Scattering for general-type Dirac systems on the semi-axis: reflection coefficients and Weyl functions

TL;DR

This paper establishes a connection between Weyl functions and reflection coefficients for general Dirac systems on the semi-axis, providing methods for their extension and explicit recovery of systems from rational reflection coefficients.

Contribution

It demonstrates that Weyl functions are unique analytic extensions of reflection coefficients and introduces procedures to recover Dirac systems from these functions.

Findings

Weyl functions are unique extensions of reflection coefficients.

Extension of Weyl functions to the real axis is achieved.

Explicit recovery of Dirac systems from rational reflection coefficients.

Abstract

We show that for general-type self-adjoint and skew-self-adjoint Dirac systems on the semi-axis Weyl functions are unique analytic extensions of the reflection coefficients. New results on the extension of the Weyl functions to the real axis and on the existence (in the skew-self-adjoint case) of the Weyl functions follow. Important procedures to recover general-type Dirac systems from the Weyl functions are applied to the recovery of Dirac systems from the reflection coefficients. We explicitly recover Dirac systems from the rational reflection coefficients as well.