Characterization of eigenfunctions of Laplacian having exponential growth using Fourier multipliers
In 1993, Robert Strichartz established a characterization for bounded eigenfunctions of the Laplacian on $\mathbb{R}^d$. Let $\left\{f_k \right\}_{k\in \mathbb{Z}}$ be a doubly infinite sequence of functions on $\mathbb{R}^d$ satisfying $Δf_k= f_{k+1}$ for all $k \in \mathbb{Z}$. If $\left\{f_k \right\}$'s are uniformly bounded, then Strichartz proved that $Δf_0= f_0$, thus generalizing a classical result of Roe on the real line. Recognizing that many physically significant eigenfunctions exhibit unbounded behavior, Howard and Reese extended this result to include functions of polynomial growth. Building upon a refined functional-analytic framework, we recently established a broader extension of Strichartz's theorem encompassing eigenfunctions of exponential growth. In the present article, we further investigate the spectral geometry of the Laplacian by replacing the differential operator with a broader class of Fourier multipliers. Specifically, we focus on radial convolution operators, including the spherical average, the ball average, and the heat operator. The central problem addressed is as follows: For a fixed multiplier $Θ$, we consider a doubly infinite sequence of exponentially growing functions $\{f_k\}_{k \in \mathbb{Z}}$ satisfying the recurrence relation $Θf_k = A f_{k+1}$ for a complex constant $A$. We demonstrate that under specific spectral conditions, the functions $f_k$ correspond precisely to the eigenfunctions of the Laplacian $Δ$ on $\mathbb{R}^d$. This result provides a unified approach to characterization theorems, linking the growth rate of eigenfunctions to the symbol of the associated multiplier.