Eigenvalue fluctuations for lattice Anderson Hamiltonians

We study the statistics of Dirichlet eigenvalues of the random Schrödinger operator $-ε^{-2}Δ^{(\text{d})}+ξ^{(ε)}(x)$, with $Δ^{(\text{d})}$ the discrete Laplacian on $\mathbb Z^d$ and $ξ^{(ε)}(x)$ uniformly bounded independent random variables, on sets of the form $D_ε:=\{x\in \mathbb Z^d\colon xε\in D\}$ for $D\subset \mathbb R^d$ bounded, open and with a smooth boundary. If $\mathbb Eξ^{(ε)}(x)=U(xε)$ holds for some bounded and continuous $U\colon D\to \mathbb R$, we show that, as $ε\downarrow0$, the $k$-th eigenvalue converges to the $k$-th Dirichlet eigenvalue of the homogenized operator $-Δ+U(x)$, where $Δ$ is the continuum Dirichlet Laplacian on $D$. Assuming further that $\text{Var}(ξ^{(ε)}(x))=V(xε)$ 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 $-Δ+U(x)$.

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