Anyon trajectories and the systematics of the three-anyon spectrum

We develop the concept of trajectories in anyon spectra, i.e. the continuous dependence of energy levels on the kinetic angular momentum. It provides a more economical and unified description, since each trajectory contains an infinite number of points corresponding to the same statistics. For a system of noninteracting anyons in a harmonic potential, each trajectory consists of two infinite straight line segments, in general connected by a nonlinear piece. We give the systematics of the three-anyon trajectories. The trajectories in general cross each other at the bosonic/fermionic points. We use the (semiempirical) rule that all such crossings are true crossings, i.e. the order of the trajectories with respect to energy is opposite to the left and to the right of a crossing.