In this paper, we study the Schr\"{o}dinger equation in the semiclassical regime and with multiscale potential function. We develop the so-called constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM), in the framework of Crank-Nicolson (CN) discretization in time. The localized multiscale basis functions are constructed by addressing the spectral problem and a constrained energy minimization problem related to the Hamiltonian norm. A first-order convergence in the energy norm and second-order convergence in the $L^2$ norm for our numerical scheme are shown, with a relation between oversampling number in the CEM-GMsFEM method, spatial mesh size and the semiclassical parameter provided. Furthermore, we demonstrate the convergence of the proposed Crank-Nicolson CEM-GMsFEM scheme with $H/\sqrt{\Lambda}$ sufficiently small (where $H$ represents the coarse size and $\Lambda$ is the minimal eigenvalue associated with the eigenvector not included in the auxiliary space). Our error bound remains uniform with respect to $\varepsilon$ (where $0 < \varepsilon\ll 1$ is the Planck constant). Several numerical examples including 1D and 2D in space, with high-contrast potential are conducted to demonstrate the efficiency and accuracy of our proposed scheme.

翻译：本文研究了半经典区域下具有多尺度势函数的薛定谔方程。我们在时间上采用Crank-Nicolson（CN）离散格式的框架内，发展了所谓的约束能量最小化广义多尺度有限元方法（CEM-GMsFEM）。通过求解与哈密顿范数相关的谱问题及约束能量最小化问题，构造了局部化的多尺度基函数。我们证明了该数值格式在能量范数上具有一阶收敛性，在$L^2$范数上具有二阶收敛性，并给出了CEM-GMsFEM方法中的过采样数、空间网格尺寸与半经典参数之间的关系。此外，我们证明了当$H/\sqrt{\Lambda}$足够小时（其中$H$表示粗网格尺寸，$\Lambda$表示未包含在辅助空间中的特征向量所对应的最小特征值），所提出的Crank-Nicolson CEM-GMsFEM格式具有收敛性。我们的误差界对$\varepsilon$保持一致（其中$0 < \varepsilon\ll 1$为普朗克常数）。通过多个包含一维和二维空间的高对比度势场数值算例，验证了所提方案的高效性与精确性。