%0 Journal Article
%T Eigenvalue fluctuations for lattice Anderson Hamiltonians
%A Marek Biskup
%A Ryoki Fukushima
%A Wolfgang Koenig
%J Mathematics
%D 2014
%I arXiv
%X We consider the random Schr\"odinger operator $-\epsilon^{-2}\Delta^{(\text{d})}+\xi^{(\epsilon)}(x)$, with $\Delta^{(\text{d})}$ the discrete Laplacian on $\mathbb Z^d$ and $\xi^{(\epsilon)}(x)$ are bounded and independent random variables, on sets of the form $D_\epsilon:=\{x\in\mathbb Z^d\colon x\epsilon\in D\}$ for $D$ bounded, open and with a smooth boundary, and study the statistics of the Dirichlet eigenvalues in the limit $\epsilon\downarrow0$. Assuming $\mathbb E\xi^{(\epsilon)}(x)=U(x\epsilon)$ holds for some bounded and continuous function $U\colon D\to\mathbb R$, the $k$-th eigenvalue converges to the $k$-th Dirichlet eigenvalue of the homogenized operator $-\Delta+U(x)$, where $\Delta$ is the continuum Laplacian on $D$. Moreover, assuming that $\text{Var}(\xi^{(\epsilon)}(x))=V(x\epsilon)$ for some positive and continuous $V\colon D\to\mathbb R$, we establish a multivariate central limit theorem for simple eigenvalues centered by their expectation and scaled by $\epsilon^{-d/2}$. The limiting covariance is expressed as integral of $V$ against the product of squares of two eigenfunctions of $-\Delta+U(x)$.
%U http://arxiv.org/abs/1406.5268v1