The Scattering Amplitude for the Schrödinger Equation with a Long-Range Potential

Abstract:

We consider the Schrödinger operator with a long-range potential V(x) in the space . Our goal is to study spectral properties of the corresponding scattering matrix and a diagonal singularity of its kernel (the scattering amplitude). It turns out that in contrast to the short-range case the Dirac-function singularity of at the diagonal disappears and the spectrum of the scattering matrix covers the whole unit circle. For an asymptotically homogeneous function V(x) of order we show that typically , where the module w and the phase ψ are asymptotically homogeneous functions, as , of orders and , respectively. Leading terms of asymptotics of w and ψ at are calculated. In the case ρ=1 our results generalize (in the limit ) the well-known formula of Gordon and Mott.

As a by-product of our considerations we show that the long-range scattering fits into the theory of smooth perturbations. This gives an elementary proof of existence and completeness of wave operators in the theory of long-range scattering. In this paper we concentrate on the case ρ>1/2 when the theory of pseudo-differential operators can be extensively used.