Matrix Sturm-Liouville equation with a Bessel-type singularity on a finite interval

TL;DR

This paper investigates the matrix Sturm-Liouville equation with a Bessel-type singularity, constructing special solutions, deriving asymptotics for spectral data, and advancing the understanding of direct and inverse spectral problems.

Contribution

It introduces new fundamental solutions and asymptotic formulas for spectral data in the context of matrix Sturm-Liouville equations with singularities.

Findings

Constructed analytic Bessel-type solutions with prescribed behavior.

Derived asymptotic formulas for Stokes multipliers.

Obtained asymptotics for eigenvalues and weight matrices.

Abstract

The matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. Special fundamental systems of solutions for this equation are constructed: analytic Bessel-type solutions with the prescribed behavior at the singular point and Birkhoff-type solutions with the known asymptotics for large values of the spectral parameter. The asymptotic formulas for Stokes multipliers, connecting these two fundamental systems of solutions, are derived. We also set boundary conditions and obtain asymptotic formulas for the spectral data (the eigenvalues and the weight matrices) of the boundary value problem. Our results will be useful in the theory of direct and inverse spectral problems.