Strichartz estimates for the Schrödinger equation in high dimensional critical electromagnetic fields
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916741148311552 |
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| author | Jia, Qiuye Zhang, Junyong |
| author_facet | Jia, Qiuye Zhang, Junyong |
| contents | We prove Strichartz estimates for the Schrödinger equation with scaling-critical electromagnetic potentials in dimensions $n\geq3$. The decay assumption on the magnetic potentials is critical, including the case of the Coulomb potential. Our approach introduces novel techniques, notably the construction of Schwartz kernels for the localized Schrödinger propagator, which separates the antipodal points of $\mathbb{S}^{n-1}$, in these scaling critical electromagnetic fields. This method enables us to prove the $L^1(\mathbb{R}^n)\to L^\infty(\mathbb{R}^n)$ for the localized Schrödinger propagator, as well as global Strichartz estimates. Our results provide a positive answer to the open problem posed in arXiv:0901.4024 arXiv:1611.04805 arXiv:0806.0778, and fill a longstanding gap left by arXiv:arXiv:0705.0546 arXiv:archive/0608699. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02296 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strichartz estimates for the Schrödinger equation in high dimensional critical electromagnetic fields Jia, Qiuye Zhang, Junyong Analysis of PDEs Mathematical Physics 35J10, 35Q41, 42B37 We prove Strichartz estimates for the Schrödinger equation with scaling-critical electromagnetic potentials in dimensions $n\geq3$. The decay assumption on the magnetic potentials is critical, including the case of the Coulomb potential. Our approach introduces novel techniques, notably the construction of Schwartz kernels for the localized Schrödinger propagator, which separates the antipodal points of $\mathbb{S}^{n-1}$, in these scaling critical electromagnetic fields. This method enables us to prove the $L^1(\mathbb{R}^n)\to L^\infty(\mathbb{R}^n)$ for the localized Schrödinger propagator, as well as global Strichartz estimates. Our results provide a positive answer to the open problem posed in arXiv:0901.4024 arXiv:1611.04805 arXiv:0806.0778, and fill a longstanding gap left by arXiv:arXiv:0705.0546 arXiv:archive/0608699. |
| title | Strichartz estimates for the Schrödinger equation in high dimensional critical electromagnetic fields |
| topic | Analysis of PDEs Mathematical Physics 35J10, 35Q41, 42B37 |
| url | https://arxiv.org/abs/2412.02296 |