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We investigate both analytically and numerically the propagation dynamic of on-axis and off-axis cosine-Gaussian (CG) beams in a linear medium with quadratic external potential. CG beam propagation evolves periodically with a period depended on the potential depth (α) and whether the beam shape is symmetrical with respect to optical axis. In each period, the CG beam first splits into two sub-beams with different accelerated direction; they then reverse the accelerated direction owing to the quadratic external potential and finally merge
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