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Let $T_{\epsilon}$ be the lifespan for the solution to the Schr\"odingerequation on $\mathbb{R}^d$ with a power nonlinearity $\lambda |u|^{2\theta/d}u$($\lambda \in \mathbb{C}$, $0<\theta<1$) and the initial data in the form$\epsilon \varphi(x)$. We provide a sharp lower bound estimate for$T_{\epsilon}$ as $\epsilon \to +0$ which can be written explicitly by$\lambda$, $d$, $\theta$, $\varphi$ and $\epsilon$. This is an improvement ofthe previous result by H.Sasaki [Adv. Diff. Eq. 14 (2009), 1021--1039].

A sharp lower bound for the lifespan of small solutions to the Schr\"odinger equation with a subcritical power nonlinearity