Exactly-solvable problems for two-dimensional excitons

TL;DR

This paper explores exactly solvable models for two-dimensional excitons, providing new integral relations, analyzing screened potentials, and deriving exact solutions for excitons in quantum Hall systems.

Contribution

It introduces new integral relations in momentum space, compares different potential models, and derives exact solutions for complex exciton systems in magnetic fields.

Findings

New integral relation for 2D hydrogen atom in momentum space

Comparison of screened exciton potentials using variable phase method

Exact solutions for excitons in fractional quantum Hall regime

Abstract

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.