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 $q$, appearing in the Dirichlet realization of the Schrödinger operator $A_q=-\Delta +q$ on a bounded domain $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, 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 $\mathrm{BSD}(q,\gamma ):=\{({\lambda }_k,{{\partial }_{\nu }{\varphi }_k}_{|\overline{\gamma }}):k\ge 1\}$, where $\{{\lambda }_k:k\ge 1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{{\varphi }_k:k\ge 1\}$ 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.

References in corpus (1)

• Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations