The GKN Theory And Spectral Theory Of Hamiltonian Systems On Time Scales

All conservative physical processes can be described by mathematical models in the Hamiltonian formalism of mechanics which is equivalent to Newtonian formalism and Lagrangian formalism of mechanics. However, the Hamiltonian formalism of mechanics has remarkable elegance and symmetry. So the Hamiltonian formalism has become one of the most useful mathematical tools in research of physical sciences. The study of fundamental theory of continuous Hamiltonian systems has a long history. In recent years, the investigation of the theory of difference systems has held a lot of attraction and assumed a greater importance as a well deserved discipline. The theory of difference equations is a lot richer than the corresponding theory of differential equations. In spite of this tendency of independence, there is a striking similarity or even duality between the theories of continuous and discrete cases. Many results in the theory of difference equations have been obtained as more or less natural discrete analogs of corresponding results of differential equations.As a tool for establishing a unified framework for explaining these analogies, a theory of dynamic equations on time scales introduced by Stefan Hilger in his Ph. D. thesis [73] in 1988 has been developed in the last decade. The theory of dynamic equations on time scales provides a unified theory for continuous and discrete equations or systems and helps avoid proving results twice in many cases, once for continuous systems. and once for discrete ones. Moreover, it explains the discrepancies which occur in parallel statements in continuous and discrete cases and therefore helps with better understanding them.The theory of dynamic systems on time scales is a fairly new subject, and research in this area is rapidly growing. An introduction to this subject is given in [21, 22]. The study of Hamiltonian systems on time scale is at beginning just these years. Ondrej Dosly and Roman Hilscher initiated the study of symplectic dynamic systems on time scales, and it was pursued by Calvin Ahlbrandt, Martin Bohner, and Jerry Ridenhour and Hilscher. So far. eigenvalue problems, Sturmian properties of the principal solution.