离散集上点相互作用的一维Dirac算子Gesztesy-Šeba实现的谱理论
On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set
摘要：
本文研究了离散集$X=\{x_n\}_{n=1}^\infty\subset \mathbb{R}.$上具有点相互作用的一维Dirac微分表达式D的Gesztesy-Šeba实现D_{X,α}和D_{X,β}的谱性质。这里$α:= \{α_{n}\}_{n=1}^\infty$和β：=\{β_{n}\}_{n=1}^ \infty \subset \mathbb{R}。Gesztesy-Šeba实现$D_{X,α}$和$D_{X,β}$是相应的薛定谔算符$H_{X,α}$和$H_{X,β}$的相对论对应物，分别具有$δ$ -和$δ'$ -相互作用。定义极小算子D_X为区间$(x_{n-1}, x_n)$上极小Dirac算子的直和。然后利用边界三元组直和的正则化方法构造了$d_*(X):=\inf\{|x_i-x_j| \,, i\not=j\} = 0$情形下极大算子$D_X^*$的边界三元组。结果表明，参数化实现D_{X,α}和D_{X,β}的边界算子$B_{X,α}$和$B_{X,β}$是雅可比矩阵。这些矩阵与具有点相互作用的薛定谔算子的谱理论中出现的矩阵有很大不同。我们表明，某些频谱特性的运营商$D_{X,α}$和$D_{X,β}$相关的Jacobi矩阵$B_{X,α}$和$B_{X,β}$，分别与相应的频谱特性。利用这种连接，我们调查的Gesztesy-{ \vS }eba实现的谱特性（自伴性，离散性，绝对连续和奇异谱）。此外，我们还研究了非相对论极限作为光速的$c\to\infty$。即使在$d_*(X)> 0.$的情况下，我们的大多数结果也是新的
英文摘要：
We investigate spectral properties of Gesztesy-Šeba realizations D_{X,α} and D_{X,β} of the 1-D Dirac differential expression D with point interactions on a discrete set $X=\{x_n\}_{n=1}^\infty\subset \mathbb{R}.$ Here $α:= \{α_{n}\}_{n=1}^\infty$ and β:=\{β_{n}\}_{n=1}^\infty \subset\mathbb{R}. The Gesztesy-Šeba realizations $D_{X,α}$ and $D_{X,β}$ are the relativistic counterparts of the corresponding Schrödinger operators $H_{X,α}$ and $H_{X,β}$ with $δ$- and $δ'$-interactions, respectively. We define the minimal operator D_X as the direct sum of the minimal Dirac operators on the intervals $(x_{n-1}, x_n)$. Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator $D_X^*$ in the case $d_*(X):=\inf\{|x_i-x_j| \,, i\not=j\} = 0$. It turns out that the boundary operators $B_{X,α}$ and $B_{X,β}$ parameterizing the realizations D_{X,α} and D_{X,β} are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schrödinger operators with point interactions. We show that certain spectral properties of the operators $D_{X,α}$ and $D_{X,β}$ correlate with the corresponding spectral properties of the Jacobi matrices $B_{X,α}$ and $B_{X,β}$, respectively. Using this connection we investigate spectral properties (self-adjointness, discreteness, absolutely continuous and singular spectra) of Gesztesy--{\vS}eba realizations. Moreover, we investigate the non-relativistic limit as the velocity of light $c\to\infty$. Most of our results are new even in the case $d_*(X)> 0.$
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