Method for computing waveguide scattering matrices in the vicinity of thresholds

Abstract

A waveguide occupies a domain $ G$ in $ \mathbb{R}^{n+1}$, $ n\geq 1$, having several cylindrical outlets to infinity. The waveguide is described by the Dirichlet problem for the Helmholtz equation. The scattering matrix $ S(\mu )$ with spectral parameter $ \mu $ changes its size when $ \mu $ crosses a threshold. To calculate $ S(\mu )$ in a neighborhood of a threshold, an ``augmented'' scattering matrix $ \mathcal {S} (\mu )$ is introduced, which keeps its size near the threshold and is analytic in $ \mu $ there. A minimizer of a quadratic functional $ J^R(\,\cdot \,, \mu )$ serves as an approximation to a row of the matrix $ \mathcal {S}(\mu )$. To construct such a functional, an auxiliary boundary-value problem is solved in the bounded domain obtained by cutting off the waveguide outlets to infinity at a distance $ R$. As $ R\to \infty $, the minimizer $ a (R, \mu )$ tends exponentially to the corresponding row of $ \mathcal {S}(\mu )$ uniformly with respect to $ \mu $ in a neighborhood of the threshold. The neighborhood may contain some waveguide eigenvalues corresponding to eigenfunctions exponentially decaying at infinity. Finally, the elements of the ``ordinary'' scattering matrix $ S(\mu )$ are expressed in terms of those of the augmented matrix $ \mathcal {S}(\mu )$.

If an interval $ [\mu _1, \mu _2]$ of the continuous spectrum contains no thresholds, the corresponding functional $ J^R(\,\cdot \,, \mu )$ should be defined for the usual matrix $ S(\mu )$ and, as $ R\to \infty $, its minimizer $ a (R, \mu )$ tends to the row of the scattering matrix at exponential rate uniformly with respect to $ \mu \in [\mu _1, \mu _2]$.