Formulation of the "simple" 1D problem

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Marek Nečada 2018-11-20 09:59:01 +00:00
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@ -3758,6 +3758,73 @@ where we used
\end_layout \end_layout
\begin_layout Section
Half-spaces and edge modes
\end_layout
\begin_layout Subsection
1D
\end_layout
\begin_layout Standard
Let us first consider the
\begin_inset Quotes eld
\end_inset
simple
\begin_inset Quotes erd
\end_inset
case without sublattices, so for example, let a set of identical particles
particles be placed with spacing
\begin_inset Formula $d$
\end_inset
on the positive
\begin_inset Formula $z$
\end_inset
-halfaxis, so their coordinates are in the set
\begin_inset Formula $C_{0}=C+\left\{ \vect 0\right\} =d\nats\hat{\vect{\mathbf{z}}}+\left\{ \vect 0\right\} $
\end_inset
.
The scattering problem on the particle placed at
\begin_inset Formula $\vect n\in C$
\end_inset
can then be described in the per-particle-matrix form as
\begin_inset Formula
\[
p_{\vect n}-p_{\vect n}^{(0)}=\sum_{\vect n'\in C_{0}\backslash\{\vect n\}}S_{\vect n\leftarrow\vect n'}Tp_{\vect n'},
\]
\end_inset
where
\begin_inset Formula $T$
\end_inset
is the
\begin_inset Formula $T$
\end_inset
-matrix,
\begin_inset Formula $S_{\vect n\leftarrow\vect n'}$
\end_inset
the translation operator and
\begin_inset Formula $p_{\vect n}^{(0)}$
\end_inset
the expansion of the external exciting fields, which can be set to zero
in order to find the system's eigenmodes.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Section \begin_layout Section
Major TODOs and open questions Major TODOs and open questions
\end_layout \end_layout
@ -3793,6 +3860,11 @@ Find a general algorithm for generating the expressions of the Hankel transforms
Three-dimensional case. Three-dimensional case.
\end_layout \end_layout
\end_inset
\end_layout
\begin_layout Section \begin_layout Section
(Appendix) Fourier vs. (Appendix) Fourier vs.
Hankel transform Hankel transform