Recent results in semiclassical approximation with rough potentials

|x−y|≤ǫ |x−y| |V (y)|dy = 0) compared to the classical Cauchy-Lipshitz condition for vector fields. On the other hand, tremendous progress have been done in the last 25 years concerning the theory of ODEs using PDE’s methods: extension of the Cauchy-Lipshitz condition to Sobolev ones (DiPerna-Lions 1989) and BV vector fields (Bouchut for the Hamiltonian case 2001 and Ambrosio for the general case 2004) have been proved to provide well-posedness of the classical flow almost everywhere, through uniqueness result for the corresponding Liouville equation in the space L+ ([0, T ];L (R)∩L(R)). Under these regularity conditions on the potential (in addition to some growing at infinity) the Schrödinger equation is well posed for all positive values of the Planck constant and it is therefore natural to ask what is happening at the classical limit. As we will see, different answers will be given, according to the choices we make first on the topology of the convergence, and secondly on the asymptotic properties of the initial datum. The general idea of the results we are going to present here can be summarized as follows: