{'fraction_non_alphanumeric': 0.07066375114622275, 'fraction_numerical': 0.03641108838259152, 'mean_word_length': 3.4573979752248003, 'pattern_counts': {'":': 0, '<': 66, '<?xml version=': 0, '>': 55, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 38, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Long Range Scattering and Modified Wave Operators for the Wave-Schr"odinger system

We study the theory of scattering for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling,in space dimension 3.We prove in particular the existence of modified wave operators for that system with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.The method consists in solving the wave equation, substituting the result into the Schr"odinger equation,which then becomes both nonlinear and nonlocal in time,and treating the latter by the method previously used for a family of generalized Hartree equations with long range interactions.


Introduction
This paper is devoted to the theory of scattering and more precisely to the existence of modified wave operators for the Wave-Schrödinger (WS) system where u and A are respectively a complex valued and a real valued function defined in space time IR 3+1 , ∆ is the Laplacian in IR 3 and ⊓ ⊔ = ∂ 2 t − ∆ is the d'Alembertian in IR 3+1 . That system is Lagrangian with Lagrangian density Formally, the L 2 norm of u is conserved, as well as the energy The Cauchy problem for the WS system (1.1) (1.2) is known to be globally well posed in the energy space X e = H 1 ⊕Ḣ 1 ⊕ L 2 for (u, A, ∂ t A) [1] [2] [4] [15]. case also for the Hartree equation with |x| −1 potential. Both are simplified models for the more complicated Maxwell-Schrödinger system in IR 3+1 , which belongs to the same case, as well as the KGS system in IR 2+1 . Whereas a well developed theory of long range scattering exists for the linear Schrödinger equation (see [3] for a recent treatment and for an extensive bibliography), there exist only few results on nonlinear long range scattering. The existence of modified wave operators in the borderline Coulomb case has been proved for the NLS equation in space dimension n = 1 [19]. That result has been extended to the NLS equation in dimensions n = 2, 3 and to the Hartree equation in dimension n ≥ 2 [5], to the derivative NLS equation in dimension n = 1 [14], to the KGS system in dimension 2 [20] and to the MS system in dimension 3 [22]. All those results are restricted to the case of small data.
In a recent series of papers, [6] [7] [8], we proved the existence of modified wave operators for a family of Hartree type equations with general (not only Coulomb) long range interactions and without any size restriction on the data. The method is strongly inspired by a previous series of papers by Hayashi et al [9] [10] [11] [12] [13] on the Hartree equation. In the latter papers it is proved first in the borderline Coulomb case and then in the whole long range case, that the global solutions of the Hartree equation with small initial data exhibit an asymptotic behaviour for large time that is typical of long range scattering and includes in particular the expected relevant phase factor. The present paper is devoted to the extension of the results of [6] [7] [8] to the WS system and in particular to the proof of the existence of modified wave operators for that system without any size restriction on the data. The method consists in eliminating the wave equation by solving it for A in terms of u and substituting the result into the Schrödinger equation, thereby obtaining a new Schrödinger equation which is both nonlinear and nonlocal in time. The latter is then treated as the Hartree equation in [6] [7] [8], namely u is expressed in terms of an amplitude w and a phase ϕ satisfying an auxiliary system similar to that introduced in [11]. Wave operators are constructed first for that auxiliary system, and then used to construct modified wave operators for the original system (1.1). The detailed construction is too complicated to allow for a more precise description at this stage, and will be described in heuristic terms in Section 2 below. In subsequent papers, the results of the present one will be extended to the case of the MS system. We now give a brief outline of the contents of this paper. A more detailed description of the technical parts will be given at the end of Section 2. After collecting some notation and preliminary estimates in Section 3, we study the asymptotic dynamics for the auxiliary system in Section 4 and uncover some difficulties due to the different propagation properties of solutions of the wave and Schrödinger equations. As a preparation for the general case, we construct in Section 5 the wave operators associated with the simplified linear system obtained by replacing (1.2) by the free wave equation ⊓ ⊔A = 0. We then solve the local Cauchy problem at infinity for the auxiliary system in Sections 6 and 7, which contain the main technical results of this paper. We finally come back from the auxiliary system to the original one (1.1) (1.2) and construct the modified wave operators for the latter in Section 8, where the final result is stated in Proposition 8.1.
We conclude this section with some general notation which will be used freely throughout this paper. We denote by · r the norm in L r ≡ L r (IR 3 ) and we define δ(r) = 3/2 − 3/r. For any interval I and any Banach space X, we denote by C(I, X) (resp. C w (I, X)) the space of strongly (resp. weakly) continuous functions from I to X and by L ∞ (I, X) (resp. L ∞ loc (I, X)) the space of measurable essentially bounded (resp. locally essentially bounded) functions from I to X. For real numbers a and b, we use the notation a ∨ b = Max(a, b), and a ∧ b = Min(a, b). Furthermore, we For any interval I ⊂ IR + , we denote byĪ the closure of I in IR + ∪ {∞} and for any interval I = [a, b) we denote by I + the interval I + = [a, ∞). In the estimates of solutions of the relevant equations, we shall use the letter C to denote constants, possibly different from an estimate to the next, depending on various parameters, but not on the solutions themselves or on their initial data. We shall use the notation C(a 1 , a 2 , · · ·) for estimating functions, also possibly different from an estimate to the next, depending in addition on suitable norms a 1 , a 2 , · · · of the solutions or of their initial data. Additional notation will be given in Section 3.

Heuristics
In this section, we discuss in heuristic terms the construction of the modifed wave operators for the system (1.1) (1.2), as it will be performed in this paper.
We refer to Section 2 of [6] [7] for general background and for a similar discussion adapted to the case of the Hartree equation. The problem that we want to address is that of classifying the possible asymptotic behaviours in time of the solutions of (1.1) (1.2) by relating them to a set of model functions V = {v = v(v + )} parametrized by some data v + and with suitably chosen and preferably simple asymptotic behaviour in time. For each v ∈ V, one tries to construct a solution (u, A) of (1.1) (1.2) such that (u, A)(t) behaves as v(t) when t → ∞ in a suitable sense. We then define the wave operator as the map Ω : v + → (u, A) thereby obtained. A similar question can be asked for t → −∞. We restrict our attention to positive time. The more standard definition of the wave operator is to defin e it as the map v + → (u, A)(0), but what really matters is the solution (u, A) in the neighborhood of infinity in time, namely in some interval [T, ∞), and continuing such a solution down to t = 0 is a somewhat different question which we shall not touch here.
In cases such as (1.1) (1.2) where the system of interest is a perturbation of a simple linear system, hereafter called the free system, a natural candidate for V is the set of solutions of the free system, parametrized by the initial data v + at time t = 0 for the Cauchy problem for that system. In the case of the system (1.1) (1.2) one is therefore tempted to consider the Cauchy problem to take v + = (u + , A + ,Ȧ + ) and to take for v(v + ) the solution (u, A) of (2.1). Cases where such a procedure yields an adequate set V are called short range cases. They require that the perturbation has sufficient decay in time or equivalently in space. This is the case for instance for the linear Schrödinger equation or for the Hartree equation with potential V (x) = |x| −γ for γ > 1. Such is not the case however for the system (1.1) (1.2). This shows up through the fact that the solution A of the wave equation ⊓ ⊔A = 0 decays at best as t −1 (in L ∞ norm), which is the borderline case of nonintegrability in time. That situation corresponds to the limiting case γ = 1 (the Coulomb case in space dimension n = 3) for the linear Schrödinger and for the Hartree equation. A similar situation prevails for the KGS system in space dimension 2 and for the MS system in space dimension 3. In the present case, which is the borderline long range case, the set of solutions of the Cauchy problem (2.1) is inadequate, and one of the tasks that will be performed in this paper (see especially Sections 7 and 8) will be to construct a better set V of model asymptotic functions.
Constructing the wave operators essentially amounts to solving the Cauchy problem with infinite initial time. The system (1.1) (1.2) in this form is not well suited for that purpose and we shall now perform a number of transformations leading to an auxiliary system for which that problem can be handled. For additional flexibility we shall first of all allow for imposing initial data at two different initial times t 0 and t 1 for the Schrödinger and wave equations respectively. With the aim of letting t 1 and t 0 tend to infinity in that order, we shall take t 0 ≤ t 1 . We shall then eliminate the wave equation by solving it and substituting the result into the Schrödinger equation. We define ω = (−∆) 1/2 , K(t) = ω −1 sin ωt ,K(t) = cos ωt and we replace the wave equation (1.2) by its solution Here A 0 is a solution of the free wave equation, with initial data (A + ,Ȧ + ) at time t = 0. For t 1 = ∞, (A + ,Ȧ + ) is naturally interpreted as the asymptotic state for A, in keeping with the previous discussion.
The Cauchy problem for the system (1.1) (2.2) with initial data u(t 0 ) = u 0 is no longer a usual PDE Cauchy problem because A 1 depends on u nonlocally in time. A convenient way to handle that difficulty is to first replace that problem by a partly linearized form thereof, namely (2.5) For given u, (2.5) is an ordinary (linear) Cauchy problem for u ′ . Solving that problem for u ′ defines a map Γ : u → u ′ , and solving the original problem then reduces to finding a fixed point of Γ, which in favourable cases can be done for instance by contraction. We shall make use of that linearization method, not for the equation for u, but for the auxiliary system to be defined below. Aside from the nonlocality in time of the nonlinear interaction term, which can be handled by the previous linearization, the system (1.1) (2.2) is rather similar to the Hartree type equations considered in [6] [7] [8], and we next perform the same change of variables, which is well adapted to the study of the asymptotic behaviour in time. The unitary group which solves the free Schrödinger equation can be written as where M(t) is the operator of multiplication by the function F is the Fourier transform and D(t) is the dilation operator normalized to be unitary in L 2 . We shall also need the operator D 0 (t) defined by We now parametrize u in terms of an amplitude w and of a real phase ϕ as (2.11) Substituting (2.11) into (1.1) yields an evolution equation for (w, ϕ), namely where we have expressed A in terms of a new function B by Corresponding to the decomposition (2.2) of A, we decompose where As in the case of the Hartree equation, we have only one evolution equation for two functions (w, ϕ) and we arbitrarily split that equation into two equations for w and ϕ. For that purpose, we split B t 1 1 into a short range and a long range parts We take 0 < β < 1 and we define The parameter β will satisfy various conditions which will appear later, all of which will be compatible with β = 1/2. We now split the equation (2.12) into the following system of two equations where we have defined for any vector field s. Since the right-hand sides of (2.18) contain ϕ only through its gradient, we can obtain from (2.18) a closed system for w and s = ∇ϕ by taking the gradient of the second equation, namely (2.20) Once the system (2.20) is solved for (w, s), one recovers ϕ easily by integrating the second equation of (2.18) over time. We refer to [6] for details. The system (2.20) will be referred to as the auxiliary system and will play an essential role in this paper. For the same reason as was explained for the partly resolved system (1.1) (2.2), we shall use at intermediate stages a partly linearized version of the system (2.20), namely to be considered as a system of equations for (w ′ , s ′ ) for given (w, s). The first question to be considered is whether the auxiliary system (2.20) defines a dynamics for large time, namely whether the Cauchy problem for that system is locally well posed in a neighborhood of infinity in time, more precisely has a unique solution defined up to infinity in time for sufficiently large t 1 and sufficiently large initial time t 0 , possibly depending on the size of the initial data. This property was satisfied by the corresponding auxiliary system associated with the Hartree equation and considered in [6] [7]. Here however we encounter serious difficulties associated with the difference of propagation properties of solutions of the Schrödinger and wave equations.
In fact a typical solution of the free Schrödinger equation behaves asymptotically in time as namely spreads by dilation by t in all directions in the support of F u + , while by the Huyghens principle A 0 remains concentrated in a neighborhood of the light cone, more precisely within a distance R of the latter if the initial data (A + ,Ȧ + ) are supported in a ball of radius R. When switching to the new variables (w, B), w tends to a limit when t → ∞ whereas B 0 concentrates in a neighborhood of the unit sphere, within a distance R/t of the latter in the previous case of compactly supported data. Note however that for t 1 = ∞, B ∞ 1 is expected to tend to a limit like w and not to concentrate like B 0 , as can be guessed from (2.15).
We shall treat the auxiliary system (2.20) by energy methods, and in particular look for w in spaces of the type C([T, ∞), H k ) where H k is the usual Sobolev space based on L 2 . In order to treat the nonlinear term B 1 (w, w), we shall need a minimal regularity, in practice k > 1. However, when taking H k norms of B 0 , the previous concentration phenomenon implies which has worse and worse asympotic behaviour in time as k increases. This difficulty manifests itself in the following way : (i) If t 0 = t 1 < ∞, the available estimates for the sytem (2.20) do not prevent finite time blow up after t 0 , even if A 0 = 0. This encourages us to take t 1 > t 0 , and actually the situation becomes slightly better in that case. Nevertheless (ii) the available estimates do not prevent finite time blow up after t 1 , which is the same fact as (i) with t 0 replaced by t 1 , and (iii) if A 0 = 0 and if t 1 is sufficiently large, the available estimates do not prevent blow up before t 1 .
A definite improvement occurs however if A 0 = 0.
(iv) If A 0 = 0, the available estimates allow for a proof of existence of solutions in [t 0 , t 1 ] for t 0 sufficiently large and arbitrary t 1 > t 0 , possibly t 1 = ∞. In particular for t 1 = ∞, the solutions are defined up to infinity in time. Furthermore, for those solutions, w(t) has a limit w + as t → ∞.
The last case brings us in the same situation as that encountered for the Hartree equation in [6] [7] and could be taken as the starting point for the construction of partial modified wave operators (restricted to the case of vanishing (A + ,Ȧ + )) by the same method as in [6] [7]. We shall however refrain from performing that construction and turn directly to the case of nonvanishing (A + ,Ȧ + ). In that case, the need to use H k norms with k > 1 for w makes the treatment of A 0 nontrivial, even if one drops the interaction term A 1 . As a preparation for the general case, we shall therefore first construct the wave operators at the same level of regularity for the simplified system namely for a linear Schrödinger equation with time dependent potential A 0 satisfying the free wave equation. After the appropriate change of variables the Schrödinger equation becomes The construction of the wave operators for that equation in L 2 , either in the form (2.22) or (2.24) can be easily performed by a simple variant of Cook's method, and the construction of the wave operators at the level of H k becomes a regularity problem for the previous wave operators. Solving that problem for k ≥ 1 (in fact for k > 1/2) requires special assumptions on the asymptotic states (w + , A + ,Ȧ + ), to the effect that the product B 0 w + decays faster in time in the relevant norms than what would naturally follow from factorized estimates. Those assumptions can be ensured for instance by imposing support properties of w + , to the effect that w + = 0 on the unit sphere, and suitable decay of (A + ,Ȧ + ) at infinity in space. They will be needed again in the treatment of the general problem. The construction of the modified wave operators in the general case follows the same pattern as for the Hartree equation. The aim is to construct solutions of the auxiliary system (2.20) with suitably prescribed asymptotic behaviour at infinity, and in particular with w(t) tending to a limit w + as t → ∞. For that purpose, we choose a suitable asymptotic pair (W, φ), and therefore a pair (W, S) with S = ∇φ with W (t) tending to w + as t → ∞, and we look for a solution (w, ϕ) of the system or equivalently for a solution (w, s) of the system (2.20) in the form where (q, ψ, σ) tends to zero as t → ∞. Actually for technical reasons, we shall need to modify the auxiliary system slightly, in the following way. When expanding , we shall replace that quantity by We furthermore define the remainders The modified auxiliary system for (q, σ) then takes the form (2.31) The construction of solutions (q, σ) tending to zero at infinity for the system (2.30) with t 1 = ∞ proceeds in several steps. We assume first that (W, S) satisfy suitable boundedness properties and that the remainders R 1 (W, S) and R 2 (W, S) satisfy suitable decay in time. We solve the linearized system (2.31) for (q ′ , σ ′ ) for given (q, σ), both with finite and infinite time t 1 and initial time t 0 . We then solve (2.30) by proving that the map Γ : (q, σ) → (q ′ , σ ′ ) is a contraction in suitable norms. We also prove that the solution of (2.30) with t 0 = t 1 < ∞ converges to the solution with t 0 = t 1 = ∞ when t 0 → ∞, a property which is natural in the framework of scattering theory. We finally construct (W, S) satisfying the required boundedness and decay properties by solving the auxiliary system (2.20) with t 1 = ∞ approximately by iteration, restricting our attention to the second iteration, which is sufficient to cover the range 1 < k < 2. The pair (W, S) or equivalently (W, φ) depends only on the limit w + of w and W as t → ∞. Solving the auxiliary system (2.30) with that (W, S) and with t 0 = t 1 = ∞ yields a solution (w, s) of the system (2.20) and therefore a solution (w, ϕ) of the system (2.18) with prescribed asymptotic behaviour characterized by (W, S) or (W, φ). That solution depends on (w + , A + ,Ȧ + ). Plugging that solution into (2.11) (2.13) and taking w + = F u + yields a map Ω : (u + , A + ,Ȧ + ) → (u, A) which is the required wave operator for the system (1.1) (1.2).
The auxiliary system (2.18) satisfies a gauge invariance property similar to that of the corresponding system for the Hartree equation used in [6] [7], and the construction of the intermediate wave operator for that system can be made in a gauge covariant way. For brevity we shall refrain from discussing that question in this paper.
We now describe the contents of the technical parts of this paper, namely Sections 3-8. In Section 3, we introduce some notation, define the relevant function spaces and collect a number of preliminary estimates. In Section 4, we study the Cauchy problem for large time for the auxiliary system (2.20). We solve the Cauchy problem with finite initial time for the linearized system (2.21) (Proposition 4.1), we prove a number of uniqueness results for the system (2.20) (Proposition 4.2), we prove the existence of a limit w(t) of w + for suitably bounded solutions of the system (2.20) (Proposition 4.3), we discuss in more quantitative terms the possible occurrence of blow up mentioned above, and we finally solve the Cauchy problem for the system (2.20) with t 1 = ∞ and large t 0 in the special case A 0 = 0 (Proposition 4.4).
In Section 5, as a preparation for the construction of the wave operators for the system (2.20) with A 0 = 0, we study the existence of wave operators for the linear problem (2.22) in the form (2.24). In particular we prove the existence of L 2wave operators by a variant of Cook's method (Proposition 5.2), we prove the H k regularity of those wave operators under suitable decay assumptions of R(W ) for the model function W (Proposition 5.3) and we finally reduce those decay properties to conditions on the asymptotic state (w + , A + ,Ȧ + ). In Section 6 and 7, we study the Cauchy problem at infinity in the general case A 0 = 0 for the auxiliary system (2.20) in the difference form (2.30). Under suitable boundedness assumptions on (W, S) and decay assumptions on R 1 (W, S) andf R 2 (W, S) we prove the existence of solutions for t 0 and t 1 finite and infinite, first for the linearized system (2.31) (Propositions 6.1 and 6.2) and then for the nonlinear system (2.30) (Proposition 6.3). We then choose appropriate (W, S), prove that they satisfy the required assumptions (Lemmas 7.1 and 7.2) and finally state the result on the Cauchy problem at infinity for the system (2.30) in H k for 1 < k < 2 (Proposition 7.1). Finally in Section 8, we construct the wave operators for the system (1.1) (1.2) from the results previously obtained for the system (2.30) and we derive the asymptotic estimates for the solutions (u, A) in their range that follow from the previous estimates (Proposition 8.1).

Notation and preliminary estimates
In this section we introduce some additional notation and we collect a number of estimates which will be used throughout this paper. We shall use the Sobolev spaces H k r defined for 1 ≤ r ≤ ∞ by where < · >= (1 + | · | 2 ) 1/2 . The subscript r will be omitted if r = 2.
We shall look for solutions of the auxiliary system (2.20) in spaces of the type C(I, X k,ℓ ) where I is an interval and and we shall use the notation Here it is understood that ∇s ∈ L 2 includes the fact that s ∈ L 6 .
We shall use extensively the following Sobolev inequalities, stated here in IR n , but to be used only for n = 3.
We shall also use extensively the following Leibnitz and commutator estimates.
Then the following estimates hold for m ≥ 0, and denotes the commutator.
The proof of those estimates is given in [17] [18] with ω replaced by < ω > and follows therefrom by a scaling argument.
We shall also need the following consequence of Lemma 3.2.
Proof. For any integer n ≥ 2, we estimate by (3.4), where b = ϕ ∞ and we can assume C ≥ 1 without loss of generality. It follows easily from (3.7) that a n ≤ n(Cb) n−1 a 1 for all n ≥ 1, from which (3.6) follows by expanding the exponential.

⊓ ⊔
We next give some estimates of B t 1 1 , B t 1 S and B t 1 L defined by (2.15) (2.17). It follows immediately from (2.17) that for m ≤ p and similarly for m ≥ p. On the other hand it follows from (2.15) that where I t 1 m is defined by (3.11) or equivalently for t > 0, t 1 > 0. Most of the subsequent estimates of B t 1 1 will follow from (3.8) (3.9) (3.10) and from an estimate of ω m (w 1w2 ) 2 . The latter follows from the HLS inequality if −3/2 < m < 0 and from (3.4) if m ≥ 0. For future reference, we quote the following special case, which will occur repeatedly and which holds for 0 < k < 3/2. The required estimate follows from the HLS inequality if 2k < 3/2 and from (3.4) if 2k ≥ 3/2, as mentioned above, and from Sobolev inequalities. We next give a special estimate of the long range part B t 1 L of B t 1 1 . (3.14) Proof. Let f = D 0 (ν)Re w 1w2 . From (2.15) (2.17), we estimate which implies (3.14).  Let ω m A + ∈ L 2 , ω m−1Ȧ + ∈ L 2 , ∇ 2 ω m A + ∈ L 1 and ∇ω mȦ + ∈ L 1 . Then the following estimate holds for all t > 0.
Proof. It suffices to prove (3.15) for m = 0, for r = 2 and r = ∞. For r = 2, it follows from (2.3) that for all t ∈ IR. For r = ∞, the result follows from the divergence theorem applied to the standard representation of solutions of the free wave equation in terms of spherical means [21]. ⊓ ⊔ The time decay expressed by (3.15) is known to be optimal, and we shall always consider solutions A 0 of the free wave equation satisfying those estimates for suitable m. In the applications, we shall use the estimates (3.15) in the equivalent form expressed in terms of B 0 defined by (2.13), namely 4 Cauchy problem and preliminary asymptotics for the auxiliary system In this section, we study the Cauchy problem for the auxiliary system (2.20) and we derive some preliminary asymptotic properties of its solutions. This section illustrates both the method of solution with the help of the linearized version (2.21) of that system and the difficulties arising from the different propagation properties of the Schrödinger and wave equations. In particular we are able to prove the existence of solutions up to infinity in time only if A 0 = 0. This section could be the starting point for the construction of partial wave operators with vanishing asymptotic states for the field A, a construction which would be very similar to that of the wave operators for the Hartree equation performed in [6] [7], but which we shall refrain from performing here. The general case of non-vanishing asymptotic states for A will be treated by a similar but more complicated method in Section 6 below.
The basic tool of this section consists of a priori estimates for suitably regular solutions of the linearized system (2.21). Those estimates can be proved by a regularisation and limiting procedure and hold in the integrated form at the available level of regularity. For brevity, we shall state them in differential form and we shall restrict the proof to the formal computation.
We first estimate a single solution of the linearized system (2.21) at the level of regularity where we shall eventually solve the auxiliary system (2.20). C(I, X k,ℓ ) with w ∈ L ∞ (I, H k ) and let (w ′ , s ′ ) be a solution of the system (2.21) in I. Then the following estimates hold for all t ∈ I : Proof. We omit the superscript t 1 in all the proof. We first estimate w ′ . It is clear from (2.21) that w ′ 2 = const. We next estimate The contribution of B 0 is estimated by Lemma 3.2 and (3.17) as with 0 < δ = δ(r) < k ∧ 3/2. This yields the first term in the RHS of (4.1) by Sobolev inequalities and interpolation. We next estimate by Lemma 3.2 and using Sobolev inequalities, we continue the previous estimate by We next estimate the contribution of B S to (4.6). By Lemma 3.2 and Sobolev inequalities, we estimate where 0 < δ = δ(r) ≤ 3/2. We choose δ = [k ∧ 3/2] and continue (4.9) as follows : If k < 3/2, so that δ = k, by Sobolev inequalities, by (3.8) (3.12) and by the definitions (4.3) (4.4).
If k > 3/2, so that δ = 3/2 and r = ∞, by ( We now turn to the estimate of s ′ , namely to the proof of (4.2). For 0 ≤ m ≤ ℓ, we estimate The first bracket in the RHS of (4.13) is estimated by Lemma 3.2 as by Sobolev inequalities.
The contribution of B L for m = ℓ is estimated by (3.9) (3.10) (3.12) and Lemma 3.2 as in all cases. The contribution of B L for m = 0 is estimated similarly as 16) in all cases, with Collecting (4.14) (4.15) (4.16) yields (4.2).
⊓ ⊔ We next estimate the difference of two solutions of the linearized system (2.21) corresponding to two different choices of (w, s). We estimate that difference at a lower level of regularity than the solutions themselves.
. Then the following estimates hold for all t ∈ I : where β 1 , m 1 and β 2 are defined by (4.3) (4.4) (4.5) and where Proof. We again omit the superscript t 1 in the proof. Taking the difference of the system (2.21) for (w ′ i , s ′ i ), we obtain the following system for (w ′ − , s ′ − ) : (4.21) We first estimate w ′ − . From (4.21) we obtain where only those terms appear that do not preserve the L 2 -norm. We estimate the first norm in the RHS by Hölder and Sobolev inequalities as follows : and in all cases We estimate the second norm in the RHS of (4.22) by (3.8) (3.10), by Lemma 3.2 and by the Hölder and Sobolev inequalities as follows : and in all cases We now turn to the estimate of s ′ − , namely to the proof of (4.19). From (4.21) we estimate for m ≥ 0 If m = 0 (a case which has to be self-estimating if ℓ 0 = 0, which is allowed if k > 3/2), we estimate the bracket in the RHS of (4.25) directly as If m > 0, we estimate that bracket by Lemma 3.2 and Sobolev inequalities as The first and third term in the RHS of (4.27) are readily controlled by the corresponding terms in (4.19) for 0 < m ≤ ℓ 0 and ℓ > 3/2. The remaining two terms are similarly controlled through Sobolev inequalities provided Those conditions are easily seen to be compatible in δ and δ ′ for all m, 0 < m ≤ ℓ 0 , provided ℓ ≥ [(ℓ 0 + 1) ∨ 3/2], which follows from ℓ > 3/2 and ℓ ≥ ℓ 0 + k. We finally estimate the contribution of B L (w + , w − ) by by (3.9) and we estimate the last norm in exactly the same way as in (4.16), thereby obtaining Collecting (4.25) (4.26), (4.27) and the discussion that follows, and (4.28) and noting that β ′ 2 + βm ≤ β 2 for m ≤ ℓ 0 ≤ ℓ − k, we obtain (4.19).

⊓ ⊔
With the estimates of Lemma 4.1 and 4.2 available, it is an easy matter to solve the Cauchy problem globally in time for the linearized system (2.21).
. That solution satisfies the estimates (4.1) (4.2) for all t ∈ I. Two such solutions Proof. The proof proceeds in the same way as that of Proposition 4.1 of [6], through a parabolic regularization and a limiting procedure, with the simplification that the We first consider the case t ≥ t 0 . The system (2.21) with a parabolic regularization added is rewritten in terms of the variables ( w ′ , s ′ ) as where the parametric dependence of F , G on (w, s) has been omitted. The Cauchy problem for the system (4.29) can be recast in the integral form From those facts and from estimates on F , G similar to and mostly contained in those of Lemma 4.1, it follows by a contraction argument that the system (4.30) has a unique solution ( w ′ η , s ′ η ) ∈ C([t 0 , t 0 + T ], X k,ℓ ) for some T > 0 depending only on |w ′ 0 | k , |s ′ 0 |˙ℓ and η. That solution satisfies the estimates (4.1) and (4.2) and can therefore be extended to I + = I ∩ {t : t ≥ t 0 } by a standard globalisation argument using Gronwall's inequality.
We next take the limit η → 0. Let η 1 , η 2 > 0 and let . By estimates similar to, but simpler than those of Lemma 4.2, since in particular (w − , s − ) = 0, we obtain Those estimates imply that (w ′ η , s ′ η ) converges in X 0,0 uniformly in time in the compact subintervals of I + , to a solution of the original system. It follows then by a standard compactness argument using the estimates (4.1) (4.2) that the limit belongs to C(I + , X k,ℓ ). This completes the proof for t ≥ t 0 . The case t ≤ t 0 is treated similarly.

⊓ ⊔
We now turn to the Cauchy problem for the auxiliary system (2.20). Because of the difficulties described in Section 2, the problem of existence of solutions is scattered with pitfalls, as the discussion below will show. On the other hand, the uniqueness problem of suitably bounded solutions is a rather easy matter and we consider that problem first. The proof relies entirely on Lemma 4.2 and therefore does not require any a priori estimate on B 0 . The snag of course is that it is difficult to prove the existence of solutions with the required boundedness properties. Proposition 4.2. Let 1 < k ≤ ℓ, ℓ > 3/2 and β > 0. Let I ⊂ [1, ∞) be an interval and let t 1 ∈Ī. Let B 0 be sufficiently regular, for instance B 0 ∈ C(I, H k 3 ). (1) Let t 0 = t 1 < ∞ and let (w 0 , s 0 ) ∈ X k,ℓ . Then the system (2.20) has at most one solution (w, s) ∈ C(I, (3) Let t 1 = ∞. Assume that w 1 − w 2 2 t β 2 and |s 1 − s 2 |˙ℓ 0 tend to zero when t → ∞. Then (w 1 , s 1 ) = (w 2 , s 2 ).
Proof. If (w i , s i ), i = 1, 2 are two solutions of the system (2.20) in C(I, X k,ℓ ), then they satisfy the estimates (4.18 , which we denote (4.18=) (4.19=) and refrain from rewriting for brevity. The proof consists in exploiting those estimates to prove that (w 1 , s 1 ) = (w 2 , s 2 ). We define y = w − 2 and We shall reduce the system (4.33) (4.34) to a standard form where Gronwall's inequality is applicable. We restrict our attention to the case t ≥ t 0 for definiteness. The case t ≤ t 0 can be treated similary. Defining z by we reduce the system (4.33) (4.34) for (y, z) to a similar system for (y, z), where f 2 , g 2 and f 1 are replaced by 0, E −1 g 2 and Ef 1 . We can therefore assume that f 2 = 0. Then which is of the same form as (4.33) with f 1 = 0. Integrating the latter yields y ≤ which together with y(t 0 ) = 0 implies y(t) = 0 for all t by an easy variant of Gronwall's inequality. Substituting that result into (4.34) (with f 2 = 0) yields z = 0 and therefore (w 1 , s 1 ) = (w 2 , s 2 ).
We now turn to the proof of Parts (2) and (3). Introducing the assumption and notation (4.31), changing the variable from ν to t ′ = νt in the definition of I t 1 m , and omitting an absolute overall constant, we can rewrite (4.18=) (4.19=) in the form where α = [k ∧ 3/2] − 1 > 0, and the goal is to prove that (4.35) (4.36) with suitable initial conditions imply y = z = 0.
Part (3). We now take t 1 = ∞. The term bz in (4.36) can be exponentiated as in the proof of Part (2). Since in addition the statement does not involve conditions on a and b, we can and shall assume without loss of generality that b = 0 and a = 1. Let Then ε(t) is nonincreasing in t and tends to zero as t → ∞. Furthermore for any Let now t 0 ∈ I (t 0 will eventually tend to ∞), y 0 = y(t 0 ) and z 0 = z(t 0 ). We estimate y and z for t ≤ t 0 by integrating (4.35) (4.36) (with t 1 = ∞, a = 1 and b = 0) between t and t 0 . Integrating (4.36) yields where we have used (4.41) and where Substituting (4.42) into (4.35), integrating and using the fact that Y (t) is decreasing in t, we obtain where Substituting (4.44) into (4.43) yields Substituting (4.46) into (4.45) yields the following inequality for Y (t) : . Note that f and g are decreasing in t and that g is integrable at infinity. Let Then (4.47) can be rewritten as and therefore by (4.47) again Now G is independent of t 0 while f tends to zero when t 0 → ∞ for fixed t under the assumptions made. Letting t 0 → ∞ then shows that which implies that y = 0, from which it follows easily that z = 0, and therefore (w 1 , s 1 ) = (w 2 , s 2 ).
⊓ ⊔ Remark 4.1. The necessity of some condition of the type (4.32) in Part (2) is easily understood on the simpler example and therefore by integration However if at 1 = π/2, (4.48) admits the nonvanishing solution y = sin at.
We next prove another property which follows easily from estimates similar to those of Lemma 4.2, namely the fact that for suitably bounded solutions (w, s) of the auxiliary system, w(t) tends to a limit w + when t → ∞.
with (w, t η−1 s) ∈ L ∞ (I, X k,ℓ 0 ) for some η > 0 and let (w, s) satisfy the first equation of the system (2.20). Then there exists w + ∈ H k such that w(t) tends to w + weakly in H k and strongly in H k ′ for 0 ≤ k ′ < k when t → ∞. Furthermore the following estimate holds for all t ∈ I : where w(t) = U(1/t)w, and Proof. Let t 0 ∈ I, w 0 = w(t 0 ) and The first equation of the system (2.20) can be rewritten as where we have omitted the superscript ∞ in B S , and therefore By exactly the same estimate as in (4.23), we obtain We next estimate (3.17) and Sobolev inequalities, and with the same δ. The last norm of B S in (4.53) is estimated exactly as in the proof of Lemma 4.1 (see (4.10) (4.11) (4.12)) as Substituting (4.51) (4.52) (4.53) (4.54) into (4.50) and integrating between t 0 and t yields from which it follows that w(t) and therefore also w(t) has a strong limit w + in L 2 when t → ∞ and that (4.49) holds. Since in addition w(t) is bounded in H k , it follows by a standard compactness argument that w + ∈ H k and that w(t) tends to w + in the other topologies stated in the Proposition.

⊓ ⊔
We now turn to the problem of existence of solutions of the auxiliary system (2.20), with the aim of proving that that system defines an asymptotic dynamics for large times and preferably up to infinity in time. Here however, we encounter the difficulties described in Section 2 and arising from the different propagation properties of the Schrödinger and wave equations. First of all for t 1 = t 0 , even if B 0 = 0, the estimates of Lemma 4.1 are insufficient to prevent blow up of the solutions in a finite time after t 0 , independently of the size of t 0 and of the initial data for (w, s) at t 0 . In fact, if in the estimates (4.1=) (4.2=) we set b 0 = 0, omit the second inequality and take s = 0 in the first one, we obtain the following stronger where m = m 1 + 1/2 > β 1 for βleq1, and p = 2. We shall prove that (4.55) does not prevent finite time blow up by showing that equality in (4.55) implies such a blow up. Taking y p instead of y as the unknown function and rescaling t and y, we can take p = 1, t 0 = 1 and C = 1 without loss of generality. We are therefore led to consider the equation or equivalently The proof will be given in Appendix A.
The previous result encourages us to take t 1 > t 0 and actually the situation improves in that case and in particular we shall prove the existence of solutions Of course for t 1 < ∞, by the previous argument, we shall be unable to exclude finite time blow up after t 1 . On the other hand, if B 0 = 0, we cannot exclude finite time blow up between t 0 and t 1 if t 1 is sufficiently large. Actually, we shall show that equality in a stronger version of (4.1) implies such a blow up. We again drop the inequality (4.2) and take s = 0 in (4.1). Omitting in addition the first term with b 0 , we are left with (4.58) Since the solution of (4.58) is increasing in time for t ≥ t 0 , blow up for (4.58) is implied by blow up for the equation Now if blow up occurs for t ≤ T * for the equation then a fortiori blow up will occur for the equation (4.59) if t 1 ≥ 2T * . It is therefore sufficient to prove blow up for (4.60), which after rescaling can be rewritten as The proof will be given in Appendix A. The condition 2k > β 1 is always satisfied in the present situation.
We now prove the main result of this section, namely the existence of solutions of the auxiliary system (2.20) defined up to t 1 , possibly with t 1 = ∞, for B 0 = 0 and for initial data given at sufficiently large t 0 < t 1 .
where β 2 is defined by (4.5). Let (w 0 , s 0 ) ∈ X k,ℓ and let y 0 = |w 0 | k and z 0 = | s 0 |˙ℓ. Then there exists T 0 < ∞ depending on (y 0 , z 0 ) such that for all t 0 ≥ T 0 , there exists T < t 0 , depending on (y 0 , z 0 ) and on t 0 , such that for all t 1 , t 0 ≤ t 1 ≤ ∞, One can take and the solution (w, s) is estimated for all t ∈ I by |w(t)| k ≤ 2y 0 , (4.64) by Proposition 4.1, is a contraction of a suitable subset of C(I, X k,ℓ ) for a suitably time rescaled norm of L ∞ (I, X 0,ℓ 0 ). We first consider the interval I = [t 0 , t 1 ) and we define the set for Y > 0, Z > 0. Let (w, s) ∈ R and (w ′ , s ′ ) = Γ(w, s). Let y = |w(t)| k , y ′ = |w ′ (t)| k , z = |s(t)|˙ℓ and z ′ = |s ′ (t)|˙ℓ. From Lemma 4.1, namely (4.1) (4.2) with b 0 = 0 and with an overall constant omitted, we obtain (4.66) (4.68) We now impose and choose thereby ensuring that Y ′ ≤ Y , Z ′ ≤ Z, so that the set R is mapped into itself by Γ. The conditions (4.69) can be rewritten as and hold for all t 0 ≥ T 0 for T 0 satisfying (4.62) with suitable C.
We next show that the map Γ is a contraction on R. We use the notation of Lemma 4.2 and in addition and a similar notation for primed quantities. From Lemma 4.2, in particular (4.18) (4.19), and again with an overall constant omitted, we obtain The second inequality in (4.74) reduces to under the first condition in (4.69) imposed previously.
We now ensure that the map Γ is a contraction for the norms defined by (4.72) in the form by taking c = 8β −1 2 Y and imposing the conditions which follow again from (4.62) for all t 0 ≥ T 0 . We have proved that Γ maps R into itself and is a contraction for the norms (4.72). By a standard compactness argument, R is easily shown to be closed for the latter norms. Therefore Γ has a unique fixed point in R, which completes the proof for t ≥ t 0 .
We now turn to the case t ≤ t 0 , namely we consider the interval I = [T, t 0 ] for some T < t 0 . The proof proceeds in exactly the same way, with however slightly different norms. In addition, one has to take into account the following fact : the various integrals I t 1 m that occur in (4.1) (4.2) and (4.18) (4.19) involve w and w 1 , w 2 up to time t 1 . In the subinterval [t 0 , t 1 ], one takes w = w 1 = w 2 = the solution constructed at the previous step (in particular w − = 0 for t ≥ t 0 , so that actually no contribution from the interval [t 0 , t 1 ] occurs in (4.18) (4.19)). In (4.1) (4.2) the contribution of the interval [t 0 , t 1 ] is taken into account by using the fact that all the integrals over time in the relevant I t 1 m are convergent at infinity and that we shall eventually use the same ansatz |w(t)| k ≤ Y = 2y 0 both for t ≤ t 0 and t ≥ t 0 . With this in mind, we complete the proof by simply giving the computational details. We consider the set For (w, s) ∈ R < , (w ′ , s ′ ) = Γ(w, s) and y, y ′ , z and z ′ defined as previously, we estimate by Lemma 4.1, again with an overall constant omitted and by integration from t to t 0 with (y ′ , z ′ )(t 0 ) = (y, z)(t 0 ) = (y 0 , z 0 ), we obtain (4.79) We now impose and choose thereby ensuring that Y ′ ≤ Y , Z ′ ≤ Z so that the set R < is mapped into itself by Γ. The conditions (4.80) can be rewritten as and hold for all t ≥ T with T defined by (4.63) and T 0 satisfying (4.62) for suitable C.
We next prove that Γ is a contraction on R < for the norm in L ∞ (I, X 0,ℓ 0 ). With the notation of Lemma 4.2 and in addition and a similar notation for primed quantities, we obtain from (4.18) (4.19) (4.84) By integration between t and t 0 , we deduce therefrom thereby ensuring the contraction in the form (4.76) which implies (4.77) by taking which hold for all t ≥ T with the choice (4.81) under the conditions (4.62) (4.63).
With the previous estimates available, the proof proceeds as in the case t ≥ t 0 .

Asymptotics and wave operators for the linear system
In this section we study the asymptotic properties of solutions of the linear equation (2.22) in the form (2.24) at the level of regularity of H k with k ≥ 1 for w.
In particular we solve the Cauchy problem at infinity, thereby constructing the wave operators in H k . For the linear equation (2.24), the wave operators in L 2 can be easily constructed by a variant of Cook's method and the construction of the wave operators in H k reduces to a regularity problem for the L 2 wave operators thereby obtained. As a preliminary to that study, we shall first solve the Cauchy problem for the equation (2.24) with finite initial time. We emphasize the fact that in this section we do not strive after any kind of optimality in the treatment of the linear equation, since we are mainly interested in a form of that treatment that can be incorporated in that of the fully interacting system.
Proof. It follows easily from standard arguments and from Lemma 3.2 that w exists and satisfies the properties stated except possibly the estimate (5.1), and we concentrate on the proof of the latter, assuming without loss of generality that |w 0 | k = 1. We first prove (5.1) by induction for integer k ≥ 1. Let 0 ≤ j ≤ k − 1 and y j = ω j w 2 . From (2.24) and from the Leibnitz formula and Sobolev inequalities, we obtain and therefore by (3.17) Substituting the induction assumption for y j and integrating (5.2) between t 0 and t, we obtain . This completes the proof for integer k. Let now k = k 0 + θ with integer k 0 ≥ 1 and 0 < θ < 1. We estimate by Lemma 3.2, with 0 < δ = δ(r) ≤ 1, by (3.17) and Sobolev inequalities. We next interpolate and obtain .
We finally substitute the estimate (5.1) for the integer values k 0 − 1, k 0 , and 1 and integrate between t 0 and t, thereby obtaining which yields (5.1) with ε = 2δ/3.

⊓ ⊔
The fact that a direct H k estimate of the solution does not prevent its H k norm to increase as a power of t is a warning of the fact that the construction of the wave operators at that level of regularity is not trivial. The same fact appeared already in Section 4 above in Warning 4.2 and compelled us to assume B 0 = 0 in Proposition 4.4.
We next construct the L 2 -wave operators for (2.24). (1) Let W ∈ C(I, L 2 ) with U(1/t)W ∈ C 1 (I, L 2 ), satisfying for some λ 0 > 0 and for all t ∈ I. Then there exists a unique solution w ∈ C(I, L 2 ) of the equation (2.24), such that w(t) − W (t) tends to zero strongly in L 2 when t → ∞. Furthermore, for all t ∈ I, The solution w is the norm limit in L ∞ (I, L 2 ) as t 0 → ∞ of the solution w t 0 of the equation (2.24) with initial condition w t 0 (t 0 ) = W (t 0 ) obtained in Proposition 5.1, and the following estimate holds for all t ∈ I : (2) Let in addition W ∈ L ∞ (I, H k ) for some k, 0 < k < 3/2. Then there exists w + ∈ H k such that W (t) tends to w + strongly in L 2 and weakly in H k when t → ∞, and the following estimate holds for all t ∈ I : Conversely let w + ∈ H k for some k, 0 < k < 3/2, and let W 1 = U * (1/t)w + . Then W 1 satisfies the assumptions of Part (1) with λ 0 = k/3.
Let W , w + and W 1 be related as above. Then the solutions of the equation (2.24) constructed in Part (1) from W and W 1 coincide.
(3) Let w + ∈ L 2 . Then the equation (2.24) has a unique solution w ∈ C(I, L 2 ) such that w(t) tends to w + strongly in L 2 when t → ∞.
Proof. Part (1). Following the sketch of Section 2, we look for w in the form w = W + q, so that q satisfies the equation and therefore the a priori estimate Define w t 0 as in Part (1) and let w t 0 = W + q t 0 so that q t 0 (t 0 ) = 0. Integrating (5.9) between t 0 and t yields and therefore, by L 2 norm conservation for the difference of two solutions for any t 0 and t 1 , 1 ≤ t 0 , t 1 < ∞. This proves convergence of q t 0 and therefore of w t 0 in norm in L ∞ (I, L 2 ). Let w be the limit of w t 0 . Taking the limit t 0 → ∞ in (5.10) yields (5.5), while taking the limit t 1 → ∞ in (5.11) yields (5.6). Clearly w satisfies the equation (2.24).
Part (2). W satisfies the equation From (3.17) and Sobolev inequalities, we obtain where a = W ; L ∞ (I, H k ) and k = δ(r). Integrating (5.12) between t 1 and t 2 and using (5.4) and (5.13), we obtain for any t 1 and t 2 , 1 ≤ t 1 , t 2 < ∞. Therefore U(1/t)W (t) and therefore also W (t) has a strong limit w + in L 2 , and from which (5.7) follows. Furthermore by a standard compactness argument, w + ∈ H k with |w + | k ≤ a and W (t) tends to w + weakly in H k . Let now w + ∈ H k and W 1 = U * (1/t)w + . Then by (5.13). The last statement follows from the fact that W and W 1 have the same limit w + in L 2 and from L 2 norm conservation for the equation (2.24). Part (3) follows from Parts (1)  (1) Let w be the solution of the equation (2.24) obtained in Proposition 5.2 part (1). Then w ∈ C(I, H k ) and w satisfies the estimates (5.5) and for all t ∈ I. Proof. Part (1) will be proved together with the limiting properties stated in Part (2). We know from Proposition 5.1 that w t 0 ∈ C(I, H k ). The main point of the proof consists in estimating q t 0 = w t 0 − W in H k uniformly in t 0 for t ≤ t 0 . We know already from (5.10) that so that by Lemma 3.2, in the same way as in (4.7), with 0 < δ = δ(r) < k ∧ 3/2. Using the estimates (3.17), Sobolev inequalities and interpolation with the help of (5.20) and the assumption (5.18), we obtain We now define Y ≡ t λ y; L ∞ ([1, t 0 ]) , substitute that definition into the RHS of (5.23), integrate the latter between t and t 0 , and obtain provided λ + ν 1 > 0 and λ + ν 2 > 0. We impose in addition ν 1 ≥ 0, ν 2 ≥ 0, multiply (5.24) by t λ , take the Supremum over t and obtain which is uniform in t 0 . The condition ν 1 ≥ 0 reduces to and can be satisfied for λ 0 > λ + k by taking δ sufficiently small. It implies ν 2 > 0.
Changing the notation to Assuming δ ≤ k − 1 without loss of generality and using for x > 0, ε > 0 and 0 < θ < 1, we obtain from (5.29) which completes the proof of the estimate of q t 0 in H k uniformly in t 0 for t ≤ t 0 .
Let nowT < ∞ and J = [1,T ]. We know from (5.30) that w t 0 is estimated in L ∞ (J, H k ) uniformly in t 0 for t 0 ≥T and that w t 0 converges to w in norm in L ∞ (J, L 2 ) by Proposition 5.2 part (1). It follows therefrom by a standard compactness argument that w ∈ (L ∞ ∩ C w )(J, H k ), that w also satisfies the estimate (5.30) and that w t 0 converges to w in the topologies described in Part (2). Strong continuity of w in H k follows from Proposition 5.1.

⊓ ⊔
In order to complete the construction of the wave operators in H k , we now have to construct model functions W satisfying the assumptions (5.4) and (5.18). In view of Proposition 5.2 part (2), we restrict our attention to W of the form for some fixed w + ∈ H k + and we take k + > 3/2. With that choice, we obtain However, with no further assumptions on w + , we are restricted to λ 0 ≤ 1/2 and consequently to k < 1/2. In fact, from (3.17) we obtain Furthermore, from Lemma 3.2 and (3.17) we obtain for k < 1/2 with 0 < δ = δ(r) ≤ 3k < 3/2. Together with an extension of Proposition 5.2 to k ≤ 1/2, which we have not performed, the estimates (5.33) (5.34) would allow us to complete the construction of the wave operators for 0 < k < 1/2, with λ 0 = 1/2 and 0 < λ < 1/2 − k.
In order to cover higher values of k, and in particular for k > 1, as will be needed for the nonlinear system (1.1) (1.2), we shall need additional conditions on w + and B 0 . We first exhibit a set of local sufficient conditions in the form of joint decay estimates for w + , and B 0 , where the nonlocal operator U * (1/t) no longer appears.
Proof. The results follow from Propositions 5.2 and 5.3 and from Lemma 5.1.

⊓ ⊔
The existence of the wave operators for u in the usual sense at the corresponding level of regularity is an easy consequence of Proposition 5.4. We refrain from giving a formal statement at this stage. The same question will be considered in Section 8 in the case of the interacting system (1.1) (1.2).

Cauchy problem at infinity for the auxiliary system
In this section we begin the construction of the wave operators for the auxiliary system (2.20) by solving the Cauchy problem at infinity for that system in the difference form (2.30), for large or infinite initial time, and for a given choice of (W, S) satisfying a number of a priori estimates. The construction of (W, S) satisfying those estimates is deferred to the next section. In the same spirit as in Section 4, we solve the system (2.30) in two steps. We first solve the linearized version of that system (2.31), thereby defining a map Γ : (q, σ) → (q ′ , σ ′ ). We then show that this map is a contraction in suitable norms on a suitable set.
The basic tool of this section again consists of a priori estimates for suitably regular solutions of the linearized system (2.31). In order to handle efficiently a non-vanishing B 0 , those estimates have to be much more elaborate than those of Section 4.
Remark 6.1. All the norms of (q, σ) and (q ′ , σ ′ ) that appear in (6.2)-(6.4) are controlled by the norms in X k,ℓ through Sobolev inequalities. Furthermore all the integrals I m are convergent if t 1 = ∞. This follows from boundedness of q in H k in all cases where m > −1/2, namely in all cases but two. The exceptions are in (6.4) and in (6.2), both of which are controlled under the additional assumption that q ∈ L 2 (I, L 2 ).
Finally it is easy to see by estimates similar to, but simpler than, those of Lemma 4.1 that all the norms of the remainders R 1 and R 2 that occur in (6.2)-(6.4) are finite under the assumptions made on (W, S).
Proof of Lemma 6.1. In all the proof, the time superscript in B S , B L and in the various I m is omitted, except in dubious cases. That time superscript is in general Proof of (6.2). From (2.31), we estimate We next estimate by (3.8) (3.10) and Sobolev inequalities Substituting (6.6) (6.7) (6.8) into (6.5) yields (6.2).
Proof of (6.3). From (2.31), we estimate and we estimate the various terms in the RHS successively. The contribution of B 0 is estimated by Lemma 3.2 and by (3.17) exactly as in Section 4 (see (4.7)) and yields The contribution of Q(s, q ′ ) is estimated by (4.8) as The contribution of Q(σ, W ) is estimated by Lemma 3.2 and Sobolev inequalities as The contribution of B S with w = W + q yields a number of terms which we order by increasing number of q or q ′ occurring therein. We first expand By Lemma 3.2 and Sobolev inequalities, we estimate where we take 0 < δ = δ(r) = (k − 1) ∧ 1/2 so that with θ = 1/(2k − 1), and by (3.10) and Lemma 3.2 which we use with m = 1, k + 1/2 and k + 1/2 − δ. Substituting those estimates into (6.13) yields (6.14) In a similar way, we estimate (6.15) By Lemma 3.2 again and by (3.10) where q 2 can be omitted for m ≤ 1/2. Substituting (6.16) with m = 1/2 and m = k − 1/2 into (6.15) yields By Lemma 3.2 and Sobolev inequalities again, we next estimate followed by (see also (3.10)) We now turn to the second contribution of B S to (6.9). By Lemma 3.2 and Sobolev inequalities again and by (6.16) with m = k − 1 and m = 0 Similarly followed by (see the proof of (6.19)) yields ω k (B S (q, q)W ) 2 ≤ C a I k−1 ω k q 2 q 3 + I 0 ( ∇q 2 q 3 ) . (6.23) Substituting (6.10) (6.11) (6.12) (6.14) (6.17) (6.19) (6.21) and (6.23) into (6.9) and reordering the contributions of B S by increasing powers of (q, q ′ ) yields (6.3).
⊓ ⊔ We next estimate the difference of two solutions of the linearized system (2.31) corresponding to two different choices of (q, σ), but to the same choice of (W, S).
and the superscript t 1 is again omitted in the estimating functions I m .
Proof. Taking the difference of the system (2.31) for (q ′ i , σ ′ i ), or equivalently and more simply rewriting (4.21) (6.32) We first estimate q ′ − . From (6.32) we obtain We expand (6.33) by using and we estimate the various terms successively. From (6.6) we obtain By the same estimates as in the proof of (4.23), we next obtain We next estimate by (6.7) (6.8) The remaining terms are new. Using (3.10) and Sobolev inequalities, we obtain Substituting (6.34)-(6.39) into (6.33) yields (6.29).
⊓ ⊔ We now begin the study of the Cauchy problem for the auxiliary system in the difference form (2.30) and for that purpose we first study that problem for the linearized version of that system. For finite initial time t 0 , that problem is solved by a minor modification of Proposition 4.1. The following proposition is simply a compilation of that result and of Lemmas 6.1 and 6.2.
We shall be eventually interested in solving the Cauchy problem for the auxiliary system (2.30) with infinite initial time t 0 . As a preliminary, we need to solve the same problem for the linearized system (2.31). This is done in the following proposition, which of course requires much stronger assumptions on the asymptotic behaviour in time of (W, S) and (q, σ). With the study of the nonlinear system in view, we already make the assumptions that will be needed for that purpose, although they could be slightly weakened for the linear problem. Since we want to take t 0 = ∞, we also take t 1 = ∞. Proposition 6.2. Let 1 < k ≤ ℓ, ℓ > 3/2. Let β, λ 0 and λ satisfy 0 < β < 1 , λ > 0 , λ 0 > λ + k , λ 0 > β(ℓ + 1) . for some η > 0 and for 0 ≤ m ≤ ℓ + 1, for all t ∈ I. Let (q, σ) ∈ C(I, X k,ℓ ) satisfy for all t ∈ I. Then the system (2.31) has a (unique) solution (q ′ , σ ′ ) ∈ C(I, X k,ℓ ) satisfying T , for all t ∈ I. That solution satisfies the estimates (6.2) (6.3) (6.4) of Lemma 6.1 for all t ∈ I. Two such solutions (q ′ i , σ ′ i ) associated with (q i , σ i ), i = 1, 2, satisfy the estimates (6.29) (6.30) of Lemma 6.2 for all t ∈ I. The solution (q ′ , σ ′ ) is actually unique in C(I, X k,ℓ ) under the condition that (q ′ , σ ′ ) tends to zero in X 0,0 norm when t → ∞.
Proof. The proof consists in showing that the solution (q ′ t 0 , σ ′ t 0 ) of the linearized system (2.31) with t 1 = ∞ and with initial data (q ′ t 0 , σ ′ t 0 )(t 0 ) = 0 for finite t 0 , obtained from Proposition 6.1, satisfies the estimates (6.51) (6.52) uniformly in t 0 for t ≤ t 0 (namely with Y ′ 0 , Y ′ and Z ′ independent of t 0 ), and that when t 0 → ∞, that solution converges on the compact subintervals of I uniformly in the norms considered in Lemma 6.2.
We next estimate y ′ , wasting part of the time decay in order to alleviate the computation. In particular when estimating s = S + σ, we use the fact that the time decay of σ is better than that of S by at least a power 1 − η. Furthermore in the contributions coming from B S , we eliminate Y 0 and λ 0 by using Y 0 ≤Ȳ = Y ∨Y 0 and λ 0 > λ + k. In particular we estimate for 0 ≤ m ≤ k, and similarly Substituting (6.47) (6.49) (6.50) into (6.3), using (6.57) (6.58) and omitting an overall constant, we obtain The terms linear in y ′ in the RHS of (6.59) can be eliminated by changing variables from y ′ to y ′ exp(−E(t)), where so that it is sufficient to estimate y ′ from (6.59) with those terms omitted and to multiply the end result by exp(E(t)). With those terms omitted, and with the help of the estimate of y ′ 0 , (6.59) can be rewritten as In particular C 1 (t) is decreasing in t.
The inequality (6.61) is essentially identical with (5.24), up to notational change and replacement of c 1 by C 1 (t). Proceeding as in Section 5, defining and reintroducing the factor exp(E(t)), we obtain (see (5.30)) an estimate which is again manifestly uniform in t 0 . This completes the proof of (6.51).
Clearly (q ′ , σ ′ ) satisfies the system (2.31). This completes the existence part of the proof.

⊓ ⊔
We now turn to the main result of this section, namely the fact that for T sufficiently large (depending on (W, S)), the auxiliary system in difference form (2.30) has a solution (q, σ) defined for all t ≥ T and decaying at infinity in a suitable sense. In the same spirit as for Proposition 4.4, this will be done by showing that the map Γ : (q, σ) → (q ′ , σ ′ ) defined by Proposition 6.2 is a contraction in suitable circumstances. According to our intuition of scattering, another natural route towards the same result would be to construct first the solution (q t 0 , σ t 0 ) of the auxiliary system (2.30) vanishing at t 0 and to take the limit of that solution as t 0 → ∞. That route can also be followed, but it is slightly more complicated than the previous one. One of the complications comes from the fact that the system (2.30) depends on t 1 . In view of Warning 4.2, for finite t 0 , we expect difficulties if we take t 1 > t 0 . This prompts us to take t 1 = t 0 . The comparison of two solutions (q t 0 , σ t 0 ) corresponding to different values of t 0 is then complicated by the fact that they do not solve exactly the same system, so that Lemma 6.2 is not directly applicable and additional terms occur in the comparison. On the other hand, for B 0 = 0, the construction of the solution (q t 0 , σ t 0 ) of (2.30) is expected to meet difficulties for t ≥ t 0 because of Warning 4.1. We shall therefore undertake it for t ≤ t 0 only, which is sufficient anyway to take the limit t 0 → ∞. That construction proceeds again by a contraction starting from the solutions obtained for the linearized system. The corresponding proof for t 0 < ∞ is not significantly simpler than for t 0 = ∞, which is another reason why the second method is more complicated than the first one, since in addition to that construction, a limiting procedure is needed.
We now state the main result and formalize the previous heuristic discussion in the following proposition. (1) For all t 0 , T ≤ t 0 < ∞, the system (2.30) with t 1 = t 0 has a unique solution (q, σ) ∈ C(I, X k,ℓ ) with I = [T, t 0 ] and (q, σ)(t 0 ) = 0. That solution satisfies the estimates (6.49) (6.50) for all t ∈ I.
Proof. Parts (1) and (2). We prove Parts (1) and (2) together, because the proof is exactly the same for both. It consists in showing that the map Γ : (q, σ) → (q ′ , σ ′ ) defined by solving the linearized system (2.31) is a contraction on a suitable subset of C(I, X k,ℓ ) in the lower norms used in Lemma 6.2. For t 0 < ∞, the map Γ is defined by Proposition 6.1, restricted to those (q, σ) satisfying (q, σ)(t 0 ) = 0, with the initial data (q ′ , σ ′ )(t 0 ) = 0. For t 0 = ∞, the map Γ is defined by Proposition 6.2. The relevant estimates on Γ are those derived in the proof of Proposition 6.2 in the case t 1 = ∞. The same estimates also apply to the case t 0 = t 1 < ∞, which is relevant for Part (1) of this proposition. They are independent of t 0 .
We next show that Γ is a contraction on R in the norms considered in Lemma 6.2. Let (q i , σ i ) ∈ R and (q ′ i , σ ′ i ) = Γ(q i , σ i ), i = 1, 2, and define (q ± , σ ± ) and (q ′ ± , σ ′ ± ) as in Lemma 6.2. We define in addition and we make similar definitions for the primed quantities. We take ℓ 0 = [3/2 − k] + and estimate y ′ − and z ′ −m by (6.29) (6.30), taking advantage of the fact that m ′ = m in (6.30) for that choice of ℓ 0 . Using the fact that Γ maps R into itself and omitting again overall constants, we obtain for 0 ≤ m ≤ ℓ 0 . Integrating (6.82) (6.83) from t to t 0 with (y ′ − , z ′ −m )(t 0 ) = 0 and using again the fact that λ 0 > λ + k > 1 and λ 0 > β(ℓ + 1) > β((ℓ 0 We now ensure that the map Γ is a contraction for the norms defined by (6.80) by taking c = 8β −1 a and T sufficiently large, depending on the parameters listed in the proposition, in part explicitly and in part through Y 0 , Y and Z defined by (6.78). (It is only at this point that we need the condition 1 + λ > β(5/2 − k), in order to ensure that the power of T in the second term in the RHS of (6.84) is negative). We have proved that for sufficiently large T , the map Γ maps R defined by (6.77) into itself and is a contraction for the norms (6.81). By a standard compactness argument, R is closed for the latter norms, and therefore Γ has a unique fixed point in R, which completes the existence part of the proof of Parts (1) and (2). The uniqueness statement of Part (1) is a special case of Proposition 4.2 part (1), while the uniqueness statement of Part (2) follows from Proposition 4.2 part (3) and from the fact that λ 0 > 1 > β 2 . Part (3). Let T < t ′ 1 < t ′ 2 < ∞ and let (q i , σ i ), i = 1, 2, be the solutions of the ssytem (2.30) obtained in part (1) and corresponding to t 0 = t 1 = t ′ i respectively.
We integrate those inequalities from t to t 1 , with initial condition at t 1 estimated by (6.88). We then substitute the result in (6.97) and omitting an overall constant, we obtain finally (see (6.84) (6.85)) (6.99) Proceeding as above, we deduce therefrom that for T sufficiently large and for a (6.100) and ) . (7.7) The remainders then become Note that the term with B 0 + B S (W, W ) in (7.1) is regarded as short range and not included in the definition of (W, S).

⊓ ⊔
We next estimate the terms in R 1 containing B 0 .
Then the following estimate holds for all m, 0 ≤ m ≤ 2, and all t ≥ 1.
Proof. The contribution of w 0 to (7.23) is estimated by Lemma 5.1 withm = 2. In order to estimate the contribution of w 1 , we decompose We first consider by (5.35) and (7.10).
We next estimate the contribution of w ′′ 1 . By the same estimates as for w 1 (see the proof of (7.11)) we obtain where we have used the fact that the factors (1 − U ( * ) (1/t)) can be replaced by t −1 ∆ for the purpose of the second estimate, and therefore ω m w ′′ 1 2 ≤ C a 3 + t −2+m/2 (1 + ℓn t) for 0 ≤ m ≤ k + − 1 (7.27) by interpolation. By Lemma 3.2 and (3.17) we then obtain (1 + ℓn t) .
⊓ ⊔ We can now collect Proposition 6.3 and Lemmas 7.1 and 7.2 to obtain the main technical result on the Cauchy problem for the auxiliary system in the difference form (2.30). We again keep the assumptions on B 0 in the implicit form of the estimates (3.17) and (5.35), which can however be replaced by sufficient conditions on (w + , A + ,Ȧ + ) by the use of Lemmas 3.5 and 5.2.
(2) All the statements of Proposition 6.3 hold.
Proof. It follows from Lemmas 7.1 and 7.2 that all the assumptions of Proposition 6.3, and in particular the estimates (6.45)-(6.48), are satisfied.

Wave operators and asymptotics for (u, A)
In this section we complete the construction of the wave operators for the system (1.1) (1.2) and we derive asymptotic properties of solutions in their range. The construction relies in an essential way on Proposition 7.1. So far we have worked with the system (2.20) for (w, s) and the first task is t o reconstruct the phase ϕ.
In order to state the regularity properties of u that follow in a natural way from the previous construction, we introduce appropriate function spaces. In addition to the operators M = M(t) and D = D(t) defined by (2.8) (2.9), we introduce the operator For any interval I ⊂ [1, ∞) and any k ≥ 0, we define the space where < λ >= (1 + λ 2 ) 1/2 for any real number or self-adjoint operator λ and where the second equality follows from (8.7).
We now collect the information obtained for the solutions of the system (1.1) (1.2) and state the main result of this paper as follows.

(8.23)
A similar result holds for k = 3/2 with a t ε loss in the decay.
⊓ ⊔ We conclude this section with some remarks on variations which can be made or attempted in the formulation of Proposition 8.1.
Remark 8.1. We have stated the assumptions on (A + ,Ȧ + ) in an implicit way in the form of conditions on the solution A 0 of the free wave equation that they generate. Sufficient conditions for (8.9) and (8.10) to hold directly expressed in terms of (A + ,Ȧ + ) and possibly w + can be found in Lemma 3.5 and Lemma 5.2, but those conditions are far from being optimal (especially those of Lemma 5.2).
Remark 8.2. We have stated the result for the partly resolved system (1.1) (2.2) instead of the original system (1.1) (1.2). This includes the fact that the initial time for (1.2) is t 1 = ∞ from the outset and makes it unnecessary to specify the uniqueness class for A, since (1.2) is solved explicitly by (2.2). On the other hand, the available regularity for A is stronger than stated, as follows from the assumption (8.9) on A 0 , from simple estimates on A ∞ 1 (|DW | 2 ) and from the remainder estimates (8.19)-(8.23).
Remark 8.3. The asymptotic behaviour of the scalar field A differs in an important way from that of a solution of the free wave equation. Omitting the remainder A 2 which tends to zero in a sense made precise by the estimates (8.19)-(8.23), one is left with A ∼ A 0 + A ∞ 1 (|DW | 2 ) and the last term is easily estimated by ω m A ∞ 1 (|DW | 2 ) 2 ≤ C(a + ) t −m+1/2 for 0 ≤ m ≤ k + . Replacing W by w + as a first approximation in that term, one obtains with B ∞ 1 (w + , w + ) constant in time. This yields a contribution to A which spreads by dilation by t and decays as t −1 in L ∞ norm. That contribution can in no obvious sense be regarded as small as compared with A 0 . Remark 8.4. One might be tempted to look for simpler asymptotic forms for u and for A by replacing for instance W by w + in (8.15) (8.16) (8.18) and/or by omitting a few factors U ( * ) (1/t) in (7.5) (7.7). This however would introduce errors at least O(t −1 ) and spoil the t −λ 0 decay in (8.12)  or equivalently y(t) ≥ y 0 exp t 1 dt ′ y(t ′ ) t ′−1−β 1 ℓn(t/t ′ ) . (A.5) We now describe the determination of (a n+1 , α n+1 ) = (a ′ , α ′ ) from (a n , α n ) = (a, α). Substituting the induction assumption into (A.5), we obtain for α > β 1 (a condition that will be ensured below) The minimum over τ of the last member of (A.6) is attained for aτ = a + 1, and it suffices to impose ℓn(a ′ /y 0 ) ≤ θ(1 − ( a + 1)ℓn( a + 1)/ a) ≡ θ(ℓn a − f ( a)) which allows us to take a ′ = y 0 a θ provided f ( a) ≡ ( a + 1)ℓn( a + 1) − a ℓn a − 1 ≤ 0 a condition which is easily seen to hold for a ≤ 1/2.
It follows from (A.10) (A.13) that u n+1 ≥ u 2 n and in particular u n ≥ 1 for all n if u 1 ≥ 1, namely if t is sufficiently large in the sense that t α 1 −2β 1 ≥ (a 1 y 0 ) −1 64(α 1 − β 1 ) 4 . (A.14) For such t, the condition u n ≥ 1 can be rewritten as y(t) ≥ a n t αn ≥ t 2β 1 y −1 0 64(α n − β 1 ) 4 ≥ 4t 2β 1 y −1 0 2 4n (α 1 − 2β 1 ) 4 . (A. 15) Since the last member of (A.15) tends to infinity with n, such a t cannot be smaller than T * , which proves finite time blow up. ⊓ ⊔ Remark A1. Since the RHS of (4.56) and (4.57) is decreasing in β 1 and increasing in m, blow up in finite time for (β 1 , m) implies blow up in finite time for (β ′ 1 , m) with β ′ 1 ≤ β 1 and for (β 1 , m ′ ) with m ′ ≥ m, while the opposite situation prevails as regards the existence of global solutions. Actually it is easy to see that (4.56) or (4.57) admits global solutions for small data if β 1 > 0 and m ≤ β 1 . When coming back to the original equation (4.55), the condition of small data becomes a condition of large t 0 .
Proof of Warning 4.2. We want to prove finite time blow up for (4.61) with y(t 0 ) = y 0 > 0. Omitting the second term in the RHS and integrating the remaining inequality, we obtain (A. 16) We next keep (A. 16), omit the first term in the RHS of (4.61) and change t to t + t 0 . It is then sufficient to prove blow up for (A.17) For that purpose, we show by induction that y satisfies y(t) ≥ y n (t) ≥ a n t αn (t + t 0 ) −(1+β 1 )γn , (A. 18) starting with a 0 = 1, α 0 = k and γ 0 = 0 given by (A.17). We obtain y ≥ a n t αn (t 0 + t) −(1+β 1 )γn ≥ (k + 1/2) 1/2 3 n/2+3/4 t −1/2 (t 0 + t) (1+β 1 )/2 . (A.27) Since the last member of (A.27) tends to infinity with n, such a t cannot be smaller than the maximal time T * of existence of the solution y of (4.61), which proves finite time blow up.