Sparse Gabor representations of metaplectic operators: controlled exponential decay and Schrödinger confinement
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918113590640640 |
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| author | Cordero, Elena Giacchi, Gianluca Pucci, Edoardo Trapasso, Salvatore Ivan |
| author_facet | Cordero, Elena Giacchi, Gianluca Pucci, Edoardo Trapasso, Salvatore Ivan |
| contents | Motivated by the phase space analysis of Schrödinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this aim, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest. As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schrödinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_02226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparse Gabor representations of metaplectic operators: controlled exponential decay and Schrödinger confinement Cordero, Elena Giacchi, Gianluca Pucci, Edoardo Trapasso, Salvatore Ivan Analysis of PDEs Mathematical Physics Functional Analysis 42B35, 81S30, 35S10, 43A65, 42B37 Motivated by the phase space analysis of Schrödinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this aim, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest. As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schrödinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms. |
| title | Sparse Gabor representations of metaplectic operators: controlled exponential decay and Schrödinger confinement |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis 42B35, 81S30, 35S10, 43A65, 42B37 |
| url | https://arxiv.org/abs/2508.02226 |