Lifshitz tail for continuous Anderson models driven by L\'{e}vy operators

TL;DR

This paper studies the asymptotic behavior of the integrated density of states near zero for a class of random Schrödinger operators involving Lévy operators, revealing how the lattice configuration influences spectral properties.

Contribution

It provides a unified analysis of the Lifshitz tail behavior for operators with local and non-local kinetic terms, extending previous results to Lévy-driven models.

Findings

Established bounds on the decay of the integrated density of states near zero.

Identified the dependence of Lifshitz tail behavior on the distribution of random potentials.

Unified treatment of local and non-local operators in spectral analysis.

Abstract

We investigate the behavior near zero of the integrated density of states $\ell$ for random Schr\"{o}dinger operators $\Phi(-\Delta) + V^{\omega}$ in $L^2(\mathbb R^d)$, $d \geq 1$, where $\Phi$ is a complete Bernstein function such that for some $\alpha \in (0,2]$, one has $ \Phi(\lambda) \asymp \lambda^{\alpha/2}$, $\lambda \searrow 0$, and $V^{\omega}(x) = \sum_{ \mathbf{i}\in \mathbb{Z}^d} q_{\mathbf{i}}(\omega) W(x-\mathbf{i})$ is a random nonnegative alloy-type potential with compactly supported single site potential $W$. We prove that there are constants $C, \widetilde C,D, \widetilde D>0$ such that $$ -C \leq\liminf_{\lambda \searrow 0} \frac{\lambda^{d/\alpha}}{|\log F_q(D \lambda)|}{\log \ell(\lambda)} \qquad \text{and} \qquad \limsup_{\lambda \searrow 0} \frac{\lambda^{d/\alpha}}{|\log F_q(\widetilde D \lambda)|}\log \ell(\lambda) \leq -\widetilde C, $$ where $F_q$ is the common cumulative distribution function of the lattice random variables $q_{\mathbf i}$. In particular, we identify how the behavior of $\ell$ at zero depends on the lattice configuration. For typical examples of $F_q$ the constants $D$ and $\widetilde D$ can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, both local and non-local kinetic terms such as the Laplace operator, its fractional powers and the quasi-relativistic Hamiltonians.