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Editor-in-Chief
Nikiforov
Vladimir O.
D.Sc., Prof.
Partners
doi: 10.17586/2226-1494-2019-19-1-166-172
SYSTEM COMPLETENESS OF RESONANCE STATES FOR GRAPHS WITH DIFFERENT GEOMETRY
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Article in Russian
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Abstract
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Blinova I.V., Van-Yun-Syan Ya.M., Popov I.Yu. System completeness of resonance states for graphs with different geometry. Scientific and Technical Journal of Information Technologies, Mechanics and Optics , 2019, vol. 19, no. 1, pp. 166–172 (in Russian). doi: 10.17586/2226-1494-2019-19-1-166-172
Abstract
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.
Keywords: quantum graph, system completeness of resonant states, Schrodinger operator
Acknowledgements. The work was partially supported by the Government of the Russian Federation (grant 08-08) and the Russian Science Foundation (grant 16-11-10330).
References
Acknowledgements. The work was partially supported by the Government of the Russian Federation (grant 08-08) and the Russian Science Foundation (grant 16-11-10330).
References
1. Leble S. Cyclic-periodic ZRP structures. Scattering problem for generalized Bloch functions and conductivity. Nanosystems: Physics, Chemistry, Mathematics, 2018, vol. 9, no. 2, pp. 225–243. doi: 10.17586/2220-8054-2018-9-2-225-243
2. Boitsev A.A., Brasche J., Neidhardt H., Popov I.Y. A model of electron transport through a boson cavity. Nanosystems: Physics, Chemistry, Mathematics, 2018, vol. 9, no. 2, pp. 171–178. doi: 10.17586/2220-8054-2018-9-2-171-178
3. Cevizovic D., Chizhov A.V., Galovic S. Vibron transport in macromolecular chains with squeezed phonons. Nanosystems: Physics, Chemistry, Mathematics, 2018, vol. 9, no. 5, pp. 597–602. doi: 10.17586/2220-8054-2018-9-5-597-602
4. Bagmutov A.S., Popov I.Yu. Current-voltage characteristics for two systems of quantum waveguides with connected quantum resonators. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2016, vol. 16, no. 4, pp. 725–730. (in Russian) doi: 10.17586/2226-1494-2016-16-4-725-730
5. Berkolaiko G., Kuchment P. Introduction to Quantum Graphs. Providence, AMS, 2012, 418 p. doi: 10.1090/surv/186
6. Popov I.Y., Popova S.L. Zero-width slit model and resonances in mesoscopic systems. Europhysics Letters, 1993, vol. 24, no. 5, pp. 373–377. doi: 10.1209/0295-5075/24/5/009
7. Borisov D., Exner P., Golovina A. Tunneling resonances in systems without a classical trapping. Journal of Mathematical Physics, 2013, vol. 54, no. 1. doi: 10.1063/1.4773098
8. Botman S.A., Leble S.B. Electrical conductivity model for qua-si-one-dimensional structures. Nanosystems: Physics, Chemistry, Mathematics, 2017, vol. 8, no. 2, pp. 231–235. doi: 10.17586/2220-8054-2017-8-2-231-235
9. Gerasimov D.A., Popov I.Y. Completeness of resonance states for quantum graph with two semi-infinite edges. Complex Variables and Elliptic Equations, 2018, vol. 63, no. 7–8, pp. 996–1010. doi: 10.1080/17476933.2017.1289517
10. Popov I.Y., Popov A.I. Line with attached segment as a model of Helmholtz resonator: Resonant states completeness. Journal of King Saud University Science, 2017, vol. 29, no. 1, pp. 133–136. doi: 10.1016/j.jksus.2016.07.004
11. Popov A.I., Popov I.Y., Gerasimov D.A. Resonance state completeness problem for quantum graph. AIP Conference Proceedings, 2017, vol. 1863, art. 390002. doi: 10.1063/1.4992567
12. Blinova I.V., Popov I.Y. Quantum graph with the Dirac operator and resonance states completeness. Operator Theory: Advances and Applications, 2018, vol. 268, pp. 111–124. doi: 10.1007/978-3-319-75996-8_5
13. Popov I.Y., Popov A.I. Quantum dot with attached wires: resonant states completeness. Reports on Mathematical Physics, 2017, vol. 80, no. 1, pp. 1–10. doi: 10.1016/s0034-4877(17)30054-x
14. Nikol'skii N. Treatise on the Shift Operator: Spectral Function Theory. Berlin, Springer, 1986, 491 p. doi: 10.1007/978-3-642-70151-1
15. Sz.-Nagy B., Foias C., Bercovici H., Kerchy L. Harmonic Analysis of Operators on Hilbert Space. 2nd ed. Berlin, Springer, 2010, 474 p. doi: 10.1007/978-1-4419-6094-8
2. Boitsev A.A., Brasche J., Neidhardt H., Popov I.Y. A model of electron transport through a boson cavity. Nanosystems: Physics, Chemistry, Mathematics, 2018, vol. 9, no. 2, pp. 171–178. doi: 10.17586/2220-8054-2018-9-2-171-178
3. Cevizovic D., Chizhov A.V., Galovic S. Vibron transport in macromolecular chains with squeezed phonons. Nanosystems: Physics, Chemistry, Mathematics, 2018, vol. 9, no. 5, pp. 597–602. doi: 10.17586/2220-8054-2018-9-5-597-602
4. Bagmutov A.S., Popov I.Yu. Current-voltage characteristics for two systems of quantum waveguides with connected quantum resonators. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2016, vol. 16, no. 4, pp. 725–730. (in Russian) doi: 10.17586/2226-1494-2016-16-4-725-730
5. Berkolaiko G., Kuchment P. Introduction to Quantum Graphs. Providence, AMS, 2012, 418 p. doi: 10.1090/surv/186
6. Popov I.Y., Popova S.L. Zero-width slit model and resonances in mesoscopic systems. Europhysics Letters, 1993, vol. 24, no. 5, pp. 373–377. doi: 10.1209/0295-5075/24/5/009
7. Borisov D., Exner P., Golovina A. Tunneling resonances in systems without a classical trapping. Journal of Mathematical Physics, 2013, vol. 54, no. 1. doi: 10.1063/1.4773098
8. Botman S.A., Leble S.B. Electrical conductivity model for qua-si-one-dimensional structures. Nanosystems: Physics, Chemistry, Mathematics, 2017, vol. 8, no. 2, pp. 231–235. doi: 10.17586/2220-8054-2017-8-2-231-235
9. Gerasimov D.A., Popov I.Y. Completeness of resonance states for quantum graph with two semi-infinite edges. Complex Variables and Elliptic Equations, 2018, vol. 63, no. 7–8, pp. 996–1010. doi: 10.1080/17476933.2017.1289517
10. Popov I.Y., Popov A.I. Line with attached segment as a model of Helmholtz resonator: Resonant states completeness. Journal of King Saud University Science, 2017, vol. 29, no. 1, pp. 133–136. doi: 10.1016/j.jksus.2016.07.004
11. Popov A.I., Popov I.Y., Gerasimov D.A. Resonance state completeness problem for quantum graph. AIP Conference Proceedings, 2017, vol. 1863, art. 390002. doi: 10.1063/1.4992567
12. Blinova I.V., Popov I.Y. Quantum graph with the Dirac operator and resonance states completeness. Operator Theory: Advances and Applications, 2018, vol. 268, pp. 111–124. doi: 10.1007/978-3-319-75996-8_5
13. Popov I.Y., Popov A.I. Quantum dot with attached wires: resonant states completeness. Reports on Mathematical Physics, 2017, vol. 80, no. 1, pp. 1–10. doi: 10.1016/s0034-4877(17)30054-x
14. Nikol'skii N. Treatise on the Shift Operator: Spectral Function Theory. Berlin, Springer, 1986, 491 p. doi: 10.1007/978-3-642-70151-1
15. Sz.-Nagy B., Foias C., Bercovici H., Kerchy L. Harmonic Analysis of Operators on Hilbert Space. 2nd ed. Berlin, Springer, 2010, 474 p. doi: 10.1007/978-1-4419-6094-8
