Email this article Asymptotics of Eigenvalues in Spectral Gaps Under Regular Perturbations of Walls of a Periodic Waveguide
- Authors: Nazarov S.A.1,2,3
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Affiliations:
- Saint-Petersburg State University
- Peter the Great Saint-Petersburg State Polytechnical University
- Institute of Problems of Mechanical Engineering RAS
- Issue: Vol 226, No 4 (2017)
- Pages: 402-444
- Section: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/239998
- DOI: https://doi.org/10.1007/s10958-017-3542-x
- ID: 239998
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Abstract
We find asymptotic representations of eigenvalues inside gaps of the continuous spectrum of a periodic waveguide with local smooth gently sloped (of depth ε ≪ 1) perturbations of walls. These eigenvalues reach the upper or lower gap edge as ε → +0. We consider several variants of the gap edge structure and obtain conditions guaranteeing the existence or absence of points of the discrete spectrum in small neighborhoods. We calculate the total number of eigenvalues in a gap for small ε. To justify the asymptotic expansions, we use elementary tools of the theory of spectral measure.
About the authors
S. A. Nazarov
Saint-Petersburg State University; Peter the Great Saint-Petersburg State Polytechnical University; Institute of Problems of Mechanical Engineering RAS
Author for correspondence.
Email: s.nazarov@spbu.ru
Russian Federation, 7-9, Universitetskaya nab., St. Petersburg, 199034; 29, Polytechnicheskaya ul., St. Petersburg, 195251; 61, V.O., Bolshoj pr., St. Petersburg, 199178
