qpms/notes/GF_vs_SWF.lyx

462 lines
11 KiB
Plaintext
Raw Normal View History

#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
This one should also be convention independent (similarly for
\begin_inset Formula $\mathcal{H}_{l}^{m}$
\end_inset
):
\begin_inset Formula
\[
\mathcal{J}_{l}^{m}\left(-\vect r\right)=\left(-1\right)^{l}\mathcal{J}_{l}^{m}\left(\vect r\right).
\]
\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
\begin_inset Marginal
status open
\begin_layout Plain Layout
\begin_inset Formula $G_{\pm}^{(\kappa)}\left(\vect x,\vect x_{0}\right)=-\frac{i\kappa}{\ush 00}\mathcal{H}_{0}^{0}\left(\kappa\left|\vect x-\vect x_{0}\right|\right)$
\end_inset
in case wacky conventions.
\end_layout
\end_inset
Lattice GF [Linton (2.3)]:
\begin_inset Formula
\begin{equation}
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}
\end{equation}
\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
and also its dual version
\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
and multiply with a dual SH + integrate
\begin_inset Formula
\begin{align}
\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 \\
& =-i\kappa\tau_{l'}^{m'}\left(\vect s,\vect k\right)j_{l'}\left(\kappa\left|\vect r\right|\right)\label{eq:tau extraction}
\end{align}
\end_inset
The expansion coefficients
\begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$
\end_inset
is then typically extracted by taking the limit
\begin_inset Formula $\left|\vect r\right|\to0$
\end_inset
.
\end_layout
\begin_layout Standard
The relation between
\begin_inset Formula $\sigma_{l}^{m}\left(\vect s,\vect k\right)$
\end_inset
and
\begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$
\end_inset
can be obtained e.g.
from the addition theorem for scalar spherical wavefunctions [Linton (C.3)],
\begin_inset Formula
\[
\mathcal{H}_{l}^{m}\left(\vect a+\vect b\right)=\sum_{l'm'}S_{ll'}^{mm'}\left(\vect b\right)\mathcal{J}_{l'}^{m'}\left(\vect a\right),\quad\left|\vect a\right|<\left|\vect b\right|
\]
\end_inset
where for the zeroth degree and order one has [Linton (C.3)]
\begin_inset Formula
\[
S_{0l'}^{0m'}\left(\vect b\right)=\sqrt{4\pi}\mathcal{H}'_{l'}^{m'}\left(-\vect b\right)
\]
\end_inset
\begin_inset Marginal
status open
\begin_layout Plain Layout
In a totally convention-independent version probably looks like
\begin_inset Formula $S_{0l'}^{0m'}\left(\vect b\right)=\ush 00\mathcal{H}'_{l'}^{m'}\left(-\vect b\right)$
\end_inset
, but the
\begin_inset Formula $Y_{0}^{0}$
\end_inset
will cancel with the expression for GF anyways, so no harm to the final
result.
\end_layout
\end_inset
From the lattice GF definition
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Lattice GF"
plural "false"
caps "false"
noprefix "false"
\end_inset
\begin_inset Formula
\begin{align*}
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}\\
& =\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}\\
& =\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}\\
& =-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}
\end{align*}
\end_inset
and mutliplying with dual SH and integrating
\begin_inset Formula
\begin{align*}
\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}\\
& =-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}\\
& =-i\kappa\sigma'_{l'}^{m'}\left(-\vect s,\vect k\right)j_{l'}\left(\kappa\left|\vect r\right|\right)
\end{align*}
\end_inset
and comparing with
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:tau extraction"
plural "false"
caps "false"
noprefix "false"
\end_inset
we have
\begin_inset Formula
\[
\tau_{l}^{m}\left(\vect s,\vect k\right)=\sigma'_{l}^{m}\left(-\vect s,\vect k\right).
\]
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
TODO maybe also define some
\begin_inset Formula $\tau'_{l}^{m}$
\end_inset
as expansion coefficients of GF into dual regular SSWFs.
\end_layout
\end_inset
\end_layout
\end_body
\end_document