Skip to main content

Rotation Numbers of Linear Schrödinger Equations with Almost Periodic Potentials and Phase Transmissions

Abstract

In this paper we study the linear Schrödinger equation with an almost periodic potential and phase transmission. Based on the extended unique ergodic theorem by Johnson and Moser, we will show for such an equation the existence of the rotation number. This extends the work of Johnson and Moser (in Commun Math Phys 84:403–438, 1982; Erratum Commun Math Phys 90:317–318, 1983) where no phase transmission is considered. The continuous dependence of rotation numbers on potentials and transmissions will be proved.

References

  1. 1

    Arnold, L.: Random Dynamical Systems. In: Springer Monographs Mathematics. Springer-Verlag, Berlin (1998)

  2. 2

    Fayad B., Krikorian R.: Exponential growth of product of matrices in \({{\rm SL}(2,\mathbb {R})}\) . Nonlinearity 21, 319–323 (2008)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  3. 3

    Feng D.-J.: Lyapunov exponents for products of matrices and multifractal analysis, I. Positive matrices. Isr. J. Math. 138, 353–376 (2003)

    MATH  Article  Google Scholar 

  4. 4

    Feng H., Zhang M.: Optimal estimates on rotation number of almost periodic systems. Z. Angew. Math. Phys. 57, 183–204 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  5. 5

    Fink A.: Almost Periodic Differential Equations. Springer, New York (1974)

    MATH  Google Scholar 

  6. 6

    Gan S., Zhang M.: Resonance pockets of Hill’s equations with two-step potentials. SIAM J. Math. Anal. 32, 651–664 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  7. 7

    Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438 (1982). Erratum, Commun. Math. Phys. 90, 317–318 (1983)

    Google Scholar 

  8. 8

    Katok A., Hasselblatt B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  9. 9

    Kronig R., Penney W.: Quantum mechanics in crystal lattices. Proc. R. Soc. Lond. 130, 499–513 (1931)

    MATH  Article  ADS  Google Scholar 

  10. 10

    Li W., Lu K.: Rotation numbers for random dynamical systems on the circle. Trans. Am. Math. Soc. 360, 5509–5528 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  11. 11

    Meng, G., Zhang, M.: Measure differential equations, I. Continuity of solutions in measures with weak* topology. Preprint. http://faculty.math.tsinghua.edu.cn/~mzhang (2009)

  12. 12

    Nemytskii V.V., Stepanov V.V.: Qualitative Theory of Differential Equations. Princeton University Press, Princeton (1960)

    MATH  Google Scholar 

  13. 13

    Niikuni H.: Absent spectral gaps of generalized Kronig-Penney Hamiltonians. Tsukuba J. Math. 31, 39–65 (2007)

    MATH  MathSciNet  Google Scholar 

  14. 14

    Niikuni H.: The rotation number for the generalized Kronig-Penney Hamiltonians. Ann. Henri Poincaré 8, 1279–1301 (2007)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  15. 15

    Novo S., Núñez C., Obaya R.: Ergodic properties and rotation number for linear Hamiltonian systems. J. Differ. Equ. 148, 148–185 (1998)

    MATH  Article  Google Scholar 

  16. 16

    Walters P.: An Introduction to Ergodic Theory. Springer-Verlag, New York (1982)

    MATH  Google Scholar 

  17. 17

    Zhang M.: The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials. J. Lond. Math. Soc. 64(2), 125–143 (2001)

    MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Meirong Zhang.

Additional information

M. Zhang is supported by the National Basic Research Program of China (Grant no. 2006CB805903), the National Natural Science Foundation of China (Grant no. 10531010), and the 111 Project of China (2007).

Communicated by Rafael D. Benguria.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zhang, M., Zhou, Z. Rotation Numbers of Linear Schrödinger Equations with Almost Periodic Potentials and Phase Transmissions. Ann. Henri Poincaré 11, 765–780 (2010). https://doi.org/10.1007/s00023-010-0045-4

Download citation

Keywords

  • Periodic Lattice
  • Lyapunov Exponent
  • Haar Measure
  • Rotation Number
  • Periodic Potential