$L^p$-mapping properties for Schrödinger operators in open sets of ${\mathbb{R}}^d$

Abstract

Let $H_V=-\Delta +V$ be a Schrödinger operator on an arbitrary open set $\Omega$ of ${\mathbb{R}}^d$, where $d\ge 3$, and $\Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $\Omega$. The purpose of this paper is to show $L^p$-boundedness of an operator $\phi (H_V)$ for any rapidly decreasing function $\phi$ on $\mathbb{R}$. $\phi (H_V)$ is defined by the spectral theorem. As a by-product, $L^p$-$L^q$-estimates for $\phi (H_V)$ are also obtained.

32 pages