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A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data

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 Schrödinger operator $H=(-{\rm i}\nabla+A)^2+V$ on a bounded domain $Ω\subset\mathbb R^n$, $n\geq2$, from partial knowledge of the boundary spectral data of $H$. The full boundary spectral data are given by the set $\{(λ_{k},{\partial_νϕ_{k}}_{|\partialΩ}):\ k\geq1\}$, where $\{ λ_k:\ k\in \mathbb N^* \}$ is the non-decreasing sequence of eigenvalues of $H$, $\{ ϕ_k:\ k\in \mathbb N^* \}$ an associated Hilbertian basis of eigenfunctions and $ν$ is the unit outward normal vector to $\partialΩ$. We prove that some asymptotic knowledge of $(λ_{k},{\partial_νϕ_{k}}_{|\partialΩ})$ with respect to $k\geq1$ determines uniquely the magnetic field $dA$ and the electric potential $V$.