Spectral analysis for differential systems with a singularity

TL;DR

This paper analyzes a class of differential systems with singularities, constructing fundamental solutions and addressing the inverse scattering problem to establish uniqueness results.

Contribution

It introduces a method for spectral analysis of differential systems with singularities and solves the inverse scattering problem with a uniqueness theorem.

Findings

Constructed a distinguished fundamental system of solutions.

Addressed the inverse scattering problem for the system.

Proved the uniqueness of the inverse scattering solution.

Abstract

We consider the differential system $y'-x^{-1}Ay-q(x)y=\rho By $ with $n\times n$ matrices $A,B, q(x)$, where $A,B$ are constant, $B$ is diagonal, $A$ and $q(x)$ are off-diagonal, $q(\cdot)\in W^1_1[0,\infty)$. Some distinguished fundamental system of solutions is constructed. Also, we discuss the inverse scattering problem and obtain the uniqueness result.