Some 1D lattice sums

Former-commit-id: 0be7e862f22ebb37523d263ad98856c1dfcc1bff

notes/ewald.lyx

@@ -3399,6 +3399,39 @@ The only diverging factor here is apparently
 \end_inset
 
 
+\end_layout
+
+\begin_layout Subsection
+1D
+\end_layout
+
+\begin_layout Standard
+One-dimensional lattice sums are provided in [REF LT, sect.
+ 3].
+ However, these are the 
+\begin_inset Quotes eld
+\end_inset
+
+non-shifted
+\begin_inset Quotes erd
+\end_inset
+
+ sums,
+\begin_inset Formula 
+\begin{eqnarray*}
+\ell_{n}\left(\beta\right) & = & \sum_{j\in\ints}^{'}e^{i\beta aj}\mathcal{H}_{n}^{0}\left(aj\hat{\vect z}\right)\\
+ & = & \sum_{j\in\ints}^{'}e^{i\beta aj}h_{n}\left(\left|aj\right|\right)Y_{n}^{0}\\
+ & = & \sqrt{\frac{2n+1}{4\pi}}\sum_{j\in\ints}^{'}P_{n}^{0}\left(\sgn j\right)h_{n}\left(\left|aj\right|\right)e^{i\beta aj}\\
+ & = & \sqrt{\frac{2n+1}{4\pi}}\sum_{j\in\ints}^{'}\left(\sgn j\right)^{n}h_{n}\left(\left|aj\right|\right)e^{i\beta aj},
+\end{eqnarray*}
+
+\end_inset
+
+where we used 
+\begin_inset Formula $P_{n}^{m}\left(\pm1\right)=\left(\pm1\right)^{n}\delta_{m0}$
+\end_inset
+
+
 \end_layout
 
 \begin_layout Section