Essential spectrum of singular discrete linear Hamiltonian systems

Abstract

The paper is concerned with the essential spectral points of singular discrete linear Hamiltonian systems. Several sufficient conditions for a real point to be in the essential spectrum are obtained in terms of the number of linearly independent square-summable solutions of the corresponding homogeneous linear system, and a sufficient and necessary condition for a real point to be in the essential spectrum is given in terms of the number of linearly independent square-summable solutions of the corresponding nonhomogeneous linear system. As a direct consequence, the corresponding results for singular higher-order symmetric vector difference expressions are given.

Essential and continuous spectrum of symmetric difference equations

“…This in turn requires the asymptotics of the solutions of (1.1). Since it has been shown in that essential and absolutely continuous spectra together with their spectral multiplicities are independent of the boundary conditions and the left regular endpoints, we will pay little attention to the left regular endpoints. To obtain the asymptotics of the eigenfunctions of (1.1), rewrite (2.4) or (2.5) in the propagator form (2.6) and hence we determine the eigenvalues of the matrix .…”

“…There are a number of papers that have looked into various components of the spectrum of Hamiltonian systems of differential operators, for example, the papers by Behncke , Sun and Shi . Similarly, the author together with others have discussed the deficiency indices as well as the location of absolutely continous spectrum of both self‐adjoint extenstion operators of differential operators as well as for self adjoint extenstion subspaces for difference equations.…”

Relationships between Perturbations of a Linear Relation and Its Operator Part

“…Motivated by the study for the adjoint of nondensely defned linear diferential operators, the concept of linear relations, a natural generalization of linear operators, was introduced in [1]. Along with the development of operator theory, the spectral theory for linear relations has been extensively studied and has important applications to several problems (cf., [2][3][4][5][6][7][8][9][10][11][12][13][14]). It is worth mentioning that the spectra of linear relations may provide some useful tools for the study of certain operators, such as the maximal and minimal operators corresponding to linear continuous Hamiltonian systems or symmetric linear diference equations [12,15], and the inverse of certain operators in the study of some Cauchy problems associated with parabolic type equations in Banach spaces [16].…”