%0 Journal Article
%A Gómez-Ullate Otaiza, David
%A Kamran, Niky
%A Milson, Robert
%T Quasi-exact solvability in a general polynomial setting
%D 2007
%@ 0266-5611
%U https://hdl.handle.net/20.500.14352/51449
%X 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.
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