Parametric correlations of scattering phase shifts and fluctuations of delay times in few-channel chaotic scattering

By using the supersymmetry method we derive an explicit expression for the parametric correlation function of densities of eigenphases $\theta_a$ of the S-matrix in a chaotic quantum system with broken time-reversal symmetry coupled to continua via M equivalent open channels;$\,\,a=1,..,M$ .We use it to find the distribution of derivatives of these eigenphases over the energy ("phaseshift times") as well as over an arbitrary external parameter.The method allows also to study parametric correlations of Wigner-Smith delay times.