We study the statistics of Dirichlet eigenvalues of the random Schr "odinger operator @math , with @math the discrete Laplacian on @math and @math uniformly bounded independent random variables, on sets of the form @math for @math bounded, open and with a smooth boundary. If @math holds for some bounded and continuous @math , we show that, as @math , the @math -th eigenvalue converges to the @math -th Dirichlet eigenvalue of the homogenized operator @math , where @math is the continuum Dirichlet Laplacian on @math . Assuming further that @math for some positive and continuous @math , 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 @math against the product of squares of the corresponding eigenfunctions of @math .
We consider random Schrodinger operators of the form ( + ), where ( ) is the lattice Laplacian on ( Z ^ d ) and ( ) is an i.i.d. random field, and study the extreme order statistics of the Dirichlet eigenvalues for this operator restricted to large but finite subsets of ( Z ^ d ). We show that, for ( ) with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class, and the corresponding eigenfunctions are exponentially localized in regions where ( ) takes large, and properly arranged, values. The picture we prove is thus closely connected with the phenomenon of Anderson localization at the spectral edge. Notwithstanding, our approach is largely independent of existing methods for proofs of Anderson localization and it is based on studying individual eigenvalue eigenfunction pairs and characterizing the regions where the leading eigenfunctions put most of their mass. We consider the spectral problem for the random Schrodinger operator on the multidimensional lattice torus increasing to the whole of lattice, with an i.i.d. potential (Anderson Hamiltonian). We obtain the explicit almost sure asymptotic expansion formulas for the extreme eigenvalues and eigenfunctions in the intermediate rank case, provided the upper distributional tails of potential decay at infinity slower than the double exponential function. For the fractional-exponential tails (including Weibull’s and Gaussian distributions), extremal type limit theorems for eigenvalues are proved, and the strong influence of parameters of the model on a specification of normalizing constants is described. In the proof we use the finite-rank perturbation arguments based on the cluster expansion for resolvents.  Let (H_V = - d^ 2 dt^ 2 + q(t, ) ) be an one-dimensional random Schrodinger operator in ℒ2(−V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x t ), wherex t is a Brownian motion on the compact Riemannian manifoldK andF:K→R1 is a smooth Morse function, ( F = 0 ). Let (N_V ( ) = Ei(V) 1 ), where Δ∈(0, ∞),E i (V) are the eigenvalues ofH V . The main result (Theorem 1) of this paper is the following. IfV→∞,E0>0,k∈Z+ anda>0 (a is a fixed constant) then @math wheren(E0) is a limit state density ofH V ,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH V ,V→∞. We consider the discrete Anderson model and prove enhanced Wegner and Minami estimates where the interval length is replaced by the IDS computed on the interval. We use these estimates to improve on the description of finite volume eigenvalues and eigenfunctions obtained in Germinet and Klopp (J Eur Math Soc http: arxiv.org abs 1011.1832, 2010). As a consequence of the improved description of eigenvalues and eigenfunctions, we revisit a number of results on the spectral statistics in the localized regime obtained in Germinet and Klopp (J Eur Math Soc http: arxiv.org abs 1011.1832, 2010) and Klopp (PTRF http: fr.arxiv.org abs 1012.0831, 2010) and extend their domain of validity, namely: the local spectral statistics for the unfolded eigenvalues; the local asymptotic ergodicity of the unfolded eigenvalues. In dimension 1, for the standard Anderson model, the improvement enables us to obtain the local spectral statistics at band edge, that is in the Lifshitz tail regime. In higher dimensions, this works for modified Anderson models. We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy @math in the localized phase. Assume the density of states function is not too flat near @math . Restrict it to some large cube @math . Consider now @math , a small energy interval centered at @math that asymptotically contains infintely many eigenvalues when the volume of the cube @math grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive * uniform Poisson behavior of the locally unfolded eigenvalues, * a.s. Poisson behavior of the joint distibutions of the unfolded energies and unfolded localization centers in a large range of scales. * the distribution of the unfolded level spacings, locally and globally, * the distribution of the unfolded localization centers, locally and globally. 