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Chamsol Park

Publications and source records attributed to Chamsol Park.

4 recordsLinked to original sources
arXiv2026

Uncertainty Principles, Spectral Localization, and Singular Schrödinger Operators on Compact Manifolds

We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schrödinger operators. Our results relate the size of the support of spectrally localized functions to the cardinality of the underlying spectral cluster and to Fourier-ratio type quantities. We obtain analogues for Schrödinger operators with singular potentials belonging to Kato and scaling-critical classes. As an application, we prove uniqueness results for recovery from incomplete spectral data on compact manifolds. Under curvature assumptions, including nonpositive and negative sectional curvatures, we also prove logarithmically improved uncertainty principles associated with shrinking spectral windows.

math.AP↗
arXiv2024

$L^q$ Estimates on the Restriction of Schrödinger Eigenfunctions with singular potentials

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.

math.AP↗
arXiv2022

Eigenfunctions restriction estimates for curves with nonvanishing geodesic curvatures in compact Riemannian surfaces with nonpositive curvature

For $2\leq p<4$, we study the $L^p$ norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact $2$-dimensional Riemannian manifolds. Burq, Gérard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp} found the eigenfunction estimates restricted to a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For $p=4$, we will also obtain a logarithmic analogous estimate, by using arguments in Xi and Zhang \cite{XiZhang2017improved}, Sogge \cite{Sogge2017ImprovedCritical}, and Bourgain \cite{Bourgain1991Besicovitch}.

math.AP↗