Content-Type: multipart/mixed; boundary="-------------0609280938509" This is a multi-part message in MIME format. ---------------0609280938509 Content-Type: text/plain; name="06-270.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-270.keywords" Nonlinear Schroedinger equations; almost invariant manifolds. ---------------0609280938509 Content-Type: application/postscript; name="BambusiSacchetti.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BambusiSacchetti.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: BS15.dvi %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Helvetica %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -t a4 -o temp.ps BS15 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2006.07.31:1511 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]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/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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b Fp(V)57 b Fs(tends)39 b(to)f(in\014nit)n(y)h(as)e Fl(j)p Fp(x)p Fl(j)42 b(!)f(1)p Fs(.)107 b(The)39 b(situation)p 456 4626 499 4 v 555 4720 a Fi(Date)5 b Fw(:)23 b(July)h(31,)g(2006.)555 4803 y(1991)j Fi(Mathematics)f(Subje)l(ct)f(Classi\014c)l(ation.)37 b Fw(Primary)21 b(35Bxx;)j(Secondary)h(35Q40,)g(35K55.)555 4883 y(This)20 b(w)n(ork)h(is)f(partially)g(supp)r(orted)i(b)n(y)f(the) g(INdAM)g(pro)t(ject)g Fi(Mathematic)l(al)j(mo)l(deling)g(and)f (numeric)l(al)456 4967 y(analysis)36 b(of)g(quantum)h(systems)e(with)i (applic)l(ations)g(to)f(nanoscienc)l(es)p Fw(.)100 b(DB)34 b(w)n(as)h(also)f(supp)r(orted)i(b)n(y)456 5050 y(MIUR)29 b(under)h(the)h(pro)t(ject)f(COFIN2005)h Fi(Sistemi)g(dinamici)h (nonline)l(ari)h(e)l(d)e(applic)l(azioni)j(\014siche)p Fw(.)79 b(AS)456 5133 y(w)n(as)20 b(also)h(supp)r(orted)h(b)n(y)f(MIUR) f(under)h(the)h(pro)t(ject)f(COFIN2005)h Fi(Sistemi)g(dinamici)h (classici,)h(quantistici)456 5216 y(e)h(sto)l(c)l(astici)p Fw(.)1933 5315 y Fm(1)p eop %%Page: 2 2 2 1 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(as)g(b)r(eating)g(motion)g(and)h(the)g(b)r(eating)f(p)r(erio)r(d)h(is) f(giv)n(en)g(b)n(y)h Fp(T)42 b Fs(:=)31 b Fp(\031)s Fo(~)p Fp(=!)s Fs(,)456 4219 y(with)d Fp(!)d Fs(=)e(\()p Fp(\025)890 4231 y Fm(2)946 4219 y Fl(\000)18 b Fp(\025)1077 4231 y Fm(1)1115 4219 y Fs(\))p Fp(=)p Fs(2.)555 4319 y(The)h(main)f (question)h(is)f(the)h(b)r(eha)n(vior)e(of)i(the)g(system)f(when)h(the) g(nonlinearit)n(y)e(is)i(restored.)456 4419 y(In)26 b(the)h(case)f(of)g (a)h(double)f(w)n(ell)g(p)r(oten)n(tial)h(the)g(problem)f(w)n(as)f(tac) n(kled)h(in)h(a)f(series)g(of)g(pap)r(ers)456 4518 y([1)o(,)h(7)o(,)g (10)o(,)f(11)o(,)h(17)o(,)f(20)o(,)h(21)o(,)f(24)o(,)h(25)o(,)f(27)o (];)h(in)g(particular,)e(it)i(w)n(as)e(sho)n(wn)h(that,)h(up)g(to)f (times)h(of)456 4618 y(order)f Fp(T)12 b Fs(,)27 b(the)h(dynamics)f(is) h(w)n(ell)f(describ)r(ed)h(b)n(y)f(an)h(Hamiltonian)f(in)n(tegrable)g (system)g(with)456 4717 y(t)n(w)n(o)g(degrees)h(of)g(freedom)g (obtained)h(b)n(y)f(restricting)g(the)h(Hamiltonian,)f(i.e.)40 b(the)29 b(energy)f(of)456 4817 y(the)22 b(system,)h(to)e(\010)1042 4829 y Fm(0)1079 4817 y Fs(.)57 b(In)22 b(particular,)g(this)g(result)g (has)f(b)r(een)h(used)g(in)g(order)e(to)i(sho)n(w)f(that)h(the)456 4917 y(b)r(eating)29 b(motion)h(is)f(generic)g(for)g(v)-5 b(alues)30 b(of)g(the)g(nonlinearit)n(y)e(strength)i Fp(\017)f Fs(b)r(elo)n(w)h(a)f(certain)456 5016 y(threshold)g(v)-5 b(alue,)31 b(while)f(new)g(lo)r(calized)f(states)h(app)r(ear)f(for)g (larger)f(nonlinearit)n(y)h(strength)456 5116 y(\(i.e.)35 b(as)21 b(the)i(n)n(um)n(b)r(er)f(of)g(particles)f(of)h(a)g (Bose-Einstein)e(condensate)h(increase\))h(and)f(for)h(ev)n(en)456 5216 y(larger)j(v)-5 b(alues)28 b(of)f(the)h(nonlinearit)n(y)f (strength)g(the)h(b)r(eating)f(motion)h(disapp)r(ears.)p eop %%Page: 3 3 3 2 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)1012 b(3)555 450 y Fs(In)37 b(the)g(case)f(of)h(a)f(m)n(ultiple)h(w)n(ell)g 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b(nonlinear)f(Sc)n(hr\177)-42 b(odinger)26 b(equation)775 1988 y Fp(i)830 1966 y Fs(_)804 1988 y Fp( )858 2000 y Fn(j)916 1988 y Fs(=)c Fp(\016)1040 2000 y Fn(j)1075 1988 y Fp( )1129 2000 y Fn(j)1183 1988 y Fs(+)c(\003)1324 2000 y Fn(j)1358 1988 y Fp( )1412 2000 y Fn(j)s Fm(+1)1550 1988 y Fs(+)g(\003)1691 2000 y Fn(j)s Fg(\000)p Fm(1)1810 1988 y Fp( )1864 2000 y Fn(j)s Fg(\000)p Fm(1)2003 1988 y Fs(+)g Fp(\021)s( )2184 2000 y Fn(j)2233 1988 y Fl(j)p Fp( )2310 2000 y Fn(j)2345 1988 y Fl(j)2368 1946 y Fm(2)p Fn(\033)2488 1988 y Fp(;)41 b(j)28 b Fs(=)23 b(1)p Fp(;)14 b(:::;)g(n;)378 b Fs(\(3\))456 2127 y(where)29 b Fp( )752 2139 y Fm(0)816 2127 y Fs(=)d Fp( )961 2139 y Fn(n)p Fm(+1)1117 2127 y Fs(:=)h(0,)j(\003)1385 2139 y Fn(j)1419 2127 y Fp(;)14 b(\016)1493 2139 y Fn(j)1528 2127 y Fp(;)g(\021)33 b Fs(are)c(suitable)h(constan)n(ts)f(and)g (where,)h(ha)n(ving)f(in)h(mind)456 2227 y(the)18 b(case)g(of)g (Bose-Einstein)f(condensates,)j Fl(j)p Fp( )1905 2239 y Fn(j)1940 2227 y Fl(j)1963 2197 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y(here)36 b(the)h(result)g(is)g(v)-5 b(alid)37 b(for)f Fh(any)i Fs(m)n(ultiple)f(w)n(ell)g(p)r(oten)n(tial,)i(whose)e (eigen)n(v)-5 b(alues)36 b(migh)n(t)456 3721 y(ful\014ll)24 b(arbitrary)d(resonance)h(conditions)h(\(i.e.)36 b(the)23 b(eigen)n(v)-5 b(alues)23 b(can)g(b)r(e)g(linearly)g(dep)r(enden)n(t) 456 3821 y(o)n(v)n(er)31 b(the)j(relativ)n(e)e(in)n(tegers\).)87 b(This)33 b(is)h(p)r(ossible)f(since)g(NLS)g(is)h(an)f(in\014nite)h (dimensional)456 3920 y(Gauge)22 b(in)n(v)-5 b(arian)n(t)22 b(Hamiltonian)h(system.)58 b(T)-7 b(o)23 b(explain)f(ho)n(w)h(this)g (prop)r(ert)n(y)f(is)h(exploited)g(w)n(e)456 4020 y(recall)e(that)i (canonical)f(p)r(erturbation)g(theory)g(allo)n(ws)g(us)g(to)h(remo)n(v) n(e)e(from)i(the)g(Hamiltonian)456 4120 y(all)k(non-resonan)n(t)f (monomials.)64 b(In)28 b(particular,)e(giv)n(en)h(an)h(arbitrary)e (monomial)h(it)h(can)f(b)r(e)456 4219 y(eliminated)d(if)h(it)g(is)f (non-resonan)n(t.)58 b(Ho)n(w)n(ev)n(er,)23 b(in)i(NLS)g(only)f(Gauge)f (in)n(v)-5 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(as)f(the)456 5016 y(domains)27 b(of)h(the)h(p)r(o)n(w)n(ers)d(of)i Fp(H)1462 5028 y Fm(0)1500 5016 y Fs(.)66 b(T)-7 b(o)28 b(use)g(suc)n(h)g(spaces)f(one)g(has)h(to)g(sho)n(w)f(that)i(they)f (form)456 5116 y(Banac)n(h)23 b(algebras)g(under)i(the)g(p)r(oin)n(t)n (wise)g(m)n(ultiplication.)61 b(Here)24 b(w)n(e)h(giv)n(e)f(a)g (detailed)h(pro)r(of)456 5216 y(of)i(suc)n(h)g(a)h(prop)r(ert)n(y)e (that)i(w)n(e)f(think)h(could)g(b)r(e)g(useful)g(in)g(further)f(in)n(v) n(estigations)f(of)h(NLS.)p eop %%Page: 4 4 4 3 bop 456 251 a Fm(4)736 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f(SA) n(CCHETTI)555 450 y Fs(The)27 b(pap)r(er)f(is)g(organized)e(as)i(follo) n(ws.)62 b(In)26 b(Section)h(2)f(w)n(e)g(state)g(our)f(main)i(results)e (\(The-)456 550 y(orem)j(2,)g(Theorem)g(3)h(and)f(their)h (corollaries\).)67 b(Section)29 b(2)f(is)h(divided)g(in)n(to)g(5)f (subsections:)456 649 y(in)f(Subsection)g(2.1)f(w)n(e)g(review)g(some)h (kno)n(wn)f(facts)h(ab)r(out)g(the)g(structure)f(of)h(the)h(lo)n(w)e 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b(section)g(is)f(also)g(divided)456 1446 y(in)n(to)f(6)g(subsections)h (that)g(corresp)r(ond)e(to)h(the)i(di\013eren)n(t)e(parts)g(of)h(the)g (pro)r(of.)61 b(The)24 b(pro)r(of)h(of)456 1546 y(Theorem)d(1)h (\(algebra)f(prop)r(ert)n(y)g(of)i(Sob)r(olev)f(lik)n(e)g(spaces\))f (and)i(of)f(some)g(tec)n(hnical)g(Lemmas)456 1646 y(are)j(left)i(to)g (app)r(endixes)f(A)h(and)g(B)f(resp)r(ectiv)n(ely)-7 b(.)555 1745 y Fh(A)l(cknow)t(le)l(dgments.)38 b Fs(DB)28 b(w)n(ould)f(lik)n(e)g(to)g(thank)g(P)n(anos)f(Kevrekidis)g(for)h(p)r (oin)n(ting)g(to)g(his)456 1845 y(atten)n(tion)i(the)h(pap)r(er)f([23)o (])h(and)f(the)h(connection)f(b)r(et)n(w)n(een)h(the)g (Gross-Pitaevskii)d(and)i(the)456 1944 y(discrete)e(NLS)h(equations.) 1622 2114 y(2.)41 b Fr(Main)32 b(resul)-6 b(ts)456 2263 y Fs(2.1.)40 b Fx(Linear)33 b(theory.)456 2409 y(Hyp)s(othesis)d(1.)41 b Fh(The)30 b(p)l(otential)h Fp(V)19 b Fs(\()p Fp(x)p Fs(\))24 b Fl(2)f Fp(C)1922 2379 y Fg(1)1993 2409 y Fs(\()p Fo(R)p Fs(\))36 b Fh(is)30 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Fh(;)636 3970 y Fs(v.)41 b Fh(The)31 b(shap)l(e)h(of)f(the)f(p)l(otential)h(at)f (the)g(b)l(ottom)g(of)h(the)f(minimum)g Fp(x)2885 3982 y Fn(j)2950 3970 y Fh(is)h(appr)l(oxima-)744 4069 y(tively)g(indep)l (endent)f(of)h Fp(j)5 b Fh(;)30 b(pr)l(e)l(cisely:)40 b(ther)l(e)30 b(exists)f Fp(r)d Fl(\025)d Fs(4)29 b Fh(such)h(that)881 4199 y Fp(d)924 4169 y Fn(k)965 4199 y Fp(V)p 881 4236 151 4 v 891 4312 a(dx)981 4288 y Fn(k)1042 4255 y Fs(\()p Fp(x)1121 4267 y Fn(j)1156 4255 y Fs(\))24 b(=)1309 4199 y Fp(d)1352 4169 y Fn(k)1393 4199 y Fp(V)p 1309 4236 V 1319 4312 a(dx)1409 4288 y Fn(k)1470 4255 y Fs(\()p Fp(x)1549 4267 y Fn(i)1577 4255 y Fs(\))p Fp(;)74 b Fl(8)p Fp(i;)14 b(j)27 b Fs(=)22 b(1)p Fp(;)14 b Fs(2)p Fp(;)g(:)g(:)g(:)f(;)h (n;)73 b Fl(8)p Fp(k)25 b Fs(=)e(2)p Fp(;)14 b(:::;)g(r)n(:)474 b Fs(\(6\))555 4424 y(Hereafter,)26 b Fp(C)33 b Fs(will)26 b(alw)n(a)n(ys)f(denote)h(a)g(p)r(ositiv)n(e,)g(t)n(ypically)g(large,)f (constan)n(t)h(whose)g(v)-5 b(alue)456 4524 y(c)n(hanges)26 b(from)h(line)h(to)f(line,)h(and)g(whic)n(h)f(is)h(indep)r(enden)n(t)g (of)g Fo(~)p Fs(,)f Fp(\017)h Fs(and)f Fp(t)p Fs(.)555 4623 y(The)32 b(op)r(erator)e Fp(H)1138 4635 y Fm(0)1207 4623 y Fs(formally)h(de\014ned)h(b)n(y)f(\(2\))h(admits)g(a)f (self-adjoin)n(t)h(realization)e(\(still)456 4723 y(denoted)h(b)n(y)g Fp(H)961 4735 y Fm(0)998 4723 y Fs(\))h(on)f Fp(L)1238 4693 y Fm(2)1275 4723 y Fs(\()p Fo(R)p Fs(\))38 b(\(Theorem)31 b(I)r(I)r(I.1.1)g(in)g([9]\))h(with)f(purely)g(discrete)g(sp)r(ectrum.) 456 4822 y(Let)c Fp(\025)652 4834 y Fn(k)694 4822 y Fs(,)g Fp(k)f Fl(2)e Fo(N)t Fs(,)33 b(b)r(e)28 b(the)g(non)g(degenerate)e (eigen)n(v)-5 b(alues)27 b(of)g Fp(H)2428 4834 y Fm(0)994 4981 y Fp(\025)1042 4993 y Fm(1)1102 4981 y Fp(<)c(\025)1238 4993 y Fm(2)1299 4981 y Fp(<)f(\025)1434 4993 y Fm(3)1495 4981 y Fp(<)h(\025)1631 4993 y Fm(4)1691 4981 y Fp(<)g(:)14 b(:)g(:)23 b(<)g(\025)2035 4993 y Fn(k)2099 4981 y Fp(<)g(:)14 b(:)g(:)f(;)42 b(\025)2410 4993 y Fn(k)2474 4930 y(k)q Fg("1)2501 4981 y Fl(!)50 b(1)p Fp(;)456 5116 y Fs(with)35 b(asso)r(ciated)f(normalized)h(\(in)g Fp(L)1681 5086 y Fm(2)1718 5116 y Fs(\))h(eigen)n(v)n(ectors)d Fp(')2312 5128 y Fn(k)2353 5116 y Fs(\()p Fp(x)p Fs(\);)40 b(the)c(set)f Fl(f)p Fp(')2911 5128 y Fn(k)2952 5116 y Fs(\()p Fp(x)p Fs(\))p Fl(g)3105 5086 y Fg(1)3105 5139 y Fn(k)q Fm(=1)3266 5116 y Fs(is)g(an)456 5216 y(orthonormal)25 b(base)i(of)h Fp(L)1268 5185 y Fm(2)1304 5216 y Fs(.)p eop %%Page: 5 5 5 4 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)1012 b(5)555 450 y Fs(The)30 b(lo)n(w)n(est)e(part)h(of)g(the)h(sp)r(ectrum) g(can)f(b)r(e)h(studied)f(in)h(the)g(semiclassical)e(limit)i(using)456 550 y(the)e(construction)e(of)i([12)o(],)g(that)g(w)n(e)f(shortly)g (recall.)555 649 y(Ha)n(ving)22 b(\014xed)h(a)f(p)r(ositiv)n(e)g (constan)n(t)g Fp(a)h(>)g Fs(1)g(=)f Fp(V)2081 661 y Fn(min)2232 649 y Fs(w)n(e)g(consider)g(the)h(set)g Fp(V)2998 619 y Fg(\000)p Fm(1)3087 649 y Fs(\(\()p Fl(\0001)p Fp(;)14 b(a)p Fs(\)\))456 749 y(and)30 b(w)n(e)g(assume)f(that)i Fp(a)f Fs(is)h(suc)n(h)f(that)g(this)h(set)f(is)h(the)f(union)h(of)f Fp(n)g Fs(disjoin)n(t)h(op)r(en)f(sets)g Fl(U)3409 761 y Fn(j)456 849 y Fs(with)e Fp(x)692 861 y Fn(j)750 849 y Fl(2)c(U)881 861 y Fn(j)916 849 y Fs(.)64 b(Ha)n(ving)27 b(\014xed)g Fp(j)33 b Fs(w)n(e)27 b(consider)g(the)h(op)r(erator)e (formally)g(de\014ned)i(on)g Fp(L)3264 818 y Fm(2)3328 849 y Fs(as)1520 1044 y Fp(H)1589 1056 y Fn(j)1647 1044 y Fs(=)23 b Fl(\000)p Fo(~)1845 1009 y Fm(2)1915 987 y Fp(d)1958 957 y Fm(2)p 1891 1024 128 4 v 1891 1101 a Fp(dx)1981 1077 y Fm(2)2048 1044 y Fs(+)18 b Fp(V)2179 1056 y Fn(j)456 1206 y Fs(where)27 b Fp(V)744 1218 y Fn(j)807 1206 y Fs(is)g(a)g(mo)r(di\014ed)h(p)r(oten)n(tial)g (de\014ned)g(as)1155 1397 y Fp(V)1203 1409 y Fn(j)1239 1397 y Fs(\()p Fp(x)p Fs(\))c(=)1462 1280 y Fq(\032)1565 1347 y Fp(V)19 b Fs(\()p Fp(x)p Fs(\))393 b(for)55 b Fp(x)24 b Fl(2)f(U)2492 1359 y Fn(j)1565 1446 y Fs(max[)p Fp(a;)14 b(V)19 b Fs(\()p Fp(x)p Fs(\)])111 b(for)55 b 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Fn(i)1417 3467 y Fl(k)1458 3495 y Fn(L)1504 3479 y Fe(1)1592 3467 y Fl(\024)22 b Fp(C)6 b(e)1783 3433 y Fg(\000)p Fm(\000)p Fn(=)p Ff(~)1951 3467 y Fp(;)42 b(i)22 b Fl(6)p Fs(=)h Fp(j;)1126 b Fs(\(9\))1204 3608 y Fl(j)p Fp(\025)1275 3620 y Fn(i)1298 3628 y Fk(j)1353 3608 y Fl(\000)1439 3586 y Fs(^)1436 3608 y Fp(\025)1484 3620 y Fn(j)1519 3608 y Fl(j)23 b(\024)g Fp(C)6 b(e)1757 3574 y Fg(\000)p Fm(\000)p Fn(=)p Ff(~)1980 3608 y Fs(for)27 b(some)55 b Fp(i)2372 3620 y Fn(j)2429 3608 y Fl(2)24 b(f)o Fs(1)p Fp(;)14 b(:::;)g(n)p Fl(g)f Fp(:)435 b Fs(\(10\))456 3763 y(Moreo)n(v)n(er,)29 b(the)j(pro)5 b(jector)1367 3742 y(^)1357 3763 y(\005)31 b(on)1579 3742 y(^)1569 3763 y(\010)1629 3775 y Fm(0)1696 3763 y Fs(:=span\()13 b(^)-55 b Fp(')2037 3775 y Fn(j)2072 3763 y Fs(\))32 b(is)f(a)g(bijection)h(among)e(\010)2970 3775 y Fm(0)3036 3763 y Fs(:=span\()p Fp(')3377 3775 y Fn(j)3412 3763 y Fs(\))456 3874 y(and)624 3853 y(^)615 3874 y(\010)675 3886 y Fm(0)737 3874 y Fs(itself.)62 b(Remark)25 b(also)f(that)h(one)g (can)g(c)n(ho)r(ose)f(the)i(functions)39 b(^)-55 b Fp(')2759 3886 y Fn(j)2819 3874 y Fs(to)25 b(b)r(e)h(real)e(v)-5 b(alued.)583 3973 y(Hence,)28 b(the)g(lo)n(w)n(est)e Fp(n)i Fs(eigen)n(v)-5 b(alues)26 b(of)i Fp(H)1916 3985 y Fm(0)1981 3973 y Fs(ful\014ll)1211 4112 y Fp(C)1276 4078 y Fg(\000)p Fm(1)1365 4112 y Fo(~)23 b Fp(<)g(\025)1569 4124 y Fn(j)1623 4112 y Fl(\000)18 b Fs(1)k Fp(<)h(C)6 b Fo(~)28 b Fp(;)97 b(j)28 b Fs(=)22 b(1)p Fp(;)14 b(:::;)g(n;)456 4251 y Fs(for)27 b(some)g Fp(C)i(>)23 b Fs(1)k(and)1419 4411 y Fp(!)e Fs(=)1594 4355 y(1)p 1594 4392 42 4 v 1594 4468 a(2)1645 4411 y(\()p Fp(\025)1725 4423 y Fn(n)1790 4411 y Fl(\000)18 b Fp(\025)1921 4423 y Fm(1)1958 4411 y Fs(\))24 b Fl(\024)e Fp(C)6 b Fo(~)2211 4377 y Fn(r)r(=)p Fm(2)3297 4411 y Fs(\(11\))456 4573 y(F)-7 b(urthermore,)26 b(making)h(use)h(of)f(the)h(same)f(argumen)n(ts,)f(it)i(follo)n(ws)f (also)g(that)1184 4712 y(inf)1107 4764 y Fn(j)s Fm(=1)p Fn(;:::;n)1670 4712 y Fs(inf)1403 4769 y Fn(\025)p Fg(2)p Fn(\033)r Fm(\()p Fn(H)1607 4777 y Fj(0)1641 4769 y Fm(\))p Fg(\000f)p Fn(\025)1792 4777 y Fj(1)1824 4769 y Fn(;:::;\025)1963 4777 y Fk(n)2003 4769 y Fg(g)2037 4712 y Fs([)p Fp(\025)19 b Fl(\000)f Fp(\025)2258 4724 y Fn(j)2294 4712 y Fs(])23 b Fl(\025)f Fp(C)2492 4678 y Fg(\000)p Fm(1)2582 4712 y Fo(~)670 b Fs(\(12\))456 4897 y(W)-7 b(e)28 b(also)e(use)i(the)g (follo)n(wing)e(pro)5 b(jectors)874 5098 y(\005)24 b(=)1087 4994 y Fn(n)1047 5019 y Fq(X)1050 5196 y Fn(j)s Fm(=1)1167 5098 y Fl(h)p Fp(')1253 5110 y Fn(j)1289 5098 y Fp(;)14 b Fl(\001i)p Fp(')1435 5110 y Fn(j)1553 5098 y Fs(and)83 b(\005)1832 5110 y Fn(c)1889 5098 y Fs(=)23 b Fo(I)6 b Fl(\000)18 b Fs(\005)83 b(and)g(\010)2533 5110 y Fm(0)2593 5098 y Fs(=)23 b(\005)p Fp(L)2800 5064 y Fm(2)2837 5098 y Fs(;)p eop %%Page: 6 6 6 5 bop 456 251 a Fm(6)736 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f(SA) n(CCHETTI)456 450 y Fx(Remark)37 b(1.)44 b Fh(As)34 b(alr)l(e)l(ady)i (emphasize)l(d)h(in)e(the)g(intr)l(o)l(duction)f(the)h(most)g(inter)l (esting)f(situ-)456 550 y(ation)i(o)l(c)l(curs)f(when)h(the)g(true)f (eigenfunctions)h Fp(')2085 562 y Fn(j)2156 550 y Fh(ar)l(e)g(delo)l(c) l(alize)l(d)i(b)l(etwe)l(en)e(the)f(wel)t(ls,)k(a)456 649 y(pr)l(op)l(erty)29 b(that)f(o)l(c)l(curs)g(if)g(the)h(p)l (otential)g(is)f(exactly)g(p)l(erio)l(dic)j(in)d(some)g(r)l(e)l(gion)h (or)f(mor)l(e)h(gen-)456 749 y(er)l(al)t(ly)38 b(if)h(the)e(or)l(der)h (of)h(magnitude)e(of)i(some)f(the)f(o\013)h(diagonal)h(elements)f(of)g (the)g(matrix)456 849 y(forme)l(d)i(by)g(the)f(c)l(onstants)f Fp(c)1417 861 y Fn(ij)1515 849 y Fh(is)h(of)h(the)g(same)f(or)l(der)h (of)g(magnitude)g(as)f(the)h(di\013er)l(enc)l(e)456 960 y(b)l(etwe)l(en)30 b(the)h(appr)l(oximate)h(eigenvalues)1809 938 y Fs(^)1805 960 y Fp(\025)1853 972 y Fn(j)1889 960 y Fh(,)f(a)g(pr)l(op)l(erty)g(that)g(is)g(quite)f(di\016cult)i(to)e (ensur)l(e)456 1060 y(in)d(a)h(gener)l(al)g(situation.)65 b(F)-6 b(or)28 b(this)g(r)l(e)l(ason,)g(essential)t(ly)h(in)e(or)l(der) h(to)g(\014x)e(ide)l(as,)k(we)e(de)l(cide)l(d)456 1159 y(to)h(state)h(our)f(assumptions)h(in)g(the)g(form)h(\(i-v\))f(ab)l (ove.)456 1298 y Fs(2.2.)40 b Fx(The)32 b(nonlinear)g(system:)40 b(Analytic)32 b(framew)m(ork)g(and)g(w)m(ell)f(p)s(osedness.)456 1448 y(De\014nition)45 b(1.)j Fh(F)-6 b(or)41 b(any)g(inte)l(ger)g Fp(s)j Fl(\025)f Fs(0)e Fh(de\014ne)g(the)g(Hilb)l(ert)g(sp)l(ac)l(e)h Fl(X)2920 1418 y Fn(s)2999 1448 y Fs(:=)h Fp(D)r Fs(\()p Fp(H)3309 1405 y Fn(s=)p Fm(2)3302 1470 y(0)3412 1448 y Fs(\))456 1547 y Fh(endowe)l(d)34 b(by)f(the)h(gr)l(aph)g(norm;)h (mor)l(e)f(pr)l(e)l(cisely)g(in)f Fl(X)2250 1517 y Fn(s)2319 1547 y Fh(we)h(wil)t(l)g(use)f(the)g(fol)t(lowing)i(norm)456 1647 y(e)l(quivalent)30 b(to)f(the)h(gr)l(aph)h(norm)828 1819 y Fl(k)o Fp(\036)p Fl(k)960 1778 y Fm(2)960 1844 y Fn(s)1021 1819 y Fs(:=)1131 1724 y Fq(\015)1131 1774 y(\015)1131 1823 y(\015)1178 1819 y Fp(H)1254 1776 y Fn(s=)p Fm(2)1247 1841 y(0)1356 1819 y Fp(\036)1405 1724 y Fq(\015)1405 1774 y(\015)1405 1823 y(\015)1451 1744 y Fm(2)1451 1877 y Fn(L)1497 1861 y Fj(2)1557 1819 y Fl(\021)22 b(h)q Fp(H)1753 1785 y Fn(s)1746 1840 y Fm(0)1788 1819 y Fp(\036;)14 b(\036)p Fl(i)1956 1844 y Fn(L)2002 1828 y Fj(2)2061 1819 y Fs(=)2149 1706 y Fq(Z)2195 1895 y Ff(R)2266 1797 y Fs(\026)2255 1819 y Fp(\036H)2380 1785 y Fn(s)2373 1840 y Fm(0)2416 1819 y Fp(\036)p Fs(d)p Fp(x;)44 b(\036)24 b Fl(2)f(X)2847 1785 y Fn(s)2883 1819 y Fp(:)391 b Fs(\(13\))555 2000 y(The)33 b(main)g(step)g(for)g(the)g (pro)r(of)g(that)g(the)g(spaces)f Fl(X)2293 1969 y Fn(s)2362 2000 y Fs(form)h(a)g(Banac)n(h)e(algebra)h(under)456 2099 y(the)c(p)r(oin)n(t)n(wise)f(m)n(ultiplication)g(is)h(the)g(follo) n(wing)e(Theorem.)456 2238 y Fx(Theorem)35 b(1.)43 b Fh(L)l(et)33 b Fp(s)h Fh(b)l(e)g(any)g(p)l(ositive)h(inte)l(ger)e(numb) l(er.)84 b(F)-6 b(or)33 b(smal)t(l)i(enough)f Fo(~)f Fh(the)h(two)456 2337 y(norms)1079 2472 y Fl(k)o Fp(\036)p Fl(k)1211 2430 y Fm(2)1211 2497 y Fn(s)1351 2472 y Fh(and)90 b Fl(k)p Fs(\()p Fl(\000)p Fo(~)1756 2437 y Fm(2)1793 2472 y Fs(\001\))1894 2437 y Fn(s=)p Fm(2)1997 2472 y Fp(\036)p Fl(k)2088 2437 y Fm(2)2088 2494 y Fn(L)2134 2477 y Fj(2)2188 2472 y Fs(+)18 b Fl(k)p Fp(V)2380 2437 y Fn(s=)p Fm(2)2482 2472 y Fp(\036)p Fl(k)2573 2437 y Fm(2)2573 2494 y Fn(L)2619 2477 y Fj(2)3297 2472 y Fs(\(14\))456 2606 y Fh(ar)l(e)30 b(e)l(quivalent)g(with)g(an)g Fo(~)g Fh(indep)l(endent)g(c)l(onstant.)555 2745 y Fs(The)f(pro)r(of,)f(whic)n (h)h(is)f(a)h(semiclassical)e(v)-5 b(arian)n(t)27 b(of)i(the)g(pro)r (of)f(of)h(Lemma)f(7.2)g(of)g([26)o(],)i(is)456 2844 y(deferred)d(to)g(App)r(endix)i(A.)456 2983 y Fx(Remark)24 b(2.)35 b Fh(In)24 b(p)l(articular)h(one)g(has)g(that)f(a)h(function)f Fp(\036)h Fh(is)g(in)f Fl(X)2571 2952 y Fn(s)2631 2983 y Fh(if)i(and)e(only)h(if)g(it)g(b)l(elongs)456 3082 y(to)k(the)h(Sob)l(olev)h(sp)l(ac)l(e)f Fp(H)1278 3052 y Fn(s)1343 3082 y Fh(and)h(it)e(de)l(c)l(ays)i(at)f(in\014nity)f(so)h (fast)h(that)e Fl(j)p Fp(\036)p Fl(j)2762 3052 y Fm(2)2800 3082 y Fp(V)2867 3052 y Fn(s)2932 3082 y Fh(is)h(inte)l(gr)l(able.)555 3221 y Fs(In)g(the)g(spaces)f Fl(X)1134 3190 y Fn(s)1200 3221 y Fs(with)h Fp(s)d Fl(\025)g Fs(1)i(the)h(system)g(\(1\))g(is)g (semilinear,)f(since,)h(using)h(\(14\))f(and)456 3320 y(Gagliardo-Nirem)n(b)r(erg)24 b(inequalit)n(y)j(one)g(has)456 3459 y Fx(Corollary)g(1.)37 b Fh(F)-6 b(or)25 b(any)h(inte)l(ger)f Fp(s)e Fl(\025)g Fs(1)h Fh(ther)l(e)i(exists)f(a)g(p)l(ositive)i(c)l (onstant)e Fp(C)2953 3471 y Fn(s)3014 3459 y Fh(indep)l(endent)456 3558 y(of)30 b Fo(~)p Fh(,)g(such)g(that)859 3695 y Fl(k)o Fp(\036)949 3707 y Fm(1)987 3695 y Fp(\036)1036 3707 y Fm(2)1074 3695 y Fl(k)1115 3720 y Fn(s)1174 3695 y Fl(\024)22 b Fp(C)1320 3707 y Fn(s)1356 3695 y Fo(~)1401 3661 y Fg(\000)p Fm(1)p Fn(=)p Fm(2)1571 3695 y Fl(k)o Fp(\036)1661 3707 y Fm(1)1699 3695 y Fl(k)1741 3720 y Fn(s)1790 3695 y Fl(k)p Fp(\036)1881 3707 y Fm(2)1918 3695 y Fl(k)1960 3720 y Fn(s)2009 3695 y Fp(;)73 b Fl(8)p Fp(\036)2201 3707 y Fm(1)2238 3695 y Fp(;)14 b(\036)2324 3707 y Fm(2)2385 3695 y Fl(2)23 b(X)2534 3661 y Fn(s)2570 3695 y Fp(;)44 b Fo(~)23 b Fl(\034)g Fs(1)p Fp(:)421 b Fs(\(15\))456 3830 y Fh(Mor)l(e)l(over,)31 b(the)f(map)1278 3964 y Fl(X)1349 3930 y Fn(s)1404 3964 y Fl(\002)18 b(X)1558 3930 y Fn(s)1617 3964 y Fl(3)23 b Fs(\()p Fp(\036;)1824 3942 y Fs(\026)1813 3964 y Fp(\036)q Fs(\))g Fl(7!)g(j)p Fp(\036)p Fl(j)2119 3930 y Fm(2)p Fn(\033)2198 3964 y Fp(\036)g Fl(2)h(X)2420 3930 y Fn(s)456 4099 y Fh(is)30 b(entir)l(e)f(analytic)i(map)g(for)f(any)g Fo(~)23 b Fp(>)g Fs(0)29 b Fh(and)h(ful\014l)t(ls)1346 4167 y Fq(\015)1346 4217 y(\015)1392 4237 y Fl(j)p Fp(\036)p Fl(j)1487 4203 y Fm(2)p Fn(\033)1565 4237 y Fp(\036)1614 4167 y Fq(\015)1614 4217 y(\015)1661 4271 y Fn(s)1719 4237 y Fl(\024)23 b Fp(C)1866 4249 y Fn(s)1902 4237 y Fo(~)1947 4203 y Fg(\000)p Fn(\033)2057 4237 y Fl(k)p Fp(\036)p Fl(k)2190 4196 y Fm(2)p Fn(\033)r Fm(+1)2190 4262 y Fn(s)2365 4237 y Fp(:)909 b Fs(\(16\))456 4380 y Fx(Remark)30 b(3.)41 b Fh(In)30 b(dimension)h Fp(d)23 b(>)f Fs(1)30 b Fh(this)g(r)l(esult)f(r)l(emains) h(valid)h(pr)l(ovide)l(d)h Fp(s)23 b(>)g(d=)p Fs(2)p Fh(.)555 4518 y Fs(Then)h(b)n(y)f(standard)f(Segal)h(theory)f(\(see)i (e.g.)35 b([16)o(]\))23 b(the)h(system)f(\(1\))h(is)f(lo)r(cally)g(w)n (ell)g(p)r(osed)456 4618 y(in)30 b(all)h(the)g(spaces)e Fl(X)1148 4588 y Fn(s)1215 4618 y Fs(with)i Fp(s)d Fl(\025)g Fs(1.)76 b(Actually)31 b Fp(s)d(>)g(d=)p Fs(2)h(is)i(enough,)g(but)g (our)f(pro)r(of)g(only)456 4717 y(applies)e(to)g(in)n(teger)g(v)-5 b(alues)29 b(of)f Fp(s)p Fs(;)i(in)e([26])g(a)h(Stric)n(hartz)e (inequalit)n(y)i(argumen)n(t)e(w)n(as)h(used)h(to)456 4817 y(sho)n(w)d(that)i(it)g(is)g(also)e(lo)r(cally)h(w)n(ell)g(p)r (osed)h(\(L)-9 b(WP\))28 b(in)f Fl(X)2283 4787 y Fn(s)2347 4817 y Fs(with)h(some)f Fp(s)h Fs(smaller)e(than)i Fp(d=)p Fs(2.)555 4917 y(F)-7 b(rom)27 b(no)n(w)g(on)f(w)n(e)h(assume)g(that)g (the)h(index)f Fp(s)g Fs(of)g(the)h(space)f(is)g(a)f(\014xed)i(p)r (ositiv)n(e)e(in)n(teger)456 5016 y(n)n(um)n(b)r(er)g(and)g(ful\014lls) h(the)g(condition)f Fp(s)d Fl(\025)f Fs(1.)63 b(In)26 b(the)h(follo)n(wing,)f(in)g(order)f(to)h(\014x)h(ideas,)f(one)456 5116 y(can)f(just)h(think)g(of)f(the)h(case)e Fp(s)f Fs(=)g(1.)61 b(W)-7 b(e)26 b(will)g(denote)f(b)n(y)g Fp(d)p Fs(\()p Fp(:)p Fs(;)14 b Fp(:)p Fs(\))27 b(the)e(distance)h(in)f (the)h(norm)456 5216 y(of)h Fl(X)621 5185 y Fn(s)657 5216 y Fs(.)p eop %%Page: 7 7 7 6 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)1012 b(7)456 450 y Fs(2.3.)40 b Fx(The)31 b(nonlinear)f(system:)40 b(Appro)m(ximativ)m(ely)30 b(in)m(v)-5 b(arian)m(t)32 b(manifold.)39 b Fs(In)27 b(order)456 550 y(to)22 b(state)h(our)f(main) g(result)h(w)n(e)f(assume)g(that)h(the)g(size)g(of)f(the)h(nonlinearit) n(y)f(is)h(small)f(enough,)456 649 y(i.e.)37 b Fl(j)p Fp(\017)p Fl(j)22 b(\034)i Fo(~)853 619 y Fn(\033)897 649 y Fs(,)k(and)f(w)n(e)h(in)n(tro)r(duce)f(the)h(small)f(parameter.) 1545 851 y Fp(\026)d Fs(:=)e Fp(!)f Fs(+)1900 794 y Fl(j)p Fp(\017)p Fl(j)p 1895 831 90 4 v 1895 907 a Fo(~)1940 883 y Fn(\033)2018 851 y Fl(\034)i Fs(1)p Fp(;)1108 b Fs(\(17\))456 1032 y(where)27 b Fp(!)j Fs(w)n(as)d(de\014ned)h(in)f (\(11\).)456 1192 y Fx(Theorem)40 b(2.)46 b Fh(Consider)40 b(the)e(system)44 b Fs(\(1\))38 b Fh(and)h(\014x)e(a)h(p)l(ositive)h Fp(s)f Fl(\025)g Fs(1)p Fh(.)63 b(Ther)l(e)38 b(exists)g(a)456 1295 y(p)l(ositive)29 b Fp(\026)803 1307 y Fg(\003)869 1295 y Fh(such)f(that,)h(if)g Fp(\026)23 b(<)g(\026)1540 1307 y Fg(\003)1578 1295 y Fo(~)1623 1264 y Fm(3)p Fn(=)p Fm(2)1727 1295 y Fh(,)29 b(then)f(ther)l(e)g(exists)g(a)g(manifold)i Fl(M)e Fh(\(dep)l(endent)g(on)456 1394 y(al)t(l)i(the)g(p)l(ar)l (ameters)h(of)f(the)g(system\))g(with)g(the)g(fol)t(lowing)i(pr)l(op)l (erties:)657 1521 y Fs(i.)41 b Fl(M)30 b Fh(is)g(close)h(to)e Fs(\010)1326 1533 y Fm(0)1363 1521 y Fh(,)i(i.e.)1512 1691 y Fp(d)p Fs(\(\010)1647 1703 y Fm(0)1685 1691 y Fp(;)14 b Fl(M)p Fs(\))23 b Fl(\024)f Fp(C)2089 1635 y(\026)p 2040 1672 150 4 v 2040 1749 a Fo(~)2085 1725 y Fm(3)p Fn(=)p Fm(2)2199 1691 y Fp(;)1075 b Fs(\(18\))744 1869 y Fh(wher)l(e)1281 2022 y Fp(d)p Fs(\(\010)1416 2034 y Fm(0)1454 2022 y Fp(;)14 b Fl(M)p Fs(\))23 b(=)45 b(sup)1734 2092 y Fn( )r Fg(2)p Fm(\010)1872 2100 y Fj(0)1951 2022 y Fs(inf)1918 2076 y Fn(')p Fg(2M)2098 2022 y Fl(k)p Fp( )21 b Fl(\000)d Fp(')p Fl(k)2394 2034 y Fn(s)2430 2022 y Fp(;)744 2236 y Fh(and)31 b(wher)l(e)f Fl(k)p Fp(')p Fl(k)1278 2248 y Fn(s)1343 2236 y Fh(is)g(the)g(norm)36 b Fs(\(13\))o Fh(.)634 2336 y Fs(ii.)41 b Fh(L)l(et)1272 2489 y Fp(d)1315 2501 y Fm(0)1376 2489 y Fs(=)22 b Fp(d)p Fs(\()p Fp( )1595 2455 y Fm(0)1633 2489 y Fp(;)14 b Fl(M)p Fs(\))23 b(=)56 b(inf)1913 2543 y Fn(')p Fg(2M)2093 2489 y Fl(k)p Fp( )2192 2455 y Fm(0)2247 2489 y Fl(\000)18 b Fp(')p Fl(k)2426 2501 y Fn(s)744 2682 y Fh(b)l(e)30 b(the)g(initial)h(distanc)l(e)f(fr)l(om)h Fl(M)e Fh(and)h(let)1311 2890 y Fp(\016)c Fs(=)d(max)1630 2773 y Fq(\032)1693 2890 y Fp(d)1736 2902 y Fm(0)1773 2890 y Fp(;)14 b Fs(exp)1951 2773 y Fq(\024)1995 2890 y Fl(\000)2070 2834 y Fp(\026)2120 2846 y Fg(\003)2157 2834 y Fo(~)2202 2804 y Fm(3)p Fn(=)p Fm(2)p 2069 2871 238 4 v 2142 2947 a Fs(2)p Fp(\026)2316 2773 y Fq(\025\033)3297 2890 y Fs(\(19\))744 3088 y Fh(Then)31 b(for)f(al)t(l)h(times)f Fp(t)g Fh(ful\014l)t(ling)1664 3276 y Fl(j)p Fp(t)p Fl(j)23 b(\024)1940 3220 y Fs(1)p 1860 3257 201 4 v 1860 3333 a Fp(C)6 b Fo(~)p Fp(\026\016)3297 3276 y Fs(\(20\))744 3469 y Fh(one)30 b(has)1705 3622 y Fp(d)p Fs(\()p Fp( )1837 3588 y Fn(t)1867 3622 y Fp(;)14 b Fl(M)p Fs(\))23 b Fl(\024)g Fp(C)6 b(\016)1048 b Fs(\(21\))1705 3697 y Fq(\015)1705 3747 y(\015)1752 3768 y Fs(\005)1814 3780 y Fn(c)1848 3768 y Fp( )1905 3734 y Fn(t)1934 3697 y Fq(\015)1934 3747 y(\015)1980 3801 y Fn(s)2039 3768 y Fl(\024)22 b Fp(C)2251 3712 y(\026)p 2201 3749 150 4 v 2201 3826 a Fo(~)2246 3802 y Fm(3)p Fn(=)p Fm(2)3297 3768 y Fs(\(22\))555 3946 y Fh(Such)30 b(a)g(manifold)i Fl(M)d Fh(is)h(c)l(al)t(le)l(d)h(an)36 b Fs(appro)n(ximativ)n(ely)26 b(in)n(v)-5 b(arian)n(t)26 b(manifold)p Fh(.)456 4106 y Fx(Remark)k(4.)41 b Fh(The)31 b(most)f(inter)l(esting)f(c)l(ases)h (ar)l(e)g(when)1178 4314 y Fp(\016)c Fs(=)c(exp)1469 4197 y Fq(\024)1513 4314 y Fl(\000)1588 4258 y Fp(\026)1638 4270 y Fg(\003)1676 4258 y Fo(~)1721 4228 y Fm(3)p Fn(=)p Fm(2)p 1587 4295 238 4 v 1660 4371 a Fs(2)p Fp(\026)1835 4197 y Fq(\025)1982 4314 y Fh(or)89 b Fp(\016)26 b Fs(=)d Fp(C)2424 4258 y(\026)p 2374 4295 150 4 v 2374 4373 a Fo(~)2419 4349 y Fm(3)p Fn(=)p Fm(2)2533 4314 y Fp(:)456 4513 y Fh(Inde)l(e)l(d)j(in)f(the)h(\014rst)e(c)l(ase)i(the)g(time)g (of)g(validity)i(of)e(al)t(l)g(the)g(estimates)g(is)g(exp)l(onential)t (ly)g(long,)456 4612 y(while)31 b(in)f(the)g(se)l(c)l(ond)g(c)l(ase)g (it)g(is)g(e)l(asy)g(to)g(obtain)g(the)g(fol)t(lowing)i(Cor)l(ol)t (lary.)456 4772 y Fx(Corollary)h(2.)40 b Fh(Assume)29 b(that)1477 4702 y Fq(\015)1477 4752 y(\015)1523 4772 y Fs(\005)1585 4784 y Fn(c)1619 4772 y Fp( )1676 4742 y Fm(0)1714 4702 y Fq(\015)1714 4752 y(\015)1760 4806 y Fn(s)1818 4772 y Fl(\024)23 b Fp(C)6 b(\026)p Fo(~)2066 4742 y Fm(3)p Fn(=)p Fm(2)2200 4772 y Fh(then,)30 b(up)g(to)g(the)g (times)1687 4997 y Fl(j)p Fp(t)p Fl(j)24 b(\024)1886 4941 y Fo(~)1931 4911 y Fm(3)p Fn(=)p Fm(2)p 1884 4978 153 4 v 1884 5054 a Fp(C)6 b(\026)1999 5030 y Fm(2)3297 4997 y Fs(\(23\))456 5195 y Fh(the)30 b(estimate)36 b Fs(\(22\))29 b Fh(holds.)p eop %%Page: 8 8 8 7 bop 456 251 a Fm(8)736 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f(SA) n(CCHETTI)456 450 y Fx(Remark)39 b(5.)45 b Fh(This)38 b(Cor)l(ol)t(lary)h(is)e(a)f(dir)l(e)l(ct)h(extension)f(of)i(the)e(r)l (esults)g(of)h Fs([11,)e(20)o(,)g(21)o(])h Fh(in)456 550 y(which)e(the)f(estimate)40 b Fs(\(22\))32 b Fh(has)i(b)l(e)l(en)e (pr)l(ove)l(d)i(\(for)g(the)f(double)h(wel)t(l)g(p)l(otential\))f(for)h (a)f(time)456 649 y(sc)l(ale)d(of)h(or)l(der)1381 847 y Fp(T)j Fs(=)1562 791 y Fp(\031)s Fo(~)p 1562 828 96 4 v 1582 904 a Fp(!)1690 847 y Fl(\024)22 b Fp(C)1855 791 y Fo(~)p 1853 828 51 4 v 1853 904 a Fp(\026)1936 847 y Fl(\034)h Fp(\026)2092 813 y Fg(\000)p Fm(2)2181 847 y Fo(~)2226 813 y Fm(3)p Fn(=)p Fm(2)2330 847 y Fp(:)456 1046 y Fh(The)29 b(impr)l(ovment)g(is)g(due)g(to)f(the)h(fact)g(that)f (our)h(c)l(onstruction)f(implies)i(that)e Fl(M)g Fh(is)h(line)l(arly) 456 1146 y(stable)h(up)g(to)f(an)h(exp)l(onential)t(ly)h(smal)t(l)g (err)l(or.)456 1310 y Fs(2.4.)40 b Fx(The)d(nonlinear)g(system:)50 b(discrete)37 b(NLS)g(and)h(suppression)d(of)i(tunneling.)456 1409 y Fs(T)-7 b(o)26 b(start)g(with)g(w)n(e)h(remark)d(that)j(NLS)g (is)f(a)g(Hamiltonian)g(system)h(\(see)f(Subsection)g(3.1)g(for)456 1509 y(a)h(precise)g(description\))g(with)h(Hamiltonian)f(function)i (giv)n(en)d(b)n(y)1295 1669 y Fl(E)7 b Fs(\()p Fp( )s(;)1490 1647 y Fs(\026)1472 1669 y Fp( )t Fs(\))23 b(:=)g Fl(E)1740 1681 y Fm(0)1777 1669 y Fs(\()p Fp( )s(;)1920 1647 y Fs(\026)1903 1669 y Fp( )s Fs(\))c(+)f Fp(\017)p Fl(P)2186 1681 y Fm(0)2223 1669 y Fs(\()p Fp( )s(;)2366 1647 y Fs(\026)2349 1669 y Fp( )s Fs(\))859 b(\(24\))456 1828 y(where)1331 2010 y Fl(E)1375 2022 y Fm(0)1412 2010 y Fs(\()p Fp( )s(;)1555 1988 y Fs(\026)1538 2010 y Fp( )t Fs(\))23 b(:=)1762 1897 y Fq(Z)1808 2085 y Ff(R)1885 1988 y Fs(\026)1868 2010 y Fp( )s Fs(\()p Fp(x)p Fs(\))p Fp(H)2105 2022 y Fm(0)2143 2010 y Fp( )s Fs(\()p Fp(x)p Fs(\))p Fp(dx)456 2211 y Fs(and)898 2392 y Fl(P)956 2404 y Fm(0)993 2392 y Fs(\()p Fp( )s(;)1136 2371 y Fs(\026)1119 2392 y Fp( )s Fs(\))h(:=)1428 2336 y(1)p 1352 2373 194 4 v 1352 2449 a Fp(\033)e Fs(+)c(1)1556 2392 y Fl(k)p Fp( )1655 2358 y Fn(\033)r Fm(+1)1783 2392 y Fl(k)1825 2358 y Fm(2)1825 2415 y Fn(L)1871 2398 y Fj(2)1930 2392 y Fs(:=)2126 2336 y(1)p 2050 2373 V 2050 2449 a Fp(\033)k Fs(+)c(1)2268 2279 y Fq(Z)2314 2468 y Ff(R)2391 2371 y Fs(\026)2374 2392 y Fp( )2431 2358 y Fn(\033)r Fm(+1)2559 2392 y Fp( )2616 2358 y Fn(\033)r Fm(+1)2745 2392 y Fp(dx)456 2594 y Fs(The)39 b(general)f(idea)h(is)g(that)h(for)f(initial)g(data)g (close)g(to)g(\010)2385 2606 y Fm(0)2462 2594 y Fs(the)h(system)f (should)g(b)r(e)h(w)n(ell)456 2697 y(describ)r(ed)24 b(b)n(y)g(the)h(Hamiltonian)f(system)g(obtained)g(b)n(y)g(restricting)g Fl(E)32 b Fs(to)2846 2676 y(^)2837 2697 y(\010)2897 2709 y Fm(0)2959 2697 y Fs(whic)n(h)24 b(is)g(close)456 2797 y(to)j(\010)617 2809 y Fm(0)654 2797 y Fs(.)37 b(Denote)1528 3015 y Fp( )26 b Fs(=)1735 2911 y Fn(n)1695 2936 y Fq(X)1698 3113 y Fn(j)s Fm(=1)1829 3015 y Fp( )1883 3027 y Fn(j)1932 3015 y Fs(^)-56 b Fp(')1972 3027 y Fn(j)2031 3015 y Fl(2)2118 2994 y Fs(^)2109 3015 y(\010)2169 3027 y Fm(0)456 3258 y Fs(then)28 b(w)n(e)f(ha)n(v)n(e)f(the)i(follo)n(wing)456 3425 y Fx(Lemma)h(1.)41 b Fh(The)31 b(r)l(estriction)f(of)48 b Fs(\(24\))29 b Fh(to)1882 3405 y Fs(^)1873 3425 y(\010)1933 3437 y Fm(0)2000 3425 y Fh(takes)h(the)g(form)596 3652 y Fl(E)7 b(j)678 3664 y Fm(^)671 3679 y(\010)718 3687 y Fj(0)838 3652 y Fs(=)1025 3548 y Fn(n)985 3573 y Fq(X)988 3750 y Fn(j)s Fm(=1)1105 3652 y Fs(\(\012)19 b(+)f Fp(\027)1340 3664 y Fn(j)1375 3652 y Fs(\))p Fl(j)p Fp( )1484 3664 y Fn(j)1519 3652 y Fl(j)1542 3617 y Fm(2)1598 3652 y Fs(+)g Fp(\017c)1804 3548 y Fn(n)1764 3573 y Fq(X)1767 3750 y Fn(j)s Fm(=1)1898 3652 y Fl(j)p Fp( )1975 3664 y Fn(j)2010 3652 y Fl(j)2033 3617 y Fm(2)p Fn(\033)r Fm(+2)2214 3652 y Fs(+)2336 3548 y Fn(n)2297 3573 y Fq(X)2299 3750 y Fn(j)s Fm(=2)2431 3652 y Fp(c)2467 3664 y Fn(j)2515 3584 y Fq(\002)2567 3630 y Fs(\026)2550 3652 y Fp( )2604 3664 y Fn(j)2639 3652 y Fp( )2693 3664 y Fn(j)s Fg(\000)p Fm(1)2831 3652 y Fs(+)g Fp( )2968 3664 y Fn(j)3021 3630 y Fs(\026)3004 3652 y Fp( )3058 3664 y Fn(j)s Fg(\000)p Fm(1)3178 3584 y Fq(\003)3231 3652 y Fs(+)1045 3880 y(+)p Fh(O)7 b Fs(\()p Fp(e)1252 3846 y Fg(\000)p Fm(2\000)p Fn(=)p Ff(~)1453 3880 y Fs(\))18 b(+)g Fp(\017)p Fh(O)8 b Fs(\()p Fp(e)1763 3846 y Fg(\000)p Fm(\000)p Fn(=)p Ff(~)1930 3880 y Fs(\))456 4040 y Fh(wher)l(e)30 b Fs(\000)g Fh(was)g(intr)l(o)l(duc)l(e)l(d)g(b)l(efor)l(e)g(e)l(quation)37 b Fs(\(8\))29 b Fh(and)959 4266 y Fs(\012)23 b(=)1144 4210 y(1)p 1140 4247 50 4 v 1140 4323 a Fp(n)1253 4162 y Fn(n)1213 4187 y Fq(X)1216 4364 y Fn(j)s Fm(=1)1347 4266 y Fp(\025)1395 4278 y Fn(j)1460 4266 y Fp(;)44 b(\027)1568 4278 y Fn(j)1626 4266 y Fs(:=)1736 4153 y Fq(Z)1783 4342 y Ff(R)1856 4244 y Fs(\026)1856 4266 y(^)-55 b Fp(')1897 4278 y Fn(j)1932 4266 y Fp(H)2001 4278 y Fm(0)2051 4266 y Fs(^)g Fp(')2092 4278 y Fn(j)2127 4266 y Fp(dx)20 b Fl(\000)e Fs(\012)23 b(=)f Fh(O)8 b Fs(\()p Fo(~)2639 4232 y Fn(r)r(=)p Fm(2)2743 4266 y Fs(\))456 4510 y Fh(and)1007 4715 y Fp(c)23 b Fs(=)f Fp(c)p Fs(\()p Fo(~)p Fs(\))i(:=)1518 4659 y(1)p 1442 4696 194 4 v 1442 4772 a Fp(\033)e Fs(+)c(1)1660 4659 y(1)p 1656 4696 50 4 v 1656 4772 a Fp(n)1769 4611 y Fn(n)1729 4636 y Fq(X)1732 4813 y Fn(j)s Fm(=1)1863 4715 y Fl(k)13 b Fs(^)-55 b Fp(')1959 4727 y Fn(j)1994 4715 y Fl(k)2036 4680 y Fm(2)p Fn(\033)r Fm(+2)2036 4744 y Fn(L)2082 4727 y Fj(2)p Fk(\033)q Fj(+2)2247 4715 y Fs(=)22 b Fh(O)2420 4623 y Fq(\020)2469 4715 y Fo(~)2514 4681 y Fg(\000)p Fn(\033)r(=)p Fm(2)2678 4623 y Fq(\021)456 4959 y Fh(and)1255 5140 y Fp(c)1291 5152 y Fn(j)1349 5140 y Fs(:=)1459 5027 y Fq(Z)1506 5216 y Ff(R)1579 5118 y Fs(\026)1579 5140 y(^)-56 b Fp(')1619 5152 y Fn(j)1655 5140 y Fp(H)1724 5152 y Fm(0)1774 5140 y Fs(^)h Fp(')1815 5152 y Fn(j)s Fg(\000)p Fm(1)1935 5140 y Fp(dx)24 b Fs(=)f Fh(O)8 b Fs(\()p Fp(e)2280 5106 y Fg(\000)p Fm(\000)p Fn(=)p Ff(~)2447 5140 y Fs(\))p eop %%Page: 9 9 9 8 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)1012 b(9)456 450 y Fh(Pr)l(o)l(of.)43 b Fs(Indeed,)28 b(w)n(e)f(ha)n(v)n(e)f (that)1440 586 y Fl(E)7 b(j)1521 598 y Fm(^)1514 613 y(\010)1561 621 y Fj(0)1621 586 y Fs(=)32 b Fl(E)1762 598 y Fm(0)1800 586 y Fl(j)1830 598 y Fm(^)1823 613 y(\010)1870 621 y Fj(0)1925 586 y Fs(+)18 b Fp(\017)23 b Fl(P)2123 598 y Fm(0)2160 586 y Fl(j)2191 598 y Fm(^)2184 613 y(\010)2231 621 y Fj(0)2281 586 y Fp(:)456 723 y Fs(The)k(\014rst)h(term)f(tak)n (es)g(the)h(form)897 904 y Fl(E)941 916 y Fm(0)978 904 y Fl(j)1008 916 y Fm(^)1001 931 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Fs(\()p Fp(e)1885 1454 y Fg(\000)p Fm(\000)p Fn(=)p Ff(~)2053 1484 y Fs(\))22 b(when)f Fl(j)p Fp(i)6 b Fl(\000)g Fp(j)f Fl(j)23 b Fs(=)f(1)f(and)h Fp(d)2880 1496 y Fn(i;j)2981 1484 y Fs(=)h Fh(O)8 b Fs(\()p Fp(e)3212 1454 y Fg(\000)p Fm(2\000)p Fn(=)p Ff(~)3412 1484 y Fs(\))456 1583 y(when)30 b Fl(j)p Fp(i)20 b Fl(\000)g Fp(j)5 b Fl(j)29 b Fp(>)e Fs(1.)76 b(F)-7 b(or)30 b(what)h(concerns)e(the)i (second)f(term,)h(follo)n(wing)f([21)o(])h(and)f(making)456 1683 y(use)d(of)34 b(\(9\),)28 b(w)n(e)f(ha)n(v)n(e)g(that)984 1885 y Fl(P)1042 1897 y Fm(0)1079 1885 y Fl(j)1109 1897 y Fm(^)1102 1912 y(\010)1149 1920 y Fj(0)1209 1885 y Fs(=)1383 1829 y(1)p 1307 1866 194 4 v 1307 1942 a Fp(\033)22 b Fs(+)c(1)1564 1781 y Fn(n)1524 1806 y Fq(X)1527 1983 y Fn(j)s Fm(=1)1658 1885 y Fl(k)13 b Fs(^)-55 b Fp(')1754 1897 y Fn(j)1789 1885 y Fl(k)1831 1850 y Fm(2)p Fn(\033)r Fm(+2)1831 1914 y Fn(L)1877 1898 y Fj(2)p Fk(\033)q Fj(+2)2019 1885 y Fl(j)p Fp( )2096 1897 y Fn(j)2131 1885 y Fl(j)2154 1851 y Fm(2)p Fn(\033)r Fm(+2)2334 1885 y Fs(+)18 b Fh(O)8 b Fs(\()p Fp(e)2560 1851 y Fg(\000)p Fm(\000)p Fn(=)p Ff(~)2727 1885 y Fs(\))456 2121 y(where)32 b Fl(k)13 b Fs(^)-55 b Fp(')797 2133 y Fn(j)832 2121 y Fl(k)874 2086 y Fm(2)p Fn(\033)r Fm(+2)874 2150 y Fn(L)920 2133 y Fj(2)p Fk(\033)q Fj(+2)1093 2121 y Fs(=)32 b Fh(O)1275 2054 y Fq(\000)1313 2121 y Fo(~)1358 2091 y Fg(\000)p Fn(\033)r(=)p Fm(2)1521 2054 y Fq(\001)1592 2121 y Fs(is)h(appro)n (ximativ)n(ely)e(indep)r(enden)n(t)i(of)g Fp(j)5 b Fs(.)53 b(Remark)32 b(also)456 2221 y(that,)22 b(since)f(the)h(functions)34 b(^)-55 b Fp(')1392 2233 y Fn(j)1448 2221 y Fs(are)20 b(real)h(v)-5 b(alued,)22 b(the)f(quan)n(tities)g Fp(c)2563 2233 y Fn(j)2619 2221 y Fs(turn)g(out)g(to)g(b)r(e)h(real.)82 b Fd(\003)555 2492 y Fs(Hence,)27 b(up)g(to)g(higher)e(order)h(terms,)g (to)h(the)f(Gauge)g(transformation)f Fp( )h Fl(!)d Fp(e)3056 2462 y Fn(i)p Fm(\012)p Fn(t=)p Ff(~)3227 2492 y Fp( )30 b Fs(and)456 2596 y(to)d(a)g(rescaling)f(of)i(time)g Fp(t)23 b Fl(!)g Fp(!)s(t=)p Fo(~)k Fs(the)h(restriction)e(of)34 b(\(24\))28 b(to)2537 2575 y(^)2528 2596 y(\010)2588 2608 y Fm(0)2653 2596 y Fs(is)f(giv)n(en)g(b)n(y)741 2810 y Fl(K)804 2822 y Fm(0)865 2810 y Fs(:=)1015 2707 y Fn(n)976 2731 y Fq(X)978 2908 y Fn(j)s Fm(=1)1109 2810 y Fp(\016)1146 2822 y Fn(j)1181 2810 y Fl(j)p Fp( )1258 2822 y Fn(j)1293 2810 y Fl(j)1316 2776 y Fm(2)1372 2810 y Fs(+)18 b Fp(\021)1553 2707 y Fn(n)1513 2731 y Fq(X)1516 2908 y Fn(j)s Fm(=1)1647 2810 y Fl(j)p Fp( )1724 2822 y Fn(j)1759 2810 y Fl(j)1782 2776 y Fm(2)p Fn(\033)r Fm(+2)1963 2810 y Fs(+)2046 2707 y Fn(n)p Fm(+1)2048 2731 y Fq(X)2051 2908 y Fn(j)s Fm(=1)2185 2810 y Fs(\003)2243 2822 y Fn(j)2277 2810 y Fs(\()2326 2788 y(\026)2309 2810 y Fp( )2363 2822 y Fn(j)2399 2810 y Fp( )2453 2822 y Fn(j)s Fg(\000)p Fm(1)2591 2810 y Fs(+)g Fp( )2728 2822 y Fn(j)2780 2788 y Fs(\026)2763 2810 y Fp( )2817 2822 y Fn(j)s Fg(\000)p Fm(1)2937 2810 y Fs(\))p Fp(;)305 b Fs(\(25\))456 3030 y(where)645 3204 y Fp(\021)26 b Fs(=)810 3148 y Fp(\017c)p 810 3185 70 4 v 817 3261 a(!)913 3204 y Fs(=)c Fh(O)1085 3087 y Fq(\022)1222 3148 y Fl(j)p Fp(\017)p Fl(j)p 1157 3185 212 4 v 1157 3263 a Fp(!)s Fo(~)1257 3239 y Fn(\033)r(=)p Fm(2)1378 3087 y Fq(\023)1480 3204 y Fp(;)42 b(\016)1582 3216 y Fn(j)1640 3204 y Fs(:=)1761 3148 y Fp(\027)1802 3160 y Fn(j)p 1761 3185 76 4 v 1771 3261 a Fp(!)1870 3204 y Fs(=)22 b Fh(O)8 b Fs(\(1\))27 b Fp(;)42 b Fs(\003)2285 3216 y Fn(j)2343 3204 y Fs(:=)2463 3148 y Fp(c)2499 3160 y Fn(j)p 2463 3185 71 4 v 2471 3261 a Fp(!)2567 3204 y Fs(=)23 b Fh(O)2740 3087 y Fq(\022)2811 3148 y Fp(e)2850 3118 y Fg(\000)p Fm(\000)p Fn(=)p Ff(~)p 2811 3185 207 4 v 2887 3261 a Fp(!)3028 3087 y Fq(\023)456 3390 y Fs(and)k(the)h(equation)f(of)h(motion)f(of)34 b(\(25\))27 b(are)g(giv)n(en)g(b)n(y)1304 3547 y(_)1278 3569 y Fp( )1332 3581 y Fn(j)1390 3569 y Fs(=)c Fl(\000)p Fp(i)1582 3513 y(@)5 b Fl(K)1694 3525 y Fm(0)p 1581 3550 150 4 v 1587 3629 a Fp(@)1652 3607 y Fs(\026)1636 3629 y Fp( )1690 3641 y Fn(j)1740 3569 y Fp(;)69 b Fs(with)56 b Fp(j)28 b Fs(=)23 b(1)p Fp(;)14 b(:::;)g(n:)456 3765 y Fs(The)29 b(system)g(\(25\))f(has)h(an)g(in)n(tegral)f(of)h(motion)g (\(the)h(restriction)e(of)h(the)g(square)f(of)h(the)h Fp(L)3408 3735 y Fm(2)456 3872 y Fs(norm)d(to)783 3851 y(^)774 3872 y(\010)834 3884 y Fm(0)871 3872 y Fs(\))h(giv)n(en)f(b)n (y)1621 4075 y Fl(I)j Fs(:=)1846 3971 y Fn(n)1806 3996 y Fq(X)1809 4173 y Fn(j)s Fm(=1)1940 4075 y Fl(j)p Fp( )2017 4087 y Fn(j)2052 4075 y Fl(j)2075 4040 y Fm(2)3297 4075 y Fs(\(26\))555 4294 y(W)-7 b(e)29 b(analyze)e(no)n(w)g(the)i (consequences)e(of)h(our)f(Theorem)g(for)h(the)g(dynamics.)66 b(F)-7 b(rom)28 b(the)456 4394 y(pro)r(of)c(of)g(Theorem)g(2)g(w)n(e)h (will)f(b)r(e)h(able)g(to)f(exactly)g(describ)r(e)g(the)h(restriction)f (of)h(the)g(system)456 4494 y(to)20 b Fl(M)p Fs(.)55 b(Actually)-7 b(,)22 b(w)n(e)e(can)g(state)g(the)h(follo)n(wing)f (Theorem)f(3,)j(whic)n(h)e(is)h(a)f(result)g(of)g(Theorem)456 4593 y(6)27 b(stated)g(b)r(elo)n(w.)456 4743 y Fx(Theorem)34 b(3.)43 b Fh(Under)34 b(the)f(same)g(assumptions)h(of)g(The)l(or)l(em)g (2)g(ther)l(e)f(exists)g(an)g(analytic)456 4843 y(c)l(anonic)l(al)27 b(tr)l(ansformation)f Fl(T)44 b Fs(:)23 b Fl(U)32 b(!)23 b(X)1775 4813 y Fn(s)1811 4843 y Fh(,)k(with)f Fl(U)34 b Fh(an)26 b(op)l(en)g(neighb)l(orho)l(o)l(d)i(of)f Fs(\010)3074 4855 y Fm(0)3137 4843 y Fh(with)f(size)456 4943 y(indep)l(endent)k(of)h Fp(\026)e Fh(and)i Fo(~)p Fh(,)f(such)f(that)1462 5080 y Fl(E)c(\016)18 b(T)45 b Fs(=)22 b(\012)p Fl(I)j Fs(+)18 b Fp(!)s Fl(K)i Fs(+)2223 5059 y(~)2202 5080 y Fl(R)456 5216 y Fh(wher)l(e)p eop %%Page: 10 10 10 9 bop 456 251 a Fm(10)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)657 450 y Fs(i.)41 b Fl(K)20 b(\016)898 429 y Fs(^)887 450 y(\005)k(=)e Fl(K)q Fh(,)31 b(i.e.)40 b Fl(K)31 b Fh(dep)l(ends)f(only)h(on)f(the)g(variables)h Fp( )2572 462 y Fm(1)2610 450 y Fp(;)14 b(:::;)g( )2807 462 y Fn(n)2852 450 y Fh(,)634 550 y Fs(ii.)41 b Fl(fI)6 b Fp(;)14 b Fl(K)q(g)23 b(\021)g Fs(0)28 b Fh(i.e.)40 b Fl(I)35 b Fh(is)30 b(an)f(inte)l(gr)l(al)g(of)h(motion)g(for)g(the)f (system)g(with)g(Hamiltonian)744 649 y Fl(K)611 749 y Fs(iii.)41 b Fl(K)25 b Fs(=)d Fl(K)982 761 y Fm(0)1038 749 y Fs(+)c Fh(O)8 b Fs(\()p Fp(\026=)p Fo(~)1362 719 y Fm(3)p Fn(=)p Fm(2)1466 749 y Fs(\))613 874 y(iv.)766 853 y(~)744 874 y Fl(R)27 b Fs(=)f Fh(O)1017 781 y Fq(h)1056 874 y Fs(exp)14 b Fl(\000)1276 781 y Fq(\020)1335 836 y Fn(\026)1375 844 y Fe(\003)1410 836 y Ff(~)1447 811 y Fj(3)p Fk(=)p Fj(2)p 1335 854 204 4 v 1400 902 a Fm(2)p Fn(\026)1548 781 y Fq(\021i)1657 874 y Fs(+)19 b Fh(O)1826 781 y Fq(\020)1876 874 y Fl(k)o Fs(\005)1979 886 y Fn(c)2013 874 y Fp( )s Fl(k)2112 832 y Fm(2)2112 898 y Fn(s)2149 781 y Fq(\021)2230 874 y Fh(and)32 b(similar)h(estimates)f(hold)h(for)f (its)744 993 y(ve)l(ctor)e(\014eld.)636 1093 y Fs(v.)41 b Fl(T)k Fs(=)22 b Fo(I)6 b Fs(+)18 b Fh(O)8 b Fs(\()p Fp(\026=)p Fo(~)1296 1063 y Fm(3)p Fn(=)p Fm(2)1400 1093 y Fs(\))30 b Fh(i.e.)39 b(the)30 b(tr)l(ansformation)h(is)f(close)h(to) e(identity.)456 1243 y Fx(Remark)c(6.)36 b Fh(In)25 b(this)g(fr)l (amework)i(the)e(manifold)i Fl(M)e Fh(turns)e(out)i(to)g(simply)h(b)l (e)f Fl(M)e Fs(=)g Fl(T)e Fs(\(\010)3349 1255 y Fm(0)3387 1243 y Fs(\))p Fh(.)555 1392 y Fs(In)k(particular)e(it)i(follo)n(ws)e (that)i(the)f(dynamics)g(on)g Fl(M)g Fs(is,)h(up)g(to)f(a)g(small)g (error,)f(the)i(same)456 1492 y(of)33 b(a)h(Hamiltonian)f(system)g (with)i(Hamiltonian)e(function)h(close)f(to)h Fl(K)2782 1504 y Fm(0)2853 1492 y Fs(with)g(an)g(in)n(tegral)456 1591 y(of)g(motion)h(giv)n(en)f(b)n(y)g Fl(I)6 b Fs(.)94 b(Th)n(us,)36 b(it)g(is)e(p)r(ossible)h(to)f(deduce)h(the)g(follo)n (wing)f(Corollary)f(of)456 1691 y(Theorem)26 b(7)i(b)r(elo)n(w,)f(whic) n(h)g(is)h(particularly)e(relev)-5 b(an)n(t)27 b(in)h(the)g(double)f(w) n(ell)h(case.)456 1840 y Fx(Corollary)33 b(3.)40 b Fh(Under)30 b(the)g(same)g(assumptions)g(of)h(The)l(or)l(em)g(2)f(one)g(also)g(has) 1029 1992 y Fl(jI)6 b Fs(\()p Fp(t)p Fs(\))19 b Fl(\000)g(I)6 b Fs(\(0\))p Fl(j)23 b(\024)g Fp(C)1716 1935 y(\026)p 1666 1973 150 4 v 1666 2050 a Fo(~)1711 2026 y Fm(3)p Fn(=)p Fm(2)1855 1992 y Fp(;)99 b Fl(jK)2063 2004 y Fm(0)2100 1992 y Fs(\()p Fp(t)p Fs(\))19 b Fl(\000)f(K)2359 2004 y Fm(0)2397 1992 y Fs(\(0\))p Fl(j)23 b(\024)f Fp(C)2761 1935 y(\026)p 2712 1973 V 2712 2050 a Fo(~)2757 2026 y Fm(3)p Fn(=)p Fm(2)3297 1992 y Fs(\(27\))456 2156 y Fh(up)29 b(to)h(the)g(times)37 b Fs(\(20\))o Fh(.)555 2305 y Fs(W)-7 b(e)37 b(fo)r(cus)f(no)n(w)f(on)h(the)h(exp)r(onen)n (tially)e(long)g(time)i(scale,)g(th)n(us)g(w)n(e)e(assume)h(that)g(the) 456 2405 y(quan)n(tit)n(y)27 b Fp(\016)j Fs(of)e(Theorem)f(2)g(is)g (giv)n(en)g(b)n(y)1517 2594 y Fp(\016)f Fs(=)c(exp)1808 2477 y Fq(\024)1852 2594 y Fl(\000)1927 2538 y Fp(\026)1977 2550 y Fg(\003)2014 2538 y Fo(~)2059 2508 y Fm(3)p Fn(=)p Fm(2)p 1926 2575 238 4 v 1999 2651 a Fs(2)p Fp(\026)2174 2477 y Fq(\025)456 2828 y Fs(and)k(from)h(\(20\))f(it)h(follo)n(ws)f (that)h(up)g(to)g(times)g(of)g(order)e Fp(e)2319 2764 y Fk(\026)2354 2772 y Fe(\003)2389 2764 y Fc(~)2422 2747 y Fj(3)p Fk(=)p Fj(2)p 2319 2778 195 3 v 2384 2811 a(2)p Fk(\026)2554 2828 y Fs(the)i(v)n(ector)e(\014eld)i Fp(X)3210 2838 y Fm(~)3192 2853 y Fg(R)3280 2828 y Fs(of)3395 2807 y(~)3374 2828 y Fl(R)456 2928 y Fs(ful\014lls)h(the)g(a)f(priori)f (estimate)1373 3117 y Fl(k)o Fp(X)1500 3127 y Fm(~)1483 3142 y Fg(R)1544 3117 y Fl(k)1586 3146 y Fn(s)1644 3117 y Fl(\024)d Fp(C)d Fs(exp)1952 3000 y Fq(\024)1995 3117 y Fl(\000)2070 3061 y Fp(\026)2120 3073 y Fg(\003)2158 3061 y Fo(~)2203 3030 y Fm(3)p Fn(=)p Fm(2)p 2070 3098 238 4 v 2164 3174 a Fp(\026)2317 3000 y Fq(\025)456 3297 y Fs(Th)n(us,)31 b(up)g(to)g(an)f(exp)r(onen)n(tially)h(small)f(drift,) i(a)f(Gauge)f(transformation)f(and)i(a)f(rescaling)456 3396 y(of)35 b(time)h(the)g(dynamics)f(on)h Fl(M)g(\021)g(T)21 b Fs(\(\010)1800 3408 y Fm(0)1837 3396 y Fs(\))36 b(is)g(that)g(of)f (the)h Fp(n)p Fs({dimensional)f(Hamiltonian)456 3496 y(system)27 b Fl(K)q Fs(,)h(whic)n(h)g(is)f(a)g(p)r(erturbation)g(of)h Fl(K)1881 3508 y Fm(0)1918 3496 y Fs(,)g(i.e.)37 b(of)28 b(the)g(discrete)f(NLS)h(\(25\))o(.)555 3595 y(It)21 b(is)g(w)n(orth)f(men)n(tioning)g(that)i(when)f Fp(\021)26 b Fs(=)c(0)f(the)g(dynamics)f(of)h Fl(K)2618 3607 y Fm(0)2676 3595 y Fs(is)g(that)g(of)g Fp(n)g Fs(decoupled)456 3695 y(harmonic)30 b(oscillators)f(corresp)r(onding)h(to)h(the)h(normal)e (mo)r(des)h(of)g(the)h(linearized)f(system.)456 3795 y(If)e(\003)598 3807 y Fn(j)657 3795 y Fl(\034)c Fp(\016)802 3807 y Fn(i)830 3795 y Fs(,)k(for)f(an)n(y)g Fp(i)g Fs(and)h Fp(j)5 b Fs(,)29 b(then)g(the)g(normal)f(mo)r(des)g(are)g(lo)r (calized,)g(i.e.)41 b(eac)n(h)27 b(normal)456 3894 y(mo)r(de)34 b(essen)n(tially)f(in)n(v)n(olv)n(es)f(only)i(one)g(of)g(the)g Fp( )2060 3906 y Fn(j)2095 3894 y Fs(;)k(on)33 b(the)i(con)n(trary)-7 b(,)34 b(in)g(the)g(m)n(uc)n(h)g(more)456 3994 y(in)n(teresting)21 b(case)h(where)g(the)g Fp(\016)1439 4006 y Fn(j)1497 3994 y Fs(are)f(of)i(the)f(same)g(order)f(of)h(magnitude)h(of)f(the)h (\003)3052 4006 y Fn(j)3086 3994 y Fs(,)h(t)n(ypically)456 4094 y(the)37 b(normal)g(mo)r(des)g(are)f(collectiv)n(e)h(motions)g(of) g(the)h(system)f(and,)i(corresp)r(ondingly)-7 b(,)38 b(in)456 4193 y(t)n(ypical)27 b(solutions)g(the)h(term)f Fl(j)p Fp( )1493 4205 y Fn(j)1528 4193 y Fs(\()p Fp(t)p Fs(\))p Fl(j)1645 4163 y Fm(2)1711 4193 y Fs(undergo)r(es)f(great)h(c)n (hanges)f(for)h(eac)n(h)g Fp(j)5 b Fs(.)555 4293 y(In)20 b(the)f(opp)r(osite)g(limit)h Fp(\021)26 b Fl(!)e(1)19 b Fs(\(an)n(ticon)n(tin)n(uum)g(limit,)j(see)d([14)o(]\),)i Fl(K)2757 4305 y Fm(0)2814 4293 y Fs(b)r(ecomes)e(a)g(system)456 4392 y(of)38 b(decoupled)h(anharmonic)e(oscillators.)107 b(Corresp)r(ondingly)37 b Fl(j)p Fp( )2628 4404 y Fn(j)2663 4392 y Fs(\()p Fp(t)p Fs(\))p Fl(j)2780 4362 y Fm(2)2857 4392 y Fs(is)h(a)h(constan)n(t)f(of)456 4492 y(motion.)60 b(One)25 b(can)g(use)f(KAM)h(or)f(Nekhoroshev)g(theory)g(in)h(order)f (to)g(study)i(the)f(dynamics)456 4592 y(of)g Fl(K)i Fs(when)f(1)d Fl(\034)g Fp(\021)j(<)d Fl(1)i Fs(and)h(to)f(deduce)h(results)f(on)g (the)h(dynamics)f(of)h(the)g(complete)f(NLS)456 4691 y(equation.)555 4791 y(Here)32 b(w)n(e)f(will)h(state)g(a)f(result)g (that)h(can)g(b)r(e)g(obtained)f(in)h(this)g(w)n(a)n(y)-7 b(.)80 b(T)-7 b(o)32 b(this)g(end,)h(for)456 4891 y(an)n(y)26 b Fp(\032)d(>)g Fs(0)k(\014xed,)h(w)n(e)f(will)h(denote)754 5120 y Fl(S)810 5086 y Fn(n)804 5141 y(\032)879 5120 y Fs(:=)989 4950 y Fq(8)989 5025 y(<)989 5174 y(:)1063 5120 y Fs(\()p Fp( )1149 5132 y Fm(1)1187 5120 y Fp(;)14 b(:::;)g( )1384 5132 y Fn(n)1429 5120 y Fs(\))23 b Fl(2)h Fo(C)1617 5086 y Fn(n)1719 5120 y Fs(:)1832 5016 y Fn(n)1793 5041 y Fq(X)1795 5218 y Fn(j)s Fm(=1)1926 5120 y Fl(j)p Fp( )2003 5132 y Fn(j)2038 5120 y Fl(j)2061 5086 y Fm(2)2122 5120 y Fs(=)e(1)28 b(and)f Fl(j)p Fp( )2517 5132 y Fn(j)2552 5120 y Fl(j)c Fp(>)g(\032)28 b(;)41 b Fl(8)p Fp(j)2907 4950 y Fq(9)2907 5025 y(=)2907 5174 y(;)p eop %%Page: 11 11 11 10 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(11)456 450 y Fs(and)27 b(w)n(e)g(will)h(denote)g(b)n(y)1279 380 y Fq(\014)1279 429 y(\014)1307 450 y Fl(S)1363 420 y Fn(n)1357 471 y(\032)1408 380 y Fq(\014)1408 429 y(\014)1464 450 y Fs(its)f(Leb)r(esgue)g(measure.)456 602 y Fx(Theorem)34 b(4.)43 b Fh(\(KAM)34 b(the)l(or)l(em)f Fl(K)q Fh(\))h(Consider)g(the)g (Hamiltonian)g(system)f Fl(K)q Fh(,)i(then)e(ther)l(e)456 702 y(exists)h(a)i(c)l(onstant)e Fp(\021)1143 714 y Fn(?)1182 702 y Fh(,)j(such)e(that,)i(for)f(any)g Fl(j)p Fp(\021)s Fl(j)d Fp(>)g(\021)2205 714 y Fn(?)2278 702 y Fh(ther)l(e)i(exists)g(a) h(set)f Fl(S)2986 714 y Fn(\021)3059 702 y Fl(\032)e(S)3213 672 y Fn(n)3207 723 y(\032)3294 702 y Fh(with)456 802 y(L)l(eb)l(esgue)c(me)l(asur)l(e)h(estimate)l(d)g(by)1481 939 y Fl(jS)1554 951 y Fn(\021)1595 939 y Fl(j)23 b(\025)1729 869 y Fq(\014)1729 918 y(\014)1756 939 y Fl(S)1812 905 y Fn(n)1806 960 y(\032)1858 869 y Fq(\014)1858 918 y(\014)1904 939 y 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Fx(Remark)c(7.)43 b Fh(A)n(l)t(l)33 b(the)g(c)l(onstants)g(in)g(the)g(ab)l(ove)h(The)l(or)l(em)g(4)f(dep)l (end)h(on)f Fp(n)p Fh(,)h(in)f Fs([2)o(,)f(8)o(])h Fh(one)456 3629 y(c)l(an)c(\014nd)h(some)g Fp(n)g Fh(indep)l(endent)g(statements.) 456 3778 y Fs(2.5.)40 b Fx(The)27 b(double)g(w)m(ell)f(p)s(oten)m (tial.)40 b Fs(In)24 b(the)f(particular)f(case)h(of)g(a)g(double)g(w)n (ell)h(p)r(oten)n(tial)456 3878 y(one)33 b(can)h(get)g(m)n(uc)n(h)f (more)h(precise)f(results,)i(b)r(oth)g(for)e(the)h(linear)g(and)g(for)f (the)h(nonlinear)456 3978 y(system.)555 4077 y(By)22 b(a)g(double)h(w)n(ell)f(p)r(oten)n(tial)g(w)n(e)g(will)h(mean)f(here)g (a)g(p)r(oten)n(tial)g Fp(V)d Fs(\()p Fp(x)p Fs(\))24 b Fl(2)f Fp(C)2929 4047 y Fg(1)3000 4077 y Fs(\()p Fo(R)p Fs(\))29 b(ful\014lling)456 4177 y(assumptions)20 b(\(i-v\))h(of)g (Section)g(2.1)g(\(with)g Fp(n)i Fs(=)g(2\))e(and)g(whic)n(h)g(moreo)n (v)n(er)d(is)j(symmetric)g(with)456 4276 y(resp)r(ect)29 b(to)h(spacial)e(re\015ection)i Fp(V)19 b Fs(\()p 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b(to)i Fp(\025)1009 4915 y Fm(1)p Fn(;)p Fm(2)1132 4903 y Fs(can)f(b)r(e)i(c)n(hosen)e(to)g(b)r(e)i(real-v)-5 b(alued)30 b(functions)j(suc)n(h)e(that)h Fp(')3117 4915 y Fm(1)3187 4903 y Fs(and)g Fp(')3407 4915 y Fm(2)456 5002 y Fs(are)26 b(resp)r(ectiv)n(ely)h(of)g(ev)n(en)h(and)f(o)r (dd-parit)n(y)-7 b(,)27 b(th)n(us,)g(de\014ning)h(the)g Fh(single)i(wel)t(l)h(states)1018 5172 y Fp(')1072 5184 y Fn(R)1150 5172 y Fs(=)1282 5115 y(1)p 1247 5152 111 4 v 1247 5169 a Fl(p)p 1317 5169 42 4 v 1317 5238 a Fs(2)1382 5172 y([)p Fp(')1459 5184 y Fm(1)1515 5172 y Fs(+)18 b Fp(')1652 5184 y Fm(2)1689 5172 y Fs(])97 b(and)83 b Fp(')2080 5184 y Fn(L)2153 5172 y Fs(=)2285 5115 y(1)p 2251 5152 111 4 v 2251 5169 a Fl(p)p 2320 5169 42 4 v 69 x Fs(2)2385 5172 y([)p Fp(')2462 5184 y Fm(1)2518 5172 y Fl(\000)18 b Fp(')2655 5184 y Fm(2)2693 5172 y Fs(])p eop %%Page: 12 12 12 11 bop 456 251 a Fm(12)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)456 450 y Fs(they)30 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b(allo)n(ws)f(us)i(to)f(describ)r(e)g(in)h(a)f(v)n(ery)456 2506 y(precise)j(w)n(a)n(y)g(the)i(tra)5 b(jectories)27 b(of)j(the)f(system)g Fl(K)i Fs(whic)n(h)e(are)g(just)h(the)f(in)n (tersection)g(of)g(the)456 2606 y(lev)n(el)e(surfaces)f(of)i(the)g (functions)g Fl(K)h Fs(and)e Fl(I)6 b Fs(.)38 b(This)27 b(is)h(p)r(ossible)f(since)g Fl(K)i Fs(is)f(close)f(to)g Fl(K)3245 2618 y Fm(0)3283 2606 y Fs(.)555 2706 y Fh(T)-6 b(o)30 b(b)l(e)g(de\014nite,)h(fr)l(om)f(now)g(on,)g(we)g(wil)t(l)h(r)l (estrict)f(to)f(the)h(c)l(ase)g Fp(\033)d Fs(=)c(2.)555 2805 y(The)29 b(system)g Fl(K)1067 2817 y Fm(0)1105 2805 y Fs(,)h(cf.)42 b(\(37\))o(,)30 b(has)e(b)r(een)i(already)e(studied)h (in)h([11)o(])f(\(see)g(also)f([17)o(,)i(24)o(,)f(25)o(]\))456 2905 y(obtaining)38 b(that,)j(for)d Fl(j)p Fp(\021)s Fl(j)k Fp(<)f Fs(2)d(almost)g(all)g(solutions)g(p)r(erform)g(b)r (eating)g(motions,)j(while)456 3004 y(at)32 b Fp(\021)i Fs(=)d Fl(\006)p Fs(2)g(a)h(bifurcation)g(o)r(ccurs)f(and)i(new)f (equilibria,)h(lo)r(calized)f(close)f(to)i(the)f(minima)456 3104 y(of)38 b(the)g(Hamiltonian)g(function,)j(app)r(ear.)105 b(As)39 b Fp(\021)i Fs(increase)c(the)h(domain)g(of)g(stabilit)n(y)g (of)456 3204 y(suc)n(h)27 b(solutions)h(increase)e(its)j(size,)f(so)f (that,)i(for)e Fp(\021)k Fs(large)c(enough)g(essen)n(tially)h(only)f (lo)r(calize)456 3303 y(motions)h(exist.)72 b(Concerning)28 b(the)i(complete)f(system)g(w)n(e)g(can)g(state)g(that)h(if)g Fp(\021)i Fs(is)d(not)h(at)f(a)456 3403 y(bifurcation)i(p)r(oin)n(t,)i (then)f(non-homo)r(clinic)e(tra)5 b(jectories)30 b(asso)r(ciated)g(to)i (the)g(Hamiltonian)456 3503 y Fl(K)519 3515 y Fm(0)584 3503 y Fs(appro)n(ximate)26 b(the)i(solution)f Fp( )1578 3472 y Fn(t)1635 3503 y Fs(for)g(times)h(of)f(the)h(order)e(\(20\).)456 3647 y Fx(Corollary)k(5.)39 b Fh(Under)27 b(the)h(same)g(assumptions)g (as)g(in)g(The)l(or)l(em)h(2,)g(assume)e(also)i Fp(\021)d Fl(6)p Fs(=)d Fl(\006)p Fs(2)p Fh(;)456 3746 y(c)l(onsider)30 b(an)g(initial)h(datum)f(such)g(that)f Fp( )1813 3758 y Fm(1)1851 3746 y Fs(\(0\))p Fp(;)14 b( )2048 3758 y Fm(2)2085 3746 y Fs(\(0\))23 b Fl(6)p Fs(=)g(0)p Fh(,)29 b(and)1424 3902 y Fl(K)1487 3914 y Fm(0)1524 3902 y Fs(\(0\))23 b Fl(6)p Fs(=)1751 3846 y Fp(\021)p 1751 3883 45 4 v 1752 3959 a Fs(2)1824 3902 y Fl(\000)18 b Fs(1)29 b Fh(if)86 b Fp(\021)26 b(>)d Fs(0)1424 4084 y Fl(K)1487 4096 y Fm(0)1524 4084 y Fs(\(0\))g Fl(6)p Fs(=)g(1)18 b Fl(\000)1894 4028 y Fp(\021)p 1894 4065 V 1895 4141 a Fs(2)1978 4084 y Fh(if)86 b Fp(\021)26 b(<)d Fs(0)456 4258 y Fh(and)29 b(also)h(such)f(that)f Fp(\016)f Fl(\024)22 b Fp(C)6 b(\026)p Fo(~)1452 4228 y Fm(3)p Fn(=)p Fm(2)1557 4258 y Fh(;)29 b(then,)h(ther)l(e)f(exists)f(a)h(p)l(ositive)h(c)l(onstant)e Fp(\026)3002 4270 y Fn(])3034 4258 y Fh(,)h(dep)l(ending)456 4358 y(only)f(on)f(how)i(much)e(the)h(ab)l(ove)g(quantities)g(di\013er) g(fr)l(om)g(the)g(c)l(onsider)l(e)l(d)h(values,)f(such)g(that,)456 4457 y(pr)l(ovide)l(d)36 b Fp(\026)c(<)g(\026)1017 4469 y Fn(])1048 4457 y Fo(~)1093 4427 y Fm(3)p Fn(=)p Fm(2)1197 4457 y Fh(,)37 b(ther)l(e)d(exists)h(a)g(solution)g(of)g(the)g (Hamiltonian)h(system)41 b Fs(\(25\))34 b Fh(with)456 4557 y(tr)l(aje)l(ctory)c Fp(\015)k Fh(such)c(that)1540 4708 y Fp(d)p Fs(\()p Fp( )1672 4674 y Fn(t)1702 4708 y Fp(;)14 b(\015)5 b Fs(\))23 b Fl(\024)g Fp(C)2054 4652 y(\026)p 2005 4689 150 4 v 2005 4767 a Fo(~)2050 4743 y Fm(3)p Fn(=)p Fm(2)3297 4708 y Fs(\(38\))456 4872 y Fh(for)30 b(the)g(times)37 b Fs(\(23\))o Fh(.)456 5016 y Fx(Remark)22 b(8.)35 b Fh(Homo)l(clinic)25 b(tr)l(aje)l(ctories)f(ar) l(e)g(absent)f(when)h Fl(j)p Fp(\021)s Fl(j)f(\024)g Fs(2)p Fh(;)i(initial)g(c)l(onditions)f(such)456 5116 y(that)33 b Fl(K)692 5128 y Fm(0)730 5116 y Fs(\(0\))d(=)970 5079 y Fn(\021)p 970 5097 37 4 v 972 5145 a Fm(2)1038 5116 y Fl(\000)20 b Fs(1)p Fh(,)35 b(for)f Fp(\021)f(>)d Fs(2)p Fh(,)k(and)g Fl(K)q Fs(\(0\))d(=)e(1)21 b Fl(\000)2250 5079 y Fn(\021)p 2250 5097 V 2252 5145 a Fm(2)2296 5116 y Fh(,)35 b(for)f Fp(\021)f(<)d Fl(\000)p Fs(2)i Fh(c)l(orr)l(esp)l (onds)j(to)e(an)456 5216 y(homo)l(clinic)e(tr)l(aje)l(ctory.)p eop %%Page: 13 13 13 12 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(13)456 450 y Fx(Remark)30 b(9.)41 b Fh(The)31 b(top)l(olo)l(gy)g(of)g (the)f(tr)l(aje)l(ctory)g Fp(\015)k Fh(is)d(determine)l(d)f(by)g(the)g (c)l(ondition)1536 612 y Fp(d)p Fs(\()p Fp( )1668 577 y Fm(0)1706 612 y Fp(;)14 b(\015)5 b Fs(\))23 b Fl(\024)g Fp(C)2058 556 y(\026)p 2009 593 150 4 v 2009 670 a Fo(~)2054 646 y Fm(3)p Fn(=)p Fm(2)3297 612 y Fs(\(39\))456 781 y Fh(in)36 b(the)g(sense)g(that)g(al)t(l)h(curves)f(ful\014l)t(ling)h (this)f(c)l(ondition)h(have)g(the)f(same)h(top)l(olo)l(gy)g(if)g(the) 456 881 y(assumptions)30 b(of)g(the)g(Cor)l(ol)t(lary)i(ar)l(e)e (ful\014l)t(le)l(d.)456 1036 y Fx(Remark)45 b(10.)i Fh(Thus)42 b(one)f(has)h(that)f(also)h(for)g(the)f(true)g(system)f(b)l(e)l(ats)h (ar)l(e)h(pr)l(esent)e(for)456 1136 y Fl(j)p Fp(\021)s Fl(j)25 b Fp(<)g Fs(2)31 b Fh(while)h(their)g(imp)l(ortanc)l(e)g(de)l (cr)l(e)l(ases)f(as)h Fl(j)p Fp(\021)s Fl(j)f Fh(incr)l(e)l(ase)h(ab)l (ove)g(2.)74 b(In)30 b(p)l(articular)i(for)456 1236 y(lar)l(ge)j (values)g(of)h Fp(\021)h Fh(only)f(motions)f(lo)l(c)l(alize)l(d)h (close)g(to)e(one)h(wel)t(l)h(ar)l(e)f(pr)l(esent)f Fs([25)o(])h Fh(at)f(le)l(ast)456 1335 y(for)c(the)g(time)g(sc)l(ales)g(c)l(ontr)l (ol)t(le)l(d)h(by)f(our)g(the)l(or)l(ems.)1333 1534 y Fs(3.)42 b Fr(Pr)n(oof)31 b(of)g(the)h(main)f(resul)-6 b(ts)555 1683 y Fh(In)28 b(or)l(der)i(to)e(simplify)j(the)e(notations)g (al)t(l)g(the)g(pr)l(o)l(ofs)h(wil)t(l)g(b)l(e)e(c)l(arrie)l(d)i(out)e (in)h(the)f(c)l(ase)i(of)456 1783 y(a)g(p)l(otential)g(with)h(only)f(2) g(wel)t(ls.)456 1953 y Fs(3.1.)40 b Fx(Hamiltonian)30 b(F)-8 b(ormalism.)37 b Fs(First)28 b(consider)e(the)i(real)f(Hilb)r (ert)h(space)1545 2098 y Fl(X)1616 2064 y Fn(s)1604 2119 y Ff(R)1675 2098 y Fs(:=)22 b Fp(D)r Fs(\(\()p Fp(H)1989 2110 y Fm(0)p Fn(;)p Ff(R)2089 2098 y Fs(\))2121 2064 y Fn(s)2157 2098 y Fs(\))456 2243 y(where)27 b Fp(H)765 2255 y Fm(0)p Fn(;)p Ff(R)891 2243 y Fs(is)h(the)g(op)r(erator)e Fp(H)1522 2255 y Fm(0)1587 2243 y Fs(restricted)g(to)i(real)f(v)-5 b(alued)27 b(functions.)555 2343 y(W)-7 b(e)22 b(mak)n(e)e Fl(X)973 2312 y Fn(s)961 2366 y Ff(R)1015 2343 y Fl(\010)6 b(X)1157 2312 y Fn(s)1145 2366 y Ff(R)1213 2343 y Fs(a)21 b(symplectic)g(space)f(b)n(y)h(in)n(tro)r(ducing)g(the)g Fh(semiclassic)l(al)32 b Fs(symplectic)456 2442 y(form)959 2609 y Fp(\013)14 b Fs(\(\()p Fp(p;)g(q)s Fs(\);)g(\()p Fp(p)1352 2574 y Fg(0)1376 2609 y Fp(;)g(q)1453 2574 y Fg(0)1476 2609 y Fs(\)\))24 b(:=)e Fo(~)1733 2496 y Fq(Z)1779 2684 y Ff(R)1839 2609 y Fs([)p Fp(p)1904 2574 y Fg(0)1927 2609 y Fs(\()p Fp(x)p Fs(\))p Fp(q)s Fs(\()p Fp(x)p Fs(\))f Fl(\000)d Fp(p)p Fs(\()p Fp(x)p Fs(\))p Fp(q)2486 2574 y Fg(0)2510 2609 y Fs(\()p Fp(x)p Fs(\)])d(d)p Fp(x:)456 2800 y Fs(Giv)n(en)24 b(a)h(smo)r(oth)f(real)g(v)-5 b(alued)25 b(function)g Fl(H)q Fs(\()p Fp(p;)14 b(q)s Fs(\),)26 b(then)g(w)n(e)e(de\014ne)h(its)g(Hamiltonian)g(v)n(ector)456 2900 y(\014eld)i Fp(X)704 2912 y Fg(H)788 2900 y Fl(2)d(X)938 2870 y Fn(s)926 2923 y Ff(R)992 2900 y Fl(\010)18 b(X)1146 2870 y Fn(s)1134 2923 y Ff(R)1210 2900 y Fs(b)n(y)27 b(the)h(prop)r(ert)n(y)1563 3045 y Fp(\013)p Fs(\()p Fp(X)1717 3057 y Fg(H)1778 3045 y Fp(;)14 b(h)p Fs(\))24 b(=)e(d)p Fl(H)q Fp(h)1126 b Fs(\(40\))456 3189 y(for)26 b(an)n(y)h Fp(h)c Fs(=)g(\()p Fp(h)978 3201 y Fn(p)1016 3189 y Fp(;)14 b(h)1101 3201 y Fn(q)1137 3189 y Fs(\))24 b Fl(2)f(X)1342 3159 y Fn(s)1330 3213 y Ff(R)1396 3189 y Fl(\010)18 b(X)1550 3159 y Fn(s)1538 3213 y Ff(R)1585 3189 y Fs(,)28 b(where)f(d)p Fl(H)h Fs(denotes)f(the)h(di\013eren)n (tial)f(of)h Fl(H)q Fs(.)64 b(It)27 b(is)h(w)n(ell)456 3289 y(kno)n(wn)d(that)i Fp(X)965 3301 y Fg(H)1053 3289 y Fs(is)f(in)h(general)e(de\014ned)i(only)f(on)g(a)g(subset)g(of)h Fl(X)2582 3259 y Fn(s)2570 3312 y Ff(R)2634 3289 y Fl(\010)16 b(X)2786 3259 y Fn(s)2774 3312 y Ff(R)2822 3289 y Fs(.)63 b(De\014ne)27 b(also)e(the)456 3389 y Fp(L)513 3359 y Fm(2)577 3389 y Fs(gradien)n(t)h Fl(r)971 3401 y Fn(p)1010 3389 y Fl(H)j Fs(of)e Fl(H)i Fs(with)f(resp)r(ect)f(to)h Fp(p)f Fs(b)n(y)815 3579 y Fl(h)q(r)917 3591 y Fn(p)955 3579 y Fl(H)q Fp(;)14 b(h)1111 3591 y Fn(p)1149 3579 y Fl(i)1181 3607 y Fn(L)1227 3590 y Fj(2)1287 3579 y Fs(:=)1397 3466 y Fq(Z)1443 3655 y Ff(R)1503 3579 y Fl(r)1572 3591 y Fn(p)1611 3579 y Fl(H)q Fs(\()p Fp(x)p Fs(\))p Fp(h)1841 3591 y Fn(p)1880 3579 y Fs(\()p Fp(x)p Fs(\))p Fp(dx)25 b Fs(=)d(d)2239 3591 y Fn(p)2278 3579 y Fl(H)q Fp(h)2397 3591 y Fn(p)2435 3579 y Fp(;)97 b Fl(8)p Fp(h)2650 3591 y Fn(p)2710 3579 y Fl(2)24 b(X)2860 3545 y Fn(s)2848 3599 y Ff(R)2896 3579 y Fp(;)378 b Fs(\(41\))456 3766 y(and)27 b(similarly)g(w)n(e)g(in)n(tro)r(duce)g(the)h(quan)n(tit)n(y)f Fl(r)1992 3778 y Fn(q)2029 3766 y Fl(H)q Fs(.)65 b(Then)1408 3915 y Fp(X)1477 3927 y Fg(H)1560 3915 y Fs(=)23 b Fo(~)1693 3880 y Fg(\000)p Fm(1)1782 3915 y Fs(\()p Fl(\000r)1948 3927 y Fn(q)1985 3915 y Fl(H)q Fp(;)14 b Fl(r)2162 3927 y Fn(p)2200 3915 y Fl(H)q Fs(\))p Fp(;)456 4060 y Fs(and)27 b(th)n(us)h(the)g(Hamilton)f(equations)g(of)h Fl(H)g Fs(are)f(giv)n(en)g(b)n(y)840 4193 y Fp(d)p 825 4231 74 4 v 825 4307 a(dt)908 4250 y Fs(\()p Fp(p;)14 b(q)s Fs(\))23 b(=)g Fp(X)1271 4262 y Fg(H)1332 4250 y Fs(\()p Fp(p;)14 b(q)s Fs(\))74 b Fl(\()-14 b(\))1815 4133 y Fq(\022)1892 4250 y Fs(_)-39 b Fp(p)23 b Fs(=)g Fl(\000)2106 4193 y Fs(1)p 2104 4231 45 4 v 2104 4307 a Fo(~)2158 4250 y Fl(r)2227 4262 y Fn(q)2264 4250 y Fl(H)28 b Fp(;)57 b Fs(_)-38 b Fp(q)26 b Fs(=)2589 4193 y(1)p 2588 4231 V 2588 4307 a Fo(~)2642 4250 y Fl(r)2711 4262 y Fn(p)2750 4250 y Fl(H)2821 4133 y Fq(\023)2896 4250 y Fp(:)378 b Fs(\(42\))456 4440 y(The)27 b(P)n(oisson)f(brac)n(k)n(ets)g(b)r(et)n (w)n(een)h(t)n(w)n(o)g(functions)h Fl(H)h Fs(and)e Fl(K)i Fs(is)f(de\014ned)g(as)812 4606 y Fl(fH)q Fp(;)14 b Fl(K)q(g)22 b Fs(:=)h Fp(\013)p Fs(\()p Fp(X)1355 4618 y Fg(H)1417 4606 y Fp(;)14 b(X)1523 4618 y Fg(K)1577 4606 y Fs(\))23 b(=)g Fl(\000)1797 4550 y Fs(1)p 1795 4587 V 1795 4663 a Fo(~)1863 4493 y Fq(Z)1909 4682 y Ff(R)1969 4606 y Fs([)p Fl(r)2061 4618 y Fn(p)2100 4606 y Fl(H)q(r)2240 4618 y Fn(q)2277 4606 y Fl(K)d(\000)e(r)2512 4618 y Fn(q)2549 4606 y Fl(H)q(r)2689 4618 y Fn(p)2727 4606 y Fl(K)q Fs(])c(d)p Fp(x)376 b Fs(\(43\))456 4793 y(whic)n(h)27 b(in)h(general)e(is)i(only) f(de\014ned)h(on)f(a)g(sub)r(domain)h(of)f Fl(X)2400 4763 y Fn(s)2388 4816 y Ff(R)2455 4793 y Fl(\010)18 b(X)2609 4763 y Fn(s)2597 4816 y Ff(R)2645 4793 y Fs(.)555 4893 y(W)-7 b(e)35 b(shall)f(use)h(complex)f(co)r(ordinates)f(in)i Fl(X)2012 4862 y Fn(s)2000 4916 y Ff(R)2071 4893 y Fl(\010)23 b(X)2230 4862 y Fn(s)2218 4916 y Ff(R)2301 4893 y Fs(iden)n(tifying)34 b(this)h(space)f(with)h Fl(X)3385 4862 y Fn(s)3421 4893 y Fs(,)456 4992 y(through)1412 5172 y(\()p Fp(p;)14 b(q)s Fs(\))23 b Fl(7!)g Fp( )j Fs(=)1937 5115 y(1)p 1902 5152 111 4 v 1902 5169 a Fl(p)p 1971 5169 42 4 v 69 x Fs(2)2023 5172 y(\()p Fp(q)21 b Fs(+)d Fp(ip)p Fs(\))p Fp(:)p eop %%Page: 14 14 14 13 bop 456 251 a Fm(14)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)456 450 y Fs(Therefore,)k(w)n(e)h(set)948 641 y Fl(r)1017 653 y Fn( )1091 641 y Fs(=)1223 585 y(1)p 1188 622 111 4 v 1188 638 a Fl(p)p 1258 638 42 4 v 1258 707 a Fs(2)1309 641 y(\()p Fl(r)1410 653 y Fn(q)1466 641 y Fl(\000)18 b Fp(i)p Fl(r)1647 653 y Fn(p)1685 641 y Fs(\))83 b(and)g Fl(r)2100 646 y Fm(\026)2086 661 y Fn( )2159 641 y Fs(=)2291 585 y(1)p 2257 622 111 4 v 2257 638 a Fl(p)p 2326 638 42 4 v 69 x Fs(2)2378 641 y(\()p Fl(r)2479 653 y Fn(q)2534 641 y Fs(+)18 b Fp(i)p Fl(r)2715 653 y Fn(p)2753 641 y Fs(\))512 b(\(44\))456 852 y(so)24 b(that,)i(if)f Fl(H)f Fs(=)f Fl(H)q Fs(\()p Fp( )s(;)1225 830 y Fs(\026)1208 852 y Fp( )s Fs(\))i(is)g(a)f(smo)r (oth)h(real)f(v)-5 b(alued)25 b(function,)h(w)n(e)f(ha)n(v)n(e)e(the)j (iden)n(ti\014cation)1346 1049 y Fp(X)1415 1061 y Fg(H)1476 1049 y Fs(\()p Fp( )s(;)1619 1027 y Fs(\026)1602 1049 y Fp( )s Fs(\))d(=)g Fl(\000)1885 993 y Fp(i)p 1877 1030 45 4 v 1877 1106 a Fo(~)1931 1049 y Fl(r)2014 1054 y Fm(\026)2000 1069 y Fn( )2051 1049 y Fl(H)q Fs(\()p Fp( )s(;)2265 1027 y Fs(\026)2248 1049 y Fp( )s Fs(\))28 b Fp(:)909 b Fs(\(45\))456 1229 y(and)27 b(in)h(complex)f(co)r(ordinates)f(the)i (P)n(oisson)e(brac)n(k)n(ets)g(are)g(computed)i(b)n(y)1067 1423 y Fl(fH)q Fp(;)14 b Fl(K)q(g)22 b Fs(:=)1474 1367 y Fp(i)p 1466 1404 V 1466 1480 a Fo(~)1535 1310 y Fq(Z)1581 1498 y Ff(R)1641 1355 y Fq(\002)1675 1423 y Fl(r)1744 1435 y Fn( )1795 1423 y Fl(H)q(r)1949 1428 y Fm(\026)1935 1443 y Fn( )1985 1423 y Fl(K)e(\000)e(r)2234 1428 y Fm(\026)2220 1443 y Fn( )2270 1423 y Fl(H)q(r)2410 1435 y Fn( )2461 1423 y Fl(K)2525 1355 y Fq(\003)2573 1423 y Fs(d)p Fp(x)631 b Fs(\(46\))555 1626 y(With)31 b(suc)n(h)e(a)g(notation)g(then)h(the)f (NLS)h(\(1\))g(can)f(b)r(e)h(written)g(in)f(the)h(form)f(of)h(a)f (Hamil-)456 1725 y(tonian)e(system,)g(the)h(corresp)r(onding)e (Hamiltonian)h(function)h(b)r(eing)g(the)g(energy)f(eq.)36 b(\(24\).)456 1887 y Fx(Remark)25 b(11.)36 b Fh(Such)25 b(a)h(Hamiltonian)h(is)e(invariant)h(under)g(the)f(action)h(of)h(the)e (Gauge)h(gr)l(oup)1542 2044 y Fp( )s Fs(\()p Fp(x)p Fs(\))f Fl(!)e Fp( )s Fs(\()p Fp(x)p Fs(\))p Fp(e)2048 2009 y Fg(\000)p Fn(i\014)2169 2044 y Fp(;)1105 b Fs(\(47\))456 2200 y Fh(for)29 b(any)g Fp(\014)k Fh(indep)l(endent)c(of)g Fp(x)p Fh(.)68 b(The)29 b(c)l(orr)l(esp)l(onding)h(c)l(onserve)l(d)f (quantity)g(is)f(the)h Fp(L)3187 2170 y Fm(2)3252 2200 y Fh(norm)1426 2394 y Fl(N)12 b Fs(\()p Fp( )s(;)1650 2372 y Fs(\026)1632 2394 y Fp( )t Fs(\))23 b(=)1833 2281 y Fq(Z)1879 2470 y Ff(R)1939 2394 y Fl(j)p Fp( )s Fs(\()p Fp(x)p Fs(\))p Fl(j)2153 2360 y Fm(2)2191 2394 y Fs(d)p Fp(x:)990 b Fs(\(48\))456 2592 y Fh(Equivalently)31 b(one)f(has)1654 2749 y Fl(f)o(E)7 b Fp(;)14 b Fl(N)e(g)23 b(\021)g Fs(0)p Fp(:)1216 b Fs(\(49\))555 2911 y(Let)24 b Fl(f)p Fp(\025)790 2923 y Fn(k)831 2911 y Fl(g)873 2881 y Fg(1)873 2934 y Fn(k)q Fm(=1)1021 2911 y Fs(and)f Fl(f)p Fp(')1274 2923 y Fn(k)1315 2911 y Fs(\()p Fp(x)p Fs(\))p Fl(g)1468 2881 y Fg(1)1468 2934 y Fn(k)q Fm(=1)1617 2911 y Fs(b)r(e)h(the)g (eigen)n(v)-5 b(alues)23 b(and)g(the)h(normalized)f(eigen)n(v)n(ectors) 456 3010 y(of)k Fp(H)619 3022 y Fm(0)656 3010 y Fs(,)h(let)1492 3229 y Fp( )s Fs(\()p Fp(x)p Fs(\))c(=)1799 3125 y Fg(1)1772 3150 y Fq(X)1772 3328 y Fn(k)q Fm(=1)1907 3229 y Fp(\020)1943 3241 y Fn(k)1984 3229 y Fp(')2038 3241 y Fn(k)2079 3229 y Fs(\()p Fp(x)p Fs(\))29 b Fp(;)1055 b Fs(\(50\))456 3469 y(and)27 b(de\014ne)h(the)g(Hilb)r(ert)g(spaces)f Fp(`)1576 3439 y Fm(2)1576 3490 y Fn(s)1640 3469 y Fs(of)h(the)g (complex)f(sequences)g(suc)n(h)g(that)1429 3642 y Fl(k)p Fp(\020)6 b Fl(k)1555 3600 y Fm(2)1555 3667 y Fn(s)1615 3642 y Fs(:=)1727 3563 y Fq(X)1726 3742 y Fn(k)q Fg(\025)p Fm(1)1861 3642 y Fp(\025)1909 3608 y Fn(s)1909 3663 y(k)1950 3642 y Fl(j)p Fp(\020)2009 3654 y Fn(k)2051 3642 y Fl(j)2074 3608 y Fm(2)2134 3642 y Fp(<)23 b Fl(1)992 b Fs(\(51\))456 3884 y(In)27 b(suc)n(h)h(a)f(w)n(a)n(y)f(w)n(e)h(ha)n(v)n(e)g (de\014ned)h(the)g(corresp)r(ondence)1101 4040 y Fp( )e Fl(2)e(X)1331 4006 y Fn(s)1390 4040 y Fl($)f(U)8 b Fs(\()p Fp( )s Fs(\))24 b(:=)e Fp(\020)30 b Fl(\021)22 b Fs(\()p Fp(\020)2032 4052 y Fm(1)2070 4040 y Fp(;)14 b(\020)2143 4052 y Fm(2)2181 4040 y Fp(:::\020)2286 4052 y Fn(j)2321 4040 y Fp(;)g(:::)p Fs(\))24 b Fl(2)f Fp(`)2596 4006 y Fm(2)2596 4061 y Fn(s)456 4196 y Fs(whic)n(h)29 b(is)g(a)h(unitary)f (isomorphism.)71 b(In)29 b(particular,)g(if)h Fl(E)37 b Fs(is)29 b(the)h(Hamiltonian)f(\(24\),)h(then)456 4296 y Fl(E)18 b(\016)11 b(U)631 4266 y Fg(\000)p Fm(1)743 4296 y Fs(\(still)25 b(denoted)f(b)n(y)f Fl(E)7 b Fs(\))24 b(is)g(the)g(Hamiltonian)g(of)g(the)g(same)f(system)h(written)g(in)g (terms)456 4396 y(of)33 b(the)h(new)g(v)-5 b(ariables)32 b Fp(\020)6 b Fs(.)88 b(In)34 b(terms)f(of)h(these)f(v)-5 b(ariables)33 b(the)h(quadratic)e(part)h Fl(E)3158 4408 y Fm(0)3229 4396 y Fs(of)g(the)456 4495 y(Hamiltonian)27 b(is)g(giv)n(en)g(b)n(y)1578 4664 y Fl(E)1622 4676 y Fm(0)1682 4664 y Fs(=)1770 4585 y Fq(X)1772 4763 y Fn(j)s Fg(\025)p Fm(1)1904 4664 y Fp(\025)1952 4676 y Fn(j)2001 4664 y Fl(j)p Fp(\020)2060 4676 y Fn(j)2095 4664 y Fl(j)2118 4622 y Fm(2)3297 4664 y Fs(\(52\))456 4905 y(and)g(the)h(P)n(oisson)e (brac)n(k)n(ets)f(can)j(b)r(e)g(written)f(as)1191 5107 y Fl(fH)q Fp(;)14 b Fl(K)q(g)23 b Fs(=)1575 5051 y Fp(i)p 1567 5088 V 1567 5164 a Fo(~)1663 5004 y Fg(1)1636 5029 y Fq(X)1636 5207 y Fn(k)q Fm(=1)1770 4990 y Fq(\024)1827 5051 y Fp(@)5 b Fl(H)p 1824 5088 126 4 v 1824 5164 a Fp(@)g(\020)1909 5176 y Fn(k)1976 5051 y Fp(@)g Fl(K)p 1970 5088 V 1970 5167 a Fp(@)2026 5145 y Fs(\026)2019 5167 y Fp(\020)2055 5179 y Fn(k)2124 5107 y Fl(\000)2220 5051 y Fp(@)g Fl(H)p 2217 5088 V 2217 5167 a Fp(@)2273 5145 y Fs(\026)2266 5167 y Fp(\020)2302 5179 y Fn(k)2369 5051 y Fp(@)g Fl(K)p 2363 5088 V 2363 5164 a Fp(@)g(\020)2448 5176 y Fn(k)2499 4990 y Fq(\025)p eop %%Page: 15 15 15 14 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(15)456 450 y Fx(Remark)37 b(12.)44 b Fh(F)-6 b(r)l(om)34 b(now)h(on)g(we)g(wil)t(l)h(work)g(in)f(the)g(sp)l(ac)l(e)g Fp(`)2520 420 y Fm(2)2520 471 y Fn(s)2592 450 y Fh(and)g(mor)l(e)l (over,)j(in)d(or)l(der)456 550 y(to)30 b(simplify,)i Fx(w)m(e)g(rescale)g(time)e(b)m(y)j(the)f(transformation)d Fp(t)23 b Fl(7!)h Fp(t=)p Fo(~)p Fh(,)30 b(thus)f(the)h(Poisson)456 649 y(br)l(ackets)g(take)g(the)g(form)1209 863 y Fl(fH)q Fp(;)14 b Fl(K)q(g)23 b Fs(=)f Fp(i)1645 759 y Fg(1)1618 784 y Fq(X)1618 963 y Fn(k)q Fm(=1)1752 746 y Fq(\024)1809 807 y Fp(@)5 b Fl(H)p 1806 844 126 4 v 1806 920 a Fp(@)g(\020)1891 932 y Fn(k)1958 807 y Fp(@)g Fl(K)p 1952 844 V 1952 923 a Fp(@)2008 901 y Fs(\026)2001 923 y Fp(\020)2037 935 y Fn(k)2106 863 y Fl(\000)2202 807 y Fp(@)g Fl(H)p 2199 844 V 2199 923 a Fp(@)2255 901 y Fs(\026)2248 923 y Fp(\020)2284 935 y Fn(k)2351 807 y Fp(@)g Fl(K)p 2345 844 V 2345 920 a Fp(@)g(\020)2430 932 y Fn(k)2481 746 y Fq(\025)3297 863 y Fs(\(53\))456 1097 y(3.2.)40 b Fx(Non)32 b(coupling)f(monomial.) 37 b Fs(It)29 b(is)f(useful)g(to)g(in)n(tro)r(duce)f(also)g(a)h (di\013eren)n(t)g(notation)456 1196 y(for)33 b(the)g(\014rst)g(t)n(w)n (o)g(v)-5 b(ariables)32 b(\(here)h(w)n(e)h(recall)e(that)i(w)n(e)f(are) f(w)n(orking)g(in)i(the)f(double)h(w)n(ell)456 1296 y(case\))27 b(and)g(for)g(the)h(remaining)f(ones,)g(th)n(us)g(let)h(us)g(denote) 1107 1448 y Fp(u)1155 1460 y Fm(1)1215 1448 y Fs(:=)23 b Fp(\020)1362 1460 y Fm(1)1427 1448 y Fp(;)97 b(u)1595 1460 y Fm(2)1655 1448 y Fs(:=)22 b Fp(\020)1801 1460 y Fm(2)1867 1448 y Fp(;)96 b(z)2025 1460 y Fn(j)2083 1448 y Fs(:=)23 b Fp(\020)2230 1460 y Fn(j)2293 1448 y Fp(;)97 b(j)28 b Fl(\025)22 b Fs(3)p Fp(:)670 b Fs(\(54\))555 1599 y(Consider)27 b(no)n(w)g(a)g(monomial)g(of)g(the)h(form)1535 1751 y Fp(\020)1577 1717 y Fn(K)1649 1729 y Fs(\026)1642 1751 y Fp(\020)1684 1717 y Fn(L)1757 1751 y Fs(=)23 b Fp(u)1893 1717 y Fn(k)1938 1751 y Fs(\026)-47 b Fp(u)1981 1717 y Fn(l)2006 1751 y Fp(z)2049 1717 y Fn(m)2116 1751 y Fs(\026)g Fp(z)2154 1717 y Fn(n)3297 1751 y Fs(\(55\))456 1903 y(where)27 b(w)n(e)g(used)g(the)h(notations)644 2033 y Fp(K)g Fs(=)23 b(\()p Fp(k)s(;)14 b(m)p Fs(\))p Fp(;)42 b(L)22 b Fs(=)h(\()p Fp(l)r(;)14 b(n)p Fs(\))644 2133 y Fp(k)26 b Fs(=)c(\()p Fp(k)875 2145 y Fm(1)913 2133 y Fp(;)14 b(k)993 2145 y Fm(2)1031 2133 y Fs(\))p Fp(;)41 b(m)23 b Fs(=)g(\()p Fp(m)1416 2145 y Fm(3)1453 2133 y Fp(;)14 b(m)1563 2145 y Fm(4)1601 2133 y Fp(;)g(m)1711 2145 y Fm(5)1748 2133 y Fp(;)g(::::)p Fs(\))p Fp(;)42 b(l)24 b Fs(=)f(\()p Fp(l)2168 2145 y Fm(1)2205 2133 y Fp(;)14 b(l)2267 2145 y Fm(2)2304 2133 y Fs(\))p Fp(;)42 b(n)23 b Fs(=)f(\()p Fp(n)2643 2145 y Fm(3)2681 2133 y Fp(;)14 b(n)2768 2145 y Fm(4)2805 2133 y Fp(;)g(n)2892 2145 y Fm(5)2929 2133 y Fp(;)g(::::)p Fs(\))644 2240 y Fp(u)692 2210 y Fn(k)755 2240 y Fl(\021)23 b Fp(u)891 2203 y Fn(k)926 2211 y Fj(1)891 2262 y Fm(1)962 2240 y Fp(u)1010 2203 y Fn(k)1045 2211 y Fj(2)1010 2262 y Fm(2)1082 2240 y Fp(;)41 b(z)1189 2210 y Fn(m)1275 2240 y Fl(\021)22 b Fp(z)1405 2203 y Fn(m)1464 2211 y Fj(3)1401 2262 y Fm(3)1500 2240 y Fp(z)1543 2203 y Fn(m)1602 2211 y Fj(4)1539 2262 y Fm(4)1652 2240 y Fp(:)14 b(:)g(:)f(z)1805 2195 y Fn(m)1864 2203 y Fk(q)1801 2249 y Fn(q)1915 2240 y Fp(:)h(:)g(:)3297 2137 y Fs(\(56\))456 2400 y Fx(Remark)22 b(13.)34 b Fh(A)22 b(monomial)i(of)g(the)f(form)30 b Fs(\(55\))22 b Fh(is)h(Gauge)h(invariant,)h(i.e.)38 b(invariant)24 b(under)456 2499 y(the)30 b(tr)l(ansformation)1293 2651 y Fp( )c Fl(7!)d Fp(e)1518 2617 y Fn(i\014)1586 2651 y Fp( )92 b Fh(that)30 b(is)90 b Fp(\020)2087 2663 y Fn(k)2151 2651 y Fl(7!)23 b Fp(e)2296 2617 y Fn(i\014)2364 2651 y Fp(\020)2400 2663 y Fn(k)456 2803 y Fh(if)30 b(and)g(only)h(if)f Fl(j)p Fp(K)6 b Fl(j)23 b Fs(=)g Fl(j)p Fp(L)p Fl(j)p Fh(,)29 b(wher)l(e)1596 2971 y Fl(j)p Fp(K)6 b Fl(j)23 b Fs(=)1829 2892 y Fq(X)1874 3069 y Fn(j)1963 2971 y Fl(j)p Fp(K)2057 2983 y Fn(j)2092 2971 y Fl(j)p Fp(:)456 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Fh(norm;)g(that)g(is)g Fl(j)p Fp(K)6 b Fl(j)22 b Fs(=)h Fl(j)p Fp(L)p Fl(j)29 b Fh(again.)555 3883 y Fs(Due)f(to)g(our)f(assumption)g(Hyp.1)g(on)h(the)f(p)r(oten)n (tial)h(one)f(has)456 4042 y Fx(Lemma)g(2.)39 b Fh(L)l(et)28 b Fp(\033)s Fs(\()p Fp(H)1199 4054 y Fm(0)1237 4042 y Fs(\))g Fh(b)l(e)g(the)h(sp)l(e)l(ctrum)e(of)i(the)f(line)l(ar)h(op)l (er)l(ator)g Fp(H)2740 4054 y Fm(0)2806 4042 y Fh(and)f(let)g Fo(~)g Fh(b)l(e)g(smal)t(l)456 4142 y(enough.)76 b(Ther)l(e)32 b(exists)f(a)h(se)l(quenc)l(e)g(of)g(\(not)f(ne)l(c)l(essarily)i(c)l (ontinuous)e(with)h(r)l(esp)l(e)l(ct)f(to)h Fo(~)p Fh(\))456 4241 y(functions)38 b Fl(f)p Fp(E)928 4253 y Fn(\015)970 4241 y Fs(\()p Fo(~)p Fs(\))p Fl(g)1121 4253 y Fn(\015)t Fg(2)p Ff(N)1251 4241 y Fh(,)j Fs(0)d Fp(<)g(E)1561 4253 y Fm(0)1637 4241 y Fp(<)h Fs(1)f Fh(and)g(ther)l(e)h(exists)f(a)h(p)l (ositive)g(c)l(onstant)f Fp(C)45 b(>)38 b Fs(1)p Fh(,)456 4341 y(indep)l(endent)30 b(of)h Fo(~)e Fh(and)h Fp(\015)5 b Fh(,)30 b(such)g(that:)657 4467 y Fs(i.)41 b([)p Fp(E)828 4479 y Fn(\015)890 4467 y Fl(\000)18 b Fp(C)1038 4436 y Fg(\000)p Fm(1)1127 4467 y Fo(~)p Fp(;)c(E)1270 4479 y Fn(\015)1331 4467 y Fs(+)k Fp(C)1479 4436 y Fg(\000)p Fm(1)1569 4467 y Fo(~)p Fs(])g Fl(\\)h Fp(\033)s Fs(\()p Fp(H)1880 4479 y Fm(0)1918 4467 y Fs(\))k(=)g Fl(;)p Fh(;)634 4566 y Fs(ii.)41 b(1)23 b Fp(<)g(E)958 4578 y Fn(\015)1019 4566 y Fl(\000)18 b Fp(E)1163 4578 y Fn(\015)t Fg(\000)p Fm(1)1314 4566 y Fp(<)k Fs(3)30 b Fh(for)g(al)t(l)h Fo(~)f Fh(and)g(al)t(l)h Fp(\015)c Fl(\025)c Fs(1)p Fh(;)611 4666 y Fs(iii.)41 b Fh(F)-6 b(or)30 b(any)g Fo(~)g Fh(\014xe)l(d,)g(we) g(c)l(onsider)h(the)e(sets)h(of)g(indexes)1078 4817 y Fp(J)1124 4829 y Fn(\015)1167 4817 y Fs(\()p Fo(~)p Fs(\))23 b(=)g Fp(J)1433 4829 y Fn(\015)1499 4817 y Fs(:=)f Fl(f)p Fp(j)28 b Fl(2)23 b Fo(N)63 b Fs(:)53 b Fp(E)2041 4829 y Fn(\015)t Fg(\000)p Fm(1)2192 4817 y Fp(<)22 b(\025)2327 4829 y Fn(j)2386 4817 y Fp(<)g(E)2534 4829 y Fn(\015)2577 4817 y Fl(g)14 b Fp(;)744 4969 y Fh(then)30 b(the)g(c)l(ar)l(dinality)h (of)g(these)f(sets)f(is)h(estimate)l(d)h(as)1624 5159 y Fs(#)p Fp(J)1739 5171 y Fn(\015)1782 5159 y Fs(\()p Fo(~)p Fs(\))23 b Fl(\024)2012 5102 y Fp(C)p 2012 5140 66 4 v 2022 5216 a Fo(~)2087 5159 y Fp(:)p eop %%Page: 16 16 16 15 bop 456 251 a Fm(16)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)456 450 y Fh(Pr)l(o)l(of.)43 b Fs(The)24 b(pro)r(of)f(is)h(an)f(immediate)h(consequence)f(of)h(the)g(follo)n (wing)f(result)h(\(see)f(Theorem)456 550 y(\(V-11\))k(in)h([18)o(],)g (see)f(also)f(Theorem)h(\(XI)r(I)r(I-81\))h(in)g([19)o(]\):)772 735 y Fp(N)839 750 y Fm([)p Fn(\013;\014)s Fm(])1007 735 y Fs(=)1152 679 y(1)p 1104 716 137 4 v 1104 792 a(2)p Fp(\031)s Fo(~)1265 668 y Fq(\002)1299 665 y(\014)1299 715 y(\014)1327 668 y(\010)1376 735 y Fs(\()p Fp(x;)14 b(p)p Fs(\))23 b Fl(2)h Fo(R)1722 701 y Fm(2)1816 735 y Fs(:)50 b Fp(\013)24 b Fl(\024)f Fp(p)2096 701 y Fm(2)2151 735 y Fs(+)18 b Fp(V)h Fs(\()p Fp(x)p Fs(\))24 b Fl(\024)f Fp(\014)2575 668 y 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b(;)14 b Fs(2)p Fp(\015)25 b Fs(+)20 b(1\))32 b(and)456 1911 y Fp(C)d(>)23 b Fs(1)k(satisfying)g(i..)64 b(F)-7 b(urthermore,)27 b(conditions)g(ii.)37 b(and)28 b(iii.)37 b(immediately)28 b(follo)n(w)e(since)689 2057 y(1)d(=)f(\(2)p Fp(\015)5 b Fs(\))18 b Fl(\000)g Fs([2\()p Fp(\015)23 b Fl(\000)18 b Fs(1\))h(+)f(1])k Fl(\024)h Fp(E)1754 2069 y Fn(\015)1815 2057 y Fl(\000)18 b Fp(E)1959 2069 y Fn(\015)t Fg(\000)p Fm(1)2110 2057 y Fl(\024)23 b Fs(\(2)p Fp(\015)g Fs(+)18 b(1\))g Fl(\000)g Fs(2\()p Fp(\015)23 b Fl(\000)18 b Fs(1\))23 b(=)f(3)456 2204 y(and)1292 2372 y(#)p Fp(J)1407 2384 y Fn(\015)1473 2372 y Fl(\024)h Fp(N)9 b Fs(\()p Fp(\015)c Fs(\))18 b(+)g Fp(N)9 b Fs(\()p Fp(\015)23 b Fl(\000)18 b Fs(1\))23 b Fl(\024)2302 2316 y Fs(2)p Fp(C)p 2302 2353 107 4 v 2333 2429 a Fo(~)2419 2372 y Fp(:)3380 2541 y Fd(\003)555 2707 y Fs(W)-7 b(e)28 b(\014x)g(a)f (sequence)g(with)h(suc)n(h)f(prop)r(erties.)555 2807 y(Hereafter,)40 b(all)d(the)h(p)r(erturbativ)n(e)e(construction)h(will) 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Fl(6)p Fs(=)22 b Fp(\015)2797 3455 y Fg(0)2820 3485 y Fh(.)555 3608 y(A)35 b(monomial)i(which)g(is)e(not)g(c)l(oupling)h(wil)t(l)h(b)l(e)e (c)l(al)t(le)l(d)45 b Fs(non)34 b(coupling)p Fh(.)89 b(A)35 b(p)l(olynomial)456 3708 y(c)l(ontaining)k Fx(only)g Fh(c)l(oupling)i(\(r)l(eps.)68 b(non)39 b(c)l(oupling\))h(monomial)h (wil)t(l)f(b)l(e)g(c)l(al)t(le)l(d)g(c)l(oupling)456 3807 y(\(r)l(eps.)f(non)29 b(c)l(oupling\).)456 3964 y Fx(Remark)35 b(14.)43 b Fh(F)-6 b(or)33 b(Gauge)h(invariant)g (monomials)g(the)g(c)l(ondition)g Fs(i.)49 b Fh(is)33 b(always)i(ful\014l)t(le)l(d.)456 4063 y(F)-6 b(urthermor)l(e,)30 b(in)f(terms)h(of)g(the)g(indexes)g Fp(k)s(;)14 b(l)r(;)g(m;)g(n)29 b Fh(c)l(ondition)i Fs(i.)38 b Fh(r)l(e)l(ads)1520 4210 y Fl(j)p Fp(k)s Fl(j)19 b(\000)f(j)p Fp(l)r Fl(j)k Fs(=)h Fl(j)p Fp(n)p Fl(j)18 b(\000)h(j)p Fp(m)p Fl(j)1083 b Fs(\(58\))456 4366 y Fx(Remark)35 b(15.)42 b Fh(A)n(ny)33 b(Gauge)h(invariant)g(analytic)g(function)f(c)l(an)g(b)l(e)h(uniquely)f (de)l(c)l(omp)l(ose)l(d)456 4466 y(into)j(the)h(sum)f(of)h(a)g(c)l (oupling)g(and)g(a)g(nonc)l(oupling)g(p)l(art.)95 b(We)37 b(r)l(e)l(c)l(al)t(l)g(that,)h(as)f(in)g(\014nite)456 4566 y(dimensional)c(sp)l(ac)l(es,)g(an)e(analytic)i(function)e(is)h(a) g(function)f(whose)i(T)-6 b(aylor)33 b(series)f(is)g(c)l(on-)456 4665 y(ver)l(gent)d(\(se)l(e)h(e.g.)39 b Fs([15)o(])p Fh(\).)555 4822 y Fs(W)-7 b(e)28 b(also)f(de\014ne)h(a)f(new)g(\()p Fo(~)h Fs(dep)r(enden)n(t\))h(norm)e(in)h(the)g(space)e Fl(X)2632 4792 y Fn(s)2696 4822 y Fs(as)h(follo)n(ws)456 4978 y Fx(De\014nition)j(3.)41 b Fh(Denote)1291 5125 y Fl(N)1359 5137 y Fn(E)1415 5125 y Fs(\()p Fp(z)t(;)19 b Fs(\026)-47 b Fp(z)s Fs(\))23 b(:=)1735 5046 y Fq(X)1776 5220 y Fn(\015)1869 5125 y Fp(E)1935 5090 y Fn(s)1930 5145 y(\015)2069 5046 y Fq(X)1986 5224 y Fn(j)s Fg(2)p Fn(J)2098 5232 y Fk(\015)2137 5224 y Fn(;j)s Fg(\025)p Fm(3)2286 5125 y Fl(j)p Fp(z)2348 5137 y Fn(j)2383 5125 y Fl(j)2406 5090 y Fm(2)3297 5125 y Fs(\(59\))p eop %%Page: 17 17 17 16 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(17)456 450 y Fh(then)29 b(the)h(quantity)1064 596 y Fl(k)p Fs(\()p Fp(u;)14 b(z)t Fs(\))p Fl(k)1338 554 y Fm(2)1338 621 y Fn(E)1418 596 y Fl(\021)22 b(k)p Fp(\020)6 b Fl(k)1631 554 y Fm(2)1631 621 y Fn(E)1710 596 y Fs(:=)22 b Fl(j)p Fp(u)1891 608 y Fm(1)1928 596 y Fl(j)1951 562 y Fm(2)2007 596 y Fs(+)c Fl(j)p Fp(u)2161 608 y Fm(2)2198 596 y Fl(j)2221 562 y Fm(2)2277 596 y Fs(+)g Fl(N)2428 608 y Fn(E)2484 596 y Fs(\()p Fp(z)t(;)g Fs(\026)-46 b Fp(z)s Fs(\))627 b(\(60\))456 739 y Fh(wil)t(l)28 b(b)l(e)f(c)l(al)t (le)l(d)h Fp(E)5 b Fh(-norm)27 b(of)h Fp(\020)6 b Fh(.)65 b(Denote)26 b Fp(`)1794 709 y Fm(2)1794 762 y Fn(E)1877 739 y Fh(the)h(sp)l(ac)l(e)h(of)f(se)l(quenc)l(es)g Fp(z)f Fs(=)d Fl(f)p Fp(z)2923 751 y Fn(j)2957 739 y Fl(g)2999 751 y Fn(j)s Fg(\025)p Fm(3)3145 739 y Fh(e)l(quipp)l(e)l(d)456 839 y(with)30 b(the)g(norm)g Fl(N)1064 851 y Fn(E)1120 839 y Fs(\()p Fp(z)t(;)18 b Fs(\026)-46 b Fp(z)r Fs(\))p Fh(.)69 b(By)31 b(abuse)f(of)g(notation)g(sometime)g(we)h(wil)t(l)g (also)f(write)1543 989 y Fl(k)o Fp(z)t Fl(k)1668 947 y Fm(2)1668 1014 y Fn(E)1747 989 y Fs(:=)23 b Fl(N)1926 1001 y Fn(E)1982 989 y Fs(\()p Fp(z)t(;)c Fs(\026)-47 b Fp(z)s Fs(\))p Fp(:)456 1143 y Fx(Remark)42 b(16.)k Fh(The)40 b Fp(E)5 b Fh(-norm)38 b(is)h(e)l(quivalent)g(to)g(the)g (standar)l(d)g(norm)g(of)g Fl(X)3008 1113 y Fn(s)3083 1143 y Fh(with)g(an)g Fo(~)456 1242 y Fh(indep)l(endent)30 b(c)l(onstant.)67 b(The)31 b(pr)l(o)l(of)f(is)g(a)g(trivial)h(c)l (omputation)f(and)g(is)g(left)g(to)f(the)h(r)l(e)l(ader.)456 1396 y Fx(Lemma)37 b(3.)45 b Fh(L)l(et)35 b Fp(\020)1113 1366 y Fn(K)1185 1374 y Fs(\026)1178 1396 y Fp(\020)1220 1366 y Fn(L)1306 1396 y Fh(b)l(e)h(a)h(non)e(c)l(oupling)i(Gauge)g (invariant)g(monomial)g(of)g(de)l(gr)l(e)l(e)f(at)456 1496 y(most)29 b Fp(tw)r(o)i Fh(in)f Fp(z)t(;)18 b Fs(\026)-46 b Fp(z)32 b Fh(\(i.e.)39 b Fl(j)p Fp(n)p Fl(j)19 b Fs(+)f Fl(j)p Fp(m)p Fl(j)23 b(\024)g Fs(2)p Fh(\),)29 b(then)h(one)g(has)969 1574 y Fq(\010)1018 1641 y Fl(N)1086 1653 y Fn(E)1142 1641 y Fp(;)14 b(\020)1221 1607 y Fn(K)1293 1619 y Fs(\026)1285 1641 y Fp(\020)1327 1607 y Fn(L)1377 1574 y Fq(\011)1449 1641 y Fs(=)23 b(0)59 b Fh(and)1842 1574 y Fq(\010)1891 1641 y Fl(j)p Fp(u)1962 1653 y Fm(1)1999 1641 y Fl(j)2022 1607 y Fm(2)2077 1641 y Fs(+)18 b Fl(j)p Fp(u)2231 1653 y Fm(2)2268 1641 y Fl(j)2291 1607 y Fm(2)2328 1641 y Fp(;)c(\020)2407 1607 y Fn(K)2479 1619 y Fs(\026)2472 1641 y Fp(\020)2514 1607 y Fn(L)2564 1574 y Fq(\011)2635 1641 y Fs(=)23 b(0)532 b(\(61\))456 1803 y Fh(Pr)l(o)l(of.)43 b Fs(First)29 b(remark)f(that)h(if)h Fp(u)1513 1773 y Fn(k)1559 1803 y Fs(\026)-48 b Fp(u)1601 1773 y Fn(l)1626 1803 y Fp(z)1669 1773 y Fn(n)1719 1803 y Fs(\026)h Fp(z)1757 1773 y Fn(m)1849 1803 y Fs(is)29 b(noncoupling)f(of)h(degree)g(at)g (most)g(t)n(w)n(o)f(in)i Fp(z)t(;)18 b Fs(\026)-46 b Fp(z)456 1902 y Fs(then)28 b(one)f(has)g Fl(j)p Fp(k)s Fl(j)c Fs(=)g Fl(j)p Fp(l)r Fl(j)k Fs(and)g(th)n(us)903 1981 y Fq(\010)952 2048 y Fp(u)1000 2060 y Fm(1)1042 2048 y Fs(\026)-48 b Fp(u)1084 2060 y Fm(1)1140 2048 y Fs(+)18 b Fp(u)1271 2060 y Fm(2)1313 2048 y Fs(\026)-48 b Fp(u)1355 2060 y Fm(2)1392 2048 y Fp(;)14 b(u)1477 2014 y Fn(k)1523 2048 y Fs(\026)-48 b Fp(u)1565 2014 y Fn(l)1590 2048 y Fp(z)1633 2014 y Fn(n)1683 2048 y Fs(\026)h Fp(z)1721 2014 y Fn(m)1783 1981 y Fq(\011)1855 2048 y Fs(=)22 b Fp(i)p Fs(\()p Fl(j)p Fp(k)s Fl(j)d(\000)f(j)p Fp(l)r Fl(j)p Fs(\))p Fp(u)2350 2014 y Fn(k)2395 2048 y Fs(\026)-47 b Fp(u)2438 2014 y Fn(l)2463 2048 y Fp(z)2506 2014 y Fn(n)2555 2048 y Fs(\026)g Fp(z)2593 2014 y Fn(m)2679 2048 y Fs(=)22 b(0)p Fp(:)466 b Fs(\(62\))456 2202 y(F)-7 b(urthermore,)29 b(either)g(it)h(is)f(of)h(degree)e(zero)h(in)g Fp(z)t(;)19 b Fs(\026)-47 b Fp(z)32 b Fs(or)d(there)g(exists)j(\026)-45 b Fp(\015)35 b Fs(and)2943 2183 y(\026)2944 2202 y Fp(j)5 b(;)3019 2183 y Fs(~)3020 2202 y Fp(j)31 b Fl(2)c Fp(J)3216 2214 y Fm(\026)-36 b Fn(\015)3285 2202 y Fs(suc)n(h)456 2302 y(that)27 b Fp(n)684 2307 y Fm(\026)685 2320 y Fn(j)743 2302 y Fs(=)c(1)k(and)h Fp(m)1134 2312 y Fm(~)1135 2325 y Fn(j)1193 2302 y Fs(=)22 b(1.)64 b(In)28 b(the)g(\014rst)f(case)g (one)g(has)g(that)1205 2395 y Fq(\010)1253 2463 y Fl(N)1321 2475 y Fn(E)1377 2463 y Fp(;)14 b(\020)1456 2428 y Fn(K)1528 2441 y Fs(\026)1521 2463 y Fp(\020)1563 2428 y Fn(L)1613 2395 y Fq(\011)1684 2463 y Fs(=)1772 2395 y Fq(\010)1820 2463 y Fl(N)1888 2475 y Fn(E)1944 2463 y Fs(\()p Fp(z)t(;)19 b Fs(\026)-47 b Fp(z)s Fs(\))p Fp(;)14 b(u)2215 2428 y Fn(k)2261 2463 y Fs(\026)-47 b Fp(u)2304 2428 y Fn(l)2329 2395 y Fq(\011)2400 2463 y Fs(=)23 b(0)456 2604 y(In)k(the)h(second)f (case,)g(then)816 2691 y Fq(\010)864 2758 y Fl(N)932 2770 y Fn(E)988 2758 y Fs(\()p Fp(z)t(;)19 b Fs(\026)-47 b Fp(z)s Fs(\))p Fp(;)14 b(u)1259 2724 y Fn(k)1305 2758 y Fs(\026)-47 b Fp(u)1348 2724 y Fn(l)1373 2758 y Fp(z)1416 2724 y Fn(n)1465 2758 y Fs(\026)g Fp(z)1503 2724 y Fn(m)1565 2691 y Fq(\011)1697 2758 y Fs(=)82 b Fp(u)1892 2724 y Fn(k)1938 2758 y Fs(\026)-47 b Fp(u)1981 2724 y Fn(l)2020 2679 y Fq(X)2060 2854 y Fn(\015)2153 2758 y Fp(E)2219 2724 y Fn(s)2214 2779 y(\015)2286 2679 y Fq(X)2271 2857 y Fn(j)s Fg(2)p Fn(J)2383 2865 y Fk(\015)2435 2691 y Fq(\010)2484 2758 y Fp(z)2523 2770 y Fn(j)2562 2758 y Fs(\026)g Fp(z)2596 2770 y Fn(j)2631 2758 y Fp(;)14 b(u)2716 2724 y Fn(k)2762 2758 y Fs(\026)-48 b Fp(u)2804 2724 y Fn(l)2829 2758 y Fp(z)2872 2724 y Fn(n)2922 2758 y Fs(\026)h Fp(z)2960 2724 y Fn(m)3022 2691 y Fq(\011)1697 3003 y Fs(=)82 b Fp(u)1892 2968 y Fn(k)1938 3003 y Fs(\026)-47 b Fp(u)1981 2968 y Fn(l)2006 3003 y Fp(z)2049 2968 y Fn(n)2098 3003 y Fs(\026)g Fp(z)2136 2968 y Fn(m)2199 3003 y Fp(E)2265 2968 y Fn(s)2263 3023 y Fm(\026)-36 b Fn(\015)2332 2924 y Fq(X)2316 3102 y Fn(j)s Fg(2)p Fn(J)2431 3110 y Fj(\026)-31 b Fk(\015)2481 3003 y Fp(i)p Fs(\()p Fp(n)2592 3015 y Fn(j)2645 3003 y Fl(\000)18 b Fp(m)2801 3015 y Fn(j)2836 3003 y Fs(\))23 b(=)g(0)3380 3233 y Fd(\003)555 3333 y Fs(Th)n(us)g(the)h Fp(E)5 b Fs({norm)23 b(is)g(in)n(v)-5 b(arian)n(t)22 b(under)i(the)g(dynamics)f (of)g(a)g(noncoupling)g(Hamiltonian)456 3432 y(of)k(degree)g(at)g(most) h(2)f(in)h Fp(z)t(;)18 b Fs(\026)-46 b Fp(z)r Fs(.)65 b(In)27 b(the)h(more)f(general)f(case)h(one)g(has)456 3586 y Fx(Corollary)37 b(6.)43 b Fh(L)l(et)33 b Fl(Z)40 b Fh(b)l(e)33 b(a)h(non)f(c)l(oupling)h(p)l(olynomial.)51 b(Assume)33 b(that)g(it)g(has)h(a)g(smo)l(oth)456 3686 y(ve)l(ctor)c(\014eld,)g(then)g(ther)l(e)g(exists)f Fp(C)36 b Fh(such)30 b(that)1179 3838 y Fl(jf)o(N)1311 3850 y Fn(E)1368 3838 y Fp(;)14 b Fl(Z)7 b(g)13 b Fs(\()p Fp(u;)19 b Fs(\026)-47 b Fp(u)o(;)14 b(z)t(;)k Fs(\026)-46 b Fp(z)r Fs(\))p Fl(j)24 b(\024)e Fp(C)e Fs([)p Fl(N)2185 3850 y Fn(E)2242 3838 y Fs(\()p Fp(z)t(;)e Fs(\026)-46 b Fp(z)r Fs(\)])2451 3796 y Fm(3)p Fn(=)p Fm(2)3297 3838 y Fs(\(63\))456 3991 y(3.3.)40 b Fx(Normal)30 b(form)h(construction.)41 b Fs(Let)28 b(us)g(rewrite)e Fl(E)7 b Fs(\()p Fp( )s(;)2517 3969 y Fs(\026)2499 3991 y Fp( )t Fs(\))28 b(as)f(follo)n(ws)1620 4133 y Fl(E)j Fs(=)23 b Fl(H)1852 4145 y Fm(0)1908 4133 y Fs(+)18 b Fp(\017)p Fl(P)2083 4145 y Fn(\017)456 4275 y Fs(where)817 4439 y Fl(H)887 4451 y Fm(0)925 4439 y Fs(\()p Fp( )s(;)1068 4417 y Fs(\026)1051 4439 y Fp( )s Fs(\))23 b(:=)g(\012\()p Fl(j)p Fp(\020)1425 4451 y Fm(1)1463 4439 y Fl(j)1486 4405 y Fm(2)1542 4439 y Fs(+)18 b Fl(j)p Fp(\020)1684 4451 y Fm(2)1721 4439 y Fl(j)1744 4405 y Fm(2)1782 4439 y Fs(\))g(+)1915 4360 y Fq(X)1917 4539 y Fn(j)s Fg(\025)p Fm(3)2049 4439 y Fp(\025)2097 4451 y Fn(j)2146 4439 y Fl(j)p Fp(\020)2205 4451 y Fn(j)2241 4439 y Fl(j)2264 4398 y Fm(2)2315 4439 y Fp(;)69 b Fs(\012)23 b(:=)2611 4383 y Fp(\025)2659 4395 y Fm(1)2715 4383 y Fs(+)18 b Fp(\025)2846 4395 y Fm(2)p 2611 4420 273 4 v 2726 4496 a Fs(2)2894 4439 y Fp(;)380 b Fs(\(64\))456 4666 y(and)1153 4809 y Fl(P)1211 4821 y Fn(\017)1242 4809 y Fs(\()p Fp( )s(;)1386 4787 y Fs(\026)1368 4809 y Fp( )t Fs(\))23 b(:=)1602 4753 y Fp(!)p 1602 4790 55 4 v 1612 4866 a(\017)1666 4809 y Fs(\()p Fl(j)p Fp(\020)1757 4821 y Fm(2)1795 4809 y Fl(j)1818 4775 y Fm(2)1874 4809 y Fl(\000)18 b(j)p Fp(\020)2016 4821 y Fm(1)2054 4809 y Fl(j)2077 4775 y Fm(2)2114 4809 y Fs(\))h(+)f Fl(P)2306 4821 y Fm(0)2343 4809 y Fs(\()p Fp( )s(;)2486 4787 y Fs(\026)2469 4809 y Fp( )s Fs(\))p Fp(:)716 b Fs(\(65\))555 4974 y(W)-7 b(e)22 b(are)f(going)f(to)i(pro)n(v)n(e)e(that)h(there)h (exists)f(a)g(canonical)g(transformation)e Fl(T)2983 4986 y Fn(\017)3037 4974 y Fs(whic)n(h)j(giv)n(es)456 5074 y(the)28 b(Hamiltonian)f(the)h(form)1455 5216 y Fl(E)e(\016)18 b(T)1630 5228 y Fn(\017)1685 5216 y Fs(=)k Fl(H)1842 5228 y Fm(0)1898 5216 y Fs(+)c Fp(\017)p Fl(Z)2075 5228 y Fn(\017)2125 5216 y Fs(+)g Fl(R)1019 b Fs(\(66\))p eop %%Page: 18 18 18 17 bop 456 251 a Fm(18)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)456 450 y Fs(where)28 b Fl(Z)757 462 y Fn(\017)817 450 y Fs(is)h(a)f(non)h(coupling)f(p)r(olynomial)g(and)g Fl(R)h Fs(has)f(a)g(smo)r(oth)h(v)n(ector)e(whic)n(h)i(is)f(exp)r(o-) 456 550 y(nen)n(tially)f(small)g(with)h Fp(\017)1237 520 y Fg(\000)p Fm(1)1326 550 y Fs(.)555 649 y(The)e(construction)f (will)g(b)r(e)h(recursiv)n(e.)60 b(T)-7 b(o)26 b(this)f(end)h(w)n(e)f (assume)g(one)g(has)g(b)r(een)h(able)g(to)456 749 y(construct)h(a)g (canonical)f(transformation)g Fl(T)1865 761 y Fn(r)1930 749 y Fs(putting)i(the)g(Hamiltonian)f(in)h(the)g(form)1076 896 y Fl(E)d(\016)18 b(T)1250 908 y Fn(r)1310 896 y Fl(\021)23 b(E)1449 862 y Fm(\()p Fn(r)r Fm(\))1561 896 y Fs(=)g Fl(H)1719 908 y Fm(0)1774 896 y Fs(+)c Fp(\017)p Fl(Z)1959 862 y Fm(\()p Fn(r)r Fm(\))2047 896 y Fs(\()p Fp(\017)p Fs(\))f(+)g Fp(\017)2280 862 y Fn(r)r Fm(+1)2401 896 y Fl(R)2471 862 y Fm(\()p Fn(r)r Fm(\))2560 896 y Fs(\()p Fp(\017)p Fs(\))639 b(\(67\))456 1044 y(with)38 b Fl(Z)722 1014 y Fm(\()p Fn(r)r Fm(\))848 1044 y Fs(b)r(eing)g(a)g(noncoupling)f (p)r(olynomial)g(and)h(where)f Fl(R)2569 1014 y Fm(\()p Fn(r)r Fm(\))2658 1044 y Fs(\()p Fp(\017)p Fs(\))i(has)e(a)h(v)n(ector) e(\014eld)456 1147 y(whic)n(h)f(is)h(b)r(ounded,)i(uniformly)d(with)h (resp)r(ect)f(to)h Fp(\017)f Fs(\()p Fl(Z)2318 1116 y Fm(\(0\))2444 1147 y Fs(=)h(0)f(and)g Fl(R)2861 1116 y Fm(\(0\))2987 1147 y Fs(=)h Fl(P)3146 1159 y Fn(\017)3177 1147 y Fs(\).)97 b(W)-7 b(e)456 1246 y(lo)r(ok)36 b(for)h(an)f (auxiliary)g(Hamiltonian)h Fl(G)1798 1258 y Fn(r)r Fm(+1)1956 1246 y Fs(suc)n(h)g(that)g(considering)f(the)h(corresp)r(onding)456 1346 y(Hamilton)27 b(equations)1645 1470 y(_)1628 1492 y Fp(\020)j Fs(=)22 b Fp(X)1850 1504 y Fg(G)1890 1512 y Fk(r)q Fj(+1)1999 1492 y Fs(\()p Fp(\020)6 b Fs(\))456 1656 y(and)28 b(the)h(corresp)r(onding)d(\015o)n(w)i Fp(\036)1519 1626 y Fn(t)1519 1677 y(r)r Fm(+1)1668 1656 y Fs(one)g(has)g(that)h Fl(E)2202 1626 y Fm(\()p Fn(r)r Fm(\))2310 1656 y Fl(\016)18 b Fp(\036)2419 1626 y Fn(\017)2447 1601 y Fk(r)q Fj(+1)2419 1677 y Fn(r)r Fm(+1)2584 1656 y Fs(is)29 b(in)f(the)h(form)f(\(67\))g(with)456 1756 y Fp(r)21 b Fs(+)d(1)27 b(in)h(place)f(of)h Fp(r)r Fs(.)555 1855 y(Explicitly)g(one)f(has)1002 2012 y Fl(E)1053 1977 y Fm(\()p Fn(r)r Fm(\))1160 2012 y Fl(\016)18 b Fp(\036)1269 1977 y Fn(\017)1297 1952 y Fk(r)q Fj(+1)1269 2032 y Fn(r)r Fm(+1)1489 2012 y Fs(=)83 b Fl(H)1707 2024 y Fm(0)1762 2012 y Fs(+)18 b Fp(\017)p Fl(Z)1946 1977 y Fm(\()p Fn(r)r Fm(\))3297 2012 y Fs(\(68\))1489 2168 y(+)83 b Fp(\017)1671 2133 y Fn(r)r Fm(+1)1805 2076 y Fq(\020)1855 2168 y Fl(R)1925 2133 y Fm(\()p Fn(r)r Fm(\))2032 2168 y Fs(+)18 b Fl(fH)2227 2180 y Fm(0)2264 2168 y Fp(;)c Fl(G)2350 2180 y Fn(r)r Fm(+1)2471 2168 y Fl(g)2513 2076 y Fq(\021)3297 2168 y Fs(\(69\))1489 2342 y(+)83 b Fl(H)1707 2354 y Fm(0)1762 2342 y Fl(\016)18 b Fp(\036)1871 2308 y Fn(\017)1899 2283 y Fk(r)q Fj(+1)1871 2362 y Fn(r)r Fm(+1)2026 2342 y Fl(\000)g(H)2179 2354 y Fm(0)2235 2342 y Fl(\000)g Fp(\017)2352 2308 y Fn(r)r Fm(+1)2487 2342 y Fl(f)o(H)2598 2354 y Fm(0)2636 2342 y Fp(;)c Fl(G)2722 2354 y Fn(r)r Fm(+1)2843 2342 y Fl(g)412 b Fs(\(70\))1489 2498 y(+)83 b Fp(\017)1685 2406 y Fq(\020)1734 2498 y Fl(Z)1801 2464 y Fm(\()p Fn(r)r Fm(\))1908 2498 y Fl(\016)18 b Fp(\036)2017 2464 y Fn(\017)2045 2439 y Fk(r)q Fj(+1)2017 2519 y Fn(r)r Fm(+1)2172 2498 y Fl(\000)g(Z)2322 2464 y Fm(\()p Fn(r)r Fm(\))2410 2406 y Fq(\021)3297 2498 y Fs(\(71\))1489 2681 y(+)83 b Fp(\017)1671 2646 y Fn(r)r Fm(+1)1805 2589 y Fq(\020)1855 2681 y Fl(R)1925 2646 y Fm(\()p Fn(r)r Fm(\))2032 2681 y Fl(\016)18 b Fp(\036)2141 2646 y Fn(\017)2169 2621 y Fk(r)q Fj(+1)2141 2701 y Fn(r)r Fm(+1)2296 2681 y Fl(\000)g(R)2449 2646 y Fm(\()p Fn(r)r Fm(\))2538 2589 y Fq(\021)3297 2681 y Fs(\(72\))456 2850 y(No)n(w,)28 b(it)h(is)f(quite)h(easy)f(to)g(understand)g(that)h(\(70{72\))e(are)g (higher)h(order)f(terms)h(\(they)h(will)456 2950 y(b)r(e)34 b(estimated)f(later\),)i(while)f(\(69\))f(is)g(the)h(term)g(of)f(order) g Fp(\017)2421 2920 y Fn(r)r Fm(+1)2541 2950 y Fs(.)55 b(Th)n(us,)35 b(if)f(one)f(is)g(able)h(to)456 3049 y(c)n(ho)r(ose)26 b Fl(G)768 3061 y Fn(r)r Fm(+1)917 3049 y Fs(so)h(that)1538 3200 y Fl(R)1608 3165 y Fm(\()p Fn(r)r Fm(\))1715 3200 y Fs(+)18 b Fl(fH)1910 3212 y Fm(0)1947 3200 y Fp(;)c Fl(G)2033 3212 y Fn(r)r Fm(+1)2154 3200 y Fl(g)1101 b Fs(\(73\))456 3344 y(is)27 b(non)h(coupling,)f(then)h(the)g(coupling)f (terms)g(are)g(pushed)h(to)f(order)f Fp(r)21 b Fs(+)e(2,)27 b(and)1205 3509 y Fl(Z)1272 3475 y Fn(r)r Fm(+1)1416 3509 y Fs(=)22 b Fl(Z)1570 3475 y Fn(r)1625 3509 y Fs(+)c Fp(\017)1742 3475 y Fn(r)1793 3417 y Fq(h)1832 3509 y Fl(R)1902 3475 y Fm(\()p Fn(r)r Fm(\))2009 3509 y Fs(+)g Fl(fH)2204 3521 y Fm(0)2241 3509 y Fp(;)c Fl(G)2327 3521 y Fn(r)r Fm(+1)2448 3509 y Fl(g)2490 3417 y Fq(i)456 3674 y Fs(is)27 b(non)h(coupling.)555 3773 y(The)21 b(main)f(step)h(of) f(the)h(pro)r(of)f(is)g(the)h(construction)f(and)g(the)h(estimate)f(of) h(suc)n(h)f(a)g(function)456 3873 y Fl(G)33 b Fs(ful\014lling)28 b(the)g(homological)d(equation)i(asso)r(ciated)g(to)g(\(73\).)456 4042 y(3.4.)40 b Fx(F)-8 b(ramew)m(ork)33 b(and)h(notations.)40 b Fs(Before)28 b(pro)r(ceeding)f(to)i(the)g(construction)f(and)g(the) 456 4142 y(estimation)36 b(of)h(suc)n(h)f(a)g Fl(G)43 b Fs(it)37 b(is)f(useful)h(to)g(extend)g(the)g(setting)g(in)g(whic)n(h) g(w)n(e)f(will)h(w)n(ork.)456 4242 y(Indeed,)i(since)e(w)n(e)g(are)f(w) n(orking)f(with)i(analytic)g(functions)g(it)g(is)g(useful)g(to)g(w)n (ork)f(in)h(the)456 4341 y Fh(c)l(omplexi\014c)l(ation)j Fs(of)32 b(our)g(space.)82 b(More)32 b(precisely)-7 b(,)33 b Fh(we)h(c)l(onsider)h(now)f(a)h(phase)g(sp)l(ac)l(e)g(in)456 4441 y(which)h(the)g(variables)h Fp(v)f Fs(:=)h(\026)-47 b Fp(u)35 b Fh(and)g Fp(w)h Fs(:=)h(\026)-47 b Fp(z)39 b Fh(ar)l(e)c(indep)l(endent)h(of)g Fp(u;)14 b(z)t Fs(.)86 b(Actually)34 b(this)g(is)456 4540 y(equiv)-5 b(alen)n(t)31 b(to)h(complexify)g(the)g(space)g(in)g(whic)n(h)g(v)-5 b(ary)31 b(the)h(v)-5 b(ariables)31 b Fp(p;)14 b(q)35 b Fs(of)d(section)f(3.1.)456 4640 y(With)d(suc)n(h)f(an)h(extension)f (the)h(P)n(oisson)d(brac)n(k)n(et)h(will)i(tak)n(e)f(the)h(form)656 4843 y Fl(f)o(H)q Fp(;)14 b Fl(K)q(g)23 b Fs(=)g Fp(i)1108 4739 y Fm(2)1064 4764 y Fq(X)1065 4943 y Fn(k)q Fm(=1)1199 4726 y Fq(\024)1261 4787 y Fp(@)5 b Fl(H)p 1252 4824 138 4 v 1252 4900 a Fp(@)g(u)1349 4912 y Fn(k)1418 4787 y Fp(@)g Fl(K)p 1409 4824 130 4 v 1409 4900 a Fp(@)g(v)1498 4912 y Fn(k)1568 4843 y Fl(\000)1673 4787 y Fp(@)g Fl(K)p 1661 4824 138 4 v 1661 4900 a Fp(@)g(u)1758 4912 y Fn(k)1823 4787 y Fp(@)g Fl(H)p 1818 4824 130 4 v 1818 4900 a Fp(@)g(v)1907 4912 y Fn(k)1957 4726 y Fq(\025)2020 4843 y Fs(+)18 b Fp(i)2172 4739 y Fg(1)2145 4764 y Fq(X)2148 4941 y Fn(j)s Fm(=3)2279 4726 y Fq(\024)2334 4787 y Fp(@)5 b Fl(H)p 2333 4824 123 4 v 2333 4900 a Fp(@)g(z)2421 4912 y Fn(j)2490 4787 y Fp(@)g Fl(K)p 2475 4824 144 4 v 2475 4900 a Fp(@)g(w)2583 4912 y Fn(j)2646 4843 y Fl(\000)2744 4787 y Fp(@)g Fl(K)p 2739 4824 123 4 v 2739 4900 a Fp(@)g(z)2827 4912 y Fn(j)2893 4787 y Fp(@)g Fl(H)p 2881 4824 144 4 v 2881 4900 a Fp(@)g(w)2989 4912 y Fn(j)3035 4726 y Fq(\025)456 5071 y Fs(Suc)n(h)27 b(a)g(phase)h(space)e(will)i(b)r(e)g(denoted)g(b)n(y)f Fl(X)1948 5041 y Fn(s)1936 5094 y Ff(C)1984 5071 y Fs(.)37 b(In)28 b Fl(X)2219 5041 y Fn(s)2207 5094 y Ff(C)2282 5071 y Fs(w)n(e)f(will)h(use)g(the)g(norm)1078 5216 y Fl(d)p Fs(\()p Fp(u;)14 b(v)s(;)g(z)t(;)g(w)r Fs(\))p Fl(e)23 b Fs(:=)g(max)13 b Fl(f)o(k)p Fs(\()p Fp(u;)h(v)s Fs(\))p Fl(k)2141 5240 y Ff(C)2177 5224 y Fj(2)2233 5216 y Fp(;)g Fl(k)o Fs(\()p Fp(z)t(;)g(w)r Fs(\))p Fl(k)2558 5240 y Fn(E)2614 5216 y Fl(g)641 b Fs(\(74\))p eop %%Page: 19 19 19 18 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(19)456 450 y Fs(with)28 b Fl(k)o Fs(\()p Fp(z)t(;)14 b(w)r Fs(\))p Fl(k)933 475 y Fn(E)1017 450 y Fs(de\014ned)28 b(b)n(y)897 618 y Fl(k)o Fs(\()p Fp(z)t(;)14 b(w)r Fs(\))p Fl(k)1185 576 y Fm(2)1185 643 y Fn(E)1264 618 y Fs(:=)23 b Fl(k)p Fp(z)t Fl(k)1500 576 y Fm(2)1500 643 y Fn(E)1575 618 y Fs(+)18 b Fl(k)o Fp(w)r Fl(k)1803 576 y Fm(2)1803 643 y Fn(E)1873 618 y Fp(;)69 b Fs(where)55 b Fl(k)17 b(\001)i(k)2376 583 y 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Fp(k)s Fl(j)14 b(\000)g(j)p Fp(l)r Fl(j)22 b Fs(=)g Fl(j)p Fp(m)p Fl(j)14 b(\000)g(j)p Fp(n)p Fl(j)23 b Fs(=)f Fl(\000)p Fs(1)j(\(due)g(to)h(the)f(fact)h(that)f Fl(G)3064 520 y Fm(01)3160 550 y Fs(is)g(linear)456 649 y(in)i Fp(w)r Fs(\),)i(and)f(therefore)1453 739 y Fq(n)1508 831 y Fl(H)1579 796 y Fm(\(1\))1668 831 y Fp(;)14 b(G)1770 796 y Fm(01)1770 851 y Fn(j)1840 739 y Fq(o)1919 831 y Fs(=)22 b Fp(i)p Fs(\012)p Fp(G)2160 796 y Fm(01)2160 851 y Fn(j)2258 831 y Fp(:)1016 b Fs(\(93\))456 1012 y(Inserting)27 b(this)g (expression)g(in)g(\(90\))h(one)f(gets)1457 1170 y Fp(i)p Fs(\012)p Fp(G)1611 1135 y Fm(01)1611 1190 y Fn(j)1700 1170 y Fl(\000)18 b Fp(i\025)1860 1182 y Fn(j)1895 1170 y Fp(G)1960 1135 y Fm(01)1960 1190 y Fn(j)2054 1170 y Fs(=)k Fp(F)2206 1135 y Fm(01)2194 1190 y Fn(j)456 1334 y Fs(whic)n(h)27 b(sho)n(ws)f(that)i(the)g(function)g Fl(G)1633 1304 y Fm(01)1731 1334 y Fs(of)34 b(\(85\))27 b(actually)g(ful\014lls)h(the)g(equation)f(\(89\))g(and)h(is)456 1434 y(coupling.)68 b(All)29 b(the)g(other)f(terms)g(can)g(b)r(e)h (studied)g(in)g(the)g(same)f(w)n(a)n(y;)g(a)g(detailed)h(pro)r(of)f(is) 456 1533 y(omitted.)2620 b Fd(\003)456 1633 y Fh(End)31 b(of)i(the)e(Pr)l(o)l(of)i(of)f(The)l(or)l(em)g(5)p Fs(.)73 b(W)-7 b(e)30 b(will)f(explicitly)h(pro)n(v)n(e)e(only)h(the)g (estimate)h(of)f(the)456 1733 y(norms)h(of)g(the)i(v)n(ector)d (\014elds)i(of)g Fp(X)1588 1748 y Fg(G)1631 1731 y Fj(01)1727 1733 y Fs(and)g Fp(X)1961 1748 y Fg(G)2004 1731 y Fj(11)2069 1733 y Fs(,)g(the)h(other)e(b)r(eing)h(similar)f(and)h(simpler.)456 1832 y(W)-7 b(e)24 b(start)g(with)g Fp(X)1045 1847 y Fg(G)1088 1830 y Fj(01)1153 1832 y Fs(.)60 b(Consider)23 b(\014rst)h(the)g(v)n(ector)f(\014eld)h(of)h Fl(F)2468 1802 y Fm(01)2538 1832 y Fs(,)g(whose)e(comp)r(onen)n(ts)h(ha)n(v)n(e) 456 1932 y(the)k(form)1477 2121 y Fp(X)1553 2083 y Fn(u)1592 2092 y Fk(l)1546 2149 y Fg(F)1600 2132 y Fj(01)1687 2121 y Fs(=)22 b Fp(i)1817 2004 y Fq(\034)1889 2064 y Fp(@)5 b(F)2003 2034 y Fm(01)p 1889 2102 184 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Fg(L)p Fm(\()p Fg(X)1354 3042 y Fk(s)1345 3082 y Fc(C)1385 3062 y Fn(;)p Ff(C)1440 3045 y Fj(4)1479 3062 y Fm(\))1532 2979 y Fl(\024)1640 2923 y Fs(1)p 1629 2960 64 4 v 1629 3036 a Fp(R)1717 2979 y Fs(sup)1731 3048 y Fn(B)1781 3056 y Fk(R)1842 2979 y Fl(d)p Fp(X)1955 2936 y Fm(\()p Fn(u;v)r Fm(\))1948 3003 y Fg(H)2104 2979 y Fs(\()p Fp(u;)g(v)s(;)g(z)t (;)g(w)r Fs(\))p Fl(e)23 b Fs(=)2643 2923 y(1)p 2632 2960 V 2632 3036 a Fp(R)2719 2883 y Fq(\014)2719 2933 y(\014)2719 2983 y(\014)2747 2979 y Fp(X)2823 2936 y Fm(\()p Fn(u;v)r Fm(\))2816 3003 y Fg(H)2973 2883 y Fq(\014)2973 2933 y(\014)2973 2983 y(\014)3000 3037 y Fn(R)3297 2979 y Fs(\(97\))456 3214 y(where)30 b(the)i(norm)f(at)g(l.h.s.)48 b(is)31 b(the)g(norm)g(as)f(a)h(linear)g(op)r(erator)e(from)i Fl(X)2867 3184 y Fn(s)2855 3237 y Ff(C)2934 3214 y Fs(to)g Fo(C)3093 3184 y Fm(4)3137 3214 y Fs(.)79 b(Using)456 3332 y(this)33 b(inequalit)n(y)f(and)g(making)g(use)h(of)39 b(\(85\))32 b(it)h(is)g(v)n(ery)e(easy)h(to)g(estimate)h Fp(X)2984 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Fq(\034)1891 4261 y Fp(@)5 b(F)2005 4231 y Fm(01)p 1891 4298 184 4 v 1937 4374 a Fp(@)g(v)2085 4317 y Fs(;)31 b(~)-59 b Fp(w)2183 4200 y Fq(\035)3297 4317 y Fs(\(99\))456 4518 y(and)27 b(similarly)g(for)g(the)h Fp(v)j Fs(comp)r(onen)n(ts,)c (and)g(th)n(us)1427 4617 y Fq(\014)1427 4667 y(\014)1427 4717 y(\014)1454 4713 y Fp(X)1530 4669 y Fm(\()p Fn(u;v)r Fm(\))1523 4741 y Fg(G)1566 4724 y Fj(01)1680 4617 y Fq(\014)1680 4667 y(\014)1680 4717 y(\014)1708 4771 y Fn(R)1785 4713 y Fl(\024)1883 4656 y Fp(C)p 1883 4694 66 4 v 1893 4770 a Fo(~)1972 4617 y Fq(\014)1972 4667 y(\014)1972 4717 y(\014)2000 4713 y Fp(X)2076 4669 y Fm(\()p Fn(u;v)r Fm(\))2069 4737 y Fg(H)2225 4617 y Fq(\014)2225 4667 y(\014)2225 4717 y(\014)2253 4771 y Fn(R)3255 4713 y Fs(\(100\))456 4893 y(for)j(some)g Fp(C)35 b(>)28 b Fs(1.)78 b(The)31 b(estimate)f(of)h(the)h Fp(z)i Fs(comp)r(onen)n(ts)c (of)h(the)g(v)n(ector)f(\014eld)h(is)g(simpler)456 4993 y(\(due)d(to)f(the)h(simpler)f(form)h(of)f(the)h Fp(z)j Fs(comp)r(onen)n(t\))d(and)f(collecting)g(the)h(t)n(w)n(o)f(one)g(gets) 1508 5192 y Fl(j)p Fp(X)1600 5207 y Fg(G)1643 5190 y Fj(01)1708 5192 y Fl(j)1731 5217 y Fn(R)1809 5192 y Fl(\024)1906 5136 y Fp(C)p 1906 5173 V 1916 5249 a Fo(~)1995 5192 y Fl(j)p Fp(X)2087 5204 y Fg(H)2148 5192 y Fl(j)2171 5217 y Fn(R)3255 5192 y Fs(\(101\))p eop %%Page: 22 22 22 21 bop 456 251 a Fm(22)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)555 450 y Fs(W)-7 b(e)35 b(come)f(no)n(w)g(to)h(the)g (estimate)f(of)h(the)g(v)n(ector)e(\014eld)i(of)g Fl(G)2553 420 y Fm(11)2623 450 y Fs(.)93 b(Preliminary)33 b(to)h(this)456 550 y(estimate)27 b(w)n(e)g(remark)g(that)g(the)h(comp)r(onen)n(ts)f (of)h(the)g(v)n(ector)e(\014eld)i(of)g Fl(F)2804 520 y Fm(11)2902 550 y Fs(are)e(giv)n(en)h(b)n(y)1253 764 y Fp(X)1329 721 y Fm(\()p Fn(u;v)r Fm(\))1322 792 y Fg(F)1376 775 y Fj(11)1501 764 y Fs(=)c Fp(d)1632 730 y Fm(2)1632 784 y Fn(z)r(w)1720 764 y Fp(X)1796 721 y Fm(\()p Fn(u;v)r Fm(\))1789 788 y Fg(H)1945 764 y Fs(\()p Fp(u;)14 b(v)s(;)g Fs(0)p Fp(;)g Fs(0\)[)p Fp(z)t(;)g(w)r Fs(])773 b(\(102\))1293 897 y Fp(X)1369 863 y Fn(w)1362 919 y Fg(F)1416 903 y Fj(11)1503 897 y Fs(=)23 b Fp(d)1634 909 y Fn(z)1672 897 y Fp(X)1748 863 y Fn(w)1741 917 y Fg(H)1802 897 y Fs(\()p Fp(u;)14 b(v)s(;)g Fs(0)p Fp(;)g Fs(0\))p Fp(z)25 b Fs(=)e Fp(F)2369 863 y Fm(11)2439 897 y Fp(z)777 b Fs(\(103\))456 1100 y(th)n(us)27 b(in)h(particular,)e(b)n(y)i(Cauc)n(h) n(y)e(inequalit)n(y)-7 b(,)1232 1337 y(sup)1246 1407 y Fn(B)1296 1415 y Fk(R)1371 1266 y Fq(\015)1371 1316 y(\015)1417 1337 y Fp(F)1482 1303 y Fm(11)1552 1337 y Fs(\()p Fp(u;)14 b(v)s Fs(\))1744 1266 y Fq(\015)1744 1316 y(\015)1790 1370 y Fg(L)p Fm(\()p Fn(`)1890 1350 y Fj(2)1890 1390 y Fk(E)1938 1370 y Fn(;`)1986 1350 y Fj(2)1986 1390 y Fk(E)2034 1370 y Fm(\))2087 1337 y Fl(\024)2195 1281 y Fs(1)p 2184 1318 64 4 v 2184 1394 a Fp(R)2272 1337 y Fl(j)p Fp(X)2371 1303 y Fn(w)2364 1357 y Fg(H)2425 1337 y Fl(j)2448 1362 y Fn(R)3255 1337 y Fs(\(104\))456 1602 y(where)27 b(the)h(norm)f(at)g(l.h.s.)37 b(is)28 b(the)g(norm)f(as)g(a)g(linear)g(op)r(erator)e(from)j Fp(`)2777 1572 y Fm(2)2777 1625 y Fn(E)2860 1602 y Fs(to)g Fp(`)2997 1572 y Fm(2)2997 1625 y Fn(E)3052 1602 y Fs(.)555 1702 y(In)36 b(order)e(to)i(estimate)g Fp(X)1419 1672 y Fn(z)1412 1728 y Fg(G)1455 1712 y Fj(11)1555 1702 y Fs(w)n(e)f(ha)n(v)n(e)g(just)h(to)g(estimate)f(the)h(norm)g(of)f(the)h (op)r(erator)456 1810 y(\(from)31 b Fp(`)723 1780 y Fm(2)723 1833 y Fn(E)810 1810 y Fs(to)g(itself)6 b(\))32 b(whose)e(matrix)h(is)g (de\014ned)h(in)f(\(87\).)78 b(T)-7 b(o)31 b(this)h(end)f(remark)f (that)h(the)456 1909 y(b)r(oundedness)24 b(of)h Fp(G)1097 1879 y Fm(11)1192 1909 y Fs(as)g(an)f(op)r(erator)f(from)h Fp(`)1964 1879 y Fm(2)1964 1932 y Fn(E)2045 1909 y Fs(to)h(itself)g (\(or)f(to)h(its)g(dual\))f(is)h(equiv)-5 b(alen)n(t)25 b(to)456 2009 y(the)g(b)r(oundedness)h(of)f(the)h(op)r(erator)e(with)i (matrix)e(elemen)n(ts)i Fp(t)2470 2021 y Fn(l)2495 2009 y Fp(G)2560 1979 y Fm(11)2560 2033 y Fn(j)s(l)2631 2009 y Fp(s)2670 2021 y Fn(j)2730 2009 y Fs(as)f(an)g(op)r(erator)f(from)456 2120 y Fp(`)491 2090 y Fm(2)554 2120 y Fs(to)i(itself,)h(where)e Fp(t)1145 2132 y Fn(l)1171 2120 y Fp(;)14 b(s)1247 2132 y Fn(j)1308 2120 y Fs(are)25 b(suitable)h(p)r(ositiv)n(e)g(n)n(um)n(b)r (ers)f(\(in)i(fact,)f Fp(`)2740 2090 y Fm(2)2803 2120 y Fs(and)g Fp(`)2998 2090 y Fm(2)2998 2143 y Fn(E)3080 2120 y Fs(are)f(spaces)456 2219 y(with)j(not)f(equiv)-5 b(alen)n(t)28 b(norms\).)64 b(Th)n(us)27 b(Lemma)g(8)h(of)f(the)h(app)r (endix)g(ensures)f(that)938 2461 y(sup)952 2531 y Fn(B)1002 2539 y Fk(R)1077 2391 y Fq(\015)1077 2440 y(\015)1123 2461 y Fp(G)1188 2427 y Fm(11)1259 2461 y Fs(\()p Fp(u;)14 b(v)s Fs(\))1451 2391 y Fq(\015)1451 2440 y(\015)1497 2494 y Fg(L)p Fm(\()p Fn(`)1597 2474 y Fj(2)1597 2514 y Fk(E)1645 2494 y Fn(;`)1693 2474 y Fj(2)1693 2514 y Fk(E)1741 2494 y Fm(\))1794 2461 y Fl(\024)1897 2405 y Fs(1)p 1891 2442 54 4 v 1891 2518 a Fp(\013)1968 2461 y Fl(j)q Fp(X)2068 2427 y Fn(w)2061 2482 y Fg(H)2121 2461 y Fl(j)2144 2486 y Fn(R)2213 2461 y Fp(;)69 b(\013)23 b Fl(\025)g Fp(C)2534 2427 y Fg(\000)p Fm(1)2623 2461 y Fo(~)2668 2427 y Fm(3)p Fn(=)p Fm(2)2773 2461 y Fp(;)459 b Fs(\(105\))456 2727 y(for)37 b(some)g Fp(C)46 b(>)39 b Fs(1,)h(since)d Fl(j)p Fp(\025)1409 2739 y Fn(j)1470 2727 y Fl(\000)25 b Fp(\025)1608 2739 y Fn(l)1633 2727 y Fl(j)40 b(\025)g Fp(C)1866 2696 y Fg(\000)p Fm(1)1955 2727 y Fo(~)e Fs(if)g Fp(j)k Fs(and)c Fp(l)h Fs(b)r(elong)e(to)h (di\013eren)n(t)f(sets)h Fp(J)3379 2739 y Fn(\015)3421 2727 y Fs(.)456 2826 y(W)-7 b(orking)38 b(in)h(a)f(similar)g(w)n(a)n(y) g(for)h(the)g(other)f(comp)r(onen)n(ts)h(and)f(the)i(other)e(parts)g (of)h(the)456 2926 y(v)n(ector)26 b(\014eld)i(of)f(the)h(function)g Fl(G)33 b Fs(one)28 b(gets)f(the)h(result.)1157 b Fd(\003)555 3025 y Fs(F)-7 b(rom)20 b(the)g(pro)r(of)g(\(esp)r(ecially)h(\(96\))o (,)h(\(102\))d(and)h(\(103\))o(\))h(also)e(the)h(follo)n(wing)f(useful) i(Lemma)456 3125 y(follo)n(ws)456 3313 y Fx(Lemma)29 b(5.)41 b Fh(The)31 b(fol)t(lowing)h(estimates)e(hold)826 3516 y Fl(j)p Fp(X)918 3528 y Fg(F)976 3516 y Fl(j)999 3541 y Fn(R)1076 3516 y Fl(\024)23 b Fs(5)14 b Fl(j)o Fp(X)1311 3528 y Fg(H)1372 3516 y Fl(j)1395 3541 y Fn(R)1493 3516 y Fp(;)99 b Fl(j)p Fp(X)1707 3528 y Fg(Z)1763 3516 y Fl(j)1787 3541 y Fn(R)1864 3516 y Fl(\024)23 b Fs(6)14 b Fl(j)o Fp(X)2099 3528 y Fg(H)2160 3516 y Fl(j)2183 3541 y Fn(R)2281 3516 y Fp(;)99 b Fl(jZ)7 b(j)2516 3541 y Fn(R)2593 3516 y Fl(\024)23 b Fs(6)14 b Fl(j)o(H)q(j)2853 3541 y Fn(R)3255 3516 y Fs(\(106\))456 3705 y(3.6.)40 b Fx(Quan)m(titativ)m(e)d(estimates.)i Fs(First)31 b(w)n(e)g(\014x)g(a) g(p)r(ositiv)n(e)f Fp(R)i Fs(in)g(suc)n(h)f(a)f(w)n(a)n(y)g(that)i Fp(X)3364 3717 y Fg(P)3411 3725 y Fk(\017)456 3804 y Fs(is)f(analytic)h(on)f Fp(B)1046 3816 y Fn(R)1101 3804 y Fs(,)i(and)f(c)n(ho)r(ose)e(constan)n(ts)h Fp(P)12 b Fs(,)33 b Fp(P)2148 3774 y Fg(\003)2218 3804 y Fs(\(that)g(dep)r(end) g(on)e Fo(~)h Fs(and)g(on)f(all)h(the)456 3904 y(other)27 b(parameters\))f(suc)n(h)h(that)1367 4107 y Fl(j)p Fp(X)1459 4119 y Fg(P)1506 4127 y Fk(\017)1540 4107 y Fl(j)1563 4131 y Fn(R)1640 4107 y Fl(\024)c Fp(P)39 b(;)97 b Fl(jP)2021 4119 y Fn(\017)2053 4107 y Fl(j)2076 4131 y Fn(R)2153 4107 y Fl(\024)23 b Fp(P)2306 4072 y Fg(\003)2344 4107 y Fp(:)888 b Fs(\(107\))456 4309 y(All)34 b(along)f(this)h(section)f(w) n(e)h(\014x)g(a)f(small)h(v)-5 b(alue)33 b(of)h Fo(~)p Fs(,)i(and)d(will)i(mak)n(e)e(explicit)h(estimates)456 4409 y(so)h(that)h(at)f(the)h(end)g(it)g(will)g(b)r(e)g(p)r(ossible)g (to)f(insert)h(the)g(dep)r(endence)g(on)f Fo(~)h Fs(of)g(the)g(\014nal) 456 4509 y(estimate.)456 4697 y Fx(Lemma)20 b(6.)33 b Fh(\(Iter)l(ative)22 b(L)l(emma\))h(Consider)g(a)g(Gauge)f(invariant)h (Hamiltonian)g(of)g(the)g(form)1324 4907 y Fl(E)1375 4873 y Fm(\()p Fn(r)r Fm(\))1487 4907 y Fs(=)f Fl(H)1644 4919 y Fm(0)1700 4907 y Fs(+)c Fp(\017)p Fl(Z)1884 4873 y Fm(\()p Fn(r)r Fm(\))1991 4907 y Fs(+)g Fp(\017)2108 4873 y Fn(r)r Fm(+1)2228 4907 y Fl(R)2298 4873 y Fm(\()p Fn(r)r Fm(\))2387 4907 y Fp(;)845 b Fs(\(108\))456 5113 y Fh(with)34 b Fl(Z)707 5083 y Fm(\()p Fn(r)r Fm(\))830 5113 y Fh(non)f(c)l(oupling)i(and)g(wher)l(e)g Fl(R)1803 5083 y Fm(\(0\))1923 5113 y Fs(=)30 b Fl(P)2076 5125 y Fn(\017)2142 5113 y Fh(and)35 b Fl(Z)2375 5083 y Fm(\(0\))2494 5113 y Fs(=)c(0)p Fh(.)85 b(Fix)34 b Fp(\016)g(<)c(R)q(=)p Fs(\()p Fp(r)24 b Fs(+)d(1\))p Fh(,)456 5216 y(assume)g(that)h(the)g (Hamiltonian)h(ve)l(ctor)f(\014elds)h(of)f Fl(Z)2100 5185 y Fm(\()p Fn(r)r Fm(\))2211 5216 y Fh(and)g(of)h Fl(R)2524 5185 y Fm(\()p Fn(r)r Fm(\))2634 5216 y Fh(ar)l(e)g(analytic) g(on)f Fp(B)3248 5228 y Fn(R)p Fg(\000)p Fn(r)r(\016)3419 5216 y Fh(,)p eop %%Page: 23 23 23 22 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(23)456 450 y Fh(and)30 b(that)1186 696 y Fl(j)p Fp(X)1278 715 y Fg(Z)1331 698 y Fj(\()p Fk(r)q Fj(\))1413 696 y Fl(j)1436 721 y Fn(R)p Fg(\000)p Fn(r)r(\016)1631 696 y Fl(\024)1718 526 y Fq(8)1718 600 y(<)1718 750 y(:)2063 586 y Fs(0)354 b(if)90 b Fp(r)26 b Fs(=)d(0)1792 755 y(6)p Fp(P)1912 692 y Fq(P)2000 713 y Fn(r)r Fg(\000)p Fm(1)2000 779 y Fn(i)p Fm(=0)2136 637 y Fq(\022)2225 698 y Fp(\017)p 2207 735 71 4 v 2207 811 a(\017)2241 823 y Fm(0)2288 637 y Fq(\023)2349 655 y Fn(i)2459 755 y Fs(if)90 b Fp(r)26 b Fl(\025)d Fs(1)2843 696 y Fp(;)389 b Fs(\(109\))1186 950 y Fq(\014)1186 1000 y(\014)1186 1050 y(\014)1214 1046 y Fl(Z)1281 1012 y Fm(\()p Fn(r)r Fm(\))1370 950 y Fq(\014)1370 1000 y(\014)1370 1050 y(\014)1397 1104 y Fn(R)p Fg(\000)p Fn(r)r(\016)1592 1046 y Fl(\024)1679 876 y Fq(8)1679 950 y(<)1679 1100 y(:)2043 936 y Fs(0)373 b(if)90 b Fp(r)26 b Fs(=)d(0)1753 1105 y(6)p Fp(P)1860 1074 y Fg(\003)1911 1042 y Fq(P)1999 1063 y Fn(r)r Fg(\000)p Fm(1)1999 1129 y Fn(i)p Fm(=0)2135 988 y Fq(\022)2224 1048 y Fp(\017)p 2206 1085 V 2206 1162 a(\017)2240 1174 y Fm(0)2286 988 y Fq(\023)2348 1005 y Fn(i)2458 1105 y Fs(if)90 b Fp(r)26 b Fl(\025)d Fs(1)3255 1046 y(\(110\))1186 1330 y Fl(j)p Fp(X)1278 1349 y Fg(R)1335 1333 y Fj(\()p Fk(r)q Fj(\))1418 1330 y Fl(j)1441 1355 y Fn(R)p Fg(\000)p Fn(r)r(\016)1635 1330 y Fl(\024)1736 1274 y Fp(P)p 1733 1311 V 1733 1387 a(\017)1767 1358 y Fn(r)1767 1409 y Fm(0)1843 1330 y Fp(;)1965 1235 y Fq(\014)1965 1284 y(\014)1965 1334 y(\014)1993 1330 y Fl(R)2063 1296 y Fm(\()p Fn(r)r Fm(\))2152 1235 y Fq(\014)2152 1284 y(\014)2152 1334 y(\014)2180 1388 y Fn(R)p Fg(\000)p Fn(r)r(\016)2374 1330 y Fl(\024)2471 1274 y Fp(P)2536 1244 y Fg(\003)p 2471 1311 103 4 v 2488 1387 a Fp(\017)2522 1358 y Fn(r)2522 1409 y Fm(0)2614 1330 y Fp(;)618 b Fs(\(111\))456 1533 y Fh(with)1089 1714 y Fp(\017)1123 1726 y Fm(0)1183 1714 y Fs(:=)1331 1657 y Fp(\013\016)p 1303 1694 148 4 v 1303 1770 a Fs(75)p Fp(P)1461 1714 y(;)103 b Fh(wher)l(e)90 b Fp(\013)23 b Fs(=)g Fp(\013)p Fs(\()p Fo(~)p Fs(\))h Fl(\025)f Fp(C)2384 1679 y Fg(\000)p Fm(1)2473 1714 y Fo(~)2518 1679 y Fm(3)p Fn(=)p Fm(2)2622 1714 y Fp(;)610 b Fs(\(112\))456 1957 y Fh(then,)33 b(if)752 1840 y Fq(\022)842 1901 y Fp(\017)p 823 1938 71 4 v 823 2014 a(\017)857 2026 y Fm(0)904 1840 y Fq(\023)993 1957 y Fp(<)28 b Fs(1)p Fp(=)p Fs(2)j Fh(ther)l(e)i(exists)f(a)g(Hamiltonian)i(function)e Fl(G)2618 1969 y Fn(r)r Fm(+1)2772 1957 y Fh(analytic)i(on)e Fp(B)3273 1969 y Fn(R)p Fg(\000)p Fn(\016)r(r)456 2127 y Fh(gener)l(ating)i(the)h(c)l(anonic)l(al)g(tr)l(ansformation)g Fp(\036)1994 2097 y Fn(\017)2022 2072 y Fk(r)q Fj(+1)1994 2148 y Fn(r)r Fm(+1)2165 2127 y Fh(such)g(that)f Fl(E)2584 2097 y Fm(\()p Fn(r)r Fm(\))2695 2127 y Fl(\016)21 b Fp(\036)2807 2097 y Fn(\017)2835 2072 y Fk(r)q Fj(+1)2807 2148 y Fn(r)r Fm(+1)2978 2127 y Fh(has)35 b(the)g(form)456 2227 y Fs(\(108\))28 b Fh(and)h(satis\014es)g(the)h(estimates)36 b Fs(\(109\))o Fh(-)p Fs(\(111\))28 b Fh(with)h Fp(r)20 b Fs(+)c(1)29 b Fh(in)g(plac)l(e)h(of)g Fp(r)r Fh(,)g(mor)l(e)l(over)g (the)456 2327 y(new)f(Hamiltonian)i(is)f(Gauge)g(invariant)h(and)f(the) g(c)l(anonic)l(al)h(tr)l(ansformation)g(ful\014l)t(ls)1284 2534 y Fs(sup)1123 2608 y Fg(d)p Fn(\030)r Fg(e\024)p Fn(R)p Fg(\000)p Fm(\()p Fn(r)r Fm(+1\))p Fn(\016)1585 2442 y Fq(l)1629 2534 y Fp(\030)22 b Fl(\000)c Fp(\036)1819 2500 y Fn(\017)1847 2475 y Fk(r)q Fj(+1)1956 2534 y Fs(\()p Fp(\030)t Fs(\))2061 2442 y Fq(m)2127 2534 y Fl(\024)23 b Fp(\017)2263 2417 y Fq(\022)2352 2478 y Fp(\017)p 2334 2515 V 2334 2591 a(\017)2368 2603 y Fm(0)2415 2417 y Fq(\023)2476 2434 y Fn(r)2536 2478 y Fp(P)p 2536 2515 65 4 v 2542 2591 a(\013)3255 2534 y Fs(\(113\))456 2776 y Fh(Pr)l(o)l(of.)43 b Fs(Decomp)r(ose)d Fl(R)1234 2746 y Fm(\()p Fn(r)r Fm(\))1364 2776 y Fs(in)n(to)g(its)h(coupling)f(part)g Fl(F)2272 2788 y Fn(r)2349 2776 y Fs(and)h(its)f(noncoupling)g(part)g Fl(Z)3384 2788 y Fn(r)3421 2776 y Fs(.)456 2875 y(De\014ne)24 b Fl(Z)776 2845 y Fm(\()p Fn(r)r Fm(+1\))972 2875 y Fs(:=)f Fl(Z)1150 2845 y Fm(\()p Fn(r)r Fm(\))1250 2875 y Fs(+)12 b Fp(\017)1361 2845 y Fn(r)1396 2875 y Fl(Z)1456 2887 y Fn(r)1517 2875 y Fs(and)24 b(use)g(Lemma)g(5)g(to)g(estimate)g Fl(j)p Fp(X)2693 2895 y Fg(Z)2746 2878 y Fj(\()p Fk(r)q Fj(+1\))2899 2875 y Fl(j)2922 2900 y Fn(R)p Fg(\000)p Fn(\016)r Fm(\()p Fn(r)r Fm(+1\))3229 2875 y Fs(.)60 b(Use)456 2981 y(Theorem)26 b(2)i(to)f(construct)g Fl(G)1391 2993 y Fn(r)r Fm(+1)1540 2981 y Fs(as)g(the)h(solution)f(of)1532 3139 y Fl(fH)1644 3151 y Fm(0)1681 3139 y Fs(;)14 b Fl(G)1767 3151 y Fn(r)r Fm(+1)1888 3139 y Fl(g)23 b Fs(=)g Fl(\000F)2166 3151 y Fn(r)456 3296 y Fs(then)1206 3405 y Fq(\014)1206 3455 y(\014)1233 3476 y Fp(X)1302 3488 y Fg(G)1342 3496 y Fk(r)q Fj(+1)1451 3405 y Fq(\014)1451 3455 y(\014)1478 3509 y Fn(R)p Fg(\000)p Fn(r)r(\016)1673 3476 y Fl(\024)1776 3420 y Fs(1)p 1770 3457 54 4 v 1770 3533 a Fp(\013)1848 3476 y Fl(j)p Fp(X)1940 3495 y Fg(R)1997 3478 y Fj(\()p Fk(r)q Fj(\))2079 3476 y Fl(j)2102 3501 y Fn(R)p Fg(\000)p Fn(r)r(\016)2296 3476 y Fl(\024)2424 3420 y Fp(P)p 2394 3457 125 4 v 2394 3533 a(\013\017)2481 3504 y Fn(r)2481 3555 y Fm(0)3255 3476 y Fs(\(114\))456 3704 y(The)i(Hamiltonian)g Fl(E)1150 3674 y Fm(\()p Fn(r)r Fm(\))1252 3704 y Fl(\016)14 b Fp(\036)1357 3674 y Fn(\017)1385 3649 y Fk(r)q Fj(+1)1518 3704 y Fs(w)n(as)24 b(computed)i(in)f(subsection)g(3.2)f(and)h(is)g (giv)n(en)g(b)n(y)f(equa-)456 3804 y(tions)j(\(68\){\(72\))o(,)g(whic)n (h)h(has)f(the)h(form)f(\(108\))g(pro)n(vided)g(one)g(de\014nes)1224 3978 y Fl(R)1294 3944 y Fm(\()p Fn(r)r Fm(+1\))1490 3978 y Fs(=)c Fl(R)1648 3944 y Fm(\()p Fn(r)r Fm(+1\))1648 3998 y Fn(a)1839 3978 y Fs(+)18 b Fl(R)1992 3935 y Fm(\()p Fn(r)r Fm(+1\))1992 4003 y Fn(b)2184 3978 y Fs(+)g Fl(R)2337 3944 y Fm(\()p Fn(r)r Fm(+1\))2337 3998 y Fn(c)456 4136 y Fs(where)561 4294 y Fl(R)631 4259 y Fm(\()p Fn(r)r Fm(+1\))631 4314 y Fn(a)828 4294 y Fs(=)k Fp(\017)949 4259 y Fg(\000)p Fm(\()p Fn(r)r Fm(+2\))1174 4294 y Fs(\(70\))o Fp(;)42 b Fl(R)1456 4250 y Fm(\()p Fn(r)r Fm(+1\))1456 4319 y Fn(b)1652 4294 y Fs(=)22 b Fp(\017)1773 4259 y Fg(\000)p Fm(\()p Fn(r)r Fm(+2\))1998 4294 y Fs(\(71\))55 b(and)g Fl(R)2460 4259 y Fm(\()p Fn(r)r Fm(+1\))2460 4314 y Fn(c)2656 4294 y Fs(=)23 b Fp(\017)2778 4259 y Fg(\000)p Fm(\()p Fn(r)r Fm(+2\))3002 4294 y Fs(\(72\))o Fp(:)456 4459 y Fs(Then,)k(from)h(Lemma)f(11,)g(with)h Fp(\026)23 b Fs(=)g Fp(\017)1706 4429 y Fm(\()p Fn(r)r Fm(+1\))1878 4459 y Fs(,)28 b(it)g(follo)n(ws)e(that)906 4556 y Fq(\014)906 4605 y(\014)906 4655 y(\014)933 4651 y Fp(X)1002 4684 y Fg(R)1059 4654 y Fj(\()p Fk(r)q Fj(+1\))1059 4704 y Fk(b)1213 4556 y Fq(\014)1213 4605 y(\014)1213 4655 y(\014)1240 4709 y Fm(\()p Fn(R)p Fg(\000)p Fn(r)r(\016)r Fm(\))p Fg(\000)p Fn(\016)1631 4651 y Fl(\024)1865 4595 y Fs(5)p 1788 4632 195 4 v 1788 4708 a Fp(\016)s(\017)1862 4684 y Fn(r)r Fm(+1)1993 4651 y Fp(\026)14 b Fl(j)p Fp(X)2149 4670 y Fg(Z)2202 4654 y Fj(\()p Fk(r)q Fj(\))2283 4651 y Fl(j)2307 4676 y Fn(R)p Fg(\000)p Fn(r)r(\016)2492 4581 y Fq(\014)2492 4630 y(\014)2519 4651 y Fp(X)2588 4663 y Fg(G)2628 4674 y Fj(\()p Fk(r)q Fj(+1\))2782 4581 y Fq(\014)2782 4630 y(\014)2809 4689 y Fn(R)p Fg(\000)p Fn(r)r(\016)1631 4906 y Fl(\024)1788 4850 y Fs(5)p 1788 4887 42 4 v 1789 4963 a Fp(\016)1853 4765 y Fq(")1902 4906 y Fs(6)p Fp(P)2023 4803 y Fn(r)r Fg(\000)p Fm(1)2022 4828 y Fq(X)2028 5004 y Fn(i)p Fm(=0)2156 4789 y Fq(\022)2246 4850 y Fp(\017)p 2227 4887 71 4 v 2227 4963 a(\017)2261 4975 y Fm(0)2308 4789 y Fq(\023)2369 4807 y Fn(i)2397 4765 y Fq(#)2459 4789 y(\022)2560 4850 y Fp(P)p 2530 4887 125 4 v 2530 4963 a(\013\017)2617 4935 y Fn(r)2617 4986 y Fm(0)2664 4789 y Fq(\023)1631 5170 y Fl(\024)1788 5114 y Fs(60)p Fp(P)1937 5083 y Fm(2)p 1788 5151 186 4 v 1799 5227 a Fp(\013\016)s(\017)1926 5198 y Fn(r)1926 5249 y Fm(0)p eop %%Page: 24 24 24 23 bop 456 251 a Fm(24)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)456 450 y Fs(since)27 b Fp(\017)c Fl(\024)813 417 y Fm(1)p 813 431 34 4 v 813 479 a(2)856 450 y Fp(\017)890 462 y Fm(0)927 450 y Fs(.)65 b(Similarly)908 574 y Fq(\014)908 624 y(\014)908 674 y(\014)936 670 y Fp(X)1005 702 y Fg(R)1062 672 y Fj(\()p Fk(r)q Fj(+1\))1062 712 y Fk(c)1215 574 y Fq(\014)1215 624 y(\014)1215 674 y(\014)1242 728 y Fm(\()p Fn(R)p Fg(\000)p Fn(r)r(\016)r Fm(\))p Fg(\000)p Fn(\016)1633 670 y Fl(\024)1806 613 y Fs(5)p 1790 650 74 4 v 1790 727 a Fp(\016)s(\017)1874 670 y(\026)14 b Fl(j)p Fp(X)2030 689 y Fg(R)2087 672 y Fj(\()p Fk(r)q Fj(\))2169 670 y Fl(j)2192 694 y Fn(R)p Fg(\000)p Fn(r)r(\016)2378 599 y Fq(\014)2378 649 y(\014)2405 670 y Fp(X)2474 682 y Fg(G)2514 693 y Fj(\()p Fk(r)q Fj(+1\))2667 599 y Fq(\014)2667 649 y(\014)2695 708 y Fn(R)p Fg(\000)p Fn(r)r(\016)1633 900 y Fl(\024)1790 844 y Fs(5)p Fp(\017)1866 814 y Fn(r)p 1790 881 112 4 v 1826 957 a Fp(\016)1926 783 y Fq(\022)2000 844 y Fp(P)p 1997 881 71 4 v 1997 957 a(\017)2031 929 y Fn(r)2031 979 y Fm(0)2078 783 y Fq(\023)g(\022)2254 844 y Fp(P)p 2224 881 125 4 v 2224 957 a(\013\017)2311 929 y Fn(r)2311 979 y Fm(0)2358 783 y Fq(\023)2443 900 y Fl(\024)2551 844 y Fs(5)p Fp(P)2658 814 y Fm(2)p 2540 881 165 4 v 2540 957 a Fp(\013\016)s(\017)2667 929 y Fn(r)2667 979 y Fm(0)2728 783 y Fq(\022)2818 844 y Fp(\017)p 2800 881 71 4 v 2800 957 a(\017)2834 969 y Fm(0)2880 783 y Fq(\023)2942 800 y Fn(r)1633 1136 y Fl(\024)1840 1080 y Fs(5)p Fp(P)1947 1050 y Fm(2)p 1790 1117 243 4 v 1790 1193 a Fs(2)1832 1169 y Fn(r)1869 1193 y Fp(\013\016)s(\017)1996 1165 y Fn(r)1996 1215 y Fm(0)456 1370 y Fs(F)-7 b(urthermore,)26 b(from)h(Lemma)h(12,)f(with)h Fl(H)c Fs(=)e Fl(R)2027 1340 y Fm(\()p Fn(r)r Fm(\))2144 1370 y Fs(and)27 b Fp(\026)c Fs(=)g Fp(\017)2500 1340 y Fm(\()p Fn(r)r Fm(+1\))2673 1370 y Fs(,)k(it)h(follo)n(ws)f(that)877 1482 y Fq(\014)877 1532 y(\014)877 1581 y(\014)904 1577 y Fp(X)973 1610 y Fg(R)1030 1580 y Fj(\()p Fk(r)q Fj(+1\))1030 1619 y Fk(a)1184 1482 y Fq(\014)1184 1532 y(\014)1184 1581 y(\014)1211 1635 y Fm(\()p Fn(R)p Fg(\000)p Fn(r)r(\016)r Fm(\))p Fg(\000)p Fn(\016)1602 1577 y Fl(\024)1815 1521 y Fs(25)p 1759 1558 195 4 v 1759 1634 a Fp(\016)s(\017)1833 1610 y Fn(r)r Fm(+2)1964 1577 y Fp(\026)2014 1543 y Fm(2)2065 1577 y Fl(j)p Fp(X)2157 1596 y Fg(R)2214 1580 y Fj(\()p Fk(r)q Fj(\))2296 1577 y Fl(j)2319 1602 y Fn(R)p Fg(\000)p Fn(r)r(\016)2504 1507 y Fq(\014)2504 1557 y(\014)2532 1577 y Fp(X)2601 1589 y Fg(G)2641 1600 y Fj(\()p Fk(r)q Fj(+1\))2794 1507 y Fq(\014)2794 1557 y(\014)2822 1615 y Fn(R)p Fg(\000)p Fn(r)r(\016)1602 1808 y Fl(\024)1759 1752 y Fs(25)p Fp(\017)1877 1722 y Fn(r)p 1759 1789 154 4 v 1816 1865 a Fp(\016)1937 1691 y Fq(\022)2011 1752 y Fp(P)p 2008 1789 71 4 v 2008 1865 a(\017)2042 1836 y Fn(r)2042 1887 y Fm(0)2089 1691 y Fq(\023)13 b(\022)2264 1752 y Fp(P)p 2235 1789 125 4 v 2235 1865 a(\013\017)2322 1836 y Fn(r)2322 1887 y Fm(0)2369 1691 y Fq(\023)2453 1808 y Fl(\024)2551 1752 y Fs(25)p Fp(P)2700 1722 y Fm(2)p 2551 1789 186 4 v 2561 1865 a Fp(\013\016)s(\017)2688 1836 y Fn(r)2688 1887 y Fm(0)2760 1691 y Fq(\022)2849 1752 y Fp(\017)p 2831 1789 71 4 v 2831 1865 a(\017)2865 1877 y Fm(0)2912 1691 y Fq(\023)2973 1708 y Fn(r)1602 2044 y Fl(\024)1788 1988 y Fs(25)p Fp(P)1937 1958 y Fm(2)p 1759 2025 243 4 v 1759 2101 a Fs(2)1801 2077 y Fn(r)1838 2101 y Fp(\013\016)s(\017)1965 2072 y Fn(r)1965 2123 y Fm(0)456 2261 y Fs(Hence)1049 2463 y Fl(j)q Fp(X)1142 2482 y Fg(R)1199 2465 y Fj(\()p Fk(r)q Fj(+1\))1352 2463 y Fl(j)1375 2487 y Fm(\()p Fn(R)p Fg(\000)p Fn(r)r(\016)r Fm(\))p Fg(\000)p Fn(\016)1765 2463 y Fl(\024)1937 2406 y Fp(P)p 1923 2443 94 4 v 1923 2520 a(\013\016)2040 2345 y Fq(\024)2095 2406 y Fs(25)p Fp(P)p 2094 2443 150 4 v 2094 2520 a Fs(2)2136 2496 y Fn(r)2172 2520 y Fp(\017)2206 2491 y Fn(r)2206 2542 y Fm(0)2271 2463 y Fs(+)2364 2406 y(60)p Fp(P)p 2364 2443 148 4 v 2403 2520 a(\017)2437 2491 y Fn(r)2437 2542 y Fm(0)2541 2463 y Fs(+)2655 2406 y(5)p Fp(P)p 2634 2443 150 4 v 2634 2520 a Fs(2)2676 2496 y Fn(r)2712 2520 y Fp(\017)2746 2491 y Fn(r)2746 2542 y Fm(0)2793 2345 y Fq(\025)1765 2695 y Fl(\024)1937 2639 y Fp(P)p 1923 2676 94 4 v 1923 2752 a(\013\016)2040 2578 y Fq(\024)2097 2639 y Fp(P)p 2094 2676 71 4 v 2094 2752 a(\017)2128 2724 y Fn(r)2128 2774 y Fm(0)2189 2578 y Fq(\022)2250 2695 y Fs(60)k(+)2444 2639 y(30)p 2444 2676 84 4 v 2446 2752 a(2)2488 2728 y Fn(r)2537 2578 y Fq(\023\025)3255 2695 y Fs(\(115\))456 2918 y(If)35 b Fp(r)k Fs(=)c(0)g(then)g(the)h(second)f(term)g(is)g(not)g(presen)n(t) g(and)g(the)g(square)f(brac)n(k)n(et)g(in)h(\(115\))g(is)456 3017 y(equal)25 b(to)h(30)p Fp(P)12 b Fs(;)25 b(if)i Fp(r)e Fl(\025)e Fs(1,)j(one)f(has)h(that)g(the)g(square)f(brac)n(k)n (et)f(in)i(\(115\))f(is)h(less)g(than)g(75)p Fp(P)7 b(=\017)3409 2987 y Fn(r)3409 3038 y Fm(0)456 3117 y Fs(and)30 b(th)n(us)h(the)g (thesis)g(on)f(the)i(v)n(ector)d(\014elds)i(follo)n(ws.)76 b(The)31 b(estimates)f(of)h(the)g(mo)r(duli)g(are)456 3217 y(obtained)c(in)h(a)f(similar)g(w)n(a)n(y)f(from)h(Lemma)h(5.)1418 b Fd(\003)555 3316 y Fs(It)31 b(is)g(clear)f(that)h(the)g(Hamiltonian)f (\(24\))h(of)g(the)g(NLS)g(ful\014lls)g(the)g(assumptions)f(of)h(the) 456 3416 y(Lemma)21 b(with)h Fp(r)k Fs(=)d(0)e(and)g(th)n(us)h(the)g (Lemma)g(allo)n(ws)e(us)h(to)h(put)g(our)f(Hamiltonian)h(in)f(normal) 456 3516 y(form)e(up)g(to)h(an)n(y)e(order)g Fp(r)r Fs(.)55 b(T)-7 b(o)19 b(obtain)g(the)g(exp)r(onen)n(tially)g(small)g(estimate)g (of)h(the)g(remainder)456 3615 y(tak)n(e)j Fp(\016)j Fs(=)d Fp(R)q(=)p Fs(2)p Fp(r)j Fs(in)e(order)f(to)h(\014x)g(the)h (domain)e(of)i(de\014nition)f(of)g(the)h(\014nal)f(Hamiltonian,)h(and) 456 3715 y(then)j(c)n(ho)r(ose)f(an)g(optimal)h(v)-5 b(alue)28 b(of)g Fp(r)r Fs(,)h(this)f(can)f(b)r(e)i(done)e(b)n(y)h (minimizing)g(the)h(estimate)f(of)456 3814 y(the)g(remainder)e(or)h (simply)g(c)n(ho)r(osing)1578 4032 y Fp(r)1615 4044 y Fg(\003)1676 4032 y Fs(:=)1787 3915 y Fq(\024)1913 3976 y Fp(R)q(\013)p 1841 4013 262 4 v 1841 4089 a Fs(150)p Fp(eP)12 b(\017)2112 3915 y Fq(\025)456 4254 y Fs(\(with)23 b([)p Fp(:)p Fs(])f(denoting)h(the)f(in)n(teger)g(part\).)57 b(One)22 b(th)n(us)h(obtains)f(the)h(follo)n(wing)e(Theorem)h(whic)n(h) 456 4357 y(is)27 b(the)h(main)g(tec)n(hnical)f(result)g(of)h(the)g(pap) r(er,)f(where)g Fl(Z)j Fs(=)22 b Fp(\017)p Fl(Z)2481 4327 y Fm(\()p Fn(r)2538 4335 y Fk(?)2572 4327 y Fm(\))2630 4357 y Fs(and)28 b Fl(R)23 b Fs(=)g Fp(\017)3007 4327 y Fn(r)3038 4335 y Fk(?)3072 4327 y Fm(+1)3160 4357 y Fl(R)3230 4327 y Fm(\()p Fn(r)3287 4335 y Fk(?)3322 4327 y Fm(\))3352 4357 y Fs(.)456 4528 y Fx(Theorem)30 b(6.)41 b Fh(Consider)31 b(the)f(Hamiltonian)h Fl(E)37 b Fh(\(cfr.)i Fs(\(24\))o Fh(\),)31 b(de\014ne)1629 4743 y Fp(\017)1663 4755 y Fg(\003)1724 4743 y Fs(:=)1900 4687 y Fp(\013R)p 1844 4724 229 4 v 1844 4800 a Fs(150)p Fp(eP)2082 4743 y(;)1150 b Fs(\(116\))456 4943 y Fh(assume)36 b Fl(j)p Fp(\017)p Fl(j)e Fp(<)g(\017)1000 4955 y Fg(\003)1037 4943 y Fp(=)p Fs(2)h Fh(then)h(ther)l(e)g(exists)g(an)g(analytic)h(c)l (anonic)l(al)g(tr)l(ansformation)g Fl(T)57 b Fh(such)456 5043 y(that)1478 5216 y Fl(E)26 b(\016)18 b(T)44 b Fs(=)23 b Fl(H)1855 5228 y Fm(0)1911 5216 y Fs(+)18 b Fl(Z)25 b Fs(+)18 b Fl(R)p Fp(;)1000 b Fs(\(117\))p eop %%Page: 25 25 25 24 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(25)456 450 y Fh(with)33 b Fl(Z)40 b Fh(non)33 b(c)l(oupling;)j(b)l (oth)d Fl(Z)40 b Fh(and)33 b Fl(R)g Fh(have)h(a)g(ve)l(ctor)f(\014eld)h (which)g(is)g(analytic)g(in)f Fp(B)3323 465 y Fn(R=)p Fm(2)456 550 y Fh(and)d(ful\014l)t(l)g(the)g(estimate)987 711 y Fl(j)p Fp(X)1079 723 y Fg(Z)1136 711 y Fl(j)1159 736 y Fn(R=)p Fm(2)1303 711 y Fl(\024)23 b Fs(12)p Fp(\017P)40 b(;)99 b Fl(j)p Fp(X)1816 723 y Fg(R)1877 711 y Fl(j)1900 736 y Fn(R=)p Fm(2)2045 711 y Fl(\024)22 b Fp(\017P)k Fs(exp)2385 619 y Fq(h)2425 711 y Fl(\000)2504 619 y Fq(\020)2562 655 y Fp(\017)2596 667 y Fg(\003)p 2562 692 72 4 v 2582 768 a Fp(\017)2644 619 y Fq(\021i)3255 711 y Fs(\(118\))1073 897 y Fl(jZ)7 b(j)1185 921 y Fn(R=)p Fm(2)1330 897 y Fl(\024)23 b Fs(12)p Fp(\017P)1601 862 y Fg(\003)1637 897 y Fp(;)99 b Fl(jRj)1876 921 y Fn(R=)p Fm(2)2020 897 y Fl(\024)23 b Fp(\017P)2207 862 y Fg(\003)2258 897 y Fs(exp)2399 804 y Fq(h)2438 897 y Fl(\000)2517 804 y Fq(\020)2576 840 y Fp(\017)2610 852 y Fg(\003)p 2576 877 V 2595 954 a Fp(\017)2658 804 y Fq(\021i)3255 897 y Fs(\(119\))456 1061 y Fh(Mor)l(e)l(over)k(the)g(tr)l(ansforme)l (d)g(Hamiltonian)g(is)g(Gauge)g(invariant)g(and)g(the)g(c)l(anonic)l (al)g(tr)l(ans-)456 1160 y(formation)k(ful\014l)t(ls)1476 1339 y Fs(sup)1407 1413 y Fg(d)p Fn(\030)r Fg(e\024)p Fn(R=)p Fm(2)1670 1339 y Fl(d)p Fp(\030)22 b Fl(\000)c(T)j Fs(\()p Fp(\030)t Fs(\))q Fl(e)i(\024)g Fs(2)p Fp(\017)2253 1283 y(P)p 2253 1320 65 4 v 2259 1396 a(\013)3255 1339 y Fs(\(120\))555 1555 y(It)34 b(is)g(also)e(imp)r(ortan)n(t)h(to)h (reform)n(ulate)e(the)i(Theorem)f(computing)g Fp(P)45 b Fs(in)34 b(terms)g(of)f(the)456 1654 y(original)26 b(quan)n(tities)h Fp(!)s(;)14 b(\017;)g Fo(~)27 b Fs(i.e.)37 b(using)28 b(\(77\),)f(th)n(us)h(one)f(easily)g(gets)g(the)h(follo)n (wing)456 1808 y Fx(Corollary)33 b(7.)40 b Fh(Fix)30 b(a)g(p)l(ositive)h Fp(R)q Fh(,)f(c)l(onsider)h(the)f(Hamiltonian)h Fl(E)36 b Fh(\(cfr.)k Fs(\(24\))o Fh(\),)30 b(de\014ne)g(the)456 1907 y(smal)t(l)g(p)l(ar)l(ameter)1631 2091 y Fp(\026)23 b Fs(:=)g Fp(!)e Fs(+)1985 2035 y Fl(j)p Fp(\017)p Fl(j)p 1981 2072 90 4 v 1981 2148 a Fo(~)2026 2124 y Fn(\033)2080 2091 y Fp(;)1152 b Fs(\(121\))456 2272 y Fh(then)24 b(ther)l(e)h (exists)g Fp(\026)1110 2284 y Fg(\003)1171 2272 y Fp(>)d Fs(0)j Fh(indep)l(endent)g(of)h Fo(~)p Fp(;)14 b(!)s(;)g(\017)p Fh(,)25 b(such)g(that,)h(if)g Fp(\026)d(<)f(\026)2792 2284 y Fg(\003)2831 2272 y Fo(~)2876 2241 y Fm(3)p Fn(=)p Fm(2)2980 2272 y Fp(=)p Fs(2)i Fh(then)g(ther)l(e)456 2371 y(exists)j(an)h(analytic)h(c)l(anonic)l(al)g(tr)l(ansformation)f Fl(T)49 b Fh(which)30 b(tr)l(ansform)e Fl(E)35 b Fh(into)e Fs(\(117\))o Fh(,)c(wher)l(e)981 2571 y Fl(j)p Fp(X)1073 2583 y Fg(Z)1129 2571 y Fl(j)1152 2596 y Fn(R=)p Fm(2)1297 2571 y Fl(\024)23 b Fp(C)6 b(\026)30 b(;)99 b Fl(j)p Fp(X)1744 2583 y Fg(R)1805 2571 y Fl(j)1828 2596 y Fn(R=)p Fm(2)1972 2571 y Fl(\024)23 b Fp(C)6 b(\026)14 b Fs(exp)2330 2454 y Fq(\024)2373 2571 y Fl(\000)2448 2515 y Fp(\026)2498 2527 y Fg(\003)2536 2515 y Fo(~)2581 2485 y Fm(3)p Fn(=)p Fm(2)p 2448 2552 238 4 v 2542 2628 a Fp(\026)2695 2454 y Fq(\025)3255 2571 y Fs(\(122\))1143 2813 y Fl(jZ)7 b(j)1256 2838 y Fn(R=)p Fm(2)1400 2813 y Fl(\024)23 b Fp(C)6 b(\026;)99 b Fl(jRj)1841 2838 y Fn(R=)p Fm(2)1986 2813 y Fl(\024)23 b Fp(C)6 b(\026)14 b Fs(exp)2344 2696 y Fq(\024)2387 2813 y Fl(\000)2462 2757 y Fp(\026)2512 2769 y Fg(\003)2550 2757 y Fo(~)2595 2727 y Fm(3)p Fn(=)p Fm(2)p 2462 2794 V 2556 2870 a Fp(\026)2709 2696 y Fq(\025)3255 2813 y Fs(\(123\))456 3000 y Fh(Mor)l(e)l(over)27 b(the)g(tr)l (ansforme)l(d)g(Hamiltonian)g(is)g(Gauge)g(invariant)g(and)g(the)g(c)l (anonic)l(al)g(tr)l(ans-)456 3099 y(formation)k(ful\014l)t(ls)1439 3257 y Fs(sup)1370 3330 y Fg(d)p Fn(\030)r Fg(e\024)p Fn(R=)p Fm(2)1632 3257 y Fl(d)p Fp(\030)23 b Fl(\000)18 b(T)j Fs(\()p Fp(\030)t Fs(\))q Fl(e)i(\024)g Fp(C)2254 3201 y(\026)p 2205 3238 150 4 v 2205 3315 a Fo(~)2250 3291 y Fm(3)p Fn(=)p Fm(2)3255 3257 y Fs(\(124\))555 3473 y(Theorem)k(3)g(is)h(a)f(direct)g(corollary)e(of)j(Corollary)d(7.) 555 3572 y(The)31 b(manifold)h Fp(z)g Fs(=)d(0)h(is)i(appro)n(ximately) d(in)n(v)-5 b(arian)n(t)30 b(for)h(the)g(dynamics)g(of)g(the)h(system) 456 3672 y(\(117\))25 b(and)g(on)h(suc)n(h)g(a)f(manifold)h(the)g (dynamics)g(is)g(that)g(of)g(a)f(Hamiltonian)h(system)f(with)i(a)456 3771 y(Hamiltonian)g(function)h(whic)n(h)g(is)f(an)g(exp)r(onen)n (tially)g(small)h(p)r(erturbation)f(of)1448 3912 y Fl(K)q Fs(\()p Fp(u;)20 b Fs(\026)-48 b Fp(u)p Fs(\))23 b(:=)g Fl(Z)7 b Fs(\()p Fp(u;)19 b Fs(\026)-47 b Fp(u)o(;)14 b Fs(0)p Fp(;)g Fs(0\))p Fp(:)968 b Fs(\(125\))456 4053 y(Suc)n(h)27 b(a)g(system)h(has)f(the)h(additional)f(in)n(tegral)f(of)i (motion)1513 4206 y Fl(I)6 b Fs(\()p Fp(u;)19 b Fs(\026)-47 b Fp(u)o Fs(\))23 b(=)1871 4127 y Fq(X)1916 4304 y Fn(j)2005 4206 y Fl(j)p Fp(u)2076 4218 y Fn(j)2110 4206 y Fl(j)2133 4172 y Fm(2)2198 4206 y Fp(:)1034 b Fs(\(126\))555 4431 y(In)28 b(the)g(true)f(nonlinear)g(system)g(one)h(has)456 4584 y Fx(Theorem)g(7.)39 b Fh(Assume)28 b Fp(\026)23 b(<)f(\026)1492 4596 y Fg(\003)1531 4584 y Fo(~)1576 4554 y Fm(3)p Fn(=)p Fm(2)1708 4584 y Fh(and)29 b(c)l(onsider)g(the)f (Cauchy)i(pr)l(oblem)f(for)g(the)g(system)456 4684 y Fs(\(117\))f Fh(which)k(is)e(e)l(quivalent)g(to)g(NLS.)59 b(De\014ne)1228 4884 y Fp(\016)1268 4850 y Fm(2)1328 4884 y Fs(=)23 b(max)1584 4767 y Fq(\032)1647 4884 y Fs(exp)1787 4767 y Fq(\024)1831 4884 y Fl(\000)1906 4828 y Fp(\026)1956 4840 y Fg(\003)1994 4828 y Fo(~)2039 4798 y Fm(3)p Fn(=)p Fm(2)p 1906 4865 238 4 v 1999 4941 a Fp(\026)2153 4767 y Fq(\025)2211 4884 y Fp(;)14 b Fl(k)o Fp(z)2328 4896 y Fm(0)2365 4884 y Fl(k)2406 4842 y Fm(2)2406 4909 y Fn(s)2444 4767 y Fq(\033)456 5075 y Fh(\()p Fp(z)529 5087 y Fm(0)595 5075 y Fh(b)l(eing)30 b(the)g(initial)h(datum)f(for)g Fp(z)t Fh(\))f(then,)h(one)g(has)1631 5216 y Fl(k)p Fp(z)t Fs(\()p Fp(t)p Fs(\))p Fl(k)1851 5240 y Fn(s)1910 5216 y Fl(\024)22 b Fp(C)6 b(\016)1156 b Fs(\(127\))p eop %%Page: 26 26 26 25 bop 456 251 a Fm(26)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)456 450 y Fh(and)1049 611 y Fl(jI)6 b Fs(\()p Fp(t)p Fs(\))19 b Fl(\000)f(I)6 b Fs(\(0\))p Fl(j)24 b(\024)f Fp(C)6 b(\016)1716 577 y Fm(2)1783 611 y Fp(;)99 b Fl(jK)q Fs(\()p Fp(t)p Fs(\))19 b Fl(\000)f(K)q Fs(\(0\))p Fl(j)24 b(\024)e Fp(C)6 b(\026\016)2647 577 y Fm(2)3255 611 y Fs(\(128\))456 772 y Fh(for)30 b(al)t(l)h(times)f Fp(t)f Fh(ful\014l)t(ling)1674 968 y Fl(j)p Fp(t)p Fl(j)24 b(\024)1928 911 y Fs(1)p 1871 949 156 4 v 1871 1025 a Fp(C)6 b(\026\016)2036 968 y(:)1196 b Fs(\(129\))456 1172 y Fh(Pr)l(o)l(of.)43 b Fs(First)25 b(remark)e(that)j(due)f(to)g (the)g(equiv)-5 b(alence)25 b(of)g(the)g Fp(E)31 b Fs(norm)24 b(and)h(the)g Fp(s)g Fs(norm)g(one)456 1271 y(has)1414 1357 y Fq(p)p 1497 1357 263 4 v 75 x Fl(N)1565 1444 y Fn(E)1621 1432 y Fs(\()p Fp(\020)1689 1444 y Fm(0)1727 1432 y Fs(\))f(=)e Fl(k)p Fp(z)1951 1444 y Fm(0)1988 1432 y Fl(k)2029 1457 y Fn(E)2108 1432 y Fl(\024)h Fp(C)6 b(\016)o(:)935 b Fs(\(130\))456 1593 y(Th)n(us,)27 b(b)n(y)h(\(63\))g (and)f(\(123\))g(one)g(has)763 1757 y Fp(d)p Fl(N)874 1769 y Fn(E)p 763 1795 168 4 v 810 1871 a Fp(dt)964 1814 y Fl(\024)22 b Fp(C)6 b(\026)p Fs(\()p Fl(N)1266 1826 y Fn(E)1323 1814 y Fs(\))1355 1779 y Fm(3)p Fn(=)p Fm(2)1478 1814 y Fs(+)18 b Fp(C)6 b(\026)14 b Fs(exp)1831 1697 y Fq(\024)1875 1814 y Fl(\000)1950 1757 y Fp(\026)2000 1769 y Fg(\003)2038 1757 y Fo(~)2083 1727 y Fm(3)p Fn(=)p Fm(2)p 1949 1795 238 4 v 2043 1871 a Fp(\026)2197 1697 y Fq(\025)2254 1737 y(p)p 2337 1737 125 4 v 77 x Fl(N)2405 1826 y Fn(E)2485 1814 y Fl(\024)22 b Fp(C)6 b(\026)p Fs(\()p Fl(N)2787 1826 y Fn(E)2844 1814 y Fs(\))2876 1779 y Fm(3)p Fn(=)p Fm(2)3255 1814 y Fs(\(131\))456 2020 y(whic)n(h)27 b(can)g(b)r(e)h(solv)n(ed)f(giving)g(the)h(estimate) 1643 2181 y Fl(N)1711 2193 y Fn(E)1767 2181 y Fs(\()p Fp(t)p Fs(\))c Fl(\024)e Fs(2)p Fp(\016)2054 2147 y Fm(2)3255 2181 y Fs(\(132\))456 2342 y(for)h(the)h(times)g(\(129\))o(.)60 b(The)24 b(estimate)g(of)f Fl(k)p Fp(z)t Fs(\()p Fp(t)p Fs(\))p Fl(k)2014 2367 y Fn(s)2074 2342 y Fs(follo)n(ws)f(from)i(equiv) -5 b(alence)23 b(of)h(the)g(norms.)456 2442 y(T)-7 b(o)33 b(obtain)g(the)h(estimate)g(of)f(the)h(di\013usion)g(of)f Fl(I)40 b Fs(simply)34 b(remark)e(that)i Fl(N)45 b Fs(=)33 b Fl(I)c Fs(+)22 b Fl(N)3299 2454 y Fn(E)3389 2442 y Fs(is)456 2541 y(an)31 b(exact)h(in)n(tegral)e(of)i(motion)g(and)g(use) g(the)g(estimate)g(\(132\))o(.)82 b(Finally)-7 b(,)33 b(to)f(estimate)g(the)456 2641 y(di\013usion)c(of)g Fl(K)i Fs(remark)c(that)j(the)f(total)g(Hamiltonian)g(is)g(an)g(in)n(tegral)f (of)h(motion,)g(but)h(one)456 2740 y(has)923 2940 y Fl(jE)d(\016)18 b(T)40 b(\000)18 b(H)1314 2952 y Fm(0)1369 2940 y Fs(+)g Fl(K)q(j)24 b(\024)f Fp(C)6 b(\026)1780 2823 y Fq(\022)1841 2940 y Fl(k)o Fp(z)t Fs(\()p Fp(t)p Fs(\))p Fl(k)2061 2898 y Fm(2)2061 2965 y Fn(s)2116 2940 y Fs(+)18 b(exp)2340 2823 y Fq(\024)2384 2940 y Fl(\000)2459 2884 y Fp(\026)2509 2896 y Fg(\003)2547 2884 y Fo(~)2592 2854 y Fm(3)p Fn(=)p Fm(2)p 2459 2921 238 4 v 2552 2997 a Fp(\026)2706 2823 y Fq(\025\023)3255 2940 y Fs(\(133\))456 3146 y(whic)n(h)27 b(using)g(the)h(previous)f(estimates)g(implies)h(the)g(thesis.)979 b Fd(\003)456 3246 y Fh(Pr)l(o)l(of)33 b(of)h(The)l(or)l(em)f(2.)94 b Fs(De\014ne)31 b(\014rst)g Fl(M)c Fs(:=)h Fl(T)21 b Fs(\(\010)2126 3258 y Fm(0)2164 3246 y Fs(\).)77 b(In)31 b(order)e(to)i(distinguish)g(b)r(et)n(w)n(een)456 3345 y(the)g(original)e(co)r(ordinates)h(of)h(the)g(NLS)g(equation)f(and)h (the)g(new)g(co)r(ordinates)f(in)n(tro)r(duced)456 3445 y(b)n(y)g(the)h(transformation)e Fl(T)52 b Fs(w)n(e)31 b(will)g(denote)f(the)i(new)e(v)-5 b(ariables)30 b(b)n(y)g(adding)h(a)f (prime,)h(i.e.)456 3545 y(w)n(e)21 b(will)i(write)f Fp(\020)30 b Fs(=)22 b Fl(T)g Fs(\()p Fp(\020)1225 3515 y Fg(0)1249 3545 y Fs(\).)57 b(Remark)22 b(that)g(since)g Fl(T)44 b Fs(is)22 b(Lipsc)n(hitz)g(together)g(with)g(its)h(in)n(v)n(erse,)456 3651 y(then)28 b(for)f(an)n(y)g(couple)g(of)g(p)r(oin)n(ts)h Fp(\020)6 b(;)1618 3629 y Fs(~)1610 3651 y Fp(\020)34 b Fs(one)28 b(has)1221 3832 y Fp(C)1286 3798 y Fg(\000)p Fm(1)1389 3737 y Fq(\015)1389 3787 y(\015)1389 3837 y(\015)1436 3832 y Fp(\020)c Fl(\000)1587 3811 y Fs(~)1579 3832 y Fp(\020)1622 3737 y Fq(\015)1622 3787 y(\015)1622 3837 y(\015)1668 3891 y Fn(s)1727 3832 y Fl(\024)1814 3737 y Fq(\015)1814 3787 y(\015)1814 3837 y(\015)1860 3832 y Fp(\020)1902 3798 y Fg(0)1945 3832 y Fl(\000)2035 3811 y Fs(~)2028 3832 y Fp(\020)2070 3798 y Fg(0)2093 3737 y Fq(\015)2093 3787 y(\015)2093 3837 y(\015)2139 3891 y Fn(s)2198 3832 y Fl(\024)f Fp(C)2365 3737 y Fq(\015)2365 3787 y(\015)2365 3837 y(\015)2411 3832 y Fp(\020)i Fl(\000)2562 3811 y Fs(~)2555 3832 y Fp(\020)2597 3737 y Fq(\015)2597 3787 y(\015)2597 3837 y(\015)2643 3891 y Fn(s)3255 3832 y Fs(\(134\))456 4018 y(and)i(therefore)1007 4179 y Fp(d)p Fs(\()p Fp( )1139 4145 y Fm(0)1177 4179 y Fp(;)14 b Fl(M)p Fs(\))23 b Fl(\024)g Fp(\016)j Fs(=)-14 b Fl(\))23 b Fp(d)p Fs(\()p Fp(\020)1794 4145 y Fm(0)5 b Fg(0)1855 4179 y Fp(;)14 b Fs(\010)1952 4191 y Fm(0)1989 4179 y Fs(\))24 b Fl(\024)e Fp(C)6 b(\016)50 b Fl(\()-14 b(\))46 b(k)p Fp(z)2567 4145 y Fg(0)2563 4199 y Fm(0)2599 4179 y Fl(k)2641 4204 y Fn(s)2700 4179 y Fl(\024)22 b Fp(C)6 b(\016)366 b Fs(\(135\))456 4341 y(where)28 b(b)n(y)g(a)g(sligh)n(t)g (abuse)g(of)g(notation)g(w)n(e)g(wrote)g Fp( )2180 4311 y Fm(0)2242 4341 y Fs(=)c Fl(T)d Fs(\()p Fp(\020)2471 4311 y Fg(0)2495 4341 y Fs(\).)68 b(Th)n(us)28 b(eq.)40 b(\(127\))27 b(implies)456 4441 y(\(21\))j(for)h(the)g(considered)f (times.)79 b(T)-7 b(o)31 b(get)f(\(22\),)i(just)g(remark)d(that)j (\(again)e(with)h(a)g(sligh)n(t)456 4540 y(abuse)c(of)g(notation\))1398 4726 y(\005)1460 4738 y Fn(c)1494 4726 y Fp( )f Fs(=)d Fp(z)j Fs(=)d Fp(z)1858 4692 y Fg(0)1899 4726 y Fs(+)18 b Fh(O)2067 4634 y Fq(\020)2177 4670 y Fp(\026)p 2127 4707 150 4 v 2127 4785 a Fo(~)2172 4761 y Fm(3)p Fn(=)p Fm(2)2286 4634 y Fq(\021)456 4917 y Fs(and)27 b(th)n(us)h(\(127\))e (implies)i(\(22\).)1911 b Fd(\003)555 5016 y Fs(The)35 b(pro)r(of)g(of)g(Corollary)e(3)i(is)g(also)f(a)h(direct)g(consequence) f(of)h(eqs.)59 b(\(128\))35 b(when)g(one)456 5116 y(considers)29 b(the)h(deformation)f(induced)i(b)n(y)f(the)h(c)n(hange)e(of)h(v)-5 b(ariables.)43 b(A)31 b(detailed)f(pro)r(of)g(is)456 5216 y(omitted.)p eop %%Page: 27 27 27 26 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(27)1243 450 y Fr(Appendix)32 b Fs(A.)41 b Fr(Pr)n(oof)32 b(of)f(Theorem)h(1)555 600 y Fs(F)-7 b(or)27 b(p)r(ossible)g(future)h (reference,)f(in)h(this)g(section)f(w)n(e)g(will)h(w)n(ork)e(in)i Fo(R)2804 569 y Fn(d)2849 600 y Fs(.)555 699 y(W)-7 b(e)30 b(start)f(b)n(y)g(recalling)f(some)h(notations)f(and)h(de\014nitions)h (from)f(Rob)r(ert)g([18)o(])h(that)f(will)456 799 y(b)r(e)f(used)f(in)h (the)g(follo)n(wing)456 957 y Fx(De\014nition)j(4.)41 b Fh(A)30 b(function)g Fp(m)24 b Fs(:)g Fo(R)1635 927 y Fm(2)p Fn(d)1737 957 y Fl(!)g Fs([0)p Fp(;)14 b Fs(+)p Fl(1)p Fs(\))29 b Fh(is)i(c)l(al)t(le)l(d)g(a)37 b Fs(temp)r(ered)29 b(w)n(eigh)n(t)g Fh(if)i(ther)l(e)456 1057 y(exist)e Fp(C)708 1069 y Fm(0)746 1057 y Fp(;)14 b(N)850 1069 y Fm(0)909 1057 y Fp(>)23 b Fs(0)29 b Fh(such)h(that)900 1208 y Fp(m)p Fs(\()p Fp(X)7 b Fs(\))23 b Fl(\024)g Fp(C)1283 1220 y Fm(0)1320 1208 y Fp(m)p Fs(\()p Fp(X)1494 1220 y Fm(1)1531 1208 y Fs(\)\(1)c(+)f Fl(j)p Fp(X)25 b Fl(\000)18 b Fp(X)2008 1220 y Fm(1)2045 1208 y Fl(j)p Fs(\))2100 1173 y Fn(N)2153 1181 y Fj(0)2219 1208 y Fp(;)99 b Fl(8)p Fp(X)r(;)14 b(X)2565 1220 y Fm(1)2624 1208 y Fl(2)23 b Fo(R)2756 1173 y Fm(2)p Fn(d)3255 1208 y Fs(\(136\))456 1366 y Fx(De\014nition)i(5.)36 b Fh(A)24 b(function)h Fp(a)e Fl(2)h Fp(C)1626 1336 y Fg(1)1696 1366 y Fs(\()p Fo(R)1783 1336 y Fm(2)p Fn(d)1860 1366 y Fs(\))i Fh(is)f(c)l(al)t(le)l (d)h(a)g(semiclassic)l(al)h(symb)l(ol)f(with)f(weight)456 1466 y Fs(\()p Fp(m;)14 b(\032)p Fs(\))p Fh(,)30 b(with)g Fp(\032)23 b Fl(\025)g Fs(0)p Fh(,)30 b(if)g(for)h(any)f(multi)f(index) h Fp(\013)h Fh(ther)l(e)e(exists)h(a)g(c)l(onstant)f Fp(C)2948 1478 y Fn(\013)3025 1466 y Fh(such)h(that)1107 1664 y Fl(j)p Fp(@)1179 1630 y Fn(\013)1226 1664 y Fp(a)p Fs(\()p Fp(X)7 b Fs(\))p Fl(j)23 b(\024)g Fp(C)1603 1676 y Fn(\013)1779 1608 y Fp(m)p Fs(\()p Fp(X)7 b Fs(\))p 1660 1645 451 4 v 1660 1723 a(\(1)19 b(+)f Fl(j)p Fp(X)7 b Fl(j)p Fs(\))1990 1699 y Fn(\032)p Fg(j)p Fn(\013)p Fg(j)2150 1664 y Fp(;)99 b Fl(8)p Fp(X)29 b Fl(2)23 b Fo(R)2549 1630 y Fm(2)p Fn(d)3255 1664 y Fs(\(137\))456 1860 y Fh(In)29 b(this)h(c)l(ase)g(we)g(write)g Fp(a)23 b Fl(2)h Fs(\006)1441 1830 y Fn(m)1441 1881 y(\032)1504 1860 y Fh(.)555 2019 y Fs(In)c(the)g(follo)n(wing)e(w)n(e)h(will)h (also)e(need)i(an)f(extension)g(to)g Fo(~)g Fs(dep)r(enden)n(t)h (families)g(of)f(sym)n(b)r(ols.)456 2178 y Fx(De\014nition)33 b(6.)43 b Fh(A)31 b(smo)l(oth)i(map)g Fo(~)27 b Fl(7!)g Fp(a)p Fs(\()p Fo(~)p Fs(\))g Fl(2)h Fs(\006)2090 2147 y Fn(m)2090 2198 y(\032)2185 2178 y Fh(is)k(c)l(al)t(le)l(d)i(an)e Fo(~)p Fh(-admissible)i(symb)l(ol)f(of)456 2280 y(class)d Fs(\()p Fp(m;)14 b(\032)p Fs(\))p Fh(,)30 b(if)h(for)g(any)f(inte)l (ger)g Fp(N)38 b Fh(lar)l(ge)31 b(enough)f(one)g(has)1062 2434 y Fp(a)p Fs(\()p Fo(~)p Fs(\))23 b(=)g Fp(a)1370 2446 y Fm(0)1426 2434 y Fs(+)18 b Fo(~)p Fp(a)1598 2446 y Fm(1)1653 2434 y Fs(+)g Fo(~)1781 2400 y Fm(2)1818 2434 y Fp(a)1862 2446 y Fm(2)1918 2434 y Fs(+)g Fp(:::)g Fs(+)g Fo(~)2216 2400 y Fn(N)2279 2434 y Fp(a)2323 2446 y Fn(N)2404 2434 y Fs(+)h Fp(r)2525 2446 y Fn(N)6 b Fm(+1)3255 2434 y Fs(\(138\))456 2601 y Fh(with)31 b Fp(a)681 2613 y Fn(j)742 2601 y Fl(2)26 b Fs(\006)883 2571 y Fn(m;)p Fg(\000)p Fm(2)p Fn(j)883 2621 y(\032)1113 2601 y Fh(and)31 b Fp(r)1312 2613 y Fn(N)6 b Fm(+1)1460 2601 y Fs(\()p Fo(~)p Fs(\))32 b Fh(b)l(ounde)l(d)f(in)g Fs(\006)2081 2558 y Fn(m;)p Fg(\000)p Fm(2\()p Fn(N)6 b Fm(+1\))2081 2610 y Fn(\032)2444 2601 y Fh(.)43 b(Her)l(e)31 b Fs(\006)2772 2571 y Fn(m;K)2772 2621 y(\032)2946 2601 y Fh(is)g(the)h(class)g(of)456 2700 y(symb)l(ols)e(with)g(weight)1513 2784 y Fq(\000)1552 2851 y Fp(m)14 b Fs(\(1)k(+)g Fl(j)p Fp(X)7 b Fl(j)p Fs(\))1968 2817 y Fn(K)t(\032)2066 2851 y Fp(;)14 b(\032)2146 2784 y Fq(\001)2197 2851 y Fp(:)555 3009 y Fs(Giv)n(en)34 b(an)g Fo(~)p Fs(-admissible)f(sym)n(b)r(ol)g Fp(a)h Fs(of)g(class)f(\()p Fp(m;)14 b(\032)p Fs(\),)36 b(one)e(de\014nes)g (the)g(corresp)r(onding)456 3109 y(W)-7 b(eyl)27 b(op)r(erator)f (acting)h(on)h Fp(L)1422 3079 y Fm(2)1486 3109 y Fs(b)n(y)724 3305 y Fp(O)r(p)831 3271 y Fn(w)885 3305 y Fs(\()p Fp(a)p Fs(\))p Fp( )s Fs(\()p Fp(x)p Fs(\))d(:=)1406 3249 y(1)p 1306 3286 240 4 v 1306 3362 a(\(2)p Fp(\031)s Fo(~)p Fs(\))1507 3338 y Fn(d)1570 3192 y Fq(Z)1616 3380 y Ff(R)1663 3364 y Fj(2)p Fk(d)1739 3305 y Fs(e)1776 3271 y Fn(i)1810 3241 y Fj(\()p Fk(x)p Fe(\000)p Fk(y)q(;\030)q Fj(\))p 1810 3257 203 3 v 1897 3291 a(2)2026 3305 y Fp(a)2084 3188 y Fq(\022)2155 3249 y Fp(x)19 b Fs(+)f Fp(y)p 2155 3286 193 4 v 2230 3362 a Fs(2)2357 3305 y Fp(;)c(\030)t(;)g Fo(~)2516 3188 y Fq(\023)2591 3305 y Fp( )s Fs(\()p Fp(y)s Fs(\)d)2803 3268 y Fn(d)2841 3305 y Fp(y)s Fs(d)2931 3268 y Fn(d)2970 3305 y Fp(\030)249 b Fs(\(139\))456 3505 y(By)21 b(the)i(theory)e(of)h([18)o(])g(one)f(has)h(that,)h(for)e (an)n(y)g Fo(~)i Fp(>)g Fs(0,)g(suc)n(h)e(an)h(op)r(erator)e(is)i(w)n (ell)f(de\014ned)i(on)456 3605 y(the)k(Sc)n(h)n(w)n(artz)e(space.)62 b(Under)27 b(suitable)g(conditions)f(it)h(extends)g(to)f(a)h (selfadjoin)n(t)f(op)r(erator)456 3704 y(on)h Fp(L)628 3674 y Fm(2)665 3704 y Fs(.)456 3863 y Fx(De\014nition)h(7.)39 b Fh(A)28 b(str)l(ongly)g(admissible)i(op)l(er)l(ator)f(of)g(weight)g Fs(\()p Fp(m;)14 b(\032)p Fs(\))28 b Fh(is)h(a)f Fp(C)2953 3833 y Fg(1)3052 3863 y Fh(applic)l(ation)1306 4018 y Fp(A)c Fs(:)f(\(0)p Fp(;)14 b Fo(~)1594 4030 y Fg(\003)1632 4018 y Fs(\))23 b Fl(!)g(L)p Fs(\()p Fp(S)5 b Fs(\()p Fo(R)2024 3983 y Fn(d)2069 4018 y Fs(\))p Fp(;)14 b(L)2195 3983 y Fm(2)2232 4018 y Fs(\()p Fo(R)2318 3983 y Fn(d)2363 4018 y Fs(\)\))456 4168 y Fh(such)29 b(that)h(ther)l(e)g(exists)f(an)h (admissible)i(symb)l(ol)f Fp(a)22 b Fl(2)i Fs(\006)2241 4138 y Fn(m)2241 4189 y(\032)2334 4168 y Fh(such)29 b(that)h Fp(A)p Fs(\()p Fo(~)p Fs(\))24 b(=)f Fp(O)r(p)3082 4138 y Fn(w)3136 4168 y Fs(\()p Fp(a)p Fs(\()p Fo(~)p Fs(\)\))p Fh(.)555 4327 y Fs(One)37 b(of)g(the)h(most)f(imp)r(ortan)n(t)g(prop)r (erties)f(of)h(strongly)f(admissible)h(op)r(erators)e(is)i(the)456 4426 y(giv)n(en)26 b(b)n(y)i(the)g(follo)n(wing)e(Theorem.)456 4585 y Fx(Theorem)38 b(8.)45 b Fh(\(The)l(or)l(em)37 b(II-32)g(of)g Fs([18)o(])p Fh(\))f(L)l(et)g Fp(A)f Fs(=)g Fp(O)r(p)2346 4555 y Fn(w)2400 4585 y Fs(\()p Fp(a)p Fs(\))h Fh(and)h Fp(B)i Fs(=)34 b Fp(O)r(p)3020 4555 y Fn(w)3075 4585 y Fs(\()p Fp(b)p Fs(\))i Fh(b)l(e)g(two)456 4685 y(str)l(ongly)d(admissible)h(op)l(er)l(ators)g(of)g(weights)g Fs(\()p Fp(m;)14 b(\032)p Fs(\))33 b Fh(and)g Fs(\()p Fp(n;)14 b(\032)p Fs(\))33 b Fh(r)l(esp)l(e)l(ctively;)j(then)d Fp(AB)k Fh(is)456 4784 y(a)32 b(str)l(ongly)h(admissible)h(op)l(er)l (ator)f(with)g(symb)l(ol)g Fp(c)p Fs(\()p Fo(~)p Fs(\))f Fh(and)h(weight)g Fs(\()p Fp(mn;)14 b(\032)p Fs(\))p Fh(.)79 b(Mor)l(e)l(over,)34 b(if)456 4884 y Fp(a)23 b Fl(\030)610 4822 y Fq(P)698 4909 y Fn(j)s Fg(\025)p Fm(0)832 4884 y Fo(~)877 4854 y Fn(j)911 4884 y Fp(a)955 4896 y Fn(j)1020 4884 y Fh(and)30 b Fp(b)23 b Fl(\030)1328 4822 y Fq(P)1415 4909 y Fn(j)s Fg(\025)p Fm(0)1549 4884 y Fo(~)1594 4854 y Fn(j)1629 4884 y Fp(b)1665 4896 y Fn(j)1729 4884 y Fh(then)30 b Fp(c)23 b Fl(\030)2060 4822 y Fq(P)2148 4909 y Fn(j)s Fg(\025)p Fm(0)2282 4884 y Fo(~)2327 4854 y Fn(j)2362 4884 y Fp(c)2398 4896 y Fn(j)2462 4884 y Fh(with)817 5123 y Fp(c)853 5135 y Fn(j)911 5123 y Fs(=)1166 5044 y Fq(X)999 5226 y Fg(j)p Fn(\013)p Fg(j)p Fm(+)p Fg(j)p Fn(\014)s Fg(j)p Fm(+)p Fn(k)q Fm(+)p Fn(l)p Fm(=)p Fn(j)1532 5067 y Fs(1)p 1478 5104 151 4 v 1478 5180 a Fp(\013)p Fs(!)p Fp(\014)t Fs(!)1652 5006 y Fq(\022)1723 5067 y Fs(1)p 1723 5104 42 4 v 1723 5180 a(2)1775 5006 y Fq(\023)1836 5023 y Fg(j)p Fn(\013)p Fg(j)1937 5006 y Fq(\022)2008 5067 y Fs(1)p 2008 5104 V 2008 5180 a(2)2059 5006 y Fq(\023)2120 5023 y Fg(j)p Fn(\014)s Fg(j)2218 5123 y Fs(\()p Fp(@)2299 5089 y Fn(\013)2294 5143 y(\030)2347 5123 y Fp(D)2418 5089 y Fn(\014)2416 5143 y(x)2462 5123 y Fp(a)2506 5135 y Fn(k)2547 5123 y Fs(\)\()p Fp(@)2660 5083 y Fn(\014)2655 5148 y(\030)2705 5123 y Fp(D)2776 5089 y Fn(\013)2774 5143 y(x)2824 5123 y Fp(b)2860 5135 y Fn(l)2885 5123 y Fs(\))338 b(\(140\))p eop %%Page: 28 28 28 27 bop 456 251 a Fm(28)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)456 450 y Fh(wher)l(e)30 b Fp(D)759 462 y Fn(x)824 450 y Fs(=)22 b Fl(\000)p Fp(i@)1049 462 y Fn(x)1090 450 y Fh(.)69 b(F)-6 b(urthermor)l(e,)30 b(for)g(any)g Fp(N)i Fl(\025)23 b Fs(0)p Fh(,)30 b(one)g(has)1401 667 y Fp(c)23 b Fs(=)1578 563 y Fn(N)1548 588 y Fq(X)1550 765 y Fn(j)s Fm(=0)1682 667 y Fo(~)1727 633 y Fn(j)1761 667 y Fp(c)1797 679 y Fn(j)1851 667 y Fs(+)18 b Fo(~)1979 633 y Fn(N)6 b Fm(+1)2126 667 y Fp(\016)2163 679 y Fn(N)g Fm(+1)2310 667 y Fp(;)922 b Fs(\(141\))456 892 y Fh(and,)35 b(for)f(any)f Fp(\013;)14 b(\014)38 b Fh(ther)l(e)c(exists)f(a)g(p)l (ositive)i(\014nite)e Fp(M)41 b Fh(such)34 b(that)f(the)g(fol)t(lowing) j(estimate)456 991 y(holds)31 b(\(for)f(simplicity)i(we)e(r)l(estrict)f (to)h(the)g(c)l(ase)g Fp(\032)23 b Fs(=)g(0)p Fh(\))1711 1061 y Fq(\014)1711 1111 y(\014)1711 1161 y(\014)1738 1157 y Fp(@)1787 1122 y Fn(\013)1782 1177 y(x)1834 1157 y Fp(@)1883 1117 y Fn(\014)1878 1182 y(\030)1928 1157 y Fp(\016)1965 1169 y Fn(N)6 b Fm(+1)2112 1157 y Fs(\()p Fp(x;)14 b(\030)t Fs(\))2300 1061 y Fq(\014)2300 1111 y(\014)2300 1161 y(\014)2351 1157 y Fl(\024)23 b Fp(C)6 b(m)p Fs(\()p Fp(x;)14 b(\030)t Fs(\))p Fp(n)p Fs(\()p Fp(x;)g(\030)t Fs(\))252 b(\(142\))901 1414 y Fl(\002)980 1247 y Fq(2)980 1397 y(4)1065 1310 y Fn(N)1034 1335 y Fq(X)1037 1512 y Fn(j)s Fm(=0)1168 1414 y Fs(\()q Fp(p)1243 1426 y Fn(mM)1375 1414 y Fs(\()p Fp(a)1451 1426 y Fn(j)1486 1414 y Fs(\))p Fp(p)1560 1426 y Fn(nM)1675 1414 y Fs(\()p Fp(r)1744 1426 y Fn(N)6 b Fm(+1)1892 1414 y Fs(\()p Fp(b)p Fs(\)\))19 b(+)f Fp(p)2168 1426 y Fn(nM)2282 1414 y Fs(\()p Fp(b)2350 1426 y Fn(j)2385 1414 y Fs(\))p Fp(p)2459 1426 y Fn(mM)2592 1414 y Fs(\()p Fp(r)2661 1426 y Fn(N)6 b Fm(+1)2808 1414 y Fs(\()p Fp(a)p Fs(\)\)\))275 b(\(143\))739 1746 y(+)944 1667 y Fq(X)818 1846 y Fn(N)6 b Fg(\024)p Fn(j)s Fm(+)p Fn(k)q Fg(\024)p Fm(2)p Fn(N)1204 1746 y Fp(p)1246 1758 y Fn(mM)1378 1746 y Fs(\()p Fp(a)1454 1758 y Fn(j)1489 1746 y Fs(\))p Fp(p)1563 1758 y Fn(q)r(M)1670 1746 y Fs(\()p Fp(b)1738 1758 y Fn(k)1778 1746 y Fs(\))19 b(+)f Fp(p)1954 1758 y Fn(mM)2087 1746 y Fs(\()p Fp(r)2156 1758 y Fn(N)6 b Fm(+1)2303 1746 y Fs(\()p Fp(a)p Fs(\)\))p Fp(p)2485 1758 y Fn(nM)2601 1746 y Fs(\()p Fp(r)2670 1758 y Fn(N)g Fm(+1)2817 1746 y Fs(\()p Fp(b)p Fs(\)\))2949 1579 y Fq(3)2949 1729 y(5)3255 1746 y Fs(\(144\))456 1986 y Fh(wher)l(e)28 b(we)g(denote)l(d)g(by)g Fp(r)1253 1998 y Fn(N)6 b Fm(+1)1401 1986 y Fs(\()p Fp(a)p Fs(\))28 b Fh(the)f(r)l(emainder)i(of)f(the)g(asymptotic)h(exp)l(ansion)f(of)g Fp(a)g Fh(trun-)456 2086 y(c)l(ate)l(d)i(at)f(or)l(der)i Fp(N)38 b Fh(and)31 b(we)f(denote)l(d)1146 2284 y Fp(p)1188 2296 y Fn(mM)1321 2284 y Fs(\()p Fp(a)p Fs(\))23 b(=)222 b(sup)1540 2358 y Fg(j)p Fn(\013)p Fg(j)p Fm(+)p Fg(j)p Fn(\014)s Fg(j\024)p Fn(M)s(;)28 b Fm(\()p Fn(x;\030)r Fm(\))2086 2215 y Fl(j)p Fp(@)2158 2185 y Fn(\013)2153 2236 y(x)2205 2215 y Fp(@)2254 2175 y Fn(\014)2249 2240 y(\030)2299 2215 y Fp(a)p Fs(\()p Fp(x;)14 b(\030)t Fs(\))p Fl(j)p 2086 2265 469 4 v 2189 2341 a Fp(m)p Fs(\()p Fp(x;)g(\030)t Fs(\))2565 2284 y Fp(:)667 b Fs(\(145\))555 2504 y(Fix)25 b(a)g(p)r(ositiv)n(e)f(in)n(teger)g Fp(s)f Fl(\025)g Fs(1)h(denote)h Fp(A)f Fs(=)e Fp(H)2074 2474 y Fn(s)2067 2525 y Fm(0)2135 2504 y Fs(\(where)i Fp(H)2473 2516 y Fm(0)2536 2504 y Fs(is)g(the)i(op)r(erator)d(\(2\)\),)j(from)456 2604 y(Theorem)k(8)h(it)g(follo)n(ws)f(that)i Fp(A)d Fs(=)f Fp(O)r(p)1720 2574 y Fn(w)1774 2604 y Fs(\()p Fp(a)p Fs(\))k(with)g(a)e(suitable)h Fp(a)g Fs(ha)n(ving)f(principal)h (sym)n(b)r(ol)456 2703 y Fp(a)500 2715 y Fm(0)560 2703 y Fs(=)22 b(\()p Fp(\030)719 2673 y Fm(2)773 2703 y Fs(+)15 b Fp(V)k Fs(\))952 2673 y Fn(s)988 2703 y Fs(.)63 b(Denote)26 b(also)f Fp(b)e Fs(:=)g Fp(\030)1732 2673 y Fm(2)p Fn(s)1816 2703 y Fs(+)15 b Fp(V)1963 2673 y Fn(s)2025 2703 y Fs(and)26 b Fp(B)h Fs(=)c Fp(O)r(p)2470 2673 y Fn(w)2524 2703 y Fs(\()p Fp(b)p Fs(\).)63 b(Since)27 b Fp(V)18 b Fs(\()p Fp(x)p Fs(\))25 b Fl(\025)d Fs(1)k(then)456 2803 y(one)h(has)967 2906 y(1)p 955 2943 66 4 v 955 3019 a Fp(C)1054 2962 y Fl(\024)1163 2906 y Fs(1)p 1151 2943 V 1151 3019 a Fp(C)1226 2962 y(b)p Fs(\()p Fp(x;)14 b(\030)t Fs(\))24 b Fl(\024)f Fp(a)1606 2974 y Fm(0)1643 2962 y Fs(\()p Fp(x;)14 b(\030)t Fs(\))24 b Fl(\024)f Fp(C)6 b(b)p Fs(\()p Fp(x;)14 b(\030)t Fs(\))p Fp(;)42 b Fl(8)p Fs(\()p Fp(x;)14 b(\030)t Fs(\))23 b Fl(2)h Fo(R)2688 2928 y Fm(2)p Fn(d)2766 2962 y Fp(:)466 b Fs(\(146\))456 3138 y Fx(Lemma)29 b(7.)41 b Fh(One)29 b(has)1195 3279 y Fl(k)p Fp(A )s Fl(k)1398 3303 y Fn(L)1444 3287 y Fj(2)1503 3279 y Fl(\024)22 b Fp(C)e Fl(k)p Fp(B)t( )s Fl(k)1877 3303 y Fn(L)1923 3287 y Fj(2)1973 3279 y Fp(;)98 b Fl(8)p Fp( )26 b Fl(2)d Fp(D)r Fs(\()p Fp(B)t Fs(\))p Fp(;)731 b Fs(\(147\))1214 3403 y Fl(k)o Fp(B)t( )s Fl(k)1421 3428 y Fn(L)1467 3411 y Fj(2)1526 3403 y Fl(\024)23 b Fp(C)d Fl(k)p Fp(A )s Fl(k)1895 3428 y Fn(L)1941 3411 y Fj(2)1991 3403 y Fp(;)99 b Fl(8)p Fp( )25 b Fl(2)f Fp(D)r Fs(\()p Fp(A)p Fs(\))p Fp(:)717 b Fs(\(148\))456 3572 y Fh(Pr)l(o)l(of.)43 b Fs(Consider)31 b Fp(b)1103 3542 y Fg(\000)p Fm(1)1192 3572 y Fs(,)i(then)g Fp(b)1478 3542 y Fg(\000)p Fm(1)1597 3572 y Fl(2)f Fs(\006)1744 3542 y Fn(b)1773 3517 y Fe(\000)p Fj(1)1744 3593 y Fm(0)1854 3572 y Fs(.)83 b(Denote)33 b Fl(B)g Fs(:=)d Fp(O)r(p)2563 3542 y Fn(w)2617 3572 y Fs(\()p Fp(b)2685 3542 y Fg(\000)p Fm(1)2774 3572 y Fs(\).)84 b(By)31 b(Theorem)h(8)456 3672 y(one)27 b(has)1341 3817 y Fp(A)p Fl(B)f Fs(=)c Fp(O)r(p)1678 3782 y Fn(w)1746 3724 y Fq(\020)1806 3760 y Fp(a)1850 3772 y Fm(0)p 1806 3798 82 4 v 1829 3874 a Fp(b)1897 3724 y Fq(\021)1965 3817 y Fs(+)c Fo(~)p Fp(O)r(p)2200 3782 y Fn(w)2254 3817 y Fs(\()p Fp(\016)2323 3829 y Fm(1)2360 3817 y Fs(\))863 b(\(149\))1679 3969 y Fp(B)t Fl(B)26 b Fs(=)c Fo(I)6 b Fs(+)18 b Fo(~)p Fp(O)r(p)2200 3935 y Fn(w)2254 3969 y Fs(\()p Fp(\016)2323 3981 y Fm(2)2360 3969 y Fs(\))863 b(\(150\))456 4110 y(with)29 b Fp(\016)683 4122 y Fm(1)p Fn(;)p Fm(2)802 4110 y Fs(estimated)g(b)n(y)h(\(142\))o(-\(144\))o(.)70 b(Since)2028 4076 y Fn(a)2064 4084 y Fj(0)p 2028 4091 69 4 v 2047 4138 a Fn(b)2135 4110 y Fs(is)29 b(b)r(ounded)h(together)e (with)h(its)g(deriv)-5 b(a-)456 4209 y(tiv)n(es,)41 b(it)e(follo)n(ws)e (that)i(\001)1324 4221 y Fm(1)1403 4209 y Fs(:=)i Fp(O)r(p)1639 4179 y Fn(w)1694 4209 y Fs(\()p Fp(\016)1763 4221 y Fm(1)1800 4209 y Fs(\))e(is)g(b)r(ounded.)109 b(Similarly)38 b(\001)2842 4221 y Fm(2)2921 4209 y Fs(:=)j Fp(O)r(p)3157 4179 y Fn(w)3211 4209 y Fs(\()p Fp(\016)3280 4221 y Fm(2)3318 4209 y Fs(\))e(is)456 4309 y(b)r(ounded.)555 4408 y(Th)n(us,)32 b(using)e(Neumann)h(form)n(ula)f(one)h(gets)f(that)i(the)f(op)r(erator) e Fo(I)8 b Fs(+)20 b Fo(~)p Fs(\001)2950 4420 y Fm(2)3018 4408 y Fs(is)31 b(in)n(v)n(ertible)456 4508 y(pro)n(vided)26 b Fo(~)i Fs(is)f(small)g(enough.)64 b(So,)28 b(from)f(\(150\))g(one)g (has)916 4650 y(\()p Fo(I)6 b Fs(+)18 b Fo(~)p Fs(\001)1196 4662 y Fm(2)1233 4650 y Fs(\))1265 4615 y Fg(\000)p Fm(1)1355 4650 y Fp(B)t Fl(B)25 b Fs(=)d Fo(I)117 b Fl(\()-14 b(\))129 b(B)2090 4615 y Fg(\000)p Fm(1)2202 4650 y Fs(=)22 b(\()p Fo(I)6 b Fs(+)18 b Fo(~)p Fs(\001)2569 4662 y Fm(2)2606 4650 y Fs(\))2638 4615 y Fg(\000)p Fm(1)2728 4650 y Fp(B)t(:)437 b Fs(\(151\))456 4790 y(Finally)27 b(one)g(has)946 4931 y Fl(k)p Fp(A )s Fl(k)1148 4956 y Fn(L)1194 4939 y Fj(2)1313 4931 y Fs(=)1461 4860 y Fq(\015)1461 4910 y(\015)1507 4931 y Fp(A)p Fl(B)s(B)1685 4896 y Fg(\000)p Fm(1)1773 4931 y Fp( )1830 4860 y Fq(\015)1830 4910 y(\015)1876 4964 y Fn(L)1922 4947 y Fj(2)1981 4931 y Fl(\024)c(k)o Fp(A)p Fl(B)s(k)2271 4956 y Fg(L)p Fm(\()p Fn(L)2389 4939 y Fj(2)2421 4956 y Fn(;L)2487 4939 y Fj(2)2519 4956 y Fm(\))2563 4860 y Fq(\015)2563 4910 y(\015)2609 4931 y Fl(B)2667 4896 y Fg(\000)p Fm(1)2755 4931 y Fp( )2812 4860 y Fq(\015)2812 4910 y(\015)2858 4964 y Fn(L)2904 4947 y Fj(2)3255 4931 y Fs(\(152\))1313 5075 y Fl(\024)83 b Fp(C)1540 5005 y Fq(\015)1540 5054 y(\015)1586 5075 y Fs(\()p Fo(I)6 b Fs(+)18 b Fo(~)p Fs(\001)1866 5087 y Fm(2)1904 5075 y Fs(\))1936 5041 y Fg(\000)p Fm(1)2025 5075 y Fp(B)t( )2149 5005 y Fq(\015)2149 5054 y(\015)2195 5108 y Fn(L)2241 5092 y Fj(2)2301 5075 y Fl(\024)k Fp(C)e Fl(k)p Fp(B)t( )s Fl(k)2675 5100 y Fn(L)2721 5083 y Fj(2)2770 5075 y Fp(:)3380 5216 y Fd(\003)p eop %%Page: 29 29 29 28 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(29)1291 450 y Fr(Appendix)32 b Fs(B.)41 b Fr(Technical)32 b(Lemmas)555 600 y Fs(W)-7 b(e)35 b(start)f(b)n(y)g(a)h(Lemma)f(whic)n (h)g(is)h(needed)g(for)f(the)h(estimate)f(of)h(the)g(solution)f(of)g (the)456 699 y(homological)24 b(equation.)62 b(In)26 b(its)h(statemen)n(t)f(w)n(e)g(will)g(denote)g(b)n(y)g Fp(`)2600 669 y Fm(2)2664 699 y Fs(the)g(real)f(Hilb)r(ert)i(space)456 799 y(of)g(the)h(sequences)f(\()p Fp(z)1141 811 y Fn(j)1176 799 y Fs(\))1208 811 y Fn(j)s Fg(\025)p Fm(3)1356 799 y Fs(endo)n(w)n(ed)f(b)n(y)i(the)g(scalar)e(pro)r(duct)1195 966 y Fl(h)p Fp(z)t Fs(;)14 b Fp(z)1350 932 y Fg(0)1372 966 y Fl(i)24 b Fs(:=)1539 887 y Fq(X)1541 1065 y Fn(j)s Fg(\025)p Fm(3)1672 966 y Fp(z)1711 978 y Fn(j)1746 966 y Fp(z)1789 932 y Fg(0)1785 987 y Fn(j)1843 966 y Fs(=)1930 887 y Fq(X)1971 1062 y Fn(\015)2161 887 y Fq(X)2064 1065 y Fn(j)s Fg(\025)p Fm(3)p Fn(;)k(j)s Fg(2)p Fn(J)2339 1073 y Fk(\015)2391 966 y Fp(z)2430 978 y Fn(j)2465 966 y Fp(z)2508 932 y Fg(0)2504 987 y Fn(j)456 1223 y Fx(Lemma)34 b(8.)43 b Fh(L)l(et)33 b Fp(F)41 b Fs(:)30 b Fp(`)1246 1193 y Fm(2)1312 1223 y Fl(!)g Fp(`)1460 1193 y Fm(2)1530 1223 y Fh(b)l(e)j(a)h(b)l(ounde)l(d)g(line)l(ar)g(op)l(er)l(ator,)h (assume)f(that)f(the)g(c)l(orr)l(e-)456 1323 y(sp)l(onding)27 b(matrix)f(elements)g Fp(F)1450 1335 y Fn(j)s(l)1533 1323 y Fh(ar)l(e)h(di\013er)l(ent)f(fr)l(om)h(zer)l(o)f(only)h(if)g Fp(j)h Fl(2)c Fp(J)2802 1335 y Fn(\015)2871 1323 y Fh(and)j Fp(l)d Fl(2)f Fp(J)3202 1335 y Fn(\015)3241 1319 y Fe(0)3294 1323 y Fh(with)456 1423 y Fp(\015)k Fl(6)p Fs(=)c Fp(\015)662 1392 y Fg(0)685 1423 y Fh(.)68 b(De\014ne)29 b(a)h(new)g(line)l(ar)h (op)l(er)l(ator)f Fp(G)g Fh(with)h(matrix)1553 1608 y Fp(G)1618 1620 y Fn(j)s(l)1698 1608 y Fs(:=)1940 1552 y Fp(F)1993 1564 y Fn(j)s(l)p 1819 1589 352 4 v 1819 1665 a Fp(i)p Fs(\()p Fp(\025)1928 1677 y Fn(l)1972 1665 y Fl(\000)18 b Fp(\025)2103 1677 y Fn(j)2138 1665 y Fs(\))3255 1608 y(\(153\))456 1803 y Fh(Then)30 b(ther)l(e)g(exists)f(a)h(p)l (ositive)i Fp(C)j Fh(such)30 b(that)g(the)g(fol)t(lowing)i(estimate)e (holds)1311 1988 y Fl(k)p Fp(G)p Fl(k)1460 2013 y Fg(L)p Fm(\()p Fn(`)1560 1996 y Fj(2)1592 2013 y Fn(;`)1640 1996 y Fj(2)1672 2013 y Fm(\))1725 1988 y Fl(\024)1864 1932 y Fp(C)p 1823 1969 150 4 v 1823 2046 a Fo(~)1868 2022 y Fm(3)p Fn(=)p Fm(2)1996 1988 y Fl(k)o Fp(F)12 b Fl(k)2143 2013 y Fg(L)p Fm(\()p Fn(`)2243 1996 y Fj(2)2276 2013 y Fn(;`)2324 1996 y Fj(2)2356 2013 y Fm(\))2399 1988 y Fp(;)833 b Fs(\(154\))456 2180 y Fh(Pr)l(o)l(of.)43 b Fs(First)29 b(w)n(e)h(recall)f(that)h Fl(j)p Fp(\025)1522 2192 y Fn(j)1577 2180 y Fl(\000)20 b Fp(\025)1710 2192 y Fn(l)1736 2180 y Fl(j)27 b(\025)f Fp(C)1942 2149 y Fg(\000)p Fm(1)2031 2180 y Fo(~)k Fs(if)h Fp(j)g Fl(2)c Fp(J)2378 2192 y Fn(\015)2451 2180 y Fs(and)j Fp(l)e Fl(2)f Fp(J)2796 2192 y Fn(\015)2835 2175 y Fe(0)2891 2180 y Fs(for)i Fp(\015)j Fl(6)p Fs(=)26 b Fp(\015)3234 2149 y Fg(0)3257 2180 y Fs(,)31 b(and)456 2279 y(that)c(#)p Fp(J)750 2291 y Fn(\015)816 2279 y Fl(\024)c Fp(C)t(=)p Fo(~)p Fs(.)64 b(Fix)28 b Fp(l)c Fl(2)g Fp(J)1464 2291 y Fn(\015)1503 2275 y Fe(0)1529 2279 y Fs(.)64 b(First)28 b(remark)e(that)1453 2368 y Fq(X)1498 2544 y Fn(j)1587 2446 y Fl(j)p Fp(F)1663 2458 y Fn(j)s(l)1720 2446 y Fl(j)1743 2412 y Fm(2)1803 2446 y Fl(\024)d(k)o Fp(F)12 b Fl(k)2039 2405 y Fm(2)2039 2471 y Fg(L)p Fm(\()p Fn(`)2139 2455 y Fj(2)2171 2471 y Fn(;`)2219 2455 y Fj(2)2251 2471 y Fm(\))3255 2446 y Fs(\(155\))456 2678 y(then,)28 b(b)n(y)f(Sc)n(h)n(w)n (artz)f(inequalit)n(y)893 2865 y Fq(X)937 3042 y Fn(j)1027 2944 y Fl(j)p Fp(G)1115 2956 y Fn(j)s(l)1171 2944 y Fl(j)d(\024)1305 2777 y Fq(0)1305 2926 y(@)1378 2865 y(X)1422 3042 y Fn(j)1511 2944 y Fl(j)p Fp(F)1587 2956 y Fn(j)s(l)1644 2944 y Fl(j)1667 2909 y Fm(2)1704 2777 y Fq(1)1704 2926 y(A)1777 2794 y Fm(1)p Fn(=)p Fm(2)1895 2777 y Fq(0)1895 2926 y(@)2068 2865 y(X)1968 3044 y Fn(j)s Fg(2)p Fn(J)2080 3052 y Fk(\015)2118 3044 y Fn(;\015)t Fg(6)p Fm(=)p Fn(\015)2267 3027 y Fe(0)2462 2888 y Fs(1)p 2312 2925 343 4 v 2312 3001 a Fl(j)p Fp(\025)2383 3013 y Fn(l)2427 3001 y Fl(\000)c Fp(\025)2559 3013 y Fn(j)2594 3001 y Fl(j)2617 2977 y Fm(2)2664 2777 y Fq(1)2664 2926 y(A)2737 2794 y Fm(1)p Fn(=)p Fm(2)456 3186 y Fs(Fix)27 b Fp(l)e Fl(2)e Fp(J)778 3198 y Fn(\015)817 3182 y Fe(0)871 3186 y Fs(and)k(estimate)1430 3297 y Fq(X)1474 3474 y Fn(j)1724 3320 y Fs(1)p 1573 3357 V 1573 3433 a Fl(j)p Fp(\025)1644 3445 y Fn(l)1689 3433 y Fl(\000)18 b Fp(\025)1820 3445 y Fn(j)1855 3433 y Fl(j)1878 3409 y Fm(2)1949 3376 y Fs(=)2051 3297 y Fq(X)2036 3476 y Fn(\015)t Fg(6)p Fm(=)p Fn(\015)2165 3459 y Fe(0)2216 3297 y Fq(X)2200 3475 y Fn(j)s Fg(2)p Fn(J)2312 3483 y Fk(\015)2525 3320 y Fs(1)p 2375 3357 V 2375 3433 a Fl(j)p Fp(\025)2446 3445 y Fn(l)2490 3433 y Fl(\000)g Fp(\025)2621 3445 y Fn(j)2656 3433 y Fl(j)2679 3409 y Fm(2)3255 3376 y Fs(\(156\))1007 3703 y(=)1095 3537 y Fq(0)1095 3686 y(@)1225 3625 y(X)1168 3803 y Fn(\015)t Fm(=)p Fn(\015)1297 3786 y Fe(0)1318 3803 y Fg(\006)p Fm(1)1417 3703 y Fs(+)1496 3596 y Fn(\015)1535 3571 y Fe(0)1556 3596 y Fg(\000)p Fm(2)1508 3625 y Fq(X)1507 3800 y Fn(\015)t Fm(=1)1655 3703 y Fs(+)1791 3625 y Fq(X)1734 3803 y Fn(\015)t Fg(\025)p Fn(\015)1864 3786 y Fe(0)1885 3803 y Fm(+2)1969 3537 y Fq(1)1969 3686 y(A)2055 3537 y(0)2055 3686 y(@)2143 3625 y(X)2128 3803 y Fn(j)s Fg(2)p Fn(J)2240 3811 y Fk(\015)2452 3647 y Fs(1)p 2302 3684 V 2302 3760 a Fl(j)p Fp(\025)2373 3772 y Fn(l)2417 3760 y Fl(\000)g Fp(\025)2548 3772 y Fn(j)2584 3760 y Fl(j)2607 3736 y Fm(2)2654 3537 y Fq(1)2654 3686 y(A)3255 3703 y Fs(\(157\))456 3946 y(but,)28 b(due)g(to)f(the)h(c)n(hoice)f(of)g (the)h(n)n(um)n(b)r(ers)g Fp(E)1913 3958 y Fn(\015)1983 3946 y Fs(one)f(has)1112 4196 y Fl(j)p Fp(\025)1183 4208 y Fn(j)1237 4196 y Fl(\000)18 b Fp(\025)1368 4208 y Fn(l)1394 4196 y Fl(j)23 b(\025)1528 4026 y Fq(8)1528 4101 y(<)1528 4250 y(:)1711 4096 y Fp(C)1776 4066 y Fg(\000)p Fm(1)1865 4096 y Fo(~)192 b Fs(if)89 b Fp(\015)28 b Fs(=)23 b Fp(\015)2446 4066 y Fg(0)2487 4096 y Fl(\006)18 b Fs(1)1644 4196 y Fp(E)1705 4208 y Fn(\015)1744 4191 y Fe(0)1789 4196 y Fl(\000)g Fp(E)1933 4208 y Fn(\015)2102 4196 y Fs(if)89 b Fp(\015)28 b Fl(\024)23 b Fp(\015)2446 4165 y Fg(0)2487 4196 y Fl(\000)18 b Fs(2)1602 4295 y Fp(E)1663 4307 y Fn(\015)t Fg(\000)p Fm(1)1809 4295 y Fl(\000)g Fp(E)1953 4307 y Fn(\015)1992 4291 y Fe(0)2102 4295 y Fs(if)89 b Fp(\015)28 b Fl(\025)23 b Fp(\015)2446 4265 y Fg(0)2487 4295 y Fs(+)18 b(2)3255 4196 y(\(158\))456 4444 y(th)n(us)27 b(\(156\))g(is)h(estimated)f(b)n(y)694 4574 y Fq(\022)755 4691 y Fs(sup)799 4757 y Fn(\015)894 4691 y Fs(#)p Fp(J)1009 4703 y Fn(\015)1052 4574 y Fq(\023)1127 4524 y(0)1127 4674 y(@)1209 4635 y Fs(2)p Fp(C)1316 4605 y Fm(2)p 1209 4672 145 4 v 1240 4748 a Fo(~)1285 4724 y Fm(2)1382 4691 y Fs(+)1465 4583 y Fn(\015)1504 4558 y Fe(0)1526 4583 y Fg(\000)p Fm(2)1478 4612 y Fq(X)1477 4788 y Fn(\015)t Fm(=1)1831 4635 y Fs(1)p 1635 4672 434 4 v 1635 4748 a(\()p Fp(E)1728 4760 y Fn(\015)1767 4744 y Fe(0)1812 4748 y Fl(\000)18 b Fp(E)1956 4760 y Fn(\015)1999 4748 y Fs(\))2031 4724 y Fm(2)2097 4691 y Fs(+)2237 4612 y Fq(X)2180 4790 y Fn(\015)t Fg(\025)p Fn(\015)2310 4774 y Fe(0)2331 4790 y Fm(+2)2677 4635 y Fs(1)p 2439 4672 519 4 v 2439 4748 a(\()p Fp(E)2532 4760 y Fn(\015)t Fg(\000)p Fm(1)2678 4748 y Fl(\000)g Fp(E)2822 4760 y Fn(\015)2861 4744 y Fe(0)2888 4748 y Fs(\))2920 4724 y Fm(2)2967 4524 y Fq(1)2967 4674 y(A)3255 4691 y Fs(\(159\))456 4938 y(Since)24 b(the)h(sums)g(in)g(\(159\))e(are)h(con)n(v)n(ergen)n(t)e (due)j(to)f(our)g(c)n(hoice)g(of)g(the)h(sequence)f Fp(E)3144 4950 y Fn(\015)3212 4938 y Fs(on)g(has)1513 5052 y Fq(X)1557 5229 y Fn(j)1807 5075 y Fs(1)p 1656 5112 343 4 v 1656 5188 a Fl(j)p Fp(\025)1727 5200 y Fn(l)1772 5188 y Fl(\000)18 b Fp(\025)1903 5200 y Fn(j)1938 5188 y Fl(j)1961 5164 y Fm(2)2032 5131 y Fl(\024)2138 5075 y Fp(C)p 2129 5112 83 4 v 2129 5188 a Fo(~)2174 5164 y Fm(3)p eop %%Page: 30 30 30 29 bop 456 251 a Fm(30)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)456 450 y Fs(whic)n(h)27 b(giv)n(es)1374 568 y Fq(X)1419 745 y Fn(j)1508 647 y Fl(j)p Fp(G)1596 659 y Fn(j)s(l)1653 647 y Fl(j)c(\024)1838 591 y Fp(C)p 1796 628 150 4 v 1796 705 a Fo(~)1841 681 y Fm(3)p Fn(=)p Fm(2)1970 647 y Fl(k)o Fp(F)12 b Fl(k)2117 672 y Fg(L)p Fm(\()p Fn(`)2217 655 y Fj(2)2250 672 y Fn(;`)2298 655 y Fj(2)2329 672 y Fm(\))3255 647 y Fs(\(160\))456 890 y(>F)-7 b(rom)27 b(this)h(one)f(has)762 1136 y Fl(k)o Fp(Gz)t Fl(k)952 1095 y Fm(2)952 1161 y Fn(`)980 1145 y Fj(2)1040 1136 y Fs(=)1127 1058 y Fq(X)1172 1234 y Fn(j)1261 991 y Fq(\014)1261 1041 y(\014)1261 1091 y(\014)1261 1141 y(\014)1261 1190 y(\014)1289 1058 y(X)1338 1236 y Fn(l)1422 1136 y Fp(G)1487 1148 y Fn(j)s(l)1544 1136 y Fp(z)1583 1148 y Fn(l)1608 991 y Fq(\014)1608 1041 y(\014)1608 1091 y(\014)1608 1141 y(\014)1608 1190 y(\014)1636 1012 y Fm(2)1696 1136 y Fl(\024)1784 1058 y Fq(X)1828 1234 y Fn(j)1917 994 y Fq( )1983 1058 y(X)2033 1236 y Fn(l)2117 1136 y Fl(j)p Fp(G)2205 1148 y Fn(j)s(l)2262 1136 y Fl(j)2285 994 y Fq(!)13 b( )2430 1058 y(X)2479 1236 y Fn(l)2564 1136 y Fl(j)p Fp(G)2652 1148 y Fn(j)s(l)2708 1136 y Fl(j)h(j)p Fp(z)2807 1148 y Fn(l)2832 1136 y Fl(j)2855 1102 y Fm(2)2893 994 y Fq(!)907 1438 y Fs(=)994 1296 y Fq( )1060 1359 y(X)1109 1538 y Fn(l)1194 1438 y Fl(j)p Fp(G)1282 1450 y Fn(j)s(l)1338 1438 y Fl(j)1361 1296 y Fq(!)1441 1359 y(X)1486 1536 y Fn(j)1575 1296 y Fq( )1641 1359 y(X)1690 1538 y Fn(l)1774 1438 y Fl(j)p Fp(G)1862 1450 y Fn(j)s(l)1919 1438 y Fl(j)g(j)p Fp(z)2018 1450 y Fn(l)2043 1438 y Fl(j)2066 1404 y Fm(2)2103 1296 y Fq(!)2192 1438 y Fl(\024)2290 1366 y Fp(C)20 b Fl(k)o Fp(F)12 b Fl(k)2517 1325 y Fm(2)2517 1391 y Fg(L)p Fm(\()p Fn(`)2617 1375 y Fj(2)2649 1391 y Fn(;`)2697 1375 y Fj(2)2729 1391 y Fm(\))p 2290 1419 470 4 v 2483 1495 a Fo(~)2528 1471 y Fm(3)2783 1438 y Fl(k)o Fp(z)t Fl(k)2908 1396 y Fm(2)2908 1463 y Fn(`)2936 1446 y Fj(2)3380 1680 y Fd(\003)555 1779 y Fs(W)-7 b(e)26 b(rep)r(ort)f(no)n(w)f(some)h(Lemmas) g(from)g([3])g(whic)n(h)h(are)e(needed)i(for)f(the)h(pro)r(of)e(of)i (Lemma)456 1879 y(6.)62 b(Here)25 b Fp(\026)h Fs(will)g(b)r(e)h(small)e (parameter)g(\(not)h(the)g(small)g(parameter)f(used)g(in)i(the)f(main)g (part)456 1979 y(of)h(the)h(text\))g(that)g(in)g(applications)f(will)h (b)r(e)g(replaced)e(b)n(y)i Fp(\017)2384 1948 y Fn(r)2448 1979 y Fs(with)g(some)f Fp(r)r Fs(.)456 2142 y Fx(Lemma)34 b(9.)43 b Fh(L)l(et)33 b Fl(G)i Fs(:)30 b Fp(B)1264 2154 y Fn(\032)1332 2142 y Fl(!)g Fo(C)15 b Fh(,)41 b Fp(\032)29 b(>)h Fs(0)i Fh(b)l(e)i(a)g(function)f(whose)i(Hamiltonian)f(ve)l(ctor) g(\014eld)456 2242 y(is)k(analytic)h(as)g(map)f(fr)l(om)h Fp(B)1448 2254 y Fn(\032)1524 2242 y Fl(!)f(X)1716 2212 y Fn(s)1704 2265 y Ff(C)1752 2242 y Fh(;)43 b(\014x)37 b(a)h(p)l(ositive)i Fp(\016)h(<)c(\032)p Fh(.)102 b(Assume)37 b Fp(\026)14 b Fl(j)p Fp(X)3155 2254 y Fg(G)3202 2242 y Fl(j)3225 2267 y Fn(\032)3302 2242 y Fp(<)37 b(\016)456 2352 y Fh(and)30 b(c)l(onsider)h(the)g(\015ow)f Fp(\036)1309 2322 y Fn(t)1369 2352 y Fh(of)h(the)g(c)l(orr)l(esp)l(onding)g (Hamiltonian)g(ve)l(ctor)g(\014eld.)70 b(Then,)31 b(for)456 2452 y Fl(j)p Fp(t)p Fl(j)23 b(\024)f Fp(\026)p Fh(,)31 b(one)f(has)1538 2576 y Fl(j)p Fp(\036)1610 2542 y Fn(t)1658 2576 y Fl(\000)19 b Fo(I)-7 b Fl(j)1803 2588 y Fn(\032)p Fg(\000)p Fn(\016)1943 2576 y Fl(\024)22 b Fp(\026)14 b Fl(j)p Fp(X)2186 2588 y Fg(G)2233 2576 y Fl(j)2257 2601 y Fn(\032)2339 2576 y Fp(:)893 b Fs(\(161\))456 2742 y Fh(Pr)l(o)l(of.)43 b Fs(It)28 b(is)f(just)h(an)g(application)f (of)g(the)h(equalit)n(y)1066 2954 y Fp(\030)t Fs(\()p Fp(t)p Fs(\))19 b Fl(\000)f Fp(\030)t Fs(\(0\))23 b(=)1559 2841 y Fq(Z)1642 2862 y Fn(t)1605 3030 y Fm(0)1695 2898 y Fp(d\030)p 1695 2935 84 4 v 1700 3011 a(dt)1788 2954 y Fs(\()p Fp(s)p Fs(\))p Fp(ds)h Fs(=)2085 2841 y Fq(Z)2168 2862 y Fn(t)2131 3030 y Fm(0)2211 2954 y Fp(X)2280 2966 y Fg(G)2327 2954 y Fs(\()p Fp(\030)t Fs(\()p Fp(s)p Fs(\)\))p Fp(ds)29 b(:)3380 3154 y Fd(\003)456 3318 y Fx(Lemma)37 b(10.)44 b Fh(Consider)37 b Fl(G)k Fh(as)36 b(ab)l(ove)h(and)f(let)g Fl(H)h Fh(b)l(e)e(an)h(analytic)h(function)f(with)g(ve)l(ctor)456 3417 y(\014eld)29 b(analytic)i(in)e Fp(B)1111 3429 y Fn(\032)1149 3417 y Fh(,)h(\014x)e Fs(0)23 b Fp(<)f(\016)k(<)d(\032)29 b Fh(assume)g Fp(\026)14 b Fl(j)p Fp(X)2139 3429 y Fg(G)2186 3417 y Fl(j)2209 3442 y Fn(\032)2271 3417 y Fl(\024)23 b Fp(\016)s(=)p Fs(3)p Fh(,)28 b(then,)i(for)g Fl(j)p Fp(t)p Fl(j)23 b(\024)g Fp(\026)p Fh(,)29 b(one)h(has)1202 3642 y Fl(j)p Fp(X)1294 3656 y Fg(H\016)p Fn(\036)1425 3639 y Fk(t)1456 3642 y Fl(j)1479 3671 y Fn(\032)p Fg(\000)p Fn(\016)1625 3642 y Fl(\024)1713 3525 y Fq(\022)1774 3642 y Fs(1)18 b(+)1927 3586 y(3)p 1927 3623 42 4 v 1928 3699 a Fp(\016)1978 3642 y(\026)c Fl(j)p Fp(X)2134 3654 y Fg(G)2181 3642 y Fl(j)2204 3667 y Fn(\032)2243 3525 y Fq(\023)2318 3642 y Fl(j)p Fp(X)2410 3654 y Fg(H)2471 3642 y Fl(j)2494 3667 y Fn(\032)456 3859 y Fh(Pr)l(o)l(of.)43 b Fs(First)27 b(remark)f(that,)i(since)g Fp(\036)1658 3829 y Fn(t)1715 3859 y Fs(is)g(a)f(canonical)f(transformation)g(one)h (has)1244 4018 y Fp(X)1313 4032 y Fg(H\016)p Fn(\036)1444 4015 y Fk(t)1475 4018 y Fs(\()p Fp(\030)t Fs(\))d(=)f Fp(d\036)1783 3984 y Fg(\000)p Fn(t)1864 4018 y Fs(\()p Fp(\036)1945 3984 y Fn(t)1975 4018 y Fs(\()p Fp(\030)t Fs(\)\))p Fp(X)2180 4030 y Fg(H)2242 4018 y Fs(\()p Fp(\036)2323 3984 y Fn(t)2353 4018 y Fs(\()p Fp(\030)t Fs(\)\))766 b(\(162\))456 4177 y(from)27 b(whic)n(h)881 4336 y Fp(X)950 4350 y Fg(H\016)p Fn(\036)1081 4333 y Fk(t)1112 4336 y Fs(\()p Fp(\030)t Fs(\))d(=)1328 4269 y Fq(\000)1366 4336 y Fp(d\036)1458 4302 y Fg(\000)p Fn(t)1539 4336 y Fs(\()p Fp(\036)1620 4302 y Fn(t)1650 4336 y Fs(\()p Fp(\030)t Fs(\)\))c Fl(\000)e Fo(I)1927 4269 y Fq(\001)1973 4336 y Fp(X)2042 4348 y Fg(H)2103 4336 y Fs(\()p Fp(\036)2184 4302 y Fn(t)2214 4336 y Fs(\()p Fp(\030)t Fs(\)\))h(+)f Fp(X)2521 4348 y Fg(H)2582 4336 y Fs(\()p Fp(\036)2663 4302 y Fn(t)2693 4336 y Fs(\()p Fp(\030)t Fs(\)\))p Fp(:)456 4496 y Fs(T)-7 b(o)27 b(estimate)g(the)h(\014rst)g(term)f(\014x)h Fp(\016)1578 4508 y Fm(1)1638 4496 y Fs(:=)23 b Fp(\016)s(=)p Fs(3;)k(w)n(e)g(ha)n(v)n(e)1014 4655 y(sup)913 4728 y Fg(d)p Fn(\030)r Fg(e\024)p Fn(\032)p Fg(\000)p Fm(3)p Fn(\016)1206 4736 y Fj(1)1254 4655 y Fl(d)p Fp(d\036)1383 4620 y Fg(\000)p Fn(t)1465 4655 y Fs(\()p Fp(\036)1546 4620 y Fn(t)1576 4655 y Fs(\()p Fp(\030)t Fs(\)\))19 b Fl(\000)f Fo(I)-7 b Fl(e)17 b(\024)124 b Fs(sup)1994 4728 y Fg(d)p Fn(\030)r Fg(e\024)p Fn(\032)p Fg(\000)p Fm(2)p Fn(\016)2287 4736 y Fj(1)2335 4655 y Fl(d)p Fp(d\036)2464 4620 y Fg(\000)p Fn(t)2545 4655 y Fs(\()p Fp(\030)t Fs(\))20 b Fl(\000)e Fo(I)-7 b Fl(e)1410 4894 y(\024)1524 4838 y Fs(1)p 1508 4875 75 4 v 1508 4951 a Fp(\016)1545 4963 y Fm(1)1690 4894 y Fs(sup)1606 4968 y Fg(d)p Fn(\030)r Fg(e\024)p Fn(\032)p Fg(\000)p Fn(\016)1866 4976 y Fj(1)1914 4894 y Fl(d)p Fp(\036)2000 4860 y Fg(\000)p Fn(t)2081 4894 y Fs(\()p Fp(\030)t Fs(\))19 b Fl(\000)f Fp(\030)t Fl(e)23 b(\024)2497 4838 y Fp(\026)p 2485 4875 V 2485 4951 a(\016)2522 4963 y Fm(1)2583 4894 y Fl(j)p Fp(X)2675 4906 y Fg(G)2722 4894 y Fl(j)2745 4919 y Fn(\032)2798 4894 y Fp(:)555 5116 y Fs(Going)28 b(bac)n(k)f(to)h Fp(\016)j Fs(and)c(adding)h(the)g(trivial)g(estimate)g(of)f(the)i(second)e(term)h (one)g(has)f(the)456 5216 y(thesis.)2697 b Fd(\003)p eop %%Page: 31 31 31 30 bop 1502 251 a Fm(NORMAL)29 b(F)n(ORMS)f(AND)i(NLS)979 b(31)456 450 y Fx(Lemma)32 b(11.)41 b Fh(L)l(et)32 b Fl(G)37 b Fh(and)32 b Fl(H)g Fh(b)l(e)g(as)g(ab)l(ove,)i(\014x)d Fs(0)26 b Fp(<)g(\016)k(<)c(\032)p Fh(,)32 b(and)h(assume)e Fp(\026)14 b Fl(j)p Fp(X)3069 462 y Fg(G)3116 450 y Fl(j)3140 475 y Fn(\032)3205 450 y Fp(<)26 b(\016)s(=)p Fs(3)p Fh(,)456 557 y(then,)k(for)g Fl(j)p Fp(t)p Fl(j)23 b(\024)g Fp(\026)30 b Fh(one)g(has)1273 731 y Fl(j)p Fp(X)1365 745 y Fg(H\016)p Fn(\036)1496 728 y Fk(t)1523 745 y Fg(\000H)1636 731 y Fl(j)1659 760 y Fn(\032)p Fg(\000)p Fn(\016)1805 731 y Fl(\024)1903 675 y Fs(5)p 1903 712 42 4 v 1904 788 a Fp(\016)1968 731 y Fl(j)p Fp(X)2060 743 y Fg(H)2121 731 y Fl(j)2144 756 y Fn(\032)2196 731 y Fp(\026)14 b Fl(j)p Fp(X)2352 743 y Fg(G)2399 731 y Fl(j)2422 756 y Fn(\032)456 904 y Fh(Pr)l(o)l(of.)43 b Fs(One)27 b(has)489 1039 y Fp(X)558 1053 y Fg(H\016)p Fn(\036)689 1036 y Fk(t)721 1039 y Fs(\()p Fp(\030)t Fs(\))19 b Fl(\000)f Fp(X)996 1051 y Fg(H)1057 1039 y Fs(\()p Fp(\030)t Fs(\))24 b(=)1272 972 y Fq(\000)1310 1039 y Fp(d\036)1402 1005 y Fg(\000)p Fn(t)1484 1039 y Fs(\()p Fp(\036)1565 1005 y Fn(t)1595 1039 y Fs(\()p Fp(\030)t Fs(\)\))19 b Fl(\000)f Fo(I)1872 972 y Fq(\001)1918 1039 y Fp(X)1987 1051 y Fg(H)2047 1039 y Fs(\()p Fp(\036)2128 1005 y Fn(t)2158 1039 y Fs(\()p Fp(\030)t Fs(\)\))i(+)2397 972 y Fq(\002)2432 1039 y Fp(X)2501 1051 y Fg(H)2561 1039 y Fs(\()p Fp(\036)2642 1005 y Fn(t)2672 1039 y Fs(\()p Fp(\030)t Fs(\)\))g Fl(\000)e Fp(X)2980 1051 y Fg(H)3041 1039 y Fs(\()p Fp(\030)t Fs(\))3145 972 y Fq(\003)3221 1039 y Fp(:)456 1174 y Fs(The)j(norm)f(of)h(the)g (square)e(brac)n(k)n(et)h(is)h(easily)f(estimated)g(using)h(Lagrange)d (Theorem)i(and)h(the)456 1274 y(Cauc)n(h)n(y)k(inequalit)n(y)h(in)g (order)f(to)h(b)r(ound)h Fp(dX)1914 1286 y Fg(H)1975 1274 y Fs(.)63 b(The)26 b(other)g(term)g(w)n(as)g(already)e(estimated) 456 1373 y(in)j(Lemma)h(10,)f(so)f(w)n(e)i(ha)n(v)n(e)e(the)i(thesis.) 1612 b Fd(\003)555 1473 y Fs(Although)38 b Fl(H)1002 1485 y Fm(0)1078 1473 y Fs(has)f(un)n(b)r(ounded)i(v)n(ector)d(\014eld) j(the)f(v)n(ector)f(\014eld)h(of)g Fl(H)2904 1485 y Fm(0)2967 1473 y Fl(\016)25 b Fp(\036)3083 1443 y Fn(\026)3153 1473 y Fl(\000)g(H)3313 1485 y Fm(0)3389 1473 y Fs(is)456 1572 y(b)r(ounded,)j(more)e(precisely)h(w)n(e)g(ha)n(v)n(e)456 1722 y Fx(Lemma)43 b(12.)48 b Fh(L)l(et)41 b Fl(H)q Fh(,)j Fl(H)1344 1734 y Fm(0)1381 1722 y Fh(,)g Fl(F)49 b Fh(and)41 b Fl(G)47 b Fh(as)41 b(in)g Fl(x)p Fh(2.3;)48 b(that)41 b(is)g Fl(G)47 b Fh(is)41 b(the)g(solution)g(of)h(the)456 1822 y(Homolo)l(gic)l(al)29 b(e)l(quation)35 b Fs(\(78\))o Fh(;)29 b(denote)f(by)h Fp(\036)1874 1792 y Fn(t)1931 1822 y Fh(the)f(\015ow)g(of)g(the)g(c)l(orr)l(esp)l(onding)h (Hamiltonian)456 1921 y(ve)l(ctor)h(\014eld)g(and)1146 2056 y Fp(`)p Fs(\()p Fp(\030)t Fs(\))24 b(:=)f Fl(H)1490 2068 y Fm(0)1527 2056 y Fs(\()p Fp(\036)1608 2022 y Fn(\026)1653 2056 y Fs(\()p Fp(\030)t Fs(\)\))d Fl(\000)e(H)1962 2068 y Fm(0)1999 2056 y Fs(\()p Fp(\030)t Fs(\))h Fl(\000)f Fp(\026)c Fl(fH)2381 2068 y Fm(0)2418 2056 y Fp(;)g Fl(G)5 b(g)13 b Fp(;)456 2191 y Fh(then)29 b(the)h(ve)l(ctor)g(\014eld)g(of)h Fp(`)e Fh(is)h(analytic)h(and,)g(for)g(any)f Fp(\016)c(<)d(\032)p Fh(,)30 b(satis\014es)1342 2361 y Fl(j)p Fp(X)1434 2373 y Fn(`)1465 2361 y Fl(j)1489 2386 y Fn(\032)p Fg(\000)p Fn(\016)1634 2361 y Fl(\024)23 b Fp(\026)1772 2327 y Fm(2)1819 2305 y Fs(25)p 1819 2342 84 4 v 1841 2418 a Fp(\016)1926 2361 y Fl(j)p Fp(X)2018 2373 y Fg(G)2065 2361 y Fl(j)2088 2386 y Fn(\032)2141 2361 y Fl(j)p Fp(X)2233 2373 y Fg(H)2294 2361 y Fl(j)2317 2386 y Fn(\032)2369 2361 y Fp(:)456 2534 y Fh(Pr)l(o)l(of.)43 b Fs(One)27 b(has)767 2711 y Fl(H)837 2723 y Fm(0)874 2711 y Fs(\()p Fp(\036)955 2677 y Fn(\026)1000 2711 y Fs(\()p Fp(\030)t Fs(\)\))20 b Fl(\000)e(H)1309 2723 y Fm(0)1346 2711 y Fs(\()p Fp(\030)t Fs(\))24 b(=)1562 2598 y Fq(Z)1645 2619 y Fn(\026)1608 2787 y Fm(0)1728 2655 y Fp(d)p 1713 2692 74 4 v 1713 2768 a(dt)1796 2711 y Fl(H)1866 2723 y Fm(0)1903 2711 y Fs(\()p Fp(\036)1984 2677 y Fn(t)2014 2711 y Fs(\()p Fp(\030)t Fs(\)\))p Fp(dt)h Fs(=)d Fl(\000)2414 2598 y Fq(Z)2497 2619 y Fn(\026)2460 2787 y Fm(0)2555 2711 y Fl(F)8 b Fs(\()p Fp(\036)2704 2677 y Fn(t)2734 2711 y Fs(\()p Fp(\030)t Fs(\)\))p Fp(dt;)456 2893 y Fs(where)27 b(w)n(e)g(used)g(the)h(homological)e(equation)h(\(78\))g (to)h(calculate)f Fl(f)o(H)2680 2905 y Fm(0)2718 2893 y Fp(;)14 b Fl(G)5 b(g)o Fs(.)65 b(Then)28 b(one)f(has)1287 3075 y Fp(`)p Fs(\()p Fp(\030)t Fs(\))c(=)1537 2962 y Fq(Z)1620 2982 y Fn(\026)1583 3150 y Fm(0)1678 3007 y Fq(\002)1713 3075 y Fl(F)8 b Fs(\()p Fp(\036)1862 3040 y Fn(t)1892 3075 y Fs(\()p Fp(\030)t Fs(\)\))19 b Fl(\000)f(F)8 b Fs(\()p Fp(\030)t Fs(\))2302 3007 y Fq(\003)2351 3075 y Fp(dt)p Fs(;)456 3252 y(Using)27 b(Lemma)g(5)h(and)f(Lemma)g(11)g (one)g(gets)g(the)h(thesis.)1064 b Fd(\003)1708 3422 y Fr(References)491 3555 y Fw([1])35 b(W.H.)23 b(Asc)n(h)n(bac)n(her,)i (J.)f(F)-6 b(r\177)-35 b(ohlic)n(h,)24 b(G.M.)f(Graf,)g(K.)h(Sc)n (hnee,)h(M.)f(T)-6 b(ro)n(y)n(er,)23 b Fi(Symmetry)j(br)l(e)l(aking)g (r)l(e)l(gime)601 3638 y(in)f(the)h(nonline)l(ar)g(Hartr)l(e)l(e)g(e)l (quation)p Fw(,)e(J.)g(Math.)f(Ph)n(ys.)h Fb(43)p Fw(,)e(3879-3891)k (\(2002\).)491 3721 y([2])35 b(D.)29 b(Bam)n(busi,)g Fi(Exp)l(onential)k(stability)d(of)i(br)l(e)l(athers)g(in)f (Hamiltonian)h(networks)f(of)h(we)l(akly)f(c)l(ouple)l(d)601 3804 y(oscil)t(lators)p Fw(,)24 b(Nonlinearit)n(y)g Fb(9)p Fw(,)e(433-457,)j(\(1996\).)491 3887 y([3])35 b(D.)25 b(Bam)n(busi,)g Fi(Nekhor)l(oshev)k(The)l(or)l(em)h(for)e(smal)t(l)g (amplitude)i(solutions)f(in)e(nonline)l(ar)j(Schr\177)-36 b(odinger)601 3970 y(e)l(quations)p Fw(,)24 b(Math.)f(Z.)h Fb(130)p Fw(,)e(345-387,)i(\(1999\).)491 4053 y([4])35 b(D.)e(Bam)n(busi,)i Fi(On)g(long)h(time)f(stability)f(in)h (Hamiltonian)i(p)l(erturb)l(ations)g(of)e(non-r)l(esonant)i(line)l(ar) 601 4136 y(PDEs)p Fw(,)23 b(Nonlinearit)n(y)h Fb(12)p Fw(,)f(823{850,)i(\(1999\).)491 4219 y([5])35 b(D.)24 b(Bam)n(busi,)g(B.)h(Greb)r(ert,)h Fi(Birkho\013)h(normal)i(form)e(for) h(PDEs)f(with)g(tame)h(mo)l(dulus)p Fw(,)f(T)-6 b(o)26 b(app)r(ear)g(in)601 4302 y(Duk)n(e)e(Math.)g(J..)491 4385 y([6])35 b(D.)27 b(Bam)n(busi,)h(S.)f(Gra\016,)i(T.)e(P)n(aul,)i Fi(Normal)i(form)f(and)h(quantization)f(formulae)p Fw(,)g(Comm)n(un.)c (Math.)601 4468 y(Ph)n(ys.)d Fb(207)p Fw(,)f(173-195,)j(\(1999\).)491 4551 y([7])35 b(D.)28 b(Bam)n(busi,)h(A.)g(Sacc)n(hetti,)j Fi(Stability)e(of)h(sp)l(e)l(ctr)l(al)h(eigensp)l(ac)l(es)g(in)f (nonline)l(ar)h(Schr\177)-36 b(odinger)31 b(e)l(qua-)601 4634 y(tions)p Fw(,)23 b(Preprin)n(t)g(\(2006\).)491 4717 y([8])35 b(D.)d(Bam)n(busi,)i(D.)e(V)-6 b(ella,)35 b Fi(Quasi)f(p)l(erio)l(dic)i(br)l(e)l(athers)f(in)f(Hamiltonian)h (lattic)l(es)g(with)f(symmetries)p Fw(,)601 4800 y(DCDS-B)23 b Fb(2)p Fw(,)f(389-399,)j(\(2002\).)491 4883 y([9])35 b(F.A.)22 b(Berezin,)i(M.A.)e(Sh)n(ubin,)i Fi(The)i(Schr\177)-36 b(odinger)26 b(e)l(quation)p Fw(,)e(\(Klu)n(w)n(er)f(Ac.)h(Publ.,)e (1991\).)456 4967 y([10])35 b(V.)26 b(Grecc)n(hi,)j(A.)d(Martinez,)i Fi(Non-line)l(ar)i(Stark)e(e\013e)l(ct)g(and)i(mole)l(cular)h(lo)l(c)l (alization)p Fw(,)f(Comm.)24 b(Math.)601 5050 y(Ph)n(ys.)f Fb(166)p Fw(,)f(533-548,)j(\(1995\).)456 5133 y([11])35 b(V.)d(Grecc)n(hi,)k(A.)c(Martinez,)k(A.)c(Sacc)n(hetti,)37 b Fi(Destruction)d(of)h(the)f(b)l(e)l(ating)h(e\013e)l(ct)e(for)i(a)f (non-line)l(ar)601 5216 y(Schr\177)-36 b(odinger)26 b(e)l(quation)p Fw(,)e(Comm.)d(Math.)j(Ph)n(ys.)f Fb(227)p Fw(,)f(191-209,)i(\(2002\).) p eop %%Page: 32 32 32 31 bop 456 251 a Fm(32)703 b(D)n(ARIO)23 b(BAMBUSI)f(AND)h(ANDREA)f (SA)n(CCHETTI)456 450 y Fw([12])35 b(B.)d(Hel\013er,)i Fi(Semi-classic)l(al)g(analysis)h(for)f(the)g(Schr\177)-36 b(odinger)34 b(op)l(er)l(ator)i(and)f(applic)l(ations)p Fw(,)h(Lecture)601 533 y(Notes)24 b(in)f(Mathematics)h(1336)h (\(Springer-V)-6 b(erlag:)31 b(Berlin)22 b(Heidelb)r(erg)i(1988\).)456 616 y([13])35 b(S.B.)26 b(Kuksin,)i Fi(Ne)l(arly)i(inte)l(gr)l(able)f (in\014nite-dimensional)h(Hamiltonian)h(systems)p Fw(,)d(Lecture)g (Notes)h(in)601 699 y(Mathematics)23 b(1556)i(\(Springer-V)-6 b(erlag:)31 b(Berlin)23 b(1993\).)456 782 y([14])35 b(R.S.)27 b(MacKa)n(y)-6 b(,)29 b(S.)e(Aubry)-6 b(,)28 b Fi(Pr)l(o)l(of)k(of)d (existenc)l(e)g(of)g(br)l(e)l(athers)h(for)g(time-r)l(eversible)e(or)i (Hamiltonian)601 865 y(networks)c(of)f(we)l(akly)h(c)l(ouple)l(d)i (oscil)t(lators)p Fw(,)c(Nonlinearit)n(y)g Fb(7)p Fw(,)f(1623-1643,)i (\(1994\).)456 948 y([15])35 b(J.)27 b(Mujica,)i Fi(Complex)h(analysis) h(in)e(Banach)i(sp)l(ac)l(es)p Fw(,)f(\(North)f(Holland)f(Mathematical) g(Studies)h(120,)601 1031 y(Amsterdam)21 b(1986\).)456 1114 y([16])35 b(A.)30 b(P)n(azy)-6 b(,)34 b Fi(Semigr)l(oups)g(of)e (line)l(ar)i(op)l(er)l(ators)g(and)g(applic)l(ations)h(to)d(p)l(artial) i(di\013er)l(ential)f(e)l(quations)p Fw(,)601 1197 y(\(Springer-V)-6 b(erlag:)30 b(New)24 b(Y)-6 b(ork,)23 b(1983\).)456 1280 y([17])35 b(S.)25 b(Ragha)n(v)l(an,)j(A.)d(Smerzi,)g(S.)g(F)-6 b(an)n(toni,)27 b(S.R.)e(Sheno)n(y)-6 b(,)28 b Fi(Coher)l(ent)g(oscil)t (lations)g(b)l(etwe)l(en)g(two)g(we)l(akly)601 1363 y(c)l(ouple)l(d)22 b(Bose-Einstein)f(c)l(ondensates:)31 b(Josephson)23 b(e\013e)l(cts,)e Fu(\031)h Fi(oscil)t(lations,)h(and)e(macr)l(osc)l(opic)i(quan-)601 1446 y(tum)i(self-tr)l(apping)p Fw(,)f(Ph)n(ys.)g(Rev.)f(A)h Fb(59)p Fw(,)e(620-633,)j(\(1999\).)456 1529 y([18])35 b(D.)50 b(Rob)r(ert,)58 b Fi(A)n(utour)52 b(de)f(l'appr)l(oximation)j (semi-classique)p Fw(,)j(Progress)51 b(in)g(Mathematics,)57 b(68.)601 1612 y(\(Birkhuser:)30 b(Boston)25 b(1987\).)456 1695 y([19])35 b(M.)c(Reed,)k(B.)d(Simon,)h Fi(Metho)l(ds)i(of)f(mo)l (dern)h(mathematic)l(al)h(physics:)49 b(IV)34 b(analysis)h(of)f(op)l (er)l(ators)p Fw(,)601 1778 y(\(Academic)23 b(Press:)31 b(New-Y)-6 b(ork)23 b(1972\).)456 1861 y([20])35 b(A.)27 b(Sacc)n(hetti,)32 b Fi(Nonline)l(ar)f(time-dep)l(endent)f (one-dimensional)h(Schr\177)-36 b(odinger)31 b(e)l(quation)f(with)g (double)601 1944 y(wel)t(l)25 b(p)l(otential)p Fw(,)g(SIAM:)e(J.)g (Math.)h(Anal.)f Fb(35)p Fw(,)f(1160-1176,)j(\(2004\).)456 2028 y([21])35 b(A.)21 b(Sacc)n(hetti,)j Fi(Nonline)l(ar)g(double)h (wel)t(l)f(Schr\177)-36 b(odinger)25 b(e)l(quations)f(in)g(the)f (semiclassic)l(al)j(limit)p Fw(,)21 b(J.)g(Stat.)601 2111 y(Ph)n(ys.)i Fb(119)p Fw(,)f(1347-1382,)j(\(2005\).)456 2194 y([22])35 b(J.)27 b(Sj\177)-35 b(ostrand,)28 b Fi(Semi-excite)l(d) h(levels)g(in)g(non-de)l(gener)l(ate)g(p)l(otential)h(wel)t(ls)p Fw(,)f(Asymptotic)e(Analysis)g Fb(6)p Fw(,)601 2277 y(29-43,)c (\(1992\).)456 2360 y([23])35 b(A.)24 b(T)-6 b(rom)n(b)r(ettoni,)26 b(A.)e(Smerzi,)g Fi(Discr)l(ete)j(solitons)h(and)g(br)l(e)l(athers)g (with)f(diluite)g(Bose{Einstein)h(c)l(on-)601 2443 y(densates)p Fw(,)23 b(Ph)n(ys.)h(Rev.)f(Lett.)i Fb(86)p Fw(,)d(2353-2356,)j (\(2001\).)456 2526 y([24])35 b(A.)26 b(V)-6 b(ardi,)27 b(J.R.)f(Anglin,)h Fi(Bose-Einstein)i(c)l(ondensates)h(b)l(eyond)f(me)l (an)g(\014eld)g(the)l(ory:)40 b(quantum)29 b(b)l(ack)601 2609 y(r)l(e)l(action)d(as)g(de)l(c)l(oher)l(enc)l(e)p Fw(,)f(Ph)n(ys.)e(Rev.)h(Lett.)g Fb(86)p Fw(,)f(568-571,)h(\(2001\).) 456 2692 y([25])35 b(A.)20 b(V)-6 b(ardi,)20 b(J.R.)g(Anglin,)g Fi(Dynamics)j(of)g(a)g(two-mo)l(de)h(Bose-Einstein)f(c)l(ondensate)h(b) l(eyond)f(me)l(an)h(\014eld)601 2775 y(the)l(ory)p Fw(,)f(Ph)n(ys.)h (Rev.)f(A)h Fb(64)p Fw(,)e(013605,)j(\(2001\).)456 2858 y([26])35 b(K.)26 b(Y)-6 b(a)t(jima,)27 b(G.)g(Zhang,)i Fi(L)l(o)l(c)l(al)h(smo)l(othing)h(pr)l(op)l(erty)f(and)g(Sctrichartz)f (ine)l(quality)g(for)h(Schr\177)-36 b(odinger)601 2941 y(e)l(quations)26 b(with)g(p)l(otentials)h(sup)l(er)l(quadr)l(atic)h (at)d(in\014nity)p Fw(,)e(J.)g(Di\013.)g(Eq.)g Fb(202)p Fw(,)f(81-101,)i(\(2004\).)456 3024 y([27])35 b(J.)e(Zhang,)38 b Fi(Stability)c(of)h(attr)l(active)g(Bose-Einstein)g(c)l(ondensates)p Fw(,)i(J.)d(Stat.)h(Ph)n(ys.)e Fb(101)p Fw(,)j(731-746,)601 3107 y(\(2000\).)555 3262 y Fv(Dip)l(ar)l(timento)19 b(di)i(Ma)l(tema)l(tica,)f(Universit)1879 3256 y(\023)1877 3262 y(a)g(degli)g(studi)h(di)g(Milano,)g(Via)e(Saldini)h(50,)h(Milano) 456 3345 y(20133,)i(It)l(al)l(y)555 3429 y Fi(E-mail)j(addr)l(ess)5 b Fw(:)33 b Fa(bambusi@mat.unimi.it)555 3570 y Fv(Dip)l(ar)l(timento)26 b(di)h(Ma)l(tema)l(tica)e(Pura)h(ed)h(Applica)l(t)l(a,)f(Universit)2559 3564 y(\023)2557 3570 y(a)h(degli)g(studi)g(di)g(Modena)f(e)456 3653 y(Reggio)e(Emilia,)h(Via)g(Campi)g(213/B,)g(Modena)f(41100,)g(It)l (al)l(y)555 3736 y Fi(E-mail)i(addr)l(ess)5 b Fw(:)33 b Fa(Sacchetti@unimore.it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0609280938509--