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We study the homogenization of a Schr\"{o}dinger equation in a locallyperiodic medium. For the time and space scaling of semi-classical analysis weconsider well-prepared initial data that are concentrated near a stationarypoint (with respect to both space and phase) of the energy, i.e. the Bloch celleigenvalue. We show that there exists a localized solution which isasymptotically given as the product of a Bloch wave and of the solution of anhomogenized Schr\"{o}dinger equation with quadratic potential.

Localization for the Schr\"{o}dinger equation in a locally periodic medium