Quasi - Exactly - Solvable Differential Equations

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 finitedimensional representation. In one-dimensional case a classification is given by algebras sl2(R) (for differential operators in R) and sl2(R)q (for finitedifference operators in R), osp(2, 2) (operators in one real and one Grassmann variable, or equivalently, 2× 2 matrix operators in R) and gl2(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.