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

TL;DR

This paper establishes new criteria for block Jacobi matrices to be selfadjoint with discrete spectra and explores their deep connection with Dirac operators with point interactions, providing novel insights into their deficiency indices and spectral properties.

Contribution

It introduces new conditions linking the deficiency indices of block Jacobi matrices and Dirac operators, and employs this connection to analyze spectral properties of both.

Findings

Conditions for selfadjointness and discreteness of spectrum of block Jacobi matrices.

Equality of deficiency indices between Jacobi matrices and Dirac operators.

Application of the connection to matrix Schrödinger 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 $\mathrm{\bf D}_{X,\alpha}$.