Density of reflection resonances in one-dimensional disordered Schr\"odinger operators

TL;DR

This paper develops an analytic method to evaluate the density of complex resonance poles in one-dimensional disordered Schrödinger operators, linking resonance density to reflection coefficients and analyzing different sample regimes.

Contribution

It introduces a novel analytic approach connecting resonance density to reflection coefficients at complex energies, providing explicit formulas for different sample sizes and disorder regimes.

Findings

Derived explicit formulas for resonance density in infinite samples.

Analyzed the crossover from narrow to broad resonances.

Validated results with numerical simulations of the Anderson model.

Abstract

We develop an analytic approach to evaluating the density $\rho ({\cal E},\Gamma)$ of complex resonance poles with real energies $\mathcal{E}$ and widths $\Gamma$ 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 $E = {\cal E} +i\eta$, identifying the parameter $\eta>0$ 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 $\rho ({\cal E},\Gamma)$, describing the crossover from narrow to broad resonances in a unified way. Similarly, our approach yields a limiting formula for $\rho ({\cal E},\Gamma)$ 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.