On Gaussian decay rates of harmonic oscillators and equivalences of related Fourier uncertainty principles

TL;DR

This paper advances understanding of Gaussian decay rates for harmonic oscillators, proving a key conjecture for specific times and exploring equivalences among Fourier uncertainty principles using advanced harmonic analysis techniques.

Contribution

It proves Vemuri's conjecture for harmonic oscillators at an arithmetic progression of times and establishes new equivalences among Fourier uncertainty principles.

Findings

Proved Gaussian decay conjecture for harmonic oscillators at specific times.

Established new equivalences among Fourier uncertainty principles.

Provided endpoint results for decay and uncertainty principles.

Abstract

We make progress on a question by Vemuri on the optimal Gaussian decay of harmonic oscillators, proving the original conjecture up to an arithmetic progression of times. The techniques used are a suitable translation of the problem at hand in terms of the free Schr\"odinger equation, the machinery developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega , and a lemma which relates decay on average to pointwise decay. Such a lemma produces many more consequences in terms of equivalences of uncertainty principles. Complementing such results, we provide endpoint results in particular classes induced by certain Laplace transforms, both to the decay Lemma and to the remaining cases of Vemuri's conjecture, shedding light on the full endpoint question.