Quantum Hamilton–Jacobi route to exceptional Laguerre polynomials and the corresponding rational potentials

Abstract

A method to construct the multi-indexed exceptional Laguerre polynomials using the isospectral deformation technique and quantum Hamilton–Jacobi (QHJ) formalism is presented. For a given potential, the singularity structure of the quantum momentum function, defined within the QHJ formalism, allows us to find its solutions. We show that this singularity structure can be exploited to construct the generalised superpotentials, which lead to rational potentials with exceptional polynomials as solutions. We explicitly construct such rational extensions of the radial oscillator and their solutions, which involve exceptional Laguerre orthogonal polynomials having two indices. The weight functions of these polynomials are also presented. We also discuss the possibility of constructing more rational potentials with interesting solutions.