A note on spectral properties of the \(p\)-adic tree (Q2795593)

Let \((\mathbb{Z}_p, \rho_p)\) be the Cantor metric space of \(p\)-adic integers equipped with the usual \(p\)-adic metric \(\rho_p\), via Michon's correspondence. The weighted rooted \(p\)-adic tree \(\{V,E\}\) associated to this space is constructed by considering the balls \(V_n\) of diameter \(p^{-n}\) in \(\mathbb{Z}_p\) as the vertices of the \(p\)-adic tree. Since \(\mathbb{Z}_p\) is a totally disconnected space, the range of \(\rho_p\) is countable and consists of numbers of the form \(p^{-n}\). The pair \(e = (v,v')\) is an edge between two vertices \(v,v'\) if \(v \in V_n, v' \in V_{n+1}\) and \(v' \subset v\). The paper under review analyzes the forward derivative \(D\) on the weighted rooted \(p\)-adic tree associated to \((\mathbb{Z}_p, \rho_p)\) and its adjoint \(D^\ast\) and studies the spectrum of the operator \(D^\ast D\). The motivation for studying the spectrum of \(D^\ast D\) is that it may have some relevance for developing the structure of \(p\)-adic quantum mechanics. The operator \(D^\ast D\), a natural analog of the Laplacian, can be taken as an alternative starting point for the theory of \(p\)-adic Schrodinger operators. The paper describes the re-parametrization of the \(p\)-adic tree that leads to a decomposition of the operator \(D^\ast D\) into a direct sum of simpler operators. The paper studies some spectral properties of the operator \(D^\ast D\) and shows that the spectrum is closely related to the roots of a certain \(q\)-hypergeometric function and discusses the analytic continuation of the zeta function associated with the operator \(D^\ast D\).