Microlocal properties of scattering matrices
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 Schrödigner operators, but it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2014
- DOI:
- 10.48550/arXiv.1407.8299
- arXiv:
- arXiv:1407.8299
- Bibcode:
- 2014arXiv1407.8299N
- Keywords:
-
- Mathematics - Analysis of PDEs;
- Mathematical Physics;
- 58J50;
- 35P25;
- 81U05