Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics

Abstract

In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefunctions of some rational extensions of well-known exactly solvable quantum potentials. Such rational extensions are most easily constructed in the framework of supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class of translationally shape invariant potentials. We review the most recent results in this field, which use higher-order SUSYQM. We also comment on some recent re-examinations of the shape invariance condition, which are independent of the EOP construction problem.

21 pages, no figure; communication at the Symposium Symmetries in Science XV, July 31-August 5, 2011, Bregenz, Austria

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• Exceptional Laguerre polynomials
• DARBOUX partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials
• Rational extension and Jacobi-type {\boldmath{$X_m$}} solutions of a quantum nonlinear oscillator
• General solution of the exceptional Hermite differential equation and its minimal surface representation
• Quantum Hamilton-Jacobi Quantization and Shape Invariance