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

Abstract

A method to construct multi-indexed exceptional Laguerre polynomials using isospectral deformation technique and quantum Hamilton-Jacobi (QHJ) formalism is presented. We construct generalized superpotentials using singularity structure analysis which lead to rational potentials with multi-indexed 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 exact expressions for the L1, L2 and L3 type polynomials, along with their weight functions are presented. We also discuss the possibility of constructing more rational potentials with interesting solutions.keywords Exceptional orthogonal polynomials, multi-indexed exceptional polynomials, exactly solvable models, rational potentials, shape invariance, isospectral deformation, quantum Hamilton-Jacobi formalism, quantum momentum function. 1

Quantum Hamilton–Jacobi quantization and shape invariance

“…We have mainly investigated conventional superpotentials that do not have an explicit dependence on ℏ. While some studies [45,46] have been done of the rational extensions [47][48][49][50][51][52][53][54] that explicitly depend on ℏ, these studies have been limited to the extensions that originate from some specific conventional potentials. An explicitly SI based analysis would be an improvement.…”

Construction of 2nd stage shape invariant potentials

“…This naturally raises the question if the process can be repeated to arrive at potentials related to next generation of exceptional polynomials. This question has been addressed in a recent article by Sree Ranjani within the frame work of QHJ [3]. She has explicitly constructed the second generation potentials and the corresponding polynomials.…”