Olivier Lenoble, and Peter Müller* We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schrödinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the ac-conductivity is bounded from above by Cν 2 (log 1 ν)d+2 at small frequencies ν. This is to be compared to Mott’s formula, which predicts the leading term to be Cν 2 (log 1 ν)d+1.

We review various formulations of conductivity for one-particle Hamiltonians and relate them to the current-current correlation measure. We prove that the current-current correlation measure for random Schrödinger operators has a density at coincident energies provided the energy lies in a localization regime. The density vanishes at such energies and an upper bound on the rate of vanishing is computed. We also relate the current-current correlation measure to the localization length.