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Lord Rayleigh, “The theory of Helmholtz resonator,” Proc. Roy. Soc. A,92, No. 638, 265–275 (1916).
J. W. Miles, “Scattering by a spherical cap,” J. Acoust. Soc. Am.,50, No. 3, Pt. 2, 892–903 (1971).
V. M. Babich, “On the analytic continuation of the resolvent of exterior three-dimensional problems for the Laplacian to the second sheet,” in: Theory of Functions, Functional Analysis and Their Applications [in Russian], Vol. 3, Khar'kov University Press, Khar'kov (1966), pp. 151–157.
P. D. Lax and R. S. Phillips, Scattering Theory, Academic Press, New York (1967).
C. G. Goldstein, “Scattering theory for elliptic differential operators in unbounded domains,” J. Math. Anal. Appl.,45, 723–745 (1974).
R. R. Gadyl'shin, “On the amplitude of oscillations in the Helmholtz resonator,” Dokl. Akad. Nauk SSSR,310, No. 5, 1094–1097 (1990).
R. R. Gadyl'shin, “Surface potentials and the method of matched asymptotic expansions in the problem of the Helmholtz resonator,” Algebra Analiz,4, No. 2, 88–115 (1992).
A. A. Arsen'ev, “On the singularities of the analytic continuation and the resonance properties of a solution to the scattering problem for the Helmholtz equation,” Zh. Vychisl. Mat. Mat. Fiz.,12, No. 1, 112–138 (1972).
J. T. Beale, “Scattering frequencies of resonators,” Comm. Pure Appl. Math.,26, No. 4, 549–564 (1973).
S. V. Petras, “On resonance series splitting on the ‘nonphysical’ sheet,” Funkts. Anal. Prilozhen.,9, No. 2, 89–90 (1975).
S. V. Petras, “On resonance series splitting on the ‘nonphysical’ sheet,” Zap. Nauchn. Sem. LOMI,51, 155–169 (1975).
A. A. Arsen'ev, “On the existence of resonance poles and resonances in the scattering problem in the case of second- and third-type boundary value conditions,” Zh. Vychisl. Mat. Mat. Fiz.,16, No. 3, 718–724 (1976).
E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer-Verlag, New York (1980).
C. A. Fernandez, “Resonances in scattering by a resonator,” Indiana Univ. Math. J.,34, No. 1, 115–125 (1985).
P. D. Hislop and A. Martinez, “Scattering resonances of a Helmholtz resonator,” Indiana Univ. Math. J.,40, No. 2, 767–788 (1991).
M. D. Van Dyke, Perturbation Method in Fluid Mechanics, Academic Press, New York (1964).
A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York (1973).
A. M. Il'in, Matching of Asymptotic Expansions for Solutions of Boundary Value Problems [in Russian], Nauka, Moscow (1989).
V. G. Maz'ya, S. A. Nazarov, and B. A. Plamenevskii, “Asymptotic expansions of eigenvalues of boundary value problems for the Laplace operator in domains with small holes,” Izv. Akad. Nauk SSSR, Ser. Mat.,48, No. 2, 347–371 (1984).
R. R. Gadyl'shin, “Asymptotics of eigenvalues for singular perturbed self-adjoint elliptic problems with small parameter in boundary conditions, Differents. Uravn.,22, No. 4, 640–652 (1986).
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York (1983).
V. Yu. Gotlib, “Scattering by a round resonator with a narrow slot as a problem of perturbation theory,” Dokl. Akad. Nauk SSSR,287, No. 5, 1109–1113 (1986).
Institute of Mathematics, The Urals Division of the Russian Academy of Sciences, Ufa. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 4, pp. 3–16, October–December, 1993.
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Gadyl'shin, R.R. On the poles of an acoustic resonator. Funct Anal Its Appl 27, 229–239 (1993). https://doi.org/10.1007/BF01078839
- Functional Analysis
- Acoustic Resonator