$L^p$-mapping properties for Schr\"odinger operators in open sets of $\mathbb R ^d$

Let $H_V=-\Delta +V$ be a Schr\"odinger operator on an arbitrary open set $\Omega$ of $\mathbb R^d$, where $d \geq 3$, 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 $\varphi(H_V)$ for any rapidly decreasing function $\varphi$ on $\mathbb R$. $\varphi(H_V)$ is defined by the spectral theorem. As a by-product, $L^p$-$L^q$-estimates for $\varphi(H_V)$ are also obtained.