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Explore the mathematical foundations of electromagnetic waveguides through a comprehensive lecture that examines the quadratic eigenvalue problem for determining axial wave numbers β at given frequencies ω. Begin with a review of fundamental concepts and establish bounds on dispersion curves for heterogeneous waveguides using min-max formulas. Delve into the core challenge of proving that eigenvalues β form a discrete set in the complex plane, learning the abstract framework methods applied to both elastic and electromagnetic waveguides. Analyze the symmetry properties of the spectrum and its location within complex plane sectors, while discovering how the augmented formulation of Maxwell equations simplifies the study of quadratic eigenvalue problems. Examine recent theoretical advances proving the existence of infinitely many eigenvalues at given frequencies, supported by completeness results based on Keldysh theorem. Conclude by investigating inverse modes - propagating modes where phase velocity and group velocity have opposite signs - and understand how heterogeneity influences their appearance through the augmented formulation approach. This advanced mathematical treatment builds upon foundational electromagnetic theory to address the non-self-adjoint characteristics typical of waveguide eigenvalue problems, drawing parallels with elastic waveguide behavior and Lamb modes.
