Scattering theory in the energy space for a class of nonlinear Schrödinger equations. (English) Zbl 0535.35069

This paper is devoted to the theory of scattering for a class of nonlinear Schrödinger (NLS) equations \((1)\quad i{\dot \phi}=- (1/2)\Delta \phi +f(\phi)\) with \(\phi\) a complex function defined in space-time \({\mathbb{R}}^{n+1}\) and \(f\) a nonlinear interaction satisfying suitable power bounds at zero and infinity. The main purpose is to prove the existence of the wave operators and, in space dimension \(n\geq 3\), asymptotic completeness for repulsive interactions, for all initial data and solutions in the energy space, namely the Sobolev space \(H^ 1\). The existence of the wave operators is proved by solving the Cauchy problem at infinity in the form of the integral equation \[ \phi(t)=\exp(\frac{i}{2}t\Delta)\phi_{\pm}-i\int^{t}_{\pm \infty}d\tau \exp(\frac{i}{2}(t-\tau)\Delta)f(\phi(\tau)). \] That equation is solved by a contraction method in a Banach space \({\mathcal X}_ 0\) of functions of time with values in a space \(X\supset H^ 1\), exhibiting a suitable time decay expressed in terms of integrability properties in time. It follows from a slight extension of inequalities due to Strichartz that the solutions of the free equation \(i{\dot \phi}=- (1/2)\Delta \phi\) with initial data in \(H^ 1\) satisfy that time decay and belong to \({\mathcal X}_ 0\). As a consequence, the previous argument yields the existence of the wave operators in all \(H^ 1\). It yields also, as a byproduct, asymptotic completeness for small data in \(H^ 1\). Asymptotic completeness (for general data) amounts to the fact that all solutions of the Cauchy problem with finite initial time satisfy the previous time decay. It is proved here for \(n\geq 3\), for repulsive interactions (in a suitable sense), for arbitrary initial data in \(H^ 1\), by an extension of a method first used by Morawetz and Strauss (1972) for the massive nonlinear Klein Gordon (NLKG) equation and by Lin and Strauss (1978) the NLS equation for more restricted interactions and initial data. The basic ingredients consist of two inequalities: one of them follows from the approximate dilational invariance of the equation (1) and the other one replaces for the NLS equation the finiteness of the propagation speed of the NLKG equation. Those inequalities are exploited through the integral equation \[ \phi(t)=\exp(\frac{i}{2}t\Delta)\phi_ 0-i\int^{t}_{0}d\tau \exp(\frac{i}{2}(t-\tau)\Delta)f(\phi(\tau)) \] which is equivalent to the Cauchy problem with finite (here zero) initial time, by using sharp estimates on the integrand, the proof of which requires the use of Besov spaces. The assumptions on f under which asymptotic completeness holds in \(H^ 1\) cover the case of a sum of powers \(\lambda \phi | \phi |^{p-1}\) with \(\lambda\geq 0\) and \(4/n<p-1<4/(n-2)\), \(n\geq 3\).


35P25 Scattering theory for PDEs
35G20 Nonlinear higher-order PDEs
35L70 Second-order nonlinear hyperbolic equations
35B45 A priori estimates in context of PDEs
47H10 Fixed-point theorems