Resonances for random highly oscillatory potentials

Abstract

We study discrete spectral quantities associated to Schrödinger 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 $N\to \infty$. 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.

37 pages

References in corpus (4)

• Pollicott-Ruelle resonances via kinetic Brownian motion
• Resonances for 1D half-line periodic operators: I. Generic case
• Resonances for 1D half-line periodic operators: II. Special case
• Vlasov-Fokker-Planck equation: stochastic stability of resonances and unstable manifold expansion

Cited by in corpus (4)

• Characterization of edge states in perturbed honeycomb structures
• Dirac operators and domain walls
• Defect resonances of truncated crystal structures
• Localization of eigenvectors of non-Hermitian banded noisy Toeplitz matrices