Coupled supersymmetry and ladder structures beyond the harmonic oscillator

ABSTRACT

The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator (QMHO) carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general – except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalise the Heisenberg uncertainty principle. Coupled SUSY is explicitly realised by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalised harmonic oscillators.

GRAPHICAL ABSTRACT

KEYWORDS:

• Sections
• supersymmetry
• uncertainty principles
• coherent states
• ladder operators
• lie algebras

View correction statement:

Corrigendum

Acknowledgments

We thank John R. Klauder for his helpful comments and suggestions in the drafting of this manuscript. We congratulate Professor Michael Baer on his 80th birthday and wish him many more years!

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Cameron L. Williams http://orcid.org/0000-0002-2025-7394

Donald Kouri http://orcid.org/0000-0002-9146-2926

Additional information

Funding

This research was supported in part under Grant E-0608 from the Robert A. Welch Foundation. B.G.B was supported in part by Division of Mathematical Sciences NSF grant DMS-1412524.