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, ..., 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 {\cal P}{^{({m})}}_m\subset{\cal P}{^{({m})}}_{m+1}\subset\cdots , where {\cal P}{^{({m})}}_n is a codimension m subspace of lang1, z, ..., zⁿrang. 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 {\cal P}{^{({1})}}_n = \langle 1,z^2,\ldots,z^n\rangle . By construction, these algebraically deformed Hamiltonians do not have an \mathfrak{sl}(2) hidden symmetry algebra structure.