Random Matrix Theory for Non-Hermitian Hamiltonians

We study the random matrix theory for non-Hermitian Hamiltonians for dissipative systems. We derive the universal behavior of the level repulsion of the complex eigenvalues of such a non-Hermitian Hamiltonian -- which deviates significantly from the level statistics of a random, Hermitian Hamiltonian -- in the absence of symmetries, or in the presence of time-reversal symmetry with T ² = ±1. We compute the density of states, which develops tails ρ(ε) ~ 1/ε² and a loss of spectral weight at low energies, proportional to the strength of non-Hermiticity. We conclude by discussing scattering experiments in which these effects can be measured.