Spectral analysis of a flat plasma sheet model

Abstract

The spectral analysis of the electromagnetic field on the background of a infinitely thin flat plasma layer is carried out. This model is loosely imitating a single base plane from graphite and it is of interest for theoretical studies of fullerenes. The model is naturally split into the TE-sector and TM-sector. Both the sectors have positive continuous spectra, but the TM-modes have in addition a bound state, namely, the surface plasmon. This analysis relies on the consideration of the scattering problem in the TE- and TM-sectors. The spectral zeta function and integrated heat kernel are constructed for different branches of the spectrum in an explicit form. As a preliminary, the rigorous procedure of integration over the continuous spectra is formulated by introducing the spectral densityin terms of the scattering phase shifts. The asymptotic expansion of the integrated heat kernel at small values of the evolution parameter is derived. By making use of the technique of integral equations, developed earlier by the same authors, the local heat kernel (Green's function or fundamental solution) is constructed also. As a by-product, a new method is demonstrated for deriving the fundamental solution to the heat conduction equation (or to the Schrödinger equation) on an infinite line with the $\delta$-like source. In particular, for the heat conduction equation on an infinite line with the $\delta$-source a nontrivial counterpart is found, namely, a spectral problem with point interaction, that possesses the same integrated heat kernel while the local heat kernels (fundamental solutions) in these spectral problems are different.

19 pages, REVTeX4, one reference is added, the version published in J. Phys. A: Math. Gen

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• On the entropy of a spherical plasma shell
• Singular perturbations with boundary conditions and the Casimir effect in the half space
• Divergences in the vacuum energy for frequency-dependent interactions
• Van der Waals interaction between an atom with spherical plasma shell
• Electromagnetic shape resonances of a dielectric sphere and radiation of portable telephones
• The Casimir-Polder interaction an atom with spherical shell