  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.
