A mathematical formulation of the Mahaux-Weidenm\"uller formula for the scattering matrix

TL;DR

This paper provides a rigorous mathematical explanation of the Mahaux-Weidenm"uller formula for the scattering matrix in waveguides, including error estimates for finite-rank approximations, bridging physics intuition with mathematical precision.

Contribution

It offers a formal mathematical derivation of the Mahaux-Weidenm"uller formula and analyzes the accuracy of finite-rank approximations of the interaction matrix.

Findings

The formula aligns with the standard scattering matrix in mathematics.

Finite-rank approximations have errors inversely proportional to a power of the rank.

An example demonstrates the optimality of the error estimate.

Abstract

The purpose of this note is to give a mathematical explanation of a formula for the scattering matrix for a manifold with infinite cylindrical ends or a waveguide. This formula, which is well known in the physics literature, is sometimes referred to as the Mahaux-Weidenm\"uller formula. We show that a version of this formula given below gives the standard scattering matrix used in the mathematics literature. We also show that the finite rank approximation of the interaction matrix gives an approximation of the scattering matrix with errors inversely proportional to a fixed dimension-dependent power of the rank. A simple example shows that this estimate is optimal.