WIP infinite sys.

Former-commit-id: e14d9ee7a04af1fa42b2a7de849de0569a2eb471

lepaper/finite.lyx

@@ -968,6 +968,13 @@ reference "eq:plane wave expansion"
 
 \begin_layout Subsection
 Multiple scattering
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:Multiple-scattering"
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard

lepaper/infinite.lyx

@@ -95,6 +95,16 @@
 
 \begin_layout Section
 Infinite periodic systems
+\begin_inset FormulaMacro
+\newcommand{\dlv}{\vect b}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\rlv}{\vect b}
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -121,13 +131,43 @@ Topology anoyne?
  scatterer arrays.
 \end_layout
 
+\begin_layout Subsection
+Notation
+\end_layout
+
 \begin_layout Subsection
 Formulation of the problem
 \end_layout
 
 \begin_layout Standard
-Assume a system of compact EM scatterers in otherwise homogeneous and isotropic
+Let us have a linear system of compact EM scatterers on a homogeneous background
- medium, and assume that the system, i.e.
+ as in Section 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "subsec:Multiple-scattering"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, but this time, system shall be periodic: let there be a 
+\begin_inset Formula $d$
+\end_inset
+
+-dimensional (
+\begin_inset Formula $d$
+\end_inset
+
+ can be 1, 2 or 3) lattice embedded into the three-dimensional real space,
+ with lattice vectors.
+ set of 
+\begin_inset Formula $d$
+\end_inset
+
+ (one to three) lattice vectorsAssume a system of compact EM scatterers
+ in otherwise homogeneous and isotropic medium, and assume that the system,
+ i.e.
  both the medium and the scatterers, have linear response.
  A scattering problem in such system can be written as
 \begin_inset Formula 
@@ -216,9 +256,9 @@ and we assume periodic solution
 \begin_inset Formula 
 \begin{eqnarray*}
 \sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
-\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect0α\leftarrow\vect bβ})A_{\vect0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
+\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect 0α\leftarrow\vect bβ})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
-\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect0\beta}\left(\vect k\right) & = & 0,\\
+\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
-A_{\vect0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect0\beta}\left(\vect k\right) & = & 0.
+A_{\vect 0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
 \end{eqnarray*}
 
 \end_inset
@@ -234,7 +274,7 @@ lattice Fourier transform
  of the translation operator,
 \begin_inset Formula 
 \begin{equation}
-W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
+W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
 \end{equation}
 
 \end_inset
@@ -255,7 +295,7 @@ reference "eq:W definition"
 \end_inset
 
  is the asymptotic behaviour of the translation operator, 
-\begin_inset Formula $S_{\vect0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
+\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
 \end_inset
 
  that makes the convergence of the sum quite problematic for any 
@@ -295,7 +335,7 @@ reference "eq:W definition"
 
  in terms of integral with a delta comb
 \begin_inset FormulaMacro
-\newcommand{\basis}[1]{\mathfrak{#1}}
+\renewcommand{\basis}[1]{\mathfrak{#1}}
 \end_inset
 
 
@@ -351,7 +391,7 @@ translation operator for spherical waves originating in
 \end_inset
 
  is in fact a function of a single 3d argument, 
-\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
+\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
 \end_inset
 
 .
@@ -365,7 +405,7 @@ reference "eq:W integral"
  can be rewritten as
 \begin_inset Formula 
 \[
-W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0))\left(\vect k\right)}
+W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
 \]
 
 \end_inset
@@ -390,10 +430,10 @@ reference "eq:Dirac comb uaFt"
  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\vect0)}\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\vect 0)}\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\vect0)}\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\vect 0)}\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\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}}\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}}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 
+ & = & \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 
 \end{eqnarray}
 
 \end_inset
@@ -495,8 +535,8 @@ 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}}(\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{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{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}
+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}
 \end{eqnarray}
 
 \end_inset