2011/02/27 by Angel Ballesteros, Ángel Ballesteros, Alberto Enciso +5 · 73 citations
Mathematics · Physics and Astronomy · #Classical mechanics #Curvature #Eigenfunction #Eigenvalues and eigenvectors #Geometry #Hamiltonian (control theory) #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Physics #Position (finance) #Quantization (signal processing) #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum harmonic oscillator #Quantum mechanics #Space (punctuation) #math-ph #math.MP #nlin.SI #quant-ph
paper · pdf · doi:10.1016/j.aop.2011.03.002
published in Annals of Physics 326(8), 2053-2073 (Elsevier BV) · 26 pages, 5 figures
arxiv created 2011/02/27 · openalex publication_date 2011/03/08 · arxiv updated 2011/06/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.