dorsal/arxiv
View SchemaThe Darboux transformation and algebraic deformations of shape-invariant potentials
| Authors | David Gomez-Ullate, Niky Kamran, Robert Milson |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0308062 |
| URL | https://arxiv.org/abs/quant-ph/0308062 |
| DOI | 10.1088/0305-4470/37/5/022 |
| Journal | J.Phys. A37 (2004) 1780-1804 |
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 $\mathcal{P}^{(m)}_m\subset\mathcal{P}^{(m)}_{m+1}\subset...$, 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.
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"abstract": "We investigate the backward Darboux transformations (addition of a lowest\nbound state) of shape-invariant potentials on the line, and classify the\nsubclass of algebraic deformations, those for which the potential and the bound\nstates are simple elementary functions. A countable family, $m=0,1,2,...$, of\ndeformations exists for each family of shape-invariant potentials. We prove\nthat the $m$-th deformation is exactly solvable by polynomials, meaning that it\nleaves invariant an infinite flag of polynomial modules\n$\\mathcal{P}^{(m)}_m\\subset\\mathcal{P}^{(m)}_{m+1}\\subset...$, where\n$\\mathcal{P}^{(m)}_n$ is a codimension $m$ subspace of $\u003c1,z,...,z^n\u003e$. In\nparticular, we prove that the first ($m=1$) algebraic deformation of the\nshape-invariant class is precisely the class of operators preserving the\ninfinite flag of exceptional monomial modules $\\mathcal{P}^{(1)}_n = \u003c\n1,z^2,...,z^n\u003e$. By construction, these algebraically deformed Hamiltonians do\nnot have an $\\mathfrak{sl}(2)$ hidden symmetry algebra structure.",
"arxiv_id": "quant-ph/0308062",
"authors": [
"David Gomez-Ullate",
"Niky Kamran",
"Robert Milson"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
"math.MP",
"nlin.SI"
],
"doi": "10.1088/0305-4470/37/5/022",
"journal_ref": "J.Phys. A37 (2004) 1780-1804",
"title": "The Darboux transformation and algebraic deformations of shape-invariant potentials",
"url": "https://arxiv.org/abs/quant-ph/0308062"
},
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