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arXiv:2311.12586

Axial Symmetry and Refined Spike Profiles of Ground States for Rotating Two-Component Bose Gases

Yongshuai Gao, Luo, Yong

Published Nov 21, 2023• 0 citations
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Abstract

This paper considers ground states of two-component Bose gases confined in an anharmonic trap rotating at the velocity $0<Ω<\infty$, where the intraspecies interaction $(-a_1,-a_2)$ and the interspecies interaction $-β$ are both attractive, i.e, $a_1, a_2$ and $β$ are all positive. We prove the axially symmetry and the refined spike profiles of ground states as $β\nearrowβ^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)}$, where $00$ is the unique positive solution of $-Δu+u-u^3=0$ in $\mathbb{R}^2$.

DOI
10.48550/arxiv.2311.12586

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As a continuation of our prior article, we study new pattern formations of ground states (u1,u2)(u_1,u_2)(u1​,u2​) for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in R2\mathbb {R}^2R2, where the intraspecies interaction (−a,−b)(-a,-b)(−a,−b) and the interspecies interaction −β-\beta−β are both attractive, i.e., aaa, bbb, and β\betaβ are all positive. If 0<b<a∗:=∥w∥220<b<a^*:=\|w\|^2_20<b<a∗:=∥w∥22​ and 0<β<a∗0<\beta <a^*0<β<a∗ are fixed, where www is the unique positive solution of Δw−w+w3=0\Delta w-w+w^3=0Δw−w+w3=0 in R2\mathbb {R}^2R2, the semi-trivial behavior of (u1,u2)(u_1,u_2)(u1​,u2​) as a↗a∗a\nearrow a^*a↗a∗ is proved in the sense that u1u_1u1​ concentrates at a unique point and while u2≡0u_2\equiv 0u2​≡0 in R2\mathbb {R}^2R2. However, if 0<b<a∗0<b<a^*0<b<a∗ and a∗≤β<β∗=a∗+(a∗−a)(a∗−b)a^*\le \beta <\beta ^*=a^*+\sqrt {(a^*-a)(a^*-b)}a∗≤β<β∗=a∗+(a∗−a)(a∗−b)​, the refined spike profile and the uniqueness of (u1,u2)(u_1,u_2)(u1​,u2​) as a↗a∗a\nearrow a^*a↗a∗ are analyzed, where (u1,u2)(u_1,u_2)(u1​,u2​) must be unique, u1u_1u1​ concentrates at a unique point, and meanwhile u2u_2u2​ can either blow up or vanish, depending on how β\betaβ approaches a∗a^*a∗.
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We study the rotational properties of an attractively interacting Bose gas in an anharmonic potential. Low-energy excitations for the two possible rotational ground-state configurations (vortex and the center-of-mass rotating state) are analyzed. The vortex excitation spectrum is all positive for weak couplings, but as the interactions become stronger, the energy of the lowest mode decreases rapidly to a negative value. The broken rotational symmetry involved in the center-of-mass rotating state induces the appearance of an extra zero-energy mode in the Bogoliubov spectrum. The excitations of the center-of-mass rotational state also demonstrate the coupling between the center of mass and relative motions.