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 .
Another way to look at Anderson localization is by analyzing the limiting spectral statistics for operators in an increasing sequence of finite volumes. In the localized regime, the statistics is expected to be given by a Poisson point process. This has so far been proved in the bulk'' (i.e., the interior) of the spectrum (Molchanov @cite_31 in @math and Minami @cite_2 for general @math ) . At spectral edges there seem to be only partial results for bounded potentials at this time (Germinet and Klopp @cite_12 @cite_34 ) although a somewhat more complete theory has been developed for some unbounded potentials (Astrauskas @cite_20 @cite_3 , Biskup and K "onig @cite_14 ). In the delocalization regime, the spectral statistics is expected to be that seen in random matrix ensembles.
@cite_14 @cite_3 @cite_2 @cite_31 @cite_34 @cite_12 @cite_20
2019358407 1895809589 2005820649 2103519107
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 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 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. We present a source localization method based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the spl lscr sub 1 -norm. A number of recent theoretical results on sparsifying properties of spl lscr sub 1 penalties justify this choice. Explicitly enforcing the sparsity of the representation is motivated by a desire to obtain a sharp estimate of the spatial spectrum that exhibits super-resolution. We propose to use the singular value decomposition (SVD) of the data matrix to summarize multiple time or frequency samples. Our formulation leads to an optimization problem, which we solve efficiently in a second-order cone (SOC) programming framework by an interior point implementation. We propose a grid refinement method to mitigate the effects of limiting estimates to a grid of spatial locations and introduce an automatic selection criterion for the regularization parameter involved in our approach. We demonstrate the effectiveness of the method on simulated data by plots of spatial spectra and by comparing the estimator variance to the Crame spl acute r-Rao bound (CRB). We observe that our approach has a number of advantages over other source localization techniques, including increased resolution, improved robustness to noise, limitations in data quantity, and correlation of the sources, as well as not requiring an accurate initialization.