The author considers the asymptotics as \(h\to0\) for Schrödinger equations in the form \[ ih\partial_tu^h=\Bigl[- {\textstyle {h^2\over2}\Delta+ U\Bigl({x\over h}}\Bigr)+ V(x)\Bigr]u^h, \] in \(\mathbb{R}^t\times \mathbb{R}^d\). The case \(U\equiv 0\) corresponds to the “standard” semi-classical problem. In the general case the author shows how the limiting evolution keeps trace of quantum effects and provides a picture of quantum scattering. The essential point is to realize the matching between the asymptotics \(h\to0\) for \(-(h^2/2)+V(x)\), describing the evolution on a macroscopic scale and the asymptotics \(|y|\to+\infty\) for the operator \(-{1\over2}\Delta_y+ U(y)\) associated with the microscopic scale.