Microlocal analysis of scattering data for nested conormal potentials (Q765921)

The author considers the potential scattering problem for the wave equation (\(n\geq 3\)) \[ (\partial_t^2-\Delta+q)u=0,\,\,\,\,(x,t)\in\mathbb R^n\times\mathbb R,\,\,\,\,u=\delta(t-x\cdot\omega),\,\,t\ll -\rho, \] where \(q\) is a compactly supported time-independent potential and \(\omega\in\mathbb S^{n-1}\) is varying, \(\rho\) being any given value such that \( \text{supp}\,q\subset\{|x|\leq\rho\}.\) The class of potentials \(q\) considered here is that of potentials having singularities conormal to a nested pair of submanifolds \(S_2\subset S_1\) of \(\mathbb R^n\) of arbitrary condimension. The inverse problem the author solves here is the one of determining \(S_1\) and \(S_2\) and the principal symbol of \(q\) (which suffices to determine the singularities of \(q\)) from the leading singularities of the back scattering distribution \(\alpha\bigr|_{\mathbb B}\), where \(\mathbb B=\{\theta=-\omega\}\subset\mathbb R\times\mathbb S^{n-1}\times\mathbb S^{n-1}\) and where \[ \alpha(t,\theta,\omega)=\lim_{r\to +\infty}r^{(n-1)/2}\partial_t u(t+r,r\theta,\omega). \] The author shows that, away from those \(\omega\)'s that are tangent to either of the submanifolds, \(\alpha\) is the sum of a paired Lagrangian distribution associated with two cleanly intersecting reflected Lagrangians, two reflected Lagrangian distributions, and a single peak Lagrangian distribution, modulo Sobolev errors. Although, as is well known in the physics literature, the strongest singularity lies on the peak Lagrangian, it is shown that it is the restriction of the reflected Lagrangians and their points of intersection to various submanifolds of scattering data in \(\mathbb R\times\mathbb S^{n-1}\times\mathbb S^{n-1}\) that determine the singularities of \(q\). The paper generalizes, closely following it, a result of \textit{A. Greenleaf} and \textit{G. Uhlmann} [Commun. Math. Phys. 157, No. 3, 549--572 (1993; Zbl 0790.35112)] to the more complicated geometry of a nested \(q\).