Non-existence of vortices in the small density region of a condensate ✩

In this paper, we answer a question raised by Lev Pitaevskii and prove that the ground state of the Gross– Pitaevskii energy describing a Bose–Einstein condensate in a rotationally symmetric trap at low rotation does not have vortices in the low density region. Therefore, the first ground state with vortices has its vortices in the bulk. In fact we prove something stronger, which is that the ground state for the model at low and moderate rotations is equal to the ground state in a condensate with no rotation. This is obtained by proving that for small rotational velocities, the ground state is multiple of the ground state with zero rotation. We rely on sharp bounds of the decay of the wave function combined with weighted Jacobian estimates. © 2010 Elsevier Inc. All rights reserved