This paper describes a new method to calculate the stationary scattering matrix and its derivatives for Euclidean waveguides. This is an adaptation and extension to a procedure developed by Levitin and Strohmaier which was used to compute the stationary scattering matrix on surfaces with hyperbolic cusps (Levitin and Strohmaier, 2019), but limited to those surfaces. At the time of writing, these procedures are the first and only means to explicitly compute such objects. In this context the challenge we faced was that on Euclidean waveguides, the scattering matrix naturally inhabits a Riemann surface with a countably infinite number of sheets making it more complicated to define and compute. We overcame this by breaking up the waveguide into compact and non-compact components, systematically describing the resolvent for the Neumann Laplace operator on both of them, giving a thorough treatment of the Riemann surface, and then using a “gluing” construction (Melrose, 1995) to define the resolvent on the whole surface. From the resolvent, we were able to obtain the scattering matrix. The algorithm we have developed to do this not only computes the scattering matrix itself on such domains, but also arbitrarily high derivatives of it directly. We have applied this, together with the finite element method, to calculate resonances for a selection of domains and will present the results of some numerical calculations in the final section. Whilst this is certainly not the first, nor only method to compute resonances on these domains, i.e. Levitin and Marletta have done so previously (Levitin and Marletta, 2008) and Aslanyan, Parnovski and Valiliev before them (Aslanyan et al., 2000) and other techniques, such as perfectly matched layers may be adapted for this purpose (Jiang and Xiang, 2020). The method described here has several advantages in terms of speed and accuracy and moreover, provides more information about the scattering phenomena.

本文介绍了一种计算欧几里得波导的固定散射矩阵及其导数的新方法。这是对Levitin和Strohmaier开发的程序的改编和扩展，该程序用于计算具有双曲线尖头的表面上的静态散射矩阵（Levitin和Strohmaier，2019），但仅限于这些表面。在撰写本文时，这些过程是显式计算此类对象的第一个也是唯一的方法。在这种情况下，我们面临的挑战是，在欧几里得波导上，散射矩阵自然地位于一个黎曼表面，该表面具有无限数量的薄片，这使定义和计算变得更加复杂。我们通过将波导分解为紧凑的和非紧凑的组件来克服了这一问题，系统地在两者上描述Neumann Laplace算子的分解物，对Riemann表面进行彻底处理，然后使用“胶合”构造（Melrose，1995）在整个表面上定义分解物。从分解物中，我们能够获得散射矩阵。我们为此目的开发的算法不仅可以在此类域上计算散射矩阵本身，还可以直接计算其任意高阶导数。我们将其与有限元方法一起用于计算选择域的共振，并将在最后一节中介绍一些数值计算的结果。虽然这当然不是在这些域上计算共振的第一个方法，也不是唯一的方法，例如，Levitin和Marletta以前已经这样做（Levitin和Marletta，2008年），但Aslanyan，为此，可以使用之前的Parnovski和Valiliev（Aslanyan等人，2000）以及其他技术，例如完美匹配的层（Jiang和Xiang，2020）。此处描述的方法在速度和准确性方面具有多个优点，此外，它还提供了有关散射现象的更多信息。