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IMA Journal of Numerical Analysis Advance Access published online on December 10, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn060
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

The spectral problem for a class of highly oscillatory Fredholm integral operators

Hermann Brunner

Department of Mathematics and Statistics, Memorial University of Newfoundland, St John's, NL, Canada A1C 5S7

Arieh Iserles{dagger}

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, UK

Syvert P. Nørsett

Department of Mathematics, Norwegian University of Science and Technology, Trondheim N-7491, Norway

{dagger} Corresponding author. Email: ai{at}damtp.cam.ac.uk

Received on 14 April 2008. Revised on 6 September 2008.


  Abstract

Let Formula be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K0(x, y)ei{omega}|x–y|. We study the spectral problem for large {omega}, showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f(x;{omega}) to the associated Fredholm integral equation f = µFormula f+a as {omega} -> {infty}, thus refining a classical result by Ursell. Possible extensions of these results to highly oscillatory Fredholm integral operators with more general highly oscillating kernels are also discussed.

Key Words: complex-symmetric Fredholm integral operator; highly oscillatory kernel; asymptotic behaviour of spectrum; Fredholm integral equations; asymptotic behaviour of highly oscillatory solutions


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