Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characterization of eigenfunctions of Laplacian having exponential growth using Fourier multipliers

Basil Paul, Pradeep Boggarapu

TL;DR

The paper extends Strichartz-type spectral characterizations of Laplacian eigenfunctions to functions with exponential growth by replacing Δ with a broad class of Fourier multipliers Θ acting as radial mean operators, including M_t, B_t, and the heat semigroup e^{-tΔ}. It develops a Schwartz-type functional-analytic framework, introducing S^a(R^d) and its dual, along with a spherical Fourier transform that extends to exponential-type distributions, and proves a One Radius Theorem linking mean-value properties to Δ-eigenfunctions. The main results (Theorems 3.3, 3.4, 3.6) show that, for Θ with real-valued symbols and a Strichartz-type recursion Θ T_k = A T_{k+1}, the initial data T_0 has Fourier support on the level set |m(ξ)| = |A| and decomposes into Θ-eigencomponents with equal spectral modulus, recovering Δ-characterizations as well as mean-operator analogues; analogous statements hold for the spherical and ball means and the heat semigroup. This unifies growth-rate spectral assertions across a family of Fourier multipliers and provides a robust framework for spectral geometry of the Laplacian in contexts beyond bounded or polynomial growth, with potential applications to harmonic analysis on R^d and related spaces.

Abstract

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.

Characterization of eigenfunctions of Laplacian having exponential growth using Fourier multipliers

TL;DR

The paper extends Strichartz-type spectral characterizations of Laplacian eigenfunctions to functions with exponential growth by replacing Δ with a broad class of Fourier multipliers Θ acting as radial mean operators, including M_t, B_t, and the heat semigroup e^{-tΔ}. It develops a Schwartz-type functional-analytic framework, introducing S^a(R^d) and its dual, along with a spherical Fourier transform that extends to exponential-type distributions, and proves a One Radius Theorem linking mean-value properties to Δ-eigenfunctions. The main results (Theorems 3.3, 3.4, 3.6) show that, for Θ with real-valued symbols and a Strichartz-type recursion Θ T_k = A T_{k+1}, the initial data T_0 has Fourier support on the level set |m(ξ)| = |A| and decomposes into Θ-eigencomponents with equal spectral modulus, recovering Δ-characterizations as well as mean-operator analogues; analogous statements hold for the spherical and ball means and the heat semigroup. This unifies growth-rate spectral assertions across a family of Fourier multipliers and provides a robust framework for spectral geometry of the Laplacian in contexts beyond bounded or polynomial growth, with potential applications to harmonic analysis on R^d and related spaces.

Abstract

In 1993, Robert Strichartz established a characterization for bounded eigenfunctions of the Laplacian on . Let be a doubly infinite sequence of functions on satisfying for all . If 's are uniformly bounded, then Strichartz proved that , 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 , we consider a doubly infinite sequence of exponentially growing functions satisfying the recurrence relation for a complex constant . We demonstrate that under specific spectral conditions, the functions correspond precisely to the eigenfunctions of the Laplacian on . This result provides a unified approach to characterization theorems, linking the growth rate of eigenfunctions to the symbol of the associated multiplier.
Paper Structure (13 sections, 33 theorems, 76 equations, 1 figure)

This paper contains 13 sections, 33 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Euclidean spherical function and its properties
  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. Convolution
  9. Multiplier operators
  10. One Radius Theorem
  11. Characterization of exponential type tempered eigendistributions
  12. Proof of Theorem \ref{['theorem1.5']}
  13. Multipliers on $\mathcal{S}(\mathbb R^d)$

Key Result

Theorem 1.1

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

Figures (1)

  • Figure 1: A covering of $\mathbb{D} \cup \{1\}$ by $\{\mathbb{D}_s \cup \{1\} | 0<s<1\}$, where $\mathbb{D}_s$ are discs bounded by circles $C_s$ of radius $s$ and center $((1-s),0)$

Theorems & Definitions (57)

  • Theorem 1.1: Roe
  • Theorem 1.2: Strichartz
  • Theorem 1.3: Howard and Reese
  • Theorem 1.4: BP
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 47 more