JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 134, 471481 (1988) Non-autonomous Classical Scattering M. A. ASTABURUAGA* AND C. FERNANDEZ* Facultad de Matemdticas, Ponttjkia Universidad Cutdlica de Chile, Casilla 6 177, Santiago, Chile AND VICTOR H. CURT& Departamento de Matemriticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile Submitted by A. G. Ramm Received February 9, 1987 1. INTRODUCTION In this paper we study the scattering theory for a classical particle moving in R” under the influence of a time dependent force f(x(t), t), where x(t) is the position of the particle at time t. We shall assume that its mass is one so that the dynamics is given by Newton’s equation, -f(t) =f(x(t), th (1) with initial conditions x(0) = x0 E R”, a(O) = u0 E R”. We essentially follow the ideas of Simon [S], who studied the case when the force depends only on the position (the autonomous case). As usual, we define a scattering state as the data (x, (0) a,(O)) such that, as t -+ +_m, the corresponding solution x+(t) is asymptotic to a solution CI + + h t f of the free equation ji = 0, with b + # 0. In the autonomous case, there is an alternative approach [ 1, 33 in which the problem is studied in L* of the phase space. In our case, one could try to formulate the problem in the context of the time-dependent Scattering theory in Hilbert spaces (see the references given in [4, Section X1.4]), but we have chosen to work directly on phase space. In Section 2 we show, under certain conditions on the force f(x, t), that, given (a, b) E R” x R” with b # 0, there exists a unique solution x(t) of (1) * Partially supported by DIUC, Pontificia UniversidadCat6lica de Chile. 471 0022-247X/88 33.00 Copynghl 0 1988 by Academic Press, Inc All righta of reproduction m any form reserved.