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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2309.01313 |
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| _version_ | 1866917582750089216 |
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| author | Black, Adam Toprak, Ebru Vergara, Bruno Zou, Jiahua |
| author_facet | Black, Adam Toprak, Ebru Vergara, Bruno Zou, Jiahua |
| contents | We study the long-time behavior of solutions to the Schrödinger equation with a repulsive Coulomb potential on $\mathbb{R}^3$ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian $H=-Δ+\frac{q}{|x|}$ on radial data $f$, which allows us to explicitly write the evolution $e^{itH}f$. A comprehensive analysis of the kernel is then used to establish that, for large times, $\|e^{i t H}f\|_{L^{\infty}} \leq C t^{-\frac{3}{2}}\|f\|_{L^1}$. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_01313 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pointwise decay for radial solutions of the Schrödinger equation with a repulsive Coulomb potential Black, Adam Toprak, Ebru Vergara, Bruno Zou, Jiahua Analysis of PDEs 35J10, 33C15 We study the long-time behavior of solutions to the Schrödinger equation with a repulsive Coulomb potential on $\mathbb{R}^3$ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian $H=-Δ+\frac{q}{|x|}$ on radial data $f$, which allows us to explicitly write the evolution $e^{itH}f$. A comprehensive analysis of the kernel is then used to establish that, for large times, $\|e^{i t H}f\|_{L^{\infty}} \leq C t^{-\frac{3}{2}}\|f\|_{L^1}$. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems. |
| title | Pointwise decay for radial solutions of the Schrödinger equation with a repulsive Coulomb potential |
| topic | Analysis of PDEs 35J10, 33C15 |
| url | https://arxiv.org/abs/2309.01313 |