# Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

1 CPT - E8 Dynamique quantique et analyse spectrale
CPT - Centre de Physique Théorique - UMR 7332
Abstract : We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schr\"odinger operator $A_q=-\Delta+q$ on a bounded domain $\Omega\subset \mathbb{R}^n$, $n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of $\partial\Omega$. More precisely, for $\gamma$ an open and non-empty subset of $\partial\Omega$, we consider the boundary spectral data on $\gamma$ given by $\mathrm{BSD}(q,\gamma):=\{(\lambda_{k},{\partial_\nu \phi_{k}}_{|\overline{\gamma}}):\ k \geq1\}$, where $\{ \lambda_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ \phi_k:\ k \geq1 \}$ an associated Hilbertian basis of eigenfunctions, and $\nu$ is the unit outward normal vector to $\partial\Omega$. We prove that the data $\mathrm{BSD}(q,\gamma)$ uniquely determine a bounded potential $q\in L^\infty(\Omega)$. Previous uniqueness results, with arbitrarily small $\gamma$, assume that $q$ is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.
Keywords :
Document type :
Journal articles

Cited literature [27 references]

https://hal.archives-ouvertes.fr/hal-01495713
Contributor : Morgan Morancey <>
Submitted on : Sunday, March 26, 2017 - 6:15:54 PM
Last modification on : Monday, March 4, 2019 - 2:04:22 PM
Document(s) archivé(s) le : Tuesday, June 27, 2017 - 1:48:38 PM

### File

KianMoranceyOksanen_BoundaryCo...
Files produced by the author(s)

### Citation

Y Kian, M Morancey, L Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control and Related Fields, AIMS, In press, ⟨10.3934/mcrf.2019015⟩. ⟨hal-01495713⟩

Record views