A note on spectral properties of the $p$-adic tree

We study the spectrum of the operator $D^*D$, where the operator $D$, introduced in \cite{KMR}, is a forward derivative on the $p$-adic tree, a weighted rooted tree associated to $\mathbb Z_p$ via Michon's correspondence. We show that the spectrum is closely related to the roots of a certain $q-$hypergeometric function and discuss the analytic continuation of the zeta function associated with $D^*D$.