Qualitative dynamics of potential systems on simply connected compact manifolds with ellipsoidal boundary in R 3

Potential systems on a simply connected compact manifold with ellipsoidal boundary in R³ are investigated in the present work for the case in which the potential of the system is an even forth-order symmetric polynomial. The sufficient conditions of the compact manifold boundary invariance are established. It is demonstrated that the interior punctured at the origin of coordinates is the region of compact manifold boundary attraction, and the qualitative dynamics is investigated. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 78–83, March, 2007.     

Giant oscillations of energy levels in mesoscopic superconductors

The interplay of geometrical and Andreev quantization in mesoscopic superconductors leads to giant mesoscopic oscillations of energy levels as functions of the Fermi momentum and/or sample size. Quantization rules are formulated for closed quasiparticle trajectories in the presence of normal scattering at the sample boundaries. Two generic examples of mesoscopic systems are studied: (i) one-dimensional Andreev states in a quantum box and (ii) a single vortex in a mesoscopic cylinder.