Spectral problems of dissipative singular $q$-Sturm-Liouville operators in limit-circle case

Abstract

We consider the dissipative singular $q$-Sturm-Liouvilleoperators acting in the Hilbert space $L_{w,q}^{2}(R_{+})$,\ that theextensions of a minimal symmetric operator\ with defect index ($2,2$) (inlimit-circle case).\ We construct a self-adjoint dilation of the dissipativeoperator and its incoming and outgoing spectral representations, which makeit possible to determine the scattering matrix of the dilation in terms ofthe Weyl-Titchmarsh function of a self-adjoint $q$-Sturm-Liouville operator.We also construct a functional model of the dissipative operator anddetermine its characteristic function in terms of the scattering matrix ofthe dilation (or of the Weyl-Titchmarsh function). Theorems on thecompleteness of the system of or root functions of the dissipative andaccumulative $q$-Sturm-Liouville operators are proved.