We develop an analytic approach to evaluating the density
ρ(E,Γ) of complex resonance poles with real energies
E and widths
Γ in the pure reflection problem from a one-dimensional disordered sample with white-noise random potential. We start with establishing a general link between the density of resonances and the distribution of the reflection coefficient
r=∣R(E,L)∣2, where
R(E,L) is the reflection amplitude, at {\it complex} energies
E=E+iη, identifying the parameter
η>0 with the uniform rate of absorption within the disordered medium. We show that leveraging this link allows for a detailed analysis of the resonance density in the weak disorder limit. In particular, for a (semi)infinite sample, it yields an explicit formula for
ρ(E,Γ), describing the crossover from narrow to broad resonances in a unified way. Similarly, our approach yields a limiting formula for
ρ(E,Γ) in the opposite case of a short disordered sample, with size much smaller than the localization length. This regime seems to have not been systematically addressed in the literature before, with the corresponding analysis requiring an accurate and rather non-trivial implementation of WKB-like asymptotics in the scattering problem. Finally, we study the resonance statistics numerically for the one-dimensional Anderson tight-binding model and compare the results with our analytic expressions.
We connect quantum compact graphs with infinite leads, and turn them into scattering systems.We derive an exact expression for the scattering matrix, and explain how it is related to the spectrum of the corresponding closed graph.The resulting expressions allow us to get a clear understanding of the phenomenon of resonance trapping due to over-critical coupling with the leads.Finally, we analyze the statistical properties of the resonance widths and compare our results with the predictions of Random Matrix Theory.Deviations appearing due to the dynamical nature of the system are pointed out and explained.
Resonance trapping appears in open many-particle quantum systems at high level density when the coupling to the continuum of decay channels reaches a critical strength. Here a reorganization of the system takes place and a separation of different time scales appears. We investigate it under the influence of additional weakly coupled channels as well as by taking into account the real part of the coupling term between system and continuum. We observe a saturation of the mean width of the trapped states. Also the decay rates saturate as a function of the coupling strength. The mechanism of the saturation is studied in detail. In any case, the critical region of reorganization is enlarged. When the transmission coefficients for the different channels are different, the width distribution is broadened as compared to a ${\ensuremath{\chi}}_{K}^{2}$ distribution where
K is the number of channels. Resonance trapping takes place before the broad state overlaps regions beyond the extension of the spectrum of the closed system.
The universal correlations which describe the response of energy levels of a disordered metallic grain to an arbitrary perturbation are shown to be equivalent to time-dependent correlations of a one-dimensional quantum Hamiltonian with inverse-square interaction. These resutls establish a direct connection between a strongly interacting quantum Hamiltonian, the nonlinear \ensuremath{\sigma} model of disordered electronic systems, and quantum chaotic spectra. As a consequence we have an expression for the correlation function of the quantum Hamiltonian which we believe to be exact for all space and time.
We introduce a bare-bone random matrix quantum impurity model, by hybridizing a localized spinless electronic level with a bath of random fermions in the Gaussian Orthogonal Ensemble (GOE). While stripped out of correlations effects, this model reproduces some salient features of the impurity charge distribution obtained in previous works on interacting disordered impurity models. Computing by numerical sampling the impurity charge distribution in our model, we find a crossover from a Gaussian distribution (centered on half a charge unit) at large hybridization, to a bimodal distribution (centered both on zero and full occupations of the charge) at small hybridization. In the bimodal regime, a universal
(−3/2) power-law is also observed. All these findings are very well accounted for by an analytic surmise computed with a single random electron level in the bath. We also derive an exact functional integral for the general probability distribution function of eigenvalues and eigenstates, that formally captures the statistical behavior of our model for any number
N of fermionic orbitals in the bath. In the Gaussian regime and in the limit
N→∞, we are able to solve exactly the random matrix theory (RMT) for the charge distribution, obtaining perfect agreement with the numerics. Our results could be tested experimentally in mesoscopic devices, for instance by coupling a small quantum dot to a chaotic electronic reservoir, and using a quantum point contact as local charge sensor for the quantum dot occupation.
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.