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

TL;DR

This paper investigates third-order differential operators with distributional coefficients, establishing uniqueness theorems for inverse spectral problems and exploring methods to reconstruct the distribution coefficient from spectral data.

Contribution

It introduces new uniqueness results for inverse spectral problems involving third-order operators with distributional coefficients and discusses reconstruction techniques.

Findings

Proved uniqueness theorems for inverse spectral problems.

Established methods for reconstructing the distribution coefficient.

Discussed open problems in the spectral theory of such operators.

Abstract

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.