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Mat. Sb., 1992, Volume 183, Number 3, Pages 3–37 (Mi msb1039)  

This article is cited in 16 scientific papers (total in 16 papers)

The Helmholtz resonator and the theory of operator extensions in a space with indefinite metric

I. Yu. Popov


Abstract: To investigate the Helmholtz resonator a model is developed based on the theory of selfadjoint extensions of symmetric operators in a space with indefinite metric. In the case of a small opening compared to the wavelength, approximations of any predetermined precision are obtained for the Green functions of the Dirichlet and Neumann problems for the Helmholtz resonator. The problem of resonances is considered in the framework of the Lax–Phillips approach. Formulae to determine the resonances with any required precision are obtained and substantiated.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1993, 75:2, 285–315

Bibliographic databases:

UDC: 517.9
MSC: 35J05, 47A20, 47B50
Received: 20.12.1990

Citation: I. Yu. Popov, “The Helmholtz resonator and the theory of operator extensions in a space with indefinite metric”, Mat. Sb., 183:3 (1992), 3–37; Russian Acad. Sci. Sb. Math., 75:2 (1993), 285–315

Citation in format AMSBIB
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\by I.~Yu.~Popov
\paper The Helmholtz resonator and the~theory of operator extensions in a~space with indefinite metric
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\yr 1992
\vol 183
\issue 3
\pages 3--37
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\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1993
\vol 75
\issue 2
\pages 285--315
\crossref{https://doi.org/10.1070/SM1993v075n02ABEH003386}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Popov I., “The Resonator with Narrow Slit and the Model Based on the Operator Extensions Theory”, J. Math. Phys., 33:11 (1992), 3794–3801  crossref  mathscinet  zmath  adsnasa  isi
    2. R. R. Gadyl'shin, “Splitting of the poles of a Helmholtz resonator”, Russian Acad. Sci. Izv. Math., 43:2 (1994), 233–260  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. A. A. Kiselev, B. S. Pavlov, “Essential spectrum of the Laplacian for the Neumann problem in a model region of complicated structure”, Theoret. and Math. Phys., 99:1 (1994), 383–395  mathnet  crossref  mathscinet  zmath  isi
    4. Gadylshin R., “On Acoustic Helmholtz Resonator and on its Electromagnetic Analog”, J. Math. Phys., 35:7 (1994), 3464–3481  crossref  mathscinet  adsnasa  isi
    5. A. A. Kiselev, I. Yu. Popov, “Indefinite metric and scattering by a domain with a small hole”, Math. Notes, 58:6 (1995), 1276–1285  mathnet  crossref  mathscinet  zmath  isi
    6. Popova S., “Unlocking of Quantum Waveguides”, Pisma Zhurnal Tek. Fiz., 22:6 (1996), 55–57  isi
    7. Alexander Kiselev, “Some Examples in One-Dimensional “Geometric” Scattering on Manifolds”, Journal of Mathematical Analysis and Applications, 212:1 (1997), 263  crossref
    8. V.A. Geyler, I.Yu. Popov, S.L. Popova, “Transmission coefficient for ballistic transport through quantum resonator”, Reports on Mathematical Physics, 40:3 (1997), 531  crossref
    9. Andronov I., “Zero-Range Potential Model of a Protruding Stiffener”, J. Phys. A-Math. Gen., 32:20 (1999), L231–L238  crossref  zmath  adsnasa  isi
    10. B. S Pavlov, I. Yu Popov, V. A Geyler, O. S Pershenko, “Possible construction of a quantum multiplexer”, Europhys Lett, 52:2 (2000), 196  crossref  elib
    11. Kurasov P. Pavlov B., “Few-Body Krein's Formula”, Operator Theory and Related Topics, Operator Theory : Advances and Applications, 118, ed. Adamyan V. Gohberg I. Gorbachuk M. Gorbachuk V. Kaashoek M. Langer H. Popov G., Birkhauser Verlag Ag, 2000, 225–254  mathscinet  zmath  isi
    12. Kurasov P. Watanabe K., “On H-4-Perturbations of Self-Adjoint Operators”, Partial Differential Equations and Spectral Theory, Operator Theory : Advances and Applications, 126, ed. Demuth M. Schulze B., Birkhauser Verlag Ag, 2001, 179–196  mathscinet  zmath  isi
    13. Kurasov P., Pavlov Y., “On Field Theory Methods in Singular Perturbation Theory”, Lett. Math. Phys., 64:2 (2003), 171–184  crossref  mathscinet  zmath  isi
    14. Kurasov P., “H-N-Perturbations of Self-Adjoint Operators and Krein's Resolvent Formula”, Integr. Equ. Oper. Theory, 45:4 (2003), 437–460  crossref  mathscinet  zmath  isi
    15. Yuri Shondin, “On approximation of high order singular perturbations”, J Phys A Math Gen, 38:22 (2005), 5023  crossref  mathscinet  zmath  isi  elib
    16. Anikevich A.S., “Spektralnaya zadacha dlya tsepochek slabosvyazannykh sharoobraznykh rezonatorov”, Nanosistemy: fizika, khimiya, matematika, 3:3 (2012), 23–30  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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