deletions | additions       diff --git a/untitled.tex b/untitled.tex  index e9a105e..0ace310 100644  --- a/untitled.tex  +++ b/untitled.tex 
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V_{\beta k} V_{\gamma q}^{*} [G^>_{\beta\gamma}(\epsilon) G^{h,<}_{qk}(\omega+\epsilon) - G^{>}_{\beta q}(\epsilon)G^{h,<}_{\gamma k}(\epsilon+\omega)]  + V_{\beta k}^{*}V_{\gamma q} [G^{>,t}_{kq}(\epsilon_1) G^{<}_{\gamma\beta}(\epsilon+\omega) - G^{h,>}_{k\gamma}(\epsilon)G^{<}_{q \beta}(\epsilon+\omega)]\Biggr\}\,,  \end{multline}  Let us treat first the following term: $\sum_{k\beta,q\gamma} V_{\beta k} V_{\gamma q}^{*} G^>_{\beta\gamma}(\epsilon) [G^>_{\beta\gamma}(\epsilon)  G^{h,<}_{qk}(\omega+\epsilon) - G^{>}_{\beta q}(\epsilon)G^{h,<}_{\gamma k}(\epsilon+\omega)$ k}(\epsilon+\omega)]$  Then,  \begin{equation}  G_{kq}^{h,<}(\omega+\epsilon) = g_{q}^{h,<}(\omega+\epsilon)\delta_{kq} + \sum_{\beta\gamma} [g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{<,t}(\omega)+g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{a,t}(\omega+\epsilon)+g_{k}^{h,a}(\omega+\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{<,t}(\omega+\epsilon)\,,  \end{equation}