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Inverse problem for interior spectral data of the Sturm-Liouville operator. (English) Zbl 1035.34008
Let ${\lambda }_{n}$ and ${y}_{n}\left(x\right)$, $n\ge 0,$ be the eigenvalues and the eigenfunctions of the Sturm-Liouville operator $ly:=-{y}^{\text{'}\text{'}}+q\left(x\right)y$, $0\le x\le 1,$ with separated boundary conditions. It is proved that the specification of the numbers ${\lambda }_{n}$, ${y}_{n}^{\text{'}}\left(1/2\right)/{y}_{n}\left(1/2\right)$, $n\ge 0,$ uniquely determines the potential $q\left(x\right)$ a.e. on the interval $\left(0,1\right)·$ Another uniqueness theorem is also proved when parts of two spectra and the logarithmic derivative of the eigenfunctions at an interior point are given.

##### MSC:
 34A55 Inverse problems of ODE 34B24 Sturm-Liouville theory 34L05 General spectral theory for OD operators 47E05 Ordinary differential operators
##### Keywords:
Sturm-Liouville operator; inverse spectral problems