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Now we enter the expression for $G_{kq}^{h,t}(t,t')$ in the frequency domain  \begin{equation}  G_{kq}^{h,t}(\omega) = G_{kq}^{h,t}(\omega)\delta_{kq} g_{q}^{h,t}(\omega)\delta_{kq}  + \sum_{\beta\gamma} g_{k}^{h,t}(\omega) V_{\gamma k} G^{t}_{\gamma\beta}(\omega)V_{\beta q}^* g_{q}^{h,t}(\omega)\,, \end{equation}  \begin{multline}