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 $\mu \in [\mu _1, \mu _2]$.