Uniqueness and stability results for an inverse spectral problem in a periodic waveguide

Let $\Omega =\omega\times\mathbb R$ where $\omega\subset \mathbb R^2$ be a bounded domain, and $V : \Omega \to\mathbb R$ a bounded potential which is $2\pi$-periodic in the variable $x_{3}\in \mathbb R$. We study the inverse problem consisting in the determination of $V$, through the boundary spectral data of the operator $u\mapsto Au := -\Delta u + Vu$, acting on $L^2(\omega\times(0,2\pi))$, with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for two potentials $V_{1}$ and $V_{2}$ we denote by $(\lambda_{1,k})_{k}$ and $(\lambda_{2,k})_{k}$ the eigenvalues associated to the operators $A_{1}$ and $A_{2}$ (that is the operator $A$ with $V := V_{1}$ or $V:=V_{2}$), then if $\lambda_{1,k} - \lambda_{2,k} \to 0$ as $k \to \infty$ we have that $V_{1} \equiv V_{2}$, provided one knows also that $\sum_{k\geq 1}\|\psi_{1,k} - \psi_{2,k}\|_{L^2(\partial\omega\times[0,2\pi])}^2 < \infty$, where $\psi_{m,k} := \partial\phi_{m,k}/\partial{\bf n}$. We establish also an optimal Lipschitz stability estimate. The arguments developed here may be applied to other spectral inverse problems, and similar results can be obtained

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

International audienceWe 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