数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (1): 409-438.doi: 10.1007/s10473-023-0123-6

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LIMIT BEHAVIOR OF GROUND STATES OF 2D BINARY BECS IN STEEP POTENTIAL WELLS*

Yuzhen kong1,2, Zhiyuan cui1, Dun zhao1,†   

  1. 1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China;
    2. College of Medical Informatics, Chongqing Medical University, Chongqing 400016, China
  • 收稿日期:2021-04-14 修回日期:2022-07-02 发布日期:2023-03-01
  • 通讯作者: †Dun ZHAO. E-mail: zhaod@lzu.edu.cn
  • 基金资助:
    *NSFC (12075102 and 11971212) and the Fundamental Research Funds for the Central Universities (lzujbky-2020-pd01).

LIMIT BEHAVIOR OF GROUND STATES OF 2D BINARY BECS IN STEEP POTENTIAL WELLS*

Yuzhen kong1,2, Zhiyuan cui1, Dun zhao1,†   

  1. 1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China;
    2. College of Medical Informatics, Chongqing Medical University, Chongqing 400016, China
  • Received:2021-04-14 Revised:2022-07-02 Published:2023-03-01
  • Contact: †Dun ZHAO. E-mail: zhaod@lzu.edu.cn
  • About author:Yuzhen kong,E-mail: 103186@cqmu.edu.cn;Zhiyuan cui,E-mail: 623821458@qq.com
  • Supported by:
    *NSFC (12075102 and 11971212) and the Fundamental Research Funds for the Central Universities (lzujbky-2020-pd01).

摘要: We study the ground states of attractive binary Bose-Einstein condensates with $N$ particles, which are trapped in the steep potential wells $\lambda V(x)$ in $\mathbb{R}^2$. We show that there exists a positive number $N_{*}$ such that if $N>N_{*}$, the system admits no ground state for any $\lambda>0$. Moreover, there exist two positive numbers, $M_{*}$ and $\lambda^*(N)$, such that if $N<M_{*}$, then for any $\lambda>\lambda^*(N)$, the system admits at least one ground state. As $\lambda\rightarrow\infty$, for any fixed $N<M_{*}$, we give a detailed description for the limit behavior of both positive and semi-trivial ground states.

关键词: ground state, binary Bose-Einstein condensate, steep potential well, limit behavior

Abstract: We study the ground states of attractive binary Bose-Einstein condensates with $N$ particles, which are trapped in the steep potential wells $\lambda V(x)$ in $\mathbb{R}^2$. We show that there exists a positive number $N_{*}$ such that if $N>N_{*}$, the system admits no ground state for any $\lambda>0$. Moreover, there exist two positive numbers, $M_{*}$ and $\lambda^*(N)$, such that if $N<M_{*}$, then for any $\lambda>\lambda^*(N)$, the system admits at least one ground state. As $\lambda\rightarrow\infty$, for any fixed $N<M_{*}$, we give a detailed description for the limit behavior of both positive and semi-trivial ground states.

Key words: ground state, binary Bose-Einstein condensate, steep potential well, limit behavior