On asymptotic stability of solitary waves for Schödinger equation coupled to nonlinear oscillator, II

The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator; mathematically the system under study is a nonlinear Schrödinger equation, whose nonlinear term includes a Dirac delta. The coupled system is invariant with respect to the phase rotation group U(1). This article, which extends the results of a previous one, provides a proof of asymptotic stability of solitary wave solutions in the case that the linearization contains a single discrete oscillatory mode satisfying a non-degeneracy assumption of the type known as the Fermi Golden Rule. 1 Supported partly by Alexander von Humboldt Research Award, by FWF grant P19138-N13, and DFG grant 436 RUS 113/929/0-1. Supported partly by RFBR grants 06-01-00096 and 07-01-00018a, and RFBR-DFG grant 08-01-91950NNIOa. Partially supported by EPSRC grant A00133/01

1 Introduction and statement of results 1

.1 Introduction
In this article we continue the study of large time asymptotics for a model U(1)-invariant nonlinear Schrödinger equation iψ(x, t) = −ψ ′′ (x, t) − δ(x)F (ψ(0, t)), x ∈ R, (1.1) which was initiated in [1]. Here ψ(x, t) is a continuous complex-valued wave function and F is a continuous function, the dots stand for the derivatives in t and the primes in x; all derivatives and the equation are understood in the distribution sense. Our main focus is on the role that certain solitary waves (also referred to as nonlinear bound states, or solitons) play in the description of the solution for large times. These solitary waves are solutions of the form e iωt ψ ω (x), where ψ ω solves a nonlinear eigenvalue problem (1.10). In [1] the asymptotic stability of these solitary waves was proved under a condition on the nonlinearity which ensures that the linearization about the solitary wave consists entirely of continuous spectrum, except for the two dimensional generalized null space which is always present due to the U(1) symmetry of the equation. In this article this result is extended to the case that the spectrum of the linearization includes an additional discrete component, which satisfies a non-degeneracy condition related to the Fermi Golden Rule. In order to explain these results fully we will introduce our conditions on the nonlinearity F , in the remainder of this section. In the following section we will discuss the basic properties of the solitary waves. We will then be able to state our main theorem precisely as theorem 1.3. For a more lengthy discussion of our motivation, and of previous results in the literature ( [2,6,7,8,10,11]) we refer the reader to the introduction of [1]. It will be convenient to rewrite (1.1) in real form: we identify a complex number ψ = ψ 1 +iψ 2 with the real two-dimensional vector (ψ 1 , ψ 2 ) ∈ R 2 and rewrite (1.1) in the vectorial form jψ(x, t) = −ψ ′′ (x, t) − δ(x)F(ψ(0, t)), (1.2) where and F(ψ) ∈ R 2 is the real vector version of F (ψ) ∈ C. We assume that the oscillator force F admits a real-valued potential which is conserved for sufficiently regular finite energy solutions. We assume that the potential U(ψ) satisfies the inequality U(z) ≥ A − B|z| 2 with some A ∈ R, B > 0. (1.6) We also assume that U(ψ) = u(|ψ| 2 ) with u ∈ C 2 (R). Therefore, by (1.4), where a(|ψ| 2 ) is real.Then F (e iθ ψ) = e iθ F (ψ), θ ∈ [0, 2π], and e iθ ψ(x, t) is a solution to (1.1) if ψ(x, t) is. Therefore, equation (1.1) is U(1)-invariant and the Nöther theorem implies the conservation of the following charge: Under these conditions the existence of global solutions to the Cauchy problem for (1.1) was proved in [4]. We work in the space H 1 (R) = H 1 , the Sobolev space of complex valued measurable functions with (|ψ ′ | 2 + |ψ| 2 dx < ∞, and C b (R, X) is the space of bounded continuous functions R → X into a Banach space X. Theorem 1.1 ([4]). Under conditions (1.4), (1.6) and (1.7), the following statements hold. i) For any ψ 0 (x) ∈ H 1 there exists a unique solution ψ(t) = ψ( · , t) ∈ C b (R, H 1 ) to the equation (1.1) with initial condition ψ(x, 0) = ψ 0 (x).
ii) The charge Q(ψ(t)) and Hamiltonian H(ψ(t)) are constant along the solution. iii) There exists Λ(ψ 0 ) > 0 such that the following a priori bound holds: (1.9) Next, in §1.2 we describe all nonzero solitary waves, and then formulate the main theorem in §1.3.

Solitary waves
Equation (1.1) admits finite energy solutions of type ψ ω (x)e iωt , called solitary waves or nonlinear eigenfunctions. The frequency ω and the amplitude ψ ω (x) solve the following nonlinear eigenvalue problem: (1.10) It is straightforward to check (see [1]) that the set of all nonzero solitary waves consists of functions ψ ω (x)e iθ , ψ ω (x) = C(ω)e − √ ω|x| , C > 0, ω > 0, where √ ω = a(C 2 )/2 > 0, and where θ ∈ [0, 2π] is arbitrary. This condition means that C is restricted to lie in a set which, in the case of polynomial F , is a finite union of one-dimensional intervals. Notice that C = 0 corresponds to the zero function ψ(x) = 0 which is always a solitary wave as F (0) = 0, and for ω ≤ 0 only the zero solitary wave exists. The real form of the solitary wave is e jθ Ψ ω where Ψ ω = (ψ ω (x), 0). We will also need the following lemma from [1]: For C > 0, a > 0 we have Linearization at the solitary wave e jθ Ψ ω leads to the operator where P 1 is the projector in R 2 acting as χ 1 χ 2 → χ 1 0 , (see [1]). Let C = j −1 B. The continuous spectrum of C coincides with C + ∪ C − where C + = [iω, i∞), and C − = (−i∞, −iω]. The discrete spectrum depend is described in [1]. Zero is always present in the discrete spectrum on account of the circular symmetry of the problem, and there is a corresponding generalized eigenspace of dimension at least two. If a ′ > a/C 2 there is a positive eigenvalue corresponding to linear instability of the solitary wave, while for a ′ < a/C 2 the discrete spectrum consists either only of zero, or contains in addition two pure imaginary eigenvalues. The presence of such imaginary discrete spectrum corresponds to the possibility of a periodic pulsation of the solitary wave at the linearized level, a possibility which has to be taken care of in the proof of asymptotic stability.

Statement of the main theorem
Previously, in [1], we considered the case when a ′ ∈ (−∞, 0) ∪ (0, a √ 2C 2 ), (1.12) in which case the operator C has no discrete spectrum except zero. Under this condition we proved asymptotic stability for initial data close to a solitary wave both in the energy norm and in the weighted Banach norm, L p β , defined by, (1.13) In the present paper we extend this understanding to allow for the presence of additional discrete spectrum in the linearization: to be precise we will consider the case when ). (1.14) In this case, there are, in addition to zero, 2 simple pure imaginary eigenvalues ±iµ, which satisfy the property 2µ > ω. If assumption (1.14) is true for a fixed value ω 0 , it also true for values of ω in a small interval centered at ω 0 . Let u(x, ω) = u 1 u 2 be the eigenvector of C associated to iµ. We choose the function u 1 (x) to be real, in which case u 2 (x) is purely imaginary. Then u * := u 1 −u 2 , is the eigenvector associated to −iµ (see appendix A). We consider the initial value ψ(x, 0) = ψ 0 (x) to be of the form where f 0 belongs to the eigenspace associated to the continuous spectrum of C(ω 0 ). We assume that z 0 and f 0 are sufficiently small, and also assume a non-degeneracy condition which we now explain. Let ·, · denote the Hermitian scalar product in L 2 of C 2 -valued functions, and be the quadratic terms coming from the Taylor expansion of the nonlinearity: The non-degeneracy condition has the form where τ + (2iµ 0 ) is the eigenfunction associated to 2iµ 0 = 2iµ(ω 0 ). This condition should be thought of as a nonlinear version of the Fermi Golden Rule of quantum mechanics ([9, Section XII.6] or [5]), and will be referred to simply as the Fermi Golden Rule; see [10,11] for the its introduction into nonlinear evolution equations. In appendix E we express (1.17) in terms of C and a(C 2 ), and hence show that the Fermi Golden Rule holds generically for polynomial nonlinearity.
Our main theorem is following: Assume further that the spectral condition (1.14) and the non-degeneracy condition (1.17) hold for the solitary wave with C = C(ω 0 ) = C 0 . Then for ε sufficiently small the solution admits the following scattering asymptotics in C b (R) ∩ L 2 (R): are the corresponding asymptotic scattering states and ϕ ± (t) = ω ± t + p ± log(1 + k ± t) + κ ± , for some constants ω ± , p ± , k ± , κ ± .
The asymptotics (1.19) can be rewritten in terms of the original complex notation as: where W (t) is the dynamical group of the free Schrödinger equation, and φ ± = Φ 1 ± + iΦ 2 ± (Φ k ± , k = 1, 2, being the components of the vector-function Φ ± ). Thus the main conclusion is that asymptotically the dynamics decomposes into a nonlinear bound state e iϕ ± (t) ψ ω ± (undergoing uniform phase rotation, modulo the logarithmic phase shift in ϕ ±(t) ) and a solution of the free Schrödiniger equation The proof is divided into the following three main steps which are carried out, respectively, in §3, §4 and §5.
Step 1 To decompose the solution ψ(t) according to the spectral decomposition of the operator C given in §2.1, and obtain a system of equations in §3.1 equivalent to (1.1). This system is then put into a canonical form, in which certain non-resonant terms are excluded. This is carried out in §3.4, the final form of the equations being given in §3.4.5.
Step 2 To use the time decay for the linearized evolution on the continuous spectral subspace established in §2.2 to prove asymptotic stability of the solitary waves in §4.6.
Step 3 Finally, to construct the wave operator and obtain the scattering asymptotics (1.19), in §5.2.

Linearization
In this section we summarize the spectral properties of the operator C which appears in the linearization of (1.2) about a soliton, and then give some estimates for the linearized evolution. The proof of these properties can be found in [1], with the exception of proposition 2.3 which is proved in appendix C.

Spectral properties of the linearization
The linearized equation readṡ The resolvent R(λ) := (C − λ) −1 is an integral operator with matrix valued integral kernel R(λ, x, y) = Γ(λ, x, y) + P (λ, x, y), (2.5) As explained already in §1.2, the continuous spectrum consists of C + ∪ C − , and correspondingly k ± (λ) = √ −ω ∓ iλ is (respectively) the square root defined with a cut in the complex λ plane so that k ± (λ) is analytic on C \ C ± and Im k ± (λ) > 0 for λ ∈ C \ C ± . The constants α, β and D = D(λ) are given by the formulas In addition to this continuous spectrum, there is discrete spectrum, which appears in this formalism as the set of poles of R(λ); these poles in turn correspond to the roots of the determinant D(λ). From the analysis in [1] we know that if a ′ ∈ (a/ √ 2C 2 , a/C 2 ), then the determinant has the following roots: λ = 0 with the multiplicity 2 and two pure imaginary roots Note that spectral condition (1.14) is more restrictive. It implies in addition that 2µ > ω, as can be verified by a simple computation. The generalised null space X 0 of the non-self-adjoint operator C is two dimensional, is spanned by jΨ ω , ∂ ω Ψ ω , and CjΨ The symplectic form Ω for the vectors ψ and η is defined by By Lemma 1. 2 Hence, the symplectic form Ω is nondegenerate on X 0 , i.e. X 0 is a symplectic subspace. There exists a symplectic projection operator P 0 from L 2 (R) onto X 0 represented by the formula Remark 2.1. Since jΨ ω , ∂ ω Ψ ω lie in H 1 (R) the operator P 0 extends uniquely to define a continuous linear map H −1 (R) → X 0 . In particular this operator can be applied to the Dirac measure δ(x).

Estimates for the linearized evolution
We now recall from [1] some estimates on the group e Ct which will be needed in §4. First we recall a bound for the action of e Ct on the Dirac distribution δ = δ(x) for small t. Lemma 8.1 from [1] gives the following small t behaviour: Second we list the large time dispersive estimates. To do this let us introduce, for β ≥ 2, a Banach space M β , which is the subset of distributions which are linear combinations of L 1 β functions and multiples of the Dirac distribution at the origin with the norm: and a Banach space Proposition 2.2. (see [1]) Assume that the spectral condition (1.14) holds. Then for h ∈ M β with β ≥ 2 the following bounds hold: 14)

15)
where Π + (resp. Π − ) is the spectral projection operator onto the spectral subspace associated to C + (resp. C − ), the positive (resp. negative) part of the continuous spectrum.
We shall also need the following bound, which is new.
Assume that the spectral condition (1.14) holds. Then for h ∈ M β with β > 2 and t > 1 the following bounds hold: We prove this proposition in appendix C.

Spectral decomposition and canonical forms
In this section we will use the spectral resolution associated to the operator C to decompose the solution ψ, obtaining evolution equations for the different spectral components. Then, following [2], we introduce normal coordinate transformations to transform the evolution equations into simpler canonical forms in which certain non-resonant terms are absent. The final form of these equations is given in §3.4.5.

Modulation equations
In this section we present the modulation equations which allow a construction of solutions ψ(x, t) of the equation (1.1) close at each time t to a soliton i.e. to one of the functions in the set S described in §1.2 with time varying ("modulating") parameters (ω, θ) = (ω(t), θ(t)).
We look for a solution to (1.2) in the form Since this is a solution of (1.2) as long as χ ≡ 0 withθ = ω andω = 0 it is natural to look for solutions in which χ is small and with γ treated perturbatively. We look for χ = w(x, t) + f (x, t) where w = zu + zu * ∈ X 1 and f ∈ X c . Now we give a system of coupled equations for ω(t), γ(t), z(t) and f (x, t).
Lemma 3.1. Given a solution of (1.2) in the form (3.1) with f ∈ X c as just described, the functions ω(t), γ(t), z(t) and f (x, t) satisfy the systeṁ

Frozen spectral decomposition
The linear part of the evolution equation (3.5) for f is non-autonomous, due to the dependence of the operator C on ω(t). In order to make use of the dispersive properties obtained in §2.2, it is convenient (following [2]) to introduce a small modification of (3.5), which leads to an autonomous linearized equation. Let us fix an interval [0, T ] and decompose f (t) ∈ X c t into the sum Here X d T = P d T X is the spectral space associated to the discrete spectrum at time T and X c T = P c T X is the spectral space associated to the continuous spectrum at time T , P c T = P c (ω(T )) and P d T = I − P c T . In the following, we denote ω T = ω(T ) and C T = C(ω T ). We will obtain estimates uniform in T , and later consider the limit T → +∞.
We now introduce a shorthand for the bounds we are about to prove: R(A, B, . . . ) (resp. R(ω, A, . . . )) is a general notation for a positive function which remain bounded as A, B, . . . approach zero (resp. if ω is close to ω 0 and A, . . . approach zero); it could be unbounded and even infinite if ω is outside some vicinity of ω 0 . The formula f = Rg implies that |f | ≤ Rg. Introducing also the notation R 1 (ω) = R( ω − ω 0 C[0,T ] ), we get Lemma 3.2. The function g is estimated in terms of h as follows: Proof. Let us use the identities T is a"small" finite dimensional operator: Applying the projection P c T to (3.5), we geṫ

Asymptotic expansion of dynamics
The preceding sections have provided a change of variables ψ → (ω, γ, z, h) under which (1.2) is mapped into the system comprising (3.2),(3.3), (3.4) and (3.8). Since we are interested in proving that for large times z, h are small it is necessary to expand the inhomogeneous terms in these equations in terms of z, h. This is carried out in this section, leading to the conclusion that the system (3.2),(3.3), (3.4) and (3.8) can be written in more detail as the system comprising (3.22),(3.29), (3.33) and (3.41).

Preliminaries
This section is devoted to some useful preliminary estimates. We start with a bound for the We also need to expand the nonlinear term F(ψ) = a(|ψ| 2 )ψ near the solitary wave since the inhomogeneous terms all involve E[χ], the nonlinear part of δ(x)F , defined using the Taylor expansion of δ(x)Fψ near Ψ: Thus E[χ] contains all the higher order terms which are at least quadratic in χ, as χ → 0, and where E j is of order j in χ and E R the remainder. It is easy to check that ) as a symmetric bilinear (resp. trilinear) form Here summation is taken over all transposition of integers 1, 2, 3. Notice also that In the remaining part of the paper we shall prove the following asymptotics: To justufy these asymptotics, we will separate leading terms and remainders in right hand side of equations (3.2)-(3.4), (3.8). Namely, we shall expand the expressions forω, γ andż up to and including terms of the order O(t −3/2 ), and forḣ up to O(t −1 ) keeping in mind the asymptotics (3.19). This choice is necessary for applicaton of the method of majorants.

The equation for γ
Using again the equality Q = jE we get

The equation for z
Denote κ = u, ju and rewrite (3.4) in the form:

Equation (3.31) can be represented in the forṁ
where, using the calculations in the previous two sections, we have in particular,

The equation for h
We now turn to equation (3.8) forḣ that we rewrite in the forṁ where remainder H R is For the H R we have, recalling (2.12), the following estimate Now we continue the isolation the leading terms in the right hand site of (3.35). Note that, from the formulae in the discussion surrounding (1.11), with σ(t) = ω − ω T +γ, and Using the identity P c T = 1 − P d T , we obtain Next we need an additional construction to combine first two terms in RHS of (3.38). Namely, lemma 3.5 below shows that the "main part" of the second term is iσ(t)(Π + T −Π − T )h, where Π + and Π − are the spectral projection operators on the spectral space associated to the positive and negative part of the continuous spectrum respectively at time T ; see the discussion preceding (2.14). Hence, we denote This lemma is proved in appendix D. Lemma 3.5 and the bound (3.39) imply Proposition 3.6. The remainderH R admits the bound

Canonical form of the equations
Our goal is to transform the evolution equations for (ω, γ, z, h) to a more simple, canonical form. We will use the idea of normal coordinates, trying to keep unchanged the estimates for the remainders as much as is possible. This means we observe that for our purposes the unknowns (ω, γ, z, h) lie in a neighbourhood of the point such that Θ is a diffeomorphism between neighbourhoods of (ω 0 , 0, 0, 0) in the space ∈ R × R × C × L ∞ −β , and DΘ(ω 0 , 0, 0, 0) is the identity. This map Θ is obtained, as usual in the normal form method, by looking for (ω 1 , γ 1 , z 1 , h 1 ) as a power series in (ω, γ, z, h), starting with the identity map at highest order -see (3.47)-(3.49),(3.57),(3.73) and (3.82) for the detailed expressions. The coefficients in these expressions are then chosen so as to put the equations for (ω 1 , γ 1 , z 1 , h 1 ) in a simpler canonical form which is suitable for our subsequent work. This final canonical form for the system is summarised in section 3.4.5.

Canonical form of the equation for h
As a starting point we expand out the middle term on the right hand side of (3.41), obtaininġ Here, the coefficients H ij are defined by We want to extract from h the term of order z 2 ∼ t −1 . For this purpose we expand h as with some a ij ≡ a ij (ω, x) satisfying a ij = a ij , and Note that k 1 is just the solution of the corresponding homogeneous equationk (3.48) such that the equation for h 1 has the forṁ whereĤ R =H R + H ′ , with estimates as in (3.44), and also Proof. (cf. Section 4.2.2 in [2]) We substitute (3.48) into (3.41) and equate the coefficients of the quadratic powers of z. In addition we replace the discrete eigenvalue µ(t) by its value at time T , i.e. µ T = µ(ω(T )), and include the correction in the remainder. Then we get The dependency in x appears here through the coefficients a ij = a ij (ω, x). Notice, from (1.16) and the formuae for the projection operators in §2.1, that each H ij ∈ X c T is the sum of a multiple of δ(x) and a function exponentially decreasing at infinity. Hence, there exists a solution a 11 in the form where C −1 T stands for regular part of the resolvent R(λ) at λ = 0 since the singular part of R(λ)H 11 vanishes for H 11 ∈ X c T . The function a 11 is exponentially decreasing at infinity. For a 20 and a 02 we choose the following inverse operators: This choice is motivated by proposition 2.3, and putting t = 0 in that proposition we have the bound (3.51). The remainder H ′ can be written as

Canonical form of the equation for ω
We want to remove all terms in the right hand side of (3.22) except the remainder Ω R . This is possible by methods of Buslaev and Sulem [2, Proposition 4.1] since Ω 11 = 0 by (3.25).
It should be noted that the resonant term zz in the equation forω 1 is absent. From the first equation of (3.60) we obtain Multiply the second equation of (3.60) by j we get jΩ ′ 10 + iµjb ′ 10 − Cjb ′ 10 = 0 since jC * = −Cj. Without loss of generality we can assume that jΩ ′ 10 ∈ X c t by Remark 3.4. Therefore, there exists the solution b ′ 10 in the form where (C − iµ) −1 stands for regular part of the resolvent R(λ) at λ = iµ since the singular part of R(λ)jΩ ′ 10 vanishes for jΩ ′ 10 ∈ X c T . The functions b ′ 10 , b ′ 01 decrease exponentially at infinity, and the equations for b 21 = b 12 , b 30 = b 03 can be easily solved.

Canonical form of the equation for z
In this section we obtain a canonical form of the equation (3.33) for z, and carry out a computation of the coefficient of the resonant "z 2 z " term, which gives the Fermi Golden Rule. Substituting (3.6) and (3.47) into (3.33) and putting the contribution of g + h 1 + k 1 in the remainderZ R , we obtaiṅ For the coefficient κ = κ(ω) we get (see [2,  for ω in some vicinity of ω 0 .
Proof. We first notice that the coefficient where we denote Using that P c T commutes with R(2iµ + 0), we have R(2iµ + 0)P c T = P c T R(2iµ + 0)P c T . We have also that (P c T ) * = −jP c T j, hence Hence by the Cauchy residue theorem we have Now we use the representation where D = D(λ + 0) and since the function α is even. Using that where p.v. is the Cauchy principal value, we have The integral terms in (3.71) is real. Thus, The non-degeneracy condition (1.17) implies that τ + (2iµ T ), E 2 [u, u] = 0 in some vicinity of ω 0 . Using also the inequality k + (2iµ T + 0) < 0, we deduce Re Z ′ 21 < 0.
We now need an estimate on the remainderZ R .
Lemma 3.10. The remainderZ R has the form Proof. The remainderZ R is given bỹ which implies (3.72. We can apply now the method of normal coordinates to equation (3.63).

Lemma 3.11. (cf. [2, Proposition 4.9])
There exist coefficients c ij such that the new function z 1 defined by

73)
satisfies an equation of the forṁ whereẐ R satisfies estimates of the same type asZ R , and Proof. Substituting z 1 in equation (3.63) for z and equating the coefficients, we get, in particular, It is easy to check that all of the coefficients Z 11 , Z 20 , Z 02 , and Z 21 defined in (3.34) are pure imaginary, and hence (3.75) follows immediately.
Denoting K T = K(ω T ), the equation for z 1 is rewritten aṡ It is easier to deal with y = |z 1 | 2 , rather than z 1 , because y decreases at infinity while z 1 is oscillating The equation satisfied by y is simply obtained by multiplying (3.74) by z 1 and taking the real part:

Canonical form of the equation for γ
The only difference between equations (3.22) and (3.29) for ω and γ is that, in general the coefficient Γ 11 = 0. We can nevertheless perform the same change of variables as for ω, obtaining: Lemma 3.12. There exist coefficients d ij (ω), 0 ≤ i, j ≤ 3, and vector functions d ′ ij (x, ω) such that the new function γ 1 defined as

82)
with d ij = d ji , is a solution of the differential equatioṅ FurthermoreΓ R satisfies the same estimate (3.28) as Γ R .

Summary of the equations in canonical form
We summarize the main formulas of §3.4.1- §3.4.3. First we recall that where k and h 1 are defined in (3.48)-(3.49). The equations satisfied by h and h 1 are, respectively, (see (3.38) and (3.50))ḣ The remainderH R is estimated in (3.44) andĤ R =H R + H ′ , where H ′ is estimated in (3.55) and (3.56). The second equation, given in Lemma 3.8, determines the evolution of ω 1 : whereΩ R is estimated in 3.59, and ω 1 and ω are related by (see (3.57)) The third equation describes the evolution of z 1 (see (3.74)): with the estimate (3.79) for Z R . ¿From (3.63), z and z 1 are related by that The fourth equation is for the evolution of y = |z 1 | 2 : where The negativity of Re iK T is a key point in the analysis and was proved in Lemma 3.9. It is clear from the last equation that this condition gives a nonlinear damping effect in the evolution of the amplitude of the discrete mode -this is the crucial dynamical consequence of the Fermi Golden Rule, with obvious relevance to the large time behaviour of the solutions.

A bound for k 1
Lemma 3.13. The function k 1 defined in (3.49) satisfies the following bound: Proof. Equalities (3.40) and (3.49) imply Denoting ν = t 0 β(τ )dτ , we obtain by expanding the exponential and using the idempotency of projections (the Euler trick):

Large time asymptotics
In this section we will make use of the dispersive estimates given in §2.2 to prove the asymptotic representation for the solution of (1.2) with initial data as in theorem 1.3. The idea is to fix an interval [0, T ] and carry out the frozen spectral decomposition relative to the operator C T = C(ω T ) at time T , as described in §3.2. We then obtain bounds for certain majorants on this interval which are uniform in T , and thus make it possible to obtain the asymptotics for the solution in the limit T → +∞. We will use the R notation explained prior to (3.7) and (3.93) to express estimates and bounds concisely.

Definition of majorants
We define the quantities which will be refered to in the following as "majorants", and denote M the 3-dimensional vector (M 0 , M 1 , M 2 ). Observe that (3.7), (3.51) and (3.97) imply that f can be bounded in terms of these: so that control of M will allow complete control of the asymptotic behaviour of the solution.
The goal of this section is to prove that if ε is sufficiently small, M is bounded uniformly in T . This is done by first bounding the initial data, and inhomogeneous terms, in the equations in section 3.4.5 in terms of the M j , and then using the estimates for the homogeneous evolutions to self-consistently bound the M j in terms of themselves, uniformly in T > 0 and ǫ ≪ 1.

Estimate of the remainders and initial data
Proof. Using again the equality f = g + h = g + k + k 1 + h 1 , lemma 3.97 and the definitions of the M j , the remainder Y R is bounded as follows: Proof. It follows from (3.44) which implies (4.6).
The second summand H ′ is represented as in (3.55) where the A m are estimated in (3.56). For the A m we now obtain: Proof. Estimate (3.56) implies which implies (4.7). Now we estimate the initial data. Referring to the formulas at the end of §3.5, we have

Integral inequalities and decay in time
This section is devoted to a study of the system: under some assumptions on the initial data, and on the inhomogeneous (or source) terms Y and H. Equation (4.10) for y is of Ricatti type, and is similar to (3.90), while (4.11) is similar to (3.85). First, for the initial data, we assume with some constant y 0 and h 0 > 0. As for the source terms, we assume that and that H( T with the following bounds: (4.14) where the quantities Y , H 1 , A m are supposed to be given positive constants. All these assumptions are motivated by the estimates of the remainders in §4.2, and by the final estimates we intend to prove on ω, z, h and h 1 . Equation (4.10) corresponds to equation (3.90) and the assumption (4.13) on the source term has the form of estimate (4.5) for the remainder Y R . Similarly, equation (4.11) corresponds to equation (3.85) and assumptions (4.14)-(4.15) correspond to the inequalities (4.6)-(4.7). Finally, corresponding to (3.75), we work under the assumption Proposition 5.6]) The solution of (4.10), with initial condition and source term satisfying (4.12) and (4.13) respectively, is bounded as follows for t > 0: (4.16) Let us consider equation (4.11) for h 1 .

Inequalities for the majorants
Proof.
Step iii) Let us now consider h 1 , the solution of (3.85). It has the form (4.11) with H =Ĥ R = H R + H ′ , whereH R and H ′ identify respectively to H 1 and H 2 . More precisely, using (4.6) and (4.7), we have Concerning the initial conditions, we know that h 1 (0) = h(0). Thus by (4.9) Applying Lemma 4.5, we deduce that which implies (4.20).

Uniform bounds for majorants
The aim of this section is to prove that if ε is sufficiently small, all the M i are bounded uniformly in T and ε.
Replacing M 2 1 in the right-hand by its bound (4.19), we get an inequality in the form where F (M) is an appropriate function . From this inequality it follows that M is bounded independent of ε ≪ 1, since M(0) is small, and M(t) is a continuous function of t.
Corollary 4.8. The function ω(t) has a limit ω + as t → ∞. Furthermore, the following estimates hold for all t > 0: , then applying this result to |ω(t 1 ) − ω(t 2 )|, we see that ω(t) is a Cauchy sequence. It thus has a limit, denoted ω + and (4.23) holds. The next two results follow immediately, while the final one is a consequence of (4.4).

Large time behaviour of the solution
In this section we deduce from corollary 4.8 a theorem which describe a large time behaviour of the solution. Notice that in the decomposition f = g + h = g + h 1 + k + k 1 , a fixed time T has been chosen, and all the components depend on ω(T ). From the above proposition, we know that ω(t) has a limit ω + as t → ∞. So we can reformulate the decomposition by choosing T = ∞ and ω T = ω + . Namely, let us denote P c ∞ = P c (ω + ) and P d ∞ = 1 − P c ∞ . We define f = g + h where g = P d ∞ f and h = P c ∞ f . We also decompose h = h 1 + k + k 1 where k = a 20 z 2 + a 11 zz + a 02 z 2 , where a ij = a ij (ω + , x) and C + = C(ω + ) + i ω(t) − ω + +γ(t) Π + ∞ − Π − ∞ . All the estimates previously obtained in §3.4.2- §4 for finite T can be extended to T = ∞ and ω T = ω + without modification. Thus we have proved the following result: Theorem 4.9. Let the conditions of theorem 1.3 hold. Then, for ε sufficiently small, there exist C 1 functions ω(t), γ(t), z(t) as in lemma 3.1, and constants ω + ∈ R and M > 0, such that for all t ≥ 0:

Scattering asymptotics
We have now obtained the representation (4.27) of the solution ψ(x, t). In order to prove statement of theorem 1.3 it remains to: • construct asymptotic expressions for ω(t), z(t), γ(t), which is done in section 5.1 following [2], and then • to prove the existence of Ψ ± , and hence obtain the scattering asymptotics (1.19); this second stage amounts to the construction of the wave operator, and is carried out in §5.2 by the study of some oscillatory integrals.
The function f (x, t) which is a solution of 5.10 can be expressed formally as where W (t) is the dynamical group of the free Schrödinger equation. To establish the asymptotic behavior (1.20), it suffices to prove that These assertions follow from the definition (5.7) of the function R, and the following two lemmas. The first lemma studies the contribution to φ + (x) and r + (x, t) from the terms in (5.7) involving δ(x) which is O(t −1 ) as t → ∞ by (4.26).
and hence φ ∈ L 2 (R), since it is already known to be bounded and continuous. It remains to prove the decay rate of r(x, t) in the norm L 2 (R). Let us represent the function r as Here ζ t (u) is the characteristic function of the interval (0, 1/t). As above ρ is bounded, but also since the Hausdorff-Young inequality implies that for any q > 2, and in fact as t → ∞: for some constant C = C(L 1 , p), for q −1 + p −1 = 1. The Young inequality then implies that if r > q. To have r > q, we must take q < 4, or equivalently p > 4/3. Hence, we have The second lemma studies the contribution to φ + (x) and r + (x, t) from terms without δ(x) in (5.7). Consider the expansions (3.22), (3.29), (3.33), forω(t),γ(t), andż(t) − iµz(t): the main (quadratic) parts of these contain the terms z 2 + (t), z 2 + (t), z + (t)z + (t), which are O(t −1 ) as t → ∞. The remainders are O(t −3/2 ), and it is straightforward (from the unitarity of W ) to bound the contribution of these to φ + in C b ∩ L 2 , and to check that these contribute O(t −1/2 ) to r + in C b ∩ L 2 . Thus, without loss of generality, we may replaceω(t),γ(t), andż(t) − iµz(t) by the main quadratic parts. We first show how to treat these terms with the phase θ(t) in (5.7) replaced by ϕ + (t) ≡ ω + t, and then consider the general case in lemma 5.5.
To bound ρ 1 there are two cases. Firstly, if x q > t then |ρ To bound ρ 2 , we notice that τ ≥ cx on J 2t , and also τ ≥ t, and then just estimate To bound ρ 3 , notice that for τ ∈ J 3t the inequalities (5.23) hold so that it is possible to integrate by parts (as in (5.22) above) since the denominator which appears is bounded below. As with φ 3 , the boundary terms arising in this integration by parts are ≤ c τ 1/2 x −p /(x + τ ) ≤ c t −1/2 x −p . Also, as in (5.24), the integral which remains after this integration by parts can be bounded as ≤ ct −1/2 x −p + cx q/2−2p ≤ ct −1/2 x −p + ct −ν x q/2−2p+νq , since t ≤ τ ≤ x q in J 3t , so that we may assume t ≤ x q in the estimation of ρ 3 . Therefore, in conclusion we have the following estimate for x > l ≫ 1 and t large: This shows that ρ(x, t) is square integrable and (5.21) holds if The conditions on the exponents can be written equivalently as Proof. The estimates in the proof of lemma 5.4 which involve estimating the integral of the absolute value are completely unaffected by change of phase, so it is only necessary to re-assess the argument involving integration by parts, i.e. the treatment of φ 3 and ρ 3 . For example, in the more difficult case when Π(t) = z 2 + (t)e iθ(t) , we proceed as follows in the estimation of ρ 3 . Writeφ = θ − φ + , and integrate by parts exactly as before, leaving along the e iφ factor: this factor then carries throught to the integrand after the integration by parts, and we need to bound: (The treatment of the boundary terms is unaffected since |e iφ | = 1.) But since by assumptioṅ φ(τ ) = O(τ −1 ) the extra contribution to the integrand can clearly be estimated for large x, τ in the same way as the term arising from differentiation of τ 3/2 , and so (5.25) still holds, as required to complete the proof.
Remark 5.6. The t → −∞ case is handled in an identical way.
Since both k + = √ −ω + µ and k − = √ −ω − µ are purely imaginary, the first component u 1 is real, while the second one u 2 is imaginary. It is easy to prove that u * = (u 1 , −u 2 ) is the eigenfunction associated to λ = −iµ. 2ik

B The eigenfunctions of the continuous spectrum
Solving this equation, we get B = −D and then C = 4βik + . Finally, we obtain II. It is easily to check that an odd solution u = s of equation (A. 1) is For λ = iν with ν < −ω we have similarly
Further, the integrand in K 3 (t) is an analytic function of λ = 0, ±iµ with the values in B β for β ≥ 0. At the points λ = 0 and λ = ±iµ the integrand has the poles of finite order. Hoverever, all the Laurent coefficients vanish when applied to P c h ∈ X c . Hence for K 3 (t) we obtain, twice integrating by parts, K 3 (t)P c h L ∞ −β ≤ c(1 + t) −3/2 h M β , completing the proof.
Writing e iζ|x| dζ = 1 i|x| de iζ|x| , and integrating by parts, we get that completing the proof of lemma 3.5.
Notice that the function c(ν) is algebraic. It follows from this that if the function a(·) is polynomial, or even real analytic, then generically the Fermi Golden Rule holds except possibly at a discrete set of values of C. To see this observe that if the set of points where F (C 2 ) ≡ a ′′ (C 2 ) − c(ν)a ′ (C 2 ) C 2 vanishes has an accumulation point, then F must be identically zero since it is real analytic. But the condition a ′′ (C 2 ) = c(ν)a ′ (C 2 ) C 2 is a second order ordinary differential equation which determines the function a(C 2 ) given its value a(C 2 0 ) and that of its first derivative a ′ (C 2 0 ) at any point C = C 0 . Clearly a generic polynomial function a(C 2 ) will not satisfy this equation, and so the set of points where the Fermi Golden Rule fails cannot have any accumulation points generically. This is also true for real analytic a in the following sense. Fix any C 0 , then there is a two parameter family of functions a(C 2 ) for which a ′′ (C 2 ) = c(ν)a ′ (C 2 ) C 2 ; (this family of exceptional functions is parameterized by a(C 0 ), a ′ (C 0 )). If a(C 2 ) is not one of these functions then the set of values for which the Fermi Golden Rule fails is at most a discrete set.