关键词: integrated density of states, random Schrödinger operator, fractional Laplace operator, potential, Dirichlet eigenvalue

Abstract: Explicit asymptotic properties of the integrated density of states $N(\lambda)$ with respect to the spectrum for the random Schrödinger operator $H^{\omega}=(-\Delta)^{\alpha/2}+V^{\omega}$ are established, where $\alpha\in (0,2]$ and $V^\omega(x)=\sum_{i \in \mathbb{Z}^{d}} \xi_i(\omega) W(x-i)$ is a random potential term generated by a sequence of independent and identically distributed random variables $\{\xi_i\}_{i\in \mathbb{Z}^d}$ and a non-negative measurable function $W(x)$. In particular, the exact order of asymptotic properties of $N(\lambda)$ depends on the decay properties of the reference function $W(x)$ and the spectrum properties of the first Dirichlet eigenvalue of $(-\Delta)^{\alpha/2}$.

Key words: integrated density of states, random Schrödinger operator, fractional Laplace operator, potential, Dirichlet eigenvalue