Applications of the random matrix ensembles to the quantum statistical systems

TL;DR

This paper explores how random matrix ensembles can be applied to quantum statistical systems, analyzing operators on various Hilbert spaces and solving related eigenproblems with physical applications.

Contribution

It introduces a framework for applying random matrix theory to quantum systems using finite-dimensional Hilbert spaces and solves associated eigenproblems.

Findings

Random matrix ensembles effectively model quantum operators.

Eigenproblems are explicitly solved in the context of quantum systems.

Connections between random operators and physical problems are demonstrated.

Abstract

The random matrix ensembles are applied to the quantum statistical systems. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The linear operators describing the systems act on these Hilbert spaces and they are treated as random matrices in generic bases of the eigenfunctions. The random eigenproblems are presented and solved. Examples of random operators are presented with connection to physical problems.