Communications in Mathematical Physics

, Volume 354, Issue 1, pp 85–113 | Cite as

Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-Periodic Potential in Dimension Two

  • Yulia Karpeshina
  • Young-Ran Lee
  • Roman Shterenberg
  • Günter Stolz


We prove the existence of ballistic transport for the Schrödinger operator with limit-periodic or quasi-periodic potential in dimension two. This is done under certain regularity assumptions on the potential which have been used in prior work to establish the existence of an absolutely continuous component and other spectral properties. The latter include detailed information on the structure of generalized eigenvalues and eigenfunctions. These allow one to establish the crucial ballistic lower bound through integration by parts on an appropriate extension of a Cantor set in momentum space, as well as through stationary phase arguments.


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  1. 1.
    Asch J., Knauf A.: Motion in periodic potentials. Nonlinearity 11, 175–200 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Avron J., Simon B.: Almost periodic Schrödinger operators, I. Limit periodic potentials. Comm. Math. Phys. 82, 101–120 (1981)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bellissard J., Schulz-Baldes H.: Subdiffusive quantum transport for 3D Hamiltonians with absolutely continuous spectra. J. Stat. Phys. 99, 587–594 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bourgain J.: Anderson localization for quasi-periodic lattice Schrödinger operators on \({\mathbb{Z}^d}\), d arbitrary. Geom. Funct. Anal. 17, 682–706 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bourgain J., Goldstein M., Schlag W.: Anderson localization for Schrödinger operators on \({\mathbb{Z}^2}\) with quasi-periodic potential. Acta Math. 188, 41–86 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chulaevsky V.A., Dinaburg E.I.: Methods of KAM-theory for long-range quasi-periodic operators on \({\mathbb{Z}^{\nu}}\). Pure Point Spectrum. Commun. Math. Phys. 153, 559–577 (1993)ADSCrossRefGoogle Scholar
  7. 7.
    Chulaevsky V., Delyon F.: Purely absolutely continuous spectrum for almost Mathieu operators. J. Stat. Phys. 55, 1279–1284 (1989)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Combes J.-M.: Connections between quantum dynamics and spectral properties of time-evolution operators. Differ. Equ. Appl. Math. Phys. Math. Sci. Eng. 192, 59–68 (1993)MathSciNetMATHGoogle Scholar
  9. 9.
    Damanik D., Lenz D., Stolz G.: Lower transport bounds for one-dimensional continuum Schrödinger operators. Math. Ann. 336, 361–389 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Damanik D., Tcheremchantsev S.: Power-law bounds on transfer matrices and quantum dynamics in one dimension. Comm. Math. Phys. 236, 513–534 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Damanik D., Tcheremchantsev S.: Scaling estimates for solutions and dynamical lower bounds on wavepacket spreading. J. Anal. Math. 97, 103–131 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Damanik D., Tcheremchantsev S.: Upper bounds in quantum dynamics. J. Am. Math. Soc. 20, 799–827 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Damanik D., Tcheremchantsev S.: A general description of quantum dynamical spreading over an orthonormal basis and applications to Schrödinger operators. Discret. Contin. Dyn. Syst. 28, 1381–1412 (2010)CrossRefMATHGoogle Scholar
  14. 14.
    Dinaburg E.I., Sinai Ya.: The one-dimensional Schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl. 9, 279–289 (1975)CrossRefMATHGoogle Scholar
  15. 15.
    Eliasson L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Comm. Math. Phys. 146, 447–482 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gel’fand I.M.: Expansion in eigenfunctions of an equation with periodic coefficients. Dokl. Akad. Nauk SSSR 73, 1117–1120 (1950) (in Russian)Google Scholar
  17. 17.
    Germinet F., Kiselev A., Tcheremchantsev S.: Transfer matrices and transport for Schrödinger operators. Ann. Inst. Fourier 54, 787–830 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Guarneri I.: Spectral properties of quantum diffusion on discrete lattices. Europhys. Lett. 10, 95–100 (1989)ADSCrossRefGoogle Scholar
  19. 19.
    Guarneri I.: On an estimate concerning quantum diffusion in the presence of a fractional spectrum. Europhys. Lett. 21, 729–733 (1993)ADSCrossRefGoogle Scholar
  20. 20.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, p. 256. Springer, Berlin (1990)Google Scholar
  21. 21.
    Jitomirskaya S., Schulz-Baldes H., Stolz G.: Delocalization in random polymer models. Comm. Math. Phys. 233, 27–48 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Johnson R., Moser J.: The rotation number for almost periodic potentials. Comm. Math. Phys. 84, 403–438 (1982)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Karpeshina Yu., Lee Y.-R.: Spectral properties of polyharmonic operators with limit-periodic potential in dimension two. J. Anal. Math. 102, 225–310 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Karpeshina Yu., Lee Y.-R.: Absolutely continuous spectrum of a polyharmonic operator with a limit-periodic potential in dimension two. Comm. Partial Differ. Equ. 33, 1711–1728 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Karpeshina Yu., Lee Y.-R.: Spectral properties of a limit-periodic Schrödinger operator in dimension two. J. Anal. Math. 120, 1–84 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Karpeshina Yu., Shterenberg R.: Multiscale analysis in momentum space for quasi-periodic potential in dimension two. J. Math. Phys. 54, 073507 (2013) 1–92ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Karpeshina, Yu., Shterenberg, R.: Extended States for the Schrödinger Operator with Quasi-Periodic Potential in Dimension Two (2014). arXiv:1408.5660
  28. 28.
    Kiselev A., Last Y.: Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains. Duke Math. J. 102, 125–150 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Last Y.: Quantum dynamics and decompositions of singular continuous spectra. J. Funct. Anal. 142, 406–445 (1996)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Molchanov S.A., Chulaevsky V.: The structure of a spectrum of lacunary-limit-periodic Schrödinger operator. Funct. Anal. Appl. 18, 343–344 (1984)CrossRefMATHGoogle Scholar
  31. 31.
    Moser J., Pöschel J.: An extension of a result by Dinaburg and Sinai on quasiperiodic potentials. Comment. Math. Helv. 59, 39–85 (1984)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Pastur L., Figotin A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992)CrossRefMATHGoogle Scholar
  33. 33.
    Radin C., Simon B.: Invariant domains for the time-dependent Schrödinger equation. J. Differ. Equ. 29, 289–296 (1978)ADSCrossRefMATHGoogle Scholar
  34. 34.
    Reed M., Simon B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)MATHGoogle Scholar
  35. 35.
    Rüssmann H.: On the one dimensional Schrödinger equation with quasi-periodic potential. Ann. N.Y. Acad. Sci. 357, 90–107 (1980)ADSCrossRefGoogle Scholar
  36. 36.
    Skriganov, M.M., Sobolev, A.V.: On the spectrum of a limit-periodic Schrödinger operator. Algebra i Analiz 17, 5 (2005); English translation: St. Petersburg Math. J. 17, 815–833 (2006)Google Scholar
  37. 37.
    Tcheremchantsev S.: Mixed lower bounds for quantum transport. J. Funct. Anal. 197, 247–282 (2003)MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Campbell HallUniversity of Alabama at BirminghamBirminghamUSA
  2. 2.Department of MathematicsSogang UniversitySeoulSouth Korea

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