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

Let
be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K₀(x, y)e^i|x–y|. We study the spectral problem for large , 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;) to the associated Fredholm integral equation f = µ
f + a as $$omega o mathrm{infty }$$, 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.

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Original source : http://imajna.oxfordjournals.org/cgi/content/short...