In this work, we solve time, space and time-space fractional Schrödinger equations based on the non-singular Caputo–Fabrizio derivative definition for 1D infinite-potential well problem. To achieve this, we first work out the fractional differential equations defined in terms of Caputo–Fabrizio derivative. Then, the eigenvalues and the eigenfunctions of the three kinds of fractional Schrödinger equations are deduced. In contrast to Laskin’s results which are based on Riesz derivative, both the obtained wave number and wave function are different from the standard ones. Moreover, the number of solutions is finite and dependent on the space derivative order. When the fractional orders of derivatives become integer numbers (one for time derivative or/and two for space), our findings collapse to the standard results.

中文翻译：

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无奇异核的 Caputo-Fabrizio 导数在分数阶薛定谔方程中的应用

在这项工作中，我们基于一维无限势阱问题的非奇异 Caputo-Fabrizio 导数定义来求解时间、空间和时空分数阶薛定谔方程。为了实现这一点，我们首先计算出根据 Caputo-Fabrizio 导数定义的分数阶微分方程。然后，推导了三种分数阶薛定谔方程的特征值和特征函数。与 Laskin 基于 Riesz 导数的结果相比，得到的波数和波函数都与标准的不同。此外，解的数量是有限的并且取决于空间导数阶。当导数的分数阶数变为整数（一个用于时间导数或/和两个用于空间）时，我们的研究结果将崩溃为标准结果。