Geodesic
Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide
Matematičeskie zametki, Tome 87 (2010) no. 5, pp. 764-786

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It is established that a small periodic singular or regular perturbation of the boundary of a cylindrical three-dimensional waveguide can open up a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator in the resulting periodic waveguide. A singular perturbation results in the formation of a periodic family of small cavities while a regular one leads to a gentle periodic bending of the boundary. If the period is short, there is no gap, while if it is long, a gap appears immediately after the first segment of the continuous spectrum. The result is obtained by asymptotic analysis of the eigenvalues of an auxiliary problem on the perturbed cell of periodicity.
Keywords: cylindrical waveguide, gap in a continuous spectrum, Laplace operator, Dirichlet problem, Helmholtz equation, cell of periodicity, Sobolev space. 
@article{MZM_2010_87_5_a9,
     author = {S. A. Nazarov},
     title = {Opening of a {Gap} in the {Continuous} {Spectrum} of a {Periodically} {Perturbed} {Waveguide}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {764--786},
     publisher = {mathdoc},
     volume = {87},
     number = {5},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_5_a9/}
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S. A. Nazarov. Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide. Matematičeskie zametki, Tome 87 (2010) no. 5, pp. 764-786. http://geodesic.mathdoc.fr/item/MZM_2010_87_5_a9/