Characterization of eigenfunctions of the Laplacian having exponential growth

In 1993, Robert Strichartz proved a characterization for the bounded eigenfunctions of Laplacian $\mathrm{\Delta }=-{\sum }_{j=1}^d\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_j^2}$ 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 $\mathrm{\Delta }{f}_k=f_{k+1}$ and $\mathrm{\parallel }{f}_k{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^d)}\le C$ for all $k\in \mathbb{Z}$, for some $C>0$, then $f_0$ is an eigenfunction of $\mathrm{\Delta }$. 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.