Additional analytically exact solutions for three-anyons

TL;DR

This paper introduces new exact solutions for three anyons in harmonic or free space, expanding the known solutions and exploring their properties related to the statistical parameter and boundary conditions.

Contribution

It provides a new family of analytically exact solutions for three anyons, including cases with finite collision probability density, and discusses their spectral properties.

Findings

Solutions satisfy hard-core conditions at specific parameters

Solutions have finite non-zero two-particle collision density

Solutions lack a one-to-one mapping between bosonic and fermionic spectra

Abstract

We present new family of exact analytic solutions for three anyons in a harmonic potential (or in free space) in terms of generalized harmonics on $S^3$, which supplement the known solutions. The new solutions satisfy the hard-core condition when $\alpha={1\over 3},1$ ($\alpha$ being the statistical parameter) but otherwise, have finite non-vanishing two-particle colliding probability density, which is consistent with self-adjointness of the Hamiltonian. These solutions, however, do not have one-to-one mapping property between bosonic and fermionic spectra.