Dynamical self-averaging for a lattice Schrödinger equation with weak random potential

2013/12/25 by Maximilian Butz, Butz, Maximilian
Mathematics · Physics and Astronomy · #80M40 #81Q10 #82D30 #Advanced Mathematical Physics Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:80M40 #msc:81Q10 #msc:82D30

paper · pdf · doi:10.48550/arxiv.1312.6979

38 pages, 7 figures Added proof for almost sure convergence

openalex publication_date 2013/12/25 · arxiv created 2015/06/20 · arxiv updated 2015/06/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study the kinetic, weak coupling limit of the dynamics governed by a discrete random Schrödinger operator on ℤ³. For sequences of ℓ²(ℤ³)-bounded initial states and convergent initial Wigner transform, we prove that the scaled Wigner transform converges to the solution of a linear Boltzmann equation in L^r(ℙ)for all r>0, thus considerably strengthening a previous result by Chen. The key ingredients for the proof are a finer classification of graphs in the expansion of the perturbed dynamics as well as a novel resolvent estimate for the unperturbed Schrödinger operator. Under some additional assumption on the sequence of initial states we even prove almost sure convergence.

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