Heat equations and wavelets on Mumford curves

TL;DR

This paper develops heat operators and wavelet analysis on Mumford curves over non-archimedean fields, solving heat equations and exploring spectral properties in this mathematical setting.

Contribution

It introduces a framework for heat operators and wavelets on Mumford curves, including invariant function spaces and solutions to associated heat equations.

Findings

Constructed invariant L2-spaces and integral operators on Mumford curves.

Diagonalized heat operators using wavelets and graph Laplacian eigenvectors.

Proved existence of solutions to heat equations and analyzed spectral gaps.

Abstract

A general class of heat operators over non-archimedean local fields acting on $L_2$-function spaces on affinoid domains in the local field are developed. $L_2$-spaces and integral operators invariant under the action of a non-archimedean Schottky group are constructed in order to have nice function spaces and operators on Mumford curves. General wavelets are constructed which, together with functions coming from certain graph Laplacian eigenvectors, diagonalise these operators. The corresponding Cauchy problems for the heat equations with these operators are solved in the affirmative, and properties of wavelet eigenvalues and the spectral gaps of the operators on Mumford curves are studied.