Supersymmetric Quantum Mechanics, Engineered Hierarchies of Integrable Potentials, and the Generalised Laguerre Polynomials

TL;DR

This paper explores supersymmetric quantum mechanics hierarchies beyond shape invariance, introducing periodic structures that generate new orthogonal polynomials, including generalized Laguerre polynomials, and offers methods for engineering quantum potentials from spectra.

Contribution

It extends supersymmetric quantum mechanics to periodic hierarchies with arbitrary level separations, leading to new classes of integrable potentials and polynomials, including explicit solutions for N=1 and N=2 cases.

Findings

Generated new classes of orthogonal polynomials related to periodic hierarchies.

Derived generalized Rodrigues formulae and recursion relations for these polynomials.

Identified and algebraically solved new integrable quantum potentials with arbitrary energy gaps.

Abstract

Within the context of Supersymmetric Quantum Mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the usual restriction of shape invariance for intertwined potentials, it is suggested to require a similar relation for Hamiltonians in the hierarchy separated by an arbitrary number of levels, N. By requiring further that these two Hamiltonians be in fact identical up to an overall shift in energy, a periodic structure is installed in the hierarchy of quantum systems which should allow for its solution. Specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated, while the methods of Supersymmetric Quantum Mechanics then lead to generalised Rodrigues formulae and recursion relations for such polynomials. The approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra. In this paper these ideas are presented and solved explicitly for the cases N=1 and N=2. The latter case is related to the generalised Laguerre polynomials, for which indeed new results are thereby obtained. At the same time new classes of integrable quantum potentials which generalise that of the harmonic oscillator and which are characterised by two arbitrary energy gaps are identified, for which a complete solution is achieved algebraically.