Study On Essential Spectrum Of Singular Discrete Matrix Systems

The basic theory of matrix system is widely used in system theory,nonlinear analysis and development equation problems.From the point of view of practical application,it applies partial differential equations to solve problems,elasticity,fluid mechanics,magnetolrydrodynamics,quantum mechanics and other mathematical physics problems have important applications.At present,for discrete matrix systems only in Sun and Zhu have been studied in references[55].This paper will further study this class of 2 × 2 diserete matrix system with singular endpoints is studied.The specific arrangement is as follows:The first chapter introduces the background and development status of discrete matrix system research,and gives the main content and innovations of this paper.The second chapter is preliminary knowledge,which mainly reviews the basic concepts and results of linear subspaces(linear relations)in Hilbert space,and gives preliminary lemmas.The third chapter studies a class of discrete matrix systems with a singular endpoint.Firstly,the definitions of the maximum subspace,domain minimum and minimum subspace of this kind of discrete matrix system are given,and the minimum subspace is proved to be Hcrmitian.Then a class of discrete Haniiltonian systems with parameters is proposed,and based on the method of[29],we give the equivalent relationship between discrete matrix systems and discrete Hamiltonian systems with parameters.Then,using this relationship,the discrete matrix limit point and the classification of limit circles are given(see Theorem 3.9).The fourth chapter is the research on the essential spectrum of discrete matrix systems.From the equivalence relationship between discrete Hamiltonian systems with parameters and discrete matrix systems,the specific characterization of the essential spectrum is obtained.