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Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide

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Abstract

We consider the problem on normal waves in an inhomogeneous waveguide structure reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The inhomogeneity of the dielectric filling and the occurrence of the spectral parameter in the transmission conditions necessitate giving a special definition of what a solution of the problem is. To find the solution, we use the variational statement of the problem. The variational problem is reduced to the study of an operator function. We study the properties of the operator function needed for the analysis of its spectral properties. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved.

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References

  1. Il’inskii, A.S. and Shestopalov, Yu.V., Primenenie metodov spektral’noi teorii v zadachakh rasprostraneniya voln (Application of Spectral Theory Methods to Wave Propagation Problems), Moscow: Mosk. Gos. Univ., 1989.

    Google Scholar 

  2. Smirnov, Yu.G., The method of operator pencils in boundary value problems of conjugation for a system of elliptic equations, Differ. Equations, 1991, vol. 27, no. 1, pp. 112–118.

    MathSciNet  MATH  Google Scholar 

  3. Smirnov, Yu.G., The application of the operator pencil method in a problem concerning the natural waves of a partially filled wave guide, Dokl. Phys., 1990, vol. 35, no. 5, pp. 430–431.

    MathSciNet  Google Scholar 

  4. Delitsyn, A. L., An approach to the completeness of normal waves in a waveguide with magnetodielectric filling, Differ. Equations, 2000, vol. 36, no. 5, pp. 695–700.

    Article  MathSciNet  MATH  Google Scholar 

  5. Veselov, G.I. and Raevskii, S.B., Sloistye metallo-dielektricheskie volnovody (Layered Metal–Dielectric Waveguides), Moscow: Radio i Svyaz’, 1988.

    Google Scholar 

  6. Yeh, P., Yariv, A., and Marom, E., Theory of Bragg fiber, J. Opt. Soc. Amer., 1978, vol. 68, no. 9, pp. 1196–1201.

    Article  Google Scholar 

  7. Sharma, P., Sharma, S.K., and Singh, B., Theory and applications of Bragg fibers: A review, Int. J. Commun. Eng. Technol., 2014, vol. 4, no. 1, pp. 1–32.

    Article  Google Scholar 

  8. Smirnov, Yu.G., Matematicheskie metody issledovaniya zadach elektrodinamiki (Mathematical Methods for Electrodynamic Problems), Penza: Penz. Gos. Univ., 2009.

    Google Scholar 

  9. Nikiforov, A.F. and Uvarov, V.B., Spetsial’nye funktsii matematicheskoi fiziki (Special Functions of Mathematical Physics), Moscow: Nauka, 1978.

    MATH  Google Scholar 

  10. Kato, T., Perturbation Theory for Linear Operators, Heidelberg: Springer-Verlag, 1966. Translated under the title Teoriya vozmushchenii lineinykh operatorov, Moscow: Mir, 1972.

    Google Scholar 

  11. Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz (Functional Analysis), Moscow: Gos. Izd. Fiz. Mat. Lit., 1977.

    MATH  Google Scholar 

  12. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Abramowitz,M. and Stegun, I., Eds., Washington, D.C.: National Bureau of Standards, 1964. Translated under the title Spravochnik po spetsial’nym funktsiyam, Moscow: Nauka, 1979.

  13. Gokhberg, I.Ts. and Krein, M.G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gil’bertovom prostranstve (Introduction to the Theory of Linear Nonself-Adjoint Operators in Hilbert Spaces), Moscow: Nauka, 1965.

    Google Scholar 

  14. Gradshtein, I.S. and Ryzhik, I.M., Tablitsy integralov, summ, ryadov i proizvedenii (Tables of Integrals, Sums, Series, and Products), Moscow: Nauka, 1971, 4th ed.

    Google Scholar 

  15. Raevskii, A.B. and Raevskii, S.B., Kompleksnye volny (Complex Waves) Moscow: Radiotekhnika, 2010.

    MATH  Google Scholar 

  16. Shestopalov, Yu. and Smirnov, Yu., Eigenwaves in waveguides with dielectric inclusions: Spectrum, Appl. Anal., 2014, vol. 93, no. 2, pp. 408–427.

    Article  MathSciNet  MATH  Google Scholar 

  17. Shestopalov, Yu. and Smirnov, Yu., Eigenwaves in waveguides with dielectric inclusions: Completeness, Appl. Anal., 2014, vol. 93, no. 9, pp. 1824–1845.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Penza State University, Penza, 440026, Russia

    Yu. G. Smirnov & E. Yu. Smolkin

Authors
  1. Yu. G. Smirnov
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  2. E. Yu. Smolkin
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Correspondence to Yu. G. Smirnov.

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Original Russian Text © Yu.G. Smirnov, E.Yu. Smolkin, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 10, pp. 1298-1308.

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Smirnov, Y.G., Smolkin, E.Y. Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide. Diff Equat 53, 1262–1273 (2017). https://doi.org/10.1134/S0012266117100032

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  • DOI: https://doi.org/10.1134/S0012266117100032

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