Some spectral problems of dissipative q-Sturm–Liouville operators in limit-point case for q > 1

Abstract

The main purpose of this study is to investigate dissipative singular q-Sturm–Liouville operators in a suitable Hilbert space and to examine the extensions of a minimal symmetric operator in limit-point case. We make a self-adjoint dilation of the dissipative operator together with its incoming and outgoing spectral components, which satisfy determining the scattering function of the dilation via Lax-Phillips theory. We also construct a functional model of the maximal dissipative operator by using the incoming spectral representation and we find its characteristic function in terms of the Weyl–Titchmarsh function of the self-adjoint q-Sturm–Liouville operator whenever q > 1. Furthermore, we present a theorem about the completeness of the system of eigenfunctions and associated functions (or root functions) of the dissipative q-Sturm–Liouville operator.