Almost standing waves in a periodic waveguide with resonator, and near-threshold eigenvalues

Research outputpeer-review

7 Citations (Scopus)

Abstract

The definition and an existence criterion are given for the standing waves at the threshold of the continuous spectrum for a periodic quantum waveguide with a resonator (the Dirichlet problem for the Laplace operator). Such waves and their linear combinations do not transfer energy to infinity, and they only differ from the standing waves with the zero Floquet parameter by an exponentially decaying term. It is shown that the almost standing and trapped waves at the threshold generate eigenvalues in the discrete spectrum of a waveguide with a regular sloping local perturbation of the wall.

Original languageEnglish
Pages (from-to)377-410
Number of pages34
JournalSt. Petersburg Mathematical Journal
Volume28
Issue number3
DOIs
Publication statusPublished - 1 Jan 2017

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Standing Wave
Resonator
Waveguide
Resonators
Waveguides
Eigenvalue
Continuous Spectrum
Discrete Spectrum
Laplace Operator
Energy Transfer
Dirichlet Problem
Linear Combination
Infinity
Perturbation
Zero
Term
Energy transfer

Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

Cite this

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AB - The definition and an existence criterion are given for the standing waves at the threshold of the continuous spectrum for a periodic quantum waveguide with a resonator (the Dirichlet problem for the Laplace operator). Such waves and their linear combinations do not transfer energy to infinity, and they only differ from the standing waves with the zero Floquet parameter by an exponentially decaying term. It is shown that the almost standing and trapped waves at the threshold generate eigenvalues in the discrete spectrum of a waveguide with a regular sloping local perturbation of the wall.

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KW - Asymptotics

KW - Discrete spectrum

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KW - Threshold scattering matrix

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