Deficiency indices and discreteness property of block Jacobi matrices and Dirac operators with point interactions

Abstract: The paper concerns with infinite symmetric block Jacobi matrices $\bf J$ with $p\times p$-matrix entries. We present new conditions for general block Jacobi matrices to be selfadjoint and have discrete spectrum. In our previous papers there was established a close relation between a class of such matrices and symmetric $2p\times 2p$ Dirac operators $\mathrm{\bf D}{X,\alpha}$ with point interactions in $L^2(\Bbb R; \Bbb C^{2p})$. In particular, their deficiency indices are related by $n\pm(\mathrm{\bf D}{X,\alpha})= n\pm({\bf J}{X,\alpha})$. For block Jacobi matrices of this class we present several conditions ensuring equality $n\pm({\bf J}{X,\alpha})=k$ with any $k \le p$. Applications to matrix Schrodinger and Dirac operators with point interactions are given. It is worth mentioning that a connection between Dirac and Jacobi operators is employed here in both directions for the first time. In particular, to prove the equality $n\pm({\bf J}{X,\alpha})=p$ for ${\bf J}{X,\alpha}$ we first establish it for Dirac operator $\mathrm{\bf D}_{X,\alpha}$.