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

A spectral analysis of discrete-time quantum walks related to the birth and death chains