Third-order inverse spectral problem with the three-point boundary conditions

In this paper, we study a new inverse spectral problem that consists in the recovery of the third-order differential equation from two spectra corresponding to the boundary conditions $y(0)=y(1)=y(2)=0$ and $y(0)=y^{\mathrm{\prime }}(0)=y(1)=0$. The uniqueness and existence theorems for the solution are obtained. To prove the results, we treat the inverse problem using a general approach that reconstructs higher-order differential operators from the Weyl-Yurko matrix.