, Volume 146, Issue 3, pp 447-482

Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation

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Abstract

We show that the 1-dimensional Schrödinger equation with a quasiperiodic potential which is analytic on its hull admits a Floquet representation for almost every energyE 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.

Communicated by Ya. G. Sinai