Spectrum Of Infinite Dimensional Hamiltonian Operators And Completeness Of The Eigenfunction Systems

This dissertation focuses on the completeness of the eigenfunctions systems(symplectic orthogonal system) of infinite dimensional Hamiltonian operators and researches into the spectral theory and on existence of maximal definite invariant subspace in Krein space,which developes the Strum-Liouville problems and the methods of eigcnfunctions expansion and provides a theoretical basis for employing the method of separation of variables based on Hamiltonian systems.In theories of infinite dimensional Hamiltonian operators and infinite dimensional Hamiltonian systelns,completeness of the eigenfunctions systems(symplectic orthogohal system) of the infinite dimensional Hamiltonian operators is very important problem. The traditional method of separation of variable is effective to solve partial differential equations which can be transformed into the Strum-Liouville problem after separating variables.However,infinite dimensional Hamiltonian operator is non-selfadjoint operator in generally,therefore,to employ the method of separation of variables based on Hamiltonian systems,the completeness of the eigenfunctions systems(symplectic orthogonal system) of the infinite dimensional Hamiltonian operators must be solved. Thus,we obtain the sufficient conditions of the completeness in the sense of Cauchy Principal Value of the eigenfunctions systems of the infinite dimensional Hamiltonian operator by taking advatage of the symplectic orthogonality of eigenfunctions systems and the property of existing real eigenvalues or pure imaginary eigenvalues only and appear pairwise according to positive and negative,consequently,we get solutions of complete in sense of Cauchy principal value.This works give a new method and new idea to solve infinite dimensional Hamiltonian system and even ordinary partial differential equations and possess high theoretical values and practical significance. To solve more general infinite dimensional Hamiltonian systems,we must study the properties of general infinite dimensional Hamiltonian operator,which belongs to areas of linear operator theory.As far as we know,spectral analysis of linear operator is important component of functional analysis and soul of linear operator theory,its centre subject is spectral decomposition.Therefore,in this paper,we also focus on spectrum of infinite dimensional Hamiltonian operator and obtain spectral properties of upper triangular infinite dimensional Hamiltonian operator,which provides necessary preparations for solving completion problem and spectral perturbation problem of upper triangular infinite dimensional Hamiltonian operator;To solve the problem of infinite dimensional Hamiltonian operator generates C₀ Semi-group,we also obtain the sufficient conditions of infinite dimensional Hamiltonian operator exists pure imaginary spectrum only;In addition,when the operator is invertible,it provide theoretical foundations for semi-analytical method and the partial differential equations can be transformed into ordinary differential equations,therefore,the problem of invertibility of infinite dimensional Hamiltonian operator is very important and the nature of problem is zero point whether belongs to regular set.Thereby,taking full advantage of structure of non-negative Hamiltonian operator,the sufficient conditions for nonnegative Hamiltonian operators exist everywhere defined bounded inverse are given. It is worth noting that the notion of numerical range is important in various applications, since it was used to as a tool in order to localize the spectrum of operators,that is to say,the closure of numerical range contains the spectral set.However.recently H.Langer found that the quadratic numerical range of bounded operator is a subset of the numerical range and that its closure still contains the spectral set.Thus,in general, it gives better information about the location of the spectrum of bounded linear operator than the numerical range.So,in this paper we study the quadratic numerical range and numerical range of a class of unbounded infinite dimensional Hamiltonian operators and the conclusion that not only the closure of the numerical range contains the spectral set but also the closure of the quadratic numerical range contains the spectral set is shown.We also investigate the spectral theory of infinite dimensional Hamiltonian operators in complete indefinite metric space.Linear operator theory in indefinite metric space is not a logical promotion of linear operator theory in Hilbert space,but has profound theoretical basis,its application including physics,mathematics and mechanics. In view of particularities of infinite dimensional Hamiltonian operators,after introducing appropriate indefinite metric,it can become anti-selfadjoint operator;therefore, we can draw meaningful conclusions.Furthermore,the sufficient conditions of infinite dimensional Hamiltonian operator exists maximal definite invariant subspace is given.In 1909,H.Weyl discovered that complement in the spectrum of the Weyl spectrum of bounded selfadjoint linear operator coincides with the isolated eigenvalue of finite multiplicity.Today this result is known as Weyl's theorem and it has been studied by numerous authors,such as J.Schwartz,S.Berberian,and extended from bounded selfadjoint operator to other class of bounded operator.But most of results are Weyl's theorem for bounded operators and about unbounded operators are very rare.Hence in this paper we consider how Weyl's theorem survives for unbounded self-adjoint operator under small perturbations and the sufficient conditions of compact operator survives Weyl's theorem are given.This paper contains seven chapters.In first chapter,we introduce the significance of topics and main results we obtained;In second chapter the spectral properties of upper triangular Hamiltonian operators are given and the eigenvalue problems of infinite dimensional Hamiltonian operators are discussed;In third chapter,completeness in the sense of Cauchy principal value of the eigenfunctions systems(symplectic orthogonal system) of the infinite dimensional Halniltonian operators is studied;In fourth chapter we investigate the invertibility of non-negative Hamiltonian operators; In chapter fifth,the properties of numerical range and quadratic numerical range of infinite dimensional Hamiltonian operators are considered;In sixth chapter,the Weyl's theorem for unbounded operators under small perturbations is studied;In last chapter we introduce spectral theory of infinite dimensional Hamiltonian operators in Krein space.