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

摘要

We consider the multidimensional Borg-Levinson problem of determining apotential $q$, appearing in the Dirichlet realization of the Schr"odingeroperator $A_q=-Delta+q$ on a bounded domain $Omegasubset mathbb{R}^n$,$ngeq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of$partialOmega$. More precisely, for $gamma$ an open and non-empty subset of$partialOmega$, we consider the boundary spectral data on $gamma$ given by$mathrm{BSD}(q,gamma):={(lambda_{k},{partial_uphi_{k}}_{|overline{gamma}}): k geq1}$, 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 $u$ is the unitoutward normal vector to $partialOmega$. We prove that the data$mathrm{BSD}(q,gamma)$ uniquely determine a bounded potential $qinL^infty(Omega)$. Previous uniqueness results, with arbitrarily small$gamma$, assume that $q$ is smooth. Our approach is based on the BoundaryControl method, and we give a self-contained presentation of the method,focusing on the analytic rather than geometric aspects of the method.