TY - JOUR

T1 - Spectral analysis of integral-differential operators applied in linear antenna modeling

AU - Bekers, D.J.

AU - Eijndhoven, van, S.J.L.

PY - 2012

Y1 - 2012

N2 - The current on a linear strip or wire solves an equation governed by a linear integro-differential operator that is the composition of the Helmholtz operator and an integral operator with a logarithmically singular displacement kernel. Investigating the spectral behaviour of this classical operator, we first consider the composition of the second-order differentiation operator and the integral operator with logarithmic displacement kernel. Employing methods of an earlier work by J. B. Reade, in particular the Weyl–Courant minimax principle and properties of the Chebyshev 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 integro-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.
Keywords: eigenvalue problems; integro-differential operators; logarithmic kernel; linear antennas

AB - The current on a linear strip or wire solves an equation governed by a linear integro-differential operator that is the composition of the Helmholtz operator and an integral operator with a logarithmically singular displacement kernel. Investigating the spectral behaviour of this classical operator, we first consider the composition of the second-order differentiation operator and the integral operator with logarithmic displacement kernel. Employing methods of an earlier work by J. B. Reade, in particular the Weyl–Courant minimax principle and properties of the Chebyshev 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 integro-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.
Keywords: eigenvalue problems; integro-differential operators; logarithmic kernel; linear antennas

U2 - 10.1017/S0013091510001331

DO - 10.1017/S0013091510001331

M3 - Article

SN - 0013-0915

VL - 55

SP - 333

EP - 354

JO - Proceedings of the Edinburgh Mathematical Society. Series II

JF - Proceedings of the Edinburgh Mathematical Society. Series II

IS - 2

ER -