Characterization of self-adjoint extensions for discrete symplectic systems

TL;DR

This paper characterizes all self-adjoint extensions of discrete symplectic systems, providing explicit descriptions, discussing uniqueness, and establishing a limit point criterion that generalizes classical results.

Contribution

It offers a comprehensive characterization of self-adjoint extensions for discrete symplectic systems, including explicit formulas and a generalized limit point criterion.

Findings

Explicit description of Krein--von Neumann extension

Uniqueness results for scalar case on finite intervals

A generalized limit point criterion for symplectic systems

Abstract

All self-adjoint extensions of minimal linear relation associated with the discrete symplectic system are characterized. Especially, for the scalar case on a finite discrete interval some equivalent forms and the uniqueness of the given expression are discussed and the Krein--von Neumann extension is described explicitly. In addition, a limit point criterion for symplectic systems is established. The result partially generalizes even a classical limit point criterion for the second order Sturm--Liouville difference equations.