### 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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