Existence theorem on spectral function for singular nonsymmetric first order differential operators

TL;DR

This paper extends spectral theory to nonsymmetric first-order differential operators on the half line, establishing spectral functions, expansion formulas, and Parseval equalities using Marchenko's method for different coefficient matrix cases.

Contribution

It introduces spectral functions for nonsymmetric operators, generalizing classical spectral theory for self-adjoint cases, with proofs via Marchenko's method.

Findings

Spectral functions are constructed for two coefficient matrix cases.

Marchenko-Parseval equality is established.

Expansion formulas for the operators are derived.

Abstract

In this paper we study spectral function for a nonsymmetric differential operator on the half line. Two cases of the coefficient matrix are considered, and for each case we prove by Marchenko's method that, to the boundary value problem, there corresponds a spectral function related to which a Marchenko-Parseval equality and an expansion formula are established. Our results extend the classical spectral theory for self-adjoint Sturm-Liouville operators and Dirac operators.