## On the singular spectrum for adiabatic quasi-periodic Schrödinger operators on the real line.(English)Zbl 1059.81057

The paper is devoted to the continuation of the author investigations on analysis of the spectrum of the family of differential operators $H_{z,\varepsilon}=-\frac{d^2}{dx^2}+V(x-z)+W(\varepsilon x)$ acting on $$L^2(\mathbb{R})$$, where $$V(x)$$ and $$W(x)$$ are periodic, sufficiently regular real valued functions, $$z\in\mathbb{R}$$ indexes the family of operators, and $$\varepsilon>0$$ is chosen so that the potential $$V(x-z)+W(\varepsilon x)$$ is quasi-periodic.
In the paper the spectral properties of the operators $$H_{z,\varepsilon}$$ for $$\varepsilon$$ positive small are studied. It is assumed that the adiabatic iso-energetic curves are extended along the momentum direction. In the energy intervals where this happens, using the monodromy matrix method as in [A. Fedotov and F. Klopp, Commun. Math. Phys. 227, 1–92 (2002; Zbl 1004.81008)] the asymptotics of the monodromy matrix for the family $$(H_{z,\varepsilon})$$ is studied. By means of this an asymptotic formulae for the Lyapunov exponent of $$H_{z,\varepsilon}$$ is obtained, and pure singularity of the spectrum is shown.

### MSC:

 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 47E05 General theory of ordinary differential operators 47A10 Spectrum, resolvent

Zbl 1004.81008
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