Spectral analysis of a flat plasma sheet model

The spectral analysis of the electromagnetic field on the background of an infinitely thin flat plasma layer is carried out. This model loosely imitates a single base plane from graphite and it is of interest for theoretical studies of fullerenes. By making use of the Hertz potentials the solutions to Maxwell equations with the appropriate matching conditions at the plasma layer are derived and on this basis the spectrum of electromagnetic oscillations is determined. 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 density in 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 also constructed. As a by-product, a new method is demonstrated for deriving the fundamental solution to the heat conduction equation (or to the Schroedinger equation) on an infinite line with the δ-like source. In particular, for the heat conduction equation on an infinite line with the δ-source a nontrivial counterpart is found, namely, a spectral problem with point interaction, which possesses the same integrated heat kernel while the local heat kernels (fundamental solutions) in these spectral problems are different