The Self-adjointness Of Products For Hamiltonian Operators

The theory of ordinary differential operators is the fundamentality of differential equations,classical physics,modern physics and other technology fields,it is one of the integrative and edging mathematics branches which combines ordinary differential equation,functional analysis,space theory,operator theory etc.. It contains many important problems,such as deficiency index theory,adjoint extension,spectral analysis,expansion with eigenfunctions,numerical method, inverse question,and so on.The self-adjointness of product of differential operators has obtained some known results(see[22],[23],[24]for details).However,for the self-adjointness of product of Hamiltonian operators,there is no systematic research up to date. This paper will discuss the self-adjointness of product of Hamiltonian operators generated by the regular and singular system.Using the general structure of selfadjoint extension theory of differential operator and analytical skills,we obtain the sufficient conditions for self-adjointness of products of two and four Hamiltonian operators on I(I =[a,b]or[a,∞)).According to the contents,the text is divided into four chapters:ChapterⅠ,Preface.ChapterⅡ,The introduction of the prior knowledge and the basic theories for the operators.ChapterⅢ,By using of the self-adjoint conditions for Hamiltonian operator, the self-adjoint extension theory and the general structure analysis techniques of differential operator,we give sufficient conditions for self-adjointness of the product of two Hamiltonian operators on I(I =[a,b]or[a,∞)).ChapterⅣ,On the basis of ChapterⅢ,we further study the self-adjointness for the product of the four Hamiltonian operator,and obtain sufficient conditions for self-adjointness of the product of four Hamiltonian operators on I(I =[a,b] or[a,∞)).