N-fold Supersymmetry and Quasi-solvability Associated with X_2-Laguerre Polynomials

TL;DR

This paper introduces a new class of quasi-solvable quantum systems with N-fold supersymmetry linked to X_2-Laguerre polynomials, expanding the understanding of exceptional polynomial subspaces and their algebraic structures.

Contribution

It constructs a novel family of N-fold supersymmetric systems preserving exceptional polynomial subspaces, including a specific case involving X_2-Laguerre polynomials and their interrelations.

Findings

Includes a new rational radial oscillator potential example.

Shows the connection between two types of X_2-Laguerre polynomials via N-fold supercharge.

Demonstrates the preservation of exceptional polynomial subspaces in the constructed systems.

Abstract

We construct a new family of quasi-solvable and N-fold supersymmetric quantum systems where each Hamiltonian preserves an exceptional polynomial subspace of codimension 2. We show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the X_2-Laguerre polynomials of the second kind. In addition, we find that the two kinds of the X_2-Laguerre polynomials are ingeniously connected with each other by the N-fold supercharge.