$L^q$ Estimates on the Restriction of Schrödinger Eigenfunctions with singular potentials
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| Format: | Preprint |
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2024
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| author | Blair, Matthew D. Park, Chamsol |
| author_facet | Blair, Matthew D. Park, Chamsol |
| contents | We consider eigenfunction estimates in $L^p$ for Schrödinger operators, $H_V=-Δ_g+V(x)$, on compact Riemannian manifolds $(M, g)$. Eigenfunction estimates over the full manifolds were already obtained by Sogge \cite{Sogge1988concerning} for $V\equiv 0$ and the first author, Sire, and Sogge \cite{BlairSireSogge2021Quasimode}, and the first author, Huang, Sire, and Sogge \cite{BlairHuangSireSogge2022UniformSobolev} for critically singular potentials $V$. For the corresponding restriction estimates for submanifolds, the case $V\equiv 0$ was considered in Burq, Gérard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp}. In this article, we will handle eigenfunction restriction estimates for some submanifolds $Σ$ on compact Riemannian manifolds $(M, g)$ with $n:=\dim M\geq 2$, where $V$ is a singular potential. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_15715 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^q$ Estimates on the Restriction of Schrödinger Eigenfunctions with singular potentials Blair, Matthew D. Park, Chamsol Analysis of PDEs Classical Analysis and ODEs Spectral Theory 58J40, 35S30, 42B37 We consider eigenfunction estimates in $L^p$ for Schrödinger operators, $H_V=-Δ_g+V(x)$, on compact Riemannian manifolds $(M, g)$. Eigenfunction estimates over the full manifolds were already obtained by Sogge \cite{Sogge1988concerning} for $V\equiv 0$ and the first author, Sire, and Sogge \cite{BlairSireSogge2021Quasimode}, and the first author, Huang, Sire, and Sogge \cite{BlairHuangSireSogge2022UniformSobolev} for critically singular potentials $V$. For the corresponding restriction estimates for submanifolds, the case $V\equiv 0$ was considered in Burq, Gérard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp}. In this article, we will handle eigenfunction restriction estimates for some submanifolds $Σ$ on compact Riemannian manifolds $(M, g)$ with $n:=\dim M\geq 2$, where $V$ is a singular potential. |
| title | $L^q$ Estimates on the Restriction of Schrödinger Eigenfunctions with singular potentials |
| topic | Analysis of PDEs Classical Analysis and ODEs Spectral Theory 58J40, 35S30, 42B37 |
| url | https://arxiv.org/abs/2406.15715 |