Dissipative second-order difference operators with general boundary conditions

Allahverdiev B.

JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, vol.10, no.1, pp.1-16, 2004 (SCI-Expanded) identifier identifier


A space of boundary values is constructed for minimal symmetric second-order difference operator in the Hilbert space I w 2 (Z) (Z:{0,+/-1,+/-2,...}) with defect index (2,2) (in Weyl's limit-circle cases at +/-infinity). A description of all maximal dissipative (accretive), selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at +/-infinity. We investigate maximal dissipative operators with, generally speaking, nonseparated boundary conditions. In particular, if we consider separated boundary conditions, that at -infinity and infinity nonselfadjoint (dissipative) boundary conditions are prescribed simultaneously. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We also construct a functional model of maximal dissipative operator and determine its characteristic function. We prove a theorem on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator.