Quartic Anharmonic Oscillator and Random Matrix Theory

TL;DR

This paper establishes a connection between the ground state energy of the quantum quartic oscillator and the mean eigenvalue of large positive definite random Hermitian matrices, providing new analytical expressions.

Contribution

It introduces a novel relationship linking quantum oscillator energies with random matrix eigenvalues, enabling new analytical approaches.

Findings

Derived closed-form expressions for oscillator energy

Presented recurrence relations using orthogonal polynomial methods

Linked quantum mechanics with random matrix theory

Abstract

In this paper the relationship between the problem of constructing the ground state energy for the quantum quartic oscillator and the problem of computing mean eigenvalue of large positively definite random hermitean matrices is established. This relationship enables one to present several more or less closed expressions for the oscillator energy. One of such expressions is given in the form of simple recurrence relations derived by means of the method of orthogonal polynomials which is one of the basic tools in the theory of random matrices.