Controlled time-decaying harmonic potentials decelerate the velocity of the charged particle but the particle never be trapped by this harmonic potentials. This physical phenomena changes threshold between the short range class of potential and long-range class of potential in the sense of the existence of physical wave operators. In this paper, we reveal such a threshold is $1/(1-\lambda)$ for some $0\leq \lambda<1/2$ , which is determined by the mass of the particle and a coefficient of harmonic potential.

中文翻译：

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时间衰减谐波振荡器的波算符的存在与不存在

受控的时间衰减谐波电位会降低带电粒子的速度，但粒子永远不会被这种谐波电位捕获。这种物理现象在物理波算符存在的意义上改变了短程势类和长程势类之间的阈值。在本文中，我们揭示了对于某些 $0\leq \lambda<1/2$ 的阈值是 $1/(1-\lambda)$，该阈值由粒子的质量和谐波势系数决定。