  The discrete quantum walk in N dimensions is analyzed from the perspective of
its dispersion relations. This allows understanding known properties, as well
as designing new ones when spatially extended initial conditions are
considered. This is done by deriving wave equations in the continuum, which are
generically of the Schr\"odinger type, and allow devising interesting
behaviors, such as ballistic propagation without deformation, or the generation
of almost flat probability distributions, what is corroborated numerically.
There are however special points where the energy surfaces display
intersections and, near them, the dynamics is entirely different. Applications
to the two- and three-dimensional Grover walks are presented.
