Universal K-matrix distribution in beta=2 Ensembles of Random Matrices

Understand

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 x 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 x N matrices taken from invariant ensembles with the Dyson symmetry index beta=1,2,4.
• For beta=2 we prove by explicit calculation a universality conjecture by P.
• Brouwer 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 ${\cal P}(K)\propto \left[\det{({\lambda}^2+(K-{\epsilon})^2)}\right]^{-M}$ in the limit $N\to \infty$, provided the parameter M is fixed and the spectral parameter E is taken within the support of the eigenvalue distribution of H.