Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators

Abstract

Abstract In this study we construct a space of boundary values of the minimal symmetric discrete Sturm-Liouville (or second-order difference) operators with defect index (1, 1) (in limit-circle case at ±∞ and limit-point case at ∓∞), acting in the Hilbert space ℓ ϱ 2 ( Z ) ℓ ϱ 2 ( Z ) ( Z := { 0 , ± 1 , ± 2 , … } ) $\ell_{\varrho}^{2}(\mathbb{Z}) (\mathbb{Z} :=\{0,\pm 1,\pm 2,\dots\})$ . Such a description of all maximal dissipative, maximal accumulative and self-adjoint extensions is given in terms of boundary conditions at ± ∞. After constructing the space of the boundary values, we investigate two classes of maximal dissipative operators. This investigation is done with the help of the boundary conditions, called “dissipative at −∞” and “dissipative at ∞”. In each of these cases we construct a self-adjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations. These representations allow us to determine the scattering matrix of dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the Weyl-Titchmarsh function of the self-adjoint operator. Finally, we prove a theorem on completeness of the system of eigenvectors and associated vectors (or root vectors) of the maximal dissipative operators.