 
Summary: Asymptotic Analysis 50 (2006) 1330 13
IOS Press
Structure of the short range amplitude
for general scattering relations
Ivana Alexandrova 1
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: alexandr@math.toronto.edu
Abstract. We consider scattering by short range perturbations of the semiclassical Laplacian. We prove that when a polynomial
bound on the resolvent holds the scattering amplitude is a semiclassical Fourier integral operator associated to the scattering
relation near a nontrapped ray. Compared to previous work, we allow the scattering relation to have more general structure.
Keywords: short range perturbations, scattering amplitude, scattering relation, semiclassical Fourier integral operators
1. Introduction and statement of results
We study the structure of the scattering amplitude associated to the semiclassical Schrödinger opera
tor with a short range potential on Rn. We prove that, when restricted away from the diagonal on Sn×Sn,
the natural scattering amplitude quantizes the scattering relation in the sense of semiclassical Fourier
integral operators. The scattering relation at energy > 0 here is given roughly by the Hamiltonian
flow of the symbol p of the operator between two hypersurfaces "at infinity" inside the energy surface
{p = }.
1.1. A survey of earlier results
The structure of the scattering matrix has been of significant interest to researchers in mathematical
