Skip to main content
SpringerLink
Log in

Continuity of the Integrated Density of States on Random Length Metric Graphs

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge lengths are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Comm. Math. Phys. 157(2), 245–278 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Adachi, T., Sunada, T.: Density of states in spectral geometry. Comment. Math. Helv. 68(3), 480–493 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aizenman, M., Sims, R., Warzel, S.: Absolutely continuous spectra of quantum tree graphs with weak disorder. Comm. Math. Phys. 264(2), 371–389 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  4. Aizenman, M., Sims, R., Warzel, S.: Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs. Probab. Theory Related Fields 136(3), 363–394 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bellissard, J., Lima, R., Testard, D.: Almost Periodic Schrödinger Operators. Mathematics + Physics, vol. 1, pp. 1–64. World Science, Singapore (1985)

    Google Scholar 

  6. Bourgain, J., Kenig, C.E.: On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math. 161, 389–426 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Brüning, J., Geyler, V., Pankrashkin, K.: Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, 1–70 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cattaneo, C.: The spectrum of the continuous Laplacian on a graph. Monatsh. Math. 124(3), 215–235 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chayes, J.T., Chayes, L., Franz, J.R., Sethna, J.P., Trugman, S.A.: On the density of states for the quantum percolation problem. J. Phys. A 19, L1173–L1177 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  10. Combes, J.-M., Hislop, P.D.: Localization for some continuous, random Hamiltonians in d-dimensions. J. Funct. Anal. 124, 149–180 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Combes, J.M., Hislop, P.D., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140(3), 469–498 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Combes, J.M., Hislop, P.D., Nakamura, S.: The L p-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators. Comm. Math. Phys. 218, 113–130 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Probability and its Applications. Birkhäuser Boston, Boston (1990)

    Google Scholar 

  14. Exner, P., Helm, M., Stollmann, P.: Localization on a quantum graph with a random potential on the edges. Rev. Math. Phys. 19(9), 923–939 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Exner, P., Keating, J.P., Kuchment, P., Sunada, T., Teplayaev, A. (eds.): Analysis on graphs and its applications. In: Proc. Symp. Pure Math., vol. 77. American Mathematical Society, Providence (2008)

    Google Scholar 

  16. Froese, R., Hasler, D., Spitzer, W.: Absolutely continuous spectrum for the Anderson model on a tree: a geometric proof of Klein’s theorem. J. Funct. Anal. 230(1), 184–221 (2006)

    MATH  MathSciNet  Google Scholar 

  17. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Comm. Math. Phys. 88, 151–184 (1983)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Gruber, M.J., Helm, M., Veselić, I.: Optimal Wegner estimates for random Schrödinger operators on metric graphs. In: Exner, P., Keating, J.P., Kuchment, P., Sunada, T., Teplayaev, A. (eds.) Proc. Symp. Pure Math., vol. 77, pp. 409–422. American Mathematical Society, Providence (2008)

    Google Scholar 

  19. Gruber, M.J., Lenz, D., Veselić, I.: Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over ℤd. J. Funct. Anal. 253(2), 515–533 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gruber, M.J., Lenz, D., Veselić, I.: Uniform existence of the integrated density of states for combinatorial and metric graphs over ℤd. In: In: Exner, P., Keating, J.P., Kuchment, P., Sunada, T., Teplayaev, A. (eds.) Proc. Symp. Pure Math., vol. 77, pp. 87–108. American Mathematical Society, Providence (2008)

    Google Scholar 

  21. Gruber, M., Veselić, I.: The modulus of continuity of the ids for random Schrödinger operators on metric graphs. Random Oper. Stochastic Equations 16, 1–10 (2008)

    Article  MATH  Google Scholar 

  22. Harmer, M.: Hermitian symplectic geometry and the factorization of the scattering matrix on graphs. J. Phys. A 33(49), 9015–9032 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Hislop, P.D., Klopp, F.: The integrated density of states for some random operators with nonsign definite potentials. J. Funct. Anal. 195(1), 12–47 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hundertmark, D., Killip, R., Nakamura, S., Stollmann, P., Veselić, I.: Bounds on the spectral shift function and the density of states. Comm. Math. Phys. 262(2–3), 489–503 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Hupfer, T., Leschke, H., Müller, P., Warzel, S.: Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials. Rev. Math. Phys. 13, 1547–1581 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hislop, P., Post, O.: Exponential localization for radial random quantum trees. Waves Random Media. math-ph/0611022 (2006)

  27. Helm, M., Veselić, I.: Linear Wegner estimate for alloy-type Schrödinger operators on metric graphs. J. Math. Phys. 48(9), 092107, 7 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  28. Kirsch, W.: Random Schrödinger operators. In: Holden, H., Jensen, A., (eds.) Schrödinger Operators, Lecture Notes in Physics, vol. 345. Springer, Berlin (1989)

    Google Scholar 

  29. Kirsch, W.: Wegner estimates and Anderson localization for alloy-type potentials. Math. Z. 221, 507–512 (1996)

    MATH  MathSciNet  Google Scholar 

  30. Kirsch, W.: An invitation to Random Schrödinger operators. arXiv:0709.3707 (2007)

  31. Klein, A.: Spreading of wave packets in the Anderson model on the Bethe lattice. Comm. Math. Phys. 177(3), 755–773 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Klein, A.: Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133(1), 163–184 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  33. Klopp, F.: Localization for some continuous random Schrödinger operators. Comm. Math. Phys. 167(3), 553–569 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. Klassert, S., Lenz, D., Peyerimhoff, N., Stollmann, S.: Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature. Proc. Amer. Math. Soc. 134(5), 1549–1559 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kirsch, W., Martinelli, F.: On the density of states of Schrödinger operators with a random potential. J. Phys. A: Math. Gen. 15, 2139–2156 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. In: Proceedings of Symposia in Pure Mathematics. Spectral Theory and Mathematical Physics, vol. 76, pp. 649–698. AMS, New York (2007)

    Google Scholar 

  37. Kuchment, P., Post, O.: On the spectra of carbon nano-structures. Comm. Math. Phys. 275(3), 805–826 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Klopp, F., Pankrashkin, K.: Localization on quantum graphs with random vertex couplings. J. Statist. Phys. 131, 561–673 (2008)

    Article  MathSciNet  Google Scholar 

  39. Klopp, F., Pankrashkin, K.: Localization on quantum graphs with random edge length. Lett. Math. Phys. 87, 99–114 (2009)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. Kotani, S., Simon, B.: Localization in general one-dimensional random systems. II. Continuum Schrödinger operators. Comm. Math. Phys. 112(1), 103–119 (1987)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Kostrykin, V., Schrader, R.: Kirchhoff’s rule for quantum wires. J. Phys. A 32(4), 595–630 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. Kuchment, P.: On the Floquet theory of periodic difference equations. In: Geometrical and algebraical aspects in several complex variables (Cetraro, 1989). Sem. Conf., vol. 8, pp. 201–209. EditEl, Rende (1991)

    Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  44. Kuchment, P.: Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A 38(22), 4887–4900 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  45. Kirsch, W., Veselić, I.: Existence of the density of states for one-dimensional alloy-type potentials with small support. In: Mathematical Results in Quantum Mechanics (Taxco, Mexico, 2001). Contemp. Math., vol. 307, pp. 171–176. American Mathematical Society, Providence (2002)

    Google Scholar 

  46. Kostrykin, V., Veselić, I.: On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials. Math. Z. 252(2), 367–392 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  47. Lenz, D.: Random operators and crossed products. Math. Phys. Anal. Geom. 2(2), 197–220 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  48. Lindenstrauss, E.: Pointwise theorems for amenable groups. Invent. Math. 146(2), 259–295 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  49. Lenz, D., Müller, P., Veselić, I.: Uniform existence of the integrated density of states for models on ℤd. Positivity 12(4), 571–589 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  50. Lledó, F., Post, O.: Eigenvalue bracketing for discrete and metric graphs. J. Math. Anal. Appl. 348(2), 806–833 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  51. Lenz, D., Peyerimhoff, N., Post, O., Veselić, I.: Continuity properties of the integrated density of states on manifolds. Japan. J. Math. 3(1), 121–161 (2008)

    Article  Google Scholar 

  52. Lenz, D., Peyerimhoff, N., Veselić, I.: Integrated density of states for random metrics on manifolds. Proc. London Math. Soc. (3) 88(3), 733–752 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  53. Lenz, D., Peyerimhoff, N., Veselić, I.: Groupoids, von Neumann algebras and the integrated density of states. Math. Phys. Anal. Geom. 10(1), 1–41 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  54. Lenz, D., Veselić, I.: Hamiltonians on discrete structures: jumps of the integrated density of states and uniform convergence. Math. Z. doi:10.1007/s00209-008-0441-3 (2008)

  55. Matsumoto, H.: On the integrated density of states for the Schrödinger operators with certain random electromagnetic potentials. J. Math. Soc. Japan. 45, 197–214 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  56. Mathai, V., Th. Schick, Yates, S.: Approximating spectral invariants of Harper operators on graphs. II. Proc. Amer. Math. Soc. 131(6), 1917–1923 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  57. Mathai, V., Yates, S.: Approximating spectral invariants of Harper operators on graphs. J. Funct. Anal. 188(1), 111–136 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  58. Pastur, L.A.: Selfaverageability of the number of states of the Schrödinger equation with a random potential. Mat. Fiz. i Funkcional. Anal. 238, 111–116 (1971)

    MathSciNet  Google Scholar 

  59. Pastur, L., Figotin, A.: Spectra of Random and Almost-periodic Operators, vol. 297. Grundlehren der Mathematischen Wissenschaften, Springer, Berlin (1992)

    Google Scholar 

  60. Post, O.: Equilateral quantum graphs and boundary triples. In: Exner, P., Keating, J.P., Kuchment, P., Sunada, T., Teplayaev, A. (eds.) Analysis on Graphs and its Applications. Proc. Symp. Pure Math., vol. 77, pp. 469–490. American Mathematical Society, Providence (2008)

    Google Scholar 

  61. Peyerimhoff, N., Veselić, I.: Integrated density of states for ergodic random Schrödinger operators on manifolds. Geom. Dedicata 91(1), 117–135 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  62. Shubin, M.A.: Spectral theory and the index of elliptic operators with almost-periodic coefficients. Russian Math. Surveys 34, 109–157 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  63. Stollmann, P.: Caught by disorder: Bound States in Random Media. Progress in Mathematical Physics, vol. 20. Birkhäuser Verlag, Basel (2001)

    Google Scholar 

  64. von Below, J.; A characteristic equation associated to an eigenvalue problem on C 2-networks. Linear Algebra Appl. 71, 309–325 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  65. Veselić, I.: Wegner estimate and the density of states of some indefinite alloy-type Schrödinger operators. Lett. Math. Phys. 59(3), 199–214 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  66. Veselić, I.: Spectral analysis of percolation Hamiltonians. Math. Ann. 331(4), 841–865 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  67. Veselić, I.: Wegner estimates for sign-changing single site potentials. arXiv:0806.0482 (2008)

  68. Veselić, I.: Lifshitz asymptotics for Hamiltonians monotone in the randomness. Oberwolfach Rep. 4(1), 380–382 (2007)

    Google Scholar 

  69. Wegner, F.: Bounds on the DOS in disordered systems. Z. Phys. B 44, 9–15 (1981)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik & Informatik, Mathematisches Institut, Friedrich-Schiller-Universität Jena, 07737, Jena, Germany

    Daniel Lenz

  2. Department of Mathematical Sciences, Durham University, Science Laboratories South Road, Durham, DH1 3LE, UK

    Norbert Peyerimhoff

  3. Institut für Mathematik, SFB 647 ‘Space – Time – Matter’, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Olaf Post

  4. Fakultät für Mathematik, TU Chemnitz, 09107, Chemnitz, Germany & Emmy-Noether Programme of the DFG

    Ivan Veselić

Authors
  1. Daniel Lenz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Norbert Peyerimhoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Olaf Post
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ivan Veselić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olaf Post.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lenz, D., Peyerimhoff, N., Post, O. et al. Continuity of the Integrated Density of States on Random Length Metric Graphs. Math Phys Anal Geom 12, 219–254 (2009). https://doi.org/10.1007/s11040-009-9059-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11040-009-9059-x

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation