  The problem of the recovery of a real-valued potential in the two-dimensional
Schrodinger equation at positive energy from the Dirichlet-to-Neumann map is
considered. It is know that this problem is severely ill-posed and the
reconstruction of the potential is only logarithmic stable in general. In this
paper a new stability estimate is proved, which is explicitly dependent on the
regularity of the potentials and on the energy. Its main feature is an
efficient increasing stability phenomenon at sufficiently high energies: in
some sense, the stability rapidly changes from logarithmic type to Holder type.
The paper develops also several estimates for a non-local Riemann-Hilbert
problem which could be of independent interest.
