On Solutions of Systems of Differential Equations on Half-Line with Summable Coefficients

TL;DR

This paper develops solutions for systems of first-order differential equations on a half-line with summable, nonlinear spectral coefficients, providing analytic solutions useful for inverse spectral problems.

Contribution

It introduces new methods for constructing analytic solutions with exponential asymptotics for systems with nonlinear spectral dependence on a half-line.

Findings

Constructed fundamental systems with sectorial analyticity

Developed non-fundamental solutions covering larger sectors

Applicable to inverse spectral problems for higher-order operators

Abstract

We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of differential operators. Here, we consider the system of first-order differential equations on a half-line with summable coefficients, containing a nonlinear dependence on the spectral parameter. We obtain fundamental systems of solutions with analyticity in certain sectors, in which it is possible to apply the method of successive approximations. We also construct non-fundamental systems of solutions with analyticity in a large sector, including two previously considered neighboring sectors. The obtained results admit applications in studying inverse spectral problems for the higher-order differential operators with distribution coefficients.