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Cantor and Band Spectra for Periodic Quantum Graphs with Magnetic Fields

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Abstract

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.

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References

  1. Abilio C.C., Butaud P., Fournier Th., Pannetier B., Vidal J., Tedesco S., Dalzotto B. (1999) Magnetic field induced localization in a two-dimensional superconducting wire network. Phys. Rev. Lett. 83, 5102–5105

    Article  ADS  Google Scholar 

  2. Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H. Solvable models in quantum mechanics. 2nd ed. Providence, R: AMS Chelsea Publ., 2005

  3. Alexander S. (1983) Superconductivity of networks. A percolation approach to the effects of disorder. Phys. Rev. B 27, 1541–1557

    Article  MathSciNet  ADS  Google Scholar 

  4. Avila A., Jitomirskaya S. The ten martini problem. Ann. Math. (to appear) http://arXiv. org:math.DS/0503363, 2005

  5. Avila A., Krikorian R. Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. To appear in Ann. of Math., http:// arXiv.org:math.DS/0306382, 2003

  6. Avron J.E., Seiler R., Simon B. (1983) Homotopy and quantization in condensed matter physics. Phys. Rev. Lett. 51, 51–54

    Article  ADS  Google Scholar 

  7. Birman M.Sh., Suslina T.A. (2000) A periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity. St. Petersburg Math. J. 11, 203–232

    MathSciNet  Google Scholar 

  8. Boca F.P., Zaharescu A. (2005) Norm estimates of almost Mathieu operators. J. Funct. Anal. 220, 76–96

    Article  MATH  MathSciNet  Google Scholar 

  9. Boon M.H. (1972) Representations of the invariance group for a Bloch electron in a magnetic field. J. Math. Phys. 13, 1268–1285

    Article  MathSciNet  Google Scholar 

  10. Coddington E.A., Levinson N., (1995) Theory of ordinary differential equations. New York etc, McGraw-Hill

    MATH  Google Scholar 

  11. de Gennes P.-G. (1981) Diamagnétisme de grains supraconduteurs près d’un seuil de percolation. C.R. Acad. Sci. Paris, Sér. II 292, 9–12

    Google Scholar 

  12. Derkach V.A., Malamud M.M. (1991) Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95, 1–95

    Article  MATH  MathSciNet  Google Scholar 

  13. Exner P. (1995) Lattice Kronig-Penney models. Phys. Rev. Lett. 74, 3503–3506

    Article  ADS  Google Scholar 

  14. Exner P.(1997) A duality between Schrödinger operators on graphs and certain Jacobi matrices. Ann. Inst. H. Poincaré 66, 359–371

    MATH  MathSciNet  Google Scholar 

  15. Gehtman M.M., Stankevich I.V. (1977) The generalized Kronig-Penney problem. Funct. Anal. Appl. 11, 51–52

    Article  MATH  Google Scholar 

  16. Geyler V.A., Margulis V.A. (1987) Anderson localization in the nondiscrete Maryland model. Theor. Math. Phys. 70, 133–140

    Article  Google Scholar 

  17. Geyler V.A., Senatorov M.M. (1997) The structure of the spectrum of the Schrödinger operator with a magnetic field in a strip and infinite-gap potentials. Sb. Math. 88(5): 657–669

    Article  Google Scholar 

  18. Gorbachuk V.I., Gorbachuk. M.A. (1991) Boundary value problems for operator differential equations. Dordrecht etc, Kluwer Acad. Publ

    MATH  Google Scholar 

  19. Helffer B., Kerdelhué P., Sjöstrand J. (1990) Le papillon de Hofstadter revisité. Mém. Soc. Math. Fr., Nouv. Sér. 43: 1–87

    Google Scholar 

  20. Helffer B., Sjöstrand J. (1989) Semi-classical analysis for Harper’s equation III: Cantor structure of the spectrum. Mém. Soc. Math. Fr., Nouv. Sér. 39, 1–124

    Google Scholar 

  21. Hofstadter D.R. (1976) Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249

    Article  ADS  Google Scholar 

  22. Hryniv R.O., Mykytyuk Ya.V. (2001) 1-D Schrödinger operators with periodic singular potentials. Methods Funct. Anal. Topology 7(4): 31–42

    MATH  MathSciNet  Google Scholar 

  23. Kohmoto M. (1993) Quantum-wire networks and the quantum Hall effect. J. Phys. Soc. Japan 62, 4001–4008

    Article  Google Scholar 

  24. Kostrykin V., Schrader R. (2003) Quantum wires with magnetic fluxes. Commun. Math. Phys. 237, 161–179

    MATH  MathSciNet  ADS  Google Scholar 

  25. Kreft Ch., Seiler R. (1996) Models of the Hofstadter type. J. Math. Phys. 37, 5207–5243

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Kuchment P., Quantum graphs I. (2004) Some basic structures. Waves Random Media 14, S107–S128

    Article  MATH  Google Scholar 

  27. Kuchment P. (2005) Quantum graphs II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A: Math. Gen. 38, 4887–4900

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Langbein D. (1969) The tight-binding and the nearly-free-electron approach to lattice electron in external magnetic field. Phys. Rev. (2)180: 633–648

    Article  ADS  Google Scholar 

  29. Levitan B.M., Sargsyan I.S. (1990) Sturm-Liouville and Dirac operators. Dordrecht etc, Kluwer

    MATH  Google Scholar 

  30. Mikhailets V.A., Sobolev A.V. (1999) Common eigenvalue problem and periodic Schrödinger operators. J. Funct. Anal. 165, 150–172

    Article  MATH  MathSciNet  Google Scholar 

  31. Naud C., Faini G., Mailly D. (2001) Aharonov–Bohm cages in 2D normal metal networks. Phys. Rev. Lett. 86, 5104–5107

    Article  ADS  Google Scholar 

  32. Naud C., Faini G., Mailly D., Vidal J., Douçot B., Montambaux G., Wieck A., Reuter D. (2002) Aharonov– Bohm cages in the GaAlAs/GaAs system. Physica E 12, 190–196

    Article  ADS  Google Scholar 

  33. Novikov S.P. (1985) Two-dimensional Schrödinger operators in periodic fields. J. Soviet Math. 28(1), 1–20

    MATH  Google Scholar 

  34. Pavlov B.S. (1987) The theory of extensions and explicitly solvable models. Russ. Math. Surv. 42, 127–168

    Article  MATH  Google Scholar 

  35. Prange R.E., Girvin S.M.(eds). (1990) The Quantum Hall Effect. New York, Springer-Verlag

    Google Scholar 

  36. Puig J. (2004) Cantor spectrum for the almost Mathieu operator. Commun. Math. Phys. 244, 297–309

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. Schenker J.H., Aizenman M. (2000) The creation of spectral gaps by graph decoration. Lett. Math. Phys. 53, 253–262

    Article  MATH  MathSciNet  Google Scholar 

  38. Shubin M.A. (1994) Discrete magnetic Laplacian. Commun. Math. Phys. 164, 259–275

    Article  MATH  MathSciNet  ADS  Google Scholar 

  39. Sobolev A.V. (1999) Absolute continity of the periodic magnetic Schrödinger operator. Invent. Math. 137, 85–112

    Article  MATH  MathSciNet  Google Scholar 

  40. Thouless D.J., Kohmoto M., Nightingale M.P., den Nijs M. (1982) Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408

    Article  ADS  Google Scholar 

  41. Zak J. (1964) Magnetic translation group. Phys. Rev. 134, A1602–A1606

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany

    Jochen Brüning & Konstantin Pankrashkin

  2. Mathematical Faculty, Mordovian State University, Saransk, 430000, Russia

    Vladimir Geyler

Authors
  1. Jochen Brüning
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  2. Vladimir Geyler
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  3. Konstantin Pankrashkin
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Correspondence to Jochen Brüning.

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Communicated by B. Simon

The work was supported by the Deutsche Forschungsgemeinschaft, the Sonderforschungsbereich “Raum, Zeit, Materie” (SFB 647), and the International Bureau of BMBF at the German Aerospace Center (IB DLR, cooperation Germany–New Zealand NZL 05/001)

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Brüning, J., Geyler, V. & Pankrashkin, K. Cantor and Band Spectra for Periodic Quantum Graphs with Magnetic Fields. Commun. Math. Phys. 269, 87–105 (2007). https://doi.org/10.1007/s00220-006-0050-0

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