najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2015-6-3-309-319najo-862Research ArticleMATHEMATICSМАТЕМАТИКАTopological damping of Aharonov-Bohm effect: quantum graphs and vertex conditionsKurasovO.

106 91 Stockholm

kurasov@math.su.seSerioA.

106 91 Stockholm

Department of Mathematics, Stockholm UniversitySweden20151508202563309–319Copyright © Kurasov O., Serio A., 20252025Kurasov O., Serio A.Kurasov O., Serio A.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/862

The magnetic Schr¨ odinger operator was studied on a figure 8-shaped graph. It is shown that for specially chosen vertex conditions, the spectrum of the magnetic operator is independent of the flux through one of the loops, provided the flux through the other loop is zero. Topological reasons for this effect are explained.

Quantum graphsMagnetic fieldTrace formulaThe work of PK was partially supported by the Swedish Research Council (Grant D0497301) and ZiF-Zentrum f¨ ur interdisziplinre Forschung, Bielefeld (Coopera tion Group Discrete and continuous models in the theory of networks). The authors would like to thank Muhammad Usman for careful reading of the manuscript.ReferencesY.Aharonov, D.Bohm. Significance of electromagnetic potentials in quantum theory. Phys. Rev., 1959, 115, P. 485–491.Y.Aharonov, D.Bohm. Significance of electromagnetic potentials in quantum theory. Phys. Rev., 1959, 115, P. 485–491.G.Berkolaiko, P.Kuchment. Introduction to quantum graphs. Mathematical Surveys and Monographs, Amer. Math. Soc., 2013, 186 Providence, RI.G.Berkolaiko, P.Kuchment. Introduction to quantum graphs. Mathematical Surveys and Monographs, Amer. Math. Soc., 2013, 186 Providence, RI.B.Gutkin and U.Smilansky. Can one hear the shape of a graph? J. Phys. A, 2001, 34, P. 6061–6068.B.Gutkin and U.Smilansky. Can one hear the shape of a graph? J. Phys. A, 2001, 34, P. 6061–6068.V.Kostrykin, R.Schrader. Quantum wires with magnetic fluxes, Dedicated to Rudolf Haag. Comm. Math. Phys., 2003, 237 (1–2), P. 161–179.V.Kostrykin, R.Schrader. Quantum wires with magnetic fluxes, Dedicated to Rudolf Haag. Comm. Math. Phys., 2003, 237 (1–2), P. 161–179.T.Kottos and U.Smilansky. Periodic orbit theory and spectral statistics for quantum graphs. Ann. Physics, 1999, 274 (1), P. 76–124.T.Kottos and U.Smilansky. Periodic orbit theory and spectral statistics for quantum graphs. Ann. Physics, 1999, 274 (1), P. 76–124.P.Kurasov. Graph Laplacians and topology. Arkiv f¨or Matematik, 2008, 46, P. 95–111.P.Kurasov. Graph Laplacians and topology. Arkiv f¨or Matematik, 2008, 46, P. 95–111.P.Kurasov. Schr¨ odinger operators on graphs and geometry. I. Essentially bounded potentials. J. Funct. Anal., 2008, 254 (4), P. 934–953.P.Kurasov. Schr¨ odinger operators on graphs and geometry. I. Essentially bounded potentials. J. Funct. Anal., 2008, 254 (4), P. 934–953.P.Kurasov. Inverse problems for Aharonov-Bohm rings. Math. Proc. Cambridge Philos. Soc., 2010, 148 (2), P. 331–362.P.Kurasov. Inverse problems for Aharonov-Bohm rings. Math. Proc. Cambridge Philos. Soc., 2010, 148 (2), P. 331–362.P.Kurasov. Quantum graphs: spectral theory and inverse problems. To appear in Birkhäuser.P.Kurasov. Quantum graphs: spectral theory and inverse problems. To appear in Birkhäuser.P.Kurasov and M.Nowaczyk. Inverse spectral problem for quantum graphs. J. Phys. A, 2005, 38 (22), P. 4901–4915.P.Kurasov and M.Nowaczyk. Inverse spectral problem for quantum graphs. J. Phys. A, 2005, 38 (22), P. 4901–4915.P.Kurasov and M.Nowaczyk. Geometric properties of quantum graphs and vertex scattering matrices. Opuscula Math., 2010, 30 (3), P. 295–309.P.Kurasov and M.Nowaczyk. Geometric properties of quantum graphs and vertex scattering matrices. Opuscula Math., 2010, 30 (3), P. 295–309.O. Post. Spectral analysis on graph-like spaces. Lecture Notes in Mathematics, 2012, 2039, 431 p.O. Post. Spectral analysis on graph-like spaces. Lecture Notes in Mathematics, 2012, 2039, 431 p.J.-P.Roth. Le spectre du laplacien sur un graphe. Theorie du potentiel, Lecture Notes in Math., 1984, 1096, P. 521–539 [in French].J.-P.Roth. Le spectre du laplacien sur un graphe. Theorie du potentiel, Lecture Notes in Math., 1984, 1096, P. 521–539 [in French].

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