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

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space â„“ 2 𠑤 ( ℤ ) ( ℤ : = { 0 , ± 1 , ± 2 , … } ), that is, the extensions of a minimal symmetric operator with defect index ( 2 , 2 ) (in the Weyl-Hamburger limit-circle cases at ± ∞ ). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at − ∞ ” and “dissipative at ∞ .” 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.