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This article is cited in 2 scientific papers (total in 2 papers)
Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Models with Quadratic Complex Interaction. I. Two-Dimensional Model
Ian Marquettea, Christiane Quesneb 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
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.
Keywords:
quantum mechanics, complex potentials, pseudo-Hermiticity, Lie algebras, Lie superalgebras.
Received: September 1, 2021; in final form January 3, 2022; Published online January 14, 2022
Citation:
Ian Marquette, Christiane Quesne, “Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Models with Quadratic Complex Interaction. I. Two-Dimensional Model”, SIGMA, 18 (2022), 004, 11 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1799 https://www.mathnet.ru/eng/sigma/v18/p4
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