Resonances for random highly oscillatory potentials

TL;DR

This paper analyzes the spectral properties of Schrödinger operators with highly oscillatory, random potentials, revealing how eigenvalues and resonances behave as the disorder scale diminishes, with convergence regimes depending on frequency effects.

Contribution

It provides a novel perturbation analysis demonstrating almost sure convergence of spectral quantities for disordered potentials in high oscillation regimes.

Findings

Eigenvalues and resonances converge almost surely as disorder scale decreases.

Identification of stochastic and deterministic regimes based on frequency effects.

Analysis of the interplay between large deviations and constructive interference.

Abstract

We study discrete spectral quantities associated to Schr\"odinger operators of the form $-\Delta_{\mathbb{R}^d}+V_N$, $d$ odd. The potential $V_N$ models a highly disordered crystal; it varies randomly at scale $N^{-1} \ll 1$. We use perturbation analysis to obtain almost sure convergence of the eigenvalues and scattering resonances of $-\Delta_{\mathbb{R}^d}+V_N$ as $N \rightarrow \infty$. We identify a stochastic and a deterministic regime for the speed of convergence. The type of regime depends whether the low frequencies effects due to large deviations overcome the (deterministic) constructive interference between highly oscillatory terms.

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