  We introduce and explore a family of self-dual models of single-particle
motion in quasiperiodic potentials, with hopping amplitudes that fall off as a
power law with exponent $p$. These models are generalizations of the familiar
Aubry-Andre model. For large enough $p$, their static properties are similar to
those of the Aubry-Andre model, although the low-frequency conductivity in the
localized phase is sensitive to $p$. For $p \leq 2.1$ the Aubry-Andre
localization transition splits into three transitions; two distinct
intermediate regimes with both localized and delocalized states appear near the
self-dual point of the Aubry-Andre model. In the intermediate regimes, the
density of states is singular continuous in much of the spectrum, and is
approximately self-similar: states form narrow energy bands, which are divided
into yet narrower sub-bands; we find no clear sign of a mobility edge. When $p
< 1$, localized states are not stable in random potentials; in the present
model, however, tightly localized states are present for relatively large
systems. We discuss the frequency-dependence and strong sample-to-sample
fluctuations of the low-frequency optical conductivity, although a suitably
generalized version of Mott's law is recovered when the power-law is slowly
decaying. We present evidence that many of these features persist in models
that are away from self-duality.
