On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials

TL;DR

This paper classifies the most general rational quantum potentials related to generalized Hermite polynomials, revealing their connection to third-order shape-invariant Hamiltonians and the fourth Painlevé equation, and detailing their spectral structure.

Contribution

It establishes a comprehensive classification of rational potentials linked to generalized Hermite polynomials via third-order shape-invariant Hamiltonians and Painlevé equations, detailing spectral and eigenfunction structures.

Findings

Spectral structure is a union of finite and infinite equidistant eigenvalues.

Eigenfunctions are products of weight functions and polynomials satisfying second-order differential equations.

Generalized Hermite polynomials determine the finite sequence dimension and spectral gap.

Abstract

This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermite polynomials. This is achieved by exploiting the intrinsic relation between third-order shape-invariant Hamiltonians and the fourth Painlev\'e equation, such that the generalized Hermite polynomials emerge from the $-1/x$ and $-2x$ hierarchies of rational solutions. Such a relation unequivocally establishes the discrete spectrum structure, which, in general, is composed as the union of a finite- and infinite-dimensional sequence of equidistant eigenvalues separated by a gap. The two indices of the generalized Hermite polynomials determine the dimension of the finite sequence and the gap. Likewise, the complete set of eigensolutions can be decomposed into two disjoint subsets. In this form, the eigensolutions within each set are written as the product of a weight function defined on the real line times a polynomial. These polynomials fulfill a second-order differential equation and are alternatively determined from a three-term recurrence relation (second-order difference equation), the initial conditions of which are also fixed in terms of generalized Hermite polynomials.