On the singular spectrum for adiabatic quasi-periodic Schrödinger Operators

Details On the singular spectrum for adiabatic quasi-periodic Schrödinger Operators On the singular spectrum for adiabatic quasi-periodic Schrödinger Operators

2008-11-23

arxiv.org/abs/0811.3745v1

Physics

Mathematical Physics

23 pages

Scientific paper

In this paper we study spectral properties of a family of quasi-periodic Schr\"odinger 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 in \cite{FEKL1}.

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