## Abstract

Let ℱ_{ω} be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K_{0}(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 = μℱ_{ω}f + 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.

Original language | English |
---|---|

Pages (from-to) | 108-130 |

Number of pages | 23 |

Journal | IMA Journal of Numerical Analysis |

Volume | 30 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2010 |

## Scopus Subject Areas

- Mathematics(all)
- Computational Mathematics
- Applied Mathematics

## User-Defined Keywords

- Asymptotic behaviour of highly oscillatory solutions
- Asymptotic behaviour of spectrum
- Complex-symmetric Fredholm integral operator
- Fredholm integral equations
- Highly oscillatory kernel