Lifshitz tails for the fractional Anderson model

Abstract

We consider the $d$-dimensional fractional Anderson model $(-\Delta {)}^{\alpha }+V_{\omega }$ on ${\ell }^2({\mathbb{Z}}^d)$ where $0<\alpha \le 1$. 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 $d/(2\alpha )$. 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 $$\frac{c_{\alpha ,d}}{|n-m{|}^{d+2\alpha }}\le -(-\Delta {)}^{\alpha }(n,m)\le \frac{C_{\alpha ,d}}{|n-m{|}^{d+2\alpha }}$$ for positive constants $c_{\alpha ,d}$, $C_{\alpha ,d}$ and all $n\ne m\in {\mathbb{Z}}^d$.

Lower bound on matrix elements of d dimensional fractional discrete Laplacian added