Perturbed Dyadic Cubes and Quantitative Estimates for Schr\"odinger Operators with Potentials in $RH^{n/2}$

Abstract

Let $L:=-\Delta+V$ be a Schr\"odinger operator on the Euclidean space $\mathbb{R}^n$ with potential $V$ in the reverse H\"older 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 $L^{-\alpha/2}$ for all $\alpha\in (0,2]$ and $p\in (1,\infty)$. As applications, some quantitative spectral estimates for $L$ are given.