Microlocal analysis of scattering data for nested conormal potentials

Abstract

Working in the time domain, we show that the location of the singularities and the principal symbol of a potential that is conormal to nested submanifolds $S_2\subset S_1\subset {\mathbb{R}}^n$, for $n\ge 3$, can be recovered from the backscattering as well as from the restriction of the far-field pattern to more general determined sets of scattering data. This extends the work of Greenleaf and Uhlmann where the potentials considered are conormal to a single submanifold $S\subset {\mathbb{R}}^n$. We utilize the microlocal analysis of the wave operator $\square ={\partial }_t^2-{△}_x$ and multiplication by a nested conormal distribution in order to study their action on spaces of conormal-like distributions.

arXiv comment52 pages, 1 figure; to appear in Journal of Functional Analysis