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 ; Alexander Sakhnovich

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018), pp. 532-548

Résumé

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 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 $\rho_k$. 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 identity matrix $I_m$.