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

Allahverdiev B. P., AYGAR KÜÇÜKEVCİLİOĞLU Y.

Filomat, cilt.38, sa.22, ss.7693-7705, 2024 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 38 Sayı: 22
  • Basım Tarihi: 2024
  • Doi Numarası: 10.2298/fil2422693a
  • Dergi Adı: Filomat
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.7693-7705
  • Anahtar Kelimeler: Characteristic func-tion, Completeness of the root functions, Dissipative operator, q-Sturm–Liouville equation, Self-adjoint dilation, Weyl–Titchmarsh function
  • Ankara Üniversitesi Adresli: Evet

Özet

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.