  Exciton dynamics, lifetimes and scattering are directly related to the
exciton dispersion, or bandstructure. While electron and phonon bandstructures
are well understood and can be easily calculated from first principles, the
exciton bandstructure is commonly conflated with the underlying electronic
bandstructure, where the exciton dispersion is assumed to follow the same
dispersion as the electron and hole bands from which it is composed (i.e., the
effective mass model). Here, we present a general theory of exciton
bandstructure within both ab initio and model Hamiltonian approaches. We show
that contrary to common assumption, the exciton bandstructure contains
non-analytical discontinuities -- a feature which is impossible to obtain from
the electronic bandstructure alone. These discontinuities are purely quantum
phenomena, arising from the exchange scattering of electron-hole pairs. We show
that the degree of these discontinuities depends on materials' symmetry and
dimensionality, with jump discontinuities occurring in 3D and different orders
of removable discontinuities in 2D and 1D. We connect these unexpected features
to the early stages of exciton dynamics, which shows remarkable correspondence
with recent experimental observations and suggests that the measured diffusion
patterns are influenced by the underlying exciton bandstructure.
