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Characterization of eigenfunctions of the Laplacian having exponential growth

Basil Paul, Pradeep Boggarapu

TL;DR

The paper addresses the problem of characterizing eigenfunctions of the Laplacian with exponential growth on $\mathbb R^d$, extending Strichartz's bounded-growth result and the polynomial-growth generalization of Howard and Reese to exponential-type growth. It develops a Schwartz-type framework, introducing $S^a(\mathbb R^d)$ and its dual, along with the even holomorphic strip space $H_e(\Omega_a)$, and extends the spherical Fourier transform to duals to study exponential-tempered eigendistributions. A Roe-type spectral criterion is established for sequences $\{T_k\}$ with $(\Delta- z_0 I)T_k = A T_{k+1}$, tying the growth-bound parameter $A$ to eigenfunction outcomes via the boundary geometry of the spectrum in the strip $\Omega_a$. The work also proves a one-dimensional analogue for $\frac{d}{dx}$, showing that exponential growth imposes a rigid eigen-distribution structure and that, under certain bounds, one recovers eigenfunctions of the derivative. Overall, the results extend classical spectral-characterization paradigms to spaces of exponential growth, providing new tools for harmonic analysis and PDEs in unbounded domains with exponential bounds.

Abstract

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.

Characterization of eigenfunctions of the Laplacian having exponential growth

TL;DR

The paper addresses the problem of characterizing eigenfunctions of the Laplacian with exponential growth on , extending Strichartz's bounded-growth result and the polynomial-growth generalization of Howard and Reese to exponential-type growth. It develops a Schwartz-type framework, introducing and its dual, along with the even holomorphic strip space , and extends the spherical Fourier transform to duals to study exponential-tempered eigendistributions. A Roe-type spectral criterion is established for sequences with , tying the growth-bound parameter to eigenfunction outcomes via the boundary geometry of the spectrum in the strip . The work also proves a one-dimensional analogue for , showing that exponential growth imposes a rigid eigen-distribution structure and that, under certain bounds, one recovers eigenfunctions of the derivative. Overall, the results extend classical spectral-characterization paradigms to spaces of exponential growth, providing new tools for harmonic analysis and PDEs in unbounded domains with exponential bounds.

Abstract

In 1993, Robert Strichartz proved a characterization for the bounded eigenfunctions of Laplacian on : If be a doubly infinite sequence of functions on such that and for all , for some , then is an eigenfunction of . 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.
Paper Structure (12 sections, 9 theorems, 63 equations, 3 figures)

This paper contains 12 sections, 9 theorems, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Characterization of the exponential type tempered eigendistributions of the Laplacian
  3. Setting the framework
  4. Schwartz type space $S^a(\mathbb R^d)$ and $H_e(\Omega_{a})$
  5. Extension of spherical Fourier transform to corresponding dual spaces $S^a (\mathbb R^d)'$ and $H_e(\Omega_{a})'$
  6. Radialization operator
  7. Translation operator
  8. Point spectrum of Laplacian in $X_a$
  9. Characterization of the exponential type tempered eigendistributions of the Laplacian
  10. Proof of Theorem \ref{['Theorem1.5']}
  11. Roe's characterization of exponential function
  12. Proof of Theorem \ref{['Theorem 1.2']}

Key Result

Theorem 1.1

Let $\left\{f_{n}\right\}_{n=-\infty}^{\infty}$ be a doubly infinite sequence of real-valued functions of a real variable with Suppose further that there exists a real $M$ such that $|f_{n}(x)| \leq M$ for all $n \in \mathbb Z$ and $x \in \mathbb R.$ Then, $f_{0}(x)=a \sin (x+\phi)$ for some real constants $a$ and $\phi$.

Figures (3)

  • Figure 1: Point spectrum of $\Delta$ on $X_a$.
  • Figure 2: Point spectrum of $\Delta$ on $X_a$ and its translation by $z_0$.
  • Figure 3: The strip $\Omega_a$ and the point spectrum of $\frac{d}{dx}$ on $X_a$.

Theorems & Definitions (13)

  • Theorem 1.1: Roe
  • Theorem 1.2: Howard
  • Theorem 1.3: Strichartz
  • Theorem 1.4: Howard and Reese
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • ...and 3 more