On the spectral theory of Gesztesy-Šeba 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 $\alpha :=\{{\alpha }_n{\}}_{n=1}^{\infty }$ and β:=\{β_{n}\}_{n=1}^\infty \subset\mathbb{R}. The Gesztesy-Šeba realizations $D_{X,\alpha }$ and $D_{X,\beta }$ are the relativistic counterparts of the corresponding Schrödinger operators $H_{X,\alpha }$ and $H_{X,\beta }$ with $\delta$- and ${\delta }^{\prime }$-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 $D_X^{\ast }$ in the case $d_{\ast }(X):=\inf \{|x_i-x_j|,i\ne j\}=0$. It turns out that the boundary operators $B_{X,\alpha }$ and $B_{X,\beta }$ 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 $D_{X,\alpha }$ 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 $d_{\ast }(X)>0.$

accepted for publication in Journal of Differential Equations

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• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Boundary triples and Weyl functions for singular perturbations of self-adjoint operators

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