The problem of recovering the singularities of a potential from backscattering data is studied. Let $\Omega$ be a smooth precompact domain in ${ℝ}^n$ which is convex (or normally accessible). Suppose $V_i=v+w_i$ with $v\in C_c^{\infty }({ℝ}^n)$ and $w_i$ conormal to the boundary of $\Omega$ and supported inside $\bar{\Omega }$ then if the backscattering data of $V_1$ and $V_2$ are equal up to smoothing, we show that $w_1-w_2$ is smooth.

2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38.We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result to describing...