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SYSTEM COMPLETENESS OF RESONANCE STATES FOR GRAPHS WITH DIFFERENT GEOMETRY

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Resonant states (quasi-intrinsic functions) play an important role in the scattering problem and in the description of transportprocesses. We consider the system completeness of resonance states on a finite subgraph for quantum graphs with both finite and infinite edges. The Schrodinger operator acts on the graph edges. The relationship of the scattering problem with the Szőkefalvi-Nagy functional model is taken into account. In particular, the scattering matrix is a characteristic function of the functional model, and the question of the system completeness of resonant states is reduced to factorization type determining of the characteristic function on the Blaschke product and the singular internal function. This fact makes it possible to use an effective sign of a singular factor absence in the decomposition of the characteristic function for the proof of completeness (incompleteness) available in the functional model. The system incompleteness of resonance states for a "ring-type" graph connected to a waveguide at one point (initial graph) is proved. The dependence of the system completeness of resonant states on initial graph geometry changes is studied.

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