%0 Journal Article
%T Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators
%A Victor Chulaevsky
%J International Journal of Statistical Mechanics
%D 2013
%R 10.1155/2013/931063
%X We propose a simple approach allowing reducing the eigenvalue concentration analysis of a class of random operator ensembles with singular probability distribution to the analysis of an auxiliary ensemble with bounded probability density. Our results apply to the Wegner- and Minami-type estimates for single- and multiparticle operators. 1. Introduction The role and importance of the eigenvalue concentration (EVC) estimates in the spectral theory of random operators hardly needs to be explained today. It is, however, interesting that the very first general estimate, established over thirty years ago by Wegner [1], is very accurate; in particular, it features the optimal dependence upon the volume of the finite-size configuration space in which the random operator at hand is considered. In the general terminology of the point random fields, the Wegner estimate is an upper bound on the first-order correlation function of the (finite) point random field on formed by the eigenvalues of the respective random operator. The next significant step in this area was made by Minami [2] who proved the regularity of, and obtained an efficient upper bound on, the second-order eigenvalue correlation function for . Although this result appeared in the remarkable paper by Minami [2] only as an intermediate (but crucial) technical estimate its importance has been immediately recognized, and what is called today the Minami estimate has been encapsulated and further extended to correlation functions of all orders , in independent works by Bellissard et al. [3] and by Graf and Vaghi [4] (the order of citation is merely alphabetical). Combes et al. [5] extended these results to a large class of marginal probability distributions. Minami focused in [2] on the local Poisson statistics of the eigenvalues; his analysis was further developed by Nakano [6]. The ideal situation for the methods of the above-mentioned works [1, 3, 4], considering discrete Schr£żdinger operators with IID random potentials, is where the common marginal distribution of the random potential has bounded density (the compactness of its support is often an additional convenient feature). Unfortunately, one encounters some analytical problems well before the marginal probability distribution of the random potential becomes singular; it suffices to take a distribution with unbounded density, for example, the so-called arcsine law with the probability density and the gentle, integrable singularities at suffice to ruin the original proof of the Wegner estimate, although it can be enhanced so as to apply to random
%U http://www.hindawi.com/journals/ijsm/2013/931063/