Weyl-Titchmarsh Theory for Hamiltonian Dynamic Systems

Abstract

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale $𝕋$, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for $𝕋$ $=ℝ$ and $𝕋=\mathbb{Z}$ within one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i) $M\left(\lambda \right)$ theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.