Spectral Properties of Integral Differential Operators Applied in Linear Antenna Modeling
article
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.
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.
TNO Identifier
463706
Source
Proceedings of the Edinburgh Mathematical Society, 55(June), pp. 333-354.
Pages
333-354
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