TY - JOUR

T1 - Method for computing waveguide scattering matrices in the vicinity of thresholds

AU - Plamenevski, B.A.

AU - Poretski, A.S.

AU - Sarafanov, O.V.

PY - 2015

Y1 - 2015

N2 - © 2014 American Mathematical Society. A waveguide occupies a domain G in Rn+1, n ≥ 1, having several cylindrical outlets to infinity. The waveguide is described by the Dirichlet problem for the Helmholtz equation. The scattering matrix S(μ) with spectral parameter μ changes its size when μ crosses a threshold. To calculate S(μ) in a neighborhood of a threshold, an "augmented" scattering matrix S(μ) is introduced, which keeps its size near the threshold and is analytic in μ there. A minimizer of a quadratic functional JR(·, μ) serves as an approximation to a row of the matrix S(μ). 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→∞, the minimizer a(R, μ) tends exponentially to the corresponding row of S(μ) uniformly with respect to μ in a neighborhood of the threshold. The neighborhood may contain some waveguide eigenvalues corresponding to eigenfunctions exponentially decaying at

AB - © 2014 American Mathematical Society. A waveguide occupies a domain G in Rn+1, n ≥ 1, having several cylindrical outlets to infinity. The waveguide is described by the Dirichlet problem for the Helmholtz equation. The scattering matrix S(μ) with spectral parameter μ changes its size when μ crosses a threshold. To calculate S(μ) in a neighborhood of a threshold, an "augmented" scattering matrix S(μ) is introduced, which keeps its size near the threshold and is analytic in μ there. A minimizer of a quadratic functional JR(·, μ) serves as an approximation to a row of the matrix S(μ). 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→∞, the minimizer a(R, μ) tends exponentially to the corresponding row of S(μ) uniformly with respect to μ in a neighborhood of the threshold. The neighborhood may contain some waveguide eigenvalues corresponding to eigenfunctions exponentially decaying at

M3 - Article

VL - 26

SP - 91

EP - 116

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 1

ER -