On the spectral theory of Gesztesy-\v{S}eba realizations of 1-D Dirac operators with point interactions on a discrete set

Abstract

We investigate spectral properties of Gesztesy-\v{S}eba realizations D_{X,\alpha} and D_{X,\beta} 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 $\alpha := \{\alpha_{n}\}_{n=1}^\infty$ and \beta :=\{\beta_{n}\}_{n=1}^\infty \subset\mathbb{R}. The Gesztesy-\v{S}eba realizations $D_{X,\alpha}$ and $D_{X,\beta}$ are the relativistic counterparts of the corresponding Schr\"odinger operators $H_{X,\alpha}$ and $H_{X,\beta}$ with $\delta$- and $\delta'$-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 $B_{X,\alpha}$ and $B_{X,\beta}$ parameterizing the realizations D_{X,\alpha} and D_{X,\beta} are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schr\"odinger operators with point interactions. We show that certain spectral properties of the operators $D_{X,\alpha}$ and $D_{X,\beta}$ correlate with the corresponding spectral properties of the Jacobi matrices $B_{X,\alpha}$ and $B_{X,\beta}$, 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 $d_*(X)> 0.$