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

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.

💡 Research Summary

The paper presents a systematic construction of a new class of quasi‑solvable quantum mechanical models that possess N‑fold supersymmetry (N‑SUSY) while preserving an exceptional polynomial subspace of codimension two. The authors begin by reviewing the concepts of ordinary supersymmetric quantum mechanics, its generalization to N‑fold supersymmetry, and the role of exceptional orthogonal polynomials (EOPs), especially the X₂‑Laguerre families, which are characterized by two missing degrees (codimension 2).

A central achievement of the work is the explicit definition of the exceptional subspace ( \mathcal{V}^{(2)}_N ), spanned by two families of X₂‑Laguerre polynomials: the first‑kind (L^{(\alpha)}_n(x)) and the second‑kind (\tilde{L}^{(\alpha)}_n(x)). By constructing a 2N‑th order differential operator (L_N) that leaves (\mathcal{V}^{(2)}_N) invariant, the authors derive the N‑fold supercharge (Q_N) as a suitably weighted version of (L_N). The adjoint (Q_N^\dagger) together with (Q_N) intertwine two partner Hamiltonians, \