Weyl理论及矩形矩阵势Dirac系统正反问题的显式解

Weyl functions of Dirac systems and of their generalizations: integral representation, inverse problem, and discrete interpolation

Self-adjoint Dirac systems and subclasses of canonical systems, which generalize Dirac systems are studied. Explicit and global solutions of direct and inverse problems are obtained. A local Borg-Marchenko-type theorem, integral representation of the Weyl function, and results on interpolation of Weyl functions are also derived.

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.

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

Discrete Dirac type self-adjoint system is equivalent to the block Szegö recurrence. Representation of the fundamental solution is obtained, inverse problems on the interval and semi-axis are solved. A Borg-Marchenko type result is obtained too. Connections with the block Toeplitz matrices are treated.

Weyl-Titchmarsh M-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on $\bbR$. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.

A sequence of zero modes of Weyl-Dirac operators and an associated sequence of solvable polynomials

It is shown that a series of solvable polynomials is attached to the series of zero modes constructed by Adam, Muratori and Nash \cite{AdamMuratoriNash1

关于具有无穷远处非零位势的Dirac、动力学Dirac及Dirac–Weyl系统的新显式解类、其性质与应用

On new classes of explicit solutions of Dirac, dynamical Dirac and Dirac--Weyl systems with non-vanishing at infinity potentials, their properties and applications

本文利用GBDT版本的Bäcklund-Darboux变换，构造了具有无穷远处非零位势的自伴与斜自伴Dirac系统、动力学Dirac及Dirac–Weyl系统的广泛新显式解，并推导了Weyl–Titchmarsh函数。

Singular Weyl-Titchmarsh-Kodaira theory for one-dimensional Dirac operators

We develop singular Weyl-Titchmarsh-Kodaira theory for one-dimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg-Marchenko and Hochstadt-Liebermann type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.

On Spectral Deformations and Singular Weyl Functions for One-Dimensional Dirac Operators

We investigate the connection between singular Weyl-Titchmarsh-Kodaira theory and the double commutation method for one-dimensional Dirac operators. In particular, we compute the singular Weyl function of the commuted operator in terms of the data from the original operator. These results are then applied to radial Dirac operators in order to show that the singular Weyl function of such an operator belongs to a generalized Nevanlinna class $N_{κ_0}$ with $κ_0=\lfloor|κ| + \frac{1}{2}\rfloor$, where $κ\in \mathbb{R}$ is the corresponding angular momentum.

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。

Discrete Dirac systems on the semiaxis: rational reflection coefficients and Weyl functions

We consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems, obtain explicit expressions for reflection coefficients and show that rational reflection coefficients and Weyl functions coincide.

Weyl functions and the boundary value problem for a matrix nonlinear Schrödinger equation on a semi-strip

Rectangular matrix solutions of the defocusing nonlinear Schrödinger equation (dNLS) are considered on a semi-strip. Evolution of the corresponding Weyl function is described in terms of the initial-boundary conditions. Then initial condition is recovered from the boundary conditions. Thus, solutions of dNLS are recovered from the boundary conditions.

Dirac operators with exponentially decaying entropy

We prove that the Weyl function of the one-dimensional Dirac operator on the half-line $\mathbb{R}_+$ with exponentially decaying entropy extends meromorphically into the horizontal strip $\{0\ge \mbox{Im}\,z > -δ\}$ for some $δ> 0$ depending on the rate of decay. If the entropy decreases very rapidly then the corresponding Weyl function turns out to be meromorphic in the whole complex plane. In this situation we show that poles of the Weyl function (scattering resonances) uniquely determine the operator.

Recovering Dirac Operator with Nonlocal Boundary Conditions

In this paper inverse problems for Dirac operator with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are provided, which are generalizations of the well-known Weyl function and Borg's inverse problem for the classical Dirac operator.