Unifying discrete and continuous Weyl-Titchmarsh theory via a class of linear Hamiltonian systems on Sturmian time scales

TL;DR

This paper develops a unified Weyl-Titchmarsh theory for linear Hamiltonian systems on Sturmian time scales, bridging continuous and discrete spectral theories with new operator and spectral analysis results.

Contribution

It introduces a comprehensive spectral theory for Hamiltonian systems on Sturmian time scales, unifying continuous and discrete cases with novel operator and matrix disk methods.

Findings

Matrix disks are nested and converge to a limit set.

Relationships among solution spaces, defect indices, and matrix radii are established.

Classification of systems into limit point and limit circle cases is provided.

Abstract

In this study, we are concerned with introducing Weyl-Titchmarsh theory for a class of dynamic linear Hamiltonian nabla systems over a half-line on Sturmian time scales. After developing fundamental properties of solutions and regular spectral problems, we introduce the corresponding maximal and minimal operators for the system. Matrix disks are constructed and proved to be nested and converge to a limiting set. Some precise relationships among the rank of the matrix radius of the limiting set, the number of linearly independent square summable solutions, and the defect indices of the minimal operator are established. Using the above results, a classification of singular dynamic linear Hamiltonian nabla systems is given in terms of the defect indices of the minimal operator, and several equivalent conditions on the cases of limit point and limit circle are obtained, respectively. These results unify and extend certain classic and recent results on the subject in the continuous and discrete cases, respectively, to Sturmian time scales.