Ivashkevych G. Over-barrier dynamics of particles in multi-well potentials: dynamical chaos and supersymmetry

The thesis is devoted to the some problems of classical over-barrier transitions in potentials with complex topology with mixed state and applications of N = 4 supersymmetric quantum mechanics to the construction of new exactly solvable stochastic models with non-trivial potentials. For the first time, the practical aspects of applications of geometrical stochasticity criteria to multi-well potentials were considered, including the applicability conditions of these criteria. The correct procedure for application of these criteria for multi-well potentials was constructed. For the first time, the classical escape problem was studied in the context of the mixed state and escape laws were obtained for uniform and point ensembles, as well as quantities, which characterize these laws, characteristic trajectories groups, and asymptotic particles distributions of particles in considered potentials. As an example of a realistic system with mixed state, the system of axially symmetric currents was constructed and all main features of escape in such a system were researched. For the first time, the procedure for construction of stochastic models in the framework of N = 4 supersymmetric quantum mechanics was proposed. As an example of proposed procedure, the generalization of Ornstein-Uhlenbeck process was considered for the case of stable and bistable stochastic models. The hierarchy of Hamiltonians was built in the framework of N=4 supersymmetric quantum mechanics starting from one of the Peschl-Teller potentials. A numerical library for high-preformance simulations of chaotic dynamics and dynamics of quantum wave packets on Graphic Processing Unit was developed and used in all simulations in present thesis. Considered methods can be used in the research of various transititon processes, many of which can be described as an evolution in some classical or quantum potential.