An \(n\)-dimensional Borg-Levinson theorem
Let \(y=y(x;\lambda,q)\) denote the solution of the Sturm-Liouville problem \(-y''+q(x)y=\lambda y\) over \(\Omega =(0,1)\), subject to initial conditions \(y(0,\lambda)=0\), \(y'(0,\lambda)=1\). It is well known that the Dirichlet eigenvalues \(\mu_ n=\mu_ n(q)\) (i.e. the \(\mu\) 's satisfying \(y(1;\mu,q)=0)\) and the ``normalizing constants'' \(k_ n=k_ n(q)=y'(1;\mu_ n(q),q)\) together determine the potential q uniquely on [0,1] [see \textit{G. Borg}, Acta Math. 78, 1-96 (1946) and \textit{N. Levinson}, Mat. Tidsskr. B 1949, 25-30 (1949; Zbl 0045.364)]. In the present paper, this is generalized to the higher dimensional case as follows. Given a bounded smooth domain \(\Omega\) in \({\mathbb{R}}^ n \)and \(q\in C^{\infty}({\bar \Omega})\), let \(\mu_ n=\mu_ n(q)\) denote the eigenvalues and \(\phi_ n=\phi_ n(x;q)\) (some) eigenfunctions of the Dirichlet problem \[ (*)\quad -\Delta u+qu=\lambda u\quad (in\quad \Omega),\quad u\equiv 0\quad (\text{on } \partial \Omega). \] Then the sequences \(\mu_ n=\mu_ n(q)\) and \(k_ n=k_ n(q)=(\partial /\partial \nu)\phi_ n(\cdot;q)|_{\partial \Omega}\) determine the potential q uniquely on \({\bar \Omega}\). The proof builds on a careful analysis of the Green's function of the ``iterates'' of (*), as well as norm estimates for ``scattering'' solutions of (*) on \({\mathbb{R}}^ n.\)
scientific article; zbMATH DE number 4050176