تقرير
On one-dimensional Bose gases with two- and (critical) attractive three-body interactions
| العنوان: | On one-dimensional Bose gases with two- and (critical) attractive three-body interactions |
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| المؤلفون: | Nguyen, Dinh-Thi, Ricaud, Julien |
| المصدر: | SIAM Journal on Mathematical Analysis (2024+) |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics Condensed Matter Mathematical Physics |
| مصطلحات موضوعية: | Mathematical Physics, Condensed Matter - Quantum Gases, 81V70, 35J10, 35Q55, 82B10, 82D05 |
| الوصف: | We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons interact with each other via both a two-body interaction potential of the form $a N^{\alpha-1} U(N^\alpha(x-y))$ and an attractive three-body interaction potential of the form $-b N^{2\beta-2} W(N^\beta(x-y,x-z))$, where $a\in\mathbb{R}$, $b,\alpha>0$, $0<\beta<1$, $U, W \geq 0$, and $\int_{\mathbb{R}}U(x) \mathop{}\!\mathrm{d}x = 1 = \iint_{\mathbb{R}^2} W(x,y) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y$. The system is stable either for any $a\in\mathbb{R}$ as long as $b<\mathfrak{b} := 3\pi^2/2$ (the critical strength of the 1D focusing quintic nonlinear Schr\"odinger equation) or for $a \geq 0$ when $b=\mathfrak{b}$. In the former case, fixing $b \in (0,\mathfrak{b})$, we prove that in the mean-field limit the many-body system exhibits the Bose$\unicode{x2013}$Einstein condensation on the cubic-quintic NLS ground states. When assuming $b=b_N \nearrow \mathfrak{b}$ and $a=a_N \to 0$ as $N \to\infty$, with the former convergence being slow enough and "not faster" than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case $b=\mathfrak{b}$ fixed, we obtain the convergence of many-body energy for small $\beta$ when $a > 0$ is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is "faster" than the slow enough convergence $0 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2210.04515 |
| رقم الأكسشن: | edsarx.2210.04515 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |