Asymptotic Stability of Multi-soliton Solutions

8/13/2019 Asymptotic Stability of Multi-soliton Solutions

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arXiv:ma

th-ph/0309021v1

9Sep2003

ASYMPTOTIC STABILITY OF MULTI-SOLITON SOLUTIONS

FOR NONLINEAR SCHRODINGER EQUATIONS

Galina Perelman

Centre de MathematiquesEcole Polytechnique

F-91128 Palaiseau CedexFrance

e-mail: [email protected]

Abstract. We consider the Cauchy problem for the nonlinear Schrodinger equa-tion it = + F(||

2), in space dimensions d 3, with initial data close toa sum ofN decoupled solitons. Under some suitable assumptions on the spectralstructure of the one soliton linearizations we prove that for large time the asymp-

totics of the solution is given by a sum of solitons with slightly modified parametersand a small dispersive term.

0. Introduction

In this paper we consider the nonlinear Schrodinger equation

(0.1) it =+ F(||2), (t, x) R Rd, d 3.

For suitable Fit possesses important solutions of special form - solitary waves (or,shortly, solitons):

ei(x b(t), E),

=t+ + 12

x v, b(t) =vt+ c, E= + |v|24

>0,

where , R, v, c Rd are constants and is a ground state that is a smoothpositive spherically symmetric, exponentially decreasing solution of the equation

(0.2) +E +F(2)= 0.

Solitary wave solutions are of special importance not only because they are simpleand sometimes explicit solutions of evolution equations, but also because of thedistinguished role they appear to play in the solution of the initial value problem.This is best known for completely integrable equations like the cubic Schrodinger

equation

(0.3) it =xx ||2.

In the general position case the solution of the Cauchy problem for this equationwith rapidly decreasing smooth initial data has in L2(R) the asymptotic behavior

N

j=1

eij(x bj (t), Ej ) + eil0tf+, j =j t +j +

xvj2

, bj =vj t +cj

1


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where l0 = 2x, and f+ is some function in L2(R). The number N, the functionf+ and the soliton parameters (j , Ej , vj , cj ) depend on the initial data. Due topossibility of explicitly integrating equation (0.3) with the help of inverse scatteringmethods, they can be described by effective formulas in terms of initial condition.

See, for example, [22] for these results.Numerical experiments have shown that even without the presence of an inverse-

scattering theory, solutions, in general, eventually resolve themselves into an ap-proximate superposition of weakly interacting solitary waves and decaying disper-sive waves (see [11], for example). While exact theory confirming the special roleof solitary waves as a nonlinear basis with respect to which it is natural to viewthe solutions in the limit of large time is not generally available, partial indicationis provided by stability theory of such waves. A considerable literature has beendevoted to the problem of orbital stability of solitons following the work of Ben-jamin [1], see also [7, 13, 14, 29, 34, 35]. The problem arises in connection with theCauchy problem for equation (0.1) with initial data of the form

(0.4)

t=0= (x, E0) +0,

where 0 is small in the Sobolev space H1(Rd). It was shown that under certain

additional conditions the solution (x, t), t 0 remains close (again in the spaceH1(Rd)) to the surface

{ei(x c, E0), R, c Rd}.

This notion of stability establishes that the shape of the wave is stable, but doesnot fully resolve the question of what the asymptotic behavior of the system is.

The first asymptotic stability results were obtained by Soffer and Weinstein in

the context of the equation

(0.5) it =+ [V(x) + ||m1],

(see [27, 28] and [30, 31, 32, 33, 36] for the further developments related to thismodel). The solitons for (0.5) arise as a perturbation of the eigenfunction of theoperator + V(x) and, in contrast to the case of equation (0.1), they have afixed center, which simplifies the analysis to some extent.

For the one- dimensional equation

(0.6) it=xx+ F(||2)

the asymptotic stability of solitons was studied in the works of Buslaev and author[4, 5]. We considered the Cauchy problem (0.6), (0.4) and proved that in the casewhere the spectrum of the linearization of equation (0.6) at the initial soliton has thesimplest possible structure in some natural sense, the solution has an asymptoticbehavior of the form

= ei+(x b+(t), E+) + eil0tf++ o(1), + = +t + ++

xv+2

, b+ = v+t + c+

as t +, where the parameters (+, E+, v+c+) of the limit soliton are closeto the initial ones (0, E0, 0, 0) and f+ is small. Some asymptotic results in the


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framework of significantly freer conditions on the linearization were obtained in [5],see also [6]. Recently the analysis of [4, 5, 6] was extended to the multidimensionalcase (0.1) by Cuccagna [8, 9].

As a natural generalization of the above situation one can consider the case

of several weakly interacting solitons. Assume that one has a set of solitonsei0j+i

xv0j2 (x b0j , E0j ), j = 1, . . . , N , that are well separated either in the orig-

inal space or in Fourier space: for j = k, either |v0jk | or mint0

|b0jk (t)| is sufficiently

large, where v0jk =v0j v0k, b0jk (t) =b0j b0k+ v

0jk t. In the second case we shall

assume that the collision time t0jk = b0jk(0)v0jk|v0jk|2 is bounded from above, see

subsection 1.4, (1.7) for the exact formulation.Consider the Cauchy problem for equation (0.1) with initial data close to a sum

Nj=1

ei0j+ixv0j2 (x b0j , E0j ).

If all the linearizations constructed independently from the solitons (E0j ) satisfythe spectral conditions introduced in the case of one soliton, one can expect thatas t + the solution looks like a sum of N soliton with slightly modifiedparameters plus a small dispersive term. In [23] this was proved in the case d =1, N= 2, see also [19] for the asymptotic stability results for the sums of solitonsin the context of KdV type equations. The goal of the present paper is to extendthe result of [23] to the multidimensional case d 3 (omitting also the restrictionN=2). The main new ingredient in the analysis is a combination of the estimatesfor the linear one soliton evolution obtained by Cuccagna in [8] with the ideas ofHagedorn [15].

The structure of this paper is briefly as follows. It consists of two sections. Inthe first section we introduce some preliminary objects and state the main result.The second contains the complete proofs of the indicated results, some technicaldetails being removed to the appendices.

1. Background and statement of the results

1.1. Assumptions on F. Consider the nonlinear Schrodinger equation

(1.1) it =+ F(||2), (t, x) R Rd, d 3.

We assume the following.

Hypothesis H0. Fis a smooth function, F(0) = 0, F satisfies the estimates

F() Cq, |F()()| C p, = 0, 1, 2,

whereC >0, 1, q < 2

d

, p < 2

d2.

Set g () =E+ F(2).

Hypothesis H1.

(i) There exists 0 > 0 such that g() > 0 for < 0, g() < 0 for > 0 andg(0)< 0.

(ii) There exists1>0 such that10

dsg(s) = 0.

Further assumptions are given in terms of the function

I(, ) =g () + ( + 2)g().

We consider 0 of (H1) and assume:


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Hypothesis H2. For any > 0 there exists a()> 0, continuously dependingon, such thatI(t, ) 0 for0< t < andI(t, ) 0 for t > .

We suppose hypotheses (H1,2) to be true for Ein some open interval A R+.Under these assumptions equation (0.2) forE A, has a unique positive spheri-

cally symmetric smooth exponentially decreasing solution(x, E), see [2, 20]. Moreprecisely, as |x|

(x, E) C e

E|x||x|(d1)

2 .

This asymptotic estimate can be differentiated any number of times with respectto x and E.

We shall call the functionsw(x, ) = exp(i+iv x/2)(xb, E), = ( ,E ,b ,v) R2d+2 by soliton states. w(x, (t)) is a solitary wave solution iff(t) satisfies thesystem:

(1.2) = E|v|2

4

, E= 0, b=v, v = 0.

1.2. One soliton linearization. Consider the linearization of equation (1.1) ona soliton w(x, (t)):

w +,

it = ( +F(|w|2)) +F(|w|2)(|w|2+ w2).

Introducing the function f:

f=

f

f

, (x, t) = exp(i)f(y, t),

=(t) + v x2

, y= x b(t),

one gets

i ft =L(E) f , L(E) =L0(E) +V(E), L0(E) = ( + E)3,

V(E) =V1(E)3+iV2(E)2, V1 = F(2) +F(2)2, V2(E) =F(2)2.

Here 2, 3 are the standard Pauli matrices

2= 0 ii 0 , 3=

1 00 1 .

We consider L as an operator in L2(Rd C2) defined on the domain where L0 is

self adjoint. L satisfies the relations

3L3=L, 1L1 = L,

where 1 =

0 11 0

. The continuous spectrum of L(E) fills up two semi-axes

(, E] and [E, ). In addition L(E) may have finite and finite dimensionalpoint spectrum on the real and imaginary axis.


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Zero is always a point of the discrete spectrum. One can indicate d+ 1 eigen-functions

0 =

1

1

, j =yj

1

1

, j= 1, . . . d ,

and d + 1 generalized eigenfunctions

d+1= E

1

1

, d+1+j =

1

2yj

1

1

, j = 1, . . . d ,

Lj = 0, Ld+1+j =j, j = 0, . . . , d .

LetMbe the generalized null space of the operator L. Under assumptions (H0,1,2),

the vectorsj , j= 0, . . . , 2d + 1, span the subspace M iff

d

dE(E)22= 0,

see [34, 20, 8].We shall assume that

Hypothesis H3. The setA0 ofE A such that(i) zero is the only eigenvalue of the operatorL(E), and the dimension of the cor-

responding generalized null space is equal to 2d + 2;(ii) Eis not a resonance forL(E);

is nonempty.

Obviously, the setA0 is open.

Remark. E is said to be a resonance of L(E) if there is a solution of theequation (L(E) E) = 0 such that < x >s L2 for any s >1/2 but not fors= 0. Ecan never be a resonance ifd 5, see lemma A4.3.

Consider the evolution operator eitL. One has the following proposition.

Proposition 1.1. ForE A0 and anyx0, x1 Rd,

(1.3) x x00 eiL(E)tP(E)f2 Ct

d/2 x x10 f2, 0 >

d

2,

whereP(E) is the spectral projection onto the subspace of the continuous spectrumofL(E):

Ker P =M, RanP = (3M).

The constant C here is uniform with respect to x0, x1 Rd and E in compact

subsets ofA0.

This proposition is an immediate consequence of theLp-Lq estimates ofeiLtP

proved by Cuccagna [8]. For the sake of completeness we sketch the proof of (1.3)in appendix 4.

1.3. The nonlinear equation. We formulate here the necessary facts about theCauchy problem for equation (1.1) with initial data in H1(Rd).


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Proposition 1.2. Suppose that F satisfies (H0). Then the Cauchy problem forequation(1.1) with initial data(x, 0) =0(x),0 H1(Rd) has a unique solution in the spaceC(R H1), and satisfies the conservation laws

dx||2 =const, H()

dx[||2 + U(||2)] =const,

whereU() =0 dsF(s). Furthermore, for all t R

(t)H1 c(0H1)0H1 ,

wherec : R+ R+ is a smooth function.

The assertion stated here can be found in [10, 11], for example.

1.4. Description of the problem. Consider the Cauchy problem for equation(1.1) with initial data

(1.4) |t=0=0 H1 L1, 0 =

Nj=1

w(, 0j ) + 0,

(1.5) 0j = (0j, E0j , b0j , v0j ), minj=k

|v0jk | v0 > 0.

Here v0jk =v0j v0k. Set b0jk =b0j b0k, j =k. Write b

0jk as the sum

(1.6) b0jk =r0jk t

0jk v

0jk , r

0jk v

0jk = 0, t

0jk =

b0jk v0jk

|v0jk |2

.

For j =k we define the effective small parameterjk :

(1.7) jk =

(min

t0|b0jk (t)| + |v

0jk |)

1, if t0jk < r0jk >,

|v0jk |1 otherwise,

where b0jk (t) =b0jk + tv

0jk , is a fixed positive constant.

Assume that(T1) max

j=kjk is sufficiently small

1;

(T2) E0j A0, j = 1, . . . , N .Our goal is to describe the asymptotic behavior of the solution as t +,

provided0 is sufficiently small in the following sense:(T3) for some m, 1m + 1m = 1, m 2p+ 2,

4d + 2 < m 0. Here+j (t) is the trajectory of(1.2) with the initial data+j (0) =

+j .

2. Proof of the theorem

Up to some technical modifications the main line of the proof repeats that of[23].

2.1. Splitting of the motions. Following [23] we decompose the solution asfollows.

(2.1) (x, t) =N

j=1

w(x, j (t)) + (x, t).

Here j (t) = (j (t), Ej (t), bj (t), vj (t)) is an arbitrary trajectory in the set of ad-missible values of parameters, it is not a solution of (1.2) in general.

We fix the decomposition (2.1) by imposing the orthogonality conditions

(2.2)

fj (t), 3k(Ej (t))

= 0, j = 1, . . . , N , k = 0, . . . , 2d+ 1.

Here

fj =

fjfj

, (x, t) = exp(ij )fj (yj, t),

j =j (t) +vj x/2, yj =x bj(t),

is the inner product in L2(Rd C2).

Geometrically these conditions mean that for each t the vector fj (t) belongs tothe subspace of the continuous spectrum of the operator L(Ej(t)).

For of the form (1.4) with minj,kj=k

(|v0jk |+ |b0jk |) sufficiently large, and with 0

sufficiently small in some Lp norm, the solvability of (2.2) is guaranteed by thenon-degeneration of the corresponding Jacobi matrix, see lemma A1.1. So, one canassume that the initial decomposition (1.4) obeys (2.2). To prove the existence ofa decomposition (2.1), (2.2) for t > 0, one can invoke a standard continuity typeargument, see appendix 1 for the details.


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Here

Nj =N0+

k,k=jV(wk) +

k,k=j

ei3k l(k)0(yk, Ek), j = 1, . . . , N ,

e= d

dE22, n=

1

222.

The right hand side of (2.4) also contain the derivative , which enters linearlyin l(k). In principle, system (2.4) can be solved with respect to derivative andtogether with equation (2.3) constitutes a complete system for and :

(2.5) it = H((t)) + N(, ),

(2.6) =G(, ), |t=0= 0, j (0) =0j .

2.2. Integral representations for . In this subsection we follow closely theconstructions of Hagedorn [15] (developed in order to prove the asymptotic com-pleteness for the charge transfer model), see also [21]. We start by rewriting (2.5)as an integral equation

(2.7) (t) =U0(t, 0)0 i

t0

U0(t, s)

N

j=1

Vj (s)(s) + N

ds,

HereU0(t, ) =ei(t)3, Vj =V(wj ).

Next we introduce the one soliton adiabatic propagatorsUAj (t, ):

iUAj t (t, ) =Lj (t)UAj (t, ), U

Aj (t, )|t= =I ,

Lj (t) =3+ Vj (t) + Rj (t), Rj (t) =iT0j (t)[Pj(t), Pj(t)]T

0j (t),

Vj(t) =T0j (t)Tj (t)V(E0j )T

j(t)T0j(t), Pj (t) =Tj (t)P(E0j )T

j(t).

HereT0j (t) =B0j(t),b0j(t),v0j , Tj (t) =Bj(t),aj(t),0,

j =

t0

ds

Ej (s) E0j+

|vj (s) v0j |2

4

, aj =

t0

ds(vj (s) v0j ),

(B,b,vf)(x) =e

i3+ivx2 3

f(x b),0j (t) = (0j (t), E0j , b0j (t), v0j ) being the solution of (1.2) with initial data 0j (0) =0j . Obviously,

PAj (t)UAj (t, ) =U

Aj (t, )P

Aj (),

wherePAj (t) =T0j (t)Pj(t)T

0j(t).

Write the solution as the sum:

(t) = hj (t) + kj(t), hj(t) =PA

j (t)(t).


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Using the adiabatic evolution UAj (t, ) one can write the following representationfor hj(t)

(2.8) hj

(t) =U

A

j (t, 0)PA

j (0)

0 i t

0 U

A

j (t, s)PA

j (s)[

m, m=j Vm(s)(s) + D

j(s)]ds,

Here

(2.9) Dj =N+ (Vj Vj) Rj .

Combining (2.7), (2.8) one gets finally

(2.10) = (I) + (II) + (III) + (IV),

where

(I) =U0

(t, 0)0

ij t0

dsU0

(t, s)Vj

(s)UAj

(s, 0)PAj

(0)0

,

(II) =j,mj=m

t0

dsKj (t, s)Vm(s)(s),

(III) =i

t0

dsU0(t, s)D,

(IV) =

j

t0

dsKj (t, s)Dj(s).

Here

(2.11) D= N+

j

Vjkj+ (Vj Vj )

,

Kj(t, s) =

ts

dU0(t, )Vj ()UAj (, s)P

Aj (s).

The relations (2.4), (2.7), (2.10) make up the final form of the equation which isused to prove theorem 1.1.

2.3. Estimates of solitons parameters. Following [4, 23] we consider (2.4),(2.7), (2.10) on some finite interval [0, t1] and then study the limit t1 +. On

the interval [0, t1] we introduce a natural system of norms for the components ofthe solution :

M0(t) =

Nj=1

|j (t) 0j| + |Ej (t) Ej0| + |cj (t) b0j | + |vj (t) v0j |,

M1(t) =N

j=1

< yj > (t)2, M2(t) =(t)2p+2, >

d + 2

2 ,

without loss of generality one can assume that m = 2p + 2.


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These norms generate the system of majorants

M0(t) = sup0t

M0(), Ml(t) = sup0t

Ml()l(), l= 1, 2, Mk = Mk(t1).

Here 1< 1 < 32 ifd= 3 and 1< 1=

dp2 for d 4, 2 =d(

12

12p+2),

(t) =< t >1 +j,kj=k

< t tjk >1,

tjk being the collision times that are defined as follows. We set tjk = 0 ift0jk 0.

For (j, k) such that t0jk >0, we define tjk by the relation,

tjk0

dsvjk (s) v0jk

|v0jk |2

=t0jk ,

where

vjk (t) =

vjk (t), if t t1,

vjk (t1), if t > t1,vjk (t) =vj (t) vk(t).

Let us mention that(i) tjk are well defined provided |vjk (t) v

0jk |< v0, 0 t t1;

(ii) the collision timestjk belonging to the interval [0, t1] do not depend ont1.It follows directly from the definition ofM0 that

(2.12) |j (t)|,|aj (t)| M0(t) + M

20 (t), |bj (t) bj (t)| M0(t),

(2.13) |j (x, t) j (x, t)| M0(t)< x bj (t)> +M0(t) t0

ds|cj(s)|,

wherebj (t) =b0j (t) + aj (t), j(x, t) =0j (t) + j (t) +v0j x/2.

It is also easy to check that bjk =bj bk admits the estimates

(2.14) |bjk (t)| c|v0jk ||t tjk |,

(2.15) |bjk (t)| c(mins0

|b0jk (s)| + |v0jk ||t tjk |) c, t

0jk < r

0jk >

providedM0(t) c for 0 t t1. Here and below c is used as a general notationof positive constants that depend only on v0, and eventually on Ej , j = 1, . . . , N ,in that case they can be chosen uniformly with respect to Ej in some finite vicinityofE0j .

Consider relations (2.4). Since

|N0| c

j,kj=k

|wj ||wk|(1 + ||) +

||2 + ||2p+1 if p > 12 ,

||2 if p 12


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and

|< ei3j l(j )0( bj , Ej ), 3ei3kl( bk, Ek)> | = O(|j|e

c|bjk||vjk |),

j=k , j = (j , Ej , cj , vj ), bjk =bj bk, one gets immediately from (2.4)

(2.16) |j (t)| W(M)[i,li=k

ec|bik(t)|+M

21(t) + M

22(t)

21(t)].

We use W(M) as a general notation for functions ofM0, M1, M2, which arebounded in some finite vicinity of the point Ml = 0, l = 0, 1, 2, and may acquire+ out some larger vicinity. They depend only on v0, 0, Ej0, j = 1, . . . , N andcan be chosen to be spherically symmetric and monotone. In all the formulas whereWappear it would not be hard to replace them by some explicit expressions butsuch expressions are useless for our aims.

Combining (2.13), (2.16) one gets

(2.17) |j (x, t) j(x, t)| W(M)M0(t)< x bj (t)> .

Integrating (2.16) and taking into account (2.14), (2.15) we obtain

(2.18) M0 W(M)[+ M21+ M

22].

Consider the vectors kj (t) = (I PAj (t))(t),

kj (x, t) =2d+1

l=0 kjl (t)eij3l(x

bj (t), E0j ). The orthogonality conditions (2.2) together with (2.12), (2.17) leadimmediately to the estimate:

(2.19) |kjl (t)| W(M)M0(t)ec|xbj(t)|(t)2 W(M)M0(t)M1(t)1(t).

2.4. Linear estimates. To study the behavior of solutions of the integral equa-tion (2.10) we need some estimates of the evolution operators UAm(t, )P

Am(). The

necessary estimates are collected in this subsection, the proofs being removed tothe appendices.

Lemma 2.1. For anyx0, x1 Rd, 0 t t1,

(2.20) x x00UAj (t, )P

Aj ()f2 W(M) t

d/2 x x10 f2.

The functionW here is independent ofx0, x1 and t1.See appendix 2 for the proof.Remark. Due to the representation

UAj (t, )PAj ()f=P

Aj (t)U0(t, )f i

t

dsUAj (t, s)PAj (s)(Vj(s) + Rj (s))U0(s, )f,

and the estimate

|(Rj (t)f)(x)| W(M)ec|xbj(t)|(|j | + |a

j |)e

c|xbj(t)|f2


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(2.21) W(M)ec|xbj(t)|M0(t)ec|xbj(t)|f2,

(2.20) leads immediately to the inequality

(2.22) x x00UAj (t, )P

Aj ()f2 W(M)

(fp1 + fp2)|t |d(

12 1p1 ) t d(

1p1 1

p2)

,

where 2 p1< 2dd2 < p2 ,

1pi

+ 1pi

= 1,i = 1, 2. Obviously, the same estimate

is valid for Kj(t, ):

(2.23) x x00 Kj(t, )f2 W(M)

(fp1 + fp2)

|t |d(12 1p1 ) t d(

1p1 1

p2)

.

The key point of our analysis is the following lemma that is essentially lemma3.6 of [15].

Lemma 2.2. Introduce the operatorsTjki (t, ), j,k, i= 1, . . . , N , i=k

Tjki (t, ) =Aj (t)Kk(t, )Ai(),

whereAj (t) is the multiplication by< x bj (t)>. Then, for0 t t1

t0

dTjki (t, ) W(M)(1ik + M0(t)),

with some1> 0. The norm here stands for theL2 L2 operator norm.

See appendix 3 for the proof.2.5. Estimates of the nonlinear terms. Here we derive the necessary estimatesofD ,Dj. We write D as the sum:

D= D0 +D1 + D2,

where

D0 =N00+

j

(Vj Vj) +Vjkj+ e

ij3 l(j )0( bj, Ej )

,

N00 =F(|s|2)

ss

j

F(|wj |2)

wjwj

+ V(s)

j

Vj ,

D1 =F(|s+ |2)

s+

s

F(|s|

2)

ss

V(s) F(||

2)

,

D2 =F(||2)

.

In a similar way,Dj =D

0j + D

1 + D2, j= 1, . . . N ,


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where

D0j =N00+ (Vj Vj ) Rj +

k

eik3 l(k)0( bk, Ek).

Estimating N00 by

(2.24) |N00| c(1 + ||)j,kk=j

|wj ||wk|,

and using (2.12), (2.17), (2.19), (2.21) one gets

|D0|, |D0j | W(M)[(1 + ||)i,ki=k

ec(|xbi|+|xbk|)

+i

ec|xbi|(|i| +M0(t)|| +M0(t)M1(t))].

which together with (2.16) leads to the inequality

(2.25) D0L1L2 ,D0j L1L2 W(M)[e

c|bjk(t)|+ (M0M1+ M21+ M22)

1(t)].

Consider D1,D2. We estimate them as follows.

|D1 +D2| W(M)[|s|||2 + ||3 + ||2p+1], if d= 3,

(2.26) |D1| W(M)|s|||2, |D2| ||2p+1, if

1

2< p d/2 .

Consider expression (II):

(2.29) < yj > (II)2 W(M)M1(t)

k,ik=i

t0

dsTjki (t, s)1(s).


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15

By lemma 2.1,Tjki (t, s) W(M)< t s >

d/2 .

So, the integral in the right hand side of (2.29) can be estimated as follows.

t0

dsTjki (t, s)1(s)

t0

dsTjki (t, s)d/2(s)

21d t

0

dsTjki (t, s)

1 21d

W(M)(M0+ 2ik )

t0

ds < t s >d/2 d/2(s) 21

d

W(M)(M0+ 2ik )

1(t),

0< = 1 21d ,2 = 1. At the second step here we have used lemma 2.2. Thus,

(2.30) < yj > (II)2 W(M)(M0+

2ik )M1(t)

1(t).

Consider the two last terms in the r.h.s. of (2.10). By (2.25), (2.23) (withp1= 2, p2= ) one has

< yj >U0(t, s)D0(s)2, < yj > Km(t, s)D0m(s)2 W(M)< t s >

d/2

(2.31) [i,ki=k

e|bik(s)|+ (M0(t)M1(t) + M21(t) + M22(t))

1(s)].

Using (2.27), (2.23) one can estimate the contribution ofD1, D2 as follows.

< yj >U0(t, s)(D1(s) + D2(s))2, < yj > Km(t, s)(D1(s) +D2(s))2

(2.32) W(M)[M21(t) + Mr12 (t)]|t s|

2 < t s >1+2 1(s).

Here 1< r1 = 1 + min{p, p1}< 2. Combining (2.31), (2.32) and integrating with

respect tos one gets

< yj > (III)2, < yj > (IV)2 W(M)[

i,ki=k

t0

ds e|bik(s)|

< t s >d/2

+(M0M1+ M21+ M

r12 )

1(t)],

or taking into account (2.14), (2.15),

(2.33)< yj >

(III)2, < yj > (IV)2 W(M)[ + M0M1+ M21+ Mr12 ]

1(t).

Combining (2.28), (2.30), (2.33), one obtains

M1 W(M)[N+ 2 + M0M1+ M

21+ M

r12 ].

Changing if necessary the coefficient function Wone can simplify this inequality:

(2.34) M1 W(M)[N+2 + Mr12 ].


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16

2.7. Closing of the estimates. Here we derive a Lm estimate of which willclose the system of the inequalities for the majorants. To estimate Lm - norm ofwe use representation (2.7). By (2.24), (2.27),

Nm W(M)[k,ik=i

ec|bik(t)|+ M21+ Mr12 )1(t)].

As a consequence,

(2.35) M2 W(M)[N+12 + M1].

Here we have made use of the inequality t0

dsec|bik(s)|

|t s|2W(M)

12ik< t tik >2

,

which is an immediate consequence of (2.14), (2.15).Combining (2.18), (2.34), (2.35) one gets

(2.36) M1, M2 W(M)(N+3), M0 W(M)(N

2 +23),

3 = min{12 , 2, 1 2} > 0, the coefficient functions W(M) being independent

of t1. These inequalities mean that for N and sufficiently small M can belongeither to a small neighborhood of zero or to some domain whose distance from zerois bounded from below uniformly with respect to N, . Since Ml are continuousfunctions of t1 and for t1 = 0 are small only the first possibility can be realized.This means that for N and in some finite vicinity of zero,

M1

(t), M2

(t) c(N+3), M0

(t) c(N2 + 23), 0 t t1

.

The constantc here is independent ofN,,t1. Sincet1 is arbitrary these estimatesare valid, in fact, for all t 0. More precisely, one has

(2.37)M0(t) c(N

2 + 23), M1(t) c(N+ 3)1 (t), M2(t) c(N+

3)2 (t),

where (t) is the weight function corresponding to t1=:

(t) =< t >1 +j,kj=k

< t tjk >1,

tjk = 0 ift0jk 0, and tjk

0

dsvjk (s) v

0jk

|v0jk |2

=t0jk ,

ift0jk >0.

By (2.15), (2.16), the estimates (2.36) imply the existence of the limit trajectories+j (t) = (+j (t), E+j , b+j (t), v+j ), j = 1, . . . , N ,

b+j (t) =v+j t +b+j , v+j =v0j+

0

dsvj(s),


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17

b+j =b0j +

0

ds(cj(s) + vj (s) v+j ),

+j (t) = (E+j|v+j |

2

4 )t + +j , E+j =E0j +

0

dsEj(s),

+j =0j+

0

ds

Ej E+j +|vj v+j |

2

4 +j

1

2vj cj

.

Obviously, as t +,

|Ej (t) E+j |, |vj (t) v+j |= 0(t21+1),

|bj (t) b+j (t)|,|j (t) +j (t)| =O(t21+2).

Appendix 1

Here we outline the arguments needed for the proof of the existence of a decom-position (2.1) satisfying (2.2) for all t 0. We begin with the following lemma.Given N solitons w(0j ), 0j = (0j , E0j , b0j , v0j ), j = 1, . . . , N , we define theeffective coupling parameter(0),0= (01, . . . , 0N),

(0) = maxj=k

(|v0jk | + |b0jk |)

1.

For Lp(Rd), = (1, . . . , N), j = (j, Ej , bj , vj ) R A R

d Rd,j = 1, . . . , N , consider the functionalsFj,l(, ; 0), j = 1, . . . , N , l= 0, . . . 2d + 1,

Fj,l(, ; 0) = +N

k=1

w(0k) w(k), 3l(j) ,where

w=

w

w

, l(x, ) =e

i3+ixv2 3l(x b, E), = ( ,E ,b ,v).

Set Fj = (Fj,0, . . . , F j,2d+1), F = (F1, . . . , F N).

Lemma A1.1. LetE0j A0, j = 1, . . . , N . There exist constantsn0> 0, 0> 0,K >0, depending only onE0j , j = 1, . . . , N such that if(0) 0 andp n0then the equation

F(, ; 0) = 0

has a unique solution= (1, . . . , N), being aC1 function of, that satisfies

(A1.1) |j b0j+1

2(vj v0j ) b0j | + |Ej E0j | + |bj b0j | + |vj v0j | Kp.

Remark. It follows directly from (A1.1) that(i) for some constantK1

+N

k=1

w(0k) w(k)p K1p,


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18

(ii) if for some pair (j, k),t0jk 0< r0jk > then the new collision time tjk =

bjk vjk|vjk|2

satisfies a similar estimate with a constant = 0(1 + O(p).

Proof of Lemma A1.1. Let us pass from to a new system of parameters =(1, . . . , N),

j = (j 0j +1

2(vj v0j ) b0j , Ej , bj b0j , vj v0j ).

We representF(, ; 0) as the sum

F =F0 +F1 +F2,

F0j = (j , E0j ), = (0, . . . 2d+1),

l(, E) =

0(E) 0(), l()

,

F1j,l= k, k=j 0(0k) 0(k), l(j )

.

At last,

F2j,l = Gl(j , fj ), (x) =ei0j+i

v0j x

2 fj (x b0j ),

Gl(, f) =< f , 3l()> is aC1 function off and .

The direct calculations give

(A1.2) | det (, E)|

=(0,E,0,0)= e2(E)n2d(E).

Set0= (01, . . . , 0N), 0j = (0, E0j , 0, 0). By (A1.2),

(A1.3) | det F0|=0 =

N

j=1

e2(E0j )n2d(E0j )

is nonzero ifE0j A0,j = 1, . . . , N .

Consider F1. It is not difficult to check that for in some finite vicinity of0the derivative F

1 satisfies the inequality

(A1.4) |F1| C (0),

constantCdepending only on E0j .By the implicit function theorem, the desired result is a direct consequence of

(A1.3), (A1.4). To prove the existence of a decomposition (2.1) satisfying (2.2) for all t >0 we

use some standard continuity type arguments. Since C(R H1) there exists

a small interval [0, t1] where the constructions of lemma A1.1 can be used. Thisleads to a representation (2.1) satisfying the orthogonality conditions for t [0, t1].For the components of such a representation estimates (2.15), (2.36) give

|E0j E| C(N2 + 23), (|vjk | + |bjk |)

1 C ,

(t)m C(N+ 3),

which allows us to extend decomposition (2.1), (2.2) on a larger interval [0, t1+ t2]with some t2 > 0. On this new interval the same estimates hold, so one cancontinue the procedure with steps of the same length t2. As a result, one gets adecomposition (2.1) satisfying (2.2) for all t 0.


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Appendix 2

Here we prove lemma 2.1. Consider the equation

(A2.1) it = L(t), L(t) = ( + E)3+ V(t) + i[P(t), P(t)],

V(t) =T(t)V(E)T(t), P(t) =T(t)P(E)T(t),

where T(t) = B(t),a(t),0. We denote the corresponding propagator by U(t, ).Clearly,

UAj (t, ) =B0j(t)U(t, )

=j, a=aj,E=E0jB0j().

We shall assume that for some positive constants n, R, 1,

(A2.2) |(t)| + |a(t)| n,

(A2.3) |(t)| + |a(t)| L

l=0

< R(t tl)>21 ,

t R+. Here L N, 0 =t0< t1< < tL. One has the following lemma.

Lemma A2.1. For anyx0, x1 Rd, t 0,0,

x b00 U(t, )P()f2 Ct

d/2 x x10 f2,

provided n is sufficiently small andR is sufficiently large: n +R1 C.

In this appendix we use Cas a general notation for constants that depend onlyon M,,Eand can be chosen uniformly with respect to E in compact subsets ofA0.

It follows from (2.12), (2.14), (2.15), (2.16) that for M in some finite vicinity ofzero the functions j , aj satisfy assumptions (A2.2), (A2.3) with 1 = 21 2, tl,l = 1, . . . , L, being the collision times tik, i, k = 1, . . . N, i = k. So, lemma A2.1implies lemma 2.1.

Proof of lemma A2.1. Lemma A2.1 follows from proposition 1.1 by a simpleperturbation argument. On the intervals [tl, tl+1], l= 0, . . . L 1 we introduce thefollowing linear approximations l(t), al(t) of(t), a(t):

l

(t) =(t) t

tlds s

tlds1

1 (

s1 tltl+1 tl )

(s1)

tl+1t

ds

tl+1s

ds1( s1 tltl+1 tl

)(s1),

al(t) =a(t)

ttl

ds

stl

ds1

1 (

s1 tltl+1 tl

)

a(s1)

tl+1t

ds

tl+1s

ds1( s1 tltl+1 tl

)a(s1).


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Here C(R), () =

1 for|| 14 ,

0 for|| 34 .For t [tL, ) we define the corresponding

L+1(t), aL+1(t) as follows.

L+1(t) =(t)

t

ds

s

ds1(s1),

aL+1(t) =a(t)

t

ds

s

ds1a(s1).

Clearly, for t [tl, tl+1], l= 0, . . . , L, tL+1=, one has

(A2.4) |(t) l(t)|, |a(t) al(t)| C R2,

(A2.5) |dl

dt

|, |dal

dt

| C(n + R1).

On the interval [tl, tl+1] one can pick out the leading term of (A2.1) in the form

(A2.6) it =Ll(t), Ll(t) = ( + E)3+ V

l(t),

Vl(t) =Tl(t)V(El)Tl

, Tl(t) =Bl(t),al(t),rl,

l(t) =l(t) rl al(t)

2 , rl =

dal

dt, El =E+

dl

dt

|rl|2

4 .

We denote the propagator corresponding to (A2.6) by Ul(t, ). Clearly,

Ul

(t, ) =Tl

(t)ei(t

)L(El)

Tl

(), Pl

(t)Ul

(t, ) =Ul

(t, )Pl

(),

where Pl(t) =Tl(t)P(El)Tl

(t).Consider the expression(t) U(t, )P()f,tl < tl+1.For tl t tl+1 we write (t) as the sum = h+k, h(t) = P

l(t)(t). Since(t) =P(t)(t), the 2d + 2 dimensional componentk is controlled by h:

e|xa(t)|k(t)2 C(|(t) l(t)| + |a(t) al(t)| + |rl| + |E El|)e|xa(t)|(t)2

(A2.7) C(R1 + n)e|xa(t)|(t)2

for some > 0, provided n, R1

are sufficiently small. In the last inequality weused (A2.4), (A2.5).

For h one can write the following integral representation

(A2.8) h(t) =Pl(t)h0(t) i

t

dsPl(t)Ul(t, s)[Vl(s)h0(s) + Rl(s)(s)],

whereh0(t) =e

i(E)(t)3P()f,

Rl(t) =V(t) Vl(t) +i[P(t), P(t)].


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Obviously,

|V(x, t) Vl(x, t)| C|(t) l(t)| + |a(t) al(t)| + |rl| + |E El|)e|xa(t)|

(A2.9) C(R1 + n)e|xa(t)|,

(A2.10) |[P(t), P(t)]f| C ne|xa(t)|e|xa(t)|f2.

Estimates (A2.7), (A2.9), (A2.10) and representation (A2.8) together with propo-sition 1.1 imply immediately that for tl t tl+1 and for any R thefollowing inequality holds

x a(t)0 (t)2

< t + >d/2

(A2.11) C supst

x a(s)0 eit3(s)P()f2< s + >d/2

,

where Cdo not depend on . (A2.11) implies in particular, that

(A2.12) x a(t)0 U(t, )P()f2 C < t >d/2 x x00 f2,

x0 Rd, tl t tl+1, l = 0, . . . , L.To prove that this estimate is in fact true for any 0 t we use the induction

arguments. Assume that one has (A2.12) for t tl < . We need to showthat then the same is true for tl < t tl+1. For t (tl, tl+1] we writeU(t, )P()f=U(t, tl)U(tl, )P()f. Using (A2.12) and the representation

U(t, )P()f=ei(t)(E)3P()f

(A2.13) i

t

dsei(ts)(E)3 (V(s) +i[P(s), P(s)]) U(s, )P()f,

one checks easily that

< x a(t)>0 ei3(ttl)U(tl, )P()f2 C < t >d/2 x x00 f2.

By (A2.11), this implies that (A2.12) is valid for 0 t tl+1 and thus, forany 0 t. Moreover, by (A2.13) one can replace a(t) in the left hand side of(A2.12) by any x1 Rd:

x x10 U(t, )P()f2 C x x0

0 f2, 0 t.


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Appendix 3

Here we prove lemma 2.2. We start by proving a similar result for the freeoperatorsT0jkl (t, ):

T0jkl (t, ) =Aj (t)

t

dU0(t, )Vk()U0(, s)Ai().

Lemma A3.1. Fori=k, t 0, one has

(A3.1)

t0

dT0jki (t, ) W(M)1ik

with some1> 0.

Proof. SinceT0jkl (t, ) C < t >

d/2,

one has

(A3.2) I(t) =

t0

dT0jki (t, ) C t

< t >.

In this appendix the constants Cdepend only on E0k.For t 2, where is a small positive number, we write the integral I(t) as a

sum of two terms I(t) =I0(t) + I1(t),

I0(t) =

t20

dTjki (t, ),

Tjki (t, ) =Aj (t)

t

+

dsU0(t, s)Vk(s)U0(s, )Ai(),

I1(t) being the rest. Obviously,

(A3.3) I1(t) C .

Consider I0(t). To estimate this expression we write T

jki (t, ) in the form

Tjki =

T11jki T

12jki

T21jki T22

jki

,

where

T11jki (t, ) =Aj(t)

t+

dsei(ts)V1k (s)ei(s)Ai(),

T12jki (t, ) =Aj (t)

t+

dsei(ts)V2k (s)ei(s)Ai(),

V1k (x, t) =V1(x bk, E0), V2k (x, t) =e

2ik(x,t)V2(x bk(t), E0),

(A3.4) T22jki (t, )f=T11

jki (t, )f , T21jki (t, )f=T

12jki (t, )

f


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B2jki (t, ) =

t+

ds(t s)d/2(s )d/2

Rd

dzei|xz+bj (t)bk(s)|

4(ts) e2i(z+bk(s),s)V2(z)e i|zy+bk(s)bi()|

4(s) ,

which implies

t20

T12jki (t, ) W(M)1d+2

t0

d

t

ds < t s >d/2+

< s >d/2+ |bjk (t)(s ) bik()(t s)|

(A3.7) W(M)1d+2ik.

Combining (A3.2), (A3.3), (A3.4), (A3.6), (A3.7) one obtains

I(t) W(M)( + 1d+2ik),

which leads immediately to (A3.1) with 1 2+d2 .

Let us introduce the operatorsT1jki (t, ):

T1jkl (t, ) =Aj (t)

t

dsU0(t, s)Vk(s)(I PA

k (s))U0(s, )Ai().

It is not difficult to check that for any 1,

Aj(t)U0(t, s)Vk(s)(I PA

k (s))U0(s, )Ai()

W(M)< t s >d/2< s >d/2+ .

As a consequence,(A3.8) t

0

dT1jkl (t, ) W(M)

t0

d < t >d/2+W(M)ik.

At the last step here we have used (2.14), (2.15).Proof of lemma 2.2. This lemma follows directly from (A3.1), (A3.8) and the

following representation

Tjkl (t, ) =T0jki (t, ) T

1jki (t, )

(A3.9) i

t

d

dsAj(t)U0(t, )Vk()PA

k ()UAk(, s)Rk(s)U0(s, )Ai()

(A3.10) i

t

dAj (t)U0(t, )Vk()PAk ()A

1k ()T

0kki(, )


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(A3.11)

t

d

dsAj (t)U0(t, )Vk()PAk ()U

Ak(, s)[Vk(s) + Rk(s)]A

1k (s)T

0kki(s, ).

We estimate the right hand side of this representation term by term. Using lemma2.1 and inequality (2.21) one gets

(A3.12)

t0

d(A3.9) W(M)M0(t).

Expression (A3.10) can be estimated as follows(A3.13) t

0

d(A3.10) W(M)

t0

d

t

d < t >d/2 T0kki(, ) W(M)ik.

In a similar way,

t0

d(A3.11) W(M) t0

d t

d

ds < t >d/2

(A3.14) < s >d/2 T0kki(s, ) W(M)ik.

Combining (A3.1), (A3.8), (A3.12), (A3.13), (A3.14) one gets lemma 2.2.

Appendix 4

Here we discuss the proof of proposition 1.1. Since only the weighted estimatesare needed, rather then follow [8, 37, 38] we use the approach of [16, 17, 18]. It turnsout that the arguments of [16, 17, 18] can be applied almost without modifications.

So, we describe only the main steps of the proof, referring the reader to [16, 17, 18]for most of the details.

We start be recalling briefly some basic properties of the free resolvent R0() =( +E )1 0

0 ( +E+ )1

. LetHt,s stand for the weighted Sobolev

spaces:Ht,s ={f, fHt,s < x >

s (1 )t/2f2< }.

We denote by B (Hs,t, Hs1,t1) the space of bounded operators from Hs,t to Hs1,t1 .Set Ls2 = H

0,s, B(Hs,t) = B(Hs,t, Hs,t). If s > 1 and t R the resolvent R0()which is originally defined as B(L2) valued analytic function of C\ (, E] [E, ) can be extended continuously to the C+ = {im 0} when consideredas a B(Hs,t, H

s,t+2) valued function. The following properties ofR

0() are well

known, see [16, 17, 18, 37, 38] and references therein.

Lemma A4.1. Let k = 0, 1, . . . . If s > k + 1/2, then the derivative R(k)0 ()

B(Hs,0, Hs,0) is continuous in C+ \ {E, E}, with

(A4.1) R(k)0 () =O(||

(k+1)/2),

in this norm as inC+.

The behavior ofR0() for close to E is described by the following lemma,see again [16,17,18].


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Lemma A4.2. As E, R0() admits the following asymptotic expansion inB(Hs,t, Hs,t+2).Form odd:

(A4.2) R0() =l

j=0

Gj,0( E)j +

lj=0

Gj,1( E)j+ 12 + O(( E)l+1),

form even:

(A4.3) R0() =l

j=0

Gj,0( E)j + ln( E)

lj=0

Gj,1( E)j +o(( E)l),

where l = 0, 1, . . . , s > C(l, d), the coefficients Gj,k belong to B(Hs,t, Hs,t+2),

Gj,1 = 0 for j < d3

2 if d is odd and for j < d2

2 if d is even. Representations(A4.2), (A4.3) can be differentiated with respect to any number of times.

Here ( E)1/2 and ln( E) are defined on the complex plane with the cutalong [E, ). The explicit expressions for the constantsC(l, d) can be found in [17,18, 19]. Similar expansions hold as E.

For [E, ), consider the operator

I+R0( +i0)V :Ls2 L

s2 ,

s >1.

Lemma A4.3. LetE A0. ThenKer(I+ R0( + i0)V) is trivial.

Proof. We start by the case = E. Let Ker(I+ G0V). This implies that belongs toL2(Rd)+< x >(d2) L(Rd) and satisfies

L= E .

Hypothesis H3 then allows us to conclude that = 0.We consider next the case > E. Let Ker(I+ R0(+ i0)V). Since V is

spherically symmetric, one can assume that (x) = f(r)Y(), r = |x|, = x|x| ,fL2(R+; r

d1 < r >2s dr) andY L2(Sd1),

Sd1Y =nY, n= n(d 2 + n),

for some n {0, 1, . . . }. Thenfhas to satisfy

(A4.4) lnf

(

d2

dr2

d 1

r

d

dr+ E+

nr2

)3+ V

f=f,

f(0) = 0 if n= 0, f(0) = 0 if n >0,

and as r ,

(A4.5) f=cr(d2)

2 H(1) (kr)

1

0

+ O(er ), >0,


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for some constantc. Herek = (E)1/2 >0, =n + (d2)2 ,H(1) is the first Hankel

function. Asymptotic representation (A4.5) can be differentiated with respect to rany number of times.

The Wronskian

w(f, g) =rd1(< f, g >R2 < f, g >R2)

does not depend on r iff and g are solutions of (A4.4). Calculating w(f, f) onegets

2ik|c|2 = 0,

which implies that L2. Since E A0, this means that = 0. Consider the full resolvent R() = (L )1. R()P (R()) is a B(L2)

valued holomorphic (meromorphic with the only pole in zero) function of C \ (, E] [E, ). R() satisfies the relations

(A4.6) 1R()1= R().

The analytic properties ofR() near the cuts (, E], [E, ) are collected inthe two following lemmas. In both of them we assume that E A0.

Lemma A4.4. For s > 1, R()P can be extended continuously to C+ as a

B(Ls2, Ls2 ) valued function. Moreover, if s > k +

12 then R

(k)()P exists and

continuous for C+ \ {E, E} and

(A4.7) R(k)()P =O(||(k+1)/2)

inB(Ls2, Ls2 ) as inC

+.

Lemma A4.5. As E, R() admits the following asymptotic expansion in

B(L

s

2, Ls

2 ).

Form odd:

(A4.8) R() =l

j=0

Bj,0( E)j +

l1j=0

Bj,1( E)j+ 12 +O(( E)l),

form even:

(A4.9) R() =l

j=0

k=0

Bj,k( E)j (ln( E))k + +o(( E)l),

where l = 0, 1, . . . , s > C(l, d), Bj,k B(Ls2, L

s2 ), Bj,k = 0 for k = 1, j 2jd2 if d is even. Representations (A4.8), (A4.9) can bedifferentiated with respect to any number of times.

These results is a standard consequence of the corresponding properties of thefree resolvent (lemmas A4.1,2) and lemma A4.3, see [16, 17, 18].

Consider the propagator eitL. Lemma A4.4, together with (A4.6), (A4.8),(A4.9) allows us to represent the expression

eitLP f , g

, f, g C0 (R

d) in the

form

(A4.10)

eitLP f g

=

E

d[eit E()f, g eit E()1f, 1g],


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28

where

E() = 1

2i(R( + i0) R( i0)).

It follows from (A4.8), (A4.9) that as E,E() admits the following asymptotic

expansion in B(Ls2, Ls2 ) with s sufficiently large.For d odd:

(A4.11) E() =E0( E)d22 + O(( E)

d2 ),

for d even:

(A4.12) E() =E0( E)d22 +

O (ln( E)( E)2) if d= 4,

O(( E)d2 ) if d 6.

E0 B(Ls2, L

s2 ). These expansions can be differentiated with respect to any

number of times.Combining (A4.10), (A4.7), (A4.11), (A4.12) one gets immediately [18]

< x >s eitLP f2 C < t >d/2 < x >s f2,

provided s is sufficiently large. To recover proposition 1.1 it is sufficient now toinject this inequality in the following representation for eitLP

eitLP = P eitL0 i t0

dsei(ts)L0P V eisL0

t0

ds t

sdei(t)L0V ei(s)LP V eisL0 .

Acknowledgement. It is a pleasure to thank F.Nier for numerous helpful discus-sions.

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Asymptotic Stability of Multi-soliton Solutions