Quasi-exactly solvable problems and random matrix theory

TL;DR

This paper explores a novel connection between quasi-exactly solvable quantum problems and random matrix models, enabling new methods for semiclassical expansions and revealing Lie algebraic structures.

Contribution

It establishes a direct relationship between quasi-exactly solvable problems and random matrix theory, facilitating the construction of topological expansions.

Findings

Relationship between solvable quantum problems and complex matrix models

Reduction of topological expansion construction to semiclassical expansions

Discussion of Lie algebraic structures in the relationship

Abstract

There exists an exact relationship between the quasi-exactly solvable problems of quantum mechanics and models of square and rectangular random complex matrices. This relationship enables one to reduce the problem of constructing topological ($1/N$) expansions in random matrix models to the problem of constructing semiclassical expansions for observables in quasi-exactly solvable problems. Lie algebraic aspects of this relationship are also discussed.