On the density of complex eigenvalues of Wigner reaction matrix in a disordered or chaotic system with absorption

In an absorptive system the Wigner reaction $K-$matrix (directly related to the impedance matrix in acoustic or electromagnetic wave scattering) is non-selfadjoint, hence its eigenvalues are complex. The most interesting regime arises when the absorption, taken into account as an imaginary part of the spectral parameter, is of the order of the mean level spacing. I show how to derive the mean density of the complex eigenvalues for reflection problems in disordered or chaotic systems with broken time-reversal invariance. The computations are done in the framework of nonlinear $\sigma -$ model approach, assuming fixed $M$ and $N\to \mathrm{\infty }$. Some explicit formulas are provided for zero-dimensional quantum chaotic system as well as for a semi-infinite quasi-1D system with fully operative Anderson localization.