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Asymptotic formula for an eigenvalue of the dirichlet problem in a cranked waveguide

Abstract

The asymptotic behavior of an eigenvalue of the Dirichlet problem for a spectral Helmholtz equation in a two-dimensional cranked acoustic waveguide with yielding walls or in a quantum waveguide is obtained. A waveguide is thought of as a cranked strip, but the boundary value problem is posed in a straight strip of unit width with wedge-shaped notches, with appropriate conjugation conditions on the edges of the notches, which provide for a smooth wave field after the initial form of the waveguide is restored. The bend angles are assumed to be small; i.e., the wedge-shaped notches are supposed to be thin, the asymptotic behavior is built from the corresponding small geometric parameter.

It is known that for any bend angles the waveguide has an eigenvalue of the discrete spectrum which lies below the continuous spectrum of the boundary value problem.

The main asymptotic term of the eigenfunction is taken as a standing wave in the intact strip. This wave provides for small residuals under the conjugation condition in question on the wedge-shaped notches, which are offset by solutions of the problem again in the straight strip subject to jump conditions at the transversal cross-sections, to which thin notches are contracted. These solutions become linearly increasing at infinity. As a result, both the initial approximation and the asymptotic corrections fail to have a natural decay property, which is characteristic of an eigenfunction that vanishes exponentially at infinity and known for this reason as a trapped mode. To remedy the behavior of the asymptotic ansatz at infinity we build an additional (outer) asymptotic expansion with good properties away from the wedge-shaped notches. The decay condition for the outer expansion leads to an asymptotic formula for an eigenvalue, the asymptotic behavior is justified by means of the spectral measure apparatus.

It should be mentioned that both the asymptotic constructions (and even the order of the main correction) and the justification scheme are substantially different from those characteristic of spectral problems in bounded domains.

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Correspondence to S. A. Nazarov.

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Original Russian Text © S.A. Nazarov, 2011, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2011, No. 3, pp. 29–35.

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Nazarov, S.A. Asymptotic formula for an eigenvalue of the dirichlet problem in a cranked waveguide. Vestnik St.Petersb. Univ.Math. 44, 190 (2011). https://doi.org/10.3103/S1063454111030046

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Keywords

  • cranked quantum and acoustic waveguides
  • trapped modes
  • localized solutions
  • discrete spectrum