Krein-Adler transformations for shape-invariant potentials and pseudo virtual states

TL;DR

This paper demonstrates that Darboux-Crum transformations involving pseudo virtual states are equivalent to Krein-Adler transformations in one-dimensional quantum systems with shape-invariant potentials, using polynomial identities of classical orthogonal polynomials.

Contribution

It establishes a novel equivalence between two transformation methods in quantum mechanics using polynomial identities, expanding understanding of shape-invariant potentials.

Findings

Equivalence between Darboux-Crum and Krein-Adler transformations for multiple examples.

Use of polynomial Wronskian identities of Hermite, Laguerre, and Jacobi polynomials.

Application to quantum systems with shape-invariant potentials.

Abstract

For eleven examples of one-dimensional quantum mechanics with shape-invariant potentials, the Darboux-Crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to Krein-Adler transformations deleting multiple eigenstates with shifted parameters. These are based upon infinitely many polynomial Wronskian identities of classical orthogonal polynomials, i.e. the Hermite, Laguerre and Jacobi polynomials, which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape-invariant potentials.