  The Diestel-Leader graph DL(q,r) is the horocyclic product of the homogeneous
trees with respective degrees q+1 and r+1. When q=r, it is the Cayley graph of
the lamplighter group (wreath product of the cyclic group of order q with the
infinite cyclic group) with respect to a natural generating set. For the
"Simple random walk" (SRW) operator on the latter group, Grigorchuk & Zuk and
Dicks & Schick have determined the spectrum and the (on-diagonal) spectral
measure (Plancherel measure). Here, we show that thanks to the geometric
realization, these results can be obtained for all DL-graphs by directly
computing an l^2-complete orthonormal system of finitely supported
eigenfunctions of the SRW. This allows computation of all matrix elements of
the spectral resolution, including the Plancherel measure. As one application,
we determine the sharp asymptotic behaviour of the N-step return probabilities
of SRW. The spectral computations involve a natural approximating sequence of
finite subgraphs, and we study the question whether the cumulative spectral
distributions of the latter converge weakly to the Plancherel measure. To this
end, we provide a general result regarding Foelner approximations; in the
specific case of DL(q,r), the answer is positive only when r=q.
