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

TL;DR

This paper proves that the potential in a periodic waveguide can be uniquely determined and stably reconstructed from boundary spectral data, with results applicable to other spectral inverse problems.

Contribution

It establishes uniqueness and stability results for an inverse spectral problem in a periodic waveguide, including an optimal Lipschitz stability estimate.

Findings

Unique determination of potential from spectral data.

Asymptotic eigenvalue difference implies potential equality.

Established Lipschitz stability estimate.

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 $\sum_{k\geq 1}\|\psi_{1,k} - \psi_{2,k}\|_{L^2(\partial\omega\times[0,2\pi])}^2 < \infty$, 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.