  We consider a new model of quantum walk on a one-dimensional momentum space
that includes both discrete jumps and continuous drift. Its time evolution has
two stages; a Markov diffusion followed by localized dynamics. As in the well
known quantum kicked rotor, this model can be mapped into a localized
one-dimensional Anderson model. For exceptional (rational) values of its scale
parameter, the system exhibits resonant behavior and reduce to the usual
discrete time quantum walk on the line.
