How many random walks correspond to a given set of return probabilities to the origin?

We consider the class of simple random walks or birth and death chains on the nonnegative integers. The set of return probabilities Pn00, n ⩾ 0, uniquely determines the spectral measure of the process. We characterize the class of simple random walks with the same spectral measure or same return probabilities to the origin. The analysis is based on the spectral theory developed by Karlin and McGregor (1959), continued fractions and canonical moments.