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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• Main Authors: Marquette, Ian, Quesne, Christiane
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211541
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 004, 11 pages

A shape-invariant, nonseparable, and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined to reveal its hidden algebraic structure. The two operators ⁺ and ⁻, coming from the shape invariant supersymmetrical approach, where ⁺ acts as a raising operator. At the same time, ⁻ annihilates all wavefunctions, are completed by introducing a novel pair of operators ⁺ and ⁻, where ⁻ acts as the missing lowering operator. These four operators then serve as building blocks for constructing (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 (4) algebra, as well as an (1/4) superalgebra. Hence, the hidden algebraic structure of the model is very similar to that known for the two-dimensional real harmonic oscillator.