Physics Letters A 182 ( 1993) 207-2 13 North-Holland zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA PHYSICS LETTERS A Reactive scattering with exact propagators B. Gaveau zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Universitk PM. Curie, Mathematiques, Tour 45-46, 5” etage, 75252 Paris Cedex 05, France and L.S. Schulman Physics Department, Clarkson University, Potsdam, NY 13699-5820, USA Received 20 August 1993; accepted Communicated by J.P. Vigier for publication 14 September 1993 We study the scattering of one particle off another while one of them is bound to a fixed potential. For a heavy, projectile, we obtain comprehensive results using a certain exact time dependent propagator, when that is available. fast moving, The scattering of one particle off another, while one of them is bound to a third particle, is the subject of a substantial literature [ 1- 15 ] and plays a role at many scales of physics, including molecular, atomic, nuclear and subnuclear. There is no completely solved example of such a scattering, which is not surprising considering the essential three-body nature of the problem and the variety of phenomena addressed: elastic and inelastic scattering, ionization and pickup or stripping. We here present a formal structure and a model in which a relatively explicit solution can be provided. The key to our method is the use of explicit time dependent propagators. This allows one of the dynamical processes to be treated as solved, allowing, under appropriate circumstances, a convergent perturbation theory for the other. In this paper, we begin from the most simplified example, and, after exploring its properties, indicate how more realistic situations can be handled. A particle A, with coordinate x (in one or more dimensions), is bound to a force center by a potential PV( x). A second particle, B, which is to be thought of as massive and rapidly moving, passes near the first. Its co- ordinate is X and it interacts with A though a potential V(x- X). In the first instance, we take X to be a fixed linear function of time; that is, we ignore dynamical features of the projectile (B) as well as changes in its trajectory due to the scattering. At this level, only A is treated dynamically and its Hamiltonian is 3% &p2+ W(x) + V(x-X(t)) ) (1) with X(t)= -L + (t + 4 T) u and u = 2L/ T. This simplified representation of the originally posed problem still has no known exact solutions. To proceed, we assume that the propagator&r the zyxwvutsrqponmlkjihgfedcbaZYXWVUTS potential V(x) alone, is known. That is, let G be the propagator for the Hamiltonian H= 1 j&-p2+ VW l (2) As usual, G(x, t; y) = S(t) (x 1 exp ( - iHt/A) 1 y) . For our purposes, we require the propagator for V(x- X( t) ) which can be obtained from G by a Galilean transformation. The Galilean transformation [ 163 ] is more subtle 0375-9601/93/$ 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved. 207