Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation

We investigate the Lie and the Strang splitting for the cubic nonlinear Schrödinger equation on the full space and on the torus in up to three spatial dimensions. We prove that the Strang splitting converges in L2L2 with order 1+θ1+θ for initial values in H2+2θH2+2θ with θ∈(0,1)θ∈(0,1) and that the Lie splitting converges with order one for initial values in H2H2.