Lower bounds for shape resonances widths of Schrodinger operators

In this lecture we present some results giving general lower bounds of shape resonance widths near positive energy levels in the semi-classical limit for Schrodinger operators in the exterior of smooth compact obstacles with Dirichlet or Neuman boundary conditions and with long range dilation analytic potentials. These lower bounds are exponentially small with respect to the Planck constant. We also give some consequences of these lower bounds on the asymptotic behaviour in large time of solutions of wave equations.