Exponential local energy decay of solutions to the wave equation with $L^\infty$ electric and magnetic potentials
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916784524754944 |
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| author | Larraín-Hubach, Andrés Shapiro, Jacob Vodev, Georgi |
| author_facet | Larraín-Hubach, Andrés Shapiro, Jacob Vodev, Georgi |
| contents | In this paper we prove sharp resolvent estimates for the magnetic Schrödinger operator in $\mathbb{R}^d$, $d\ge 3$, with $L^\infty$ short-range electric and magnetic potentials. We also show that these resolvent estimates still hold for the Dirichlet self-adjoint realization of the Schrödinger operator in the exterior of a non-trapping obstacle in $\mathbb{R}^d$, $d\ge 2$, provided the magnetic potential is supposed identically zero. As an application of the resolvent estimates, we obtain an exponential decay of the local energy of solutions to the wave equation with $L^\infty$ electric and magnetic potentials which decay exponentially at infinity, in all odd and even dimensions, provided the low frequencies are cut off in a suitable way. We also show that in odd dimensions there is no need to cut off the low frequencies in order to get an exponential local energy decay, provided we assume that zero is neither an eigenvalue nor a resonance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential local energy decay of solutions to the wave equation with $L^\infty$ electric and magnetic potentials Larraín-Hubach, Andrés Shapiro, Jacob Vodev, Georgi Analysis of PDEs In this paper we prove sharp resolvent estimates for the magnetic Schrödinger operator in $\mathbb{R}^d$, $d\ge 3$, with $L^\infty$ short-range electric and magnetic potentials. We also show that these resolvent estimates still hold for the Dirichlet self-adjoint realization of the Schrödinger operator in the exterior of a non-trapping obstacle in $\mathbb{R}^d$, $d\ge 2$, provided the magnetic potential is supposed identically zero. As an application of the resolvent estimates, we obtain an exponential decay of the local energy of solutions to the wave equation with $L^\infty$ electric and magnetic potentials which decay exponentially at infinity, in all odd and even dimensions, provided the low frequencies are cut off in a suitable way. We also show that in odd dimensions there is no need to cut off the low frequencies in order to get an exponential local energy decay, provided we assume that zero is neither an eigenvalue nor a resonance. |
| title | Exponential local energy decay of solutions to the wave equation with $L^\infty$ electric and magnetic potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.07058 |