Summary: ON POINT SPECTRUM AT CRITICAL COUPLING
ARTUR AVILA
Abstract. We give a short proof of absence of point spectrum at critical
coupling for the almost Mathieu operator, for any irrational frequency, except
(possibly) for countably many values of the phase.
1. Introduction
Here we consider the almost Mathieu operator with critical coupling H = H, :
l2
(Z) l2
(Z),
(1.1) (Hu)n = un+1 + un-1 + 2 cos(2( + n))un,
where R \ Q (the frequency) and R (the phase) are parameters. The
spectrum of H is a independent set = .
It is expected that H, has purely singular continuous spectrum for every
R\Q and every R. Since the Lebesgue measure of is zero [AK1], there can not
be absolutely continuous spectrum anyway, so singular continuous spectrum follows
from absence of point spectrum in this context. It is known that eigenfunctions, if
they exist, can not be in l1
(Z) [D].
The first results on absence of point spectrum were obtained under certain topo-

Source: Avila, Artur - Instituto Nacional de Matemática Pura e Aplicada (IMPA)

Collections: Mathematics