Time splitting method for nonlinear Schrödinger equation with rough initial data in L^2
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We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schrödinger equation with rough initial data in L^2, {[ i∂_t u +Δ u = λ |u|^p u, (x,t) ∈ℝ^d ×ℝ_+, u (x,0) =ϕ (x), x∈ℝ^d, ]. where λ∈{-1,1} and p >0. While the Lie approximation Z_L is known to converge to the solution u when the initial datum ϕ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data ϕ∈ L^2 (ℝ^d), we prove the convergence of the Lie approximation Z_L to the solution u in the mass-subcritical range, max{1,2/d}≤ p < 4/d. Furthermore, our argument can be extended to the case of initial data ϕ∈ H^s (ℝ^d) (0<s≤1), for which we obtain a convergence rate of order s/2-s that breaks the natural order barrier s/2.
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