We consider scattering of a free quantum particle on a singular potential with rather arbitrary geometry of the boundaries of its support. In the classical limit ¯h = 0 this problem reduces to the well known problem of chaotic scattering. The universal estimates for the stability of the scattering amplitudes are derived. The application of the obtained results to the mesoscopic systems and quantum chaos are discussed. We also discuss a possibility of experimental verification of the obtained results.
Quantized, compact graphs are excellent paradigms for quantum chaos in bounded systems. Connecting them with leads to infinity, we show that they display all the features which characterize quantum chaotic scattering. We derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. A statistical analysis of the cross sections and resonance parameters compares well with the predictions of random matrix theory. Hence, this system is proposed as a convenient tool to study the generic behavior of chaotic scattering systems and their semiclassical description.
The leading quantum corrections for the scattering of a particle in one dimension from a pulsed potential are discussed. Depending on the model parameters, the scattering may be either regular or chaotic. The quantum corrections are obtained by applying a semiclassical formalism that gives the distribution of scattered momenta down to a resolution of order ${\mathrm{\ensuremath{\Elzxh}}}^{4/5}$. At this resolution, the main quantum effect is the softening of the classical rainbow singularities into peaks with widths of order ${\mathrm{\ensuremath{\Elzxh}}}^{2/3}$. Quantum corrections for examples of both regular and chaotic scattering are compared with exact quantum results, and a criterion is given for estimating the resolution at which the corrections first become significant.
In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between phase-space structures, reflecting the distinction between chaotic, mixed or integrable classical dynamics, and the corresponding quantum properties. Well established techniques dealing with ergodic wave interference in the usual semiclassical limit
ℏ→0, where the classical limit is given by Hamiltonian mechanics of particles, constitute a now standard part of the toolkit of theoretical physics. During the last years, these ideas have been extended into the field theoretical domain of systems composed of
N indistinguishable particles, aka quantum fields, displaying a different type of semiclassical limit
ℏeff=1/N→0 and accounting for genuine many-body quantum interference. The foundational concept behind this idea of many-body interference, the many-body version of the van Vleck-Gutzwillers semiclassical propagator, is explained in detail. Based on this the corresponding semiclassical many-body theory is reviewed. It provides a unified framework for understanding a variety of quantum chaotic phenomena addressed, including random-matrix spectral correlations in many-body systems, the universal morphology of many-body eigenstates, interference effects kin to mesoscopic weak localization, and the key to the scrambling of many-body correlations characterized by out-of-time-order correlators.