Quasi-exact solvability in a general polynomial setting

Our goal in this paper is to extend the theory of quasi-exactly solvable Schrodinger operators beyond the Lie-algebraic class. Let P-n be the space of nth degree polynomials in one variable. We first analyze exceptional polynomial subspaces M subset of P-n, which are those proper subspaces of Pn invariant under second-order differential operators which do not preserve Pn. We characterize the only possible exceptional subspaces of codimension one and we describe the space of second-order differential operators that leave these subspaces invariant. We then use equivalence under changes of variable and gauge transformations to achieve a complete classification of these new, non-Lie algebraic Schrodinger operators. As an example, we discuss a finite gap elliptic potential which does not belong to the Treibich-Verdier class.