Hardy's Uncertainty principle for Schr\"odinger equations with quadratic Hamiltonians

TL;DR

This paper extends Hardy's uncertainty principle to Schr"odinger equation propagators with quadratic Hamiltonians, broadening its applicability to various metaplectic operators and providing concrete examples like fractional Fourier transforms.

Contribution

It generalizes Hardy's uncertainty principle to all propagators of Schr"odinger equations with quadratic Hamiltonians, including anisotropic cases and specific decay conditions.

Findings

Extended Hardy's principle to all quadratic Hamiltonian propagators

Provided examples such as fractional Fourier transforms

Suggested directional Gaussian decay conditions

Abstract

Hardy's uncertainty principle is a classical result in harmonic analysis, stating that a function in $L^2(\mathbb{R}^d)$ and its Fourier transform cannot both decay arbitrarily fast at infinity. In this paper, we extend this principle to the propagators of Schr\"odinger equations with quadratic Hamiltonians, known in the literature as metaplectic operators. These operators generalize the Fourier transform and have captured significant attention in recent years due to their wide-ranging applications in time-frequency analysis, quantum harmonic analysis, signal processing, and various other fields. However, the involved structure of these operators requires careful analysis, and most results obtained so far concern special propagators that can basically be reduced to rescaled Fourier transforms. The main contributions of this work are threefold: (1) we extend Hardy's uncertainty principle, covering all propagators of Schr\"odinger equations with quadratic Hamiltonians, (2) we provide concrete examples, such as fractional Fourier transforms, which arise when considering anisotropic harmonic oscillators, (3) we suggest Gaussian decay conditions in certain directions only, which are related to the structure of the corresponding symplectic projection.