The decay of quantum systems with a small number of open channels

Abstract

Recent developments in the quantum mechanical description of open systems are presented. In particular, the specific properties of the deep quantum region are pointed out, where the number of open channels is small. Based on the statistical assumptions of Random Matrix Theory, the solution of the Schrödinger equation describing generic open systems is reviewed. The connection between the scattering matrix and the decay behavior is established. On its basis, the non–exponential decay law for systems with a small number of open channels is derived. As a simple physical system possessing both a theoretical description and an experimental realization, scattering systems with leads are studied. Recent experimental results obtained from the scattering of microwaves on such cavities are shown to provide a detailed confirmation of the theoretical predictions for the decay law. A second main topic is the behavior of quantum systems with overlapping resonances. For systems with a small number of open channels it is shown how this regime can be reached by enlarging the coupling strength between system and environment. Using Random Matrix considerations, the essential features of such systems are presented, and the differences between the decay of chaotic and of regular systems are pointed out. A class of simple scattering systems with overlapping resonances is introduced and its S-matrix properties are discussed. Finally, an extremum principle determining the gross features of the scattering matrix is suggested and investigated for the case of one open channel.