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

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.