Abstract
We consider the dissipative singular q-Sturm-Liouville operators acting in the Hilbert space L2 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.
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