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 ( 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.