From afac9706afe67ab1e6c1a7b1c93bd1e2575f4676 Mon Sep 17 00:00:00 2001 From: pfloos Date: Mon, 3 Oct 2022 21:20:06 +0200 Subject: [PATCH] OK with appendix --- g.tex | 25 ++++++++++++++----------- 1 file changed, 14 insertions(+), 11 deletions(-) diff --git a/g.tex b/g.tex index d016540..aaa56dd 100644 --- a/g.tex +++ b/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 -Eq.~\eqref{final}, thus confirming independently the validity of this equation. +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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibliography{g}