We prove the existence of complex eigenfrequencies of open waveguide resonators in the form of parallel-plate waveguides and waveguides of rectangular crosssection containing layered dielectric inclusions. It is shown that complex eigenfrequencies are finite-multiplicity poles of the analytical continuation of the operator of the initial diffraction problem and its Green's function to a multi-sheet Riemann surface, and also of the transmission coefficient extended to the complex plane of some of the problem parameters. The eigenfrequencies are associated with resonant states (RSs) and eigenvalues of distinct families of Sturm-Liouville problems on the line; they form countable sets of points in the complex plane with the only accumulation point at infinity and depend continuously on the problem parameters. The set of complex eigenfrequencies is similar in its structure to the set of eigenvalues of a Laplacian in a rectangle. The presence of a resonance domain in the form of a parallel-plane layered dielectric insert removes the continuous frequency spectrum and gives rise to a discrete set of points shifted to (upper half of) the complex plane.
2. Shestopalov, Y., Y. Smirnov, and E. Derevyanchuk, "Inverse problem method for complex permittivity reconstruction of layered media in a rectangular waveguide," Phys. Status Solidi C, 1-6, 2014.
3. Gamow, G., "Zur quantentheorie des atomkernes," Z. Phys., Vol. 51, 204-212, 1928.
4. Armitage, L. J., et al., "Resonant-state expansion applied to planar waveguides," Phys. Rev. A, Vol. 89, 053831, 2014.
5. Shestopalov, Y. and V. Shestopalov, Spectral Theory and Excitation of Open Structures, The IEE Peter Peregrinus, London, 1996.
6. Brovenko, A., P. Melezhik, and A. Poedinchuk, "Spectral problems in the theory of diffraction of waves on layered media," Telecommunications and Radio Engineering, Vol. 72, 1821-1838, 2013.
7. Colton, A. and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998.
8. Shestopalov, V. and Y. Sirenko, Dynamical Theory of Gratings, Naukova Dumka, Kiev, 1989.