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

Guo, Yujin; Lin, Chang–Shou; Wei, Juncheng

doi:10.1137/16m1100290

Cited by 63 publications

(69 citation statements)

References 26 publications

(70 reference statements)

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“…We also investigate the borderline situation C(a, b) = 1. It differs markedly from the purely non-dipolar situation thoroughly investigated in [18] and following papers [17,19,25] (see also [2,28,15,32] for related topics), for then the next order of the original, physical, 2D interaction matters in the high-frequency limit.…”

Section: Introductionmentioning

confidence: 83%

On the Stability of 2D Dipolar Bose--Einstein Condensates

Eychenne¹,

Rougerie

2019

SIAM J. Math. Anal.

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We study the existence of energy minimizers for a Bose-Einstein condensate with dipole-dipole interactions, tightly confined to a plane. The problem is critical in that the kinetic energy and the (partially attractive) interaction energy behave the same under mass-preserving scalings of the wave-function. We obtain a sharp criterion for the existence of ground states, involving the optimal constant of a certain generalized Gagliardo-Nirenberg inequality.

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Section: Introductionmentioning

confidence: 83%

On the Stability of 2D Dipolar Bose--Einstein Condensates

Eychenne¹,

Rougerie

2019

SIAM J. Math. Anal.

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“…Secondly, since the existing argument only gives that ε k u 2k (ε k x + x 1k ) → 0 uniformly in R 2 as k → ∞, one needs to investigate an approach of addressing that u 2k ≡ 0 for sufficiently large k > 0. As shown in Lemma 3.3, we shall achieve this purpose by analyzing the more refined energy estimates of e(a k , b, β k ), for which we make full use of the refined spike profiles proved in [13,Theorem 1.2]. Once u 2k ≡ 0 holds for sufficiently large k > 0, we definē…”

Section: Semi-trivial Limit Behavior Of Bec Systemsmentioning

confidence: 99%

Ground states of two-component attractive Bose-Einstein condensates II: Semi-trivial limit behavior

Guo

Wei

et al. 2019

Trans. Amer. Math. Soc.

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As a continuation of [14], we study new pattern formations of ground states (u 1 , u 2 ) for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in R 2 , where the intraspecies interaction (−a, −b) and the interspecies interaction −β are both attractive, i.e, a, b and β are all positive. If 0 < b < a * := w 2 2 and 0 < β < a * are fixed, where w is the unique positive solution of ∆w − w + w 3 = 0 in R 2 , the semi-trivial behavior of (u 1 , u 2 ) as a ր a * is proved in the sense that u 1 concentrates at a unique point and while u 2 ≡ 0 in R 2 . However, if 0 < b < a * and a * ≤ β < β * = a * + (a * − a)(a * − b), the refined spike profile and the uniqueness of (u 1 , u 2 ) as a ր a * are analyzed, where (u 1 , u 2 ) must be unique, u 1 concentrates at a unique point, and meanwhile u 2 can either blow up or vanish, depending on how β approaches to a * .

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“…The variational problem e(N ) is essentially in the class of L 2 −critical constraint minimization problems, which were studied recently in the nonrelativistic cases, e.g. [5,19,20,22] and references therein. Comparing with these mentioned works, it however deserves to remark that the analysis of e(N ) is more complicated in a substantial way, which is mainly due to the nonlocal nature of the pseudo-differential operator √ −∆ + m 2 , and the convolution-type nonlinearity as well.…”

Section: Introductionmentioning

confidence: 99%

“…where Q ∞ > 0 is the unique positive minimizer of (2.1) described below. Based on the refined estimates of Section 2, motivated by [9,18,19], we shall employ the nondegenerancy and uniqueness of Q ∞ to complete the proof of Theorem 1.2 by establishing Pohozaev identities. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

The Lieb-Yau conjecture for ground states of pseudo-relativistic Boson stars

Guo

Zeng

2020

Journal of Functional Analysis

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It is known that ground states of the pseudo-relativistic Boson stars exist if and only if the stellar mass N > 0 satisfies N < N * , where the finite constant N * is called the critical stellar mass. Lieb and Yau conjecture in [Comm. Math. Phys., 1987] that ground states of the pseudo-relativistic Boson stars are unique for each N < N * . In this paper, we prove that the above uniqueness conjecture holds for the particular case where N > 0 is small enough.By making full use of (1.3), Lenzmann in [26] established the following interesting existence and analytical characters of minimizers for e(N ):Theorem A ([26, Theorem 1]) Under the assumption m > 0, the following results hold for e(N ):1. e(N ) has minimizers if and only if 0 < N < N * , where the finite constant N * is independent of m. Any minimizer3. Any nonnegative minimizer of e(N ) must be strictly positive and radially symmetricdecreasing, up to phase and translation.We remark that the existence of the critical constant N * > 0 stated in Theorem A, which is called the critical stellar mass of boson stars, was proved earlier in [14,29]. Further, the dynamics and some other analytic properties of minimizers for e(N ) were also investigated by Lenzmann and his collaborators in [12,13,14,15,25,26] and references therein. Stimulated by [20,22], the related limit behavior of minimizers for e(N ) as N ր N * were studied more recently in [23,33,37], where the Gagliardo-Nirenberg type inequality (1.3) also played an important role. Note also from the variational theory that any minimizer u of e(N ) satisfies the Euler-Lagrange equation −∆ + m 2 − m u − 1 |x| * |u| 2 u = −µu in R 3 , (1.5)

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

Cited by 63 publications

References 26 publications

On the Stability of 2D Dipolar Bose--Einstein Condensates

On the Stability of 2D Dipolar Bose--Einstein Condensates

Ground states of two-component attractive Bose-Einstein condensates II: Semi-trivial limit behavior

The Lieb-Yau conjecture for ground states of pseudo-relativistic Boson stars

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