Work

Hardy's uncertainty principle for Schrödinger equations with quadratic Hamiltonians

Year: 2025

Type: article

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 geometry of the corresponding Hamiltonian flow.

Source: Journal of the London Mathematical Society

DOI: 10.1112/jlms.70134

Authors: Elena Cordero, Gianluca Giacchi, Eugenia Malinnikova

Institutions: University of Turin, Università della Svizzera italiana, Norwegian University of Science and Technology, Stanford University

Language: English

Topic: Mathematical Analysis and Transform Methods

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,930

Funders: National Science Foundation, Istituto Nazionale di Alta Matematica "Francesco Severi"