Homogenization of the Schrödingerequation with a time oscillating potential

Grégoire Allaire; M. Vanninathan

doi:10.3934/dcdsb.2006.6.1

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2006, Volume 6, Issue 1: 1-16. Doi: 10.3934/dcdsb.2006.6.1

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Homogenization of the Schrödinger equation with a time oscillating potential

Grégoire Allaire^1, and
M. Vanninathan^2,

1.
Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 PALAISEAU Cedex
2.
TIFR Center, P.O. Box 1234, Bangalore - 560012

Received: February 28, 2005

Revised: July 31, 2005

Published: December 2005

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Abstract

We study the homogenization of a Schrödinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous "Fermi golden rule" for the transition probability between two such states which is at the basis of various optical properties of semiconductors. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves theory.

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Mathematics Subject Classification: Primary: 35B27; Secondary: 35Q40.

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