扰动二进方体与位势属于$RH^{n/2}$的薛定谔算子的定量估计
Perturbed Dyadic Cubes and Quantitative Estimates for Schrödinger Operators with Potentials in $RH^{n/2}$
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中文总结 AI 辅助
本文针对位势属$RH^{n/2}$的薛定谔算子，构造反映位势扰动内蕴几何的新二进方体系$\r D^V$，借此刻画里斯位势的$L^p$算子范数，并给出相关定量谱估计。
中文摘要 AI 辅助
设$L:=-Δ+V$是欧氏空间$\r^n$上的薛定谔算子，其位势$V$属于反向赫尔德类$RH^{n/2}$，满足$V$在无穷远处既不衰减过快也不剧烈振荡的温和假设。本文构造了一套新的二进方体系$\r D^V$，它反映了被$V$扰动的内蕴几何结构。随后利用$\r D^V$的定量几何信息，刻画了所有$α\in(0,2]$且$p\in(1,\infty)$时里斯位势$L^{-α/2}$的$L^p$算子范数。作为应用，还给出了$L$的若干定量谱估计。
英文摘要
Let $L:=-Δ+V$ be a Schrödinger operator on the Euclidean space $\mathbb{R}^n$ with potential $V$ in the reverse Hölder class $RH^{n/2}$ satisfying some mild assumptions that $V$ neither decays too rapidly nor oscillates violently at infinity. In this paper, the authors construct a new system of dyadic cubes $\mathcal{D}^V$ that reflects the intrinsic geometry perturbed by $V$. Then using the quantitative geometric information of $\mathcal{D}^V$, the authors characterize the $L^p$ operator norm of the Riesz potential $L^{-α/2}$ for all $α\in (0,2]$ and $p\in (1,\infty)$. As applications, some quantitative spectral estimates for $L$ are given.
