Fractional supersymmetry algebra and lacunary Hermite polynomials

TL;DR

This paper explores fractional supersymmetric quantum mechanics of order r, revealing that its Hamiltonian has an r-fold degenerate spectrum and its eigenfunctions involve lacunary Hermite polynomials, connecting algebraic structures with special functions.

Contribution

It introduces a realization of fractional supersymmetry involving reflection operators and links the eigenvalues and eigenfunctions to Hermite and lacunary Hermite polynomials.

Findings

Hamiltonian exhibits r-fold degeneracy

Eigenvalues of supercharges are zeros of Hermite polynomials

Eigenfunctions involve lacunary Hermite polynomials

Abstract

We consider a realization of fractional supersymmetric of quantum mechanics of order $r$, where the Hamiltonian and supercharges involve reflection operators. It is shown that the Hamiltonian has $r$-fold degenerate spectrum and the eigenvalues of hermitian supercharges are zeros of the associated Hermite polynomials of Askey and Wimp. Also it is shown that the associated eigenfunctions involve lacunary Hermite polynomials.