Connection Between Type A and E Factorizations and Construction of Satellite Algebras

TL;DR

This paper explores the connection between type A and E factorizations and introduces a general method for constructing satellite algebras, which link wavefunctions of different potentials and energies in quantum systems.

Contribution

It proposes a universal procedure to determine satellite algebras for Hamiltonians with type E factorizations, expanding the algebraic tools for quantum potential analysis.

Findings

Constructed satellite algebras for various potentials

Identified conserved quantities in example systems

Demonstrated the generality of the method

Abstract

Recently, we introduced a new class of symmetry algebras, called satellite algebras, which connect with one another wavefunctions belonging to different potentials of a given family, and corresponding to different energy eigenvalues. Here the role of the factorization method in the construction of such algebras is investigated. A general procedure for determining an so(2,2) or so(2,1) satellite algebra for all the Hamiltonians that admit a type E factorization is proposed. Such a procedure is based on the known relationship between type A and E factorizations, combined with an algebraization similar to that used in the construction of potential algebras. It is illustrated with the examples of the generalized Morse potential, the Rosen-Morse potential, the Kepler problem in a space of constant negative curvature, and, in each case, the conserved quantity is identified. It should be stressed that the method proposed is fairly general since the other factorization types may be considered as limiting cases of type A or E factorizations.