A multidimensional Borg-Levinson theorem for magnetic Schr\"odinger operators with partial spectral data

TL;DR

This paper proves that partial spectral data of a magnetic Schrödinger operator in a bounded domain uniquely determine the magnetic field and electric potential, advancing inverse spectral problem understanding.

Contribution

It establishes a multidimensional Borg-Levinson theorem using asymptotic boundary spectral data to uniquely identify magnetic and electric potentials.

Findings

Unique determination of magnetic field and electric potential from partial spectral data.

Asymptotic knowledge of spectral data suffices for reconstruction.

Advances inverse spectral theory for magnetic Schrödinger operators.

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 $V$.