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 ${\mathcal{T}}$, we completely characterize the point spectrum of operators $A_{{\mathcal{T}}}$ on ${\mathcal{T}}$ 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 $A_{{\mathcal{T}}}$ 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 $A_{{\mathcal{T}}}$ with purely absolutely continuous spectrum is open, and its complement has large codimension.