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 Schr\"{o}dinger 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 $$\lim_{s\to 0} s^{d/\alpha}\ln\ell([0,s]) = -C \left(\lambda_d^{(\alpha)}\right)^{d/\alpha},$$ 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 $\mathbb R^d.$