L^p-MAPPING PROPERTIES FOR SCHRӦDINGER OPERATORS IN OPEN SETS OF R^d

Let $H_V = - \\Delta +V$ be a Schr\\"odinger operator on an arbitrary open set $\\Omega \\subset \\mathbb{R}^d$ ($d \\ge 3$), where $\\Delta$ 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 resolution theorem. As a by-product, $L^p$--$L^q$--estimates for $\\varphi(H_V)$ are also obtained