Uniqueness and stability results for an inverse spectral problem in a periodic waveguide
Résumé
Let Ω = ω × R where ω ⊂ R 2 be a bounded domain, and V : Ω −→ R a bounded potential which is 2π-periodic in the variable x3 ∈ R. We study the inverse problem consisting in the determination of V , through the boundary spectral data of the operator u → Au := −∆u + V u, acting on L 2 (ω × (0, 2π)), with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for two potentials V1 and V2 we denote by (λ 1,k) k and (λ 2,k) k the eigenvalues associated to the operators A1 and A2 (that is the operator A with V := V1 or V := V2), then if λ 1,k −λ 2,k → 0 as k → ∞ we have that V1 ≡ V2, provided one knows also that k≥1 ψ 1,k − ψ 2,k 2 L 2 (∂ω×[0,2π]) < ∞, where ψ m,k := ∂ϕ m,k /∂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.
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