Content-Type: multipart/mixed; boundary="-------------0011090452633" This is a multi-part message in MIME format. ---------------0011090452633 Content-Type: text/plain; name="00-443.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-443.keywords" Localization, dynamical, pure point ---------------0011090452633 Content-Type: application/postscript; name="dls.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dls.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: DLS.DVI %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips DLS.DVI -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.11.02:1520 %%BeginProcSet: texc.pro %! /TeXDict 300 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 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y Fl(p=d)1662 1790 y Fj(\024)28 b Fn(d)1818 1805 y Fl(k)1861 1790 y Fo(\()p Fn(p)p Fo(\))f Fj(\024)h Fn(C)2188 1805 y Fm(2)2228 1790 y Fn(k)2282 1749 y Fl(p=d)2393 1790 y Fn(:)0 1963 y Fb(The)h(simplest)f(example)g(is)h(the)g(c)-5 b(anonic)g(al)28 b(b)-5 b(asis)29 b(of)g Fn(l)1979 1927 y Fm(2)2018 1963 y Fo(\()p Fp(Z)2124 1927 y Fl(d)2165 1963 y Fo(\))g Fb(or,)i(mor)-5 b(e)28 b(gener)-5 b(al)5 b(ly,)30 b(c)-5 b(omplete)28 b(systems)0 2084 y(with)35 b(SULE)g([5],)f(wher)-5 b(e)35 b(the)f(functions)h Fn(e)1581 2099 y Fl(k)1624 2084 y Fo(\()p Fn(n)p Fo(\))g Fb(ar)-5 b(e)34 b(wel)5 b(l)35 b(lo)-5 b(c)g(alise)g(d)33 b(and)h(fast)h(de)-5 b(c)g(aying)34 b(at)h(in\014nity.)146 2285 y Fo(One)40 b(can)g(observ)m(e)i(that)e(if) e(the)i(system)h(is)e(not)h(complete,)h(then)f Fn(d)2735 2300 y Fl(k)2778 2285 y Fo(\()p Fn(p)p Fo(\))f(can)h(gro)m(w)g(as)g (fast)g(as)0 2405 y(y)m(ou)32 b(will)d(\(taking,)i(for)g(example,)h Fn(e)1333 2420 y Fl(k)1403 2405 y Fo(=)c Fn(\016)1550 2421 y Fl(m)p Fm(\()p Fl(k)r Fm(\))1741 2405 y Fo(with)k(fast)f(gro)m (wing)g Fj(j)p Fn(m)p Fo(\()p Fn(k)s Fo(\))p Fj(j)p Fo(\).)42 b(Ev)m(en)34 b(if)c(the)i(system)g(is)0 2525 y(complete)j(but)h(the)g (functions)f Fn(e)1236 2540 y Fl(k)1315 2525 y Fo(deca)m(y)i(not)e(to)s (o)g(fast)g(at)h(in\014nit)m(y)-8 b(,)35 b(it)g(is)g(p)s(ossible)g (that)g Fn(d)3447 2540 y Fl(k)3490 2525 y Fo(\()p Fn(p)p Fo(\))g(are)0 2646 y(fast)29 b(gro)m(wing)f(\(in)g(particular,)g Fn(d)1219 2661 y Fl(k)1261 2646 y Fo(\()p Fn(p)p Fo(\))g(=)g(+)p Fj(1)g Fo(for)g(an)m(y)i Fn(k)s Fo(\).)42 b(An)29 b(in)m(teresting)g (problem:)40 b(is)29 b(it)f(p)s(ossible)0 2766 y(that)k Fn(d)262 2781 y Fl(k)304 2766 y Fo(\()p Fn(p)p Fo(\))g(gro)m(w)g (faster)h(than)f Fn(k)1246 2730 y Fl(p=d)1389 2766 y Fo(for)f(some)h(complete)g(systems)h(where)h(all)c(the)i(functions)g Fn(e)3603 2781 y Fl(k)3646 2766 y Fo(\()p Fn(n)p Fo(\))0 2887 y(deca)m(y)i(fast)e(\(for)g(example,)h(exp)s(onen)m(tially\))e(as) i Fj(j)p Fn(n)p Fj(j)27 b(!)g Fo(+)p Fj(1)p Fo(?)0 3219 y Fk(3)165 b(Lo)5 b(calization)66 b(for)c(a)f(giv)-5 b(en)62 b(v)-5 b(ector:)86 b(necessary)61 b(condi-)247 3402 y(tions)0 3621 y Fo(Let)35 b Fn(H)41 b Fo(b)s(e)35 b(a)f(self-adjoin)m(t)f(op)s(erator)g(in)h Fj(H)e Fo(=)e Fn(l)1784 3584 y Fm(2)1824 3621 y Fo(\()p Fp(Z)1930 3584 y Fl(d)1971 3621 y Fo(\),)69 b Fn( )34 b Fj(2)d(H)q Fn(;)51 b Fj(k)p Fn( )t Fj(k)31 b Fo(=)f(1,)69 b Fn( )t Fo(\()p Fn(t)p Fo(\))31 b(=)f(exp)r(\()p Fj(\000)p Fn(itH)8 b Fo(\))p Fn( )t Fo(.)0 3741 y(F)-8 b(or)32 b(an)m(y)h Fn(p)28 b(>)f Fo(0)p Fn(;)17 b(t)28 b Fj(2)g Fp(R)k Fo(de\014ne)i(the)f (momen)m(ts)f(of)g(the)h(p)s(osition)e(op)s(erator)1172 3958 y Fj(j)p Fn(X)8 b Fj(j)1317 3911 y Fl(p)1317 3983 y( )1369 3958 y Fo(\()p Fn(t)p Fo(\))28 b(=)1641 3875 y Fi(X)1611 4067 y Fl(n)p Fh(2)p Fg(Z)1755 4048 y Ff(d)1808 3958 y Fj(j)p Fn( )t Fo(\()p Fn(t;)17 b(n)p Fo(\))p Fj(j)2144 3917 y Fm(2)2183 3958 y Fo(\()p Fj(j)p Fn(n)p Fj(j)k Fo(+)h(1\))2541 3917 y Fl(p)2581 3958 y Fn(:)0 4259 y Fo(W)-8 b(e)44 b(prefer)g(to)f(tak)m(e)h(\()p Fj(j)p Fn(n)p Fj(j)30 b Fo(+)f(1\))1195 4223 y Fl(p)1278 4259 y Fo(rather)44 b(than)f Fj(j)p Fn(n)p Fj(j)1934 4223 y Fl(p)2017 4259 y Fo(to)g(a)m(v)m(oid)g(some)h(tec)m(hnical)f (problems)g(in)g(the)0 4380 y(pro)s(ofs.)0 4605 y Fp(De\014nition)36 b(3.1)49 b Fd(One)33 b(has)g(dynamical)e(lo)s(calization)d(for)k Fn( )37 b Fd(and)c(the)g(momen)m(t)e(of)h(order)h Fn(p)p Fd(,)g(if)1487 4822 y Fo(sup)1548 4896 y Fl(t)1651 4822 y Fj(j)p Fn(X)8 b Fj(j)1796 4774 y Fl(p)1796 4847 y( )1848 4822 y Fo(\()p Fn(t)p Fo(\))27 b Fn(<)h Fo(+)p Fj(1)p Fn(:)0 5084 y Fd(W)-8 b(e)33 b(shall)e(denote)i(it)f(as)h Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))p Fd(.)146 5204 y(One)33 b(has)g(Cesaro)h(dynamical)c(lo)s(calization)f Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))32 b Fd(if)924 5470 y Fo(sup)972 5549 y Fl(T)1071 5470 y Fj(hj)p Fn(X)8 b Fj(j)1255 5423 y Fl(p)1255 5495 y( )1307 5470 y Fj(i)p Fo(\()p Fn(T)14 b Fo(\))27 b Fj(\021)h Fo(sup)1673 5549 y Fl(T)1788 5470 y Fo(1)p Fn(=T)1973 5353 y Fi(Z)2056 5379 y Fl(T)2019 5542 y Fm(0)2128 5470 y Fj(j)p Fn(X)8 b Fj(j)2273 5423 y Fl(p)2273 5495 y( )2325 5470 y Fo(\()p Fn(t)p Fo(\))p Fn(dt)27 b(<)h Fo(+)p Fj(1)p Fn(:)1866 5806 y Fo(7)p eop %%Page: 8 8 8 7 bop 0 390 a Fo(Clearly)-8 b(,)32 b Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))27 b Fj(\))g Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\).)0 580 y Fp(De\014nition)36 b(3.2)49 b Fd(One)d(has)f(dynamical)e(lo)s(calization)e(\(Cesaro)46 b(dynamical)d(lo)s(calization\))e(for)k Fn( )0 700 y Fd(if)39 b Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))40 b Fd(\(resp)s(ectiv)m(ely)-8 b(,)43 b Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))p Fd(\))40 b(holds)f(for)h(an)m(y)h Fn(p)g(>)f Fo(0)p Fd(.)66 b(W)-8 b(e)41 b(shall)e(write)h Fn(D)s(L)p Fo(\()p Fn( )t Fo(\))g Fd(and)0 820 y Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t Fo(\))32 b Fd(resp)s(ectiv)m(ely)-8 b(.)0 1010 y Fo(Again,)32 b Fn(D)s(L)p Fo(\()p Fn( )t Fo(\))27 b Fj(\))g Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t Fo(\).)146 1130 y(Let)29 b Fj(H)401 1145 y Fl(c)464 1130 y Fo(b)s(e)g(the)f (subspace)j(of)d(con)m(tin)m(uum)g(sp)s(ectrum)h(of)f Fn(H)35 b Fo(and)29 b Fn(P)2639 1145 y Fl(c)2702 1130 y Fo(the)g(orthogonal)d(pro)5 b(jection)0 1251 y(on)38 b(it.)59 b(It)38 b(is)g(w)m(ell)f(kno)m(wn)i(that)f(if)f Fn(P)1400 1266 y Fl(c)1435 1251 y Fn( )k Fj(6)p Fo(=)36 b(0,)k(then)e Fj(j)p Fn(X)8 b Fj(j)2139 1203 y Fl(p)2139 1276 y( )2191 1251 y Fo(\()p Fn(t)p Fo(\))37 b Fj(!)g Fo(+)p Fj(1)g Fo(as)i Fn(t)e Fj(!)f(1)i Fo(for)g(an)m(y)g Fn(p)f(>)g Fo(0.)0 1371 y(So,)k(the)f(dynamical)d(lo)s(calization)e(is) k(p)s(ossible)g(only)f(if)h Fn( )k Fj(2)c(H)2436 1386 y Fl(pp)2551 1371 y Fo(-)f(the)i(subspace)h(of)e(pure)h(p)s(oin)m(t)0 1491 y(sp)s(ectrum)33 b(of)e Fn(H)8 b Fo(.)43 b(W)-8 b(e)33 b(shall)e(denote)i(b)m(y)g Fn(\025)f Fo(the)h(eigen)m(v)-5 b(alues)32 b(of)g Fn(H)39 b Fo(and)33 b(b)m(y)g Fj(H)2941 1506 y Fl(\025)3018 1491 y Fo(the)g(corresp)s(onding)0 1612 y(eigenspaces:)1432 1732 y Fj(H)1516 1747 y Fl(\025)1589 1732 y Fo(=)28 b Fj(f)p Fn(')p Fj(j)k Fn(H)8 b(')27 b Fo(=)g Fn(\025')p Fj(g)p Fn(:)0 1892 y Fo(Clearly)-8 b(,)35 b(the)g(subspaces)i Fj(H)1063 1907 y Fl(\025)1143 1892 y Fo(and)e Fj(H)1419 1907 y Fl(\026)1501 1892 y Fo(are)g(m)m(utually)e(orthogonal)g(for)h Fn(\025)e Fj(6)p Fo(=)f Fn(\026)j Fo(and)h Fj(H)3286 1907 y Fl(pp)3393 1892 y Fo(=)d Fj(\010)3578 1907 y Fl(\025)3623 1892 y Fj(H)3707 1907 y Fl(\025)3753 1892 y Fo(.)0 2012 y(W)-8 b(e)38 b(shall)e(denote)j(b)m(y)f Fn(P)929 2027 y Fl(\025)1012 2012 y Fo(orthogonal)e(pro)5 b(jection)38 b(on)f Fj(H)2198 2027 y Fl(\025)2244 2012 y Fo(.)59 b(F)-8 b(or)36 b(an)m(y)j Fn( )h Fj(2)d(H)2989 2027 y Fl(pp)3102 2012 y Fo(consider)h(the)g(set)0 2133 y(\(at)32 b(most)g(coun)m(table\))1428 2253 y Fn(I)8 b Fo(\()p Fn( )t Fo(\))27 b(=)h Fj(f)p Fn(\025)p Fj(j)k Fn(P)1983 2268 y Fl(\025)2028 2253 y Fn( )g Fj(6)p Fo(=)27 b(0)p Fj(g)p Fn(:)0 2413 y Fo(Then)34 b Fn( )i Fo(can)d(b)s(e)g (written)f(as)989 2663 y Fn( )g Fo(=)1189 2555 y Fm(+)p Fh(1)1192 2580 y Fi(X)1187 2764 y Fl(k)r Fm(=1)1333 2663 y Fn( )1396 2678 y Fl(k)1439 2663 y Fn(;)49 b( )1578 2678 y Fl(k)1649 2663 y Fj(6)p Fo(=)27 b(0)p Fn(;)49 b( )1940 2678 y Fl(k)2011 2663 y Fj(2)28 b(H)2189 2678 y Fl(\025)2230 2690 y Ff(k)2273 2663 y Fn(;)49 b(\025)2406 2678 y Fl(k)2476 2663 y Fj(2)28 b Fn(I)8 b Fo(\()p Fn( )t Fo(\))p Fn(;)0 2921 y Fo(where)33 b Fn(\025)338 2936 y Fl(k)408 2921 y Fj(6)p Fo(=)28 b Fn(\025)569 2936 y Fl(s)637 2921 y Fo(for)j Fn(k)g Fj(6)p Fo(=)c Fn(s)p Fo(.)43 b(\(It)32 b(is)f(p)s(ossible)g(that)h(the)g(sum)f(is)h (\014nite,)f(in)g(this)g(case)i(the)f(problem)e(of)0 3041 y(dynamical)h(lo)s(calization)d(for)k(the)h(v)m(ector)h Fn( )j Fo(is)32 b(rather)g(trivial\).)41 b(F)-8 b(or)32 b(an)m(y)h Fn(k)j Fo(de\014ne)1564 3227 y Fn(e)1609 3242 y Fl(k)1679 3227 y Fo(=)28 b Fj(k)p Fn( )1896 3242 y Fl(k)1939 3227 y Fj(k)1989 3186 y Fh(\000)p Fm(1)2083 3227 y Fn( )2146 3242 y Fl(k)2189 3227 y Fn(:)0 3413 y Fo(As)h Fj(H)224 3428 y Fl(\025)265 3440 y Ff(k)335 3413 y Fo(and)f Fj(H)604 3428 y Fl(\025)645 3436 y Ff(s)711 3413 y Fo(are)g(m)m(utually)f(orthogonal)f(for)i Fn(k)i Fj(6)p Fo(=)e Fn(s)p Fo(,)h(the)f(system)h Fj(M)p Fo(\()p Fn( )t Fo(\))f Fj(\021)g(f)p Fn(e)3165 3428 y Fl(k)3207 3413 y Fj(g)g Fo(is)g(orthonor-)0 3534 y(mal)j(in)g Fj(H)387 3549 y Fl(pp)463 3534 y Fo(.)43 b(Finally)-8 b(,)30 b(an)m(y)j Fn( )f Fj(2)c(H)1341 3549 y Fl(pp)1449 3534 y Fo(can)33 b(b)s(e)g(written)f(as)1619 3720 y Fn( )f Fo(=)1817 3637 y Fi(X)1857 3821 y Fl(k)1953 3720 y Fn(\015)2004 3735 y Fl(k)2047 3720 y Fn(e)2092 3735 y Fl(k)2134 3720 y Fn(;)0 3978 y Fo(where)49 b Fn(\015)348 3993 y Fl(k)443 3978 y Fo(=)k Fj(h)p Fn( )t(;)17 b(e)767 3993 y Fl(k)810 3978 y Fj(i)p Fn(;)63 b(H)8 b(e)1073 3993 y Fl(k)1169 3978 y Fo(=)53 b Fn(\025)1355 3993 y Fl(k)1398 3978 y Fn(e)1443 3993 y Fl(k)1533 3978 y Fo(and)48 b Fj(M)p Fo(\()p Fn( )t Fo(\))k(=)h Fj(f)p Fn(e)2277 3993 y Fl(k)2320 3978 y Fj(g)47 b Fo(is)g(some)h(orthonormal)d(system)j(of)0 4099 y(eigenfunctions)32 b(of)g Fn(H)40 b Fo(dep)s(ending)33 b(on)g Fn( )t Fo(.)43 b(It)33 b(is)f(ob)m(vious)h(that)1328 4285 y Fn( )t Fo(\()p Fn(t)p Fo(\))27 b(=)1637 4202 y Fi(X)1678 4386 y Fl(k)1774 4285 y Fo(exp)q(\()p Fj(\000)p Fn(it\025)2163 4300 y Fl(k)2206 4285 y Fo(\))p Fn(\015)2295 4300 y Fl(k)2337 4285 y Fn(e)2382 4300 y Fl(k)2425 4285 y Fn(:)1127 b Fo(\(3)p Fn(:)p Fo(1\))0 4538 y(One)33 b(can)g(also)e(note)i(that)g(the)g(sp)s(ectral)f(measure)h(of)f Fn( )k Fo(is)d(equal)f(to)1577 4724 y Fn(\026)1636 4739 y Fl( )1716 4724 y Fo(=)1819 4641 y Fi(X)1860 4825 y Fl(k)1956 4724 y Fn(a)2007 4739 y Fl(k)2050 4724 y Fn(\016)2093 4739 y Fl(\025)2134 4751 y Ff(k)2176 4724 y Fn(;)1376 b Fo(\(3)p Fn(:)p Fo(2\))0 4986 y(where)34 b Fn(a)333 5001 y Fl(k)403 4986 y Fo(=)28 b Fj(j)p Fn(\015)586 5001 y Fl(k)628 4986 y Fj(j)656 4950 y Fm(2)723 4986 y Fo(=)f Fj(k)p Fn( )939 5001 y Fl(k)982 4986 y Fj(k)1032 4950 y Fm(2)1099 4986 y Fn(>)h Fo(0)p Fn(;)1328 4920 y Fi(P)1415 5007 y Fl(k)1474 4986 y Fn(a)1525 5001 y Fl(k)1596 4986 y Fo(=)g Fj(k)p Fn( )t Fj(k)1867 4950 y Fm(2)1933 4986 y Fo(=)g(1.)146 5107 y(Let)33 b Fn(d)372 5122 y Fl(k)414 5107 y Fo(\()p Fn(p)p Fo(\))g(b)s(e)g(the)g(momen)m(ts)f(of)g(the)h (functions)f Fn(e)2032 5122 y Fl(k)2075 5107 y Fo(:)1246 5293 y Fn(d)1297 5308 y Fl(k)1339 5293 y Fo(\()p Fn(p)p Fo(\))c(=)1625 5210 y Fi(X)1595 5402 y Fl(n)p Fh(2)p Fg(Z)1739 5383 y Ff(d)1792 5293 y Fj(j)p Fn(e)1865 5308 y Fl(k)1908 5293 y Fo(\()p Fn(n)p Fo(\))p Fj(j)2070 5252 y Fm(2)2109 5293 y Fo(\()p Fj(j)p Fn(n)p Fj(j)21 b Fo(+)i(1\))2468 5252 y Fl(p)2507 5293 y Fn(:)146 5557 y Fo(The)34 b(\014rst)f(result)f (w)m(e)i(shall)d(pro)m(v)m(e)j(is)e(a)g(necessary)j(condition)c(for)i (dynamical)d(lo)s(calization.)1866 5806 y(8)p eop %%Page: 9 9 9 8 bop 0 390 a Fp(Theorem)37 b(3.3)49 b Fd(1.)43 b(F)-8 b(or)32 b(an)m(y)h Fn(p)28 b(>)f Fo(0)p Fd(,)1249 645 y Fo(lim)17 b(inf)1288 704 y Fl(T)10 b Fh(!1)1520 645 y Fj(hj)p Fn(X)e Fj(j)1704 597 y Fl(p)1704 670 y( )1755 645 y Fj(i)p Fo(\()p Fn(T)14 b Fo(\))27 b Fj(\025)2102 537 y Fh(1)2078 562 y Fi(X)2073 746 y Fl(k)r Fm(=1)2219 645 y Fn(a)2270 660 y Fl(k)2312 645 y Fn(d)2363 660 y Fl(k)2406 645 y Fo(\()p Fn(p)p Fo(\))0 919 y Fd(\(with)32 b(the)h(con)m(v)m(en)m(tion)h(that)e(if)g Fn(d)1268 934 y Fl(k)1310 919 y Fo(\()p Fn(p)p Fo(\))c(=)f(+)p Fj(1)p Fd(,)32 b(for)g(some)h Fn(k)s Fd(,)f(then)2531 853 y Fi(P)2618 940 y Fl(k)2688 919 y Fo(=)c(+)p Fj(1)p Fd(\).)146 1040 y(2.)44 b Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))27 b Fj(\))g Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))27 b Fj(\))h Fn(d)1475 1055 y Fl(k)1517 1040 y Fo(\()p Fn(p)p Fo(\))g Fn(<)f Fo(+)p Fj(1)32 b Fd(for)g(an)m(y)h Fn(k)j Fd(and)2591 973 y Fi(P)2678 1060 y Fl(k)2738 1040 y Fn(a)2789 1055 y Fl(k)2831 1040 y Fn(d)2882 1055 y Fl(k)2925 1040 y Fo(\()p Fn(p)p Fo(\))27 b Fn(<)h Fo(+)p Fj(1)p Fd(.)0 1228 y Fc(Pro)s(of.)53 b Fo(It)33 b(follo)m(ws)e(from)g (\(3.1\))h(that)775 1488 y Fj(j)p Fn( )t Fo(\()p Fn(t;)17 b(n)p Fo(\))p Fj(j)1111 1447 y Fm(2)1177 1488 y Fo(=)1351 1380 y Fh(1)1326 1405 y 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b(;)17 b(T)d Fo(\))27 b Fj(\021)i Fo(1)p Fn(=T)848 2384 y Fi(Z)931 2411 y Fl(T)894 2573 y Fm(0)1002 2501 y Fn(dt)1145 2418 y Fi(X)1105 2607 y Fh(j)p Fl(n)p Fh(j\024)p Fl(N)1321 2501 y Fj(j)p Fn( )t Fo(\()p Fn(t;)17 b(n)p Fo(\))p Fj(j)1657 2460 y Fm(2)1696 2501 y Fo(\()p Fj(j)p Fn(n)p Fj(j)22 b Fo(+)g(1\))2055 2460 y Fl(p)2122 2501 y Fo(=)2295 2393 y Fh(1)2271 2418 y Fi(X)2225 2603 y Fl(k)r(;m)p Fm(=1)2453 2501 y Fn(b)2494 2516 y Fl(k)r(m)2599 2501 y Fo(\()p Fn(T)14 b Fo(\))p Fn(\015)2797 2516 y Fl(k)p 2839 2449 118 4 v 2839 2501 a Fn(\015)2890 2516 y Fl(m)2956 2501 y Fn(d)3007 2516 y Fl(k)r(m)3112 2501 y Fo(\()p Fn(p;)j(N)10 b Fo(\))p Fn(;)183 b Fo(\(3)p Fn(:)p Fo(4\))0 2813 y(where)41 b Fn(d)340 2828 y Fl(k)r(m)445 2813 y Fo(\()p Fn(p;)17 b(N)10 b Fo(\))39 b(=)857 2747 y Fi(P)944 2834 y Fh(j)p Fl(n)p Fh(j\024)p Fl(N)1165 2813 y Fn(e)1210 2828 y Fl(k)1253 2813 y Fo(\()p Fn(n)p Fo(\))p 1387 2728 246 4 v Fn(e)1432 2828 y Fl(m)1499 2813 y Fo(\()p Fn(n)p Fo(\)\()p Fj(j)p Fn(n)p Fj(j)26 b Fo(+)h(1\))2001 2777 y Fl(p)2040 2813 y Fo(,)81 b Fn(b)2189 2828 y Fl(k)r(m)2295 2813 y Fo(\()p Fn(T)14 b Fo(\))39 b(=)g(1)h(for)f Fn(k)j Fo(=)e Fn(m)g Fo(and)f Fn(b)3412 2828 y Fl(k)r(m)3518 2813 y Fo(\()p Fn(T)14 b Fo(\))39 b(=)0 2933 y(\(exp)q(\()p Fj(\000)p Fn(iT)14 b Fo(\()p Fn(\025)501 2948 y Fl(k)569 2933 y Fj(\000)26 b Fn(\025)729 2948 y Fl(m)796 2933 y Fo(\)\))f Fj(\000)h Fo(1\))p Fn(=)p Fo(\()p Fj(\000)p Fn(iT)14 b Fo(\()p Fn(\025)1450 2948 y Fl(k)1517 2933 y Fj(\000)26 b Fn(\025)1677 2948 y Fl(m)1744 2933 y Fo(\))37 b(for)g Fn(k)h Fj(6)p Fo(=)e Fn(m)p Fo(.)58 b(As)38 b Fn(A)p Fo(\()p Fn(N)5 b(;)17 b(T)d Fo(\))35 b Fj(\024)h(hj)p Fn(X)8 b Fj(j)3172 2886 y Fl(p)3172 2958 y( )3224 2933 y Fj(i)p Fo(\()p Fn(T)14 b Fo(\),)38 b(for)e(an)m(y)0 3054 y Fn(N)i(>)28 b Fo(0)k(w)m(e)i(ha)m(v)m(e)g(the)f (inequalit)m(y)977 3256 y(liminf)1230 3271 y Fl(T)10 b Fh(!1)1426 3256 y Fj(hj)p Fn(X)e Fj(j)1610 3209 y Fl(p)1610 3281 y( )1662 3256 y Fj(i)p Fo(\()p Fn(T)14 b Fo(\))27 b Fj(\025)h Fo(liminf)2234 3271 y Fl(T)10 b Fh(!1)2430 3256 y Fn(A)p Fo(\()p Fn(N)5 b(;)17 b(T)d Fo(\))p Fn(:)775 b Fo(\(3)p Fn(:)p Fo(5\))0 3458 y(On)31 b(the)g(other)g(hand,)g(due)h (to)e(the)h(dominated)f(con)m(v)m(ergence)j(theorem,)e(one)g(can)g(tak) m(e)g(the)h(limit)27 b(in)0 3579 y(\(3.4\))32 b(for)g(a)g(\014xed)i Fn(N)43 b Fo(as)33 b Fn(T)41 b Fj(!)28 b(1)p Fo(:)934 3781 y(lim)905 3840 y Fl(T)10 b Fh(!1)1114 3781 y Fn(A)p Fo(\()p Fn(N)5 b(;)17 b(T)d Fo(\))27 b(=)1592 3698 y Fi(X)1632 3882 y Fl(k)1728 3781 y Fj(j)p Fn(\015)1807 3796 y Fl(k)1849 3781 y Fj(j)1877 3740 y Fm(2)1973 3698 y Fi(X)1933 3886 y Fh(j)p Fl(n)p Fh(j\024)p Fl(N)2149 3781 y Fj(j)p Fn(e)2222 3796 y Fl(k)2265 3781 y Fo(\()p Fn(n)p Fo(\))p Fj(j)2427 3740 y Fm(2)2466 3781 y Fo(\()p Fj(j)p Fn(n)p Fj(j)22 b Fo(+)g(1\))2825 3740 y Fl(p)2864 3781 y Fn(:)688 b Fo(\(3)p Fn(:)p Fo(6\))0 4080 y(As)33 b Fn(a)195 4095 y Fl(k)266 4080 y Fo(=)27 b Fj(j)p Fn(\015)448 4095 y Fl(k)490 4080 y Fj(j)518 4044 y Fm(2)557 4080 y Fo(,)33 b(it)e(follo)m(ws)h(from)f(\(3.5\))h(and)h(\(3.6\))f(that)799 4283 y(liminf)1053 4298 y Fl(T)10 b Fh(!1)1249 4283 y Fj(hj)p Fn(X)e Fj(j)1433 4235 y Fl(p)1433 4308 y( )1484 4283 y Fj(i)p Fo(\()p Fn(T)14 b Fo(\))27 b Fj(\025)1803 4199 y Fi(X)1843 4384 y Fl(k)1939 4283 y Fn(a)1990 4298 y Fl(k)2090 4199 y Fi(X)2050 4388 y Fh(j)p Fl(n)p Fh(j\024)p Fl(N)2266 4283 y Fj(j)p Fn(e)2339 4298 y Fl(k)2382 4283 y Fo(\()p Fn(n)p Fo(\))p Fj(j)2544 4241 y Fm(2)2583 4283 y Fo(\()p Fj(j)p Fn(n)p Fj(j)22 b Fo(+)g(1\))2942 4241 y Fl(p)0 4572 y Fo(for)28 b(an)m(y)h Fn(N)39 b(>)27 b Fo(0.)42 b(T)-8 b(aking)28 b(the)h(limit)c Fn(N)39 b Fj(!)27 b Fo(+)p Fj(1)p Fo(,)i(w)m(e)h(obtain)d(the)i(\014rst)g (statemen)m(t)g(of)f(the)i(Theorem.)0 4692 y(The)k(second)f(statemen)m (t)g(follo)m(ws)f(directly)g(from)f(the)i(\014rst.)146 4813 y(As)44 b(a)e(corollary)f(of)i(Theorem)g(3.3,)i(w)m(e)f(shall)d (pro)m(v)m(e)j(a)f(necessary)i(condition)c(for)i(dynamical)0 4933 y(lo)s(calization)29 b(for)j(v)m(ector)h Fn( )k Fo(in)32 b(terms)g(of)g(its)g(sp)s(ectral)h(measure)f Fn(\026)2496 4948 y Fl( )2581 4933 y Fo(giv)m(en)h(b)m(y)h(\(3.2\).)0 5121 y Fp(Theorem)j(3.4)49 b Fd(The)34 b(follo)m(wing)29 b(statemen)m(ts)34 b(hold:)146 5242 y(1.)44 b(F)-8 b(or)31 b(an)m(y)j Fn(p)27 b(>)h Fo(0)p Fd(,)995 5481 y Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))27 b Fj(\))g Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))27 b Fj(\))2153 5398 y Fi(X)2193 5582 y Fl(k)2289 5481 y Fn(a)2392 5383 y Fe(1)p 2350 5395 115 4 v 2350 5437 a(1+)p Ff(\014)2340 5506 y Fl(k)2506 5481 y Fn(<)h Fo(+)p Fj(1)1866 5806 y Fo(9)p eop %%Page: 10 10 10 9 bop 0 390 a Fd(for)32 b(all)f Fo(0)c Fn(<)h(\014)33 b(<)27 b(p=d)p Fd(.)146 511 y(2.)1135 631 y Fn(D)s(L)p Fo(\()p Fn( )t Fo(\))g Fj(\))h Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t Fo(\))27 b Fj(\))2108 548 y Fi(X)2148 732 y Fl(k)2244 631 y Fn(a)2295 590 y Fl(\027)2295 656 y(k)2366 631 y Fn(<)h Fo(+)p Fj(1)0 872 y Fd(for)k(all)f Fn(\027)j(>)27 b Fo(0)p Fd(.)0 1076 y Fc(Pro)s(of.)53 b Fo(Assume)33 b(that)g Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))32 b(holds.)43 b(Theorem)33 b(3.3)f(implies)1499 1213 y Fi(X)1540 1397 y Fl(k)1635 1296 y Fn(a)1686 1311 y Fl(k)1729 1296 y Fn(d)1780 1311 y Fl(k)1823 1296 y Fo(\()p Fn(p)p Fo(\))27 b Fn(<)h Fo(+)p Fj(1)p Fn(:)0 1588 y Fo(One)g(can)g(apply)f (the)h(Theorem)g(2.2)f(to)h(the)g(orthonormal)d(system)k Fj(f)p Fn(e)2605 1603 y Fl(k)2647 1588 y Fj(g)f Fo(and)f(reorder)i Fn(d)3292 1603 y Fl(k)3334 1588 y Fo(\()p Fn(p)p Fo(\))e(so)h(that)0 1708 y Fn(d)51 1723 y Fl(k)93 1708 y Fo(\()p Fn(p)p Fo(\))g Fj(\025)g Fn(D)s(k)489 1672 y Fl(p=d)600 1708 y Fo(.)44 b(Therefore,)33 b(after)g(reordering,)1525 1845 y Fi(X)1566 2030 y Fl(k)1662 1928 y Fn(a)1713 1943 y Fl(k)1756 1928 y Fn(k)1810 1887 y Fl(p=d)1949 1928 y Fn(<)27 b Fo(+)p Fj(1)p Fn(:)0 2221 y Fo(Let)32 b(0)27 b Fn(<)h(\014)33 b(<)28 b(p=d;)47 b(r)31 b Fo(=)c(1)20 b(+)h Fn(\014)6 b(;)17 b(r)1265 2185 y Fh(0)1315 2221 y Fo(=)27 b(1)20 b(+)h(1)p Fn(=\014)6 b Fo(.)42 b(Applying)31 b(the)h(H\177)-49 b(older)31 b(inequalit)m(y)-8 b(,)31 b(one)h(can)g(easily)0 2341 y(see)i(that)738 2449 y Fi(X)778 2633 y Fl(k)874 2532 y Fn(a)977 2435 y Fe(1)p 935 2447 115 4 v 935 2488 a(1+)p Ff(\014)925 2557 y Fl(k)1092 2532 y Fj(\024)1197 2386 y Fi( )1262 2449 y(X)1303 2633 y Fl(k)1399 2532 y Fn(a)1450 2547 y Fl(k)1493 2532 y Fn(k)1547 2491 y Fl(p=d)1658 2386 y Fi(!)1776 2382 y Fe(1)p 1734 2394 V 1734 2435 a(1+)p Ff(\014)1879 2386 y Fi( )1945 2449 y(X)1985 2633 y Fl(k)2081 2532 y Fn(k)2135 2491 y Fh(\000)p Fl(p=)p Fm(\()p Fl(\014)s(d)p Fm(\))2399 2386 y Fi(!)2465 2409 y Fl(\014)s(=)p Fm(\(1+)p Fl(\014)s Fm(\))2763 2532 y Fn(<)28 b Fo(+)p Fj(1)0 2833 y Fo(The)k(fact)f(that)600 2767 y Fi(P)688 2854 y Fl(k)747 2833 y Fn(a)850 2736 y Fe(1)p 808 2748 V 808 2789 a(1+)p Ff(\014)798 2858 y Fl(k)965 2833 y Fn(<)c Fo(+)p Fj(1)p Fo(,)k(do)s(es)h(not)f(dep)s (end)i(of)d(reordering)h(of)g Fj(f)p Fn(a)2816 2848 y Fl(k)2859 2833 y Fj(g)p Fo(.)43 b(The)32 b(second)g(part)f(of)0 2954 y(the)i(Theorem)g(is)f(ob)m(vious.)0 3157 y Fp(Corollary)k(3.5)49 b Fb(If)36 b Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t Fo(\))36 b Fb(holds,)g(then)h(the)f(numb)-5 b(ers)36 b Fn(a)2303 3172 y Fl(k)2382 3157 y Fb(ar)-5 b(e)37 b(fast)f(de)-5 b(c)g(aying:)47 b(for)37 b(any)f Fn(s)31 b(>)f Fo(0)p Fb(,)0 3277 y(one)k(c)-5 b(an)35 b(r)-5 b(e)g(or)g(der)34 b Fj(f)p Fn(a)797 3292 y Fl(k)840 3277 y Fj(g)h Fb(so)f(that)1591 3398 y Fn(a)1642 3413 y Fl(k)1712 3398 y Fj(\024)29 b Fn(C)7 b Fo(\()p Fn(s)p Fo(\))p Fn(k)2071 3357 y Fh(\000)p Fl(s)2162 3398 y Fn(:)146 3601 y Fo(One)39 b(should)e(stress)j(that)e (the)g(statemen)m(ts)h(of)f(Theorem)g(3.4)f(and)h(Corollary)f(3.5)g (are)h(w)m(eak)m(er)0 3722 y(than)26 b(that)f(of)h(Theorem)g(3.3,)g(b)s (ecause)i(they)e(do)g(not)f(dep)s(end)i(on)f(the)g(system)h Fj(f)p Fn(e)3009 3737 y Fl(k)3052 3722 y Fj(g)p Fo(,)f(and)g (inevitably)-8 b(,)0 3842 y(one)33 b(loses)g(some)g(information)d(ab)s (out)i(the)i(momen)m(ts)e Fn(d)2096 3857 y Fl(k)2139 3842 y Fo(\()p Fn(p)p Fo(\).)44 b(If)33 b Fn(d)2484 3857 y Fl(k)2526 3842 y Fo(\()p Fn(p)p Fo(\))g(gro)m(w)g(v)m(ery)i(fast)e (as)g Fn(k)e Fj(!)d(1)p Fo(,)0 3962 y(it)k(is)g(ev)m(en)i(p)s(ossible)e (that)996 3896 y Fi(P)1084 3983 y Fl(k)1143 3962 y Fn(a)1194 3926 y Fl(\027)1194 3987 y(k)1265 3962 y Fn(<)c Fo(+)p Fj(1)k Fo(for)g(all)e Fn(\027)k(>)28 b Fo(0,)k(but)2334 3896 y Fi(P)2422 3983 y Fl(k)2481 3962 y Fn(a)2532 3977 y Fl(k)2575 3962 y Fn(d)2626 3977 y Fl(k)2668 3962 y Fo(\()p Fn(p)p Fo(\))c(=)f(+)p Fj(1)32 b Fo(for)g(all)f Fn(p)c(>)h Fo(0.)146 4083 y(Finally)-8 b(,)44 b(in)g(this)f(section)h (w)m(e)h(shall)e(giv)m(e)h(necessary)i(conditions)d(for)h Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))43 b(in)g(terms)h(of)0 4203 y(pro)5 b(jections)33 b(of)f Fn( )k Fo(on)d Fj(H)932 4218 y Fl(\025)973 4230 y Ff(k)1015 4203 y Fo(.)0 4407 y Fp(Lemma)k(3.6)49 b Fd(Let)35 b Fj(M)30 b Fo(=)h Fj(f)p Fn(e)1118 4422 y Fl(k)1161 4407 y Fj(g)j Fd(b)s(e)g(an)m(y)i (orthonormal)c(system)j(in)f Fj(H)h Fd(\(in)f(particular,)f(the)i (system)0 4527 y Fj(M)p Fo(\()p Fn( )t Fo(\))p Fd(\),)65 b Fn( )31 b Fj(2)d(H)34 b Fd(De\014ne)f(the)g(follo)m(wing)d(function:) 1057 4747 y Fn(R)1131 4762 y Fl( )r(;)p Fh(M)1287 4747 y Fo(\()p Fn(n)p Fo(\))e(=)g(sup)1607 4826 y Fl(k)1716 4747 y Fj(j)p Fn(\015)1795 4762 y Fl(k)1837 4747 y Fn(e)1882 4762 y Fl(k)1925 4747 y Fo(\()p Fn(n)p Fo(\))p Fj(j)p Fn(;)81 b(\015)2246 4762 y Fl(k)2316 4747 y Fo(=)28 b Fj(h)p Fn( )t(;)17 b(e)2615 4762 y Fl(k)2657 4747 y Fj(i)p Fn(:)0 5012 y Fd(Then)1470 5049 y Fi(X)1440 5241 y Fl(n)p Fh(2)p Fg(Z)1584 5222 y Ff(d)1637 5132 y Fn(R)1712 5091 y Fm(2)1711 5157 y Fl( )r(;)p Fh(M)1867 5132 y Fo(\()p Fn(n)p Fo(\))28 b Fj(\024)g(k)p Fn( )t Fj(k)2301 5091 y Fm(2)1841 5806 y Fo(10)p eop %%Page: 11 11 11 10 bop 0 390 a Fc(Pro)s(of.)53 b Fo(As)33 b(the)g(system)h Fj(f)p Fn(e)1105 405 y Fl(k)1147 390 y Fj(g)f Fo(is)f(orthonormal,)1913 324 y Fi(P)2001 411 y Fl(k)2060 390 y Fj(j)p Fn(\015)2139 405 y Fl(k)2181 390 y Fj(j)2209 354 y Fm(2)2276 390 y Fj(\024)c(k)p Fn( )t Fj(k)2548 354 y Fm(2)2587 390 y Fn(:)k Fo(Therefore,)1107 610 y Fn(S)i Fo(=)1304 527 y Fi(X)1314 711 y Fl(k)r(;n)1441 610 y Fj(j)p Fn(\015)1520 625 y Fl(k)1562 610 y Fn(e)1607 625 y Fl(k)1650 610 y Fo(\()p Fn(n)p Fo(\))p Fj(j)1812 568 y Fm(2)1879 610 y Fo(=)1982 527 y Fi(X)2023 711 y Fl(k)2119 610 y Fj(j)p Fn(\015)2198 625 y Fl(k)2240 610 y Fj(j)2268 568 y Fm(2)2334 610 y Fj(\024)29 b(k)p Fn( )t Fj(k)2607 568 y Fm(2)2646 610 y Fn(:)906 b Fo(\(3)p Fn(:)p Fo(7\))0 909 y(On)33 b(the)g(other)f(hand,)712 1128 y Fn(S)i Fo(=)909 1045 y Fi(X)948 1219 y Fl(n)1046 1045 y Fi(X)1087 1230 y Fl(k)1182 1128 y Fj(j)p Fn(\015)1261 1143 y Fl(k)1303 1128 y Fn(e)1348 1143 y Fl(k)1391 1128 y Fo(\()p Fn(n)p Fo(\))p Fj(j)1553 1087 y Fm(2)1620 1128 y Fj(\025)1725 1045 y Fi(X)1764 1219 y Fl(n)1862 1128 y Fo(sup)1916 1207 y Fl(k)2025 1128 y Fj(j)p Fn(\015)2104 1143 y Fl(k)2146 1128 y Fn(e)2191 1143 y Fl(k)2234 1128 y Fo(\()p Fn(n)p Fo(\))p Fj(j)2396 1087 y Fm(2)2463 1128 y Fo(=)2567 1045 y Fi(X)2605 1219 y Fl(n)2703 1128 y Fn(R)2778 1087 y Fm(2)2777 1153 y Fl( )r(;)p Fh(M)2934 1128 y Fo(\()p Fn(n)p Fo(\))511 b(\(3)p Fn(:)p Fo(8\))0 1420 y(The)34 b(result)e(of)g(the)h(Lemma)e (follo)m(ws)h(from)f(\(3.7\)-\(3.8\).)146 1540 y(The)k(function)e Fn(R)805 1555 y Fl( )r(;)p Fh(M)961 1540 y Fo(\()p Fn(n)p Fo(\))h(\(esp)s(ecially)e(its)h(deca)m(y)i(prop)s(erties\))e(will)e (pla)m(y)i(an)h(imp)s(ortan)m(t)d(role)i(in)0 1661 y(the)g(next)g(part) f(of)g(the)g(pap)s(er.)44 b(Lemma)31 b(3.6)h(implies)d(that)k Fn(R)2309 1676 y Fl( )r(;)p Fh(M)2497 1661 y Fo(alw)m(a)m(ys)g(lies)e (in)h Fn(l)3122 1624 y Fm(2)3162 1661 y Fo(\()p Fp(Z)3268 1624 y Fl(d)3308 1661 y Fo(\).)44 b(W)-8 b(e)32 b(shall)0 1781 y(see)42 b(b)s(elo)m(w)f(that)g(if)f Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))41 b(holds,)i(then)f Fn(R)1794 1796 y Fl( )r(;)p Fh(M)p Fm(\()p Fl( )r Fm(\))2095 1781 y Fo(deca)m(ys)h(faster)e(at)g(in\014nit)m(y)-8 b(.)69 b(On)41 b(the)h(other)0 1901 y(hand,)d(in)d(the)h(next)h (section)g(w)m(e)g(shall)d(pro)m(v)m(e)k(that)d(a)h(su\016cien)m(tly)h (fast)f(deca)m(y)h(of)f Fn(R)3221 1916 y Fl( )r(;)p Fh(M)3415 1901 y Fo(for)f(some)0 2022 y Fj(M)c Fo(implies)e Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\).)0 2249 y Fp(De\014nition)36 b(3.7)49 b Fd(W)-8 b(e)35 b(sa)m(y)g(that)f(a)h(function)f Fn(f)41 b Fo(:)31 b Fp(Z)1951 2213 y Fl(d)2023 2249 y Fj(!)f Fp(C)35 b Fd(\()p Fn(f)41 b Fo(:)31 b Fp(Z)2522 2213 y Fl(d)2594 2249 y Fj(!)f Fp(R)p Fd(\))k(is)g(fast)g(deca)m(ying)h (if)e(for)0 2370 y(an)m(y)g Fn(s)28 b(>)f Fo(0)1320 2490 y(sup)1372 2559 y Fl(n)1483 2490 y Fj(j)p Fn(f)11 b Fo(\()p Fn(n)p Fo(\))p Fj(j)p Fo(\()p Fj(j)p Fn(n)p Fj(j)21 b Fo(+)h(1\))2090 2449 y Fl(s)2154 2490 y Fn(<)28 b Fo(+)p Fj(1)p Fn(:)0 2752 y Fp(Theorem)37 b(3.8)49 b Fd(The)34 b(follo)m(wing)29 b(statemen)m(ts)34 b(hold:)146 2873 y(1.)44 b Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))27 b Fj(\))g Fn(C)7 b(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))27 b Fj(\))1424 2806 y 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Fo(so)36 b(w)m(e)h(obtain)e(the)h(\014rst)h(statemen)m(t) f(of)g(the)g(Theorem.)54 b(The)37 b(second)g(statemen)m(t)g(follo)m(ws) e(directly)0 4587 y(from)c(the)i(\014rst.)0 4919 y Fk(4)165 b(Lo)5 b(calization)72 b(for)67 b(a)h(giv)-5 b(en)68 b(v)-5 b(ector:)98 b(su\016cien)-5 b(t)68 b(condi-)247 5102 y(tions)0 5557 y Fo(In)33 b(this)f(section)h(w)m(e)g(shall)f(giv)m (e)g(some)h(su\016cien)m(t)g(conditions)f(for)g Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\))32 b(and)h Fn(D)s(L)p Fo(\()p Fn( )t Fo(\).)1841 5806 y(11)p eop %%Page: 12 12 12 11 bop 146 390 a Fo(Let)23 b Fj(M)k Fo(=)h Fj(f)p Fn(e)657 405 y Fl(k)700 390 y Fj(g)22 b Fo(b)s(e)h(an)m(y)g (orthonormal)e(system)i(of)f(eigenfunctions)h(of)f Fn(H)30 b Fo(and)23 b Fj(H)3136 405 y Fh(M)3247 390 y Fo(the)g(subspace)0 511 y(of)32 b Fj(H)195 526 y Fl(pp)303 511 y Fo(spanned)i(b)m(y)f Fj(M)p Fo(.)43 b(F)-8 b(or)32 b(an)m(y)h Fn( )f Fj(2)c(H)1640 526 y Fh(M)1728 511 y Fo(,)33 b(w)m(e)h(ha)m(v)m(e)g(the)f(iden)m(tit)m (y)1330 731 y Fn( )f 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Fl(k)1594 4049 y Fo(\()p Fn(p)p Fo(\))p Fn(d)1770 4064 y Fl(m)1836 4049 y Fo(\()p Fn(p)p Fo(\))27 b(=)h(\()2130 3966 y Fi(X)2170 4151 y Fl(k)2266 3944 y Fi(q)p 2349 3944 313 4 v 105 x Fn(a)2400 4064 y Fl(k)2443 4049 y Fn(d)2494 4064 y Fl(k)2536 4049 y Fo(\()p Fn(p)p Fo(\)\))2699 4008 y Fm(2)2739 4049 y Fn(:)146 4303 y Fo(If)43 b(the)g(functions)g Fn(e)908 4318 y Fl(k)994 4303 y Fo(and)g Fn(e)1239 4318 y Fl(m)1349 4303 y Fo(ha)m(v)m(e)h(only)e(small)f(o)m(v)m(erlapping)h (for)g Fn(k)48 b Fj(6)p Fo(=)e Fn(m)p Fo(,)f(one)e(can)h(b)s(etter)0 4424 y(estimate)32 b(the)h(sums)1142 4544 y Fn(d)1193 4559 y Fl(k)r(m)1297 4544 y Fo(\()p Fn(p)p Fo(\))28 b(=)1553 4461 y Fi(X)1592 4635 y Fl(n)1690 4544 y Fj(j)p Fn(e)1763 4559 y Fl(k)1806 4544 y Fo(\()p Fn(n)p Fo(\))p Fn(e)1985 4559 y Fl(m)2051 4544 y Fo(\()p Fn(n)p Fo(\))p Fj(j)p Fo(\()p Fj(j)p Fn(n)p Fj(j)22 b Fo(+)g(1\))2572 4503 y Fl(p)2611 4544 y Fn(:)0 4781 y Fo(If)28 b Fn(d)144 4796 y Fl(k)r(m)248 4781 y Fo(\()p Fn(p)p Fo(\))g(deca)m(y)h(fast)f (when)g Fj(j)p Fn(k)15 b Fj(\000)d Fn(m)p Fj(j)29 b(!)e(1)p Fo(,)h(the)g(sum)g(on)f(the)i(r.h.s)f(of)f(\(4.3\))g(can)h(b)s(e)g(ma)5 b(jorated)27 b(b)m(y)0 4901 y Fn(C)94 4835 y Fi(P)181 4922 y Fl(k)240 4901 y Fn(a)291 4916 y Fl(k)334 4901 y Fn(d)385 4916 y Fl(k)428 4901 y Fo(\()p Fn(p)p Fo(\))k(\(or)g(b)m(y)h (something)e(close)i(to)f(it\).)42 b(In)31 b(this)g(case)i(one)e (obtains)g(a)g(su\016cien)m(t)h(condition)0 5021 y(whic)m(h)k(is)e (close)h(to)g(\(or)g(ev)m(en)i(iden)m(tical)c(with\))i(the)g(necessary) j(condition)c(giv)m(en)h(b)m(y)h(Theorem)f(3.3.)0 5142 y(In)e(particular,)e(this)h(is)g(the)h(case)h(when)f Fj(M)f Fo(is)h(the)g(canonical)e(basis)h(in)g Fn(l)2747 5106 y Fm(2)2787 5142 y Fo(\()p Fp(Z)2893 5106 y Fl(d)2934 5142 y Fo(\).)146 5262 y(The)39 b(su\016cien)m(t)g(condition)e(of)h (Lemma)e(4.1)i(is,)h(ho)m(w)m(ev)m(er,)j(di\016cult)37 b(to)g(apply)h(in)f(the)i(concrete)0 5383 y(cases,)30 b(b)s(ecause)f(one)e(should)g(con)m(trol)g(at)g(the)h(same)f(time)f (the)h(gro)m(wth)h(of)f Fn(d)2799 5398 y Fl(k)2841 5383 y Fo(\()p Fn(p)p Fo(\))g(and)h(the)f(decreasing)1841 5806 y(12)p eop %%Page: 13 13 13 12 bop 0 390 a Fo(of)33 b Fn(a)163 405 y Fl(k)206 390 y Fo(.)45 b(More)33 b(practical)f(su\016cien)m(t)i(condition)e(can) h(b)s(e)h(giv)m(en)f(in)f(terms)h(of)g(the)h(function)e Fn(R)3462 405 y Fl( )r(;)p Fh(M)3619 390 y Fo(\()p Fn(n)p Fo(\),)0 511 y(de\014ned)i(b)m(y)672 631 y Fn(R)746 646 y Fl( )r(;)p Fh(M)902 631 y Fo(\()p Fn(n)p Fo(\))28 b(=)f(sup)1222 710 y Fl(k)1331 631 y Fj(j)p Fn(\015)1410 646 y Fl(k)1452 631 y Fn(e)1497 646 y Fl(k)1540 631 y Fo(\()p Fn(n)p Fo(\))p Fj(j)p Fn(;)81 b(\015)1861 646 y Fl(k)1931 631 y Fo(=)28 b Fj(h)p Fn( )t(;)17 b(e)2230 646 y Fl(k)2272 631 y Fj(i)p Fn(;)2419 548 y Fi(X)2460 732 y Fl(k)2556 631 y Fj(j)p Fn(\015)2635 646 y Fl(k)2677 631 y Fj(j)2705 590 y Fm(2)2772 631 y Fo(=)27 b Fj(k)p Fn( )t Fj(k)3042 590 y Fm(2)3081 631 y Fn(:)0 865 y Fo(Later)32 b(on)h(in)f(this)g (section,)h Fn( )j Fo(and)d Fj(M)f Fo(are)h(\014xed)g(and)g(w)m(e)h (omit)c(them)j(in)e(notations.)146 986 y(T)-8 b(o)33 b(pro)m(v)m(e)h Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\),)32 b(one)h(shall)e(use)j(the)f(trivial)c(b)s(ound)k(\(4.2\):) 1389 1189 y Fj(j)p Fn( )t Fo(\()p Fn(t;)17 b(n)p Fo(\))p Fj(j)27 b(\024)1857 1106 y Fi(X)1898 1291 y Fl(k)1994 1189 y Fj(j)p Fn(\015)2073 1204 y Fl(k)2114 1189 y Fn(e)2159 1204 y Fl(k)2202 1189 y Fo(\()p Fn(n)p Fo(\))p Fj(j)p Fn(:)0 1465 y Fo(Clearly)-8 b(,)30 b(an)m(y)g(term)g(in)f(the)h(sum)g (is)g(ma)5 b(jorated)29 b(b)m(y)i Fn(R)q Fo(\()p Fn(n)p Fo(\),)f(and)g(if)f(the)i(sum)e(has)i(a)e(\014nite)h(n)m(um)m(b)s(er)g (of)0 1586 y(terms,)38 b(the)g(su\016cien)m(tly)g(fast)f(deca)m(y)i(of) d Fn(R)q Fo(\()p Fn(n)p Fo(\))i(implies)d Fn(D)s(L)p Fo(\()p Fn( )t(;)17 b(p)p Fo(\).)56 b(Ho)m(w)m(ev)m(er,)41 b(t)m(ypically)36 b(it)g(is)h(not)0 1706 y(the)f(case,)i(and)f(one)f (needs)i(some)d(deca)m(y)j(in)d Fn(k)k Fo(so)e(that)e(the)i(sum)f(con)m (v)m(erge.)56 b(The)37 b(k)m(ey)g(momen)m(t)e(is)0 1826 y(the)g(follo)m(wing:)45 b(one)35 b(can)g(sacrify)f(some)h(deca)m(y)h (in)e Fn(n)h Fo(to)f(obtain)g(a)g(necessary)j(deca)m(y)f(in)e Fn(k)s Fo(.)50 b(This)34 b(is)0 1947 y(p)s(ossible)29 b(due)h(to)g(the)g(gro)m(wth)f(of)h(the)g(momen)m(ts)f Fn(d)1888 1962 y Fl(k)1930 1947 y Fo(\()p Fn(p)p Fo(\))h(giv)m(en)f(b)m (y)i(Theorem)f(2.2.)42 b(Surprisingly)-8 b(,)29 b(one)0 2067 y(can)40 b(ev)m(en)i(allo)m(w)c(some)i(gro)m(wth)g(in)f Fn(k)k Fo(in)c(the)i(b)s(ounds)f(for)g Fn(\015)2328 2082 y Fl(k)2370 2067 y Fn(e)2415 2082 y Fl(k)2458 2067 y Fo(\()p Fn(n)p Fo(\).)66 b(Namely)-8 b(,)40 b(supp)s(ose)i(that)d(for)0 2188 y(some)32 b Fn(\013)d(>)e Fo(0)33 b(the)g(momen)m(ts)f Fn(d)1154 2203 y Fl(k)1196 2188 y Fo(\()p Fn(\013)q Fo(\))g(are)h (\014nite)f(for)g(an)m(y)h Fn(k)s Fo(.)44 b(Consider)33 b(the)g(function)1376 2446 y Fn(R)q Fo(\()p Fn(n;)17 b(\013)q Fo(\))27 b(=)h(sup)1877 2526 y Fl(k)1996 2379 y Fj(j)p Fn(\015)2075 2394 y Fl(k)2117 2379 y Fn(e)2162 2394 y Fl(k)2205 2379 y Fo(\()p Fn(n)p Fo(\))p Fj(j)p 1996 2423 371 4 v 2066 2515 a Fn(d)2117 2530 y Fl(k)2159 2515 y Fo(\()p Fn(\013)q Fo(\))2377 2446 y Fn(:)0 2706 y Fo(F)-8 b(or)34 b Fn(\013)d Fo(=)g(0)k(one)f(has)h Fn(d)868 2721 y Fl(k)911 2706 y Fo(\(0\))30 b(=)h(1)k(\(so,)g Fn(d)1495 2721 y Fl(k)1537 2706 y Fo(\()p Fn(\013)q Fo(\))f(are)h(alw)m (a)m(ys)g(\014nite\))f(and)h Fn(R)q Fo(\()p Fn(n;)17 b Fo(0\))34 b(coincides)g(with)h(the)0 2826 y(function)d Fn(R)q Fo(\()p Fn(n)p Fo(\))h(considered)g(ab)s(o)m(v)m(e.)45 b(F)-8 b(or)31 b(an)m(y)j Fn(s)27 b(>)h Fo(0)k(de\014ne)i(the)f(momen)m (ts)1249 3030 y Fn(L)1315 3045 y Fl(\013)1365 3030 y Fo(\()p Fn(s)p Fo(\))27 b(=)1618 2947 y Fi(X)1656 3121 y Fl(n)1754 3030 y Fn(R)1829 2989 y Fm(2)1869 3030 y Fo(\()p Fn(n;)17 b(\013)q Fo(\)\()p Fj(j)p Fn(n)p Fj(j)k Fo(+)h(1\))2468 2989 y Fl(s)2504 3030 y Fn(;)0 3296 y Fo(where)34 b Fn(L)348 3311 y Fl(\013)398 3296 y Fo(\()p Fn(s)p Fo(\))e(dep)s(end)i(also)d(on)i Fn( )t(;)17 b Fj(M)32 b Fo(and)g(it)g(is)g(p)s(ossible)g(that)g Fn(L)2514 3311 y Fl(\013)2564 3296 y Fo(\()p Fn(s)p Fo(\))c(=)f(+)p Fj(1)p Fo(.)0 3485 y Fp(Theorem)37 b(4.2)49 b Fd(Let)31 b Fn( )h Fj(2)c(H)1113 3500 y Fh(M)1201 3485 y Fn(;)48 b(\013)28 b(>)g Fo(0)p Fd(.)42 b(Supp)s(ose)32 b(that)f Fn(d)2230 3500 y Fl(k)2272 3485 y Fo(\()p Fn(\013)q Fo(\))c Fn(<)h Fo(+)p Fj(1)i Fd(for)g(an)m(y)i Fn(k)s Fd(.)43 b(The)32 b(follo)m(wing)0 3606 y(statemen)m(ts)i(hold:)146 3726 y(1.)44 b(Let)32 b Fn(\016)g Fj(2)c Fo(\(0)p Fn(;)17 b Fo(1\))p Fn(;)48 b(")28 b(>)f Fo(0)p Fd(.)43 b(Then)621 3992 y Fo(sup)682 4066 y Fl(t)785 3992 y Fj(j)p Fn( )t Fo(\()p Fn(t;)17 b(n)p Fo(\))p Fj(j)27 b(\024)h Fn(C)7 b Fo(\()p Fn(";)17 b(\016)n(;)g(d)p Fo(\))p Fn(L)1698 3951 y Fl(\016)r(=)p Fm(2)1698 4017 y Fl(\013)1822 3846 y Fi( )1898 3924 y Fo(2)p Fn(\013)23 b Fo(+)f Fn(d)p Fo(\(2)f Fj(\000)i Fn(\016)t Fo(\))p 1898 3969 576 4 v 2162 4060 a Fn(\016)2506 3992 y Fo(+)f Fn(")2650 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y Fl(\013)2607 4582 y Fo(\(2)p Fn(\013)23 b Fo(+)f Fn(p)g Fo(+)g(2)p Fn(d)f Fo(+)h Fn(")p Fo(\))230 b(\(4)p Fn(:)p Fo(5\))146 4848 y Fd(3.)44 b(If)32 b Fn(R)q Fo(\()p Fn(n;)17 b(\013)q Fo(\))32 b Fd(is)g(fast)h(deca)m(ying)g(in)f Fn(n)p Fd(,)g(then)i(so)e(is)g Fo(sup)2216 4871 y Fl(t)2263 4848 y Fj(j)p Fn( )t Fo(\()p Fn(t;)17 b(n)p Fo(\))p Fj(j)31 b Fd(and)66 b Fn(D)s(L)p Fo(\()p Fn( )t Fo(\))32 b Fd(holds.)0 5046 y Fc(Pro)s(of.)53 b Fo(Let)32 b Fn(r)f Fo(=)736 5000 y Fm(2)p Fl(\013)p Fm(+)p Fl(d)p Fm(\(2)p Fh(\000)p Fl(\016)r Fm(\))p 736 5023 351 4 v 895 5081 a Fl(\016)1118 5046 y Fo(+)22 b Fn(")p Fo(.)43 b(If)32 b Fn(L)1495 5061 y Fl(\013)1545 5046 y Fo(\()p Fn(r)s Fo(\))27 b(=)h(+)p Fj(1)p Fo(,)k(the)g(b)s(ound)h(\(4.4\))f(is)f(trivially)f(true)i(with)g (an)m(y)0 5167 y(\014nite)g(constan)m(t)i Fn(C)7 b Fo(.)43 b(Supp)s(ose)34 b(that)e Fn(L)1450 5182 y Fl(\013)1500 5167 y Fo(\()p Fn(r)s Fo(\))27 b Fn(<)h Fo(+)p Fj(1)p Fo(.)146 5287 y(It)33 b(follo)m(ws)e(from)h(de\014nition)f(of)h 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Fn(R)q Fo(\()p Fn(n;)17 b(\013)q Fo(\))1163 b(\(4)p Fn(:)p Fo(7\))0 1122 y(directly)32 b(from)f(de\014nition)h(of)g Fn(R)q Fo(\()p Fn(n;)17 b(\013)q Fo(\).)43 b(Using)32 b(the)h(b)s(ounds)g(\(4.6\)-\(4.7\),)f(one)g(can)h(estimate:)672 1326 y Fj(j)p Fn( )t Fo(\()p Fn(t;)17 b(n)p Fo(\))p Fj(j)27 b(\024)1141 1243 y Fi(X)1181 1428 y Fl(k)1277 1326 y Fj(j)p Fn(\015)1356 1341 y Fl(k)1398 1326 y Fn(e)1443 1341 y Fl(k)1486 1326 y Fo(\()p Fn(n)p Fo(\))p Fj(j)g(\024)1780 1243 y Fi(X)1821 1428 y Fl(k)1900 1326 y Fo(\()p Fn(d)1989 1341 y Fl(k)2032 1326 y Fo(\()p Fn(\013)q Fo(\))p Fn(R)2245 1341 y Fl(\013)2294 1326 y Fo(\()p Fn(n)p Fo(\)\))2466 1285 y Fm(1)p Fh(\000)p Fl(\016)2594 1326 y Fj(j)p Fn(\015)2673 1341 y Fl(k)2715 1326 y Fn(e)2760 1341 y Fl(k)2803 1326 y Fo(\()p Fn(n)p Fo(\))p Fj(j)2965 1285 y Fl(\016)3030 1326 y Fj(\024)988 1629 y Fn(R)1063 1588 y Fm(1)p Fh(\000)p Fl(\016)1191 1629 y Fo(\()p Fn(n;)17 b(\013)q Fo(\))p Fn(L)1498 1588 y Fl(\016)r(=)p Fm(2)1498 1653 y Fl(\013)1606 1629 y Fo(\()p Fn(r)s Fo(\))1746 1546 y Fi(X)1786 1730 y Fl(k)1882 1629 y Fj(j)p Fn(e)1955 1644 y Fl(k)1997 1629 y Fo(\()p Fn(n)p Fo(\))p Fj(j)2159 1588 y Fl(\016)2197 1629 y Fn(d)2248 1644 y Fl(k)2290 1629 y Fo(\()p Fn(\013)q Fo(\))p Fn(d)2480 1578 y Fh(\000)p Fl(\016)r(=)p Fm(2)2480 1654 y Fl(k)2643 1629 y Fo(\()p Fn(r)s Fo(\))p Fn(;)786 b Fo(\(4)p Fn(:)p Fo(8\))0 1869 y(where)37 b(the)g(summation)d(is)i (carried)g(only)g(o)m(v)m(er)h Fn(k)i Fo(suc)m(h)f(that)e Fn(a)2406 1884 y Fl(k)2483 1869 y Fn(>)e Fo(0,)j(so)f Fn(d)2880 1884 y Fl(k)2923 1869 y Fo(\()p Fn(r)s Fo(\))d Fn(<)h Fo(+)p Fj(1)p Fo(.)54 b(Let)36 b Fn(s)e Fo(=)0 1989 y(2)p Fn(=)p Fo(\(2)12 b Fj(\000)g Fn(\016)t Fo(\))p Fn(;)45 b(s)489 1953 y Fh(0)540 1989 y Fo(=)28 b(2)p Fn(=\016)t Fo(.)41 b(Applying)27 b(the)h(H\177)-49 b(older)27 b(inequalit)m(y)-8 b(,)28 b(and)g(using)g(the)g(fact)g(that)3211 1923 y Fi(P)3299 2010 y Fl(k)3358 1989 y Fj(j)p Fn(e)3431 2004 y Fl(k)3474 1989 y Fo(\()p Fn(n)p Fo(\))p Fj(j)3636 1953 y Fm(2)3703 1989 y Fj(\024)0 2110 y Fo(1,)k(one)h(obtains:)587 2373 y Fn(S)g Fj(\021)785 2226 y Fi( )851 2290 y(X)892 2474 y Fl(k)988 2373 y Fj(j)p Fn(e)1061 2388 y Fl(k)1103 2373 y Fo(\()p Fn(n)p Fo(\))p Fj(j)1265 2331 y Fl(\016)1303 2373 y Fn(d)1354 2388 y Fl(k)1396 2373 y Fo(\()p Fn(\013)q Fo(\))p Fn(d)1586 2322 y Fh(\000)p Fl(\016)r(=)p Fm(2)1586 2398 y Fl(k)1749 2373 y Fo(\()p Fn(r)s Fo(\))1872 2226 y Fi(!)1937 2249 y Fl(s)2002 2373 y Fj(\024)2107 2290 y Fi(X)2147 2474 y Fl(k)2243 2373 y Fn(d)2294 2322 y Fm(2)p Fl(=)p Fm(\(2)p Fh(\000)p Fl(\016)r Fm(\))2294 2398 y Fl(k)2547 2373 y Fo(\()p Fn(\013)q Fo(\))p Fn(d)2737 2322 y Fh(\000)p Fl(\016)r(=)p Fm(\(2)p Fh(\000)p Fl(\016)r Fm(\))2737 2398 y Fl(k)3044 2373 y Fo(\()p Fn(r)s Fo(\))p Fn(:)385 b Fo(\(4)p Fn(:)p Fo(9\))146 2649 y(Let)40 b Fn(w)j(<)d(q)t Fo(.)66 b(Applying)39 b(the)h(H\177)-49 b(older)40 b(inequalit)m(y)f(with)g Fn(s)h Fo(=)h Fn(q)t(=w)s(;)17 b(s)2778 2613 y Fh(0)2840 2649 y Fo(=)40 b Fn(q)t(=)p Fo(\()p Fn(q)31 b Fj(\000)c Fn(w)s 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Fo(\()p Fn(q)t Fo(\)\))2897 3096 y Fl(w)r(=q)3022 3137 y Fn(:)482 b Fo(\(4)p Fn(:)p Fo(10\))0 3367 y(The)34 b(b)s(ound)e(\(4.10\))g(with)g Fn(w)e Fo(=)e Fn(\013)q(;)49 b(q)31 b Fo(=)d Fn(r)35 b Fo(and)e(\(4.9\))f(imply)1345 3571 y Fn(S)i Fj(\024)1544 3488 y Fi(X)1585 3673 y Fl(k)1680 3571 y Fn(d)1731 3586 y Fl(k)1774 3571 y Fo(\()p Fn(r)s Fo(\))1897 3530 y Fm(\(2)p Fl(\013=r)r Fh(\000)p Fl(\016)r Fm(\))p Fl(=)p Fm(\(2)p Fh(\000)p Fl(\016)r Fm(\))2408 3571 y Fn(:)1096 b Fo(\(4)p Fn(:)p Fo(11\))0 3848 y(No)m(w)33 b(w)m(e)h(shall)d(use)i(the)g(result)g(of)f(Theorem)h(2.2.)43 b(One)33 b(can)g(reorder)g Fn(d)2692 3863 y Fl(k)2734 3848 y Fo(\()p Fn(r)s Fo(\))f(so)h(that)1473 4052 y Fn(d)1524 4067 y Fl(k)1566 4052 y Fo(\()p Fn(r)s Fo(\))27 b Fj(\025)h Fn(D)s Fo(\()p Fn(r)m(;)17 b(d)p Fo(\))p Fn(k)2171 4010 y Fl(r)r(=d)2280 4052 y Fn(:)1224 b Fo(\(4)p Fn(:)p Fo(12\))0 4256 y(The)34 b(c)m(hoice)e(of)h Fn(r)i Fo(and)d(\(4.9\)-\(4.12\))f (yield)1448 4460 y Fn(S)j Fj(\024)28 b Fn(C)7 b Fo(\()p 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Fo(\()p Fn(r)s Fo(\))2220 525 y Fi( )2285 588 y(X)2326 772 y Fl(k)2422 671 y Fn(d)2473 686 y Fl(k)2515 671 y Fo(\()p Fn(r)s Fo(\))2648 598 y Fe(2)p Ff(\013)p Fe(+)p Ff(p)p Fa(\000)p Ff(r)p 2647 615 228 4 v 2731 656 a Fe(2)p Ff(r)2889 525 y Fi(!)2955 548 y Fm(2)3011 671 y Fn(:)493 b Fo(\(4)p Fn(:)p Fo(15\))0 939 y(Again,)31 b(b)m(y)i(Theorem)g(2.2)f (and)g(the)h(c)m(hoice)f(of)g Fn(r)s Fo(,)g(the)g(sum)h(con)m(v)m (erges)h(and)e(the)h(second)g(statemen)m(t)0 1060 y(of)39 b(the)g(Theorem)h(follo)m(ws)e(directly)h(from)e(\(4.15\).)63 b(The)40 b(third)f(statemen)m(t)h(follo)m(ws)e(directly)g(from)0 1180 y(the)33 b(\014rst.)146 1301 y(As)23 b(a)f(direct)g(application)e (of)i(Theorem)h(4.2)f(w)m(e)h(obtain)e(the)i(necessary)i(and)d (su\016cien)m(t)i(conditions)0 1421 y(for)32 b Fn(D)s(L)p Fo(\()p Fn( )t Fo(\).)0 1609 y Fp(Corollary)k(4.3)49 b Fd(Let)32 b Fn( )g Fj(2)c(H)1137 1624 y Fl(pp)1244 1609 y Fd(and)k Fn(R)q Fo(\()p Fn(n)p Fo(\))c(=)f Fn(R)q Fo(\()p Fn(n;)17 b Fo(0\))32 b 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Fo(\()p Fn(n;)17 b(m)p Fo(\))28 b Fj(\024)g Fn(G)1757 4410 y Fl(\013)1807 4395 y Fo(\()p Fn(n;)17 b(m)p Fo(\))27 b Fj(\024)i Fn(U)2269 4410 y Fl(\013)2318 4395 y Fo(\()p Fn(n;)17 b(m)p Fo(\))p Fn(:)971 b Fo(\(5)p Fn(:)p Fo(1\))0 4563 y(The)50 b(\014rst)f (inequalit)m(y)f(in)g(\(5.1\))g(is)g(ob)m(vious.)92 b(T)-8 b(o)49 b(pro)m(v)m(e)h(the)f(second,)55 b(for)48 b(an)m(y)h Fn(g)59 b Fj(2)c(L)49 b Fo(de\014ne)5 4677 y Fi(e)0 4684 y Fn(g)s Fo(\()p Fn(n)p Fo(\))28 b(=)g(\()p Fn(d)405 4699 y Fl(g)444 4684 y Fo(\()p Fn(\013)q Fo(\)\))621 4647 y Fh(\000)p Fm(1)p Fl(=)p Fm(2)786 4684 y Fn(g)t Fo(\()p Fn(n)p Fo(\),)k(so)h(that)1360 4880 y Fj(j)p Fn(g)t Fo(\()p Fn(n)p Fo(\))p Fn(g)t Fo(\()p Fn(m)p Fo(\))p Fj(j)p 1360 4924 V 1471 5016 a Fn(d)1522 5031 y Fl(g)1561 5016 y Fo(\()p Fn(\013)q Fo(\))1848 4947 y(=)28 b Fj(j)1985 4940 y Fi(e)1980 4947 y Fn(g)s Fo(\()p Fn(n)p Fo(\))2169 4940 y Fi(e)2164 4947 y Fn(g)s Fo(\()p Fn(m)p Fo(\))p Fj(j)p Fn(:)0 5206 y Fo(One)33 b(v)m(eri\014es)h(that)1375 5244 y Fi(X)1413 5418 y Fl(n)1511 5327 y Fj(j)1544 5320 y Fi(e)1539 5327 y Fn(g)s Fo(\()p Fn(n)p Fo(\))p Fj(j)1751 5286 y Fm(2)1790 5327 y Fo(\()p Fj(j)p Fn(n)p Fj(j)22 b Fo(+)g(1\))2149 5286 y Fl(\013)2226 5327 y Fo(=)27 b(1)p Fn(;)0 5557 y Fo(so)125 5550 y Fi(e)120 5557 y Fn(g)k Fj(2)d(K)34 b Fo(and)e(the)h(second)h(inequalit)m(y)e(in)g (\(5.1\))g(holds.)1841 5806 y(16)p eop %%Page: 17 17 17 16 bop 0 390 a Fp(Lemma)37 b(5.2)49 b Fd(Let)33 b Fn(\013)28 b Fj(\025)g Fo(0)p Fn(;)17 b( )32 b Fj(2)c(H)q Fn(;)49 b(')28 b Fo(=)f Fn(P)1660 405 y Fl(I)1700 390 y Fo(\()p Fn(H)8 b Fo(\))p Fn( )t Fd(.)43 b(The)33 b(inequalit)m(y)f (holds:)1131 610 y Fn(R)1205 625 y Fl(';)p Fh(M)1359 610 y Fo(\()p Fn(n;)17 b(\013)q Fo(\))27 b Fj(\024)1772 527 y Fi(X)1732 719 y Fl(m)p Fh(2)p Fg(Z)1895 700 y Ff(d)1948 610 y Fn(N)2026 625 y Fl(\013)2076 610 y Fo(\()p Fn(n;)17 b(m)p Fo(\))p Fj(j)p Fn( )t Fo(\()p Fn(m)p Fo(\))p Fj(j)p Fn(;)929 b Fo(\(5)p Fn(:)p Fo(2\))0 909 y Fd(where)34 b Fn(N)360 924 y Fl(\013)442 909 y Fd(is)e(one)h(of)f(the)h(three)g (functions)g Fn(F)1731 924 y Fl(\013)1780 909 y Fn(;)17 b(G)1901 924 y Fl(\013)1951 909 y Fn(;)g(U)2061 924 y Fl(\013)2110 909 y Fd(.)0 1137 y Fc(Pro)s(of.)53 b Fo(As)33 b Fn(')28 b Fo(=)f Fn(P)776 1152 y Fl(I)816 1137 y Fo(\()p Fn(H)8 b Fo(\))p Fn( )36 b Fo(and)d(the)g(system)g Fj(M)f Fo(is)g(complete)g(in)g Fj(H)q Fo(\()p Fn(I)8 b Fo(\),)1113 1357 y Fn(')28 b Fo(=)1308 1274 y Fi(X)1349 1458 y Fl(k)1445 1357 y Fn(\015)1496 1372 y Fl(k)1538 1357 y Fn(e)1583 1372 y Fl(k)1626 1357 y Fn(;)82 b(\015)1786 1372 y Fl(k)1856 1357 y Fo(=)27 b Fj(h)p Fn(';)17 b(e)2151 1372 y Fl(k)2194 1357 y Fj(i)27 b Fo(=)h Fj(h)p Fn( )t(;)17 b(e)2559 1372 y Fl(k)2601 1357 y Fj(i)p Fn(:)0 1644 y Fo(Therefore,)1471 1764 y Fn(\015)1522 1779 y Fl(k)1592 1764 y Fo(=)1695 1681 y Fi(X)1724 1856 y Fl(m)1832 1764 y Fn( )t Fo(\()p Fn(m)p Fo(\))p 2060 1680 250 4 v Fn(e)2105 1779 y Fl(k)2148 1764 y Fo(\()p Fn(m)p Fo(\))0 1995 y(and)1142 2078 y Fj(j)p Fn(\015)1221 2093 y Fl(k)1262 2078 y 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b(is)0 3993 y(fast)d(deca)m(ying)g(in)f Fn(n)p Fo(.)50 b(The)36 b(third)e(statemen)m(t)h(of)g(Theorem)g(4.2)f(implies) f Fn(D)s(L)p Fo(\()p Fn(')p Fo(\),)i(so)g Fj(B)s Fo(-dynamical)0 4113 y(lo)s(calization)29 b(holds)j(on)g Fn(I)8 b Fo(.)146 4234 y(T)-8 b(o)41 b(pro)m(v)m(e)g(the)g(second)h(statemen)m(t)f(of)e (the)i(Theorem,)i(observ)m(e)f(that)e(since)h(the)g(sp)s(ectrum)f(of)0 4354 y Fn(H)55 b Fo(is)46 b(simple)g(on)h Fn(I)8 b Fo(,)50 b(the)e(system)g Fj(M)e Fo(is)h(unique)g(and)g(coincides)g(with)g(the)g (set)h(of)f(normalised)0 4475 y(eigenfunctions)29 b(of)f Fn(H)37 b Fo(with)29 b(eigen)m(v)-5 b(alues)29 b(from)e Fn(I)8 b Fo(.)43 b(Therefore,)31 b Fn(F)2455 4490 y Fl(\013)2504 4475 y Fo(\()p Fn(n;)17 b(m)p Fo(\))28 b(=)f Fn(G)2975 4490 y Fl(\013)3025 4475 y Fo(\()p Fn(n;)17 b(m)p Fo(\).)42 b(Moreo)m(v)m(er,)0 4595 y(one)33 b(sees)i(easily)d(that)h(for)f(an)m (y)i Fn(')28 b Fj(2)h(H)q Fo(\()p Fn(I)8 b Fo(\),)33 b Fj(M)p Fo(\()p Fn(')p Fo(\))g(is)f(a)h(subset)i(of)d Fj(M)p 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5806 y Fo(17)p eop %%Page: 18 18 18 17 bop 0 390 a Fo(Let)34 b Fn( )g Fj(2)d(B)s Fo(,)k(so)f(that)f Fn(D)s(L)p Fo(\()p Fn(')p Fo(\))h(holds.)48 b(By)34 b(the)h(second)g (statemen)m(t)f(of)g(Theorem)g(3.8,)g(the)g(function)0 511 y Fn(R)74 526 y Fl(';)p Fh(M)228 511 y Fo(\()p Fn(n;)17 b Fo(0\))35 b(is)g(fast)h(decying)g(in)e Fn(n)p Fo(.)53 b(In)36 b(particular,)e(if)h Fn( )h Fo(=)d Fn(\016)2343 526 y Fl(m)2410 511 y Fn(;)52 b(m)33 b Fj(2)g Fp(Z)2774 474 y Fl(d)2815 511 y Fo(,)k(then)f Fn(\015)3155 526 y Fl(k)3230 511 y Fo(=)p 3338 426 250 4 v 32 w Fn(e)3383 526 y Fl(k)3426 511 y Fo(\()p Fn(m)p Fo(\))g(and)0 631 y Fn(R)74 646 y Fl(';)p Fh(M)228 631 y Fo(\()p Fn(n;)17 b Fo(0\))27 b(=)h Fn(F)649 646 y Fm(0)688 631 y Fo(\()p Fn(n;)17 b(m)p Fo(\))33 b(is)f(fast)h(decain)m(y)g(in)f Fn(n)p Fo(,)h(so)f(the)h(second)h(statemen)m(t)f(of)f(Theorem)h(holds.) 146 751 y(As)e(to)f(the)g Fj(A)p Fo(-dynamical)d(lo)s(calization)f(on)k Fn(I)8 b Fo(,)31 b(there)f(are)g(man)m(y)g(p)s(ossible)g(su\016cien)m 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b(sup)1475 4262 y Fl(t)1561 4188 y Fj(h)p Fn(')p Fo(\()p Fn(t;)g(\022)s Fo(\))p Fn(;)g Fj(j)p Fn(X)8 b Fj(j)2056 4147 y Fl(p)2094 4188 y Fn(')p Fo(\()p Fn(t;)17 b(\022)s Fo(\))p Fj(i)28 b Fn(<)f Fo(+)p Fj(1)p Fn(;)0 4437 y Fo(where)1157 4557 y Fn(')p Fo(\()p Fn(t;)17 b(\022)s Fo(\))27 b(=)h(exp)q(\()p Fj(\000)p Fn(itH)8 b Fo(\()p Fn(\022)s Fo(\)\))p Fn(P)2201 4572 y Fl(I)2241 4557 y Fo(\()p Fn(H)g Fo(\()p Fn(\022)s Fo(\)\))p Fn( )t(:)0 4727 y Fo(One)33 b(can)g(giv)m(e)g(the)g(follo)m(wing)d(b)s(ound)j (whic)m(h)g(can)g(b)s(e)g(used)h(to)e(pro)m(v)m(e)i(strong)f(dynamical) d(lo)s(caliza-)0 4847 y(tion.)0 5041 y Fp(Theorem)37 b(5.5)49 b Fd(Let)39 b Fn(\013)h Fj(\025)f Fo(0)g Fd(b)s(e)h(suc)m(h)g (that)f(for)g Fn(\026)p Fd(-almost)d(all)i Fn(\022)s Fd(,)2604 4975 y Fi(P)2691 5062 y Fl(n)2755 5041 y Fj(j)p Fn(g)t Fo(\()p Fn(n)p Fo(\))p Fj(j)2996 5005 y Fm(2)3034 5041 y Fo(\()p Fj(j)p Fn(n)p Fj(j)26 b Fo(+)h(1\))3402 5005 y Fl(\013)3490 5041 y Fn(<)39 b Fo(+)p Fj(1)0 5161 y Fd(for)34 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511 y Fo(Therefore,)34 b(for)e(an)m(y)h Fn(\027)h(>)28 b Fo(0,)1044 645 y Fi(X)1004 834 y Fh(j)p Fl(n)p Fh(j)p Fl(>N)1221 728 y Fj(j)p Fn(e)1294 743 y Fl(k)1336 728 y Fo(\()p Fn(n)p Fo(\))p Fj(j)1498 687 y Fm(2)1565 728 y Fj(\024)g Fn(C)7 b Fo(\()p Fn(\027)q(;)17 b(d)p Fo(\))p Fn(L)g Fo(exp)q(\()p Fj(\000)p Fo(\()p Fn(\015)27 b Fj(\000)c Fn(\027)6 b Fo(\))p Fn(N)k Fo(\))p Fn(:)0 1039 y Fo(Let)33 b Fn(L)g Fo(b)s(e)f(suc)m(h)i(that)f Fn(C)7 b Fo(\()p Fn(\027)q(;)17 b(d)p Fo(\))p Fn(L)g Fo(exp\()p Fj(\000)p Fo(\()p Fn(\015)28 b Fj(\000)22 b Fn(\027)6 b Fo(\))p Fn(N)k Fo(\))29 b(=)e(1)p Fn(=)p Fo(2.)43 b(Then)34 b Fn(I)8 b Fo(\()p Fn(L)p Fo(\))28 b Fj(\032)g Fn(J)9 b Fo(\()p Fn(N)h Fo(\))33 b(and)574 1257 y(Card\()p Fn(I)8 b Fo(\()p Fn(L)p Fo(\)\))28 b Fj(\024)g Fo(Card\()p Fn(J)9 b Fo(\()p Fn(N)h Fo(\)\))28 b Fj(\024)g Fn(C)7 b Fo(\()p Fn(d)p Fo(\)\()p Fn(N)32 b Fo(+)22 b(1\))2371 1215 y Fl(d)2439 1257 y Fj(\024)28 b Fn(C)7 b Fo(\()p Fn(\015)e(;)17 b(\027)q(;)g(d)p Fo(\))g(log)3083 1214 y Fl(d)3140 1257 y Fn(L)373 b Fo(\(6)p Fn(:)p Fo(1\))0 1474 y(for)32 b(an)m(y)h Fn(L)28 b Fj(\025)h Fn(L)599 1489 y Fm(0)638 1474 y Fo(\()p Fn(\015)5 b(;)17 b(\027)q(;)g(d)p Fo(\).)43 b(The)33 b(result)g(of)f(the)h(theorem)f(follo)m(ws)g(directly)g(from)f (\(6.1\).)146 1595 y(With)42 b(this)f(result)h(w)m(e)h(can)f(obtain)f (a)g(necessary)k(condition)40 b(for)h Fn(E)6 b(D)s(L)p Fo(\()p Fn( )t Fo(\))42 b(in)f(terms)h(of)g(pro-)0 1715 y(jections)48 b Fn( )434 1730 y Fl(k)525 1715 y Fo(and)g(in)f(terms)h (of)g(co)s(e\016cien)m(ts)h(of)e(the)i(sp)s(ectral)f(measure)g(of)f Fn( )t Fo(.)90 b(Let)48 b Fj(M)p Fo(\()p Fn( )t Fo(\))g(b)s(e)0 1836 y(the)43 b(orthonormal)c(system)44 b(of)d(eigenfunctions)h(of)g Fn(H)50 b Fo(de\014ned)43 b(in)f(section)g(2)g(and)g Fn(R)3266 1851 y Fl( )r(;)p Fh(M)p Fm(\()p Fl( )r Fm(\))3526 1836 y Fo(\()p Fn(n)p Fo(\))i(=)0 1956 y(sup)147 1979 y Fl(k)206 1956 y Fj(j)p Fn(\015)285 1971 y Fl(k)327 1956 y Fn(e)372 1971 y Fl(k)415 1956 y Fo(\()p Fn(n)p Fo(\))p Fj(j)p Fo(,)51 b(where)e Fn(\015)1003 1971 y Fl(k)1098 1956 y Fo(=)k Fj(h)p Fn( )t(;)17 b(e)1422 1971 y Fl(k)1464 1956 y Fj(i)p Fn(;)64 b(H)8 b(e)1728 1971 y Fl(k)1823 1956 y Fo(=)53 b Fn(\025)2009 1971 y Fl(k)2052 1956 y Fn(e)2097 1971 y Fl(k)2139 1956 y Fo(.)88 b(The)49 b(sp)s(ectral)e(measure)h(of)f Fn( )k Fo(can)d(b)s(e)0 2076 y(written)32 b(as)1577 2197 y Fn(\026)1636 2212 y Fl( )1716 2197 y Fo(=)1819 2114 y Fi(X)1860 2298 y Fl(k)1956 2197 y Fn(a)2007 2212 y Fl(k)2050 2197 y Fn(\016)2093 2212 y Fl(\025)2134 2224 y Ff(k)2176 2197 y Fn(:)0 2489 y Fp(Theorem)37 b(6.2)49 b Fd(Supp)s(ose)34 b(that)1296 2707 y Fo(sup)1357 2781 y Fl(t)1460 2707 y Fj(j)p Fn( )t Fo(\()p Fn(t;)17 b(n)p Fo(\))p Fj(j)27 b(\024)h Fn(C)c Fo(exp)q(\()p Fj(\000)p Fn(\013)q Fj(j)p Fn(n)p Fj(j)p Fo(\))1078 b(\(6)p Fn(:)p Fo(2\))0 2964 y Fd(for)32 b(some)g Fn(\013)d(>)e Fo(0)p Fd(.)44 b(Then)1417 3085 y Fo(sup)1471 3164 y Fl(k)1580 3085 y Fj(j)p Fn(\015)1659 3100 y Fl(k)1701 3085 y Fj(j)1729 3044 y Fm(2)1768 3085 y 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b(it)d(follo)m(ws)h(from)f(\(6.3\))h (and)h(Theorem)g(6.1)f(that)g(after)g(reordering)976 605 y Fn(a)1027 620 y Fl(k)1098 605 y Fj(\021)c(j)p Fn(\015)1282 620 y Fl(k)1324 605 y Fj(j)1352 564 y Fm(2)1419 605 y Fj(\024)g Fn(C)7 b Fo(\()p Fn(\021)1687 620 y Fl(k)1730 605 y Fo(\(2)p Fn(\013)q Fo(\)\))1956 564 y Fh(\000)p Fm(1)2077 605 y Fj(\024)28 b Fn(C)c Fo(exp)q(\()p Fj(\000)p Fn(\014)6 b(k)2655 564 y Fm(1)p Fl(=d)2766 605 y Fo(\))0 821 y(with)32 b(some)h(p)s(ositiv)m(e)f Fn(C)r(;)17 b(\014)6 b Fo(.)146 941 y(In)42 b(the)g(follo)m(wing)d(statemen)m(t)j(w)m(e)h (shall)d(use)j(the)f(same)g(notations)e(as)i(in)f(Theorem)h(4.2.)70 b(As)0 1061 y(usually)-8 b(,)28 b Fj(M)f Fo(=)g Fj(f)p Fn(e)694 1076 y Fl(k)737 1061 y Fj(g)g Fo(is)g(an)m(y)h(orthonormal)d (system)j(of)f(eigenfunctions)g(of)g Fn(H)8 b Fo(.)41 b(Moreo)m(v)m(er,)30 b(for)d Fn(\016)32 b(>)27 b Fo(0)0 1182 y(w)m(e)34 b(de\014ne)1306 1332 y Fn(R)1380 1347 y Fl( )r(;)p Fh(M)1537 1332 y Fo(\()p Fn(n;)17 b(\016)t Fo(\))27 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Fm(1)p Fl(=d)2795 5385 y Fo(\))p Fn(:)719 b Fo(\(6)p Fn(:)p Fo(7\))0 5557 y(T)-8 b(aking)32 b(in)g(\(6.7\))g Fn(B)38 b Fo(so)32 b(that)h Fn(\014)6 b(=)p Fo(2)p Fn(B)1353 5521 y Fm(1)p Fl(=d)1491 5557 y Fo(=)27 b Fn(\013)q Fo(,)33 b(w)m(e)g(obtain)f(the)h(\014rst)g (statemen)m(t)g(of)f(the)h(Theorem.)1841 5806 y(21)p eop %%Page: 22 22 22 21 bop 146 390 a Fo(T)-8 b(o)35 b(pro)m(v)m(e)h(the)g(second)g (statemen)m(t)f(of)f(the)i(Theorem)f(w)m(e)h(shall)d(need)j(a)f(b)s (ound)g(relating)e Fn(\021)3599 405 y Fl(k)3641 390 y Fo(\()p Fn(\013)q Fo(\))0 511 y(and)g Fn(\021)238 526 y Fl(k)280 511 y Fo(\()p Fn(\027)6 b Fo(\))33 b(for)f Fn(\027)j Fj(\024)28 b Fn(\013)q Fo(.)43 b(It)32 b(follo)m(ws)g(from)f (de\014nition)h(of)g Fn(\021)2162 526 y Fl(k)2205 511 y Fo(\()p Fn(\013)q Fo(\))g(that)1303 704 y Fj(j)p Fn(e)1376 719 y Fl(k)1418 704 y Fo(\()p Fn(n)p Fo(\))p Fj(j)1580 663 y Fm(2)1647 704 y Fj(\024)c Fn(\021)1800 719 y Fl(k)1843 704 y Fo(\()p Fn(\013)q Fo(\))17 b(exp\()p Fj(\000)p Fn(\013)q 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4606 y Fm(2)3139 4642 y Fo(\()p Fn(R)q Fo(\),)i(de\014ne)g (the)0 4763 y(momen)m(ts)1228 4883 y Fn(d)1279 4898 y Fl(k)1321 4883 y Fo(\()p Fn(p)p Fo(\))27 b(=)1577 4766 y Fi(Z)1623 4954 y Fg(R)1710 4883 y Fj(j)p Fn(e)1783 4898 y Fl(k)1826 4883 y Fo(\()p Fn(x)p Fo(\))p Fj(j)1985 4842 y Fm(2)2024 4883 y Fo(\()p Fj(j)p Fn(x)p Fj(j)22 b Fo(+)g(1\))2380 4842 y Fl(p)2419 4883 y Fn(dx:)0 5076 y Fo(It)28 b(is)f(su\016cien)m(t)i(to)e(tak)m(e)h(an)m(y)h(orthonormal) c(system)j Fj(f)p Fn(e)2065 5091 y Fl(k)2108 5076 y Fo(\()p Fn(x)p Fo(\))p Fj(g)g Fo(in)f Fn(L)2492 5040 y Fm(2)2532 5076 y Fo(\([)p Fj(\000)p Fo(1)p Fn(;)17 b Fo(1]\))27 b(and)h(to)f(put)h Fn(e)3426 5091 y Fl(k)3469 5076 y Fo(\()p Fn(x)p Fo(\))g(=)f(0)0 5196 y(for)35 b Fj(j)p Fn(x)p Fj(j)f Fn(>)f Fo(1.)53 b(F)-8 b(or)35 b(suc)m(h)j(a)d(system)i Fn(d)1399 5211 y Fl(k)1442 5196 y Fo(\()p Fn(p)p Fo(\))c Fj(\024)h Fo(2)1760 5160 y Fl(p)1835 5196 y Fo(for)h(an)m(y)i Fn(k)s Fo(.)54 b(Ho)m(w)m(ev)m(er,)39 b(if)c(the)h(functions)g Fn(e)3467 5211 y Fl(k)3510 5196 y Fo(\()p Fn(x)p Fo(\))g(do)0 5316 y(not)e(oscillate)f(fast,)i(the)g(same)f(phenomenon)h(of)f ("repulsion")f(of)i(eigenfunctions)f(o)s(ccurs)h(and)g(one)0 5437 y(can)c(sho)m(w)h(the)g(result)f(similar)d(to)i(that)h(of)g (Theorem)g(2.2.)42 b(The)32 b(main)e(result)h(of)f(this)h(section)g(is) g(the)0 5557 y(follo)m(wing.)1841 5806 y(24)p eop %%Page: 25 25 25 24 bop 0 390 a Fp(Theorem)37 b(7.1)49 b Fd(Let)33 b Fj(f)p Fn(e)937 405 y Fl(k)980 390 y Fj(g)p Fn(;)17 b(k)30 b Fj(2)e Fp(N)k Fd(b)s(e)h(an)f(orthonormal)e(system)k(in)e Fn(L)2699 354 y Fm(2)2739 390 y Fo(\()p Fp(R)2861 354 y Fl(d)2901 390 y Fo(\))g Fd(suc)m(h)i(that)1299 631 y Fo(lim)1242 690 y Fl(R)p Fh(!)p Fm(+)p Fh(1)1508 631 y Fo(sup)1562 710 y Fl(k)1672 514 y Fi(Z)1718 702 y Fh(j)p Fl(u)p Fh(j)p Fl(>R)1927 631 y Fj(j)1960 624 y Fi(b)1955 631 y Fn(e)2000 646 y Fl(k)2043 631 y Fo(\()p Fn(u)p Fo(\))p Fj(j)2203 590 y Fm(2)2241 631 y Fn(du)27 b Fo(=)h(0)p Fn(;)1024 b Fo(\(7)p Fn(:)p Fo(1\))0 851 y Fd(where)284 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Ff(d)2172 2776 y Fi(Z)2218 2964 y Fl(K)2278 2972 y Ff(n)2320 2964 y Fm(\()p Fl(")p Fm(\))2411 2893 y Fo(\()p Fj(j)p Fn(x)p Fj(j)22 b Fo(+)g(1\))2767 2852 y Fl(p)2807 2893 y Fj(j)p Fn(e)2880 2908 y Fl(k)2922 2893 y Fo(\()p Fn(x)p Fo(\))p Fj(j)3081 2852 y Fm(2)3121 2893 y Fn(dx;)0 3191 y Fo(w)m(e)34 b(obtain)d(that)1159 3432 y Fn(C)1236 3385 y Fl(p)1229 3454 y Fm(1)1276 3432 y Fo(\()p Fn(")p Fo(\))p Fn(w)1468 3447 y Fl(k)1510 3432 y Fo(\()p Fn(p)p Fo(\))c Fj(\024)h Fn(d)1818 3447 y Fl(k)1861 3432 y Fo(\()p Fn(p)p Fo(\))f Fj(\024)h Fn(C)2195 3385 y Fl(p)2188 3454 y Fm(2)2235 3432 y Fo(\()p Fn(")p Fo(\))p Fn(w)2427 3447 y Fl(k)2469 3432 y Fo(\()p Fn(p)p Fo(\))p Fn(;)958 b Fo(\(7)p Fn(:)p Fo(2\))146 3606 y(where)602 3727 y Fn(w)672 3742 y Fl(k)715 3727 y Fo(\()p Fn(p)p Fo(\))27 b(=)971 3644 y Fi(X)1009 3818 y Fl(n)1091 3727 y Fo(\()p Fj(j)p Fn(n)p Fj(j)21 b Fo(+)h(1\))1449 3686 y Fl(p)1505 3610 y Fi(Z)1551 3798 y Fl(K)1611 3806 y Ff(n)1653 3798 y Fm(\()p Fl(")p Fm(\))1762 3727 y Fj(j)p Fn(e)1835 3742 y Fl(k)1877 3727 y Fo(\()p Fn(x)p Fo(\))p Fj(j)2036 3686 y Fm(2)2076 3727 y Fn(dx)27 b Fj(\021)2314 3644 y Fi(X)2353 3818 y Fl(n)2434 3727 y Fo(\()p Fj(j)p Fn(n)p Fj(j)22 b Fo(+)g(1\))2793 3686 y Fl(p)2832 3727 y Fj(j)p Fn(g)2907 3742 y Fl(k)2950 3727 y Fo(\()p Fn(n)p Fo(\))p Fj(j)3112 3686 y Fm(2)3151 3727 y Fn(:)0 3968 y Fo(One)30 b(could)f(try)h(to)f (apply)g(Lemma)g(2.1)g(taking)f Fn(a)1846 3983 y Fl(k)r(n)1960 3968 y Fo(=)f Fj(j)p Fn(g)2138 3983 y Fl(k)2180 3968 y Fo(\()p Fn(n)p Fo(\))p Fj(j)2342 3932 y Fm(2)2382 3968 y Fo(.)42 b(It)30 b(is)f(ob)m(vious)h(that)3207 3901 y Fi(P)3295 3988 y Fl(n)3358 3968 y Fj(j)p Fn(g)3433 3983 y Fl(k)3475 3968 y Fo(\()p Fn(n)p Fo(\))p Fj(j)3637 3932 y Fm(2)3704 3968 y Fo(=)0 4088 y Fj(k)p Fn(e)95 4103 y Fl(k)138 4088 y Fj(k)188 4052 y Fm(2)255 4088 y Fo(=)d(1,)e(so)e(the)h(condition)e(\(2.2\))g(is)h(satis\014ed.)41 b(Ho)m(w)m(ev)m(er,)27 b(it)22 b(is)h(not)g(clear)f(whether)3210 4022 y Fi(P)3297 4109 y Fl(k)3357 4088 y Fj(j)p Fn(g)3432 4103 y Fl(k)3474 4088 y Fo(\()p Fn(n)p Fo(\))p Fj(j)3636 4052 y Fm(2)3703 4088 y Fj(\024)0 4208 y Fn(A)28 b(<)f Fo(+)p Fj(1)p Fo(.)43 b(T)-8 b(o)33 b(a)m(v)m(oid)f(this)h(problem,)e (w)m(e)j(shall)d(consider)i(rather)f(the)h(quan)m(tities)1404 4453 y Fn(h)1460 4468 y Fl(k)1502 4453 y Fo(\()p Fn(n)p Fo(\))28 b(=)1768 4336 y Fi(Z)1814 4524 y Fl(K)1874 4532 y Ff(n)1916 4524 y Fm(\()p Fl(")p Fm(\))2024 4453 y Fn(e)2069 4468 y Fl(k)2112 4453 y Fo(\()p Fn(x)p Fo(\))p Fn(dx:)0 4718 y Fo(By)33 b(the)g(Cauc)m(h)m(y-Sc)m(h)m(w)m(artz)j(inequalit)m(y) -8 b(,)1444 4959 y Fj(j)p Fn(h)1528 4974 y Fl(k)1570 4959 y Fo(\()p Fn(n)p Fo(\))p Fj(j)1732 4918 y Fm(2)1799 4959 y Fj(\024)29 b Fn(")1951 4918 y Fl(d)1991 4959 y Fj(j)p Fn(g)2066 4974 y Fl(k)2108 4959 y Fo(\()p Fn(n)p Fo(\))p Fj(j)2270 4918 y Fm(2)2309 4959 y Fn(:)1243 b Fo(\(7)p Fn(:)p Fo(3\))146 5133 y(Therefore,)34 b(\(7.2\))e(implies) 1080 5353 y Fn(d)1131 5368 y Fl(k)1174 5353 y Fo(\()p Fn(p)p Fo(\))27 b Fj(\025)h Fn(")1477 5312 y Fh(\000)p Fl(d)1572 5353 y Fn(C)1649 5306 y Fl(p)1642 5375 y Fm(1)1689 5353 y Fo(\()p Fn(")p Fo(\))1828 5270 y Fi(X)1866 5445 y Fl(n)1947 5353 y Fo(\()p Fj(j)p Fn(n)p Fj(j)22 b Fo(+)g(1\))2306 5312 y Fl(p)2345 5353 y Fj(j)p Fn(h)2429 5368 y Fl(k)2471 5353 y Fo(\()p Fn(n)p Fo(\))p Fj(j)2633 5312 y Fm(2)2673 5353 y Fn(;)879 b Fo(\(7)p Fn(:)p Fo(4\))1841 5806 y(25)p eop %%Page: 26 26 26 25 bop 0 390 a Fo(and)34 b(to)g(pro)m(v)m(e)h(the)f(Theorem)g(it)f (is)g(su\016cien)m(t)i(to)f(sho)m(w)h(that)f(the)g(n)m(um)m(b)s(ers)h Fn(a)2959 405 y Fl(k)r(n)3074 390 y Fo(=)30 b Fj(j)p Fn(h)3264 405 y Fl(k)3307 390 y Fo(\()p Fn(n)p Fo(\))p Fj(j)3469 354 y Fm(2)3542 390 y Fo(v)m(erify)0 511 y(the)j(conditions)f (of)g(Lemma)f(2.1)h(for)g(some)h Fn(")27 b(>)g Fo(0.)44 b(One)33 b(can)f(represen)m(t)j Fn(h)2802 526 y Fl(k)2845 511 y Fo(\()p Fn(n)p Fo(\))d(as)0 631 y Fn(h)56 646 y Fl(k)99 631 y Fo(\()p Fn(n)p Fo(\))h(=)h Fj(h)p Fn(e)460 646 y Fl(k)502 631 y Fn(;)17 b(\021)594 646 y Fl(n)641 631 y Fj(i)680 650 y Fl(L)728 631 y Fe(2)762 650 y Fm(\()p Fg(R)855 631 y Ff(d)892 650 y Fm(\))924 631 y Fo(,)37 b(where)g Fn(\021)1321 646 y Fl(n)1404 631 y Fo(is)e(the)h (caracteristic)g(function)f(of)h Fn(K)2811 646 y Fl(n)2858 631 y Fo(\()p Fn(")p Fo(\).)53 b(Since)36 b(the)g(system)0 751 y Fj(f)p Fn(e)95 766 y Fl(k)138 751 y Fj(g)c Fo(is)g(orthonormal,) 1352 789 y Fi(X)1393 973 y Fl(k)1489 872 y Fj(j)p Fn(h)1573 887 y Fl(k)1615 872 y Fo(\()p Fn(n)p Fo(\))p Fj(j)1777 831 y Fm(2)1844 872 y Fj(\024)c(k)p Fn(\021)2047 887 y Fl(n)2094 872 y Fj(k)2144 831 y Fm(2)2211 872 y Fo(=)g Fn(")2361 831 y Fl(d)2401 872 y Fn(;)0 1126 y Fo(so)33 b(\(2.1\))f(holds)g(with)g Fn(A)c Fo(=)g Fn(")1081 1090 y Fl(d)1121 1126 y Fo(.)146 1246 y(T)-8 b(o)32 b(pro)m(v)m(e)g(\(2.2\)) f(is)f(more)h(di\016cult.)42 b(W)-8 b(e)31 b(shall)f(sho)m(w)i(that)f (the)h(n)m(um)m(b)s(ers)g Fn(")2964 1210 y Fh(\000)p Fl(d)3059 1246 y Fj(j)p Fn(h)3143 1261 y Fl(k)3185 1246 y Fo(\()p Fn(n)p Fo(\))p Fj(j)3347 1210 y Fm(2)3418 1246 y Fo(are)f(close)0 1367 y(to)f Fj(j)p Fn(g)192 1382 y Fl(k)234 1367 y Fo(\()p Fn(n)p Fo(\))p Fj(j)396 1331 y Fm(2)465 1367 y Fo(for)g Fn(")f Fo(small)f(enough)j(if)e(the)h (condition)f(\(7.1\))g(is)h(satis\014ed.)43 b(Using)2962 1300 y Fi(P)3049 1388 y Fl(n)3113 1367 y Fj(j)p Fn(g)3188 1382 y Fl(k)3230 1367 y Fo(\()p Fn(n)p Fo(\))p Fj(j)3392 1331 y Fm(2)3459 1367 y Fo(=)27 b(1,)k(w)m(e)0 1487 y(shall)g(pro)m(v)m (e)j(\(2.2\))e(for)h(some)f Fn(B)5 b Fo(\()p Fn(")p Fo(\))28 b Fn(>)f Fo(0)33 b(if)e Fn(")i Fo(is)f(small)e(enough.)44 b(W)-8 b(e)34 b(need)f(the)g(follo)m(wing)d(tec)m(hnical)0 1608 y(result.)0 1811 y Fp(Lemma)37 b(7.2)49 b Fd(Let)33 b Fn( )f Fj(2)c Fn(L)1019 1775 y Fm(2)1058 1811 y Fo(\()p Fp(R)1180 1775 y Fl(d)1220 1811 y Fo(\))p Fn(;)50 b Fj(k)p Fn( )t Fj(k)27 b Fo(=)g(1)p Fd(.)44 b(F)-8 b(or)31 b(an)m(y)j Fn(n)28 b Fj(2)g Fp(Z)2359 1775 y Fl(d)2400 1811 y Fn(;)17 b(")27 b(>)g Fo(0)32 b Fd(de\014ne)904 2110 y Fo(\001)985 2125 y Fl(n)1033 2110 y Fo(\()p Fn(")p Fo(\))27 b(=)1285 1993 y Fi(Z)1331 2181 y Fl(K)1391 2189 y Ff(n)1433 2181 y Fm(\()p Fl(")p Fm(\))1542 2110 y Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))p Fj(j)1796 2069 y Fm(2)1835 2110 y Fn(dx)22 b Fj(\000)h Fn(")2109 2069 y Fh(\000)p Fl(d)2220 1960 y Fi(\014)2220 2010 y(\014)2220 2060 y(\014)2220 2110 y(\014)2220 2160 y(\014)2248 1993 y(Z)2294 2181 y Fl(K)2354 2189 y Ff(n)2396 2181 y Fm(\()p Fl(")p Fm(\))2504 2110 y Fn( )t Fo(\()p Fn(x)p Fo(\))p Fn(dx)2808 1960 y Fi(\014)2808 2010 y(\014)2808 2060 y(\014)2808 2110 y(\014)2808 2160 y(\014)2836 1987 y Fm(2)0 2392 y Fd(\()p Fo(\001)119 2407 y Fl(n)166 2392 y Fo(\()p Fn(")p Fo(\))k Fj(\025)i Fo(0)j Fd(b)m(y)h(Cauc)m(h)m(y-Sc)m(h)m(w)m(artz)j(inequalit)m(y\).)146 2512 y(There)e(exists)f(some)g(univ)m(ersal)f(constan)m(t)i Fn(C)7 b Fo(\()p Fn(d)p Fo(\))32 b Fd(suc)m(h)i(that)e(for)g(an)m(y)h Fn(")28 b(>)f Fo(0)p Fn(;)17 b(R)29 b(>)e Fo(0)770 2818 y(0)g Fj(\024)981 2735 y Fi(X)951 2927 y Fl(n)p Fh(2)p Fg(Z)1095 2908 y Ff(d)1148 2818 y Fo(\001)1229 2833 y Fl(n)1276 2818 y Fo(\()p Fn(")p Fo(\))g Fj(\024)i Fn(C)7 b Fo(\()p Fn(d)p Fo(\))1752 2672 y Fi( )1816 2818 y Fn(R)1891 2777 y Fm(2)1931 2818 y Fn(")1977 2777 y Fm(2)2038 2818 y Fo(+)2136 2701 y Fi(Z)2182 2890 y Fh(j)p Fl(u)p Fh(j)p Fl(>R)2392 2818 y Fj(j)2440 2785 y Fi(b)2420 2818 y Fn( )s Fo(\()p Fn(u)p Fo(\))p Fj(j)2646 2777 y Fm(2)2685 2818 y Fn(du)2792 2672 y Fi(!)2857 2695 y Fm(1)p Fl(=)p Fm(2)2983 2818 y Fn(:)0 3131 y Fc(Pro)s(of.)53 b Fo(One)33 b(can)g(represen)m(t)h (\001)1260 3146 y Fl(n)1308 3131 y Fo(\()p Fn(")p Fo(\))e(as)812 3371 y(\001)893 3386 y Fl(n)940 3371 y Fo(\()p Fn(")p Fo(\))27 b(=)h Fn(")1239 3330 y Fh(\000)p Fl(d)1350 3254 y Fi(Z)1397 3443 y Fl(K)1457 3451 y Ff(n)1499 3443 y Fm(\()p Fl(")p Fm(\))1607 3371 y Fn(dx)k( )t Fo(\()p Fn(x)p Fo(\))1960 3254 y Fi(Z)2007 3443 y Fl(K)2067 3451 y Ff(n)2109 3443 y Fm(\()p Fl(")p Fm(\))2217 3371 y Fn(dy)j Fo(\()p 2389 3287 515 4 v Fn( )t Fo(\()p Fn(x)p Fo(\))22 b Fj(\000)h Fn( )t Fo(\()p Fn(y)t Fo(\))o(\))p Fn(:)611 b Fo(\(7)p Fn(:)p Fo(5\))0 3596 y(Applying)28 b(t)m(wice)h(the)g(Cauc)m (h)m(y-Sc)m(h)m(w)m(artz)j(inequalit)m(y)c(\(to)g(the)h(in)m(tegral)e (o)m(v)m(er)j Fn(y)h Fo(and)e(to)f(the)i(in)m(tegral)0 3717 y(o)m(v)m(er)k Fn(x)p Fo(\),)f(w)m(e)g(obtain)f(from)f(\(7.5\):) 606 3967 y(\001)687 3926 y Fm(2)687 3991 y Fl(n)734 3967 y Fo(\()p Fn(")p Fo(\))c Fj(\024)h Fn(")1034 3926 y Fh(\000)p Fl(d)1146 3850 y Fi(Z)1192 4038 y Fl(K)1252 4046 y Ff(n)1294 4038 y Fm(\()p Fl(")p Fm(\))1402 3967 y Fn(dx)p Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))p Fj(j)1762 3926 y Fm(2)1818 3850 y Fi(Z)1864 4038 y Fl(K)1924 4046 y Ff(n)1966 4038 y Fm(\()p Fl(")p Fm(\))2074 3850 y Fi(Z)2120 4038 y Fl(K)2180 4046 y Ff(n)2222 4038 y Fm(\()p Fl(")p Fm(\))2331 3967 y Fn(dxdy)t Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))21 b Fj(\000)h Fn( )t Fo(\()p Fn(y)t Fo(\))p Fj(j)3109 3926 y Fm(2)3147 3967 y Fn(:)0 4232 y Fo(Therefore,)86 4449 y Fi(X)124 4624 y Fl(n)222 4532 y Fo(\001)303 4547 y Fl(n)351 4532 y Fo(\()p Fn(")p Fo(\))27 b Fj(\024)h Fn(")651 4491 y Fh(\000)p Fl(d=)p Fm(2)833 4386 y Fi( )899 4449 y(X)937 4624 y Fl(n)1035 4415 y Fi(Z)1081 4604 y Fl(K)1141 4612 y Ff(n)1183 4604 y Fm(\()p Fl(")p Fm(\))1292 4532 y Fn(dx)p Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))p Fj(j)1652 4491 y Fm(2)1691 4386 y Fi(!)1757 4409 y Fm(1)p Fl(=)p Fm(2)1883 4386 y Fi( )1949 4449 y(X)1987 4624 y Fl(n)2085 4415 y Fi(Z)2132 4604 y Fl(K)2192 4612 y Ff(n)2234 4604 y Fm(\()p Fl(")p Fm(\))2342 4415 y Fi(Z)2388 4604 y Fl(K)2448 4612 y Ff(n)2490 4604 y Fm(\()p Fl(")p Fm(\))2598 4532 y Fn(dxdy)t Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))21 b Fj(\000)i Fn( )t Fo(\()p Fn(y)t Fo(\))p Fj(j)3377 4491 y Fm(2)3415 4386 y Fi(!)3481 4409 y Fm(1)p Fl(=)p Fm(2)3618 4532 y Fo(=)900 4913 y Fn(")946 4872 y Fh(\000)p Fl(d=)p Fm(2)1128 4767 y Fi( )1194 4830 y(X)1233 5004 y Fl(n)1331 4796 y Fi(Z)1377 4984 y Fl(K)1437 4992 y Ff(n)1479 4984 y Fm(\()p Fl(")p Fm(\))1587 4796 y Fi(Z)1633 4984 y Fl(K)1693 4992 y Ff(n)1735 4984 y Fm(\()p Fl(")p Fm(\))1843 4913 y Fn(dxdy)t Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))f Fj(\000)g Fn( )t Fo(\()p Fn(y)t Fo(\))p Fj(j)2622 4872 y Fm(2)2660 4767 y Fi(!)2726 4790 y Fm(1)p Fl(=)p Fm(2)2853 4913 y Fn(:)699 b Fo(\(7)p Fn(:)p Fo(6\))0 5167 y(One)33 b(can)g(observ)m(e) h(that)f Fj(j)p Fn(x)22 b Fj(\000)g Fn(y)t Fj(j)27 b(\024)h Fn(")1406 5083 y Fj(p)p 1489 5083 51 4 v 84 x Fn(d)k Fo(for)g(an)m(y)h Fn(x;)17 b(y)31 b Fj(2)d Fn(K)2260 5182 y Fl(n)2307 5167 y Fo(\()p Fn(")p Fo(\).)43 b(Therefore,)19 5334 y Fi(X)57 5508 y Fl(n)155 5300 y Fi(Z)201 5489 y Fl(K)261 5497 y Ff(n)303 5489 y Fm(\()p Fl(")p Fm(\))411 5300 y Fi(Z)458 5489 y Fl(K)518 5497 y Ff(n)560 5489 y Fm(\()p Fl(")p Fm(\))668 5417 y Fn(dxdy)t Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))21 b Fj(\000)h Fn( )t Fo(\()p Fn(y)t Fo(\))p Fj(j)1446 5376 y Fm(2)1512 5417 y Fj(\024)1617 5334 y Fi(X)1656 5508 y Fl(n)1754 5300 y Fi(Z)1800 5489 y Fl(K)1860 5497 y Ff(n)1902 5489 y Fm(\()p Fl(")p Fm(\))2010 5417 y Fn(dx)2133 5300 y Fi(Z)2179 5489 y Fg(R)2245 5470 y Ff(d)2302 5417 y Fn(dy)t Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))f Fj(\000)i Fn( )t Fo(\()p Fn(y)t Fo(\))p Fj(j)2975 5376 y Fm(2)3013 5417 y Fn(F)14 b Fo(\()p Fj(j)p Fn(x)22 b Fj(\000)h Fn(y)t Fj(j)j(\024)i Fn(")3590 5328 y Fj(p)p 3673 5328 V 89 x Fn(d)p Fo(\))1841 5806 y(26)p eop %%Page: 27 27 27 26 bop 887 424 a Fo(=)990 307 y Fi(Z)1036 496 y Fg(R)1102 477 y Ff(d)1159 307 y Fi(Z)1206 496 y Fg(R)1272 477 y Ff(d)1329 424 y Fn(dxdy)t Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))21 b Fj(\000)h Fn( )t Fo(\()p Fn(y)t Fo(\))p Fj(j)2107 383 y Fm(2)2146 424 y Fn(F)14 b Fo(\()p Fj(j)p Fn(x)21 b Fj(\000)i Fn(y)t Fj(j)k(\024)h Fn(")2723 335 y Fj(p)p 2805 335 51 4 v 2805 424 a Fn(d)p Fo(\))p Fn(;)658 b Fo(\(7)p Fn(:)p Fo(7\))0 641 y(where)42 b Fn(F)55 b Fo(is)41 b(the)g(caracteristic)g(function)g(of)f(the)i(set)g Fj(f)p Fo(\()p Fn(x;)17 b(y)t Fo(\))40 b Fj(j)g(j)p Fn(x)29 b Fj(\000)f Fn(y)t Fj(j)41 b(\024)i Fn(")2985 557 y Fj(p)p 3068 557 V 84 x Fn(d)o Fj(g)p Fo(.)69 b(The)42 b(b)s(ounds)0 761 y(\(7.6\)-\(7.7\))31 b(imply)1370 799 y Fi(X)1408 973 y Fl(n)1506 882 y Fo(\001)1587 897 y Fl(n)1635 882 y Fo(\()p Fn(")p Fo(\))c Fj(\024)h Fn(")1935 841 y Fh(\000)p Fl(d=)p Fm(2)2101 882 y Fn(L)2167 841 y Fm(1)p Fl(=)p Fm(2)2277 882 y Fo(\()p Fn(\016)t Fo(\))p Fn(;)1152 b Fo(\(7)p Fn(:)p Fo(8\))0 1127 y(where)36 b Fn(L)p Fo(\()p Fn(\016)t Fo(\))31 b(=)610 1056 y Fi(R)649 1152 y Fg(R)715 1133 y Ff(d)773 1056 y Fi(R)812 1152 y Fg(R)878 1133 y Ff(d)935 1127 y Fj(j)p Fn( )t Fo(\()p Fn(x)p Fo(\))23 b Fj(\000)h Fn( )t Fo(\()p Fn(y)t Fo(\))p Fj(j)1508 1090 y Fm(2)1546 1127 y Fn(F)14 b Fo(\()p Fj(j)p Fn(x)23 b Fj(\000)h Fn(y)t Fj(j)30 b(\024)h Fn(\016)t Fo(\))j(and)h Fn(\016)f Fo(=)d Fn(")2627 1042 y Fj(p)p 2710 1042 V 85 x Fn(d)o Fo(.)49 b(Changing)34 b(the)h(v)-5 b(ariable)0 1247 y Fn(z)32 b Fo(=)c Fn(y)d Fj(\000)e Fn(x)33 b Fo(in)f(the)h(in)m (tegral,)e(w)m(e)i(obtain)f(in)g(F)-8 b(ourier)31 b(represen)m(tation) 1013 1469 y Fn(L)p Fo(\()p Fn(\016)t Fo(\))d(=)1334 1352 y Fi(Z)1380 1541 y Fh(j)p Fl(z)s Fh(j\024)p Fl(\016)1564 1469 y Fn(dz)1681 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b(particular,)g(this)g(is)f(the)i(case)g(if)e Fn(H)41 b Fo(=)34 b Fj(\000)p Fo(\001)25 b(+)f Fn(V)e Fo(\()p Fn(x)p Fo(\))36 b(with)g Fn(V)22 b Fo(\()p Fn(x)p Fo(\))1841 5806 y(29)p eop %%Page: 30 30 30 29 bop 0 390 a Fo(b)s(ounded)39 b(from)e(b)s(elo)m(w)i(and)f Fn(I)46 b Fo(=)37 b(\()p Fj(\0001)p Fn(;)17 b(K)7 b Fo(].)61 b(The)39 b(result)g(similar)c(to)j(that)g(of)g(Theorem)h(5.5)f(can)0 511 y(b)s(e)33 b(pro)m(v)m(ed)g(in)f(the)h(case)g Fn(H)8 b Fo(\()p Fn(\022)s Fo(\))28 b(=)f Fj(\000)p Fo(\001)c(+)e Fn(V)h Fo(\()p Fn(x;)17 b(\022)s Fo(\),)33 b(where)g Fn(V)22 b Fo(\()p Fn(x;)17 b(\022)s Fo(\))28 b Fj(\025)g(\000)p Fn(M)43 b Fo(for)32 b Fn(\026)p Fo(-a.e.)43 b Fn(\022)36 b Fo(and)c(a.e.)p Fn(x)p Fo(.)0 631 y(The)i(constan)m(ts)f(in)f(the)h (b)s(ounds)h(will)c(dep)s(end)j(on)g Fn(";)17 b(p;)g(d)31 b Fo(and)i Fn(K)c Fo(+)22 b Fn(M)10 b Fo(.)146 751 y(The)26 b(main)c(results)j(of)f(Section)g(6)h(can)f(b)s(e)h(also)e(generalized) h(to)g(the)h(con)m(tin)m(uous)g(case)g(in)f(a)g(similar)0 872 y(w)m(a)m(y)-8 b(.)146 992 y Fb(A)j(cknow)5 b(le)-5 b(dgments)p Fo(.)51 b(I)36 b(thank)g(F.)f(Germinet)f(for)h(stim)m (ulating)e(discussions)j(on)f(the)h(sub)5 b(ject)37 b(of)0 1112 y(the)c(pap)s(er.)0 1445 y Fk(References)0 1664 y Fo([1])670 b(M.)29 b(Aizenman:)40 b(Lo)s(calization)25 b(at)i(w)m(eal)h(disorder:)41 b(Some)27 b(elemen)m(tary)h(b)s(ounds.) 773 1785 y(Rev.)34 b(Math.)f(Ph)m(ys.)h Fp(6)p Fo(,)f(1163-1182)d (\(1994\))0 1988 y([2])670 b(M.)30 b(Aizenman,)f(J.H.)h(Sc)m(henk)m (er,)i(R.M.)e(F)-8 b(riedric)m(h,)28 b(D.)h(Hundertmark:)42 b(Finite-)773 2108 y(v)m(olume)28 b(fractional-momen)m(t)d(criteria)h (for)i(Anderson)h(lo)s(calization:)37 b(to)28 b(app)s(ear)773 2229 y(in)k(Comm.)g(Math.)h(Ph)m(ys.)0 2432 y([3])670 b(D.)39 b(Damanik)d(and)j(P)-8 b(.)39 b(Stollman:)52 b(Multi-scale)37 b(analysis)h(implies)e(strong)j(dy-)773 2553 y(namical)31 b(lo)s(calization.)d(Preprin)m(t)33 b(\(1999\))0 2756 y([4])670 b(S.)34 b(De)e(Bi)m(\022)-46 b(evre)33 b(and)g(F.)g(Germinet:)43 b(Dynamical)30 b(lo)s(calization)f (for)k(the)g(random)773 2876 y(dimer)f(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erator.)f(J.)g(Stat.)h(Ph)m(ys.)h Fp(98)p Fo(,)f(1135-1148)d(\(2000\))0 3080 y([5])670 b(R.)43 b(Del)e(Rio,)j(S.)f(Jitomirsk)-5 b(a)m(y)m(a,)43 b(Y.)g(Last)g(and)f(B.)h(Simon:)62 b(Op)s(erators)42 b(with)773 3200 y(singular)33 b(con)m(tin)m(uous)h(sp)s(ectrum)g(IV:)g (Hausdor\013)g(dimensions,)f(rank)h(one)g(p)s(er-)773 3321 y(turbation)e(and)h(lo)s(calization.)28 b(J.)33 b(d'Analyse)g(Math.)g Fp(69)p Fo(,)g(153-200)e(\(1996\))0 3524 y([6])670 b(F.)31 b(Germinet)e(and)i(S.)g(De)f(Bi)m(\022)-46 b(evre:)43 b(Dynamical)28 b(lo)s(calization)f(for)j(discrete)h(and)773 3644 y(con)m(tin)m(uous)e(random)e(Sc)m(hr\177)-49 b(odinger)29 b(op)s(erators.)f(Comm)m(un.)f(Math.)i(Ph)m(ys.)h Fp(194)p Fo(,)773 3765 y(323-341)h(\(1998\))0 3968 y([7])670 b(F.)37 b(Germinet:)49 b(Dynamical)34 b(lo)s(calization)e(I)s(I)37 b(with)f(an)g(application)e(to)i(the)h(al-)773 4089 y(most)32 b(Mathieu)h(op)s(erator.)f(J.)h(Stat.)f(Ph)m(ys.)j Fp(95)p Fo(,)d(273-286)f(\(1999\))0 4292 y([8])670 b(F.)27 b(Germinet)f(and)h (S.)g(Jitomirsk)-5 b(a)m(y)m(a:)39 b(Strong)26 b(dynamical)f(lo)s (calization)e(for)k(the)773 4412 y(almost)k(Mathieu)i(mo)s(del.)e (Preprin)m(t)i(\(2000\))1841 5806 y(30)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0011090452633--