Weyl matrix functions and inverse problems for discrete Dirac type self-adjoint system: explicit and general solutions

Scattering for general-type Dirac systems on the semi-axis: reflection coefficients and Weyl functions

We show that for general-type self-adjoint and skew-self-adjoint Dirac systems on the semi-axis Weyl functions are unique analytic extensions of the reflection coefficients. New results on the extension of the Weyl functions to the real axis and on the existence (in the skew-self-adjoint case) of the Weyl functions follow. Important procedures to recover general-type Dirac systems from the Weyl functions are applied to the recovery of Dirac systems from the reflection coefficients. We explicitly recover Dirac systems from the rational reflection coefficients as well.

Inverse problems for self-adjoint Dirac systems: explicit solutions and stability of the procedure

A procedure to recover explicitly self-adjoint matrix Dirac systems on semi-axis (with both discrete and continuous components of spectrum) from rational Weyl functions is considered. Its stability is proved. GBDT version of Baecklund-Darboux transformation and various important results on Riccati equations are used for this purpose.

On accelerants and their analogs, and on the characterization of the rectangular Weyl functions for Dirac systems with locally square-integrable potentials on a semi-axis

We characterize the set of rectangular Weyl matrix functions corresponding to Dirac systems with locally square-integrable potentials on a semi-axis and demonstrate a new way to recover the locally square-integrable potential from the Weyl function. Important interconnections between our approach and accelerants of convolution operators are discussed as well.

离散自伴Dirac系统：渐近关系、Weyl函数与Toeplitz矩阵

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

本文用离散自伴Dirac系统替代Szegő递推与矩阵正交多项式来研究分块Toeplitz矩阵，证明Christoffel–Darboux公式的类似物并导出再生核渐近关系，进而刻画有理Weyl–Titchmarsh函数情形下出现的分块对角加分块半可分离Toeplitz矩阵。

Weyl理论及矩形矩阵势Dirac系统正反问题的显式解

Weyl theory and explicit solutions of direct and inverse problems for a Dirac system with rectangular matrix potential

针对矩形矩阵势的Dirac系统，发展了非经典Weyl理论，引入Weyl函数并处理正问题，同时获得了有理Weyl矩阵函数情形下正反问题的显式解。