462 lines
11 KiB
Plaintext
462 lines
11 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 584
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\begin_document
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\begin_header
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\secnumdepth 3
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\tocdepth 3
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\end_header
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\begin_body
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\begin_layout Title
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Periodic Green's functions vs.
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VSWF lattice sums
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\end_layout
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\ud}{\mathrm{d}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\abs}[1]{\left|#1\right|}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\mathbf{#1}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\uvec}[1]{\hat{\mathbf{#1}}}
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\end_inset
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\lang english
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\begin_inset FormulaMacro
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\newcommand{\ush}[2]{Y_{#1}^{#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ushD}[2]{Y'_{#1}^{#2}}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\vsh}{\vect A}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\vshD}{\vect{A'}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wfkc}{\vect y}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wfkcout}{\vect u}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wfkcreg}{\vect v}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wckcreg}{a}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\wckcout}{f}
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\end_inset
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\end_layout
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\begin_layout Section
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Some definitions and useful relations
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\mathcal{H}_{l}^{m}\left(\vect d\right)\equiv h_{l}^{+}\left(\left|\vect d\right|\right)\ush lm\left(\uvec d\right)
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\]
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\end_inset
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\begin_inset Formula
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\[
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\mathcal{J}_{l}^{m}\left(\vect d\right)\equiv j_{l}\left(\left|\vect d\right|\right)\ush lm\left(\uvec d\right)
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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Dual spherical harmonics and waves
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\int\ush lm\ushD{l'}{m'}\,\ud\Omega=\delta_{l,l'}\delta_{m,m'}
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\]
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\end_inset
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\begin_inset Formula
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\[
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\mathcal{J}'_{l}^{m}\left(\vect d\right)\equiv j_{l}\left(\left|\vect d\right|\right)\ushD lm\left(\uvec d\right)
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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Expansion of plane wave (CHECKME whether this is really convention-independent,
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but it seems so)
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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e^{i\kappa\vect r\cdot\uvec r'}=4\pi\sum_{l,m}i^{n}\mathcal{J}'_{l}^{m}\left(\kappa\vect r\right)\ush lm\left(\uvec r'\right)=4\pi\sum_{l,m}i^{n}\mathcal{J}{}_{l}^{m}\left(\kappa\vect r\right)\ushD lm\left(\uvec r'\right)
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\]
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\end_inset
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This one should also be convention independent (similarly for
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\begin_inset Formula $\mathcal{H}_{l}^{m}$
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\end_inset
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):
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\begin_inset Formula
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\[
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\mathcal{J}_{l}^{m}\left(-\vect r\right)=\left(-1\right)^{l}\mathcal{J}_{l}^{m}\left(\vect r\right).
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\]
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\end_inset
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\end_layout
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\begin_layout Section
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Helmholtz equation and Green's functions (in 3D)
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\end_layout
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\begin_layout Standard
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Note that the notation does not follow Linton's (where the wavenumbers are
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often implicit)
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\left(\nabla^{2}+\kappa^{2}\right)G^{(\kappa)}\left(\vect x,\vect x_{0}\right)=\delta\left(\vect x-\vect x_{0}\right)
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\]
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\end_inset
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\begin_inset Formula
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\begin{align*}
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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|}\\
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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)
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\end{align*}
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\end_inset
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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\begin_inset Formula $G_{\pm}^{(\kappa)}\left(\vect x,\vect x_{0}\right)=-\frac{i\kappa}{\ush 00}\mathcal{H}_{0}^{0}\left(\kappa\left|\vect x-\vect x_{0}\right|\right)$
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\end_inset
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in case wacky conventions.
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\end_layout
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\end_inset
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Lattice GF [Linton (2.3)]:
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\begin_inset Formula
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\begin{equation}
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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}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Section
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GF expansion and lattice sum definition
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\end_layout
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\begin_layout Standard
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Let's define
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\begin_inset Formula
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\[
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\sigma_{l}^{m}\left(\vect s,\vect k\right)=\sum_{\vect R\in\Lambda}\mathcal{H}_{l}^{m}\left(\kappa\left(\vect s+\vect R\right)\right)e^{i\vect k\cdot\vect R},
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\]
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\end_inset
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and also its dual version
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\begin_inset Formula
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\[
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\sigma'_{l}^{m}\left(\vect s,\vect k\right)=\sum_{\vect R\in\Lambda}\mathcal{H}'_{l}^{m}\left(\kappa\left(\vect s+\vect R\right)\right)e^{i\vect k\cdot\vect R}.
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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Inspired by [Linton (4.1)]; assuming that
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\begin_inset Formula $\vect s\notin\Lambda$
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\end_inset
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, let's expand the lattice Green's function around
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\begin_inset Formula $\vect s$
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\end_inset
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:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)=-i\kappa\sum_{l,m}\tau_{l}^{m}\left(\vect s,\vect k\right)\mathcal{J}_{l}^{m}\left(\kappa\vect r\right)
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\]
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\end_inset
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and multiply with a dual SH + integrate
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\begin_inset Formula
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\begin{align}
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\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 \\
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& =-i\kappa\tau_{l'}^{m'}\left(\vect s,\vect k\right)j_{l'}\left(\kappa\left|\vect r\right|\right)\label{eq:tau extraction}
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\end{align}
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\end_inset
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The expansion coefficients
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\begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$
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\end_inset
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is then typically extracted by taking the limit
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\begin_inset Formula $\left|\vect r\right|\to0$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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The relation between
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\begin_inset Formula $\sigma_{l}^{m}\left(\vect s,\vect k\right)$
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\end_inset
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and
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\begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$
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\end_inset
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can be obtained e.g.
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from the addition theorem for scalar spherical wavefunctions [Linton (C.3)],
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\begin_inset Formula
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\[
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\mathcal{H}_{l}^{m}\left(\vect a+\vect b\right)=\sum_{l'm'}S_{ll'}^{mm'}\left(\vect b\right)\mathcal{J}_{l'}^{m'}\left(\vect a\right),\quad\left|\vect a\right|<\left|\vect b\right|
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\]
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\end_inset
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where for the zeroth degree and order one has [Linton (C.3)]
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\begin_inset Formula
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\[
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S_{0l'}^{0m'}\left(\vect b\right)=\sqrt{4\pi}\mathcal{H}'_{l'}^{m'}\left(-\vect b\right)
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\]
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\end_inset
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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In a totally convention-independent version probably looks like
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\begin_inset Formula $S_{0l'}^{0m'}\left(\vect b\right)=\ush 00\mathcal{H}'_{l'}^{m'}\left(-\vect b\right)$
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\end_inset
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, but the
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\begin_inset Formula $Y_{0}^{0}$
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\end_inset
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will cancel with the expression for GF anyways, so no harm to the final
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result.
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\end_layout
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\end_inset
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From the lattice GF definition
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Lattice GF"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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\begin_inset Formula
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\begin{align*}
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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}\\
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& =\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}\\
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& =\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}\\
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& =-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}
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\end{align*}
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\end_inset
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and mutliplying with dual SH and integrating
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\begin_inset Formula
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\begin{align*}
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\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}\\
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& =-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}\\
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& =-i\kappa\sigma'_{l'}^{m'}\left(-\vect s,\vect k\right)j_{l'}\left(\kappa\left|\vect r\right|\right)
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\end{align*}
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\end_inset
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and comparing with
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:tau extraction"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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we have
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\begin_inset Formula
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\[
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\tau_{l}^{m}\left(\vect s,\vect k\right)=\sigma'_{l}^{m}\left(-\vect s,\vect k\right).
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\]
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\end_inset
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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TODO maybe also define some
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\begin_inset Formula $\tau'_{l}^{m}$
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\end_inset
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as expansion coefficients of GF into dual regular SSWFs.
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\end_layout
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\end_inset
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\end_layout
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\end_body
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\end_document
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