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

SIGMA 18 (2022), 005, 24 pages      arXiv:2010.15276      https://doi.org/10.3842/SIGMA.2022.005

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-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 three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied. We present a set of six operators $\{A^{\pm},B^{\pm},C^{\pm}\}$ that can be combined to build a ${\mathfrak{gl}}(3)$ hidden algebra. The latter can be embedded in an ${\mathfrak{sp}}(6)$ algebra, as well as in an ${\mathfrak{osp}}(1/6)$ superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.

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

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