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

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

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