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saving work in second order

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Antoine Marie 2022-10-24 14:41:19 +02:00
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Notes/PerturbativeAnalysis.tex
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@ -500,10 +500,12 @@ Recalling that $\bHod{0} = \bO$ and $\bHd{1} = \bO$, we derive
\dv{\bV{}{(2),\dagger}}{s} & \dv{\bC{}{(2)}}{s} \dv{\bV{}{(2),\dagger}}{s} & \dv{\bC{}{(2)}}{s}
\end{pmatrix} \\ \end{pmatrix} \\
\dv{\bF^{(2)}}{s} &= \bF^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF^{(0)} - 2 \bV{}{(1)}\bC{\text{d}}{(0)}\bV{}{(1),\dagger}\\ \dv{\bF^{(2)}}{s} &= \bF^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF^{(0)} - 2 \bV{}{(1)}\bC{\text{d}}{(0)}\bV{}{(1),\dagger}\\
\dv{\bC{}{(2)}}{s} &= \\ \dv{\bC{}{(2)}}{s} &= 2 \bC{\text{d}}{(0)}\bC{\text{od}}{(2)}\bC{\text{d}}{(0)}- (\bC{\text{d}}{(0)})^2\bC{\text{od}}{(2)} - \bC{\text{od}}{(2)}(\bC{\text{d}}{(0)})^2 \\
&-2 \bC{\text{d}}{(1)}\bC{\text{od}}{(0)}\bC{\text{d}}{(1)}- (\bC{\text{d}}{(1)})^2\bC{\text{od}}{(0)} - \bC{\text{od}}{(0)}(\bC{\text{d}}{(1)})^2 \notag \\
\dv{\bV{}{(2)}}{s} &= 2 \bF^{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF^{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\ \dv{\bV{}{(2)}}{s} &= 2 \bF^{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF^{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
&- 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} + \bF^{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \\ &- 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} + \bF^{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \notag \\
\dv{\bV{}{(2),\dagger}}{s} &= \dv{\bV{}{(2),\dagger}}{s} &= 2 \bC{\text{d}}{(0)}\bV{}{(2),\dagger}\bF^{(0)} - \bV{}{(2),\dagger}(\bF^{(0)})^2 - (\bC{\text{d}}{(0)})^2\bV{}{(2),\dagger} \\
&- 2 \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \bV{}{(1),\dagger} + \bC{\text{od}}{(1)} \bV{}{(1),\dagger} \bF^{(0)} + \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} \bV{}{(1),\dagger} \notag
\end{align} \end{align}
\begin{align} \begin{align}
@ -516,6 +518,13 @@ Recalling that $\bHod{0} = \bO$ and $\bHd{1} = \bO$, we derive
&\color{red}{\boxed{\color{black}{- \sum_R \frac{\eps^{(0)}_{p} + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R}{(\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2}(1 - e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]})}}} \notag &\color{red}{\boxed{\color{black}{- \sum_R \frac{\eps^{(0)}_{p} + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R}{(\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2}(1 - e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]})}}} \notag
\end{align} \end{align}
\begin{align}
(\dv{\bV{}{(2)}}{s})_{pQ} &= (2 \bF^{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF^{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
& - 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} + \bF^{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)})_{pQ} \notag \\
v^{(2)}_{pQ}(s) &= v^{(2)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} + \text{Non-homogeneous solution} \notag \\
v^{(2)}_{pQ}(s) &= \text{Non-homogeneous solution}
\end{align}
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\subsection{Downfolding the SRG-transformed matrix} \subsection{Downfolding the SRG-transformed matrix}
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