  We consider the multidimensional Borg-Levinson theorem of determining both
the magnetic field $dA$ and the electric potential $V$, appearing in the
Dirichlet realization of the magnetic Schr\"odinger operator $H=(-{\rm
i}\nabla+A)^2+V$ on a bounded domain $\Omega\subset\mathbb R^n$, $n\geq2$, from
partial knowledge of the boundary spectral data of $H$. The full boundary
spectral data are given by the set $\{(\lambda_{k},{\partial_\nu
\phi_{k}}_{|\partial\Omega}):\ k\geq1\}$, where $\{ \lambda_k:\ k\in \mathbb
N^* \}$ is the non-decreasing sequence of eigenvalues of $H$, $\{ \phi_k:\ k\in
\mathbb N^* \}$ an associated Hilbertian basis of eigenfunctions and $\nu$ is
the unit outward normal vector to $\partial\Omega$. We prove that some
asymptotic knowledge of $(\lambda_{k},{\partial_\nu
\phi_{k}}_{|\partial\Omega})$ with respect to $k\geq1$ determines uniquely the
magnetic field $dA$ and the electric potential $V$.
