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| Let $H:=-\Delta+V$ be a Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V\in L^r_{{\rm loc}}({\bf R}^N)$ with $r>N/2$. In this talk we focus on a nonnegative Schr\"odinger operator $H:=-\Delta+V$ with a radially symmetric potential $V=V(|x|)$ behaving like $V(r)=\omega r^{-2}(1+o(1))$ as $r\to \infty$ with $\omega>-(N-2)^2/4$. We show the exact and optimal decay rates of the operator norm of the Schr\"odinger heat semigroup $e^{-tH}$ in Lorentz spaces as $t\to\infty$. |
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