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$L^p$-mapping properties for Schrödinger operators in open sets of $\mathbb R ^d$ (1602.08208v1)

Published 26 Feb 2016 in math.FA and math.SP

Abstract: Let $H_V=-\Delta +V$ be a Schr\"odinger operator on an arbitrary open set $\Omega$ of $\mathbb Rd$, 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 $Lp$-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, $Lp$-$Lq$-estimates for $\varphi(H_V)$ are also obtained.

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