Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dirac lattices, zero-range potentials and self adjoint extension   We consider the electromagnetic field in the presence of polarizable point
dipoles. In the corresponding effective Maxwell equation these dipoles are
described by three dimensional delta function potentials. We review the
approaches handling these: the selfadjoint extension,
regularization/renormalisation and the zero range potential methods. Their
close interrelations are discussed in detail and compared with the
electrostatic approach which drops the contributions from the self fields. For
a homogeneous two dimensional lattice of dipoles we write down the complete
solutions, which allow, for example, for an easy numerical treatment of the
scattering of the electromagnetic field on the lattice or for investigating
plasmons. Using these formulas, we consider the limiting case of vanishing
lattice spacing. For a scalar field and for the TE polarization of the
electromagnetic field this transition is smooth and results in the results
known from the continuous sheet. Especially for the TE polarization, we
reproduce the results known from the hydrodynamic model describing a two
dimensional electron gas. For the TM polarization, for polarizability parallel
and perpendicular to the lattice, in both cases, the transition is singular.
For the parallel polarizability this is surprising and different from the
hydrodynamic model. For perpendicular polarizability this is what was known in
literature. We also investigate the case when the transition is done with
dipoles described by smeared delta function, i.e., keeping a regularization.
Here, for TM polarization for parallel polarizability, when subsequently doing
the limit of vanishing lattice spacing, we reproduce the result known from the
hydrodynamic model. In case of perpendicular polarizability we need an
additional renormalization to reproduce the result obtained previously by
stepping back from the dipole approximation.

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The categories of this paper are: econ
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an