Complex Scaling for the Helmholtz Equation with Dirichlet Boundary Conditions in a Perturbed Half-Space

Abstract

SIAM Journal on Numerical Analysis, Volume 64, Issue 2, Page 510-536, April 2026.
Abstract. We present a new analysis of complex scaling applied to 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 fields 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. No other modifications of the domain or the governing equations are introduced. We illustrate the performance of the scheme with two and three dimensional examples and discuss its extension to other boundary conditions and open waveguides.