Unitarization of the classical statistical s matrix for systems with localization

It has recently been observed that a large reverberant cavity admits a classically motivated random s matrix that is not unitary but that can be made so in a minimally invasive manner. A random process with an envelope s2 (t) ∼exp(-t TH) representing reflection from a structure having no internal time scales other than Heisenberg time TH was shown to lead to a unitary S matrix exhibiting mesoscopic behaviors not present in the classically inspired original s2 (t). These included enhanced backscatter, quantum echo, power law tails, and level repulsion. Here the procedure is extended to two systems having, in addition to Heisenberg times, internal time scales corresponding to conduction and diffusion. The repaired S matrices for coupled rooms and one-dimensional random structures with multiple scattering are found to correspond to Wigner K matrices with signatures of localization.