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Mathematical theory of normal waves in an anisotropic rod. (English) Zbl 1450.78008

Summary: The problem on normal waves in an anisotropic inhomogeneous dielectric rod is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study of an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-function on the complex plane is found.

MSC:

78A50 Antennas, waveguides in optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
78A25 Electromagnetic theory (general)
78M30 Variational methods applied to problems in optics and electromagnetic theory
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
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References:

[1] Smirnov, Yu., Application of the operator pencil method in the eigenvalue problem, Dokl. Akad. Nauk SSSR, 312, 597-599 (1990)
[2] Smirnov, Yu., The method of operator pencils in the boundary transmission problems for elliptic system of equations, Differ. Equat., 27, 140-147 (1991) · Zbl 0768.35022
[3] Yu. Smirnov, Mathematical Methods for Electromagnetic Problems (PenzGU, Penza, 2009) [in Russian].
[4] Shestopalov, Y.; Smirnov, Y., Eigenwaves in waveguides with dielectric inclusions: Spectrum, Applicable Anal., 93, 408-427 (2014) · Zbl 1294.78013 · doi:10.1080/00036811.2013.778980
[5] Shestopalov, Y.; Smirnov, Y., Eigenwaves in waveguides with dielectric inclusions: Completeness, Applicable Anal., 93, 1824-1845 (2014) · Zbl 1301.78006 · doi:10.1080/00036811.2013.850494
[6] Keldysh, M. V., On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators, Dokl. Akad. Nauk SSSR, 77, 11-14 (1951) · Zbl 0045.39402
[7] Delitsyn, A. L., An approach to the completeness of normal waves in a waveguide with magnitodielectric filling, Differ. Equat., 36, 695-700 (2000) · Zbl 1033.78512 · doi:10.1007/BF02754228
[8] Krasnushkin, P. E.; Moiseev, E. I., On the excitation of oscillations in layered radiowaveguide, Dokl. Akad. Nauk SSSR, 264, 1123-1127 (1982)
[9] Zilbergleit, A. S.; Kopilevich, Yu. I., Spectral Theory of Guided Waves (1966), London: Inst. of Phys, London
[10] Lozhechko, V. V.; Shestopalov, Yu. V., Problems of the excitation of open cylindrical resonators with an irregular boundary, Comput. Math. Math. Phys., 35, 53-61 (1995) · Zbl 0844.65093
[11] Veselov, G. I.; Raevskii, S. B., Metal-Dielectric Waveguides Formed by Layers (1988), Moscow: Radio Svyaz, Moscow
[12] Levin, L., Theory of Waveguides (1975), London: Newnes-Butterworths, London
[13] Smirnov, Yu. G.; Smolkin, E., Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide, Differ. Equat., 53, 1168-1179 (2017) · Zbl 1384.78008
[14] Smirnov, Yu. G.; Smolkin, E.; Snegur, M. O., 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) · Zbl 1416.78015 · doi:10.1134/S096554251811012X
[15] Smirnov, Yu. G.; Smolkin, E., Operator function method in the problem of normal waves in an inhomogeneous waveguide, Differ. Equat., 54, 1262-1273 (2018) · Zbl 1384.78008
[16] Smirnov, Yu. G.; Smolkin, E., 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) · Zbl 1397.78043 · doi:10.1134/S1064562418010271
[17] Smirnov, Yu. G.; Smolkin, E., Eigenwaves in a lossy metal-dielectric waveguide, Applicable Anal., 97, 1-12 (2018) · Zbl 1394.35490 · doi:10.1080/00036811.2017.1343467
[18] Snyder, A. W.; Love, J., Optical Waveguide Theory (1983), Berlin: Springer, Berlin
[19] Costabel, M., Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19, 613-626 (1988) · Zbl 0644.35037 · doi:10.1137/0519043
[20] L. L. Helms, Introduction to Potential Theory (R. E. Krieger, New York, 1975).
[21] Kolmogorov, A. N.; Fomin, S. V., Elements of the Theory of Functions and Functional Analysis (1999)
[22] Watson, G. N., A Treatise on the Theory of Bessel Functions (1995), London: Cambridge Univ. Press, London · Zbl 0849.33001
[23] Kato, T., Perturbation Theory for Linear Operators (1980), New York: Springer, New York · Zbl 0435.47001
[24] Shestopalov, Yu. V.; Smirnov, Yu. G.; Chernokozhin, E. V., Logarithmic Integral Equations in Electromagnetics (2000), Holland: De Gruyter, Holland
[25] Adams, R. A., Sobolev Spaces (1975), London: Academic, London · Zbl 0314.46030
[26] Il’inskii, A. S.; Smirnov, Yu. G., Diffraction of Electromagnetic Waves on Conductive Thin Screens: Pseudodifferential Operators in Diffraction Problems (1996), Moscow: Radiotehnika, Moscow
[27] Triebel, H., Theory of Function Spaces (1983), Amsterdam: Birkhäuser, Amsterdam · Zbl 0546.46028
[28] Abramowitz, M.; Stegun, I., Handbook of Mathematical Functions with Formulas (1965), Dover, New York: Graphs, and Mathematical Tables, Dover, New York
[29] Gohberg, I. C.; Krein, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space (1965), Moscow: Nauka, Moscow · Zbl 0138.07803
[30] Vladimirov, V. S., Equations of Mathematical Physics (1985), Moscow: Nauka, Moscow
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