On the Joint Distribution of Energy Levels of Random Schroedinger Operators

We consider operators with random potentials on graphs, such as the lattice version of the random Schroedinger operator. The main result is a general bound on the probabilities of simultaneous occurrence of eigenvalues in specified distinct intervals, with the corresponding eigenfunctions being separately localized within prescribed regions. The bound generalizes the Wegner estimate on the density of states. The analysis proceeds through a new multiparameter spectral averaging principle.