We consider a wide variety of scattering problems including scattering by Dirichlet, Neumann, and penetrable obstacles. We consider a radial perfectly-matched layer (PML) and show that for any PML width and a steep-enough scaling angle, the PML solution is exponentially close, both in frequency and the tangent of the scaling angle, to the true scattering solution. Moreover, for a fixed scaling angle and large enough PML width, the PML solution is exponentially close to the true scattering solution in both frequency and the PML width. In fact, the exponential bound holds with rate of decay $c(w\tan\theta -C) k$ where $w$ is the PML width and $\theta$ is the scaling angle. More generally, the results of the paper hold in the framework of black-box scattering under the assumption of an exponential bound on the norm of the cutoff resolvent, thus including problems with strong trapping. These are the first results on the exponential accuracy of PML at high-frequency with non-trivial scatterers.

翻译：我们考虑了一系列广泛散射问题，包括Dirichlet、Neumann以及可穿透障碍物的散射。研究采用径向完美匹配层（PML），并证明：对于任意PML宽度和足够大的缩放角度，PML解在频率和缩放角正切值两个维度上均指数逼近真实散射解。此外，当固定缩放角度且PML宽度足够大时，PML解在频率和宽度两个维度上同样指数逼近真实散射解。事实上，该指数界服从衰减率$c(w\tan\theta -C) k$，其中$w$为PML宽度，$\theta$为缩放角度。更一般地，本文结果适用于黑箱散射框架——在截断预解式范数指数有界的假设下，可涵盖强束缚问题。这是首个针对非平凡散射体在高频条件下证明PML指数精度的研究成果。