Eigenvalue fluctuations for lattice Anderson Hamiltonians

Abstract

We study the statistics of Dirichlet eigenvalues of the random Schrödinger operator $-epsilon^{-2}Delta^{(text{rm d}mkern0.5mu)}+xi^{(epsilon)}(x)$, with $Delta^{(text{rm d}mkern0.5mu)}$ the discrete Laplacian on ${Bbb Z}^d$ and $xi^{(epsilon)}(x)$ uniformly bounded independent random variables, on sets of the form $D_epsilon:={xin{Bbb Z}^dcolon xepsilonin D}$ for $Dsubset{Bbb R}^d$ bounded, open, and with a smooth boundary. If ${Bbb E}xi^{(epsilon)}(x)=U(xepsilon)$ holds for some bounded and continuous $Ucolon Dto{Bbb R}$, we show that, as $epsilondownarrow0$, 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 $text{rm Var}(xi^{(epsilon)}(x))=V(xepsilon)$ for some positive and continuous $Vcolon Dto{Bbb R}$, 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 $-Delta+U(x)$.