  The total spontaneous emission rate of a quantum emitter in the presence of
an infinite MoS\textsubscript{2} monolayer is enhanced by several orders of
magnitude, compared to its free-space value, due to the excitation of surface
exciton polariton modes and lossy modes. The spectral and distance dependence
of the spontaneous emission rate are analyzed and the lossy-surface-wave,
surface exciton polariton mode and radiative contributions are identified. The
transverse magnetic and transverse electric exciton polariton modes can be
excited for different emission frequencies of the quantum emitter, and their
contributions to the total spontaneous emission rate are different. To
calculate these different decay rates, we use the non-Hermitian description of
light-matter interactions, employing a Green's tensor formalism. The distance
dependence follows different trends depending on the emission energy of quantum
emitter. For the case of the lossy surface waves, the distance dependence
follows a $z^{-n}$, $n=2,3,4$, trend. When transverse magnetic exciton
polariton modes are excited, they dominate and characterize the distance
dependence of the spontaneous emission rate of a quantum emitter in the
presence of the MoS\textsubscript{2} layers. The interaction between a quantum
emitter and a MoS\textsubscript{2} superlattice is investigated and we observe
a splitting of the modes supported by the superlattice. Moreover, a blue shift
of the peak values of the spontaneous emission rate of a quantum emitter is
observed as the number of layers is increased. The field distribution profiles,
created by a quantum emitter, are used to explain this behavior.
