Quantum graph with the Dirac operator and resonance states completeness

Abstract

Quantum graphs with the Dirac operator at the edges are considered. Resonances (quasi-eigenvalues) and resonance states are found for certain star-like graphs and graphs with loops. Completeness of the resonance states on finite subgraphs is studied. Due to use of a functional model, the problem reduces to factorization of the characteristic matrixfunction. The result is compared with the corresponding completeness theorem for the Schrödinger quantum graph.