  The realization of mixtures of excitons and charge carriers in van-der-Waals
materials presents a new frontier for the study of the many-body physics of
strongly interacting Bose-Fermi mixtures. In order to derive an effective
low-energy model for such systems, we develop an exact diagonalization approach
based on a discrete variable representation that predicts the scattering and
bound state properties of three charges in two-dimensional transition metal
dichalcogenides. From the solution of the quantum mechanical three-body problem
we thus obtain the bound state energies of excitons and trions within an
effective mass model which are in excellent agreement with Quantum Monte Carlo
predictions. The diagonalization approach also gives access to excited states
of the three-body system. This allows us to predict the scattering phase shifts
of electrons and excitons that serve as input for a low-energy theory of
interacting mixtures of excitons and charge carriers at finite density. To this
end we derive an effective exciton-electron scattering potential that is
directly applicable for Quantum Monte-Carlo or diagrammatic many-body
techniques. As an example, we demonstrate the approach by studying the
many-body physics of exciton Fermi polarons in transition-metal
dichalcogenides, and we show that finite-range corrections have a substantial
impact on the optical absorption spectrum. Our approach can be applied to a
plethora of many-body phenomena realizable in atomically thin semiconductors
ranging from exciton localization to induced superconductivity.
