Localization of the discrete one-dimensional quasi-periodic Schrödinger operators

In this paper, we study the spectral properties of a family of discrete one-dimensional quasi-periodic Schrodinger operators (depending on a phase theta). In large disorder, under some suitable conditions on v$$ v $$ and a diophantine rotation number, we prove using basically KAM theory that the spectrum of this operator is pure point for all theta is an element of[0,2 pi)$$ \theta \in \left[0,2\pi \right) $$ with exponential decaying eigenfunctions.