Spherical functions and spectrum of the Laplacian on semi-homogeneous trees

We determine the ℓ^p-spectrum of the isotropic nearest-neighbor stochastic transition operator µ_1 acting on functions on the set V of vertices of a semi-homogeneous tree; in the much simpler setting of homogeneous trees, the spectrum has been known for a long time. The spectrum is given by the eigenvalues of spherical functions, normalized at a reference vertex v_0. We first show that spherical functions are boundary integrals of generalized Poisson kernels that, unlike the homogeneous setting, are not complex powers of the usual Poisson kernel. Then we compute these generalized Poisson kernels via Markov chains and their generating functions, whence we work out explicit expressions for spherical functions, that turn out to have an ℓ^p behavior different from the homogeneous setting; indeed, one of them, for an appropriate choice of v_0, belongs to ℓ^p for some p<2. Up to normalization, on each of the two homogeneity classes, that is, on each orbit V_+, V_− of the Markov chain induced by µ_1, the operator µ_1^2 differs from the step-2 isotropic operator µ_2 only by a shift. On the other hand, the recurrence relation associated to the semi-homogeneous µ_2 is that of a polygonal graph, akin to that of µ1 on a homogeneous tree. By this token, we compute the spectra of µ_1^2 on ℓ^p(V_+) and ℓ^p(V_−), hence, by extracting square roots, the spectrum of µ_1 on ℓ^p(V) for 1⩽p<∞. We show that this spectrum is disconnected for p in an interval containing 2 but connected for all other values of p, whereas in the homogeneous setting it is connected for every p.