Property C and uniqueness theorems for multidimensional inverse spectral problems (Q923273)

Property C, introduced by the author, is used to prove the following uniqueness theorem: let \(\ell_ qu:=[-\nabla^ 2+q(x)]\psi_ n=\mu_ n\psi_ n\) in \(D\), \(\psi_{nN}+\sigma (s)\psi_ n=0\) on \(\Gamma\), \(D\subset\mathbb R^ 3\) is a bounded domain with a smooth boundary \(\Gamma \subset C^ 2\), \(q(x)=\overline{q(x)}\in L^ 2(D)\), \(0<c\leq \sigma (s)\in C^ 1(\Gamma)\), \(N\) is the outer normal to \(\Gamma\). Define the spectral data \(\{\mu_ n,\psi_ n(s)|_{\Gamma}\}\forall n=1,2,...\), the eigenvalues are counted according to their multiplicities. Theorem. These spectral data determine \(q(x)\) and \(\sigma(s)\) uniquely. If the boundary condition is \(\phi_ n=0\) on \(\Gamma\) then the spectral data \(\{\lambda_ n,\phi_{nN}|_{\Gamma}\} \forall n\) determine \(q(x)\) uniquely. Here \(\ell_ q\phi_ n=\lambda_ n\phi_ n\) in \(D\), \(\phi_ n|_{\Gamma}=0\). This last result has been obtained by the author earlier [e.g., see J. Math. Anal. Appl. 134, 211--253 (1988; Zbl 0632.35076)] by a different method.