Eigenfunctions expansion for discrete symplectic systems with general linear dependence on spectral parameter

TL;DR

This paper extends the eigenfunction expansion theory for discrete symplectic systems with general spectral dependence, providing explicit integral representations and discussing extensions to half-line cases.

Contribution

It generalizes the eigenfunction expansion theorem for discrete symplectic systems to include arbitrary linear spectral dependence, expanding the theoretical framework.

Findings

Established eigenfunction expansion for generalized discrete symplectic systems

Derived explicit integral representation of the Weyl--Titchmarsh M-function

Discussed extension to half-line cases with illustrative examples

Abstract

Eigenfunctions expansion for discrete symplectic systems on a finite discrete interval is established in the case of a general linear dependence on the spectral parameter as a significant generalization of the Expansion theorem given by Bohner, Do\v{s}l\'{y} and Kratz in [Trans. Amer. Math. Soc. 361 (2009), 3109--3123]. Subsequently, an integral representation of the Weyl--Titchmarsh $M({\lambda})$-function is derived explicitly by using a suitable spectral function and a possible extension to the half-line case is discussed. The main results are illustrated by several examples.