Some spectral problems of dissipative q-Sturm–Liouville operators in limit-point case for q > 1
Filomat, vol.38, no.22, pp.7693-7705, 2024 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 38 Issue: 22
- Publication Date: 2024
- Doi Number: 10.2298/fil2422693a
- Journal Name: Filomat
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.7693-7705
- Keywords: Characteristic func-tion, Completeness of the root functions, Dissipative operator, q-Sturm–Liouville equation, Self-adjoint dilation, Weyl–Titchmarsh function
- Ankara University Affiliated: Yes
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.