Quasi-Exactly-Solvable Differential Equations

TL;DR

This paper classifies linear differential and difference operators with finite-dimensional polynomial invariant subspaces, linking them to universal enveloping algebras of certain Lie algebras, and explores operators with infinitely many such subspaces.

Contribution

It provides a comprehensive classification of quasi-exactly-solvable operators using Lie algebra representations and extends to operators with infinitely many polynomial invariant subspaces.

Findings

Operators are represented as polynomial elements of universal enveloping algebras.

Classification includes algebras sl_2(R), sl_2(R)_q, osp(2,2), and gl_2(R)_K.

Infinite families of operators with polynomial invariant subspaces are characterized.

Abstract

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of the algebra of differential (difference) operators in finite-dimensional representation. In one-dimensional case a classification is given by algebras $sl_2({\bold R})$ (for differential operators in ${\bold R}$) and $sl_2({\bold R})_q$ (for finite-difference operators in ${\bold R}$), $osp(2,2)$ (operators in one real and one Grassmann variable, or equivalently, $2 \times 2$ matrix operators in ${\bold R}$) and $gl_2 ({\bold R})_K$ ( for the operators containing the differential operators and the parity operator). A classification of linear operators possessing infinitely many finite-dimensional invariant subspaces with a basis in polynomials is presented.