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

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 and has m 1 entries 1 and m 2 entries −1 (m 1 + m 2 = m) on the main diagonal. We construct systems with rational Weyl functions and explicitly solve 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 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 C k tend to the indentity matrix I m . MSC(2010): 34B20, 39A12, 39A30, 47A57Keywords. Discrete self-adjoint Dirac system, Weyl function, inverse problem, explicit solution, stability of solving inverse problem, asymptotics of the potential, Verblunsky-type coefficient.

Discrete self-adjoint Dirac systems: asymptotic relations, Weyl functions and Toeplitz matrices

“…Self-adjoint discrete Dirac systems have been introduced in [10] and studied further in [11,29,33,34] following the case of skew-self-adjoint discrete Dirac systems in [17]. In particular, the paper [33] is dedicated to the interrelations between self-adjoint discrete Dirac systems and block Toeplitz matrices, which (in the scalar case) are in many respects similar to the interrelations between the famous Szegő recurrences and Toeplitz matrices.…”