From 98ffdfc8740cbb29409838ead51df3844ac6a162 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Tue, 15 Aug 2017 12:17:43 +0300 Subject: [PATCH] =?UTF-8?q?[ewald]=20Pokra=C4=8Dov=C3=A1n=C3=AD?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Former-commit-id: 5c163530c176c0eb5a9e372414898830df65a189 --- notes/ewald.lyx | 188 +++++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 179 insertions(+), 9 deletions(-) diff --git a/notes/ewald.lyx b/notes/ewald.lyx index 371c868..386b8ad 100644 --- a/notes/ewald.lyx +++ b/notes/ewald.lyx @@ -187,6 +187,16 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\hgfr}{\mathbf{F}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\ph}{\mathrm{ph}} +\end_inset + + \end_layout \begin_layout Title @@ -685,7 +695,7 @@ h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\righ \end_inset so if we find a way to deal with the radial functions -\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ +\begin_inset Formula $s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ \end_inset , @@ -755,7 +765,7 @@ Here \begin_layout Standard Obviously, all the terms -\begin_inset Formula $\propto s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ +\begin_inset Formula $\propto s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ \end_inset , @@ -778,7 +788,7 @@ reference "eq:spherical Hankel function series" \begin_layout Standard The remaining task is therefore to find a suitable decomposition of -\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ +\begin_inset Formula $s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ \end_inset , @@ -786,16 +796,16 @@ The remaining task is therefore to find a suitable decomposition of \end_inset into short-range and long-range parts, -\begin_inset Formula $s_{q}(r)=s_{q}^{\textup{S}}(r)+s_{q}^{\textup{L}}(r)$ +\begin_inset Formula $s_{k_{0},q}(r)=s_{k_{0},q}^{\textup{S}}(r)+s_{k_{0},q}^{\textup{L}}(r)$ \end_inset , such that -\begin_inset Formula $s_{q}^{\textup{L}}(r)$ +\begin_inset Formula $s_{k_{0},q}^{\textup{L}}(r)$ \end_inset contains all the slowly decaying asymptotics and its Hankel transforms decay desirably fast as well, -\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$ +\begin_inset Formula $\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$ \end_inset , @@ -810,7 +820,7 @@ The remaining task is therefore to find a suitable decomposition of must be sufficiently smooth in the origin, so that \begin_inset Formula \begin{equation} -\pht n{s_{q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement} +\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{k_{0},q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{k_{0},q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement} \end{equation} \end_inset @@ -852,8 +862,9 @@ The electrostatic Ewald summation uses regularisation with \end_inset . - However, such choice does not seem to lead to an analytical solution for - the current problem + However, such choice does not seem to lead to an analytical solution (really? + could not something be dug out of DLMF 10.22.54?) for the current problem + \begin_inset CommandInset ref LatexCommand eqref reference "eq:2d long range regularisation problem statement" @@ -876,6 +887,165 @@ leads to satisfactory results, as will be shown below. Hankel transforms of the long-range parts \end_layout +\begin_layout Standard +Let +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{eqnarray} +\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & \equiv & \int_{0}^{\infty}\frac{e^{ik_{0}r}}{\left(k_{0}r\right)^{q}}J_{n}\left(kr\right)\left(1-e^{-cr}\right)^{\kappa}r\,\ud r\nonumber \\ + & = & k_{0}^{-q}\int_{0}^{\infty}r^{1-q}J_{n}\left(kr\right)\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}e^{-(\sigma c-ik_{0})r}\ud r\nonumber \\ + & \underset{\equiv}{\textup{form.}} & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\pht n{s_{q,k_{0}}^{\textup{L}1,\sigma c}}\left(k\right).\label{eq:2D Hankel transform of regularized outgoing wave, decomposition} +\end{eqnarray} + +\end_inset + +From [REF DLMF 10.22.49] one digs +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula +\begin{eqnarray*} +\mu & \leftarrow & 2-q\\ +\nu & \leftarrow & n\\ +b & \leftarrow & k\\ +a & \leftarrow & c-ik_{0} +\end{eqnarray*} + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{multline} +\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right),\\ +\Re\left(2-q+n\right)>0,\Re(c-ik_{0}\pm k)\ge0,\label{eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1} +\end{multline} + +\end_inset + +and from [REF DLMF 15.9.17] +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula +\begin{eqnarray*} +a & \leftarrow & \frac{2-q+n}{2}\\ +c & \leftarrow & 1+n\\ +z & \leftarrow & \frac{-k^{2}}{\left(c-ik_{0}\right)^{2}} +\end{eqnarray*} + +\end_inset + + +\begin_inset Formula +\begin{eqnarray*} +\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right) & = & \frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}2^{n}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1-\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right)^{-\frac{2-q+n}{2}+\frac{n}{2}}P_{2-q+n-(1+n)}^{1-(1+n)}\left(\frac{1}{\sqrt{1-\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)}}\right)\\ + & = & \frac{k^{n}Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{1-q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right) +\end{eqnarray*} + +\end_inset + + +\begin_inset Formula +\[ +\left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|<\pi,\quad\left|\ph\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right|<\pi +\] + +\end_inset + +in other words, neither +\begin_inset Formula $-k^{2}/\left(c-ik_{0}\right)^{2}$ +\end_inset + + nor +\begin_inset Formula $1+k^{2}/\left(c-ik_{0}\right)^{2}$ +\end_inset + + can be non-positive real number. + For assumed positive +\begin_inset Formula $k_{0}$ +\end_inset + + and non-negative +\begin_inset Formula $c$ +\end_inset + + and +\begin_inset Formula $k$ +\end_inset + +, the former case can happen only if +\begin_inset Formula $k=0$ +\end_inset + + and the latter only if +\begin_inset Formula $c=0\wedge k_{0}=k$ +\end_inset + +. + +\begin_inset Formula +\begin{eqnarray*} +\left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|<\pi & \Leftrightarrow & \left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|\neq\pi\\ +\varphi & \equiv & \ph\left(c-ik_{0}\right)<0,\\ +\ph k & \equiv & 0\\ +\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}} & = & 2\varphi\\ +\rightsquigarrow\left|\varphi\right| & \neq & \pi/2\\ +\rightsquigarrow c & \neq & k_{0}\\ +\left|\ph\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right| & = & \left|-2\varphi+\ph\left(\left(c-ik_{0}\right)^{2}+k^{2}\right)\right| +\end{eqnarray*} + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{multline} +\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{1-q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right),\\ +k>0\wedge k_{0}>0\wedge c\ge0\wedge\lnot\left(c=0\wedge k_{0}=k\right)\label{eq:2D Hankel transform of exponentially suppressed outgoing wave expanded} +\end{multline} + +\end_inset + +with principal branches of the hypergeometric functions, associated Legendre + functions, and fractional powers. + The conditions from +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1" + +\end_inset + + should hold, but we will use +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded" + +\end_inset + + formally even if they are violated, with the hope that the divergences + eventually cancel in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:2D Hankel transform of regularized outgoing wave, decomposition" + +\end_inset + +. +\end_layout + \begin_layout Subsection 3d \end_layout