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 $\mathbf{J}$ 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 ${\mathbf{D}}_{X,\alpha }$ with point interactions in $L^2(\mathbb{R};{\mathbb{C}}^{2p})$. In particular, their deficiency indices are related by $n_{\pm }({\mathbf{D}}_{X,\alpha })=n_{\pm }({\mathbf{J}}_{X,\alpha })$. For block Jacobi matrices of this class we present several conditions ensuring equality $n_{\pm }({\mathbf{J}}_{X,\alpha })=k$ 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 $n_{\pm }({\mathbf{J}}_{X,\alpha })=p$ for ${\mathbf{J}}_{X,\alpha }$ we first establish it for Dirac operator ${\mathbf{D}}_{X,\alpha }$.

typos corrected; Section "Application to Schrödinger and Dirac operators with $\delta$-interactions" added