Demkov-Fradkin tensor for curved harmonic oscillators

Abstract

In this work, we obtain the Demkov-Fradkin tensor of symmetries for the quantum curved harmonic oscillator in a space with constant curvature given by a parameter $\kappa$. In order to construct this tensor we have firstly found a set of basic operators which satisfy the following conditions: i) their products give symmetries of the problem; in fact the Hamiltonian is a combination of such products; ii) they generate the space of eigenfunctions as well as the eigenvalues in an algebraic way; iii) in the limit of zero curvature, they come into the well known creation/annihilation operators of the flat oscillator. The appropriate products of such basic operators will produce the curved Demkov-Fradkin tensor. However, these basic operators do not satisfy Heisenberg commutators but close another Lie algebra. As a by-product, the classical Demkov-Fradkin tensor for the classical curved harmonic oscillator has been obtained by the same method. The case of two dimensions has been worked out in detail: the operators close a $s{o}_{\kappa }(4)$ Lie algebra; the spectrum and eigenfunctions are explicitly solved in an algebraic way and in the classical case the trajectories have been computed.

25 pages, 10 figures

References in corpus (18)

• Classical and Quantum Superintegrability with Applications
• Completeness of superintegrability in two-dimensional constant curvature spaces
• Deformed oscillator algebras for two dimensional quantum superintegrable systems
• Exact Solvability of Superintegrable Systems
• Factorizations of one dimensional classical systems
• A New Superintegrable Hamiltonian
• Periodic orbits for an infinite family of classical superintegrable systems
• Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds
• The anisotropic oscillator on curved spaces: A new exactly solvable model
• Curvature-dependent formalism, Schrödinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces
• Superintegrable quantum u(3)--systems and higher rank factorizations
• Intertwining symmetry algebras of quantum superintegrable systems on the hyperboloid
• Infinite families of superintegrable systems separable in subgroup coordinates
• The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• Quantum, classical symmetries and action-angle variables by factorization of superintegrable systems

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