  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.
