[Paper Review] Around Uncertainty Principles of Ingham-type on $\R^n$, $\T^n$ and Two Step Nilpotent Lie Groups

Mithun Bhowmik, Swagato K. Ray+1 more|arXiv (Cornell University)|May 31, 2016

Mathematical Analysis and Transform Methods16 references3 citations

TL;DR

This paper establishes uncertainty principles of Ingham-type on $ ^n$, $ ^n$, and two-step nilpotent Lie groups, generalizing classical Fourier decay conditions for compactly supported or half-space supported functions. It proves that such functions vanish if their Fourier transforms decay faster than certain exponential rates, and applies these results to establish unique continuation properties for solutions to time-dependent Schrödinger equations on Euclidean space and nilpotent groups.

ABSTRACT

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. We view these results as uncertainty principles for Fourier transforms. We prove certain analogues of these uncertainty principles on the $n$-dimensional Euclidean space, the $n$-dimensional torus and connected, simply connected two step nilpotent Lie groups. We also use these results to show a unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on the Euclidean space and a class of connected, simply connected two step nilpotent Lie groups.

Motivation & Objective

• To extend classical Ingham and Paley-Wiener uncertainty principles from the real line to higher-dimensional Euclidean spaces, the torus, and two-step nilpotent Lie groups.
• To characterize the existence of nonzero functions vanishing on open sets or half-spaces via pointwise decay conditions on their Fourier transforms.
• To establish connections between such uncertainty principles and unique continuation properties for solutions to time-dependent Schrödinger equations on $ ^n$ and nilpotent Lie groups.
• To generalize results from the Euclidean setting to noncommutative settings using harmonic analysis on nilpotent groups and representation theory.

Proposed method

• Prove a multivariable analogue of Ingham's uncertainty principle on $ ^n$ for functions vanishing on open sets, using integrability conditions on decay functions $ heta(t)$.
• Establish a multivariable analogue of the Paley-Wiener theorem on $ ^n$ for functions supported on half-spaces, using integrability of $\psi(t)/(1+t^2)$.
• Adapt these results to the $n$-dimensional torus $ ^n$ by considering functions vanishing on open subsets, leveraging periodicity and Fourier analysis on compact abelian groups.
• Extend the framework to connected, simply connected two-step nilpotent Lie groups $G$ by analyzing the center $rak{z}$ and using the Schrödinger equation's solution structure via the Weyl transform and Fourier inversion.
• Use the representation-theoretic decomposition of $L^2(G)$ into Schrödinger representations to analyze decay in the $v$- and $z$-variables separately.
• Apply the uncertainty principles to the initial value problem for the Schrödinger equation by analyzing the Fourier transform of the solution in the $z$-variable and using decay estimates on $w((v,z),t_0)$.

Experimental results

Research questions

• RQ1 Under what decay conditions on the Fourier transform does a nonzero function on $ ^n$ that vanishes on an open set necessarily vanish identically?
• RQ2 What integrability condition on a function $\psi$ ensures the existence of a nonzero $L^2$ function on $ ^n$ supported on a half-space whose Fourier transform decays like $e^{-\psi(|y|)}$?
• RQ3 How can Ingham-type uncertainty principles be extended to the $n$-dimensional torus $ ^n$, where compactly supported or half-space supported functions do not naturally arise?
• RQ4 Can uncertainty principles on two-step nilpotent Lie groups be used to prove unique continuation for solutions to the time-dependent Schrödinger equation?
• RQ5 What conditions on the decay of the initial data and the solution at a fixed time ensure that the solution vanishes identically on $G \times \mathbb{R}$?

Key findings

• A function $f \in L^1(\mathbb{R}^n)$ that vanishes on an open set and satisfies $|\widehat{f}(y)| = O(e^{-\theta(|y|)|y|})$ for large $|y|$ must be identically zero if $\int_1^\infty \frac{\theta(t)}{t} dt < \infty$.
• A nonzero $L^2(\mathbb{R}^n)$ function supported on a half-space with Fourier transform satisfying $|\widehat{f}(y)| = O(e^{-\psi(|y|)})$ exists if and only if $\int_{\mathbb{R}} \frac{\psi(t)}{1+t^2} dt < \infty$.
• On the $n$-dimensional torus $\mathbb{T}^n$, a nonzero function vanishing on an open subset has a Fourier transform that decays like $e^{-\theta(|y|)|y|}$ only if $\int_1^\infty \frac{\theta(t)}{t} dt < \infty$.
• For a connected, simply connected two-step nilpotent Lie group $G$, if the initial data $f(v,z)$ decays like $e^{-|z|\theta(|z|)}$ and the solution $w((v,z),t_0)$ decays like $e^{-\psi(|v|)}$, then $w \equiv 0$ on $G \times \mathbb{R}$ provided $\int_\mathbb{R} \frac{\psi(r)}{1+r^2} dr = \infty$ and $\int_1^\infty \frac{\theta(r)}{r} dr = \infty$.
• The unique continuation property for the time-dependent Schrödinger equation on $\mathbb{R}^n$ and such Lie groups follows from the interplay between the decay of the initial data and the solution at a fixed time, as governed by the uncertainty principles.
• The proof technique relies on lifting the problem to the Schrödinger representation, using the Weyl transform and Fourier inversion, and applying the uncertainty principles to the $v$- and $z$-variables separately.