Third-order differential operators with a second-order distribution coefficient

In this paper, we study differential operators associated with the formal expression $y''' + s(\sigma' y)' + s \sigma' y' + \kappa \sigma'' y$ with distribution coefficient $\sigma'' \in W_3^{-2}$, where $s$ and $\kappa$ are constants. The uniqueness theorems are proved for the inverse spectral problems that consist in the recovery of $\sigma$ from the Weyl-Yurko matrix on a finite interval and on the half-line. In addition, we discuss the reconstruction of $\sigma$ and formulate some open problems.