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Annales Henri Poincaré

, Volume 16, Issue 5, pp 1155–1189 | Cite as

Zero Modes of Quantum Graph Laplacians and an Index Theorem

  • Jens Bolte
  • Sebastian EggerEmail author
  • Frank Steiner
Article

Abstract

We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper, the number of zero modes is expressed in terms of the trace of a unitary matrix \({\mathfrak{S}}\) that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part, a Dirac operator is defined whose square is related to the Laplacian. To accommodate Laplacians with negative eigenvalues, it is necessary to define the Dirac operator on a suitable Kreĭn space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kreĭn-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator.

Keywords

Dirac Operator Zero Mode Trace Formula Hermitian Form Index Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Thaller B.: The Dirac Equation. Springer, Berlin (1992)CrossRefGoogle Scholar
  2. 2.
    Kottos T., Smilansky U.: Quantum chaos on graphs. Phys. Rev. Lett. 79, 4794–4797 (1997)CrossRefADSGoogle Scholar
  3. 3.
    Kottos T., Smilansky U.: Periodic orbit theory and spectral statistics for quantum graphs. Ann. Phys. 274, 76–124 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  4. 4.
    Exner, P., Keating, J.P., Kuchment, P., Sunada, T., Teplyaev, A. (eds.): Analysis on graphs and its applications. In: Volume 77 of Proceedings of Symposia in Pure Mathematics. American Mathematical Society, Providence, RI, 2008. Papers from the program held in Cambridge, January 8–June 29 (2007)Google Scholar
  5. 5.
    Berkolaiko, G., Kuchment, P.: Introduction to quantum graphs. American Mathematical Society, Providence, RI (2013)Google Scholar
  6. 6.
    Alonso V., De Vincenzo S.: General boundary conditions for a Dirac particle in a box and their non-relativistic limits. J. Phys. A 30, 8573–8585 (1997)CrossRefADSzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bulla W., Trenkler T.: The free Dirac operator on compact and noncompact graphs. J. Math. Phys. 31, 1157–1163 (1990)CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bolte J., Harrison J.: Spectral statistics for the Dirac operator on graphs. J. Phys. A 36, 2747–2769 (2003)CrossRefADSzbMATHMathSciNetGoogle Scholar
  9. 9.
    Gaveau B., Okada M.: Differential forms and heat diffusion on one-dimensional singular varities. Bull. Sci. Math. 115, 61–79 (1991)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Fulling S.A., Kuchment P., Wilson J.H.: Index theorems for quantum graphs. J. Phys. A 40, 14165–14180 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  11. 11.
    Post O.: First order approach and index theorems for discrete and metric graphs. Ann. Henri Poincaré 10, 823–866 (2009)CrossRefADSzbMATHMathSciNetGoogle Scholar
  12. 12.
    Bolte J., Endres S.: The trace formula for quantum graphs with general self adjoint boundary conditions. Ann. Henri Poincaré 10, 189–223 (2009)CrossRefADSzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kostrykin V., Schrader R.: Kirchhoff’s rule for quantum wires. J. Phys. A 32, 595–630 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kostrykin V., Schrader R.: Laplacians on metric graphs: eigenvalues, resolvents and semigroups. In Quantum graphs and their applications. Am. Math. Soc. 415, 201–225 (2006)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kuchment P.: Quantum graphs. I. Some basic structures. Waves Random Media 14, S107–S128 (2004)CrossRefADSzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kostrykin, V., Schrader, R.: The inverse scattering problem for metric graphs and the traveling salesman problem. arxiv:math-ph/0603010 (2006)
  17. 17.
    Kurasov P., Nowaczyk M.: Geometric properties of quantum graphs and vertex scattering matrices. Opuscula Math. 30, 295–309 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Lancaster P., Tismenetsky M.: The Theory of Matrices. Academic Press, San Diego (1985)zbMATHGoogle Scholar
  19. 19.
    Schrader R.: Finite propagation speed and causal free quantum fields on networks. J. Phys. A 42, 495401–39 (2009)Google Scholar
  20. 20.
    Albeverio S., Kostenko A., Malamud M.: Spectral theory of semibounded Sturm-Liouville operators with local interactions on a discrete set. J. Math. Phys. 51, 102102–24 (2010)Google Scholar
  21. 21.
    Ya Azizov T., Iokhvidov I.S.: Linear Operators in Spaces with an Indefinite Metric. Wiley, New York (1989)Google Scholar
  22. 22.
    Brézis H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973)zbMATHGoogle Scholar
  23. 23.
    Brüning J., Geyler V., Pankrashkin K.: Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, 1–70 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Royal HollowayUniversity of LondonEghamUK
  2. 2.Institut für Theoretische PhysikUniversität UlmUlmGermany
  3. 3.Observatoire de Lyon Centre de Recherche Astrophysique de Lyon, Université de Lyon CNRS UMR 5574: Université Lyon 1 and École Normale Supérieure de LyonSaint-Genis-LavalFrance

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