In this paper, the Crank-Nicolson Fourier spectral method is proposed for solving the space fractional Schrödinger equation with wave operators. The equation is treated with the conserved Crank-Nicolson Fourier Galerkin method and the conserved Crank-Nicolson Fourier collocation method, respectively. In addition, the ability of the constructed numerical method to maintain the conservation of mass and energy is studied in detail. Meanwhile, the convergence with spectral accuracy in space and second-order accuracy in time is verified for both Galerkin and collocation approximations. Finally, the numerical experiments verify the properties of the conservative difference scheme and demonstrate the correctness of theoretical results.

中文翻译：

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带波动算子的空间分数阶薛定谔方程的一种保守的 Crank-Nicolson Fourier 谱方法

本文提出了用波算子求解空间分数阶薛定谔方程的Crank-Nicolson Fourier谱法。该方程分别用守恒的 Crank-Nicolson Fourier Galerkin 方法和守恒的 Crank-Nicolson Fourier 搭配方法处理。此外，详细研究了所构造的数值方法保持质量和能量守恒的能力。同时，验证了伽辽金和搭配近似的空间光谱精度和时间二阶精度的收敛性。最后，数值实验验证了保守差分格式的性质，证明了理论结果的正确性。