Spectral Analysis of Integral-differential Operators Applied in Linear Antenna Modeling

The current on a linear strip or wire solves an equation governed by a linear integraldifferential operator that is the composition of the Helmholtz operator and an integral operator with logarithmically singular displacement kernel. We investigate the spectral behavior of this classical operator, particularly because various methods of analysis and solution rely on asymptotic properties of spectra, while no investigations of the spectrum of this operator seem to exist. In our approach, we first consider the composition of the second order differentiation operator and the integral operator with logarithmic displacement kernel. Employing the Weyl-Courant minimax principle and properties of the Čebysev polynomials of the first and second kind, we derive index-dependent bounds for the ordered sequence of eigenvalues of this operator and specify their ranges of validity. Additionally we derive bounds for the eigenvalues of the integral operator with logarithmic kernel. With slight modification our result extends to kernels that are the sum of the logarithmic displacement kernel and a real displacement kernel whose second derivative is square integrable. Employing this extension we derive bounds for the eigenvalues of the integral-differential operator of a linear strip with the complex kernel replaced by its real part. Finally, for specific geometry and frequency settings, we present numerical results for the eigenvalues of the considered operators using Ritz’s methods with respect to finite bases.