Scattering matrices and Weyl functions

TL;DR

This paper derives explicit formulas for scattering matrices and spectral shift functions using Weyl functions for selfadjoint extensions of symmetric operators, with applications to various differential operators.

Contribution

It provides new explicit formulas for scattering matrices and spectral shift functions in terms of Weyl functions, including a simple proof of the Krein-Birman formula.

Findings

Explicit formulas for scattering matrices and spectral shift functions.

Application to Sturm-Liouville, Dirac, and Schrödinger operators.

Simplified proof of the Krein-Birman formula.

Abstract

For a scattering system $\{A_\Theta,A_0\}$ consisting of selfadjoint extensions $A_\Theta$ and $A_0$ of a symmetric operator $A$ with finite deficiency indices, the scattering matrix $\{S_\gT(\gl)\}$ and a spectral shift function $\xi_\Theta$ are calculated in terms of the Weyl function associated with the boundary triplet for $A^*$ and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar and matrix potentials, to Dirac operators and to Schr\"odinger operators with point interactions.