diff --git a/BSEdyn.tex b/BSEdyn.tex
index b501aa2..59e72ba 100644
--- a/BSEdyn.tex
+++ b/BSEdyn.tex
@@ -1143,28 +1143,27 @@ The data that support the findings of this study are available within the articl
 
 \appendix
 
-\section{$L_0(1,4; 1',3)$ $(t_1)$-time Fourier transform}
+\section{Fourier transform of $L_0(1,4; 1',3)$}
 \label{app:A}
 
 In this Appendix, we derive Eqs.~\eqref{eq:iL0} to \eqref{eq:iL0bis}.
-Defining the $t_1$-time Fourier transform of $L_0(1,3;4,1')$ with
-$(t_{1'} = t_1^{+})$
+Combining the Fourier transform (with respect to $t_1$) of $L_0(1,3;4,1')$
 \begin{align}  
-	[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 ) = -i
-	\int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1')
+	[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 ) 
+	= -i \int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1'),
 \end{align}
- we plug-in the Fourier expansion of the Green's function, e.g.
-\begin{align*}
-     G(1,3) =  \int \frac{ d\omega }{ 2\pi } G(\bx_1,\bx_3;\omega) e^{-i \omega \tau_{13} }
-\end{align*}
-with $\tau_{13} = (t_1-t_3)$ to obtain:
+(where $t_{1'} = t_1^{+}$) with the Fourier expansion of the Green's function, \eg,
+\begin{align}
+     G(1,3) =  \int \frac{ d\omega }{ 2\pi } G(\bx_1,\bx_3;\omega) e^{-i \omega \tau_{13} },
+\end{align}
+(where $\tau_{13} = t_1-t_3$), we obtain
 \begin{multline}  
 	[L_0](\bx_1,4;\bx_{1'},3 \;| \;  \omega_1 )   =
 	\\
 	\int \frac{ d\omega }{ 2i\pi } \;  G(\bx_1,\bx_3;\omega) \; G(\bx_4,\bx_{1'};\omega-\omega_1)  
-	  e^{ i \omega t_3 }	e^{-i (\omega-\omega_1) t_4 } \nonumber
+	  e^{ i \omega t_3 }	e^{-i (\omega-\omega_1) t_4 }.
 \end{multline}
-With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readily
+Applying the change of variable $\omega \ra \omega + \omega_1/2$, one gets
 \begin{multline}  
 	[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 )     = 
 	\\
@@ -1172,9 +1171,9 @@ With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readi
 	\int \frac{ d\omega }{ 2i\pi } \;   G\qty(\bx_1,\bx_3;\omega+ \frac{\omega_1}{2} )  G\qty(\bx_4,\bx_{1'};\omega-\frac{\omega_1}{2} )  \;
   e^{ i \omega \tau_{34} }	
 \end{multline}
-   with $\tau_{34} = t_3 - t_4$ and $t^{34}= (t_3+t_4)/2$.
-  Using now the Lehman representation of the Green's functions (Eq.~\eqref{eq:G-Lehman}), and picking up the poles associated with the occupied (virtual) states in the upper (lower)  half-plane for $\tau_{34} > 0$ ($\tau_{34} < 0$), one obtains using the residue theorem (with $\tau = \tau_{34})$
-\begin{equation}
+with $\tau_{34} = t_3 - t_4$ and $t^{34}= (t_3+t_4)/2$.
+Finally, using the Lehman representation of the Green's functions [see Eq.~\eqref{eq:G-Lehman}], and picking up the poles associated with the occupied (virtual) states in the upper (lower) half-plane for $\tau_{34} > 0$ ($\tau_{34} < 0$), one obtains, using the residue theorem,
+\begin{equation} \label{eq:A}
 \begin{split}
     & \int \frac{ d \omega }{2i\pi}    \; G\qty(\bx_1,\bx_3; \omega +  \homu  )    G\qty(\bx_4,\bx_{1'}; \omega - \homu  )  e^{  i    \omega  \tau    } 
       \\
@@ -1185,14 +1184,14 @@ With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readi
   & -  \sum_{bj} \frac{ \phi_j(\bx_1) \phi_j^*(\bx_3) \phi_b(\bx_4) \phi_b^*(\bx_{1'})} {  \omega_1   + (\e{b} - \e{j} ) -i\eta } 
      \qty[  \theta(\tau)  e^{i ( \e{j} - \homu ) \tau } + \theta(-\tau)  e^{i ( \e{b} + \homu ) \tau }    ] 
      \\
- & +  \sum_{ab} \text{ pp terms }  + \sum_{ij} \text{ hh terms }
+ & +  \sum_{ab} \text{pp}  + \sum_{ij} \text{hh},
 \end{split}
 \end{equation}
-where (pp) and (hh) labels particle-particle and hole-hole channels neglected here. 
-Projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$   selects the first line  of the RHS, leading to Eq.~\eqref{eq:iL0bis}
-with $ (\Om{1}{} \to \Om{s}{} )$.
+with $\tau = \tau_{34}$, and where pp and hh label the particle-particle and hole-hole channels (respectively) that are neglected here. \cite{Strinati_1988}
+Projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ selects the first line  of the right-hand-side of Eq.~\eqref{eq:A}, yielding Eq.~\eqref{eq:iL0bis}
+with $\Om{1}{} \to \Om{s}{}$.
 
-\section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}$ in the electron/hole product basis  }
+\section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}$ in the electron-hole product basis}
 \label{app:B}
 
 We now derive in some more details Eq.~\eqref{eq:spectral65}.