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We develop the Riemann-Hilbert problem approach to inverse scattering for thetwo-dimensional Schrodinger equation at fixed energy. We obtain global orgeneric versions of the key results of this approach for the case of positiveenergy and compactly supported potentials. In particular, we do not assume thatthe potential is small or that Faddeev scattering solutions do not havesingularities (i.e. we allow the Faddeev exceptional points to exist).Applications of these results to the Novikov-Veselov equation are alsoconsidered.

A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy