Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 004, 11 pages      arXiv:2010.15273      https://doi.org/10.3842/SIGMA.2022.004

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model

Ian Marquette ^a and Christiane Quesne ^b
a) School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia
^b) Physique Nucléaire Théorique et Physique Mathématique,Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium

Received September 01, 2021, in final form January 03, 2022; Published online January 14, 2022

Abstract
A shape invariant nonseparable and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined with the purpose of exhibiting its hidden algebraic structure. The two operators $A^+$ and $A^-$, coming from the shape invariant supersymmetrical approach, where $A^+$ acts as a raising operator while $A^-$ annihilates all wavefunctions, are completed by introducing a novel pair of operators $B^+$ and $B^-$, where $B^-$ acts as the missing lowering operator. These four operators then serve as building blocks for constructing ${\mathfrak{gl}}(2)$ generators, acting within the set of associated functions belonging to the Jordan block corresponding to a given energy eigenvalue. This analysis is extended to the set of Jordan blocks by constructing two pairs of bosonic operators, finally yielding an ${\mathfrak{sp}}(4)$ algebra, as well as an ${\mathfrak{osp}}(1/4)$ superalgebra. Hence, the hidden algebraic structure of the model is very similar to that known for the two-dimensional real harmonic oscillator.

Key words: quantum mechanics; complex potentials; pseudo-Hermiticity; Lie algebras; Lie superalgebras.

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