Does Atkinson – Wilcox expansion converges for any convex obstacle?

The Atkinson-Wilcox theorem claims that any scattered eld in the exterior of a sphere can be expanded into a uniformly and absolutely convergent series in inverse powers of the radial variable and that once the leading coecient of the expansion is known the full series can be recovered uniquely through a recurrence relation. The leading coecient of the series is known as the scattering amplitude or the far eld pattern of the radiating eld. In this work we give a simple characterization of the strictly convex domains, such that a reasonable generalization of the Atkinson- Wilcox expansion converges uniformly in the corresponding exterior domain. All these strictly convex domains are spheres.