We present a new complexification scheme based on the classical double layer potential for the solution of the Helmholtz equation with Dirichlet boundary conditions in compactly perturbed half-spaces in two and three dimensions. The kernel for the double layer potential is the normal derivative of the free-space Green's function, which has a well-known analytic continuation into the complex plane as a function of both target and source locations. Here, we prove that - when the incident data are analytic and satisfy a precise asymptotic estimate - the solution to the boundary integral equation itself admits an analytic continuation into specific regions of the complex plane, and satisfies a related asymptotic estimate (this class of data includes both plane waves and the field induced by point sources). We then show that, with a carefully chosen contour deformation, the oscillatory integrals are converted to exponentially decaying integrals, effectively reducing the infinite domain to a domain of finite size. Our scheme is different from existing methods that use complex coordinate transformations, such as perfectly matched layers, or absorbing regions, such as the gradual complexification of the governing wavenumber. More precisely, in our method, we are still solving a boundary integral equation, albeit on a truncated, complexified version of the original boundary. In other words, no volumetric/domain modifications are introduced. The scheme can be extended to other boundary conditions, to open wave guides and to layered media. We illustrate the performance of the scheme with two and three dimensional examples.

翻译：针对二维和三维紧致扰动半空间中Dirichlet边界条件下的Helmholtz方程求解，我们提出了一种基于经典双层势的新型复化方案。该双层势的核是自由空间格林函数关于法向导数，作为目标点与源点位置的函数，其具有众所周知的复平面解析延拓性质。本文证明：当入射数据解析且满足精确的渐近估计时，边界积分方程的解本身允许向复平面特定区域进行解析延拓，并满足相应的渐近估计（此类数据包括平面波和点源诱导场）。进一步研究表明，通过精心选择的围道变形，振荡积分可转化为指数衰减积分，从而将无限域有效约化为有限尺寸区域。本方案不同于采用复坐标变换的现有方法（如完全匹配层），也不同于采用吸收区域的方法（如控制波数的渐进复化）。更准确地说，本方法仍在求解边界积分方程，尽管是在原始边界经截断复化后的版本上进行。换言之，未引入任何体积/区域修改。该方案可推广至其他边界条件、开放波导及分层介质。我们通过二维和三维算例展示了该方案的性能。