Abstract
For the problem of plane waves scattered by a domain with a small hole, we suggest a model based on the theory of self-adjoint extensions of symmetric operators in a space with indefinite metric. For two-dimensional problems of scattering on a line with a hole and on a semi-ellipse connected by a hole with a half-plane, we justify the choice of extension that guarantees the coincidence of the model solution with the solution of the “actual” problem in the far zone with a high degree of accuracy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. S. Pavlov, “Theory of extensions and explicitly solvable models,”Uspekhi Mat. Nauk [Russian Math. Surveys],42, No. 6, 99–131 (1987).
I. Yu. Popov, “The extension theory and diffraction problems,”Lect. Notes Phys.,324, 218–229 (1989).
B. S. Pavlov and I. Yu. Popov, “Scattering on a resonator with a small or point hole,”Vestnik Leningrad. Univ., ser. Mat.-Mekh.-Astronom. [Vestnik Leningrad Univ. Math.], No. 13, 116–118 (1984).
M. M. Zimnev and I. Yu. Popov, “Parameter choice for the zero-width slot model,”Zh. Vychisl. Mat. i Mat. Fiz. [U.S.S.R. Comput. Math. and Math. Phys.],24, No. 3, 466–470 (1987).
Yu. G. Shondin, “Quantum mechanics models in ℝ n related to the extension of the energy operators in Pontryagin space,”Zh. Vychisl. Mat. i Mat. Fiz. [U.S.S.R. Comput. Math. and Math. Phys.],74, No. 3, 331–344 (1988).
F. A. Berezin, “On the Lie model,”Mat. Sb. [Math. USSR-Sb.],60, No. 4, 425–446 (1963).
M. G. Krein and G. K. Langer, “Deficiency subspaces and generalized resolvents of an Hermitian operator in the space Π x ,”Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],5, No. 2, 59–71; No. 3, 54–69 (1971).
G. N. Watson,A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge (1952).
Ph. M. Morse and H. Feshbach,Methods of Theoretical Physics, McGraw-Hill Book Company, New York-London (1953).
H. Bateman and A. Erdélyi,Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York-Toronto-London (1955).
N. W. McLachlan,Theory and Application of the Mathieu Functions, Oxford (1947).
V. M. Babich, “Analytic continuation of the resolvents of exterior problems for the Laplace operator to the second sheet,” in:Teoriya Funktsii i Ikh Prilozh. [in Russian], Izd. Khar'kov Gos. Univ., Khar'kov (1966), pp. 151–157.
I. Yu. Popov, “The Helmholtz resonator and extension theory for operators in a space with indefinite metric,”Mat. Sb. [Math. USSR-Sb.],183, No. 3, 3–37 (1992).
I. Yu. Popov, “The resonator with narrow slit and the model based on operator extension theory,”J. Math. Phys.,33, No. 11, 3794–3801 (1992).
R. R. Gadyl'shin, “The method of matched asymptotic expansions in the problem on the acoustic Helmholtz resonator,”Prikl. Mat. Mekh. [J. Appl. Math. Mekh.],56, No. 3, 412–418 (1992).
Author information
Authors and Affiliations
Additional information
Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 837–850, December, 1995.
The authors are grateful to B. S. Pavlov and L. M. Grigoryan for useful discussion.
The work was partially supported by the State Commission on Higher Education of the Russian Federation under grant No. 94-2.7-1067.
Rights and permissions
About this article
Cite this article
Kiselev, A.A., Popov, I.Y. Indefinite metric and scattering by a domain with a small hole. Math Notes 58, 1276–1285 (1995). https://doi.org/10.1007/BF02304886
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02304886