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Normal Waves in an Open Anisotropic Regular Waveguide of Arbitrary Cross Section

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Abstract

The problem on normal waves in an open regular waveguide of arbitrary cross-section is considered. The permittivity and permeability of the waveguide are described by tensors. The determination of normal waves is reduced to a boundary eigenvalue problem for longitudinal components of the electromagnetic field in Sobolev spaces. A variational formulation corresponding to this problem is obtained. The variational relation leads to an eigenvalue problem for the operator-function. Properties of the operator-function are investigated. Discreteness of the spectrum is proved. Numerical results are presented.

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REFERENCES

  1. Yu. G. Smirnov, ‘‘Application of the operator pencil method in the eigenvalue problem for partially,’’ Dokl. Akad. Nauk SSSR 312, 597–599 (1990).

    Google Scholar 

  2. Yu. G. Smirnov, ‘‘The method of operator pencils in the boundary transmission problems for elliptic system of equations,’’ Differ. Equat. 27, 140–147 (1991).

    Google Scholar 

  3. Yu. G. Smirnov, Mathematical Methods for Electromagnetic Problems (PGu Press, Penza, 2009) [in Russian].

  4. Y. Shestopalov and Y. Smirnov, ‘‘Eigenwaves in waveguides with dielectric inclusions: Spectrum,’’ Applic. Anal. 93, 408–427 (2014).

    Article  MathSciNet  Google Scholar 

  5. Y. Shestopalov and Y. Smirnov, ‘‘Eigenwaves in waveguides with dielectric inclusions: Completeness,’’ Applic. Anal. 93, 1824–1845 (2014).

    Article  MathSciNet  Google Scholar 

  6. M. V. Keldysh, ‘‘On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators,’’ Dokl. Akad. Nauk SSSR 77, 11–14 (1951).

    Google Scholar 

  7. Yu. G. Smirnov and E. Smolkin, ‘‘Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide,’’ Differ. Equat. 53, 1262–1273 (2017).

    Article  MathSciNet  Google Scholar 

  8. Yu. G. Smirnov, E. Smolkin, and M. O. Snegur, ‘‘Analysis of the spectrum of azimuthally symmetric waves of an open inhomogeneous anisotropic waveguide with longitudinal magnetization,’’ Comput. Math. Math. Phys. 58, 1887–1901 (2018).

    Article  MathSciNet  Google Scholar 

  9. Yu. G. Smirnov and E. Smolkin, ‘‘Operator function method in the problem of normal waves in an inhomogeneous waveguide,’’ Differ. Equat. 54, 1262–1273 (2018).

    MathSciNet  MATH  Google Scholar 

  10. Yu. G. Smirnov and E. Smolkin, ‘‘Investigation of the spectrum of the problem of normal waves in a closed regular inhomogeneous dielectric waveguide of arbitrary cross section,’’ Dokl. Math. 97, 86–89 (2017).

    Article  Google Scholar 

  11. Yu. G. Smirnov and E. Smolkin, ‘‘Eigenwaves in a lossy metal-dielectric waveguide,’’ Applic. Anal. 97 (1), 1–12 (2018).

    Article  MathSciNet  Google Scholar 

  12. A. W. Snyder and J. Love, Optical Waveguide Theory (Springer, Berlin, 1983).

    Google Scholar 

  13. Yu. G. Smirnov and E. Smolkin, ‘‘Mathematical theory of normal waves in an open metal-dielectric regular waveguide of arbitrary cross section,’’ Math. Model. Anal. 25, 391–408 (2020).

    Article  MathSciNet  Google Scholar 

  14. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, Cambridge, 1995).

    MATH  Google Scholar 

  15. T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1980).

    MATH  Google Scholar 

  16. Yu. V. Shestopalov, Yu. G. Smirnov, and E. V. Chernokozhin, Logarithmic Integral Equations in Electromagnetics (De Gruyter, Holland, 2000).

    Book  Google Scholar 

  17. R. A. Adams, Sobolev Spaces (Academic, New York, 1975).

    MATH  Google Scholar 

  18. A. S. Ilyinsky and Yu. G. Smirnov, Electromagnetic Wave Diffraction by Conducting Screens (VSP, Utrecht, Netherlands, 1998).

    Google Scholar 

  19. H. Triebel, Theory of Function Spaces (Birkhäuser, Basel, 1983).

    Book  Google Scholar 

  20. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965).

    MATH  Google Scholar 

  21. I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space (Am. Math. Soc., Philadelphia, 1969).

    Book  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation, project no. 20-11-20087.

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Authors and Affiliations

  1. Penza State University, 440026, Penza, Russia

    E. Smolkin & M. A. Moskaleva

Authors
  1. E. Smolkin
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  2. M. A. Moskaleva
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Correspondence to E. Smolkin or M. A. Moskaleva.

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(Submitted by E. E. Tyrtyshnikov)

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Smolkin, E., Moskaleva, M.A. Normal Waves in an Open Anisotropic Regular Waveguide of Arbitrary Cross Section. Lobachevskii J Math 42, 1453–1464 (2021). https://doi.org/10.1134/S1995080221060299

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  • DOI: https://doi.org/10.1134/S1995080221060299

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