Lie-algebraic approach to the theory of polynomial solutions. I. Ordinary differential equations and finite-difference equations in one variable

TL;DR

This paper introduces a Lie-algebraic framework for classifying differential and finite-difference equations with polynomial solutions, linking to quasi-exactly-solvable problems and expanding the understanding of the generalized Bochner problem.

Contribution

It presents a novel Lie-algebraic method for classifying equations with polynomial solutions using spectral problems in the universal enveloping algebra of sl_2(R) and its q-deformation.

Findings

Classifies equations with polynomial solutions via Lie algebra representations.

Connects polynomial solutions to quasi-exactly-solvable problems.

Provides a unified algebraic approach to differential and finite-difference equations.

Abstract

A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial element of the universal enveloping algebra of $sl_2({\bf R})$ (for differential equations) or $sl_2({\bf R})_q$ (for finite-difference equations) in the "projectivized" representation possessing an invariant subspace. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.