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Spectral analysis of integro-differential operators applied in linear antenna modelling

Part of: Ordinary differential operators Eigenvalue problems

Published online by Cambridge University Press:  12 April 2012

Dave J. Bekers  and
Stephanus J. L. van Eijndhoven
Dave J. Bekers
Affiliation:
TNO Defence, Security and Safety, Business Unit Observation Systems, 2597 AK Den Haag, The Netherlands ([email protected])
Stephanus J. L. van Eijndhoven
Affiliation:
Faculty of Mathematics and Computer Science, Technische Universiteit Eindhoven, 5600 MB, Eindhoven, The Netherlands ([email protected])
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Abstract

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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 problemsintegro-differential operatorslogarithmic kernellinear antennas

MSC classification

Secondary: 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 45C05: Eigenvalue problems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 333 - 354
Copyright
Copyright © Edinburgh Mathematical Society 2012

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