Spectral analysis of a flat plasma sheet model

TL;DR

This paper conducts a detailed spectral analysis of electromagnetic fields on a thin plasma sheet, deriving explicit formulas for spectral functions and heat kernels, with applications to models like graphene and fullerenes.

Contribution

It introduces a rigorous spectral analysis method for a flat plasma sheet, including explicit constructions of spectral zeta functions, heat kernels, and a novel approach to fundamental solutions.

Findings

Explicit spectral zeta functions and heat kernels derived.

Surface plasmon bound states identified in TM-sector.

New method for fundamental solutions with delta-like sources demonstrated.

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 Schr\"odinger 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.