  We study the inverse problem of determining a real-valued potential in the
two-dimensional Schr\"odinger equation at negative energy from the
Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a
stability estimate of logarithmic type holds. In this paper we prove three new
stability estimates. The main feature of the first one is that the stability
increases exponentially with respect to the smoothness of the potential, in a
sense to be made precise. The others show how the first estimate depends on the
energy, for low and high energies (in modulus). In particular it is found that
for high energies the stability estimate changes, in some sense, from
logarithmic type to Lipschitz type: in this sense the ill-posedness of the
problem decreases when increasing the energy (in modulus).
