From d640176d621a4d2f4c84d2985965c4ed3ca07e75 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= <marek@necada.org>
Date: Tue, 29 Mar 2022 14:32:06 +0300
Subject: [PATCH] Expressing VSWF in terms of SSWF

---
 notes/VSWF_from_SSWF.md      | 33 +++++++++++++++++++++++++++++++++
 notes/mathjax_newcommands.js | 11 +++++++++--
 2 files changed, 42 insertions(+), 2 deletions(-)
 create mode 100644 notes/VSWF_from_SSWF.md

diff --git a/notes/VSWF_from_SSWF.md b/notes/VSWF_from_SSWF.md
new file mode 100644
index 0000000..e3b5e1a
--- /dev/null
+++ b/notes/VSWF_from_SSWF.md
@@ -0,0 +1,33 @@
+VSWF expansions in terms of SSWF
+================================
+
+From
+\cite necada_multiple-scattering_2021, eq. (2.19)
+\f[
+	\wfkcout_{\tau lm}\left(\kappa (\vect r - \vect r_1) \right) =
+		\sum_{\tau'l'm'} \tropSr{\kappa(\vect r_2 - \vect r_1)}_{\tau l m;\tau'l'm} \wfkcreg_{\tau'l'm'}(\vect r -\vect r_2),
+	\qquad |\vect r -\vect r_2| < |\vect r_1 - \vect r_2|,
+\f]
+setting \f$ \vect r = \vect r_2\f$ and considering that 
+ \f$ \wfkcreg_{\tau'l'm'}(\vect 0) \ne \vect 0 \f$ only for electric dipole waves (\f$ \tau = \mathrm{E}, l=1 \f$),
+ we have
+\f[
+	\wfkcout_{\tau lm}\left(\kappa (\vect r - \vect r_1) \right) =
+		\sum_{m'} \tropSr{\kappa(\vect r - \vect r_1)}_{\tau l m;\mathrm{E}1m} \wfkcreg_{\mathrm{E}1m'}(\vect 0),
+	\qquad \vect r \ne \vect r_1 .
+\f]
+Combining this with
+\cite necada_multiple-scattering_2021, eq. (2.25)
+\f[
+	\tropSr{\vect d}_{\tau l m; \tau' l' m'} = \sum_{\lambda =|l-l'|+|\tau-\tau'|}^{l+l'} 
+		C^{\lambda}_{\tau l m;\tau' l'm'} \underbrace{ \spharm{\lambda}{m-m'}(\uvec d) h_\lambda^{(1)}(d)}_{\sswfout_\lambda^{m-m'}(\vect d)},
+\f]
+we get 
+\f[
+	\wfkcout_{\tau lm}(\vect d) = \sum_{m'=-1}^1 \wfkcreg_{\mathrm{E}1m'}(\vect 0) 
+		\sum_{\lambda=l-1+|\tau-\tau'|}^{l+1} 
+		C^\lambda_{\tau l m;\mathrm{E}1m'} \sswfout_\lambda^{m-m'}(\vect d).
+\f]
+Note that the VSWF components in this expression are given in global "cartesian" basis, 
+*not* the local orthonormal basis derived from spherical coordinates.
+(This is mostly desirable, because in lattices we need to work with flat coordinates anyway.)
diff --git a/notes/mathjax_newcommands.js b/notes/mathjax_newcommands.js
index 06e0a81..f0a07b9 100644
--- a/notes/mathjax_newcommands.js
+++ b/notes/mathjax_newcommands.js
@@ -31,8 +31,15 @@ MathJax.Hub.Config({
 	    spharm: ["{{Y_{\\mathrm{#1}}}_{#2}^{#3}}", 3, ""], // Spherical harmonics
 	    spharmR: ["{{Y_{\\mathrm{#1}}}_{\\mathrm{#1}{#2}{#3}}", 4, ""], // Spherical harmonics
 	    csphase: "\\mathsf{C_{CS}}", // Condon-Shortley phase
-	    tropSrr: ["{{S^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator singular
-	    tropRrr: ["{{R^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator regular
+	    tropS: "{\\mathcal{S}}", // Translation operator singular
+	    tropR: "{\\mathcal{R}}", // Translation operator regular
+	    tropSr: ["{{\\mathcal{S}^\\mathrm{#1}}\\pr{{#2}}}", 2, ""], // Translation operator singular
+	    tropRr: ["{{\\mathcal{R}^\\mathrm{#1}}\\pr{{#2}}}", 2, ""], // Translation operator regular
+	    tropSrr: ["{{\\mathcal{S}^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator singular
+	    tropRrr: ["{{\\mathcal{R}^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator regular
+            sswfout: "{\\psi}", // outgoing SSWF
+            sswfreg: "{\\phi}", // regular SSWF
+
 
             // Kristensson's VSWFs, complex version (2014 notes)
 	    wfkc: "{\\vect{y}}", // any wave