The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the
Stability of Solving of the Inverse Problem

Inna Roitberg

University of Leipzig, 10 Augustusplatz, Leipzig, 04109, Germany
E-mail: innaroitberg@gmail.com

Alexander Sakhnovich

Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, A-1090 Vienna,
Austria
E-mail: oleksandr.sakhnovych@univie.ac.at

Received February 8, 2018.

To V.A. Marchenko with admiration

Abstract

We consider discrete self-adjoint Dirac systems determined by the potentials
(sequences) {C_{k}} such that the matrices C_{k} are positive definite and j-unitary, where j is a diagonal m × m matrix which has m_{1} entries 1 and m_{2} entries –1 (m_{1} +m_{2} = m) on the main diagonal. We construct systems with the rational Weyl functions and explicitly solve the inverse problem to recover systems from the contractive rational Weyl functions. Moreover, we study the stability of this procedure. The matrices C_{k} (in the potentials)
are the so-called Halmos extensions of the Verblunsky-type coefficients ρ_{k}. We show that in the case of the contractive rational Weyl functions the coefficients ρ_{k} tend to zero and the matrices Ck tend to the identity matrix
I_{m}.

Mathematics Subject Classification 2000: 34B20, 39A12, 39A30, 47A57. Key words: discrete self-adjoint Dirac system, Weyl function, inverse problem, explicit solution, stability of solution of the inverse problem, asymptotics of the potential, Verblunsky-type coefficient.