Uniformly distributed eigenfunctions on tori with random impurities

2015/02/17 by Ueberschaer, Henrik
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1502.05010

Abstract

We study a random Schroedinger operator, the Laplacian with N independently uniformly distributed random delta potentials on flat tori T^d_L = R^d/LZ^d, d = 2, 3, where L > 0 is large. We determine a condition in terms of the size of the torus L, the density of the potentials ρ= N/L^d and the energy of the eigenfunction E such any such eigenfunctions will with nonzero probability be uniformly distributed on the entire torus. We remark that the equidistribution we prove here is still consistent with a localized regime, where the localization length is much larger than the size of the torus. In fact our result implies a certain polynomial lower bound on the localization length, so the localization length becomes infinitely large as E tends to infinity.

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• Spectral geometry of flat tori with random impurities
• Delocalization for random displacement models with Dirac masses
• Quantitative equidistribution of eigenfunctions for toral Schrödinger operators
• Uniform Distribution of Eigenstates on a Torus with Two Point Scatterers
• Spectral statistics for random Schrödinger operators in the localized regime