Microlocal properties of scattering matrices

TL;DR

This paper develops a microlocal analysis framework for scattering matrices associated with differential operators on manifolds, showing they are pseudodifferential operators with explicit principal symbols, applicable to various physical models.

Contribution

It constructs a time-dependent scattering theory for operators on manifolds and characterizes the scattering matrix as a pseudodifferential operator with a Born approximation principal symbol.

Findings

Scattering matrices are pseudodifferential operators on energy surfaces.

Principal symbols of scattering matrices are given by a Born approximation.

Applicable to discrete Schrödinger operators and differential operators with short-range perturbations.

Abstract

We consider scattering theory for a pair of operators $H_0$ and $H=H_0+V$ on $L^2(M,m)$, where $M$ is a Riemannian manifold, $H_0$ is a multiplication operator on $M$ and $V$ is a pseudodifferential operator of order $-\mu$, $\mu>1$. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schr\"odigner operators, but it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.