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Singular continuous spectrum for a class of nonprimitive substitution Schrödinger operators


Authors: César R. de Oliveira and Marcus V. Lima
Journal: Proc. Amer. Math. Soc. 130 (2002), 145-156
MSC (1991): Primary 81Q10; Secondary 11B85, 47B39
DOI: https://doi.org/10.1090/S0002-9939-01-06148-2
Published electronically: June 8, 2001
MathSciNet review: 1855631
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Abstract | References | Similar Articles | Additional Information

Abstract: We present a class of discrete Schrödinger operators, with potentials derived from nonprimitive substitutions, that has purely singular continuous spectrum. We give sufficient conditions on the substitution rule assuring singular continuous spectrum, either for a generic set in the hull of the potential or for a set of total invariant measure.


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  • 1. Axel, F., Gratias, D. (eds.), Beyond Quasicrystals, Berlin: Les Editions de Physique/Springer-Verlag: 1995. MR 97e:00004
  • 2. Bellissard, J.: Spectral properties of Schrödinger's operator with Thue-Morse potential. In: Luck, J.-M., Moussa, P., Waldschmidt, M. (eds.) Number Theory and Physics, Springer Proceedings in Physics, Vol. 47, Berlin: Springer 1990, pp 140-150. MR 92b:82007
  • 3. Bellissard, J., Bovier, A., Ghez, J.-M.: Spectral properties of a tight binding Hamiltonian with period doubling potential. Commun. Math. Phys. 135, 379-399 (1991). MR 91m:81042
  • 4. Bellissard, J., Bovier, A., Ghez, J.-M.: Gap labelling theorems for one dimensional discrete Schrödinger operators. Rev. Math. Phys. 4, 1-37 (1992).MR 93f:47090
  • 5. Bovier, A., Ghez, J.-M.: Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions. Commun. Math. Phys. 158, 45-66 (1993). MR 94k:82064
  • 6. Bovier, A., Ghez, J.-M.: Erratum to Ref. [5], Commun. Math. Phys. 166, 431-432 (1994). MR 95j:82033
  • 7. Damanik, D.: Singular continuous spectrum for the period doubling Hamiltonian on a set of full measure. Commun. Math. Phys. 196, 477-483 (1998). MR 99i:81035
  • 8. Damanik, D.: Singular continuous spectrum for a class of substitution Hamiltonians. Lett. Math. Phys. 46, 303-311 (1998). MR 99m:81053
  • 9. Delyon, F., Petritis, D.: Absence of localization in a class of Schrödinger operators with quasiperiodic potential. Commun. Math. Phys. 103, 441-444 (1986). MR 87e:81030
  • 10. Delyon, F., Peyrière, J.: Recurrence of the eigenstates of a Schrödinger operator with automatic potential. J. Stat. Phys. 64, 363-368 (1991). MR 92h:82066
  • 11. Hof, A.: Some remarks on discrete aperiodic Schrödinger operators. J. Stat. Phys. 72, 1353-1374 (1993). MR 94j:82007
  • 12. Hof, A., Knill, O., Simon, B.: Singular continuous spectrum for palindromic Schrödinger operators. Commun. Math. Phys. 174, 149-159 (1995). MR 97c:47038
  • 13. Kotani, S.: Jacobi matrices with random potential taking finitely many values. Rev. Math. Phys. 1, 129-133 (1989). MR 91b:81023
  • 14. Süto, A.: The spectrum of a quasiperiodic Schrödinger operator. Commun. Math. Phys. 111, 409-415 (1987). MR 88m:81032
  • 15. Süto, A.: Singular Continuous Spectrum on a Cantor Set of Zero Lebesgue Measure for the Fibonacci Hamiltonian. J. Stat. Phys. 56, 525-531 (1989). MR 90e:82046
  • 16. Iochum, B., Testard, D.: Power law growth for the resistance in the Fibonacci model. J. Stat. Phys. 65, 715-723 (1991). MR 93f:82006
  • 17. de Oliveira, C. R., Lima, M. V.: A nonprimitive substitution Schrödinger operator with generic singular continuous spectrum. Rep. Math. Phys. 45, 431-436 (2000). CMP 2000:16
  • 18. Cycon, H. L., Forese, R. G., Kirsch, W., Simon, B.: Schrödinger Operators. Berlin: Springer-Verlag, 1987. MR 88g:35003
  • 19. Queffélec, M.: Substitution Dynamical Systems - Spectral Analysis, LNM 1294. Berlin: Springer-Verlag, 1987. MR 89g:54094
  • 20. Wen, Z.-Y.: Singular words, invertible substitutions and local isomorphisms. In Ref. [1] pp. 433-440. MR 98g:11027
  • 21. Simon, B.: Operators with singular continuous spectrum: I. General operators. Ann. Math. 141, 131-145 (1995). MR 96a:47038
  • 22. Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math. 135, 329-367 (1999). MR 2000f:47060
  • 23. Petersen, K.: Ergodic Theory, Cambridge: University Press, 1983. MR 87i:28002

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Additional Information

César R. de Oliveira
Affiliation: Departamento de Matemática – UFSCar, São Carlos, SP, 13560-970 Brazil
Email: oliveira@dm.ufscar.br

Marcus V. Lima
Affiliation: Departamento de Matemática – UFSCar, São Carlos, SP, 13560-970 Brazil, and AFA, Pirassununga, SP, 13630–000 Brazil
Email: lima@dm.ufscar.br

DOI: https://doi.org/10.1090/S0002-9939-01-06148-2
Received by editor(s): May 30, 2000
Published electronically: June 8, 2001
Additional Notes: The first author was partially supported by CNPq (Brazil).
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society

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