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

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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 $d *X:=\inf\{|x i-x j| \,, i ot=j\} = 0$. 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 $d *X 0.$

Autor: Raffaele Carlone; Mark Malamud; Andrea Posilicano

Fuente: https://archive.org/