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

Abstract

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