On Eisenbud's and Wigner's R-matrix: A general approach

TL;DR

This paper provides a rigorous, generalized framework for Wigner's and Eisenbud's R-matrix method in scattering theory, applicable to Schrödinger operators, using boundary triplets and Weyl functions.

Contribution

It introduces an abstract generalization of the R-matrix method within the boundary triplet framework, extending its applicability to a broader class of operators.

Findings

Develops a rigorous mathematical foundation for the R-matrix method.

Applies the generalized method to Schrödinger operators on the real axis.

Provides a unified approach for scattering matrices of selfadjoint extensions.

Abstract

The main objective of this paper is to give a rigorous treatment of Wigner's and Eisenbud's $R$-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the $R$-matrix method is developed and the results are applied to Schr\"odinger operators on the real axis.