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

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 $I_m$.Comment: This paper is a generalization and further development of the topics discussed in arXiv:math/0703369, arXiv:1206.2915, arXiv:1508.07954, arXiv:1510.0079

Recovery of Dirac system from the rectangular Weyl matrix function

Weyl theory for Dirac systems with rectangular matrix potentials is non-classical. The corresponding Weyl functions are rectangular matrix functions. Furthermore, they are non-expansive in the upper semi-plane. Inverse problems are treated for such Weyl functions, and some results are new even for the square Weyl functions. High energy asymptotics of Weyl functions and Borg-Marchenko type uniqueness results are derived too

Inverse problems and nonlinear evolution equations: solutions, Darboux matrices and Weyl-Titchmarsh functions

This monograph fits theclearlyneed for books with a rigorous treatment of theinverse problems for non-classical systems and that of initial-boundary-value problems for integrable nonlinear equations. The authorsdevelop a unified treatment of explicit and global solutions via the transfer matrix function in a form due to Lev A. Sakhnovich. The book primarily addresses specialists in the field. However, it is self-contained andstarts with preliminaries and examples, and hencealso serves as an introduction for advanced graduate students in the field