Extensions and spectral problems of 1D discrete Hamiltonian systems

Source abstract

In this paper, we construct a space of boundary values of the minimal symmetric discrete Hamiltonian operator with defect index (2,2), which is known as Weyl's limit‐circle cases at ± ∞ , acting in the Hilbert space , where . With the help of the space of the boundary values, we describe all maximal dissipative (accretive), self‐adjoint, and other extensions of such a symmetric operator. In these descriptions, we investigate maximal dissipative operators with general boundary conditions. For maximal dissipative operator, a self‐adjoint dilation is constructed. Further, following the scattering theory, its incoming and outgoing spectral representations are set. These representations allow us to determine the scattering matrix of the dilation. Moreover, we construct a functional model of the maximal dissipative operator, and we define its characteristic function in terms of the scattering matrix of the dilation. Finally, we prove a completeness theorem about the system of root vectors of the maximal dissipative operator. Copyright © 2013 John Wiley & Sons, Ltd.