Anyon spectra and the third virial coefficient

The third virial coefficient of a gas of anyons in a harmonic potential is studied using two different approximations to obtain the three-anyon spectrum. A semiclassical (weak-coupling) approximation which provides us with the overall behavior of the spectrum leads to a divergent third virial coefficient. As a second approximation the low-lying three-anyon spectrum at the fermion point is constructed up to second order in the statistical parameter. The analogue spectrum at the boson point follows by symmetry. In this case the results are consistent with a finite third virial coefficient. The semiclassical approximation is also applied to the four-anyon spectrum and can be readily generalized to many anyons. Within their range of validity, both types of approximate spectra are in complete agreement with previously obtained exact numerical spectra.