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Supplementary (derivation of 1D and 2D lattice sums) Former-commit-id: bf57f560e9eeb960e92251393dddd33ebbfd1d14
2020-06-23 11:28:01 +03:00
#LyX 2.4 created this file. For more info see https://www.lyx.org/
\lyxformat 584
\begin_document
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\options supplementary
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\end_header
\begin_body
\begin_layout Title
Derivation of the 1D and 2D lattice sums for the 3D Helmholtz equation with
general lattice offset
\end_layout
\begin_layout Author
Marek Nečada
\end_layout
\begin_layout Section
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 Subsection
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
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
Expansion of a plane wave:
\begin_inset Note Note
status open
\begin_layout Plain Layout
(CHECKME whether this is really convention-independent, but it seems so)
\end_layout
\end_inset
\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 is also 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 Subsection
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{e^{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}^{+}\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
In case of wacky conventions,
\begin_inset Formula $G_{0}^{(\kappa)}\left(\vect x,\vect x_{0}\right)=-\frac{i\kappa}{\ush 00}\mathcal{H}_{0}^{0}\left(\kappa\left|\vect x-\vect x_{0}\right|\right)$
\end_inset
.
\end_layout
\begin_layout Standard
Lattice GF
\begin_inset CommandInset citation
LatexCommand cite
after "(2.3)"
key "linton_lattice_2010"
literal "false"
\end_inset
:
\begin_inset Formula
\begin{equation}
G_{\Lambda}^{(\kappa)}\left(\vect s,\vect k\right)\equiv\sum_{\vect R\in\Lambda}G_{0}^{\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 Subsection
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
\begin_inset CommandInset citation
LatexCommand cite
after "(4.1)"
key "linton_lattice_2010"
literal "false"
\end_inset
: 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
\begin_inset CommandInset citation
LatexCommand cite
after "(C.3)"
key "linton_lattice_2010"
literal "false"
\end_inset
,
\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
\begin_inset CommandInset citation
LatexCommand cite
after "(C.3)"
key "linton_lattice_2010"
literal "false"
\end_inset
\begin_inset Foot
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
\begin_inset Formula
\[
S_{0l'}^{0m'}\left(\vect b\right)=\sqrt{4\pi}\mathcal{H}'_{l'}^{m'}\left(-\vect b\right).
\]
\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 a 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
\begin_layout Section
Derivation of the 1D and 2D lattice sum
\end_layout
\begin_layout Standard
With
\begin_inset CommandInset citation
LatexCommand cite
key "linton_lattice_2010"
literal "false"
\end_inset
in hand, the short-range part is rather easy.
Let's get the long-range part.
\end_layout
\begin_layout Standard
We first need to find the long-range part of the expansion coefficient
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{i}{\kappa j_{l'}\left(\kappa\left|\vect r\right|\right)}\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)\ushD{l'}{m'}\left(\uvec r\right).\label{eq:tau extraction formula}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
We take
\begin_inset CommandInset citation
LatexCommand cite
after "(2.24)"
key "linton_lattice_2010"
literal "false"
\end_inset
with slightly modified notation
\begin_inset Formula $\left(\vect k_{\vect K}\equiv\vect K+\vect k\right)$
\end_inset
\begin_inset Formula
\[
G_{\Lambda}^{(1;\kappa)}\left(\vect r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect r}\int_{1/\eta}^{\infty e^{i\pi/4}}e^{-\kappa^{2}\gamma^{2}t^{2}/4}e^{-\left|\vect r^{\bot}\right|^{2}/t^{2}}t^{1-d_{c}}\ud t
\]
\end_inset
or, evaluated at point
\begin_inset Formula $\vect s+\vect r$
\end_inset
instead
\begin_inset Formula
\[
G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\left(\vect s+\vect r\right)}\int_{1/\eta}^{\infty e^{i\pi/4}}e^{-\kappa^{2}\gamma^{2}t^{2}/4}e^{-\left|\vect s^{\bot}+\vect r^{\bot}\right|^{2}/t^{2}}t^{1-d_{c}}\ud t.
\]
\end_inset
The integral can be by substitutions taken into the form
\begin_inset Note Note
status open
\begin_layout Plain Layout
\lang english
\begin_inset Formula
\[
G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{2\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta}^{\infty\exp\left(i\pi/4\right)}e^{-\kappa^{2}\gamma_{m}^{2}\zeta^{2}/4}e^{-\left|\vect r_{\bot}\right|^{2}/\zeta^{2}}\zeta^{1-d_{c}}\ud\zeta
\]
\end_inset
Try substitution
\begin_inset Formula $t=\zeta^{2}$
\end_inset
: then
\begin_inset Formula $\ud t=2\zeta\,\ud\zeta$
\end_inset
(
\begin_inset Formula $\ud\zeta=\ud t/2t^{1/2}$
\end_inset
) and
\begin_inset Formula
\[
G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\kappa^{2}\gamma_{m}^{2}t/4}e^{-\left|\vect r_{\bot}\right|^{2}/t}t^{\frac{-d_{c}}{2}}\ud t
\]
\end_inset
Try subst.
\begin_inset Formula $\tau=k^{2}\gamma_{m}^{2}/4$
\end_inset
\end_layout
\begin_layout Plain Layout
\lang english
\begin_inset Formula
\[
G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\left(\frac{\kappa\gamma_{m}}{2}\right)^{d_{c}}\int_{\kappa^{2}\gamma_{m}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{m}^{2}/4\tau}\tau^{\frac{-d_{c}}{2}}\ud\tau
\]
\end_inset
\end_layout
\end_inset
\begin_inset Formula
\[
G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\left(\vect s+\vect r\right)}\int_{\kappa^{2}\gamma_{m}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{m}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}}\ud\tau.
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Foot
status open
\begin_layout Plain Layout
[Linton, (2.25)] with slightly modified notation:
\begin_inset Formula
\[
G_{\Lambda}^{(1;\kappa)}\left(\vect r\right)=-\frac{1}{\sqrt{4\pi}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect r}\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\left|\vect r^{\bot}\right|^{2j}}{j!}\left(\frac{\kappa\gamma_{\vect{\vect k_{\vect K}}}}{2}\right)^{2j-1}\Gamma_{j\vect k_{\vect K}}
\]
\end_inset
We want to express an expansion in a shifted point, so let's substitute
\begin_inset Formula $\vect r\to\vect s+\vect r$
\end_inset
\begin_inset Formula
\[
G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)=-\frac{1}{\sqrt{4\pi}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\left(\vect s+\vect r\right)}\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\left|\vect s^{\bot}+\vect r^{\bot}\right|^{2j}}{j!}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j-1}\Gamma_{j\vect k_{\vect K}}
\]
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
Let's do the integration to get
\begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$
\end_inset
\begin_inset Formula
\[
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\left(\vect s+\vect r\right)}\int_{\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}}\ud\tau.
\]
\end_inset
The
\begin_inset Formula $\vect r$
\end_inset
-dependent plane wave factor can be also written as
\begin_inset Formula
\begin{align*}
e^{i\vect k_{\vect K}\cdot\vect r} & =e^{i\left|\vect k_{\vect K}\right|\vect r\cdot\uvec{\vect k_{\vect K}}}=4\pi\sum_{lm}i^{l}\mathcal{J}'_{l}^{m}\left(\left|\vect k_{\vect K}\right|\vect r\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\\
& =4\pi\sum_{lm}i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ushD lm\left(\uvec{\vect r}\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right),
\end{align*}
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
or the other way around
\begin_inset Formula
\[
e^{i\vect k_{\vect K}\cdot\vect r}=4\pi\sum_{lm}i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect r}\right)\ushD lm\left(\uvec{\vect k_{\vect K}}\right)
\]
\end_inset
\end_layout
\end_inset
so
\begin_inset Formula
\begin{multline*}
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\frac{1}{2\pi\mathcal{A}}\times\\
\times\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ushD lm\left(\uvec r\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\int_{\kappa^{2}\gamma_{\vect{\vect k_{\vect K}}}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect{\vect k_{\vect K}}}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}}\ud\tau.
\end{multline*}
\end_inset
\end_layout
\begin_layout Standard
We also have
\begin_inset Formula
\begin{align*}
e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau} & =e^{-\left(\left|\vect s_{\bot}\right|^{2}+\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\\
& =e^{-\left|\vect s_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\sum_{j=0}^{\infty}\frac{1}{j!}\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}}{4\tau}\right)^{j},
\end{align*}
\end_inset
hence
\begin_inset Formula
\begin{align*}
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right) & =-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ushD lm\left(\uvec r\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\times\\
& \quad\times\sum_{j=0}^{\infty}\frac{1}{j!}\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect{\vect k_{\vect K}}}^{2}}{4}\right)^{j}\underbrace{\int_{\kappa^{2}\gamma_{\vect K}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}-j}\ud\tau}_{\Delta_{j}^{\left(d_{\Lambda}\right)}}\\
& =-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\sum_{j=0}^{\infty}\frac{\Delta_{j}^{\left(d_{\Lambda}\right)}}{j!}\times\\
& \quad\times\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}}{4}\right)^{j}\\
& =-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\times\\
& \quad\times\left(\frac{\kappa\gamma_{\vect{\vect k_{\vect K}}}}{2}\right)^{2j}\sum_{k=0}^{j}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left|\vect r_{\bot}\right|^{2(j-k)}\left(2\vect r_{\bot}\cdot\vect s_{\bot}\right)^{k}.
\end{align*}
\end_inset
The integral
\begin_inset Formula $\Delta_{j}^{\left(d_{\Lambda}\right)}$
\end_inset
is (for the 2D case) equivalent to that in
\begin_inset CommandInset citation
LatexCommand cite
key "kambe_theory_1968"
literal "false"
\end_inset
.
\end_layout
\begin_layout Standard
If we label
\begin_inset Formula $\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|\cos\varphi\equiv\vect r_{\bot}\cdot\vect s_{\bot}$
\end_inset
, we have
\begin_inset Formula
\begin{multline*}
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\times\\
\times\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\sum_{k=0}^{j}\left(2\left|\vect s_{\bot}\right|\right)^{k}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left|\vect r_{\bot}\right|^{2j-k}\left(\cos\varphi\right)^{k},
\end{multline*}
\end_inset
and if we label
\begin_inset Formula $\left|\vect r\right|\sin\vartheta\equiv\left|\vect r_{\bot}\right|$
\end_inset
\begin_inset Formula
\begin{multline*}
\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\times\\
\times\sum_{k=0}^{j}\left|\vect r\right|^{2j-k}\left(2\left|\vect s_{\bot}\right|\right)^{k}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\sin\vartheta\right)^{2j-k}\left(\cos\varphi\right)^{k}.
\end{multline*}
\end_inset
Now let's put the RHS into
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:tau extraction formula"
plural "false"
caps "false"
noprefix "false"
\end_inset
and try eliminating some sum by taking the limit
\begin_inset Formula $\left|\vect r\right|\to0$
\end_inset
.
We have
\begin_inset Formula $j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\sim\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)^{l}/\left(2l+1\right)!!$
\end_inset
; the denominator from
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:tau extraction formula"
plural "false"
caps "false"
noprefix "false"
\end_inset
behaves like
\begin_inset Formula $j_{l'}\left(\kappa\left|\vect r\right|\right)\sim\left(\kappa\left|\vect r\right|\right)^{l'}/\left(2l'+1\right)!!.$
\end_inset
The leading terms are hence those with
\begin_inset Formula $\left|\vect r\right|^{l-l'+2j-k}$
\end_inset
.
So
\begin_inset Formula
\begin{multline*}
\tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa^{1+l'}}\left(2l'+1\right)!!\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\ush lm\left(\uvec{\vect k_{\vect K}}\right)\times\\
\times\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\sum_{k=0}^{j}\delta_{l'-l,2j-k}\left(2\left|\vect s_{\bot}\right|\right)^{k}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\sin\vartheta\right)^{l'-l}\left(\cos\varphi\right)^{k}.
\end{multline*}
\end_inset
Let's now focus on rearranging the sums; we have
\begin_inset Formula
\[
S(l')\equiv\sum_{l=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{j}\delta_{l'-l,2j-k}f(l',l,j,k)=\sum_{l=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{j}\delta_{l'-l,2j-k}f(l',l,j,2j-l'+l).
\]
\end_inset
We have
\begin_inset Formula $0\le k\le j$
\end_inset
, hence
\begin_inset Formula $0\le2j-l'+l\le j$
\end_inset
, hence
\begin_inset Formula $-2j\le-l'+l\le-j$
\end_inset
, hence also
\begin_inset Formula $l'-2j\le l\le l'-j$
\end_inset
, which gives the opportunity to swap the
\begin_inset Formula $l,j$
\end_inset
sums and the
\begin_inset Formula $l$
\end_inset
-sum becomes finite; so also consuming
\begin_inset Formula $\sum_{k=0}^{j}\delta_{l'-l,2j-k}$
\end_inset
we get
\begin_inset Formula
\[
S(l')=\sum_{j=0}^{\infty}\sum_{l=\max(0,l'-2j)}^{l'-j}f(l',l,j,2j-l'+l).
\]
\end_inset
Finally, we see that the interval of valid
\begin_inset Formula $l$
\end_inset
becomes empty when
\begin_inset Formula $l'-j<0$
\end_inset
, i.e.
\begin_inset Formula $j>l'$
\end_inset
; so we get a finite sum
\begin_inset Formula
\[
S(l')=\sum_{j=0}^{l'}\sum_{l=\max(0,l'-2j)}^{l'-j}f(l',l,j,2j-l'+l).
\]
\end_inset
Applying rearrangement,
\begin_inset Formula
\begin{multline*}
\tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{j=0}^{l'}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\times\sum_{l=\max\left(0,l'-2j\right)}^{l'-j}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l'+l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\\
\times\sum_{m=-l}^{l}\ush lm\left(\uvec{\vect k_{\vect K}}\right)\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\sin\vartheta\right)^{l'-l}\left(\cos\varphi\right)^{2j-l'+l},
\end{multline*}
\end_inset
or replacing the angles with their original definition,
\begin_inset Formula
\begin{multline*}
\tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{j=0}^{l'}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2j}\times\sum_{l=\max\left(0,l'-2j\right)}^{l'-j}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l'+l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\\
\times\sum_{m=-l}^{l}\ush lm\left(\uvec K\right)\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\frac{\left|\vect r_{\bot}\right|}{\left|\vect r\right|}\right)^{l'-l}\left(\frac{\vect r_{\bot}\cdot\vect s_{\bot}}{\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|}\right)^{2j-l'+l},
\end{multline*}
\end_inset
and if we want a
\begin_inset Formula $\sigma_{l'}^{m'}\left(\vect s,\vect k\right)$
\end_inset
instead, we reverse the sign of
\begin_inset Formula $\vect s$
\end_inset
and replace all spherical harmonics with their dual counterparts:
\begin_inset Formula
\begin{multline*}
\sigma_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\sum_{j=0}^{l'}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\sum_{l=\max\left(0,l'-2j\right)}^{l'-j}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l'+l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\times\\
\times\sum_{m=-l}^{l}\ushD lm\left(\uvec{\vect k_{\vect K}}\right)\int\ud\Omega_{\vect r}\,\ush{l'}{m'}\left(\uvec r\right)\ush lm\left(\uvec r\right)\left(\frac{\left|\vect r_{\bot}\right|}{\left|\vect r\right|}\right)^{l'-l}\left(\frac{-\vect r_{\bot}\cdot\vect s_{\bot}}{\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|}\right)^{2j-l'+l},
\end{multline*}
\end_inset
and remembering that in the plane wave expansion the
\begin_inset Quotes eld
\end_inset
duality
\begin_inset Quotes erd
\end_inset
is interchangeable,
\begin_inset Formula
\begin{multline*}
\sigma_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\sum_{j=0}^{l'}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\sum_{l=\max\left(0,l'-2j\right)}^{l'-j}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l'+l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\times\\
\times\sum_{m=-l}^{l}\ush lm\left(\uvec{\vect k_{\vect K}}\right)\underbrace{\int\ud\Omega_{\vect r}\,\ush{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\frac{\left|\vect r_{\bot}\right|}{\left|\vect r\right|}\right)^{l'-l}\left(\frac{-\vect r_{\bot}\cdot\vect s_{\bot}}{\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|}\right)^{2j-l'+l}}_{\equiv A_{l',l,m',m,j}^{\left(d_{\Lambda}\right)}}.
\end{multline*}
\end_inset
The angular integral is easier to evaluate when
\begin_inset Formula $d_{\Lambda}=2$
\end_inset
, because then
\begin_inset Formula $\vect r_{\bot}$
\end_inset
is parallel (or antiparallel) to
\begin_inset Formula $\vect s_{\bot}$
\end_inset
, which gives
\begin_inset Formula
\[
A_{l',l,m',m,j}^{\left(2\right)}=\left(-\frac{\vect r_{\bot}\cdot\vect s_{\bot}}{\left|\vect r_{\bot}\cdot\vect s_{\bot}\right|}\right)^{2j-l'+l}\int\ud\Omega_{\vect r}\,\ush{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\frac{\left|\vect r_{\bot}\right|}{\left|\vect r\right|}\right)^{2j}
\]
\end_inset
and if we set the normal of the lattice correspond to the
\begin_inset Formula $z$
\end_inset
axis, the azimuthal part of the integral will become zero unless
\begin_inset Formula $m'=m$
\end_inset
for any meaningful spherical harmonics convention, and the polar part for
the only nonzero case has a closed-form expression, see e.g.
\begin_inset CommandInset citation
LatexCommand cite
after "(A.15)"
key "linton_lattice_2010"
literal "false"
\end_inset
, so one arrives at an expression similar to
\begin_inset CommandInset citation
LatexCommand cite
after "(3.15)"
key "kambe_theory_1968"
literal "false"
\end_inset
\lang english
\begin_inset Formula
\begin{multline}
\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{2}\mathcal{A}}\pi^{3/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
\times\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\ush lm\left(\vect k_{\vect K}\right)\sum_{j=0}^{l-\left|m\right|}\left(-1\right)^{j}\gamma_{\vect k_{\vect K}}^{2}{}^{2j+1}\times\\
\times\Delta_{j}\left(\frac{\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}}{4\eta^{2}},-i\kappa\gamma_{\vect k_{\vect K}}^{2}s_{\perp}\right)\times\\
\times\sum_{\substack{s\\
j\le s\le\min\left(2j,l-\left|m\right|\right)\\
l-j+\left|m\right|\,\mathrm{even}
}
}\frac{1}{\left(2j-s\right)!\left(s-j\right)!}\frac{\left(-\kappa s_{\perp}\right)^{2j-s}\left(\left|\vect k_{\vect K}\right|/\kappa\right)^{l-s}}{\left(\frac{1}{2}\left(l-m-s\right)\right)!\left(\frac{1}{2}\left(l+m-s\right)\right)!}\label{eq:Ewald in 3D long-range part 1D 2D-1}
\end{multline}
\end_inset
where
\begin_inset Formula $s_{\perp}\equiv\vect s\cdot\uvec z=\vect s_{\bot}\cdot\uvec z$
\end_inset
.
If
\begin_inset Formula $d_{\Lambda}=1$
\end_inset
, the angular becomes more complicated to evaluate due to the different
behaviour of the
\begin_inset Formula $\vect r_{\bot}\cdot\vect s_{\bot}/\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|$
\end_inset
factor.
The choice of coordinates can make most of the terms dissapear: if the
lattice is set parallel to the
\begin_inset Formula $z$
\end_inset
axis,
\begin_inset Formula $A_{l',l,m',m,j}^{\left(1\right)}$
\end_inset
is zero unless
\begin_inset Formula $m=0$
\end_inset
, but one still has
\begin_inset Formula
\[
A_{l',l,m',0,j}^{\left(1\right)}=\pi\delta_{m',l'-l-2j}\lambda'_{l0}\lambda_{l'm'}\int_{-1}^{1}\ud x\,P_{l'}^{m'}\left(x\right)P_{l}^{0}\left(x\right)\left(1-x^{2}\right)^{\frac{l'-l}{2}}
\]
\end_inset
where
\begin_inset Formula $\lambda_{lm}$
\end_inset
are constants depending on the conventions for spherical harmonics.
This does not seem to have such a nice closed-form expression as in the
2D case, but it can be evaluated e.g.
using the common recurrence relations for associated Legendre polynomials.
Of course when
\begin_inset Formula $\vect s=0$
\end_inset
, one gets relatively nice closed expressions, such as those in
\begin_inset CommandInset citation
LatexCommand cite
key "linton_lattice_2010"
literal "false"
\end_inset
.
\end_layout
\begin_layout Standard
\lang english
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "Tmatrix"
options "siam"
encoding "default"
\end_inset
\end_layout
\end_body
\end_document