Heavy tails and one-dimensional localization

TL;DR

This paper investigates the spectral properties of a one-dimensional Schrödinger operator with heavy-tailed random potentials, focusing on Lyapunov exponents, density of states, and spectrum nature under stable law distributions.

Contribution

It introduces new models of random potentials with heavy tails and analyzes their spectral characteristics, extending understanding of localization phenomena in such systems.

Findings

Existence of Lyapunov exponents for heavy-tailed potentials

Characterization of the integrated density of states

Insights into the spectrum's nature under stable law distributions

Abstract

We address the fundamental questions concerning the operator \begin{eqnarray*} 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 the random potential $V$ has a variety of forms. In one example, it is composed of width one bumps of random heights where the square root of the heights are in the domain of attraction of a stable law with index $\alpha\in(0,1)$ or in another it is composed of width one bumps of height one where the distance between bumps is in the domain of attraction of a stable law with index $\alpha\in(0,1).$ We consider the existence of Lyapunov exponents, integrated density of states and the nature of the spectrum of the operator.