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Analytic families of reducible linear quasi-periodic differential equations

Published online by Cambridge University Press:  17 March 2006

JOAQUIM PUIG
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain (e-mail: joaquim.puig@upc.edu, carles@maia.ub.es) Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
CARLES SIMÓ
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain (e-mail: joaquim.puig@upc.edu, carles@maia.ub.es)

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 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 Schrödinger operators with quasi-periodic forcing are derived. In particular, we prove that, generically, the spectrum of Schrödinger 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 $so(3,\mathbb{R})$ and $sp(n,\mathbb{R})$.

Type
Research Article
Copyright
2006 Cambridge University Press

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