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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Leningrad University, Leningrad, 1980; D. Reidel, Boston, 1987).
C. Wilcox, Scattering Theory for Diffraction Gratings (Springer, Berlin, 1980).
V. P. Maslov, “Asymptotics of the Eigenfunctions of the Equation Δu + k 2 u = 0 with Boundary Value Conditions on Equidistant Curves and Scattering of Electromagnetic Waves in a Waveguide,” Dokl. AN SSSR 123(4), 631–633 (1958).
Y. Avishai, D. Bessis, B. G. Giraud, and G. Mantica, “Quantum Bound States in Open Geometries,” Physical Review B 44(15), 8028–8034 (1991).
P. Duclos and P. Exner, “Curvature-Induced Bound States in Quantum Waveguides in Two and Three Dimensions,” Review Math. Phys. 7(1), 73–102 (1995).
S. A. Nazarov, “Discrete Spectrum of Cranked, Branchy, and Periodic Waveguides,” Algebra i Analiz 23(2), 206–247 (2011).
S. A. Nazarov, “Confined Waves in a T-Shaped Waveguide,” Akust. Zh. 56(6), 747–758 (2010).
W. Bulla, P. Gesetesy, W. Renter, and B. Simon, “Weakly Coupled Bound States in Quantum Waveguides,” Proc. Amer. Math. Soc. 125(8), 1487–1495 (1997).
I. V. Kamotskii and S. A. Nazarov, “Wood’s Anomalies and Surface Waves in Problem of Scattering by a Periodic Boundary, I, II,” Mat. Sb. 190(1), 109–138; Mat. Sb. 190 (2), 43–70 (1999).
R. R. Gadyl’shin, “Local Perturbations of Quantum Waveguides,” Teoret. Mat. Fiz. 145(3), 358–371 (2005).
S. A. Nazarov, “Variational and Asymptotic Methods for Finding Eigenvalues below the Continuous Spectrum Thresholds,” Sibirsk. Mat. Zh. 51(5), 1086–1101 (2010).
V. G. Maz’ya, S. A. Nazarov, and B. A. Plamenevskii, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vols. 1–2 (Izd. Tbil. Gos. Univ., Tbilisi, 1981; Akademie, Berlin, 1991; Birkhäser, Basel, 2000).
Original Russian Text © S.A. Nazarov, 2011, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2011, No. 3, pp. 29–35.
About this article
Cite this article
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
- cranked quantum and acoustic waveguides
- trapped modes
- localized solutions
- discrete spectrum