Inverse scattering problem for a third-order differential operator with double potential

TL;DR

This work addresses the inverse scattering problem for a self-adjoint third-order differential operator on the real line with a double potential $p$ and $q$, constructing a rigorous framework based on Jost solutions and complex-analytic methods. It develops a Jost-solution theory and unitarity relations for the scattering data, and reduces the inverse problem to two closed Marchenko-type systems of linear singular integral equations that recover $p(x)$ and $q(x)$ on $\mathbb{R}_+$ and $\mathbb{R}_-$ from spectral data. The approach relies on Riemann boundary value problems in multiple sectors, with bound states identified via finite double zeros of a transmission coefficient, whose cubes give the eigenvalues $\lambda^3$ of $L_{p,q}$. The results provide explicit reconstruction formulas and address reflectionless cases, contributing a general methodology for inverse scattering of higher-order operators with nontrivial cross-derivative potentials.

Abstract

Direct and inverse scattering problems for a third-order self-adjoint differential operator on the whole axis are studied. This operator is the sum of three summands: operator of third derivative, operator of multiplication by a function, and operator of multiplication by derivative of a function. For the solution of the inverse scattering problem, two closed systems of linear integral equations are obtained. Knowing solutions to these systems, using explicit formulas, methods of restoration of both potentials on half-axes $\mathbb{R}_\pm$ are specified.

Theorems & Definitions (20)

• Remark 1.1
• Remark 1.2
• Lemma 1.1
• Theorem 1.1
• Remark 1.3
• Remark 1.4
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 1.2