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

TL;DR

This paper studies discrete self-adjoint Dirac systems with positive definite, j-unitary potentials, providing explicit solutions for inverse problems, analyzing asymptotics of Verblunsky-type coefficients, and examining the stability of the inverse solution process.

Contribution

It constructs systems with rational Weyl functions, explicitly solves the inverse problem, and analyzes the asymptotic behavior and stability of the solutions.

Findings

Weyl functions can be rational and contractive, enabling explicit inverse solutions.

Verblunsky-type coefficients tend to zero, and potentials approach the identity matrix.

The inverse problem stability is established for the constructed systems.

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 $I_m$.