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WIP tweaking infinite lattice text 

Former-commit-id: 6d28b1a6a255c787245b822a8db51f1b6bab1cb7
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18
lepaper/Tmatrix.bib
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@ -172,6 +172,24 @@
number = {4}
}
@article{gavin_feast_2018,
title = {{{FEAST}} Eigensolver for Nonlinear Eigenvalue Problems},
author = {Gavin, Brendan and Mi{\k{e}}dlar, Agnieszka and Polizzi, Eric},
year = {2018},
month = jul,
volume = {27},
pages = {107--117},
issn = {1877-7503},
doi = {10.1016/j.jocs.2018.05.006},
url = {http://www.sciencedirect.com/science/article/pii/S1877750318302096},
urldate = {2020-06-03},
abstract = {The linear FEAST algorithm is a method for solving linear eigenvalue problems. It uses complex contour integration to calculate the eigenvectors associated with eigenvalues that are located inside some user-defined region in the complex plane. This makes it possible to parallelize the process of solving eigenvalue problems by simply dividing the complex plane into a collection of disjoint regions and calculating the eigenpairs in each region independently of the eigenpairs in the other regions. In this paper we present a generalization of the linear FEAST algorithm that can be used to solve nonlinear eigenvalue problems. Like its linear progenitor, the nonlinear FEAST algorithm can be used to solve nonlinear eigenvalue problems for the eigenpairs corresponding to eigenvalues that lie in a user-defined region in the complex plane, thereby allowing for the calculation of large numbers of eigenpairs in parallel. We describe the nonlinear FEAST algorithm, and use several physically motivated examples to demonstrate its properties.},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/DCVYC9IH/Gavin ym. - 2018 - FEAST eigensolver for nonlinear eigenvalue problem.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/LJHSWVFQ/S1877750318302096.html},
journal = {Journal of Computational Science},
keywords = {Contour integration,FEAST,Nonlinear eigenvalue problem,Polynomial eigenvalue problem,Quadratic eigenvalue problem,Residual inverse iteration},
language = {en}
}
@article{guo_lasing_2019,
title = {Lasing at \${{K}}\$ {{Points}} of a {{Honeycomb Plasmonic Lattice}}},
author = {Guo, R. and Ne{\v c}ada, M. and Hakala, T. K. and V{\"a}kev{\"a}inen, A. I. and T{\"o}rm{\"a}, P.},

472
lepaper/infinite.lyx
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@ -451,7 +451,12 @@ noprefix "false"
\end_inset
.
The gain in the system introduces some challenges, which we will discuss
\begin_inset Note Note
status open
\begin_layout Plain Layout
The gain in the system introduces some challenges, which we will discuss
in Section
\begin_inset CommandInset ref
LatexCommand eqref
@ -465,6 +470,11 @@ noprefix "false"
.
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Numerical solution
\end_layout
@ -628,16 +638,17 @@ noprefix "false"
\end_layout
\begin_layout Standard
An alternative, more robust approach to generic minimisation algorithms
is Beyn's contour integral method
An alternative, faster and more robust approach to generic minimisation
algorithms are eigensolvers for nonlinear eigenvalue problems based on
contour integration
\begin_inset CommandInset citation
LatexCommand cite
key "beyn_integral_2012"
key "beyn_integral_2012,gavin_feast_2018"
literal "false"
\end_inset
which finds the roots of
which are able to find the roots of
\begin_inset Formula $M\left(\omega,\vect k\right)=0$
\end_inset
@ -753,7 +764,7 @@ empty
as defined in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lilgamma"
reference "eq:lilgamma_old"
plural "false"
caps "false"
noprefix "false"
@ -1091,7 +1102,24 @@ noprefix "false"
\end_layout
\begin_layout Standard
For reader's reference, we list the Ewald-type formulae for lattice sums
The lattice sums
\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)$
\end_inset
are related to what is also called
\emph on
structural constants
\emph default
in some literature
\begin_inset CommandInset citation
LatexCommand cite
key "kambe_theory_1967,kambe_theory_1967-1,kambe_theory_1968"
literal "false"
\end_inset
, but the phase and normalisation differ.
For reader's reference, we list the Ewald-type formulae for lattice sums
\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)$
\end_inset
@ -1129,7 +1157,7 @@ FP: Check sign of s
\begin_inset Formula
\begin{multline}
\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{\kappa^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect{R_{n}+\vect s}\right|^{l}\ush lm\left(\vect{R_{n}+\vect s}\right)e^{i\vect k\cdot\left(\vect{R_{n}+\vect s}\right)}\\
\times\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-\kappa/4\xi^{2}}\xi^{2l}\ud\xi\\
\times\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-\kappa^{2}/4\xi^{2}}\xi^{2l}\ud\xi\\
+\delta_{\vect{R_{n}},-\vect s}\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{\kappa^{2}}{4\eta^{2}}\right)\ush lm\left(\vect{R_{n}+\vect s}\right).\label{eq:Ewald in 3D short-range part}
\end{multline}
@ -1209,6 +1237,32 @@ Poznámka ohledně zahození radiální části u kulových fcí?
\end_layout
\begin_layout Standard
In practice, the integrals in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Ewald in 3D short-range part"
plural "false"
caps "false"
noprefix "false"
\end_inset
can be easily evaluated by numerical quadrature and the incomplete
\begin_inset Formula $\Gamma$
\end_inset
-functions using the series or continued fraction representations from
\begin_inset CommandInset citation
LatexCommand cite
key "NIST:DLMF"
literal "false"
\end_inset
.
\end_layout
\begin_layout Standard
The explicit form of the long-range part of the lattice sum depends on the
lattice dimensionality.
@ -1221,7 +1275,11 @@ The explicit form of the long-range part of the lattice sum depends on the
\begin_inset Formula $\left\{ \vect b_{i}\right\} _{i=1}^{d}$
\end_inset
lying in the same subspace as the direct lattice vectors
lying in the same
\begin_inset Formula $d$
\end_inset
-dimensional subspace as the direct lattice vectors
\begin_inset Formula $\left\{ \vect a_{i}\right\} _{i=1}^{d}$
\end_inset
@ -1233,12 +1291,15 @@ The explicit form of the long-range part of the lattice sum depends on the
\end_layout
\begin_layout Standard
If
\begin_layout Paragraph
Case
\begin_inset Formula $d=3$
\end_inset
,
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{\kappa\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\frac{\left(\left|\vect k+\vect K\right|/\kappa\right)^{l}}{\kappa^{2}-\left|\vect k+\vect K\right|^{2}}e^{\left(\kappa^{2}-\left|\vect k+\vect K\right|^{2}\right)/4\eta^{2}}\ush lm\left(\vect k+\vect K\right)\label{eq:Ewald in 3D long-range part 3D}
@ -1255,12 +1316,16 @@ regardless of chosen coordinate axes.
cases).
\end_layout
\begin_layout Standard
If
\begin_layout Paragraph
Case
\begin_inset Formula $d=2$
\end_inset
, reasonable explicit forms assume that the lattice lies inside the
\end_layout
\begin_layout Standard
Reasonable explicit forms assume that the lattice lies inside the
\begin_inset Formula $xy$
\end_inset
@ -1325,7 +1390,7 @@ FP: check sign of
\begin_inset Formula
\begin{multline}
\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{2}\mathcal{A}}\pi^{3/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
\times\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{l-\left|m\right|}\left(-1\right)^{j}\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+1}\Delta_{j}\left(\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)}{4\eta^{2}},-i\kappa\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)s_{\perp}\right)\times\\
\times\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{l-\left|m\right|}\left(-1\right)^{j}\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+1}\Delta_{j}\left(\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)^{2}}{4\eta^{2}},-i\kappa\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)s_{\perp}\right)\times\\
\times\sum_{\begin{array}{c}
s\\
j\le s\le\min\left(2j,l-\left|m\right|\right)\\
@ -1343,7 +1408,7 @@ status open
\begin_inset Formula
\begin{multline}
\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{d}\mathcal{A}}\pi^{2+\left(3-d\right)/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
\times\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-2j}\Gamma\left(\frac{d-1}{2}-j,\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\times\\
\times\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-2j}\Gamma\left(\frac{d-1}{2}-j,\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)^{2}}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\times\\
\times\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+3-d}\label{eq:Ewald in 3D long-range part 1D 2D-1}
\end{multline}
@ -1355,18 +1420,155 @@ status open
\end_inset
where
\begin_inset Formula $\gamma\left(z\right)=\left(z^{2}-1\right)^{\frac{1}{2}}$
\begin_inset Formula
\begin{equation}
\gamma\left(z\right)=\left(z^{2}-1\right)^{\frac{1}{2}},\label{eq:lilgamma}
\end{equation}
\end_inset
and
\begin_inset Formula
\begin{equation}
\Delta_{j}\left(x,z\right)=\int_{x}^{\infty}t^{\frac{-1}{2}-n}\exp\left(-t+\frac{z^{2}}{4t}\right)\ud t.\label{eq:Delta_j}
\end{equation}
\end_inset
If the normal component
\begin_inset Formula $s_{\bot}$
\end_inset
is zero, in the last sum in eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Ewald in 3D long-range part 1D 2D"
plural "false"
caps "false"
noprefix "false"
\end_inset
only one term (
\begin_inset Formula $s=2j$
\end_inset
) will remain if
\begin_inset Formula $l-\left|m\right|$
\end_inset
is even; for
\begin_inset Formula $l-\left|m\right|$
\end_inset
odd, the sum will vanish completely.
Moreover,
\begin_inset Formula $\Delta_{j}\left(x,0\right)=\Gamma\left(1/2-j,x\right)$
\end_inset
.
If
\begin_inset Formula $s_{\bot}\ne0$
\end_inset
, the integral
\begin_inset Formula $\Delta_{j}\left(x,z\right)$
\end_inset
can be evaluated e.g.
using the Taylor series
\lang finnish
\begin_inset Formula
\[
\Delta_{j}\left(x,z\right)=\int_{x}^{\infty}t^{\frac{-1}{2}-n}\exp\left(-t+\frac{z^{2}}{4t}\right)\ud t
\Delta_{j}\left(x,z\right)=\sum_{k=0}^{\infty}\Gamma\left(\frac{1}{2}-j-k,x\right)\frac{\left(z/2\right)^{2k}}{k!}
\]
\end_inset
and if the normal component
which is the first choice for small
\begin_inset Formula $z$
\end_inset
\lang english
.
Kambe
\begin_inset CommandInset citation
LatexCommand cite
key "kambe_theory_1968"
literal "false"
\end_inset
mentions a recurrence formula that can be obtained integrating
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Delta_j"
plural "false"
caps "false"
noprefix "false"
\end_inset
by parts (note that the signs are wrong in
\begin_inset CommandInset citation
LatexCommand cite
key "kambe_theory_1968"
literal "false"
\end_inset
)
\begin_inset Formula
\begin{equation}
\Delta_{j+1}\left(x,z\right)=\frac{4}{z^{2}}\left(\left(\frac{1}{2}-j\right)\Delta_{j}\left(x,z\right)-\Delta_{j-1}\left(x,z\right)+x^{\frac{1}{2}-j}e^{-x+\frac{z^{2}}{4x}}\right)\label{eq:Delta_j recurrent}
\end{equation}
\end_inset
with the first two terms
\begin_inset Formula
\begin{align*}
\Delta_{0}\left(x,z\right) & =\frac{\sqrt{\pi}}{2}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)+w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),\\
\Delta_{1}\left(x,z\right) & =i\frac{\sqrt{\pi}}{z}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)-w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),
\end{align*}
\end_inset
where
\begin_inset Formula $w\left(z\right)=e^{-z^{2}}\left(1+2i\pi^{-1/2}\int_{0}^{z}e^{t^{2}}\ud t\right)$
\end_inset
is the Faddeeva function.
However, the recurrence formula
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Delta_j recurrent"
plural "false"
caps "false"
noprefix "false"
\end_inset
is unsuitable for numerical evaluation if
\begin_inset Formula $z$
\end_inset
is small or
\begin_inset Formula $j$
\end_inset
is large due to its numerical instability.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
and if the normal component
\begin_inset Formula $s_{\perp}$
\end_inset
@ -1380,7 +1582,17 @@ where
\end_inset
The function
\end_layout
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
The function
\begin_inset Formula $\gamma\left(z\right)$
\end_inset
@ -1397,71 +1609,11 @@ noprefix "false"
is defined as
\begin_inset Formula
\begin{equation}
\gamma\left(z\right)=\left(z-1\right)^{\frac{1}{2}}\left(z+1\right)^{\frac{1}{2}},\quad-\frac{3\pi}{2}<\arg\left(z-1\right)<\frac{\pi}{2},-\frac{\pi}{2}<\arg\left(z+1\right)<\frac{3\pi}{2}.\label{eq:lilgamma}
\gamma\left(z\right)=\left(z-1\right)^{\frac{1}{2}}\left(z+1\right)^{\frac{1}{2}},\quad-\frac{3\pi}{2}<\arg\left(z-1\right)<\frac{\pi}{2},-\frac{\pi}{2}<\arg\left(z+1\right)<\frac{3\pi}{2}.\label{eq:lilgamma_old}
\end{equation}
\end_inset
The Ewald parameter
\begin_inset Formula $\eta$
\end_inset
determines the pace of convergence of both parts.
The larger
\begin_inset Formula $\eta$
\end_inset
is, the faster
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
converges but the slower
\begin_inset Formula $\sigma_{l,m}^{\left(L,\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
converges.
Therefore (based on the lattice geometry) it has to be adjusted in a way
that a reasonable amount of terms needs to be evaluated numerically from
both
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
and
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
.
For one-dimensional, square, and cubic lattices, the optimal choice is
\begin_inset Formula $\eta=\sqrt{\pi}/p$
\end_inset
where
\begin_inset Formula $p$
\end_inset
is the direct lattice period
\begin_inset CommandInset citation
LatexCommand cite
key "linton_lattice_2010"
literal "false"
\end_inset
.
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Marginal
status open
\begin_layout Plain Layout
Whatabout different geometries?
\end_layout
\end_inset
\end_layout
@ -1491,7 +1643,12 @@ FP: I have some error estimates derived in my notes.
\end_layout
\begin_layout Standard
For a two-dimensional lattice, the incomplete
One pecularity of the two-dimensional case is the two-branchedness of the
function
\begin_inset Formula $\gamma\left(z\right)$
\end_inset
and the incomplete
\begin_inset Formula $\Gamma$
\end_inset
@ -1499,12 +1656,26 @@ For a two-dimensional lattice, the incomplete
\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
\end_inset
in the long-range part has a branch point at
\begin_inset Formula $z=0$
appearing in the long-range part.
As a consequence, if we now explicitly label the dependence on the wavenumber,
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\kappa,\vect k,\vect s\right)$
\end_inset
and special care has to be taken when choosing the appropriate branch.
If the wavenumber of the medium has a positive imaginary part,
has branch points at
\begin_inset Formula $\kappa=\left|\vect k+\vect K\right|$
\end_inset
for every reciprocal lattice vector
\begin_inset Formula $\vect K$
\end_inset
.
If the wavenumber
\begin_inset Formula $\kappa$
\end_inset
of the medium has a positive imaginary part,
\begin_inset Formula $\Im\kappa>0$
\end_inset
@ -1527,17 +1698,19 @@ noprefix "false"
\end_inset
converges absolutely even in the direct space, and it is equal to the Ewald
sum with the principal value of the incomplete
\begin_inset Formula $\Gamma$
sum with the principal branches used both in
\begin_inset Formula $\gamma\left(z\right)$
\end_inset
function being used in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Ewald in 3D long-range part 1D 2D z = 0"
plural "false"
caps "false"
noprefix "false"
and
\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
\end_inset
\begin_inset CommandInset citation
LatexCommand cite
key "linton_lattice_2010"
literal "false"
\end_inset
@ -1546,7 +1719,7 @@ noprefix "false"
\begin_inset Formula $\kappa$
\end_inset
, the branch choice is made in such way that
, we typically choose the branch in such way that
\begin_inset Formula $W_{\alpha\beta}\left(\vect k\right)$
\end_inset
@ -1650,34 +1823,102 @@ status open
\end_inset
\end_layout
\begin_layout Paragraph
Case
\begin_inset Formula $d=1$
\end_inset
\end_layout
\begin_layout Standard
In practice, the integrals in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Ewald in 3D short-range part"
plural "false"
caps "false"
noprefix "false"
For one-dimensional chains, the easiest choice is to align the lattice with
the
\begin_inset Formula $z$
\end_inset
can be easily evaluated by numerical quadrature and the incomplete
\begin_inset Formula $\Gamma$
axis.
\end_layout
\begin_layout Subsubsection
Choice of Ewald parameter and high-frequency breakdown
\end_layout
\begin_layout Standard
The Ewald parameter
\begin_inset Formula $\eta$
\end_inset
-functions using the series 8.7.3 from
determines the pace of convergence of both parts.
The larger
\begin_inset Formula $\eta$
\end_inset
is, the faster
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
converges but the slower
\begin_inset Formula $\sigma_{l,m}^{\left(L,\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
converges.
Therefore (based on the lattice geometry) it has to be adjusted in a way
that a reasonable amount of terms needs to be evaluated numerically from
both
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
and
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
.
For one-dimensional, square, and cubic lattices, the optimal choice for
small frequencies (wavenumbers) is
\begin_inset Formula $\eta=\sqrt{\pi}/p$
\end_inset
where
\begin_inset Formula $p$
\end_inset
is the direct lattice period
\begin_inset CommandInset citation
LatexCommand cite
key "NIST:DLMF"
key "linton_lattice_2010"
literal "false"
\end_inset
.
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Marginal
status open
\begin_layout Plain Layout
Whatabout different geometries?
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Subsection
Physical interpretation of wavenumber with negative imaginary part; screening
\begin_inset CommandInset label
@ -1687,6 +1928,15 @@ name "subsec:Physical-interpretation-of"
\end_inset
\end_layout
\begin_layout Plain Layout
left out for the time being
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
@ -1749,7 +1999,7 @@ noprefix "false"
\begin_layout Standard
Ewald summation can be used for evaluating scattered field intensities outside
scatterers' circumscribing spheres: thes requires expressing VSWF cartesian
scatterers' circumscribing spheres: this requires expressing VSWF cartesian
components in terms of scalar spherical wavefunctions defined in
\begin_inset CommandInset ref
LatexCommand eqref