Point Spectrum of Periodic Operators on Universal Covering Trees

For any multi-graph G with edge weights and vertex potential, and its universal covering tree T , we completely characterize the point spectrum of operators A_T on T arising as pull-backs of local, self-adjoint operators A_G on G. This builds on work of Aomoto, and includes an alternative proof of the necessary condition for point spectrum derived in [5]. Our result gives a finite time algorithm to compute the point spectrum of A_T from the graph G, and additionally allows us to show that this point spectrum is itself contained in the spectrum of A_G. Finally, we prove that typical pull-back operators have a spectral delocalization property: the set of edge weight and vertex potential parameters of A_G giving rise to A_T with purely absolutely continuous spectrum is open, and its complement has large codimension.