Volume 56, Issue 7, July 2015
Index of content:
- Quantum Mechanics
On square-integrability of solutions of the stationary Schrödinger equation for the quantum harmonic oscillator in two dimensional constant curvature spaces56(2015); http://dx.doi.org/10.1063/1.4922813View Description Hide Description
In this work, one provides a justification of the condition that is usually imposed on the parameters of the hypergeometric equation, related to the solutions of the stationary Schrödinger equation for the harmonic oscillator in two-dimensional constant curvature spaces, in order to determine the solutions which are square-integrable. One proves that in case of negative curvature, it is a necessary condition of square integrability and in case of positive curvature, a necessary condition of regularity. The proof is based on the analytic continuation formulas for the hypergeometric function. It is observed also that the same is true in case of a slightly more general potential than the one for harmonic oscillator.
- Methods of Mathematical Physics
56(2015); http://dx.doi.org/10.1063/1.4922939View Description Hide Description
The Lax-Phillips scattering theory is an appealing abstract framework for the analysis of scattering resonances. Quantum mechanical adaptations of the theory have been proposed. However, since these quantum adaptations essentially retain the original structure of the theory, assuming the existence of incoming and outgoing subspaces for the evolution and requiring the spectrum of the generator of evolution to be unbounded from below, their range of applications is rather limited. In this paper, it is shown that if we replace the assumption regarding the existence of incoming and outgoing subspaces by the assumption of the existence of Lyapunov operators for the quantum evolution (the existence of which has been proved for certain classes of quantum mechanical scattering problems), then it is possible to construct a structure analogous to the Lax-Phillips structure for scattering problems for which the spectrum of the generator of evolution is bounded from below.