SOME SPECTRAL PROBLEMS OF DISSIPATIVE q-STURM--LIOUVILLE OPERATORS IN LIMIT-POINT CASE FOR q>1

Abstract

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.