Lie-algebraic approach to the theory of polynomial solutions. III. Differential equations in two real variables and general outlook

Abstract

Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the eigenvalue problem for a polynomial elements of the universal enveloping algebras of the algebras $s{l}_3(\mathbf{R})$, $s{l}_2(\mathbf{R})\oplus s{l}_2(\mathbf{R})$ and $g{l}_2(\mathbf{R})▹<{\mathbf{R}}^{r+1},r>0$ taken in the "projectivized" representations (in differential operators of the first order in two real variables) possessing an invariant subspace. General insight to the problem of a description of linear differential operators possessing an invariant sub-space with a basis in polynomials is presented. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

28pp

Cited by in corpus (3)

• Lie Algebras of Differential Operators and Partial Integrability
• The Lie Algebraic Structure of Differential Operators Admitting Invariant Spaces of Polynomials
• Quasi-exactly solvable problems and the dual (q-)Hahn polynomials