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arXiv 0912.0418math-phmath.MP
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三体系统的零能基态

Zero Energy Ground State in the Three-Body System

Dmitry K. Gridnev

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摘要：

考虑$\mathbb{R}^3$中具有非正势和非负本质谱的三体系统.在对势下降的一定条件下，证明了如果至少有一对粒子具有零能量共振，则不存在三粒子的平方可积零能量基态。这补充了\cite{1}中的分析，其中证明了平方可积零能基态是可能的，因为在所有两体子系统中没有负能量束缚态和零能共振。作为推论，证明了可以调节对势的耦合常数，使得对于任何给定的$R, ε>0$：（a）本质谱的底部为零;（b）存在负能量基态$ψ(ξ)$，其中$\int |ψ(ξ)|^2 = 1$ ;（c）$\int_{|ξ| \leq R} |ψ(ξ)|^2 < ε$。

英文摘要：

We consider a 3--body system in $\mathbb{R}^3$ with non--positive potentials and non--negative essential spectrum. Under certain requirements on the fall off of pair potentials it is proved that if at least one pair of particles has a zero energy resonance then a square integrable zero energy ground state of three particles does not exist. This complements the analysis in \cite{1}, where it was demonstrated that square integrable zero energy ground states are possible given that in all two--body subsystems there is no negative energy bound states and no zero energy resonances. As a corollary it is proved that one can tune the coupling constants of pair potentials so that for any given $R, ε>0$: (a) the bottom of the essential spectrum is at zero; (b) there is a negative energy ground state $ψ(ξ)$, where $\int |ψ(ξ)|^2 = 1$; (c) $\int_{|ξ| \leq R} |ψ(ξ)|^2 < ε$.

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