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

Abstract: 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 $V$.