  We prove that in dimension $n\ge 2$ the main singularities of a complex
potential $q$ having a certain a priori regularity are contained in the Born
approximation $q_{B}$ constructed from backscattering data. This is archived
using a new explicit formula for the multiple dispersion operators in the
Fourier transform side. We also show that ${q-q_{B}}$ can be up to one
derivative more regular than $q$ in the Sobolev scale. On the other hand, we
construct counterexamples showing that in general it is not possible to have
more than one derivative gain, sometimes even strictly less, depending on the a
priori regularity of $q$.
