  We investigate the effects of environmental dielectric screening on the
electronic dispersion and the band gap in the atomically-thin, quasi
two-dimensional (2D) semiconductor WS$_2$ using correlative angle-resolved
photoemission and optical spectroscopies, along with first-principles
calculations. We find the main effect of increased environmental screening to
be a reduction of the band gap, with little change to the electronic dispersion
of the band structure. These essentially rigid shifts of the bands results from
the special spatial structure of the changes in the Coulomb potential induced
by the dielectric environment in the 2D limit. Our results suggest dielectric
engineering as a non-invasive method of tailoring the band structure of 2D
semiconductors and provide guidance for understanding the electronic properties
of 2D materials embedded in multilayer heterostructures.
