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


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.


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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.

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Zhang, M., Zhou, Z. Rotation Numbers of Linear Schrödinger Equations with Almost Periodic Potentials and Phase Transmissions. Ann. Henri Poincaré 11, 765–780 (2010).

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