On Gaussian random matrices coupled to the discrete Laplacian

TL;DR

This paper investigates the spectral properties of operators formed by coupling Gaussian random matrices with the discrete Laplacian, providing insights into quantum scattering models with complex physical systems.

Contribution

It introduces a novel mathematical model coupling Gaussian random matrices to the discrete Laplacian and derives the joint distribution of eigenvalues and resonances.

Findings

Derived the joint distribution of eigenvalues and resonances.

Provided a mathematical framework for quantum scattering in complex systems.

Linked random matrix theory with physical models of quantum leads.

Abstract

We study operators obtained by coupling an $n \times n$ random matrix from one of the Gaussian ensembles to the discrete Laplacian. We find the joint distribution of the eigenvalues and resonances of such operators. This is one of the possible mathematical models for quantum scattering in a complex physical system with one semi-infinite lead attached.