Notes: periodic Greens functions vs SWF lattice sums
Former-commit-id: 342ee89b71d416f4f452222a69d439738d522fab
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Marek Nečada
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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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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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Dual spherical harmonics and waves
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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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\[
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\mathcal{J}'_{l}^{m}\left(\vect d\right)\equiv j_{l}\left(\left|\vect d\right|\right)\ushD lm\left(\uvec d\right)
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\]
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\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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\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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Lattice GF [Linton (2.3)]:
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\begin_inset Formula
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\[
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G_{\Lambda}^{(\kappa)}\left(\vect s,\vect k\right)\equiv\sum_{\vect R\in\Lambda}G_{+}^{\kappa}\left(\vect s-\vect R\right)e^{i\vect k\cdot\vect R}
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\]
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\end_inset
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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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\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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\end_body
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\end_document
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