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

Abstract

We consider the multidimensional Borg-Levinson problem of determining a potential \begin{document}$q$\end{document} , appearing in the Dirichlet realization of the Schrodinger operator \begin{document}$A_q = -\Delta+q$\end{document} on a bounded domain \begin{document}$\Omega\subset\mathbb{R}^n$\end{document} , \begin{document}$n\geq2$\end{document} , from the boundary spectral data of \begin{document}$A_q$\end{document} on an arbitrary portion of \begin{document}$\partial\Omega$\end{document} . More precisely, for \begin{document}$\gamma$\end{document} an open and non-empty subset of \begin{document}$\partial\Omega$\end{document} , we consider the boundary spectral data on \begin{document}$\gamma$\end{document} given by \begin{document}${\rm BSD}(q, \gamma): = \{(\lambda_{k}, {\partial_\nu \varphi_{k}}_{|\gamma}): k \geq1\}$\end{document} , where \begin{document}$\{ \lambda_k: k \geq1\}$\end{document} is the non-decreasing sequence of eigenvalues of \begin{document}$A_q$\end{document} , \begin{document}$\{ \varphi_k: k \geq1 \}$\end{document} an associated orthonormal basis of eigenfunctions, and \begin{document}$\nu$\end{document} is the unit outward normal vector to \begin{document}$\partial\Omega$\end{document} . Our main result consists of determining a bounded potential \begin{document}$q\in L^\infty(\Omega)$\end{document} from the data \begin{document}${\rm BSD}(q, \gamma)$\end{document} . Previous uniqueness results, with arbitrarily small \begin{document}$\gamma$\end{document} , assume that \begin{document}$q$\end{document} 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.