Recovering the total singularity of a conormal potential from backscattering data
It is proved that in special cases the total singularity of the potential with a singularity conormal to a hypersurface bounding a smooth convex domain is determined by backscattering data. Using the calculus of paired Lagrangian distributions the author manages to refine the results by \textit{A. Greenleaf} and \textit{G. Uhlmann} [J. Func. Anal. 89, 202-232 (1990; Zbl 0717.44001)]. In particular the following theorem is proved: Let \( \Omega \subset {\mathbb{R}}^n \) be a normally accessible, precompact domain with smooth boundary, \( \partial \Omega\), in \( {\mathbb{R}}^n \) with \( n \geq 3.\) Suppose \( q_i = v + w_i, i=1,2 \) with \( v \in C^\infty_c ({\mathbb{R}}^n)\), \( w_i \in I^\mu_{phg} ({\mathbb{R}}^n, N^* (\partial \Omega)), \text{ supp} (w_i) \subset \overline{\Omega} \) with \[ \mu + 1/2 - n/4 < \gamma_n \] and let \( \beta_i (s,w) \) be the associated back-scattering kernels. If \( \beta_1 (s,w) - \beta_2 (s,w) \) is smooth, then \( w_1 - w_2 \in C^\infty\).
scientific article; zbMATH DE number 1227220
scientific article