  This article is concerned with uniqueness and stability issues for the
inverse spectral problem of recovering the magnetic field and the electric
potential in a Riemannian manifold from some asymptotic knowledge of the
boundary spectral data of the corresponding Schr\"odinger operator under
Dirichlet boundary conditions. The spectral data consist of some asymptotic
knowledge of a subset of eigenvalues and Neumann traces of the associated
eigenfunctions of the magnetic Laplacian. We also address the same question for
Schr\"odinger operators under Neumann boundary conditions, in which case we
measure the Dirichlet traces of eigenfunctions. In our results we characterize
the uniqueness of the magnetic field from a rate of growth of the eigenvalues,
combined with suitable asymptotic properties of boundary observation of
eigenfunctions, of the associated magnetic Schr\"odinger operator. To our best
knowledge this is the first result proving uniqueness from such general
asymptotic behavior of boundary spectral data.
