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The Darboux transformation and algebraic deformations of shape-invariant potentials | | David Gomez-Ullate
; Niky Kamran
; Robert Milson
; | | Date: |
11 Aug 2003 | | Journal: | J.Phys. A37 (2004) 1780-1804 | | Subject: | Quantum Physics; Mathematical Physics; Exactly Solvable and Integrable Systems | quant-ph hep-th math-ph math.MP nlin.SI | | 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 $mathcal{P}^{(m)}_msubsetmathcal{P}^{(m)}_{m+1}subset...$, where $mathcal{P}^{(m)}_n$ 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 $mathcal{P}^{(1)}_n = < 1,z^2,...,z^n>$. By construction, these algebraically deformed Hamiltonians do not have an $mathfrak{sl}(2)$ hidden symmetry algebra structure. | | Source: | arXiv, quant-ph/0308062 | | Services: | Forum | Review | PDF | Favorites | |
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