Gómez-Ullate Otaiza, DavidKamran, NikyMilson, Robert2023-06-202023-06-202004-02-060305-447010.1088/0305-4470/37/5/022https://hdl.handle.net/20.500.14352/51463©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. The authors would like to thank Prof. González-López and Prof. Gesztesy for interesting discussions, as well as the referees, who made very interesting remarks on the first version of the paperWe investigate the backward Darboux transformations (addition of the lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, m = 0, 1,.2,..., of deformations exists for each family of shape-invariant potentials. We prove that the m_th deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules P_(m)^(m) Ϲ P_(-m+1)^(m) Ϲ (...) , where P_n^(m) is a codimension m subspace of <1, z,..., z_(n)>. In particular, we prove that the first (m = 1) algebraic deformation of the shape-invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules P_n^(1) = <1, z_(2),..., z_(n)>. By construction, these algebraically deformed Hamiltonians do not have an sl(2) hidden symmetry algebra structure.engjournal articlehttp://dx.doi.org/10.1088/0305-4470/37/5/022http://iopscience.iop.orghttp://arxiv.org/abs/quant-ph/0308062open access51-73Differential-operatorsSchrodinger-operatorsFactorization methodQuantum-mechanicsSupersymmetryEquationFísica-Modelos matemáticosFísica matemática