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In this paper, we consider the convergence problem of Schrödinger equation. Firstly, we show the almost everywhere pointwise convergence of Schrödinger equation in Fourier-Lebesgue spaces H ^ 1 p , p 2 ( R ) ( 4 ≤ p > ∞ ) , \hat {H}^{\frac {1}{p},\frac {p}{2}}(\mathbf {R})(4\leq p>\infty ), H ^ 3 s 1 p , 2 p 3 ( R 2 ) ( s 1 > 1 3 , 3 ≤ p > ∞ ) , \hat {H}^{\frac {3 s_{1}}{p},\frac {2p}{3}}(\mathbf {R}^{2})(s_{1}>\frac {1}{3},3\leq p>\infty ), H ^ 2 s 2 p , p ( R n ) ( s 2 > n 2 ( n + 1 ) , 2 ≤ p > ∞ , n ≥ 3 ) \hat {H}^{\frac {2 s_{2}}{p},p}(\mathbf {R}^{n})(s_{2}>\frac {n}{2(n+1)},2\leq p>\infty ,n\geq 3) with rough data. Secondly, we show that the maximal function estimate related to one dimensional Schrödinger equation can fail with data in H ^ s , p 2 ( R ) ( s > 1 p ) \hat {H}^{s,\frac {p}{2}}(\mathbf {R})(s>\frac {1}{p}) . Finally, we show the stochastic continuity of Schrödinger equation with random data in L ^ r ( R n ) ( 2 ≤ r > ∞ ) \hat {L}^{r}(\mathbf {R}^{n})(2\leq r>\infty ) almost surely. The main ingredients are maximal function estimates and density theorem in Fourier-Lebesgue spaces as well as some large deviation estimates.

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