Radon transforms and lamplighter random walks

We compute the spectrum of the lamplighter random walk in the case where the underlying graph is a path, representing the state space as a product of two rooted q-ary trees and using suitable Radon transforms. We analyze with the same techniques two additional examples in which the action of the automorphism group on the state space has orbits and the restriction on each orbit is not multiplicity free. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.