I will present a localization result for the Schrödinger equation in a locally periodic medium. For the time and space scaling of semi-classical analysis we consider well-prepared initial data that are concentrated near a stationary point of the energy. By the method of two-scale convergence, we show that there exists a localized solution which is asymptotically given as the product of a Bloch wave and of the solution of an homogenized Schrödinger equation with quadratic potential. In the last part of the talk I will also discuss some results, obtained by the same method, on the diffraction of Bloch wave packets in periodic media over long times.
The results have been obtained in joint work with Grégoire Allaire and Jeffrey Rauch.