The Multi-Dimensional Hardy Uncertainty Principle and its Interpretation in Terms of the Wigner Distribution; Relation With the Notion of Symplectic Capacity

2008/03/06 by Maurice A. de Gosson, Maurice de Gosson, de Gosson, Maurice +2
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #NA #Quantum chaos and dynamical systems #math-ph #math.CA #math.MP

paper · pdf · doi:10.48550/arxiv.0803.0910

arxiv created 2008/03/06 · openalex publication_date 2008/03/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend Hardy's uncertainty principle for a square integrable function and its Fourier transform to the multidimensional case using a symplectic diagonalization. We use this extension to show that Hardy's uncertainty principle is equivalent to a statement on the Wigner distribution of the function. We give a geometric interpretation of our results in terms of the notion of symplectic capacity of an ellipsoid. Furthermore, we show that Hardy's uncertainty principle is valid for a general Lagrangian frame of the phase space. Finally, we discuss an extension of Hardy's theorem for the Wigner distribution for exponentials with convex exponents.

Citations

• Geometric quantization and multiplicities of group representations

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