Universal K-matrix distribution in β = 2 ensembles of random matrices

Abstract

The K-matrix, also known as the ‘Wigner reaction matrix’ in nuclear scattering or the ‘impedance matrix’ in 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 Brouwer (1995 Phys. Rev. B 51 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 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.