The Atkinson-Wilcox theorem in ellipsoidal geometry

The famous Atkinson-Wilcox theorem claims that any scattered field, no matter what the boundary conditions on the surface of the scatterer are, can be expanded into a uniformly and absolutely convergent series in inverse powers of distance and that once the leading coefficient of the expansion is known the full series can be recovered up to the smallest sphere containing the scatterer in its interior. The leading coefficient of the series is nothing else but the scattering amplitude. This is a very useful theorem, which provides the exact analogue of the Sommerfeld radiation condition, but it has the disadvantage of recovering the scattered field only outside the sphere circumscribing the scatterer. This means that an elongated obstacle which has a very large, as it compares to its volume, circumscribing sphere leaves a lot of exterior space where the scattered field cannot be recovered from its scattering amplitude. In the present work the Atkinson-Wilcox theorem has been extended to the ellipsoidal system where the theorem as well as the relative recovering algorithm holds true all the way down to the smallest circumscribing ellipsoid. Considering the anisotropic character of the ellipsoidal geometry it is obvious that an appropriately chosen ellipsoid can fit almost every smooth convex obstacle. Furthermore, such a result offers the best opportunity to develop a hybrid method based on the theory of infinite elements. Two orientations dependent differential operators are introduced in the recurrence scheme which, as the ellipsoid degenerates to a sphere, one of them vanishes, while the other reduces to the Beltrami operator. A reduction to spherical geometry is also included.