Gómez-Ullate Otaiza, DavidKamran, NikyMilson, Robert2023-06-202023-06-202005-03-040305-447010.1088/0305-38/9/011https://hdl.handle.net/20.500.14352/51461©IOP science. The research of DGU is supported in part by a CRM-ISM Postdoctoral Fellowship and the Spanish Ministry of Education under grant EX2002-0176. The research of NK and RM is supported by the National Science and Engineering Research Council of Canada. DGU would like to thank the Department of Mathematics and Statistics of alhousie University for their warm hospitality.We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: we show that the generalized Lame potential possesses four algebraic sectors, and describe a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic.engjournal articlehttp://dx.doi.org/10.1088/0305-4470/38/9/011http://iopscience.iop.org/http://arxiv.org/abs/nlin/0401030open access51-73Differential-operatorsLame equationPotentialsAlgebraFísica-Modelos matemáticosFísica matemática