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

Nazarov, S.

doi:10.1090/spmj/1455

Almost standing waves in a periodic waveguide with resonator, and near-threshold eigenvalues
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by S. A. Nazarov
Translated by: A. Plotkin

St. Petersburg Math. J. 28 (2017), 377-410

DOI: https://doi.org/10.1090/spmj/1455

Published electronically: March 29, 2017

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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.

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