Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients

In this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order (Formula presented.). We show that every matrix function of this class is associated with some differential expression. Moreover, we construct the family of associated matrices for a fixed differential expression. Furthermore, our regularization results are applied to inverse spectral theory. We study a new type of inverse spectral problems, which consist of the recovery of distribution coefficients from the spectral data independently of the associated matrix. The uniqueness theorems are proved for the inverse problems by the Weyl–Yurko matrix and by the discrete spectral data. As examples, we consider the cases (Formula presented.) and (Formula presented.) in more detail. © 2023 by the author.