American Institute of Mathematical Sciences

June  2019, 9(2): 289-312. doi: 10.3934/mcrf.2019015

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

 1 Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France 2 Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France 3 Department of Mathematics, University College London, London, UK

* Corresponding author

Received  April 2017 Revised  January 2018 Published  November 2018

We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schrödinger 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 ${\rm BSD}(q, \gamma): = \{(\lambda_{k}, {\partial_\nu \varphi_{k}}_{|\gamma}):\ k \geq1\}$, where $\{ \lambda_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ \varphi_k:\ k \geq1 \}$ an associated orthonormal basis of eigenfunctions, and $\nu$ is the unit outward normal vector to $\partial\Omega$. Our main result consists of determining a bounded potential $q\in L^\infty(\Omega)$ from the data ${\rm BSD}(q, \gamma)$. 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.

Citation: Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control & Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015
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References:
Geometric condition (3.10)
Sets $A_{x, \varepsilon}$
Support of the geometric optics solution
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