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

TL;DR

This paper investigates the inverse scattering problem for a third-order differential operator with a double potential, providing explicit formulas and integral equations to reconstruct potentials on the entire real axis.

Contribution

It introduces a novel approach with two closed systems of linear integral equations for solving the inverse scattering problem of a third-order operator with a double potential.

Findings

Derived explicit formulas for potential reconstruction

Established two systems of linear integral equations

Provided methods for restoring potentials on half-axes

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.