Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Ladder operators and hidden algebras for shape invariant nonseparable and nondiagonalizable models with quadratic complex interaction. I. Two-dimensional model

Ian Marquette, C. Quesne

Published Oct 28, 2020• 2 citations
Publisher

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 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.

DOI
10.3842/sigma.2022.004

Related papers

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

Ian Marquette, C. Quesne, Université Libre de Bruxelles, Belgium

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 ± , B ± , C ± } that can be combined to build a gl(3) hidden algebra.The latter can be embedded in an sp(6) algebra, as well as in an 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.

arXiv:2010.15276•Symmetry Integrability and Geometry Methods and Applications•Jan 14, 2022•2 citations

Exactly solvable nonseparable and nondiagonalizable two-dimensional model with quadratic complex interaction

F. Cannata, M. V. Ioffe, D. N. Nishnianidze

We study a quantum model with nonisotropic two-dimensional oscillator potential but with additional quadratic interaction x1x2 with imaginary coupling constant. It is shown that for a specific connection between coupling constant and oscillator frequencies, the model is not amenable to a conventional separation of variables. The property of shape invariance allows to find analytically all eigenfunctions and the spectrum is found to be equidistant. It is shown that the Hamiltonian is nondiagonalizable, and the resolution of the identity must include also the corresponding associated functions. These functions are constructed explicitly, and their properties are investigated. The problem of R-separation of variables in two-dimensional systems is discussed.

arXiv:0910.0590•Journal of Mathematical Physics•Feb 1, 2010•20 citations

Hidden $\mathfrak{u}(2,1)$ symmetry and Jordan chains in a resonant ghostly three-dimensional model

Andreas Fring, Ian Marquette

We investigate a three-dimensional ghostly Hamiltonian realisation of the fully degenerate resonant sixth-order Pais-Uhlenbeck oscillator. On the classical level, the phase-space flow is non-diagonalisable and decomposes into two complex-conjugate Jordan chains of length three, explaining the appearance of oscillatory solutions with secular terms. Upon quantisation, we construct intertwining operators whose quadratic combinations generate a hidden spectrum-generating $\mathfrak{u}(2,1)$-algebra. The associated descendant spaces are finite-dimensional invariant subspaces carrying non-trivial Jordan structure. Although these spaces admit a natural decomposition into irreducible modules of a distinguished $\mathfrak{sl}_2$-subalgebra, this decomposition does not in general coincide with the Jordan decomposition of the Hamiltonian. We further derive a tri-Hamiltonian formulation from Lie point symmetries of the classical flow and show that the corresponding Hamiltonians are naturally encoded by the same hidden algebra. Nevertheless, unlike in the non-resonant case, no positive-definite linear combination of them generates the same dynamics. Finally, we analyse the common centraliser of the tri-Hamiltonian family in $U(\mathfrak u(2,1))$, showing that the natural higher-order candidate $Q$ is reducible and yields no independent classical or quantum integral. The model thus provides a resonant higher-derivative system in which hidden $\mathfrak{u}(2,1)$ symmetry, classical and quantum Jordan structures, and multi-Hamiltonian geometry coexist.

quant-pharXiv:2606.02290•Jun 2, 2026

The supersymmetric Jaynes–Cummings model and its solutions

Abdulaziz D Alhaidari

The super-algebraic structure of a generalized version of the Jaynes–Cummings model is investigated. We find that a Z 2 graded extension of the so (2,1) Lie algebra is the underlying symmetry of this model. It is isomorphic to the four-dimensional super-algebra u (1/1) with two odd and two even elements. Differential matrix operators are taken as realization of the elements of the superalgebra to which the model Hamiltonian belongs. Several examples with various choices of superpotentials are presented. The energy spectrum and corresponding wavefunctions are obtained analytically.

arXiv:physics/0609022•Journal of Physics A Mathematical and General•Nov 30, 2006•5 citations

Curvature-induced modifications in the algebraic structure of the two-dimensional Mathews–Lakshmanan nonlinear oscillator and the associated coherent states

V. Chithiika Ruby, M. Lakshmanan

Abstract The two-dimensional Mathews–Lakshmanan oscillator, which can be viewed as the planar representation of a harmonic oscillator on a space of constant curvature expressed in Euclidean coordinates, is characterized by a non-standard Hamiltonian. And it can be equivalently interpreted as a position-dependent mass system. Its quantum counterpart is exactly solvable. In this paper, we factorize the associated Hamiltonian and construct the corresponding deformed ladder operators, which do annihilation and creation operations on the eigenstates of the system and also relate them with the eigenstates of the superpartners. We analyze the shape-invariance property of the supersymmetric partner potentials and identify the underlying algebra that leads to the construction of the nonlinear coherent states for the system.

Physica Scripta•Jul 27, 2026•0 citations

Weyl algebra and a realization of the unitary symmetry

Aleksander Strasburger

summary:In the paper the origins of the intrinsic unitary symmetry encountered in the study of bosonic systems with finite degrees of freedom and its relations with the Weyl algebra (1979, Jacobson) generated by the quantum canonical commutation relations are presented. An analytical representation of the Weyl algebra formulated in terms of partial differential operators with polynomial coefficients is studied in detail. As a basic example, the symmetry properties of the $d$-dimensional quantum harmonic oscillator are considered. In this case the usual set of the creation and annihilation operators is used as basis of the Weyl algebra under consideration. From this model are deduced the underlying symplectic structure and some of its consequences. In particular, the transitions from the initial $\text{Sp}(d,\bbfC)$-invariant model to the $\text{GL}(d,\bbfC)$- and, at last, $\text{U}(d)$-invariant models are performed.

•Jan 1, 1998•2 citations
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 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.
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 ± , B ± , C ± } that can be combined to build a gl(3) hidden algebra.The latter can be embedded in an sp(6) algebra, as well as in an 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.
We study a quantum model with nonisotropic two-dimensional oscillator potential but with additional quadratic interaction x1x2 with imaginary coupling constant. It is shown that for a specific connection between coupling constant and oscillator frequencies, the model is not amenable to a conventional separation of variables. The property of shape invariance allows to find analytically all eigenfunctions and the spectrum is found to be equidistant. It is shown that the Hamiltonian is nondiagonalizable, and the resolution of the identity must include also the corresponding associated functions. These functions are constructed explicitly, and their properties are investigated. The problem of R-separation of variables in two-dimensional systems is discussed.
Hidden u(2,1)\mathfrak{u}(2,1)u(2,1) symmetry and Jordan chains in a resonant ghostly three-dimensional model
We investigate a three-dimensional ghostly Hamiltonian realisation of the fully degenerate resonant sixth-order Pais-Uhlenbeck oscillator. On the classical level, the phase-space flow is non-diagonalisable and decomposes into two complex-conjugate Jordan chains of length three, explaining the appearance of oscillatory solutions with secular terms. Upon quantisation, we construct intertwining operators whose quadratic combinations generate a hidden spectrum-generating u(2,1)\mathfrak{u}(2,1)u(2,1)-algebra. The associated descendant spaces are finite-dimensional invariant subspaces carrying non-trivial Jordan structure. Although these spaces admit a natural decomposition into irreducible modules of a distinguished sl2\mathfrak{sl}_2sl2​-subalgebra, this decomposition does not in general coincide with the Jordan decomposition of the Hamiltonian. We further derive a tri-Hamiltonian formulation from Lie point symmetries of the classical flow and show that the corresponding Hamiltonians are naturally encoded by the same hidden algebra. Nevertheless, unlike in the non-resonant case, no positive-definite linear combination of them generates the same dynamics. Finally, we analyse the common centraliser of the tri-Hamiltonian family in U(u(2,1))U(\mathfrak u(2,1))U(u(2,1)), showing that the natural higher-order candidate QQQ is reducible and yields no independent classical or quantum integral. The model thus provides a resonant higher-derivative system in which hidden u(2,1)\mathfrak{u}(2,1)u(2,1) symmetry, classical and quantum Jordan structures, and multi-Hamiltonian geometry coexist.
The super-algebraic structure of a generalized version of the Jaynes–Cummings model is investigated. We find that a Z 2 graded extension of the so (2,1) Lie algebra is the underlying symmetry of this model. It is isomorphic to the four-dimensional super-algebra u (1/1) with two odd and two even elements. Differential matrix operators are taken as realization of the elements of the superalgebra to which the model Hamiltonian belongs. Several examples with various choices of superpotentials are presented. The energy spectrum and corresponding wavefunctions are obtained analytically.
Abstract The two-dimensional Mathews–Lakshmanan oscillator, which can be viewed as the planar representation of a harmonic oscillator on a space of constant curvature expressed in Euclidean coordinates, is characterized by a non-standard Hamiltonian. And it can be equivalently interpreted as a position-dependent mass system. Its quantum counterpart is exactly solvable. In this paper, we factorize the associated Hamiltonian and construct the corresponding deformed ladder operators, which do annihilation and creation operations on the eigenstates of the system and also relate them with the eigenstates of the superpartners. We analyze the shape-invariance property of the supersymmetric partner potentials and identify the underlying algebra that leads to the construction of the nonlinear coherent states for the system.
summary:In the paper the origins of the intrinsic unitary symmetry encountered in the study of bosonic systems with finite degrees of freedom and its relations with the Weyl algebra (1979, Jacobson) generated by the quantum canonical commutation relations are presented. An analytical representation of the Weyl algebra formulated in terms of partial differential operators with polynomial coefficients is studied in detail. As a basic example, the symmetry properties of the ddd-dimensional quantum harmonic oscillator are considered. In this case the usual set of the creation and annihilation operators is used as basis of the Weyl algebra under consideration. From this model are deduced the underlying symplectic structure and some of its consequences. In particular, the transitions from the initial $\text{Sp}(d,\bbfC)$-invariant model to the $\text{GL}(d,\bbfC)$- and, at last, U(d)\text{U}(d)U(d)-invariant models are performed.