Wave scattering in random media.

Abstract

We study the scattering theory for the descrete Schrodinger equation with a random potential having large finite support. We consider in one dimension a wave packet incoming from one side of the disordered section. We prove that the transmission of a wave packet is improbable if the disordered section is large, and that a fluctuation deep within the disordered section has a very small effect on the scattering of wave packets. The scattering theory for the discrete random Schrodinger equation in a strip in two dimensions is also considered. We derive large deviation bounds on the elements of the transmission matrix uniform in the energy parameter. These uniform bounds are used to show that the probability of a significant portion of a wave packet is transmitted is small as the length of the disordered section becomes large. We also study the time delay in potential scattering. We consider the situation when the potential becomes a white noise. The time delay is related to the energy derivative of the phase shift. We derive stochastic differential equations for the phase shift and the frequency derivative of the phase shift. We find that there is no time delay in the low frequency limit. However in the high frequency limit we find the time delay is a random function of the depth of the disordered section.