Spectral data asymptotics for the matrix Sturm-Liouville operator

TL;DR

This paper derives asymptotic formulas for eigenvalues and weight matrices of matrix Sturm-Liouville operators, advancing inverse spectral theory and extending results to operators on star-shaped graphs with various boundary conditions.

Contribution

It provides new asymptotic formulas for spectral characteristics of matrix Sturm-Liouville operators, including on star-shaped graphs with different matching conditions.

Findings

Asymptotic formulas for eigenvalues and weight matrices derived

Techniques involve analysis of analytic functions and contour integration

Formulas adapted to Sturm-Liouville operators on star-shaped graphs

Abstract

The self-adjoint matrix Sturm-Liouville operator on a finite interval with a boundary condition in the general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm-Liouville operators on a star-shaped graph with two different types of matching conditions.