najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2021-12-4-425-429najo-457Research ArticleMATHEMATICSМАТЕМАТИКАDirac operator with different potentials on edges of quantum graph: resonance asymptoticsОператор Дирака с различными потенциалами на ребрах квантового графа: асимптотика резонансаBelolipetskaiaA. G.BelolipetskaiaA. G.

Kronverkskiy, 49, Saint Petersburg, 197101.

Kronverkskiy, 49, Saint Petersburg, 197101.

annabel1502@mail.ruPopovI. Y.PopovI. Y.

Kronverkskiy, 49, Saint Petersburg, 197101.

Kronverkskiy, 49, Saint Petersburg, 197101.

popov1955@gmail.comITMO UniversityРоссияITMO UniversityRussian Federation202104082025124425429Copyright © Belolipetskaia A.G., Popov I.Y., 20252025Belolipetskaia A.G., Popov I.Y.Belolipetskaia A.G., Popov I.Y.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/457

Asymptotics of resonances for the Dirac operator with different potentials on edges of a quantum graph with the Kirchhoff coupling conditions at vertices is studied. The results are obtained for a quantum graph that consists of a compact interior and a finite number of exterior edges of infinite length connected to the interior.

Исследуется асимптотика резонансов для оператора Дирака с различными потенциалами на ребрах квантового графа с условиями связи Кирхгофа в вершинах. Результаты получены для квантового графа, состоящего из компактной внутренности и конечного числа внешних ребер бесконечной длины, соединенных с внутренностью.

квантовый графоператор Диракаасимптотикарезонансquantum graphDirac operatorasymptoticsresonanceReferencesBerkolaiko G., Kuchment P. Introduction to quantum graphs. American Mathematical Soc., 2013.Berkolaiko G., Kuchment P. Introduction to quantum graphs. American Mathematical Soc., 2013.Exner P. et al. (ed.). Analysis on Graphs and Its Applications: Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007. American Mathematical Soc., 2008.Exner P. et al. (ed.). Analysis on Graphs and Its Applications: Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007. American Mathematical Soc., 2008.Davies E.B., Pushnitski A. Non-Weyl resonance asymptotics for quantum graphs. Analysis and PDE, 2011, 4, P. 729–756.Davies E.B., Pushnitski A. Non-Weyl resonance asymptotics for quantum graphs. Analysis and PDE, 2011, 4, P. 729–756.Exner P., Lipovsky J. Resonances from perturbations of quantum graphs with rationally related edges.´ Journal of Physics A: Mathematical and Theoretical, 2010, 43, P. 105301.Exner P., Lipovsky J. Resonances from perturbations of quantum graphs with rationally related edges.´ Journal of Physics A: Mathematical and Theoretical, 2010, 43, P. 105301.Davies E.B., Exner P., Lipovsky J. Non-Weyl asymptotics for quantum graphs with general coupling conditions. J. Phys. A: Math. Theor., 2010, 43, P. 474013.Davies E.B., Exner P., Lipovsky J. Non-Weyl asymptotics for quantum graphs with general coupling conditions. J. Phys. A: Math. Theor., 2010, 43, P. 474013.Exner P., Lipovsky J. Non-Weyl resonance asymptotics for quantum graphs in a magnetic field. Phys. Lett. A, 2011, 375, P. 805–807.Exner P., Lipovsky J. Non-Weyl resonance asymptotics for quantum graphs in a magnetic field. Phys. Lett. A, 2011, 375, P. 805–807.Aslanyan A., Parnovski L., Vassiliev D. Complex resonances in acoustic waveguides. Q. J. Mech. Appl. Math., 2000, 53(3), P. 429–447.Aslanyan A., Parnovski L., Vassiliev D. Complex resonances in acoustic waveguides. Q. J. Mech. Appl. Math., 2000, 53(3), P. 429–447.Blinova I.V., Popov I.Y. Quantum graph with the Dirac operator and resonance states completeness. Operator Theory: Adv. Appl., 2018, 268, P. 111–124.Blinova I.V., Popov I.Y. Quantum graph with the Dirac operator and resonance states completeness. Operator Theory: Adv. Appl., 2018, 268, P. 111–124.Christiansen T. Some upper bounds on the number of resonances for manifolds with infinite cylindrical ends. Ann. Henri Poincare, 2002, 3, P. 895–920.Christiansen T. Some upper bounds on the number of resonances for manifolds with infinite cylindrical ends. Ann. Henri Poincare, 2002, 3, P. 895–920.Edward J. On the resonances of the Laplacian on waveguides. J. Math. Anal. Appl., 2002, 272, P. 89–116.Edward J. On the resonances of the Laplacian on waveguides. J. Math. Anal. Appl., 2002, 272, P. 89–116.Gerasimov D.A., Popov I.Y. Completeness of resonance states for quantum graph with two semi-infinite edges. Compl. Var. Elliptic Eq., 2018, 63, P. 996–1010.Gerasimov D.A., Popov I.Y. Completeness of resonance states for quantum graph with two semi-infinite edges. Compl. Var. Elliptic Eq., 2018, 63, P. 996–1010.Popov I.Y., Blinova I.V. Quantum graph in a magnetic field and resonance states completeness. Operator Theory: Adv. Appl., 2020, 276, P. 540–553.Popov I.Y., Blinova I.V. Quantum graph in a magnetic field and resonance states completeness. Operator Theory: Adv. Appl., 2020, 276, P. 540–553.Sjostrand J.,Zworski M. Complex scaling and the distribution of scattering poles.¨ J.Amer. Math. Soc., 1991, 4, P. 729–769.Sjostrand J.,Zworski M. Complex scaling and the distribution of scattering poles.¨ J.Amer. Math. Soc., 1991, 4, P. 729–769.Blinova I.V., Popov I.Y., Popov A.I. Resonance states completeness for relativistic particle on a sphere with two semi-infinite lines attached. Journal of King Saud University - Science, 2020, 32(1), P. 836–841.Blinova I.V., Popov I.Y., Popov A.I. Resonance states completeness for relativistic particle on a sphere with two semi-infinite lines attached. Journal of King Saud University - Science, 2020, 32(1), P. 836–841.Popov I.Y., Popov A.I. Quantum Dot with Attached Wires: Resonant States Completeness. Reports on Mathematical Physics, 2017, 80(1), P. 1–10.Popov I.Y., Popov A.I. Quantum Dot with Attached Wires: Resonant States Completeness. Reports on Mathematical Physics, 2017, 80(1), P. 1–10.Popov I., Gerasimov D., Blinova I., Popov A. Incompleteness of resonance states for quantum ring with two semi-infinite edges. Analysis and Mathematical Physics, 2019, 9(3), P. 1287–1302.Popov I., Gerasimov D., Blinova I., Popov A. Incompleteness of resonance states for quantum ring with two semi-infinite edges. Analysis and Mathematical Physics, 2019, 9(3), P. 1287–1302.Duclos P., Exner P., Meller B. Open quantum dots: Resonances from perturbed symmetry and bound states in strong magnetic fields. Rep. Math. Phys., 2001, 47(2), P. 253–267.Duclos P., Exner P., Meller B. Open quantum dots: Resonances from perturbed symmetry and bound states in strong magnetic fields. Rep. Math. Phys., 2001, 47(2), P. 253–267.

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