#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