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

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

1992-09-22

arxiv.org/abs/hep-th/9209082v1

Physics

High Energy Physics

High Energy Physics - Theory

28pp

Scientific paper

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 $sl_3({\bf R})$, $sl_2({\bf R})\oplus sl_2({\bf R})$ and $gl_2 ({\bf R})\ \triangleright\!\!\!< {\bf R}^{r+1}\ , r>0$ 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.

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