04-180 Joaquim Puig, Carles Simo

Analytic families of reducible linear quasi-periodic differential equations (1402K, Postscript) Jun 7, 04

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Abstract. 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 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 which 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 Schr\"odinger operators with quasi-periodic forcing are derived. In particular, we prove that, generically, the spectrum of Schr\"odinger 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 $so(3,\mR)$ and $sp(n,\mR)$.

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