Multidimensionnel Borg-Levinson uniqueness and stability results for the Robin Laplacian with unbounded potential

This article deals with the uniqueness and stability issues in the inverse problem of determining the unbounded potential of the Schr\"odinger operator in a bounded domain of dimension 3 or greater, endowed with Robin boundary condition, from knowledge of its boundary spectral data. These data are defined by the pairs formed by the eigenvalues and either full or partial Dirichlet measurement of the eigenfunctions on the boundary of the domain

Stability of determining a Dirichlet-Laplace-Beltrami operator from its boundary spectral data

We establish stability inequalities for the problem of determining a Dirichlet-Laplace-Beltrami operator from its boundary spectral data. We study the case of complete spectral data as well as the case of partial spectral data

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

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

Quantitative strong unique continuation for elliptic operators -- application to an inverse spectral problem

Based on the three-ball inequality and the doubling inequality established in [24], we quantify the strong unique continuation for an elliptic operator with unbounded lower order coefficients. This result is then used to improve the quantitative unique continuation from a set of positive measure obtained in [24]. We also derive a uniform quantitative strong unique continuation for eigenfunctions that we use to prove that two Dirichlet-Laplace-Beltrami operators are gauge equivalent whenever their corresponding metrics coincide in the vicinity of the boundary and their boundary spectral data coincide on a subset of positive measure

Stability in Non-Normal Periodic Jacobi Operators: Advancing B\"org's Theorem

Periodic Jacobi operators naturally arise in numerous applications, forming a cornerstone in various fields. The spectral theory associated with these operators boasts an extensive body of literature. Considered as discretized counterparts of Schr\"odinger operators, widely employed in quantum mechanics, Jacobi operators play a crucial role in mathematical formulations. The classical uniqueness result by G. B\"org in $1946$ occupies a significant place in the literature of inverse spectral theory and its applications. This result is closely intertwined with M. Kac's renowned article, 'Can one hear the shape of a drum?' published in $1966$. Since $1975,$ discrete versions of B\"org's theorem have been available in the literature. In this article, we concentrate on the non-normal periodic Jacobi operator and the discrete versions of B\"org's Theorem. We extend recently obtained stability results to encompass non-normal cases. The existing stability findings establish a correlation between the oscillations of the matrix entries and the size of the spectral gap. Our result encompasses the current self-adjoint versions of B\"org's theorem, including recent quantitative variations. Here, the oscillations of the matrix entries are linked to the path-connectedness of the pseudospectrum. Additionally, we explore finite difference approximations of various linear differential equations as specific applications