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arXiv · 1502.02455

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

Grégoire Allaire·Mariapia Palombaro
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Abstract

We study the homogenization of a Schr\"{o}dinger 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 (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 Schr\"{o}dinger equation with quadratic potential.

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Grégoire Allaire, Mariapia Palombaro. 2015-02-09. Localization for the Schr\"{o}dinger equation in a locally periodic medium. https://doi.org/10.1137/050635572

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