The author takes into account two families of nonlinearity, depending on the value of the parameter alpha, which measures the strength of nonlinear interactions: 1) nonlocal nonlinearity, in the case \(\alpha =0\); 2) local or nonlocal nonlinearity, in the case \(\alpha>= 1\). The Schrödinger-Poisson system in a space of dimension \(d>=3\), is included in first case. The second case includes the cubic nonlinearity and some explanations why the case of the cubic nonlinearity is not treated for the case \(\alpha=0\). The author makes an analysis which is limited to bounded time interval, such that the exponentials are controlled and this is the reason why he don’t keep track of such factors when they are included in a uniform constant. Since the solution of the Euler-Poisson equation generically develops a singularity in finite time, the mentioned analysis is bound to finite time intervals, in the case \(\alpha=0\). Another reason: in the case \(\alpha>=1\), the solution of the Burgers equation generically develops a singularity in finite time. The main result of the paper measures the accuracy of the Lie-Trotter splitting operator associate to the Schrödinger-Poisson equation in the semi-classical regime, when the Wentzel-Kramers-Brillouin approximation is valid.