  Several problems in mathematical physics relating to excitons in two
dimensions are considered. First, a fascinating numerical result from a
theoretical treatment of screened excitons stimulates a re-evaluation of the
familiar two-dimensional hydrogen atom. Formulating the latter problem in
momentum space leads to a new integral relation in terms of special functions,
and fresh insights into the dynamical symmetry of the system are also obtained.
A discussion of an alternative potential to model screened excitons is given,
and the variable phase method is used to compare bound-state energies and
scattering phase shifts for this potential with those obtained using the
two-dimensional analogue of the Yukawa potential. The second problem relates to
excitons in a quantising magnetic field in the fractional quantum Hall regime.
An exciton against the background of an incompressible quantum liquid is
modelled as a few-particle neutral composite consisting of a positively-charged
hole and several quasielectrons with fractional negative charge. A complete set
of exciton basis functions is derived, and these functions are classified using
a result from the theory of partitions. Some exact results are obtained for
this complex few-particle problem.
