Mapping Properties of Functions of Schrödinger Operators between $L^{p}$-Spaces and Besov Spaces

Abstract

Sufficient conditions are given for the boundedness of $f(H)$, $H = -\triangle + V$, in $L^p({\mathbf{R}}^d)$, $1 \le p \le \infty$. Optimal results with respect to the decay of $f$ are obtained for $L^p$-boundedness of $e^{-itH} f(H)$ and the nearly-optimal norm-estimate $\|e^{-itH}f(H)\|_{\mathcal{B}(L^p)} \le C(1 + |t|)^\gamma$, $t \in \mathbf{R}$, $\gamma > d|1/2 - 1/p|$ is proved. Results are also obtained on the mapping properties of $e^{-itH}$ between certain Besov spaces.