Uniform stability of higher-order inverse spectral problems

Abstract

In this paper, the reconstruction of a linear differential operator of arbitrary order $n \ge 2$ is studied by using two types of spectral characteristics: (i) eigenvalues and weight numbers, (ii) $(2n-2)$ spectra. We prove the unconditional uniform stability of these inverse problems, generalizing the results of Savchuk and Shkalikov [Funct. Anal. Appl. 44 (2010), no. 4, 270--285] to $n > 2$. Furthermore, we for the first time obtain sufficient conditions of solvability for the higher-order inverse problem by $(2n-2)$ spectra. By applying our main results, we get new theorems on the necessary and sufficient conditions of solvability and on the uniform stability of the inverse problems for $n = 3$ and $n = 4$. Our approach is based on the method of spectral mappings, which provides a constructive solution of the inverse problems.