Communications in Mathematical Physics

, Volume 330, Issue 3, pp 1095–1113 | Cite as

Phase Transition and Semi-Global Reducibility

  • Jiangong You
  • Qi ZhouEmail author


For 1D continuous Schödinger operators with large analytic quasi-periodic potentials of two frequencies, one knows that the spectral measure is singular at the bottom of the spectrum and purely absolutely continuous in the upper part of the spectrum, so there is a phase transition when energy increases. In this paper, we obtain the exact power-law for the phase transition in energy by the semi-global reducibility theory of analytic quasi-periodic linear systems.


Lyapunov Exponent Rotation Number Absolute Continuity Pure Point Homotopy Method 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.Laboratoire de Probabilités et Modèles aléatoiresUniversité Pierre et Marie CurieParis Cedex 05France

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