# 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 :
Type de document :
Pré-publication, Document de travail
2017

Littérature citée [29 références]

https://hal.archives-ouvertes.fr/hal-01495713
Contributeur : Morgan Morancey <>
Soumis le : dimanche 26 mars 2017 - 18:15:54
Dernière modification le : mardi 20 mars 2018 - 16:55:00
Document(s) archivé(s) le : mardi 27 juin 2017 - 13:48:38

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• HAL Id : hal-01495713, version 1

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Y Kian, M Morancey, L Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. 2017. 〈hal-01495713〉

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