Scattering for anisotropic potentials

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.