Multidimensional Almost-Periodic Schrödinger Operators with Cantor Spectrum

  • David Damanik
  • Jake FillmanEmail author
  • Anton Gorodetski


We construct multidimensional almost-periodic Schrödinger operators whose spectrum has zero lower box-counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.

Mathematics Subject Classification



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We are grateful to Leonid Parnovski for useful comments on an earlier version of this manuscript.


  1. 1.
    Avila, A.: On the spectrum and Lyapunov exponent of limit-periodic Schrödinger operators. Commun. Math. Phys. 288, 907–918 (2009)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Avron, J., Simon, B.: Almost periodic Schrödinger operators. I. Limit periodic potentials. Commun. Math. Phys. 82, 101–120 (1981)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Damanik, D., Fillman, J., Gorodetski, A.: Continuum Schrödinger operators associated with aperiodic subshifts. Ann. Henri Poincaré 15, 1123–1144 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Damanik, D., Fillman, J., Lukic, M.: Limit-periodic continuum Schrödinger operators with zero-measure Cantor spectrum. J. Spectr. Theory 7, 1101–1118 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Embree, M., Fillman, J.: Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials. J. Spectr. Theory. (in press). arXiv:1701.00863
  7. 7.
    Exner, P., Turek, O.: Periodic quantum graphs from the Bethe–Sommerfeld perspective. J. Phys. A Math. Theor. 50, 455201 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fillman, J., Han, R.: Discrete Bethe–Sommerfeld conjecture for triangular, square, and hexagonal lattices. Preprint arXiv:1806.01988
  9. 9.
    Fillman, J., Lukic, M.: Spectral homogeneity of limit-periodic Schrödinger operators. J. Spectr. Theory 7, 387–406 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gordon, A.: On the point spectrum of the one-dimensional Schrödinger operator. Usp. Math. Nauk 31, 257–258 (1976)Google Scholar
  11. 11.
    Hadj Amor, S.: Hölder continuity of the rotation number for quasi-periodic co-cycles in \(SL(2, \mathbb{R})\). Commun. Math. Phys. 287, 565–588 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Han, R., Jitomirskaya, S.: Discrete Bethe–Sommerfeld conjecture. Commun. Math. Phys. 361, 205–216 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Helffer, B., Mohamed, A.: Asymptotics of the density of states for the Schrödinger operator with periodic electric potential. Duke Math. J. 92, 1–60 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Karpeshina, Y.: Perturbation Theory for the Schrödinger Operator with a Periodic Potential, Lecture Notes in Mathematics, vol. 1663. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  15. 15.
    Karpeshina, Y., Lee, Y.-R.: Spectral properties of a limit-periodic Schrödinger operator in dimension two. J. Anal. Math. 120, 1–84 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Karpeshina, Y., Shterenberg, R.: Extended states for the Schrödinger operator with quasi-periodic potential in dimension two. To appear in Mem. Am. Math. Soc. arXiv:1408.5660
  17. 17.
    Krüger, H.: Periodic and limit-periodic discrete Schrödinger operators. Preprint arXiv:1108.1584
  18. 18.
    Kuchment, P.: An overview of periodic elliptic operators. Bull. Am. Math. Soc. 53, 343–414 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Molchanov, S., Chulaevsky, V.: The structure of a spectrum of the lacunary-limit-periodic Schrödinger operator. Funct. Anal. Appl. 18, 343–344 (1984)CrossRefGoogle Scholar
  20. 20.
    Moser, J.: An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum. Comment. Math. Helv. 56, 198–224 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Parnovski, L.: Bethe–Sommerfeld conjecture. Ann. Henri Poincaré 9, 457–508 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pastur, L., Tkachenko, V.A.: On the spectral theory of the one-dimensional Schrödinger operator with limit-periodic potential (Russian). Dokl. Akad. Nauk SSSR 279, 1050–1053 (1984)MathSciNetGoogle Scholar
  23. 23.
    Pastur, L., Tkachenko, V.: Spectral theory of a class of one-dimensional Schrödinger operators with limit-periodic potentials. Trudy Moskov. Mat. Obshch. 51, 114–168 (1988)Google Scholar
  24. 24.
    Popov, V.N., Skriganov, M.: A remark on the spectral structure of the two dimensional Schrödinger operator with a periodic potential. Zap. Nauchn. Sem. LOMI AN SSSR 109, 131–133 (1981). (in Russian)zbMATHGoogle Scholar
  25. 25.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)zbMATHGoogle Scholar
  26. 26.
    Simon, B.: On the genericity of nonvanishing instability intervals in Hill’s equation. Ann. Inst. H. Poincaré Sect. A (N.S.) 24, 91–93 (1976)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Skriganov, M.: Proof of the Bethe–Sommerfeld conjecture in dimension two. Sov. Math. Dokl. 20, 89–90 (1979)zbMATHGoogle Scholar
  28. 28.
    Skriganov, M.: Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators. Proc. Steklov Math. Inst. 171, 3–122 (1984)Google Scholar
  29. 29.
    Skriganov, M.: The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential. Inv. Math. 80, 107–121 (1985)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Veliev, O.A.: Spectrum of multidimensional periodic operators. Teor. FunktsiĭFunktsional. Anal. i Prilozhen 49, 17–34 (1988). (in Russian)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsRice UniversityHoustonUSA
  2. 2.Department of MathematicsVirginia TechBlacksburgUSA
  3. 3.Department of MathematicsUniversity of CaliforniaIrvineUSA

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