Diagrammatic technique for calculating nonlinear susceptibilities

Rules of a diagrammatic technique are formulated for calculating quadratic susceptibilities at $T=0$ and $T\not=0$. When $T\not=0$, the nonlinear susceptibilities are obtained by analytic continuation of the Matsubara function $K^c(\omega_{n_1},\omega_{n_2})$. Analytic continuation with respect to two frequencies can be made in each diagram as in [1]. Rules of the diagrammatic technique are also formulated for the double spectral densities, which determine all possible constructions of three pairs of operators: three-particle correlation functions, cross sections of three-quantum processes, nonlinear susceptibilities, etc. Unitarity relations for the double spectral densities are obtained.