  In this paper, we consider the convergence problem of Schr\"odinger equation.
Firstly, we show the almost everywhere pointwise convergence of Schr\"odinger
equation in Fourier-Lebesgue spaces
$\hat{H}^{\frac{1}{p},\frac{p}{2}}(\mathbb{R})(4\leq p<\infty),$
$\hat{H}^{\frac{3 s_{1}}{p},\frac{2p}{3}}(\mathbb{R}^2)(s_{1}>\frac{1}{3},3\leq
p<\infty),$ $\hat{H}^{\frac{2
s_{1}}{p},p}(\mathbb{R}^n)(s_{1}>\frac{n}{2(n+1)},2\leq p<\infty,n\geq3)$ with
rough data. Secondly, we show that the maximal function estimate related to one
Schr\"odinger equation can fail with data in
$\hat{H}^{s,\frac{p}{2}}(\mathbb{R})(s<\frac{1}{p})$. Finally, we show the
stochastic continuity of Schr\"odinger equation with random data in
$\hat{L}^{r}(\mathbb{R}^n)(2\leq r<\infty)$ almost surely. The main ingredients
are Lemmas 2.4, 2.5, 3.2-3.4.
