Based on fractional Schrödinger equation (FSE), we have emphatically investigated the evolution of Airy-Gaussian (AiG) beams under Gaussian potential by split-step Fourier method. Only when beam is incident vertically can splitting occur. When taking Gaussian potentials into account, periodic evolution can be achieved by assigning appropriate values to barrier parameters, avoiding the transmission phenomenon. Transmission and reflection (TR) ratio varies with values of barrier width and height, and period changes accordingly. Critical barrier height increases with the larger Lévy index as a whole, while it eventually tends to be stable with the larger barrier width. In addition, compared with Gaussian wells, it is easier to achieve total reflection in Gaussian barriers.

基于分数薛定ding方程（FSE），我们通过分步傅立叶方法着重研究了高斯势下的艾里-高斯（AiG）光束的演化。仅当光束垂直入射时，才会发生分裂。当考虑高斯势能时，可以通过为势垒参数分配适当的值来实现周期性演化，从而避免传输现象。透射和反射（TR）比率随势垒宽度和高度的值而变化，并且周期也相应地变化。整体而言，临界势垒高度随Lévy指数的增大而增加，而最终随着势垒宽度的增大，临界势垒高度趋于稳定。另外，与高斯井相比，在高斯势垒中更容易实现全反射。