Exact multiplicities in the three-anyon spectrum.

Using the symmetry properties of the three-anyon spectrum, we obtain exactly the multiplicities of states with given energy and angular momentum. The results are shown to be in agreement with the proper quantum mechanical and semiclassical considerations, and the unexplained points are indicated. email: [email protected] It is well known that the quantum mechanical spectrum ofN non-interacting bosons or fermions can be obtained given just the single-particle spectrum, but for anyons this does not hold because the N -anyon problem is essentially many-particle. For arbitrary N there are two classes of exact solutions [1, 2, 3, 4] but their relative number decreases rapidly with N increasing. The only case in which one can proceed with the exact analysis more or less far ahead is that of N = 3. In the previous work [5] we have shown that it is possible to calculate exactly all the degeneracies in the threeanyon spectrum using certain symmetry properties of the latter. Here we will carry out an analogous calculation, taking into account in addition the angular momentum. Our present consideration will allow us to shed at least some light on the problem of quantum mechanical description of anyonic spectra, which at the moment is far from being closed. As it was done earlier, we consider the problem of three non-interacting anyons in a harmonic potential, with the particle mass and the frequency set to unity. The Hamiltonian is Ĥ = ∑3 j=1 Ĥj with the one-particle Hamiltonian Ĥj = 1 2 (−∆j + rj). The single-particle state is uniquely determined by the two quantum numbers – energy E which may equal 1, 2, . . . , and angular momentum L = −(E−1),−(E−3), . . . , E− 3, E − 1. Formally, the number of single-particle states with energy E and momentum L is given by g1(E,L) = 