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

Abstract

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 𝐴⁺ and 𝐴⁻, coming from the shape invariant supersymmetrical approach, where 𝐴⁺ acts as a raising operator. At the same time, 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 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 𝔰𝔭(4) algebra, as well as an 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.