najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2019-10-2-131-140najo-627Research ArticlePHYSICSФИЗИКАA model of an electron in a quantum graph interacting with a two-level systemBoitsevA. A.

Kronverkskiy, 49, St. Petersburg, 197101 

boitsevanton@gmail.comPopovI. Y.

Kronverkskiy, 49, St. Petersburg, 197101 

popov1955@gmail.comITMO UniversityRussian Federation201912082025102131140Copyright © Boitsev A.A., Popov I.Y., 20252025Boitsev A.A., Popov I.Y.Boitsev A.A., Popov I.Y.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/627

A model of an electron in a quantum graph interacting with a two-level system is considered. The operator describing the model has the form of sum of tensor products. Self-adjoint extensions and a scattering matrix are written in terms of a boundary triplet, corresponding to the considered symmetric operator. Diagrams of reflection are calculated and numerical results are discussed .

Aharonov-Bohm ringnanostructurequantum graphscatteringsolvable modelThis work was partially financially supported by the Government of the Russian Federation (grant 08-08), by grant 16-11-10330 of Russian Science Foundation.ReferencesDauber J., Oellers M., Venn F., Epping A., Watanabe K., Taniguchi T., Hassler F., Stampfer C. Aharonov-Bohm oscillations and magnetic focusing in ballistic graphene rings. Phys. Rev. B, 2017, 96, P. 205407.Dauber J., Oellers M., Venn F., Epping A., Watanabe K., Taniguchi T., Hassler F., Stampfer C. Aharonov-Bohm oscillations and magnetic focusing in ballistic graphene rings. Phys. Rev. B, 2017, 96, P. 205407.Chakraborty T., Manaselyan A., Barseghyan M. Irregular Aharonov-Bohm effect for interacting electrons in a ZnO quantum ring. J Phys Condens Matter, 2017, 29(7), P. 075605.Chakraborty T., Manaselyan A., Barseghyan M. Irregular Aharonov-Bohm effect for interacting electrons in a ZnO quantum ring. J Phys Condens Matter, 2017, 29(7), P. 075605.Caudrelier V., Mintchev M., Ragoucy E. Exact scattering matrix of graphs in magnetic field and quantum noise. J. Math. Phys., 2014, 55, P. 083524.Caudrelier V., Mintchev M., Ragoucy E. Exact scattering matrix of graphs in magnetic field and quantum noise. J. Math. Phys., 2014, 55, P. 083524.Exner P., Lotoreichik V., Perez-Obiol A. On the bound states of magnetic Laplacians on wedges. Rep. Math. Phys, 2018, 82, P. 161–185.Exner P., Lotoreichik V., Perez-Obiol A. On the bound states of magnetic Laplacians on wedges. Rep. Math. Phys, 2018, 82, P. 161–185.Kurasov P., Serio A. Topological damping of Aharonov-Bohm effect: quantum graphs and vertex conditions. Nanosystems: Phys. Chem. Math., 2015, 6(3), P. 309–319.Kurasov P., Serio A. Topological damping of Aharonov-Bohm effect: quantum graphs and vertex conditions. Nanosystems: Phys. Chem. Math., 2015, 6(3), P. 309–319.Eremin D.A., Grishanov E.N., Nikiforov D.S., Popov I.Y. Wave dynamics on time-depending graph with Aharonov-Bohm ring. Nanosystems: Phys. Chem. Math., 2018, 9(4), P. 457–463.Eremin D.A., Grishanov E.N., Nikiforov D.S., Popov I.Y. Wave dynamics on time-depending graph with Aharonov-Bohm ring. Nanosystems: Phys. Chem. Math., 2018, 9(4), P. 457–463.Chatterjee A., Smolkina M.O., Popov I.Y. Persistent current in a chain of two Holstein-Hubbard rings in the presence of Rashba spin-orbit interaction. Nanosystems: Phys. Chem. Math., 2019, 10(1), P. 50–62.Chatterjee A., Smolkina M.O., Popov I.Y. Persistent current in a chain of two Holstein-Hubbard rings in the presence of Rashba spin-orbit interaction. Nanosystems: Phys. Chem. Math., 2019, 10(1), P. 50–62.G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs. AMS, Providence, 2012.G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs. AMS, Providence, 2012.Boitsev A.A., Brasche J., Malamud M., Neidhardt H., Popov I.Y. Boundary Triplets, Tensor Products and Point Contacts to Reservoirs Annales Henri Poincare, IET, 2018, 19(9), P. 2783–2837.Boitsev A.A., Brasche J., Malamud M., Neidhardt H., Popov I.Y. Boundary Triplets, Tensor Products and Point Contacts to Reservoirs Annales Henri Poincare, IET, 2018, 19(9), P. 2783–2837.Derkach V. A., Malamud M. M., 1991, Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal., 95(1), P. 1–95.Derkach V. A., Malamud M. M., 1991, Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal., 95(1), P. 1–95.Malamud M. M. Some classes of extensions of a Hermitian operator with lacunae, Ukra¨ın. Mat. Zh., 1992, 44(2), P. 215–233.Malamud M. M. Some classes of extensions of a Hermitian operator with lacunae, Ukra¨ın. Mat. Zh., 1992, 44(2), P. 215–233.Malamud M. M., 1992, Some classes of extensions of a Hermitian operator with lacunae, Ukra¨ın. Mat. Zh., 44(2), P. 215–233.Malamud M. M., 1992, Some classes of extensions of a Hermitian operator with lacunae, Ukra¨ın. Mat. Zh., 44(2), P. 215–233.Boitsev A. A., Brasche J., Neidhardt H., Popov I. Y., A model of electron transport through a boson cavity. Nanosystems: Phys. Chem. Math., 2018, 9, P. 171–178.Boitsev A. A., Brasche J., Neidhardt H., Popov I. Y., A model of electron transport through a boson cavity. Nanosystems: Phys. Chem. Math., 2018, 9, P. 171–178.Behrndt J., Malamud M.M., Neidhardt H. Scattering matrices and Weyl functions. Proc. Lond. Math. Soc., 2008, 97, P. 568–598.Behrndt J., Malamud M.M., Neidhardt H. Scattering matrices and Weyl functions. Proc. Lond. Math. Soc., 2008, 97, P. 568–598.Kato T. Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York 1966.Kato T. Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York 1966.Yafaev D.R. Mathematical Scattering Theory: General Theory, Translations of Mathematical Monographs, 105. American Mathematical Society, Providence, RI, 1992.Yafaev D.R. Mathematical Scattering Theory: General Theory, Translations of Mathematical Monographs, 105. American Mathematical Society, Providence, RI, 1992.Baumgartel H., Wollenberg M. Mathematical Scattering Theory, Akademie-Verlag, Berlin, 1983.Baumgartel H., Wollenberg M. Mathematical Scattering Theory, Akademie-Verlag, Berlin, 1983.Donoghue W.F. Monotone Matrix Functions and Analytic Continuation, Springer Verlag, Berlin and New York, 1974.Donoghue W.F. Monotone Matrix Functions and Analytic Continuation, Springer Verlag, Berlin and New York, 1974.

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