Geodesic
Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 3, pp. 521-538

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Perturbations of an eigenvalue in the continuous spectrum of the Neumann problem for the Laplacian in a strip waveguide with an obstacle symmetric about the midline are studied. Such an eigenvalue is known to be unstable, and an arbitrarily small perturbation can cause it to leave the spectrum to become a complex resonance point. Conditions on the perturbation of the obstacle boundary are found under which the eigenvalue persists in the continuous spectrum. The result is obtained via the asymptotic analysis of an auxiliary object, namely, an augmented scattering matrix. 
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     author = {S. A. Nazarov},
     title = {Enforced stability of an eigenvalue in the continuous spectrum of a~waveguide with an~obstacle},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {521--538},
     publisher = {mathdoc},
     volume = {52},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_3_a10/}
}
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S. A. Nazarov. Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 3, pp. 521-538. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_3_a10/