Conductivity and the current-current correlation measure

TL;DR

This paper reviews conductivity formulations for one-particle Hamiltonians, establishes the existence and properties of the current-current correlation measure in localization regimes, and relates it to localization length.

Contribution

It proves the existence of a density for the current-current correlation measure at coincident energies in localization regimes and relates it to localization length.

Findings

Density of the correlation measure vanishes at certain energies.

Upper bound on the rate of vanishing of the density.

Relation between correlation measure and localization length.

Abstract

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 Schr\"odinger 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.