We consider the d -dimensional fractional Anderson model $$(-\Delta )^\alpha + V_\omega $$ ( - Δ ) α + V ω on $$\ell ^2({\mathbb {Z}}^d)$$ ℓ 2 ( Z d ) where $$0<\alpha \leqslant 1$$ 0 < α ⩽ 1 . Here $$-\Delta $$ - Δ is the negative discrete Laplacian and $$V_\omega $$ V ω 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 $$ d/ (2\alpha )$$ d / ( 2 α ) . 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 $$ \frac{c_{\alpha ,d}}{|n-m|^{d+2\alpha }} \leqslant -(-\Delta )^\alpha (n,m)\leqslant \frac{C_{\alpha ,d}}{|n-m|^{d+2\alpha }}$$ c α , d | n - m | d + 2 α ⩽ - ( - Δ ) α ( n , m ) ⩽ C α , d | n - m | d + 2 α for positive constants $$c_{\alpha ,d}$$ c α , d , $$C_{\alpha ,d}$$ C α , d and all $$n\ne m\in {\mathbb {Z}}^d$$ n ≠ m ∈ Z d .

中文翻译：

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分数安德森模型的 Lifshitz 尾

我们考虑 d 维分数阶 Anderson 模型 $$(-\Delta )^\alpha + V_\omega $$ ( - Δ ) α + V ω on $$\ell ^2({\mathbb {Z}}^d )$$ ℓ 2 ( Z d ) 其中 $$0<\alpha \leqslant 1$$ 0 < α ⩽ 1 。这里 $$-\Delta $$ - Δ 是负离散拉普拉斯算子，$$V_\omega $$ V ω 是由 iid 随机变量组成的随机安德森势。我们证明该模型在频谱的下边缘显示 Lifshitz 尾，指数为 $$ d/ (2\alpha )$$ d / ( 2 α ) 。为此，我们证明了负离散分数拉普拉斯算子的非对角矩阵元素为负且满足两侧边界 $$ \frac{c_{\alpha ,d}}{|nm|^{ d+2\alpha }} \leqslant -(-\Delta )^\alpha (n,m)\leqslant \frac{C_{\alpha ,d}}{|nm|^{d+2\alpha }}$ $ c α , d | n - m | d + 2 α ⩽ - ( - Δ ) α ( n , m ) ⩽ C α , d | n - m | d + 2 α 对于正常数 $$c_{\alpha ,d}$$ c α , d , $$C_{\alpha ,d}$$ C α , d 和所有 $$n\ne m\in {\ mathbb {Z}}^d$$ n ≠ m ∈ Z d 。