Ladder Operators and Rational
Extensions
David Gómez-Ullate, Yves Grandati, Zoé McIntyre, and Robert Milson
Abstract This note presents the classification of ladder operators corresponding
to the class of rational extensions of the harmonic oscillator. We show that it is
natural to endow the class of rational extensions and the corresponding intertwining
operators with the structure of a category REXT. The combinatorial data for this
interpretation is realized as a functor MD → REXT, where MD refers to the set
of Maya diagrams appropriately endowed with categorical structure. Our formalism
allows us to easily reproduce and extend earlier results on ladder operators.
Keywords Rational extensions · Ladder operators · Maya diagrams
1 Introduction
Supersymmetric quantum mechanics (SUSYQM) has proven to be a key technique
in the construction of exactly solvable potentials and in the understanding of shapeinvariance. The supersymmetric partners of the harmonic oscillators are known
as rational extensions because the corresponding potentials have the form of a
harmonic oscillator plus a rational term that vanishes at infinity.
D. Gómez-Ullate
Escuela Superior de Ingeniería, U. Cádiz, Puerto Real, Spain
Departamento de Física Teórica, U. Complutense, Madrid, Spain
e-mail: david.gomezullate@uca.es
Y. Grandati
L. Physique et Chimie Théoriques, U. de Lorraine, Metz, France
e-mail: yves.grandati@univ-lorraine.fr
Z. McIntyre
Department of Physics, McGill U., Montréal, QC, Canada
e-mail: zoe.mcintyre@mail.mcgill.ca
R. Milson ()
Department of Mathematics and Statistics, Dalhousie U., Halifax, NS, Canada
e-mail: rmilson@dal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_11
121
Extensions
David Gómez-Ullate, Yves Grandati, Zoé McIntyre, and Robert Milson
Abstract This note presents the classification of ladder operators corresponding
to the class of rational extensions of the harmonic oscillator. We show that it is
natural to endow the class of rational extensions and the corresponding intertwining
operators with the structure of a category REXT. The combinatorial data for this
interpretation is realized as a functor MD → REXT, where MD refers to the set
of Maya diagrams appropriately endowed with categorical structure. Our formalism
allows us to easily reproduce and extend earlier results on ladder operators.
Keywords Rational extensions · Ladder operators · Maya diagrams
1 Introduction
Supersymmetric quantum mechanics (SUSYQM) has proven to be a key technique
in the construction of exactly solvable potentials and in the understanding of shapeinvariance. The supersymmetric partners of the harmonic oscillators are known
as rational extensions because the corresponding potentials have the form of a
harmonic oscillator plus a rational term that vanishes at infinity.
D. Gómez-Ullate
Escuela Superior de Ingeniería, U. Cádiz, Puerto Real, Spain
Departamento de Física Teórica, U. Complutense, Madrid, Spain
e-mail: david.gomezullate@uca.es
Y. Grandati
L. Physique et Chimie Théoriques, U. de Lorraine, Metz, France
e-mail: yves.grandati@univ-lorraine.fr
Z. McIntyre
Department of Physics, McGill U., Montréal, QC, Canada
e-mail: zoe.mcintyre@mail.mcgill.ca
R. Milson ()
Department of Mathematics and Statistics, Dalhousie U., Halifax, NS, Canada
e-mail: rmilson@dal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_11
121