The fractional Schr\"{o}dinger equation (FSE) on the real line arises in a broad range of physical settings and their numerical simulation is challenging due to the nonlocal nature and the power law decay of the solution at infinity. In this paper, we propose a new spectral discretization scheme for the FSE in space based upon Malmquist-Takenaka functions. We show that this new discretization scheme achieves much better performance than existing discretization schemes in the case where the underlying FSE involves the square root of the Laplacian, while in other cases it also exhibits comparable or even better performance. Numerical experiments are provided to illustrate the effectiveness of the proposed method.

翻译：实际线上的分数 Schr\"{o}dinger 等式(FSE) 出现在广泛的物理环境中,由于无穷无尽的解决方案的非本地性质和动力法的衰落,它们的数值模拟具有挑战性。在本文中,我们提议根据马尔姆奎斯特- 竹纳卡的功能为空间的FSE制定一个新的光谱分解计划。我们表明,在基础FSE涉及拉普拉西亚的平方根的情况下,这种新的分解计划比现有的离散计划取得更好的效果,而在另一些情况下,它也显示出可比较或甚至更好的效果。我们提供了数字实验,以说明拟议方法的有效性。