We propose a novel framework, called moving window method, for solving the linear Schr\"odinger equation with an external potential in $\mathbb{R}^d$. This method employs a smooth cut-off function to truncate the equation from Cauchy boundary conditions in the whole space to a bounded window of scaled torus, which is itself moving with the solution. This allows for the application of established schemes on this scaled torus to design algorithms for the whole-space problem. Rigorous analysis of the error in approximating the whole-space solution by numerical solutions on a bounded window is established. Additionally, analytical tools for periodic cases are used to rigorously estimate the error of these whole-space algorithms. By integrating the proposed framework with a classical first-order exponential integrator on the scaled torus, we demonstrate that the proposed scheme achieves first-order convergence in time and $\gamma/2$-order convergence in space for initial data in $H^\gamma(\mathbb{R}^d) \cap L^2(\mathbb{R}^d;|x|^{2\gamma} dx)$ with $\gamma \geq 2$. In the case where $\gamma = 1$, the numerical scheme is shown to have half-order convergence under an additional CFL condition. In practice, we can dynamically adjust the window when waves reach its boundary, allowing for continued computation beyond the initial window. Extensive numerical examples are presented to support the theoretical analysis and demonstrate the effectiveness of the proposed method.

翻译：我们提出了一种称为移动窗口法的新框架，用于求解 $\mathbb{R}^d$ 上含外部势函数的线性薛定谔方程。该方法采用光滑截断函数，将整个空间中具有柯西边界条件的方程截断至一个缩放环面上的有界窗口，该窗口本身随解一起移动。这使得可以在该缩放环面上应用成熟的数值格式，从而为全空间问题设计算法。我们严格分析了在有界窗口上通过数值解逼近全空间解所产生的误差。此外，利用周期情形的分析工具，我们严格估算了这些全空间算法的误差。通过将所提框架与缩放环面上的经典一阶指数积分器相结合，我们证明了对于 $\gamma \geq 2$ 且属于 $H^\gamma(\mathbb{R}^d) \cap L^2(\mathbb{R}^d;|x|^{2\gamma} dx)$ 空间的初始数据，所提格式在时间上达到一阶收敛，在空间上达到 $\gamma/2$ 阶收敛。当 $\gamma = 1$ 时，在附加的 CFL 条件下，数值格式被证明具有半阶收敛性。在实际计算中，当波前抵达窗口边界时，我们可以动态调整窗口，从而能够在初始窗口之外继续进行模拟。我们提供了大量的数值算例以支持理论分析，并证明了所提方法的有效性。