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

TL;DR

This paper generalizes the Poisson kernel to non-unitary scattering matrices, providing a framework for chaotic scattering with direct processes in systems lacking flux conservation, such as lossy or gain media.

Contribution

It introduces a squared Poisson kernel for non-unitary matrices, extending the invariant measure relationship to systems with absorption or amplification.

Findings

The invariant measure for non-unitary matrices is related by the Poisson kernel squared.

The transformation maps systems with direct processes to those without, simplifying analysis.

Results apply to chaotic systems with gain or loss, like microwave cavities and elastic resonators.

Abstract

The problem of chaotic scattering in presence of direct processes or prompt responses is mapped via a transformation to the case of scattering in absence of such processes for non-unitary scattering matrices, \tilde S. In the absence of prompt responses, \tilde S is uniformly distributed according to its invariant measure in the space of \tilde S matrices with zero average, < \tilde S > =0. In the presence of direct processes, the distribution of \tilde S is non-uniform and it is characterized by the average < \tilde S > (\neq 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, here we show that for non-unitary 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 example, transport experiments in chaotic systems, where gains or losses are present, like microwave chaotic cavities or graphs, and acoustic or elastic resonators.