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We establish improved uniform error bounds for the time-splitting methods forthe long-time dynamics of the Schr\"odinger equation with small potential andthe nonlinear Schr\"odinger equation (NLSE) with weak nonlinearity. For theSchr\"odinger equation with small potential characterized by a dimensionlessparameter $\varepsilon \in (0, 1]$ representing the amplitude of the potential,we employ the unitary flow property of the (second-order) time-splittingFourier pseudospectral (TSFP) method in $L^2$-norm to prove a uniform errorbound at $C(T)(h^m +\tau^2)$ up to the long time $T_\varepsilon= T/\varepsilon$for any $T>0$ and uniformly for $0<\varepsilon\le1$, while $h$ is the meshsize, $\tau$ is the time step, $m \ge 2$ depends on the regularity of the exactsolution, and $C(T) =C_0+C_1T$ grows at most linearly with respect to $T$ with$C_0$ and $C_1$ two positive constants independent of $T$, $\varepsilon$, $h$and $\tau$. Then by introducing a new technique of {\sl regularity compensationoscillation} (RCO) in which the high frequency modes are controlled byregularity and the low frequency modes are analyzed by phase cancellation andenergy method, an improved uniform error bound at $O(h^{m-1} + \varepsilon\tau^2)$ is established in $H^1$-norm for the long-time dynamics up to the timeat $O(1/\varepsilon)$ of the Schr\"odinger equation with$O(\varepsilon)$-potential with $m \geq 3$, which is uniformly for$\varepsilon\in(0,1]$. Moreover, the RCO technique is extended to prove animproved uniform error bound at $O(h^{m-1} + \varepsilon^2\tau^2)$ in$H^1$-norm for the long-time dynamics up to the time at $O(1/\varepsilon^2)$ ofthe cubic NLSE with $O(\varepsilon^2)$-nonlinearity strength, uniformly for$\varepsilon \in (0, 1]$. Extensions to the first-order and fourth-ordertime-splitting methods are discussed.

Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schr\"odinger equation