论文标题
在具有两体和（临界）有吸引力的三体相互作用的一维玻色气体上
On one-dimensional Bose gases with two- and (critical) attractive three-body interactions
论文作者
论文摘要
我们考虑了一种一维，被困的，聚焦的bose气，其中$ n $ bosons通过两种形式的两体相互作用的潜在互动$ a n^{α-1} u（n^α（x-y））$ and $ -b n^{2β-2} w（n^{2β-2）w（n^2β-2} w（n^^β（x-n^β（x-y，x-y，x-y，x-y，x-y，x-y）， $ a \ in \ in \ mathbb {r} $，$ b，α> 0 $，$ 0 <β<1 $，$ u，$ u，w \ geq 0 $，和$ \ int _ {\ intbb {r}} u（x）u（x） w（x，y）\ mathop {} \！\ mathrm {d} x \ mathop {} \！\ mathrm {d} y $。该系统对于任何$ a \ in \ Mathbb {r} $都可以稳定，只要$ b <\ mathfrak {b}：=3π^2/2 $（1d focusing Quintic nonlinearearearearearearearearequdinger方程的关键强度）或$ a \ geq 0 $ a \ geq 0 $ b = \ b = \ mathfrak $ a \ geq 0 $。在前一种情况下，在（0，\ mathfrak {b}）$中修复$ b \，我们证明，在均值场上限制了多体系统在Cubintic-Quintic NLS地面上显示了Bose $ \ Unicode {X2013} $ Einstein Condensation。当假设$ b = b_n \ nearRow \ mathfrak {b} $和$ a = a_n \ to 0 $ as $ n \ to \ to \ infty $时，前者的融合足够慢，不是后者的速度不快，我们证明系统的基础状态完全凝结在（独特的）nls nls nls nls equintic nls equintic nls equintic nls equintic nls。在后一种情况下，$ b = \ mathfrak {b} $固定，当固定$ a> 0 $时，我们获得了小$β$的多体能量的收敛。最后，我们分析当收敛$ b_n \附近\ mathfrak {b} $是“快”比足够慢的收敛$ 0 <a_n \ searrow 0 $时，我们分析了多体基础状态的行为。
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 $a N^{α-1} U(N^α(x-y))$ and an attractive three-body interaction potential of the form $-b N^{2β-2} W(N^β(x-y,x-z))$, where $a\in\mathbb{R}$, $b,α>0$, $0<β<1$, $U, W \geq 0$, 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 $b<\mathfrak{b} := 3π^2/2$ (the critical strength of the 1D focusing quintic nonlinear Schrödinger 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 $β$ 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 $0<a_N \searrow 0$.
