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Rosa edited untitled.tex
about 8 years ago
Commit id: 84c514b8141455009250c48e74b0b1ae9e949106
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\end{eqnarray}
On the other hand we have for $G^>_{\beta\gamma}(\epsilon)$ (accordingly with J.S note)
\begin{eqnarray}
G^>_{\beta\gamma}(\epsilon) = \sum_{k\alpha\delta} \int
\frac{d\omega}{2\pi} \frac{d\epsilon}{2\pi} G^r_{\beta \alpha} [V^*_{k\alpha}
g^<_{k}(\omega)V_{k\delta} g^<_{k}(\epsilon)V_{k\delta} + V_{k\alpha}
g^{h,<}_{k}(\omega) g^{h,<}_{k}(\epsilon) V^*_{k\delta}]G^a_{\delta
\gamma}] \gamma}
\end{eqnarray}
We need to compute the following product of Green functions: $G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$
\begin{eqnarray}
&&G^>_{\beta\gamma}(\epsilon) G_{kq}^{h,<}(\omega+\epsilon) = \sum_{k\alpha\delta} \int \frac{d\omega}{2\pi} G^r_{\beta \alpha} [V^*_{k\alpha} g^<_{k}(\omega)V_{k\delta} + V_{k\alpha} g^{h,<}_{k}(\omega) V^*_{k\delta}]G^a_{\delta \gamma}] g_{q}^{h,<}(\omega+\epsilon)\delta_{kq}
\\ \nonumber
&+& \sum_{p\beta\gamma\alpha\gamma}\int \frac{d\omega}{2\pi} [G^r_{\beta \alpha} [V^*_{k\alpha} g^<_{k}(\epsilon)V_{k\delta} +
\sum_{\beta\gamma} V_{k\alpha} g^{h,<}_{k}(\epsilon) V^*_{k\delta}]G^a_{\delta \gamma}] [g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^*
g_{q}^{h,<}(\omega)\nonumber g_{q}^{h,<}(\omega) \nonumber
\\
&&+g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)+g_{k}^{h,<}(\omega+\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)\,,
\end{eqnarray}