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

Abstract EN

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space lw2(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.