Recovering the singularities of a potential from Backscattering data

Cristóbal J. Meroño

Published 2017-09-03Version 1

We prove that, up to dimension 7, the main singularities of a complex potential $q$ having some a priori regularity are contained in the Born approximation $q_B$ constructed from backscattering data. In particular in dimension 2,3,4 we show that under certain assumptions $q-q_B$ is one derivative more regular than $q$ and that this result is optimal. These results are based on new Sobolev estimates of the double, triple and quadruple dispersion operators in general dimension $n$. We construct low regularity counterexamples which among other things show that when $n>4$ the double dispersion operator can have worse Sobolev regularity than the potential, even if $q$ is compactly supported, radial and real. We also give H\"older estimates of the double dispersion operator for $n\ge3$, known up to now only in dimension 2.