Eigensystem multiscale analysis for the Anderson model via the Wegner estimate

We present a new approach to the eigensystem multiscale analysis (EMSA) for random Schr\"odinger operators that relies on the Wegner estimate. The EMSA treats all energies of the finite volume operator in an energy interval at the same time, simultaneously establishing localization of all eigenfunctions with eigenvalues in the energy interval with high probability. It implies all the usual manifestations of localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization). The new method removes the restrictive level spacing hypothesis used in the previous versions of the EMSA. The method is presented in the context of the Anderson model, allowing for single site probability distributions that are H\"older continuous of order $\alpha \in (0,1]$.Comment: arXiv admin note: substantial text overlap with arXiv:1611.02650, arXiv:1509.0852

The conductivity measure for the Anderson model

We study the ac-conductivity in linear response theory for the Anderson tight-binding model. We define the electrical ac-conductivity and calculate the linear-response current at zero temperature for arbitrary Fermi energy. In particular, the Fermi energy may lie in a spectral region where extended states are believed to exist