Linear differential operators with distribution coefficients of various singularity orders

TL;DR

This paper studies linear differential operators with distribution coefficients of different singularity levels, providing matrix regularization and new inverse spectral problem solutions for recovering operators from spectral data.

Contribution

It introduces a matrix regularization method and proves uniqueness theorems for inverse spectral problems involving distribution coefficients.

Findings

Derived the associated matrix for regularization.

Established uniqueness theorems for inverse problems.

Developed the spectral mappings method for these inverse problems.

Abstract

In this paper, the linear differential expression of order $n \ge 2$ with distribution coefficients of various singularity orders is considered. We obtain the associated matrix for the regularization of this expression. Furthermore, we present the new statements of inverse spectral problems that consist in the recovery of differential operators with distribution coefficients from the Weyl matrix on the half-line and on a finite interval. The uniqueness theorems for these inverse problems are proved by developing the method of spectral mappings.