Spectral problems of non-self-adjoint singular q-Sturm-Liouville problem with an eigenparameter in the boundary condition

In this paper, a non-self-adjoint (dissipative) q-Sturm-Liouville boundary-value problem in the limit-circle case with an eigenparameter in the boundary condition is investigated. The method is based on the use of the dissipative operator whose spectral analysis is su ffi cient for boundary value problem. A selfadjoint dilation of the dissipative operator together with its incoming and outgoing spectral representations is established and so it becomes possible to determine the scattering function of the dilation. A functional model of the dissipative operator is constructed and its characteristic function in terms of scattering function of dilation is defined. Theorems on the completeness of the system of eigenvectors and the associated vectors of the dissipative operator and the q-Sturm-Liouville boundary value problem are presented.