Momentum Dynamics of One Dimensional Quantum Walks
  We derive the momentum space dynamic equations and state functions for one dimensional quantum walks by using linear systems and Lie group theory. The momentum space provides an analytic capability similar to that contributed by the z transform in discrete systems theory. The state functions at each time step are expressed as a simple sum of three Chebyshev polynomials. The functions provide an analytic expression for the development of the walks with time. 
Ian Fuss, Langord B. White, Peter J. Sherman and Sanjeev Naguleswaran