小速度下带调和外势的Boson Star方程的极限行为

Limit Behavior of the Boson Star Equation with a Harmonic External Potential at Small Velocity

摘要: 在本文中，我们考虑Boson star方程$i{\partial }_t\psi =\left(\sqrt{-\Delta +m^2}\right)\psi -\left(\frac{1}{\left|x\right|}\ast {\left|\psi \right|}^2\right)\psi +V\left(x\right)\psi$ 在${ℝ}^3$ 上。其中$\psi \left(t,x\right)=e^{it\mu }\phi \left(x-vt\right)$ 为行波孤立子，$v\in {ℝ}^3$ 表示速度。众所周知，Fröhlich、Jonsson和Lenzmann证明了当$\left|v\right|<1$ 时，存在一个临界常数$N_c\left(v\right)$ ，使得行波存在当且仅当$0<N<N_c\left(v\right)$ ，其中$N$ 表示粒子数。在本文中，我们考虑$v=\left(\beta ,0,0\right)$ (其中$0<\beta <1$ )，并令$N_c\left(\beta \right)={N_c\left(v\right)|}_{v=\left(\beta ,0,0\right)}$ 。基于这一事实，当$\beta \to 0^{+}$ 时，对于满足${\Vert {\phi }_{\beta }\Vert }_{L^2}^2=\left(1-\beta \right)N_c\left(\beta \right)$ 的推进基态${\phi }_{\beta }$ ，我们计算其极限行为。

Abstract: In this paper, we consider the Boson star equation $i{\partial }_t\psi =\left(\sqrt{-\Delta +m^2}\right)\psi -\left(\frac{1}{\left|x\right|}\ast {\left|\psi \right|}^2\right)\psi +V\left(x\right)\psi$ , on ${ℝ}^3$ where $\psi \left(t,x\right)=e^{it\mu }\phi \left(x-vt\right)$ refers to the travelling solitary waves, $v\in {ℝ}^3$ denotes the velocity. It is well known that Fröhlich, Jonsson, and Lenzmann proved that for $\left|v\right|<1$ , there exists a critical constant $N_c\left(v\right)$ such that travelling waves exist if and only if $0<N<N_c\left(v\right)$ , where $N$ denotes the particle number. In the present paper, we consider $v=\left(\beta ,0,0\right)$ (where $0<\beta <1$ ) and set $N_c\left(\beta \right)={N_c\left(v\right)|}_{v=\left(\beta ,0,0\right)}$ . Based on this fact, as $\beta \to 0^{+}$ , we compute the limit behavior of the boosted ground state ${\phi }_{\beta }$ satisfying ${\Vert {\phi }_{\beta }\Vert }_{L^2}^2=\left(1-\beta \right)N_c\left(\beta \right)$ .