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On the singular spectrum for adiabatic quasiperiodic Schrödinger operators. (English) Zbl 1201.81053

Summary: We study spectral properties of a family of quasiperiodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp (2005).

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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References:

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