Static Vortex Solutions in a Planar Particle System Obeying a Generalized Exclusion Principle

We consider a planar particle system obeying a generalized Pauli exclusion principle. In the mean field approximation, this system is described by a Schrödinger equation we recently introduced, containing a complex nonlinearity. The particle number, the total energy, and the angular momentum are conserved in such a system. We consider vortexlike stationary solutions of the form $\psi(\mathbf r)= \rho(r)^{1/2}e^{in\theta}$ and write the differential equation for the vortex shape. We find an analytic solution of this equation and obtain a closed expression for the vortex profile. We investigate some mean properties and, in particular, calculate the energy spectrum and angular momentum of the vortex.