Spectral data asymptotics for the higher-order differential operators with distribution coefficients

TL;DR

This paper derives spectral data asymptotics for higher-order differential operators with distribution coefficients, extending known results to more generalized, non-regular cases, with implications for inverse spectral problems.

Contribution

It provides the first asymptotic analysis of spectral data for higher-order operators with distribution coefficients, using regularization and Birkhoff solutions.

Findings

Spectral data asymptotics obtained for operators with distribution coefficients.

Estimates of spectral data differences when coefficients coincide.

Extension of spectral asymptotics to non-regular, distribution-based operators.

Abstract

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author's knowledge, there were no results in this direction for the higher-order differential operators with distribution coefficients (generalized functions) in a general form. The technique of this paper relies on the recently obtained regularization and the Birkhoff-type solutions for differential operators with distribution coefficients. Our results have applications to the theory of inverse spectral problems as well as a separate significance.