Regularization and inverse spectral problems for differential operators with distribution coefficients

TL;DR

This paper develops a framework linking regularization matrices to differential operators with distribution coefficients, and applies it to inverse spectral problems, proving uniqueness results for recovering distribution coefficients from spectral data.

Contribution

It introduces a new class of matrix functions associated with differential expressions and applies this to inverse spectral problems, establishing uniqueness theorems.

Findings

Established a correspondence between matrix functions and differential expressions.

Proved uniqueness theorems for inverse spectral problems using Weyl-Yurko matrices.

Analyzed the case n=2 in detail as an example.

Abstract

In this paper, we consider a class of matrix functions, which contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order $n \ge 2$. 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 in 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 an example, we consider the case $n = 2$ in detail.