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Heavy tails and one-dimensional localization

Banff International Research Station for Mathematical Innovation and Discovery

In this talk we address a question posed several years ago by G. Zaslovski: what is the effect of heavy tails of one-dimensional random potentials on the standard objects of localization theory: Lyapunov exponents, density of states, statistics of eigenvalues, etc. We'll consider several models of potentials constructed by the use of $iid$ random variables which belong to the domain of attraction of the stable distribution with parameter $\alpha<1.$ In order to put our results in context, we'll recall the "regular theory" as presented in Carmona-Lacroix or Figotin-Pastur. We consider the one-dimensional Schr\"{o}dinger operator on the half line with boundary condition: \begin{eqnarray}\label{seqn} H^{\theta_0}\psi(x)=-\psi''(x)+V(x,\omega)\psi(x),\,\psi(0)\cos\theta_0-\psi'(0)\sin\theta_0=0. \end{eqnarray} where for each $x\in [0,\infty),\,V(x,\cdot)$ is a random variable on a basic probability space $(\Omega,\mathcal{F}, P)$ and $\theta_0\in[0,\pi]$ is fixed. Our potentials $V(x,\omega)$ will be piecewise constant, these are the so-called Kr\"{o}nig-Penny type potentials. As opposed to the regular theory, the large tails of the probability distribution of the potential $V$ will lead to random Lyapunov exponents and a different rate of decay of eigenfunctions from the standard case. The talk is based on joint work with S. Molchanov and N. Squartini.

Mathematics; Probability theory and stochastic processes; Potential theory; Probability / statistical mechanics

video/mp4

Author affiliation: UC Irvine

2019-03-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68600

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/