Dirac lattices, zero-range potentials and self adjoint extension

Abstract

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.

25 pages

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Cited by in corpus (10)

• Filtering Spins by Scattering from a Lattice of Point Magnets
• Casimir Energy for concentric $\delta$-${\delta }^{\prime }$ spheres
• Casimir effect for Dirac lattices
• Casimir and Casimir-Polder forces with dissipation from first principles
• Band spectra of periodic hybrid $\delta$-${\delta }^{\prime }$ structures
• Vacuum polarization with zero-range potentials on a hyperplane
• The Casimir energy anomaly for a point interaction
• The role of scalar current coupling along surfaces
• On finite temperature Casimir effect for Dirac lattices
• Vacuum and thermal energies for two oscillators interacting through a field