General-type discrete self-adjoint Dirac systems: explicit solutions of direct and inverse problems, asymptotics of Verblunsky-type coefficients and stability of solving 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 $m\times m$ 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 ${\rho }_k$. We show that in the case of the contractive rational Weyl functions the coefficients ${\rho }_k$ tend to zero and the matrices $C_k$ tend to the indentity matrix $I_m$.

This paper is a generalization and further development of the topics discussed in arXiv:math/0703369, arXiv:1206.2915, arXiv:1508.07954, arXiv:1510.00793