Local solvability and stability of an inverse spectral problem for higher-order differential operators

TL;DR

This paper establishes the local solvability and stability of an inverse spectral problem for higher-order differential operators with distribution coefficients, using a constructive method that reduces the problem to a linear equation.

Contribution

It is the first to prove local solvability and stability for inverse spectral problems of higher-order operators with distribution coefficients, providing explicit estimates.

Findings

Proved local solvability under small spectral data perturbations

Established stability estimates for coefficient recovery

Reduced nonlinear inverse problem to a linear equation in Banach space

Abstract

In this paper, we for the first time prove local solvability and stability of an inverse spectral problem for higher-order ($n > 3$) differential operators with distribution coefficients. The inverse problem consists in the recovery of differential equation coefficients from $(n-1)$ spectra and the corresponding weight numbers. The proof method is constructive. It is based on the reduction of the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation remains uniquely solvable. Furthermore, we estimate the differences of the coefficients in the corresponding functional spaces.