$L^p$ spectral theory and heat dynamics of locally symmetric spaces

TL;DR

This paper investigates the $L^p$ spectral properties and heat dynamics on arithmetic locally symmetric spaces with $ ext{Q}$-rank one, revealing distinct behaviors for different $p$ values and characterizing eigenfunctions via Eisenstein series.

Contribution

It provides new insights into the $L^p$ spectrum of the Laplacian on these spaces and links spectral properties to heat semigroup dynamics for different $p$ regimes.

Findings

Open subset of eigenvalues for $p<2$ with Eisenstein series eigenfunctions

Discrete real eigenvalues for $p>2$

Different heat semigroup behaviors depending on whether $p<2$ or $p extgreater=2$

Abstract

In this paper we first derive several results concerning the $L^p$ spectrum of arithmetic locally symmetric spaces whose $\Q$-rank equals one. In particular, we show that there is an open subset of $\C$ consisting of eigenvalues of the $L^p$ Laplacian if $p <2$ and that corresponding eigenfunctions are given by certain Eisenstein series. On the other hand, if $p>2$ there is at most a discrete set of real eigenvalues of the $L^p$ Laplacian. These results are used in the second part of this paper in order to show that the dynamics of the $L^p$ heat semigroups for $p<2$ is very different from the dynamics of the $L^p$ heat semigroups if $p\geq 2$.