Resonances near thresholds in slightly twisted waveguides
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- by Vincent Bruneau, Pablo Miranda and Nicolas Popoff PDF
- Proc. Amer. Math. Soc. 146 (2018), 4801-4812 Request permission
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.References
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Additional Information
- Vincent Bruneau
- Affiliation: Université de Bordeaux, IMB, UMR 5251, 33405 Talence cedex, France
- MR Author ID: 607313
- 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
- MR Author ID: 807084
- Email: pablo.miranda.r@usach.cl
- Nicolas Popoff
- Affiliation: Université de Bordeaux, IMB, UMR 5251, 33405 Talence cedex, France
- MR Author ID: 978275
- Email: Nicolas.Popoff@u-bordeaux.fr
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4801-4812
- MSC (2010): Primary 35J10, 81Q10, 35P20
- DOI: https://doi.org/10.1090/proc/14141
- MathSciNet review: 3856147