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Asymptotic Analysis 50 (2006) 1330 13 Structure of the short range amplitude

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
E-mail: alexandr@math.toronto.edu
Abstract. We consider scattering by short range perturbations of the semi-classical Laplacian. We prove that when a polynomial
bound on the resolvent holds the scattering amplitude is a semi-classical Fourier integral operator associated to the scattering
relation near a non-trapped ray. Compared to previous work, we allow the scattering relation to have more general structure.
Keywords: short range perturbations, scattering amplitude, scattering relation, semi-classical Fourier integral operators
1. Introduction and statement of results
We study the structure of the scattering amplitude associated to the semi-classical 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 semi-classical 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


Source: Alexandrova, Ivana - Department of Mathematics and Statistics, State University of New York at Albany


Collections: Mathematics