H)d Fv(\))30 b Fr(ar)-5 b(e)83 4912 y(of)30 b(\034nite)f(multiplicity)g(and)h(c)-5 b(an)31 b(ac)-5 b(cumulate)32 b(only)d(at)h(p)-5 b(oints)30 b(b)-5 b(elonging)31 b(to)f Fw(\034)10 b Fv(\()p Fw(H)d Fv(\))p Fr(,)31 b(and)f Fw(\026)3238 4879 y Fs(A)3294 4912 y Fv(\()p Fw(H)7 b Fv(\))31 b Fr(is)e(e)-5 b(qual)83 5025 y(to)32 b Fv(I)-18 b(R)20 b Fu(n)h Fw(\024)p Fv(\()p Fw(H)7 b Fv(\))p Fr(.)42 b(Mor)-5 b(e)g(over,)34 b(for)e(e)-5 b(ach)34 b Fw(\025)25 b Fu(2)g Fv(I)-18 b(R)o Fr(,)1177 5229 y Fv(~)-53 b Fw(\045)1216 5192 y Fs(A)1216 5252 y(H)1283 5229 y Fv(\()p Fw(\025)p Fv(\))26 b(=)f(inf)6 b Fu(f)p Fw(\025)20 b Fu(\000)g Fw(\026)1935 5152 y Fl(\014)1935 5206 y(\014)1998 5229 y Fw(\026)25 b Fu(2)g Fw(\034)10 b Fv(\()p Fw(H)d Fv(\))p Fw(;)15 b(\026)26 b Fu(\024)f Fw(\025)p Fu(g)83 5433 y Fr(with)32 b(the)h(c)-5 b(onvention)33 b(that)f(the)h(in\034mum)e(over)j(an)e(empty)g(set)h(is)f Fv(+)p Fu(1)p Fr(.)1880 5836 y Fx(12)p eop %%Page: 13 13 bop 249 482 a Fr(Pr)-5 b(o)g(of:)51 b Fx(a\))35 b(W)-8 b(e)36 b(b)s(egin)g(with)g(some)f(preliminary)i(observ)-5 b(ations.)56 b(One)36 b(notices)g(that)g(for)f(eac)m(h)h Fw(\013)e Fu(2)83 594 y(L)p Fx(,)41 b Fw(\034)10 b Fv(\()p Fw(H)380 561 y Fs(\013)429 594 y Fv(\))40 b(=)f Fu([)675 609 y Fs(\014)s(<\013)852 594 y Fw(\033)904 608 y Fs(p)944 594 y Fv(\()p Fw(H)1062 561 y Fs(\014)1109 594 y Fv(\))g Fx(and)h Fw(\024)p Fv(\()p Fw(H)1538 561 y Fs(\013)1588 594 y Fv(\))g(=)f Fu([)1834 609 y Fs(\014)s FC(\024)p Fs(\013)2011 594 y Fw(\033)2063 608 y Fs(p)2102 594 y Fv(\()p Fw(H)2220 561 y Fs(\014)2268 594 y Fv(\))g Fx(are)g(coun)m(table)i (sets.)65 b(F)-8 b(urthermore,)43 b(if)91 707 y Fv(~)-53 b Fw(\045)130 674 y Fs(A)183 651 y Fn(\013)130 734 y Fs(H)193 715 y Fn(\013)241 707 y Fv(\()p Fw(\025)p Fv(\))26 b Fw(>)f Fv(0)j Fx(for)f(all)g Fw(\025)f Fu(2)e Fv(I)-18 b(R)14 b Fu(n)g Fw(\034)c Fv(\()p Fw(H)1303 674 y Fs(\013)1353 707 y Fv(\))p Fx(,)28 b(then)g Fw(\033)1696 721 y Fs(p)1736 707 y Fv(\()p Fw(H)1854 674 y Fs(\013)1903 707 y Fv(\))g Fx(can)f(only)h(accum)m(ulate)h (at)e(p)s(oin)m(ts)h(of)f Fw(\034)10 b Fv(\()p Fw(H)3439 674 y Fs(\013)3489 707 y Fv(\))p Fx(,)28 b(since)83 820 y(otherwise)j(it)g(w)m (ould)g(con)m(tradict)h Fr(i\))e Fx(of)g(Prop)s(osition)h(5.3.)41 b(Since)31 b Fw(\024)p Fv(\()p Fw(H)2539 787 y Fs(\013)2589 820 y Fv(\))g Fx(is)e(equal)i(to)f Fw(\034)10 b Fv(\()p Fw(H)3262 787 y Fs(\013)3312 820 y Fv(\))21 b Fu([)e Fw(\033)3500 834 y Fs(p)3540 820 y Fv(\()p Fw(H)3658 787 y Fs(\013)3708 820 y Fv(\))p Fx(,)83 933 y(this)30 b(implies)g(that)g(if)g Fw(\034)10 b Fv(\()p Fw(H)1011 900 y Fs(\013)1061 933 y Fv(\))30 b Fx(is)f(closed)h(and) 39 b Fv(~)-54 b Fw(\045)1704 900 y Fs(A)1757 877 y Fn(\013)1704 960 y Fs(H)1767 941 y Fn(\013)1816 933 y Fv(\()p Fw(\025)p Fv(\))26 b Fw(>)f Fv(0)30 b Fx(for)f(all)h Fw(\025)c Fu(2)f Fv(I)-18 b(R)18 b Fu(n)i Fw(\034)10 b Fv(\()p Fw(H)2896 900 y Fs(\013)2946 933 y Fv(\))p Fx(,)30 b(then)g Fw(\024)p Fv(\()p Fw(H)3411 900 y Fs(\013)3462 933 y Fv(\))f Fx(is)h(also)83 1046 y(closed.)249 1159 y(b\))f(The)g(pro)s(of)f(is)g(going)h(to)f(b)s(e)g(p) s(erformed)h(b)m(y)g(induction)h(o)m(v)m(er)f Fu(L)p Fx(.)40 b(W)-8 b(e)28 b(ha)m(v)m(e)i(already)f(noticed)g(that)83 1272 y(for)k(eac)m(h)i Fw(\013)c Fu(2)g(L)p Fx(,)j Fw(V)807 1239 y Fs(\013)890 1272 y Fx(is)f(a)h(cartesian)g(p)s(oten)m(tial)h(relativ)m(e)g (to)f Fu(L)2335 1239 y Fs(\013)2418 1272 y Fx(and)g Fw(H)2680 1239 y Fs(\013)2763 1272 y Fx(is)f(of)g(class)g Fw(C)3250 1239 y Ft(1)3243 1294 y Fs(u)3289 1272 y Fv(\()p Fw(A)3392 1239 y Fs(\013)3442 1272 y Fv(\))p Fx(.)51 b(First,)83 1385 y(for)34 b Fu(j)p Fw(\013)p Fu(j)f Fv(=)f(0)p Fx(,)j Fw(H)657 1352 y Fs(\013)741 1385 y Fx(is)f(equal)g(to)h(the)g(real)f(n)m(um)m(b)s(er)i Fw(V)1939 1352 y Fs(\013)1988 1385 y Fx(,)f Fw(\034)10 b Fv(\()p Fw(H)2216 1352 y Fs(\013)2266 1385 y Fv(\))32 b(=)g Fu(;)j Fx(and)f Fw(\024)p Fv(\()p Fw(H)2865 1352 y Fs(\013)2916 1385 y Fv(\))e(=)f Fu(f)p Fw(V)3204 1352 y Fs(\013)3254 1385 y Fu(g)p Fx(.)52 b(Moreo)m(v)m(er,)91 1498 y Fv(~)-53 b Fw(\045)130 1465 y Ft(0)130 1525 y Fs(H)193 1506 y Fn(\013)241 1498 y Fv(\()p Fw(\025)p Fv(\))45 b(=)e(+)p Fu(1)d Fx(for)h(all)h Fw(\025)h Fu(2)g Fv(I)-18 b(R)p Fx(,)43 b(while)f Fw(\045)1655 1465 y Ft(0)1655 1525 y Fs(H)1718 1506 y Fn(\013)1767 1498 y Fv(\()p Fw(V)1875 1465 y Fs(\013)1925 1498 y Fv(\))h(=)g(0)f Fx(and)f Fw(\045)2437 1465 y Ft(0)2437 1525 y Fs(H)2500 1506 y Fn(\013)2549 1498 y Fv(\()p Fw(\025)p Fv(\))j(=)f(+)p Fu(1)e Fx(for)g Fw(\025)i Fu(6)p Fv(=)g Fw(V)3465 1465 y Fs(\013)3515 1498 y Fx(.)72 b(The)83 1611 y(prop)s(osition)29 b(is)f(th)m(us)h(v)m(eri\034ed)h(in)e(this) h(sp)s(ecial)f(case.)40 b(Next,)28 b(for)h(an)m(y)g(\034xed)f Fw(\013)p Fx(,)h Fu(j)p Fw(\013)p Fu(j)e(6)p Fv(=)e(0)p Fx(,)k(w)m(e)g (assume)f(that)83 1724 y(the)33 b(assertions)f(of)g(the)g(prop)s(osition)i (are)e(pro)m(v)m(ed)i(for)e(eac)m(h)h Fw(H)2272 1691 y Fs(\014)2351 1724 y Fx(with)g Fw(\014)27 b Fk(l)21 b Fw(\013)33 b Fx(and)g(w)m(e)g(pro)m (v)m(e)g(it)f(for)h Fw(H)3694 1691 y Fs(\013)3743 1724 y Fx(.)83 1836 y(F)-8 b(or)31 b(simplicit)m(y)g(but)g(with)g(no)g(loss)e(of)h (generalit)m(y)-8 b(,)33 b(w)m(e)e(c)m(ho)s(ose)g Fw(\013)25 b Fv(=)g Fw(o)p Fx(.)249 1949 y(c\))h(W)-8 b(e)25 b(start)h(b)m(y)f (determining)i Fw(\045)1376 1916 y Fs(A)1376 1976 y(H)1434 1988 y Fn(\014)1505 1949 y Fx(for)e(eac)m(h)i Fw(\014)15 b Fk(l)10 b Fw(o)p Fx(.)39 b(If)24 b Fw(j)31 b Fu(2)25 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)i(;)e(n)p Fu(g)26 b Fx(suc)m(h)g(that)g Fw(\014)3190 1963 y Fs(j)3252 1949 y Fu(6)p Fv(=)f(0)p Fx(,)h(then)g Fw(H)3721 1964 y Fs(\014)83 2085 y Fx(is)i(equal)h(to)f Fu(\000)p Fv(\001)664 2099 y Fs(j)716 2085 y Fu(\012)16 b Fw(I)23 b Fv(+)16 b Fw(I)24 b Fu(\012)15 b Fw(H)1186 2052 y Fs(\014)1234 2085 y Fx(.)39 b(The)29 b Fw(\045)p Fx(-function)g(for)f(op)s(erators)h(of)f(this) g(form)h(is)f(extensiv)m(ely)g(studied)83 2232 y(in)37 b([1)q(],)i(Theorem)g (8.3.6)f(:)54 b Fw(\045)1093 2199 y Fs(A)1093 2259 y(H)1151 2271 y Fn(\014)1196 2232 y Fv(\()p Fw(\025)p Fv(\))38 b(=)f(inf)1575 2247 y Fs(\025)1616 2256 y Fm(1)1651 2247 y Ft(+)p Fs(\025)1747 2256 y Fm(2)1782 2247 y Ft(=)p Fs(\025)1897 2131 y Fl(n)1958 2232 y Fw(\045)2005 2199 y Fs(A)2058 2175 y Fn(\014)2005 2267 y Fs(H)2068 2248 y Fn(\014)2113 2232 y Fv(\()p Fw(\025)2201 2246 y Ft(1)2241 2232 y Fv(\))21 b(+)f Fw(\045)2435 2199 y Fs(A)2488 2175 y Fn(\014)2526 2155 y Fq(?)2435 2259 y FC(\000)p Ft(\001)2549 2269 y Fn(j)2585 2232 y Fv(\()p Fw(\025)2673 2246 y Ft(2)2713 2232 y Fv(\))2748 2131 y Fl(o)2809 2232 y Fx(.)61 b(But)38 b Fw(\045)3129 2199 y Fs(A)3182 2175 y Fn(\014)3220 2155 y Fq(?)3129 2259 y FC(\000)p Ft(\001)3243 2269 y Fn(j)3279 2232 y Fv(\()p Fw(\025)3367 2246 y Ft(2)3407 2232 y Fv(\))g(=)e Fu(1)h Fx(if)83 2411 y Fw(\025)136 2425 y Ft(2)211 2411 y Fw(<)f Fv(0)g Fx(and)h Fw(\045)628 2378 y Fs(A)681 2355 y Fn(\014)719 2334 y Fq(?)628 2438 y FC(\000)p Ft(\001)742 2448 y Fn(j)779 2411 y Fv(\()p Fw(\025)867 2425 y Ft(2)907 2411 y Fv(\))f(=)f Fw(\025)1137 2425 y Ft(2)1213 2411 y Fx(if)h Fw(\025)1355 2425 y Ft(2)1430 2411 y Fu(\025)g Fv(0)p Fx(,)i(and)f(hence)g Fw(\045)2131 2378 y Fs(A)2131 2438 y(H)2189 2450 y Fn(\014)2235 2411 y Fv(\()p Fw(\025)p Fv(\))f(=)g(inf)2611 2426 y Fs(\026)p FC(\024)p Fs(\025)2769 2310 y Fl(n)2830 2411 y Fw(\045)2877 2378 y Fs(A)2930 2355 y Fn(\014)2877 2446 y Fs(H)2940 2427 y Fn(\014)2986 2411 y Fv(\()p Fw(\026)p Fv(\))20 b(+)g(\()p Fw(\025)h Fu(\000)f Fw(\026)p Fv(\))3512 2310 y Fl(o)3572 2411 y Fx(.)59 b(By)83 2570 y(assumption,)52 b Fv(~)-54 b Fw(\045)645 2537 y Fs(A)698 2513 y Fn(\014)645 2605 y Fs(H)708 2586 y Fn(\014)754 2570 y Fv(\()p Fw(\025)p Fv(\))42 b(=)g(inf)5 b Fu(f)p Fw(\025)27 b Fu(\000)g Fw(\026)1459 2492 y Fl(\014)1459 2547 y(\014)1530 2570 y Fw(\026)41 b Fu(2)g Fw(\034)10 b Fv(\()p Fw(H)1896 2537 y Fs(\014)1943 2570 y Fv(\))p Fw(;)15 b(\026)43 b Fu(\024)e Fw(\025)p Fu(g)p Fx(,)i(and)d(so)48 b Fv(~)-53 b Fw(\045)2747 2537 y Fs(A)2800 2513 y Fn(\014)2747 2605 y Fs(H)2810 2586 y Fn(\014)2895 2570 y Fx(is)40 b(zero)g(on)h Fw(\034)10 b Fv(\()p Fw(H)3500 2537 y Fs(\014)3547 2570 y Fv(\))40 b Fx(and)83 2692 y(strictly)e(p)s(ositiv)m(e)g(on)g Fv(I)-18 b(R)25 b Fu(n)h Fw(\034)10 b Fv(\()p Fw(H)1221 2659 y Fs(\014)1269 2692 y Fv(\))38 b Fx(\(since)g Fw(\034)10 b Fv(\()p Fw(H)1774 2659 y Fs(\014)1822 2692 y Fv(\))38 b Fx(is)f(assumed)i(to)f(b)s(e)f (closed\).)64 b(Th)m(us,)41 b(in)d(view)g(of)g Fr(ii\))83 2814 y Fx(of)f(Prop)s(osition)h(5.3,)i Fw(\045)913 2781 y Fs(A)966 2758 y Fn(\014)913 2849 y Fs(H)976 2830 y Fn(\014)1022 2814 y Fv(\()p Fw(\025)p Fv(\))d(=)g(0)h Fx(if)f Fw(\025)g Fu(2)f Fw(\024)p Fv(\()p Fw(H)1820 2781 y Fs(\014)1868 2814 y Fv(\))h Fx(and)h 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Fv(\))p Fx(.)40 b(T)-8 b(o)28 b(c)m(hec)m(k)h(this)e(assertion,)h(w)m(e)g(\034rst)g(observ)m (e)f(that)h(if)f Fw(U)37 b Fx(is)27 b(a)h(self-adjoin)m(t)g(op)s(erator)g(in) 83 1037 y Fu(H)160 1051 y Fs(\013)233 1037 y Fx(of)23 b(class)g Fw(C)606 1004 y Ft(1)p Fs(;)p Ft(1)700 1037 y Fv(\()p Fw(A)803 1004 y Fs(\013)853 1037 y Fv(\))h Fx(for)f(some)h Fw(\013)h Fu(2)g(L)p Fx(,)g(then)f Fw(U)33 b Fx(is)23 b(also)h(of)f(class)g Fw(C)2471 1004 y Ft(1)p Fs(;)p Ft(1)2565 1037 y Fv(\()p Fw(A)p Fv(\))p Fx(.)39 b(F)-8 b(urthermore,)27 b(the)d(pro)s(duct)83 1150 y(of)37 b(a)g(\034nite)g(n)m(um)m(b)s(er)i(of)d(b)s(ounded)i(p)s(oten)m (tials)g(of)f(class)f Fw(C)2157 1117 y Ft(1)p Fs(;)p Ft(1)2251 1150 y Fv(\()p Fw(A)p Fv(\))i Fx(b)s(elongs)f(to)g(the)g(same)h(class)e (\(this)h(is)83 1263 y(easily)d(pro)m(v)m(ed)h(b)m(y)f(using)g(Lemma)h(6.2.1) g(and)f(Prop)s(osition)h(5.2.3)f(of)g([1]\).)51 b(Finally)-8 b(,)36 b(b)s(oth)e(requiremen)m(t)83 1376 y(of)f(de\034nition)j(4.3)e(are)g (satis\034ed)g(b)m(y)g(the)g(p)s(oten)m(tial)h Fw(V)20 b Fx(,)35 b(whic)m(h)g(is)e(therefore)i(cartesian.)51 b(Note)34 b(that)h(the)83 1489 y(example)c(giv)m(en)g(in)g(the)f(In)m(tro)s(duction)i(is)e(of)g(this)g (t)m(yp)s(e.)83 1774 y Fy(6)131 b(The)44 b(Asymptotic)h(V)-11 b(elo)t(cit)l(y)83 1977 y Fx(In)32 b(this)g(section,)h(w)m(e)g(pro)m(v)m(e)h (the)e(existence)h(of)f(the)g(asymptotic)h(v)m(elo)s(cit)m(y)g(and)g(state)f (some)g(of)g(its)g(prop-)83 2090 y(erties.)46 b(This)32 b(v)m(elo)s(cit)m(y)h (is)e(going)i(to)f(pla)m(y)h(an)f(essen)m(tial)h(role)f(in)h(the)f (de\034nition)i(of)d(the)i(w)m(a)m(v)m(e)g(op)s(erators.)83 2203 y(W)-8 b(e)31 b(refer)f(to)h([4])f(for)g(further)h(commen)m(ts)h(and)f (other)g(applications.)83 2409 y Fj(Prop)s(osition)i(6.1)45 b Fr(L)-5 b(et)38 b Fw(V)58 b Fr(b)-5 b(e)38 b(a)g(c)-5 b(artesian)38 b(p)-5 b(otential)38 b(such)g(that)g Fw(H)j Fv(=)34 b Fu(\000)p Fv(\001)24 b(+)f Fw(V)58 b Fr(is)37 b(of)g(class)h Fw(C)3563 2376 y Ft(1)3556 2431 y Fs(u)3602 2409 y Fv(\()p Fw(A)p Fv(\))p Fr(.)83 2521 y(Assume)33 b(also)f(that)h(for)f(e)-5 b(ach)33 b Fw(j)e Fu(2)25 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)j(;)d(n)p Fu(g)32 b Fr(and)h(for)f(some)g(asymptotic)g(lo)-5 b(c)g(alization)33 b(function)f Fw(\030)t Fr(,)1089 2645 y Fl(Z)1180 2671 y FC(1)1140 2851 y Ft(0)1270 2637 y Fl(\015)1270 2692 y(\015)1270 2746 y(\015)1270 2801 y(\015)1321 2769 y Fw(\030)1380 2641 y Fl(\022)1447 2769 y Fu(\006)1528 2707 y Fw(Q)1600 2721 y Fs(j)p 1528 2748 109 4 v 1560 2831 a Fw(r)1646 2641 y Fl(\023)1728 2769 y Fu(r)1804 2783 y Fs(j)1840 2769 y Fw(V)20 b Fv(\()p Fw(Q)p Fv(\))2055 2637 y Fl(\015)2055 2692 y(\015)2055 2746 y(\015)2055 2801 y(\015)2106 2859 y FC(H)2166 2841 y Fm(2)2201 2859 y FC(!H)2332 2841 y Fq(\000)p Fm(2)2434 2769 y Fw(dr)28 b(<)d Fu(1)p Fw(:)891 b Fx(\(5\))83 3015 y Fr(Then,)199 3197 y(i\))45 b(ther)-5 b(e)27 b(exists)f(a)g Fw(n)p Fr(-tuple)g Fu(P)33 b Fr(of)25 b(c)-5 b(ommuting)27 b(self-adjoint)e(op)-5 b(er)g(ators)27 b(such)f(that)g(for)f (al)5 b(l)26 b Fw(f)34 b Fu(2)25 b Fw(C)3502 3211 y Ft(0)3541 3197 y Fv(\(I)-18 b(R)3658 3161 y Fs(n)3705 3197 y Fv(\))p Fr(:)1330 3444 y(s)20 b Fu(\000)68 b Fv(lim)1478 3500 y Fs(t)p FC(!)p Ft(+)p FC(1)1715 3444 y Fw(e)1757 3407 y Fs(iH)5 b(t)1874 3444 y Fw(f)1943 3316 y Fl(\022)2024 3383 y Fw(Q)p 2020 3423 79 4 v 2020 3507 a Fv(2)p Fw(t)2109 3316 y Fl(\023)2191 3444 y Fw(e)2233 3407 y FC(\000)p Fs(iH)g(t)2430 3444 y Fv(=)25 b Fw(f)10 b Fv(\()p Fu(P)d Fv(\))p Fw(;)172 3728 y Fr(ii\))44 b(for)31 b(e)-5 b(ach)33 b Fw(\021)c Fu(2)c Fw(C)879 3742 y Ft(0)918 3728 y Fv(\(I)-18 b(R\))31 b Fr(and)g(e)-5 b(ach)33 b Fw(f)i Fu(2)25 b Fw(C)1708 3742 y Ft(0)1747 3728 y Fv(\(I)-18 b(R)1864 3692 y Fs(n)1911 3728 y Fv(\))p Fr(,)31 b Fv([)p Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(;)15 b(f)10 b Fv(\()p Fu(P)d Fv(\)])28 b(=)d(0)p Fr(,)32 b(i.e.)f(the)g(asymptotic)h(velo)-5 b(city)309 3841 y Fu(P)40 b Fr(c)-5 b(ommutes)33 b(with)f(the)h(Hamiltonian)e Fw(H)7 b Fr(,)144 4026 y(iii\))44 b(the)34 b(subsp)-5 b(ac)g(e)37 b(of)c(the)i(states)f(with)f(zer)-5 b(o)35 b(asymptotic)e(velo)-5 b(city)36 b(is)d(e)-5 b(qual)36 b(to)d(the)i(subsp)-5 b(ac)g(e)36 b(sp)-5 b(anne)g(d)309 4139 y(by)33 b(the)f(eigenve)-5 b(ctors)35 b(of)d Fw(H)7 b Fr(.)83 4345 y Fx(Since)32 b(the)g(limit)g Fw(t)27 b Fu(!)g(\0001)k Fx(could)h(b)s(e)f(handled)i(similarly)-8 b(,)32 b(w)m(e)h(simply)e(do)h(not)g(consider)g(it.)44 b(The)32 b(en)m(tire)83 4458 y(section)f(is)f(dev)m(oted)i(to)e(the)h(pro)s(of)f(of)g (this)h(prop)s(osition)g(and)g(therefore,)g(unless)f(otherwise)i(stated,)f (it)f(is)83 4571 y(alw)m(a)m(ys)25 b(assumed)f(that)h(the)f(p)s(oten)m(tial)i Fw(V)44 b Fx(satis\034es)23 b(its)h(h)m(yp)s(otheses.)39 b(The)24 b(pro)s(ofs)g(in)m(v)m(olv)m(e)h(a)f(considerable)83 4684 y(n)m(um)m(b)s(er)k (of)e(comm)m(utator)i(computations,)h(whic)m(h)f(will)f(of)e(course)i(not)g (b)s(e)f(presen)m(ted)h(in)g(full)f(details.)40 b(W)-8 b(e)83 4797 y(start)29 b(with)g(some)g(considerations)h(on)f(the)g(notations)h(and)f (with)g(a)g(tec)m(hnical)h(lemma)g(that)f(will)g(b)s(e)g(used)83 4910 y(freely)h(subsequen)m(tly)-8 b(.)249 5023 y(Let)29 b(us)f(consider)h (an)f(op)s(erator-v)-5 b(alued)30 b(mapping)f Fv(\010)c(:)h([1)p Fw(;)15 b Fu(1)p Fv(\))27 b Fu(3)d Fw(t)h Fu(7!)h Fv(\010\()p Fw(t)p Fv(\))f Fu(2)g(B)s Fv(\()p Fu(H)q Fv(\))p Fx(.)39 b(If)28 b(there)h(exists)83 5135 y(some)22 b(constan)m(t)g Fw(c)k(<)f Fu(1)c Fx(suc)m(h)h(that)g Fu(k)p Fv(\010\()p Fw(t)p Fv(\))p Fu(k)k(\024)f Fw(c)d Fx(for)f(all)h Fw(t)j Fu(\025)g Fv(1)p Fx(,)e(then)f Fv(\010)f Fx(is)g(said)g(to)h(b)s(e)f(a)h Fr(b)-5 b(ounde)g(d)23 b Fx(op)s(erator-)83 5248 y(v)-5 b(alued)35 b(mapping.)56 b(W)-8 b(e)35 b(write)h Fv(\010)c Fu(2)h Fw(B)5 b(C)1526 5215 y Ft(1)1564 5175 y Fl(\000)1606 5248 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))2093 5175 y Fl(\001)2171 5248 y Fx(if)34 b(the)h(mapping)i(is)d(b)s(ounded)i(and)f (di\033eren-)83 5361 y(tiable)27 b(in)g(norm)g(with)f(b)s(ounded)h(deriv)-5 b(ativ)m(e.)41 b(Let)27 b Fw(m)e Fx(b)s(e)h(an)m(y)h(p)s(ositiv)m(e)g(in)m (teger)g(and)g Fw(t)e Fu(2)g Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fx(.)40 b(Then)27 b Fv(\010)83 5401 y Fl(\000)125 5474 y Fx(or)32 b(b)m(y)h(a)g(sligh)m(t)g(abuse)g(of)f(notation)i Fv(\010\()p Fw(t)p Fv(\))1581 5401 y Fl(\001)1655 5474 y Fx(b)s(elongs)f(to)g Fw(L)2157 5441 y Fs(m)2223 5401 y Fl(\000)2265 5474 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)2631 5401 y Fl(\001)2707 5474 y Fx(if)32 b Fu(k)p Fv(\010\()p Fu(\001)p Fv(\))p Fu(k)f(2)d Fw(L)3225 5441 y Fs(m)3292 5401 y Fl(\000)3333 5474 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)3699 5401 y Fl(\001)3743 5474 y Fx(,)83 5587 y(to)34 b Fw(o)p Fv(\()p Fw(t)309 5554 y FC(\000)p Fs(m)431 5587 y Fv(\))f Fx(if)h Fv(lim)712 5601 y Fs(t)p FC(!1)898 5587 y Fw(t)931 5554 y Fs(m)997 5587 y Fu(k)p Fv(\010\()p Fw(t)p Fv(\))p Fu(k)e Fv(=)f(0)p Fx(,)k(or)e(to)h Fw(O)s Fv(\()p Fw(t)1862 5554 y FC(\000)p Fs(m)1984 5587 y Fv(\))g Fx(if)f Fw(t)2172 5554 y Fs(m)2238 5587 y Fu(k)p Fv(\010\()p Fw(t)p Fv(\))p Fu(k)f(\024)f Fw(c)g(<)f Fu(1)k Fx(for)f(all)h Fw(t)d Fu(\025)f Fv(1)p Fx(.)51 b(W)-8 b(e)34 b(sa)m(y)1880 5836 y(14)p eop %%Page: 15 15 bop 83 482 a Fx(that)34 b Fv(\010)f Fx(is)g Fr(inte)-5 b(gr)g(able)37 b(along)f(the)g(evolution)e Fx(\(with)g(resp)s(ect)f(to)h Fw(H)7 b Fx(\))33 b(if)g(there)h(exists)f(a)h(constan)m(t)g Fw(c)d(<)f Fu(1)83 594 y Fx(suc)m(h)h(that)483 521 y Fl(R)543 547 y FC(1)526 626 y Ft(1)633 594 y Fu(jh)p Fw(e)735 561 y FC(\000)p Fs(iH)5 b(t)908 594 y Fw(';)15 b Fv(\010\()p Fw(t)p Fv(\))p Fw(e)1218 561 y FC(\000)p Fs(iH)5 b(t)1391 594 y Fw(')p Fu(ij)p Fw(dt)26 b Fu(\024)f Fw(c)p Fu(k)p Fw(')p Fu(k)1900 561 y Ft(2)1972 594 y Fx(for)30 b(all)g Fw(')c Fu(2)f(H)q Fx(.)249 707 y(F)-8 b(or)31 b(eac)m(h)g Fw(\013)26 b Fu(2)f(L)30 b Fx(w)m(e)h(de\034ne)g(the)g (op)s(en)f(subset)h(of)e Fv(I)-18 b(R)2107 672 y Fs(n)2154 707 y Fx(:)960 912 y Fw(Y)1013 926 y Fs(\013)1088 912 y Fv(:=)25 b Fu(f)p Fw(x)g Fu(2)g Fv(I)-18 b(R)1499 874 y Fs(n)1576 834 y 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b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)i(;)e(n)p Fu(g)p Fr(,)47 b(e)-5 b(ach)45 b Fw(\013)i Fu(2)f(L)d Fr(and)h(al)5 b(l)44 b Fw(t)i Fu(\025)g Fv(1)p Fr(,)h(the)83 1893 y(fol)5 b(lowing)32 b(statements)g(ar)-5 b(e)33 b(true:)199 2081 y(i\))45 b(if)34 b Fw(f)44 b Fr(is)34 b(a)h(twic)-5 b(e)35 b(di\033er)-5 b(entiable)37 b(function)d(on)h Fv(I)-18 b(R)2009 2045 y Fs(n)2090 2081 y Fr(with)35 b(b)-5 b(ounde)g(d)36 b(derivatives)g(of)f (or)-5 b(der)36 b(0,)e(1)h(and)309 2216 y(2,)d(then)616 2115 y Fl(h)659 2216 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(;)15 b(f)970 2115 y Fl(\020)1037 2175 y Fs(Q)p 1035 2195 61 4 v 1035 2247 a Ft(2)p Fs(t)1105 2115 y Fl(\021)q(i)1218 2216 y Fv(\()p Fw(H)27 b Fv(+)20 b Fw(i)p Fv(\))26 b Fu(2)f Fw(O)s Fv(\()p Fw(t)1765 2183 y FC(\000)p Ft(1)1859 2216 y Fv(\))p Fr(,)172 2463 y(ii\))44 b(if)32 b Fw(f)i Fu(2)25 b Fw(C)634 2430 y Ft(2)627 2485 y Fs(c)673 2463 y Fv(\(I)-18 b(R)790 2427 y Fs(n)837 2463 y Fv(\))32 b Fr(with)g(supp)-5 b(ort)33 b(in)f Fw(Y)1603 2425 y 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Fl(\001)3242 2709 y Fu(\\)20 b Fw(o)p Fv(\()p Fw(t)3435 2676 y Ft(0)3474 2709 y Fv(\))p Fr(,)158 2956 y(iv\))45 b(if)32 b Fw(f)i Fu(2)25 b Fw(C)627 2970 y Fs(c)662 2956 y Fv(\(I)-18 b(R)778 2920 y Fs(n)825 2956 y Fv(\))33 b Fr(with)f(supp)-5 b(ort)33 b(in)f Fw(Y)1572 2970 y Fs(\013)1621 2956 y Fr(,)g(then)h Fw(f)1951 2855 y Fl(\020)2018 2915 y Fs(Q)p 2015 2935 V 2015 2987 a Ft(2)p Fs(t)2086 2855 y Fl(\021)2155 2882 y(\000)2197 2956 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))21 b Fu(\000)f Fw(\021)s Fv(\()p Fw(H)2669 2970 y Fs(\013)2719 2956 y Fv(\))2754 2882 y Fl(\001)2821 2956 y Fu(2)25 b Fw(o)p Fv(\()p Fw(t)3019 2923 y Ft(0)3059 2956 y Fv(\))p Fr(,)185 3181 y(v\))46 b(the)33 b(op)-5 b(er)g(ator)34 b Fv(\()p Fw(P)907 3195 y Fs(j)964 3181 y Fv(+)20 b Fw(i)p Fv(\))p Fw(Q)1193 3195 y Fs(j)1230 3181 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\)\(1)22 b(+)e Fw(Q)1696 3148 y Ft(2)1736 3181 y Fv(\))1771 3148 y FC(\000)p Ft(1)p Fs(=)p Ft(2)1968 3181 y Fr(b)-5 b(elongs)34 b(to)e Fu(B)s Fv(\()p Fu(H)q Fv(\))p Fr(.)83 3394 y Fx(The)27 b(pro)s(of)f(of)g(this)h(lemma)g(is)f (giv)m(en)h(in)g(the)g(App)s(endix.)39 b(Let)27 b(us)f(ho)m(w)m(ev)m(er)i (men)m(tion)h(that)e(the)f(statemen)m(t)83 3506 y Fr(i\))e Fx(requires)h(only)g(the)g(h)m(yp)s(othesis)g(that)h Fw(V)45 b Fx(b)s(e)24 b Fv(\001)p Fx(-b)s(ounded)i(with)f(relativ)m(e)h(b)s(ound)f (less)f(than)i(one.)39 b(In)24 b(line)83 3619 y(with)i Fr(iv\))p Fx(,)g(one)f(could)h(also)f(pro)m(v)m(e)h(some)f(anisotropic)h (non-propagation)i(estimates)d(at)g(suitable)h(energies.)83 3732 y(More)31 b(general)g(results)f(of)g(this)g(t)m(yp)s(e)h(can)g(b)s(e)f (found)g(in)h([2].)249 3845 y(F)-8 b(or)23 b(eac)m(h)g(op)s(erator-v)-5 b(alued)24 b(mapping)f Fv(\010)i Fu(2)g Fw(B)5 b(C)1924 3812 y Ft(1)1962 3772 y Fl(\000)2004 3845 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))2491 3772 y Fl(\001)2556 3845 y Fx(w)m(e)23 b(de\034ne)g(its)f Fr(Heisenb)-5 b(er)g(g)27 b(deriva-)83 3958 y(tive)k Fb(D)p Fv(\010)f Fx(:)40 b(let)31 b Fw(';)15 b( )34 b Fx(in)d Fu(H)1002 3925 y Ft(2)1071 3958 y Fx(\(the)g(domain)g(of)f Fw(H)7 b Fx(\))30 b(and)h Fw(t)25 b Fu(\025)g Fv(1)p Fx(,)31 b(then)720 4205 y Fu(h)p Fw( )s(;)15 b Fb(D)p Fv(\010\()p Fw(t)p Fv(\))p Fw(')p Fu(i)27 b Fv(:=)f Fw(i)p Fu(h)p Fw(H)7 b( )s(;)15 b Fv(\010\()p Fw(t)p Fv(\))p Fw(')p Fu(i)22 b(\000)e Fw(i)p Fu(h)p Fw( )s(;)15 b Fv(\010\()p Fw(t)p Fv(\))p Fw(H)7 b(')p Fu(i)23 b Fv(+)d Fu(h)p Fw( )s(;)2768 4144 y(d)p 2752 4184 81 4 v 2752 4267 a(dt)2842 4205 y Fv(\010\()p Fw(t)p Fv(\))p Fw(')p Fu(i)p Fw(:)523 b Fx(\(6\))83 4449 y(One)33 b(notices)g(that)g Fu(h)p Fw( )s(;)15 b(e)957 4416 y Fs(iH)5 b(t)1075 4449 y Fb(D)p Fv(\010\()p Fw(t)p Fv(\))p Fw(e)1366 4416 y FC(\000)p Fs(iH)g(t)1538 4449 y Fw(')p Fu(i)33 b Fx(is)f(equal)h(to) 2134 4413 y Fs(d)p 2121 4428 62 4 v 2121 4480 a(dt)2193 4449 y Fu(h)p Fw( )s(;)15 b(e)2372 4416 y Fs(iH)5 b(t)2490 4449 y Fv(\010\()p Fw(t)p Fv(\))p Fw(e)2701 4416 y FC(\000)p Fs(iH)g(t)2873 4449 y Fw(')p Fu(i)33 b Fx(for)f(eac)m(h)i Fw(';)15 b( )33 b Fu(2)28 b(H)3704 4416 y Ft(2)3743 4449 y Fx(.)83 4562 y(If,)43 b(for)e(eac)m(h)h Fw(t)i Fu(\025)f Fv(1)p Fx(,)h Fb(D)p Fv(\010\()p Fw(t)p Fv(\))d Fx(extends)h(con)m(tin)m(uously)h(to)e(a)g(b)s(ounded)h(op)s (erator)g(\(w)m(e)g(k)m(eep)g(the)g(same)83 4675 y(notation)33 b(for)f(this)h(extension\),)g(and)f(if)g(there)h(exists)e(some)i(constan)m(t) g Fw(c)c(<)f Fu(1)k Fx(suc)m(h)g(that)h Fu(k)p Fb(D)p Fv(\010\()p Fw(t)p Fv(\))p Fu(k)d(\024)e Fw(c)83 4788 y Fx(for)35 b(all)h Fw(t)d Fu(\025)g Fv(1)p Fx(,)k(then)f(w)m(e)g(write)g Fv(\010)d Fu(2)h Fw(B)5 b(C)1559 4755 y Ft(1)1552 4815 y Fs(H)1618 4714 y Fl(\000)1660 4788 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))2147 4714 y Fl(\001)2190 4788 y Fx(.)55 b(In)35 b(our)h(applications)h Fv(\010\()p Fw(t)p Fv(\))e Fx(will)h(often)f(b)s(e)83 4932 y(equal)g Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(f)597 4831 y Fl(\020)664 4891 y Fs(Q)p 661 4911 61 4 v 661 4964 a Ft(2)p Fs(t)732 4831 y Fl(\021)802 4932 y Fw(\021)s Fv(\()p Fw(H)g Fv(\))35 b Fx(with)g Fw(\021)g Fu(2)d Fw(C)1493 4899 y FC(1)1486 4955 y Fs(c)1567 4932 y Fv(\(I)-18 b(R\))34 b Fx(and)h Fw(f)43 b Fx(a)35 b(b)s(ounded)g(function)g(de\034ned)g (on)g Fv(I)-18 b(R)3359 4897 y Fs(n)3406 4932 y Fx(.)52 b(Then)35 b(a)83 5077 y(su\036cien)m(t)h(condition)g(suc)m(h)g(that)f Fv(\010)d Fu(2)h Fw(B)5 b(C)1615 5044 y Ft(1)1608 5104 y Fs(H)1674 5003 y Fl(\000)1716 5077 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))2203 5003 y Fl(\001)2281 5077 y Fx(is)34 b(that)h Fw(f)42 b Fu(2)33 b Fw(C)2829 5044 y Ft(2)2867 5077 y Fv(\(I)-18 b(R)2984 5041 y Fs(n)3031 5077 y Fv(\))35 b Fx(and)h(that)f(all)g(\034rst)83 5190 y(order)c(partial)g(deriv) -5 b(ativ)m(es)31 b(of)f Fw(f)40 b Fx(ha)m(v)m(e)31 b(a)g(b)s(ounded)g(supp)s (ort.)249 5303 y(As)e(men)m(tioned)j(in)e(the)g(In)m(tro)s(duction,)i(w)m(e)e (shall)g(pro)m(v)m(e)h(v)-5 b(arious)30 b(propagation)i(estimates.)40 b(F)-8 b(or)31 b(this)83 5416 y(purp)s(ose,)g(w)m(e)h(review)g(t)m(w)m(o)g (standard)g(results)f(that)g(will)h(b)s(e)f(constan)m(tly)h(used)f(\(pro)s (ofs)f(can)i(b)s(e)e(found)i(in)83 5529 y(the)f(App)s(endix\).)1880 5836 y(15)p eop %%Page: 16 16 bop 83 482 a Fj(Lemma)35 b(6.3)45 b Fr(Consider)36 b(a)h(self-adjoint)g(op)-5 b(er)g(ator-value)g(d)40 b(mapping)d Fv(\010)g Fr(in)f Fw(B)5 b(C)2939 449 y Ft(1)2977 408 y Fl(\000)3019 482 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))3506 408 y Fl(\001)3549 482 y Fr(.)55 b(As-)83 594 y(sume)29 b(that)f(ther)-5 b(e)30 b(exist)f(some)f(b)-5 b(ounde)g(d)30 b(op)-5 b(er)g(ator-value)g(d)33 b(mappings)28 b Fw(B)5 b(;)15 b(L;)g(F)e Fr(,)29 b(with)e Fw(L)f Fu(2)e Fw(L)3277 561 y Ft(1)3317 521 y Fl(\000)3358 594 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)3724 521 y Fl(\001)83 707 y Fr(and)29 b Fw(F)42 b Fr(inte)-5 b(gr)g(able)31 b(along)f(the)g(evolution,) g(such)g(that)f(one)g(of)g(the)h(fol)5 b(lowing)28 b(two)h(ine)-5 b(qualities)30 b(is)f(satis\034e)-5 b(d)83 820 y(for)32 b(al)5 b(l)32 b Fw(t)25 b Fu(\025)g Fv(1)33 b Fr(and)f(e)-5 b(ach)34 b Fw(')26 b Fu(2)f(H)1215 787 y Ft(2)1254 820 y Fr(:)754 1024 y Fu(\006h)p Fw(';)15 b Fb(D)p Fv(\010\()p Fw(t)p Fv(\))p Fw(')p Fu(i)28 b(\025)d(h)p Fw(';)15 b(B)1634 987 y FC(\003)1674 1024 y Fv(\()p Fw(t)p Fv(\))p Fw(B)5 b Fv(\()p Fw(t)p Fv(\))p Fw(')p Fu(i)21 b Fv(+)f Fu(h)p Fw(';)15 b(F)e Fv(\()p Fw(t)p Fv(\))p Fw(')p Fu(i)23 b Fv(+)d Fu(h)p Fw(';)15 b(L)p Fv(\()p Fw(t)p Fv(\))p Fw(')p Fu(i)p Fw(:)83 1229 y Fr(Then)32 b(ther)-5 b(e)34 b(exists)f Fw(c)25 b(<)g Fu(1)32 b Fr(such)h(that)1463 1155 y Fl(R)1523 1182 y FC(1)1506 1261 y Ft(1)1613 1229 y Fu(k)p Fw(B)5 b Fv(\()p Fw(t)p Fv(\))p Fw(e)1877 1196 y FC(\000)p Fs(iH)g(t)2050 1229 y Fw(')p Fu(k)2154 1196 y Ft(2)2194 1229 y Fw(dt)26 b Fu(\024)f Fw(c)p Fu(k)p Fw(')p Fu(k)2584 1196 y Ft(2)2657 1229 y Fr(for)32 b(al)5 b(l)32 b Fw(')26 b Fu(2)f(H)3179 1196 y Ft(2)3218 1229 y Fr(.)83 1441 y Fj(Lemma)35 b(6.4)45 b Fr(Consider)33 b(an)h(op)-5 b(er)g(ator-value)g(d)38 b(mapping)33 b Fv(\010)h Fr(in)f Fw(B)5 b(C)2499 1408 y Ft(1)2538 1368 y Fl(\000)2579 1441 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))3066 1368 y Fl(\001)3109 1441 y Fr(,)34 b(and)g(let)g Fw(H)3552 1455 y Ft(1)3625 1441 y Fr(and)83 1554 y Fw(H)159 1568 y Ft(2)234 1554 y Fr(b)-5 b(e)38 b(two)e(self-adjoint)g (op)-5 b(er)g(ators)38 b(in)e Fu(H)h Fr(of)f(domain)g Fu(D)2125 1568 y Ft(1)2201 1554 y Fr(and)g Fu(D)2450 1568 y Ft(2)2526 1554 y Fr(r)-5 b(esp)g(e)g(ctively.)56 b(Assume)37 b(that)g(ther)-5 b(e)83 1667 y(exist)37 b(a)g(\034nite)f(inte)-5 b(ger)38 b Fw(N)46 b Fr(and)37 b(some)g(b)-5 b(ounde)g(d)39 b(op)-5 b(er)g(ator-value)g (d)40 b(mappings)d Fw(E)2908 1682 y Fs(k)2950 1667 y Fw(;)15 b(F)3048 1682 y Fs(k)3128 1667 y Fr(and)37 b Fw(L)f Fr(such)i(that)83 1780 y(for)32 b(al)5 b(l)32 b Fw(')26 b Fu(2)f(D)597 1794 y Ft(1)636 1780 y Fr(,)32 b Fw( )d Fu(2)c(D)940 1794 y Ft(2)1012 1780 y Fr(and)32 b(al)5 b(l)32 b Fw(t)25 b Fu(\025)g Fv(1)p Fr(:)147 2065 y Fu(j)p Fw(i)p Fu(h)p Fw(H)314 2079 y Ft(2)353 2065 y Fw( )s(;)15 b Fv(\010\()p Fw(t)p Fv(\))p Fw(')p Fu(i)23 b(\000)c Fw(i)p Fu(h)p Fw( )s(;)c Fv(\010\()p Fw(t)p Fv(\))p Fw(H)1244 2079 y Ft(1)1285 2065 y Fw(')p Fu(i)21 b Fv(+)f Fu(h)p Fw( )s(;)1656 2004 y(d)p 1639 2044 81 4 v 1639 2128 a(dt)1729 2065 y Fv(\010\()p Fw(t)p Fv(\))p Fw(')p Fu(ij)27 b(\024)2174 1952 y Fs(N)2140 1979 y Fl(X)2141 2177 y Fs(k)r Ft(=1)2286 2065 y Fu(k)p Fw(E)2398 2080 y Fs(k)2442 2065 y Fv(\()p Fw(t)p Fv(\))p Fw( )s Fu(kk)p Fw(F)2755 2080 y Fs(k)2800 2065 y Fv(\()p Fw(t)p Fv(\))p Fw(')p Fu(k)21 b Fv(+)f Fu(k)p Fw(L)p Fv(\()p Fw(t)p Fv(\))p Fu(kk)p Fw( )s Fu(kk)p Fw(')p Fu(k)p Fw(:)3653 2273 y Fx(\(7\))83 2484 y Fr(Assume)45 b(furthermor)-5 b(e)46 b(that)e Fw(L)g Fr(b)-5 b(elongs)46 b(to)e Fw(L)1746 2451 y Ft(1)1786 2410 y Fl(\000)1827 2484 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)2193 2410 y Fl(\001)2281 2484 y Fr(and)45 b(that)f(ther)-5 b(e)46 b(is)e(a)g(c)-5 b(onstant)45 b Fw(c)i(<)g Fu(1)83 2597 y Fr(such)d(that)f(for)h(e)-5 b(ach)44 b Fw(k)49 b Fu(2)c(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)i(;)e(N)10 b Fu(g)p Fr(,)1560 2524 y Fl(R)1620 2550 y FC(1)1602 2629 y Ft(1)1710 2597 y Fu(k)p Fw(E)1822 2612 y Fs(k)1865 2597 y Fv(\()p Fw(\034)g Fv(\))p Fw(e)2027 2564 y FC(\000)p Fs(iH)2164 2573 y Fm(2)2200 2564 y Fs(\034)2243 2597 y Fw( )s Fu(k)2350 2564 y Ft(2)2390 2597 y Fw(d\034)56 b Fu(\024)45 b Fw(c)2688 2564 y Ft(2)2728 2597 y Fu(k)p Fw( )s Fu(k)2880 2564 y Ft(2)2964 2597 y Fr(for)e(al)5 b(l)43 b Fw( )49 b Fu(2)44 b(D)3542 2611 y Ft(2)3625 2597 y Fr(and)83 2636 y Fl(R)144 2663 y FC(1)126 2742 y Ft(1)234 2710 y Fu(k)p Fw(F)337 2725 y Fs(k)380 2710 y Fv(\()p Fw(\034)10 b Fv(\))p Fw(e)542 2677 y FC(\000)p Fs(iH)679 2686 y Fm(1)715 2677 y Fs(\034)758 2710 y Fw(')p Fu(k)862 2677 y Ft(2)902 2710 y Fw(d\034)36 b Fu(\024)25 b Fw(c)1160 2677 y Ft(2)1199 2710 y Fu(k)p Fw(')p Fu(k)1348 2677 y Ft(2)1422 2710 y Fr(for)32 b(al)5 b(l)32 b Fw(')25 b Fu(2)g(D)1935 2724 y Ft(1)1975 2710 y Fr(.)41 b(Then)32 b(s)20 b Fu(\000)g Fv(lim)2553 2724 y Fs(t)p FC(!1)2739 2710 y Fw(e)2781 2677 y Fs(iH)2863 2686 y Fm(2)2898 2677 y Fs(t)2928 2710 y Fv(\010\()p Fw(t)p Fv(\))p Fw(e)3139 2677 y FC(\000)p Fs(iH)3276 2686 y Fm(1)3311 2677 y Fs(t)3373 2710 y Fr(exists.)83 2922 y Fx(In)25 b(most)h(of)f(our)h(applications)i Fw(H)1237 2936 y Ft(1)1301 2922 y Fx(and)e Fw(H)1548 2936 y Ft(2)1613 2922 y Fx(are)g(equal)f(to)h Fw(H)7 b Fx(,)27 b(and)f(therefore)g (the)g(l.h.s.)g(of)g(\(7\))f(is)h(nothing)83 3035 y(but)31 b Fu(jh)p Fw( )s(;)15 b Fb(D)p Fv(\010\()p Fw(t)p Fv(\))p Fw(')p Fu(ij)p Fx(.)249 3148 y(The)29 b(next)g(lemma)g(con)m(tains)h(t)m(w)m(o)g (statemen)m(ts)f(usually)g(called)g Fr(maximal)i(velo)-5 b(city)32 b(estimates)p Fx(.)39 b(Both)83 3261 y(are)23 b(pro)m(v)m(ed)h(under)f(the)f (single)h(assumption)g(that)g(the)g(p)s(oten)m(tial)h Fw(V)42 b Fx(b)s(e)22 b Fv(\001)p Fx(-b)s(ounded)h(with)g(relativ)m(e)g(b)s(ound)83 3374 y(less)30 b(than)h(one.)41 b(It)30 b(sligh)m(tly)g(extends)h(the)g(v)-5 b(alidit)m(y)31 b(of)f(similar)g(results)h(obtained)g(in)g([4].)249 3487 y(T)-8 b(o)35 b(shorten)g(some)f(equations)h(b)s(elo)m(w)g(and)f(when)h (the)g(con)m(text)g(lea)m(v)m(es)g(no)g(doubt,)h(the)e(argumen)m(ts)83 3600 y(of)c(certain)h(functions)g(are)g(not)f(rep)s(eated)i(all)e(along)h (the)g(pro)s(ofs.)83 3812 y Fj(Lemma)k(6.5)45 b Fr(F)-7 b(or)32 b(e)-5 b(ach)34 b Fw(\021)29 b Fu(2)c Fw(C)1232 3779 y FC(1)1225 3835 y Fs(c)1306 3812 y Fv(\(I)-18 b(R\))p Fr(,)32 b(ther)-5 b(e)34 b(exists)e(a)h(c)-5 b(onstant)32 b Fw(c)2466 3826 y Fs(\021)2534 3812 y Fw(>)25 b Fv(0)32 b Fr(with)g(the)h(pr)-5 b(op)g(erty)34 b(that)249 3925 y(i\))26 b(for)h(e)-5 b(ach)28 b Fw(f)34 b Fu(2)25 b Fw(C)913 3892 y FC(1)906 3948 y Fs(c)987 3925 y Fv(\(I)-18 b(R\))27 b Fr(with)f(supp)-5 b(ort)28 b(in)e Fv(\()p Fw(c)1849 3939 y Fs(\021)1892 3925 y Fw(;)15 b Fu(1)p Fv(\))p Fr(,)28 b(ther)-5 b(e)28 b(exists)f Fw(c)e(<)g Fu(1)i Fr(such)g(that)g(for)g(al)5 b(l)26 b Fw(')g Fu(2)f(H)q Fr(,)1101 4076 y Fl(Z)1192 4102 y FC(1)1152 4282 y Ft(1)1282 4068 y Fl(\015)1282 4122 y(\015)1282 4177 y(\015)1282 4231 y(\015)1333 4200 y Fw(f)1402 4071 y Fl(\022)1479 4138 y Fu(j)p Fw(Q)p Fu(j)p 1479 4179 123 4 v 1501 4262 a Fv(2)p Fw(t)1611 4071 y Fl(\023)1693 4200 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)1936 4162 y FC(\000)p Fs(iH)e(t)2109 4200 y Fw(')2168 4068 y Fl(\015)2168 4122 y(\015)2168 4177 y(\015)2168 4231 y(\015)2219 4094 y Ft(2)2284 4138 y Fw(dt)p 2284 4179 81 4 v 2308 4262 a(t)2399 4200 y Fu(\024)25 b Fw(c)p Fu(k)p Fw(')p Fu(k)2683 4162 y Ft(2)2724 4200 y Fw(;)249 4454 y Fr(ii\))35 b(if)h Fw(f)45 b Fr(is)36 b(a)g Fw(C)814 4421 y FC(1)888 4454 y Fr(-function)g(on)g Fv(I)-18 b(R)35 b Fr(with)h(supp)-5 b(ort)37 b(in)e Fv(\()p Fw(c)2240 4468 y Fs(\021)2283 4454 y Fw(;)15 b Fu(1)p Fv(\))36 b Fr(and)g(such)h(that)f Fw(f)42 b Fv(=)31 b(1)37 b Fr(in)e(a)h(neigh-)83 4567 y(b)-5 b(ourho)g(o)g(d)35 b(of)d Fu(1)p Fr(,)g(then)1096 4816 y(s)20 b Fu(\000)40 b Fv(lim)1244 4872 y Fs(t)p FC(!1)1426 4816 y Fw(e)1468 4778 y Fs(iH)5 b(t)1585 4816 y Fw(\021)s Fv(\()p Fw(H)i Fv(\))p Fw(f)1856 4688 y Fl(\022)1933 4754 y Fu(j)p Fw(Q)p Fu(j)p 1933 4795 123 4 v 1955 4878 a Fv(2)p Fw(t)2065 4688 y Fl(\023)2147 4816 y Fw(\021)s Fv(\()p Fw(H)g Fv(\))p Fw(e)2390 4778 y FC(\000)p Fs(iH)e(t)2589 4816 y Fv(=)25 b(0)p Fw(:)898 b Fx(\(8\))249 5080 y Fr(Pr)-5 b(o)g(of:)75 b Fx(W)-8 b(e)48 b(\034x)g(a)f(n)m(um)m(b)s(er)j Fw(c)1358 5047 y FC(0)1358 5103 y Fs(\021)1454 5080 y Fu(2)k Fv(I)-18 b(R)47 b Fx(suc)m(h)h(that)g(supp)q Fw(\021)57 b Fu(\032)d Fv(\()p Fu(\0001)p Fw(;)15 b(c)2822 5047 y FC(0)2822 5103 y Fs(\021)2865 5080 y Fv(\))48 b Fx(and)g(a)f(function)55 b Fv(~)-51 b Fw(\021)57 b Fu(2)83 5193 y Fw(C)155 5160 y FC(1)148 5215 y Fs(c)229 5119 y Fl(\000)271 5193 y Fv(\()p Fu(\0001)p Fw(;)15 b(c)547 5160 y FC(0)547 5215 y Fs(\021)590 5193 y Fv(\))625 5119 y Fl(\001)708 5193 y Fx(satisfying)48 b Fv(~)-51 b Fw(\021)s(\021)48 b Fv(=)d Fw(\021)g Fx(and)d Fv(0)k Fu(\024)51 b Fv(~)-52 b Fw(\021)48 b Fu(\024)d Fv(1)p Fx(.)75 b(W)-8 b(e)43 b(let)f Fw(c)2566 5160 y FC(00)2566 5215 y Fs(\021)2650 5193 y Fx(b)s(e)g(a)g(p)s (ositiv)m(e)g(constan)m(t)h(suc)m(h)83 5306 y(that)38 b Fv(sup)423 5328 y Fs(j)474 5306 y Fu(k)p Fw(\037)576 5341 y Fm(\()p Fq(\0001)p Fn(;c)756 5324 y Fq(0)756 5355 y Fn(\021)794 5341 y Fm(])820 5306 y Fv(\()p Fw(H)7 b Fv(\))p Fw(P)1031 5320 y Fs(j)1069 5306 y Fu(k)37 b(\024)g Fw(c)1298 5273 y FC(00)1298 5328 y Fs(\021)1340 5306 y Fw(=n)p Fx(,)i(where)f Fw(\037)1829 5331 y Fn(I)1906 5306 y Fx(is)f(the)g(c)m(haracteristic)j(function)e(of)f(the)g (in)m(terv)-5 b(al)39 b Fw(I)7 b Fx(.)83 5443 y(W)-8 b(e)31 b(no)m(w)g(\034x)f(the)h(constan)m(t)g Fw(c)1114 5457 y Fs(\021)1187 5443 y Fx(suc)m(h)g(that)g Fw(c)1626 5457 y Fs(\021)1693 5443 y Fw(>)25 b(c)1828 5410 y FC(0)q(0)1828 5466 y Fs(\021)1871 5443 y Fx(.)1880 5836 y(16)p eop %%Page: 17 17 bop 249 482 a Fx(a\))32 b(F)-8 b(or)32 b(the)h(pro)s(of)e(of)h Fr(i\))p Fx(,)f(c)m(ho)s(ose)1453 458 y Fv(~)1433 482 y Fw(f)37 b Fu(2)28 b Fw(C)1676 449 y FC(1)1669 504 y Fs(c)1750 482 y Fv(\(I)-18 b(R\))32 b Fx(with)g(supp)s(ort)g(in)g Fv(\()p Fw(c)2658 496 y Fs(\021)2701 482 y Fw(;)15 b Fu(1)p Fv(\))32 b Fx(and)g(suc)m(h)h(that) 3499 458 y Fv(~)3479 482 y Fw(f)9 b(f)37 b Fv(=)28 b Fw(f)83 621 y Fx(and)36 b Fv(0)e Fu(\024)466 597 y Fv(~)447 621 y Fw(f)42 b Fu(\024)34 b Fv(1)p Fx(.)55 b(F)-8 b(or)36 b Fw(t)d Fu(\025)g Fv(1)p Fx(,)k(set)e Fv(\010\()p Fw(t)p Fv(\))f(:=)f Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(F)1974 520 y Fl(\020)2039 576 y FC(j)p Fs(Q)p FC(j)p 2039 600 95 4 v 2056 652 a Ft(2)p Fs(t)2143 520 y Fl(\021)2213 621 y Fw(\021)s Fv(\()p Fw(H)g Fv(\))36 b Fx(with)g Fw(F)13 b Fv(\()p Fw(s)p Fv(\))34 b(=)2983 548 y Fl(R)3043 574 y Fs(s)3026 653 y FC(\0001)3170 621 y Fw(f)3225 588 y Ft(2)3264 621 y Fv(\()p Fw(\034)10 b Fv(\))p Fw(d\034)g Fx(.)56 b(Then)83 766 y Fv(\010)25 b Fu(2)g Fw(B)5 b(C)406 733 y Ft(1)399 793 y Fs(H)466 692 y Fl(\000)507 766 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))994 692 y Fl(\001)1068 766 y Fx(and)30 b(one)h(has)381 1022 y Fu(\000)p Fb(D)p Fv(\010\()p Fw(t)p Fv(\))25 b(=)832 961 y(1)p 832 1001 46 4 v 838 1084 a Fw(t)887 1022 y(\021)951 894 y Fl(\032)1019 1022 y Fu(\000)1100 961 y Fv(1)p 1100 1001 V 1100 1084 a(2)1155 1022 y Fw(P)33 b Fu(\001)1327 961 y Fw(Q)p 1302 1001 123 4 v 1302 1084 a Fu(j)p Fw(Q)p Fu(j)1434 1022 y Fw(f)1489 985 y Ft(2)1543 894 y Fl(\022)1620 961 y Fu(j)p Fw(Q)p Fu(j)p 1620 1001 V 1642 1084 a Fv(2)p Fw(t)1752 894 y Fl(\023)1839 1022 y Fu(\000)1940 961 y Fv(1)p 1940 1001 46 4 v 1940 1084 a(2)1996 1022 y Fw(f)2051 985 y Ft(2)2105 894 y Fl(\022)2182 961 y Fu(j)p Fw(Q)p Fu(j)p 2182 1001 123 4 v 2204 1084 a Fv(2)p Fw(t)2314 894 y Fl(\023)2431 961 y Fw(Q)p 2406 1001 V 2406 1084 a Fu(j)p Fw(Q)p Fu(j)2558 1022 y(\001)21 b Fw(P)33 b Fv(+)2796 961 y Fu(j)p Fw(Q)p Fu(j)p 2796 1001 V 2818 1084 a Fv(2)p Fw(t)2928 1022 y(f)2983 985 y Ft(2)3037 894 y Fl(\022)3114 961 y Fu(j)p Fw(Q)p Fu(j)p 3114 1001 V 3136 1084 a Fv(2)p Fw(t)3246 894 y Fl(\023\033)3397 1022 y Fw(\021)s(:)83 1269 y Fx(By)d(using)g Fa(i)9 b Fv(\))31 b Fx(of)f(Lemma)i(6.2,)e(one)h(can)g(c)m(hec)m(k)g(that)463 1517 y Fv(~)-51 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(P)34 b Fu(\001)831 1455 y Fw(Q)p 805 1496 V 805 1579 a Fu(j)p Fw(Q)p Fu(j)938 1517 y Fw(f)993 1479 y Ft(2)1052 1517 y Fv(+)20 b Fw(f)1198 1479 y Ft(2)1272 1455 y Fw(Q)p 1246 1496 V 1246 1579 a Fu(j)p Fw(Q)p Fu(j)1399 1517 y(\001)g Fw(P)g Fv(~)-52 b Fw(\021)t Fv(\()p Fw(H)7 b Fv(\))26 b(=)f Fw(f)1908 1389 y Fl(\032)1983 1517 y Fv(~)-52 b Fw(\021)t(P)33 b Fu(\001)2196 1455 y Fw(Q)p 2171 1496 V 2171 1579 a Fu(j)p Fw(Q)p Fu(j)2323 1493 y Fv(~)2304 1517 y Fw(f)c Fv(+)2489 1493 y(~)2469 1517 y Fw(f)2558 1455 y(Q)p 2533 1496 V 2533 1579 a Fu(j)p Fw(Q)p Fu(j)2686 1517 y(\001)20 b Fw(P)g Fv(~)-52 b Fw(\021)2851 1389 y Fl(\033)2934 1517 y Fw(f)29 b Fv(+)20 b Fw(O)s Fv(\()p Fw(t)3239 1479 y FC(\000)p Ft(1)3334 1517 y Fv(\))p Fw(:)83 1801 y Fx(Our)40 b(c)m(hoice)i(of)d Fw(c)708 1768 y FC(0)q(0)708 1823 y Fs(\021)791 1801 y Fx(implies)h(that)g Fu(\000)p Fw(c)1422 1768 y FC(0)q(0)1422 1823 y Fs(\021)1506 1801 y Fu(\024)1628 1765 y Ft(1)p 1628 1780 36 4 v 1628 1832 a(2)1688 1700 y Fl(n)1756 1801 y Fv(~)-52 b Fw(\021)s(P)34 b Fu(\001)1964 1760 y Fs(Q)p 1944 1780 95 4 v 1944 1833 a FC(j)p Fs(Q)p FC(j)2068 1777 y Fv(~)2049 1801 y Fw(f)29 b Fv(+)2234 1777 y(~)2214 1801 y Fw(f)2298 1760 y Fs(Q)p 2278 1780 V 2278 1833 a FC(j)p Fs(Q)p FC(j)2403 1801 y Fu(\001)21 b Fw(P)e Fv(~)-51 b Fw(\021)2568 1700 y Fl(o)2670 1801 y Fu(\024)41 b Fw(c)2821 1768 y FC(00)2821 1823 y Fs(\021)2864 1801 y Fw(:)f Fx(F)-8 b(urthermore)42 b(one)e(has)93 1928 y FC(j)p Fs(Q)p FC(j)p 93 1952 V 110 2004 a Ft(2)p Fs(t)198 1973 y Fw(f)253 1940 y Ft(2)307 1872 y Fl(\020)371 1928 y FC(j)p Fs(Q)p FC(j)p 371 1952 V 388 2004 a Ft(2)p Fs(t)475 1872 y Fl(\021)557 1973 y Fu(\025)28 b Fw(c)695 1987 y Fs(\021)737 1973 y Fw(f)792 1940 y Ft(2)846 1872 y Fl(\020)910 1928 y FC(j)p Fs(Q)p FC(j)p 910 1952 V 927 2004 a Ft(2)p Fs(t)1015 1872 y Fl(\021)1069 1973 y Fx(,)k(and)h(hence)f Fu(\000)p Fb(D)p Fv(\010\()p Fw(t)p Fv(\))c Fu(\025)2012 1937 y Ft(1)p 2012 1952 36 4 v 2017 2004 a Fs(t)2058 1973 y Fw(\021)s(f)10 b Fu(f)p Fw(c)2245 1987 y Fs(\021)2308 1973 y Fu(\000)21 b Fw(c)2439 1940 y FC(0)q(0)2439 1995 y Fs(\021)2482 1973 y Fu(g)p Fw(f)10 b(\021)24 b Fv(+)d Fw(O)s Fv(\()p Fw(t)2883 1940 y FC(\000)p Ft(2)2978 1973 y Fv(\))p Fx(.)45 b(Since)32 b Fw(c)3359 1987 y Fs(\021)3423 1973 y Fu(\000)21 b Fw(c)3554 1940 y FC(00)3554 1995 y Fs(\021)3624 1973 y Fw(>)28 b Fv(0)83 2116 y Fx(and)k(b)m(y)h(observing)f(that)g(for)g(an) m(y)h Fw(')28 b Fu(2)g(H)q Fx(,)38 b Fv(~)-51 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(')33 b Fx(b)s(elongs)f(to)g Fu(H)2424 2083 y Ft(2)2463 2116 y Fx(,)g(the)h(statemen)m(t)g Fr(i\))f Fx(is)f(seen)h(to)g (b)s(e)g(a)83 2229 y(consequence)f(of)f(Lemma)i(6.3.)249 2342 y(b\))d(F)-8 b(or)30 b Fw(f)38 b Fx(as)28 b(in)h Fr(i\))g Fx(or)g Fr(ii\))p Fx(,)f(let)1318 2318 y Fv(~)1298 2342 y Fw(f)35 b Fu(2)25 b Fw(C)1536 2309 y FC(1)1529 2364 y Fs(c)1610 2342 y Fv(\(I)-18 b(R\))29 b Fx(with)g(supp)s(ort)g(in)g Fv(\()p Fw(c)2506 2356 y Fs(\021)2549 2342 y Fw(;)15 b Fu(1)p Fv(\))29 b Fx(b)s(e)g(suc)m(h)h(that)3283 2318 y Fv(~)3263 2342 y Fw(f)9 b(f)3372 2309 y FC(0)3420 2342 y Fv(=)25 b Fw(f)3571 2309 y FC(0)3622 2342 y Fx(and)83 2481 y Fv(0)h Fu(\024)269 2457 y Fv(~)250 2481 y Fw(f)34 b Fu(\024)25 b Fv(1)p Fx(.)39 b(F)-8 b(or)24 b Fw(r)k Fu(\025)d Fv(1)f Fx(set)f Fv(\010)1123 2495 y Fs(r)1161 2481 y Fv(\()p Fw(t)p Fv(\))j(:=)f Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(f)1682 2380 y Fl(\020)1747 2436 y FC(j)p Fs(Q)p FC(j)p 1747 2460 95 4 v 1747 2512 a Ft(2)p Fs(r)r(t)1851 2380 y Fl(\021)1921 2481 y Fw(\021)s Fv(\()p Fw(H)g Fv(\))24 b Fx(and)g(observ)m(e)h(that)f Fv(\010)2886 2495 y Fs(r)2947 2481 y Fx(is)f(an)h(op)s(erator-v)-5 b(alued)83 2626 y(mapping)32 b(b)s(elonging)f(to)f Fw(B)5 b(C)1120 2593 y Ft(1)1113 2653 y Fs(H)1180 2552 y Fl(\000)1222 2626 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))1709 2552 y Fl(\001)1752 2626 y Fx(.)40 b(As)30 b(in)g(a\))h(ab)s(o)m(v)m(e,)g(one)g(\034nds)f(that) 565 2882 y Fb(D)p Fv(\010)711 2896 y Fs(r)748 2882 y Fv(\()p Fw(t)p Fv(\))c(=)983 2821 y(1)p 983 2861 46 4 v 989 2944 a Fw(t)1039 2882 y(\021)1106 2858 y Fv(~)1087 2882 y Fw(f)1156 2754 y Fl(\032)1256 2821 y Fv(1)p 1235 2861 89 4 v 1235 2944 a(2)p Fw(r)1340 2882 y Fv(~)-52 b Fw(\021)t(P)33 b Fu(\001)1554 2821 y Fw(Q)p 1528 2861 123 4 v 1528 2944 a Fu(j)p Fw(Q)p Fu(j)1661 2882 y Fw(f)1716 2845 y FC(0)1759 2882 y Fv(+)1881 2821 y(1)p 1859 2861 89 4 v 1859 2944 a(2)p Fw(r)1958 2882 y(f)2013 2845 y FC(0)2071 2821 y Fw(Q)p 2046 2861 123 4 v 2046 2944 a Fu(j)p Fw(Q)p Fu(j)2199 2882 y(\001)20 b Fw(P)f Fv(~)-51 b Fw(\021)24 b Fu(\000)2484 2821 y(j)p Fw(Q)p Fu(j)p 2484 2861 V 2484 2944 a Fv(2)p Fw(r)s(t)2617 2882 y(f)2672 2845 y FC(0)2694 2754 y Fl(\033)2797 2858 y Fv(~)2778 2882 y Fw(f)9 b(\021)24 b Fv(+)19 b Fw(O)s Fv(\()p Fw(t)3131 2845 y FC(\000)p Ft(2)3226 2882 y Fv(\))p Fw(:)83 3124 y Fx(All)34 b(terms)h(b)s(et)m(w)m(een)h(brac)m(k)m (ets)g(are)f(norm)g(b)s(ounded)h(indep)s(enden)m(tly)g(of)e Fw(t)g Fx(for)h Fw(t)d Fu(\025)g Fv(1)p Fx(.)54 b(So)34 b(b)m(y)h(applying)83 3237 y(Lemma)d(6.4)e(and)h(b)m(y)g(using)f Fr(i\))g Fx(one)h(gets)f(the)h (existence)f(of)g(s)20 b Fu(\000)g Fv(lim)2452 3251 y Fs(t)p FC(!1)2638 3237 y Fw(e)2680 3204 y Fs(iH)5 b(t)2797 3237 y Fv(\010)2863 3251 y Fs(r)2901 3237 y Fv(\()p Fw(t)p Fv(\))p Fw(e)3046 3204 y FC(\000)p Fs(iH)g(t)3218 3237 y Fx(.)249 3350 y(c\))33 b(Let)h(us)f(sho)m(w)h(that)f(this)h(limit)f(is)g(zero)h(if)e Fw(f)43 b Fx(satis\034es)33 b(the)g(h)m(yp)s(othesis)g(of)g Fr(i\))p Fx(.)48 b(Indeed,)35 b(one)e(ma)m(y)83 3463 y(then)38 b(c)m(ho)s(ose)609 3439 y Fv(~)590 3463 y Fw(f)46 b Fu(2)36 b Fw(C)850 3430 y FC(1)843 3485 y Fs(c)924 3463 y Fv(\(I)-18 b(R\))38 b Fx(with)g(supp)s(ort)f(in)g Fv(\()p Fw(c)1854 3477 y Fs(\021)1897 3463 y Fw(;)15 b Fu(1)p Fv(\))38 b Fx(and)f(suc)m(h)h(that) 2716 3439 y Fv(~)2697 3463 y Fw(f)9 b(f)46 b Fv(=)37 b Fw(f)10 b Fx(.)60 b(Then)38 b(b)m(y)g Fr(i\))e Fx(there)83 3576 y(exists)30 b Fw(c)25 b(<)g Fu(1)30 b Fx(suc)m(h)h(that)g(for)f(all)h Fw(')25 b Fu(2)g(H)q Fx(,)319 3711 y Fl(Z)410 3737 y FC(1)369 3917 y Ft(1)499 3757 y Fl(\014)499 3812 y(\014)530 3834 y Fu(h)p Fw(';)15 b(e)706 3797 y Fs(iH)5 b(t)824 3834 y Fv(\010)890 3848 y Fs(r)928 3834 y Fv(\()p Fw(t)p Fv(\))p Fw(e)1073 3797 y FC(\000)p Fs(iH)g(t)1245 3834 y Fw(')p Fu(i)1339 3757 y Fl(\014)1339 3812 y(\014)1395 3773 y Fw(dt)p 1395 3814 81 4 v 1419 3897 a(t)1511 3834 y Fu(\024)25 b(k)p Fw(f)10 b Fu(k)1752 3848 y Fs(L)1800 3830 y Fq(1)1884 3711 y Fl(Z)1975 3737 y FC(1)1935 3917 y Ft(1)2065 3703 y Fl(\015)2065 3757 y(\015)2065 3812 y(\015)2065 3866 y(\015)2135 3810 y Fv(~)2115 3834 y Fw(f)2185 3706 y Fl(\022)2262 3773 y Fu(j)p Fw(Q)p Fu(j)p 2262 3814 123 4 v 2262 3897 a Fv(2)p Fw(r)s(t)2394 3706 y Fl(\023)2476 3834 y Fw(\021)s Fv(\()p Fw(H)d Fv(\))p Fw(e)2719 3797 y FC(\000)p Fs(iH)e(t)2892 3834 y Fw(')2951 3703 y Fl(\015)2951 3757 y(\015)2951 3812 y(\015)2951 3866 y(\015)3002 3729 y Ft(2)3067 3773 y Fw(dt)p 3067 3814 81 4 v 3090 3897 a(t)3182 3834 y Fu(\024)25 b Fw(c)p Fu(k)p Fw(')p Fu(k)3466 3797 y Ft(2)3507 3834 y Fw(:)121 b Fx(\(9\))83 4092 y(This)35 b(implies)g(that)h Fv(lim)932 4106 y Fs(t)p FC(!1)1103 4092 y Fu(h)p Fw(';)15 b(e)1279 4059 y Fs(iH)5 b(t)1397 4092 y Fv(\010)1463 4106 y Fs(r)1500 4092 y Fv(\()p Fw(t)p Fv(\))p Fw(e)1645 4059 y FC(\000)p Fs(iH)g(t)1818 4092 y Fw(')p Fu(i)35 b Fx(m)m(ust)h(b)s(e)e(zero)i(for)e(eac)m(h)i Fw(')d Fu(2)g(H)q Fx(,)i(and)g(hence)h(w)23 b Fu(\000)83 4205 y Fv(lim)209 4219 y Fs(t)p FC(!1)395 4205 y Fw(e)437 4172 y Fs(iH)5 b(t)554 4205 y Fv(\010)620 4219 y Fs(r)658 4205 y Fv(\()p Fw(t)p Fv(\))p Fw(e)803 4172 y FC(\000)p Fs(iH)g(t)1001 4205 y Fv(=)25 b(0)30 b Fx(b)m(y)h(p)s(olarization.)249 4318 y(d\))40 b(Let)h Fw(f)49 b Fx(satisfy)39 b(the)h(h)m(yp)s(othesis)g Fr(ii\))f 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y(existence)j(of)f(s)20 b Fu(\000)f Fv(lim)844 978 y Fs(t)p FC(!1)1030 964 y Fw(e)1072 931 y Fs(iH)5 b(t)1189 964 y Fv(\010)1255 978 y Fs(r)1292 964 y Fv(\()p Fw(t)p Fv(\))p Fw(e)1437 931 y FC(\000)p Fs(iH)g(t)1640 964 y Fx(w)m(as)31 b(sho)m(wn)g(in)f(b\),)h(the)g(inequalities)g(in)f(\(10\))h(imply)g(that)1152 1171 y(s)20 b Fu(\000)45 b Fv(lim)1299 1225 y Fs(r)r FC(!1)1489 1070 y Fl(\020)1544 1171 y Fx(s)20 b Fu(\000)40 b Fv(lim)1690 1227 y Fs(t)p FC(!1)1872 1171 y Fw(e)1914 1133 y Fs(iH)5 b(t)2031 1171 y Fv(\010)2097 1185 y Fs(r)2135 1171 y Fv(\()p Fw(t)p Fv(\))p Fw(e)2280 1133 y FC(\000)p Fs(iH)g(t)2452 1070 y Fl(\021)2532 1171 y Fv(=)25 b(0)p Fw(:)909 b Fx(\(11\))249 1420 y(No)m(w)34 b(set)f Fv(\010)668 1434 y Ft(1)707 1420 y Fv(\()p Fw(t)p Fv(\))e(=)f(\010) 1008 1434 y Fs(r)1046 1420 y Fv(\()p Fw(t)p Fv(\))22 b(+)1274 1397 y(~)1264 1420 y(\010)1330 1434 y Fs(r)1368 1420 y Fv(\()p Fw(t)p Fv(\))34 b Fx(with)1724 1397 y Fv(~)1713 1420 y(\010)1779 1434 y Fs(r)1817 1420 y Fv(\()p Fw(t)p 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Fr(ii\))d Fx(without)j(assuming)f(an)g(additional)h(condition)g(for)e Fw(f)10 b Fx(,)32 b(c)m(ho)s(ose)h(an)m(y)g(real)g(function)83 1904 y Fw(g)c Fu(2)c Fw(C)313 1871 y FC(1)306 1926 y Fs(c)387 1904 y Fv(\(I)-18 b(R\))31 b Fx(suc)m(h)g(that)g(supp)p Fw(g)e Fu(\032)c Fv(\()p Fw(c)1398 1918 y Fs(\021)1441 1904 y Fw(;)15 b Fu(1)p Fv(\))30 b Fx(and)1813 1830 y Fl(R)1874 1857 y FC(1)1856 1936 y(\0001)2001 1904 y Fw(g)2047 1871 y Ft(2)2087 1904 y Fv(\()p Fw(y)s Fv(\))p Fw(dy)f Fv(=)c(1)p Fx(,)31 b(and)g(set)f Fw(F)13 b Fv(\()p Fw(x)p Fv(\))26 b(=)3155 1830 y Fl(R)3216 1857 y Fs(x)3198 1936 y FC(\0001)3343 1904 y Fw(g)3389 1871 y Ft(2)3429 1904 y Fv(\()p Fw(y)s Fv(\))p Fw(dy)s Fx(.)41 b(If)83 2017 y Fw(f)h Fx(satis\034es)32 b(the)i(h)m(yp)s(othesis)e(of)h Fr(ii\))p Fx(,)f(then)i Fw(f)d Fu(\000)21 b Fw(F)46 b Fx(satis\034es)33 b(that)g(of)f Fr(i\))p Fx(,)h(and)g(\(8\))h(follo)m(ws)f(b)m(y)g(com)m (bining)83 2129 y(the)e(results)f(of)g(c\))g(and)h(d\).)2677 b Fc(\004)249 2242 y Fx(In)38 b(the)g(next)g(lemma,)j(a)d(certain)h (distortion)g(of)f(the)h(mapping)g Fv(I)-18 b(R)2618 2207 y Fs(n)2703 2242 y Fu(3)37 b Fw(x)i Fu(7!)3031 2207 y Ft(1)p 3031 2222 36 4 v 3031 2274 a(2)3076 2242 y Fw(x)3128 2209 y Ft(2)3206 2242 y Fu(2)e Fv(I)-18 b(R)38 b Fx(will)g(pla)m(y)83 2355 y(a)h(crucial)h(tec)m(hnical)g(role.)67 b(W)-8 b(e)39 b(refer)g(to)g([9])g(and)g(esp)s(ecially)g(to)g([4])g(for)g(similar)g (constructions)h(used)83 2468 y(in)f(the)h(N-b)s(o)s(dy)e(problem.)68 b(F)-8 b(or)39 b Fw(\016)44 b(>)c Fv(0)p Fx(,)h(let)f Fw(m)1806 2494 y Fn(\016)1884 2468 y Fu(2)f Fw(C)2056 2435 y FC(1)2130 2468 y Fv(\(I)-18 b(R\))40 b Fx(b)s(e)e(a)h(real)h(con)m(v)m(ex)g(function)g (suc)m(h)f(that)83 2581 y Fw(m)163 2606 y Fn(\016)201 2581 y Fv(\()p Fw(y)s Fv(\))45 b(=)488 2545 y Ft(1)p 488 2560 V 488 2612 a(2)533 2581 y Fw(\016)576 2548 y Ft(2)658 2581 y Fx(if)c Fu(j)p Fw(y)s Fu(j)j(\024)g Fw(\016)h Fx(and)d Fw(m)1361 2606 y Fn(\016)1399 2581 y Fv(\()p Fw(y)s Fv(\))j(=)1686 2545 y Ft(1)p 1686 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b Fx(and)83 2875 y Fw(@)136 2842 y Ft(2)131 2901 y Fs(j)t(j)200 2875 y Fw(r)241 2901 y Fn(\016)279 2875 y Fv(\()p Fw(x)p Fv(\))26 b Fu(\025)f Fw(\030)567 2842 y Ft(2)622 2802 y Fl(\000)664 2875 y Fu(\000)745 2827 y Fs(x)785 2837 y Fn(j)p 744 2855 73 4 v 746 2907 a Ft(2)p Fs(\016)826 2802 y Fl(\001)888 2875 y Fv(+)20 b Fw(\030)1023 2842 y Ft(2)1078 2802 y Fl(\000)1129 2827 y Fs(x)1169 2837 y Fn(j)p 1129 2855 V 1131 2907 a Ft(2)p Fs(\016)1212 2802 y Fl(\001)1283 2875 y Fx(for)30 b(an)m(y)h(asymptotic)g(lo) s(calization)h(function)f Fw(\030)j Fx(and)d(all)f Fw(x)c Fu(2)f Fv(I)-18 b(R)3511 2840 y Fs(n)3558 2875 y Fx(.)83 3068 y Fj(Lemma)35 b(6.6)45 b Fr(F)-7 b(or)36 b(e)-5 b(ach)37 b Fw(\021)d Fu(2)d Fw(C)1250 3035 y FC(1)1243 3091 y Fs(c)1324 3068 y Fv(\(I)-18 b(R\))p Fr(,)36 b(e)-5 b(ach)37 b Fw(j)g Fu(2)30 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)j(;)d(n)p Fu(g)36 b Fr(and)f(e)-5 b(ach)37 b Fw(f)k Fu(2)30 b Fw(C)2973 3035 y FC(1)2966 3091 y Fs(c)3047 3068 y Fv(\(I)-18 b(R)3164 3032 y Fs(n)3211 3068 y Fv(\))36 b Fr(with)f(supp)-5 b(ort)83 3181 y(in)32 b Fw(Y)267 3143 y FC(\006)247 3208 y Fs(j)326 3181 y Fr(,)g(ther)-5 b(e)33 b(exists)g(a)f(c)-5 b(onstant)33 b Fw(c)26 b(<)f Fu(1)32 b Fr(such)h(that)f(for)g(al)5 b(l)33 b Fw(')25 b Fu(2)g(H)q Fr(,)882 3326 y Fl(Z)973 3352 y FC(1)932 3532 y Ft(1)1063 3318 y Fl(\015)1063 3372 y(\015)1063 3427 y(\015)1063 3481 y(\015)1113 3449 y Fw(f)1183 3321 y Fl(\022)1263 3388 y Fw(Q)p 1259 3429 79 4 v 1259 3512 a Fv(2)p Fw(t)1348 3321 y Fl(\023)15 b(\022)1497 3449 y Fw(P)1555 3463 y Fs(j)1612 3449 y Fu(\000)1713 3387 y Fw(Q)1785 3401 y Fs(j)p 1713 3429 109 4 v 1728 3512 a Fv(2)p Fw(t)1831 3321 y Fl(\023)1913 3449 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)2156 3412 y FC(\000)p Fs(iH)e(t)2329 3449 y Fw(')2388 3318 y Fl(\015)2388 3372 y(\015)2388 3427 y(\015)2388 3481 y(\015)2439 3344 y Ft(2)2504 3388 y Fw(dt)p 2504 3429 81 4 v 2527 3512 a(t)2619 3449 y Fu(\024)25 b Fw(c)p Fu(k)p Fw(')p Fu(k)2903 3412 y Ft(2)2944 3449 y Fw(:)638 b Fx(\(12\))83 3678 y Fr(F)-7 b(urthermor)i(e,)1057 3836 y(s)20 b Fu(\000)40 b Fv(lim)1205 3892 y Fs(t)p FC(!1)1386 3836 y Fw(f)1456 3708 y Fl(\022)1536 3775 y Fw(Q)p 1533 3815 79 4 v 1533 3899 a Fv(2)p Fw(t)1621 3708 y Fl(\023)15 b(\022)1770 3836 y Fw(P)1828 3850 y Fs(j)1885 3836 y Fu(\000)1986 3774 y Fw(Q)2058 3788 y Fs(j)p 1986 3815 109 4 v 2001 3899 a Fv(2)p Fw(t)2104 3708 y Fl(\023)2186 3836 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)2429 3799 y FC(\000)p Fs(iH)e(t)2628 3836 y Fv(=)25 b(0)p Fw(:)249 4081 y Fr(Pr)-5 b(o)g(of:)63 b Fx(Let)42 b(us)f(\034x)g(a)g(n)m(um)m(b)s(er)i Fw(c)1453 4048 y FC(0)1453 4103 y Fs(\021)1540 4081 y Fu(2)g Fv(I)-18 b(R)41 b Fx(suc)m(h)h(that)g(supp) p Fw(\021)47 b Fu(\032)d Fv(\()p Fu(\0001)p Fw(;)15 b(c)2858 4048 y FC(0)2858 4103 y Fs(\021)2900 4081 y Fv(\))42 b Fx(and)g(a)f(function) 49 b Fv(~)-52 b Fw(\021)47 b Fu(2)83 4194 y Fw(C)155 4161 y FC(1)148 4216 y Fs(c)229 4120 y Fl(\000)271 4194 y Fv(\()p Fu(\0001)p Fw(;)15 b(c)547 4161 y FC(0)547 4216 y Fs(\021)590 4194 y Fv(\))625 4120 y Fl(\001)694 4194 y Fx(satisfying)33 b Fv(~)-51 b Fw(\021)s(\021)29 b Fv(=)c Fw(\021)31 b Fx(and)d Fv(0)d Fu(\024)32 b Fv(~)-52 b Fw(\021)29 b Fu(\024)c Fv(1)p Fx(.)40 b(Let)28 b Fw(c)2205 4208 y Fs(\021)2274 4194 y Fx(b)s(e)g(the)g(p)s (ositiv)m(e)f(constan)m(t)i(dep)s(ending)g(on)83 4306 y Fw(\021)34 b Fx(giv)m(en)d(b)m(y)f(the)h(previous)f(lemma.)249 4419 y(a\))40 b(Consider)g Fw(h)i Fu(2)e Fw(C)1022 4386 y FC(1)1015 4442 y Fs(c)1096 4419 y Fv(\(I)-18 b(R\))40 b Fx(suc)m(h)g(that)g Fw(h)p Fv(\()p Fu(j)p Fw(x)p Fu(j)p Fv(\))p Fw(f)10 b Fv(\()p Fw(x)p Fv(\))43 b(=)d Fw(f)10 b Fv(\()p Fw(x)p Fv(\))40 b Fx(for)f(all)h Fw(x)h Fu(2)f Fv(I)-18 b(R)3036 4384 y Fs(n)3122 4419 y Fx(and)40 b Fw(h)p Fv(\()p Fw(y)s Fv(\))i(=)e(1)g Fx(if)83 4532 y Fw(y)28 b Fu(2)d Fv([0)p Fw(;)15 b(c)391 4546 y Fs(\021)455 4532 y Fv(+)k(1\))p Fx(.)42 b(F)-8 b(or)31 b Fw(\016)e(>)c Fv(0)30 b Fx(and)h Fw(t)25 b Fu(\025)g Fv(1)p Fx(,)31 b(set)520 4771 y Fw(L)p Fv(\()p Fw(t)p Fv(\))26 b(:=)842 4709 y(1)p 842 4750 46 4 v 842 4833 a(2)913 4642 y Fl(\022)979 4771 y Fw(P)34 b Fu(\000)1175 4709 y Fw(Q)p 1171 4750 79 4 v 1171 4833 a Fv(2)p Fw(t)1260 4642 y Fl(\023)1347 4771 y Fu(\001)20 b(r)p Fw(r)1509 4796 y Fn(\016)1562 4642 y Fl(\022)1643 4709 y Fw(Q)p 1639 4750 V 1639 4833 a Fv(2)p Fw(t)1728 4642 y Fl(\023)1815 4771 y Fv(+)1916 4709 y(1)p 1916 4750 46 4 v 1916 4833 a(2)1971 4771 y Fu(r)p Fw(r)2088 4796 y Fn(\016)2141 4642 y Fl(\022)2221 4709 y Fw(Q)p 2218 4750 79 4 v 2218 4833 a Fv(2)p Fw(t)2306 4642 y Fl(\023)2393 4771 y Fu(\001)2439 4642 y Fl(\022)2506 4771 y Fw(P)33 b Fu(\000)2701 4709 y Fw(Q)p 2698 4750 V 2698 4833 a Fv(2)p Fw(t)2786 4642 y Fl(\023)2873 4771 y Fv(+)20 b Fw(r)3005 4796 y Fn(\016)3059 4642 y Fl(\022)3139 4709 y Fw(Q)p 3135 4750 V 3135 4833 a Fv(2)p Fw(t)3224 4642 y Fl(\023)3306 4771 y Fw(;)83 5041 y Fx(and)32 b Fv(\010\()p Fw(t)p Fv(\))c(:=)f Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(h)848 4941 y Fl(\020)914 4997 y FC(j)p Fs(Q)p FC(j)p 914 5021 95 4 v 931 5073 a Ft(2)p Fs(t)1019 4941 y Fl(\021)1088 5041 y Fw(L)p Fv(\()p Fw(t)p Fv(\))p Fw(h)1320 4941 y Fl(\020)1385 4997 y FC(j)p Fs(Q)p FC(j)p 1385 5021 V 1402 5073 a Ft(2)p Fs(t)1490 4941 y Fl(\021)1559 5041 y Fw(\021)s Fv(\()p Fw(H)g Fv(\))p Fx(.)46 b(One)32 b(can)g(c)m(hec)m(k) h(that)f Fv(\010)27 b Fu(2)g Fw(B)5 b(C)2959 5008 y Ft(1)2952 5068 y Fs(H)3019 4968 y Fl(\000)3061 5041 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))3548 4968 y Fl(\001)3622 5041 y Fx(and)83 5171 y(that)780 5356 y Fb(D)p Fv(\010\()p Fw(t)p Fv(\))83 b(=)g Fw(\021)s Fu(f)p Fb(D)p Fw(h)p Fu(g)p Fw(Lh\021)26 b Fv(+)20 b Fw(\021)s(hL)p Fu(f)p Fb(D)p Fw(h)p Fu(g)p Fw(\021)25 b Fu(\000)20 b Fw(\021)s(h)p Fu(r)p Fw(V)h Fv(\()p Fw(Q)p Fv(\))g Fu(\001)f(r)p Fw(r)2932 5381 y Fn(\016)2970 5356 y Fw(h\021)1266 5546 y Fv(+)1347 5485 y(1)p 1347 5525 46 4 v 1353 5609 a Fw(t)1402 5546 y(\021)s(h)1517 5418 y Fl(\024)1566 5546 y Fw(P)33 b Fu(\000)1761 5485 y Fw(Q)p 1758 5525 79 4 v 1758 5609 a Fv(2)p Fw(t)1846 5418 y Fl(\025)1894 5452 y Fn(T)1962 5546 y Fw(@)2015 5509 y Ft(2)2055 5546 y Fw(r)2096 5572 y Fn(\016)2149 5418 y Fl(\024)2197 5546 y Fw(P)h Fu(\000)2393 5485 y Fw(Q)p 2390 5525 V 2390 5609 a Fv(2)p Fw(t)2478 5418 y Fl(\025)2541 5546 y Fw(h\021)24 b Fv(+)c Fw(O)s Fv(\()p Fw(t)2893 5509 y FC(\000)p Ft(3)2987 5546 y Fv(\))p Fw(;)560 b Fx(\(13\))1880 5836 y(18)p eop %%Page: 19 19 bop 83 482 a Fx(where)29 b Fw(@)395 449 y Ft(2)434 482 y Fw(r)475 507 y Fn(\016)541 482 y Fx(is)e(the)h(matrix)h(of)e(second)h(deriv)-5 b(ativ)m(es)29 b(of)e Fw(r)2052 507 y Fn(\016)2090 482 y Fx(.)40 b(In)27 b(order)h(to)g(b)s(e)g(able)g(to)g(use)f(Lemma)i(6.3,)g(w)m(e)83 594 y(obtain)i(no)m(w)h(some)e(estimates)h(for)f(eac)m(h)h(term)g(of)f (\(13\).)249 707 y(Some)41 b(comm)m(utator)h(calculations,)i(using)c(rep)s (eatedly)g Fr(i\))g Fx(of)f(Lemma)i(6.2,)i(sho)m(w)e(that)f(the)g(term)83 843 y Fw(\021)s Fu(f)p Fb(D)p Fw(h)p Fu(g)p Fw(Lh\021)32 b Fv(+)26 b Fw(\021)s(hL)p Fu(f)p Fb(D)p Fw(h)p Fu(g)p Fw(\021)44 b Fx(can)c(b)s(e)f(rewritten)i(as)1952 807 y Ft(1)p 1952 822 36 4 v 1957 874 a Fs(t)1997 843 y Fw(\021)s(g)2107 742 y Fl(\020)2171 798 y FC(j)p Fs(Q)p FC(j)p 2171 822 95 4 v 2188 874 a Ft(2)p Fs(t)2276 742 y Fl(\021)2345 843 y Fv(\011\()p Fw(t)p Fv(\))p Fw(g)2581 742 y Fl(\020)2646 798 y FC(j)p Fs(Q)p FC(j)p 2646 822 V 2663 874 a Ft(2)p Fs(t)2750 742 y Fl(\021)2820 843 y Fw(\021)29 b Fv(+)d Fw(O)s Fv(\()p Fw(t)3131 810 y FC(\000)p Ft(2)3226 843 y Fv(\))p Fx(,)41 b(where)f Fv(\011)f Fx(is)83 978 y(a)j(b)s(ounded)g(op)s(erator-v)-5 b(alued)44 b(mapping)f(and)f Fw(g)k Fx(b)s(elongs)c(to)g Fw(C)2399 946 y FC(1)2392 1001 y Fs(c)2473 978 y Fv(\(I)-18 b(R\))42 b Fx(and)g(satis\034es)g(the)g (conditions)83 1091 y(supp)p Fw(g)30 b Fu(\032)c Fv(\()p Fw(c)513 1105 y Fs(\021)556 1091 y Fw(;)15 b Fu(1)p Fv(\))31 b Fx(and)g Fw(g)s Fv(\()p Fw(y)s Fv(\))p Fw(h)1145 1058 y FC(0)1170 1091 y Fv(\()p Fw(y)s Fv(\))c(=)f Fw(h)1464 1058 y FC(0)1488 1091 y Fv(\()p Fw(y)s Fv(\))31 b Fx(for)f(all)i Fw(y)d Fu(\025)d Fv(0)p Fx(.)42 b(The)31 b(statemen)m(t)i Fr(i\))d Fx(of)h(the)g(previous)g (lemma)83 1204 y(implies)f(that)h(the)f(term)965 1169 y Ft(1)p 965 1184 36 4 v 970 1236 a Fs(t)1010 1204 y Fw(\021)s(g)s Fv(\011)p Fw(g)s(\021)35 b Fx(is)29 b(in)m(tegrable)j(along)e(the)h(ev)m(olution.)41 b(W)-8 b(e)30 b(also)g(observ)m(e)g(that)h Fw(h)3537 1171 y Ft(2)3577 1204 y Fu(r)3653 1218 y Fs(j)3689 1204 y Fw(r)3730 1230 y Fn(\016)83 1317 y Fx(b)s(elongs)h(to)f Fw(C)592 1284 y FC(1)585 1340 y Fs(c)667 1317 y Fv(\(I)-18 b(R)784 1281 y Fs(n)831 1317 y Fv(\))31 b Fx(and)h(has)g(its)f(supp)s(ort)h(in)g Fw(Y)1880 1279 y FC(\000)1860 1344 y Fs(j)1960 1317 y Fu([)20 b Fw(Y)2114 1279 y Ft(+)2094 1344 y Fs(j)2173 1317 y Fx(.)45 b(Therefore,)33 b(using)e Fr(ii\))i Fx(of)e(Lemma)i(6.2,)g(the)83 1443 y(term)f Fw(\021)s(h)p Fu(r)p Fw(V)42 b Fu(\001)21 b(r)p Fw(r)734 1468 y Fn(\016)772 1443 y Fw(h\021)35 b Fx(b)s(elongs)d(to)f Fw(L)1403 1410 y Ft(1)1442 1369 y Fl(\000)1484 1443 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)1850 1369 y Fl(\001)1894 1443 y Fx(.)43 b(Finally)-8 b(,)33 b(the)f(prop)s(erties)g(of)f Fw(r)3017 1468 y Fn(\016)3086 1443 y Fx(men)m(tioned)j(b)s(efore)83 1556 y(this)c(lemma)i(imply)e(that)288 1738 y Fv(1)p 288 1778 46 4 v 294 1862 a Fw(t)343 1799 y(\021)s(h)458 1671 y Fl(\024)507 1799 y Fw(P)j Fu(\000)702 1738 y Fw(Q)p 699 1778 79 4 v 699 1862 a Fv(2)p Fw(t)787 1671 y Fl(\025)835 1705 y Fn(T)903 1799 y Fw(@)956 1762 y Ft(2)996 1799 y Fw(r)1037 1825 y Fn(\016)1090 1671 y Fl(\024)1138 1799 y Fw(P)h Fu(\000)1334 1738 y Fw(Q)p 1330 1778 V 1330 1862 a Fv(2)p Fw(t)1419 1671 y Fl(\025)1482 1799 y Fw(h\021)29 b Fu(\025)1714 1738 y Fv(1)p 1714 1778 46 4 v 1720 1862 a Fw(t)1769 1799 y(\021)s(h)1884 1671 y Fl(\022)1952 1799 y Fw(P)2010 1813 y Fs(j)2067 1799 y Fu(\000)2168 1737 y Fw(Q)2240 1751 y Fs(j)p 2168 1778 109 4 v 2183 1862 a Fv(2)p Fw(t)2286 1671 y Fl(\023)2368 1799 y Fw(\030)2412 1762 y Ft(2)2467 1671 y Fl(\022)2534 1799 y Fu(\006)2621 1737 y Fw(Q)2693 1751 y Fs(j)p 2615 1778 123 4 v 2615 1862 a Fv(4)p Fw(\016)s(t)2747 1671 y Fl(\023)15 b(\022)2896 1799 y Fw(P)2954 1813 y Fs(j)3011 1799 y Fu(\000)3112 1737 y Fw(Q)3184 1751 y Fs(j)p 3112 1778 109 4 v 3127 1862 a Fv(2)p Fw(t)3230 1671 y Fl(\023)3312 1799 y Fw(h\021)198 b Fx(\(14\))83 2048 y(for)28 b(an)m(y)g(asymptotic)g(lo)s (calization)h(function)g Fw(\030)i Fx(and)d(eac)m(h)h Fw(j)i Fu(2)25 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)i(;)e(n)p Fu(g)p Fx(.)40 b(Th)m(us,)29 b(b)m(y)f(applying)h(Lemma)83 2161 y(6.3)35 b(with)f(the)h (term)g Fw(B)896 2128 y FC(\003)935 2161 y Fv(\()p Fw(t)p Fv(\))p Fw(B)5 b Fv(\()p Fw(t)p Fv(\))35 b Fx(equal)f(to)h(the)f(r.h.s.)h(of)f (\(14\),)h(one)g(obtains)g(that)g(there)g(exists)e Fw(c)g(<)e Fu(1)83 2273 y Fx(suc)m(h)g(that)g(for)f(eac)m(h)h Fw(j)g Fu(2)25 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)j(;)d(n)p Fu(g)30 b Fx(and)h(all)g Fw(')25 b Fu(2)g(H)q Fx(:)650 2415 y Fl(Z)741 2441 y FC(1)701 2621 y Ft(1)831 2407 y Fl(\015)831 2461 y(\015)831 2516 y(\015)831 2570 y(\015)882 2538 y Fw(\030)941 2410 y Fl(\022)1008 2538 y Fu(\006)1095 2476 y Fw(Q)1167 2490 y Fs(j)p 1089 2518 123 4 v 1089 2601 a Fv(4)p Fw(\016)s(t)1220 2410 y Fl(\023)15 b(\022)1369 2538 y Fw(P)1427 2552 y Fs(j)1484 2538 y Fu(\000)1585 2476 y Fw(Q)1657 2490 y Fs(j)p 1585 2518 109 4 v 1600 2601 a Fv(2)p Fw(t)1704 2410 y Fl(\023)1786 2538 y Fw(h)1853 2410 y Fl(\022)1930 2477 y Fu(j)p Fw(Q)p Fu(j)p 1930 2518 123 4 v 1952 2601 a Fv(2)p Fw(t)2062 2410 y Fl(\023)2145 2538 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)2388 2501 y FC(\000)p Fs(iH)e(t)2561 2538 y Fw(')2620 2407 y Fl(\015)2620 2461 y(\015)2620 2516 y(\015)2620 2570 y(\015)2671 2433 y Ft(2)2735 2477 y Fw(dt)p 2735 2518 81 4 v 2759 2601 a(t)2850 2538 y Fu(\024)25 b Fw(c)p Fu(k)p Fw(')p Fu(k)3134 2501 y Ft(2)3176 2538 y Fw(:)406 b Fx(\(15\))249 2797 y(Since)32 b Fw(f)40 b Fx(has)31 b(a)g(compact)h(supp)s(ort)f(in)g Fw(Y)1685 2759 y FC(\006)1665 2824 y Fs(j)1744 2797 y Fx(,)g(w)m(e)h(ma)m(y)f (no)m(w)h(\034x)f Fw(j)37 b Fx(and)31 b Fw(\016)k Fx(suc)m(h)c(that)h Fw(\030)3233 2723 y Fl(\000)3274 2797 y Fu(\006)3355 2749 y Fs(x)3395 2759 y Fn(j)p 3355 2776 73 4 v 3357 2828 a Ft(2)p Fs(\016)3437 2723 y Fl(\001)3494 2797 y Fw(f)10 b Fv(\()p Fw(x)p Fv(\))26 b(=)83 2910 y Fw(f)10 b Fv(\()p Fw(x)p Fv(\))27 b Fx(for)h(all)f Fw(x)f Fu(2)e Fv(I)-18 b(R)790 2874 y Fs(n)837 2910 y Fx(.)39 b(Then)29 b(the)e(\034rst)h(assertion)g(of)f(the)h(lemma)g(is) f(a)h(simple)g(consequence)g(of)f(\(15\))h(and)83 3046 y(of)i(the)h(estimate) 704 2945 y Fl(h)747 3046 y Fw(P)805 3060 y Fs(j)842 3046 y Fw(;)15 b(h)949 2945 y Fl(\020)1014 3001 y FC(j)p Fs(Q)p FC(j)p 1014 3025 95 4 v 1031 3077 a Ft(2)p Fs(t)1118 2945 y Fl(\021)q(i)1241 3046 y Fu(2)25 b Fw(O)s Fv(\()p Fw(t)1467 3013 y FC(\000)p Ft(1)1561 3046 y Fv(\))p Fx(.)249 3218 y(b\))d(F)-8 b(or)23 b Fw(t)i Fu(\025)g Fv(1)p Fx(,)f(set)e Fv(\010\()p Fw(t)p Fv(\))j(:=)h Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))1423 3117 y Fl(n)1485 3218 y Fw(P)1543 3232 y Fs(j)1600 3218 y Fu(\000)1700 3170 y Fs(Q)1756 3180 y Fn(j)p 1700 3197 89 4 v 1714 3249 a Ft(2)p Fs(t)1798 3117 y Fl(o)1874 3218 y Fw(f)1944 3117 y Fl(\020)2011 3176 y Fs(Q)p 2008 3197 61 4 v 2008 3249 a Ft(2)p Fs(t)2079 3117 y Fl(\021)2155 3218 y Fv(~)-52 b Fw(\021)2196 3185 y Ft(2)2236 3218 y Fv(\()p Fw(H)7 b Fv(\))p Fw(f)2459 3117 y Fl(\020)2526 3176 y Fs(Q)p 2523 3197 V 2523 3249 a Ft(2)p Fs(t)2594 3117 y Fl(\021)15 b(n)2724 3218 y Fw(P)2782 3232 y Fs(j)2839 3218 y Fu(\000)2940 3170 y Fs(Q)2996 3180 y Fn(j)p 2940 3197 89 4 v 2954 3249 a Ft(2)p Fs(t)3038 3117 y Fl(o)3114 3218 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fx(,)25 b(and)d(notice)83 3362 y(that)31 b Fv(\010)f Fx(b)s(elongs)g(to)h Fw(B)5 b(C)956 3329 y Ft(1)949 3389 y Fs(H)1015 3289 y Fl(\000)1057 3362 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))1544 3289 y Fl(\001)1587 3362 y Fx(.)41 b(One)30 b(can)h(c)m(hec)m(k)h(that)224 3620 y Fb(D)p Fv(\010\()p Fw(t)p Fv(\))83 b(=)g Fu(\000)791 3558 y Fv(2)p 791 3599 46 4 v 797 3682 a Fw(t)846 3620 y Fv(\010\()p Fw(t)p Fv(\))20 b Fu(\000)g Fw(\021)s Fu(r)1250 3634 y Fs(j)1287 3620 y Fw(V)g Fv(\()p Fw(Q)p Fv(\))p Fw(f)d Fv(~)-52 b Fw(\021)1605 3582 y Ft(2)1645 3620 y Fw(f)1714 3492 y Fl(\032)1783 3620 y Fw(P)1841 3634 y Fs(j)1898 3620 y Fu(\000)1998 3558 y Fw(Q)2070 3572 y Fs(j)p 1998 3599 109 4 v 2014 3682 a Fv(2)p Fw(t)2117 3492 y Fl(\033)2200 3620 y Fw(\021)24 b Fu(\000)c Fw(\021)2423 3492 y Fl(\032)2491 3620 y Fw(P)2549 3634 y Fs(j)2607 3620 y Fu(\000)2707 3558 y Fw(Q)2779 3572 y Fs(j)p 2707 3599 V 2722 3682 a Fv(2)p Fw(t)2826 3492 y Fl(\033)2909 3620 y Fw(f)c Fv(~)-51 b Fw(\021)3012 3582 y Ft(2)3051 3620 y Fw(f)10 b Fu(r)3182 3634 y Fs(j)3218 3620 y Fw(V)20 b Fv(\()p Fw(Q)p Fv(\))p Fw(\021)710 3871 y Fv(+)807 3810 y(1)p 791 3850 79 4 v 791 3933 a(2)p Fw(t)879 3871 y(\021)942 3743 y Fl(\032)1011 3871 y Fw(P)1069 3885 y Fs(j)1126 3871 y Fu(\000)1227 3809 y Fw(Q)1299 3823 y Fs(j)p 1227 3850 109 4 v 1242 3933 a Fv(2)p Fw(t)1345 3743 y Fl(\033)15 b(\032\022)1563 3871 y Fw(P)33 b Fu(\000)1759 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Fu(\001)20 b(r)p Fw(f)30 b Fv(+)20 b Fu(r)p Fw(f)29 b Fu(\001)2570 3994 y Fl(\022)2637 4123 y Fw(P)k Fu(\000)2833 4061 y Fw(Q)p 2829 4102 V 2829 4185 a Fv(2)p Fw(t)2918 3994 y Fl(\023)o(\033)16 b(\032)3136 4123 y Fw(P)3194 4137 y Fs(j)3251 4123 y Fu(\000)3352 4061 y Fw(Q)3424 4075 y Fs(j)p 3352 4102 109 4 v 3367 4185 a Fv(2)p Fw(t)3470 3994 y Fl(\033)3554 4123 y Fw(\021)s(:)83 4365 y Fx(Ha)m(ving)31 b(in)f(mind)h(the)g(use)f(of)g(Lemma)i(6.4,)f(w)m(e)g(collect)g(some)f (estimates)h(for)f(eac)m(h)i(of)e(these)g(terms.)249 4478 y(The)23 b(second)h(and)f(the)g(third)h(terms)f(on)g(the)g(r.h.s.)g(b)s(elong)h(to)f Fw(L)2423 4445 y Ft(1)2462 4405 y Fl(\000)2504 4478 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)2870 4405 y Fl(\001)2936 4478 y Fx(b)s(ecause)23 b Fw(\021)s Fu(r)3387 4492 y Fs(j)3424 4478 y Fw(V)d(f)c Fv(~)-51 b Fw(\021)25 b Fx(and)90 4591 y Fv(~)-52 b Fw(\021)s(f)10 b Fu(r)262 4605 y Fs(j)298 4591 y Fw(V)20 b(\021)30 b Fx(are)c(in)m(tegrable)i(in)e(norm)h(and)g(the)f 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Fv(2)p Fw(t)1640 5215 y Fl(\033)1724 5344 y Fw(\021)s(')p Fu(i)1866 5212 y Fl(\014)1866 5266 y(\014)1866 5321 y(\014)1866 5375 y(\014)1922 5344 y Fu(\024)2028 5282 y Fw(c)p 2028 5323 40 4 v 2031 5406 a(t)2093 5212 y Fl(\015)2093 5266 y(\015)2093 5321 y(\015)2093 5375 y(\015)2143 5344 y Fw(g)2205 5215 y Fl(\032)2273 5344 y Fw(P)2331 5358 y Fs(j)2388 5344 y Fu(\000)2489 5282 y Fw(Q)2561 5296 y Fs(j)p 2489 5323 109 4 v 2504 5406 a Fv(2)p Fw(t)2607 5215 y Fl(\033)2691 5344 y Fw(\021)s( )2801 5212 y Fl(\015)2801 5266 y(\015)2801 5321 y(\015)2801 5375 y(\015)2867 5212 y(\015)2867 5266 y(\015)2867 5321 y(\015)2867 5375 y(\015)2918 5344 y Fw(g)2980 5215 y Fl(\032)3048 5344 y Fw(P)3106 5358 y Fs(j)3163 5344 y Fu(\000)3264 5282 y Fw(Q)3336 5296 y Fs(j)p 3264 5323 V 3279 5406 a Fv(2)p Fw(t)3382 5215 y Fl(\033)3465 5344 y Fw(\021)s(')3572 5212 y Fl(\015)3572 5266 y(\015)3572 5321 y(\015)3572 5375 y(\015)83 5587 y Fx(for)30 b(some)h(p)s(ositiv)m(e)f (constan)m(t)i Fw(c)25 b(<)g Fu(1)30 b Fx(and)h(all)g Fw( )s(;)15 b(')26 b Fu(2)f(H)q Fx(.)1880 5836 y(19)p eop %%Page: 20 20 bop 249 482 a Fx(Since)35 b(the)g(term)g Fu(\000)950 446 y Ft(2)p 950 461 36 4 v 955 513 a Fs(t)995 482 y Fv(\010\()p Fw(t)p Fv(\))g Fx(satis\034es)f(a)g(similar)h(estimate,)i(w)m(e)e(can)g (apply)g(Lemma)g(6.4)g(with)g Fw(E)3622 497 y Fs(k)3697 482 y Fv(=)83 626 y Fw(F)141 641 y Fs(k)212 626 y Fv(=)311 554 y Fl(p)p 402 554 51 4 v 412 590 a Fs(c)p 412 605 31 4 v 415 657 a(t)452 626 y Fw(g)514 525 y Fl(n)575 626 y Fw(P)633 640 y Fs(j)690 626 y Fu(\000)791 578 y Fs(Q)847 588 y Fn(j)p 791 605 89 4 v 804 657 a Ft(2)p Fs(t)889 525 y Fl(o)964 626 y Fw(\021)s Fx(.)46 b(By)31 b(taking)i(in)m(to)g(accoun)m(t)g(the)g(\034rst)f(statemen)m (t)h(of)f(the)g(presen)m(t)h(lemma)g(w)m(e)83 771 y(obtain)39 b(the)g(existence)g(of)f(s)25 b Fu(\000)h Fv(lim)1324 785 y Fs(t)p FC(!1)1510 771 y Fw(e)1552 739 y Fs(iH)5 b(t)1669 771 y Fv(\010\()p Fw(t)p Fv(\))p Fw(e)1880 739 y FC(\000)p Fs(iH)g(t)2052 771 y Fx(.)65 b(It)38 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2664 1058 61 4 v 2664 1110 a Ft(2)p Fs(t)2735 978 y Fl(\021)15 b(\020)2859 1079 y Fw(P)2917 1093 y Fs(j)2974 1079 y Fu(\000)3075 1031 y Fs(Q)3131 1041 y Fn(j)p 3075 1058 89 4 v 3088 1110 a Ft(2)p Fs(t)3173 978 y Fl(\021)3242 1079 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)3485 1046 y FC(\000)p Fs(iH)e(t)3658 1079 y Fw(')3717 974 y Fl(\015)3717 1029 y(\015)3717 1084 y(\015)83 1215 y Fx(exists)30 b(for)g(eac)m(h)h Fw(')26 b Fu(2)f(H)q Fx(.)40 b(But)30 b(this)h(limit)g(has)f(to)g(b)s(e)g(equal)h(to)f(zero)h(b)s (ecause)g(of)f(\(12\).)580 b Fc(\004)249 1327 y Fx(W)-8 b(e)29 b(no)m(w)h(pro)m(v)m(e)g(the)f(existence)g(of)f(the)h(asymptotic)g(v)m(elo)s (cit)m(y)-8 b(.)41 b(One)29 b(ma)5 b(jor)29 b(ingredien)m(t)i(is)d(a)h(class) f(of)83 1440 y(functions)f(with)h(some)f(sp)s(ecial)g(prop)s(ert)m(y)-8 b(.)40 b(Let)28 b Fo(F)e Fx(b)s(e)g(the)i(set)e(of)h(all)g(functions)g Fw(f)35 b Fu(2)25 b Fw(C)3077 1407 y FC(1)3070 1463 y Fs(c)3151 1440 y Fv(\(I)-18 b(R)3268 1405 y Fs(n)3315 1440 y Fv(\))27 b Fx(suc)m(h)g(that,)83 1553 y(for)34 b(eac)m(h)h Fw(j)j Fu(2)31 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)j(;)d(n)p Fu(g)p Fx(,)35 b(there)g(exists)f (a)g(neigh)m(b)s(ourho)s(o)s(d)h(of)f(the)h(h)m(yp)s(ersurface)g Fw(x)3092 1567 y Fs(j)3160 1553 y Fv(=)c(0)k Fx(in)f(whic)m(h)i Fw(f)83 1666 y Fx(do)s(es)30 b(not)h(dep)s(end)g(on)f Fw(x)939 1680 y Fs(j)976 1666 y Fx(.)40 b(One)31 b(can)f(c)m(hec)m(k)i(that)f Fo(F)f Fx(is)f(dense)i(in)g Fw(C)2431 1680 y Ft(0)2470 1666 y Fv(\(I)-18 b(R)2587 1630 y Fs(n)2634 1666 y Fv(\))p Fx(.)83 1902 y Fj(Lemma)35 b(6.7)45 b Fr(F)-7 b(or)32 b(e)-5 b(ach)34 b Fw(f)h Fu(2)25 b Fw(C)1231 1916 y Ft(0)1270 1902 y Fv(\(I)-18 b(R)1387 1866 y Fs(n)1434 1902 y Fv(\))p Fr(,)32 b(the)h(limit)e(s)20 b Fu(\000)g Fv(lim)2167 1916 y Fs(t)p FC(!1)2353 1902 y Fw(e)2395 1869 y Fs(iH)5 b(t)2512 1902 y Fw(f)2582 1801 y Fl(\020)2649 1861 y Fs(Q)p 2646 1881 61 4 v 2646 1934 a Ft(2)p Fs(t)2717 1801 y Fl(\021)2786 1902 y Fw(e)2828 1869 y FC(\000)p Fs(iH)g(t)3033 1902 y Fr(exists.)249 2142 y(Pr)-5 b(o)g(of:)39 b Fx(Since)26 b(the)h(set)f(of)g(v)m(ectors)g(of)g(the)g(form)h Fw(\021)1954 2109 y Ft(2)1993 2142 y Fv(\()p Fw(H)7 b Fv(\))p Fw(')27 b Fx(with)g Fw(\021)i Fu(2)c Fw(C)2666 2109 y FC(1)2659 2164 y Fs(c)2740 2142 y Fv(\(I)-18 b(R\))26 b Fx(and)h Fw(')e Fu(2)g(H)i Fx(is)e(dense)i(in)83 2277 y Fu(H)q Fx(,)e(it)g(is)f(enough)i(to)f(pro)m(v)m (e)g(the)g(existence)g(of)g(s)8 b Fu(\000)h Fv(lim)1906 2291 y Fs(t)p FC(!1)2092 2277 y Fw(e)2134 2245 y Fs(iH)c(t)2251 2277 y Fw(f)2320 2177 y Fl(\020)2387 2236 y Fs(Q)p 2384 2257 V 2384 2309 a Ft(2)p Fs(t)2455 2177 y Fl(\021)2525 2277 y Fw(e)2567 2245 y FC(\000)p Fs(iH)g(t)2739 2277 y Fw(\021)2787 2245 y Ft(2)2826 2277 y Fv(\()p Fw(H)i Fv(\))p Fx(,)27 b(whic)m(h)f(is)e(equiv)-5 b(alen)m(t)83 2413 y(b)m(y)31 b Fr(i\))e Fx(of)h(Lemma)i(6.2)f(with)f(the)h (existence)g(of)1202 2664 y(s)20 b Fu(\000)40 b Fv(lim)1349 2720 y Fs(t)p FC(!1)1530 2664 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)1773 2627 y Fs(iH)e(t)1891 2664 y Fw(f)1961 2536 y Fl(\022)2041 2603 y Fw(Q)p 2038 2643 79 4 v 2038 2727 a Fv(2)p Fw(t)2126 2536 y Fl(\023)2208 2664 y Fw(e)2250 2627 y FC(\000)p Fs(iH)g(t)2422 2664 y Fw(\021)s Fv(\()p Fw(H)i Fv(\))p Fw(:)959 b Fx(\(16\))83 2915 y(Since)35 b Fo(F)f Fx(is)g(dense)h(in)g Fw(C)934 2929 y Ft(0)973 2915 y Fv(\(I)-18 b(R)1090 2879 y Fs(n)1137 2915 y Fv(\))p Fx(,)36 b(there)f(is)f(also)h(no)f(loss)g(of)h (generalit)m(y)h(in)e(assuming)h(that)g Fw(f)42 b Fu(2)32 b Fo(F)h Fx(.)53 b(F)-8 b(or)83 3051 y Fw(t)31 b Fu(\025)g Fv(1)p Fx(,)k(set)f Fv(\010\()p Fw(t)p Fv(\))d(:=)g Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))1042 2950 y Fl(n)1114 3015 y Ft(1)p 1114 3030 36 4 v 1114 3082 a(2)1174 2950 y Fl(\020)1229 3051 y Fw(P)33 b Fu(\000)1423 3010 y Fs(Q)p 1421 3030 61 4 v 1421 3082 a Ft(2)p Fs(t)1492 2950 y Fl(\021)1566 3051 y Fu(\001)20 b(r)p Fw(f)1757 2950 y Fl(\020)1824 3010 y Fs(Q)p 1821 3030 V 1821 3082 a Ft(2)p Fs(t)1892 2950 y Fl(\021)1966 3051 y Fv(+)2067 3015 y Ft(1)p 2067 3030 36 4 v 2067 3082 a(2)2112 3051 y 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b(o)g(of:)60 b Fx(a\))41 b(Let)g Fw(f)51 b Fu(2)41 b Fw(C)1109 2116 y FC(1)1102 2171 y Fs(c)1183 2149 y Fv(\(I)-18 b(R\))41 b Fx(b)s(e)f(suc)m(h)g(that)h Fw(f)51 b Fv(=)42 b(1)e Fx(in)g(a)h(neigh)m(b)s(ourho)s(o)s(d)g(of)f Fv(0)p Fx(.)70 b(Then,)44 b(since)83 2183 y Fl(n)144 2284 y Fw(f)213 2183 y Fl(\020)277 2239 y FC(j\001j)p 277 2263 59 4 v 290 2316 a Fs(r)346 2183 y Fl(\021o)461 2347 y Fs(r)r FC(\025)p Ft(1)631 2284 y Fx(con)m(v)m(erges)g(b)s(oundedly)e(to)g(the)h(function)g Fv(I)-18 b(R)2238 2248 y Fs(n)2330 2284 y Fu(3)45 b Fw(x)g Fu(7!)f Fv(1)p Fx(,)i(the)c(relation)h Fv(1)-21 b(I)3352 2299 y Ft(I)j(R)3411 2274 y Fn(n)3503 2284 y Fv(=)44 b Fw(I)49 b Fx(is)83 2471 y(satis\034ed)31 b(if)f(s)19 b Fu(\000)h Fv(lim)787 2485 y Fs(r)r FC(!1)981 2471 y Fx(s)g Fu(\000)g Fv(lim)1254 2485 y Fs(t)p FC(!1)1440 2471 y Fw(e)1482 2438 y Fs(iH)5 b(t)1599 2471 y Fw(f)1668 2371 y Fl(\020)1733 2427 y FC(j)p Fs(Q)p FC(j)p 1733 2451 95 4 v 1733 2503 a Ft(2)p Fs(r)r(t)1837 2371 y Fl(\021)1907 2471 y Fw(e)1949 2438 y FC(\000)p Fs(iH)g(t)2146 2471 y Fv(=)25 b Fw(I)7 b Fx(.)249 2607 y(Let)31 b Fw(\021)e Fu(2)c Fw(C)643 2574 y FC(1)636 2630 y Fs(c)717 2607 y Fv(\(I)-18 b(R\))30 b Fx(and)h Fw(r)d Fu(\025)d Fv(1)p Fx(.)41 b(Using)30 b Fr(i\))f Fx(of)h(Lemma)i(6.2,)f(one)f(observ)m(es)h(that)157 2861 y(s)20 b Fu(\000)40 b Fv(lim)304 2917 y Fs(t)p FC(!1)486 2733 y Fl(\032)554 2861 y Fw(I)27 b Fu(\000)20 b Fw(e)754 2824 y Fs(iH)5 b(t)871 2861 y Fw(f)941 2733 y Fl(\022)1018 2800 y Fu(j)p Fw(Q)p Fu(j)p 1018 2840 123 4 v 1018 2924 a Fv(2)p Fw(r)s(t)1150 2733 y Fl(\023)1232 2861 y Fw(e)1274 2824 y FC(\000)p Fs(iH)g(t)1446 2733 y Fl(\033)1529 2861 y Fw(\021)1577 2824 y Ft(2)1617 2861 y Fv(\()p Fw(H)i Fv(\))26 b(=)f Fx(s)20 b Fu(\000)40 b Fv(lim)2039 2917 y Fs(t)p FC(!1)2221 2861 y Fw(e)2263 2824 y Fs(iH)5 b(t)2380 2861 y Fw(\021)s Fv(\()p Fw(H)i Fv(\))2596 2733 y Fl(\032)2665 2861 y Fw(I)28 b Fu(\000)19 b Fw(f)2893 2733 y Fl(\022)2970 2800 y Fu(j)p Fw(Q)p Fu(j)p 2970 2840 V 2970 2924 a Fv(2)p Fw(r)s(t)3102 2733 y Fl(\023\033)3252 2861 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)3495 2824 y FC(\000)p Fs(iH)e(t)3668 2861 y Fw(:)83 3115 y Fx(The)32 b(r.h.s.)g(is)f(zero)h(for)f Fw(r)j Fx(large)f(enough)f(\(dep)s (ending)h(on)f Fw(\021)s Fx(\))g(b)m(y)f(the)h(second)g(assertion)g(of)g (Lemma)g(6.5.)83 3251 y(Hence)g(s)21 b Fu(\000)g Fv(lim)629 3265 y Fs(r)r FC(!1)823 3251 y Fx(s)g Fu(\000)g Fv(lim)1098 3265 y Fs(t)p FC(!1)1284 3150 y Fl(n)1345 3251 y Fw(I)27 b Fu(\000)20 b Fw(e)1545 3218 y Fs(iH)5 b(t)1662 3251 y Fw(f)1732 3150 y Fl(\020)1796 3206 y FC(j)p Fs(Q)p FC(j)p 1796 3230 95 4 v 1796 3282 a Ft(2)p Fs(r)r(t)1900 3150 y Fl(\021)1970 3251 y Fw(e)2012 3218 y FC(\000)p Fs(iH)g(t)2184 3150 y Fl(o)2260 3251 y Fw(\021)2308 3218 y Ft(2)2347 3251 y Fv(\()p Fw(H)i Fv(\))29 b(=)f(0)p Fx(.)46 b(Since)33 b(the)f(set)g(of)g(v)m(ectors)g(of)83 3394 y(the)f(form)f Fw(\021)500 3361 y Ft(2)540 3394 y Fv(\()p Fw(H)7 b Fv(\))p Fw(')31 b Fx(with)g Fw(\021)e Fu(2)c Fw(C)1221 3361 y FC(1)1214 3417 y Fs(c)1295 3394 y Fv(\(I)-18 b(R\))30 b Fx(and)h Fw(')26 b Fu(2)f(H)31 b Fx(is)e(dense)i(in)g Fu(H)q Fx(,)e(this)i(\034nishes)f(the)h(pro)s(of)f(of)g Fr(i\))p Fx(.)249 3507 y(b\))i(The)h(pro)s(of)f(that)h Fv([)p Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(;)15 b(f)10 b Fv(\()p Fu(P)d Fv(\)])31 b(=)d(0)k Fx(for)g(eac)m(h)h Fw(\021)f Fu(2)c Fw(C)2266 3474 y FC(1)2259 3530 y Fs(c)2340 3507 y Fv(\(I)-18 b(R\))32 b Fx(and)h(eac)m(h)g Fw(f)38 b Fu(2)28 b Fw(C)3151 3474 y FC(1)3144 3530 y Fs(c)3225 3507 y Fv(\(I)-18 b(R)3342 3471 y Fs(n)3389 3507 y Fv(\))32 b Fx(is)g(easily)83 3620 y(obtained)37 b(with)f(the)g(help)g(of)f(the)h (statemen)m(t)h Fr(i\))d Fx(of)h(Lemma)i(6.2.)56 b(The)36 b(assertion)g Fr(ii\))f Fx(follo)m(ws)h(then)g(b)m(y)83 3733 y(densit)m(y)31 b(of)f Fw(C)570 3700 y FC(1)563 3755 y Fs(c)644 3733 y Fv(\(I)-18 b(R)761 3697 y Fs(n)808 3733 y Fv(\))31 b Fx(in)f Fw(C)1044 3747 y Ft(0)1083 3733 y Fv(\(I)-18 b(R)1200 3697 y Fs(n)1247 3733 y Fv(\))31 b Fx(and)g(densit)m(y)g(of)f Fw(C)1976 3700 y FC(1)1969 3755 y Fs(c)2050 3733 y Fv(\(I)-18 b(R\))30 b Fx(in)h Fw(C)2403 3747 y Ft(0)2442 3733 y Fv(\(I)-18 b(R\))p Fx(.)1078 b Fc(\004)249 3846 y Fx(The)33 b(next)g(statemen)m(t)h(is)e(usually)h(called) g Fr(minimal)g(velo)-5 b(city)36 b(estimate)p Fx(.)47 b(It)32 b(is)g(a)h(non-trivial)g(conse-)83 3959 y(quence)d(of)g(the)g(Mourre)g (estimate)g(obtained)i(in)d(Section)i(5.)41 b(W)-8 b(e)30 b(p)s(oin)m(t)g (out)g(that)g(the)g Fw(C)3193 3926 y Ft(1)3186 3981 y Fs(u)3232 3959 y Fv(\()p Fw(A)p Fv(\))p Fx(-condition)83 4072 y(of)e(Prop)s(osition)h (6.1)g(is)f(required)h(in)f(order)i(to)e(ful\034l)g(the)h(h)m(yp)s(otheses)g (of)f(Prop)s(osition)h(5.4.)40 b(Let)29 b(us)f(recall)83 4185 y(that)j Fw(\024)p Fv(\()p Fw(H)7 b Fv(\))31 b Fx(is)f(a)g(closed)h(coun)m (table)h(set)e(whic)m(h)i(con)m(tains)g(the)e(thresholds)h(and)g(the)g(eigen) m(v)-5 b(alues)31 b(of)f Fw(H)7 b Fx(.)83 4397 y Fj(Lemma)35 b(6.9)45 b Fr(F)-7 b(or)32 b(e)-5 b(ach)34 b Fw(\021)29 b Fu(2)c Fw(C)1232 4364 y FC(1)1225 4420 y Fs(c)1306 4323 y Fl(\000)1348 4397 y Fv(I)-18 b(R)19 b Fu(n)i Fw(\024)p Fv(\()p Fw(H)7 b Fv(\))1720 4323 y Fl(\001)1762 4397 y Fr(,)32 b(ther)-5 b(e)34 b(exists)f Fw(")2336 4411 y Fs(\021)2403 4397 y Fw(>)25 b Fv(0)33 b Fr(such)g(that)g(for)f(some)g Fw(c)26 b(<)f Fu(1)32 b Fr(and)83 4510 y(al)5 b(l)32 b Fw(')26 b Fu(2)f(H)q Fr(,)1020 4561 y Fl(Z)1111 4587 y FC(1)1071 4767 y Ft(1)1201 4552 y Fl(\015)1201 4607 y(\015)1201 4662 y(\015)1201 4716 y(\015)1251 4684 y Fw(\037)1308 4712 y Fm([0)p Fn(;")1404 4720 y(\021)1442 4712 y Fm(])1483 4556 y Fl(\022)1560 4623 y Fu(j)p Fw(Q)p Fu(j)p 1560 4663 123 4 v 1582 4747 a Fv(2)p Fw(t)1693 4556 y Fl(\023)1775 4684 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)2018 4647 y FC(\000)p Fs(iH)e(t)2191 4684 y Fw(')2250 4552 y Fl(\015)2250 4607 y(\015)2250 4662 y(\015)2250 4716 y(\015)2301 4579 y Ft(2)2365 4623 y Fw(dt)p 2365 4663 81 4 v 2389 4747 a(t)2481 4684 y Fu(\024)25 b Fw(c)p Fu(k)p Fw(')p Fu(k)2765 4647 y Ft(2)2806 4684 y Fw(:)249 4957 y Fr(Pr)-5 b(o)g(of:)43 b Fx(Let)c Fv(~)-51 b Fw(\021)31 b Fu(2)c Fw(C)929 4924 y FC(1)922 4979 y Fs(c)1003 4883 y Fl(\000)1044 4957 y Fv(I)-18 b(R)21 b Fu(n)h Fw(\024)p Fv(\()p Fw(H)7 b Fv(\))1419 4883 y Fl(\001)1493 4957 y Fx(b)s(e)31 b(suc)m(h)h(that)39 b Fv(~)-51 b Fw(\021)s(\021)31 b Fv(=)c Fw(\021)35 b Fx(and)d Fv(0)c Fu(\024)34 b Fv(~)-52 b Fw(\021)31 b Fu(\024)c Fv(1)p Fx(.)45 b(Let)32 b Fw(\022)e Fv(:=)d(min)3470 4972 y Fs(\025)3530 4957 y Fw(\045)3577 4924 y Fs(A)3577 4984 y(H)3644 4957 y Fv(\()p Fw(\025)p Fv(\))83 5070 y Fx(for)37 b Fw(\025)g Fu(2)f Fx(supp)7 b Fv(~)-52 b Fw(\021)s Fx(.)61 b(It)37 b(follo)m(ws)g(from)h(Prop)s(ositions) f(5.3)h(and)f(5.4)h(that)g Fw(\022)h Fx(is)d(strictly)i(p)s(ositiv)m(e)f(and) g(that)90 5183 y Fv(~)-52 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\)[)p Fw(iH)r(;)15 b(A)p Fv(])7 b(~)-52 b Fw(\021)6 b Fv(\()p Fw(H)h Fv(\))26 b Fu(\025)f Fw(\022)9 b Fv(~)-51 b Fw(\021)971 5150 y Ft(2)1010 5183 y Fv(\()p Fw(H)7 b Fv(\))p Fx(.)249 5308 y(W)-8 b(e)31 b(\034x)g(the)h(constan)m(ts)g Fw(")1134 5322 y Fs(\021)1207 5308 y Fx(and)g Fw(")1426 5275 y FC(0)1426 5330 y Fs(\021)1499 5308 y Fx(suc)m(h)f(that)h Fv(0)27 b Fw(<)f(")2111 5322 y Fs(\021)2180 5308 y Fw(<)g(")2319 5275 y FC(0)2319 5330 y Fs(\021)2388 5308 y Fw(<)2517 5272 y Fs(\022)p 2495 5287 79 4 v 2495 5339 a Ft(2)p Fs(n)2599 5234 y Fl(\000)2640 5308 y Fv(sup)2777 5329 y Fs(j)2829 5308 y Fu(k)7 b Fv(~)-52 b Fw(\021)t Fv(\()p Fw(H)7 b Fv(\))p Fw(P)3134 5322 y Fs(j)3171 5308 y Fu(k)3216 5234 y Fl(\001)3258 5257 y FC(\000)p Ft(1)3353 5308 y Fx(.)42 b(The)32 b(case)90 5420 y Fv(~)-52 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))26 b(=)f(0)j Fx(is)g(excluded)g(b)s(ecause)g(the)g(statemen)m(t)i(is)d(then)i(trivial.)40 b(If)26 b Fw(B)5 b Fv(\(0)p Fw(;)15 b(\016)s Fv(\))30 b Fx(denotes)e(the)h (op)s(en)e(ball)i(in)83 5533 y Fv(I)-18 b(R)165 5498 y Fs(n)233 5533 y Fx(of)21 b(cen)m(ter)i Fv(0)f Fx(and)g(radius)g Fw(\016)30 b(>)25 b Fv(0)p Fx(,)e(let)f Fw(f)35 b Fu(2)25 b Fo(F)c Fx(b)s(e)g(suc)m(h)h (that)g(supp)q Fw(f)34 b Fu(\032)25 b Fw(B)5 b Fv(\(0)p Fw(;)15 b(")2802 5500 y FC(0)2802 5556 y Fs(\021)2845 5533 y Fv(\))22 b Fx(and)g Fw(f)35 b Fv(=)25 b(1)d Fx(on)g Fw(B)5 b Fv(\(0)p Fw(;)15 b(")3665 5547 y Fs(\021)3708 5533 y Fv(\))p Fx(.)1880 5836 y(21)p eop %%Page: 22 22 bop 83 482 a Fx(Let)38 b(also)461 458 y Fv(~)441 482 y Fw(f)47 b Fu(2)38 b Fw(C)704 449 y FC(1)697 504 y Fs(c)778 482 y Fv(\(I)-18 b(R)895 446 y Fs(n)942 482 y Fv(\))38 b Fx(with)g(supp)s(ort)g(in)g Fw(B)5 b Fv(\(0)p Fw(;)15 b(")1919 449 y FC(0)1919 504 y Fs(\021)1962 482 y Fv(\))38 b Fx(b)s(e)g(suc)m(h)g(that)2600 458 y Fv(~)2581 482 y Fw(f)9 b(f)47 b Fv(=)37 b Fw(f)47 b Fx(and)39 b Fv(0)f Fu(\024)3322 458 y Fv(~)3302 482 y Fw(f)48 b Fu(\024)37 b Fv(1)p Fx(.)63 b(F)-8 b(or)83 630 y Fw(t)25 b Fu(\025)g Fv(1)p Fx(,)i(set)e Fw(M)10 b Fv(\()p Fw(t)p Fv(\))26 b(:=)818 529 y Fl(\020)873 630 y Fw(P)33 b Fu(\000)1067 589 y Fs(Q)p 1065 609 61 4 v 1065 661 a Ft(2)p Fs(t)1135 529 y Fl(\021)1200 630 y Fu(\001)10 b(r)p Fw(f)1381 529 y Fl(\020)1448 589 y Fs(Q)p 1445 609 V 1445 661 a Ft(2)p Fs(t)1516 529 y Fl(\021)1580 630 y Fv(+)g Fw(f)1731 529 y Fl(\020)1798 589 y Fs(Q)p 1795 609 V 1795 661 a Ft(2)p Fs(t)1866 529 y Fl(\021)1946 630 y Fx(and)25 b Fv(\010\()p Fw(t)p Fv(\))h(:=)f Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(M)18 b Fv(~)-53 b Fw(\021)5 b Fv(\()p Fw(H)i Fv(\))2944 594 y Fs(A)p 2944 609 53 4 v 2958 661 a(t)3014 630 y Fv(~)-52 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(M)3306 597 y FC(\003)3347 630 y Fw(\021)s Fv(\()p Fw(H)g Fv(\))p Fx(.)39 b(The)83 765 y(statemen)m(t)32 b Fr(v\))f Fx(of)f(Lemma)i(6.2)f(assures)f(that)h Fv(\010)f Fx(is)h(a)f(b)s(ounded)i(op)s(erator-v)-5 b(alued)32 b(mapping.)42 b(Moreo)m(v)m(er,)83 878 y(one)31 b(can)f(easily)g(c)m(hec)m(k)i(that)f Fv(\010)f Fx(is)f(di\033eren)m(tiable)k(in)d(norm)h(with)g(b)s(ounded)g (deriv)-5 b(ativ)m(e.)42 b(It)30 b(follo)m(ws)g(then)83 991 y(that)h Fv(\010)25 b Fu(2)g Fw(B)5 b(C)602 958 y Ft(1)595 1018 y Fs(H)661 918 y Fl(\000)703 991 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))1190 918 y Fl(\001)1233 991 y Fx(.)41 b(So,)30 b(let)h(us)f(calculate)i(the)e(Heisen)m(b)s(erg)h (deriv)-5 b(ativ)m(e)32 b(of)e Fv(\010)p Fx(:)114 1233 y Fb(D)p Fv(\010\()p Fw(t)p Fv(\))83 b(=)g Fu(\000)p Fw(\021)s Fu(r)p Fw(f)29 b Fu(\001)21 b(r)p Fw(V)f Fv(\()p Fw(Q)p Fv(\))7 b(~)-52 b Fw(\021)1265 1171 y(A)p 1265 1212 69 4 v 1282 1295 a(t)1349 1233 y Fv(~)h Fw(\021)s(M)1489 1195 y FC(\003)1529 1233 y Fw(\021)23 b Fu(\000)d Fw(\021)s(M)d Fv(~)-52 b Fw(\021)1893 1171 y(A)p 1893 1212 V 1911 1295 a(t)1978 1233 y Fv(~)g Fw(\021)t Fu(r)p Fw(f)29 b Fu(\001)20 b(r)p Fw(V)g Fv(\()p Fw(Q)p Fv(\))p Fw(\021)600 1460 y Fv(+)681 1399 y(1)p 681 1439 46 4 v 687 1523 a Fw(t)736 1460 y(\021)799 1332 y Fl(\024)847 1460 y Fw(P)34 b Fu(\000)1043 1399 y Fw(Q)p 1039 1439 79 4 v 1039 1523 a Fv(2)p Fw(t)1128 1332 y Fl(\025)1176 1366 y Fn(T)1244 1460 y Fw(@)1297 1423 y Ft(2)1336 1460 y Fw(f)1406 1332 y Fl(\024)1454 1460 y Fw(P)f Fu(\000)1649 1399 y Fw(Q)p 1646 1439 V 1646 1523 a Fv(2)p Fw(t)1734 1332 y Fl(\025)1804 1460 y Fv(~)-52 b Fw(\021)1856 1399 y(A)p 1856 1439 69 4 v 1873 1523 a(t)1940 1460 y Fv(~)h Fw(\021)s(M)2080 1423 y FC(\003)2120 1460 y Fw(\021)23 b Fv(+)2289 1399 y(1)p 2289 1439 46 4 v 2295 1523 a Fw(t)2345 1460 y(\021)s(M)17 b Fv(~)-52 b Fw(\021)2549 1399 y(A)p 2549 1439 69 4 v 2567 1523 a(t)2634 1460 y Fv(~)h Fw(\021)2691 1332 y Fl(\024)2739 1460 y Fw(P)33 b Fu(\000)2934 1399 y Fw(Q)p 2931 1439 79 4 v 2931 1523 a Fv(2)p Fw(t)3019 1332 y Fl(\025)3067 1366 y Fn(T)3135 1460 y Fw(@)3188 1423 y Ft(2)3228 1460 y Fw(f)3298 1332 y Fl(\024)3346 1460 y Fw(P)g Fu(\000)3541 1399 y Fw(Q)p 3538 1439 V 3538 1523 a Fv(2)p Fw(t)3626 1332 y Fl(\025)3689 1460 y Fw(\021)600 1712 y Fv(+)p Fw(\021)s(M)17 b Fv(~)-52 b Fw(\021)881 1583 y Fl(\032\024)997 1712 y Fw(iH)r(;)1156 1650 y(A)p 1156 1691 69 4 v 1174 1774 a(t)1234 1583 y Fl(\025)1303 1712 y Fu(\000)1406 1650 y Fw(A)p 1404 1691 73 4 v 1404 1774 a(t)1437 1748 y Ft(2)1486 1583 y Fl(\033)1576 1712 y Fv(~)g Fw(\021)s(M)1715 1674 y FC(\003)1755 1712 y Fw(\021)24 b Fv(+)c Fw(O)s Fv(\()p Fw(t)2055 1674 y FC(\000)p Ft(2)2149 1712 y Fv(\))p Fw(:)1398 b Fx(\(18\))83 1947 y(In)30 b(order)h(to)g(b)s(e)f(able)g(to)h(use)f(Lemma)i(6.3)e(w)m(e)h (obtain)h(no)m(w)f(some)g(estimates)f(for)g(eac)m(h)i(of)e(these)g(terms.)249 2060 y(Since)25 b Fw(f)34 b Fu(2)25 b Fo(F)p Fx(,)g(the)f(statemen)m(t)h Fr(ii\))e Fx(of)g(Lemma)i(6.2)f(implies)h(that)f(the)g(\034rst)g(t)m(w)m(o)h (terms)f(on)g(the)g(r.h.s.)g(of)83 2173 y(\(18\))34 b(b)s(elong)g(to)g Fw(L)745 2140 y Ft(1)785 2099 y Fl(\000)826 2173 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)1192 2099 y Fl(\001)1236 2173 y Fx(.)50 b(Some)35 b(further)f(comm)m(utator)i(calculations)f(\(based)f (on)g(Lemma)h(6.2\))83 2286 y(sho)m(w)c(that)g(the)g(third)g(and)f(the)h (fourth)g(terms)f(on)h(the)g(r.h.s.)f(of)g(\(18\))h(can)g(b)s(e)f(rewritten)i (as)273 2432 y Fs(n)229 2459 y Fl(X)206 2657 y Fs(j;k)r Ft(=1)408 2484 y Fv(1)p 408 2525 46 4 v 414 2608 a Fw(t)464 2546 y(\021)527 2418 y Fl(\022)594 2546 y Fw(P)652 2560 y Fs(j)709 2546 y Fu(\000)810 2484 y Fw(Q)882 2498 y Fs(j)p 810 2525 109 4 v 825 2608 a Fv(2)p Fw(t)928 2418 y Fl(\023)1010 2546 y Fw(g)1053 2560 y Fs(j)1105 2418 y Fl(\022)1186 2484 y Fw(Q)p 1182 2525 79 4 v 1182 2608 a Fv(2)p Fw(t)1271 2418 y Fl(\023)1353 2546 y Fv(\011)1424 2561 y Fs(j)t(k)1498 2546 y Fv(\()p Fw(t)p Fv(\))p Fw(g)1644 2561 y Fs(k)1703 2418 y Fl(\022)1783 2484 y Fw(Q)p 1780 2525 V 1780 2608 a Fv(2)p Fw(t)1868 2418 y Fl(\023)15 b(\022)2017 2546 y Fw(P)2075 2561 y Fs(k)2138 2546 y Fu(\000)2239 2484 y Fw(Q)2311 2499 y Fs(k)p 2239 2525 115 4 v 2257 2608 a Fv(2)p Fw(t)2364 2418 y Fl(\023)2446 2546 y Fw(\021)24 b Fv(+)19 b Fw(O)s Fv(\()p Fw(t)2745 2508 y FC(\000)p Ft(2)2840 2546 y Fv(\))h(+)2996 2484 y(1)p 2996 2525 46 4 v 3002 2608 a Fw(t)3052 2546 y(L)3114 2508 y Ft(1)3153 2472 y Fl(\000)3195 2546 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)3561 2472 y Fl(\001)3604 2546 y Fw(;)3607 2766 y Fx(\(19\))83 2968 y(where)26 b Fv(\011)410 2983 y Fs(j)t(k)510 2968 y Fv(=)f(\()p Fw(@)694 2935 y Ft(2)689 2995 y Fs(j)t(k)765 2968 y Fw(f)10 b Fv(\))d(~)-52 b Fw(\021)913 2932 y Fs(A)p 913 2947 53 4 v 927 2999 a(t)982 2968 y Fv(~)h Fw(\021)s(M)1122 2935 y FC(\003)1168 2968 y Fv(~)g Fw(\021)14 b Fv(+)j(~)-51 b Fw(\021)s(M)17 b Fv(~)-52 b Fw(\021)1507 2932 y Fs(A)p 1507 2947 V 1521 2999 a(t)1577 2968 y Fv(~)g Fw(\021)t Fv(\()p Fw(@)1707 2935 y Ft(2)1702 2995 y Fs(j)t(k)1777 2968 y Fw(f)10 b Fv(\))25 b Fx(is)h(a)f(b)s(ounded)i(op)s(erator-v)-5 b(alued)27 b(mapping)g(and)f(eac)m(h)83 3095 y Fw(g)126 3109 y Fs(j)199 3095 y Fx(b)s(elongs)36 b(to)h Fw(C)718 3062 y FC(1)711 3117 y Fs(c)792 3095 y Fv(\(I)-18 b(R)909 3059 y Fs(n)956 3095 y Fv(\))p Fx(,)38 b(has)e(its)g(supp)s(ort)g(in)h Fw(Y)1878 3057 y FC(\000)1858 3122 y Fs(j)1961 3095 y Fu([)24 b Fw(Y)2119 3057 y Ft(+)2099 3122 y Fs(j)2214 3095 y Fx(and)37 b(satis\034es)f Fw(g)2778 3109 y Fs(j)2814 3095 y Fu(r)2890 3109 y Fs(j)2927 3095 y Fw(f)44 b Fv(=)35 b Fu(r)3198 3109 y Fs(j)3234 3095 y Fw(f)10 b Fx(.)58 b(By)35 b(taking)83 3208 y(Lemma)29 b(6.6)e(in)m(to)h (accoun)m(t,)i(one)d(\034nds)h(that)g(the)f(\034rst)g(term)h(of)f(\(19\))h (is)f(in)m(tegrable)i(along)f(the)f(ev)m(olution.)249 3321 y(F)-8 b(or)31 b(the)g(\034fth)f(term)h(on)g(the)f(r.h.s.)h(of)f(\(18\))h (one)f(can)h(c)m(hec)m(k)h(that)399 3562 y Fw(\021)s(M)17 b Fv(~)-52 b Fw(\021)609 3433 y Fl(\024)657 3562 y Fw(iH)r(;)816 3500 y(A)p 816 3541 69 4 v 834 3624 a(t)894 3433 y Fl(\025)964 3562 y Fv(~)g Fw(\021)t(M)1104 3524 y FC(\003)1143 3562 y Fw(\021)29 b Fu(\025)1323 3500 y Fw(\022)p 1323 3541 46 4 v 1329 3624 a(t)1378 3562 y(\021)s(M)17 b Fv(~)-52 b Fw(\021)1572 3524 y Ft(2)1613 3562 y Fw(M)1711 3524 y FC(\003)1750 3562 y Fw(\021)29 b Fv(=)1930 3500 y Fw(\022)p 1930 3541 V 1936 3624 a(t)1985 3562 y(\021)s(M)10 b(M)2229 3524 y FC(\003)2269 3562 y Fw(\021)23 b Fv(+)d Fw(O)s Fv(\()p Fw(t)2568 3524 y FC(\000)p Ft(2)2663 3562 y Fv(\))g(+)2819 3500 y(1)p 2819 3541 V 2825 3624 a Fw(t)2875 3562 y(L)2937 3524 y Ft(1)2976 3488 y Fl(\000)3018 3562 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)3384 3488 y Fl(\001)3427 3562 y Fw(;)83 3802 y Fx(where)29 b(the)g(equalit)m(y)g(has)g(b)s(een)f (obtained)i(with)f(the)g(help)g(of)f Fr(i\))g Fx(and)h Fr(iii\))e Fx(of)i(Lemma)g(6.2.)41 b(F)-8 b(urthermore,)83 3915 y(since)26 b Fw(A)g Fv(=)500 3879 y Ft(1)p 500 3894 36 4 v 500 3947 a(4)545 3915 y Fv(\()p Fw(P)f Fu(\001)12 b Fw(Q)g Fv(+)g Fw(Q)g Fu(\001)g Fw(P)h Fv(\))p Fx(,)27 b(one)g(\034nds)f(that)h(\(b)m(y)g(comm)m(uting)h(the) f(\034rst)2688 3891 y Fv(~)2668 3915 y Fw(f)35 b Fx(to)27 b(the)f(righ)m(t)i (for)e(the)g(second)83 4028 y(inequalit)m(y\))167 4137 y Fl(\015)167 4192 y(\015)167 4246 y(\015)167 4301 y(\015)237 4245 y Fv(~)217 4269 y Fw(f)16 b Fv(~)-51 b Fw(\021)330 4207 y(A)p 330 4248 69 4 v 348 4331 a(t)415 4269 y Fv(~)f Fw(\021)476 4245 y Fv(~)456 4269 y Fw(f)511 4137 y Fl(\015)511 4192 y(\015)511 4246 y(\015)511 4301 y(\015)586 4269 y Fu(\024)682 4137 y Fl(\015)682 4192 y(\015)682 4246 y(\015)682 4301 y(\015)753 4245 y Fv(~)733 4269 y Fw(f)16 b Fv(~)-52 b Fw(\021)t(P)33 b Fu(\001)985 4207 y Fw(Q)p 982 4248 79 4 v 982 4331 a Fv(2)p Fw(t)1077 4269 y Fv(~)-52 b Fw(\021)1138 4245 y Fv(~)1119 4269 y Fw(f)1173 4137 y Fl(\015)1173 4192 y(\015)1173 4246 y(\015)1173 4301 y(\015)1249 4269 y Fu(\024)25 b Fw(n)15 b Fv(sup)1467 4346 y Fs(j)1567 4141 y Fl(\032)1635 4269 y Fu(k)7 b Fv(~)-52 b Fw(\021)t(P)1787 4283 y Fs(j)1824 4269 y Fu(k)1884 4137 y Fl(\015)1884 4192 y(\015)1884 4246 y(\015)1884 4301 y(\015)1954 4245 y Fv(~)1935 4269 y Fw(f)1999 4207 y(Q)2071 4221 y Fs(j)p 1999 4248 109 4 v 2014 4331 a Fv(2)p Fw(t)2117 4137 y Fl(\015)2117 4192 y(\015)2117 4246 y(\015)2117 4301 y(\015)2168 4141 y(\033)2256 4269 y Fv(+)20 b Fw(O)s Fv(\()p Fw(t)2487 4231 y FC(\000)p Ft(1)2581 4269 y Fv(\))26 b Fu(\024)f Fw(n")2835 4231 y FC(0)2835 4291 y Fs(\021)2892 4269 y Fv(sup)2944 4346 y Fs(j)3044 4269 y Fu(k)7 b Fv(~)-51 b Fw(\021)s(P)3196 4283 y Fs(j)3233 4269 y Fu(k)20 b Fv(+)g Fw(O)s Fv(\()p Fw(t)3529 4231 y FC(\000)p Ft(1)3624 4269 y Fv(\))p Fw(;)83 4539 y Fx(whic)m(h)30 b(is)e(less)f(than)813 4503 y Fs(\022)p 812 4518 36 4 v 812 4570 a Ft(2)874 4539 y Fv(+)16 b Fw(O)s Fv(\()p Fw(t)1101 4506 y FC(\000)p Ft(1)1195 4539 y Fv(\))p Fx(.)40 b(It)28 b(follo)m(ws)h(that)g Fu(\000)1961 4503 y Ft(1)p 1961 4518 V 1966 4570 a Fs(t)2006 4539 y Fw(\021)s(M)2172 4515 y Fv(~)2152 4539 y Fw(f)16 b Fv(~)-51 b Fw(\021)2265 4503 y Fs(A)p 2265 4518 53 4 v 2279 4570 a(t)2334 4539 y Fv(~)g Fw(\021)2396 4515 y Fv(~)2376 4539 y Fw(f)9 b(M)2528 4506 y FC(\003)2568 4539 y Fw(\021)29 b Fu(\025)c(\000)2831 4503 y Fs(\022)p 2819 4518 61 4 v 2819 4570 a Ft(2)p Fs(t)2889 4539 y Fw(\021)s(M)10 b(M)3133 4506 y FC(\003)3173 4539 y Fw(\021)20 b Fv(+)c Fw(O)s Fv(\()p Fw(t)3465 4506 y FC(\000)p Ft(2)3559 4539 y Fv(\))28 b Fx(and)83 4651 y(hence)j(one)g(gets)f(that)h(:)516 4887 y Fw(\021)s(M)17 b Fv(~)-52 b Fw(\021)726 4759 y Fl(\032\024)842 4887 y Fw(iH)r(;)1001 4826 y(A)p 1001 4866 69 4 v 1019 4950 a(t)1079 4759 y Fl(\025)1148 4887 y Fu(\000)1251 4826 y Fw(A)p 1249 4866 73 4 v 1249 4950 a(t)1282 4923 y Ft(2)1331 4759 y Fl(\033)1421 4887 y Fv(~)g Fw(\021)s(M)1560 4850 y FC(\003)1600 4887 y Fw(\021)29 b Fu(\025)1796 4826 y Fw(\022)p 1780 4866 79 4 v 1780 4950 a Fv(2)p Fw(t)1868 4887 y(\021)s(M)10 b(M)2112 4850 y FC(\003)2152 4887 y Fw(\021)24 b Fv(+)19 b Fw(O)s Fv(\()p Fw(t)2451 4850 y FC(\000)p Ft(2)2546 4887 y Fv(\))h(+)2702 4826 y(1)p 2702 4866 46 4 v 2708 4950 a Fw(t)2758 4887 y(L)2820 4850 y Ft(1)2859 4814 y Fl(\000)2901 4887 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)3267 4814 y Fl(\001)3310 4887 y Fw(:)272 b Fx(\(20\))83 5138 y(F)-8 b(rom)31 b(the)f(inequalit)m(y)i Fv(\()p Fw(a)20 b Fv(+)f Fw(b)p Fv(\)\()p Fw(a)i Fv(+)e Fw(b)p Fv(\))1430 5105 y FC(\003)1495 5138 y Fu(\025)25 b Fv(\(1)20 b Fu(\000)g Fw(\027)6 b Fv(\))p Fw(aa)1964 5105 y FC(\003)2023 5138 y Fv(+)19 b(\(1)i Fu(\000)2316 5103 y Ft(1)p 2314 5118 40 4 v 2314 5170 a Fs(\027)2363 5138 y Fv(\))p Fw(bb)2476 5105 y FC(\003)2516 5138 y Fx(,)30 b(v)-5 b(alid)30 b(for)g(an)m(y)h Fw(\027)k Fx(with)c Fv(0)26 b Fw(<)f(\027)31 b(<)25 b Fv(1)83 5251 y Fx(one)31 b(deduces)g(that)g(\(tak)m(e)g Fw(\027)g Fv(=)1195 5215 y Ft(1)p 1195 5230 36 4 v 1195 5283 a(2)1240 5251 y Fx(\))827 5501 y Fw(\021)s(M)10 b(M)1071 5463 y FC(\003)1111 5501 y Fw(\021)28 b Fu(\025)1290 5439 y Fv(1)p 1290 5480 46 4 v 1290 5563 a(2)1346 5501 y Fw(\021)s(f)1449 5463 y Ft(2)1488 5501 y Fw(\021)23 b Fu(\000)d Fw(\021)1711 5373 y Fl(\032)1779 5501 y Fw(P)33 b Fu(\000)1974 5439 y Fw(Q)p 1971 5480 79 4 v 1971 5563 a Fv(2)p Fw(t)2059 5373 y Fl(\033)2148 5501 y Fu(\001)20 b(r)p Fw(f)10 b Fu(r)p Fw(f)29 b Fu(\001)2519 5373 y Fl(\032)2587 5501 y Fw(P)k Fu(\000)2782 5439 y Fw(Q)p 2779 5480 V 2779 5563 a Fv(2)p Fw(t)2867 5373 y Fl(\033)2951 5501 y Fw(\021)s(:)583 b Fx(\(21\))1880 5836 y(22)p eop %%Page: 23 23 bop 83 495 a Fx(One)30 b(can)h(c)m(hec)m(k)g(with)f(Lemma)h(6.6)g(that)f(the) g(term)1931 459 y Ft(1)p 1931 474 36 4 v 1936 526 a Fs(t)1976 495 y Fw(\021)2039 394 y Fl(n)2100 495 y Fw(P)j Fu(\000)2295 454 y Fs(Q)p 2292 474 61 4 v 2292 526 a Ft(2)p Fs(t)2363 394 y Fl(o)2443 495 y Fu(\001)19 b(r)p Fw(f)10 b Fu(r)p Fw(f)28 b Fu(\001)2811 394 y Fl(n)2872 495 y Fw(P)33 b Fu(\000)3066 454 y Fs(Q)p 3064 474 V 3064 526 a Ft(2)p Fs(t)3135 394 y Fl(o)3210 495 y Fw(\021)g Fx(is)d(in)m(tegrable)83 630 y(along)k(the)f(ev)m(olution.)49 b(Th)m(us)34 b(b)m(y)f(inserting)h(\(21\))f(in)m(to)h(\(20\),)g(b)m(y)f (applying)h(then)g(Lemma)g(6.3)f(with)h(the)83 743 y(term)26 b Fw(B)368 710 y FC(\003)407 743 y Fv(\()p Fw(t)p Fv(\))p Fw(B)5 b Fv(\()p Fw(t)p Fv(\))27 b Fx(equal)f(to)1076 707 y Fs(\022)p 1063 722 V 1063 775 a Ft(4)p Fs(t)1134 743 y Fw(\021)s(f)1237 710 y Ft(2)1276 743 y Fw(\021)j Fx(and)d(b)m(y)g(taking)h(in)m(to)f(accoun)m (t)i(all)e(previous)g(estimates,)h(one)f(obtains)83 856 y(the)31 b(exp)s(ected)f(result.)2836 b Fc(\004)249 969 y Fx(In)34 b(the)h(next)f (lemma,)j(w)m(e)e(pro)m(v)m(e)h(that)f(the)f(subspace)h(of)f Fu(H)h Fx(of)f(the)h(states)f(with)h(zero)g(asymptotic)83 1082 y(v)m(elo)s(cit)m(y)c(is)f(equal)h(to)f(the)h(subspace)f(spanned)h(b)m(y)g (the)g(eigen)m(v)m(ectors)h(of)e Fw(H)7 b Fx(.)83 1294 y Fj(Lemma)35 b(6.10)45 b Fr(The)32 b(r)-5 b(ange)34 b(of)e Fw(E)1284 1323 y Fq(f)p Fm(0)p Fq(g)1386 1294 y Fv(\()p Fu(P)7 b Fv(\))34 b Fr(is)e(e)-5 b(qual)34 b(to)e(the)h(r)-5 b(ange)34 b(of)e Fw(E)2562 1308 y Ft(p)2605 1294 y Fv(\()p Fw(H)7 b Fv(\))p Fr(.)249 1507 y(Pr)-5 b(o)g(of:)41 b Fx(First,)30 b(let)h Fw(')25 b Fu(2)g(H)31 b Fx(suc)m(h)g(that)g Fw(H)7 b(')26 b Fv(=)f Fw(\025')p Fx(.)40 b(F)-8 b(or)31 b(eac)m(h)h Fw(f)i Fu(2)25 b Fw(C)2618 1521 y Ft(0)2657 1507 y Fv(\(I)-18 b(R)2774 1471 y Fs(n)2821 1507 y Fv(\))31 b Fx(and)g Fw(t)25 b Fu(\025)g Fv(1)p Fx(,)30 b(one)h(has)178 1629 y Fl(\015)178 1684 y(\015)178 1738 y(\015)178 1793 y(\015)229 1761 y Fw(e)271 1724 y Fs(iH)5 b(t)388 1761 y Fw(f)457 1633 y Fl(\022)537 1700 y Fw(Q)p 534 1740 79 4 v 534 1824 a Fv(2)p Fw(t)622 1633 y Fl(\023)704 1761 y Fw(e)746 1724 y FC(\000)p Fs(iH)g(t)918 1761 y Fw(')21 b Fu(\000)f Fw(f)10 b Fv(\(0\))p Fw(')1318 1629 y Fl(\015)1318 1684 y(\015)1318 1738 y(\015)1318 1793 y(\015)1395 1761 y Fv(=)1491 1629 y Fl(\015)1491 1684 y(\015)1491 1738 y(\015)1491 1793 y(\015)1541 1761 y Fw(e)1583 1724 y Fs(i)p Ft(\()p Fs(H)5 b FC(\000)p Fs(\025)p Ft(\))p Fs(t)1851 1761 y Fw(f)1921 1633 y Fl(\022)2001 1700 y Fw(Q)p 1997 1740 V 1997 1824 a Fv(2)p Fw(t)2086 1633 y Fl(\023)2168 1761 y Fw(')20 b Fu(\000)g Fw(f)10 b Fv(\(0\))p Fw(')2567 1629 y Fl(\015)2567 1684 y(\015)2567 1738 y(\015)2567 1793 y(\015)2644 1761 y Fu(\024)2740 1629 y Fl(\015)2740 1684 y(\015)2740 1738 y(\015)2740 1793 y(\015)2790 1633 y(\022)2857 1761 y Fw(f)2927 1633 y Fl(\022)3007 1700 y Fw(Q)p 3004 1740 V 3004 1824 a Fv(2)p Fw(t)3092 1633 y Fl(\023)3179 1761 y Fu(\000)20 b Fw(f)10 b Fv(\(0\))3440 1633 y Fl(\023)3523 1761 y Fw(')3582 1629 y Fl(\015)3582 1684 y(\015)3582 1738 y(\015)3582 1793 y(\015)3648 1761 y Fw(:)83 2015 y Fx(The)40 b(r.h.s.)h(tends)f(to)g(zero)g(as)g Fw(t)g Fx(increases,)i(and)f(therefore)g Fw(f)10 b Fv(\()p Fu(P)d Fv(\))p Fw(')42 b Fv(=)f Fw(f)10 b Fv(\(0\))p Fw(')p Fx(,)44 b(or)c(equiv)-5 b(alen)m(tly)41 b Fw(')g Fu(2)83 2128 y Fw(E)150 2156 y Fq(f)p Fm(0)p Fq(g)253 2128 y Fv(\()p Fu(P)7 b Fv(\))p Fu(H)q Fx(.)51 b(Since)35 b Fw(E)854 2142 y Ft(p)897 2128 y Fv(\()p Fw(H)7 b Fv(\))35 b Fx(and)f Fw(E)1331 2156 y Fq(f)p Fm(0)p Fq(g)1434 2128 y Fv(\()p Fu(P)7 b Fv(\))35 b Fx(are)f(closed)g(subspaces,)h(it)f(follo)m(ws)g(that)h (an)m(y)f Fw(')e Fu(2)e Fw(E)3493 2142 y Ft(p)3537 2128 y Fv(\()p Fw(H)7 b Fv(\))p Fu(H)83 2241 y Fx(b)s(elongs)30 b(to)h Fw(E)585 2269 y Fq(f)p Fm(0)p Fq(g)687 2241 y Fv(\()p Fu(P)7 b Fv(\))p Fu(H)q Fx(.)249 2354 y(Let)23 b(us)f(no)m(w)h(sho)m(w)g(that)g Fw(E)1162 2382 y Fq(f)p Fm(0)p Fq(g)1264 2354 y Fv(\()p Fu(P)7 b Fv(\))p Fu(H)24 b Fx(is)e(orthogonal)i(to)f(the)f(con)m(tin)m(uous)j (subspace)d(of)g Fu(H)h Fx(with)g(resp)s(ect)83 2467 y(to)33 b Fw(H)7 b Fx(.)47 b(So)33 b(let)g Fw(')c Fu(2)g Fw(E)857 2495 y Fq(f)p Fm(0)p Fq(g)959 2467 y Fv(\()p Fu(P)7 b Fv(\))p Fu(H)q Fx(,)34 b Fr(i.e.)e Fw(f)10 b Fv(\()p Fu(P)d Fv(\))p Fw(')31 b Fv(=)d Fw(f)10 b Fv(\(0\))p Fw(')34 b Fx(for)e(eac)m(h)i Fw(f)k Fu(2)29 b Fw(C)2624 2481 y Ft(0)2663 2467 y Fv(\(I)-18 b(R)2780 2431 y Fs(n)2827 2467 y Fv(\))p Fx(.)48 b(It)32 b(is)g(enough)i(to)f (pro)m(v)m(e)83 2593 y(that)e Fu(h)p Fw( )s(;)15 b(')p Fu(i)28 b Fv(=)e(0)31 b Fx(for)g(an)m(y)g Fw( )j Fx(satisfying)c Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw( )31 b Fv(=)26 b Fw( )34 b Fx(with)d Fw(\021)e Fu(2)d Fw(C)2437 2560 y FC(1)2430 2616 y Fs(c)2511 2520 y Fl(\000)2553 2593 y Fv(I)-18 b(R)20 b Fu(n)h Fw(\024)p Fv(\()p Fw(H)7 b Fv(\))2926 2520 y Fl(\001)2969 2593 y Fx(.)41 b(Let)32 b Fw(")3240 2607 y Fs(\021)3313 2593 y Fx(b)s(e)e(giv)m(en)i(b)m(y) 83 2706 y(the)c(previous)g(lemma,)h(and)f(let)g Fw(f)35 b Fu(2)25 b Fw(C)1446 2673 y FC(1)1439 2729 y Fs(c)1520 2706 y Fv(\(I)-18 b(R)1637 2670 y Fs(n)1684 2706 y Fv(\))28 b Fx(with)g(supp)s(ort)g(in)f Fw(B)5 b Fv(\(0)p Fw(;)15 b(")2620 2720 y Fs(\021)2663 2706 y Fv(\))28 b Fx(b)s(e)g(suc)m(h)g(that)g Fw(f)10 b Fv(\(0\))26 b Fu(6)p Fv(=)f(0)p Fx(.)40 b(W)-8 b(e)83 2845 y(observ)m(e)34 b(that)f(s)22 b Fu(\000)f Fv(lim)882 2859 y Fs(t)p FC(!1)1068 2845 y Fw(\021)s(e)1158 2812 y Fs(iH)5 b(t)1276 2845 y Fw(f)1331 2812 y Ft(2)1385 2745 y Fl(\020)1452 2804 y Fs(Q)p 1449 2825 61 4 v 1449 2877 a Ft(2)p Fs(t)1520 2745 y Fl(\021)1589 2845 y Fw(e)1631 2812 y FC(\000)p Fs(iH)g(t)1803 2845 y Fw(\021)33 b Fv(=)c(0)34 b Fx(since)f(the)g(limit)g(exists)g(and)g(since)g(there)h (exists)83 3034 y Fw(c)26 b(<)f Fu(1)30 b Fx(suc)m(h)h(that)764 2961 y Fl(R)825 2987 y FC(1)807 3066 y Ft(1)915 2929 y Fl(\015)915 2984 y(\015)915 3038 y(\015)965 3034 y Fw(f)1035 2933 y Fl(\020)1102 2993 y Fs(Q)p 1099 3013 V 1099 3065 a Ft(2)p Fs(t)1170 2933 y Fl(\021)1239 3034 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)1482 3001 y FC(\000)p Fs(iH)e(t)1655 3034 y Fw(\035)1707 2929 y Fl(\015)1707 2984 y(\015)1707 3038 y(\015)1758 2956 y Ft(2)1823 2998 y Fs(dt)p 1823 3013 62 4 v 1841 3065 a(t)1920 3034 y Fu(\024)25 b Fw(c)p Fu(k)p Fw(\035)s Fu(k)2197 3001 y Ft(2)2268 3034 y Fx(for)30 b(all)h Fw(\035)d Fu(2)d(H)q Fx(.)40 b(W)-8 b(e)31 b(\034nally)f(ha)m(v)m(e)376 3324 y Fu(h)p Fw( )s(;)15 b(')p Fu(i)27 b Fv(=)823 3263 y(1)p 740 3304 210 4 v 740 3387 a Fw(f)795 3361 y Ft(2)834 3387 y Fv(\(0\))960 3324 y Fu(h)p Fw(\021)s( )s(;)15 b(f)1200 3287 y Ft(2)1241 3324 y Fv(\()p Fu(P)7 b Fv(\))p Fw(\021)s(')p Fu(i)27 b Fv(=)1739 3263 y(1)p 1656 3304 V 1656 3387 a Fw(f)1711 3361 y Ft(2)1750 3387 y Fv(\(0\))1912 3324 y(lim)1891 3380 y Fs(t)p FC(!1)2058 3324 y Fu(h)p Fw(\021)s(e)2183 3287 y Fs(iH)5 b(t)2301 3324 y Fw(f)2356 3287 y Ft(2)2410 3196 y Fl(\022)2490 3263 y Fw(Q)p 2487 3304 79 4 v 2487 3387 a Fv(2)p Fw(t)2575 3196 y Fl(\023)2657 3324 y Fw(e)2699 3287 y FC(\000)p Fs(iH)g(t)2871 3324 y Fw(\021)s( )s(;)15 b(')p Fu(i)27 b Fv(=)e(0)p Fw(:)3697 3572 y Fc(\004)83 3858 y Fy(7)131 b(Asymptotic)45 b(Completeness)83 4061 y Fx(In)31 b(this)g(last)h(section,)g(w)m(e)g(pro)m(v)m(e)h(the)e (existence)h(of)f(some)h(suitably)f(de\034ned)i(w)m(a)m(v)m(e)g(op)s(erators) f(and,)g(as)f(a)83 4174 y(b)m(y-pro)s(duct,)k(obtain)f(the)f(asymptotic)h (completeness.)49 b(F)-8 b(or)34 b(this)f(purp)s(ose,)h(w)m(e)f(assume)g (\034rstly)g(that)h Fw(V)83 4287 y Fx(is)27 b(a)g(cartesian)g(p)s(oten)m (tial)i(suc)m(h)e(that)h Fw(H)k Fv(=)25 b Fu(\000)p Fv(\001)13 b(+)g Fw(V)47 b Fx(is)26 b(of)h(class)g Fw(C)2409 4254 y Ft(1)2402 4310 y Fs(u)2448 4287 y Fv(\()p Fw(A)p Fv(\))p Fx(,)h(and)f(secondly)h(that)f Fw(V)47 b Fx(satis\034es)83 4400 y(the)31 b(requiremen)m(t)779 4525 y Fl(Z)870 4552 y FC(1)829 4732 y Ft(0)960 4517 y Fl(\015)960 4572 y(\015)960 4626 y(\015)960 4681 y(\015)1010 4649 y Fw(\030)1069 4521 y Fl(\022)1136 4649 y Fv(\()p Fw(\014)26 b Fu(\000)20 b Fw(\013)p Fv(\))h Fu(\001)1508 4588 y Fw(Q)p 1508 4628 72 4 v 1522 4711 a(r)1590 4521 y Fl(\023)1672 4548 y(\020)1727 4649 y Fw(V)1800 4612 y Fs(\013)1849 4649 y Fv(\()p Fw(Q)p Fv(\))g Fu(\000)f Fw(V)2176 4612 y Fs(\014)2223 4649 y Fv(\()p Fw(Q)p Fv(\))2365 4548 y Fl(\021)2420 4517 y(\015)2420 4572 y(\015)2420 4626 y(\015)2420 4681 y(\015)2471 4740 y FC(H)2531 4721 y Fm(2)2531 4756 y Fn(\013)2575 4740 y FC(!H)2706 4748 y Fn(\013)2769 4649 y Fw(dr)28 b(<)d Fu(1)535 b Fx(\(22\))83 4919 y(for)28 b(eac)m(h)h Fw(\013)c Fu(2)g(L)p Fx(,)j(eac)m(h)h Fw(\014)21 b Fk(l)15 b Fw(\013)28 b Fx(and)g(some)g(asymptotic)h(lo)s (calization)g(function)g Fw(\030)t Fx(.)39 b(This)28 b(last)g(assumption)83 5032 y(means)j(that)g(eac)m(h)g(function)g Fw(V)1186 4999 y Fs(\013)1266 5032 y Fx(tends)f(to)h(its)f(asymptotic)h(limits)g(in)f(a)h (short-range)g(w)m(a)m(y)-8 b(.)249 5145 y(Let)34 b(us)g(sho)m(w)g(that)g (the)g(ab)s(o)m(v)m(e)h(h)m(yp)s(otheses)e(are)h(su\036cien)m(t)h(for)f(the)g (existence)g(of)f(the)h(asymptotic)83 5258 y(v)m(elo)s(cit)m(y)25 b(for)e(eac)m(h)i(op)s(erator)g Fw(H)1180 5225 y Fs(\013)1230 5258 y Fx(.)38 b(Indeed,)26 b(let)e Fw(\013;)15 b(\014)31 b Fu(2)25 b(L)f Fx(and)g Fw(j)31 b Fu(2)25 b(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)i(;)e(n)p Fu(g)24 b Fx(b)s(e)g(suc)m(h)h(that)f Fw(\014)12 b Fk(l)7 b Fw(\013)p Fx(,)26 b Fw(\013)3635 5272 y Fs(j)3697 5258 y Fv(=)83 5371 y(0)j Fx(and)g Fw(\014)382 5385 y Fs(j)445 5371 y Fv(=)c Fu(\006)p Fv(1)p Fx(.)39 b(In)29 b(suc)m(h)g(a)g(situation)h Fw(V)1558 5338 y Fs(\014)1634 5371 y Fx(do)s(es)e(not)h(dep)s(end)h(on)f Fw(x)2484 5385 y Fs(j)2520 5371 y Fx(,)g(and)h(th)m(us)f Fu(r)3022 5385 y Fs(j)3058 5371 y Fw(V)3132 5338 y Fs(\013)3181 5371 y Fv(\()p Fw(Q)p Fv(\))g Fx(is)g(formally)1880 5836 y(23)p eop %%Page: 24 24 bop 83 482 a Fx(equal)25 b(to)h Fw(iP)511 496 y Fs(j)563 408 y Fl(\000)605 482 y Fw(V)678 449 y Fs(\013)728 482 y Fv(\()p Fw(Q)p Fv(\))20 b Fu(\000)g Fw(V)1054 449 y Fs(\014)1102 482 y Fv(\()p Fw(Q)p Fv(\))1244 408 y Fl(\001)1296 482 y Fu(\000)10 b Fw(i)1423 408 y Fl(\000)1464 482 y Fw(V)1538 449 y Fs(\013)1587 482 y Fv(\()p Fw(Q)p Fv(\))21 b Fu(\000)f Fw(V)1914 449 y Fs(\014)1961 482 y Fv(\()p Fw(Q)p Fv(\))2103 408 y Fl(\001)2161 482 y Fw(P)2219 496 y Fs(j)2255 482 y Fx(.)39 b(Then)26 b(some)f(simple)h(calculations,)i (using)83 594 y(this)i(observ)-5 b(ation)31 b(and)g(the)g(assumption)g (\(22\),)g(sho)m(w)g(that)g(for)f(eac)m(h)h Fw(j)36 b Fx(suc)m(h)31 b(that)g Fw(\013)3057 608 y Fs(j)3119 594 y Fv(=)25 b(0)p Fx(:)1060 729 y Fl(Z)1151 755 y FC(1)1110 935 y Ft(0)1241 721 y Fl(\015)1241 775 y(\015)1241 830 y(\015)1241 884 y(\015)1291 852 y Fw(\030)1350 724 y Fl(\022)1417 852 y Fu(\006)1498 790 y Fw(Q)1570 804 y Fs(j)p 1498 832 109 4 v 1530 915 a Fw(r)1616 724 y Fl(\023)1698 852 y Fu(r)1774 866 y Fs(j)1810 852 y Fw(V)1884 815 y Fs(\013)1933 852 y Fv(\()p Fw(Q)p Fv(\))2075 721 y Fl(\015)2075 775 y(\015)2075 830 y(\015)2075 884 y(\015)2126 943 y FC(H)2186 924 y Fm(2)2186 960 y Fn(\013)2231 943 y FC(!H)2362 913 y Fq(\000)p Fm(2)2362 953 y Fn(\013)2463 852 y Fw(dr)j(<)d Fu(1)p Fw(:)816 b Fx(\(23\))83 1127 y(Let)33 b(us)f(notice)h(that)g(in)g(the)g(sp)s(ecial)f(case)g Fw(\013)e Fv(=)e Fw(o)p Fx(,)33 b(\(23\))g(is)f(nothing)h(but)g(\(5\).)47 b(So,)34 b(for)e(eac)m(h)h Fw(\013)c Fu(2)g(L)p Fx(,)j Fw(V)3719 1094 y Fs(\013)83 1240 y Fx(is)h(a)f(cartesian)i(p)s(oten)m(tial)h(relativ)m (e)f(to)f Fu(L)1524 1207 y Fs(\013)1606 1240 y Fx(whic)m(h)h(satis\034es)e (\(23\))i(and)f(suc)m(h)h(that)f Fw(H)3062 1207 y Fs(\013)3141 1240 y Fv(=)c Fu(\000)p Fv(\001)3388 1207 y Fs(\013)3459 1240 y Fv(+)22 b Fw(V)3625 1207 y Fs(\013)3707 1240 y Fx(is)83 1353 y(of)36 b(class)f Fw(C)481 1320 y Ft(1)474 1376 y Fs(u)520 1353 y Fv(\()p Fw(A)623 1320 y Fs(\013)673 1353 y Fv(\))h Fx(\()p Fr(cf.)g Fx(Lemma)h(5.2\).)58 b(A)35 b(prop)s(osition)i(similar)g(to)f(Prop)s (osition)h(6.1)f(can)g(therefore)h(b)s(e)83 1466 y(stated)h(for)g Fw(H)591 1433 y Fs(\013)640 1466 y Fx(,)i(with)e(the)g(only)g(di\033erence)h (that)f(the)g(corresp)s(onding)h(asymptotic)f(v)m(elo)s(cit)m(y)h Fu(P)3537 1433 y Fs(\013)3625 1466 y Fx(is)e(a)83 1579 y Fu(j)p Fw(\013)p Fu(j)p Fx(-tuple)32 b(instead)f(of)f(a)g Fw(n)p Fx(-tuple.)41 b(F)-8 b(or)31 b(the)g(case)f Fw(\013)c Fv(=)f Fw(o)p Fx(,)30 b(w)m(e)h(k)m(eep)g(the)g(notation)h Fu(P)37 b Fx(instead)31 b(of)f Fu(P)3524 1546 y Fs(o)3563 1579 y Fx(.)249 1692 y(Moreo)m(v)m(er,)41 b(a)c(direct)h(consequence)g(of)f(Lemma)i(9.4.8)f(of)f([1])g(is)g(that)h(if)f (the)g(requiremen)m(t)i(\(22\))f(is)83 1805 y(assumed,)h(then)f(this)f (condition)i(is)e(also)g(satis\034ed)h(with)g(the)f(norm)h Fu(k)26 b(\001)f(k)2704 1825 y FC(H)2764 1806 y Fm(2)2799 1825 y FC(!H)2971 1805 y Fx(instead)38 b(of)f(the)g(norm)83 1918 y Fu(k)25 b(\001)f(k)247 1937 y FC(H)307 1919 y Fm(2)307 1954 y Fn(\013)352 1937 y FC(!H)483 1945 y Fn(\013)531 1918 y Fx(.)58 b(It)36 b(follo)m(ws)g(that)h(\(23\))g(is)f(ful\034lled)h(with)g(the)f(norm)h Fu(k)25 b(\001)f(k)2625 1937 y FC(H)2685 1919 y Fm(2)2720 1937 y FC(!H)2851 1919 y Fq(\000)p Fm(2)2974 1918 y Fx(instead)37 b(of)f(the)g(norm)83 2031 y Fu(k)26 b(\001)h(k)251 2061 y FC(H)311 2042 y Fm(2)311 2078 y Fn(\013)355 2061 y FC(!H)486 2031 y Fq(\000)p Fm(2)486 2072 y Fn(\013)573 2031 y Fx(.)65 b(W)-8 b(e)40 b(notice)f(furthermore)i(that)e(\(23\))g(is)g(trivially)g(satis\034ed) g(for)f Fw(k)k Fx(suc)m(h)e(that)f Fw(\013)3530 2046 y Fs(k)3612 2031 y Fu(6)p Fv(=)h(0)83 2154 y Fx(\(since)35 b Fw(V)417 2121 y Fs(\013)501 2154 y Fx(do)s(es)g(not)g(dep)s(end)g(on)g(the)g(v)-5 b(ariable)36 b Fw(x)1879 2169 y Fs(k)1921 2154 y Fx(\))f(and)g(that)h Fw(V)2445 2121 y Fs(\013)2529 2154 y Fx(is)e(a)h(cartesian)h(p)s(oten)m(tial) g(relativ)m(e)83 2267 y(to)f Fu(L)p Fx(.)53 b(Hence,)36 b(since)f Fw(H)940 2281 y Fs(\013)1021 2267 y Fv(=)e Fu(\000)p Fv(\001)22 b(+)h Fw(V)1461 2234 y Fs(\013)1545 2267 y Fx(is)35 b(of)f(class)g Fw(C)2036 2234 y Ft(1)2029 2290 y Fs(u)2075 2267 y Fv(\()p Fw(A)p Fv(\))p Fx(,)j(the)e(h)m(yp)s(otheses)g(of)f(Prop)s(osition)i(6.1)f (are)83 2380 y(ful\034lled)d(with)g Fw(H)691 2394 y Fs(\013)771 2380 y Fx(instead)g(of)f Fw(H)7 b Fx(,)31 b(and)h(all)f(dev)m(elopmen)m(ts)j (of)d(the)g(previous)h(section)g(can)f(b)s(e)g(rewritten)83 2493 y(in)36 b(terms)h(of)e Fw(H)636 2507 y Fs(\013)685 2493 y Fx(.)58 b(Let)36 b Fu(P)998 2507 y Fs(\013)1084 2493 y Fx(denote)h(the)f (asymptotic)h(v)m(elo)s(cit)m(y)g(obtained)g(for)f(the)g(op)s(erator)h Fw(H)3481 2507 y Fs(\013)3530 2493 y Fx(.)57 b(The)83 2606 y(relation)32 b(b)s(et)m(w)m(een)g Fu(P)827 2620 y Fs(\013)906 2606 y Fx(and)f Fu(P)1152 2573 y Fs(\013)1232 2606 y Fx(will)g(b)s(e)f(giv)m (en)h(later)g(on.)249 2719 y(Let)45 b(us)e(men)m(tion)j(that)e(there)h(exist) f(situations)g(for)g(whic)m(h)h(the)f(condition)i(\(22\))e(can)g(easily)g(b)s (e)83 2832 y(c)m(hec)m(k)m(ed.)83 3019 y Fj(Example)35 b(7.1)45 b Fx(If)28 b Fw(V)49 b Fx(is)29 b(the)g(sum)g(of)g(a)g(Mourre)g(p)s(oten)m (tial)i Fw(V)2242 3033 y Fs(m)2337 3019 y Fx(constructed)g(in)e(Example)h (5.5)g(and)f(of)g(a)83 3150 y(short-range)35 b(p)s(oten)m(tial)g Fw(V)1002 3164 y Fh(SR)1118 3150 y Fx(suc)m(h)f(that)g Fv(lim)1650 3164 y Fs(r)r FC(!1)1845 3045 y Fl(\015)1845 3100 y(\015)1845 3154 y(\015)1895 3150 y Fw(\030)1954 3049 y Fl(\020)2018 3105 y FC(j)p Fs(Q)p FC(j)p 2018 3129 95 4 v 2049 3181 a Fs(r)2123 3049 y Fl(\021)2193 3150 y Fw(V)2246 3164 y Fh(SR)2327 3150 y Fv(\()p Fw(Q)p Fv(\))2469 3045 y Fl(\015)2469 3100 y(\015)2469 3154 y(\015)2521 3213 y FC(H)2581 3194 y Fm(2)2615 3213 y FC(!H)2781 3150 y Fv(=)d(0)j Fx(for)f(some)h(asymptotic)83 3290 y(lo)s(calization)29 b(function)f Fw(\030)t Fx(,)g(then)g(\(22\))g(is)f(automatically)i (satis\034ed.)39 b(A)27 b(similar)h(conclusion)h(is)e(obtained)h(if)83 3403 y(in)23 b(Example)g(5.6)g(eac)m(h)g Fw(V)934 3417 y Fs(j)993 3403 y Fx(reac)m(hes)h(its)e(limits)g(in)h(a)f(short-range)i(w)m(a)m(y)-8 b(.)39 b(In)23 b(b)s(oth)f(situations)h(all)g(h)m(yp)s(otheses)83 3515 y(of)30 b(Theorems)h(1.1)g(and)g(1.2)f(are)h(ful\034lled.)83 3728 y Fj(Prop)s(osition)i(7.2)45 b Fr(L)-5 b(et)38 b Fw(V)58 b Fr(b)-5 b(e)38 b(a)g(c)-5 b(artesian)38 b(p)-5 b(otential)38 b(such)g(that)g Fw(H)j Fv(=)34 b Fu(\000)p Fv(\001)24 b(+)f Fw(V)58 b Fr(is)37 b(of)g(class)h Fw(C)3563 3695 y Ft(1)3556 3750 y Fs(u)3602 3728 y Fv(\()p Fw(A)p Fv(\))p Fr(.)83 3841 y(Assume)30 b(mor)-5 b(e)g(over)31 b(that)e(the)h(c)-5 b(ondition)29 b(\(22\))g(is)g(satis\034e)-5 b(d.)40 b(Then)29 b(for)g(e)-5 b(ach)31 b Fw(\013)26 b Fu(2)f(L)p Fr(,)k(the)h(fol)5 b(lowing)28 b(wave)83 3954 y(op)-5 b(er)g(ators)34 b(exist:)1129 4158 y Fw(W)1228 4120 y Ft(+)1215 4180 y Fs(\013o)1382 4158 y Fv(:=)83 b Fr(s)20 b Fu(\000)67 b Fv(lim)1709 4214 y Fs(t)p FC(!)p Ft(+)p FC(1)1945 4158 y Fw(e)1987 4120 y Fs(iH)2069 4128 y Fn(\013)2114 4120 y Fs(t)2143 4158 y Fw(e)2185 4120 y FC(\000)p Fs(iH)5 b(t)2357 4158 y Fw(E)2424 4183 y Fn(Y)2460 4191 y(\013)2513 4158 y Fv(\()p Fu(P)i Fv(\))p Fw(;)1129 4346 y(W)1228 4309 y Ft(+)1215 4369 y Fs(o\013)1382 4346 y Fv(:=)83 b Fr(s)20 b Fu(\000)67 b Fv(lim)1709 4402 y Fs(t)p FC(!)p Ft(+)p FC(1)1945 4346 y Fw(e)1987 4309 y Fs(iH)5 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Fx(In)39 b(order)g(to)g(pro)m(v)m (e)h(the)g(existence)f(of)g Fw(W)2050 5168 y Ft(+)2037 5224 y Fs(\013o)2120 5201 y Fx(,)i(it)e(is)g(enough)h(to)f(sho)m(w)g(that)h(for)e (eac)m(h)i Fw(\021)j Fu(2)83 5314 y Fw(C)155 5281 y FC(1)148 5337 y Fs(c)229 5314 y Fv(\(I)-18 b(R\))36 b Fx(and)h(eac)m(h)g Fw(f)45 b Fx(in)36 b(a)g(dense)g(subset)g(of)f(all)i Fw(C)1930 5328 y Ft(0)1969 5314 y Fv(\(I)-18 b(R)2086 5278 y Fs(n)2133 5314 y Fv(\))p Fx(-functions)37 b(with)f(supp)s(ort)g(in)g Fw(Y)3309 5328 y Fs(\013)3359 5314 y Fx(,)h(the)f(limit)83 5427 y(s)20 b Fu(\000)g Fv(lim)356 5441 y Fs(t)p FC(!1)542 5427 y Fw(e)584 5394 y Fs(iH)666 5402 y Fn(\013)711 5394 y Fs(t)740 5427 y Fw(e)782 5394 y FC(\000)p Fs(iH)5 b(t)954 5427 y Fw(f)10 b Fv(\()p Fu(P)d Fv(\))p Fw(\021)1197 5394 y Ft(2)1238 5427 y Fv(\()p Fw(H)g Fv(\))30 b Fx(exists.)1880 5836 y(24)p eop %%Page: 25 25 bop 249 495 a Fx(a\))25 b(Let)f Fw(f)35 b Fu(2)25 b Fo(F)e Fx(b)s(e)h(suc)m(h)h(that)g(supp)p Fw(f)35 b Fu(\032)25 b Fw(Y)1674 509 y Fs(\013)1723 495 y Fx(.)38 b(F)-8 b(or)25 b Fw(t)g Fu(\025)g Fv(1)p Fx(,)h(set)e Fw(M)10 b Fv(\()p Fw(t)p Fv(\))26 b(:=)f Fw(f)2745 394 y Fl(\020)2812 454 y Fs(Q)p 2809 474 61 4 v 2809 526 a Ft(2)p Fs(t)2880 394 y Fl(\021)2942 495 y Fv(+)8 b Fu(r)p Fw(f)3166 394 y Fl(\020)3233 454 y Fs(Q)p 3230 474 V 3230 526 a Ft(2)p Fs(t)3301 394 y Fl(\021)3363 495 y Fu(\001)3396 394 y Fl(\020)3451 495 y Fw(P)33 b Fu(\000)3646 454 y Fs(Q)p 3643 474 V 3643 526 a Ft(2)p Fs(t)3714 394 y Fl(\021)83 639 y Fx(and)c Fv(\010\()p Fw(t)p Fv(\))d(:=)f Fw(\021)s Fv(\()p Fw(H)732 653 y Fs(\013)782 639 y Fv(\))p Fw(M)10 b Fv(\()p Fw(t)p Fv(\))p Fw(\021)s Fv(\()p Fw(H)d Fv(\))p Fx(.)42 b(One)29 b(can)g(c)m(hec)m(k)i(that) e Fv(\010)c Fu(2)g Fw(B)5 b(C)2399 606 y Ft(1)2437 566 y Fl(\000)2479 639 y Fv([1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g Fu(B)s Fv(\()p Fu(H)q Fv(\))2966 566 y Fl(\001)3009 639 y Fx(.)40 b(Let)30 b(us)e(assume)h(for)83 752 y(a)h(while)h(that)g(the)g(limit)g(s)20 b Fu(\000)g Fv(lim)1235 766 y Fs(t)p 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Fl(\020)1467 1076 y Fs(Q)p 1464 1096 V 1464 1148 a Ft(2)p Fs(t)1535 1016 y Fl(\021)1605 1117 y Fw(\021)s Fv(\()p Fw(H)g Fv(\))p Fw(e)1848 1084 y FC(\000)p Fs(iH)e(t)2046 1117 y Fv(=)25 b Fx(s)20 b Fu(\000)g Fv(lim)2415 1131 y Fs(t)p FC(!1)2601 1117 y Fw(e)2643 1084 y Fs(iH)2725 1092 y Fn(\013)2769 1084 y Fs(t)2799 1117 y Fw(e)2841 1084 y FC(\000)p Fs(iH)5 b(t)3013 1117 y Fw(f)10 b Fv(\()p Fu(P)d Fv(\))p Fw(\021)3256 1084 y Ft(2)3297 1117 y Fv(\()p Fw(H)g Fv(\))p Fw(;)83 1344 y Fx(where)28 b(w)m(e)g(ha)m(v)m(e)h(used)e(successiv)m (ely)g Fr(iv\))j Fx(of)d(Lemma)h(6.2,)h(the)e(second)h(assertion)g(of)f (Lemma)h(6.6)g(and)g(the)83 1457 y(h)m(yp)s(othesis)33 b Fw(f)38 b Fu(2)28 b Fo(F)p Fx(,)33 b(and)g(\034nally)g(the)g(existence)g(of)f(the)h (asymptotic)g(v)m(elo)s(cit)m(y)g Fu(P)7 b Fx(.)47 b(Th)m(us)34 b(the)e(existence)83 1570 y(of)h(s)21 b Fu(\000)h Fv(lim)465 1584 y Fs(t)p FC(!1)651 1570 y Fw(e)693 1537 y Fs(iH)775 1545 y Fn(\013)819 1537 y Fs(t)849 1570 y Fw(e)891 1537 y FC(\000)p 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Fs(\013)2813 2000 y Fv(\()p Fw(Q)p Fv(\))2955 1926 y Fl(\001)2997 2000 y Fw(M)3095 1926 y Fl(\011)3164 2000 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))304 2169 y(=)25 b Fw(\021)s Fv(\()p Fw(H)559 2183 y Fs(\013)609 2169 y Fv(\))659 2068 y Fl(n)720 2169 y Fu(\000r)p Fw(f)k Fu(\001)21 b(r)p Fw(V)40 b Fv(+)1257 2134 y Ft(1)p 1257 2149 36 4 v 1262 2201 a Fs(t)1317 2068 y Fl(h)1360 2169 y Fw(P)33 b Fu(\000)1555 2128 y Fs(Q)p 1552 2148 61 4 v 1552 2201 a Ft(2)p Fs(t)1623 2068 y Fl(i)1666 2102 y Fn(T)1734 2169 y Fw(@)1787 2136 y Ft(2)1826 2169 y Fw(f)1896 2068 y Fl(h)1939 2169 y Fw(P)g Fu(\000)2134 2128 y Fs(Q)p 2131 2148 V 2131 2201 a Ft(2)p Fs(t)2202 2068 y Fl(i)2265 2169 y Fu(\000)20 b Fw(i)p Fv(\()p Fw(V)41 b Fu(\000)20 b Fw(V)2680 2136 y Fs(\013)2729 2169 y Fv(\))p Fw(M)2862 2068 y Fl(o)2939 2169 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))21 b(+)f Fw(O)s Fv(\()p Fw(t)3392 2136 y FC(\000)p Ft(2)3486 2169 y Fv(\))p Fw(:)61 b Fx(\(25\))249 2406 y(Since)25 b Fu(r)555 2420 y Fs(j)591 2406 y Fw(f)34 b Fx(has)24 b(supp)s(ort)g(in)g Fw(Y)1325 2368 y FC(\000)1305 2434 y Fs(j)1392 2406 y Fu([)8 b Fw(Y)1534 2368 y Ft(+)1514 2434 y Fs(j)1593 2406 y Fx(,)25 b(the)g(term)f Fw(\021)s Fv(\()p Fw(H)2161 2420 y Fs(\013)2211 2406 y Fv(\))p Fu(r)p Fw(f)18 b Fu(\001)8 b(r)p Fw(V)20 b(\021)s Fv(\()p Fw(H)7 b Fv(\))25 b Fx(b)s(elongs)f(to)g Fw(L)3277 2373 y Ft(1)3317 2333 y Fl(\000)3358 2406 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)3724 2333 y Fl(\001)83 2519 y Fx(\(the)36 b(pro)s(of)e(of)h(this)g(fact)g(is)g(similar) g(to)g(that)h(of)f(the)g(statemen)m(t)i Fr(ii\))d Fx(of)g(Lemma)j(6.2\).)55 b(Let)36 b(us)e(observ)m(e)83 2632 y(that)d(the)g(second)f(term)h(of)f (\(25\))h(is)f(equal)h(to)644 2792 y Fs(n)600 2820 y Fl(X)577 3017 y Fs(j;k)r Ft(=1)779 2844 y Fv(1)p 779 2885 46 4 v 785 2968 a Fw(t)834 2906 y(\021)s Fv(\()p Fw(H)993 2920 y Fs(\013)1043 2906 y Fv(\))1093 2778 y Fl(\022)1161 2906 y Fw(P)1219 2920 y Fs(j)1276 2906 y Fu(\000)1376 2844 y Fw(Q)1448 2858 y Fs(j)p 1376 2885 109 4 v 1392 2968 a Fv(2)p Fw(t)1495 2778 y Fl(\023)1577 2906 y Fw(g)1620 2920 y Fs(j)1672 2778 y Fl(\022)1752 2844 y Fw(Q)p 1749 2885 79 4 v 1749 2968 a Fv(2)p Fw(t)1837 2778 y Fl(\023)1919 2906 y Fu(f)p Fw(@)2017 2868 y Ft(2)2012 2929 y Fs(j)t(k)2088 2906 y Fw(f)10 b Fu(g)p Fw(g)2231 2921 y Fs(k)2289 2778 y Fl(\022)2369 2844 y Fw(Q)p 2366 2885 V 2366 2968 a Fv(2)p Fw(t)2454 2778 y Fl(\023)15 b(\022)2603 2906 y Fw(P)2661 2921 y Fs(k)2724 2906 y Fu(\000)2825 2844 y Fw(Q)2897 2859 y Fs(k)p 2825 2885 115 4 v 2843 2968 a Fv(2)p Fw(t)2950 2778 y Fl(\023)3032 2906 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(;)349 b Fx(\(26\))83 3217 y(where)36 b(eac)m(h)f Fw(g)599 3231 y Fs(j)671 3217 y Fx(b)s(elongs)g(to)f Fw(C)1186 3184 y FC(1)1179 3240 y Fs(c)1261 3217 y Fv(\(I)-18 b(R)1378 3181 y Fs(n)1425 3217 y Fv(\))p Fx(,)36 b(has)e(supp)s(ort)h(in)g Fw(Y)2208 3179 y FC(\000)2188 3244 y Fs(j)2290 3217 y Fu([)23 b Fw(Y)2447 3179 y Ft(+)2427 3244 y Fs(j)2540 3217 y Fx(and)36 b(satis\034es)e Fw(g)3101 3231 y Fs(j)3138 3217 y Fu(r)3214 3231 y Fs(j)3250 3217 y Fw(f)42 b Fv(=)32 b Fu(r)3516 3231 y Fs(j)3553 3217 y Fw(f)10 b Fx(.)52 b(In)83 3330 y(relation)34 b(with)f(\(26\),)g(it)g(is)f(useful)g(to)h(recall) g(\(from)g(Lemma)g(6.6\))g(that)g(for)f(eac)m(h)i Fw(k)e Fu(2)c(f)p Fv(1)p Fw(;)15 b(:)g(:)g(:)j(;)d(n)p Fu(g)p Fx(,)33 b(there)83 3482 y(exists)f Fw(c)d(<)g Fu(1)k Fx(suc)m(h)g(that)1031 3409 y Fl(R)1092 3435 y FC(1)1074 3514 y Ft(1)1182 3378 y Fl(\015)1182 3432 y(\015)1182 3487 y(\015)1232 3482 y Fw(g)1275 3497 y Fs(k)1333 3381 y Fl(\020)1400 3441 y Fs(Q)p 1397 3461 61 4 v 1397 3514 a Ft(2)p Fs(t)1468 3381 y Fl(\021)16 b(\020)1592 3482 y Fw(P)1650 3497 y Fs(k)1713 3482 y Fu(\000)1814 3441 y Fs(Q)1870 3453 y Fn(k)p 1814 3461 94 4 v 1830 3514 a Ft(2)p Fs(t)1918 3381 y Fl(\021)1987 3482 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(e)2230 3449 y FC(\000)p Fs(iH)e(t)2403 3482 y Fw(')2462 3378 y Fl(\015)2462 3432 y(\015)2462 3487 y(\015)2513 3404 y Ft(2)2578 3446 y Fs(dt)p 2578 3461 62 4 v 2596 3514 a(t)2678 3482 y Fu(\024)29 b Fw(c)p Fu(k)p Fw(')p Fu(k)2966 3449 y Ft(2)3040 3482 y Fx(for)j(all)h Fw(')d Fu(2)f(H)q Fx(,)j(and)83 3618 y(that)27 b(a)g(similar)g(result)f(can)h (b)s(e)g(obtained)g(with)h Fw(H)1815 3632 y Fs(\013)1890 3618 y Fx(instead)f(of)f Fw(H)33 b Fx(\(see)27 b(the)g(discussion)f(b)s(efore)h (Example)83 3731 y(7.1\).)249 3866 y(Let)39 b(us)g(\034nally)g(notice)h(that) f(for)g Fw(\016)k(>)c Fv(0)p Fx(,)1756 3793 y Fl(R)1817 3819 y FC(1)1799 3898 y Ft(1)1907 3762 y Fl(\015)1907 3816 y(\015)1907 3871 y(\015)1957 3798 y(Q)2043 3893 y FC(f)p Fs(j)t FC(j)p Fs(\013)2176 3903 y Fn(j)2208 3893 y FC(6)p Ft(=0)p FC(g)2353 3866 y Fw(\030)2412 3765 y Fl(\020)2467 3866 y Fw(\013)2525 3880 y Fs(j)2571 3818 y(Q)2627 3828 y Fn(j)p 2571 3845 89 4 v 2581 3898 a Fs(\016)r(r)2669 3765 y Fl(\021)2739 3866 y Fv(\()p Fw(V)h Fu(\000)20 b Fw(V)3032 3833 y Fs(\013)3081 3866 y Fv(\))3116 3762 y Fl(\015)3116 3816 y(\015)3116 3871 y(\015)3167 3929 y FC(H)3227 3911 y Fm(2)3262 3929 y FC(!H)3412 3866 y Fw(dr)42 b(<)d Fu(1)p Fx(.)83 4006 y(This)h(follo)m(ws)g(directly)h(from)f(\(22\))g (and)h(Lemma)g(9.4.8)f(of)g([1])g(b)m(y)g(inserting)h(and)f(remo)m(ving)h (suitable)83 4119 y(terms)34 b(b)s(et)m(w)m(een)i Fw(V)54 b Fx(and)34 b Fw(V)1050 4086 y Fs(\013)1100 4119 y Fx(.)51 b(Then)34 b(b)m(y)g(c)m(ho)s(osing)h Fw(\016)i Fx(suc)m(h)e(that)2399 4051 y Fl(Q)2485 4146 y FC(f)p Fs(j)t FC(j)p Fs(\013)2618 4156 y Fn(j)2650 4146 y FC(6)p Ft(=0)p FC(g)2795 4119 y Fw(\030)2854 4045 y Fl(\000)2896 4119 y Fw(\013)2954 4133 y Fs(j)3000 4071 y(x)3040 4081 y Fn(j)p 3000 4098 73 4 v 3020 4150 a Fs(\016)3083 4045 y Fl(\001)3139 4119 y Fw(f)10 b Fv(\()p Fw(x)p Fv(\))32 b(=)e Fw(f)10 b Fv(\()p Fw(x)p Fv(\))34 b Fx(for)83 4251 y(all)d Fw(x)25 b Fu(2)g Fv(I)-18 b(R)453 4215 y Fs(n)500 4251 y Fx(,)31 b(one)f(can)h(easily)f(sho)m(w)h(that)g Fw(\021)s Fv(\()p Fw(H)1715 4265 y Fs(\013)1765 4251 y Fv(\)\()p Fw(V)41 b Fu(\000)20 b Fw(V)2093 4218 y Fs(\013)2143 4251 y Fv(\))p Fw(M)10 b(\021)s Fv(\()p Fw(H)d Fv(\))31 b Fx(b)s(elongs)g(to)f Fw(L)3005 4218 y Ft(1)3044 4177 y Fl(\000)3086 4251 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)3452 4177 y Fl(\001)3496 4251 y Fx(.)249 4364 y(In)32 b(view)h(of)f(the)g(preceding)i(estimates,)g(the)e(existence)h (of)f(s)21 b Fu(\000)h Fv(lim)2597 4378 y Fs(t)p FC(!1)2783 4364 y Fw(e)2825 4331 y Fs(iH)2907 4339 y Fn(\013)2951 4331 y Fs(t)2981 4364 y Fv(\010\()p Fw(t)p Fv(\))p Fw(e)3192 4331 y FC(\000)p Fs(iH)5 b(t)3396 4364 y Fx(is)32 b(seen)h(to)83 4477 y(b)s(e)d(a)g(direct)i(consequence)f(of)f(Lemma)h(6.4.)249 4590 y(c\))g(By)g(similar)g(argumen)m(ts,)i(w)m(e)f(can)g(sho)m(w)g(the)f (existence)g(of)g Fw(W)2531 4557 y Ft(+)2518 4612 y Fs(o\013)2602 4590 y Fx(.)42 b(F)-8 b(or)32 b(the)f(last)g(assertion)h(of)f(the)83 4703 y(prop)s(osition,)j(one)e(easily)g(observ)m(es)h(that)f Fw(W)1660 4670 y Ft(+)1647 4725 y Fs(o\013)1731 4703 y Fw(f)10 b Fv(\()p Fu(P)1884 4717 y Fs(\013)1933 4703 y Fv(\))29 b(=)f Fw(f)10 b Fv(\()p Fu(P)d Fv(\))p Fw(W)2390 4670 y Ft(+)2377 4725 y Fs(o\013)2493 4703 y Fx(and)33 b(that)f Fw(W)2967 4670 y Ft(+)2954 4725 y Fs(\013o)3038 4703 y Fw(f)10 b Fv(\()p Fu(P)d Fv(\))29 b(=)f Fw(f)10 b Fv(\()p Fu(P)3514 4717 y Fs(\013)3563 4703 y Fv(\))p Fw(W)3697 4670 y Ft(+)3684 4725 y Fs(\013o)83 4816 y Fx(for)34 b(eac)m(h)i Fw(f)41 b Fu(2)32 b Fw(C)677 4830 y Ft(0)717 4816 y Fv(\(I)-18 b(R)834 4780 y Fs(n)881 4816 y Fv(\))34 b Fx(with)h(supp)s(ort)g(in)g Fw(Y)1662 4830 y Fs(\013)1711 4816 y Fx(.)53 b(The)35 b(conclusion)g(follo)m(ws)g(then)g(b)m(y)g(taking)g (a)g(sequence)83 4929 y Fu(f)p Fw(f)173 4944 y Fs(k)216 4929 y Fu(g)261 4944 y Fs(k)r FC(2)p Fd(N)421 4929 y Fx(of)22 b(functions)h(in)g Fw(C)1061 4943 y Ft(0)1100 4929 y Fv(\(I)-18 b(R)1217 4893 y Fs(n)1264 4929 y Fv(\))23 b Fx(with)g(supp)q Fw(f)1752 4944 y Fs(k)1819 4929 y Fu(\032)i Fw(Y)1968 4943 y Fs(\013)2040 4929 y Fx(suc)m(h)e(that)g(this)f(sequence)h(con)m(v)m(erges)h(b)s(oundedly) 83 5042 y(to)31 b Fw(\037)251 5066 y Fm(\004)299 5042 y Fx(.)3373 b Fc(\004)249 5154 y Fr(Pr)-5 b(o)g(of)47 b(of)e(The)-5 b(or)g(em)47 b(1.2:)69 b Fx(Since)46 b Fw(Z)1542 5168 y Fs(\013)1636 5154 y Fx(is)e(a)h(subset)g(of)g Fw(Y)2294 5168 y Fs(\013)2343 5154 y Fx(,)k(the)c(existence)g(of)g Fv(\012)3170 5121 y Ft(+)3170 5177 y Fs(\013)3273 5154 y Fx(follo)m(ws)h(from)83 5267 y(Prop)s(osition)37 b(7.2.)56 b(One)36 b(gets)g(from)g(the)g(relations)g(\(24\))g(that)g(Ran)2470 5194 y Fl(\000)2512 5267 y Fw(W)2611 5234 y Ft(+)2598 5290 y Fs(o\013)2681 5267 y Fw(E)2748 5292 y Fn(Z)2790 5300 y(\013)2843 5267 y Fv(\()p Fu(P)2941 5281 y Fs(\013)2991 5267 y Fv(\))3026 5194 y Fl(\001)3102 5267 y Fu(\032)e Fw(E)3274 5292 y Fn(Z)3316 5300 y(\013)3368 5267 y Fv(\()p Fu(P)7 b Fv(\))p Fu(H)37 b Fx(and)83 5380 y(Ran)245 5307 y Fl(\000)287 5380 y Fw(W)386 5347 y Ft(+)373 5403 y Fs(\013o)456 5380 y Fw(E)523 5405 y Fn(Z)565 5413 y(\013)617 5380 y Fv(\()p Fu(P)7 b Fv(\))757 5307 y Fl(\001)826 5380 y Fu(\032)25 b Fw(E)989 5405 y Fn(Z)1031 5413 y(\013)1084 5380 y Fv(\()p Fu(P)1182 5394 y Fs(\013)1232 5380 y Fv(\))p Fu(H)q Fx(.)40 b(The)31 b(Lemma)h(B.5.1)f(of)f([4])h(implies)g (then)g(the)f(second)h(part)g(of)g(the)83 5493 y(assertion)g Fr(i\))p Fx(.)1880 5836 y(25)p eop %%Page: 26 26 bop 249 482 a Fx(Clearly)34 b Fw(I)k Fv(=)745 413 y Fl(P)841 508 y Fs(\013)p FC(2L)1001 482 y Fw(E)1068 506 y Fn(Z)1110 514 y(\013)1163 482 y Fv(\()p Fu(P)7 b Fv(\))35 b Fx(and)f(b)m(y)f(using)h (Lemma)h(6.10)f(and)g(the)g(absence)g(of)f(singularly)h(con-)83 594 y(tin)m(uous)d(sp)s(ectrum,)g(one)g(obtains)g(the)g(equalities:)1148 781 y Fu(H)1225 795 y Ft(ac)1296 781 y Fv(\()p Fw(H)7 b Fv(\))26 b(=)1571 694 y Fl(M)1572 892 y Fs(\013)p FC(6)p Ft(=)p Fs(o)1723 781 y Fw(E)1790 806 y Fn(Z)1832 814 y(\013)1885 781 y Fv(\()p Fu(P)7 b Fv(\))p Fu(H)27 b Fv(=)2225 694 y Fl(M)2226 892 y Fs(\013)p FC(6)p Ft(=)p Fs(o)2377 781 y Fx(Ran)14 b Fv(\012)2619 743 y Ft(+)2619 803 y Fs(\013)2678 781 y Fw(:)83 1078 y Fx(The)27 b(statemen)m(t)h Fr(iii\))d Fx(is)g(due)i(to)g(the)f(relation)i(b)s(et)m(w)m (een)g Fu(P)2081 1045 y Fs(\013)2157 1078 y Fx(and)f Fu(P)2392 1092 y Fs(\013)2441 1078 y Fx(.)39 b(Since)27 b Fw(H)2813 1092 y Fs(\013)2888 1078 y Fv(=)e Fu(\000)p Fv(\001)3131 1045 y Fs(\013)3176 1022 y Fq(?)3244 1078 y Fu(\012)12 b Fw(I)19 b Fv(+)12 b Fw(I)19 b Fu(\012)12 b Fw(H)3694 1045 y Fs(\013)3743 1078 y Fx(,)83 1191 y(and)35 b(since)f(the)g(asymptotic)h(v)m(elo)s(cit)m(y)g (for)f(the)g(op)s(erator)h Fu(\000)p Fv(\001)2270 1205 y Fs(j)2340 1191 y Fx(is)f Fw(P)2493 1205 y Fs(j)2530 1191 y Fx(,)h(one)g(has)f Fv(\()p Fu(P)3023 1205 y Fs(\013)3073 1191 y Fv(\))3108 1205 y Fs(j)3176 1191 y Fv(=)d Fw(P)3336 1205 y Fs(j)3407 1191 y Fx(if)j Fw(\013)3552 1205 y Fs(j)3620 1191 y Fu(6)p Fv(=)e(0)83 1304 y Fx(and)g Fv(\()p Fu(P)358 1318 y Fs(\013)408 1304 y Fv(\))443 1318 y Fs(j)507 1304 y Fv(=)26 b(\()p Fu(P)709 1271 y Fs(\013)760 1304 y Fv(\))795 1318 y Fs(j)863 1304 y Fx(if)31 b Fw(\013)1005 1318 y Fs(j)1069 1304 y Fv(=)26 b(0)p Fx(.)44 b(Then)32 b Fw(E)1584 1329 y Fn(Z)1626 1337 y(\013)1678 1304 y Fv(\()p Fu(P)1776 1318 y Fs(\013)1826 1304 y Fv(\))g Fx(is)f(obtained)i (from)e(the)h(fact)f(that)h Fw(Z)3167 1318 y Fs(\013)3248 1304 y Fx(is)f(a)g(cartesian)83 1417 y(pro)s(duct)g(and)g(from)f(the)h(relation)g Fw(E)1365 1445 y Fq(f)p Fm(0)p Fq(g)1468 1417 y Fv(\()p Fu(P)1573 1384 y Fs(\013)1623 1417 y Fv(\))26 b(=)f Fw(E)1847 1431 y Ft(p)1890 1417 y Fv(\()p Fw(H)2008 1384 y Fs(\013)2058 1417 y Fv(\))p Fx(.)1579 b Fc(\004)83 1699 y Fy(App)t(endix)83 1902 y Fr(Pr)-5 b(o)g(of)38 b(of)e(L)-5 b(emma)37 b(6.2:)50 b Fx(If)34 b Fw(z)k Fu(2)33 b Fk(C)47 b Fu(n)24 b Fw(\033)s Fv(\()p Fw(H)7 b Fv(\))p Fx(,)37 b(let)f(us)f(write)h Fw(R)q Fv(\()p Fw(z)t Fv(\))g Fx(for)f Fv(\()p Fw(H)c Fu(\000)23 b Fw(z)t Fv(\))2834 1869 y FC(\000)p Ft(1)2929 1902 y Fx(.)55 b(In)35 b(order)h(to)f(express)83 2015 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))28 b Fx(in)g(terms)g(of)f(the)g (resolv)m(en)m(t)i(of)e Fw(H)34 b Fx(w)m(e)29 b(shall)e(mak)m(e)h(use)g(of)f (the)g(form)m(ula)i(\(6.1.26\))g(of)e([1])g(for)g Fw(m)f Fu(\025)e Fv(2)p Fx(:)172 2276 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))26 b(=)510 2215 y(1)p 505 2256 56 4 v 505 2339 a Fw(\031)570 2276 y Fu(=)651 2153 y Fl(Z)702 2359 y Ft(I)-18 b(R)780 2121 y Fl(\(")906 2163 y Fs(m)p FC(\000)p Ft(1)917 2190 y Fl(X)918 2388 y Fs(k)r Ft(=0)1085 2215 y Fw(i)1116 2182 y Fs(k)p 1084 2256 76 4 v 1084 2339 a Fw(k)s Fv(!)1169 2276 y Fw(\021)1217 2239 y Ft(\()p Fs(k)r Ft(\))1315 2276 y Fv(\()p Fw(\025)p Fv(\))1438 2121 y Fl(#)1507 2276 y Fw(R)q Fv(\()p Fw(\025)21 b Fv(+)f Fw(i)p Fv(\))g(+)2082 2215 y Fw(i)2113 2182 y Fs(m)p 1964 2256 333 4 v 1964 2339 a Fv(\()p Fw(m)h Fu(\000)f Fv(1\)!)2307 2276 y Fw(\021)2355 2239 y Ft(\()p Fs(m)p Ft(\))2476 2276 y Fv(\()p Fw(\025)p Fv(\))2614 2153 y Fl(Z)2706 2179 y Ft(1)2666 2359 y(0)2761 2276 y Fw(R)q Fv(\()p Fw(\025)g Fv(+)g Fw(i\026)p Fv(\))p Fw(\026)3206 2239 y Fs(m)p FC(\000)p Ft(1)3363 2276 y Fw(d\026)3465 2121 y Fl(\))3554 2276 y Fw(d\025:)3607 2484 y Fx(\(27\))83 2654 y(W)-8 b(e)31 b(shall)f(also)g(use)h(the)f(\034rst)h (resolv)m(en)m(t)g(equation)966 2849 y Fw(R)q Fv(\()p Fw(\025)20 b Fv(+)g Fw(i\026)p Fv(\))26 b(=)1478 2748 y Fl(h)1521 2849 y Fw(I)h Fv(+)1679 2776 y Fl(\000)1720 2849 y Fw(\025)21 b Fv(+)f Fw(i)p Fv(\()p Fw(\026)g Fu(\000)g Fv(1\))2197 2776 y Fl(\001)2240 2849 y Fw(R)q Fv(\()p Fw(\025)g Fv(+)g Fw(i\026)p Fv(\))2630 2748 y Fl(i)2673 2849 y Fw(R)q Fv(\()p Fw(i)p Fv(\))p Fw(;)738 b Fx(\(28\))83 3045 y(and)31 b(the)f(fact)h(that)g(if)f Fw(p;)15 b(k)s(;)g(m)30 b Fx(are)h(p)s(ositiv)m(e)f(in)m(tegers,)i(with)f Fw(m)25 b Fu(\025)g Fw(p)20 b Fv(+)g(1)p Fx(,)31 b(then)f(the)h(in)m(tegrals) 162 3152 y Fl(Z)212 3358 y Ft(I)-18 b(R)291 3276 y Fu(j)p Fw(\021)364 3238 y Ft(\()p Fs(k)r Ft(\))462 3276 y Fv(\()p Fw(\025)p Fv(\))p Fu(j)610 3202 y Fl(\002)649 3276 y Fv(1)20 b(+)g Fu(j)p Fw(\025)p Fu(j)908 3202 y Fl(\003)947 3225 y Fs(p)986 3276 y Fw(d\025)121 b Fx(and)1489 3152 y Fl(Z)1539 3358 y Ft(I)-18 b(R)1618 3152 y Fl(Z)1709 3178 y Ft(1)1668 3358 y(0)1763 3276 y Fu(j)p Fw(\021)1836 3238 y Ft(\()p Fs(m)p Ft(\))1958 3276 y Fv(\()p Fw(\025)p Fv(\))p Fu(j)2121 3147 y Fl(\024)2170 3276 y Fv(1)21 b(+)2327 3202 y Fl(\000)2369 3276 y Fu(j)p Fw(\025)p Fu(j)f Fv(+)g Fu(j)p Fw(\026)g Fu(\000)g Fv(1)p Fu(j)2844 3202 y Fl(\001)2927 3214 y Fv(1)p 2897 3255 106 4 v 2897 3338 a Fu(j)p Fw(\026)p Fu(j)3012 3147 y Fl(\025)3060 3170 y Fs(p)3115 3276 y Fw(\026)3170 3238 y Fs(m)p FC(\000)p Ft(1)3326 3276 y Fw(d\025d\026)79 b Fx(\(29\))83 3498 y(are)31 b(\034nite.)249 3611 y(a\))g(Let)g(us)f(replace)h Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))31 b Fx(b)m(y)g(\(27\))g(in)f(the)h (statemen)m(t)h Fr(i\))d Fx(and)i(observ)m(e)g(that)287 3710 y Fl(\024)335 3838 y Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(;)15 b(f)632 3710 y Fl(\022)712 3777 y Fw(Q)p 709 3818 79 4 v 709 3901 a Fv(2)p Fw(t)797 3710 y Fl(\023\025)927 3838 y Fv(\()p Fw(H)28 b Fv(+)20 b Fw(i)p Fv(\))25 b(=)1354 3777 y(1)p 1354 3818 46 4 v 1360 3901 a Fw(t)1410 3838 y(R)q Fv(\()p Fw(z)t Fv(\))1611 3710 y Fl(\032)1680 3838 y Fw(iP)33 b Fu(\001)21 b(r)p Fw(f)1993 3710 y Fl(\022)2073 3777 y Fw(Q)p 2070 3818 79 4 v 2070 3901 a Fv(2)p Fw(t)2158 3710 y Fl(\023)2245 3838 y Fu(\000)2363 3777 y Fv(1)p 2346 3818 V 2346 3901 a(4)p Fw(t)2434 3838 y Fv(\001)p Fw(f)2580 3710 y Fl(\022)2660 3777 y Fw(Q)p 2656 3818 V 2656 3901 a Fv(2)p Fw(t)2745 3710 y Fl(\023\033)2895 3838 y Fw(R)q Fv(\()p Fw(z)t Fv(\)\()p Fw(H)28 b Fv(+)20 b Fw(i)p Fv(\))p Fw(:)205 b Fx(\(30\))83 4066 y(By)31 b(taking)h Fw(z)g Fv(=)c Fw(\025)21 b Fv(+)g Fw(i\026)31 b Fx(and)h(using)g(\(28\))h (for)e(the)h(resolv)m(en)m(ts)h(on)f(the)g(r.h.s.)g(of)f(\(30\),)i(one)f (obtains)g(that)83 4179 y(the)f(norm)g(of)f(\(30\))h(is)e(less)h(than)242 4345 y Fv(1)p 242 4386 46 4 v 248 4469 a Fw(t)298 4306 y Fl(\020)352 4407 y Fv(1)21 b(+)509 4333 y Fl(\000)550 4407 y Fu(j)p Fw(\025)p Fu(j)g Fv(+)f Fu(j)p Fw(\026)g Fu(\000)g Fv(1)p Fu(j)1026 4333 y Fl(\001)1108 4345 y Fv(1)p 1079 4386 106 4 v 1079 4469 a Fu(j)p Fw(\026)p Fu(j)1194 4306 y Fl(\021)1248 4328 y Ft(2)1303 4275 y Fl(\015)1303 4329 y(\015)1303 4384 y(\015)1303 4439 y(\015)1353 4407 y Fw(R)q Fv(\()p Fw(i)p Fv(\))1539 4279 y Fl(\032)1608 4407 y Fw(iP)34 b Fu(\001)20 b(r)p Fw(f)1921 4279 y Fl(\022)2002 4345 y Fw(Q)p 1998 4386 79 4 v 1998 4469 a Fv(2)p Fw(t)2087 4279 y Fl(\023)2174 4407 y Fu(\000)2291 4345 y Fv(1)p 2275 4386 V 2275 4469 a(4)p Fw(t)2363 4407 y Fv(\001)p Fw(f)2508 4279 y Fl(\022)2588 4345 y Fw(Q)p 2585 4386 V 2585 4469 a Fv(2)p Fw(t)2673 4279 y Fl(\023\033)2808 4275 y(\015)2808 4329 y(\015)2808 4384 y(\015)2808 4439 y(\015)2874 4407 y Fu(k)p Fw(R)q Fv(\()p Fw(i)p Fv(\)\()p Fw(H)29 b Fv(+)19 b Fw(i)p Fv(\))p Fu(k)p Fw(:)151 b Fx(\(31\))83 4634 y(Since)25 b(the)f(t)m(w)m(o)i(norms)e(are)h(b)s (ounded)g(uniformly)f(in)h Fw(t)e Fx(for)h Fw(t)h Fu(\025)g Fv(1)p Fx(,)h(the)f(statemen)m(t)g Fr(i\))f Fx(is)f(no)m(w)i(easily)f(pro)m (v)m(ed)83 4747 y(b)m(y)31 b(taking)f(in)m(to)i(accoun)m(t)g(\(29\))e(with)h Fw(m)25 b Fv(=)g(3)p Fx(.)249 4860 y(b\))30 b(F)-8 b(or)30 b(the)f(pro)s(of)g(of)g(the)h(\034rst)g(statemen)m(t)g(of)f Fr(iii\))p Fx(,)g(let)h Fw(\030)j Fx(b)s(e)c(an)g(asymptotic)h(lo)s (calization)h(function)83 4973 y(and)39 b Fw(\016)k(>)c Fv(0)g Fx(suc)m(h)h(that)f Fw(\030)1020 4900 y Fl(\000)1062 4973 y Fu(\006)1143 4925 y Fs(x)1183 4935 y Fn(j)p 1142 4952 73 4 v 1162 5005 a Fs(\016)1225 4900 y Fl(\001)1281 4973 y Fw(f)10 b Fv(\()p Fw(x)p Fv(\))40 b(=)f Fw(f)10 b Fv(\()p Fw(x)p Fv(\))38 b Fx(for)h(all)g Fw(x)g Fu(2)g Fv(I)-18 b(R)2377 4937 y Fs(n)2424 4973 y Fx(.)65 b(W)-8 b(e)39 b(start)g(b)m(y)g(replacing)h Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))40 b Fx(b)m(y)83 5086 y(\(27\))33 b(in)h(the)f(statemen)m(t)h Fr(iii\))p Fx(.)47 b(By)33 b(using)g(that)g Fw(\030)t Fv(\(2)p Fw(y)s Fv(\))p Fw(\030)1991 5053 y FC(0)2016 5086 y Fv(\()p Fw(y)s Fv(\))d(=)f Fw(\030)2308 5053 y FC(0)2331 5086 y Fv(\()p Fw(y)s Fv(\))34 b Fx(for)e(all)i Fw(y)e Fu(2)d Fv(I)-18 b(R)p Fx(,)33 b(some)g(comm)m(utator)83 5199 y(calculations)f(sho)m (w)f(that)952 5377 y Fv([)p Fw(P)1035 5391 y Fs(j)1072 5377 y Fw(;)15 b(R)q Fv(\()p Fw(z)t Fv(\)])p Fw(\030)1383 5276 y Fl(\020)1438 5377 y Fu(\006)1522 5329 y Fs(Q)1578 5339 y Fn(j)p 1518 5356 95 4 v 1518 5408 a Ft(2)p Fs(\016)r(t)1623 5276 y Fl(\021)1702 5377 y Fv(=)25 b Fw(iR)q Fv(\()p Fw(z)t Fv(\))p Fw(\030)2076 5276 y Fl(\020)2130 5377 y Fu(\006)2214 5329 y Fs(Q)2270 5339 y Fn(j)p 2211 5356 V 2211 5408 a Ft(2)p Fs(\016)r(t)2315 5276 y Fl(\021)2384 5377 y Fu(r)2460 5391 y Fs(j)2497 5377 y Fw(V)20 b Fv(\()p Fw(Q)p Fv(\))p Fw(R)q Fv(\()p Fw(z)t Fv(\))762 5574 y Fu(\007)855 5538 y Ft(1)p 843 5553 60 4 v 843 5605 a Fs(\016)r(t)912 5574 y Fw(\030)956 5541 y FC(0)995 5473 y Fl(\020)1049 5574 y Fu(\006)1133 5526 y Fs(Q)1189 5536 y Fn(j)p 1130 5553 95 4 v 1130 5605 a Ft(2)p Fs(\016)r(t)1234 5473 y Fl(\021)1304 5574 y Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(\030)1549 5473 y Fl(\020)1604 5574 y Fu(\006)1685 5526 y Fs(Q)1741 5536 y Fn(j)p 1684 5553 89 4 v 1699 5605 a Fs(\016)r(t)1782 5473 y Fl(\021)1852 5574 y Fu(r)1928 5588 y Fs(j)1964 5574 y Fw(V)g Fv(\()p Fw(Q)p Fv(\))p Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(P)2423 5588 y Fs(j)2461 5574 y Fw(R)q Fv(\()p Fw(z)t Fv(\))h(+)2759 5500 y Fl(\000)2840 5538 y Ft(1)p 2811 5553 95 4 v 2811 5605 a(2)p Fs(\016)r(t)2915 5500 y Fl(\001)2957 5521 y Ft(2)3012 5574 y Fu(R)518 b Fx(\(32\))1880 5836 y(26)p eop %%Page: 27 27 bop 83 482 a Fx(where)31 b Fu(R)f Fx(is)g(equal)h(to)489 714 y Fw(iR)q Fv(\()p Fw(z)t Fv(\))p Fu(r)782 728 y Fs(j)820 714 y Fw(V)20 b Fv(\()p Fw(Q)p Fv(\))p Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(\030)1265 676 y FC(0)q(0)1324 586 y Fl(\022)1391 714 y Fu(\006)1478 652 y Fw(Q)1550 666 y Fs(j)p 1472 693 123 4 v 1472 776 a Fv(2)p Fw(\016)s(t)1603 586 y Fl(\023)1685 714 y Fw(R)q Fv(\()p Fw(z)t Fv(\))336 965 y(+)82 b Fw(R)q Fv(\()p Fw(z)t Fv(\))690 837 y Fl(\032)759 965 y Fu(\000)p Fv(4)p Fw(iP)964 979 y Fs(j)1002 965 y Fw(\030)1046 928 y FC(00)1103 837 y Fl(\022)1170 965 y Fu(\006)1258 903 y Fw(Q)1330 917 y Fs(j)p 1251 945 V 1251 1028 a Fv(2)p Fw(\016)s(t)1383 837 y Fl(\023)1470 965 y Fu(\006)1587 904 y Fv(1)p 1571 945 77 4 v 1571 1028 a Fw(\016)s(t)1658 965 y(\030)1702 928 y FC(000)1778 837 y Fl(\022)1845 965 y Fu(\006)1933 903 y Fw(Q)2005 917 y Fs(j)p 1926 945 123 4 v 1926 1028 a Fv(2)p Fw(\016)s(t)2058 837 y Fl(\023\033)2208 965 y Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(\030)2454 837 y Fl(\022)2521 965 y Fu(\006)2602 903 y Fw(Q)2674 917 y Fs(j)p 2602 945 109 4 v 2618 1028 a Fw(\016)s(t)2720 837 y Fl(\023)2802 965 y Fu(r)2878 979 y Fs(j)2914 965 y Fw(V)21 b Fv(\()p Fw(Q)p Fv(\))p Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(P)3374 979 y Fs(j)3412 965 y Fw(R)q Fv(\()p Fw(z)t Fv(\))336 1217 y(+)82 b Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fu(r)751 1231 y Fs(j)788 1217 y Fw(V)21 b Fv(\()p Fw(Q)p Fv(\))p Fw(R)q Fv(\()p Fw(z)t Fv(\))1205 1089 y Fl(\032)1274 1217 y Fu(\000)p Fv(4)p Fw(i\030)1465 1179 y FC(0)q(0)1523 1089 y Fl(\022)1590 1217 y Fu(\006)1678 1155 y Fw(Q)1750 1169 y Fs(j)p 1671 1196 123 4 v 1671 1279 a Fv(2)p Fw(\016)s(t)1803 1089 y Fl(\023)1885 1217 y Fw(P)1943 1231 y Fs(j)2000 1217 y Fu(\007)2117 1155 y Fv(1)p 2101 1196 77 4 v 2101 1279 a Fw(\016)s(t)2188 1217 y(\030)2232 1179 y FC(000)2308 1089 y Fl(\022)2375 1217 y Fu(\006)2463 1155 y Fw(Q)2535 1169 y Fs(j)p 2456 1196 123 4 v 2456 1279 a Fv(2)p Fw(\016)s(t)2588 1089 y Fl(\023\033)2738 1217 y Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(P)2982 1231 y Fs(j)3020 1217 y Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(:)83 1454 y Fx(Again,)38 b(b)m(y)d(taking)i Fw(z)h Fv(=)c Fw(\025)24 b Fv(+)g Fw(i\026)35 b Fx(and)h(using)g(\(28\))h (for)e(the)h(resolv)m(en)m(ts)h(of)e(the)i(\034rst)e(t)m(w)m(o)j(terms)e(on)g (the)83 1567 y(r.h.s.)31 b(of)f(\(32\),)h(one)f(obtains)h(that)g(their)g (norm)g(is)f(less)g(than)733 1704 y Fl(\020)787 1805 y Fv(1)21 b(+)944 1731 y Fl(\000)986 1805 y Fu(j)p Fw(\025)p Fu(j)f Fv(+)g Fu(j)p Fw(\026)g Fu(\000)g Fv(1)p Fu(j)1461 1731 y Fl(\001)1544 1743 y Fv(1)p 1514 1784 106 4 v 1514 1867 a Fu(j)p Fw(\026)p Fu(j)1629 1704 y Fl(\021)1683 1727 y Ft(2)1738 1673 y Fl(\015)1738 1728 y(\015)1738 1782 y(\015)1738 1837 y(\015)1788 1805 y Fw(R)q Fv(\()p Fw(i)p Fv(\))p Fw(\030)2019 1677 y Fl(\022)2086 1805 y Fu(\006)2174 1743 y Fw(Q)2246 1757 y Fs(j)p 2167 1784 123 4 v 2167 1867 a Fv(2)p Fw(\016)s(t)2299 1677 y Fl(\023)2381 1805 y Fu(r)2457 1819 y Fs(j)2493 1805 y Fw(V)g Fv(\()p Fw(Q)p Fv(\))p Fw(R)q Fv(\()p Fw(i)p Fv(\))2879 1673 y Fl(\015)2879 1728 y(\015)2879 1782 y(\015)2879 1837 y(\015)579 2056 y Fv(+)83 b Fw(c)772 1955 y Fl(\020)827 2056 y Fv(1)20 b(+)983 1983 y Fl(\000)1025 2056 y Fu(j)p Fw(\025)p Fu(j)h Fv(+)f Fu(j)p Fw(\026)g Fu(\000)g Fv(1)p Fu(j)1501 1983 y Fl(\001)1583 1995 y Fv(1)p 1553 2035 106 4 v 1553 2119 a Fu(j)p Fw(\026)p Fu(j)1668 1955 y Fl(\021)1723 1978 y Ft(3)1777 1924 y Fl(\015)1777 1979 y(\015)1777 2033 y(\015)1777 2088 y(\015)1828 2056 y Fw(R)q Fv(\()p Fw(i)p Fv(\))p Fw(\030)2059 1928 y Fl(\022)2125 2056 y Fu(\006)2206 1994 y Fw(Q)2278 2008 y Fs(j)p 2206 2035 109 4 v 2222 2119 a Fw(\016)s(t)2324 1928 y Fl(\023)2407 2056 y Fu(r)2483 2070 y Fs(j)2519 2056 y Fw(V)g Fv(\()p Fw(Q)p Fv(\))p Fw(R)q Fv(\()p Fw(i)p Fv(\))2905 1924 y Fl(\015)2905 1979 y(\015)2905 2033 y(\015)2905 2088 y(\015)2972 2056 y Fu(k)p Fw(P)3075 2070 y Fs(j)3112 2056 y Fw(R)q Fv(\()p Fw(i)p Fv(\))p Fu(k)p Fw(;)254 b Fx(\(33\))83 2304 y(where)41 b Fw(c)h Fv(=)557 2268 y Ft(1)p 557 2283 36 4 v 558 2336 a Fs(\016)602 2304 y Fu(k)p Fw(\030)691 2271 y FC(0)715 2304 y Fu(k)760 2318 y Fs(L)808 2299 y Fq(1)918 2304 y Fx(is)d(indep)s(enden)m(t)j(of)e Fw(t;)15 b(\021)43 b Fx(and)e Fw(\026)f Fx(\(one)g(has)g(used)g(that)2908 2268 y Ft(1)p 2908 2283 V 2913 2336 a Fs(t)2995 2304 y Fu(\024)h Fv(1)f Fx(for)g(all)h Fw(t)g Fu(\025)g Fv(1)p Fx(\).)83 2448 y(Similarly)-8 b(,)28 b(one)e(can)f(c)m(hec)m(k)i(that)f(the)g(norm)g(of)f Fu(R)g Fx(is)g(less)g(than)h(a)f(p)s(olynomial)i(in)2901 2347 y Fl(\020)2955 2448 y Fv(1)10 b(+)3091 2375 y Fl(\000)3134 2448 y Fu(j)p Fw(\025)p Fu(j)g Fv(+)g Fu(j)p Fw(\026)g Fu(\000)g Fv(1)p Fu(j)3569 2375 y Fl(\001)3645 2413 y Ft(1)p 3622 2428 82 4 v 3622 2481 a FC(j)p Fs(\026)p FC(j)3714 2347 y Fl(\021)83 2579 y Fx(of)35 b(order)g(4)g(with)h(co)s(e\036cien)m(ts)g(indep)s(enden)m(t) h(of)d Fw(t;)15 b(\025)35 b Fx(and)g Fw(\026)p Fx(.)54 b(One)36 b(\034nishes)f(the)g(pro)s(of)f(b)m(y)i(taking)f(in)m(to)83 2692 y(accoun)m(t)g(\(29\))e(with)h Fw(m)c Fv(=)f(5)34 b Fx(and)f(b)m(y)h (observing)f(that)h(the)f(t)m(w)m(o)i(norms)e(in)h(\(33\))f(and)h(the)f (factor)3531 2618 y Fl(\000)3612 2656 y Ft(1)p 3583 2671 95 4 v 3583 2723 a(2)p Fs(\016)r(t)3687 2618 y Fl(\001)3729 2640 y Ft(2)83 2805 y Fx(in)e(\(32\))f(b)s(elong)h(to)g Fw(L)841 2772 y Ft(1)880 2731 y Fl(\000)922 2805 y Fv(\(1)p Fw(;)15 b Fu(1)p Fv(\))p Fw(;)g(dt)1288 2731 y Fl(\001)1331 2805 y Fx(.)249 2918 y(c\))38 b(F)-8 b(or)37 b(the)h(second)g(statemen)m(t)h(of)e Fr(iii\))f Fx(let)h(us)g(denote)i(b)m(y)e Fw(\013)h Fx(the)f(only)h(elemen)m (t)h(of)e Fu(L)f Fx(suc)m(h)i(that)83 3031 y Fw(\013)23 b Fk(l)g Fw(o)34 b Fx(and)h Fw(\013)574 3045 y Fs(j)642 3031 y Fv(=)d Fu(\006)p Fv(1)p Fx(.)52 b(One)35 b(observ)m(es)f(that)h Fv([)p Fw(P)1777 3045 y Fs(j)1814 3031 y Fw(;)15 b(V)21 b Fv(\()p Fw(Q)p Fv(\)])33 b(=)e([)p Fw(P)2313 3045 y Fs(j)2350 3031 y Fw(;)15 b(V)21 b Fv(\()p Fw(Q)p Fv(\))i Fu(\000)g Fw(V)2796 2998 y Fs(\013)2846 3031 y Fv(\()p Fw(Q)p Fv(\)])34 b Fx(since)h Fw(V)3346 2998 y Fs(\013)3430 3031 y Fx(do)s(es)e(not)83 3143 y(dep)s(end)e(on)f Fw(x)574 3157 y Fs(j)611 3143 y Fx(.)40 b(Hence,)31 b(one)g(has)361 3331 y Fv([)p Fw(P)444 3345 y Fs(j)481 3331 y Fw(;)15 b(R)q Fv(\()p Fw(z)t Fv(\)])26 b(=)854 3257 y Fl(\010)907 3331 y Fw(R)q Fv(\()p Fw(z)t Fv(\))1093 3257 y Fl(\000)1136 3331 y Fw(V)20 b Fv(\()p Fw(Q)p Fv(\))h Fu(\000)f Fw(V)1536 3294 y Fs(\013)1585 3331 y Fv(\()p Fw(Q)p Fv(\))1727 3257 y Fl(\001)q(\011)1823 3331 y Fw(P)1881 3345 y Fs(j)1918 3331 y Fw(R)q Fv(\()p Fw(z)t Fv(\))g Fu(\000)g Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(P)2459 3345 y Fs(j)2497 3257 y Fl(\010\000)2592 3331 y Fw(V)g Fv(\()p Fw(Q)p Fv(\))h Fu(\000)f Fw(V)2992 3294 y Fs(\013)3041 3331 y Fv(\()p Fw(Q)p Fv(\))3183 3257 y Fl(\001)3225 3331 y Fw(R)q Fv(\()p Fw(z)t Fv(\))3411 3257 y Fl(\011)3465 3331 y Fw(:)83 3519 y Fx(Then,)35 b(the)f(same)g(metho)s(d)g(and)g(the)g (same)f(argumen)m(ts)j(already)e(used)f(in)h(a\))g(and)g(b\))f(ma)m(y)h(b)s (e)g(applied.)83 3654 y(Since)k Fw(\030)385 3553 y Fl(\020)439 3654 y Fu(\006)523 3606 y Fs(Q)579 3616 y Fn(j)p 520 3633 V 520 3686 a Ft(2)p Fs(\016)r(t)624 3553 y Fl(\021)694 3581 y(\000)735 3654 y Fw(V)20 b Fv(\()p Fw(Q)p Fv(\))26 b Fu(\000)e Fw(V)1144 3621 y Fs(\013)1194 3654 y Fv(\()p Fw(Q)p Fv(\))1336 3581 y Fl(\001)1378 3654 y Fw(R)q Fv(\()p Fw(i)p Fv(\))37 b Fu(2)f Fw(o)p Fv(\()p Fw(t)1795 3621 y Ft(0)1834 3654 y Fv(\))i Fx(b)m(y)f (de\034nition)h(of)f(the)g(cartesian)h(p)s(oten)m(tial)g Fw(V)20 b Fx(,)38 b(one)83 3826 y(obtains)31 b(that)g Fv([)p Fw(P)679 3840 y Fs(j)716 3826 y Fw(;)15 b(\021)s Fv(\()p Fw(H)7 b Fv(\)])p Fw(f)1053 3725 y Fl(\020)1120 3785 y Fs(Q)p 1117 3805 61 4 v 1117 3858 a Ft(2)p Fs(t)1188 3725 y Fl(\021)1268 3826 y Fu(2)24 b Fw(o)p Fv(\()p Fw(t)1465 3793 y Ft(0)1505 3826 y Fv(\))p Fx(.)249 3962 y(d\))31 b(F)-8 b(or)31 b(the)f(statemen)m(t)i Fr(v\))p Fx(,)f(let)f(us)g(observ)m(e)h(that)83 4149 y Fv(\()p Fw(P)176 4163 y Fs(j)223 4149 y Fv(+)10 b Fw(i)p Fv(\))p Fw(Q)442 4163 y Fs(j)479 4149 y Fw(R)q Fv(\()p Fw(z)t Fv(\)\(1)g(+)g Fw(Q)908 4112 y Ft(2)949 4149 y Fv(\))984 4112 y FC(\000)p Ft(1)p Fs(=)p Ft(2)1174 4149 y Fv(=)25 b(\()p Fw(P)1363 4163 y Fs(j)1410 4149 y Fv(+)10 b Fw(i)p Fv(\))p Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(Q)1815 4163 y Fs(j)1853 4149 y Fv(\(1)g(+)g Fw(Q)2096 4112 y Ft(2)2136 4149 y Fv(\))2171 4112 y FC(\000)p Ft(1)p Fs(=)p Ft(2)2346 4149 y Fu(\000)g Fv(2)p Fw(i)p Fv(\()p Fw(P)2596 4163 y Fs(j)2644 4149 y Fv(+)g Fw(i)p Fv(\))p Fw(R)q Fv(\()p Fw(z)t Fv(\))p Fw(P)3035 4163 y Fs(j)3073 4149 y Fw(R)q Fv(\()p Fw(z)t Fv(\)\(1)g(+)g Fw(Q)3502 4112 y Ft(2)3543 4149 y Fv(\))3578 4112 y FC(\000)p Ft(1)p Fs(=)p Ft(2)3743 4149 y Fw(:)83 4337 y Fx(In)32 b(order)h(to)g(mak)m(e)g(the)g(calculation)i(of)d (the)h(comm)m(utator)i Fv([)p Fw(Q)2277 4351 y Fs(j)2313 4337 y Fw(;)15 b(R)q Fv(\()p Fw(z)t Fv(\)])34 b Fx(prop)s(erly)-8 b(,)34 b(one)f(has)f(in)m(v)m(ok)m(ed)j(the)83 4450 y(in)m(v)-5 b(ariance)36 b(of)d Fu(H)698 4417 y Ft(2)771 4450 y Fx(under)i(the)f(group)h Fu(f)p Fw(e)1537 4417 y Fs(iy)r(Q)1654 4427 y Fn(j)1691 4450 y Fu(g)1736 4464 y Fs(y)r FC(2)p Ft(I)-18 b(R)1918 4450 y Fx(and)34 b(the)h(statemen)m(t)g(\(a\))f(of)g(Theorem)h(6.3.4)g(of)e([1)q(].)83 4563 y(Then)e(the)g(end)f(of)g(the)h(pro)s(of)f(follo)m(ws)h(the)g(sc)m(heme) g(of)f(the)h(previous)f(p)s(oin)m(ts.)249 4676 y(e\))g(F)-8 b(or)30 b(the)g(statemen)m(t)h Fr(ii\))p Fx(,)e(let)h Fw(\016)f(>)c Fv(0)30 b Fx(and)g Fw(\030)k Fx(b)s(e)29 b(an)h(asymptotic)h(lo)s(calization) g(function)f(suc)m(h)g(that)83 4789 y Fw(\030)142 4715 y Fl(\000)184 4789 y Fu(\006)265 4741 y Fs(x)305 4751 y Fn(j)p 264 4768 73 4 v 284 4820 a Fs(\016)347 4715 y Fl(\001)403 4789 y Fw(f)10 b Fv(\()p Fw(x)p Fv(\))26 b(=)f Fw(f)10 b Fv(\()p Fw(x)p Fv(\))30 b Fx(for)g(all)g Fw(x)c Fu(2)f Fv(I)-18 b(R)1417 4753 y Fs(n)1464 4789 y Fx(.)40 b(One)31 b(has)675 5008 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(f)946 4907 y Fl(\020)1013 4967 y Fs(Q)p 1011 4987 61 4 v 1011 5039 a Ft(2)p Fs(t)1081 4907 y Fl(\021)1151 5008 y Fu(r)1227 5022 y Fs(j)1263 5008 y Fw(V)20 b Fv(\()p Fw(Q)p Fv(\))p Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))27 b(=)e Fw(f)1872 4907 y Fl(\020)1938 4967 y Fs(Q)p 1936 4987 V 1936 5039 a Ft(2)p Fs(t)2007 4907 y Fl(\021)2076 5008 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(\030)2337 4907 y Fl(\020)2392 5008 y Fu(\006)2476 4960 y Fs(Q)2532 4970 y Fn(j)p 2472 4987 95 4 v 2472 5039 a Ft(2)p Fs(\016)r(t)2577 4907 y Fl(\021)2646 5008 y Fu(r)2722 5022 y Fs(j)2758 5008 y Fw(V)21 b Fv(\()p Fw(Q)p Fv(\))p Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))784 5205 y(+)870 5104 y Fl(h)913 5205 y Fw(\021)s Fv(\()p Fw(H)g Fv(\))p Fw(;)15 b(f)1224 5104 y Fl(\020)1291 5164 y Fs(Q)p 1289 5184 61 4 v 1289 5236 a Ft(2)p Fs(t)1359 5104 y Fl(\021)q(i)1472 5205 y Fv(\(\001)20 b(+)g Fw(i)p Fv(\)\(\001)h(+)f Fw(i)p Fv(\))2049 5172 y FC(\000)p Ft(1)2144 5205 y Fw(\030)2203 5104 y Fl(\020)2257 5205 y Fu(\006)2341 5157 y Fs(Q)2397 5167 y Fn(j)p 2338 5184 95 4 v 2338 5236 a Ft(2)p Fs(\016)r(t)2442 5104 y Fl(\021)2512 5205 y Fu(r)2588 5219 y Fs(j)2624 5205 y Fw(V)g Fv(\()p Fw(Q)p Fv(\))p Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(:)83 5451 y Fx(One)27 b(observ)m(es)g(that)g Fw(f)886 5351 y Fl(\020)952 5410 y Fs(Q)p 950 5431 61 4 v 950 5483 a Ft(2)p Fs(t)1021 5351 y Fl(\021)1101 5451 y Fx(and)1273 5351 y Fl(h)1316 5451 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))p Fw(;)15 b(f)1628 5351 y Fl(\020)1695 5410 y Fs(Q)p 1692 5431 V 1692 5483 a Ft(2)p Fs(t)1763 5351 y Fl(\021i)1875 5451 y Fv(\(\001)e(+)g Fw(i)p Fv(\))27 b Fx(are)g(norm)g(b)s(ounded)g(indep)s(enden) m(tly)i(of)d Fw(t)g Fx(for)83 5587 y Fw(t)f Fu(\025)g Fv(1)31 b Fx(\(use)f Fr(i\))g Fx(for)g(the)h(second)f(term\).)42 b(By)29 b(taking)i(\(5\))g(in)m(to)g(accoun)m(t,)h(one)f(gets)f(the)h(exp)s(ected)g (result.)1880 5836 y(27)p eop %%Page: 28 28 bop 249 482 a Fx(f)7 b(\))29 b(The)h(pro)s(of)f(of)h Fr(iv\))f Fx(is)g(similar)h(to)g(the)g(one)g(giv)m(en)g(in)g(the)g(paragraph)h(b\))f (of)f(Lemma)i(4.1.)41 b(If)29 b Fw(\016)g(>)c Fv(0)83 617 y Fx(and)34 b Fw(\030)j Fx(is)c(an)g(asymptotic)h(lo)s(calization)h(function,)g (one)f(observ)m(es)f(that)2623 516 y Fl(n)2684 617 y Fv(1)20 b Fu(\000)2840 549 y Fl(Q)2926 644 y FC(f)p Fs(j)t FC(j)p Fs(\013)3059 654 y Fn(j)3091 644 y FC(6)p Ft(=0)p FC(g)3236 617 y Fw(\030)3295 543 y Fl(\000)3337 617 y Fw(\013)3395 631 y Fs(j)3453 569 y(x)3493 579 y Fn(j)p 3442 596 95 4 v 3442 648 a Ft(2)p Fs(\016)r(t)3546 543 y Fl(\001)3588 516 y(o)3648 680 y Fs(t)p FC(\025)p Ft(1)83 768 y Fx(is)28 b(an)h(appro)m(ximate)h(unit)f(for)f(the)g(ideal)h Fu(h)p Fw(J)1593 782 y Fs(\013)1660 768 y Fu(\001)16 b Fw(C)1766 782 y Ft(0)1806 768 y Fv(\(I)-18 b(R)1923 732 y Fs(n)p FC(\003)2005 768 y Fv(\))p Fu(i)p Fx(.)40 b(Since)29 b Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))18 b Fu(\000)e Fw(\021)s Fv(\()p Fw(H)2839 782 y Fs(\013)2888 768 y Fv(\))29 b Fx(b)s(elongs)f(to)h(this)f(ideal,)83 904 y(it)i(follo)m(ws)h(that)g Fv(lim)792 918 y Fs(t)p FC(!1)978 799 y Fl(\015)978 854 y(\015)978 908 y(\015)1029 835 y(Q)1115 930 y FC(f)p Fs(j)t FC(j)p Fs(\013)1248 940 y Fn(j)1280 930 y FC(6)p Ft(=0)p FC(g)1425 904 y Fw(\030)1484 803 y Fl(\020)1538 904 y Fw(\013)1596 918 y Fs(j)1646 856 y(Q)1702 866 y Fn(j)p 1643 883 V 1643 935 a Ft(2)p Fs(\016)r(t)1747 803 y Fl(\021)1816 830 y(\000)1858 904 y Fw(\021)s Fv(\()p Fw(H)7 b Fv(\))21 b Fu(\000)f Fw(\021)s Fv(\()p Fw(H)2330 918 y Fs(\013)2380 904 y Fv(\))2415 830 y Fl(\001)2457 799 y(\015)2457 854 y(\015)2457 908 y(\015)2533 904 y Fv(=)25 b(0)30 b Fx(.)968 b Fc(\004)249 1047 y Fr(Pr)-5 b(o)g(of)28 b(of)g(L)-5 b(emma)27 b(6.3:)37 b Fx(Since)26 b(the)f(follo)m(wing)i(inequalit)m(y)f(is)f(satis\034ed)g(for)g (all)g Fw(t)g Fu(\025)g Fv(1)g Fx(and)h(all)f Fw(')h Fu(2)f(H)3704 1014 y Ft(2)3743 1047 y Fx(:)754 1245 y Fu(h)p Fw(';)15 b(B)962 1208 y FC(\003)1002 1245 y Fv(\()p Fw(t)p Fv(\))p Fw(B)5 b Fv(\()p Fw(t)p Fv(\))p Fw(')p Fu(i)27 b(\024)e(\006h)p Fw(';)15 b Fb(D)p Fv(\010\()p Fw(t)p Fv(\))p Fw(')p Fu(i)22 b(\000)e(h)p Fw(';)15 b(F)e Fv(\()p Fw(t)p Fv(\))p Fw(')p Fu(i)23 b(\000)d(h)p Fw(';)15 b(L)p Fv(\()p Fw(t)p Fv(\))p Fw(')p Fu(i)p Fw(;)83 1444 y Fx(one)31 b(obtains)g(that)342 1546 y Fl(Z)433 1573 y FC(1)393 1752 y Ft(1)523 1670 y Fu(k)p Fw(B)5 b Fv(\()p Fw(t)p Fv(\))p Fw(e)787 1632 y FC(\000)p Fs(iH)g(t)959 1670 y Fw(')p Fu(k)1063 1632 y Ft(2)1104 1670 y Fw(dt)83 b Fu(\024)1421 1538 y Fl(\014)1421 1593 y(\014)1421 1647 y(\014)1421 1702 y(\014)1451 1546 y(Z)1542 1573 y FC(1)1501 1752 y Ft(1)1616 1670 y Fu(h)p Fw(e)1693 1632 y FC(\000)p Fs(iH)5 b(t)1866 1670 y Fw(';)15 b Fb(D)p Fv(\010\()p Fw(t)p Fv(\))p Fw(e)2256 1632 y FC(\000)p Fs(iH)5 b(t)2429 1670 y Fw(')p Fu(i)p Fw(dt)2603 1538 y Fl(\014)2603 1593 y(\014)2603 1647 y(\014)2603 1702 y(\014)1421 1916 y Fv(+)1507 1792 y Fl(Z)1597 1818 y FC(1)1557 1998 y Ft(1)1687 1916 y Fu(jh)p Fw(e)1789 1878 y FC(\000)p Fs(iH)g(t)1962 1916 y Fw(';)15 b(F)e Fv(\()p Fw(t)p Fv(\))p Fw(e)2277 1878 y FC(\000)p Fs(iH)5 b(t)2450 1916 y Fw(')p Fu(ij)p Fw(dt)22 b Fv(+)e Fu(k)p Fw(')p Fu(k)2911 1878 y Ft(2)2967 1792 y Fl(Z)3057 1818 y FC(1)3017 1998 y Ft(1)3147 1916 y Fu(k)p Fw(L)p Fv(\()p Fw(t)p Fv(\))p Fu(k)p Fw(dt:)83 2174 y Fx(But)25 b(one)g(has)g(already)h(noticed)g(that)f Fu(h)p Fw(e)1465 2141 y FC(\000)p Fs(iH)5 b(t)1638 2174 y Fw(';)15 b Fb(D)p Fv(\010\()p Fw(t)p Fv(\))p Fw(e)2028 2141 y FC(\000)p Fs(iH)5 b(t)2201 2174 y Fw(')p Fu(i)26 b Fv(=)2440 2138 y Fs(d)p 2427 2153 62 4 v 2427 2205 a(dt)2499 2174 y Fu(h)p Fw(e)2576 2141 y FC(\000)p Fs(iH)5 b(t)2748 2174 y Fw(';)15 b Fv(\010\()p Fw(t)p Fv(\))p Fw(e)3058 2141 y FC(\000)p Fs(iH)5 b(t)3231 2174 y Fw(')p Fu(i)p Fx(,)27 b(and)e(hence)83 2210 y Fl(\014)83 2264 y(\014)113 2214 y(R)174 2240 y FC(1)156 2319 y Ft(1)249 2287 y Fu(h)p Fw(e)326 2254 y FC(\000)p Fs(iH)5 b(t)498 2287 y Fw(';)15 b Fb(D)p Fv(\010\()p Fw(t)p Fv(\))p Fw(e)888 2254 y FC(\000)p Fs(iH)5 b(t)1061 2287 y Fw(')p Fu(i)p Fw(dt)1235 2210 y Fl(\014)1235 2264 y(\014)1292 2287 y Fu(\024)25 b Fv(2)p Fu(k)p Fv(\010)p Fu(k)1589 2301 y Fs(L)1637 2282 y Fq(1)1707 2287 y Fu(k)p Fw(')p Fu(k)1856 2254 y Ft(2)1897 2287 y Fx(.)40 b(The)31 b(conclusion)g(is)f(no)m(w)h(straigh)m(tforw)m(ard.)205 b Fc(\004)249 2400 y Fr(Pr)-5 b(o)g(of)33 b(of)f(L)-5 b(emma)33 b(6.4:)39 b Fx(Let)31 b Fw( )e Fu(2)c(D)1510 2414 y Ft(2)1549 2400 y Fx(,)31 b Fw(')25 b Fu(2)g(D)1845 2414 y Ft(1)1915 2400 y Fx(and)31 b Fv(1)25 b Fu(\024)g Fw(s)g(<)g(t)p Fx(.)40 b(One)31 b(has)1042 2521 y Fl(\014)1042 2576 y(\014)1072 2525 y(\012)1115 2599 y Fw( )s(;)1217 2525 y Fl(\010)1271 2599 y Fw(e)1313 2566 y Fs(iH)1395 2575 y Fm(2)1430 2566 y Fs(t)1459 2599 y Fv(\010\()p Fw(t)p Fv(\))p Fw(e)1670 2566 y FC(\000)p Fs(iH)1807 2575 y Fm(1)1842 2566 y Fs(t)1892 2599 y Fu(\000)20 b Fw(e)2025 2566 y Fs(iH)2107 2575 y Fm(2)2142 2566 y Fs(s)2179 2599 y Fv(\010\()p Fw(s)p Fv(\))p Fw(e)2400 2566 y FC(\000)p Fs(iH)2537 2575 y Fm(1)2572 2566 y Fs(s)2608 2525 y Fl(\011)2677 2599 y Fw(')2736 2525 y Fl(\013)2779 2521 y(\014)2779 2576 y(\014)743 2751 y Fu(\024)839 2678 y Fl(R)900 2704 y Fs(t)882 2783 y(s)945 2674 y Fl(\014)945 2729 y(\014)975 2678 y(\012)1018 2751 y Fw( )s(;)15 b(e)1162 2718 y Fs(iH)1244 2727 y Fm(2)1280 2718 y Fs(\034)1338 2678 y Fl(\000)1380 2751 y Fw(iH)1487 2765 y Ft(2)1526 2751 y Fv(\010\()p Fw(\034)10 b Fv(\))21 b Fu(\000)e Fw(i)p Fv(\010\()p Fw(\034)10 b Fv(\))p Fw(H)2116 2765 y Ft(1)2176 2751 y Fv(+)2297 2716 y Fs(d)p 2277 2731 76 4 v 2277 2783 a(d\034)2362 2751 y Fv(\010\()p Fw(\034)g Fv(\))2548 2678 y Fl(\001)2606 2751 y Fw(e)2648 2718 y FC(\000)p Fs(iH)2785 2727 y Fm(1)2819 2718 y Fs(\034)2863 2751 y Fw(')2922 2678 y Fl(\013)2965 2674 y(\014)2965 2729 y(\014)3010 2751 y Fw(d\034)188 2954 y Fu(\024)284 2886 y Fl(P)380 2912 y Fs(N)380 2981 y(k)r Ft(=1)528 2853 y Fl(\020)582 2881 y(R)642 2907 y Fs(t)625 2986 y(s)687 2954 y Fu(k)p Fw(E)799 2969 y Fs(k)842 2954 y Fv(\()p Fw(\034)g Fv(\))p Fw(e)1004 2921 y FC(\000)p Fs(iH)1141 2930 y Fm(2)1177 2921 y Fs(\034)1220 2954 y Fw( )s Fu(k)1327 2921 y Ft(2)1368 2954 y Fw(d\034)1465 2853 y Fl(\021)1529 2849 y Fm(1)p 1529 2861 31 3 v 1529 2902 a(2)1589 2853 y Fl(\020)1643 2881 y(R)1704 2907 y Fs(t)1686 2986 y(s)1749 2954 y Fu(k)p Fw(F)1852 2969 y Fs(k)1895 2954 y Fv(\()p Fw(\034)g Fv(\))p Fw(e)2057 2921 y FC(\000)p Fs(iH)2194 2930 y Fm(1)2230 2921 y Fs(\034)2273 2954 y Fw(')p Fu(k)2377 2921 y Ft(2)2417 2954 y Fw(d\034)2514 2853 y Fl(\021)2579 2849 y Fm(1)p 2579 2861 V 2579 2902 a(2)2644 2954 y Fv(+)2734 2853 y Fl(\020)2789 2881 y(R)2849 2907 y Fs(t)2832 2986 y(s)2894 2954 y Fu(k)p Fw(L)p Fv(\()p Fw(\034)g Fv(\))p Fu(k)p Fw(d\034)3263 2853 y Fl(\021)3334 2954 y Fu(k)p Fw( )s Fu(kk)p Fw(')p Fu(k)p Fw(:)83 3170 y Fx(Th)m(us,)31 b(one)g(gets)f(that)1147 3291 y Fl(\015)1147 3346 y(\015)1197 3368 y Fw(e)1239 3335 y Fs(iH)1321 3344 y Fm(2)1356 3335 y Fs(t)1386 3368 y Fv(\010\()p Fw(t)p Fv(\))p Fw(e)1597 3335 y FC(\000)p Fs(iH)1734 3344 y Fm(1)1769 3335 y Fs(t)1798 3368 y Fw(')21 b Fu(\000)f Fw(e)2011 3335 y Fs(iH)2093 3344 y Fm(2)2128 3335 y Fs(s)2165 3368 y Fv(\010\()p Fw(s)p Fv(\))p Fw(e)2386 3335 y FC(\000)p Fs(iH)2523 3344 y Fm(1)2558 3335 y Fs(s)2594 3368 y Fw(')2653 3291 y Fl(\015)2653 3346 y(\015)709 3506 y Fv(=)25 b(sup)942 3528 y Fs( )r FC(2D)1092 3537 y Fm(2)1126 3528 y Fs(;)p FC(k)p Fs( )r FC(k)p Ft(=1)1375 3429 y Fl(\014)1375 3484 y(\014)1405 3433 y(\012)1448 3506 y Fw( )s(;)1550 3433 y Fl(\010)1604 3506 y Fw(e)1646 3473 y Fs(iH)1728 3482 y Fm(2)1763 3473 y Fs(t)1792 3506 y Fv(\010\()p Fw(t)p Fv(\))p Fw(e)2003 3473 y FC(\000)p Fs(iH)2140 3482 y Fm(1)2175 3473 y Fs(t)2225 3506 y Fu(\000)20 b Fw(e)2358 3473 y Fs(iH)2440 3482 y Fm(2)2475 3473 y Fs(s)2512 3506 y Fv(\010\()p Fw(s)p Fv(\))p Fw(e)2733 3473 y FC(\000)p Fs(iH)2870 3482 y Fm(1)2905 3473 y Fs(s)2941 3433 y Fl(\011)3009 3506 y Fw(')3068 3433 y Fl(\013)3112 3429 y(\014)3112 3484 y(\014)768 3715 y Fu(\024)25 b Fw(c)918 3647 y Fl(P)1014 3673 y Fs(N)1014 3742 y(k)r Ft(=1)1162 3614 y Fl(\020)1216 3642 y(R)1277 3668 y Fs(t)1259 3747 y(s)1322 3715 y Fu(k)p Fw(F)1425 3730 y Fs(k)1468 3715 y Fv(\()p Fw(\034)10 b Fv(\))p Fw(e)1630 3682 y FC(\000)p Fs(iH)1767 3691 y Fm(1)1803 3682 y Fs(\034)1846 3715 y Fw(')p Fu(k)1950 3682 y Ft(2)1990 3715 y Fw(d\034)2087 3614 y Fl(\021)2152 3610 y Fm(1)p 2152 3622 V 2152 3663 a(2)2217 3715 y Fv(+)2308 3614 y Fl(\020)2362 3642 y(R)2423 3668 y Fs(t)2405 3747 y(s)2467 3715 y Fu(k)p Fw(L)p Fv(\()p Fw(\034)g Fv(\))p Fu(k)p Fw(d\034)2836 3614 y Fl(\021)2908 3715 y Fu(k)p Fw(')p Fu(k)p Fw(:)525 b Fx(\(34\))83 3937 y(Since)29 b(\(34\))g(can)g(b)s(e)f(made)h(arbitrarily)g (small)g(b)m(y)g(c)m(ho)s(osing)g Fw(s)f Fx(large)h(enough,)h(one)e(gets)h (the)f(existence)h(of)83 4050 y(s)20 b Fu(\000)g Fv(lim)356 4064 y Fs(t)p FC(!1)542 4050 y Fw(e)584 4017 y Fs(iH)666 4026 y Fm(2)701 4017 y Fs(t)731 4050 y Fv(\010\()p Fw(t)p Fv(\))p Fw(e)942 4017 y FC(\000)p Fs(iH)1079 4026 y Fm(1)1114 4017 y Fs(t)1143 4050 y Fw(')31 b Fx(for)f(all)g Fw(')c Fu(2)f(D)1737 4064 y Ft(1)1777 4050 y Fx(,)30 b(whic)m(h)i(implies)e(the)h(result)g(of)f (this)g(lemma.)328 b Fc(\004)249 4163 y Fj(A)m(c)m(kno)m(wledgmen)m(ts)44 b(:)55 b Fx(The)38 b(author)h(thanks)f(M.)f(M)p Fv(\025)-45 b Fx(an)m(toiu)39 b(for)f(suggesting)g(to)g(him)g(this)g(w)m(ork)83 4275 y(and)28 b(for)g(his)g(helpful)g(advice.)40 b(He)28 b(also)g(expresses)f (his)h(gratitude)h(to)f(W.O.)h(Amrein)f(for)g(critical)h(reading)83 4388 y(of)h(the)h(text)f(and)h(for)f(his)g(constan)m(t)i(supp)s(ort.)83 4674 y Fy(References)286 4877 y Fx([1])46 b(W.O.)25 b(Amrein,)i(A.)e(Boutet)h (de)f(Mon)m(v)m(el,)j(V.)d(Georgescu,)i Fr(C)2513 4891 y Ft(0)2552 4877 y Fr(-Gr)-5 b(oups,)29 b(Commutator)e(Metho)-5 b(ds)427 4990 y(and)32 b(Sp)-5 b(e)g(ctr)g(al)34 b(The)-5 b(ory)33 b(of)f(N-Bo)-5 b(dy)34 b(Hamiltonians)p Fx(,)29 b(Birkh\344user)h(V)-8 b(erlag)32 b(\(1996\).)286 5175 y([2])46 b(W.O.)30 b(Amrein,)h(M.)f(M)p Fv(\025)-45 b Fx(an)m(toiu,)32 b(R.)e(Purice,)h Fr(Pr)-5 b(op)g(agation)34 b(Pr)-5 b(op)g(erties)34 b(for)e(Schr)-5 b(\366dinger)34 b(Op)-5 b(er-)427 5288 y(ators)32 b(A\036liate)-5 b(d)33 b(to)f(Certain)g(C)1563 5255 y FC(\003)1602 5288 y Fr(-A)n(lgebr)-5 b(as)p Fx(,)32 b(Ann.)e(Henri)h(P)m(oincar\351)h Fj(3)p Fx(,)f(1215-1232)g(\(2002\).)286 5474 y([3])46 b(E.B.)31 b(Da)m(vies,)g(B.)f(Simon,)i Fr(Sc)-5 b(attering)34 b(The)-5 b(ory)33 b(for)f(Systems)g(with)g(Di\033er)-5 b(ent)34 b(Sp)-5 b(atial)32 b(Asymp-)427 5587 y(totics)g(on)g(the)h(L)-5 b(eft)33 b(and)f(R)n(ight)p Fx(,)e(Comm)m(un.)i(Math.)f(Ph)m(ys.)f Fj(63)p Fx(,)h(277-301)g(\(1978\).)1880 5836 y(28)p eop %%Page: 29 29 bop 286 482 a Fx([4])46 b(J.)d(Derezi\253ski,)k(C.)d(G\351rard,)j Fr(Sc)-5 b(attering)45 b(The)-5 b(ory)44 b(of)g(Classic)-5 b(al)44 b(and)g(Quantum)h(N-Particle)427 594 y(Systems)p Fx(,)30 b(Springer-V)-8 b(erlag)32 b(Berlin)f(Heidelb)s(erg)g(\(1997\).)286 782 y([5])46 b(Y.)37 b(Dermenjian,)j(V.)d(Iftimie,)i Fr(Nouve)-5 b(aux)41 b(c)-5 b(anaux)41 b(de)e(di\033usion)f(p)-5 b(our)40 b(les)f(milieux)g(str)-5 b(ati\034\351s)427 895 y(p)g(erturb)g(\351s)p Fx(,)33 b(preprin)m(t)f(LA)-8 b(TP)g(,)32 b(Univ)m(ersit\351)f(de)g(Pro)m(v)m (ence,)h(Marseille.)286 1083 y([6])46 b(R.)30 b(F)-8 b(ro)s(ese,)31 b(I.)g(Herbst,)g Fr(A)h(New)i(Pr)-5 b(o)g(of)33 b(of)g(the)g(Mourr)-5 b(e)34 b(Estimate)p Fx(,)d(Duk)m(e)g(Math.)g(J.)g Fj(49)p Fx(,)g(No)g(4,)427 1196 y(1075)g(-)f(1085)h(\(1982\).)286 1383 y([7])46 b(V.)35 b(Georgescu,)i(A.)e(Iftimo)m(vici,)i Fr(On)g(the)g(Sp)-5 b(e)g(ctr)g(al)39 b(A)n(nalysis)d(of)g(A)n(nisotr)-5 b(opic)37 b(Quantum)g(Sys-)427 1496 y(tems)p Fx(,)30 b(unpublished.)286 1684 y([8])46 b(V.)34 b(Georgescu,)j(A.)d(Iftimo)m(vici,)i Fr(Cr)-5 b(osse)g(d)38 b(Pr)-5 b(o)g(ducts)38 b(of)e(C)2484 1651 y FC(\003)2523 1684 y Fr(-A)n(lgebr)-5 b(as)38 b(and)e(Sp)-5 b(e)g(ctr)g(al)38 b(A)n(nalysis)427 1797 y(of)32 b(Quantum)h(Hamiltonians)p Fx(,)c(Comm)m(un.)j (Math.)f(Ph)m(ys.)f Fj(228)p Fx(,)g(519-560)i(\(2002\).)286 1984 y([9])46 b(G.M.)30 b(Graf,)g Fr(Asymptotic)h(Completeness)i(for)f(N-Bo) -5 b(dy)33 b(Short-R)-5 b(ange)33 b(Quantum)g(Systems:)40 b(A)427 2097 y(New)33 b(Pr)-5 b(o)g(of)p Fx(,)31 b(Comm)m(un.)h(Math.)f(Ph)m(ys.)f Fj(132)p Fx(,)h(73-101)g(\(1990\))241 2285 y([10])46 b(I.)80 b(Herbst,)92 b(E.)81 b(Skibsted,)93 b Fr(Quantum)78 b(Sc)-5 b(attering)79 b(for)f(Potentials)g(Indep)-5 b(endent)79 b(of)427 2398 y Fu(j)p Fw(x)p Fu(j)p Fr(:)94 b(Asymptotic)58 b(Completeness)g(for)h (High)f(and)g(L)-5 b(ow)59 b(Ener)-5 b(gies)p Fx(,)67 b(preprin)m(t)60 b(02-149)g(at)427 2511 y(www.ma.utexas.edu/mp_arc.)241 2698 y([11])46 b(T.)30 b(Kato,)i Fr(Perturb)-5 b(ation)33 b(The)-5 b(ory)33 b(for)f(Line)-5 b(ar)32 b(Op)-5 b(er)g(ators)p Fx(,)32 b(Springer-V)-8 b(erlag)33 b(\(1966\).)241 2886 y([12])46 b(M.)28 b(M)p Fv(\025)-45 b Fx(an)m(toiu,)30 b Fr(C)1027 2853 y FC(\003)1066 2886 y Fr(-A)n(lgebr)-5 b(as,)32 b(Dynamic)-5 b(al)31 b(Systems)f(at)h (In\034nity)d(and)j(the)g(Essential)f(Sp)-5 b(e)g(ctrum)427 2999 y(of)32 b(Gener)-5 b(alize)g(d)34 b(Schr)-5 b(\366)g(dinger)34 b(Op)-5 b(er)g(ators)p Fx(,)32 b(J.)e(reine)h(angew.)h(Math.)e Fj(550)p Fx(,)g(211-229)i(\(2002\).)241 3186 y([13])46 b(M.)24 b(M)p Fv(\025)-45 b Fx(an)m(toiu,)27 b Fr(C)1020 3153 y FC(\003)1059 3186 y Fr(-A)n(lgebr)-5 b(as)28 b(and)f(T)-7 b(op)i(olo)g(gic)g(al)29 b(Dynamics)d(in)g(Sp)-5 b(e)g(ctr)g(al)29 b(A)n(nalysis)p Fx(,)24 b(preprin)m(t)j(01-)427 3299 y(476)22 b(at)h(www.ma.utexas.edu/mp_arc,)k(to)c (app)s(ear)f(in)h(the)f(Pro)s(ceedings)h(of)f(the)g(4th)h(Op)s(erator)427 3412 y(Algebras)31 b(In)m(ternational)h(Conference,)g(O)m(AMP)f(2001.)241 3600 y([14])46 b(E.)31 b(Mourre,)g Fr(A)n(bsenc)-5 b(e)34 b(of)e(Singular)g (Continuous)g(Sp)-5 b(e)g(ctrum)34 b(for)e(Certain)g(Self-A)-5 b(djoint)32 b(Op)-5 b(er-)427 3713 y(ators)p Fx(,)30 b(Comm)m(un.)j(Math.)d (Ph)m(ys.)h Fj(78)p Fx(,)f(391-408)i(\(1981\).)241 3900 y([15])46 b(E.)26 b(Mourre,)h Fr(Op)-5 b(\351r)g(ateurs)31 b(c)-5 b(onjugu\351s)30 b(et)e(pr)-5 b(opri\351t\351s)29 b(de)g(pr)-5 b(op)g(agation)p Fx(,)28 b(Comm)m(un.)f(Math.)f(Ph)m(ys.)427 4013 y Fj(91)p Fx(,)k(279-300)h(\(1983\).)241 4201 y([16])46 b(P)-8 b(.)38 b(P)m(erry)-8 b(,)40 b(I.M.)d(Sigal,)i(B.)e(Simon,)j Fr(Sp)-5 b(e)g(ctr)g(al)40 b(A)n(nalysis)d(of)i(N-Bo)-5 b(dy)39 b(Schr)-5 b(\366)g(dinger)40 b(Op)-5 b(er)g(ators)p Fx(,)427 4314 y(Ann.)30 b(Math.)h Fj(114)p Fx(,)f(519)h(-)f(567)h(\(1981\).)241 4501 y([17])46 b(M.)33 b(Reed,)i(B.)e(Simon,)j Fr(Metho)-5 b(ds)36 b(of)f(Mo)-5 b(dern)36 b(Mathematic)-5 b(al)36 b(Physics)f(IV)g(:)g(A)n (nalysis)f(of)i(Op-)427 4614 y(er)-5 b(ators)p Fx(,)31 b(A)m(cademic)h(Press) e(\(1978\).)241 4802 y([18])46 b(I.M.)35 b(Sigal,)i(A.)e(So\033er,)i Fr(The)g(N-Particle)h(Sc)-5 b(attering)38 b(Pr)-5 b(oblem:)52 b(Asymptotic)37 b(Completeness)427 4915 y(for)32 b(Short-R)-5 b(ange)33 b(Systems)p Fx(,)d(Ann.)h(Math.)f Fj(126)p Fx(,)g(35-108)i (\(1987\).)241 5103 y([19])46 b(H.)35 b(T)-8 b(ak)j(ai,)37 b Fr(On)g(a)g(Duality)g(for)g(Cr)-5 b(osse)g(d)38 b(Pr)-5 b(o)g(ducts)38 b(of)f(C)2448 5070 y FC(\003)2487 5103 y Fr(-A)n(lgebr)-5 b(as)p Fx(,)38 b(Journal)f(of)e(F)-8 b(unctional)427 5215 y(Analysis)29 b Fj(19)p Fx(,)i(25-39)g(\(1975\).)1880 5836 y(29)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0302210546245--