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Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schrödinger equation

We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schrödinger equation with small potential and the nonlinear Schrödinger equation (NLSE) with weak nonlinearity. For the Schrödinger equation with small potential characterized by a dimensionless parameter $\varepsilon \in (0, 1]$ representing the amplitude of the potential, we employ the unitary flow property of the (second-order) time-splitting Fourier pseudospectral (TSFP) method in $L^2$-norm to prove a uniform error bound at $C(T)(h^m +τ^2)$ up to the long time $T_\varepsilon= T/\varepsilon$ for any $T>0$ and uniformly for $0<\varepsilon\le1$, while $h$ is the mesh size, $τ$ is the time step, $m \ge 2$ depends on the regularity of the exact solution, 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 $τ$. Then by introducing a new technique of {\sl regularity compensation oscillation} (RCO) in which the high frequency modes are controlled by regularity and the low frequency modes are analyzed by phase cancellation and energy method, an improved uniform error bound at $O(h^{m-1} + \varepsilon τ^2)$ is established in $H^1$-norm for the long-time dynamics up to the time at $O(1/\varepsilon)$ of the Schrödinger 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 an improved uniform error bound at $O(h^{m-1} + \varepsilon^2τ^2)$ in $H^1$-norm for the long-time dynamics up to the time at $O(1/\varepsilon^2)$ of the cubic NLSE with $O(\varepsilon^2)$-nonlinearity strength, uniformly for $\varepsilon \in (0, 1]$. Extensions to the first-order and fourth-order time-splitting methods are discussed.