Localization for the Schrödinger Equation in a Locally Periodic Medium
Grégoire Allaire, Mariapia Palombaro · SIAM Journal on Mathematical Analysis · 2006
We study the homogenization of a Schrödinger equation in a locally periodic medium. For the time and space scaling of semiclassical analysis we consider well-prepared initial data that are concentrated near a stationary point (with respect to both space and phase) of the energy, i.e., the Bloch cell eigenvalue. 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.