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 DX,α and DX,β of the 1-D Dirac differential expression D with point interactions on a discrete set X={xn}n=1∞⊂R. Here α:={αn}n=1∞ and β:={βn}n=1∞⊂R. The Gesztesy–Šeba realizations DX,α and DX,β are the relativistic counterparts of the corresponding Schrödinger operators HX,α and HX,β with δ- and δ′-interactions, respectively. We define the minimal operator DX as the direct sum of the minimal Dirac operators on the intervals (xn−1,xn). Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator DX⁎ in the case d⁎(X):=inf{|xi−xj|,i≠j}=0. It turns out that the boundary operators BX,α and BX,β parameterizing the realizations DX,α and DX,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 DX,α and DX,β correlate with the corresponding spectral properties of the Jacobi matrices BX,α and BX,β, respectively. Using this connection we investigate spectral properties (self-adjointness, discreteness, absolutely continuous and singular spectra) of Gesztesy–Šeba realizations. Moreover, we investigate the non-relativistic limit as the velocity of light c→∞. Most of our results are new even in the case d⁎(X)>0.