Y. Saito, T. Umeda
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
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Weyl-Titchmarsh M-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators
我们明确地确定了半直线上和$\bbR$上与一般Dirac型算子相关的Weyl-Titchmarsh矩阵的高能渐近性。我们还证明了新的局部唯一性结果Dirac-type算子的指数小差Weyl-Titchmarsh矩阵。作为渐近高能展开式的具体应用，我们导出了Dirac算子的一个迹公式，并用它证明了一个Borg型定理。
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.
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.
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.
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.
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。
A transfer matrix function representation of the fundamental solution of the general-type discrete Dirac system, corresponding to rectangular Schur coefficients and Weyl functions, is obtained. Connections with Szegö recurrence, Schur coefficients and structured matrices are treated. Borg-Marchenko-type uniqueness theorem is derived. Inverse problems on the interval and semiaxis are solved.
Weyl matrix functions and inverse problems for discrete Dirac type self-adjoint system: explicit and general solutions
离散Dirac型自伴系统等价于块Szegö递推。得到了该方程的基本解的表达式，并求解了区间和半轴上的反问题。还得到了Borg-Marchenko型结果。与块Toeplitz矩阵的连接处理。
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.
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.
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.
Inverse problem for Dirac systems with locally square summable potentials and rectangular Weyl functions is solved. For that purpose we use a new result on the linear similarity between operators from a subclass of triangular integral operators and the operator of integration.
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.
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。
A non-classical Weyl theory is developed for skew-self-adjoint Dirac systems with rectangular matrix potentials. The notion of the Weyl function is introduced and direct and inverse problems are solved. A Borg-Marchenko type uniqueness result and the evolution of the Weyl function for the corresponding focusing nonlinear Schrödinger equation are also derived.
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。
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.
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