Eigenvalue Fluctuations for Lattice Anderson Hamiltonians

We consider the random Schrödinger operator−ε−2∆(d) +ξ (ε)(x), with ∆(d) the discrete Laplacian on Zd and ξ (ε)(x) are bounded and independent random variables, on sets of the form Dε := {x ∈ Zd : xε ∈ D} for D bounded, open and with a smooth boundary, and study the statistics of the Dirichlet eigenvalues in the limit ε ↓ 0. Assuming Eξ (ε)(x) = U(xε) holds for some bounded and continuous function U : D→ R, the k-th eigenvalue converges to the k-th Dirichlet eigenvalue of the homogenized operator−∆+U(x), where ∆ is the continuum Laplacian on D. Moreover, assuming that Var(ξ (ε)(x)) = V (xε) for some positive and continuous V : D→ R, we establish a multivariate central limit theorem for simple eigenvalues centered by their expectation and scaled by ε−d/2. The limiting covariance is expressed as integral of V against the product of squares of two eigenfunctions of −∆+U(x).