NONSELFADJOINT SINGULAR STURM-LIOUVILLE OPERATORS IN LIMIT-CIRCLE CASE

Abstract

In this paper, we study the maximal dissipative singular Sturm-Liouville operators (in Weyl's limit-circle case at singular point $b$) acting in the Hilbert space $L_{w}^{2}\left[ a,b\right)$ ($-\infty \lt a\lt b\leq \infty$ ). In fact, we consider all extensions of a minimal symmetric operator and we investigate two classes of maximal dissipative operators with separated boundary conditions, called 'dissipative at $a$' and 'dissipative at $b$'. In both cases, we construct a selfadjoint dilation of the maximal dissipative operator and determine its incoming and outgoing spectral representations. This representations make it possible to determine the scattering matrix ofthe dilation in terms of the Titchmarsh-Weyl function of a selfadjoint Sturm-Liouville operator. We also construct a functional model of the maximal dissipative operator and determine its characteristic function in terms of the scattering matrix of the dilation (or of the Titchmarsh-Weyl function). Finally we prove theorems on the completeness of the eigenfunctions and associated functions of the maximal dissipative Sturm-Liouville operators.