Dissipative Sturm-Liouville Operators with Nonseparated Boundary Conditions

Abstract

A space of boundary values is constructed for the minimal symmetric singular Sturm-Liouville operator in the Hilbert space \(L_{w}^{2}(a,b)(-\infty \le a < b \le \infty )\) with defect index (2,2) (in Weyl’s limit-circle cases at singular points a and b). A description of all maximal dissipative, maximal accretive, selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at a and b. We investigate maximal dissipative operators with, generally speaking, nonseparated boundary conditions. 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 a dissipative operator and define its characteristic function. We prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operators.