@article{gomez-ullateDarbouxTransformationAlgebraic2004,
 abstract = {We 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, \dots, of deformations exists for each family of shape-invariant potentials. We prove that the mth deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules , where  is a codimension m subspace of $łangle$1, z, \dots, zn$ångle$. 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 . By construction, these algebraically deformed Hamiltonians do not have an  hidden symmetry algebra structure.},
 author = {Gómez-Ullate, D and Kamran, N and Milson, R},
 doi = {10.1088/0305-4470/37/5/022},
 file = {/Users/david/Zotero/storage/LWHBP7YH/Gómez-Ullate et al. - 2004 - The Darboux transformation and algebraic deformations of shape-invariant potentials.pdf},
 issn = {0305-4470, 1361-6447},
 journal = {Journal of Physics A: Mathematical and General},
 month = {February},
 number = {5},
 pages = {1789--1804},
 title = {The Darboux Transformation and Algebraic Deformations of Shape-Invariant Potentials},
 urldate = {2025-10-19},
 volume = {37},
 year = {2004}
}