Critical Rotational Speeds in the Gross-Pitaevskii Theory on a Disc with Dirichlet Boundary Conditions

Abstract

We study the two-dimensional Gross-Pitaevskii theory of a rotating Bose gas in a disc-shaped trap with Dirichlet boundary conditions, generalizing and extending previous results that were obtained under Neumann boundary conditions. The focus is on the energy asymptotics, vorticity and qualitative properties of the minimizers in the parameter range $|\log\eps| \ll \Omega \lesssim \eps^{-2}|\log\eps|^{-1}$ where $ \Omega $ is the rotational velocity and the coupling parameter is written as $ \eps^{-2} $ with $ \eps \ll 1 $. Three critical speeds can be identified. At $ \Omega = \Omega_{\mathrm{c_1}} \sim|\log\eps| $ vortices start to appear and for $ |\log\eps| \ll \Omega < \Omega_{\mathrm{c_2}} \sim \eps^{-1} $ the vorticity is uniformly distributed over the disc. For $ \Omega \geq \Omega_{\mathrm{c_2}} $ the centrifugal forces create a hole around the center with strongly depleted density. For $ \Omega \ll \eps^{-2}|\log\eps|^{-1} $ vorticity is still uniformly distributed in an annulus containing the bulk of the density, but at $ \Omega = \Omega_{\mathrm{c_3}} \sim \eps^{-2}|\log\eps|^{-1} $ there is a transition to a giant vortex state where the vorticity disappears from the bulk. The energy is then well approximated by a trial function that is an eigenfunction of angular momentum but one of our results is that the true minimizers break rotational symmetry in the whole parameter range, including the giant vortex phase.