Scattering for anisotropic potentials

TL;DR

This paper studies scattering theory for operators with anisotropic potentials, establishing existence and completeness of wave operators, spectral properties, and eigenvalue behavior under various conditions.

Contribution

It introduces new conditions on anisotropic potentials and unperturbed operators to analyze spectral and scattering properties of the operator H.

Findings

Wave operators exist and are complete under certain conditions.

H has no singular continuous spectrum.

Eigenvalues can only accumulate at zero, with finitely many under stronger conditions.

Abstract

We consider the scattering for the operator $H=H_o+V$, where the unperturbed operator $H_o$ is not assumed to be elliptic and the potential $V$ is anisotropic. Under some conditions on $H_o$ and $V$ we show that the wave operators for $H_o, H$ exist and are complete, $H$ has no singular continuous spectrum and the eigenvalues of $H$ can accumulate only to zero. For stronger conditions on $V$ the operator $H$ has finite number of eigenvalues only. Moreover, these results are applied to the invariance principle and for time-dependent potentials.