Density of reflection resonances in one-dimensional disordered Schrödinger operators

TL;DR

This work addresses the density of complex resonance poles, $\rho({\cal E},\Gamma)$, for pure reflection from a one-dimensional disordered medium with white-noise potential. It introduces a central link to the reflection-coefficient statistics by expressing $\rho$ as a second-derivative of the disorder-averaged log-reflection, $\langle\ln r({\cal E}+i\eta)\rangle$, evaluated at $\eta=\Gamma$, leveraging FP dynamics and invariant embedding. The authors derive explicit results in the semiclassical, weak-disorder regime for both semi-infinite and short-sample limits, including a closed-form resonance-density formula that captures the crossover from narrow to broad resonances, and they validate these predictions with numerical simulations of the Anderson model. The approach provides a unified framework for resonance statistics in open 1D disordered systems and suggests avenues for extension to quasi-1D and higher-dimensional settings, with potential insights into Anderson localization phenomena and time-delay statistics.

Abstract

We develop an analytic approach to evaluating the density $ρ({\cal E},Γ)$ of complex resonance poles with real energies $\mathcal{E}$ 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 $E = {\cal E} +iη$, 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 $ρ({\cal E},Γ)$, describing the crossover from narrow to broad resonances in a unified way. Similarly, our approach yields a limiting formula for $ρ({\cal E},Γ)$ 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.

Figures (9)

• Figure 1: Sketch of the scattering setup. The incident wave (blue) enters the sample (grey) from the right, interacts with the disorder $V(x)$ (green), is back-reflected on its left end and eventually escapes the disordered region through its right end. Together with the reflecting left end of the sample, the disorder generates the scattered, outgoing wave (red), with reflection coefficient $R(k,L)$.
• Figure 2: Comparison between numerical simulations of the stochastic dynamics \ref{['eq:rIto']} ($10^8$ realizations) and theory. (a) Probability distribution $P(r,L)$ for $\varepsilon=10^{-1}$ as function of $r$ from numerical simulations for $\hat{\gamma} = 0.1$ (red), $\hat{\gamma} = 0.2$ (blue), $\hat{\gamma} = 0.5$ (yellow), $\hat{\gamma} = 1$ (green), $\hat{\gamma} = 2$ (orange), and $\hat{\gamma} = 5$ (purple) together with the WKB approximation (solid black lines) and the Gaussian approximation \ref{['eq:pdfGauss']} (dotted lines). (b) Same as in (a) but on a logarithmic $r$-axis. (c) $\langle r\rangle$ from numerical simulations as function of $\hat{\gamma}$ for $\varepsilon=10^{-1}$ (red), $\varepsilon=10^{-2}$ (blue), and $\varepsilon = 10^{-3}$ (green), together with the WKB (solid black lines) and from the Gaussian approximation (dotted lines). (d) Variance $\text{Var}(r)$, from the same data as in (c).
• Figure 3: Resonance densities in the limits of short and long samples (solid lines). The dash-dotted and dashed lines show $\Gamma^{-1}$ and $\Gamma^{-2}$ scalings, respectively. (a) $\rho^{(\infty)}_k(\gamma)$ as function of $\gamma$ (blue). (b) $\rho^{(\infty)}_k(\gamma)$ as function of $\hat{\gamma}$ for $\varepsilon=10^{-3}$ (red), $\varepsilon=10^{-2}$ (blue), and $\varepsilon=5\cdot 10^{-2}$ (green).
• Figure 4: Comparison between exact resonances $\tilde{E}$ and the two different kinds of parametric resonances $\tilde{E}_\text{par}$ and $\tilde{E}^*_\text{par}$ described in the main text, for different $\varepsilon$ [(a)--(e)] obtained from a single realization of the disorder with $N=10^3$, $a=t=1$.
• Figure 5: Distribution of differences $\Delta \tilde{E}^{(*)}$ between exact and parametric energy values for $N=10^3$ and $t=a=1$, obtained from $10^4$ realizations. (a) Distribution $\rho$ of differences between exact and parametric resonances based on Eq. \ref{['eq:Heffpareig']} for $\varepsilon=2.5\cdot 10^{-4}$, $\varepsilon=2.5\cdot 10^{-3}$, $\varepsilon=2.5\cdot 10^{-2}$, $\varepsilon=0.25$, and $\varepsilon=2.5$ in blue colours from light to dark. (b) Same as in (a) but for differences between exact and parametric resonances based on Eq. \ref{['eq:Heffconeig']}. The $\varepsilon$ values are the same as in (a) but shown in red colours from light to dark. (c) Locations ${\Delta \tilde{E}^{(*)}}_\text{max}$ of the maxima of the distributions in (a) [blue, dashed] and (b) [red, solid] as function of the energy $\mathcal{E}$ for $\varepsilon=2.5\cdot 10^{-4}$, $\varepsilon=2.5\cdot 10^{-3}$, $\varepsilon=2.5\cdot 10^{-2}$, and $\varepsilon=0.25$ from light to dark colours.