Scattering for the Wave Equation on de Sitter Space in All Even Spatial Dimensions
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910439089111040 |
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| author | Cicortas, Serban |
| author_facet | Cicortas, Serban |
| contents | For any $n\geq4$ even, we establish a complete scattering theory for the linear wave equation on the $(n+1)$-dimensional de Sitter space. We prove the existence and uniqueness of scattering states, and asymptotic completeness. Moreover, we construct the scattering map taking asymptotic data at past infinity $\mathscr{I}^-$ to asymptotic data at future infinity $\mathscr{I}^+$. Identifying $\mathscr{I}^-$ and $\mathscr{I}^+$ with $S^n,$ we prove that the scattering map is a Banach space isomorphism on $H^{s+n}(S^n)\times H^{s}(S^n),$ for any $s\geq1.$
The main analysis is carried out at the level of the model equation obtained by differentiating the linear wave equation $\frac{n}{2}$ times in the time variable. The main result of the paper follows from proving a scattering theory for this equation. In particular, for the model equation we construct a scattering isomorphism from asymptotic data in $H^{s+\frac{1}{2}}(S^n)\times H^s(S^n)\times H^s(S^n)$ to Cauchy initial data in $H^{s+\frac{1}{2}}(S^n)\times H^{s+\frac{1}{2}}(S^n)\times H^{s-\frac{1}{2}}(S^n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07342 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scattering for the Wave Equation on de Sitter Space in All Even Spatial Dimensions Cicortas, Serban General Relativity and Quantum Cosmology Analysis of PDEs For any $n\geq4$ even, we establish a complete scattering theory for the linear wave equation on the $(n+1)$-dimensional de Sitter space. We prove the existence and uniqueness of scattering states, and asymptotic completeness. Moreover, we construct the scattering map taking asymptotic data at past infinity $\mathscr{I}^-$ to asymptotic data at future infinity $\mathscr{I}^+$. Identifying $\mathscr{I}^-$ and $\mathscr{I}^+$ with $S^n,$ we prove that the scattering map is a Banach space isomorphism on $H^{s+n}(S^n)\times H^{s}(S^n),$ for any $s\geq1.$ The main analysis is carried out at the level of the model equation obtained by differentiating the linear wave equation $\frac{n}{2}$ times in the time variable. The main result of the paper follows from proving a scattering theory for this equation. In particular, for the model equation we construct a scattering isomorphism from asymptotic data in $H^{s+\frac{1}{2}}(S^n)\times H^s(S^n)\times H^s(S^n)$ to Cauchy initial data in $H^{s+\frac{1}{2}}(S^n)\times H^{s+\frac{1}{2}}(S^n)\times H^{s-\frac{1}{2}}(S^n)$. |
| title | Scattering for the Wave Equation on de Sitter Space in All Even Spatial Dimensions |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs |
| url | https://arxiv.org/abs/2309.07342 |