Characterization of eigenfunctions of the Laplacian having exponential growth
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| Format: | Preprint |
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2026
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| _version_ | 1866912832184909824 |
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| author | Paul, Basil Boggarapu, Pradeep |
| author_facet | Paul, Basil Boggarapu, Pradeep |
| contents | In 1993, Robert Strichartz proved a characterization for the bounded eigenfunctions of Laplacian $Δ=-\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 $Δf_k=f_{k+1}$ 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 $Δ$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_13017 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterization of eigenfunctions of the Laplacian having exponential growth Paul, Basil Boggarapu, Pradeep Classical Analysis and ODEs Primary: 42B10, Secondary: 46E10 In 1993, Robert Strichartz proved a characterization for the bounded eigenfunctions of Laplacian $Δ=-\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 $Δf_k=f_{k+1}$ 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 $Δ$. 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. |
| title | Characterization of eigenfunctions of the Laplacian having exponential growth |
| topic | Classical Analysis and ODEs Primary: 42B10, Secondary: 46E10 |
| url | https://arxiv.org/abs/2601.13017 |