Semiclassical Theory for Two-anyon System

TL;DR

This paper develops a semiclassical quantization framework for two-anyon systems under magnetic fields, addressing singular interactions and boundary ambiguities, and validates the approach with cases where exact solutions are known.

Contribution

It introduces a generalized WKB method for two-anyon bound states, incorporating self-adjoint extension parameters to handle boundary ambiguities.

Findings

Semiclassical formulas match exact solutions across various quantum numbers.

The method effectively accounts for Aharonov-Bohm interactions in the quantization.

Self-adjoint extension parameters influence energy level calculations.

Abstract

The semiclassical quantization conditions for all partial waves are derived for bound states of two interacting anyons in the presence of a uniform background magnetic field. Singular Aharonov-Bohm-type interactions between the anyons are dealt with by the modified WKB method of Friedrich and Trost. For s-wave bound state problems in which the choice of the boundary condition at short distance gives rise to an additional ambiguity, a suitable generalization of the latter method is required to develop a consistent WKB approach. We here show how the related self-adjoint extension parameter affects the semiclassical quantization condition for energy levels. For some simple cases admitting exact answers, we verify that our semiclassical formulas in fact provide highly accurate results over a broad quantum number range.