Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases

Abstract

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space ${\mathcal{l}}_w^2\left(\mathbb{Z}\right)$ ($\mathbb{Z}$:$=\{0,\pm 1,\pm 2,\dots \}$), that is, the extensions of a minimal symmetric operator with defect index ($\mathrm{2,2}$) (in the Weyl-Hamburger limit-circle cases at $\pm \infty$). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at $-\infty$” and “dissipative at $\infty$.” In each case, we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also establish a functional model of the maximal dissipative operator and determine its characteristic function through the Titchmarsh-Weyl function of the self-adjoint operator. We prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.