  The most elementary quantum walk is characterized by a 2-dimensional unitary
coin flip matrix, which can be parameterized by 4 real variables. The influence
of the choice of the coin flip matrix on the time evolution operator is
analysed in a systematic way. By changing the coin parameters, the dispersion
and asymmetry of eigenvalues of the time evolution operator can be tuned in a
controlled way. The reduced eigenvectors in coin space are distributed along
trajectories on the surface or inside the Bloch sphere, depending on the
degeneracy of the spectrum. At certain values of the coin parameters the
spectrum of the time evolution operator becomes 2-fold degenerate, but there
might exist unique eigenvalues at the top and bottom of each quasi-energy band.
The eigenstates corresponding to such eigenvalues are robust against arbitrary
temporal variations in the bias parameter of the coin, as long as rest of the
parameters remain unchanged.
