Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux–Crum transformations

Abstract

A simple derivation is presented of the four families of infinitely many shape-invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. The Darboux–Crum transformations are applied to connect the well-known shape-invariant Hamiltonians of the radial oscillator and the Darboux–Pöschl–Teller potential to the shape-invariant potentials of Odake–Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of the generalized Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.