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

TL;DR

This paper studies the spectral properties of relativistic 1-D Dirac operators with point interactions, connecting their spectral characteristics to Jacobi matrices and exploring the non-relativistic limit.

Contribution

It introduces a novel boundary triplet construction for Dirac operators with point interactions and relates their spectral properties to Jacobi matrices, extending previous Schrödinger operator results.

Findings

Spectral properties are linked to Jacobi matrices.

Constructed boundary triplet for operators with zero minimal distance.

Analyzed non-relativistic limit as speed of light tends to infinity.

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 $d_*(X)> 0.$