  The extended Harper's model, proposed by D.J. Thouless in 1983, generalizes
the famous almost Mathieu operator, allowing for a wider range of lattice
geometries (parametrized by three coupling parameters) by permitting 2D
electrons to hop to both nearest and next nearest neighboring (NNN) lattice
sites, while still exhibiting its characteristic symmetry (Aubry duality).
Previous understanding of the spectral theory of this model was restricted to
two dual regions of the parameter space, one of which is characterized by the
positivity of the Lyapunov exponent. In this paper, we complete the picture
with a description of the spectral measures over the entire remaining
(self-dual) region, for all irrational values of the frequency parameter (the
magnetic flux in the model). Most notably, we prove that in the entire interior
of this regime, the model exhibits a collapse from purely ac spectrum to purely
sc spectrum when the NNN interaction becomes symmetric. In physics literature,
extensive numerical analysis had indicated such "spectral collapse," however so
far not even a heuristic argument for this phenomenon could be provided. On the
other hand, in the remaining part of the self-dual region, the spectral
measures are singular continuous irrespective of such symmetry. The analysis
requires some rather delicate number theoretic estimates, which ultimately
depend on the solution of a problem posed by Erd\H{o}s and Szekeres.
