OK with appendix

g.tex

@@ -724,11 +724,11 @@ Let us define the energy-dependent Green's matrix
 The denomination ``energy-dependent'' is chosen here since 
 this quantity is the discrete version of the Laplace transform of the time-dependent Green's function in a continuous space,
 usually known under this name.\cite{note}
-The remarkable property is that, thanks to the summation over $N$ up to infinity, the constrained multiple sums appearing in Eq.~\eqref{eq:Gt} can be factorized in terms of a product of unconstrained single sums, as follows
+The remarkable property is that, thanks to the summation over $N$ up to infinity, the constrained multiple sums appearing in Eq.~\eqref{eq:Gt} can be factorized in terms of a product of unconstrained sums, as follows
 \begin{multline}
 	\sum_{N=1}^\infty \sum_{p=1}^N \sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1} \delta_{\sum_{k=0}^p n_k,N+1} F(n_0,\ldots,n_N)
 	\\
-	= \sum_{p=1}^{\infty} \sum_{n_0=1}^{\infty} \cdots \sum_{n_p=1}^{\infty} F(n_0,\ldots,n_N).
+	= \sum_{p=1}^{\infty} \sum_{n_0=1}^{\infty} \cdots \sum_{n_p=1}^{\infty} F(n_0,\ldots,n_N),
 \end{multline}
 where $F$ is some arbitrary function of the trapping times.
 Using the fact that $G^E_{ij}= \tau \sum_{N=0}^{\infty} G^{(N)}_{ij}$, where $G^{(N)}_{ij}$ is given by Eq.~\eqref{eq:Gt}, and summing over the variables $n_k$, we get
@@ -738,7 +738,7 @@ Using the fact that $G^E_{ij}= \tau \sum_{N=0}^{\infty} G^{(N)}_{ij}$, where $G^
 	= {G}^{E,\cD}_{i_0 i_N}
 	+ \sum_{p=1}^{\infty} \sum_{I_1 \notin \cD_0, \hdots , I_p \notin \cD_{p-1}} \\
 	\qty[ \prod_{k=0}^{p-1} \mel{ I_k }{ {\qty[ P_k \qty( H-E \Id ) P_k ] }^{-1} (-H)(\Id-P_k) }{ I_{k+1} } ]
-       {G}^{E,\cD}_{I_p i_N}
+       {G}^{E,\cD}_{I_p i_N},
 \end{multline}
 where, ${G}^{E,\cD}$ is the energy-dependent domain's Green matrix defined as ${G}^{E,\cD}_{ij} = \tau \sum_{N=0}^{\infty} \mel{ i }{ \titou{T^N_i} }{ j}$.
 
@@ -1361,6 +1361,7 @@ and
 \ee
 For a $2\times2$ matrix of the form
 \be
+\label{eq:2x2_matrix}
 	H = 
 	\begin{pmatrix}
 	H_{11}	&	H_{12}	\\
@@ -1372,7 +1373,7 @@ Using Eqs.~\eqref{eq:defA1} and \eqref{eq:defA2}, one gets, for $i = 1$ or $2$,
 \begin{align}
 	A_i & = -\frac{H_{12}}{H_{ii}-E},
 	&
-	C_i & = \frac{1}{H_{ii}-E} \Psi_i.
+	C_i & = \frac{1}{H_{ii}-E} \Psi_i,
 \end{align}
 which finally yields
 \be
@@ -1382,18 +1383,20 @@ which finally yields
 %                                          { (H_{11}-E)(H_{22}-E) - H^2_{12}}
 %	\\
 	= \frac{ H_{22}-E}{\Delta}  \Psi_1 - \frac{H_{12}}{\Delta} \Psi_2,
-\label{final}
+\label{eq:final}
 \ee
 where $\Delta$ is the determinant of $H$. 
 
-On the other hand, the quantity $\mel{ 1 }{ \qty(H-E \Id)^{-1} }{ \Psi}$ of the L.H.S of Eq.(\ref{final}) can be directly calculated using the inverse of the $2\times2$ matrix 
+Alternatively, the quantity $\mel{ 1 }{ \qty(H-E \Id)^{-1} }{ \Psi}$ in the left-hand-side of Eq.~\eqref{eq:final} can be directly calculated using the inverse of the matrix defined in Eq.~\eqref{eq:2x2_matrix}, yielding
 \be
-{ \qty(H-E \Id)^{-1} }{ |\Psi\rangle}=
+\begin{split}
+\qty(H-E \Id)^{-1} \ket{\Psi}& =
               \frac{1}{H-E \Id}      \begin{pmatrix}
          \Psi_1 \\
          \Psi_2   \\
         \end{pmatrix}
- = \frac{1}{\Delta}
+        \\
+ & = \frac{1}{\Delta}
         \begin{pmatrix}
         H_{22}-E  &      - H_{21}  \\
         - H_{21}  &       H_{11}-E  \\
@@ -1401,10 +1404,10 @@ On the other hand, the quantity $\mel{ 1 }{ \qty(H-E \Id)^{-1} }{ \Psi}$ of the
 \begin{pmatrix}
          \Psi_1 \\
          \Psi_2   \\
-        \end{pmatrix}
+        \end{pmatrix}.
+\end{split}
 \ee
-As seen, the first component of the vector ${ \qty(H-E \Id)^{-1} }{ |\Psi\rangle}$ is identical to the one given in 
+As readily seen, the first component of the vector $\qty(H-E \Id)^{-1}\ket{\Psi}$ is identical to the one given in Eq.~\eqref{eq:final}, thus confirming independently the validity of this equation.
-Eq.~\eqref{final}, thus confirming independently the validity of this equation.
 
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