Hardy's Uncertainty principle for Schrödinger equations with quadratic Hamiltonians
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| Format: | Preprint |
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2024
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| author | Cordero, Elena Giacchi, Gianluca Malinnikova, Eugenia |
| author_facet | Cordero, Elena Giacchi, Gianluca Malinnikova, Eugenia |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_13818 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hardy's Uncertainty principle for Schrödinger equations with quadratic Hamiltonians Cordero, Elena Giacchi, Gianluca Malinnikova, Eugenia Analysis of PDEs 42A38, 35S30, 35B05 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. |
| title | Hardy's Uncertainty principle for Schrödinger equations with quadratic Hamiltonians |
| topic | Analysis of PDEs 42A38, 35S30, 35B05 |
| url | https://arxiv.org/abs/2410.13818 |