Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability

Abstract

The full spectrum and eigenfunctions of the quantum version of a nonlinear oscillator defined on an N-dimensional space with nonconstant curvature are rigorously found. Since the underlying curved space generates a position-dependent kinetic energy, three different quantization prescriptions are worked out by imposing that the maximal superintegrability of the system has to be preserved after quantization. The relationships among these three Schroedinger problems are described in detail through appropriate similarity transformations. These three approaches are used to illustrate different features of the quantization problem on N-dimensional curved spaces or, alternatively, of position-dependent mass quantum Hamiltonians. This quantum oscillator is, to the best of our knowledge, the first example of a maximally superintegrable quantum system on an N-dimensional space with nonconstant curvature.

26 pages, 5 figures

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• Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
• An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric
• Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature
• Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• Coalgebra symmetry for discrete systems
• The ${\mathfrak{sl}}_2(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems
• Shannon information entropy for a quantum nonlinear oscillator on a space of non-constant curvature
• Extended Hamiltonians and shift, ladder functions and operators
• General Nth order integrals of the motion
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Discrete-time systems in quasi-standard form and the ${\mathfrak{h}}_6$ coalgebra symmetry
• Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators
• Curvature as an integrable deformation
• The geometry of real reducible polarizations in quantum mechanics
• Superintegrability of 3-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems
• Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians
• Exactly solvable deformations of the oscillator and Coulomb systems and their generalization
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$
• Symmetry operators and separation of variables for Dirac's equation on two-dimensional spin manifolds with external fields