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arXiv · 1605.00118

Eigenvectors of the critical 1-dimensional random Schroedinger operator

Ben Rifkind·Balint Virag
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Abstract

The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schroedinger operator H = Delta+V. Here Delta is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to Z_n and consider the critical model H_n. We show that the shape of a uniformly chosen eigenvector of H_n converges in law to exp (-|t|/4 + Z_t/sqrt(2)), where Z is two-sided Brownian motion.

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Ben Rifkind, Balint Virag. 2016-04-30. Eigenvectors of the critical 1-dimensional random Schroedinger operator. https://doi.org/10.1007/s00039-018-0460-0

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