Inverse fixed angle scattering and backscattering problems in two dimensions

The inverse backscattering and fixed angle scattering problems for the two-dimensional Schrodinger operator are studied. We prove new formulae for the first nonlinear terms in both problems and sharpen the estimates of this term. We prove also that all other terms in the Born series are C-alpha functions for the fixed angle scattering problem and they are H-t functions for the backscattering problem. These formulae and estimates allow us to conclude that all main singularities of the unknown potential can be recovered from the Born approximation. Especially, we show for the potentials in L-p for certain values of p that the approximation agrees with the true potential up to H-t function.