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Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. (English) Zbl 0753.34055

Summary: We show that the one-dimensional Schrödinger equation with a quasi-periodic potential which is analytic on its hull admits a Floquet representation for almost every energy \(E\) in the upper part of the spectrum. We prove that the upper part of the spectrum is purely absolutely continuous and that, for a generic potential, it is a Cantor set. We also show that for a small potential these results extend to the whole spectrum.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34C25 Periodic solutions to ordinary differential equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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