Analytic families of reducible linear quasi-periodic differential equations

In this paper we study the existence of analytic families of reducible linear quasi-periodic differential equations in matrix Lie algebras. Under suitable conditions we show, by means of a Kolmogorov-Arnold-Moser (KAM) scheme, that a real analytic quasi-periodic system close to a constant matrix can be modified by the addition of a time-free matrix that makes it reducible to constant coefficients. If the system depends analytically on external parameters, then this modifying term is also analytic. As a major application, we prove the analyticity of resonance tongue boundaries in Hill's equation with a small quasi-periodic forcing. Several consequences for the spectrum of Schrodinger operators with quasi-periodic forcing are derived. In particular, we prove that, generically, the spectrum of Schrodinger operators with a small real analytic and quasi-periodic potential has all spectral gaps open and, therefore, it is a Cantor set. Some other applications are included for linear quasi-periodic systems on so(3, R) and sp(n, R).