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...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2004 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/51463 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/51463 |
| Access Level: | acceso abierto |
| Palabra clave: | 51-73 Differential-operators Schrodinger-operators Factorization method Quantum-mechanics Supersymmetry Equation Física-Modelos matemáticos Física matemática |
| Sumario: | 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 m_th deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules P_(m)^(m) Ϲ P_(-m+1)^(m) Ϲ (...) , where 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 P_n^(1) = <1, z_(2),..., z_(n)>. By construction, these algebraically deformed Hamiltonians do not have an sl(2) hidden symmetry algebra structure. |
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