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

Abstract

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 ${\sum }_{k\ge 1}\Vert {\psi }_{1,k}-{\psi }_{2,k}{\Vert }_{L^2(\partial \omega \times [0,2\pi ])}^2<\infty$, where ${\psi }_{m,k}:=\partial {\varphi }_{m,k}/\partial \mathbf{n}$. 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.