Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2613710.22363/2658-4670-2021-29-1-14-21Research ArticleNormal modes of a waveguide as eigenvectors of a self-adjoint operator pencilMalykhMikhail D.<p>Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics</p>malykh_md@pfur.ruPeoples’ Friendship University of Russia (RUDN University)30032021291142130032021Copyright © 2021, Malykh M.D.2021<p style="text-align: justify;">A waveguide with a constant, simply connected section <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math>, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.</p>waveguidenormal modeshybridization of normal modeseigenvalue problemquadratic operator pencilsволноводнормальные модыгибридизация нормальных модзадача на собственные значенияквадратичные пучки<p>Introduction Consider a waveguide representing a cylinder of constant cross-section</p>[A. G. Sveshnikov and I. E. Mogilevsky, Mathematical problems in the theory of diffraction [Matematicheskiye zadachi teorii difraktsii]. Moscow: MSU, 2010, in Russian.][K. Zhang and D. Li, Electromagnetic theory for microwaves and optoelectronics, 2nd ed. Berlin: Springer, 2008.][A. N. Bogolyubov, A. L. Delitsyn, and A. G. Sveshnikov, “On the completeness of the set of eigen- and associated functions of a waveguide,” Computational Mathematics and Mathematical Physics, vol. 38, no. 11, pp. 1815-1823, 1998.][A. N. 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