论文部分内容阅读
Bertrands theorem asserts that any spherically symmetric natural Hamiltonian system in R3 that has a stable circular trajectory and all whose bounded trajectories are closed is either a harmonic oscillator or a Kepler system.This statement finds a natural extension through the definition of Bertrand spacetime introduced by Perlick [1]; this is a spherically symmetric and static spacetime whose timelike geodesics satisfy properties analogous to those of the trajectories of the harmonic oscillator or Kepler systems.Furthermore,if one writes the Lorentzian metric as where g is a Riemannian metric on a 3-manifold,the timelike geodesics in spacetime are related to the trajectories of the classical Hamiltonian system.Perlicks classification of Bertrand spacetimes consists of two multi-parametric families that correspond to either an oscillator or a Kepler system on a curved manifold [2] and,moreover,they are superintegrable.The corresponding quantum N-dimensional Hamiltonians are constructed by applying two different but "gauge-equivalent" procedures: either through the "direct" Schroedinger quantization or by means of the Laplace-Beltrami one (with a scalar curvature term!).Several properties of the resulting quantum superintegrable Bertrand Hamiltonians [3] are also analyzed.