Microlocal analysis of scattering data for nested conormal potentials

TL;DR

This paper demonstrates how to recover the location and principal symbol of nested conormal potentials from scattering data using microlocal analysis, extending previous work on single submanifold conormal potentials.

Contribution

It extends microlocal analysis techniques to nested conormal potentials, enabling recovery of singularities and symbols from various scattering data sets.

Findings

Recovery of singularities and principal symbols from backscattering data.

Extension of previous single submanifold results to nested submanifolds.

Application of microlocal analysis to wave operators and conormal distributions.

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 $n \geq 3$, 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 $\square=\partial_t^2 - \triangle_x$ and multiplication by a nested conormal distribution in order to study their action on spaces of conormal-like distributions.