Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory, the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices. A long-standing problem was the computation of the distribution of the off-diagonal scattering-matrix elements. We report here an exact solution to this problem and present analytical results for systems with preserved and with violated time-reversal invariance. Our derivation is based on a new variant of the supersymmetry method. We also validate our results with scattering data obtained from experiments with microwave billiards.

Correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices from the Gaussian orthogonal ensemble (GOE) frequently arise in applications of random matrix theory (RMT) to physics of quantum chaotic systems, and beyond. We provide an explicit evaluation of the large-N limits of a few nontrivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method. As one of the applications we derive the distribution of an off-diagonal entry K ab of the resolvent (or Wigner K -matrix) of GOE matrices which, among other things, is of relevance for experiments on chaotic wave scattering in electromagnetic resonators.

The recently derived distributions for the scattering-matrix elements in quantum chaotic systems are not accessible in the majority of experiments, whereas the cross sections are. We analytically compute distributions for the off-diagonal cross sections in the Heidelberg approach, which is applicable to a wide range of quantum chaotic systems. We thus eventually fully solve a problem which already arose more than half a century ago in compound-nucleus scattering. We compare our results with data from microwave and compound-nucleus experiments, particularly addressing the transition from isolated resonances towards the Ericson regime of strongly overlapping ones.

The K−matrix, also known as the "Wigner reaction matrix" in nuclear scattering or "impedance matrix" in the electromagnetic wave scattering, is given essentially by an M × M diagonal block of the resolvent (E − H) −1 of a Hamiltonian H. For chaotic quantum systems the Hamiltonian H can be modelled by random Hermitian N × N matrices taken from invariant ensembles with the Dyson symmetry index β = 1, 2 or 4. For β = 2 we prove by explicit calculation a universality conjecture by P. Brouwer (Phys. Rev. B 51 (1995), 16878-84) which is equivalent to the claim that the probability distribution of K, for a broad class of invariant ensembles of random Hermitian matrices H, converges to a matrix Cauchy distribution with density P(K) ∝ det (λ 2 + (K − ǫ) 2 ) −M in the limit N → ∞, provided the parameter M is fixed and the spectral parameter E is taken within the support of the eigenvalue distribution of H. In particular, we show that for a broad class of unitary invariant ensembles of random matrices finite diagonal blocks of the resolvent are Cauchy distributed. The cases β = 1 and β = 4 remain outstanding.