Quantum fluctuations and random matrix theory

TL;DR

This paper explores the application of random matrix theory to quantum two-dimensional electron systems, analyzing eigenfunctions and operators in various Hilbert spaces to connect mathematical models with physical phenomena.

Contribution

It introduces a framework for modeling quantum electron systems using random matrices across different Hilbert spaces, providing solutions to random eigenproblems and linking to physical applications.

Findings

Solutions to random eigenproblems in quantum systems

Connection between random matrices and physical electron models

Analysis of operators in real, complex, and quaternion Hilbert spaces

Abstract

The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. 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.