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

Abstract

In this paper, we answer a question raised by Lev Pitaevskii and prove that the ground state of the GrossPitaevskii 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.

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

“…These have been studied in various contexts (see for instance [8,10,65,125,163]). As we will see in this paper, some are actually far from optimal.…”

“…for some constant C > 0, provided ε is small (see [8,10,130], and Remark 18 herein). (We remark that the logarithmic term appears because (6), (7) imply that ∇ √ A + is not square-integrable near ∂D 0 .)…”

Resonance Phenomena in a Singular Perturbation Problem in the Case of Exchange of Stabilities

“…In these situations, the limit algebraic equation (the analog of (1.4)) typically undergoes a pitchfork or saddle-node bifurcation as the parameter y crosses a curve Γ. The case of pitchfork bifurcation has received a lot of attention recently, as it occurs when minimizing a Gross-Pitaevskii functional under the unit mass constraint (see [1], [2], [3], [19], and [25]). Due to the irregular nature of the singular limit (it does not belong in the Sobolev space H 1 ), standard weak convergence arguments are not applicable.…”

“…Typically, they can be shown outside an ε-dependent tubular neighborhood of the bifurcation curve Γ, by constructing suitable upper and lower solutions. Then, taking advantage of the radial symmetry, one shows that the solution is monotone in this tubular neighborhood and, therefore, is able to complete the estimate in the entire domain (see [2]). A class of slow-fast Hamiltonian systems, in which the slow manifold loses normal hyperbolicity due to a pitchfork bifurcation, arises in the study of crystalline grain boundaries, see [4].…”

“…We believe that the perturbation approach we introduce in the current paper can be useful in these problems, since it provides very accurate estimates up to the the bifurcation curve Γ, of the form (1.17) below. In turn, these are important since many interesting phenomena, such as the appearance of vortices (see [2]), or small peak solutions (see [13]), occur close to Γ.…”