We consider the localized region of the Anderson model and study the distribution of eigenfunctions simultaneously in space and energy. In a natural scaling limit, we prove convergence to a Poisson process. This provides a counterpoint to recent work, [9], which proves repulsion of the localization centres in a subtly different regime.

We study the level statistics of one-dimensional Schrödinger operator with random potential decaying like x −α at infinity. We consider the point process ξ L consisting of the rescaled eigenvalues and show that : (i)(ac spectrum case) for α > 1 2 , ξ L converges to a clock process, and the fluctuation of the eigenvalue spacing converges to Gaussian.(ii)(critical case) for α = 1 2 , ξ L converges to the limit of the circular β-ensemble.

We study the level statistics for two classes of 1-dimensional random Schrödinger operators : (1) for operators whose coupling constants decay as the system size becomes large, and (2) for operators with critically decaying random potential. As a byproduct of (2) with our previous result [2] imply the coincidence of the limits of circular and Gaussian beta ensembles.

No abstract

We prove Anderson localization near the bottom of the spectrum for twodimensional discrete Schrödinger operators with random magnetic fields and no scalar potentials. We suppose the magnetic fluxes vanish in pairs, and the magnetic field strength is bounded from below by a positive constant. Main lemmas are the Lifshitz tail and the Wegner estimate on the integrated density of states. Then, Anderson localization, i.e., pure point spectrum with exponentially decreasing eigenfunctions, is proved by the standard multiscale argument.