najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2017-8-4-420-425najo-590Research ArticleMATHEMATICSМАТЕМАТИКАTime dependent quantum graph with loopTime dependent quantum graph with loopEreminD. A.EreminD. A.

Department of Mathematics and IT

Bolshevistskaya Str. 68; Saransk

Department of Mathematics and IT

Bolshevistskaya Str. 68; Saransk

GrishanovE. N.GrishanovE. N.

Department of Mathematics and IT

Bolshevistskaya Str. 68; Saransk

Department of Mathematics and IT

Bolshevistskaya Str. 68; Saransk

KostrovO. G.KostrovO. G.

Department of Mathematics and IT

Bolshevistskaya Str. 68; Saransk

Department of Mathematics and IT

Bolshevistskaya Str. 68; Saransk

NikiforovD. S.NikiforovD. S.

Department of Higher Mathematics

197101; Kroverkskiy pr. 49; St. Petersburg

Department of Higher Mathematics

197101; Kroverkskiy pr. 49; St. Petersburg

PopovI. Y.PopovI. Y.

Department of Higher Mathematics

197101; Kroverkskiy pr. 49; St. Petersburg

Department of Higher Mathematics

197101; Kroverkskiy pr. 49; St. Petersburg

popov1955@gmail.comOgarev Mordovia State UniversityOgarev Mordovia State UniversityITMO UniversityITMO University20171308202584420425Copyright © Eremin D.A., Grishanov E.N., Kostrov O.G., Nikiforov D.S., Popov I.Y., 20252025Eremin D.A., Grishanov E.N., Kostrov O.G., Nikiforov D.S., Popov I.Y.Eremin D.A., Grishanov E.N., Kostrov O.G., Nikiforov D.S., Popov I.Y.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/590

   A quantum graph, consisting of a ring and segment is considered. We deal with the free Schr¨odinger ooperator at the edges and Kirchhoff conditions at the internal vertex. The lengths of the graph edges varies in time. Time evolution of wave packet is studied for different parameters of length varying law.

   A quantum graph, consisting of a ring and segment is considered. We deal with the free Schr¨odinger ooperator at the edges and Kirchhoff conditions at the internal vertex. The lengths of the graph edges varies in time. Time evolution of wave packet is studied for different parameters of length varying law.

quantum graphspectrumtime evolutionquantum graphspectrumtime evolutionThis work was partially financially supported by the Government of the Russian Federation (grant 074-U01), by grant MK-5161.2016.1 of the President of the Russian Federation, DFG Grant NE 1439/3-1, by grant 16-11-10330 of Russian Science FoundationThis work was partially financially supported by the Government of the Russian Federation (grant 074-U01), by grant MK-5161.2016.1 of the President of the Russian Federation, DFG Grant NE 1439/3-1, by grant 16-11-10330 of Russian Science FoundationReferencesPauling L. The Diamagnetic Anisotropy of Aromatic Molecules. J. Chem. Phys., 1936, 4, P. 673.Pauling L. The Diamagnetic Anisotropy of Aromatic Molecules. J. Chem. Phys., 1936, 4, P. 673.Gerasimenko N.I., Pavlov B.S. Scattering problems on noncompact graphs. Theoret. Math. Phys., 1988, 74, P. 230–240.Gerasimenko N.I., Pavlov B.S. Scattering problems on noncompact graphs. Theoret. Math. Phys., 1988, 74, P. 230–240.Albeverio S., Gesztesy F., Hoegh-Krohn R., Holden H. with an appendix by P. Exner. Solvable Models in Quantum Mechanics: Second Edition, 2005. (Providence, R.I.: AMS Chelsea Publishing).Albeverio S., Gesztesy F., Hoegh-Krohn R., Holden H. with an appendix by P. Exner. Solvable Models in Quantum Mechanics: Second Edition, 2005. (Providence, R.I.: AMS Chelsea Publishing).Pavlov B.S. The theory of extensions and explicity-solvable models. Russ. Math. Surv., 1987, 42(6), P. 127–168.Pavlov B.S. The theory of extensions and explicity-solvable models. Russ. Math. Surv., 1987, 42(6), P. 127–168.Popov I.Y., Kurasov P.A., Naboko S.N., Kiselev A.A., Ryzhkov A.E., Yafyasov A.M., Miroshnichenko G.P., Karpeshina Yu.E., Kruglov V.I., Pankratova T.F., Popov A.I. A distinguished mathematical physicist Boris S. Pavlov. Nanosystems: Physics, Chemistry, Mathematics, 2016, 7, P. 782–788.Popov I.Y., Kurasov P.A., Naboko S.N., Kiselev A.A., Ryzhkov A.E., Yafyasov A.M., Miroshnichenko G.P., Karpeshina Yu.E., Kruglov V.I., Pankratova T.F., Popov A.I. A distinguished mathematical physicist Boris S. Pavlov. Nanosystems: Physics, Chemistry, Mathematics, 2016, 7, P. 782–788.Exner P., Keating P., Kuchment P., Sumada T., Teplyaev A. Analysis on graph and its applications. Proc. Symp. Pure Math. Providence, RI, 2008, 77.Exner P., Keating P., Kuchment P., Sumada T., Teplyaev A. Analysis on graph and its applications. Proc. Symp. Pure Math. Providence, RI, 2008, 77.Duclos P., Exner P., Turek O. On the spectrum of a bent chain graph. J. Phys. A: Math. Theor., 2008, 41, P. 415206/1-18.Duclos P., Exner P., Turek O. On the spectrum of a bent chain graph. J. Phys. A: Math. Theor., 2008, 41, P. 415206/1-18.Popov I.Y., Skorynina A.N., Blinova I.V. On the existence of point spectrum for branching strips quantum graph. J. Math. Phys., 2014, 55, P. 033504/1-20.Popov I.Y., Skorynina A.N., Blinova I.V. On the existence of point spectrum for branching strips quantum graph. J. Math. Phys., 2014, 55, P. 033504/1-20.Popov I.Y., Smirnov P.I. Spectral problem for branching hain quantum graph. Phys. Lett., 2013, A 377, P. 439–442.Popov I.Y., Smirnov P.I. Spectral problem for branching hain quantum graph. Phys. Lett., 2013, A 377, P. 439–442.Jose J.V., Gordery R. Study of a quantum Fermi-acceleration model. Phys. Rev. Lett., 1986, 56, P. 290.Jose J.V., Gordery R. Study of a quantum Fermi-acceleration model. Phys. Rev. Lett., 1986, 56, P. 290.Makowski A.J., Dembinski S.T. Exactly solvable models with ime-dependent boundary conditions. Phys. Lett., 1991, A, 154(5-6), P. 217–220.Makowski A.J., Dembinski S.T. Exactly solvable models with ime-dependent boundary conditions. Phys. Lett., 1991, A, 154(5-6), P. 217–220.Cacciapuoti C., Mantile A., Posilicano A. Time dependent delta-prime interactions in dimension one. Nanosystems: Phys. Chem. Math., 2016, 7(2), P. 303–314.Cacciapuoti C., Mantile A., Posilicano A. Time dependent delta-prime interactions in dimension one. Nanosystems: Phys. Chem. Math., 2016, 7(2), P. 303–314.Matrasulov D.U., Yusupov J.R., Sabirov K.K., Sobirov Z.A. Time-dependent quantum graph. Nanosystems: Phys. Chem. Math., 2015, 6(2), P. 173–181.Matrasulov D.U., Yusupov J.R., Sabirov K.K., Sobirov Z.A. Time-dependent quantum graph. Nanosystems: Phys. Chem. Math., 2015, 6(2), P. 173–181.Karpova O., Sabirov K., Otajanov D., Ruzmetov A., Saidov A.A. Absorbing boundary conditions for Schr¨odinger equation in a time-dependent interval. Nanosystems: Phys. Chem. Math., 2017, 8(1), P. 13–19.Karpova O., Sabirov K., Otajanov D., Ruzmetov A., Saidov A.A. Absorbing boundary conditions for Schr¨odinger equation in a time-dependent interval. Nanosystems: Phys. Chem. Math., 2017, 8(1), P. 13–19.

The authors declare that there are no conflicts of interest present.