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

Bilender P Allahverdiev

We consider the dissipative singular q-Sturm–Liouville operators acting in the Hilbert space L 2 w,q (R +), that the extensions of a minimal symmetric operator with deficiency indices (2, 2) (in limit-circle case). We construct a self-adjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation in terms of the Weyl–Titchmarsh function of a self-adjoint q-Sturm-Liouville operator. We also construct a functional model of the dissipative operator and determine its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl–Titchmarsh function). Theorems on the completeness of the system of or root functions of the dissipative and accumulative q-Sturm–Liouville operators are proved.