  In this paper, we consider a spectral analysis of discrete time quantum walks
on the path. For isospectral coin cases, we show that the time averaged
distribution and stationary distributions of the quantum walks are described by
the pair of eigenvalues of the coins as well as the eigenvalues and
eigenvectors of the corresponding random walks which are usually referred as
the birth and death chains. As an example of the results, we derive the time
averaged distribution of so-called Szegedy's walk which is related to the
Ehrenfest model. It is represented by Krawtchouk polynomials which is the
eigenvectors of the model and includes the arcsine law.
