Lifschitz tail for alloy-type models driven by the fractional Laplacian
英文摘要：
We establish precise asymptotics near zero of the integrated density of states for the random Schrödinger operators $(-Δ)^{α/2} + V^ω$ in $L^2(\mathbb R^d)$ for the full range of $α\in(0,2]$ and a fairly large class of random nonnegative alloy-type potentials $V^ω$. The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limit $$\lim_{s\to 0} s^{d/α}\ln\ell([0,s]) = -C \left(λ_d^{(α)}\right)^{d/α},$$ with $C \in (0,\infty]$. The constant $C$ is is finite if and only if the common distribution of the lattice random variables charges $\left\{0\right\}$. In this case, the constant $C$ is expressed explicitly in terms of such a probability. In the limit formula, $λ_d^{(α)}$ denotes the Dirichlet ground-state eigenvalue of the operator $(-Δ)^{α/2}$ in the unit ball in $\mathbb R^d.$
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