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Characteristic frequencies of bodies with thin spikes. I. Convergence and estimates

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References

  1. J. T. Beale, “Scattering frequencies of resonators,” Comm. Pure Appl. Math.,26, No. 4, 549–564 (1973).

    Google Scholar 

  2. A. A. Arsen'ev, “Singularities of analytic continuation and resonance properties of the solution of the scattering problem for the Helmholtz equation,” Zhurn. Vychisl. Mat. Mat. Fiz.,12, No. 1, 112–138 (1972).

    Google Scholar 

  3. S. V. Petras, “Decomposition of the series of resonances on a “nonphysical” sheet,” Funkts. Analiz Prilozhen.,9, No. 2, 89–90 (1975).

    Google Scholar 

  4. S. V. Petras, “Decomposition of the series of resonances on a “nonphysical sheet”,” Zapiski Nauchn. Semin. Leningr. Otdel. Mat. Inst.,51, 155–169 (1975).

    Google Scholar 

  5. A. A. Arsen'ev, “Existence of resonance poles and resonances for scattering in the case of boundary conditions of the II and III kinds,” Zhurn. Vychisl. Mat. Mat. Fiz.,16, No. 3, 718–724 (1976).

    Google Scholar 

  6. E. Sanchez-Palencia, Inhomogeneous Media and the Theory of Oscillations [Russian translation], Mir, Moscow (1984).

    Google Scholar 

  7. C. A. Fernandez, “Resonances in scattering by a resonator,” Indiana Univ. Math. J.,34, No. 1, 115–125 (1985).

    Google Scholar 

  8. P. D. Hislop and A. Martinez, “Scattering resonances of a Helmholtz resonator,” Indiana Univ. Math. J.,40, No. 2, 767–788 (1991).

    Google Scholar 

  9. S. Jimbo, “Characterization of the eigenfunctions in the singularly perturbed domain,” Proc. Japan Acad., Ser. A,63, No. 8, 285–288.

  10. M. Van Dyke, Perturbation Methods in Fluid Mechanics [Russian translation], Mir, Moscow (1967).

    Google Scholar 

  11. A. Nayfeh, Perturbation Methods [Russian translation], Mir, Moscow (1976).

    Google Scholar 

  12. A. M. Il'in, Matching of Asymptotic Expansions of Solutions of Boundary Problems [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  13. V. G. Maz'ya, S. A. Nazarov, and B. A. Plamanevskii, “Asymptotic expansions of eigenvalues of boundary problems for Laplace operators in domains with small apertures,” Izv. Akad. Nauk SSSR. Ser. Mat.,48, No. 2, 347–371 (1984).

    Google Scholar 

  14. R. R. Gadyl'shin, “Decomposition of a multiple eigenvalue of the Dirichlet problem for the Laplace operator under a singularly perturbed boundary condition,” Mat. Zametki,52, No. 4 (1992).

    Google Scholar 

  15. R. R. Gadyl'shin, “Asymptotics of an eigenvalue of a singularly perturbed elliptic problem with small parameter in the boundary condition,” Differents. Uravnen.,22, No. 4, 640–652 (1986).

    Google Scholar 

  16. R. R. Gadyl'shin, “Spectra of elliptic boundary problems under singularly perturbed boundary conditions,” in: Asymptotic Properties of Solutions of Differential Equations [in Russian], Computing Center, Urals Section of the Academy of Sciences of the USSR, Ufa (1988), pp. 3–15.

    Google Scholar 

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Translated from Matematicheskie Zametki, Vol. 54, No. 6, pp. 10–21, December, 1993.

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Gadyl'shin, R.R. Characteristic frequencies of bodies with thin spikes. I. Convergence and estimates. Math Notes 54, 1192–1199 (1993). https://doi.org/10.1007/BF01209080

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