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

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.MSC(2010): 34A55, 34B20, 34L25, 34A05.

GBDT and explicit solutions for the matrix coupled dispersionless equations (local and nonlocal cases)

“…When (2.68) holds (and so S(0, t) > 0), one can combine [28, Theorem 2.5] (see also the references therein) and [29,Theorem 3.7] in order to obtain the existence of the Jost solutions…”

“…The asymptotics of w A may be studied in a way similar to the case p = 0 (see [29]) although the result is somewhat more complicated.…”

“…which corresponds [29] to the simplest case of the Weyl function (reflection coefficient) with a pole of the order more than one (so called multipole case).…”

Discrete Dirac systems on the semiaxis: rational reflection coefficients and Weyl functions

“…It is known that Weyl-Titchmarsh (or simply Weyl) functions of continuous Dirac systems on the semi-axis are closely related to the scattering data. Some particular results for the self-adjoint systems are contained, for instance, in [1,11] and the general cases of continuous self-adjoint and skew-self-adjoint systems were treated in the recent paper [23]. The present article may be considered as the continuation of the paper [23], where the important discrete case is dealt with.…”

“…Some particular results for the self-adjoint systems are contained, for instance, in [1,11] and the general cases of continuous self-adjoint and skew-self-adjoint systems were treated in the recent paper [23]. The present article may be considered as the continuation of the paper [23], where the important discrete case is dealt with. We consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems, obtain explicit expressions for reflection coefficients and show that rational reflection coefficients and Weyl functions coincide.…”