Letters in Mathematical Physics

, Volume 46, Issue 4, pp 303–311 | Cite as

Singular Continuous Spectrum for a Class of Substitution Hamiltonians

  • David Damanik
Article

Abstract

We consider discrete one-dimensional Schrödinger operators with potentials generated by primitive substitutions. A purely singular continuous spectrum with probability one is established provided that the potentials have a local four-block structure.

Schrödinger operators substitution potentials singular continuous spectrum. 

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References

  1. 1.
    Bellissard, J., Bovier, A. and Ghez, J.-M.: Spectral properties of a tight binding Hamiltonian with period doubling potential, Comm. Math. Phys. 135 (1991), 379–399.Google Scholar
  2. 2.
    Bellissard, J., Bovier, A. and Ghez, J.-M.: Gap labelling theorems for one-dimensional discrete Schrödinger operators, Rev. Math. Phys. 4 (1992), 1–37.Google Scholar
  3. 3.
    Bellissard, J., Iochum, B., Scoppola, E. and Testard, D.: Spectral properties of one-dimensional quasi-crystals, Comm. Math. Phys. 125 (1989), 527–543.Google Scholar
  4. 4.
    Bovier, A. and Ghez, J.-M.: Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Comm. Math. Phys. 158 (1993), 45–66.Google Scholar
  5. 5.
    Bovier, A. and Ghez, J.-M.: Erratum. Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Comm. Math. Phys. 166 (1994), 431–432.Google Scholar
  6. 6.
    Damanik, D.: α-continuity properties of one-dimensional quasicrystals, Comm. Math. Phys. 192 (1998), 169–182.Google Scholar
  7. 7.
    Damanik, D.: Singular continuous spectrum for the period doubling Hamiltonian on a set of full measure, Comm. Math. Phys. 196 (1998), 477–483.Google Scholar
  8. 8.
    Del Rio, R., Jitomirskaya, S., Last, Y. and Simon, B.: Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69 (1996), 153–200.Google Scholar
  9. 9.
    Del Rio, R., Makarov, N. and Simon, B.: Operators with singular continuous spectrum, II. Rank one operators, Comm. Math. Phys. 165 (1994), 59–67.Google Scholar
  10. 10.
    Delyon, F. and Petritis, D.: Absence of localization in a class of Schrödinger operators with quasiperiodic potential, Comm. Math. Phys. 103 (1986), 441–444.Google Scholar
  11. 11.
    Delyon, F. and Peyrière, J.: Recurrence of the eigenstates of a Schrödinger operator with automatic potential, J. Stat. Phys. 64 (1991), 363–368.Google Scholar
  12. 12.
    Gordon, A.: On the point spectrum of the one-dimensional Schrödinger operator, Uspekhi Mat. Nauk 31 (1976), 257.Google Scholar
  13. 13.
    Hof, A.: Some remarks on discrete aperiodic Schrödinger operators, J. Statist. Phys. 72 (1993), 1353–1374.Google Scholar
  14. 14.
    Hof, A., Knill, O. and Simon, B.: Singular continuous spectrum for palindromic Schrödinger operators, Comm. Math. Phys. 174 (1995), 149–159.Google Scholar
  15. 15.
    Jitomirskaya, S. and Simon, B.: Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators, Comm. Math. Phys. 165 (1994), 201–205.Google Scholar
  16. 16.
    Kaminaga, M.: Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential, Forum Math. 8 (1996), 63–69.Google Scholar
  17. 17.
    Kotani, S.: Jacobi matrices with random potentials taking finitely many values, Rev.Math. Phys. 1 (1989), 129–133.Google Scholar
  18. 18.
    Last, Y. and Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math., to appear.Google Scholar
  19. 19.
    Queffélec, M.: Substitution Dynamical Systems – Spectral Analysis, Lecture Notes in Math. 1284, Springer, Berlin, 1987.Google Scholar
  20. 20.
    Shechtman, D., Blech, I., Gratias, D. and Cahn, J.V.: Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951–1953.Google Scholar
  21. 21.
    Simon, B.: Operators with singular continuous spectrum: I. General operators, Ann. of Math. 141 (1995), 131–145.Google Scholar
  22. 22.
    Simon, B.: Operators with singular continuous spectrum: VI. Graph Laplacians and Laplace–Beltrami operators, Proc. Amer. Math. Soc. 124 (1996), 1177–1182.Google Scholar
  23. 23.
    Simon, B.: Operators with singular continuous spectrum: VII. Examples with borderline time decay, Comm. Math. Phys. 176 (1996), 713–722.Google Scholar
  24. 24.
    Simon, B. and Stolz, G.: Operators with singular continuous spectrum: V. Sparse potentials, Proc. Amer. Math. Soc. 124 (1996), 2073–2080.Google Scholar
  25. 25.
    Sütö, A.: The spectrum of a quasiperiodic Schrödinger operator, Comm. Math. Phys. 111 (1987), 409–415.Google Scholar
  26. 26.
    Sütö, A.: Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Statist. Phys. 56 (1989), 525–531.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • David Damanik
    • 1
  1. 1.Fachbereich MathematikJohann Wolfgang Goethe-UniversitätFrankfurt/MainGermany e-mail

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