An interesting feature of extended Harper's model (EHM), a generalization of the
almost Mathieu operator popularized by DJ Thouless, is the appearance of a large
regime of coupling parameters invariant under Aubry duality (``self-dual regime'').
In this regime, extensive numerical analysis in physics literature conjecture a
``strange collapse'' from purely singular continuous to purely absolutely continuous
spectrum, determined by the symmetries of the model.

Based on earlier work on the model [2], we have recently proven this conjecture [1]
by excluding eigenvalues in the self-dual regime for a full measure set of phases
and frequencies. The work is joint with S. Jitomirskaya.

[1] S. Jitomirskaya, C. A. Marx, On the spectral theory of Extended Harper's Model,
preprint (2013).

[2] S. Jitomirskaya, C. A. Marx, Analytic quasi-periodic cocycles with singularities
and the Lyapunov Exponent of Extended Harper's Model, Commun. Math. Phys. 316,
237-267 (2012).}