Spectral gaps for star-like quantum graph and for two coupled rings
https://doi.org/10.17586/2220-8054-2022-13-3-245-249
Аннотация
The spectral problems for two types of quantum graphs are considered. We deal with star-like graph and a graph consisting of two rings connected through a segment. The spectral gap, i.e. the difference between the second and the rst eigenvalues of the free Schro¨ dinger operator, is studied. The dependence of the gap on the geometric parameters of the graph is investigated. Particularly, it is shown that the maximal gap is observed for the symmetric quantum graph.
Об авторах
I. Blinova
ITMO University
Россия
Россия
A. Popov
ITMO University
Россия
Россия
A. Bosova
ITMO University
Россия
Россия
Список литературы
1. Berkolaiko G. and Kuchment P. Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths. Spectral geometry. Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, 2012, 84, P. 117-137.
2. Lipovsky J. Quantum Graphs And Their Resonance Properties. Acta Physica Slovaca, 2016, 66(4), P. 265-363.
3. Bru¨ning J., Geiler V. and Lobanov I. Spectral properties of Schrodinger operators on decorated graphs. Mat. notes, 2005, 77(6), P. 932-935.
4. Sobirov Z.A., Rakhimov K.U., Ergashov R.E. Green’s function method for time-fractional diffusion equation on the star graph with equal bonds. Nanosystems: Phys. Chem. Math., 2021, 12(3), P. 271-278.
5. Post O. Spectral Analysis on Graph-Like Spaces. Springer, 2010.
6. Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H. Solvable models in quantum mechanics. AMS Chelsea Publishing, Providence, RI, 2005.
7. Pavlov B.S. The theory of extensions, and explicitly solvable models. Uspekhi Mat. Nauk, 1987, 42(6), P. 99-131.
8. Popov I.Y. The resonator with narrow slit and the model based on the operator extensions theory. Journal of Mathematical Physics, 1992, 3311), P. 3794-3801.
9. Popov I.Y. Model of point-like window for electromagnetic Helmholtz resonator. Zeitschrift fur Analysis und ihre Anwendungen, 2013, 32(2), P. 155-162.
10. Geyler V.A., Popov I.Yu. Ballistic transport in nanostructures: explicitly solvable model. Theor. Math. Phys., 1996, 107(1), P. 427-434.
11. Behrndt J., Frank R.L., Ku¨hn Ch., Lotoreichik V., Rohleder J. Spectral Theory for Schro¨dinger Operators with delta-Interactions Supported on Curves in R3. Ann. Henri Poincare, 2017, 18, P. 1305-1347.
12. Exner P., Keating P., Kuchment P., Sunada T. and Teplyaev A. Analysis on graph and its applications, AMS, Providence, 2008.
13. Berkolaiko G. and Kuchment P.Introduction to Quantum Graphs, AMS, Providence, 2012.
14. Exner P., Kostenko A., Malamud M., Neidhardt H. Spectral theory of in nite quantum graphs. Ann. H. Poincare, 2018, 19, P. 3457-3510.
15. Boitsev A.A., Neidhardt H., Malamud M., Brasche J., Popov I.Y. Boundary Triplets, Tensor Products and Point Contacts to Reservoirs. Annales Henri Poincare, 2018, 19(9), P. 2783-2837.
16. Popov I.Y., Popov A.I. Quantum Dot with Attached Wires: Resonant States Completeness. Reports on Mathematical Physics, 2017, 80(1), P. 1-10.
17. Kurasov P., Malenova´ G. and Naboko S. Spectral gap for quantum graphs and their edge connectivity. Journal of Physics A: Mathematical and Theoretical, 2013, 46(27), article id. 275309, 16 pp.
18. Kurasov P. and Naboko S. Rayleigh estimates for differential operators on graphs. J. Spectr. Theory, 2014, 4(2), P. 211-219.
19. Kennedy J., Kurasov P. and Malenova´ G.; et al. On the Spectral Gap of a Quantum Graph. Ann. Henri Poincare´, 2016, 17, P. 2439.
20. Exner P., Khrabustovskyi A. Gap control by singular Schrodinger operators in a periodically structured metamaterial. Journal of Mathematical Physics, Analysis, Geometry, 2018, 14(3), P. 270-285. ′
21. Barseghyan D., Khrabustovskyi A. Gaps in the spectrum of a periodic quantum graph with periodically distributed δ-type interactions. J. Phys. A, 2015, 48, P. 255201.
Рецензия
Для цитирования:
, , . Наносистемы: физика, химия, математика. 2022;13(3):245-249. https://doi.org/10.17586/2220-8054-2022-13-3-245-249
For citation:
Blinova I.V., Popov A.I., Bosova A.A. Spectral gaps for star-like quantum graph and for two coupled rings. Nanosystems: Physics, Chemistry, Mathematics. 2022;13(3):245-249. https://doi.org/10.17586/2220-8054-2022-13-3-245-249
Просмотров:
2
Послать статью по эл. почте 