Spectral analysis for discontinuous non-self-adjoint singular Dirac operators with eigenparameter dependent boundary condition

In this paper, a discontinuous non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition, and with two singular endpoints is studied. The interface conditions are imposed on the discontinuous point. Firstly, we pass the considered problem to a maximal dissipative operator Lh by using operator theoretic formulation. The self-adjoint dilation Th of Lh in the space H is constructed, furthermore, the incoming and outgoing representations of Th and functional model are also constructed, hence in light of Lax-Phillips theory, we derive the scattering matrix. Using the equivalence between scattering matrix and characteristic function, completeness theorem on the eigenvectors and associated vectors of this dissipative operator is proved.