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

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.