On the Mass Concentration for Bose–Einstein Condensates with Attractive Interactions

Abstract

We consider two-dimensional attractive Bose-Einstein condensates in a trap V (x) rotating at the velocity Ω, which can be described by the complex-valued Gross-Pitaevskii energy functional. We prove that there exists a critical rotational velocity 0 < Ω * := Ω * (V ) ≤ ∞, depending on the general potential V (x), such that for any small rotational velocity 0 ≤ Ω < Ω * , minimizers exist if and only if the attractive interaction strength a satisfies a < a * = w 2 2 , where w is the unique positive radial solution of ∆w − w + w 3 = 0 in R 2 . For any 0 < Ω < Ω * , we also analyze the limit behavior of the minimizers as a ր a * , where all the mass concentrates at a global minimum point of VΩ(x) := V (x) − Ω 2 4 |x| 2 . Specially, if V (x) = |x| 2 and 0 < Ω < Ω * (= 2), we further prove that, up to the phase rotation, all minimizers are real-valued, unique and free of vortices as a ր a * .

Local Uniqueness and Refined Spike Profiles of Ground States for Two-Dimensional Attractive Bose-Einstein Condensates

“…In this paper positive minimizers of e(a) are called ground states of attractive BEC. Applying energy estimates and blow-up analysis, the spike profiles of positive minimizers for e(a) as a ր a * were recently discussed in [10,11,12] under different types of potentials V (x), see our Proposition 2.1 for some related results. In spite of these facts, it remains open to discuss the refined spike profiles of positive minimizers.…”

“…Bose-Einstein condensates (BEC) of a dilute gas with attractive interactions in R 2 can be described ( [2,5,10]) by the following Gross-Pitaevskii (GP) energy functional…”

“…As addressed recently in [10,11], ground states of attractive BEC in R 2 can be described by the constraint minimizers of the GP energy e(a) := inf {u∈H, u 2 2 =1}…”

“…The minimization problem e(a) was analyzed recently in [2,10,11,12,26] and references therein. Existing results show that e(a) is an L 2 −critical constraint variational problem.…”

“…Existing results show that e(a) is an L 2 −critical constraint variational problem. Actually, it was shown in [2,10] that e(a) admits minimizers if and only if a < a * := w 2 2 , where w = w(|x|) is the unique (up to translations) radial positive solution (cf. [7,19,14]) of the following nonlinear scalar field equation ∆w − w + w 3 = 0 in R 2 , where w ∈ H 1 (R 2 ) .…”