• Citer

Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics

par Quesne, Christiane
Référence Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 5, 084
Publication Publié, 2009

Article révisé par les pairs

• Accès en ligne
• Détails
• Contenu
• Statistiques

Résumé :

New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g. The cases where g is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same charac- teristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are con- structed. In the linear case, they contain (ν + 1)th-degree polynomials with ν = 0,1,2,..., which are shown to be X1-Laguerre or X1-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of (ν + 2)th-degree Laguerre-type polynomials and a single one of (ν + 2)th-degree Jacobi-type polynomials with ν = 0,1,2,... are identified. They are candidates for the still unknown X2-Laguerre and X2-Jacobi exceptional orthogonal polynomials, respectively.