1. Infinitely many shape-invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
- Author
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Ryu Sasaki and Satoru Odake
- Subjects
High Energy Physics - Theory ,Pure mathematics ,FOS: Physical sciences ,Mathematical proof ,DISCRETE QUANTUM-MECHANICS ,symbols.namesake ,SYSTEMS ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Quantum ,SUPERSYMMETRY ,Mathematical Physics ,ANNIHILATION-CREATION OPERATORS ,Mathematics ,Quantum Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Stochastic process ,Statistical and Nonlinear Physics ,Supersymmetry ,Mathematical Physics (math-ph) ,Invariant (physics) ,High Energy Physics - Theory (hep-th) ,Mathematics - Classical Analysis and ODEs ,Orthogonal polynomials ,Laguerre polynomials ,symbols ,Jacobi polynomials ,ORTHOGONAL POLYNOMIALS ,Exactly Solvable and Integrable Systems (nlin.SI) ,Quantum Physics (quant-ph) - Abstract
We provide analytic proofs for the shape invariance of the recently discovered [Odake and Sasaki, Phys. Lett. B 679, 414 (2009)] two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux-Poschl-Teller potential by a degree l (l = 1,2,...) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3l involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3371248], Article, JOURNAL OF MATHEMATICAL PHYSICS. 51(5):053513 (2010)
- Published
- 2010