Content-Type: multipart/mixed; boundary="-------------0211210308405" This is a multi-part message in MIME format. ---------------0211210308405 Content-Type: text/plain; name="02-476.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-476.comments" Thierry Jecko IRMAR, Universit de Rennes I, Campus Beaulieu, F-35042 Rennes C dex, France. jecko@maths.univ-rennes1.fr ---------------0211210308405 Content-Type: text/plain; name="02-476.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-476.keywords" Schr\"odinger operators, semiclassical resolvent estimates, resonances, semiclassical measure, global escape function. ---------------0211210308405 Content-Type: application/postscript; name="non-capt-scal.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="non-capt-scal.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: non-capt-scal.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o non-capt-scal.ps non-capt-scal %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2002.11.21:0936 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB 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b(self-adjoin)o(t)g(on)h(the)f(domain)g(of)h(the)129 2480 y(Laplacian)13 b(\(see)g([18]\).)20 b(Denoting)13 b(its)g(resolv)o(en)o(t)f(b)o(y)g Fm(R)p Fn(\()p Fm(z)r Fn(\))j(:=)e(\()p Fm(P)f Fj(\000)5 b Fm(z)r Fn(\))1440 2462 y Fq(\000)p Fp(1)1486 2480 y Fn(,)13 b(for)g Fm(z)i Fn(in)e(the)g(resolv)o(en)o(t)f(set)129 2540 y(of)h Fm(P)7 b Fn(,)14 b(w)o(e)f(are)h(in)o(terested)e(in)h(its)g(b)q(oundary)h(v)m (alues)g(on)g Fi(R)1198 2522 y Fp(+)1238 2540 y Fn(as)g(b)q(ounded)g (op)q(erators)h(from)d(L)1846 2522 y Fp(2)1846 2552 y Fk(s)1866 2540 y Fn(\()p Fi(R)1924 2522 y Fk(d)1942 2540 y Fn(;)c Fi(C)g Fn(\))129 2600 y(to)18 b(L)220 2582 y Fp(2)220 2613 y Fq(\000)p Fk(s)267 2600 y Fn(\()p Fi(R)324 2582 y Fk(d)342 2600 y Fn(;)8 b Fi(C)h Fn(\),)21 b(for)d Fm(s)f(>)g Fn(1)p Fm(=)p Fn(2.)29 b(By)17 b(L)845 2582 y Fp(2)845 2613 y Fk(s)865 2600 y Fn(\()p Fi(R)923 2582 y Fk(d)941 2600 y Fn(;)8 b Fi(C)g Fn(\))22 b(w)o(e)17 b(mean)g(the)h(w)o(eigh)o(ted)f(L)1562 2582 y Fp(2)1601 2600 y Fn(space)h(of)h(mesurable,)129 2660 y Fi(C)9 b Fn(-v)m(alued)17 b(functions)e Fm(f)20 b Fn(on)15 b Fi(R)690 2642 y Fk(d)722 2660 y Fn(suc)o(h)g(that)g Fm(x)e Fj(7!)h(h)p Fm(x)p Fj(i)1106 2642 y Fk(s)1125 2660 y Fm(f)5 b Fn(\()p Fm(x)p Fn(\))14 b(b)q(elongs)i(to)f(L)1496 2642 y Fp(2)1516 2660 y Fn(\()p Fi(R)1574 2642 y Fk(d)1591 2660 y Fn(;)8 b Fi(C)h Fn(\).)24 b(The)14 b(previously)129 2721 y(men)o(tioned)f (Mourre)j(theory)f(ensures)h(that,)g(for)g(an)o(y)f Fm(\025)f(>)g Fn(0)i(outside)g(the)f(pure)h(p)q(oin)o(t)g(sp)q(ectrum)e(of)129 2781 y Fm(P)7 b Fn(,)16 b(the)g(b)q(oundary)h(v)m(alues)f Fm(R)p Fn(\()p Fm(\025)c Fj(\006)f Fm(i)p Fn(0\))17 b(exist)e(\(see)h ([16,)g(13)q(]\).)129 2841 y(W)l(e)22 b(denote)h(b)o(y)f Fm(p)p Fn(\()p Fm(x;)8 b(\030)r Fn(\))26 b(:=)e Fj(j)p Fm(\030)r Fj(j)745 2823 y Fp(2)781 2841 y Fn(+)16 b Fm(V)11 b Fn(\()p Fm(x)p Fn(\),)24 b(\()p Fm(x;)8 b(\030)r Fn(\))25 b Fj(2)g Fm(T)1208 2823 y Fq(\003)1227 2841 y Fi(R)1266 2823 y Fk(d)1283 2841 y Fn(,)f(the)f(sym)o(b)q(ol)e(of)i Fm(P)30 b Fn(and)24 b(b)o(y)e Fm(\036)1913 2823 y Fk(t)1951 2841 y Fn(the)p eop %%Page: 3 3 3 2 bop 129 133 a Fl(Non-trapping)17 b(b)q(eha)o(viour,)f(21-11-02)1147 b Fn(3)129 286 y(asso)q(ciated)17 b(Hamilton)d(\015o)o(w)j(on)g Fm(T)782 268 y Fq(\003)801 286 y Fi(R)840 268 y Fk(d)857 286 y Fn(.)k(Recall)15 b(that)i(an)g(energy)e Fm(\025)i Fn(is)f(non-trapping)i(for)e Fm(p)h Fn(if)346 395 y Fj(8)p Fn(\()p Fm(x;)8 b(\030)r Fn(\))13 b Fj(2)h Fm(p)570 375 y Fq(\000)p Fp(1)617 395 y Fn(\()p Fm(\025)p Fn(\))p Fm(;)38 b Fn(lim)714 423 y Fk(t)p Fq(!\0001)833 395 y Fj(j)p Fm(\036)876 375 y Fk(t)891 395 y Fn(\()p Fm(x;)8 b(\030)r Fn(\))p Fj(j)22 b Fn(=)g(+)p Fj(1)16 b Fn(and)46 b(lim)1305 423 y Fk(t)p Fq(!)p Fp(+)p Fq(1)1424 395 y Fj(j)p Fm(\036)1467 375 y Fk(t)1482 395 y Fn(\()p Fm(x;)8 b(\030)r Fn(\))p Fj(j)22 b Fn(=)g(+)p Fj(1)11 b Fm(:)154 b Fn(\(2\))129 518 y(Notice)20 b(that)h(this)g(prop)q(ert)o(y)g(is)g (\\op)q(en":)32 b(if)21 b Fm(\025)h Fn(is)f(non-trapping,)i(there)d (exists)h(some)f(in)o(terv)m(al)g Fm(I)129 578 y Fn(ab)q(out)g Fm(\025)f Fn(suc)o(h)g(that)g(eac)o(h)g Fm(\026)g Fj(2)f Fm(I)23 b Fn(is)c(non-trapping.)30 b(Suc)o(h)19 b(an)g(in)o(terv)m(al)f (is)h(called)f(an)h(in)o(terv)m(al)f(of)129 638 y(non-trapping)f (energies.)k(W)l(e)16 b(can)g(no)o(w)h(state)f(our)h(main)e(result.)129 739 y Fg(Theorem)h(1.)24 b Ff(\([20,)17 b(7,)g(22]\))g(Under)h(the)f (pr)n(evious)g(assumptions,)h(let)g Fm(I)f Fj(\032)p Fn(]0;)8 b(+)p Fj(1)p Fn([)16 b Ff(b)n(e)i(a)f(c)n(omp)n(act)129 799 y(interval)h(of)e(non-tr)n(apping)i(ener)n(gies.)23 b(Then,)18 b(for)e(smal)r(l)i(enough)g Fm(h)1399 806 y Fp(0)1419 799 y Ff(,)e(the)i(b)n(oundary)e(values)i Fm(R)p Fn(\()p Fm(\025)11 b Fj(\006)129 859 y Fm(i)p Fn(0\))21 b Ff(exist)g(for)g Fm(\025)g Fj(2)f Fm(I)k Ff(and,)e(for)f(any)g Fm(s)f(>)g Fn(1)p Fm(=)p Fn(2)p Ff(,)j(ther)n(e)e(exists)h Fm(C)1357 866 y Fk(s)1395 859 y Fm(>)f Fn(0)g Ff(such)g(that,)h(uniformly)f(for)129 919 y Fm(\025)14 b Fj(2)g Fm(I)21 b Ff(and)d Fm(h)13 b Fj(2)p Fn(]0;)8 b Fm(h)518 926 y Fp(0)538 919 y Fn(])p Ff(,)694 1028 y Fj(kh)p Fm(x)p Fj(i)785 1007 y Fq(\000)p Fk(s)839 1028 y Fm(R)p Fn(\()p Fm(\025)k Fj(\006)f Fm(i)p Fn(0\))d Fj(h)p Fm(x)p Fj(i)1119 1007 y Fq(\000)p Fk(s)1166 1028 y Fj(k)31 b(\024)g Fm(C)1327 1035 y Fk(s)1353 1028 y Fm(h)1381 1007 y Fq(\000)p Fp(1)1440 1028 y Fm(:)502 b Fn(\(3\))129 1172 y(The)17 b(pap)q(er)h(is)f(organized)g(as)h(follo)o (ws.)24 b(In)17 b(Section)g(2,)g(w)o(e)g(deriv)o(e)f(some)g(prop)q (erties)h(of)h(the)f(\\semi-)129 1232 y(classically)i(resonan)o(t")i (situation,)h(in)e(particular)h(the)f(results)h(men)o(tioned)d(ab)q(o)o (v)o(e,)j(and)h(consider)129 1293 y(the)15 b(in)o(teraction)g(with)g (the)h(non-trapping)h(condition.)k(The)15 b(new)h(pro)q(of)h(of)f (Theorem)e(1)i(is)g(giv)o(en)f(in)129 1353 y(Section)g(3.)129 1448 y(Ac)o(kno)o(wledgmen)n(t)p 129 1465 362 2 v -3 w(:)20 b(the)13 b(author)g(thanks)h(M.)e(Klein)g(for)h(hospitalit)o(y)f (at)h(the)g(univ)o(ersit)o(y)e(of)i(P)o(otsdam)129 1509 y(\(German)o(y\))h(and)j(for)g(fruitful)e(discussions)i(and)f(advices.) 129 1710 y Fo(2)81 b(\\Semiclassicall)o(y)24 b(resonan)n(t")i (situation.)129 1855 y Fn(In)13 b(this)h(section,)f(w)o(e)g(w)o(an)o(t) h(to)g(consider)f(the)h(\\semiclassically)d(resonan)o(t")j(situation.) 21 b(Precisely)l(,)12 b(w)o(e)129 1915 y(assume)e(the)h(existence)e(of) i(a)g(sequence)f(\()p Fm(f)898 1922 y Fk(n)921 1915 y Fn(\))940 1922 y Fk(n)975 1915 y Fn(of)h(nonzero)g(functions)g(of)g (the)g(domain)f Fj(D)q Fn(\()p Fm(P)d Fn(\))12 b(of)f Fm(P)c Fn(,)12 b(of)f(a)129 1976 y(sequence)f(\()p Fm(h)373 1983 y Fk(n)396 1976 y Fn(\))415 1983 y Fk(n)453 1976 y Fj(2)p Fn(]0;)e Fm(h)574 1983 y Fp(0)593 1976 y Fn(])607 1958 y Fe(N)644 1976 y Fn(tending)j(to)h(zero,)f(and)h(of)f(a)h (sequence)e(\()p Fm(z)1397 1983 y Fk(n)1420 1976 y Fn(\))1439 1983 y Fk(n)1476 1976 y Fj(2)k Fi(C)1556 1958 y Fe(N)1596 1976 y Fn(with)d Fj(<)p Fn(\()p Fm(z)1780 1983 y Fk(n)1804 1976 y Fn(\))i Fj(!)h Fm(\025)g(>)g Fn(0)129 2036 y(and)f Fj(=)p Fn(\()p Fm(z)298 2043 y Fk(n)321 2036 y Fn(\))p Fm(=h)392 2043 y Fk(n)430 2036 y Fj(!)h Fm(r)h Fj(\025)e Fn(0,)h(suc)o(h)e(that)i Fj(kh)p Fm(x)p Fj(i)935 2018 y Fq(\000)p Fk(s)981 2036 y Fm(f)1005 2043 y Fk(n)1028 2036 y Fj(k)g Fn(=)g(1)f(and)g Fj(kh)p Fm(x)p Fj(i)1338 2018 y Fk(s)1357 2036 y Fn(\()p Fm(P)7 b Fn(\()p Fm(h)1461 2043 y Fk(n)1485 2036 y Fn(\))t Fj(\000)t Fm(z)1574 2043 y Fk(n)1597 2036 y Fn(\))p Fm(f)1640 2043 y Fk(n)1663 2036 y Fj(k)14 b Fn(=)g Fm(o)p Fn(\()p Fm(h)1824 2043 y Fk(n)1848 2036 y Fn(\),)f(where)129 2096 y Fm(P)7 b Fn(\()p Fm(h)214 2103 y Fk(n)237 2096 y Fn(\))14 b(:=)g Fj(\000)p Fm(h)403 2078 y Fp(2)403 2108 y Fk(n)426 2096 y Fn(\001)467 2103 y Fk(x)496 2096 y Fn(+)8 b Fm(V)k Fn(\()p Fm(x)p Fn(\).)20 b(As)15 b(w)o(e)f(shall)h(see)f(in)g(Section)h (3,)g(suc)o(h)f(a)h(situation)g(app)q(ears)h(when)f(one)129 2156 y(assumes,)d(in)h(view)f(of)i(the)e(pro)q(of)i(ad)g(absurdum)f(of) g(Theorem)f(1,)h(that)h(the)e(b)q(ound)j(\(3\))e(with)g Fm(s)h(>)f Fn(1)p Fm(=)p Fn(2)129 2216 y(for)18 b(\\+")h(is)e(false.)27 b(Ho)o(w)o(ev)o(er,)16 b(w)o(e)h(do)i(not)f(need)g(here)f(to)i(require) d Fm(s)h(>)g Fn(1)p Fm(=)p Fn(2)i(but)f(only)g Fm(s)f Fj(\025)g Fn(0.)26 b(The)129 2277 y(aim)15 b(of)i(this)g(section)f(is)g (to)h(sho)o(w)h(that)f Fj(h)p Fm(x)p Fj(i)942 2259 y Fp(2)p Fk(s)995 2277 y Fn(times)e(an)o(y)h(semiclassical)e(measure)i (of)h(\()p Fj(h)p Fm(x)p Fj(i)1833 2259 y Fq(\000)p Fk(s)1879 2277 y Fm(f)1903 2284 y Fk(n)1927 2277 y Fn(\))1946 2284 y Fk(n)1986 2277 y Fn(is)129 2337 y(in)o(v)m(arian)o(t)c(under)h(the)f (\015o)o(w)h Fm(\036)676 2319 y Fk(t)691 2337 y Fn(,)g(has)g(compact)f (supp)q(ort,)i(and)f(that,)h(for)f Fm(s)g(>)f Fn(1)p Fm(=)p Fn(2,)i(this)f(semiclassical)129 2397 y(measure)g(is)g(nonzero.) 22 b(The)15 b(situation)g(here)g(is)g(similar)e(to)i(those)h(in)e([3],) h(the)g(strategy)g(of)g(whic)o(h)g(w)o(e)129 2457 y(follo)o(w.)21 b(Ho)o(w)o(ev)o(er,)14 b(new)i(ingredien)o(ts)f(and)i(new)f(results)g (app)q(ear)h(in)f(the)g(presen)o(t)g(pap)q(er.)129 2553 y(Let)d(us)g(\014rst)h(recall)e(some)g(w)o(ell)g(kno)o(wn)h(facts)h(ab) q(out)g(semiclassical)d(measures,)h(whic)o(h)g(can)i(b)q(e)f(found)129 2613 y(in)18 b([6,)f(17)q(].)27 b(Let)19 b Fm(\026)480 2620 y Fk(s)517 2613 y Fn(b)q(e)f(a)h(semiclassical)c(measure)i(of)i (the)f(b)q(ounded)h(sequence)e(\()p Fj(h)p Fm(x)p Fj(i)1738 2595 y Fq(\000)p Fk(s)1785 2613 y Fm(f)1809 2620 y Fk(n)1832 2613 y Fn(\))1851 2620 y Fk(n)1893 2613 y Fn(in)h Fm(L)1985 2595 y Fp(2)2005 2613 y Fn(.)129 2673 y(Then)g Fm(\026)287 2680 y Fk(s)325 2673 y Fn(is)g(a)h(\014nite,)f(nonnegativ)o(e)h(Radon)g (measure)f(on)h(the)f(cotangen)o(t)h(space)g Fm(T)1735 2655 y Fq(\003)1754 2673 y Fi(R)1793 2655 y Fk(d)1829 2673 y Fn(\(its)f(total)129 2733 y(mass)i(is)h(less)f(than)i(1\).)35 b(F)l(urthermore,)20 b(there)g(exists)g(a)h(sequence)f Fm(h)1462 2740 y Fk(n)1507 2733 y Fj(!)i Fn(0)f(suc)o(h)g(that,)g(for)h (an)o(y)129 2794 y Fm(a)13 b Fj(2)h Fm(C)254 2776 y Fq(1)250 2806 y Fp(0)291 2794 y Fn(\()p Fm(T)346 2776 y Fq(\003)365 2794 y Fi(R)404 2776 y Fk(d)421 2794 y Fn(\),)327 2917 y(lim)315 2942 y Fk(n)p Fq(!1)415 2869 y Fh(D)441 2917 y Fm(a)467 2897 y Fk(w)494 2917 y Fn(\()p Fm(x;)8 b(h)591 2924 y Fk(n)615 2917 y Fm(D)q Fn(\))p Fj(h)p Fm(x)p Fj(i)741 2897 y Fq(\000)p Fk(s)788 2917 y Fm(f)812 2924 y Fk(n)835 2917 y Fm(;)g Fj(h)p Fm(x)p Fj(i)923 2897 y Fq(\000)p Fk(s)969 2917 y Fm(f)993 2924 y Fk(n)1017 2869 y Fh(E)1072 2917 y Fn(=)1140 2859 y Fh(Z)1163 2953 y Fk(T)1189 2944 y Fd(\003)1207 2953 y Fe(R)1234 2944 y Fc(d)1259 2917 y Fm(a)p Fn(\()p Fm(x;)g(\030)r Fn(\))g Fm(\026)1433 2924 y Fk(s)1452 2917 y Fn(\()p Fm(dx)g(d\030)r Fn(\))24 b(=:)d Fm(\026)1725 2924 y Fk(s)1744 2917 y Fn(\()p Fm(a)p Fn(\))11 b Fm(:)123 b Fn(\(4\))p eop %%Page: 4 4 4 3 bop 129 133 a Fl(Non-trapping)17 b(b)q(eha)o(viour,)f(21-11-02)1147 b Fn(4)129 286 y(Here)18 b Fj(h\001)p Fm(;)8 b Fj(\001i)20 b Fn(denotes)f(the)g Fm(L)654 268 y Fp(2)693 286 y Fn(scalar)h(pro)q (duct)g(and)g Fm(a)1145 268 y Fk(w)1173 286 y Fn(\()p Fm(x;)8 b(hD)q Fn(\))19 b(the)h(W)l(eyl)e Fm(h)p Fn(-quan)o(tization)h (of)h(the)129 347 y(sym)o(b)q(ol)15 b Fm(a)p Fn(\()p Fm(x;)8 b(\030)r Fn(\))16 b(de\014ned)g(b)o(y)143 457 y Fm(C)182 436 y Fq(1)178 469 y Fp(0)219 457 y Fn(\()p Fi(R)277 436 y Fk(d)294 457 y Fn(;)8 b Fi(C)h Fn(\))17 b Fj(3)d Fm(u)30 b Fj(7!)570 408 y Fh(\020)595 457 y Fm(a)621 436 y Fk(w)649 457 y Fn(\()p Fm(x;)8 b(hD)q Fn(\))p Fm(u)834 408 y Fh(\021)858 457 y Fn(\()p Fm(x)p Fn(\))22 b(=)g(\(2)p Fm(\031)r(h)p Fn(\))1126 436 y Fq(\000)p Fk(n)1185 398 y Fh(Z)1209 492 y Fe(R)1235 483 y Fc(n)1264 457 y Fm(e)1287 436 y Fk(i\030)q Fq(\001)p Fp(\()p Fk(x)p Fq(\000)p Fk(y)q Fp(\))p Fk(=h)1459 457 y Fm(a)1485 408 y Fh(\020)1509 457 y Fn(\()p Fm(x)11 b Fn(+)g Fm(y)r Fn(\))p Fm(=)p Fn(2)p Fm(;)d(\030)1754 408 y Fh(\021)1780 457 y Fm(u)p Fn(\()p Fm(y)r Fn(\))g Fm(dy)r(d\030)13 b(:)129 566 y Fn(F)l(or)k(the)f(W)l(eyl)g Fm(h)p Fn(-pseudo)q (di\013eren)o(tial)h(calculus)g(w)o(e)f(shall)h(use)g(at)g(man)o(y)f (places)g(in)h(this)g(pap)q(er,)g(w)o(e)129 626 y(refer)h(to)i([19,)f (17,)g(5].)30 b(W)l(e)19 b(shall)g(also)h(use)f(the)g(functional)g (calculus)f(of)h(Hel\013er-Sj\177)-24 b(ostand,)20 b(whic)o(h)129 686 y(can)c(b)q(e)g(found)h(in)f([9,)g(5].)129 782 y(Since)h Fj(kh)p Fm(x)p Fj(i)349 764 y Fk(s)367 782 y Fn(\()p Fm(P)7 b Fn(\()p Fm(h)471 789 y Fk(n)495 782 y Fn(\))12 b Fj(\000)g Fm(z)600 789 y Fk(n)623 782 y Fn(\))p Fm(f)666 789 y Fk(n)690 782 y Fj(k)k Fn(=)h Fm(o)p Fn(\()p Fm(h)856 789 y Fk(n)880 782 y Fn(\),)h(the)f(sequence)g(\()p Fm(f)1263 789 y Fk(n)1287 782 y Fn(\))1306 789 y Fk(n)1347 782 y Fn(should)h(accum)o(ulate)e(microlo)q(cally)129 842 y(on)g(the)g(energy)g(shell)g Fm(p)572 824 y Fq(\000)p Fp(1)619 842 y Fn(\()p Fm(\025)p Fn(\).)22 b(This)16 b(is)g(the)g(purp)q(ose)h(of)129 931 y Fg(Prop)r(osition)g(2.)24 b Ff(The)18 b(me)n(asur)n(e)f Fm(\026)817 938 y Fk(s)853 931 y Ff(is)g(supp)n(orte)n(d)g(in)g Fm(p)1206 913 y Fq(\000)p Fp(1)1254 931 y Fn(\()p Fm(\025)p Fn(\))p Ff(.)129 1055 y Fg(Pro)r(of:)47 b Fn(Let)26 b Fm(a)j Fj(2)h Fm(C)582 1037 y Fq(1)578 1067 y Fp(0)618 1055 y Fn(\()p Fm(T)673 1037 y Fq(\003)693 1055 y Fi(R)731 1037 y Fk(d)749 1055 y Fn(\))25 b(with)g(supp)q(ort)i(disjoin)o(t)e(from)f Fm(p)1434 1037 y Fq(\000)p Fp(1)1482 1055 y Fn(\()p Fm(\025)p Fn(\).)49 b(Since)25 b Fj(<)p Fn(\()p Fm(z)1826 1062 y Fk(n)1849 1055 y Fn(\))k Fj(!)g Fm(\025)p Fn(,)129 1115 y Fm(a)p Fj(h)p Fm(x)p Fj(i)221 1097 y Fq(\000)p Fp(2)p Fk(s)284 1115 y Fn(\()p Fm(p)12 b Fj(\000)f Fm(z)412 1122 y Fk(n)435 1115 y Fn(\))454 1097 y Fq(\000)p Fp(1)517 1115 y Fn(is)16 b(a)h(b)q(ounded)g(sym)o(b)q(ol)e(for)h Fm(n)h Fn(large)f(enough.)22 b(Therefore)648 1216 y(lim)636 1241 y Fk(n)p Fq(!1)737 1168 y Fh(D)762 1216 y Fm(a)788 1196 y Fk(w)816 1216 y Fn(\()p Fm(x;)8 b(h)913 1223 y Fk(n)936 1216 y Fm(D)q Fn(\))p Fj(h)p Fm(x)p Fj(i)1062 1196 y Fq(\000)p Fk(s)1109 1216 y Fm(f)1133 1223 y Fk(n)1157 1216 y Fm(;)g Fj(h)p Fm(x)p Fj(i)1245 1196 y Fq(\000)p Fk(s)1291 1216 y Fm(f)1315 1223 y Fk(n)1338 1168 y Fh(E)1394 1216 y Fn(=)30 b(0)11 b Fm(;)129 1322 y Fn(as)17 b Fm(n)d Fj(!)f(1)p Fn(,)j(since)f Fj(kh)p Fm(x)p Fj(i)585 1304 y Fk(s)604 1322 y Fn(\()p Fm(P)7 b Fn(\()p Fm(h)708 1329 y Fk(n)732 1322 y Fn(\))k Fj(\000)g Fm(z)835 1329 y Fk(n)858 1322 y Fn(\))p Fm(f)901 1329 y Fk(n)924 1322 y Fj(k)j Fn(=)g Fm(o)p Fn(\()p Fm(h)1085 1329 y Fk(n)1109 1322 y Fn(\).)21 b(Th)o(us)16 b Fm(\026)1315 1329 y Fk(s)1334 1322 y Fn(\()p Fm(a)p Fn(\))e(=)f(0)k(b)o(y)f(\(4\).)p 1983 1322 2 33 v 1985 1290 30 2 v 1985 1322 V 2014 1322 2 33 v 129 1471 a Fg(Remark)g(3.)24 b Ff(Thanks)18 b(to)g(this)f(lo)n (c)n(alization)h(pr)n(op)n(erty)d(of)j Fm(\026)1263 1478 y Fk(s)1281 1471 y Ff(,)g(\(4\))f(holds)g(true)h(for)f Fm(a)c Fj(2)h Fm(C)1826 1453 y Fq(1)1822 1483 y Fp(0)1863 1471 y Fn(\()p Fi(R)1921 1453 y Fk(d)1938 1471 y Fn(\))p Ff(.)129 1595 y Fn(No)o(w)k(w)o(e)g(w)o(an)o(t)g(to)h(follo)o(w)f (Burq's)f(argumen)o(ts)h(\(in)g([3]\))f(to)i(deriv)o(e)e(that)i(the)f (P)o(oisson)h(brac)o(k)o(et)e(\(in)129 1655 y(the)g(distributional)f (sense\))h Fj(f)p Fm(p;)8 b Fj(h)p Fm(x)p Fj(i)800 1637 y Fp(2)p Fk(s)837 1655 y Fm(\026)866 1662 y Fk(s)885 1655 y Fj(g)17 b Fn(equals)f Fm(r)q Fj(h)p Fm(x)p Fj(i)1164 1637 y Fp(2)p Fk(s)1201 1655 y Fm(\026)1230 1662 y Fk(s)1249 1655 y Fn(.)24 b(But)17 b(it)f(turns)i(out)f(that)h Fm(r)e Fn(=)g(0)h(in)g(our)129 1715 y(case.)k(Indeed,)15 b(writing)h Fm(P)23 b Fn(instead)17 b(of)f Fm(P)7 b Fn(\()p Fm(h)954 1722 y Fk(n)978 1715 y Fn(\),)252 1810 y Fm(o)p Fn(\()p Fm(h)322 1817 y Fk(n)346 1810 y Fn(\))41 b(=)h Fj(hh)p Fm(x)p Fj(i)571 1790 y Fk(s)590 1810 y Fn(\()p Fm(P)18 b Fj(\000)11 b Fm(z)731 1817 y Fk(n)754 1810 y Fn(\))p Fm(f)797 1817 y Fk(n)821 1810 y Fm(;)d Fj(h)p Fm(x)p Fj(i)909 1790 y Fq(\000)p Fk(s)955 1810 y Fm(f)979 1817 y Fk(n)1002 1810 y Fj(i)31 b Fn(=)f Fj(h)p Fn(\()p Fm(P)18 b Fj(\000)11 b Fm(z)1280 1817 y Fk(n)1303 1810 y Fn(\))p Fm(f)1346 1817 y Fk(n)1370 1810 y Fm(;)d(f)1416 1817 y Fk(n)1439 1810 y Fj(i)31 b Fn(=)f Fj(h)p Fm(f)1600 1817 y Fk(n)1624 1810 y Fm(;)8 b Fn(\()p Fm(P)18 b Fj(\000)j Fn(\026)-35 b Fm(z)1786 1817 y Fk(n)1810 1810 y Fn(\))p Fm(f)1853 1817 y Fk(n)1876 1810 y Fj(i)406 1883 y Fn(=)42 b Fj(h)p Fm(f)529 1890 y Fk(n)552 1883 y Fm(;)8 b Fn(\()p Fm(P)19 b Fj(\000)10 b Fm(z)715 1890 y Fk(n)739 1883 y Fn(\))p Fm(f)782 1890 y Fk(n)805 1883 y Fj(i)20 b Fn(+)f Fj(h)p Fm(f)944 1890 y Fk(n)968 1883 y Fm(;)8 b Fn(2)p Fm(i)p Fj(=)p Fn(\()p Fm(z)1109 1890 y Fk(n)1132 1883 y Fn(\))p Fm(f)1175 1890 y Fk(n)1198 1883 y Fj(i)31 b Fn(=)f Fm(o)p Fn(\()p Fm(h)1386 1890 y Fk(n)1410 1883 y Fn(\))19 b(+)h Fj(h)p Fm(f)1549 1890 y Fk(n)1572 1883 y Fm(;)8 b Fn(2)p Fm(i)p Fj(=)p Fn(\()p Fm(z)1713 1890 y Fk(n)1736 1883 y Fn(\))p Fm(f)1779 1890 y Fk(n)1803 1883 y Fj(i)j Fm(;)129 1978 y Fn(yielding)k Fj(k)p Fm(f)361 1985 y Fk(n)385 1978 y Fj(k)410 1960 y Fp(2)429 1978 y Fj(=)p Fn(\()p Fm(z)507 1985 y Fk(n)530 1978 y Fn(\))p Fm(=h)601 1985 y Fk(n)640 1978 y Fn(=)g Fm(o)p Fn(\(1\).)24 b(If)17 b Fm(r)f Fj(6)p Fn(=)f(0)i(then)g Fj(k)p Fm(f)1159 1985 y Fk(n)1182 1978 y Fj(k)1207 1960 y Fp(2)1242 1978 y Fj(!)e Fn(0.)23 b(Since)16 b Fm(s)f Fj(\025)g Fn(0,)i(1)e(=)g Fj(kh)p Fm(x)p Fj(i)1826 1960 y Fq(\000)p Fk(s)1873 1978 y Fm(f)1897 1985 y Fk(n)1920 1978 y Fj(k)1945 1960 y Fp(2)1980 1978 y Fj(\024)129 2038 y(k)p Fm(f)178 2045 y Fk(n)201 2038 y Fj(k)226 2020 y Fp(2)262 2038 y Fn(and)i(w)o(e)f (arriv)o(e)f(at)h(a)h(con)o(tradiction.)k(Therefore,)15 b(w)o(e)h(get)g(the)g(follo)o(wing)g(stronger)h(result.)129 2127 y Fg(Prop)r(osition)g(4.)24 b Ff(The)18 b(me)n(asur)n(e)f Fj(h)p Fm(x)p Fj(i)854 2109 y Fp(2)p Fk(s)890 2127 y Fm(\026)919 2134 y Fk(s)955 2127 y Ff(is)h(invariant)g(under)g Fm(\036)1387 2109 y Fk(t)1401 2127 y Ff(,)g(that)f(is)h Fj(f)p Fm(p;)8 b Fj(h)p Fm(x)p Fj(i)1723 2109 y Fp(2)p Fk(s)1759 2127 y Fm(\026)1788 2134 y Fk(s)1807 2127 y Fj(g)14 b Fn(=)g(0)p Ff(.)129 2251 y Fg(Pro)r(of:)29 b 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Fk(n)1084 2436 y Fm(a)1110 2415 y Fk(w)1138 2436 y Fn(\()p Fm(x;)g(h)1235 2443 y Fk(n)1258 2436 y Fm(D)q Fn(\)\()p Fm(P)19 b Fj(\000)11 b Fm(z)1460 2443 y Fk(n)1483 2436 y Fn(\))p Fm(f)1526 2443 y Fk(n)1549 2436 y Fm(;)d(f)1595 2443 y Fk(n)1619 2436 y Fj(i)846 2508 y Fn(=)42 b(\(2)p Fm(r)12 b Fn(+)f Fm(o)p Fn(\(1\)\))p Fj(h)p Fm(a)1201 2488 y Fk(w)1230 2508 y Fn(\()p Fm(x;)d(h)1327 2515 y Fk(n)1350 2508 y Fm(D)q Fn(\))p Fm(f)1434 2515 y Fk(n)1459 2508 y Fm(;)g(f)1505 2515 y Fk(n)1528 2508 y Fj(i)20 b Fn(+)f Fm(o)p Fn(\(1\))31 b(=)f Fm(o)p Fn(\(1\))12 b Fm(;)37 b Fn(\(5\))129 2603 y(on)16 b(one)h(hand,)f(and)379 2650 y Fh(D)405 2698 y Fm(ih)450 2678 y Fq(\000)p Fp(1)450 2711 y Fk(n)497 2698 y Fn([)p Fm(P)q(;)8 b(a)591 2678 y Fk(w)619 2698 y Fn(\()p Fm(x;)g(h)716 2705 y Fk(n)739 2698 y Fm(D)q Fn(\)])p Fm(f)837 2705 y Fk(n)861 2698 y Fm(;)g(f)907 2705 y Fk(n)930 2650 y Fh(E)997 2698 y Fn(=)41 b Fj(hf)p Fm(p;)8 b(a)p Fj(g)1217 2678 y Fk(w)1246 2698 y Fn(\()p Fm(x;)g(h)1343 2705 y Fk(n)1366 2698 y Fm(D)q Fn(\))p Fm(f)1450 2705 y Fk(n)1474 2698 y Fm(;)g(f)1520 2705 y Fk(n)1544 2698 y Fj(i)19 b Fn(+)h Fm(O)q Fn(\()p Fm(h)1725 2705 y Fk(n)1749 2698 y Fn(\))997 2790 y(=)41 b Fm(\026)1105 2797 y Fk(s)1124 2742 y Fh(\020)1149 2790 y Fj(h)p Fm(x)p Fj(i)1215 2769 y Fp(2)p Fk(s)1251 2790 y Fj(f)p Fm(p;)8 b(a)p Fj(g)1373 2742 y Fh(\021)1417 2790 y Fn(+)20 b Fm(o)p Fn(\(1\))11 b Fm(;)371 b Fn(\(6\))129 2893 y(on)16 b(the)g(other)h(hand.)22 b(Th)o(us)16 b Fj(f)p Fm(p;)8 b Fj(h)p Fm(x)p Fj(i)809 2875 y Fp(2)p Fk(s)846 2893 y Fm(\026)875 2900 y Fk(s)894 2893 y Fj(g)13 b Fn(=)h(0.)p 1983 2893 V 1985 2862 30 2 v 1985 2893 V 2014 2893 2 33 v eop %%Page: 5 5 5 4 bop 129 133 a Fl(Non-trapping)17 b(b)q(eha)o(viour,)f(21-11-02)1147 b Fn(5)129 286 y Fg(Remark)16 b(5.)24 b Ff(Notic)n(e)13 b(that)g(\(4\))g(holds)g(true)g(with)g Fj(h)p Fm(x)p Fj(i)1114 268 y Fq(\000)p Fk(s)1160 286 y Fm(f)1184 293 y Fk(n)1220 286 y Ff(and)g Fm(\026)1339 293 y Fk(s)1371 286 y Ff(r)n(eplac)n(e)n(d)f(by)h Fm(f)1634 293 y Fk(n)1670 286 y Ff(and)g Fm(\026)h Fn(:=)f Fj(h)p Fm(x)p Fj(i)1934 268 y Fp(2)p Fk(s)1971 286 y Fm(\026)2000 293 y Fk(s)129 347 y Ff(r)n(esp)n(e)n(ctively.)27 b(If)19 b(the)g(se)n(quenc)n(e)i Fn(\()p Fm(f)786 354 y Fk(n)809 347 y Fn(\))828 354 y Fk(n)870 347 y Ff(is)e(b)n(ounde)n(d)g(in)h Fm(L)1206 329 y Fp(2)1245 347 y Ff(then)f Fm(\026)h Ff(is)f(a)f(semiclassic)n(al) i(me)n(asur)n(e)f(of)129 407 y(this)h(se)n(quenc)n(e)h(\(se)n(e)f ([6]\).)30 b(This)19 b(is)h(true)h(for)e Fm(s)f Fn(=)h(0)h Ff(but,)h(so)f(far,)g Fn(\()p Fm(f)1448 414 y Fk(n)1472 407 y Fn(\))1491 414 y Fk(n)1534 407 y Ff(might)g(b)n(e)g(unb)n(ounde)n (d)h(in)129 467 y Fm(L)162 449 y Fp(2)200 467 y Ff(if)e Fm(s)e(>)f Fn(0)j Ff(\(such)g(a)g(situation)g(app)n(e)n(ars)f(for)g Fm(V)28 b Fn(=)16 b(0)j Ff(and)g Fm(s)e Fj(2)p Fn(]0;)8 b(1)p Fm(=)p Fn(2[)p Ff(,)19 b(se)n(e)g(\(11\)\).)27 b(In)19 b(this)f(c)n(ase,)129 527 y(the)e(me)n(asur)n(e)g Fm(\026)g Ff(plays)g(the)h(r)n(ole)f(of)g(a)g(semiclassic)n(al)h(me)n (asur)n(e)e(of)h Fn(\()p Fm(f)1405 534 y Fk(n)1429 527 y Fn(\))1448 534 y Fk(n)1487 527 y Ff(\(sinc)n(e)h(\(4\))f(holds)g (true\))h(but)129 587 y(might)g(not)h(b)n(elong)h(to)f(the)g(limit)g(p) n(oints)f(of)g(the)h(Wigner)g(tr)n(ansform)e(of)h Fn(\()p Fm(f)1532 594 y Fk(n)1556 587 y Fn(\))1575 594 y Fk(n)1616 587 y Ff(\(se)n(e)g([6]\).)129 721 y Fn(Lo)q(oking)e(for)e(other)h (prop)q(erties)g(of)g Fm(\026)819 728 y Fk(s)837 721 y Fn(,)g(w)o(e)f(learn)g(from)g([6])g(that)h Fm(\026)1363 728 y Fk(s)1395 721 y Fn(=)g(0)g(roughly)g(means)f(that)h(\()p Fm(f)1953 728 y Fk(n)1976 721 y Fn(\))1995 728 y Fk(n)129 781 y Fn(accum)o(ulates)g(at)j(in\014nit)o(y)e(\(in)g(the)h(p)q (osition)h(v)m(ariable)f Fm(x)p Fn(\).)21 b(More)16 b(precisely)l(,)e (if)695 903 y(lim)667 932 y Fk(R)p Fq(!)p Fp(+)p Fq(1)800 903 y Fn(lim)8 b(sup)829 937 y Fk(n)p Fq(!1)966 844 y Fh(Z)989 938 y Fq(j)p Fk(x)p Fq(j)p Fk(>R)1085 903 y Fj(h)p Fm(x)p Fj(i)1151 882 y Fq(\000)p Fp(2)p Fk(s)1215 903 y Fm(f)1244 882 y Fp(2)1239 915 y Fk(n)1272 903 y Fm(dx)30 b Fn(=)g(0)12 b Fm(;)483 b Fn(\(7\))129 1036 y(the)22 b(total)h(mass)f(of)h Fm(\026)559 1043 y Fk(s)600 1036 y Fn(equals)f(lim)822 1043 y Fk(n)p Fq(!1)924 1036 y Fj(kh)p Fm(x)p Fj(i)1015 1018 y Fq(\000)p Fk(s)1062 1036 y Fm(f)1086 1043 y Fk(n)1109 1036 y Fj(k)1134 1018 y Fp(2)1176 1036 y Fn(and,)j(in)d(particular,)h Fm(\026)1631 1043 y Fk(s)1673 1036 y Fn(is)f(nonzero.)40 b(Let)129 1097 y Fm(\022)23 b Fj(2)f Fm(C)269 1079 y Fq(1)265 1109 y Fp(0)306 1097 y Fn(\()p Fi(R)p Fn(\))c(supp)q(orted)k(near)f Fm(\025)g Fn(and)h(let)e(1)-13 b Fb(I)996 1104 y Fq(fj)p Fk(x)p Fq(j)p Fk(>R)p Fq(g)1149 1097 y Fn(b)q(e)21 b(the)g(c)o (haracteristic)f(function)g(of)i(the)f(set)129 1157 y Fj(f)p Fn(\()p Fm(x;)8 b(\030)r Fn(\))16 b Fj(2)h Fm(T)367 1139 y Fq(\003)386 1157 y Fi(R)425 1139 y Fk(d)442 1157 y Fn(;)8 b Fj(j)p Fm(x)p Fj(j)16 b Fm(>)g(R)p Fj(g)h Fn(=:)e Fm(T)772 1139 y Fq(\003)792 1157 y Fi(R)830 1139 y Fk(d)860 1157 y Fj(n)d Fm(B)937 1139 y Fq(\003)934 1169 y Fk(R)963 1157 y Fn(.)25 b(T)l(o)19 b(b)q(ound)f(ab)q(o)o(v)o(e)g (the)g(quan)o(tit)o(y)1649 1121 y Fh(R)1669 1169 y Fq(j)p Fk(x)p Fq(j)p Fk(>R)1765 1157 y Fj(h)p Fm(x)p Fj(i)1831 1139 y Fq(\000)p Fp(2)p Fk(s)1894 1157 y Fm(f)1923 1139 y Fp(2)1918 1169 y Fk(n)1952 1157 y Fm(dx)p Fn(,)129 1221 y(whic)o(h)13 b(should)h(b)q(e)g(close)f(to)i Fj(k)p Fn(1)-13 b Fb(I)703 1229 y Fq(fj)p Fk(x)p Fq(j)p Fk(>R)p Fq(g)834 1221 y Fm(\022)q Fn(\()p Fm(P)7 b Fn(\()p Fm(h)962 1228 y Fk(n)986 1221 y Fn(\)\))p Fj(h)p Fm(x)p Fj(i)1090 1203 y Fq(\000)p Fk(s)1136 1221 y Fm(f)1160 1228 y Fk(n)1184 1221 y Fj(k)1209 1203 y Fp(2)1242 1221 y Fn(b)o(y)14 b(energy)f(lo)q(calization)h(of)g Fm(f)1795 1228 y Fk(n)1818 1221 y Fn(,)g(w)o(e)f(w)o(an)o(t)129 1282 y(to)j(use)h(a)f (semiclassical)e(Mourre)i(estimate)e(\\at)j(in\014nit)o(y")363 1388 y Fm(\022)q Fn(\()p Fm(P)7 b Fn(\)1)-13 b Fb(I)487 1396 y Fq(fj)p Fk(x)p Fq(j)p Fk(>R)p Fq(g)619 1388 y Fm(ih)664 1367 y Fq(\000)p Fp(1)711 1388 y Fn([)p Fm(P)q(;)8 b(A)p Fn(]1)-13 b Fb(I)854 1396 y Fq(fj)p Fk(x)p Fq(j)p Fk(>R)p Fq(g)985 1388 y Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))30 b Fj(\025)g Fm(c)8 b(\022)q Fn(\()p Fm(P)f Fn(\)1)-13 b Fb(I)1337 1396 y Fq(fj)p Fk(x)p Fq(j)p Fk(>R)p Fq(g)1470 1388 y Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))19 b(+)1658 1375 y(~)1647 1388 y Fm(O)q Fn(\()p Fm(h)p Fn(\))11 b Fm(;)180 b Fn(\(8\))129 1502 y(where)15 b Fm(c)f(>)f Fn(0,)j Fm(A)f Fn(is)g(a)h(w)o(ell)e(c)o(hosen)h(op)q(erator,)i(and)1117 1489 y(~)1106 1502 y Fm(O)q Fn(\()p Fm(h)p Fn(\))f(stands)g(for)g(a)g (b)q(ounded)g(op)q(erator)h(on)f Fm(L)1985 1484 y Fp(2)2005 1502 y Fn(,)129 1562 y(the)h(norm)f(of)i(whic)o(h)f(is)g Fm(O)q Fn(\()p Fm(h)p Fn(\).)26 b(Suc)o(h)17 b(an)h(inequalit)o(y)d (follo)o(ws,)j(via)f(G)-6 b(\027)-30 b(arding)17 b(inequalit)o(y)l(,)e (from)i(the)129 1622 y(existence)12 b(of)j(an)f(escap)q(e)h(function)f (at)g(in\014nit)o(y)l(.)20 b(Under)13 b(\(1\),)i(w)o(e)f(do)g(ha)o(v)o (e)g(suc)o(h)g(a)g(function,)g(namely)129 1682 y Fm(x)c Fj(\001)h Fm(\030)r Fn(.)22 b(Indeed,)15 b(since)h Fm(\025)e(>)g Fn(0,)i(there)g(exists)f Fm(c)f(>)g Fn(0)j(suc)o(h)f(that)385 1789 y Fj(f)p Fm(p;)8 b(x)j Fj(\001)g Fm(\030)r Fj(g)31 b Fn(=)f(2)p Fj(j)p Fm(\030)r Fj(j)742 1768 y Fp(2)774 1789 y Fj(\000)10 b Fm(x)h Fj(\001)g(r)p Fm(V)g Fn(\()p Fm(x)p Fn(\))30 b(=)g(2)p Fm(p)12 b Fj(\000)e Fn(2)p Fm(V)i Fn(\()p Fm(x)p Fn(\))f Fj(\000)g Fm(x)f Fj(\001)h(r)p Fm(V)g Fn(\()p Fm(x)p Fn(\))30 b Fj(\025)g Fm(c)129 1895 y Fn(on)16 b Fm(p)220 1877 y Fq(\000)p Fp(1)268 1895 y Fn(\()p Fm(\025)p Fn(\))h(for)f Fj(j)p Fm(x)p Fj(j)g Fn(large)g(enough.)22 b(T)l(o)17 b(con)o(trol)f(the)g(l.h.s.)k(of)d (the)f(semiclassical)d(Mourre)j(estimate)129 1955 y(\(8\),)j(w)o(e)f (need)h(a)g(b)q(ounded)h(escap)q(e)f(function)f(at)i(in\014nit)o(y)l(.) 27 b(If)19 b(w)o(e)f(could)h(do)g(this,)g(w)o(e)f(w)o(ould)h(ha)o(v)o (e)129 2015 y(a)e(Mourre)g(estimate)e(for)i Fm(P)24 b Fn(with)17 b(a)h(b)q(ounded)f(conjugate)h(op)q(erator)g Fm(A)p Fn(,)e(whic)o(h)h(is)g(imp)q(ossible)e(\(see)129 2075 y([2]\).)20 b(So)c(w)o(e)f(seek)g(for)h(a)g(kind)f(of)h(w)o(eigh)o (ted)f(semiclassical)e(Mourre)i(estimate,)f(that)i(is)f(an)h(estimate) 129 2136 y(lik)o(e)e(\(8\))j(where)e Fm(A)h Fn(w)o(ould)g(b)q(e)h(the)e (W)l(eyl)h Fm(h)p Fn(-quan)o(tization)g(of)g Fm(a)p Fn(,)f(a)i(b)q (ounded)g(\\escap)q(e)g(function")f(at)129 2196 y(in\014nit)o(y)l(,)c (satisfying)i Fj(f)p Fm(p;)8 b(a)p Fj(g)13 b(\025)h Fm(c)p Fj(j)p Fm(x)p Fj(j)784 2178 y Fq(\000)p Fk(b)841 2196 y Fn(with)g Fm(b)g(>)f Fn(0)h(for)g Fj(j)p Fm(x)p Fj(j)f Fn(large.)20 b(T)l(o)15 b(this)e(end,)h(let)e Fm(\037)1716 2203 y Fp(0)1736 2196 y Fm(;)c(\037)1789 2203 y Fp(1)1822 2196 y Fj(2)14 b Fm(C)1908 2178 y Fq(1)1945 2196 y Fn(\()p Fi(R)o Fn(\))129 2256 y(with)h(0)f Fj(\024)g Fm(\037)361 2263 y Fp(0)380 2256 y Fm(;)8 b(\037)433 2263 y Fp(1)466 2256 y Fj(\024)14 b Fn(1,)i Fm(\037)604 2263 y Fp(0)623 2256 y Fn(\()p Fm(t)p Fn(\))d(=)h(0)i(for)g Fm(t)e Fj(\024)f Fn(1)p Fm(=)p Fn(3,)k Fm(\037)1076 2263 y Fp(1)1095 2256 y Fn(\()p Fm(t)p Fn(\))c(=)h(0)i(for)g Fm(t)e Fj(\024)f Fn(1,)j Fm(\037)1499 2263 y Fp(0)1518 2256 y Fn(\()p Fm(t)p Fn(\))e(=)f(1)j(for)g Fm(t)e Fj(\025)f Fn(2)p Fm(=)p Fn(3,)k(and)129 2316 y Fm(\037)160 2323 y Fp(1)179 2316 y Fn(\()p Fm(t)p Fn(\))c(=)h(1)j(for)f Fm(t)e Fj(\025)f Fn(2.)22 b(Let)16 b Fm(R)683 2323 y Fp(0)703 2316 y Fm(;)8 b(\017)745 2323 y Fp(0)764 2316 y Fm(;)g(\017)14 b(>)g Fn(0)i(suc)o(h)g(that)549 2380 y Fh(\020)574 2429 y Fn(\()p Fm(x;)8 b(\030)r Fn(\))14 b Fj(2)g Fm(p)770 2408 y Fq(\000)p Fp(1)818 2429 y Fn(\(])p Fm(\025)d Fj(\000)f Fm(\017)p Fn(;)e Fm(\025)k Fn(+)f Fm(\017)p Fn([\))f Fj(n)h Fm(B)1209 2408 y Fq(\003)1206 2441 y Fk(R)1233 2446 y Fa(0)1252 2380 y Fh(\021)1307 2429 y Fn(=)-8 b Fj(\))29 b(j)p Fm(\030)r Fj(j)14 b(\025)g Fm(\017)1554 2436 y Fp(0)1585 2429 y Fm(:)129 2541 y Fn(F)l(or)i(an)o(y)g Fm(\016)g Fj(2)p Fn(]0;)8 b(min)n(\(1;)g Fm(\032)p Fn(\)[,)15 b(where)h Fm(\032)h Fn(discrib)q(es)f(the)g(deca)o(y)f(of)i Fm(V)27 b Fn(at)17 b(in\014nit)o(y)e(b)o(y)g(\(1\),)i(w)o(e)e(set)271 2678 y Fm(a)297 2685 y Fq(1)334 2678 y Fn(\()p Fm(x;)8 b(\030)r Fn(\))30 b(:=)14 b Fm(\037)572 2685 y Fp(1)591 2630 y Fh(\020)616 2678 y Fj(j)p Fm(x)p Fj(j)p Fm(=R)733 2685 y Fp(0)752 2630 y Fh(\021)777 2678 y Fm(\037)808 2685 y Fp(1)827 2630 y Fh(\020)852 2678 y Fj(j)p Fm(\030)r Fj(j)p Fm(=\017)947 2685 y Fp(0)967 2630 y Fh(\021)992 2605 y( )1028 2678 y Fn(^)-27 b Fm(x)11 b Fj(\001)1094 2665 y Fn(^)1089 2678 y Fm(\030)i Fj(\000)e(j)p Fm(x)p Fj(j)1229 2658 y Fq(\000)p Fk(\016)1275 2618 y Fh(\022)1306 2678 y Fm(\037)1337 2685 y Fp(0)1356 2630 y Fh(\020)1384 2678 y Fn(^)-27 b Fm(x)10 b Fj(\001)1449 2665 y Fn(^)1444 2678 y Fm(\030)1468 2630 y Fh(\021)1504 2678 y Fj(\000)h Fm(\037)1585 2685 y Fp(0)1604 2630 y Fh(\020)1629 2678 y Fj(\000)s Fn(^)-27 b Fm(x)10 b Fj(\001)1736 2665 y Fn(^)1731 2678 y Fm(\030)1755 2630 y Fh(\021)1780 2618 y(\023)1810 2605 y(!)1854 2678 y Fm(;)88 b Fn(\(9\))129 2816 y(where)19 b(^)-27 b Fm(x)13 b Fn(:=)g Fm(x=)p Fj(j)p Fm(x)p Fj(j)p Fn(.)21 b(It)16 b(is)g(easy)g(to)h(sho)o(w)g(that)g Fm(a)1041 2823 y Fq(1)1094 2816 y Fn(is)f(a)g(smo)q(oth,)g(b)q(ounded)h (function)g(suc)o(h)f(that)315 2928 y Fj(8)p Fn(\()p Fm(\013;)8 b(\014)s Fn(\))k Fj(2)i Fi(N)560 2908 y Fp(2)p Fk(d)598 2928 y Fm(;)j Fj(9)p Fm(C)692 2935 y Fk(\013\014)751 2928 y Fm(>)d Fn(0;)j Fj(8)p Fn(\()p Fm(x;)8 b(\030)r Fn(\))k Fj(2)i Fm(T)1093 2908 y Fq(\003)1112 2928 y Fi(R)1151 2908 y Fk(d)1169 2928 y Fm(;)1199 2878 y Fh(\014)1199 2903 y(\014)1199 2928 y(\014)p Fm(@)1242 2908 y Fk(\013)1239 2941 y(x)1266 2928 y Fm(@)1295 2905 y Fk(\014)1292 2941 y(\030)1318 2928 y Fm(a)1344 2935 y Fq(1)1381 2928 y Fn(\()p Fm(x;)8 b(\030)r Fn(\))1492 2878 y Fh(\014)1492 2903 y(\014)1492 2928 y(\014)30 b Fj(\024)g Fm(C)1640 2935 y Fk(\013\014)1694 2928 y Fj(h)p Fm(x)p Fj(i)1760 2908 y Fq(\000j)p Fk(\013)p Fq(j)1932 2928 y Fn(\(10\))p eop %%Page: 6 6 6 5 bop 129 133 a Fl(Non-trapping)17 b(b)q(eha)o(viour,)f(21-11-02)1147 b Fn(6)129 286 y(and)21 b(suc)o(h)f(that,)i(for)f Fm(\017)g(>)g Fn(0)g(small)e(enough,)k(w)o(e)d(can)h(\014nd)g Fm(c)g(>)g Fn(0,)h Fm(R)g(>)f Fn(max)o(\(1;)8 b Fm(R)1755 293 y Fp(0)1775 286 y Fn(\))21 b(suc)o(h)f(that)129 347 y Fj(f)p Fm(p;)8 b(a)226 354 y Fq(1)263 347 y Fj(g)16 b(\025)g Fm(c)p Fj(h)p Fm(x)p Fj(i)446 329 y Fq(\000)p Fp(1)p Fq(\000)p Fk(\016)555 347 y Fn(on)i Fm(p)648 329 y Fq(\000)p Fp(1)696 347 y Fn(\(])p Fm(\025)12 b Fj(\000)f Fm(\017)p Fn(;)d Fm(\025)k Fn(+)g Fm(\017)p Fn([\))f Fj(n)h Fm(B)1092 329 y Fq(\003)1089 359 y Fk(R)p Fq(\000)p Fp(1)1163 347 y Fn(.)25 b(Let)18 b(us)g(men)o(tion)d(that)j(the)f(global)h(escap)q(e) 129 407 y(function)e(used)g(in)g([22])g(satis\014es)h(similar)d (conditions.)129 467 y(Notice)k(that)j Fm(P)27 b Fn(ma)o(y)18 b(b)q(e)i(seen)f(as)i(the)e(W)l(eyl)g Fm(h)p Fn(-quan)o(tization)h(of)g (the)g(sym)o(b)q(ol)e Fm(p)j Fn(for)f(whic)o(h)f(\(10\))129 527 y(holds)c(true)f(on)h Fm(p)446 509 y Fq(\000)p Fp(1)493 527 y Fn(\(])p Fm(\025)7 b Fj(\000)g Fm(\017)p Fn(;)h Fm(\025)f Fn(+)g Fm(\017)p Fn([\).)21 b(In)14 b(particular,)g(it)g (follo)o(ws)h(from)e(Hel\013er-Sj\177)-24 b(ostrand)15 b(functional)129 587 y(calculus)23 b(that)i Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))24 b(is)g(a)h Fm(h)p Fn(-pseudo)q(di\013eren)o (tial)f(op)q(erator)i(whose)e(sym)o(b)q(ol)f Fm(s)i Fn(is)f(asym)o (totic)e(to)129 614 y Fh(P)173 658 y Fk(j)r Fq(\025)p Fp(0)244 648 y Fm(h)272 629 y Fk(j)290 648 y Fm(s)313 655 y Fk(j)332 648 y Fn(,)15 b(where)h(eac)o(h)g Fm(s)635 655 y Fk(j)669 648 y Fn(satis\014es)g(\(10\),)h(and)f(whose)h (principal)e(sym)o(b)q(ol)g Fm(s)1596 655 y Fp(0)1631 648 y Fn(is)h Fm(\022)q Fn(\()p Fm(p)p Fn(\))h(\(see)e([9,)h(5]\).)129 708 y(Let)g(us)h(no)o(w)f(deriv)o(e)f(consequences)g(of)i(this)f (non-trapping)i(b)q(eha)o(viour)e(at)h(in\014nit)o(y)l(.)129 795 y Fg(Lemm)o(a)f(6.)24 b Fn(lim)463 802 y Fk(n)p Fq(!1)574 795 y Fj(k)p Fn(1)-13 b Fb(I)623 802 y Fq(fj)p Fk(x)p Fq(j)p Fk(>R)p Fq(g)754 795 y Fm(\022)q Fn(\()p Fm(P)7 b Fn(\()p Fm(h)882 802 y Fk(n)906 795 y Fn(\)\))p Fj(h)p Fm(x)p Fj(i)1010 777 y Fq(\000)p Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1173 795 y Fm(f)1197 802 y Fk(n)1221 795 y Fj(k)1246 777 y Fp(2)1296 795 y Fn(=)32 b(0)p Ff(.)129 917 y Fg(Pro)r(of:)c Fn(Let)16 b Fm(\034)j Fj(2)14 b Fm(C)522 899 y Fq(1)518 929 y Fp(0)559 917 y Fn(\()p Fi(R)p Fn(\))e(with)j(0)f Fj(\024)g Fm(\034)20 b Fj(\024)13 b Fn(1,)i Fm(\034)20 b Fn(=)14 b(1)h(on)h([)p Fj(\000)p Fm(R)9 b Fn(+)f(1;)g Fm(R)i Fj(\000)f Fn(1],)14 b(and)i(supp)9 b Fm(\034)20 b Fj(\032)13 b Fn([)p Fj(\000)p Fm(R)p Fn(;)8 b Fm(R)p Fn(].)129 977 y(Let)19 b Fm(\037)p Fn(\()p Fm(x)p Fn(\))e(:=)h Fm(\034)6 b Fn(\()p Fj(j)p Fm(x)p Fj(j)p Fn(\).)29 b(W)l(e)18 b(ha)o(v)o(e)g Fj(f)p Fm(p;)8 b Fn(\(1)14 b Fj(\000)f Fm(\037)p Fn(\))998 959 y Fp(2)1017 977 y Fm(a)1043 984 y Fq(1)1080 977 y Fj(g)18 b(\025)g Fn(\(1)13 b Fj(\000)g Fm(\037)p Fn(\))1338 959 y Fp(2)1357 977 y Fj(f)p Fm(p;)8 b(a)1454 984 y Fq(1)1492 977 y Fj(g)18 b(\025)g Fm(c)p Fn(\(1)13 b Fj(\000)g Fm(\037)p Fn(\))1771 959 y Fp(2)1790 977 y Fj(h)p Fm(x)p Fj(i)1856 959 y Fq(\000)p Fp(1)p Fq(\000)p Fk(\016)1967 977 y Fn(on)129 1037 y Fm(p)153 1019 y Fq(\000)p Fp(1)200 1037 y Fn(\(])p Fm(\025)e Fj(\000)g Fm(\017)p Fn(;)d Fm(\025)j Fn(+)g Fm(\017)p Fn([\).)21 b(By)15 b(G)-6 b(\027)-30 b(arding)16 b(inequalit)o(y)f (\(see)g([11)q(,)g(5)q(]\),)g(w)o(e)h(deduce)g(from)f(this)h(that)196 1138 y Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))304 1089 y Fh(n)332 1138 y Fm(p;)h Fn(\(1)k Fj(\000)f Fm(\037)p Fn(\))533 1117 y Fp(2)552 1138 y Fm(a)578 1145 y Fq(1)615 1089 y Fh(o)642 1101 y Fk(w)671 1138 y Fn(\()p Fm(x;)d(h)768 1145 y Fk(n)791 1138 y Fm(D)q Fn(\))g Fm(\022)q Fn(\()p Fm(P)f Fn(\))43 b Fj(\025)e Fm(c)p Fj(h)p Fm(x)p Fj(i)1169 1117 y Fq(\000)p Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1332 1138 y Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))h(\(1)k Fj(\000)f Fm(\037)p Fn(\))1595 1117 y Fp(2)1622 1138 y Fm(\022)q Fn(\()p Fm(P)c Fn(\))h Fj(h)p Fm(x)p Fj(i)1796 1117 y Fq(\000)p Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1082 1225 y Fn(+)g Fj(h)p Fm(x)p Fj(i)1194 1204 y Fq(\000)p Fp(1)1250 1225 y Fm(\022)q Fn(\()p Fm(P)f Fn(\))1370 1212 y(~)1358 1225 y Fm(O)r Fn(\()p Fm(h)1444 1232 y Fk(n)1468 1225 y Fn(\))h Fm(\022)q Fn(\()p Fm(P)f Fn(\))h Fj(h)p Fm(x)p Fj(i)1669 1204 y Fq(\000)p Fp(1)1728 1225 y Fm(;)129 1317 y Fn(where)16 b Fm(P)23 b Fn(stands)17 b(for)g Fm(P)7 b Fn(\()p Fm(h)636 1324 y Fk(n)660 1317 y Fn(\).)21 b(Since)225 1418 y Fm(ih)270 1397 y Fq(\000)p Fp(1)270 1430 y Fk(n)317 1369 y Fh(h)336 1418 y Fm(P)q(;)8 b Fn(\(\(1)k Fj(\000)f Fm(\037)p Fn(\))564 1397 y Fp(2)583 1418 y Fm(a)609 1425 y Fq(1)646 1418 y Fn(\))665 1397 y Fk(w)694 1418 y Fn(\()p Fm(x;)d(h)791 1425 y Fk(n)814 1418 y Fm(D)q Fn(\))874 1369 y Fh(i)908 1418 y Fn(=)960 1369 y Fh(n)987 1418 y Fm(p;)g Fn(\(1)k Fj(\000)f Fm(\037)p Fn(\))1188 1397 y Fp(2)1207 1418 y Fm(a)1233 1425 y Fq(1)1270 1369 y Fh(o)1298 1381 y Fk(w)1326 1418 y Fn(\()p Fm(x;)d(h)1423 1425 y Fk(n)1446 1418 y Fm(D)q Fn(\))k(+)f Fj(h)p Fm(x)p Fj(i)1633 1397 y Fq(\000)p Fp(1)1692 1405 y Fn(~)1681 1418 y Fm(O)q Fn(\()p Fm(h)1766 1425 y Fk(n)1790 1418 y Fn(\))p Fj(h)p Fm(x)p Fj(i)1875 1397 y Fq(\000)p Fp(1)129 1519 y Fn(and)17 b Fj(k)p Fm(\037\022)q Fn(\()p Fm(P)7 b Fn(\))p Fj(h)p Fm(x)p Fj(i)446 1501 y Fk(s)p Fq(\000)p Fp(1)517 1519 y Fj(h)p Fm(x)p Fj(i)583 1501 y Fq(\000)p Fk(s)630 1519 y Fm(f)654 1526 y Fk(n)677 1519 y Fj(k)14 b Fn(=)g Fm(O)q Fn(\(1\),)j(w)o(e)e(obtain,)i(using)f Fm(\016)g Fj(\024)d Fn(1,)489 1569 y Fh(\034)520 1630 y Fm(ih)565 1612 y Fq(\000)p Fp(1)565 1642 y Fk(n)612 1569 y Fh(\024)634 1630 y Fm(P)q(;)688 1582 y Fh(\020)713 1630 y Fn(\(1)e Fj(\000)g Fm(\037)p Fn(\))867 1612 y Fp(2)887 1630 y Fm(a)913 1637 y Fq(1)949 1582 y Fh(\021)974 1593 y Fk(w)1003 1630 y Fn(\()p Fm(x;)d(h)1100 1637 y Fk(n)1123 1630 y Fm(D)q Fn(\))1183 1569 y Fh(\025)1214 1630 y Fm(\022)q Fn(\()p Fm(P)f Fn(\))h Fm(f)1346 1637 y Fk(n)1378 1630 y Fm(;)16 b(\022)q Fn(\()p Fm(P)7 b Fn(\))h Fm(f)1540 1637 y Fk(n)1564 1582 y Fh(E)1620 1630 y Fj(\025)625 1694 y Fh(\020)650 1742 y Fm(c)j Fn(+)g Fm(O)q Fn(\()p Fm(h)816 1749 y Fk(n)840 1742 y Fn(\))859 1694 y Fh(\021)892 1692 y(\015)892 1717 y(\015)892 1742 y(\015)q Fn(\(1)g Fj(\000)g Fm(\037)p Fn(\))d Fm(\022)q Fn(\()p Fm(P)f Fn(\))h Fj(h)p Fm(x)p Fj(i)1252 1724 y Fq(\000)p Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1407 1742 y Fm(f)1431 1749 y Fk(n)1454 1692 y Fh(\015)1454 1717 y(\015)1454 1742 y(\015)1477 1705 y Fp(2)1508 1742 y Fm(:)129 1841 y Fn(Since)20 b Fm(a)287 1848 y Fq(1)345 1841 y Fn(is)h(a)g(b)q(ounded)h(sym)o(b)q(ol,)f(the)f(l.h.s)g(of)i(the) f(last)g(inequalit)o(y)e(tends)i(to)g(zero)g(as)h(in)f(\(5\),)129 1901 y(yielding)15 b(the)h(desired)f(result.)p 1983 1901 2 33 v 1985 1870 30 2 v 1985 1901 V 2014 1901 2 33 v 129 2057 a(Notice)d(that)i(the)g(previous)f(pro)q(of)i(sho)o(ws)f(that) h(\()p Fj(k)p Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))p Fj(h)p Fm(x)p Fj(i)1231 2039 y Fq(\000)p Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1385 2057 y Fm(f)1409 2064 y Fk(n)1433 2057 y Fj(k)p Fn(\))1477 2064 y Fk(n)1514 2057 y Fn(is)13 b(b)q(ounded.)22 b(Let)13 b Fm(\026)1891 2065 y Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)129 2117 y Fn(b)q(e)i(a)g(semiclassical)e (measure)g(of)j(this)f(sequence.)k(F)l(or)c(all)g Fm(a)e Fj(2)h Fm(C)1350 2099 y Fq(1)1346 2130 y Fp(0)1387 2117 y Fn(\()p Fm(T)1442 2099 y Fq(\003)1461 2117 y Fi(R)1500 2099 y Fk(d)1517 2117 y Fn(\),)h(w)o(e)g(ha)o(v)o(e)f Fm(\026)1776 2125 y Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1903 2117 y Fn(\()p Fm(a)p Fn(\))g(=)129 2177 y Fm(\026)158 2184 y Fk(s)176 2177 y Fn(\()p Fj(h)p Fm(x)p Fj(i)261 2159 y Fp(2)p Fk(s)p Fq(\000)p Fp(1)p Fq(\000)p Fk(\016)387 2177 y Fm(a)p Fn(\).)29 b(By)18 b(Prop)q(osition)i(2,)g Fm(\026)906 2185 y Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1052 2177 y Fn(is)e(supp)q(orted)i(in)f Fm(p)1419 2159 y Fq(\000)p Fp(1)1466 2177 y Fn(\()p Fm(\025)p Fn(\))h(and)f (Lemma)e(6)i(implies)129 2238 y(that)c Fm(\026)262 2245 y Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)404 2238 y Fn(is)g(supp)q(orted)h(in)f Fm(B)776 2220 y Fq(\003)773 2250 y Fk(R)801 2238 y Fn(.)21 b(Th)o(us,)15 b Fm(\026)1001 2245 y Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1143 2238 y Fn(and)h Fm(\026)1266 2245 y Fk(s)1300 2238 y Fn(ha)o(v)o(e)e (compact)g(supp)q(ort)i(included)e(in)129 2298 y Fm(p)153 2280 y Fq(\000)p Fp(1)200 2298 y Fn(\()p Fm(\025)p Fn(\))f Fj(\\)g Fm(B)365 2280 y Fq(\003)362 2310 y Fk(R)391 2298 y Fn(.)27 b(By)18 b(Lemma)e(6,)j(\(7\))g(holds)f(true)g(for)h(the)f (sequence)f(\()p Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))p Fj(h)p Fm(x)p Fj(i)1611 2280 y Fq(\000)p Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1766 2298 y Fm(f)1790 2305 y Fk(n)1814 2298 y Fn(\))1833 2305 y Fk(n)1856 2298 y Fn(.)28 b(Th)o(us,)129 2358 y(there)14 b(exists)g(lim)453 2365 y Fk(n)p Fq(!1)555 2358 y Fj(k)p Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))p Fj(h)p Fm(x)p Fj(i)746 2340 y Fq(\000)p Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)901 2358 y Fm(f)925 2365 y Fk(n)949 2358 y Fj(k)974 2340 y Fp(2)1008 2358 y Fn(and)15 b(it)g(is)f(the)h (total)g(mass)f(of)h Fm(\026)1595 2366 y Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1722 2358 y Fn(.)21 b(F)l(or)15 b Fm(s)f(>)g Fn(1)p Fm(=)p Fn(2,)129 2418 y(w)o(e)f(can)g(c)o(ho)q(ose)h Fm(\016)h Fj(\024)f Fm(s)p Fn(,)f(yielding)g Fj(k)p Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))p Fj(h)p Fm(x)p Fj(i)948 2400 y Fq(\000)p Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1102 2418 y Fm(f)1126 2425 y Fk(n)1150 2418 y Fj(k)1175 2400 y Fp(2)1208 2418 y Fj(\025)14 b(k)p Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))p Fj(h)p Fm(x)p Fj(i)1452 2400 y Fq(\000)p Fk(s)1498 2418 y Fm(f)1522 2425 y Fk(n)1546 2418 y Fj(k)1571 2400 y Fp(2)1590 2418 y Fn(.)21 b(But,)13 b(b)o(y)g(the)g(energy)129 2478 y(lo)q(calization)h(of)g(the)h Fm(f)546 2485 y Fk(n)569 2478 y Fn(,)f Fj(k)p Fn(\(1)7 b Fj(\000)g Fm(\022)q Fn(\()p Fm(P)g Fn(\)\))p Fj(h)p Fm(x)p Fj(i)903 2460 y Fq(\000)p Fk(s)951 2478 y Fm(f)975 2485 y Fk(n)998 2478 y Fj(k)14 b Fn(=)g Fm(O)q Fn(\()p Fm(h)1174 2485 y Fk(n)1198 2478 y Fn(\).)20 b(Th)o(us,)15 b(the)f(total)h(mass)f(of)g Fm(\026)1784 2486 y Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1912 2478 y Fn(,)g(and)129 2539 y(therefore)h(of)i Fm(\026)418 2546 y Fk(s)437 2539 y Fn(,)e(is)h(p)q(ositiv)o(e.)21 b(W)l(e)16 b(ha)o(v)o(e)f(pro)o(v)o(en)h(the)129 2625 y Fg(Prop)r(osition)h(7.)24 b Fm(\026)528 2632 y Fk(s)565 2625 y Ff(has)17 b(c)n(omp)n(act)g(supp)n(ort)f(in)i Fm(p)1096 2607 y Fq(\000)p Fp(1)1144 2625 y Fn(\()p Fm(\025)p Fn(\))11 b Fj(\\)g Fm(B)1305 2607 y Fq(\003)1302 2638 y Fk(R)1331 2625 y Ff(.)23 b(If)17 b Fm(s)d(>)f Fn(1)p Fm(=)p Fn(2)p Ff(,)19 b Fm(\026)1643 2632 y Fk(s)1679 2625 y Ff(is)e(nonzer)n(o.)129 2712 y Fg(Remark)f(8.)24 b Ff(A)o(lthough)g(it)f(is)g(not)g(app)n(ar)n(ent)f(in)i(the)f(pr)n(o)n (of)e(of)i(Pr)n(op)n(osition)f(7,)i(we)g(did)e(use)i(the)129 2773 y(invarianc)n(e)17 b(enonc)n(e)n(d)g(in)g(Pr)n(op)n(osition)e(4.) 22 b(By)17 b(\(6\),)f(this)g(invarianc)n(e)i(is)e(e)n(quivalent)j(to)d (the)h(fact)g(that)129 2833 y(the)h(l.h.s)f(of)h(\(6\))f(tends)h(to)g (zer)n(o.)129 2893 y(Of)h(c)n(ourse,)g(Pr)n(op)n(osition)f(7)h(is)g (true,)h(if)f Fm(\026)925 2900 y Fk(s)962 2893 y Ff(is)g(r)n(eplac)n(e) n(d)g(by)g(the)h(invariant)f(me)n(asur)n(e)g Fm(\026)e Fn(:=)f Fj(h)p Fm(x)p Fj(i)1934 2875 y Fp(2)p Fk(s)1971 2893 y Fm(\026)2000 2900 y Fk(s)129 2953 y Ff(c)n(onsider)n(e)n(d)g(in) i(R)n(emark)f(5.)p eop %%Page: 7 7 7 6 bop 129 133 a Fl(Non-trapping)17 b(b)q(eha)o(viour,)f(21-11-02)1147 b Fn(7)129 286 y(In)15 b(view)f(of)i(Prop)q(osition)h(7,)e(w)o(e)g(exp) q(ect)g(that)h(the)f Fm(f)1107 293 y Fk(n)1146 286 y Fn(are)g(not)h(lo)q(calized)e(on)i(non-trapping)h(regions)129 347 y(of)f Fm(p)208 329 y Fq(\000)p Fp(1)256 347 y Fn(\()p Fm(\025)p Fn(\).)21 b(Therefore,)16 b(the)g(follo)o(wing)g(prop)q (osition)h(is)f(not)h(surprising.)129 446 y Fg(Prop)r(osition)g(9.)24 b Ff(Assume)18 b(that)g Fm(\025)g Ff(is)f(a)g(non-tr)n(apping)i(ener)n (gy.)k(Then)18 b Fm(\026)1540 453 y Fk(s)1572 446 y Fn(=)c(0)p Ff(.)129 581 y Fg(Pro)r(of:)33 b Fn(By)18 b(Prop)q(osition)h(7,)g Fm(\026)743 588 y Fk(s)780 581 y Fn(has)g(compact)e(supp)q(ort)j(in)e Fm(p)1332 563 y Fq(\000)p Fp(1)1379 581 y Fn(\()p Fm(\025)p Fn(\))13 b Fj(\\)g Fm(B)1544 563 y Fq(\003)1541 593 y Fk(R)1570 581 y Fn(.)27 b(Let)18 b Fm(g)i Fj(2)d Fm(C)1832 563 y Fq(1)1828 593 y Fp(0)1869 581 y Fn(\()p Fm(T)1924 563 y Fq(\003)1943 581 y Fi(R)1982 563 y Fk(d)1999 581 y Fn(\))129 641 y(with)f Fm(g)g Fn(=)e(1)i(on)h Fm(p)463 623 y Fq(\000)p Fp(1)511 641 y Fn(\()p Fm(\025)p Fn(\))11 b Fj(\\)g Fm(B)672 623 y Fq(\003)669 653 y Fk(R)698 641 y Fn(.)21 b(By)16 b(the)g(non-trapping)i(condition,)738 770 y Fm(a)p Fn(\()p Fm(x;)8 b(\030)r Fn(\))30 b(:=)f Fj(\000)1033 711 y Fh(Z)1075 724 y Fp(+)p Fq(1)1056 806 y Fp(0)1148 770 y Fm(g)5 b Fj(\016)s Fm(\036)1233 749 y Fk(t)1247 770 y Fn(\()p Fm(x;)j(\030)r Fn(\))g Fm(dt)129 898 y Fn(is)16 b(a)g(w)o(ell-de\014ned,)f(smo)q(oth)h(function)g(on)h Fm(T)966 880 y Fq(\003)985 898 y Fi(R)1024 880 y Fk(d)1041 898 y Fn(.)k(By)16 b(Prop)q(osition)h(4,)570 1005 y(0)22 b(=)g Fm(\026)705 1012 y Fk(s)724 957 y Fh(\020)749 1005 y Fj(h)p Fm(x)p Fj(i)815 985 y Fp(2)p Fk(s)851 1005 y Fj(f)p Fm(p;)8 b(a)p Fj(g)973 957 y Fh(\021)1020 1005 y Fn(=)22 b Fm(\026)1109 1012 y Fk(s)1128 957 y Fh(\020)1153 1005 y Fj(h)p Fm(x)p Fj(i)1219 985 y Fp(2)p Fk(s)1255 957 y Fh(\021)1291 1005 y Fm(:)p 1542 1005 2 33 v 1544 974 30 2 v 1544 1005 V 1573 1005 2 33 v 129 1119 a Fg(Remark)16 b(10.)24 b Ff(Notic)n(e)18 b(that)g(we)h(do)e(not)i(use)f(a)f(glob)n (al)i(esc)n(ap)n(e)f(function,)h(like)g(in)f([7,)f(22].)23 b(In)18 b(fact,)129 1179 y(sinc)n(e)h(we)g(work)g(on)f(me)n(asur)n(es,) g(we)h(c)n(an)g(de)n(c)n(ouple)g(the)g(e\013e)n(ct)g(of)g(the)g(non-tr) n(apping)g(b)n(ehaviours)f(at)129 1239 y(in\014nity)g(\(Pr)n(op)n (osition)e(7\))h(and)h(on)g(a)f(c)n(omp)n(act)g(set)h(\(Pr)n(op)n (osition)e(9\).)129 1374 y Fn(F)l(or)f Fm(s)f(>)g Fn(1)p Fm(=)p Fn(2)i(and)g Fm(\025)g Fn(non-trapping,)g(Prop)q(ositions)h(7)e (and)h(9)g(giv)o(e)e(the)h(con)o(tradiction)g(w)o(e)f(shall)h(use)129 1434 y(in)h(Section)f(3)i(for)f(the)g(pro)q(of)i(of)e(Theorem)f(1.)129 1494 y(What)g(happ)q(ens,)h(if)f Fm(s)e Fj(\024)h Fn(1)p Fm(=)p Fn(2?)22 b(If)15 b Fm(\025)g Fn(is)g(non-trapping,)h Fm(\026)1191 1501 y Fk(s)1224 1494 y Fn(=)e(0)h(b)o(y)g(Prop)q(osition) h(9)f(but)g(the)g(pro)q(of)h(of)129 1554 y(Prop)q(osition)e(7)f(only)f (sho)o(ws)i(that)f(lim)834 1561 y Fk(n)p Fq(!1)937 1554 y Fj(k)p Fm(\022)q Fn(\()p Fm(P)7 b Fn(\))p Fj(h)p Fm(x)p Fj(i)1128 1536 y Fq(\000)p Fp(\(1+)p Fk(\016)q Fp(\))p Fk(=)p Fp(2)1283 1554 y Fm(f)1307 1561 y Fk(n)1330 1554 y Fj(k)1355 1536 y Fp(2)1389 1554 y Fn(=)13 b(0)g(whic)o(h)f(do)q(es)i (not)f(con)o(tradict)129 1614 y(a)j(priori)g Fj(kh)p Fm(x)p Fj(i)393 1596 y Fq(\000)p Fk(s)439 1614 y Fm(f)463 1621 y Fk(n)487 1614 y Fj(k)e Fn(=)f(1.)129 1675 y(This)f(situation)h (really)e(app)q(ears)j(for)f Fm(V)25 b Fn(=)14 b(0,)f(the)f(free)g (case)h(for)f(whic)o(h)g(eac)o(h)g Fm(\025)j(>)e Fn(0)g(is)g (non-trapping,)129 1735 y(and)k(for)f Fm(s)e Fj(2)g Fn([0;)8 b(1)p Fm(=)p Fn(2[.)22 b(Consider,)16 b(for)g Fm(k)g Fj(2)e Fi(R)982 1717 y Fk(d)1010 1735 y Fj(n)d(f)p Fn(0)p Fj(g)p Fn(,)532 1864 y Fm(f)556 1871 y Fk(n)580 1864 y Fn(\()p Fm(x)p Fn(\))30 b(:=)802 1830 y(1)p 762 1853 105 2 v 762 1866 a Fj(p)p 804 1866 63 2 v 32 x Fm(m)847 1905 y Fk(d)880 1864 y Fm(e)903 1844 y Fk(ih)935 1829 y Fd(\000)p Fa(1)935 1850 y Fc(n)976 1844 y Fk(k)q Fq(\001)p Fk(x)1027 1864 y Fj(h)p Fm(x)p Fj(i)1093 1844 y Fk(s)1112 1864 y Fj(j)p Fm(x)p Fj(j)1168 1844 y Fp(\(1)p Fq(\000)p Fk(d)p Fp(\))p Fk(=)p Fp(2)1332 1830 y Fn(1)p 1309 1853 71 2 v 1309 1863 a Fj(p)p 1350 1863 30 2 v 1350 1898 a Fm(n)1393 1864 y(\037)1424 1816 y Fh(\020)1448 1864 y Fj(j)p Fm(x)p Fj(j)p Fm(=n)1557 1816 y Fh(\021)1593 1864 y Fm(;)325 b Fn(\(11\))129 2005 y(where)17 b Fm(\037)g Fj(2)g Fm(C)408 1987 y Fq(1)404 2017 y Fp(0)445 2005 y Fn(\(]1;)8 b(2[;)g Fi(R)p Fn(\))15 b(with)770 1969 y Fh(R)789 2018 y Fe(R)824 2005 y Fm(\037)855 1987 y Fp(2)891 2005 y Fn(=)i(1)i(and)f Fm(m)1128 2012 y Fk(d)1166 2005 y Fn(is)g(the)g(Leb)q(esgue)h(measure)e(of)h(the)g(\()p Fm(d)13 b Fj(\000)f Fn(1\)-)129 2065 y(dimensional)i(unit)i(sphere.)22 b(By)15 b(direct)h(calculation,)f(w)o(e)h(see)g(that)h Fj(kh)p Fm(x)p Fj(i)1495 2047 y Fq(\000)p Fk(s)1541 2065 y Fm(f)1565 2072 y Fk(n)1588 2065 y Fj(k)d Fn(=)g(1,)j Fj(h)p Fm(x)p Fj(i)1800 2047 y Fk(s)1819 2065 y Fn(\()p Fj(\000)p Fm(h)1905 2047 y Fp(2)1905 2077 y Fk(n)1928 2065 y Fn(\001)11 b Fj(\000)129 2125 y(j)p Fm(k)r Fj(j)184 2107 y Fp(2)203 2125 y Fn(\))p Fm(f)246 2132 y Fk(n)283 2125 y Fn(=)j Fm(o)p Fn(\()p Fm(h)405 2132 y Fk(n)429 2125 y Fn(\),)e(lim)541 2132 y Fk(n)p Fq(!1)644 2125 y Fj(k)p Fm(\037)700 2132 y Fp(0)719 2125 y Fn(\()p Fm(x)p Fn(\))p Fj(h)p Fm(x)p Fj(i)851 2107 y Fq(\000)p Fk(s)897 2125 y Fm(f)921 2132 y Fk(n)944 2125 y Fj(k)i Fn(=)g(0)d(for)g(an)o(y)g Fm(\037)1257 2132 y Fp(0)1290 2125 y Fj(2)j Fm(C)1376 2107 y Fq(1)1372 2138 y Fp(0)1413 2125 y Fn(\()p Fi(R)1471 2107 y Fk(d)1488 2125 y Fn(\),)e(and)f(lim)1690 2132 y Fk(n)p Fq(!1)1792 2125 y Fj(kh)p Fm(x)p Fj(i)1883 2107 y Fq(\000)p Fk(s)1926 2095 y Fd(0)1940 2125 y Fm(f)1964 2132 y Fk(n)1988 2125 y Fj(k)j Fn(=)129 2185 y(0)i(for)h(an)o(y)f Fm(s)359 2167 y Fq(0)384 2185 y Fm(>)e(s)p Fn(.)21 b(The)16 b(t)o(w)o(o)h(\014rst)f(prop)q(erties)g(sho)o(w)h(that)g(w)o(e)e(ha)o (v)o(e)h(a)g(\\semiclassically)e(resonan)o(t")129 2246 y(situation.)23 b(The)17 b(third)g(one)g(pro)o(v)o(es)f(that)h Fm(\026)953 2253 y Fk(s)987 2246 y Fn(=)e(0,)i(whic)o(h)f(also)h(follo) o(ws)g(from)f(Prop)q(osition)i(9.)23 b(The)129 2306 y(fourth)e(one)h (implies)d(that,)j(for)g(an)o(y)f Fm(s)872 2288 y Fq(0)906 2306 y Fm(>)h Fn(1)p Fm(=)p Fn(2,)i(lim)1143 2313 y Fk(n)p Fq(!1)1246 2306 y Fj(kh)p Fm(x)p Fj(i)1337 2288 y Fq(\000)p Fk(s)1380 2276 y Fd(0)1394 2306 y Fm(f)1418 2313 y Fk(n)1442 2306 y Fj(k)e Fn(=)g(0,)h(whic)o(h)d(follo)o(ws)i(from)129 2366 y(the)c(pro)q(of)j(of)e(Prop)q(osition)h(7.)30 b(In)19 b(particular,)g(the)g(second)g(result)g(of)g(Prop)q(osition)h(7)g(is)e (false)h(for)129 2426 y Fm(s)j(<)g Fn(1)p Fm(=)p Fn(2.)37 b(In)21 b(view)g(of)g(Remark)f(5,)i(notice)f(that)g(the)g(sequence)f (\(11\))i(is)f(un)o(b)q(ounded)h(in)f Fm(L)1919 2408 y Fp(2)1960 2426 y Fn(for)129 2486 y Fm(s)13 b Fj(2)p Fn(]0;)8 b(1)p Fm(=)p Fn(2[.)129 2688 y Fo(3)81 b(Pro)r(of)27 b(of)g(the)g(main)g(result.)129 2833 y Fn(This)22 b(section)g(is)g(dev) o(oted)f(to)i(the)f(pro)q(of)h(of)g(Theorem)d(1.)40 b(W)l(e)22 b(start)h(b)o(y)e(pro)o(ving)h(that)h(a)g(non-)129 2893 y(trapping)14 b(energy)f Fm(\025)i Fn(m)o(ust)d(b)q(e)i(outside)g(the)f (pure)h(p)q(oin)o(t)g(sp)q(ectrum)e(of)i Fm(P)21 b Fn(for)14 b Fm(h)g Fn(small)e(enough.)21 b(This)129 2953 y(allo)o(ws)16 b(us)h(to)g(consider)g(the)f(b)q(oundary)i(v)m(alues)e Fm(R)p Fn(\()p Fm(\025)d Fj(\006)e Fm(i)p Fn(0\))16 b(of)h(the)g (resolv)o(en)o(t.)k(Assuming)15 b(that)i(\(3\))p eop %%Page: 8 8 8 7 bop 129 133 a Fl(Non-trapping)17 b(b)q(eha)o(viour,)f(21-11-02)1147 b Fn(8)129 286 y(is)16 b(false)g(for)g(\\+")h(\(for)g(instance\),)e(w)o (e)h(sho)o(w)h(that)f(this)g(is)g(a)h(\\semiclassically)c(resonan)o(t") k(situation)129 347 y(for)f Fm(s)e(>)g Fn(1)p Fm(=)p Fn(2.)22 b(No)o(w)16 b(Prop)q(ositions)i(7)f(and)f(9)h(yield)e(the)h (desired)f(con)o(tradiction.)129 442 y(Supp)q(ose)i(that)g(there)f (exists)g Fm(h)714 449 y Fk(n)752 442 y Fj(!)f Fn(0)i(suc)o(h)f(that)h Fm(\025)e Fj(2)g Fm(I)20 b Fn(b)q(elongs)d(to)g(the)g(pure)f(p)q(oin)o (t)h(sp)q(ectrum)e(of)129 502 y Fm(P)7 b Fn(\()p Fm(h)214 509 y Fk(n)237 502 y Fn(\),)17 b(for)g(all)g Fm(n)p Fn(.)24 b(Then,)16 b(for)i(all)e Fm(n)p Fn(,)h(w)o(e)g(can)g(\014nd)g(eigen)o (v)m(alues)f Fm(\025)1385 509 y Fk(n)1426 502 y Fn(of)h Fm(P)7 b Fn(\()p Fm(h)1567 509 y Fk(n)1591 502 y Fn(\))17 b(suc)o(h)g(that)g Fm(\025)1872 509 y Fk(n)1911 502 y Fj(!)e Fm(\025)p Fn(.)129 563 y(F)l(or)k(all)g Fm(n)p Fn(,)i(let)d Fm(u)455 570 y Fk(n)498 563 y Fn(b)q(e)i(a)g(normalized)e (eigenfunction)g(of)i Fm(P)7 b Fn(\()p Fm(h)1311 570 y Fk(n)1335 563 y Fn(\))20 b(with)f(eigen)o(v)m(alue)f Fm(\025)1752 570 y Fk(n)1776 563 y Fn(.)32 b(Th)o(us,)20 b(w)o(e)129 623 y(ha)o(v)o(e)h(a)h(\\semiclassically)d(resonan)o(t")j (situation)g(with)g Fm(s)h Fn(=)g(0)f(and)h(w)o(e)e(ma)o(y)f(use)i(the) g(results)f(of)129 683 y(Section)d(2.)30 b(F)l(rom)18 b(Prop)q(osition)j(9,)e(w)o(e)g(deriv)o(e)e(that)j(an)o(y)f (semiclassical)d(measure)i Fm(\026)1756 690 y Fp(0)1795 683 y Fn(of)i(\()p Fm(u)1901 690 y Fk(n)1924 683 y Fn(\))1943 690 y Fk(n)1986 683 y Fn(is)129 743 y(zero.)26 b(Let)18 b Fm(\037)e Fj(2)h Fm(C)481 725 y Fq(1)477 756 y Fp(0)518 743 y Fn(\()p Fi(R)p Fn(\))e(with)j Fm(\037)e Fn(=)h(1)h(on)g([)p Fj(\000)p Fn(1;)8 b(1])18 b(and,)g(for)h(an)o(y)f Fm(R)f(>)g Fn(0,)h(let)f Fm(\037)1640 750 y Fk(R)1668 743 y Fn(\()p Fm(x)p Fn(\))g(:=)f Fm(\037)p Fn(\()p Fj(j)p Fm(x)p Fj(j)p Fm(=R)p Fn(\).)129 803 y(F)l(or)i(an)o(y)g Fm(R)f(>)g Fn(0,)i(this)f(means)f(that)h(lim)901 810 y Fk(n)p Fq(!1)995 803 y Fj(h)p Fn(\(1)13 b Fj(\000)f Fm(\037)1152 810 y Fk(R)1180 803 y Fn(\))p Fm(u)1227 810 y Fk(n)1251 803 y Fm(;)c(u)1301 810 y Fk(n)1324 803 y Fj(i)17 b Fn(=)g(1.)27 b(But,)18 b(since)f Fm(\025)1742 810 y Fk(n)1783 803 y Fj(!)g Fm(\025)p Fn(,)h(there)129 864 y(exists,)e(b)o(y)h(the)g(exp)q (onen)o(tial)g(deca)o(y)g(of)h(eigenfunctions)e(\(see)h([1,)g(10)q (]\),)g Fm(c;)8 b(C)19 b(>)d Fn(0)i(suc)o(h)f(that,)h(for)f Fm(n)129 924 y Fn(large)f(enough,)g Fj(k)8 b Fn(exp\()p Fm(c)p Fj(h)p Fm(x)p Fj(i)p Fn(\))p Fm(u)690 931 y Fk(n)714 924 y Fj(k)14 b(\024)f Fm(C)t Fn(.)21 b(Therefore,)15 b(uniformly)f(w.r.t.)21 b Fm(n)p Fn(,)340 977 y Fh(D)366 1025 y Fn(\(1)11 b Fj(\000)g Fm(\037)501 1032 y Fk(R)529 1025 y Fn(\))p Fm(u)576 1032 y Fk(n)600 1025 y Fm(;)d(u)650 1032 y Fk(n)673 977 y Fh(E)728 1025 y Fn(=)796 977 y Fh(D)822 1025 y Fn(\(1)j Fj(\000)g Fm(\037)957 1032 y Fk(R)985 1025 y Fn(\))p Fm(e)1027 1004 y Fq(\000)p Fp(2)p Fk(c)p Fq(h)p Fk(x)p Fq(i)1145 1025 y Fm(e)1168 1004 y Fk(c)p Fq(h)p Fk(x)p Fq(i)1232 1025 y Fm(u)1260 1032 y Fk(n)1292 1025 y Fm(;)16 b(e)1345 1004 y Fk(c)p Fq(h)p Fk(x)p Fq(i)1409 1025 y Fm(u)1437 1032 y Fk(n)1460 977 y Fh(E)1516 1025 y Fj(\024)30 b Fm(C)1624 1004 y Fp(2)1643 1025 y Fm(e)1666 1004 y Fq(\000)p Fp(2)p Fk(c)p Fq(h)p Fk(R)p Fq(i)1793 1025 y Fm(:)129 1120 y Fn(T)l(aking)16 b(the)g(lim)443 1127 y Fk(R)p Fq(!)p Fp(+)p Fq(1)578 1120 y Fn(lim)8 b(sup)728 1132 y Fk(n)p Fq(!1)838 1120 y Fn(of)17 b(this)f(term,)e(w)o(e)i(arriv)o(e)f(at)h(a)h(con)o (tradiction.)129 1180 y(Th)o(us)h(Mourre)g(theory)g(\(see)g([16)q(,)f (13)q(]\))h(ensures)g(that,)h(for)g Fm(n)f Fn(large)g(enough,)h(the)f (b)q(oundary)i(v)m(alue)129 1240 y Fm(R)p Fn(\()p Fm(\025)12 b Fn(+)f Fm(i)p Fn(0\))16 b(is)g(w)o(ell)f(de\014ned)h(on)h Fm(I)i Fn(as)e(a)g(b)q(ounded)g(op)q(erator)g(from)f Fm(L)1420 1222 y Fp(2)1420 1253 y Fk(s)1456 1240 y Fn(to)g Fm(L)1548 1222 y Fp(2)1548 1253 y Fq(\000)p Fk(s)1611 1240 y Fn(for)g Fm(s)e(>)g Fn(1)p Fm(=)p Fn(2.)129 1301 y(No)o(w,)19 b(w)o(e)f(supp)q(ose)i(that)g(\(3\))f(is)g(false)f(for)i (\\+")f(and)h(some)e Fm(s)g(>)g Fn(1)p Fm(=)p Fn(2.)31 b(So,)19 b(w)o(e)g(can)g(\014nd)g Fm(h)1870 1308 y Fk(n)1912 1301 y Fj(!)g Fn(0,)129 1361 y Fm(N)j(>)16 b Fn(0,)i(\()p Fm(z)342 1368 y Fk(n)365 1361 y Fn(\))384 1368 y Fk(n)424 1361 y Fj(2)f Fi(C)507 1343 y Fe(N)536 1361 y Fn(,)h(and)h(a)f (sequence)f(\()p Fm(v)954 1368 y Fk(n)977 1361 y Fn(\))996 1368 y Fk(n)1037 1361 y Fn(of)h(nonzero)g Fm(L)1310 1343 y Fp(2)1348 1361 y Fn(functions,)g(suc)o(h)g(that)g Fj(<)p Fn(\()p Fm(z)1871 1368 y Fk(n)1894 1361 y Fn(\))f Fj(2)f Fm(I)t Fn(,)129 1421 y(0)e Fj(\024)f(=)p Fn(\()p Fm(z)297 1428 y Fk(n)321 1421 y Fn(\))g Fj(\024)h Fm(N)5 b(h)478 1428 y Fk(n)502 1421 y Fn(,)16 b(and)619 1516 y Fj(kh)p Fm(x)p Fj(i)710 1495 y Fq(\000)p Fk(s)764 1516 y Fm(R)p Fn(\()p Fm(z)843 1523 y Fk(n)867 1516 y Fm(;)8 b(h)917 1523 y Fk(n)940 1516 y Fn(\))g Fj(h)p Fm(x)p Fj(i)1033 1495 y Fq(\000)p Fk(s)1088 1516 y Fm(v)1112 1523 y Fk(n)1135 1516 y Fj(k)30 b Fm(>)g(n)8 b(h)1323 1495 y Fq(\000)p Fp(1)1323 1528 y Fk(n)1379 1516 y Fj(k)p Fm(v)1428 1523 y Fk(n)1451 1516 y Fj(k)1476 1495 y Fp(2)1507 1516 y Fm(;)411 b Fn(\(12\))129 1611 y(where)18 b Fm(R)p Fn(\()p Fm(z)351 1618 y Fk(n)375 1611 y Fm(;)8 b(h)425 1618 y Fk(n)448 1611 y Fn(\))18 b(:=)g(\()p Fm(P)7 b Fn(\()p Fm(h)659 1618 y Fk(n)683 1611 y Fn(\))13 b Fj(\000)f Fm(z)789 1618 y Fk(n)813 1611 y Fn(\))832 1593 y Fq(\000)p Fp(1)879 1611 y Fn(.)29 b(F)l(urthermore,)17 b Fj(=)p Fn(\()p Fm(z)1299 1618 y Fk(n)1322 1611 y Fn(\))p Fm(=h)1393 1618 y Fk(n)1435 1611 y Fj(!)h Fm(r)i Fj(2)e Fn([0;)8 b Fm(N)d Fn(])19 b(and)g Fj(<)p Fn(\()p Fm(z)1908 1618 y Fk(n)1931 1611 y Fn(\))g Fj(!)129 1671 y Fm(\025)29 b Fj(2)g Fm(I)t Fn(,)d(p)q(ossibly)f(after)g(extracting)g(a)g (subsequence.)47 b(W)l(e)24 b(write)h(the)g(l.h.s)f(of)h(\(12\))h(as)f (\(1)18 b(+)129 1731 y Fm(\024)157 1738 y Fk(n)180 1731 y Fn(\)\()p Fm(n=h)299 1738 y Fk(n)323 1731 y Fn(\))p Fj(k)p Fm(v)391 1738 y Fk(n)414 1731 y Fj(k)439 1713 y Fp(2)477 1731 y Fn(=:)h Fm(\021)572 1738 y Fk(n)595 1731 y Fn(,)g(for)h(some)e Fm(\024)859 1738 y Fk(n)901 1731 y Fm(>)h Fn(0.)30 b(Let)19 b Fm(w)1151 1738 y Fk(n)1193 1731 y Fn(:=)f Fm(v)1287 1738 y Fk(n)1310 1731 y Fm(=\021)1358 1738 y Fk(n)1382 1731 y Fn(.)30 b(Then,)19 b Fj(k)p Fm(w)1630 1738 y Fk(n)1653 1731 y Fj(k)g Fn(=)f Fm(o)p Fn(\()p Fm(h)1823 1738 y Fk(n)1847 1731 y Fn(\).)30 b(No)o(w,)129 1791 y(w)o(e)21 b(set)g Fm(f)311 1798 y Fk(n)358 1791 y Fn(:=)h Fm(R)p Fn(\()p Fm(z)511 1798 y Fk(n)535 1791 y Fm(;)8 b(h)585 1798 y Fk(n)608 1791 y Fn(\))p Fj(h)p Fm(x)p Fj(i)693 1773 y Fq(\000)p Fk(s)739 1791 y Fm(w)774 1798 y Fk(n)798 1791 y Fn(,)22 b(whic)o(h)f(b)q(elongs)i(to)f(the)f (domaine)f(of)i Fm(P)7 b Fn(\()p Fm(h)1660 1798 y Fk(n)1684 1791 y Fn(\).)37 b(Clearly)l(,)22 b(this)129 1852 y(sequence)e(\()p Fm(f)379 1859 y Fk(n)403 1852 y Fn(\))422 1859 y Fk(n)467 1852 y Fn(satis\014es)h Fj(kh)p Fm(x)p Fj(i)742 1834 y Fq(\000)p Fk(s)789 1852 y Fm(f)813 1859 y Fk(n)836 1852 y Fj(k)i Fn(=)g(1)f(and)g Fj(kh)p Fm(x)p Fj(i)1182 1834 y Fk(s)1200 1852 y Fn(\()p Fm(P)7 b Fn(\()p Fm(h)1304 1859 y Fk(n)1328 1852 y Fn(\))15 b Fj(\000)f Fm(z)1438 1859 y Fk(n)1461 1852 y Fn(\))p Fm(f)1504 1859 y Fk(n)1528 1852 y Fj(k)22 b Fn(=)h Fm(o)p Fn(\()p Fm(h)1706 1859 y Fk(n)1730 1852 y Fn(\),)f(that)g(is)g(the)129 1912 y(prop)q(erties)d(whic)o(h)f(c)o(haracterise)g(the)g(situation)h(w)o(e) g(called)f(\\semiclassically)e(resonan)o(t".)30 b(There-)129 1972 y(fore)16 b(w)o(e)g(can)h(use)f(the)g(results)g(of)h(Section)f(2.) 22 b(Since)16 b Fm(s)e(>)g Fn(1)p Fm(=)p Fn(2,)j(b)q(oth)h(Prop)q (ositions)f(7)g(and)g(9)g(apply)l(,)129 2032 y(yielding)f(a)i(con)o (tradiction.)26 b(Th)o(us)18 b(the)f(resolv)o(en)o(t)f(estimates)h (\(3\))h(m)o(ust)e(hold)i(true)f(and)i(the)e(pro)q(of)129 2092 y(of)f(Theorem)f(1)i(is)f(complete.)129 2292 y Fo(References)129 2420 y Fn([1])25 b(S.)20 b(Agmon:)29 b Ff(L)n(e)n(ctur)n(es)20 b(on)i(exp)n(onential)h(de)n(c)n(ay)e(of)g(solutions)h(of)f(se)n(c)n (ond-or)n(der)g(el)r(liptic)j(e)n(qua-)206 2480 y(tions.)p Fn(,)16 b(Princeton)g(Univ)o(ersit)o(y)e(Press,)i(1982.)129 2578 y([2])25 b(W.O.)13 b(Amrein,)f(A.)i(Boutet)f(de)i(Mon)o(v)o(el,)d (I.V.)h(Georgescu:)20 b Fm(C)1374 2585 y Fp(0)1394 2578 y Ff(-gr)n(oups,)c(c)n(ommutator)f(metho)n(ds)206 2638 y(and)j(sp)n(e)n(ctr)n(al)e(the)n(ory)h(of)g Fm(N)5 b Ff(-b)n(o)n(dy)17 b(hamiltonians.)p Fn(,)g(Birkh\177)-24 b(auser)15 b(1996.)129 2735 y([3])25 b(N.)20 b(Burq:)31 b Ff(Semiclassic)n(al)23 b(estimates)g(for)e(the)h(r)n(esolvent)h(in)g (non)f(tr)n(apping)g(ge)n(ometries.)g Fn(In)o(t.)206 2796 y(Math.)16 b(Res.)f(Notices)h(2002,)h(no)g(5,)f(221-241.)129 2893 y([4])25 b(J.)19 b(Derezi)q(\023)-26 b(nski:)27 b Ff(Asymptotic)21 b(c)n(ompleteness)h(for)e Fm(N)5 b Ff(-p)n(article)21 b(long-r)n(ange)h(quantum)g(systems.)206 2953 y Fn(Ann.)15 b(of)i(Math.)f(138)h(\(1993\))h(427-476.)p eop %%Page: 9 9 9 8 bop 129 133 a Fl(Non-trapping)17 b(b)q(eha)o(viour,)f(21-11-02)1147 b Fn(9)129 286 y([5])25 b(J.)18 b(Derezi)q(\023)-26 b(nski,)19 b(C.)g(G)o(\023)-23 b(erard:)26 b Ff(Sc)n(attering)21 b(the)n(ory)e(of)h(classic)n(al)h(and)f(quantum)h(N-p)n(article)g(sys-) 206 347 y(tems.)16 b Fn(Springer-V)l(erlag)g(1997.)129 446 y([6])25 b(P)l(.)16 b(G)o(\023)-23 b(erard,)17 b(E.)g(Leic)o(h)o (tnam:)j Ff(Er)n(go)n(dic)d(pr)n(op)n(erties)f(of)i(eigenfunctions)j (for)c(the)h(Dirichlet)h(pr)n(ob-)206 506 y(lem.)e Fn(Duk)o(e)f(Math.)f (J.,)h(v)o(ol.)f(71,)i(No)f(2,)g(1993.)129 605 y([7])25 b(C.)33 b(G)o(\023)-23 b(erard,)37 b(A.)c(Martinez:)54 b Ff(Princip)n(e)33 b(d'absorption)g(limite)g(p)n(our)f(des)i(op)o (\023)-24 b(er)n(ateurs)32 b(de)206 665 y(Schr\177)-25 b(odinger)18 b(\022)-25 b(a)17 b(longue)j(p)n(ort)o(\023)-24 b(ee.)16 b Fn(C.R.)f(Acad.)h(Sci.)f(306,)i(121-123,)h(1988.)129 764 y([8])25 b(B.)13 b(Hel\013er,)g(J.)g(Sj\177)-24 b(ostrand:)22 b Ff(R)o(\023)-24 b(esonanc)n(es)16 b(en)g(limite)g(semi-classique.)g Fn(Bull.)c(So)q(c.)i(Math.)g(F)l(rance)206 824 y(114,)j(Nos)f(24-25)i (\(1986\).)129 923 y([9])25 b(B.)16 b(Hel\013er,)g(J.)h(Sj\177)-24 b(ostrand:)24 b Ff(Op)o(\023)-24 b(er)n(ateurs)18 b(de)h(Schr\177)-25 b(odinger)19 b(ave)n(c)g(champs)f(magn)o(\023)-24 b(etiques)20 b(faibles)206 983 y(et)e(c)n(onstants.)f Fn(Exp)q(os)o(\023)-23 b(e)17 b(No.)f(XI)q(I,)f(S)o(\023)-23 b(eminaire)13 b(EDP)l(,)k(f)o (\023)-23 b(evrier)15 b(1989,)i(Ecole)f(P)o(olytec)o(hnique.)129 1082 y([10])24 b(P)l(.D.)34 b(Hislop,)39 b(I.M.)33 b(Sigal:)58 b Ff(Intr)n(o)n(duction)35 b(to)f(sp)n(e)n(ctr)n(al)g(the)n(ory)g(with) h(applic)n(ations)f(to)206 1142 y(Schr\177)-25 b(odinger)18 b(op)n(er)n(ators.)d Fn(Applied)g(Mathematical)f(Sciences)h(13,)h (Springer)g(V)l(erlag)g(1996.)129 1241 y([11])24 b(L.)g(H\177)-24 b(ormander:)35 b Ff(The)25 b(analysis)g(of)f(line)n(ar)h(p)n(artial)e (di\013er)n(ential)i(op)n(er)n(ators)e(III.)g Fn(Springer-)206 1301 y(V)l(erlag)16 b(Berlin)e(Heidelb)q(erg)h(1983.)129 1400 y([12])24 b(Th.)19 b(Jec)o(k)o(o:)27 b Ff(Semiclassic)n(al)22 b(r)n(esolvent)f(estimates)g(for)f(Schr\177)-25 b(odinger)21 b(matrix)f(op)n(er)n(ators)f(with)206 1461 y(eigenvalues)h(cr)n (ossing.)p Fn(,)15 b(to)h(app)q(ear)h(in)e(Mathematisc)o(he)e(Nac)o (hric)o(h)o(ten.)g(Preprin)o(t)i(a)o(v)m(ailable)g(on)206 1521 y(01-428)j(at)f Fm(http==r)q(ene:ma:utexas:)o(edu=mp)p 1080 1521 15 2 v 15 w(ar)q(c=index)11 b Fj(\000)g Fn(01)p Fm(:html)q Fn(#)p Fm(end)129 1620 y Fn([13])24 b(A.)10 b(Jensen,)i(E.)f(Mourre,)h(P)l(.)f(P)o(erry:)18 b Ff(Multiple)c(c)n (ommutator)e(estimates)i(and)f(r)n(esolvent)h(smo)n(oth-)206 1680 y(ness)19 b(in)g(quantum)h(sc)n(attering)g(the)n(ory.)c Fn(Ann.)h(Inst.)g(Henri)g(P)o(oincar)o(\023)-23 b(e,)17 b(v)o(ol.)f(41,)j(no)f(2,)g(1984,)h(p.)206 1740 y(207-225.)129 1839 y([14])24 b(M.)14 b(Klein:)20 b Ff(On)d(the)f(absenc)n(e)i(of)e(r) n(esonanc)n(es)h(for)e(Schr\177)-25 b(odinger)17 b(op)n(er)n(ators)e (with)i(non-tr)n(apping)206 1899 y(p)n(otentials)h(in)g(the)g(classic)n (al)g(limit.)f Fn(Comm.)c(Math.)j(Ph)o(ys.)g(106,)h(485-494)i (\(1986\).)129 1998 y([15])24 b(A.)12 b(Martinez:)18 b Ff(R)n(esonanc)n(e)d(fr)n(e)n(e)f(domains)g(for)g(non)h(glob)n(al)r (ly)h(analytic)f(p)n(otentials.)e Fn(Ann.)g(Henri)206 2058 y(P)o(oincar)o(\023)-23 b(e)16 b(4)g(\(2002\))i(739-756.)129 2157 y([16])24 b(E.)19 b(Mourre:)27 b Ff(A)o(bsenc)n(e)21 b(of)f(singular)h(c)n(ontinuous)g(sp)n(e)n(ctrum)f(for)f(c)n(ertain)i (self-adjoint)g(op)n(er)n(a-)206 2218 y(tors.)16 b Fn(Comm)o(un.)d(in)j (Math.)g(Ph)o(ys.)g(78,)g(391-408,)j(1981.)129 2317 y([17])24 b(F.)17 b(Nier:)23 b Ff(A)18 b(semiclassic)n(al)i(pictur)n(e)f(of)f (quantum)i(sc)n(attering.)e Fn(Ann.)f(Sci.)1644 2304 y(\023)1639 2317 y(Ecole)h(Norm.)e(Sup.)206 2377 y(\(4\))g(29)i (\(1996\),)f(no)g(2,)f(149-183.)129 2476 y([18])24 b(M.)19 b(Reed,)g(B.)g(Simon:)27 b Ff(Metho)n(ds)20 b(of)g(Mo)n(dern)g (Mathematic)n(al)h(Physics,)g(T)l(ome)g(II)f(:)h(F)l(ourier)206 2536 y(A)o(nalysis,)c(Self-adjointness.)i Fn(Academic)14 b(Press)i(1979.)129 2635 y([19])24 b(D.)16 b(Rob)q(ert:)22 b Ff(A)o(utour)17 b(de)h(l'appr)n(oximation)f(semi-classique.)h Fn(Birkh\177)-24 b(auser)16 b(1987.)129 2734 y([20])24 b(D.)c(Rob)q(ert,)i(H.)e(T)l(am)o(ura:)29 b Ff(Semiclassic)n(al)23 b(estimates)g(for)d(r)n(esolvents)j(and)f(asymptotics)f(for)206 2794 y(total)d(cr)n(oss-se)n(ction.)f Fn(Ann.)e(IHP)h(V)l(ol.)f(46,)i (1987,)g(p.)f(415-442.)129 2893 y([21])24 b(I.M.)14 b(Sigal,)i(A.)f (So\013er:)22 b Ff(The)17 b Fm(N)5 b Ff(-p)n(article)18 b(sc)n(attering)h(pr)n(oblem:)j(asymtotic)17 b(c)n(ompleteness)i(for) 206 2953 y(short-r)n(ange)e(quantum)i(systems.)d Fn(Ann.)g(of)g(Math.)g (125)i(\(1987\))f(35-108.)p eop %%Page: 10 10 10 9 bop 129 133 a Fl(Non-trapping)17 b(b)q(eha)o(viour,)f(21-11-02) 1123 b Fn(10)129 286 y([22])24 b(A.)15 b(V)l(asy)l(,)h(M.)g(Zw)o (orski:)21 b Ff(Semiclassic)n(al)e(estimates)f(in)g(asymptotic)n(al)r (ly)g(euclide)n(an)h(sc)n(attering.)206 347 y Fn(Comm.)14 b(Math.)i(Ph)o(ys.)f(212,)i(no)g(1,)f(205-217)j(\(2000\).)129 448 y([23])24 b(X.P)l(.)18 b(W)l(ang:)28 b Ff(Semiclassic)n(al)22 b(r)n(esolvent)g(estimates)f(for)e Fm(N)5 b Ff(-b)n(o)n(dy)20 b(Schr\177)-25 b(odinger)21 b(op)n(er)n(ators.)d Fn(J.)206 509 y(F)l(unct.)d(Anal.)h(97,)g(466-483)j(\(1991\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0211210308405--