Band Edge Localization and the Density of States for Acoustic and Electromagnetic Waves in Random Media

The propagation of sound waves and electromagnetic radiation in a background medium is strongly innuenced by the addition of random scatterers. We study random perturbations of a background medium which has a spectral gap in its permissible energy spectrum. We prove that the randomness localizes waves at energies near the band edges of the spectral gap of the background medium. The perturbations of the dielectric function or the sound speed are described by Anderson-type potentials, which include random displacements from the equilibrium positions which model thermal vibrations. The waves with energies near the band edges are almost-surely exponentially localized. We prove a Wegner estimate valid at all energies in the spectral gap of the unperturbed operator. It follows that the integrated density of states is Lipschitz continuous at energies in the unperturbed spectral gaps.