On a Method for Computing Waveguide Scattering Matrices in the Presence of Point Spectrum

A waveguide occupies a domain $G$ in $\mathbb R^{n+1}$, $n\ge 1$, having several cylindrical outlets to infinity. The waveguide is described by a general elliptic boundary value problem that is self-adjoint with respect to the Green formula and contains a spectral parameter $\mu$. As an approximation to a row of the scattering matrix $S(\mu)$ we suggest a minimizer of a quadratic functional $J^R(\,\cdot\,,\mu)$. To construct such a functional, we solve an auxiliary boundary value problem in the bounded domain obtained by cutting off, at a distance $R$, the waveguide outlets to infinity. It is proved that, if a finite interval $[\mu_1,\mu_2]$ of the continuous spectrum contains no thresholds, then, as $R\to\infty$, the minimizer tends to the row of the scattering matrix at an exponential rate uniformly with respect to $\mu\in[\mu_1,\mu_2]$. The interval may contain some waveguide eigenvalues whose eigenfunctions exponentially decay at infinity.