  We consider the multidimensional Borg-Levinson problem of determining a
potential $q$, appearing in the Dirichlet realization of the Schr\"odinger
operator $A_q=-\Delta+q$ on a bounded domain $\Omega\subset \mathbb{R}^n$,
$n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of
$\partial\Omega$. More precisely, for $\gamma$ an open and non-empty subset of
$\partial\Omega$, we consider the boundary spectral data on $\gamma$ given by
$\mathrm{BSD}(q,\gamma):=\{(\lambda_{k},{\partial_\nu
\phi_{k}}_{|\overline{\gamma}}):\ k \geq1\}$, where $\{ \lambda_k:\ k \geq1\}$
is the non-decreasing sequence of eigenvalues of $A_q$, $\{ \phi_k:\ k \geq1
\}$ an associated Hilbertian basis of eigenfunctions, and $\nu$ is the unit
outward normal vector to $\partial\Omega$. We prove that the data
$\mathrm{BSD}(q,\gamma)$ uniquely determine a bounded potential $q\in
L^\infty(\Omega)$. Previous uniqueness results, with arbitrarily small
$\gamma$, assume that $q$ is smooth. Our approach is based on the Boundary
Control method, and we give a self-contained presentation of the method,
focusing on the analytic rather than geometric aspects of the method.
