Eigenvalue fluctuations for lattice Anderson Hamiltonians

Abstract: We study the statistics of Dirichlet eigenvalues of 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 $\xi^{(\epsilon)}(x)$ uniformly bounded independent random variables, on sets of the form $D_\epsilon:={x\in \mathbb Z^d\colon x\epsilon\in D}$ for $D\subset \mathbb R^d$ bounded, open and with a smooth boundary. If $\mathbb E\xi^{(\epsilon)}(x)=U(x\epsilon)$ holds for some bounded and continuous $U\colon D\to \mathbb R$, we show that, as $\epsilon\downarrow0$, the $k$-th eigenvalue converges to the $k$-th Dirichlet eigenvalue of the homogenized operator $-\Delta+U(x)$, where $\Delta$ is the continuum Dirichlet Laplacian on $D$. Assuming further 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. The limiting covariance for a given pair of simple eigenvalues is expressed as an integral of $V$ against the product of squares of the corresponding eigenfunctions of $-\Delta+U(x)$.