Weyl-Titchmarsh Theory for Hamiltonian Dynamic Systems

and Applied Analysis 3 and spectral theory for the system 1.2 . Shi studied Weyl-Titchmarsh theory and spectral theory for the system 1.2 in 33, 34 ; Clark and Gesztesy established the Weyl-Titchmarsh theory for a class of discrete Hamiltonian systems that include system 1.2 23 . Sun et al. established the GKN-theory for the system 1.2 35 . 1.3. Dynamic Equations A time scale T is an arbitrary nonempty closed subset of the real numbers. The theory of time scales was introduced by Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis 36 . Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. 37, 38 and references cited therein. A book on the subject of time scales, by Bohner and Peterson 39 , summarizes and organizes much of the time scale calculus. We refer also to the book by Bohner and Peterson 40 for advances in dynamic equations on time scales and to the book by Lakshmikantham et al. 41 . This paper is devoted to the Weyl-Titchmarsh theory for linear Hamiltonian dynamic systems xΔ t A t x t B t λW2 t u t , uΔ t C t − λW1 t x t −A∗ t u t , 1.3 where t takes values in a time scale T, σ t : inf{s ∈ T | s > t} is the forward jump operator on T, x x ◦ σ, and Δ denotes the Hilger derivative. A universal method we provided here allows one to treat both continuous and discrete linear Hamiltonian systems as special cases within one theory and to explain the discrepancies between them. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems on time scales. Some ideas in this paper are motivated by some works in 5–9, 11, 18–20, 34, 35, 42 . The paper is organized as follows. Some fundamental theory for Hamiltonian systems is given in Section 2. Some regular spectral problems are considered in Section 3. The Weyl matrix disks are constructed and their properties are studied in Section 4. These matrix disks are nested and converge to a limiting set of the matrix circle. The results are some generalizations of the Weyl-Titchmarsh theory for both Hamiltonian differential systems 6, 9, 18, 20, 26 and discrete Hamiltonian systems 34 . These investigations are part of a larger program which includes the following: i M λ theory for singular Hamiltonian systems, ii on the spectrum of Hamiltonian systems, iii on boundary value problems for Hamiltonian dynamic systems. 2. Assumptions and Preliminary Results Throughout we use the following assumption. Assumption 1. ̃ T is a time scale that is unbounded above, that is, ̃ T is a closed subset of R such that sup ̃ T ∞. We let a ∈ ̃ T and define T ̃ T ∩ a,∞ . 4 Abstract and Applied Analysis In this section, we shall study the fundamental theory and properties of solutions for the Hamiltonian dynamic system 1.3 , that is, JyΔ t : λW t P t ỹ t for t ∈ T, 2.1 where