Quasi-Exactly Solvable Schr\"odinger Operators in Three Dimensions

TL;DR

This paper classifies certain algebraic structures of differential operators in three variables and uses this classification to construct new quasi-exactly solvable Schrödinger operators in three dimensions.

Contribution

It provides a partial classification of quasi-exactly solvable Lie algebras in three variables and applies this to develop new solvable Schrödinger operators.

Findings

Partial classification of Lie algebras of differential operators

Construction of new three-dimensional Schrödinger operators

Application of algebraic classification to quantum mechanics

Abstract

The main contribution of our paper is to give a partial classification of the quasi-exactly solvable Lie algebras of first order differential operators in three variables, and to show how this can be applied to the construction of new quasi-exactly solvable Schr\"odinger operators in three dimensions.