Incompleteness of resonance states for quantum ring with two semi-infinite edges

In this paper we investigate scattering problem for a quantum graph (a ring), connected with two semi-infinite leads via a Dirac delta function at boundary. We prove incompleteness of the system of resonance states in \(L_2\) on finite subgraph for the Kirchhoff coupling condition at the vertex and discuss a relation with the factorization of the characteristic function in Sz-Nagy functional model. The sensitivity of the incompleteness property to variation of the operator or the graph structure is considered. The cases of the Landau and the Dirac operators at the graph edges demonstrate the same completeness/incompleteness property as the Schrodinger case. At the same time, small variation of the graph structure restores the completeness property.