Group theoretical approach to the intertwined Hamiltonians

TL;DR

This paper uses group theory to analyze intertwined Hamiltonians, providing a new framework for understanding transformations and generating exactly solvable potentials in quantum mechanics.

Contribution

It introduces a group theoretical approach to intertwining Hamiltonians, generalizes the finite difference algorithm, and finds new potentials with known eigenfunctions and eigenvalues.

Findings

Finite difference Bäcklund formula as a group element

Group theoretical explanation for Hamiltonians related by first order operators

New potentials with exactly known eigenfunctions and eigenvalues

Abstract

We show that the finite difference B\"acklund formula for the Schr\"odinger Hamiltonians is a particular element of the transformation group on the set of Riccati equations considered by two of us in a previous paper. Then, we give a group theoretical explanation to the problem of Hamiltonians related by a first order differential operator. A generalization of the finite difference algorithm relating eigenfunctions of {\emph three} different Hamiltonians is found, and some illustrative examples of the theory are analyzed, finding new potentials for which one eigenfunction and its corresponding eigenvalue is exactly known.