Another homogeneous model but periodic in time in which delocalization
takes place was considered in [51]. It is the first example of
a model which is classically quite similar to the Chirikov standard map
but where in the quantum case the excitation is unlimited
in a striking difference from the kicked rotator. Moreover, the results
of [51,52] show that the quantum excitation can be even ballistic
when the classical motion is completely chaotic and diffusive.
Generally, the excitation for the number of levels
can be characterized by anomalous diffusion
with the exponent dependent on the parameters and changing between
0 and 2 [51,52,57]. In the delocalized phase the quasienergy
spectrum is fractal and its Hausdorff dimension changes with the
parameter. In the localized phase eigenstates are exponentially
decaying. Another interesting feature of the model is that
in the delocalized phase some part of eigenstates are localized
that is in some sense analogous to existence of discrete states in
the continuum. More recently, it was found that at the same
time the spectrum of the Harper model may have purepoint, multifractal
and continuous components [72].
For the small amplitudes of kicks the kicked Harper model is reduced to the well known Harper model which describes also the electron dynamics of a square lattice in magnetic field. In the Harper model electron is moving in an effective potential incommensurate with the lattice size. However, the underlying classical Hamiltonian is integrable. The kicked Harper model was considered as the first example[51] of motion in incommensurate potential when the underlying classical dynamics is chaotic. The localization and delocalization, for the kicked rotator and the kicked Harper model correspondingly, clearly demonstrate that the systems with quite similar classical chaotic dynamics may have rather different quantum behaviour. Some similar type of behaviour was observed also in the dynamical model of quasicrystal [58].