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Summary: THE ABSOLUTELY CONTINUOUS SPECTRUM OF THE
ALMOST MATHIEU OPERATOR
ARTUR AVILA
Abstract. We prove that the spectrum of the almost Mathieu operator is
absolutely continuous if and only if the coupling is subcritical. This settles
Problem 6 of Barry Simon's list of Schr¨odinger operator problems for the
twenty-first century.
1. Introduction
This work is concerned with the almost Mathieu operator H = H,, defined
on 2
(Z)
(1.1) (Hu)n = un+1 + un-1 + 2 cos(2[ + n])un
where = 0 is the coupling, R \ Q is the frequency and R is the phase.
This is the most studied quasiperiodic Schr¨odinger operator, arising naturally as a
physical model (see [L3] for a recent historical account and for the physics back-
ground).
We are interested on the decomposition of the spectral measures in atomic (corre-
sponding to point spectrum), singular continuous and absolutely continuous parts.
Our main result is the following.
Main Theorem 1. The spectral measures of the almost Mathieu operator are
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