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 Schrödinger operator $H=(-i\nabla +\mathrm{A}{)}^2+\mathrm{V}$ on a bounded domain $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, from partial knowledge of the boundary spectral data of $H$. The full boundary spectral data are given by the set $\{({\lambda }_k,{{\partial }_{\nu }{\varphi }_k}_{|\partial \Omega }):k\ge 1\}$, where $\{{\lambda }_k:k\ge 1\}$ is the non-decreasing sequence of eigenvalues of $H$, $\{{\varphi }_k:k\ge 1\}$ an associated Hilbertian basis of eigenfunctions and $\nu$ is the unit outward normal vector to $\partial \Omega$. We prove that some asymptotic knowledge of $({\lambda }_k,{{\partial }_{\nu }{\varphi }_k}_{|\partial \Omega })$ with respect to $k\ge 1$ determines uniquely the magnetic field $dA$ and the electric potential $V$.