From e2584e3163337f5e39f54e04de16915badb80213 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Thu, 11 Jun 2020 16:26:02 +0300 Subject: [PATCH] Notes: periodic Greens functions vs SWF lattice sums Former-commit-id: 342ee89b71d416f4f452222a69d439738d522fab --- notes/GF_vs_SWF.lyx | 299 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 299 insertions(+) create mode 100644 notes/GF_vs_SWF.lyx diff --git a/notes/GF_vs_SWF.lyx b/notes/GF_vs_SWF.lyx new file mode 100644 index 0000000..4f0b5ff --- /dev/null +++ b/notes/GF_vs_SWF.lyx @@ -0,0 +1,299 @@ +#LyX 2.4 created this file. For more info see https://www.lyx.org/ +\lyxformat 584 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\use_default_options true +\maintain_unincluded_children false +\language finnish +\language_package default +\inputencoding utf8 +\fontencoding auto +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_roman_osf false +\font_sans_osf false +\font_typewriter_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\use_lineno 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style english +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tablestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +Periodic Green's functions vs. + VSWF lattice sums +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\ud}{\mathrm{d}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\abs}[1]{\left|#1\right|} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\vect}[1]{\mathbf{#1}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\uvec}[1]{\hat{\mathbf{#1}}} +\end_inset + + +\lang english + +\begin_inset FormulaMacro +\newcommand{\ush}[2]{Y_{#1}^{#2}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\ushD}[2]{Y'_{#1}^{#2}} +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\vsh}{\vect A} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\vshD}{\vect{A'}} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\wfkc}{\vect y} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\wfkcout}{\vect u} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\wfkcreg}{\vect v} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\wckcreg}{a} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\wckcout}{f} +\end_inset + + +\end_layout + +\begin_layout Section +Some definitions and useful relations +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\mathcal{H}_{l}^{m}\left(\vect d\right)\equiv h_{l}^{+}\left(\left|\vect d\right|\right)\ush lm\left(\uvec d\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\mathcal{J}_{l}^{m}\left(\vect d\right)\equiv j_{l}\left(\left|\vect d\right|\right)\ush lm\left(\uvec d\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Dual spherical harmonics and waves +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\ush lm\ushD{l'}{m'}\,\ud\Omega=\delta_{l,l'}\delta_{m,m'} +\] + +\end_inset + + +\begin_inset Formula +\[ +\mathcal{J}'_{l}^{m}\left(\vect d\right)\equiv j_{l}\left(\left|\vect d\right|\right)\ushD lm\left(\uvec d\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Expansion of plane wave (CHECKME whether this is really convention-independent, + but it seems so) +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +e^{i\kappa\vect r\cdot\uvec r'}=4\pi\sum_{l,m}i^{n}\mathcal{J}'_{l}^{m}\left(\kappa\vect r\right)\ush lm\left(\uvec r'\right)=4\pi\sum_{l,m}i^{n}\mathcal{J}{}_{l}^{m}\left(\kappa\vect r\right)\ushD lm\left(\uvec r'\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Section +Helmholtz equation and Green's functions (in 3D) +\end_layout + +\begin_layout Standard +Note that the notation does not follow Linton's (where the wavenumbers are + often implicit) +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\left(\nabla^{2}+\kappa^{2}\right)G^{(\kappa)}\left(\vect x,\vect x_{0}\right)=\delta\left(\vect x-\vect x_{0}\right) +\] + +\end_inset + + +\begin_inset Formula +\begin{align*} +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|}\\ +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) +\end{align*} + +\end_inset + +Lattice GF [Linton (2.3)]: +\begin_inset Formula +\[ +G_{\Lambda}^{(\kappa)}\left(\vect s,\vect k\right)\equiv\sum_{\vect R\in\Lambda}G_{+}^{\kappa}\left(\vect s-\vect R\right)e^{i\vect k\cdot\vect R} +\] + +\end_inset + + +\end_layout + +\begin_layout Section +GF expansion and lattice sum definition +\end_layout + +\begin_layout Standard +Let's define +\begin_inset Formula +\[ +\sigma_{l}^{m}\left(\vect s,\vect k\right)=\sum_{\vect R\in\Lambda}\mathcal{H}_{l}^{m}\left(\kappa\left(\vect s-\vect R\right)\right)e^{i\vect k\cdot\vect R}. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Inspired by [Linton (4.1)]; assuming that +\begin_inset Formula $\vect s\notin\Lambda$ +\end_inset + +, let's expand the lattice Green's function around +\begin_inset Formula $\vect s$ +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)=-i\kappa\sum_{l,m}\tau_{l}^{m}\left(\vect s,\vect k\right)\mathcal{J}_{l}^{m}\left(\kappa\vect r\right). +\] + +\end_inset + + +\end_layout + +\end_body +\end_document