The Darboux Transformation and Algebraic Deformations of Shape-Invariant Potentials

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.