Spherically symmetric Dirac operators with variable mass and potentials infinite at infinity

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。

Dirac-Sobolev inequalities and estimates for the zero modes of massless Dirac operators

The paper analyses the decay of any zero modes that might exist for a massless Dirac operator $H:= \ba \cdot (1/i) \bgrad + Q, $ where $Q$ is $4 \times 4$-matrix-valued and of order $O(|\x|^{-1})$ at infinity. The approach is based on inversion with respect to the unit sphere in $\R^3$ and establishing embedding theorems for Dirac-Sobolev spaces of spinors $f$ which are such that $f$ and $Hf$ lie in $(L^p(\R^3))^4, 1\le p<\infty.$

Resolvent estimates of the Dirac operator

We shall investigate the asymptotic behavior of the extended resolvent R(s) of the Dirac operator as |s| increases to infinity, where s is a real parameter. It will be shown that the norm of R(s), as a bounded operator between two weighted Hilbert spaces of square integrable functions on the 3-dimensional Euclidean space, stays bounded. Also we shall show that R(s) converges 0 strongly as |s| increases to infinity. This result and a result of Yamada [15] are combined to indicate that the extended resolvent of the Dirac operator decays much more slowly than those of Schroedinger operators.

Pseudomodes for non-self-adjoint Dirac operators

Depending on the behaviour of the complex-valued electromagnetic potential in the neighbourhood of infinity, pseudomodes of one-dimensional Dirac operators corresponding to large pseudoeigenvalues are constructed. This is a first systematic non-semi-classical approach, which results in substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including superexponential ones.

Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit

A perturbation decaying to 0 at infinity and not too irregular at 0 introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for power-decaying perturbations is equivalent to the large-coupling limit. This asymptotic behaviour elucidates the origin of the dense point spectrum observed in spherically symmetric, radially periodic three-dimensional…

具有虚势的一维Dirac算子的预解估计

Resolvent estimates for one-dimensional Dirac operators with imaginary potentials

研究具有对角虚势的一维Dirac算子，利用广义强制性和Schur补主导技术建立L^2实现，并推导Airy-Dirac情形下预解式的渐近行为，进而得到一般势的预解范数渐近式，优化伪谱区域。