EXAMINING SINGULAR DISSIPATIVE QUANTUM STURM–LIOUVILLE OPERATORS IN LIMIT-CIRCLE CASE WHENEVER q > 1

In this article, we examine dissipative singular quantum (q) Sturm–Liouville operators (q > 1) acting in a suitable Hilbert space, where the extensions of a minimal symmetric operator in limit-circle case (with deficiency indices (2, 2)) are presented. We create a self-adjoint dilation of the dissipative operator along with its incoming and outgoing spectral representations. These constructions enable us to find the scattering matrix of the dilation using the Weyl–Titchmarsh function associated with a self-adjoint q-Sturm–Liouville operator . Additionally, we establish a functional model for the dissipative operator and derive its characteristic function using the scattering matrix of the dilation (or the Weyl–Titchmarsh function). We prove theorems related to the completeness of the system of eigenfunctions and associated functions (root functions) for both dissipative and accumulative q-Sturm–Liouville operators.