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

TL;DR

This paper investigates the spectral characteristics of a differential operator on the $p$-adic tree, revealing connections to hypergeometric functions and zeta functions, advancing understanding of $p$-adic spectral theory.

Contribution

It provides a detailed analysis of the spectrum of $D^*D$ on the $p$-adic tree, linking it to roots of $q$-hypergeometric functions and exploring the associated zeta function.

Findings

Spectrum related to roots of a $q$-hypergeometric function

Analytic continuation of the zeta function of $D^*D$

Spectral properties linked to $p$-adic analysis

Abstract

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 $D^*D$.