Lifshitz tails for the fractional Anderson model

We consider the $d$-dimensional fractional Anderson model $(-\Delta)^\alpha+ V_\omega$on$\ell^2(\mathbb Z^d)$where$0<\alpha\leq 1$\mathrm{.}Here$-\Delta$ is the negative discrete Laplacian and $V_{\omega }$ is the random Anderson potential consisting of iid random variables. We prove that the model exhibits Lifshitz tails at the lower edge of the spectrum with exponent $d\mathrm{/}(2\alpha )$. To do so, we show among other things that the non-diagonal matrix elements of the negative discrete fractional Laplacian are negative and satisfy the two-sided bound $$\frac{c_{\alpha ,d}}{\mathrm{\mid }n-m{\mathrm{\mid }}^{d+2\alpha }}\le -(-\mathrm{\Delta }{)}^{\alpha }(n,m)\le \frac{C_{\alpha ,d}}{\mathrm{\mid }n-m{\mathrm{\mid }}^{d+2\alpha }}$$ for positive constants $c_{\alpha ,d}$, $C_{\alpha ,d}$ and all $n\mathrm{\ne }m\in {\mathbb{Z}}^d$.