Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides. (English) Zbl 1278.35170
The paper under review is concerned with the study of the propagation of time-harmonic acoustic or transverse magnetic polarized electromagnetic waves in a periodic waveguide lying in the lalf-strip \((0,\infty)\times (0,L)\). The main results establishes that there exists a Riesz basis of the space of solutions to the time-harmonic wave equation such that the translation operator shifting a function by one periodicity length to the left is represented by an infinite Jordan matrix which contains at most a finite number of Jordan blocks of size bigger than 1. Both the cases of frequencies in a band gap and frequencies in the spectrum and a variety of boundary conditions on the top and bottom are considered. A generalization of Rouché’s theorem is a basic tool in the qualitative analysis developed in this paper. This paper also contains two appendices on radiation conditions and on uniqueness results.
Reviewer: Teodora-Liliana Rădulescu (Craiova)
MSC:
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
Keywords:
photonic crystal; photonic band gap; periodic dielectric medium; Floquet theory; analytic theory of operatorsReferences:
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