# Method for computing waveguide scattering matrices in the vicinity of thresholds

Research output

4 Citations (Scopus)

### Abstract

© 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
Original language English 91-116 St. Petersburg Mathematical Journal 26 1 Published - 2015

### Fingerprint

Scattering Matrix
Waveguide
Waveguides
Scattering
Computing
Minimizer
Augmented matrix
Infinity
Helmholtz equation
Helmholtz Equation
Eigenvalues and eigenfunctions
Dirichlet Problem
Boundary value problems
Eigenfunctions
Bounded Domain
Boundary Value Problem
Tend
Eigenvalue
Calculate

### Cite this

title = "Method for computing waveguide scattering matrices in the vicinity of thresholds",
abstract = "{\circledC} 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",
author = "B.A. Plamenevski and A.S. Poretski and O.V. Sarafanov",
year = "2015",
language = "English",
volume = "26",
pages = "91--116",
journal = "St. Petersburg Mathematical Journal",
issn = "1061-0022",
publisher = "American Mathematical Society",
number = "1",

}

In: St. Petersburg Mathematical Journal, Vol. 26, No. 1, 2015, p. 91-116.

Research output

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 -