Reconstruction of higher-order differential operators by their spectral data

TL;DR

This paper presents a new method for reconstructing higher-order differential operators from spectral data, using a linear equation approach in Banach spaces, applicable to various classes of operators with proven convergence and uniqueness.

Contribution

It introduces a general linear framework for inverse spectral problems of higher-order operators, enabling reconstruction formulas and convergence proofs for diverse operator classes.

Findings

Established a linear equation approach for inverse spectral problems.

Proved unique solvability of the main reconstruction equation.

Derived convergent reconstruction formulas for operator coefficients.

Abstract

This paper is concerned with inverse spectral problems for higher-order ($n > 2$) ordinary differential operators. We develop an approach to the reconstruction from the spectral data for a wide range of differential operators with either regular or distribution coefficients. Our approach is based on the reduction of an inverse problem to a linear equation in the Banach space of bounded infinite sequences. This equation is derived in a general form that can be applied to various classes of differential operators. The unique solvability of the linear main equation is also proved. By using the solution of the main equation, we derive reconstruction formulas for the differential expression coefficients in the form of series and prove the convergence of these series for several classes of operators. The results of this paper can be used for constructive solution of inverse spectral problems and for investigation of their solvability and stability.