minor corrections

Manuscript/SRGGW.tex

@@ -562,9 +562,12 @@ while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends t
 \begin{equation}
   \lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = \bO.
 \end{equation}
-Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}^{\GW}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
+Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
 As illustrated in Fig.~\ref{fig:flow}, this transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
 
+\titou{For a fixed value of the energy cutoff $\Lambda = s^{-1/2}$, it is important to notice that if $\abs*{\Delta_{pr}^{\nu}} \gg \Lambda$, then $W_{p,r\nu}(s) = W_{p,r\nu} e^{-(\Delta_{pr\nu})^2 s} \approx 0$ (\ie decoupled), while for $\abs*{\Delta_{pr}^{\nu}} \ll \Lambda$, we have $W_{p,r\nu}(s) \approx W_{p,r\nu}$ (\ie remains coupled).}
+\PFL{To reformulate and maybe move around.}
+
 %%% FIG 2 %%% 
 \begin{figure*}
   \includegraphics[width=0.8\linewidth]{fig1.pdf}
@@ -602,10 +605,11 @@ The fact that the $s\to\infty$ static limit does not always converge when used i
 Therefore, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states.
 
 It is instructive to plot the functional form of both regularizing functions, this is done in Fig.~\ref{fig:plot}.
-The surfaces are plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/2\eta^2$ such that the first order Taylor expansion around $(0,0)$ of both functional form are equal.
+The surfaces are plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/2\eta^2$ such that the first order Taylor expansion around $(0,0)$ of both functional forms is equal.
 One can observe that the SRG surface is much smoother than its qs counterpart.
-This is due to the fact that the SRG functional has less irregularities for $\eta=0$, in fact there is a single singularity at $x=y=0$.
-On the other hand the function $f_{\text{qs}}(x,y;0)$ is singular on two entire axis, $x=0$ and $y=0$.
+This is due to the fact that the SRG functional at $\eta=0$, $f^{\SRGqsGW}(x,y;0)$, has fewer irregularities. 
+In fact, there is a single singularity at $x=y=0$.
+On the other hand, the function $f^{\qsGW}(x,y;0)$ is singular on the two entire axes, $x=0$ and $y=0$.
 The smoothness of the SRG surface might induce a smoother convergence of SRG-qs$GW$ compared to qs$GW$.
 The convergence properties and the accuracy of both static approximations will be quantitatively gauged in Sec.~\ref{sec:results}.