We study the lattice Green's function (LGF) of the screened Poisson equation on a two-dimensional rectangular lattice. This LGF arises in numerical analysis, random walks, solid-state physics, and other fields. Its defining characteristic is the screening term, which defines different regimes. When its coefficient is large, we can accurately approximate the LGF with an exponentially converging asymptotic expansion, and its convergence rate monotonically increases with the coefficient of the screening term. To tabulate the LGF when the coefficient is not large, we derive a one-dimensional integral representation of the LGF. We show that the trapezoidal rule can approximate this integral with exponential convergence, and we propose an efficient algorithm for its evaluation via the Fast Fourier Transform. We discuss applications including computing the LGF of the three-dimensional Poisson equation with one periodic direction and the return probability of a two-dimensional random walk with killing.

翻译：我们研究了二维矩形晶格上屏蔽泊松方程的晶格格林函数（LGF）。该LGF在数值分析、随机游走、固态物理等领域均有应用。其核心特征在于屏蔽项定义了不同的物理区间。当屏蔽项系数较大时，可使用指数收敛的渐近展开精确近似LGF，且收敛速率随屏蔽项系数单调递增。针对系数较小时的LGF制表需求，我们推导了该LGF的一维积分表示。研究表明，梯形法则可实现该积分的指数收敛近似，并提出基于快速傅里叶变换的高效评估算法。最后讨论了包括计算含一个周期方向的三维泊松方程LGF以及具有灭杀效应的二维随机游走返回概率在内的应用场景。