Lifschitz tail for alloy-type models driven by the fractional Laplacian

Abstract

We establish precise asymptotics near zero of the integrated density of states for the random Schrödinger operators $(-\Delta {)}^{\alpha /2}+V^{\omega }$ in $L^2({\mathbb{R}}^d)$ for the full range of $\alpha \in (0,2]$ and a fairly large class of random nonnegative alloy-type potentials $V^{\omega }$. The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limit $$\underset{s\to 0}{\lim }{s}^{d/\alpha }\ln \ell ([0,s])=-C{\left({\lambda }_d^{(\alpha )}\right)}^{d/\alpha },$$ with $C\in (0,\infty ]$. The constant $C$ is is finite if and only if the common distribution of the lattice random variables charges $\left\{0\right\}$. In this case, the constant $C$ is expressed explicitly in terms of such a probability. In the limit formula, ${\lambda }_d^{(\alpha )}$ denotes the Dirichlet ground-state eigenvalue of the operator $(-\Delta {)}^{\alpha /2}$ in the unit ball in ${\mathbb{R}}^d.$

23 pages