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Resonances near thresholds in slightly twisted waveguides


Authors: Vincent Bruneau, Pablo Miranda and Nicolas Popoff
Journal: Proc. Amer. Math. Soc. 146 (2018), 4801-4812
MSC (2010): Primary 35J10, 81Q10, 35P20
DOI: https://doi.org/10.1090/proc/14141
Published electronically: July 23, 2018
MathSciNet review: 3856147
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Abstract: We consider the Dirichlet Laplacian in a straight three dimensional waveguide with non-rotationally invariant cross section, perturbed by a twisting of small amplitude. It is well known that such a perturbation does not create eigenvalues below the essential spectrum. However, around the bottom of the spectrum, we provide a meromorphic extension of the weighted resolvent of the perturbed operator and show the existence of exactly one pole near this point. Moreover, we obtain the asymptotic behavior of this resonance as the size of the twisting goes to 0. We also extend the analysis to the upper eigenvalues of the transversal problem, showing that the number of resonances is bounded by the multiplicity of the eigenvalue and obtaining the corresponding asymptotic behavior.


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Additional Information

Vincent Bruneau
Affiliation: Université de Bordeaux, IMB, UMR 5251, 33405 Talence cedex, France
Email: Vincent.Bruneau@u-bordeaux.fr

Pablo Miranda
Affiliation: Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Santiago, Chile
Email: pablo.miranda.r@usach.cl

Nicolas Popoff
Affiliation: Université de Bordeaux, IMB, UMR 5251, 33405 Talence cedex, France
Email: Nicolas.Popoff@u-bordeaux.fr

DOI: https://doi.org/10.1090/proc/14141
Keywords: Twisted waveguide, Dirichlet Laplacian, resonances near thresholds
Received by editor(s): November 4, 2017
Received by editor(s) in revised form: February 26, 2018
Published electronically: July 23, 2018
Additional Notes: The second author was partially supported by Conicyt-Fondecyt Iniciación 11150865 and PAI 79160144
Communicated by: Michael Hitrik
Article copyright: © Copyright 2018 American Mathematical Society