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Resonances for random highly oscillatory potentials. (English) Zbl 1400.81090

Summary: We study discrete spectral quantities associated with 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 \(N \rightarrow \infty\). We identify a stochastic and a deterministic regime for the speed of convergence. The type of regime depends whether the low frequency effects due to large deviations overcome the (deterministic) constructive interference between highly oscillatory terms.{
©2018 American Institute of Physics}

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
81U05 \(2\)-body potential quantum scattering theory
81Q15 Perturbation theories for operators and differential equations in quantum theory
35B34 Resonance in context of PDEs
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