Heat Traces and Spectral Zeta Functions for p-adic Laplacians

TL;DR

This paper explores the spectral properties of p-adic Laplacians by analyzing heat traces and zeta functions, revealing their integral representations and asymptotic behaviors in the p-adic setting.

Contribution

It introduces the study of heat traces and spectral zeta functions for p-adic Laplacians, providing integral representations and asymptotic estimates.

Findings

Heat traces are p-adic Laplace-type integrals.

Spectral zeta functions are p-adic Igusa-type integrals.

Established asymptotic behavior of heat traces and eigenvalue counting functions.

Abstract

In this article we initiate the study of the heat traces and spectral zeta functions for certain p-adic Laplacians. We show that the heat traces are given by p-adic integrals of Laplace type, and that the spectral zeta functions are p-adic integrals of Igusa-type. We find good estimates for the behaviour of the heat traces when the time tends to infinity, and for the asymptotics of the function counting the eigenvalues less than or equal to a given quantity.