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.


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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