Third-order differential operators with a second-order distribution coefficient
Natalia P. Bondarenko
TL;DR
This work develops a framework for third-order differential operators with a second-order distribution coefficient, formulating the problem via an associated matrix and a first-order system to enable inverse spectral analysis from the Weyl–Yurko matrix. It establishes uniqueness results for recovering the distribution coefficient $\sigma$ from spectral data on a finite interval and on the half-line, under fixed boundary-structure, and demonstrates Borg-type identifiability with a finite set of spectra. The reconstruction is approached through the method of spectral mappings, yielding a linear main equation whose solvability guarantees retrieval of Weyl solutions and the coefficient $\sigma$ in significant regimes (notably when $s=1$). The paper also outlines reconstruction formulas, discusses self-adjoint cases, and presents open problems including regularization for more singular models, higher-order extensions, and convergence and stability issues. These results contribute to the theory of inverse problems for differential expressions with distribution coefficients and highlight the role of the Weyl–Yurko matrix as a complete spectral descriptor in this setting.
Abstract
In this paper, we study differential operators associated with the formal expression $y''' + s(σ' y)' + s σ' y' + κσ'' y$ with distribution coefficient $σ'' \in W_3^{-2}$, where $s$ and $κ$ are constants. The uniqueness theorems are proved for the inverse spectral problems that consist in the recovery of $σ$ from the Weyl-Yurko matrix on a finite interval and on the half-line. In addition, we discuss the reconstruction of $σ$ and formulate some open problems.