Inverse Spectral Problems for Arbitrary-Order Differential Operators with Distribution Coefficients

Abstract

In this paper, we propose an approach to inverse spectral problems for the n-th order (n≥2) ordinary differential operators with distribution coefficients. The inverse problems which consist in the reconstruction of the differential expression coefficients by the Weyl matrix and by several spectra are studied. We prove the uniqueness of solution for these inverse problems, by developing the method of spectral mappings. The results of this paper generalize the previously known results for the second-order differential operators with singular potentials and for the higher-order differential operators with regular coefficients. In the future, the approach of this paper can be used for constructive solution and for investigation of solvability of the considered inverse problems.

Linear differential operators with distribution coefficients of various singularity orders

“…However, for the differential operators (1.1) with distribution coefficients, there is still no general inverse problem theory. The first steps in this direction have been taken in [47], where the uniqueness theorem has been proved for the inverse problem on a finite interval.…”

“…By using the method of spectral mappings, we prove the uniqueness theorem for Inverse Problem 1.1. Furthermore, we consider the inverse problem on a finite interval previously studied in [47]. We discuss the usage of regularizations with various (i ν ) n−2 ν=0 and the recovery of the boundary condition coefficients in a finite interval case.…”

“…Proof. The proof is based on the method of spectral mappings (see [44,47]). Define the matrix of spectral mappings…”

“…The inverse spectral problem for the differential expression (1.1) on the finite interval (0, 1) has been considered in [47] for the Mirzoev-Shkalikov case: i 2k+j = m − k − j, j = 0, 1. In contrast to the half-line, for a finite interval…”

“…Therefore, the case of arbitrary I ∈ I n can be reduced to the Mirzoev-Shkalikov case, and the results of [47] can be applied. However, in this section, we show that the regularization of Section 2 for any I ∈ I n can be used for investigating inverse problems.…”