Department of Mathematics
The spectral problem for a class of highly oscillatory Fredholm integral operators
Let ℱω be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K0(x, y)e iωx-y. We study the spectral problem for large ω, 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;ω) to the associated Fredholm integral equation f = μℱωf + a as ω→∞, 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.
Asymptotic behaviour of highly oscillatory solutions, Asymptotic behaviour of spectrum, Complex-symmetric Fredholm integral operator, Fredholm integral equations, Highly oscillatory kernel
Source Publication Title
IMA Journal of Numerical Analysis
Oxford University Press
Link to Publisher's Edition
Brunner, Hermann, Arieh Iserles, and Syvert P. Nørsett. "The spectral problem for a class of highly oscillatory Fredholm integral operators." IMA Journal of Numerical Analysis 30.1 (2010): 108-130.