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Method for computing waveguide scattering matrices in the vicinity of thresholds. / Plamenevski, B.A.; Poretski, A.S.; Sarafanov, O.V.
In: St. Petersburg Mathematical Journal, Vol. 26, No. 1, 2015, p. 91-116.Research output: Contribution to journal › Article › peer-review
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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 -
ID: 3929907