No abstract

This is a first attempt to investigate the asymptotic behavior of the integrated density of states at the infimum of the spectrum for Schrödinger operators with magnetic fields which are Gaussian random fields. In simple examples it is shown that the integrated density of states decays exponentially. These examples shall give a hint to consider in more general framework. Résumé Ceci est un premier essai pour rechercher le comportement asymptotique, a la borne inférieure du spectre, de la densité d'état intégrée pour l'opérateur de Schrödinger avec champ magnétique qui est un champ aléatoire gaussiens. Sur des exemples simples nous allons montrer que la densité d'état intégrée décroît exponentiellement. Ces exemples donneront une indication pour considérer des cas plus généraux.

dedicated to professor shinzo watanabe on his 60th birthdayThe spectra of quadratic Schro dinger operators in general dimensional Euclidean spaces are determined by considering the corresponding classical mechanics and the corresponding Dirac operators. The results are applied to the study on the semiclassical approximation of the eigenvalues of Schro dinger operators with magnetic fields.

The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.

A Wegner estimate is proven for a Schrödinger operator with a bounded random vector potential and a Gaussian random scalar potential. The estimate is used to prove the strong dynamical localization and the exponential decay of the eigenfunctions. For the proof, Klopp's method using a vector field on a probability space and Germinet and Klein's bootstrap multiscale analysis are applied. Moreover Germinet and Klein's characterization of the Anderson metal-insulator transport transition is extended to the above operator.