Gómez-Ullate Otaiza, DavidKamran, NikyMilson, Robert2023-06-202023-06-202007-100266-561110.1088/0266-5611/23/5/008https://hdl.handle.net/20.500.14352/51449© IOP Publishing. The research of DGU is supported in part by the Ramón y Cajal program of the Ministerio de Ciencia y Tecnología and by the DGI under grants FIS2005-00752 and MTM2006-00478. The research of NK and RM is supported in part by the NSERC grants RGPIN 105490-2004 and RGPIN-228057-2004, respectivelyOur 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.engjournal articlehttp://dx.doi.org/10.1088/0266-5611/23/5/008http://iopscience.iop.orghttp://arxiv.org/abs/nlin/0610065open access51-73Solvable schrodinger-operatorsTangential coversCalogeroPotentialsMonomialsEquationsAlgebrasSpacesFísica-Modelos matemáticosFísica matemática