Research On Spectral Theory Of Singular Linear Hamiltonian Systems And Its Applications

In this paper,we study the spectral theory of singular continuous Hamiltonian systems and singular discrete Hamiltonian systems.For singular continuous Hamiltonian systems,we firstly research their Friedrichs extensions（F-extensions）.Secondly,we study the regular approximation of spectra of J-symmetric Hamiltonian systems with two singular endpoints.Thirdly,we research the essential spectra of singular continuous Hamiltonian systems of arbitrary order under a class of perturbations.For singular discrete Hamiltonian systems,we study the essential spectra of singular discrete Hamiltonian systems under a class of perturbations.This dissertation is divided into six sections.In Section 1,we shall introduce the research background and current status of the above issues,and give the problems and main tasks that will be considered in this paper.In Section 2,we shall introduce some basic concepts and useful results,including linear operators,linear subspaces,and sesquilinear forms in a Hilbert space and give some lemmas.In Section 3,F-extensions of a class of singular Hamiltonian systems including nonsymmetric cases are characterized by imposing some constraints on each element of D（H）.Further,F-extensions of symmetric,J-symmetric,regular Hamiltonian systems,and S-L operators with matrix-valued coefficients are characterized.In addition,a.result is given for elements of D（H）,which makes the expression of the F-extension domain simpler.Note that these characterizations are independent of principal solutions.It is interesting that by the results in the paper the F-extension of each of a large class of non-symmetric Hamiltonian systems has similar form to that of a symmetric Hamiltonian system.Some results are new and some of them improve or extend previous results obtained by[27,71,79,84,137].In Section 4,we study the regular approximation of spectra of J-symmetric Hamiltonian systems with two singular endpoints.For any given J-self-adjoint Hamiltonian extension of systems,we firstly construct the inherited restriction operators using Jlimit-circle solutions and solutions of systems.Then,we obtain the generalized strongly resolvent convergence of the inherited restriction operators and establish local spectral exactness results.Moreover,when both endpoints are J-limit circle,the sequence of inherited restriction operators is spectrally exact for any given J-self-adjoint Hamiltonian extension of systems.The results improve or extend previous results obtained by[18,24].In Section 5,we study the essential spectrum of a continuous Hamiltonian system of arbitrary order under a class of perturbations.Some sufficient conditions for the invariance of essential spectra of some systems are obtained in terms of coefficients of systems and perturbations terms.Further,essential spectra of Hamiltonian systems with different weight functions are discussed.Here,the Hamiltonian system is of arbitrary order,including even order and odd order.In addition,the Hamiltonian system may be non-symmetric.It is noted that these perturbations are given by using the associated preminimal operator H₀₀,which provides great convenience in the study of essential spectra of continuous Hamiltonian systems since each element of the domain D(H₀₀)of H₀₀ has compact support.In Section 6,we study the essential spectrum of a discrete Hamiltonian system under a class of perturbations.We first present a new characterization of the essential spectrum in terms of singular sequences and then give the concept of perturbations small at infinity for discrete Hamiltonian systems.Based on the above characterization,the invariance of essential spectra of discrete Hamiltonian systems under these perturbations is shown.As applications,some sufficient conditions for the invariance of essential spectra of some systems are obtained in terms of coefficients of systems and perturbations terms.It is noted that the study of discrete Hamiltonian systems is more complex than that of continuous Hamiltonian systems,and our assumptions are weaker and easier to verify compared with[99,Theorem 5.1]and[124,Theorem 4.4].