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

TL;DR

This paper reveals the hidden algebraic structure of a complex, nonseparable 2D quantum model using ladder operators, showing it possesses symmetries akin to those of the harmonic oscillator.

Contribution

It introduces new operators to complete the algebraic structure of the model, uncovering hidden ${rak{gl}}(2)$, ${rak{sp}}(4)$, and ${rak{osp}}(1/4)$ algebras, extending understanding of its symmetry properties.

Findings

Constructed ${rak{gl}}(2)$ generators from shape invariant operators.

Extended algebraic structure to ${rak{sp}}(4)$ and ${rak{osp}}(1/4)$ superalgebra.

Revealed similarity to the algebraic structure of the 2D harmonic oscillator.

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.