Resolvent and spectrum for discrete symplectic systems in the limit point case

TL;DR

This paper characterizes the spectrum of self-adjoint extensions of discrete symplectic systems in the limit point case using Weyl--Titchmarsh functions, linking spectral properties to boundary conditions and extending Sturmian theory.

Contribution

It provides a complete spectral characterization for these systems and explores how boundary conditions influence the spectrum, advancing the understanding of discrete symplectic systems.

Findings

Spectrum characterized by Weyl--Titchmarsh $M_+(\lambda)$-function

Dependence of spectrum on boundary conditions analyzed

Several results extended from singular Sturmian theory

Abstract

The spectrum of an arbitrary self-adjoint extension of the minimal linear relation associated with the discrete symplectic system in the limit point case is completely characterized by using the limiting Weyl--Titchmarsh $M_+(\lambda)$-function. Furthermore, a dependence of the spectrum on a boundary condition is investigated and, consequently, several results of the singular Sturmian theory are derived.