Perturbed Dyadic Cubes and Quantitative Estimates for Schrödinger Operators with Potentials in $R{H}^{n/2}$

Abstract

Let $L:=-\hat{I}”+V$ be a Schrödinger operator on the Euclidean space ${\mathbb{R}}^n$ with potential $V$ in the reverse Hölder class $R{H}^{n/2}$ 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 $L^{-\hat{I}\pm /2}$ for all $\hat{I}\pm \in (0,2]$ and $p\in (1,\infty )$. As applications, some quantitative spectral estimates for $L$ are given.

42 pages, 5 figures. All comments are welcome