Geodesic
Discrete spectrum of cranked quantum and elastic waveguides
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 5, pp. 879-895

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The spectrum of quantum and elastic waveguides in the form of a cranked strip is studied. In the Dirichlet spectral problem for the Laplacian (quantum waveguide), in addition to well-known results on the existence of isolated eigenvalues for any angle $\alpha$ at the corner, a priori lower bounds are established for these eigenvalues. It is explained why methods developed in the scalar case are frequently inapplicable to vector problems. For an elastic isotropic waveguide with a clamped boundary, the discrete spectrum is proved to be nonempty only for small or close-to-$\pi$ angles $\alpha$. The asymptotics of some eigenvalues are constructed. Elastic waveguides of other shapes are discussed. 
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     author = {S. A. Nazarov},
     title = {Discrete spectrum of cranked quantum and elastic waveguides},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {879--895},
     publisher = {mathdoc},
     volume = {56},
     number = {5},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_5_a12/}
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S. A. Nazarov. Discrete spectrum of cranked quantum and elastic waveguides. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 5, pp. 879-895. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_5_a12/