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

Yavar Kian (1)

1 CPT - Centre de Physique Théorique - UMR 7332

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∇ + A) 2 + V on a bounded domain Ω ⊂ R n , n ≥ 2, from partial knowledge of the boundary spectral data of H. The full boundary spectral data are given by the set {(λ k , ∂ν ϕ k |∂Ω) : k ≥ 1}, where {λ k : k ∈ N * } is the non-decreasing sequence of eigenvalues of H, {ϕ k : k ∈ N * } an associated Hilbertian basis of eigenfunctions and ν is the unit outward normal vector to ∂Ω. We prove that some asymptotic knowledge of (λ k , ∂ν ϕ k |∂Ω) with respect to k ≥ 1 determines uniquely the magnetic field dA and the electric potential V .

Mots clés: 35J10; secondary: 35P99; Borg-Levinson theorem; magnetic Schrödinger operators 2010 AMS Subject Classification: Primary: 35R30; Inverse spectral problem

Domaines: Equations aux dérivées partielles [math.AP]

Yavar Kian. A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data. 2015. ⟨hal-01143482v2⟩