  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.
