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 \( \Vert f_k\Vert _{L^{\infty }(\mathbb {R}^d)} \le C\) 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 Laplacian, Howard and Reese generalized Strichartz’s theorem to characterize eigenfunctions of 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.
