On the Three-Anyon Harmonics

TL;DR

This paper analyzes the three-anyon problem using a new set of variables, deriving boundary conditions for self-adjoint Hamiltonians, and proposing a numerical approach to find wavefunctions in harmonic potentials.

Contribution

It introduces a novel variable set for the 3-anyon problem, explores boundary conditions for self-adjointness, and proposes a numerical method for solving the wavefunction equations.

Findings

Boundary conditions for 3-anyon wavefunctions are derived.

MMOR boundary conditions are one of the self-adjointness solutions.

A series expansion reduces the problem to algebraic equations for numerical analysis.

Abstract

The 3-anyon problem is studied using a set of variables recently proposed in an anyon gauge analysis by Mashkevich, Myrheim, Olaussen, and Rietman (MMOR). Boundary conditions to be satisfied by the wave functions in order to render the Hamiltonian self-adjoint are derived, and it is found that the boundary conditions adopted by MMOR are one of the ways to satisfy these general self-adjointness requirements. The possibility of scale-dependent boundary conditions is also investigated, in analogy with the corresponding analyses of the 2-anyon case. The structure of the known solutions of the 3-anyon in harmonic potential problem is discussed in terms of the MMOR variables. Within a series expansion in a boson gauge framework the problem of finding any anyon wavefunction is reduced to a (possibly infinite) set of algebraic equations, whose numerical analysis is proposed as an efficient way to study anyon physics.