Cones with convoluted geometry that always scatter or radiate
Abstract
We investigate fixed energy scattering from conical potentials having an irregular crosssection. The incident wave can be any arbitrary nontrivial Herglotz wave. We show that a large number of such local conical scatterers scatter all incident waves, meaning that the farfield will always be nonzero. In essence there are no incident waves for which these potentials would seem transparent at any given energy. We show more specifically that there is a large collection of starshaped cones whose local geometries always produce a scattered wave. In fact, except for a countable set, all cones from a family of deformations between a circular and a starshaped cone will always scatter any nontrivial incident Herglotz wave. Our methods are based on the use of spherical harmonics and a deformation argument. We also investigate the related problem for sources. In particular if the support of the source is locally a thin cone, with an arbitrary crosssection, then it will produce a nonzero farfield.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.06044
 Bibcode:
 2021arXiv211006044B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 35Q40 35P25 81U40