Spectral problems of non-self-adjoint q-Sturm-Liouville operators in limit-point case

Abstract

In this study, dissipative singular q-Sturm-Liouville operators are studied in the Hilbert space $\mathscr{L}_{r,q}^{2}$(Rq,+), that the extensions of a minimal symmetric operator in limit-point case. We construct a self-adjoint dilation of the dissipative operator together with its incoming and outgoing spectral representations so that we can determine the scattering function of the dilation as stated in the scheme of Lax-Phillips. Then, we create a functional model of the maximal dissipative operator via the incoming spectral representation and define its characteristic function in terms of the Weyl-Titchmarsh function (or scattering function of the dilation) of a self-adjoint q-Sturm-Liouville operator. Finally, we prove the theorem on completeness of the system of eigenfunctions and associated functions (or root functions) of the dissipative q-Sturm-Liouville operator.