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On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit. (English) Zbl 1101.34069

Summary: We study the spectral properties of families of quasi-periodic Schrödinger operators on the real line in the adiabatic limit in the case when the adiabatic iso-energetic curves are extended along the position direction. We prove that, in energy intervals where this is the case, most of the spectrum is purely absolutely continuous in the adiabatic limit, and that the associated generalized eigenfunctions are Bloch-Floquet solutions.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
47A10 Spectrum, resolvent
47E05 General theory of ordinary differential operators
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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