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Scattering for anisotropic potentials

Evgeny Korotyaev

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.

Scattering for anisotropic potentials

Abstract

We consider the scattering for the operator , where the unperturbed operator is not assumed to be elliptic and the potential is anisotropic. Under some conditions on and we show that the wave operators for exist and are complete, has no singular continuous spectrum and the eigenvalues of can accumulate only to zero. For stronger conditions on the operator has finite number of eigenvalues only. Moreover, these results are applied to the invariance principle and for time-dependent potentials.
Paper Structure (13 sections, 148 equations)

This paper contains 13 sections, 148 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Time-dependent potentials
  5. Related works
  6. Preliminaries
  7. Examples
  8. Preliminary results
  9. Proof of Theorem \ref{['T1.1']}
  10. The invariance principle and time periodic potentials
  11. The invariance principle
  12. Time depending potentials
  13. Appendix