Abstract and Applied Analysis
Volume 2010 (2010), Article ID 514760, 18 pages
doi:10.1155/2010/514760
Weyl-Titchmarsh theory for Hamiltonian dynamic systems
Shurong Sun1
, Martin Bohner2
and Shaozhu Chen3
1School of Science, University of Jinan, Jinan, Shandong 250022, China
2Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA
3Department of Mathematics, Shandong University in Weihai, Weihai, Shandong 264209, China
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=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(λ) theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.