Implement some of Javier's notes.

Former-commit-id: 3881eccd2bbca4975d50c4a749751b7c134d6698

lepaper/finite.lyx

@@ -481,7 +481,11 @@ The single-particle scattering problem at frequency
 \end_inset
 
 .
- Inside this volume, the electric field can be expanded as
+ Inside 
+\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
+\end_inset
+
+, the electric field can be expanded as
 \begin_inset Note Note
 status open
 
@@ -770,7 +774,7 @@ literal "false"
 
  its (maximum) refractive index.
 \begin_inset Note Note
-status open
+status collapsed
 
 \begin_layout Plain Layout
 \begin_inset Formula 
@@ -1281,7 +1285,7 @@ In practice, the multiple-scattering problem is solved in its truncated
 \begin_inset Formula $l\le L_{p}$
 \end_inset
 
-, laeving only 
+, leaving only 
 \begin_inset Formula $N_{p}=2L_{p}\left(L_{p}+2\right)$
 \end_inset
 
@@ -1428,11 +1432,7 @@ Let
 
 \end_inset
 
-where an explicit formula for the (regular) 
+where an explicit formula for the regular translation operator 
-\emph on
-translation operator
-\emph default
- 
 \begin_inset Formula $\tropr$
 \end_inset
 
@@ -1547,7 +1547,7 @@ reference "eq:regular vswf translation"
 , 
 \begin_inset Formula 
 \[
-\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{q}\right)
+\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(k\left(\vect r-\vect r_{q}\right)\right)
 \]
 
 \end_inset
@@ -1579,7 +1579,12 @@ reference "eq:regular vswf coefficient translation"
 
 \end_inset
 
-(note the reversed indices; TODO redefine them in 
+(note the reversed indices
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+; TODO redefine them in 
 \begin_inset CommandInset ref
 LatexCommand eqref
 reference "eq:regular vswf translation"
@@ -1593,7 +1598,12 @@ reference "eq:singular vswf translation"
 
 \end_inset
 
-? Similarly, if we had only outgoing waves in the original expansion around
+?
+\end_layout
+
+\end_inset
+
+) Similarly, if we had only outgoing waves in the original expansion around
  
 \begin_inset Formula $\vect r_{p}$
 \end_inset

lepaper/infinite.lyx

@@ -329,6 +329,16 @@ noprefix "false"
 
 \begin_layout Standard
 As in the case of a finite system, eq.
+ 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:Multiple-scattering problem unit cell"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
  can be written in a shorter block-matrix form,
 \begin_inset Formula 
 \begin{equation}
@@ -526,7 +536,17 @@ noprefix "false"
 \begin_inset Formula $\left|\vect k\right|=\sqrt{\epsilon\mu}\omega/c_{0}$
 \end_inset
 
- (modulo lattice points; TODO write this a clean way).
+ (modulo reciprocal lattice points
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO write this in a clean way
+\end_layout
+
+\end_inset
+
+).
  A somehow challenging step is to distinguish the different bands that can
  all be very close to the empty lattice approximation, especially if the
  particles in the systems are small.
@@ -687,7 +707,7 @@ translation operator for spherical waves originating in
 \end_inset
 
  is in fact a function of a single 3d argument, 
-\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
+\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
 \end_inset
 
 .
@@ -701,7 +721,7 @@ reference "eq:W integral"
  can be rewritten as
 \begin_inset Formula 
 \[
-W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
+W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0))\left(\vect k\right)}
 \]
 
 \end_inset
@@ -735,10 +755,10 @@ reference "eq:Dirac comb uaFt"
 (REF?) for the Fourier transform of Dirac comb)
 \begin_inset Formula 
 \begin{eqnarray}
-W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
+W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)(\vect k)\nonumber \\
- & = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
+ & = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)\left(\vect k\right)\nonumber \\
- & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
+ & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
- & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\nonumber 
+ & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\nonumber 
 \end{eqnarray}
 
 \end_inset
@@ -840,8 +860,8 @@ reference "eq:W sum in reciprocal space"
 \begin_inset Formula 
 \begin{eqnarray}
 W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
-W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
+W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
-W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
 \end{eqnarray}
 
 \end_inset
@@ -879,7 +899,7 @@ CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS
 
 \begin_inset Formula 
 \begin{equation}
-\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect 0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums}
+\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums}
 \end{equation}
 
 \end_inset

lepaper/symmetries.lyx

@@ -98,11 +98,11 @@ If the system has nontrivial point group symmetries, group theory gives
 \end_layout
 
 \begin_layout Standard
-As an example, if our system has a 
+As an example, if the system has a 
 \begin_inset Formula $D_{2h}$
 \end_inset
 
- symmetry and our truncated 
+ symmetry and the corresponding truncated 
 \begin_inset Formula $\left(I-T\trops\right)$
 \end_inset
 
@@ -961,7 +961,7 @@ where
 \begin_inset Formula $1$
 \end_inset
 
- through 
+ to 
 \begin_inset Formula $d_{n}$
 \end_inset
 
@@ -969,7 +969,7 @@ where
 \begin_inset Formula $i$
 \end_inset
 
- goes from 1 through the multiplicity of irreducible representation 
+ goes from 1 to the multiplicity of irreducible representation 
 \begin_inset Formula $\Gamma_{n}$
 \end_inset
 
@@ -1328,8 +1328,8 @@ horizontal
  the same unit cell, e.g.
 \begin_inset Formula 
 \begin{align*}
-\outcoeffp{\vect 0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0E},\\
+\outcoeffp{\vect0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0E},\\
-\outcoeff_{\vect 0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0C},
+\outcoeff_{\vect0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0C},
 \end{align*}
 
 \end_inset
@@ -1374,8 +1374,8 @@ vertical
 ,
 \begin_inset Formula 
 \begin{align*}
-\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\
+\outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\
-\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C},
+\outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C},
 \end{align*}
 
 \end_inset
@@ -1385,22 +1385,22 @@ but we want
 \end_inset
 
  to operate only inside one unit cell, so we use the Bloch condition 
-\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
+\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
 \end_inset
 
 : in this case, we have 
-\begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect 0\alpha}e^{i0}=\outcoeffp{\vect 0\alpha}$
+\begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect0\alpha}e^{i0}=\outcoeffp{\vect0\alpha}$
 \end_inset
 
 , 
-\begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect 0\alpha}=e^{i\pi}\outcoeffp{\vect 0\alpha}=-\outcoeffp{\vect 0\alpha},$
+\begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect0\alpha}=e^{i\pi}\outcoeffp{\vect0\alpha}=-\outcoeffp{\vect0\alpha},$
 \end_inset
 
 so 
 \begin_inset Formula 
 \begin{align*}
-\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0E},\\
+\outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0E},\\
-\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0C}.
+\outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0C}.
 \end{align*}
 
 \end_inset
@@ -1439,19 +1439,19 @@ the original
  rotation, as an example we have
 \begin_inset Formula 
 \begin{align*}
-\outcoeffp{\vect 0A} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(0,-1\right)E}=e^{2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0E},\\
+\outcoeffp{\vect0A} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(0,-1\right)E}=e^{2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect0E},\\
-\outcoeff_{\vect 0C} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)A}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0A},\\
+\outcoeff_{\vect0C} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)A}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect0A},\\
-\outcoeff_{\vect 0B} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)B}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0B},
+\outcoeff_{\vect0B} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)B}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect0B},
 \end{align*}
 
 \end_inset
 
 because in this case, the Bloch condition gives 
-\begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect 0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect 0\alpha}e^{2\pi i/3}=\outcoeffp{\vect 0\alpha}$
+\begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect0\alpha}e^{2\pi i/3}=\outcoeffp{\vect0\alpha}$
 \end_inset
 
 , 
-\begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect 0\alpha}.$
+\begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect0\alpha}.$
 \end_inset