  In this paper, we study the discrete-time quantum walks on 1D Chain with the
moving and swapping shift operators. We derive analytical solutions for the
eigenvalues and eigenstates of the evolution operator $\hat{U}$ using the
Chebyshev polynomial technique, and calculate the long-time averaged
probabilities for the two different shift operators respectively. It is found
that the probability distributions for the moving and swapping shift operators
display completely different characteristics. For the moving shift operator,
the probability distribution exhibits high symmetry where the probabilities at
mirror positions are equal. The probabilities are inversely proportional to the
system size $N$ and approach to zero as $N\rightarrow \infty$. On the contrary,
for the swapping shift operator, the probability distribution is not symmetric,
the probability distribution approaches to a power-law stationary distribution
as $N\rightarrow \infty$ under certain coin parameter condition. We show that
such power-law stationary distribution is determined by the eigenstates of the
eigenvalues $\pm1$ and calculate the intrinsic probability for different
starting positions. Our findings suggest that the eigenstates corresponding to
eigenvalues $\pm1$ play an important role for the swapping shift operator.
