Dispersive estimates for Schrödinger's and wave equations on Riemannian manifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910979110993920 |
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| author | Beceanu, Marius |
| author_facet | Beceanu, Marius |
| contents | This paper proves $L^p$ decay estimates for Schrödinger's and wave equations with scalar potentials on three-dimensional Riemannian manifolds.
The main result regards small perturbations of a metric with constant negative sectional curvature. We also prove estimates on $\mathbb S^3$, the three-dimensional sphere, and $\mathbb H^3$, the three-dimensional hyperbolic space.
Most of the estimates hold for the perturbed Hamiltonian $H=H_0+V$, where $H_0$ is the shifted Laplacian $H_0=-Δ+κ_0$, $κ_0$ is the constant (or asymptotic) sectional curvature, and $V$ is a small scalar potential.
The results are based on direct estimates of the wave propagator.
All results hold in three space dimensions. The metric is required to have four derivatives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06957 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dispersive estimates for Schrödinger's and wave equations on Riemannian manifolds Beceanu, Marius Analysis of PDEs 35L05, 35Q41, 35B40, 53B20, 58J37, 58J45 This paper proves $L^p$ decay estimates for Schrödinger's and wave equations with scalar potentials on three-dimensional Riemannian manifolds. The main result regards small perturbations of a metric with constant negative sectional curvature. We also prove estimates on $\mathbb S^3$, the three-dimensional sphere, and $\mathbb H^3$, the three-dimensional hyperbolic space. Most of the estimates hold for the perturbed Hamiltonian $H=H_0+V$, where $H_0$ is the shifted Laplacian $H_0=-Δ+κ_0$, $κ_0$ is the constant (or asymptotic) sectional curvature, and $V$ is a small scalar potential. The results are based on direct estimates of the wave propagator. All results hold in three space dimensions. The metric is required to have four derivatives. |
| title | Dispersive estimates for Schrödinger's and wave equations on Riemannian manifolds |
| topic | Analysis of PDEs 35L05, 35Q41, 35B40, 53B20, 58J37, 58J45 |
| url | https://arxiv.org/abs/2501.06957 |