Inverse problem for interior spectral data of the Sturm-Liouville operator (Q2777926)

Let \(\lambda_n\) and \(y_n(x)\), \(n\geq 0,\) be the eigenvalues and the eigenfunctions of the Sturm-Liouville operator \(ly:=-y''+q(x)y\), \(0\leq x\leq 1,\) with separated boundary conditions. It is proved that the specification of the numbers \(\lambda_n\), \(y_n'(1/2)/y_n(1/2)\), \(n\geq 0,\) uniquely determines the potential \(q(x)\) a.e. on the interval \((0,1).\) Another uniqueness theorem is also proved when parts of two spectra and the logarithmic derivative of the eigenfunctions at an interior point are given.