Expressions for translation operators (to be checked)

Former-commit-id: f19217f0f3f66a79d724840f633731f44bfd755f

lepaper/arrayscat.lyx

@@ -713,7 +713,7 @@ Abstract.
 \end_layout
 
 \begin_layout Itemize
-Translation operators: explicit expression, also in sph.
+Translation operators: rewrite in sph.
  harm.
  convention independent form.
 \end_layout

lepaper/finite.lyx

@@ -1535,10 +1535,37 @@ outside.
 
 \begin_layout Standard
 In our convention, the regular translation operator can be expressed explicitly
- as 
+ as (TODO CHECK CAREFULLY FOR POSSIBLE 
+\begin_inset Formula $(-1)^{m'}$
+\end_inset
+
+ AND SIMILAR FACTORS AND REWRITE IN TERMS OF SPHERICAL HARMONICS)
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Teďka jsou tam zkopírovány výrazy pro C a D z Kristenssona, chybějí fase
+\end_layout
+
+\end_inset
+
+
 \begin_inset Formula 
 \begin{multline}
-\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\dots.\label{eq:translation operator}
+\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
+\times\begin{pmatrix}l & l' & \lambda\\
+0 & 0 & 0
+\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
+m & -m' & m'-m
+\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
+\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\
+\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=-i\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
+\times\begin{pmatrix}l & l' & \lambda-1\\
+0 & 0 & 0
+\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
+m & -m' & m'-m
+\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
+\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}.\label{eq:translation operator}
 \end{multline}
 
 \end_inset