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：自由薛定谔方程在矩形计算域上的透明边界条件需要实现形如$\sqrt{\partial_t-i\triangle_{\Gamma}}$的算子，其中$\triangle_{\Gamma}$为拉普拉斯-贝尔特拉米算子。已知该算子在时间和空间上均具有非局部性，这给高效数值求解方法的研发带来了重大挑战。现有方法的计算复杂度随时间步数增长，这归因于边界算子的非局部特性。本文提出一种有效的边界算子局部近似方法，使得复杂度不依赖于时间步数。该算法的核心是基于Padé逼近的分式算子有理近似，能够恰当处理计算域角点。空间离散采用Legendre-Galerkin谱方法配合新型边界适配基函数，确保生成的线性系统具有带状结构。同时给出兼容的边界提升过程，可处理边界线段与角点。所提新方案可在任意单步时间推进框架下实现。我们特别针对两种单步法——一阶向后差分公式（BDF1）和梯形法则（TR）——进行演示。为便于比较，还给出基于卷积求积的兼容单步法方案，该方案计算成本较高但可作为黄金标准。最后通过多项数值测试验证新方法的有效性，并经验性地确认收敛阶。