Characterization of eigenfunctions of the Laplacian having exponential growth

TL;DR

This paper extends Strichartz's theorem to characterize Laplacian eigenfunctions with exponential growth, broadening the understanding of eigenfunction behavior beyond bounded and polynomial growth cases.

Contribution

It provides a new characterization of eigenfunctions of the Laplacian that exhibit exponential growth, expanding previous results limited to bounded and polynomial growth functions.

Findings

Extended Strichartz's theorem to exponential growth eigenfunctions

Characterized eigenfunctions with exponential growth behavior

Generalized previous polynomial growth results

Abstract

In 1993, Robert Strichartz proved a characterization for the bounded eigenfunctions of Laplacian $\Delta=-\sum_{j=1}^d \frac{\partial^2}{\partial x_j^2} $ on $\mathbb{R}^d$: If $\left\{f_k \right\}_{k\in \mathbb{Z}}$ be a doubly infinite sequence of functions on $\mathbb{R}^d$ such that $\Delta f_k=f_{k+1}$ and $ \|f_k\|_{L^{\infty}(\mathbb{R}^d)} \leq C$ for all $ k \in \mathbb{Z}$, for some $C>0$, then $f_0$ is an eigenfunction of $\Delta$. Observing the existence of unbounded eigenfunctions of the Laplacian, Howard and Reese generalized Strichartz's theorem to characterize eigenfunctions of the Laplacian having at most polynomial growth. In this article, we shall prove an extended version of Strichartz's theorem to characterize eigenfunctions of the Laplacian having exponential growth.