  In this paper we improve an earlier result by Bukhgeim and Uhlmann, by
showing that in dimension larger than or equal to three, the knowledge of the
Cauchy data for the Schr\"odinger equation measured on possibly very small
subsets of the boundary determines uniquely the potential. We follow the
general strategy of Bukhgeim and Uhlmann but use a richer set of solutions to
the Dirichlet problem.
