[ewald] Hankel vs. Fourier transform (appendix)
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Marek Nečada
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notes/ewald.lyx
153
notes/ewald.lyx
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@ -122,6 +122,21 @@
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\usht}[2]{\mathbb{S}_{#1}#2}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\bsht}[2]{\mathrm{S}_{#1}#2}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\pht}[2]{\mathfrak{\mathbb{H}}_{#1}#2}
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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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@ -162,6 +177,11 @@
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\end_inset
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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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\end_layout
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\begin_layout Title
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@ -623,7 +643,7 @@ The translation operator
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for compact scatterers in 3d can be expressed as
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\begin_inset Formula
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\[
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S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}Y_{p,m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(\left|\vect r\right|\right)
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S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(\left|\vect r\right|\right)
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\]
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\end_inset
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@ -640,25 +660,146 @@ where
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\begin_inset Formula $c_{p}^{l,m,t\leftarrow l',m',t'}$
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\end_inset
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are some ugly but known coefficients (Xu 1996, eqs.
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are some ugly but known coefficients (REF Xu 1996, eqs.
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76,77).
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\end_layout
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\begin_layout Section
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(Appendix) Hankel transform
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(Appendix) Fourier vs.
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Hankel transform
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\end_layout
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\begin_layout Subsection
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Three dimensions
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\end_layout
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\begin_layout Standard
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Acording to (Baddour 2010, eq.
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13) (CHECK FACTORS)
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Given a nice enough function
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\begin_inset Formula $f$
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\end_inset
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of a real 3d variable, assume its factorisation into radial and angular
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parts
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\begin_inset Formula
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\[
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\uaft f(\vect k)=
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f(\vect r)=\sum_{l,m}f_{l,m}(\left|\vect r\right|)\ush lm\left(\theta_{\vect r},\phi_{\vect r}\right).
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\]
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\end_inset
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Acording to (REF Baddour 2010, eqs.
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13, 16), its Fourier transform can then be expressed in terms of Hankel
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transforms (CHECK normalisation of
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\begin_inset Formula $j_{n}$
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\end_inset
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, REF Baddour (1))
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\begin_inset Formula
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\[
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\uaft f(\vect k)=\frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sum_{l,m}\left(-i\right)^{l}\left(\bsht{f_{l,m}}{}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)
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\]
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\end_inset
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where the spherical Hankel transform
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\begin_inset Formula $\bsht l{}$
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\end_inset
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of degree
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\begin_inset Formula $l$
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\end_inset
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is defined as (REF Baddour eq.
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2)
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\begin_inset Formula
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\[
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\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).
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\]
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\end_inset
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Using this convention, the inverse spherical Hankel transform is given by
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(REF Baddour eq.
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3)
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\begin_inset Formula
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\[
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g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k),
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\]
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\end_inset
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so it is not unitary.
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\end_layout
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\begin_layout Standard
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An unitary convention would look like this:
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\begin_inset Formula
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\[
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\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).
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\]
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\end_inset
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Then
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\begin_inset Formula $\usht l{}^{-1}=\usht l{}$
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\end_inset
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and the unitary, angular-momentum Fourier transform reads
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\begin_inset Formula
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\begin{eqnarray*}
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\uaft f(\vect k) & = & \frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sqrt{\frac{\pi}{2}}\sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)\\
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& = & \sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right).
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\end{eqnarray*}
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\end_inset
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Cool.
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\end_layout
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\begin_layout Subsection
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Two dimensions
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\end_layout
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\begin_layout Standard
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Similarly in 2d, let the expansion of
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\begin_inset Formula $f$
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\end_inset
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be
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\begin_inset Formula
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\[
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f\left(\vect r\right)=\sum_{m}f_{m}\left(\left|\vect r\right|\right)e^{im\phi_{\vect r}},
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\]
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\end_inset
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its Fourier transform is then (CHECK this, it is taken from the Wikipedia
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article on Hankel transform)
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\begin_inset Formula
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\[
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\uaft f\left(\vect k\right)=\sum_{m}i^{m}e^{im\theta_{\vect k}}\pht mf\left(\left|\vect k\right|\right)
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\]
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\end_inset
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where the Hankel transform of order
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\begin_inset Formula $m$
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\end_inset
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is defined as
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\begin_inset Formula
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\[
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\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\, J_{m}(kr)r
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\]
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\end_inset
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which is already self-inverse,
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\begin_inset Formula $\pht m{}^{-1}=\pht m{}$
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\end_inset
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(hence also unitary).
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\end_layout
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\begin_layout Section
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