The Darboux transformation and algebraic deformations of shape-invariant potentials



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Gómez-Ullate Otaiza, David and Kamran, Niky and Milson, Robert (2004) The Darboux transformation and algebraic deformations of shape-invariant potentials. Journal of physics A: Mathematical and general, 37 (5). pp. 1789-1804. ISSN 0305-4470

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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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©Iop science.
The research of DGU is supported in part by a CRM-ISM Postdoctoral Fellowship and the Spanish Ministry of Education under grant EX2002-0176. The research of NK and RM is supported by the National Science and Engineering Research Council of Canada. The authors would like to thank Prof. González-López and Prof. Gesztesy for interesting discussions, as well as the referees, who made very interesting remarks on the first version of the paper

Uncontrolled Keywords:Differential-operators; Schrodinger-operators; Factorization method; Quantum-mechanics; Supersymmetry; Equation
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:30915
Deposited On:17 Jun 2015 14:02
Last Modified:10 Dec 2018 15:09

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