Distributions of off-diagonal scattering matrix elements: Exact results

•Scattering problem in complex or chaotic systems.•Heidelberg approach to model the chaotic nature of the scattering center.•A novel route to the nonlinear sigma model based on the characteristic function.•Exact results for the distributions of off-diagonal scattering-matrix elements.•Universal aspects of the scattering-matrix fluctuations.

Scattering is a ubiquitous phenomenon which is observed in a variety of physical systems which span a wide range of length scales. The scattering matrix is the key quantity which provides a complete description of the scattering process. The universal features of scattering in chaotic systems is most generally modeled by the Heidelberg approach which introduces stochasticity to the scattering matrix at the level of the Hamiltonian describing the scattering center. The statistics of the scattering matrix is obtained by averaging over the ensemble of random Hamiltonians of appropriate symmetry. We derive exact results for the distributions of the real and imaginary parts of the off-diagonal scattering matrix elements applicable to orthogonally-invariant and unitarily-invariant Hamiltonians, thereby solving a long standing problem.