On a method for computing waveguide scattering matrices

Abstract:

A waveguide lies in $\mathbb R^2$ and, outside a large disk, coincides with the union of finitely many nonoverlapping semistrips (“cylindrical ends”). It is described by a Dirichlet problem for the Helmholtz equation. As an approximation for a row of the scattering matrix $S(\mu )$, where $\mu$ is the spectral parameter, the minimizer of a quadratic functional $J^R(\cdot , \mu )$ is used. To construct this functional, an auxiliary boundary value problem is solved in the bounded domain obtained by truncating the cylindrical ends of the waveguide at a distance $R$. As $R\to \infty$, the minimizer $a (R, \mu )$ tends with exponential rate to the corresponding row of the scattering matrix uniformly on every finite closed interval of the continuous spectrum containing no thresholds. Such an interval may contain eigenvalues of the waveguide (with eigenfunctions exponentially decaying at infinity). The applicability of this method goes far beyond the simplest model considered in the paper.