Exact Results for a Quantum Many-Body Problem in One Dimension. II

We continue our previous investigation of the properties of a system of either fermions or bosons interacting in one dimension by a two-body potential $V\left(\mathcal{r}\right)=\frac{g}{{\mathcal{r}}^2}$ with periodic boundary conditions. A method is introduced which provides a physically intuitive description of the excited states. The energies are given explicitly for finite and infinite systems, including all corrections due to quasiparticle interaction. The method is shown to apply to other one-dimensional systems as well. In particular, the results are applied to a classical one-dimensional $N$-body model of Dyson—the Coulomb gas in a Brownian medium.