Chaotic scattering with direct processes:: a generalization of Poisson's kernel for non-unitary scattering matrices

The problem of chaotic scattering in the presence of direct processes or prompt responses is mapped via a transformation to the case of scattering in the absence of such processes for non-unitary scattering matrices (S) over tilde. When prompt responses are absent, (S) over tilde S is uniformly distributed according to its invariant measure in the space of (S) over tilde (S) over tilde matrices with zero average <(S) over tilde > = 0. When direct processes occur, the distribution of (S) over tilde is non- uniform and is characterized by an average <(S) over tilde > not equal 0. In contrast to the case of unitary matrices S, where the invariant measures of S for chaotic scattering with and without direct processes are related through the well- known Poisson kernel, we show that for nonunitary scattering matrices the invariant measures are related by the Poisson kernel squared. Our results are relevant to situations where flux conservation is not satisfied, for transport experiments in chaotic systems where gains or losses are present, for example in microwave chaotic cavities or graphs, and acoustic or elastic resonators.