  We present a new method to solve the dynamical Bethe-Salpeter Equation
numerically. The method allows one to investigate the effects of dynamical
dielectric screening on the spectral position of excitons in transition-metal
dichalcogenide monolayers. The dynamics accounts for the response of optical
phonons in the materials below and on top the monolayer to the electric field
lines between the electron and hole of the exciton. The inclusion of this
effect unravels the origin of a counterintuitive energy blueshift of the
exciton resonance, observed recently in monolayer semiconductors that are
supported on ionic crystals with large dielectric constants. A surprising
result is that while energy renormalization of a free electron in the
conduction band or a free hole in the valence band is controlled by the
low-frequency dielectric constant, the bandgap energy introduces a phase
between the photoexcited electron and hole, rendering contributions from the
high-frequency dielectric constant also important when evaluating self-energies
of the exciton components. As a result, bandgap renormalization of the exciton
is not the sum of independent contributions from energy shifts of the
conduction and valence bands. The theory correctly predicts the energy shifts
of exciton resonances in various dielectric environments that embed
two-dimensional semiconductors.
