Few-particle anyon exciton: Exact solutions

The anyon exciton model, which describes an exciton against the background of an incompressible quantum liquid, is generalized to the case of an arbitrary number of anyons. Some mathematical aspects of this quantum-mechanical few-particle problem are considered and several exact solutions are obtained. The four-particle case is also considered in the classical limit in both planar and spherical geometries. Such a classical approach gives an adequate description of an anyon exciton at large separation between the valence hole and the two-dimensional electron gas. It is shown that in this limit in a planar geometry the anyon exciton is always energetically more favorable than a charged anyon ion. This indicates that the appearance of fractionally-charged anyon ions reported in recent numerical calculations is an artefact apparently caused by finite-size effects in a spherical geometry.