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

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 −1/x and −2x hierarchies of rational solutions of the fourth Painlevé equation. Such a relation unequivocally establishes the discrete spectrum structure, 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 define the dimension of the finite sequence and the gap. Likewise, the complete set of eigensolutions decomposes into two disjoint subsets, whose elements are written as the product of a polynomial times a weight function supported on the real line. These polynomials fulfill a second-order differential equation and are alternatively determined from a three-term recurrence relation, the initial conditions of which are also fixed in terms of generalized Hermite polynomials.