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Hardy's Uncertainty principle for Schrödinger equations with quadratic Hamiltonians

Elena Cordero, Gianluca Giacchi, Eugenia Malinnikova

TL;DR

This work extends Hardy's uncertainty principle from functions and their Fourier transforms to the propagators of Schrödinger equations with quadratic Hamiltonians, i.e., metaplectic operators. It provides a general Hardy-type uncertainty principle valid for all metaplectic operators, regardless of the invertibility of the projection block $B$, and identifies directional decay governed by the kernels and ranges of $B$, via the eigenvalues of $MB^TNB$. Concrete instances are given, including fractional Fourier transforms and tensor products, along with a dynamical version for Schrödinger evolutions that connects time evolution to symplectic geometry and Gaussian decay properties. The results yield sharp, directionally selective uncertainty criteria, and are applied to anisotropic harmonic oscillators and the standard harmonic oscillator, highlighting the strong interplay between harmonic analysis, symplectic geometry, and quantum dynamics with implications for time-frequency analysis and quantum harmonic analysis.

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 Schrödinger 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 Schrödinger 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.

Hardy's Uncertainty principle for Schrödinger equations with quadratic Hamiltonians

TL;DR

This work extends Hardy's uncertainty principle from functions and their Fourier transforms to the propagators of Schrödinger equations with quadratic Hamiltonians, i.e., metaplectic operators. It provides a general Hardy-type uncertainty principle valid for all metaplectic operators, regardless of the invertibility of the projection block , and identifies directional decay governed by the kernels and ranges of , via the eigenvalues of . Concrete instances are given, including fractional Fourier transforms and tensor products, along with a dynamical version for Schrödinger evolutions that connects time evolution to symplectic geometry and Gaussian decay properties. The results yield sharp, directionally selective uncertainty criteria, and are applied to anisotropic harmonic oscillators and the standard harmonic oscillator, highlighting the strong interplay between harmonic analysis, symplectic geometry, and quantum dynamics with implications for time-frequency analysis and quantum harmonic analysis.

Abstract

Hardy's uncertainty principle is a classical result in harmonic analysis, stating that a function in and its Fourier transform cannot both decay arbitrarily fast at infinity. In this paper, we extend this principle to the propagators of Schrödinger 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 Schrödinger 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.

Paper Structure

This paper contains 21 sections, 25 theorems, 150 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Linear algebra notation and change of variables
  4. The symplectic group
  5. Metaplectic operators
  6. A representation formula for $\hat{S}f$
  7. Integral representation of ter Morsche and Oonincx
  8. Metaplectic operators as partial Fourier transforms
  9. The uncertainty principle
  10. Formulation of the main result
  11. Proof of the main result
  12. Considerations about the decay directions
  13. Examples for Hardy's uncertainty principle
  14. Metaplectic multipliers
  15. Fractional Fourier transforms
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $f\in L^2(\mathbb{R}^d)$, and $a,b>0$. Assume that: If $ab>1$, then $f\equiv0$. If $ab=1$ then $f(x)=Ce^{-\pi a|x|^2}$.

Figures (1)

  • Figure 1: The directional selectivity of Hardy's uncertainty principle for metaplectic operators, involving $\ker(B)^\perp$ and $R(B)$, illustrated in Example \ref{['exIntro1']}.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Example 1.6
  • Theorem 1.7
  • Remark 2.1
  • Remark 2.2
  • Example 2.3
  • ...and 41 more