The transparent boundary condition for the free Schr\"{o}dinger equation on a rectangular computational domain requires implementation of an operator of the form $\sqrt{\partial_t-i\triangle_{\Gamma}}$ where $\triangle_{\Gamma}$ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Pad\'e approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.

翻译：针对矩形计算域上的自由薛定谔方程，透明边界条件需实现形式为 $\sqrt{\partial_t-i\triangle_{\Gamma}}$ 的算子，其中 $\triangle_{\Gamma}$ 为拉普拉斯-贝尔特拉米算子。该算子在时间与空间上均具有非局部性，这为发展高效数值求解方法带来了显著挑战。现有方法的计算复杂度随时间步数增长，这可归因于边界算子的非局部特性。本工作提出一种边界算子的有效局部近似方法，使得最终计算复杂度保持与时间步数无关。该算法的核心是基于帕德逼近的有理近似方法，对特定分数阶算子进行逼近，并能妥善处理计算域的角点问题。在空间离散方面，采用勒让德-伽辽金谱方法配合新构建的边界适应基函数，确保所得线性系统为带状结构。同时提出相容的边界提升程序，可同时处理边界线段与角点。所提出的新颖方案可在任何单步时间推进算法的框架内实施。特别地，我们以两种单步方法——一阶后向差分公式（BDF1）与梯形法则（TR）——具体演示这些思想。为便于比较，同时给出符合单步方法框架的卷积求积方案，该方案虽计算代价较高，但可作为基准参照。最后通过若干数值实验验证新方法的有效性，并实证检验收敛阶数。