title: Method for computing waveguide scattering matrices in the vicinity of thresholds
metadata.name: dc.contributor.author
metadata.value: Plamenevskii, B.A.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Poretskii, A.S.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Sarafanov, O.V.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2016-05-27T14:58:10Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2016-05-27T14:58:10Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2015-01
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: St. Petersburg Math. J. 26 (2015), 91-116
metadata.lang: -
metadata.name: dc.identifier.issn
metadata.value: 1061-0022
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/2288
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: 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]$.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: en
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Method for computing waveguide scattering matrices in the vicinity of thresholds
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: PPSeng (2).pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/2288/1/PPSeng%20%282%29.pdf
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