Abstract
We study the statistics of Dirichlet eigenvalues of the random Schrodinger operator -epsilon(-2)Delta((d)) + xi((epsilon))(x), with Delta((d)) the discrete Laplacian on Z(d) and xi((epsilon))(x) uniformly bounded independent random variables, on sets of the form D-epsilon := {x is an element of Z(d) : x epsilon is an element of D} for D subset of R-d bounded, open, and with a smooth boundary. If E xi((epsilon))(x) = U(x epsilon) holds for some bounded and continuous U : D -> R, we show that, as epsilon down arrow 0, the kth eigenvalue converges to the kth Dirichlet eigenvalue of the homogenized operator -Delta + U(x), where Delta is the continuum Dirichlet Laplacian on D.
Assuming further that Var(xi((epsilon))(x)) = V (x epsilon) for some positive and continuous V : D -> 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).