Notes on τ vs. σ
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Marek Nečada
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@ -216,6 +216,18 @@ e^{i\kappa\vect r\cdot\uvec r'}=4\pi\sum_{l,m}i^{n}\mathcal{J}'_{l}^{m}\left(\ka
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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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@ -245,11 +257,24 @@ G_{\pm}^{(\kappa)}\left(\vect x,\vect x_{0}\right) & =G_{\pm}^{(\kappa)}\left(\v
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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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\[
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G_{\Lambda}^{(\kappa)}\left(\vect s,\vect k\right)\equiv\sum_{\vect R\in\Lambda}G_{+}^{\kappa}\left(\vect s-\vect R\right)e^{i\vect k\cdot\vect R}
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\]
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\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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@ -264,7 +289,15 @@ GF expansion and lattice sum definition
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Let's define
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\begin_inset Formula
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\[
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\sigma_{l}^{m}\left(\vect s,\vect k\right)=\sum_{\vect R\in\Lambda}\mathcal{H}_{l}^{m}\left(\kappa\left(\vect s-\vect R\right)\right)e^{i\vect k\cdot\vect R}.
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\sigma_{l}^{m}\left(\vect s,\vect k\right)=\sum_{\vect R\in\Lambda}\mathcal{H}_{l}^{m}\left(\kappa\left(\vect s+\vect R\right)\right)e^{i\vect k\cdot\vect R},
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\]
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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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@ -287,11 +320,140 @@ Inspired by [Linton (4.1)]; assuming that
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\begin_layout Standard
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\begin_inset Formula
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\[
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G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)=-i\kappa\sum_{l,m}\tau_{l}^{m}\left(\vect s,\vect k\right)\mathcal{J}_{l}^{m}\left(\kappa\vect r\right).
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G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)=-i\kappa\sum_{l,m}\tau_{l}^{m}\left(\vect s,\vect k\right)\mathcal{J}_{l}^{m}\left(\kappa\vect r\right)
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\]
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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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