Scattering Properties and Dispersion Estimates for a One-Dimensional Discrete Dirac Equation

Dispersion Estimates for One-dimensional Discrete Schrödinger and Wave Equations

We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schrödinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials. To this end we also provide new results concerning scattering for one-dimensional discrete perturbed Schrödinger operators which are of independent interest. Most notably we show that the reflection and transmission coefficients belong to the Wiener algebra.

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。

二维方格上离散 Dirac 算子的连续统极限

Continuum limits for discrete Dirac operators on 2D square lattices

本文研究了二维方格上离散 Dirac 算子在网格趋于零时的连续统极限，通过提出自然嵌入方法，证明了其在强预解意义下收敛于连续统算子，但在范数预解意义下不收敛。

Bound states in semi-Dirac semi-metals

New insights into transport properties of nanostructures with a linear dispersion along one direction and a quadratic dispersion along another are obtained by analysing their spectral stability properties under small perturbations. Physically relevant sufficient and necessary conditions to guarantee the existence of discrete eigenvalues are derived under rather general assumptions on external fields. One of the most interesting features of the analysis is the evident spectral instability of the systems in the weakly coupled regime. The rigorous theoretical results are illustrated by numerical …

First order approach and index theorems for discrete and metric graphs

The aim of the present paper is to introduce the notion of first order (supersymmetric) Dirac operators on discrete and metric (``quantum'') graphs. In order to cover all self-adjoint boundary conditions for the associated metric graph Laplacian, we develop systematically a new type of discrete graph operators acting on a decorated graph. The decoration at each vertex of degree-d is given by a subspace of $\C^d$, generalising the fact that a function on the standard vertex space has only a scalar value. We develop the notion of exterior derivative, differential forms, Dirac and Laplace operato…

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 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.

Discrete Dirac system: rectangular Weyl functions, direct and inverse problems

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.