狄拉克算符和畴壁
Dirac operators and domain walls
摘要：
本文研究了具有空间变化质量项的一维Dirac算子的本征值问题。众所周知，当质量函数具有扭结或\emph{domain wall}的形式，在严格正和严格负渐近质量$\pmκ_\infty$之间过渡时，在$\pm\infty$处，狄拉克算子在连续谱的间隙内具有能量为零的简单本征值（几何重数等于1），对应的\emph{zero mode}是指数定域本征函数。我们证明了当质量函数具有$2 δ$畴壁的形式时，Dirac算子具有两个符号相反的实单本征值，其阶为$e^{- 2 |κ_\infty| δ}$ .相关联的本征函数，直到$L^2$误差的顺序$e^{- 2 |κ_\infty| δ}$，单畴壁零模式的移位副本的线性组合。对于三个畴壁的情况，有两个非零的简单本征值如上所述和一个简单的本征值在能量为零。我们的方法是基于一个Lyapunov-施密特减少策略，我们概述了他们的自然扩展的情况下，$n$域墙的域墙之间的最小距离是足够大的。我们考虑的狄拉克算子类控制拓扑保护的"边缘状态“从狄拉克点（线性带交叉）的薛定谔算子类域壁调制的周期性电位在一个和两个空间维度的分歧。本研究结果可用于构造丰富的缺陷模式类的周期性结构调制的多个畴壁。
英文摘要：
We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying ``mass'' term. It is well-known that when the mass function has the form of a kink, or \emph{domain wall}, transitioning between strictly positive and strictly negative asymptotic mass, $\pmκ_\infty$, at $\pm\infty$, the Dirac operator has a simple eigenvalue of zero energy (geometric multiplicity equal to one) within a gap in the continuous spectrum, with corresponding \emph{zero mode}, an exponentially localized eigenfunction. We prove that when the mass function has the form of \emph{two} domain walls separated by a sufficiently large distance $2 δ$, the Dirac operator has two real simple eigenvalues of opposite sign and of order $e^{- 2 |κ_\infty| δ}$. The associated eigenfunctions are, up to $L^2$ error of order $e^{- 2 |κ_\infty| δ}$, linear combinations of shifted copies of the single domain wall zero mode. For the case of three domain walls, there are two non-zero simple eigenvalues as above and a simple eigenvalue at energy zero. Our methods are based on a Lyapunov-Schmidt reduction strategy and we outline their natural extension to the case of $n$ domain walls for which the minimal distance between domain walls is sufficiently large. The class of Dirac operators we consider controls the bifurcation of topologically protected ``edge states'' from Dirac points (linear band crossings) for classes of Schrödinger operators with domain-wall modulated periodic potentials in one and two space dimensions. The present results may be used to construct a rich class of defect modes in periodic structures modulated by multiple domain walls.
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