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

Let $Ω=ω\times\mathbb R$ where $ω\subset \mathbb R^2$ be a bounded domain, and $V : Ω\to\mathbb R$ a bounded potential which is $2π$-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 $u\mapsto Au := -Δu + Vu$, acting on $L^2(ω\times(0,2π))$, with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for two potentials $V_{1}$ and $V_{2}$ we denote by $(λ_{1,k})_{k}$ and $(λ_{2,k})_{k}$ 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 $λ_{1,k} - λ_{2,k} \to 0$ as $k \to \infty$ we have that $V_{1} \equiv V_{2}$, provided one knows also that $\sum_{k\geq 1}\|ψ_{1,k} - ψ_{2,k}\|_{L^2(\partialω\times[0,2π])}^2 < \infty$, where $ψ_{m,k} := \partialϕ_{m,k}/\partial{\bf n}$. 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.