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\begin_body

\begin_layout Title
Periodic Green's functions vs.
 VSWF lattice sums
\end_layout

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\ud}{\mathrm{d}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\abs}[1]{\left|#1\right|}
\end_inset

 
\begin_inset FormulaMacro
\newcommand{\vect}[1]{\mathbf{#1}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\uvec}[1]{\hat{\mathbf{#1}}}
\end_inset


\lang english

\begin_inset FormulaMacro
\newcommand{\ush}[2]{Y_{#1}^{#2}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\ushD}[2]{Y'_{#1}^{#2}}
\end_inset


\end_layout

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\vsh}{\vect A}
\end_inset


\begin_inset FormulaMacro
\newcommand{\vshD}{\vect{A'}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\wfkc}{\vect y}
\end_inset


\begin_inset FormulaMacro
\newcommand{\wfkcout}{\vect u}
\end_inset


\begin_inset FormulaMacro
\newcommand{\wfkcreg}{\vect v}
\end_inset


\begin_inset FormulaMacro
\newcommand{\wckcreg}{a}
\end_inset


\begin_inset FormulaMacro
\newcommand{\wckcout}{f}
\end_inset


\end_layout

\begin_layout Section
Some definitions and useful relations
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\mathcal{H}_{l}^{m}\left(\vect d\right)\equiv h_{l}^{+}\left(\left|\vect d\right|\right)\ush lm\left(\uvec d\right)
\]

\end_inset


\begin_inset Formula 
\[
\mathcal{J}_{l}^{m}\left(\vect d\right)\equiv j_{l}\left(\left|\vect d\right|\right)\ush lm\left(\uvec d\right)
\]

\end_inset


\end_layout

\begin_layout Standard
Dual spherical harmonics and waves
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\int\ush lm\ushD{l'}{m'}\,\ud\Omega=\delta_{l,l'}\delta_{m,m'}
\]

\end_inset


\begin_inset Formula 
\[
\mathcal{J}'_{l}^{m}\left(\vect d\right)\equiv j_{l}\left(\left|\vect d\right|\right)\ushD lm\left(\uvec d\right)
\]

\end_inset


\end_layout

\begin_layout Standard
Expansion of plane wave (CHECKME whether this is really convention-independent,
 but it seems so)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
e^{i\kappa\vect r\cdot\uvec r'}=4\pi\sum_{l,m}i^{n}\mathcal{J}'_{l}^{m}\left(\kappa\vect r\right)\ush lm\left(\uvec r'\right)=4\pi\sum_{l,m}i^{n}\mathcal{J}{}_{l}^{m}\left(\kappa\vect r\right)\ushD lm\left(\uvec r'\right)
\]

\end_inset

This one should also be convention independent (similarly for 
\begin_inset Formula $\mathcal{H}_{l}^{m}$
\end_inset

):
\begin_inset Formula 
\[
\mathcal{J}_{l}^{m}\left(-\vect r\right)=\left(-1\right)^{l}\mathcal{J}_{l}^{m}\left(\vect r\right).
\]

\end_inset


\end_layout

\begin_layout Section
Helmholtz equation and Green's functions (in 3D)
\end_layout

\begin_layout Standard
Note that the notation does not follow Linton's (where the wavenumbers are
 often implicit)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\left(\nabla^{2}+\kappa^{2}\right)G^{(\kappa)}\left(\vect x,\vect x_{0}\right)=\delta\left(\vect x-\vect x_{0}\right)
\]

\end_inset


\begin_inset Formula 
\begin{align*}
G_{0}^{(\kappa)}\left(\vect x,\vect x_{0}\right) & =G_{0}^{(\kappa)}\left(\vect x-\vect x_{0}\right)=-\frac{\cos\left(\kappa\left|\vect x-\vect x_{0}\right|\right)}{4\pi\left|\vect x-\vect x_{0}\right|}\\
G_{\pm}^{(\kappa)}\left(\vect x,\vect x_{0}\right) & =G_{\pm}^{(\kappa)}\left(\vect x-\vect x_{0}\right)=-\frac{e^{\pm i\kappa\left|\vect x-\vect x_{0}\right|}}{4\pi\left|\vect x-\vect x_{0}\right|}=-\frac{i\kappa}{4\pi}h_{0}^{\pm}\left(\kappa\left|\vect x-\vect x_{0}\right|\right)=-\frac{i\kappa}{\sqrt{4\pi}}\mathcal{H}_{0}^{0}\left(\kappa\left|\vect x-\vect x_{0}\right|\right)
\end{align*}

\end_inset


\begin_inset Marginal
status open

\begin_layout Plain Layout
\begin_inset Formula $G_{\pm}^{(\kappa)}\left(\vect x,\vect x_{0}\right)=-\frac{i\kappa}{\ush 00}\mathcal{H}_{0}^{0}\left(\kappa\left|\vect x-\vect x_{0}\right|\right)$
\end_inset

 in case wacky conventions.
\end_layout

\end_inset

Lattice GF [Linton (2.3)]:
\begin_inset Formula 
\begin{equation}
G_{\Lambda}^{(\kappa)}\left(\vect s,\vect k\right)\equiv\sum_{\vect R\in\Lambda}G_{+}^{\kappa}\left(\vect s-\vect R\right)e^{i\vect k\cdot\vect R}\label{eq:Lattice GF}
\end{equation}

\end_inset


\end_layout

\begin_layout Section
GF expansion and lattice sum definition
\end_layout

\begin_layout Standard
Let's define 
\begin_inset Formula 
\[
\sigma_{l}^{m}\left(\vect s,\vect k\right)=\sum_{\vect R\in\Lambda}\mathcal{H}_{l}^{m}\left(\kappa\left(\vect s+\vect R\right)\right)e^{i\vect k\cdot\vect R},
\]

\end_inset

and also its dual version
\begin_inset Formula 
\[
\sigma'_{l}^{m}\left(\vect s,\vect k\right)=\sum_{\vect R\in\Lambda}\mathcal{H}'_{l}^{m}\left(\kappa\left(\vect s+\vect R\right)\right)e^{i\vect k\cdot\vect R}.
\]

\end_inset


\end_layout

\begin_layout Standard
Inspired by [Linton (4.1)]; assuming that 
\begin_inset Formula $\vect s\notin\Lambda$
\end_inset

, let's expand the lattice Green's function around 
\begin_inset Formula $\vect s$
\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)=-i\kappa\sum_{l,m}\tau_{l}^{m}\left(\vect s,\vect k\right)\mathcal{J}_{l}^{m}\left(\kappa\vect r\right)
\]

\end_inset

and multiply with a dual SH + integrate
\begin_inset Formula 
\begin{align}
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)\ushD{l'}{m'}\left(\uvec r\right) & =-i\kappa\sum_{l,m}\tau_{l}^{m}\left(\vect s,\vect k\right)j_{l}\left(\kappa\left|\vect r\right|\right)\delta_{ll'}\delta_{mm'}\nonumber \\
 & =-i\kappa\tau_{l'}^{m'}\left(\vect s,\vect k\right)j_{l'}\left(\kappa\left|\vect r\right|\right)\label{eq:tau extraction}
\end{align}

\end_inset

The expansion coefficients 
\begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$
\end_inset

 is then typically extracted by taking the limit 
\begin_inset Formula $\left|\vect r\right|\to0$
\end_inset

.
\end_layout

\begin_layout Standard
The relation between 
\begin_inset Formula $\sigma_{l}^{m}\left(\vect s,\vect k\right)$
\end_inset

 and 
\begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$
\end_inset

 can be obtained e.g.
 from the addition theorem for scalar spherical wavefunctions [Linton (C.3)],
 
\begin_inset Formula 
\[
\mathcal{H}_{l}^{m}\left(\vect a+\vect b\right)=\sum_{l'm'}S_{ll'}^{mm'}\left(\vect b\right)\mathcal{J}_{l'}^{m'}\left(\vect a\right),\quad\left|\vect a\right|<\left|\vect b\right|
\]

\end_inset

where for the zeroth degree and order one has [Linton (C.3)]
\begin_inset Formula 
\[
S_{0l'}^{0m'}\left(\vect b\right)=\sqrt{4\pi}\mathcal{H}'_{l'}^{m'}\left(-\vect b\right)
\]

\end_inset


\begin_inset Marginal
status open

\begin_layout Plain Layout
In a totally convention-independent version probably looks like 
\begin_inset Formula $S_{0l'}^{0m'}\left(\vect b\right)=\ush 00\mathcal{H}'_{l'}^{m'}\left(-\vect b\right)$
\end_inset

, but the 
\begin_inset Formula $Y_{0}^{0}$
\end_inset

 will cancel with the expression for GF anyways, so no harm to the final
 result.
\end_layout

\end_inset

From the lattice GF definition 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Lattice GF"
plural "false"
caps "false"
noprefix "false"

\end_inset


\begin_inset Formula 
\begin{align*}
G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right) & \equiv\frac{-i\kappa}{\sqrt{4\pi}}\sum_{\vect R\in\Lambda}\mathcal{H}_{0}^{0}\left(\kappa\left(\vect s+\vect r-\vect R\right)\right)e^{i\vect k\cdot\vect R}\\
 & =\frac{-i\kappa}{\sqrt{4\pi}}\sum_{\vect R\in\Lambda}\mathcal{H}_{0}^{0}\left(\kappa\left(\vect s+\vect r-\vect R\right)\right)e^{i\vect k\cdot\vect R}\\
 & =\frac{-i\kappa}{\sqrt{4\pi}}\sum_{\vect R\in\Lambda}\sum_{l'm'}S_{0l'}^{0m'}\left(\kappa\left(\vect s-\vect R\right)\right)\mathcal{J}_{l'}^{m'}\left(\kappa\vect r\right)e^{i\vect k\cdot\vect R}\\
 & =-i\kappa\sum_{\vect R\in\Lambda}\sum_{lm}\mathcal{H}'_{l}^{m}\left(-\kappa\left(\vect s-\vect R\right)\right)\mathcal{J}_{l}^{m}\left(\kappa\vect r\right)e^{i\vect k\cdot\vect R}
\end{align*}

\end_inset

and mutliplying with dual SH and integrating 
\begin_inset Formula 
\begin{align*}
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)\ushD{l'}{m'}\left(\uvec r\right) & =-i\kappa\sum_{\vect R\in\Lambda}\sum_{lm}\mathcal{H}'_{l}^{m}\left(-\kappa\left(\vect s-\vect R\right)\right)j_{l}\left(\kappa\left|\vect r\right|\right)\delta_{ll'}\delta_{mm'}e^{i\vect k\cdot\vect R}\\
 & =-i\kappa\sum_{\vect R\in\Lambda}\mathcal{H}'_{l'}^{m'}\left(\kappa\left(-\vect s+\vect R\right)\right)j_{l'}\left(\kappa\left|\vect r\right|\right)e^{i\vect k\cdot\vect R}\\
 & =-i\kappa\sigma'_{l'}^{m'}\left(-\vect s,\vect k\right)j_{l'}\left(\kappa\left|\vect r\right|\right)
\end{align*}

\end_inset

and comparing with 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:tau extraction"
plural "false"
caps "false"
noprefix "false"

\end_inset

 we have
\begin_inset Formula 
\[
\tau_{l}^{m}\left(\vect s,\vect k\right)=\sigma'_{l}^{m}\left(-\vect s,\vect k\right).
\]

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO maybe also define some 
\begin_inset Formula $\tau'_{l}^{m}$
\end_inset

 as expansion coefficients of GF into dual regular SSWFs.
\end_layout

\end_inset


\end_layout

\end_body
\end_document
