A space of boundary values is constructed for minimal symmetric Sturm-Liouville operator acting in L-e(2)(a, b) with defect index (1, 1) (in limit-circle case at a (b) and limit-point case at b (a)). All maximal dissipative, maximal accumulative and self-adjoint extensions of such a symmetric operator are described in terms of boundary conditions at a (b). In each case, we construct a self-adjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which allows us to determine the scattering matrix. We establish a functional model of the dissipative operator and construct its characteristic function in terms of the Weyl-Titchmarsh function on the self-adjoint operator. We also prove the completeness of the root functions of the dissipative Sturm-Liouville operators.