  We prove that the Dirichlet eigenvalues and Neumann boundary data of the
corresponding eigenfunctions of the operator $-\Delta + q$, determine the
potential $q$, when $q \in L^{n/2}(\Omega,\mathbb{R})$ and $n \geq 3$. We also
consider the case of incomplete spectral data, in the sense that the above
spectral data is unknown for some finite number of eigenvalues. In this case we
prove that the potential $q$ is uniquely determined for $q \in
L^p(\Omega,\mathbb{R})$ with $p=n/2$, for $n\geq4$ and $p>n/2$, for $n=3$.
