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

Let $\mathrm{\Omega }=\omega \times \mathbb{R}$ where $\omega \subset {\mathbb{R}}^2$ be a bounded domain, and $V:\mathrm{\Omega }\to \mathbb{R}$ 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 $u\mapsto Au:=-\mathrm{\Delta }u+Vu$, 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$k \to \infty$wehavethat$V_{1} \equiv V_{2}$, 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.