Energy estimates and symmetry breaking in attractive Bose–Einstein condensates with ring-shaped potentials

Abstract

This paper is concerned with the properties of L 2 -normalized minimizers of the Gross-Pitaevskii (GP) functional for a two-dimensional Bose-Einstein condensate with attractive interaction and ring-shaped potential. By establishing some delicate estimates on the least energy of the GP functional, we prove that symmetry breaking occurs for the minimizers of the GP functional as the interaction strength a > 0 approaches a critical value a * , each minimizer of the GP functional concentrates to a point on the circular bottom of the potential well and then is non-radially symmetric as a ր a * . However, when a > 0 is suitably small we prove that the minimizers of the GP functional are unique, and this unique minimizer is radially symmetric.

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

“…By (2.9), we can follow Lemma 4 in [10] to derive that there exists a positive constant K, independent of a, such that 10) where u a > 0 is any minimizer of e(a). Applying (2.9) and (2.10), a proof similar to that of Theorem 2.1 in [12] then gives that there exist two positive constants m < M , independent of a, such that…”

“…Since the proof of Proposition 2.1 is similar to those in [10,11,12], which handle (1.1) with different potentials V (x), we shall briefly sketch the structure of the proof.…”

“…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.…”

“…In Section 4 we shall extend the refined spike behavior of Theorem 1.2 to more general potentials V (x) = g(x)h(x), where h(−x) = h(x) is homogeneous and satisfies (1.14) and 0 ≤ C ≤ g(x) ≤ 1 C holds in R 2 , see Theorem 4.4 for details. To establish Theorem 1.2 and Theorem 4.4, our Proposition 2.1 shows that the arguments of [10,11,12] give the leading expansion terms of the minimizer u a and the associated Lagrange multiplier µ a satisfying (1.6) as well. In order to get (1.17) for the rest terms of u a , the difficulty is to obtain the more precise estimate of µ a , which is overcome by the very delicate analysis of the associated equation (1.6), together with the constraint condition of u a .…”

“…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.…”