Tail bounds for the Dyson series of random Schrödinger equations

2025/02/04 by Adam Black, Black, Adam, Reuben Drogin +3 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2502.02566

openalex publication_date 2025/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study Schrödinger equations on ℤ^d and ℝ^d, d≥ 2 with random potentials of strength λ. Our main result gives tail bounds for the terms of the Dyson series that are effective at time scales on the order of λ^-2+ε. As corollaries, we obtain estimates on the frequency localization and spatial delocalization of approximate eigenfunctions in the spirit of works by Schlag-Shubin-Wolff and T. Chen. These estimates also apply to Floquet states associated to time-periodic potentials. Our proof is elementary in that we use neither sophisticated harmonic analysis nor diagrammatic arguments. Instead, we use only the noncommutative Khintchine inequality from random matrix theory combined with pointwise dispersive estimates for the free Schrödinger equation.

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