Local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields

Abstract

We review the proof of the existence of a fundamental solution for a pseudo-differential operator with polynomial symbol based on the existence of a meromorphic continuation for the local zeta function attached to the symbol. We compute fundamental solutions for quasielliptic and Schrodinger-type pseudo-differential operators. As an application we solve certain initial value problems for Schrodinger-type pseudo-differential equations. We pose several questions and problems about the connection between local zeta functions and pseudo-differential operators.