Exactly-Solvable Problems for Two-Dimensional Excitons

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 two-dimensional hydrogen atom. This yields a new integral relation in terms of special functions, and fresh insights into the dynamical symmetry of the system are also obtained. The second problem relates to excitons in a quantizing magnetic field in the fractional quantum Hall regime. An exciton against the background of an incompressible quantum liquid is modeled as a few-particle neutral composite. A complete set of exciton basis functions is derived and classified, and some exact results are obtained for this complex few-particle problem.