Symplectic Geometry Of Linear Hamiltonian Operator Symmetric Expansion Portrayed

Along with the development of modern physics and applied mathematics, the theory of Hamiltonian system has aroused people's widespread attention day by day,and it applied in physical discipline and other engineering disciplines more and more. The extension theory of Hamiltonian system as an important branch in the theory of Hamiltonian system, has abtained widespread attention of many physicists and other engineering scientists and it is an important tool to solve many problems.Many useful methods and conclusions on the self-adjoint extension prob-lems of Hamiltonian system have obtained which include GKN theory for linear Hamiltonian systems, the self-adjoint extensions of minimal operator for singular Hamiltonian systems which are determined by using Weyl's solutions, self-adjoint extensions for linear Hamiltonian systems with one singular endpoint and self-adjoint extensions for linear Hamiltonian systems with two singular endpoints. However, for introducing methods of linear complex symplect'ic geometry to the extension theory of Hamiltonian system,there is no systematic research up to date.In this paper, we give complex symplectic geometry characterizations for the symmetric extensions of regular or sigular linear Hamiltonian operators, together with the symplectic geometry description of the self-adjoint domains of regular linear Hamiltonian operators. A symplectic space is constructed by the domain of maximal operator and minimal operator of linear Hamiltonian systems. By using Lagrangian sub-manifold, the symmetric extensions of Hamiltonian operator are described and the classification related to the self-adjoint domains are presented, the symmetric extensions of Hamiltonian operator and the classification related to the self-adjoint domains are studied in terms of the symplectic geometry.According to the contents, the thesis is divided into three chapters.Chapter 1, preface.In Chapter 2, we give complex symplectic geometry characterizations for the symmetric extensions of the minimal operator, which is generated by reg-ular linear Hamiltonian systems. A symplectic space is constructed by the do-main of maximal operator and minimal operator of regular linear Hamiltonian systems. By using Lagrangian sub-manifold, the symmetric extensions of the minimal Hamiltonian operator are described and the classification related to the self-adjoint domains are presented.we give the correspondence between the set of symmetic extensions of the minimal operator and the set of Lagrangian sub-manifold,and the necessary and sufficient conditions which ensure that its exten-sions is symmetic extensions are obtained.Further, give the complete characteri-zations for the self-adjoint domains with complex symplectic geometry.In Chapter 3, we give complex symplectic geometry characterizations for the symmetric extensions of the minimal operator, which is generated by singu-lar linear Hamiltonian systems.A symplectic space is constructed by the domain of maximal operator and minimal operator of singular linear Hamiltonian sys-tems. By using Lagrangian sub-manifold,the symmetric extensions of the mini-mal Hamiltonian operator is described. We give the correspondence between the set of symmetic extensions of the minimal operator and the set of Lagrangian sub-manifold, the necessary and sufficient conditions which ensure that its extensions is symmetic extensions are obtained.