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Let $\Omega =\omega\times\mathbb R$ where $\omega\subset \mathbb R^2$ be abounded 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 inverseproblem consisting in the determination of $V$, through the boundary spectraldata of the operator $u\mapsto Au := -\Delta u + Vu$, acting on$L^2(\omega\times(0,2\pi))$, with quasi-periodic and Dirichlet boundaryconditions. More precisely we show that if for two potentials $V_{1}$ and$V_{2}$ we denote by $(\lambda_{1,k})_{k}$ and $(\lambda_{2,k})_{k}$ theeigenvalues associated to the operators $A_{1}$ and $A_{2}$ (that is theoperator $A$ 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 Lipschitzstability estimate. The arguments developed here may be applied to otherspectral inverse problems, and similar results can be obtained.

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