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

Abstract

The first part of the paper concerns with infinite symmetric block Jacobi matrices J with p×p-matrix entries. We present new conditions for general block Jacobi matrices to be self-adjoint and have discrete spectrum. In the second part of the paper a special classes of block Jacobi matrices JX,α are investigated. Our approach here substantially relies on a close connection between Jacobi matrices JX,α from this class and symmetric 2p×2p Dirac operators DX,α with point interactions in L2((a,b);C2p) established in our previous papers. In particular, their deficiency indices are related by n±(DX,α)=n±(JX,α) and under a simple additional assumption they are discrete only simultaneously. For block Jacobi matrices of this class we present several conditions ensuring equality n±(JX,α)=k with any k≤p. 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±(JX,α)=p for JX,α it is first established for Dirac operator DX,α. We also find several conditions for matrix Schrödinger and Dirac operators with point interactions on finite or infinite intervals to have discrete spectrum.