Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering for systems with broken time reversal invariance

Assuming the validity of random matrices for describing the statistics of a closed chaotic quantum system, we study analytically some statistical properties of the S-matrix characterizing scattering in its open counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the so-called stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken time-reversal invariance coupled to continua via M open channels. By using the supersymmetry method we derive: (i) an explicit expression for the density of S-matrix poles (resonances) in the complex energy plane (ii) an explicit expression for the parametric correlation function of densities of eigenphases of the S-matrix. We use it to find the distribution of derivatives of these eigenphases with respect to the energy ("partial delay times" ) as well as with respect to an arbitrary external parameter.