A note on spectral properties of the p-adic tree

Abstract

We study the spectrum of the operator D∗D, where the operator D, introduced in Klimek et al. [e-print arXiv:1403.7263v2], is a forward derivative on the p-adic tree, a weighted rooted tree associated to ℤp 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.