Weyl–Titchmarsh theory for a class of discrete linear Hamiltonian systems

Abstract

This paper is concerned with establishing the Weyl–Titchmarsh theory for a class of discrete linear Hamiltonian systems over a half-line. Fundamental properties of solutions, regular spectral problems, and the corresponding maximal and minimal operators are first studied. Matrix disks are constructed and proved to be nested and converge to a limiting set. Some precise relationships among the rank of the matrix radius of the limiting set, the number of linearly independent square summable solutions, and the defect indices of the minimal operator are established. Based on the above results, a classification of singular discrete linear Hamiltonian systems is given in terms of the defect indices of the minimal operator, and several equivalent conditions on the cases of limit point and limit circle are obtained, respectively. Especially, several problems in the limit point case are more carefully investigated, including fundamental properties of square summable solutions, properties of the Weyl function, which is the unique element in the limiting set in this case, and inhomogeneous boundary problems, self-adjointness of the corresponding Hamiltonian operator, relationship between the spectrum of the Hamiltonian operator and the analyticity of the Weyl function, as well as the dependence of the spectrum on the boundary data, in which some interesting separation results for the spectrum are obtained. Finally, another set of four equivalent conditions on the limit point case are established.