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

Let ℱ_ω 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 = μℱ_ωf + 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.