The Darboux transformation and algebraic deformations of shape-invariant potentials

Abstract

We investigate the backward Darboux transformations (addition of a 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 ${\mathcal{P}}_m^{(m)}\subset {\mathcal{P}}_{m+1}^{(m)}\subset ...$, where ${\mathcal{P}}_n^{(m)}$ 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 ${\mathcal{P}}_n^{(1)}=<1,z^2,...,z^n>$. By construction, these algebraically deformed Hamiltonians do not have an $\mathfrak{sl}(2)$ hidden symmetry algebra structure.

18 pages, 3 figures. Paper has been considerably extended and revised. References added

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