On the spectral theory of Gesztesy-Å eba realizations of 1-D Dirac operators with point interactions on a discrete set

Abstract

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 $d_*(X)> 0.$

accepted for publication in Journal of Differential Equations