On the point spectrum of quasi-periodic Schrödinger operators

In this paper we study the spectral properties of a family of discrete one-dimensional quasi-periodic Schrödinger operators with discontinuous potential. In large disorder with piecewise analytic potential, under some suitable transversality condition and a Diophantine rotation number, we prove using basically KAM theory that the spectrum of this operator have a large punctual component. Moreover, the most of its eigenvectors are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -localized in one site. We give an explicit estimation of the measure of the punctual spectrum and we prove that the set of phases for which this spectrum is empty have zero Hausdorff dimension.