Based on the second-order cosine spectral differentiation matrix deduced in this paper, we present an efficient and high accurate numerical scheme for solving two-dimensional Schrödinger equation with Neumann boundary conditions. The Crank–Nicolson finite difference scheme is utilized in temporal discretization and the cosine spectral-collocation method is employed for the approximation of Laplacian operator. The new scheme conserves the mass and energy in discrete level and is implemented by a fast algorithm in terms of the relations between the cosine spectral differentiation matrix and fast cosine transform. More importantly, this strategy can be extended to address the problems with high (even)-order derivatives in space. Numerical examples and applications are listed to confirm the validity and high accuracy of the method.

基于本文推导的二阶余弦谱微分矩阵，我们提出了一种有效和高精度的数值方案，用于求解带有Neumann边界条件的二维Schrödinger方程。Crank–Nicolson有限差分方案用于时间离散化，余弦谱配置方法用于拉普拉斯算子的逼近。该新方案在离散级上节省了质量和能量，并且根据余弦谱微分矩阵与快速余弦变换之间的关系，通过一种快速算法来实现。更重要的是，可以扩展该策略以解决空间中具有高（偶数）阶导数的问题。数值例子和应用被列出来确认该方法的有效性和高精度。