# QHJ route to multi-indexed exceptional Laguerre polynomials and corresponding rational potentials

## TL;DR

This paper introduces a novel method combining isospectral deformation and quantum Hamilton-Jacobi formalism to construct multi-indexed exceptional Laguerre polynomials and their rational potentials, expanding the toolkit for quantum exactly solvable models.

## Contribution

It presents a new approach to generate multi-indexed exceptional Laguerre polynomials and rational potentials using QHJ formalism and singularity analysis, with explicit constructions and potential extensions.

## Key findings

- Explicit expressions for L1, L2, L3 type polynomials and weight functions.
- Construction of rational extensions of the radial oscillator.
- Discussion on potential for more rational potentials with interesting solutions.

## 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.