Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

Abstract

It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered, transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. Stäckel systems are studied in particular details. Examples in conformally and non-conformally flat manifolds are given.

18 pages

References in corpus (7)

• Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
• The Tremblay-Turbiner-Winternitz system as extended Hamiltonian
• Superintegrable Extensions of Superintegrable Systems
• A new integrable system on the sphere and conformally equivariant quantization
• Separable quantizations of Stäckel systems
• Classical and Quantum Superintegrability of Stäckel Systems
• Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

Cited by in corpus (3)

• On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces
• Extended Hamiltonians and shift, ladder functions and operators
• Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures