From e2584e3163337f5e39f54e04de16915badb80213 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= <marek@necada.org>
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