Extensions, Dilations and Functional Models of Singular Sturm-Liouville Operators

Abstract

A space of boundary values is constructed for minimal symmetric singular Sturm-Liouville operator acting in the Hilbert space Lw [a, b), −∞ < a < b ≤ ∞, with deficiency indices (2, 2) (in Weyl’s limit-circle case). A description of all maximal dissipative, maximal accretive, self-adjoint, and other extensions of such a symmetric operator is given in terms of boundary conditions at end points a and b. We investigate maximal dissipative operators with general (coupled or separated) boundary conditions. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and determine its characteristic function. We prove the theorem on completeness of the system of eigenfunctions and associated functions of the maximal dissipative operators.