Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability. (English) Zbl 1220.81109
Summary: 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 Schrödinger 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.
MSC:
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
Keywords:
nonlinear oscillator; superintegrability; deformation; hyperbolic; nonconstant curvature; position-dependent massReferences:
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