Theory of potential scattering, taking into account spatial anisotropy

Abstract

We get new tests for the existence and completeness of wave operators under perturbation of a pseudodifferential operator with constant symbol P(ξ) by a bounded potential v(x). The term anisotropic is understood in the sense that the growth of P(ξ) as ξ→∞ and the decrease of v(x) as x→∞ can depend essentially on the direction of the vectors ξ and x respectively. This permits us to include in the sphere of applications of the abstract scattering theory of a nonelliptic unperturbed operator the D'Alembert operator, an ultrahyperbolic operator, nonstationary Schrödinger operator, etc. In view of the anisotropic character of the assumptions on the potential, the results obtained are new even in the elliptic case. As an example we consider a Schrödinger operator with potential close to the energy of a pair of interacting systems of many particles.