Lie-algebraic approach to the theory of polynomial solutions. II. Differential equations in one real and one Grassmann variables and 2x2 matrix differential equations

Details Lie-algebraic approach to the theory of polynomial solutions. II. Differential equations in one real and one Grassmann variables and 2x2 matrix differential equations Lie-algebraic approach to the theory of polynomial solutions. II. Differential equations in one real and one Grassmann variables and 2x2 matrix differential equations

1992-09-22

arxiv.org/abs/hep-th/9209080v1

Physics

High Energy Physics

High Energy Physics - Theory

24pp

Scientific paper

A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solutions(the generalized Bochner problem) is given. The main result is based on the consideration of the eigenvalue problem for a polynomial element of the universal enveloping algebra of the algebra $osp(2,2)$ in the "projectivized" representation (in differential operators of the first order) possessing an invariant subspace. A classification of 2 x 2 matrix differential equations in one real variable possessing polynomial solutions is described. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

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