Inverse spectral problem for a third-order differential operator

TL;DR

This paper solves the inverse spectral problem for a third-order differential operator with a potential, providing a method to uniquely determine the potential from four spectral datasets using a system of integral equations.

Contribution

It introduces a new approach to reconstruct the potential of a third-order differential operator from spectral data, establishing uniqueness and explicit formulas.

Findings

Potential is uniquely determined by four spectra.

A closed system of integral equations is derived for potential reconstruction.

Main parameters of the system are expressed via spectral data.

Abstract

Inverse spectral problem for a self-adjoint differential operator, which is the sum of the operator of the third derivative on a finite interval and of the operator of multiplication by a real function (potential), is solved. Closed system of integral linear equations is obtained. Via solution to this system, the potential is calculated. It is shown that the main parameters of the obtained system of equations are expressed via spectral data of the initial operator. It is established that the potential is unambiguously defined by the four spectra.